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ffcb8f77-81e2-4e6e-97f5-50415e5c69cb.102
|
**3.2 Perception of the implementation of the National health policy in Ghana**
The study findings showed that participants agreed on the implementation of the National Health Policy of Ghana. Participants were asked questions about their perception of health in the implementation of the Persons with Disability Act and on the NHIS (NHIS). Some of the perceptions revealed in this study were the ineffectiveness of the Persons with Disability Act, poor quality health service, and less health service coverage by the NHIS. It also showed that participants were unanimous on the health service coverage by the NHIS. All participants acknowledged that the NHIS does cover some aspects of healthcare services like consultation fee, patient card, half of the lap fees, part of medication, and one-month cost of physiotherapy but does not cover tertiary services like surgery and some specialized services like speech and language therapy, occupational therapy among others. Participants revealed in the study that these tertiary and specialized services are constantly needed by children with CP.
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{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"book_id": "ffcb8f77-81e2-4e6e-97f5-50415e5c69cb",
"url": "https://mts.intechopen.com/storage/books/11893/authors_book/authors_book.pdf",
"author": "",
"title": "Cerebral Palsy",
"publisher": "IntechOpen",
"isbn": "9781803565828",
"section_idx": 102
}
|
ffcb8f77-81e2-4e6e-97f5-50415e5c69cb.103
|
**3.3 Parental factors**
The choices made by parents as to whether to seek healthcare for their children with CP is dependent on other factors. According to Khatri and Karkee [14], parental factors are more likely to shape the health-seeking behavior of individuals. Study participants were therefore asked questions on parental factors that can influence access
*Access to Healthcare Services Among Children with Cerebral Palsy in the Greater Accra Region… DOI: http://dx.doi.org/10.5772/intechopen.106684*
#### **Table 2.**
*Dimensions of access to health care services.*
#### **Figure 2.**
*Conceptual framework- Factors influencing access to healthcare (Adapted from the WHO; International classification of functioning, disability, and health [8, 9].*
to healthcare services for children with CP. From the study findings, participants mentioned knowledge of disability, perception about disability (CP), income spent on child's health care, transportation cost, and satisfaction as factors that influence access to healthcare services for children with CP (**Figure 2**).
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{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"book_id": "ffcb8f77-81e2-4e6e-97f5-50415e5c69cb",
"url": "https://mts.intechopen.com/storage/books/11893/authors_book/authors_book.pdf",
"author": "",
"title": "Cerebral Palsy",
"publisher": "IntechOpen",
"isbn": "9781803565828",
"section_idx": 103
}
|
ffcb8f77-81e2-4e6e-97f5-50415e5c69cb.104
|
*3.3.1 Causes of disability*
Concerning knowledge of the causes of disability, some participants attributed it to spirituality, brain damage (medical condition), and diseases. Some participants believe that haters from families can make a child have a disability through witchcraft and voodoo especially when the child has a bright future. The study revealed that parents with such beliefs do not seek medical care for their children but rather seek spiritual healing for their children with CP.
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2025-04-07T04:13:03.918799
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|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"book_id": "ffcb8f77-81e2-4e6e-97f5-50415e5c69cb",
"url": "https://mts.intechopen.com/storage/books/11893/authors_book/authors_book.pdf",
"author": "",
"title": "Cerebral Palsy",
"publisher": "IntechOpen",
"isbn": "9781803565828",
"section_idx": 104
}
|
ffcb8f77-81e2-4e6e-97f5-50415e5c69cb.105
|
*3.3.2 Symptoms of CP*
Regarding the signs and symptoms of CP, the following were mentioned: poor eye contact, poor posture, and balance, communication, as well as impaired fine and gross motor function.
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{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"book_id": "ffcb8f77-81e2-4e6e-97f5-50415e5c69cb",
"url": "https://mts.intechopen.com/storage/books/11893/authors_book/authors_book.pdf",
"author": "",
"title": "Cerebral Palsy",
"publisher": "IntechOpen",
"isbn": "9781803565828",
"section_idx": 105
}
|
ffcb8f77-81e2-4e6e-97f5-50415e5c69cb.106
|
*3.3.3 Treatment*
Considering treatment, the majority of the respondents believed that CP cannot be treated (cured) but can be managed through the following means: Specialized healthcare services including physiotherapy, occupational therapy, speech and language therapy, and generalized healthcare services including primary healthcare like screening and health checkups. The majority of the respondents with this knowledge seek healthcare for their children with CP as revealed in this study.
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{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"book_id": "ffcb8f77-81e2-4e6e-97f5-50415e5c69cb",
"url": "https://mts.intechopen.com/storage/books/11893/authors_book/authors_book.pdf",
"author": "",
"title": "Cerebral Palsy",
"publisher": "IntechOpen",
"isbn": "9781803565828",
"section_idx": 106
}
|
ffcb8f77-81e2-4e6e-97f5-50415e5c69cb.107
|
*3.3.4 Perceptions of disability*
The results of the study also revealed that respondents share different perceptions of disability. Some of the respondents mentioned that disability can result in stigmatization, discrimination disrespect, loneliness, depression, and even suicide. Respondents agreed that when someone has a disability stigmatization and discrimination inevitably lead to depression, loneliness, and suicidal ideation. The study showed that respondents with such perceptions feel reluctant to seek healthcare for their wards with CP with the fear of being stigmatized and discriminated against. Some participants also mentioned that disability can be frustrating and lead to the loss of a job.
Participants believed that negative perception about disability may lead to less access to healthcare services for children with CP while the positive perception may lead to frequent access to healthcare services. Respondents with a negative perception of disability do not seek regular healthcare for their children with CP but rather go to seek spiritual healing for their wards with CP. Also, it was revealed that respondents with positive perceptions frequently seek healthcare for their children with CP.
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2025-04-07T04:13:03.918930
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{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"book_id": "ffcb8f77-81e2-4e6e-97f5-50415e5c69cb",
"url": "https://mts.intechopen.com/storage/books/11893/authors_book/authors_book.pdf",
"author": "",
"title": "Cerebral Palsy",
"publisher": "IntechOpen",
"isbn": "9781803565828",
"section_idx": 107
}
|
ffcb8f77-81e2-4e6e-97f5-50415e5c69cb.108
|
*3.3.5 The financial burden on a child with CP health care*
The study showed that the majority of the respondents are not able to access healthcare for their wards with CP as they spend a greater part of their income on *Access to Healthcare Services Among Children with Cerebral Palsy in the Greater Accra Region… DOI: http://dx.doi.org/10.5772/intechopen.106684*
their child's health. Participants agreed that the healthcare needs of children with CP are a lot and most expensive. They stated that specialized services like physiotherapy, occupational therapy, speech and language therapy, and special nutrition help to manage children with CP but these services are expensive and sometimes they find it difficult to access (**Figure 2**). However, some participants also specified that because of the expensive nature of specialized services they tend to do basic physiotherapy for their wards in their homes since they cannot keep up with the services in the hospitals. Participants were unanimous on how they spend so much money on both primary and secondary healthcare services for their children with CP and this discourages them from accessing healthcare services for their children with CP. Parents of children with CP all agreed that income level may influence access to healthcare services in either a positive or negative way depending on the kind of job a parent has (**Table 2**). The study revealed that access to healthcare services for children with CP is higher among parents with a higher income than those with a lower income. Some participants also stated that the kind of job one does determine the level of income.
#### **3.4 Transportation**
The study revealed that parents of children with CP do experience transportation problems and that discourages them from accessing healthcare services for their children with CP. All participants agreed that transportation is another factor that influences access to healthcare services. The majority of the participants stated that their means of transportation to the hospital is public transport (trotro, taxi, uber). All participants mentioned that transportation problems include access to public transport, transportation cost, and stigmatization in public transport.
Concerning access to public transport majority of the respondents agreed that most public transport designs are not accessible and friendly to children with CP. This makes it difficult for them to access public transport to a health facility. It was also revealed in this study that all respondents agreed there is no designated space for persons with disabilities inside public transport. This makes them feel very uncomfortable when using public transport to a health facility. Findings from this study showed that respondents who use public transport do not frequently access healthcare services for their children with CP as they are discouraged from all the hustle and frustration, they face from accessing public transport.
Some of the participants asserted that transportation cost to the closest health facility is expensive and, in most cases, uber or taxi drivers do not want to render services to them due to their children with disabilities. Participants also believed that commercial drivers (uber and taxi) charge them a higher cost because of their children with disabilities. The study found that passengers, conductors (mate), and commercial drivers (trotro) do stigmatize parents of children with CP when boarding a car to a health facility. Study participants specified that passengers do not want to sit by them with the belief that they will end up having a child with a disability and also bus conductors (mate) and commercial drivers (trotro) also ask them to pay for the entire seat or else they will not pick them up. This discourages parents from accessing healthcare for their children with CP.
Also, some respondents mentioned that stigmatization from passengers, bus conductors, and drivers do put them off sometimes and not access healthcare for their child with CP.
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|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"book_id": "ffcb8f77-81e2-4e6e-97f5-50415e5c69cb",
"url": "https://mts.intechopen.com/storage/books/11893/authors_book/authors_book.pdf",
"author": "",
"title": "Cerebral Palsy",
"publisher": "IntechOpen",
"isbn": "9781803565828",
"section_idx": 108
}
|
ffcb8f77-81e2-4e6e-97f5-50415e5c69cb.109
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*3.4.1 Satisfaction*
The results of the study showed that participants agreed that satisfaction with health care services does influence their access to healthcare. The participants attributed satisfaction to quality health care, waiting time for treatment, and costeffectiveness of healthcare services. Respondents explained quality health care as the one that is considered to be safe, efficient, inclusive, patient-centered, timely, and that makes customers happy. Respondent also believed that when patients do not get quality healthcare, they may reduce the number of times they visit a particular health facility or they may stop accessing health care from those particular health facilities.
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{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"book_id": "ffcb8f77-81e2-4e6e-97f5-50415e5c69cb",
"url": "https://mts.intechopen.com/storage/books/11893/authors_book/authors_book.pdf",
"author": "",
"title": "Cerebral Palsy",
"publisher": "IntechOpen",
"isbn": "9781803565828",
"section_idx": 109
}
|
ffcb8f77-81e2-4e6e-97f5-50415e5c69cb.110
|
**3.5 Identification of health service needs (health care factors)**
From the study findings, participants were generally able to identify healthcare factors that include specialized services availability, proximity to an available health facility, availability of specialist healthcare providers, accessibility of building and equipment, and healthcare provider's attitude do influence access to healthcare services for children with CP (**Figure 2**).
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2025-04-07T04:13:03.919274
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|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"book_id": "ffcb8f77-81e2-4e6e-97f5-50415e5c69cb",
"url": "https://mts.intechopen.com/storage/books/11893/authors_book/authors_book.pdf",
"author": "",
"title": "Cerebral Palsy",
"publisher": "IntechOpen",
"isbn": "9781803565828",
"section_idx": 110
}
|
ffcb8f77-81e2-4e6e-97f5-50415e5c69cb.111
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*3.5.1 Availability of specialized services*
Respondents mentioned some specialized services that health facilities need to make available for children with CP, these include speech and language therapy, physiotherapy, occupational therapy, behavioral therapy, augmentative communication, and dietary approaches. Respondents believed that these specialized services help children with talking, walking, participating in the activities of daily life (such as brushing teeth and getting dressed), interacting with others, learning social skills, and managing their emotions. Study participants mentioned that they feel encouraged to access health care from health facilities that provide specialized services.
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2025-04-07T04:13:03.919314
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|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"book_id": "ffcb8f77-81e2-4e6e-97f5-50415e5c69cb",
"url": "https://mts.intechopen.com/storage/books/11893/authors_book/authors_book.pdf",
"author": "",
"title": "Cerebral Palsy",
"publisher": "IntechOpen",
"isbn": "9781803565828",
"section_idx": 111
}
|
ffcb8f77-81e2-4e6e-97f5-50415e5c69cb.112
|
*3.5.2 Proximity to an available health facility*
All study participants mentioned that distance from a patient's (clients) home to the nearest health facility can influence access to health care. Participants believed that clients are discouraged to access the closest health facility when they face a transportation problem and have to travel far distances. However, clients whose home is not too far from the nearest hospital is encouraged to access healthcare for their child with CP.
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2025-04-07T04:13:03.919372
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|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"book_id": "ffcb8f77-81e2-4e6e-97f5-50415e5c69cb",
"url": "https://mts.intechopen.com/storage/books/11893/authors_book/authors_book.pdf",
"author": "",
"title": "Cerebral Palsy",
"publisher": "IntechOpen",
"isbn": "9781803565828",
"section_idx": 112
}
|
ffcb8f77-81e2-4e6e-97f5-50415e5c69cb.113
|
*3.5.3 Availability of specialist healthcare providers*
From the study findings, participants acknowledged and stated that the availability of rehabilitation specialists in health centers influences their access to healthcare for their children with CP. Respondents asserted that some hospitals have specialized services written on their signboards but do not provide such services because specialists are scarce. This influences their decision to access healthcare services from certain health facilities. Respondents believed that rehabilitation specialists like physiotherapists, speech and language therapists, and occupational therapists are very few and scarce in Ghana, especially in the rural areas. Respondents also asserted that they have to travel long distances to access these specialized services in the urban centers which sometimes transportation cost and rehabilitation cost becomes a challenge to them.
#### **Table 3.**
*Reasonable accommodation and suggested approaches.*
## *3.5.4 Accessibility of building and equipment*
Study participants agreed that many health centers are not disability-friendly. They linked the accessibility of the building to the physical environment of the health facility including entrance to consulting rooms, OPD, and top floors. Some participants also stated that equipment like standing frames, power tables, parallel bars, and stand-assist devices in most hospitals are not friendly to children with CP (**Table 3**). Participants believed that most rehabilitation equipment in certain hospitals is meant for stroke patients and not for children with CP.
### *3.5.5 Healthcare providers attitude*
Attitudes of healthcare providers play a significant role in parents' decisions to access health care for their children with disabilities. Some participants acknowledged the fact that not all healthcare providers' attitude is bad. However, the majority of the respondents stated that their experience with healthcare providers has been bad and that discourages them from accessing healthcare for their children with CP. It was strongly perceived that bad behavior like discrimination from healthcare providers towards parents of children with CP would rather discourage them from accessing health care.
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2025-04-07T04:13:03.919413
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|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"book_id": "ffcb8f77-81e2-4e6e-97f5-50415e5c69cb",
"url": "https://mts.intechopen.com/storage/books/11893/authors_book/authors_book.pdf",
"author": "",
"title": "Cerebral Palsy",
"publisher": "IntechOpen",
"isbn": "9781803565828",
"section_idx": 113
}
|
ffcb8f77-81e2-4e6e-97f5-50415e5c69cb.115
|
**4.1 Knowledge of National health policy**
The level of knowledge of the National Health Policy was well known among almost all the participants as it was considered a policy that aims to promote health for everyone in Ghana. This view is similar to Vartan and Montuschi's [15]'s assertion that the national health policy has been developed to promote, restore, and maintain good health for all people living in Ghana. Participants also explained their knowledge of the national health policy from the persons with disabilities Act perspective as a means of getting access to healthcare services without any barrier or discrimination. This finding in this study is consistent with the Americans with Disability Act (1990) Section 504 of the Rehabilitation Act that forbids discrimination based on disability and also the Person with Disability Act (2006) of Ghana Section 4 (1) and (6) that prohibit discriminate, exploit or to subject a person with disabilities to abusive or degrading treatment.
The present study findings indicate that the majority of the respondents had sufficient knowledge regarding the NHIS (NHIS) describing it as a safety net that replaces the cash and system of service delivery in Ghana. This finding is in agreement with a study conducted by Akande and Akande [16] on "The Awareness and Attitude of Practitioners on NHIS in Llorin showed that all respondents were aware of the scheme but only a few did not know. Another study conducted by Dixon et al. [17] on "Ghana's NHIS: a national level investigation of members' perceptions on service provision in Ghana" found that the NHIS replaces cash-and-carry, which required individuals to make a payment from their pockets at service usage. However, another study by Gopalan and Durairaj [18] showed that better-educated individuals can access diverse sources of information, correctly process and take advantage of benefits than those who are less educated and those without formal education. Those who could not afford to spend more on the healthcare needs of children with CP may adopt other coping mechanisms such as alternative care, presenting late at the health facilities, or not receiving care at all.
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2025-04-07T04:13:03.919544
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{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"book_id": "ffcb8f77-81e2-4e6e-97f5-50415e5c69cb",
"url": "https://mts.intechopen.com/storage/books/11893/authors_book/authors_book.pdf",
"author": "",
"title": "Cerebral Palsy",
"publisher": "IntechOpen",
"isbn": "9781803565828",
"section_idx": 115
}
|
ffcb8f77-81e2-4e6e-97f5-50415e5c69cb.116
|
**4.2 Perception of implementation of National health policy**
Findings from this current study revealed that respondents are influenced in accessing health care for their wards even when they are insured as they perceived the implementation of the National Health Policy to be ineffective concerning poorquality health service for the insured. This finding is similar to Bruce et al. [19] study results on "The perceptions and experiences of health care providers and clients in two districts in Ghana" which showed that insured clients are not satisfied with the healthcare they received and perceived that they were given poorer quality services and tend to wait longer as compared to those making Out of Pocket Payment (OOP). The present study also revealed that the implementation of the National Health Policy is ineffective as respondents perceived that it does not cover major specialized services and treatment for children with CP. This study finding is in contrast to Dalinjog and Laar's [20] study on "The effectiveness of the National Health Policy in Ghana" which found that both insured and uninsured respondents had positive perceptions and were satisfied with the care provided.
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doab
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2025-04-07T04:13:03.919691
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5-9-2023 17:54
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"book_id": "ffcb8f77-81e2-4e6e-97f5-50415e5c69cb",
"url": "https://mts.intechopen.com/storage/books/11893/authors_book/authors_book.pdf",
"author": "",
"title": "Cerebral Palsy",
"publisher": "IntechOpen",
"isbn": "9781803565828",
"section_idx": 116
}
|
ffcb8f77-81e2-4e6e-97f5-50415e5c69cb.117
|
**4.3 Parental factors**
The study discovered that participants were generally aware of some of the factors that influence access to healthcare services among children with CP. A study conducted by Khartri and Karkee [14] showed that distance to health facilities, social support, age, the behavior of health workers, and access to quality health services shapes how parents access healthcare for their children and themselves. This finding is in agreement with Boz et al. [21] study on "The affecting factors of healthcare
#### *Access to Healthcare Services Among Children with Cerebral Palsy in the Greater Accra Region… DOI: http://dx.doi.org/10.5772/intechopen.106684*
services demand in terms of health services use: A field application in Edirne city" found that personal income, gender, attitudes, and behaviors of physicians affect access to healthy demand. In the same study, it was reported that family members, perception of economic level, attitudes, and behaviors of physicians were found to influence health demand. Findings from this current study revealed that participants know about disability regarding the causes, signs, and symptoms as well as treatment for children with CP. This positively influenced parents' access to healthcare for their children with CP. This finding is consistent with Matt's [22] study on "Perception of disability among caregivers of children with disabilities in Nicaragua" which found that parents with higher education have a better understanding and knowledge of their children's disability and frequently access health care services for their wards than parents with lower or no education. The current study findings also agree with Khatri and Karkee's [14] assertion that illiterate parents who belong to the lowest wealth quintile have lower access and use of healthcare for their children with CP.
Also, considering the causes of disability, this current study revealed that respondents know the causes of disability. Findings from the study showed that disability can be caused by disease and neurological problems leading to brain damage. This finding is in agreement with the Center for Diseases Control and Prevention [23] assertion that risk factors such as infections during pregnancy, premature birth, and diseases like jaundice can cause CP.
Respondents in this study linked signs and symptoms of CP to poor eye contact, poor posture and balance, communication difficulties, and impaired fine and gross motor function. This finding agrees with the CP Alliance [24] assertion that children with CP show signs and symptoms like swallowing difficulties, poor muscle spasms, low muscle tone, poor muscle control, reflexes, and posture, drooling, developmental delay, gastrointestinal problems, and not walking by 12–18 months. It was found from this study that treatment for children with CP includes physiotherapy, occupational therapy, and speech and language therapy. This finding is consistent with the CP Guide [25] statement that children with CP can improve their motor skills with alternative therapy, medication, and surgery through multidisciplinary teams such as neurologists, orthopedic surgeons, developmental pediatricians, physiotherapists, occupational therapists, nutritionists, respiratory therapists, psychologists to assess ability and behavior and speech and language, therapist.
It was also found in this study that perceptions of parents about their children's disability (CP) also influence their decision to access healthcare for their children with CP. It was also established that stigmatization, discrimination, disrespect, loneliness, depression, and suicidal ideation are linked to disability (CP). This is similar to Physioplus's [26] assertion that families with a child with a disability are more prone to depression, suicide, financial problem, relationship challenges, divorce, and bankruptcy. This statement is also in line with another study by Butchner [27] on "society's attitude towards persons with disabilities" which found that society perceives that disability is a curse and punishment from ancestors and gods. Also, Duran and Ergun [28] in their study on "The stigma perceived by parents of children with disability: an interpretative phenomenological analysis study in Balikesir found that the majority of people often have negative perceptions and stigmatizing attitudes towards children with disabilities (CP) and their families. In the same study, it was found that parents of children with disabilities cope with insults and rude behaviors from community members while they struggle with the challenges of their children with CP. This is similar to a previous study by Opoku et al. [6] who affirmed that persons with disabilities are severely stigmatized, discriminated against, and excluded from all forms
of the development process resulting in limiting their opportunities to be engaged in decision-making and accessing healthcare.
Income was found from this current study to influence parents' access to health care services for their children with CP. Respondents emphasized that the level of income of a parent is dependent on the kind of job the parent does. This present study revealed that parents of children with CP who have no jobs find it difficult to access healthcare services for their wards due to the cost of treatment. This finding is in line with DeVoe et al. [29] assertion that children with CP from lower-income families experience more gaps in healthcare than children with CP from higher-income families.
Moreover, this study discovered that transportation is another factor that influences access to health care. Findings from this study indicated that transportation cost, distance to the nearest health facility, and stigmatization from drivers, passengers, and bus conductors (mate) influence respondents' ability to access healthcare services for their children with CP. A previous study showed that healthcare utilization is influenced by the direct costs of healthcare services, travel time, and patient income [30]. This is in line with another study conducted by Bulamu Healthcare [31] in Uganda which specified that patients complain about poor sanitation, lack of drugs and equipment, long waiting times, rude behavior of health workers, and poor referrals. However. in that same study it was revealed that over 8000 rural Ugandans travel as far as 50 miles to attend a Bulamu weeklong medical camp for healthcare.
Also, findings from this present study showed that respondents are not satisfied with the waiting time and cost of health care services. Respondents from this present study linked healthcare satisfaction to quality health care, waiting time for treatment, and cost-effectiveness of healthcare services. This assertion is consistent with Khatri and Karkee's [14] statement that quality health care accounts for patient satisfaction especially in terms of waiting time, cost of service, coordination, information, and physician's behavior. This finding agrees with Janzek-hawlat's [3] findings that some physicians in public health facilities can be very rude due to the workload mounted on them.
#### **4.4 Healthcare factors**
Considering specialized services available, the present study found that respondents were informed about the available specialized services for their children with CP. It was discovered that speech and language therapy, physiotherapy, occupational therapy, behavioral therapy, augmentative communication, and dietary were some of the specialized services available but scarce and that makes it difficult to access healthcare for their children due to waiting time for treatment. Respondents in this present study believed that children who can access these services will be able to walk, interact with others through play, learn social skills, seat properly, and have good muscle and neck control as well as good balance and body posture. This finding agrees with Balcı's [32] study on "Current Rehabilitation Methods for Cerebral Palsy" which found that children with CP that undergo muscle strengthening training, manual stretching, massage, neurodevelopmental treatment, conductive education, speech and language therapy, occupational therapy, and dieting have good body posture, balancing, neck coordination, strong muscle control and can walk sometimes. However, the lack of appropriate services for individuals with CP is a significant barrier to health care. For instance, qualitative research in Uttar Pradesh and Tamil Nadu states of India revealed that after the cost, the lack of services in the area was the second most significant barrier to using health facilities [8, 9].
#### *Access to Healthcare Services Among Children with Cerebral Palsy in the Greater Accra Region… DOI: http://dx.doi.org/10.5772/intechopen.106684*
Also, it was found from this current study that proximity to an available health facility is another factor that influences access to healthcare for children with CP. From the study, respondents were discouraged to access the nearest health facility when transportation and distance to the health facility are problems. This finding is similar to Awoyemi et al. [33] study on "Effect of Distance on Utilization of Health Care Services in Rural Kogi State in Nigeria" which found that distance and total cost of healthcare affects the utilization of both public and private hospitals. This finding also agrees with Nesbitt et al. [34] study on "Barriers and facilitating factors in access to health services in the Republic of Moldova" which found that distance from a health service provider, travel time, and waiting time to see a health professional is are strong factors that influence access to health care.
Moreover, the availability of specialists in hospitals was found in this present study to be scarce in most hospitals, and because of that, only a few respondents travel a long distance to access these specialized services for their wards. This finding is consistent with WHO [35] reported that the registered number of rehabilitation specialists is far below the required minimum of 750 per 1 million even in developed countries. In addition to this, the present study revealed that most public and private hospitals are not environmentally friendly for children with CP who use wheelchairs. Respondents asserted that most public hospitals do not have elevators and ramps making it difficult to access healthcare services for their wards. This finding is in line with Jamshidi et al. [36] study on "The effects of environmental factors on the patients' outcomes in hospital environments: A review of the literature" found that medical equipment adaptability, unit layout, room features, ramps, and elevators affect patients' access to healthcare. Another study by Douglas and Douglas, [37] revealed that patients' need for personal space, a homely welcoming atmosphere, a supportive environment, ramps, and elevators influence access to their healthcare. Cristina and Candidate [11] also found in their study that out of 256 respondents, 9 (4%) were not able to access the building and 47(18%) were not able to be transferred from their wheelchair to the examination table.
It was also found from this study that some healthcare providers discriminate against parents of children with CP when seeking primary healthcare services and physiotherapy. This finding agrees with the WHO [8, 9] report that parents of children with disabilities face stigmatization and discrimination in most health facilities. This finding is also in line with Rogers et al. [38] study on "Discrimination in healthcare settings is associated with disability in Older Adults: Health and Retirement Study, 2008-2012" which revealed that 12.6% experienced discrimination infrequently whilst 5.9% experienced discrimination frequently.
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**5. Conclusion**
The study established that many respondents believed that most healthcare facilities are not physically accessible due to the absence of ramps and elevators. Respondents are discriminated against and stigmatized both in hospitals and on public transport. It was also found that many public means of transport are not accessible to children with CP. Respondents believed that the National Health Policy is ineffective and the NHIS does not cover a wide range of services for children with CP. Moreover, the study also pointed out that the majority of the respondents seek medical care for their children with CP however others also seek spiritual healing for their children with CP.
*Cerebral Palsy - Updates*
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**Author details**
Nathaniel Larbi Andah Social and Behavioral Science Department, School of Public Health - University of Ghana, Greater Accra, Ghana
\*Address all correspondence to: [email protected]
© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
*Access to Healthcare Services Among Children with Cerebral Palsy in the Greater Accra Region… DOI: http://dx.doi.org/10.5772/intechopen.106684*
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The Japan Obstetric Compensation System for Cerebral Palsy: Novel System to Improve Quality and Safety in Perinatal Care and Mitigate Conflict
*Shin Ushiro*
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**Abstract**
The Japan Obstetric Compensation System for Cerebral Palsy was launched in 2009 in response to a shortage of obstetricians and a surge in disputes. The system is characterized by the provision of no-fault compensation, investigation, and prevention. We have certified more than 3000 cerebral palsy cases for compensation and have delivered investigative reports, prevention reports, and educative media for professionals and expectant mothers. We have also produced recapitulation of individual investigative report to be uploaded on the webpage of the system to enhance transparency. The disclosure is reviewed to be consistent with lately revised Personal Information Protection Law in 2020. In order to expand the system by revising eligibility criteria, the system was and will be reviewed in 2015 and 2022. The new criteria that were crafted in ad-hoc committee in 2019–2020 will be applied in 2022 and later. As such, the system has been a significant part of perinatal care delivery system in Japan.
**Keywords:** cerebral palsy, the Japan Obstetric Compensation System for Cerebral Palsy, Japan Council for Quality Healthcare, no-fault compensation
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**1. Introduction**
The Japan Obstetric Compensation System for Cerebral Palsy [1–4] was launched in 2009 by the Japan Council for Quality Health Care (JQ ) as operating organization, with the background of a shortage of obstetricians in Japan and a surge in disputes particularly caused by occurrence of cerebral palsy (CP). More than 10 years have passed since the system commenced, and it has given rise to enormous achievements such as early resolution of disputes displayed in the rapid decrease in the number of lawsuit statistics and quality improvement of perinatal care. It is of note that the system was designed in an introductory committee in the presence of range of stakeholders such as professional organizations, academic organizations, insurance firm, lawyers, and patient representatives. They have been involved in implementing the
system that was helpful in obtaining confidence in the system. Here, in this article, current status of the system and challenges ahead are described.
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**2.1 Perinatal care and conflict**
Among the disputes, those related to cerebral palsy were said to be a heavy burden for obstetricians because the cause of cerebral palsy was often unknown. In discussions with perinatal care professionals in Japan and abroad, it has been said that cerebral palsy is one of the causes of disputes, including court cases, and of obstetricians leaving their jobs, not only in high-income countries but also in middle-low-income countries (LMICs). In cases where a child is born in a distressed status despite normal course of pregnancy and delivery, or where the child's neurological deficits become apparent to develop profound cerebral palsy in spite of little or no findings on hypoxic condition during delivery, the family's sentiment may become complicated due to uncertainty on the cause of cerebral palsy and a dispute may arise. Therefore, obstetricians had been discussing for many years the establishment of a compensation system that runs on no-fault basis in anticipation for mitigating the conflict.
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**2.2 Deliberation on establishing no-fault compensation system**
Discussions on the establishment of a no-fault compensation system have been held by the Japan Medical Association (JMA) since the 1960s. In the report entitled "Report on the Legal Response to Medical Accidents and Its Basic Theory" published in 1972, the following recommendations were set forth [5].
The JMA physician liability insurance system was established in 1973 in response to (i), but the other two items were not materialized for decades that followed. With decades passing by, the shortage of physicians in obstetrics and gynecology and the declining birthrate became social problems. In January 2006, the JMA made a proposition in its report entitled "Aiming at the establishment of a disability compensation system for medical care" stating "Ideally, it is desirable to implement a no-fault compensation system for entire medical specialties" and "however, neurological sequelae related to childbirth (so-called cerebral palsy) is prioritized for no-fault compensation." In August of the same year, they presented the details on the novel system [6]. Furthermore, in November of the year, "A Framework for a No-Fault Compensation System in Obstetrics" (Study Group on Medical Dispute Resolution, Social Security
*The Japan Obstetric Compensation System for Cerebral Palsy: Novel System to Improve Quality… DOI: http://dx.doi.org/10.5772/intechopen.106760*
#### **Figure 1.**
*No-fault compensation/investigation/prevention system for cerebral palsy, 2009.*
System Study Group, Political Research Committee of the Liberal Democratic Party (LDP)) was published that was followed by growing anticipation to launch the system. It depicted that the novel system is equipped with two pillars such as compensation on no-fault basis and investigation and prevention. At the same time, relevant organizations and groups expressed their concern and requested that the JQ should be an operator of the system. Accordingly, the Preparatory Office for the novel system was installed in the JQ in February 2007 that served as secretariat for the Introductory Committee for the novel system. In March 2008, the Board of Directors of the JQ formally decided to be the operating organization. All in this way, the system has been in operation since January 2009 (**Figure 1**).
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**2.3 Registration of childbirth facility**
The system was launched and being hailed by professional societies such as the Japan Association of Obstetricians and Gynecologists (JAOG) and Japanese Midwife Association (JMA). They, therefore, helped the JQ to involve childbirth facilities across Japan for registration in the system. Although there is no legislation that mandates them participate in the system, the JQ successfully observed extremely high registration rate in the system as high as 99.9% achieved in close cooperation with the societies and relevant stakeholders [7] (**Table 1**).
#### **Table 1.**
*Registration rate by type of facility. As of November 30, 2021.*
#### **2.4 Review and compensation**
The scope of those eligible for compensation must meet the general criteria, which consists of birth weight and weeks of pregnancy, or the individual criteria when the weeks of pregnancy are less than the general criteria : 28 weeks or more of pregnancy, case-by-case criteria: umbilical artery blood pH less than 7.1, meeting one of the prescribed patterns in the fetal heart rate labor diagram (CTG) that indicate hypoxia in the fetus, etc., meets severity criteria: degree equivalent to level 1 or 2 of the physical disability grade defined in the Welfare for the Disabled Act, and does not meet the exclusion criteria such as cerebral palsy obviously caused by congenital factors or factors taking place in neonatal period and later [8]. Even if congenital factors (brain malformation, genetic abnormality, chromosomal abnormality, etc.) exist, patients are not necessarily excluded because the factors may not be the obvious cause of profound CP. Decision for approval is made based on medical examination as to what caused profound CP that applicant suffers. The general criteria and caseby-case criteria were revised in reference to aggregated data and scientific progress on cerebral palsy. The latest criteria that applied to cerebral palsy who were born in 2022 or later does not include case-by-case criteria due to expansion of general criteria so that more cerebral palsy would be covered (**Table 2**). The revision of the criteria is described later.
As of June 2021, 3374 cases have been approved for compensation, and payment for the cases have been swiftly made or in progress. The annual number of persons eligible for compensation to such criteria as 2009 and 2015 criteria that have been confirmed so far is 419 in 2009, 382 in 2010, 355 in 2011, 362 in 2012, 351 in 2013, 326 in 2014, and 376 in 2015. For children born in later years, the application is still allowed until they are 5 years old (**Table 3**). It should be noted that the number of approved patients rose to a certain extent in 2015 because new criteria that could cover more cerebral palsy were applied. In addition, applicants of uncompensated cases may apply to the Appeal Committee if they are not convinced on the results of the review.
A uniform compensation of 30 million yen is paid for each case once approved by the Review Committee. There are two different ways applied to the payment such as lumpsum payment and annual installment that continue 20 times (**Figure 2**). If childbirth facility is liable for the development of cerebral palsy, the compensation and the damage payment are adjusted to eliminate duplicative payments. In other words, the child and the family cannot receive both the compensation and the damage payment in the system [9].
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**3. Investigation**
#### **3.1 Production of investigative report**
The purpose of the investigation is (i) to analyze the case from a medical point of view based on record and data on the cerebral palsy to learn knowledge for prevention and (ii) prevent conflict between childbirth facility and patient/family and bring it to early settlement by sharing investigative report for mutual understanding on the childbirth event. Unlike court system, this is a process of analysis genuinely from medical and midwifery point of view [10].
*The Japan Obstetric Compensation System for Cerebral Palsy: Novel System to Improve Quality… DOI: http://dx.doi.org/10.5772/intechopen.106760*
#### **Table 2.**
*Eligibility criteria: criteria of 2009, 2015 and 2022.*
The Investigation Committee holds seven subcommittees to compile a draft report (**Figure 3**). One committee is composed of 14 members: nine obstetricians including the chairperson, two pediatricians, one midwife, and two lawyers. The role of the medical members is to analyze the case from medical viewpoint, while the lawyers provide views so that the report will be easy for patient/family to understand. A working manual was crafted to ensure that the reports are compiled in standardized fashion. The draft report compiled therein is reviewed in the Investigative Committee for approval. At the same time, a "summarized edition" of the investigative report is issued with personal identifiers deleted and held available on the system's website for prevention and improvement of perinatal care.
#### **Table 3.**
*Statistics of eligible case by birth year.*
#### **Figure 2.**
*Sum of compensation payment (30 million JPY = 342,000 USD).*
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**3.2 The relationship between the disclosure of the "Synthesized edition" of the investigative report and the latest revision of the Personal Information Protection Law and relevant administrative guidelines**
The "Synthesized edition" of the investigative report has been published and posted on the website as one of the products of the system [11]. They have been referred by parents, patient groups, and medical professionals for various purposes such as confirming transparency and improving quality of perinatal care through scientific research. The revised Act on the Protection of Personal Information was enacted and promulgated in 2015 and fully enforced on May 30, 2017, which unprecedentedly forced the "Donor rule" applicable when we consider if the data we disclose on the web is "Personal information." "Personal information" shall be provided to third parties through the prior consent of an individual to which the data belong with some exceptions such as the data provision for promoting public health. The "Donor rule" states that the data is defined as "Personal information" when the
*The Japan Obstetric Compensation System for Cerebral Palsy: Novel System to Improve Quality… DOI: http://dx.doi.org/10.5772/intechopen.106760*
#### **Figure 3.**
*Production of standardized investigative report.*
donor of the data, i.e., the JQ can identify an individual from whom the data derive even if recipient of the data, i.e., the general public does not know whose data it is. Accordingly, the "Synthesized edition" that had been available on the website turned out to be "Personal information" that could be transferred to third parties principally through prior consent procedure. Therefore, the "Synthesized edition" posted on the website, for which consent of family and childbirth facility for the disclosure had not been obtained, had to be temporarily withdrawn from the website, and the Steering Committee took deeper dive into the issue from a broad perspectives such as purpose and impact of the disclosure and procedures required for the disclosure in consistent with the revised Personal Information Protection Act [12].
In January 2019, the JQ consulted with legal experts and the government officials again on this issue. In light of their comments and guidance, the JQ decided to make efforts in obtaining the consent of the guardians, the childbirth institutions, and the relevant medical institutions on all the "Synthesized editions" in response to the public concern on the system and the changing public view with regard to the handling of personal information, although the JQ believed that it fell under the exceptions for obtaining prior consent to the provision of personal information to third parties in the revised Personal Information Protection Act (**Table 4**). Later, when the JQ's policy on disclosure of the "Synthesized edition" was proposed at the 40th Steering Committee meeting held in January 2019, comments such as "all synthesized editions should be disclosed on the web as they were" and "The JQ should clarify the reasons for no-consent by guardians and/or childbirth facilities in detail" were raised from many of committee members.
#### **Table 4.**
*Article 23, paragraph 1, item 3 of the Personal Information Protection Act.*
Article 23
A business operator handling personal information shall not provide personal data to a third party without obtaining the prior consent of the individual, except in the following cases
<sup>(</sup>i)-(ii) Omitted. (iii) When it is particularly necessary for the improvement of public health or the promotion of the sound growth of children, and it is difficult to obtain the consent of the person concerned.
In February 2019, the JQ conducted a questionnaire survey targeting guardians and childbirth facilities to get hold of the reasons why they answered "agree" or "disagree" on the disclosure of the "Synthesized edition." At the 41st Steering Committee meeting held in August 2019, the JQ reviewed the aim and value of this system to consider if we should disclose all the "Synthesized edition" that achieves public good such as quality improvement in perinatal care as only about 3/4 of the "Synthesized edition" is agreed on the disclosure [13]. The JQ concluded at that time that it continued its efforts to improve the rate of consent on disclosure and consulted with the relevant ministries and the government to explore measures to disclose more "Synthesized editions" on the web.
In December 2019, the Personal Information Protection Committee in the government published the "Outline of the Amendment of the System for the So-called Triennial Review of the Personal Information Protection Act" (**Table 5**), and in January 2020, the Ministry of Health, Labor and Welfare (MHLW) presented a new commentary (**Table 6**). At the 42nd Steering Committee held in February 2020 and the 43rd Steering Committee meeting held on July 3, this issue was discussed to eventually compile an audacious policy on releasing all the "Synthesized editions" on the web. In the meantime, at the 94th meeting of the Investigation Committee
With regard to this point, the current Personal Information Protection Act has exceptions to the limitation of the purpose of use and provision to third parties, such as "when it is necessary for the protection of the life, body, or property of an individual and it is difficult to obtain the consent of the individual" and "when it is particularly necessary for the improvement of public health or the promotion of the sound growth of children and it is difficult to obtain the consent of the individual." The use of personal information for public benefit is also considered acceptable in certain cases. However, since there is a tendency that these exceptions have been strictly applied so far, it is necessary to provide specific examples in guidelines and Q&As according to the expected needs. Therefore, we will promote the utilization of personal information that benefits the entire nation, such as the resolution of social issues, by providing specific examples in the guidelines and Q&As according to the expected needs.
For example, a case in which a medical institution or a pharmaceutical manufacturer uses the information for the purpose of contributing to the development of medical research in order to realize healthcare services, drugs, and medical devices of high quality in terms of safety and effectiveness.
#### **Table 5.**
*The Personal Information Protection Law, dubbed as Triennial Review, Outline of System Revisions (December 13, 2019) (excerpt).*
In December 2019, the Personal Information Protection Committee released the "Personal Information Protection Act: dubbed as Triennial Review: Outline of Revisions," which also states that "the handling of personal information in the private sector is a matter for each business operator to decide. Therefore, it would be desirable for the JQ to consider the policy again, taking into account the balance between the promotion of public health and the protection of personal information. In addition, the MHLW has no objection if it is widely accepted by the society to disclose the summarized edition of all investigative reports as they were."
#### **Table 6.**
<sup>3.</sup> Clarification of the exception defined in the law pertaining to the handling of personal information for the purpose of public interest
With the rapid progress of information and communication technology, it has become possible to collect and analyze big data such as customer information. In this context, Japan is aiming to realize Society 5.0, which is a new society in which advanced technologies such as big data analysis are incorporated into all industries and social life to achieve both economic development and solutions to social issues. As social issues become more diverse, it is desirable to support an environment in which businesses can make use of data in order to efficiently and effectively solve these issues.
*Commentary issued to the JQ by the Ministry of Health, Labor and Welfare.*
*The Japan Obstetric Compensation System for Cerebral Palsy: Novel System to Improve Quality… DOI: http://dx.doi.org/10.5772/intechopen.106760*
held on June 10, the following comments to support the full disclosure were proposed: "In the Investigative Report, the causes of cerebral palsy are analyzed in detail and carefully for each case. So, they are worthwhile to disclose, and "All the "Synthesized editions" need to be disclosed on the web as they were. Accordingly, it was unanimously agreed that the "Synthesized edition" is published for all the Investigation Reports. From the viewpoint of improving public health, the publication of the "Synthesized edition" falls under the exceptions of Article 23, Paragraph 1, Item 3 of the Personal Information Protection Act as described above (**Table 4**). In addition, in order to prevent the recurrence of CPs, which is the purpose of the system, and to widely commit to quality improvement in perinatal care, the JQ believed that it was incredibly important to disclose all "Synthesized edition" on the web after a year-long argument over the disclosure under the revised Personal Information Protection Act. As there were some opinions that a certain level of agreement has been formed between the JQ and the family and childbirth facility who had disagreed on the disclosure, the JQ needed to make efforts to carefully convince families and childbirth facilities to agree on the new policy on disclosure. After all those discussions, the new disclosure policy was agreed in the Steering Committee [14].
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**4.1 Publication of annual prevention reports, educational materials on fetal heart rate monitoring, and leaflets for professionals and expectant mothers**
The investigative reports are collectively analyzed in order to prevent recurrence and to improve quality in perinatal care [15]. Here, we applied the knowledge and procedure devised through the medical adverse event reporting and learning system that the JQ had run for more than a decade to produce materials for effective prevention of CP through collective analysis. Specifically, the JQ conducted a quantitative and epidemiological analysis of aggregated cases to produce report for prevention on annual basis based on such data as status of pregnancy, clinical courses of pregnancy, delivery and neonatal condition, and the local context of healthcare delivery system [16]. The JQ also produced educational materials such as fetal heart rate monitoring textbook of profound CPs and leaflets for medical professionals and pregnant women [17].
#### **4.2 Scientific achievements of the Prevention Working Group**
Under the Prevention Committee, a working group for prevention, which consists of obstetricians nominated by the Japan Society of Obstetrics and Gynecology (JSOG) and the Japan Association of Obstetricians and Gynecologists (JAOG), as well as academic experts such as epidemiologists, was established in May 2014 that has carried out data analysis of the aggregated Investigative Reports. With the data, comparative study between the data of CPs that were subject to compensation in this system and that of the "Japan Society of Obstetrics and Gynecology Perinatal Registration Database" was conducted. In addition, an analysis of intrauterine infections and fetal heart rate patterns in children with CP was conducted in response to the requests mentioned in the Prevention Report to the relevant academic societies and organizations. The analyses have been
#### **Figure 4.**
*Educative book on CTG pattern of CPs.*
implemented in the working group from such multifaceted viewpoints as obstetrics and public health.
As the system requests that childbirth facility submit application with relevant documents such as medical chart, cardiotocogram (CTG) recording and so on, the system happened to provided experts an opportunity to scientifically look into CTG data through collective analysis. It is normally difficult in Japan to obtain CTG data of profound cerebral palsy as it suddenly happens at any childbirth facility across the country. Taking advantage of the considerable number of CTG recordings, the experts published an educational book on CTG interpretation, which is available on the website of the system (**Figure 4**) [18].
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**4.3 Impact on lawsuit statistics**
The purpose of the system is to prevent disputes and to improve quality in perinatal care through no-fault compensation, investigation/prevention. The lawsuit statistics of obstetrics and gynecology as a breakdown of "the number of completed lawsuit of entire medical specialty" published by the Committee on Medical Lawsuit of the Supreme Court of Japan shows a remarkable decreasing trend (**Figure 5**). The "Report on the Verification of the Speeding Up of Trials" published in July 2013 by the Supreme Court of Japan stated that: [19]
*"It is noteworthy that the Japan Obstetric Compensation System for Cerebral Palsy has brought investigative system by a third party and system of equally imposing financial burden for monetary compensation in Japanese society sharing the idea that perinatal care inherently holds potential risk.*
*It is concerned whether the system expands to cover other medical specialties.*
*The system having approved significant number of CP cases supposedly has affected to a certain extent statistics of lawsuit cases of medicine."*
As such, the system was hailed not only by medical society but by legal circle in Japan. As described in the report, medical professionals anticipated to expand
*The Japan Obstetric Compensation System for Cerebral Palsy: Novel System to Improve Quality… DOI: http://dx.doi.org/10.5772/intechopen.106760*
**Figure 5.** *Lawsuit statistics 2004–2020.*
the system or launch a similar system to cover more clinical specialties. However, there has never been emergence of desire in medical society comparable to the one observed in late 2000s that led to the launch of the system for cerebral palsy. Therefore, the expansion is still under discussion in academic society such as the Japan Surgical Society.
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**5. Review and overhaul of the system: 2015 and 2022**
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**5.1 Timetable for review and overhaul agreed at the inception of the system**
The Japan Obstetric Compensation System for Cerebral Palsy was launched in expedited manner in the wake of deteriorating perinatal care delivery system with challenges difficult to address at the time of the inception. Therefore, the report of the Introductory Committee stated that "the system will be verified in five years at the latest, and necessary revisions will be made to the scope of eligible patients, the amount of monetary compensation, price of insurance premium, and the governing structure of the system as appropriate. As such, periodical review and overhaul has been systematically embedded in the system" [20–24].
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**5.2 2015 Overhaul**
Accordingly, the Steering Committee of the system began deliberation over the review of the system in February 2012. The committee conducted fact-finding research including the collection and analysis of population-based data on the incidence of cerebral palsy, which is necessary for estimating the number of eligible patient and is crucial for re-designing the system. The results were reported to the Steering Committee in July 2013, and the committee and the Medical Insurance Subcommittee of the Social Security Council of the MHLW reviewed to revamp the system in terms of the scope of eligibility, the amount of monetary compensation, the price of insurance premium, and the way to spend surplus that had aggregated
since the launch of the system. The review concluded that the system expanded the scope of eligibility with the same amount of monetary compensation to be applied in January 2015 and later. As to how to spend growing surplus, the Medical Insurance Subcommittee agreed that the insurance premium was reduced to the price that work with the surplus to sustainably ensure budget for compensation. It was planned that the surplus was spent for the next 10 years by 2024 to consume it.
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**5.3 2022 Overhaul**
With the system being run carrying out the revised eligibility criteria, the Steering Committee meeting held on July 20, 2018, found that such issues had arisen as "more than 50% of the patients on case-by-case review were not covered by the system", "a sense of unfairness is spreading because some patients were covered and others were not despite of suffering commonly from CP with similar clinical course," and "the revised criteria has already been inconsistent with the latest knowledge from scientific viewpoint." Therefore, it was concluded that the system needed to be revised as soon as possible. In July of the same year, the Steering Committee submitted a request to the MHLW that the committee commenced the review of the system to overhaul because the MHLW is authorized to fix the price of childbirth lumpsum payment under the government regulation that substantially gives rise to financial source of compensation. Later in the year, the MHLW responded to the JQ claiming that the JQ listens to the voices of relevant parties such as healthcare-related entities, patient groups, and insurers, proposes the blueprint to reform the system, and reports the conclusion to the MHLW so that the MHLW would take necessary action for the reform.
With those dialogs between the JQ and the MHLW, the first round of the Committee on the Review of the Japan Obstetric Compensation System for Cerebral Palsy was held on September 2020. At the meeting, the items to be examined and reviewed were presented to the committee members such as "Efficiency in running the system," "Latest estimates of the number of eligible patients," "Price of insurance premium," "Eligibility criteria for compensation," "Financial resource for compensation," and "Price of compensation." The JQ engaged in Q&A session in the committee in exploring the expansion of the system, which was in line with the views of the most committee members who engaged in perinatal care. In addition to the agreement with members with healthcare background, it was necessary to make efforts to reach a unanimous agreement of the stakeholders including public health insurers involved in the meeting. Therefore, the JQ requested committee members and all those involved in perinatal care across the country for attention and support for the direction, i.e., expansion of the system that JQ proposed in response to the difficult reality in perinatal care delivery system.
The Committee compiled a report on the blueprint of the revision to submit to the MHLW subcommittee on healthcare insurance that works under the MHLW Social Security Panel. The subcommittee includes members such as healthcare insurers, academic experts, and representatives of healthcare professionals, industries, and labor unions. It endorsed the report in December 2020 that led to the launch of the revised system in January 2022 (**Table 2**).
#### **5.4 Future implication of the no-fault compensation system**
The Japan Obstetrics Compensation System for Cerebral Palsy, which was launched in 2009, celebrated its tenth anniversary in 2019. During this period, the *The Japan Obstetric Compensation System for Cerebral Palsy: Novel System to Improve Quality… DOI: http://dx.doi.org/10.5772/intechopen.106760*
system has made enormous achievements such as delivery of no-fault compensation for profound CPs, provision of investigative report to share both with families and childbirth facilities, prevention activities through collective analysis of aggregated investigative reports, and sharing plenty of scientific data on CPs gained through the system on a national scale. The system was reviewed 5 years after it was launched on a planned timetable produced initially. The review concluded that the system was run appropriately in line with the original objectives, such as provision of monetary compensation on no-fault basis, early resolution of disputes, and quality improvement of perinatal care through investigation and prevention. Then, the revised system was partly initiated in January 2014 on such details as procedure of investigation and adjustment of monetary compensation and damage payment, and in January 2015 on the rest of the details such as scope of eligibility to cover more CPs and other issues relevant to insurance. The JQ completed another review of the system to explore further expansion to cover more cerebral palsy cases in January 2022 and later. As seen above, the JQ believes that it is vital to improve the system in continued fashion through periodical review in cooperation with stakeholders.
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**Author details**
Shin Ushiro1,2
1 Japan Council for Quality Health Care (JQ ), Japan
2 Division of Patient Safety, Kyushu University Hospital, Japan
\*Address all correspondence to: [email protected]
© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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# Plastics in the Environment
*Edited by Alessio Gomiero*
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Plastics in the Environment
*Edited by Alessio Gomiero*
Published in London, United Kingdom
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*Supporting open minds since 2005*
Plastics in the Environment http://dx.doi.org/10.5772 /intechopen.75849 Edited by Alessio Gomiero
#### Contributors
Katrin Schuhen, Michael Sturm, Adrian Herbort, Alessio Gomiero, Pierluigi Strafella, Gianna Fabi, Daniela Berto, Federico Rampazzo, Claudia Gion, Seta Noventa, Malgorzata Formalewicz, Francesca Ronchi, Umberto Traldi, Giordano Giorgi, Elen Pacheco, Denis Ribeiro Dias, Maria Guimarães, Christine Nacimento, Célio Albano, Geovanio Oliveira, Mônica Andrade, Ana Sousa, Ana Silva, John A Glaser
#### © The Editor(s) and the Author(s) 2019
The rights of the editor(s) and the author(s) have been asserted in accordance with the Copyright, Designs and Patents Act 1988. All rights to the book as a whole are reserved by INTECHOPEN LIMITED. The book as a whole (compilation) cannot be reproduced, distributed or used for commercial or non-commercial purposes without INTECHOPEN LIMITED's written permission. Enquiries concerning the use of the book should be directed to INTECHOPEN LIMITED rights and permissions department ([email protected]).
Violations are liable to prosecution under the governing Copyright Law.
Individual chapters of this publication are distributed under the terms of the Creative Commons Attribution 3.0 Unported License which permits commercial use, distribution and reproduction of the individual chapters, provided the original author(s) and source publication are appropriately acknowledged. If so indicated, certain images may not be included under the Creative Commons license. In such cases users will need to obtain permission from the license holder to reproduce the material. More details and guidelines concerning content reuse and adaptation can be found at http://www.intechopen.com/copyright-policy.html.
#### Notice
Statements and opinions expressed in the chapters are these of the individual contributors and not necessarily those of the editors or publisher. No responsibility is accepted for the accuracy of information contained in the published chapters. The publisher assumes no responsibility for any damage or injury to persons or property arising out of the use of any materials, instructions, methods or ideas contained in the book.
First published in London, United Kingdom, 2019 by IntechOpen IntechOpen is the global imprint of INTECHOPEN LIMITED, registered in England and Wales, registration number: 11086078, The Shard, 25th floor, 32 London Bridge Street London, SE19SG – United Kingdom Printed in Croatia
British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library
Additional hard copies can be obtained from [email protected]
Plastics in the Environment Edited by Alessio Gomiero p. cm. Print ISBN 978-1-83880-492-3 Online ISBN 978-1-83880-493-0 eBook (PDF) ISBN 978-1-78984-045-2
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We are IntechOpen, the world's leading publisher of Open Access books Built by scientists, for scientists
4,100+ 116,000+ 125M+
Open access books available International authors and editors Downloads
Our authors are among the
151 Top 1% 12.2% Countries delivered to most cited scientists Contributors from top 500 universities
Selection of our books indexed in the Book Citation Index in Web of Science™ Core Collection (BKCI)
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Interested in publishing with us? Contact [email protected]
Numbers displayed above are based on latest data collected. For more information visit www.intechopen.com
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Meet the editor
Alessio Gomiero holds a PhD in Environmental Science. As a senior researcher, he has contributed to the development of integrated chemical and biological tools to routinely assess the environmental pollution associated with different anthropogenic activities. Emphasis is placed on the characterization of chemical-induced adverse effects by cellular and subcellular endpoints on several marine and freshwater biological models.
Contents
from the Adriatic Sea
*and Giordano Giorgi*
*by John A. Glaser*
in Composites for Automotive Industry
**Preface III**
**Chapter 1 1**
**Chapter 2 21**
**Chapter 3 37**
**Chapter 4 55**
**Chapter 5 73**
From Macroplastic to Microplastic Litter: Occurrence, Composition, Source Identification and Interaction with Aquatic Organisms. Experiences
Technological Approaches for the Reduction of Microplastic Pollution
Elemental Analyzer/Isotope Ratio Mass Spectrometry (EA/IRMS) as a Tool to Characterize Plastic Polymers in a Marine Environment *by Daniela Berto, Federico Rampazzo, Claudia Gion, Seta Noventa, Malgorzata Formalewicz, Francesca Ronchi, Umberto Traldi*
Study of the Technical Feasibility of the Use of Polypropylene Residue
*by Denis R. Dias, Maria José O. C. Guimarães, Christine R. Nascimento,*
*Ana Maria F. de Sousa, Ana Lúcia N. da Silva and Elen B. A. Vasques Pacheco*
*Celio A. Costa, Giovanio L. de Oliveira, Mônica C. de Andrade,*
Biological Degradation of Polymers in the Environment
*by Alessio Gomiero, Pierluigi Strafella and Gianna Fabi*
in Seawater Desalination Plants and for Sea Salt Extraction *by Katrin Schuhen, Michael Toni Sturm and Adrian Frank Herbort*
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"author": "",
"title": "Plastics in the Environment",
"publisher": "IntechOpen",
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"section_idx": 5
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Contents
*by John A. Glaser*
Preface
This book is a collection of reviewed and relevant research chapters concerning developments within the plastics in the environment field of study. The book includes scholarly contributions by various authors and is edited by experts pertinent to plastic pollution. Each contribution comes as a separate chapter complete in
The book consists of five chapters: (Chapter 1) From Macroplastic to Microplastic Litter: Occurrence, Composition, Source Identification and Interaction with Aquatic Organisms. Experiences from the Adriatic Sea, (Chapter 2) Technological Approaches for the Reduction of Microplastic Pollution in Seawater Desalination Plants and for Sea Salt Extraction, (Chapter 3) Elemental Analyzer/Isotope Ratio Mass Spectrometry (EA/IRMS) as a Tool to Characterize Plastic Polymers in a Marine Environment , (Chapter 4) Study of the Technical Feasibility of the Use of Polypropylene Residue in Composites for Automotive Industry and (Chapter 5)
This book will be interesting to various readers, researchers, scholars, and specialists in the field, who will find this information useful for the advancement of their
IntechOpen
itself but directly related to the book's topics and objectives.
Biological Degradation of Polymers in the Environment.
research work.
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"isbn": "9781838804930",
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Preface
This book is a collection of reviewed and relevant research chapters concerning developments within the plastics in the environment field of study. The book includes scholarly contributions by various authors and is edited by experts pertinent to plastic pollution. Each contribution comes as a separate chapter complete in itself but directly related to the book's topics and objectives.
The book consists of five chapters: (Chapter 1) From Macroplastic to Microplastic Litter: Occurrence, Composition, Source Identification and Interaction with Aquatic Organisms. Experiences from the Adriatic Sea, (Chapter 2) Technological Approaches for the Reduction of Microplastic Pollution in Seawater Desalination Plants and for Sea Salt Extraction, (Chapter 3) Elemental Analyzer/Isotope Ratio Mass Spectrometry (EA/IRMS) as a Tool to Characterize Plastic Polymers in a Marine Environment , (Chapter 4) Study of the Technical Feasibility of the Use of Polypropylene Residue in Composites for Automotive Industry and (Chapter 5) Biological Degradation of Polymers in the Environment.
This book will be interesting to various readers, researchers, scholars, and specialists in the field, who will find this information useful for the advancement of their research work.
IntechOpen
**1**
**Chapter 1**
**Abstract**
Adriatic Sea
addressing marine litter.
distribution models
**1. Introduction**
From Macroplastic to Microplastic
Litter: Occurrence, Composition,
Organisms. Experiences from the
Marine litter is human-created waste that has been discharged into the coastal or marine environment. "Marine debris" is defined as anthropogenic, manufactured, or processed solid material discarded, disposed of, or abandoned in the environment, including all materials discarded into the sea, on the shore, or brought indirectly to the sea by rivers, sewage, storm water, waves, or winds. A large fraction of marine debris is made up of plastic items. Plastic marine debris has become one of the most prevalent pollution related problems affecting the marine environment globally. The widespread challenge of managing marine litter is a useful illustration of the global and transboundary nature of many marine environmental problems. At a global level, plastic litter constitutes 83–87% of all marine litter. Land-based sources are estimated to be responsible for approximately 80% of marine litter. The largest portion of plastic associated with marine pollution is often linked to the contribution from terrestrial sources associated with accidental or deliberate spills as well as inefficient waste management systems in heavily anthropized coastal regions. This chapter is intended to serve as a catalyst for further discussion to explore the potential for developing a Mediterranean regional framework for
*Alessio Gomiero, Pierluigi Strafella and Gianna Fabi*
**Keywords:** plastic debris, Adriatic Sea, sediments, floating litter, sediments,
We live in the "Plastic Age". From its creation in the early 1870, plastic material has largely contributed to the society development making everyday life easier. Plastic material offer good advantages as it can be customized with specific shapes and chemical and physical properties i.e., elasticity, hardness, lightness, transparency and durability. Due to this, the production has dramatically boosted annual plastic production from 0.5 million tons in the 40s to 550 million tons in 2018 [1].
Source Identification and
Interaction with Aquatic
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From Macroplastic to Microplastic Litter: Occurrence, Composition, Source Identification and Interaction with Aquatic Organisms. Experiences from the Adriatic Sea
*Alessio Gomiero, Pierluigi Strafella and Gianna Fabi*
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2025-04-07T04:13:03.926933
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ffcbf675-2a25-491a-a3fe-16b210576719.10
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**Abstract**
Marine litter is human-created waste that has been discharged into the coastal or marine environment. "Marine debris" is defined as anthropogenic, manufactured, or processed solid material discarded, disposed of, or abandoned in the environment, including all materials discarded into the sea, on the shore, or brought indirectly to the sea by rivers, sewage, storm water, waves, or winds. A large fraction of marine debris is made up of plastic items. Plastic marine debris has become one of the most prevalent pollution related problems affecting the marine environment globally. The widespread challenge of managing marine litter is a useful illustration of the global and transboundary nature of many marine environmental problems. At a global level, plastic litter constitutes 83–87% of all marine litter. Land-based sources are estimated to be responsible for approximately 80% of marine litter. The largest portion of plastic associated with marine pollution is often linked to the contribution from terrestrial sources associated with accidental or deliberate spills as well as inefficient waste management systems in heavily anthropized coastal regions. This chapter is intended to serve as a catalyst for further discussion to explore the potential for developing a Mediterranean regional framework for addressing marine litter.
**Keywords:** plastic debris, Adriatic Sea, sediments, floating litter, sediments, distribution models
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**1. Introduction**
We live in the "Plastic Age". From its creation in the early 1870, plastic material has largely contributed to the society development making everyday life easier. Plastic material offer good advantages as it can be customized with specific shapes and chemical and physical properties i.e., elasticity, hardness, lightness, transparency and durability. Due to this, the production has dramatically boosted annual plastic production from 0.5 million tons in the 40s to 550 million tons in 2018 [1].
However, plastics sturdiness presents some negative implications as the increasing rate of plastic consumption worldwide its release in the environment associated with a low degradation rate is resulting in its accumulation in coastal and marine sediments, pelagic and benthic biota from coastal to open ocean areas at each latitude from the poles to the equator. Depending on sources and formation mechanisms plastic fragments are split into "primary" and "secondary". Primary plastics are resulting from the direct input of freshly manmade emissions, adding new micronized size by-design plastic material to the environment. According to this definition, major sources primary plastics are: (A) polymers intentionally produced and used as such. In this group belong i.e., personal care consumer products, industrial or commercial products and other specialty chemicals with plastic microbeads; (B) inherent collateral products of other industrial activities or (C) plastic sourced as accidental or deliberate spillage i.e., pellets loss from plastic factories and transport. In contrast, secondary plastics are associated as secondary pollution sources where larger plastic items undergo degradation and subsequent fragmentation leads to the formation of smaller plastic pieces as they start to break down by photooxidative degradation followed by thermal and/or chemical degradation [2].
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"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"book_id": "ffcbf675-2a25-491a-a3fe-16b210576719",
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"publisher": "IntechOpen",
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**2. Sources, degradation processes, detection of plastic debris in marine environments**
While addressing the comprehension of plastics degradation mechanisms in marine aquatic environments it is useful to divide them into plastics with a carboncarbon backbone and plastics with heteroatoms in the main chain. Some of the most environmentally recurrent polymers like polyethylene, polypropylene, polystyrene and polyvinylchloride have a pure carbon-based backbone. On the contrary, polyethylene terephthalate and polyurethane plastics have heteroatoms in the main chain. Most packaging materials are made of plastics with a carbon-carbon backbone structure. As they are very often discarded after a short period of time, there is a high potential to observe significant loading in the environment. All these polymers are susceptible to photo-initiated oxidative degradation, which is believed to be their most important abiotic degradation pathway in aerobic outdoor environments. This degradation pathway consists of a complex sequential multi-step process where initially chemical bonds in the main polymer chain are broken down by light, by heat or by a combination of both to produce a free radical formation [3, 4]. Polymer radicals react with oxygen and form a peroxy-radical species. As a side effect, the co-occurring formation of hydroperoxides promotes a further complex pathway of radical reactions leading to significant autoxidation of the target polymer. These processes ultimately lead to chain scission, branching and creation of oxygencontaining functional groups. As the molecular weight of the polymers is reduced, the material becomes fragile and is more vulnerable to fragmentation, which makes a higher surface area reactive to further degradation. Nevertheless, anti-oxidants and stabilizers used as additives inhibit the degradation of the polymer. Thus, degradation rates depend strongly on used additives and plasticizers [4]. In most cases these are well-known toxic chemicals not covalently bonded to the polymer and therefore capable of leaching out from the plastic during the degradation process, and easily enters into the aquatic environment representing a further point of concern for eco-toxicologists. On the other hand, different degradation mechanisms cause degradation of plastics with heteroatoms in the main chain. They show an increased thermal stability compared to polymers with a simple carbon backbone. Under marine environmental conditions the degradation processes of plastics like polyethylene terephthalate (PET) or polyurethane (PU) are normally controlled by hydrolytic
#### *From Macroplastic to Microplastic Litter: Occurrence, Composition, Source Identification… DOI: http://dx.doi.org/10.5772/intechopen.81534*
cleavage. Similar to carbon-carbon backbone plastic polymers, PET can undergo photo-induced autoxidation via radical reactions leading to the ultimate formation of a carboxylic acid end groups, which show a promoting effect on thermo- as well as photo-oxidative degradation. Weathering of PET in the marine environment occurs mainly by photo-induced oxidation and secondly by hydrolytic degradation processes which cause the yellowing of the polymer. For thermo-oxidative degradation the consequences are an in the content of the some end groups i.e., carboxylic acid as well as a general decrease in molecular weight of the main polymer [4]. Hydrolysis also leads to a reduction in molecular weight and an increase in carboxylic acid end groups. PET is highly resistant to environmental biodegradation because of its compact structure [4]. On the other hand, polyurethane-like compounds show carbon, oxygen and nitrogen in the main chain demonstrating enhanced susceptibility to degradation via photo-oxidation, hydrolysis and biodegradation. Plastic floating on the ocean surface is exposed to moderate temperatures, solar radiation at wavelengths of 300 nm and longer, as well as oxidizing conditions. Since temperatures are moderate, the most important factors initiating abiotic degradation are oxygen and sunlight. According to recent studies, fragmentation patterns first occur at the plastic surface, which is exposed and available for chemical or photo-chemical attack. The process is more efficient with smaller plastic fragments as they show a higher surface to volume ratio [5]. Changes in color and crazing of the surface are the initial visual effects of polymer degradation. Surface cracking makes the inside of the plastic material available for further degradation, which eventually leads to embrittlement and disintegration. Furthermore, almost all commercial plastics include additives. These co-production chemicals embedded in the polymers can also leach into the aquatic environment, which is an additional point of concern. As these substances enhance plastics' resistance to degradation, it becomes difficult to quantitatively estimate the fragmentation patterns since different plastic products can vary in their composition. On the other hand, additional factors can significantly influence degradation rates as floating plastic may develop biofilms that shield it from UV radiation. The formation of biofilm in plastic microliter collected from the marine aquatic environment has been previously documented worldwide [6–8]. Such phenomena could lead to a reduction in photo-initiated degradation. So far, there have been very few studies of degradation mechanisms for plastic polymers in the marine environment although some promising early findings have been reported by ongoing joint research initiatives (e.g., JPI-Weather Mic and JPI-PlasTox). The biofilm formation can also affect the vertical distribution of plastic fragments largely affecting their distribution in the water column or in the sedimentary environment. Most synthetic polymers are buoyant in water and substantial quantities of plastic debris that are buoyant enough to float in seawater are transported and potentially washed ashore. The polymers that are denser than seawater tend to settle near the point where they entered the environment; however, they can still be transported by underlying currents. **Table 1** resumes the theoretical densities of the most recurring polymers found in the environment. Microbial films rapidly develop on submerged plastics and change their physicochemical properties such as surface hydrophobicity and buoyancy [9, 10]. All in all, plastic debris is a mixture of molecules and chemicals, its size ranging from some meters to a few micrometers and probably nanometers. It is derived from a broad variety of origins, such as fishing gear, nets, bottles, bags, food packaging, taps, straws, cigarette butts and cosmetic microbeads and the associated fragmentation of all of these. Plastic debris has become ubiquitous in all environmental compartments of the marine ecosystem form sediments to sea surface. Thus, the observed loadings floating in the ocean represents only a limited portion of the total input. It has been previously reported that most plastic litter ends up on the seabed with a remaining fraction distributed on beaches or floating on the seawater surface leading one to
#### **Table 1.**
*Theoretical densities of the most recurring polymers found in the environment.*
consider that merely quantifying floating plastic debris may lead to a significant underestimation of the actual amount of plastics in aquatic environments [11].
### **2.1 The interaction of plastic debris with aquatic life**
Overall ecosystem health can be significantly affected by the accumulation of trash and plastics in our seas. Ingestion of and entanglement in marine debris directly impacts marine life. Laboratory studies provide a strong proof of evidence for the effects of microplastic ingestion observed in organisms collected from the natural environment. Indeed, in laboratories, under natural like conditions, microplastics have been shown to be ingested by amphipods, barnacles, lugworms and bivalves [12–14]. In the same organisms, the uptake of microplastics caused notable ultrastructural changes in the investigated tissues including histological changes as well as cell functioning impairments [15]. In field observations, the occurrence of MPs in the gastrointestinal tract and gills of pelagic and demersal fish and marine mammals has been documented [16, 17]. Past reports have shown that many marine organisms wrongly identify plastic debris for food. Ingestion of marine debris induce different deleterious effects such as pathological alteration, starvation and mechanical blockages of digestive processes. Furthermore, the interaction of plastic fragments, especially those at micrometric and nanometric scales, with organic pollutants are of importance in relation to environmental contamination and biological effects on organisms in the water column as well as in the sedimentary environment [18, 19]. Hydrophobic pollutants co-occurring in the aquatic environment may in fact adsorb onto MP debris. According to the different sizes, plastic fragments have the potential to transport contaminants more effectively through biological membranes and ultimately inside cells of aquatic organisms. The presence of organic pollutants on marine plastics has been illustrated for a wide range of chemicals in natural aquatic conditions [20, 21]. The exposure routes
*From Macroplastic to Microplastic Litter: Occurrence, Composition, Source Identification… DOI: http://dx.doi.org/10.5772/intechopen.81534*
of organic pollutant-enriched MPs are varied, while the toxicity is largely inversely correlated to the size of the particles, as the smaller the particle the further into the organism it can penetrate releasing toxic chemicals under acidic gut conditions [22]. According to the properties of the adsorbed chemicals, several toxicity mechanisms are represented by increased oxidative stress, genotoxicity, depletion of immune competence, impairment of key cell functioning, loss in reproductive performance, disorders in energy metabolism, and changes in liver physiology [23–25].
#### **2.2 Extracting microplastics from environmental matrices**
Different methods have been developed for identifying plastics, including meso, micro and nanoplastics in water, sediments and biota as well as to a lesser extent in soil. The percentage of organic matter (OM) in general as well as some recurring specific macromolecules, such as fats and proteins may hamper the analysis, thus hiding plastic fragments in visual analyses and distort signals in Fourier transformed infrared (FT-IR) and Raman spectroscopy, two of the most frequently used methods for plastic identification [26, 27]. Hence, identifying and quantifying plastic materials in organic matter enriched samples may be a challenge. In sediments, several available protocols recommend a preliminary sorting of plastic size grounding and sieving. After sieving, the mineral phase of soils might be removed easily using density fractionation methods. Different density solutions have been used including NaCl, ZnCl2, NaI and more recently 3Na2WO4 9WO3 H2O to obtain dense floating solutions [28, 29]. However, it has been shown that simple density fractionations will not succeed in separating organic matter from plastic materials in sediments because most of the OM show densities between 1.0 and 1.4 g/cm3 , similar to that of several environmentally recurring plastic types like PET, PP, PE and Nylon. Sufficient removal of OM without destroying small plastic polymers is challenging because large parts of OM are refractory. At the same time, polymers show strong sensitivity to acidic or strong oxidizing treatment conditions, which induce permanent modifications (e.g. yellowing), thus hampering their classification by microscope-oriented techniques. To efficiently remove OM, multistep extraction, purification processes based on alkaline treatments possibly combined with multi-enzymatic digestion steps have been suggested for the analyses of biota water or sediments. Enzymatic digestion has been promising for the removal of organic as well as other interferents, such as chitin, agar and lipid enriched samples [27]. Strong alkali digestions have been pointed out as being effective for sediments as well as biological samples, without altering the plastic itself [30]. While on the contrary and as previously mentioned, strong acidic conditions induce partial dissolution of polycarbonate as well as partial digestion of polyethylene and polypropylene [13]. Another largely exploited strategy to remove organic matter relies on the application of concentrated hydrogen peroxide [26]. However, its use must be critically evaluated in terms of digestion conditions as treatments with incubation exceeding 48 h with temperatures exceeding 50C, which may degrade plastic polymers like polyethylene and polypropylene [31]. In this context, some authors have recently suggested an effective combined multistep method based on a sequence of enzymatic digestions followed by a short hydrogen peroxide treatment for the removal of organic matter from complex environmental matrices (e.g., wastewater samples). In summary, several promising methods have been tested for extracting, purifying and pre-concentrating plastic materials from sediments and marine biota, all of them having potential limitations. More research is needed to develop a standard protocol for isolating plastics from a range of different environmental matrices, ideally at low cost and without altering plastic properties.
### **2.3 Overview of the most applied detection and quantification methods**
Once isolated, plastic fragments can be tracked and characterized by different analytical techniques. Some are defined as "surface oriented" methods like Raman spectroscopy, Fourier Transformed Infra-Red (FTIR), Scanning Electron Microscopy/Energy Dispersive X-Ray Spectroscopy (SEM-EDS) and environmental scanning electron microscope (ESEM) with an attached X-ray energy dispersive system (ESEM-EDS). Plastic fragments are visually sorted and analyzed coupled with microscopy. However, as discussed above, the use of strong oxidant/acidic agents applied during the extraction from sometimes complex environmental matrices (e.g., organic matter enriched marine sediments, or fat rich marine biota) may induce alteration in the plastic surface like partial dissolution, yellowing and polymer structure disruption leading to erroneous characterization of microparticles. Furthermore, some compounds of natural origin occurring in marine samples (e.g., chitin) have shown spectroscopic properties similar to those of the most recurrent plastic polymers leading to inaccurate polymer characterizations and overall abundance estimation. In addition, these microscopy-based techniques are time consuming and unable to process large numbers of samples. However, significant advances in the automatic and semi-automatic FTIR spectra recognition have been recently presented as promising time saving solutions (Jes recent paper). Alternatively, promising solutions include the Pyrolysis-gas chromatography in combination with mass spectrometry (Pyr-GC-MS) as well as the Thermogravimetric analysis coupled with mass spectrometry (TGA-MS). Pyr-GC-MS in particular can be used to assess the chemical composition of potential microplastic particles by analyzing their thermal degradation products. The polymer origin of particles is identified by comparing their characteristic combustion products with reference pyrograms of known virgin-polymer samples. Py-GC/ MS had the advantage of being able to analyze the polymer type and OPA content in one run without using any solvents and with few background contaminations. Additionally, the Pyr-GC/MS method has an appropriate degree of sensitivity for analyzing plasticizers in microplastic particles with limited sample masses. However, although the pyrolysis-GC/MS approach allows for a good assignment of potential microplastics to polymer type it has the disadvantage of being a "destructive" technique as the sample is burned to obtain the pyrolytic products. Furthermore, due to limitations in the quantity of sample loaded in the pyrolysis cup only particles of a certain minimum size can be processed resulting in a lower size limitation of particles that can be analyzed. Each of these methods have their own limitations and advantages, therefore, their combined use, especially for the analysis of complex environmental samples, is a recommended strategy to reduce the effect of interferents in the analysis and obtain reliable results.
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**3. The Mediterranean and the Adriatic Sea**
With some of the most significant amounts of solid waste generated annually per person (208–760 kg/year), the Mediterranean Sea is one of the world's areas most affected by litter [32]. The estimated amount is 62 million of macrolitter items floating on the surface of the whole basin [33]. Litter enters the seas from land-based sources, ships and other infrastructure at sea and can travel long distances before being deposited on the seabed or along the coasts. Mean densities of floating microplastics in the Mediterranean Sea of more than 100,000 items/km2 [34] indicate the importance of this threat for the basin. In this context, the Adriatic Sea represents a hot spot for plastic litter both because of peculiarities in its oceanographic
*From Macroplastic to Microplastic Litter: Occurrence, Composition, Source Identification… DOI: http://dx.doi.org/10.5772/intechopen.81534*
conditions as well as the high degree of anthropogenic pressure related to tourism, artisanal and industrial activities coexisting in a narrow area. The Adriatic Sea is an elongated basin, located in the central Mediterranean, between the Italian peninsula and the Balkans, with its major axis in the NW-SE direction. The northern area is very shallow, gently sloping, with an average depth of about 35 m, while the central part is on average 140 m deep, with the two Pomo depressions reaching 260 m. The northern and central parts of the basin are affected by a great number of rivers along the Italian coast, of which the Po river is the most relevant. River discharge and wind stress are the main drivers of the water circulation. West Adriatic Current (WAC), flowing SE along the western coast, and East Adriatic Current (EAC), flowing NE along the eastern coast are the main currents affecting the Adriatic circulation. There are two main cyclonic gyres, one in the northern part and the other in the south. The Bora wind (from NE) causes free sea surface to rise close to the coast enhancing the WAC and the Sirocco wind (from SE), which is the major wind affecting the Adriatic Sea, leads flood events in the shallow lagoons along the basin coast [35]. A vertical thermohaline front parallel to the coast and extending throughout the water mass, divides the coastal waters from the open sea. This retains the materials flowing from rivers and other water sources within the coastal area. A stratification characterizes the water column separating the warmer surface waters with lower salinity from deeper, colder and more saline ones during summer [35].
### **3.1 Marine sources of plastic pollution**
### *3.1.1 Plastic products in aquaculture and fishery*
Across the Mediterranean, but in the Adriatic Sea in particular, there is a continued demand to increase aquaculture production to fulfill the increasing market demand. Mussels, clams, sea bass and seabream production has become a significant source of regional income. Aquaculture was developed to support consumers' demand for seafood and the methods of production have continued to expand with the growing consumer market. As the need for fish and mussel aquaculture has increased, the development and expansion of aquaculture facilities in coastal and open water locations has increased accordingly. The expansion of the industry and the diversity of materials used to build and maintain aquaculture systems have paralleled the development of synthetic polymers over recent decades. Synthetic fibers offer greater strength and durability than natural fiber ropes; they are cheap, durable and easier to handle compared to their natural counterparts. Most modern aquaculture activities use plastic-based lines, cages, or nets suspended from buoyant or submergible structures (in part made of plastic) and have nanotech plastic-based biofouling and paint applied. Today, tanks, pens, nets, floats, pontoons as well as the pipes of the fish feed supplying systems are made of plastic materials. All plastic material within an aquaculture site is maintained and controlled for chemical degradation, biofouling and corrosion, and is regularly inspected to ensure strength and stability. In the context of global plastic pollution to the oceans, aquaculture may be a contributor to this. However, the estimation of their contribution remains a knowledge gap and lost or derelict gear as well as other possible plastics emissions from aquaculture can be a locally important contributor especially in coastal areas with intensive activity. New reports also point out a potential micro and nanoplastic contamination in wild and cultured seafood products even if the extent of such phenomena is still unknown. There is also concern regarding fisheries as a source of microplastics to the marine environment because both sectors use plastics that may degrade/fragment into microplastics. The coastal areas of Emilia Romagna and the Croatian coast represent sites of intense mussel
and fish aquaculture production with hundreds of tons produced yearly. On the other hand, intense fishing activities coexist with a variety of fishing gear and methods being used in industrial and small-scale fisheries. Fishing gear for capture fisheries includes trawl nets, dredges, surrounding nets, lift nets, seine nets, traps, hook and lines. Nets and floats are made from a range of plastics including PP, PET, NyL, PVC, polyamide (PA) and PS.
#### *3.1.2 Offshore oil and gas production activities*
In oil and gas exploration, drilling fluids based on plastic microbeads were introduced a decade ago. Teflon strengthened particles have been largely applied for drilling purposes internationally. Despite the use of Teflon and other polymers with specific features being used extensively in production, waste treatment processes are not designed for, and give no mention of how to handle plastic particles, so this has clearly not been addressed as an issue in the past. Therefore, there is a substantial lack of information on potential loadings of microplastics used in this sector. To date, few fragmentary studies have addressed this topic. CEFAS's report entitled, "The discharge of plastic materials during offshore oil and gas operations" suggests that 532 tons of plastics and 7475 tons of "possible plastics" have been released from the UK offshore oil sector. Although knowledge about microplastic from oil and gas extraction activities is limited, it is very likely they represent a potential contributor in the emissions of plastics in aquatic environments, including microplastic and fibers, emphasizing that it should certainly be considered in future source assessments. The mapping of the distribution of rigs and platforms in the Adriatic Sea where tens of oil fields with hundreds of medium sized oil rigs occur, may provide estimations about the geographic distribution of the potential input related to these industrial activities.
#### *3.1.3 Decommissioning of ships and oil rigs*
Ships and maritime installations contain many plastic items, like insulation, coating, electrical wiring, furniture and textiles. Ideally, installations should be stripped of all potentially hazardous materials before dismantling. However, plastics items are not identified in the list of harmful materials. Therefore, polymerbased coatings and several kinds of insulation and wiring are rarely stripped.
#### *3.1.4 Transportation and logistics*
The distribution of products can contribute to the release of plastics in the environment. Most transferring of stock will occur alongside the transport infrastructure network. However, even if recognized as an important source of pollution, the contribution from releases during transportation, and as is the case for shipping, a map of the main transportation network including roads and harbors is still lacking. Systematic mapping in the Adriatic context has been suggested to improve the understanding of the areas where potential inputs can occur, providing a proxy for the potential intensity for release. The Adriatic Ship Traffic Database also contains information on ports in the Adriatic Sea that could be used to gauge the intensity of port activity to identify which of the port areas could potentially be receiving the largest inputs. Furthermore, the cruise ship industry is pointed out as a significant contributor to the problem of plastic pollution in the Adriatic sea. However, very limited data are available and no specific regulations in place for their plastic waste management and/or assessment of their environmental impact [36].
*From Macroplastic to Microplastic Litter: Occurrence, Composition, Source Identification… DOI: http://dx.doi.org/10.5772/intechopen.81534*
#### **3.2 Land-based sources**
### *3.2.1 Waste management*
At a global level, the major challenge to tackle the input of plastic debris from land into the ocean is the lack of adequate waste management in coastal regions with a high and growing population density. Due to a generally high population density in coastal areas of the Adriatic, the pressure resulting from land-based inputs should be relatively high overall. Given such levels of anthropogenic pressure, the lack of, or deficient local waste management systems may lead to locally high inputs linked to industrial or domestic waste management.
There are no studies looking specifically at the leakage and marine input of plastic debris linked to these waste management systems, but ongoing work to quantify and characterize beach litter here points toward potential input from inadequate waste management on the eastern shores of Croatia where the islands of the Quarnero natural park present high loadings of plastic fragments. The composition of the waste accumulated resembles the composition of surveys carried out in the mid-Adriatic region where influence from higher population densities along the coastline is being registered. In addition, a study looking into microplastics near Venice has detected exceptionally high concentrations of small plastic fragments and microplastics in a nearby sandy beach [52]. Though not specified in this report, this exceptionally high concentration of microplastics, including large amounts of plastic fibers and film, could be linked to this location being close to the harbor as well as the lack of waste management facilities. To gain further insight into the potential release of plastics associated with waste management, it would be useful to map the distribution of population density as well as the location of urban agglomerations and settlements as this information will provide an indication of potential localized points of release of plastic waste into the environment. This kind of information is readily available at a sufficient resolution to allow identification of the areas within the Adriatic Sea that need more attention to this potential source of plastic pollution.
#### *3.2.2 Sewage treatment plants*
A rough estimation predicts that 70–80% of marine litter, composed primarily of plastics, originate from inland sources, ending in rivers and oceans. However, inland deposition of MP has not been investigated thoroughly. Potential sources include sewage treatment plants (STPs) and runoff from urban, agricultural, tourist, and industrial areas. As the retention capacity of conventional wastewater treatment processes to MPs appears to be variable in both magnitude and specificity, a characterization of MP emission by STPs and other sources is needed to map major sources of freshwater and terrestrial MPs. A relevant input to the terrestrial ecosystem is by fertilizers obtained by processing sewage sludge, as it typically contains more MPs than liquid effluents. Such fertilizers are frequently used in agriculture, implying a potential accumulation of plastic particles in the soil with continued use, and a systematic examination and quantification has been addressed by several research groups around the world. However, due to runoff, deposited plastic items are most likely transported to rivers and other waterways and ultimately discharged into estuarine and marine environments.
#### *3.2.3 Agricultural production*
The north of Italy and Croatia represent areas of intense horticultural activities where the agricultural practice of plastic mulching is prevalent. Plastic sheets are
used to cover soil in order to preserve moisture, improve fertility and reduce weed infestation. Very often, fragments of plastic films are left behind after use and may accumulate in the soil, further fragmenting to produce nanometric particles. It has been estimated that 125–850 tons of microplastic per million inhabitants are added each year to agricultural soils in Europe, with an annual total of 63,000– 430,000 tons of microplastic added to European farmlands. The northern part of Italy and Croatia is an area of significant agricultural and horticultural activities, therefore representing a potential hot spot for the release of plastic fragments in the terrestrial ecosystem. However, due to runoff phenomena these plastic items are most likely transported to rivers and other waterways and ultimately discharged into the estuarine and marine environments.
#### *3.2.4 City dust and road wear*
The first pilot studies of microplastic abundance in confined areas of heavily populated areas like the Oslo fjord noted that a large fraction of particles may be related to city dust (e.g. asphalt and car tires). City dust in urban runoff is known as a significant source of pollution to waterways. Plastics, such as styrene-butadiene, styrene-ethylene-butylene-styrene copolymer, are also used in road materials to make the asphalt more elastic [37]. Another potential contributor to the emissions of plastic fragments is road marking paint as these paints have a variable fraction (1–10%) of thermoplastic component (e.g. styrene-isoprene-styrene, ethylenevinyl acetate, polyamide and acryl-monomer). On the other hand, the tread of car tires is largely based on styrene-butadiene rubber, a synthetic polymer formulation. Therefore, road dust entering the sea through air or storm water carries a significant fraction of microplastic from road materials, marking paint and car tires.
#### **3.3 Pathways and distribution**
The description and understanding of the pathways of the entry of marine plastic pollution into the Adriatic Sea is a central element in tracing the pollution back to its sources and developing effective plastic pollution preventing policies. A complete understanding of the input of plastic pollution into the aquatic environment needs to consider the source sectors and the mechanisms of transportation, distribution and partition through different environmental matrices. If the release occurs in the terrestrial environment, rivers and wind or atmospheric circulation constitute the logic pathways. When considering the presence of plastic debris and microplastics in a part of the global Mediterranean Sea there is a need to consider the transfer of marine plastic pollution into the relevant part of the large water bodies through the regional circulation pathway like the Adriatic Sea. The understanding of the input through these pathways is crucial in gauging the relative importance of local sea-based or coastal sources versus remote sources within the Arctic watershed or from other parts of the ocean.
#### *3.3.1 Riverine input*
The Adriatic Sea has a limited watershed. The largest rivers in the area are mostly located in the northern sector and include the Po, Adige, Tagliamento, and Arsa rivers. In terms of discharge, the Po River has the largest discharge with 1540 m3 /s followed closely by the Adige River with 235 m3 /s. The Po Basin is home to some 14 million people and extends over 24% of Italy's territory. The Po catchment is densely populated and subjected to high anthropogenic pressure heavily anthropized. Indeed, it represents the largest cultivated area in Italy and accounts for one third of national's
#### *From Macroplastic to Microplastic Litter: Occurrence, Composition, Source Identification… DOI: http://dx.doi.org/10.5772/intechopen.81534*
agricultural production. The area account also for one of the highest concentrations of economic activities. Such massive river discharges make terrestrial influences particularly strong in the Adriatic Sea. However, to date there is no monitoring of the flux of plastics from rivers into the Adriatic Sea and though it has been identified as a possible pathway, the contribution of riverine discharge to plastic input is expected to be high because these rivers flow through densely populated and anthropized watersheds.
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*3.3.2 Atmospheric input*
It has been speculated that at the global level much less plastic debris is transported by wind than by rivers [38, 39]. However, wind transport of plastic debris may be significant, particularly in coastal areas dominated by strong periodic winds. Wind may be a significant contributor in lightweight debris distribution. During intense storms wind can mobilize debris that would not normally be available for transport and carry it directly into rivers and the sea. Wind-blown litter is likely to be considerable as the Adriatic Sea is characterized by periodically windy shorelines. Atmospheric circulation has been proven to provide an efficient pathway for the transportation of floating microfibers and small plastic particles in the Mediterranean Sea as well as in other areas [33, 40]. Furthermore, some preliminary transport models tailored to the Adriatic oceanographic conditions, considering the contribution of waves and wind in the surface plastic distribution, define the Adriatic Sea as a highly "dissipative" system with respect to floating plastics with a calculated half-life of floating condition of 43.1 days [41, 42]. The authors conclude by pointing out that by construction the Adriatic coastline may be responsible for the main sink of floating plastic debris.
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*3.3.3 Oceanic input*
The contribution of inputs through the movement of marine water masses by currents also needs to be considered in the global distribution model. The Adriatic region is poorly connected to the Mediterranean through the southern edges of the Otranto strait and the Ionian Sea exchanging with the Mediterranean Sea. The exchange of water, and possibly any moving plastic pollution, from and to the Mediterranean Sea has recently been addressed by the modeling work of Liubartseva et al. [40] and partially by the results of Pasquini et al., [40] which pointed out the formation of an accumulation zone corresponding to the three well known gyres located northside, central and in the southern sector of the Adriatic Sea.
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**4. Occurrence of plastic litter in the Adriatic Sea**
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**4.1 Levels of macro- and microlitter in beaches**
Some key research projects have recently addressed the need of defining the baseline levels of litter (macro-, meso- and microplastics) in the intertidal areas of beaches within the Adriatic Sea. Blašković et al. [41] investigated the occurrence of plastic debris in several sites of the Natural Park of Telaščica (Croatia). In all analyzed sites, fibers were the most recurring shape (90%) within the identified plastic debris while films where the second most common plastic fragment observed (7%) followed by pellet, foams, granules and unrecognized plastic pieces. Most of the plastic debris belonged to the size fraction from 1 mm and 64 μm (88%) followed by the fraction between 1 and 2 mm (11%). These results confirm previous characterization efforts of Laglbauer et al. [43] in six Slovenian beaches located in the gulf
of Trieste (North-East Adriatic Sea). Within this assessment the authors sorted out a total of 5870 macro-debris units, yielding a median density of 1.25 items/m2 . The detailed analyses of the processed samples revealed a dominant secondary microplastics source being fibers the 85% of the total observed plastics and a number of 155 particles m2 in the infralittoral zone, and 133 particles m2 on the shoreline. On the Adriatic beaches surveyed, plastic dominated in terms of abundance, followed by paper and other groups. The average density was 0.2 litter items m2 , but at one beach it raised to 0.57 items m2 . Among plastic, cigarette butts were the most frequently found type of litter, and other plastic items with the highest occurrence were: small fragments, bottles and bottle caps, cutlery, and mesh bags. Their presence is a good indicator of pollution from beach users [44]. Most of the beached marine litter are from land-based sources, but with different sources and contributors. The main source of litter was primarily touristic activities, accounting for 37.9% of found litter which is lower than r the Mediterranean average (52%; [45, 46]). Filter cigarette were the second litter origin, but with a value (25.5%) lower than indicated for the Mediterranean (40%) [44]. The high percentages of in situ deposited litter found in the investigated sites are caused by the high number of visitors, more than 700,000 annually mainly during the touristic season (see i.e., http://statistica.regione.veneto. it; http://imprese.regione.emilia-romagna.it).
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**4.2 Levels of macro- and microlitter in surface waters**
Few studies have addressed the occurrence of floating plastic debris in the surface water of the Adriatic Sea. Suaria et al. [33] reported by a larger study addressing the Mediterranean Sea and partially the Adriatic sector a clear prevalence of smaller particles. Quantitative estimations collected by a 400 μm net mesh pointed out values ranging from 0.4 ± 0.7 to 1.0 ± 1.8 items/m3 . The overall result the study pointed out that, within a total no. of 14,106 scored particles, 26% of all counted particles were smaller than 300 μm while 51% were smaller than 500 μm being the mean abundance of these meso-particles of 0.016 ± 0.028 particles/m<sup>2</sup> . PE was the predominant form with an overall frequency of 52%, followed by PP (16%) and synthetic paints (7.7%). Polyamides (PA) accounted for 4.7% of all categorized particles which accounted alone for 2%), while PVC, PS and PVA represented equally contributed with 3% of the total. Other less frequent polymers (<1%) included: PET, polyisoprene, poly(vinyl stearate) (PVS), ethylene-vinyl acetate (EVA) and cellulose acetate. Noteworthy the authors concluded that the composition of western Mediterranean samples was dominated by low-density polymers such as polyethylene and polypropylene while the processed Adriatic samples instead were more heterogeneous and rather characterized by a higher presence of paint chips, PS, PVC, PVA and PAs. Within the "Derelict Fishing Gear Management System project – "DeFishGear" project co-funded by IPA-Adriatic Cross-border Cooperation Programme and the European Union, 120 visuals transect surveys were conducted during three cruises, covering a total length of 922.2 km [47]. A total of 1364 macro marine debris objects were observed floating on the Adriatic. The densities of the recorded floating debris were 5.66 items/km2 . The authors estimated that the observed floating marine debris was mostly originated from coastal segments close the high-density population cities and major rivers and transported by cyclonic surface circulation until either stranding. They calculated an average time from source to the sighting point of 22.8 days. These outcomes support Carlson and co-workers [48] previous assessment where an average residence time of 22.9 days but with also an average transit times of 20–60 days from a coastal region in the northwest Adriatic to a coastal region in the southwest [47]. The transport pathways, residence times, and probable sources and sinks identified further
support with previous studies of the Adriatic Sea surface circulation and marine debris published by Liubartseva et al., [40].
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**4.3 Levels of macro- and microlitter in sediments**
Data regarding macro- and mesolitter on the sea-floor in the Adriatic Sea are also available from the "SoleMon" Project (Solea Monitoring—Rapido trawl survey in the Northern Adriatic Sea), carried out since 2005 in the Northern and Central Adriatic Sea [49]. Plastic litter was divided by the authors in three sub-categories based on its source: fishing nets, mussel culture debris and other plastic e.g., bottles, plastic glasses, bags. Lost fishing nets and mussel culture debris accounted for 50% of the overall plastic litter collected over the investigated period. The remaining plastic comprised a wide range of objects such as garbage bags, shopping bags, cups, bottles, food packaging, dishes, other kitchen stuffs and industrial packaging [40, 48]. Results of this study indicated that the largest amount of mussel culture debris was found close to the coast and its distribution was constant over the years. These nets might have been accidentally lost/ abandoned at sea during the collection and preparation of the product [50]. In the meantime, the fishing nets were found mainly close to the coast within 3 nm. This distribution was explained as fishing nets were mainly set-nets used by small scale fisheries that usually fish not further than 3 nm where there is not trawl fishing that can destroy these nets. A significant contribution of plastic litter found close to the coast was represented by food packaging, plastic bags, bottles and dishes or kitchen tools. The land origin is due to the municipal solid waste [48]. The authors concluded considering that the distribution varied among the years, but the occurrence was mostly related to both the close position of the sampling site to large cities along the coast, where the population density increases during the touristic season as well as the contribution of river [40, 50, 51]. As regards the microliter in the sedimentary environment, a preliminary assessment of microplastics in marine sediments along a coast- off-shore transect in the Central Adriatic was performed by Munari et al. [44]. Plastic fragments recollected from 64 samples were scored, weighted and identified by FTIR. Microplastics ranging 1–30 mm were found in all analyzed samples. The most recurring shapes were filaments-like (69.3%), followed by fragments-like (16.4%), and film-like (14.3%). In term of size distribution, plastic fragments in a range from 1 to 5 mm accounted for 65.1% of debris, while larger fragments (5–20 mm) contributed with the 30.3% of total amount, while larger fragments >20 mm represented the 4.6% of total. Six were the most recurring polymer types: nylon, polyethylene and ethylene vinyl alcohol copolymer. Furthermore, sediments from several sampling sites located in Italy, Slovenia, Croatia, and Greece were also analyzed for plastic debris content by the "DeFishGear" project. Plastic fragments in beach sediments were ranked into large sized particles (1–5 mm) and small microplastic particles (<1 mm). In general, microplastic from 1 to 5 mm ranged from 11 to 710 items/m<sup>2</sup> . On the other hand, the fraction of smaller size scored from 70 to 6724 items/kg of dry sediments. The mean concentration for all Adriatic region was calculated as 113 ± 101 items/ kg for the larger sized fragments and 1133 ± 1271 items/kg of dry sediments for the smaller ones. In detail, the selected Croatian beaches showed considerably greater presence of smaller microplastic per kg of sediment with value of approx. 227 items/kg of sediment while the larger sized fragments sored values approx. Ten times lower (17–28 items/kg of dry sediments). The composition of sorted fragments <1 mm showed the prevalence of plastic fragments as fragments represented approx. 70% of the total while filaments represented the left 29% of the total while a limited amount (1.8 and 0.9%) were film and foams. The chemical
characterization of microplastic of the larger particles was performed on foams, pellets, fragments and filaments, while filaments and films were analyzed among the smaller sized particles. Beside the PE and PP in a few percent also PA, PET, PES, PS, PO, nylon and acrylic fibers were present among larger particles, while among the smaller viscose was detected. In the Greek sector data were obtained from three sites: the Halikounas, Issos and Acharavi beaches. The mean concentration of 1–5 mm sized debris varied from 68 items/m<sup>2</sup> (Halikounas) to 58 items/m<sup>2</sup> (Acharavi) while the small sized fraction of Ø > 1 mm showed values from 19 to 7 items/m<sup>2</sup> respectively for Halikounas and Acharavi. The most abundant categories on Halikounas beach were fragments and foam, while on the contrary pellets were the most abundant in Issos and Acharavi beaches. Chemical characterization of fragments, for Halikounas beach were done being both PE and PP the most recurring polymers in the larger particles while PP was the most occurring polymer in the smaller size fraction. The same project also addressed the occurrence in the Italian sector. High amount of small microplastic particles (<1 mm), up to 2526 items/kg of sediment, was found in the Cesenatico area. In the meantime, a limited amount corresponding to 0.56–1.02 items/kg of large particles (1–5 mm) were reported. Overall, 73% of the small microplastic particles were characterized by fragments while the remaining 26% as filaments. On the other hand, the large microplastic particles had different amount of all categories; however, fragments resulted the most abundant category (44%). The chemical identification showed PE as the most abundant material, followed by PP, PO, PES, PS and PAN. In the Slovenian coastline the selected sampling site showed a higher abundance of small microplastic particles (615 items/kg) respect of large microplastic particles (516 items/kg). In detail, the analysis of the small size fraction reported filaments being the predominant type of the microplastic composition, with representation of approx., 76% of the total. The second most common type of microplastic category were fragments and the third were films, with occurrence high as 9.5%. The chemical identification pointed out PE as the most recurring polymer type in the analyzed sediment samples, followed by PP, PET and PVC. Finally, Vianello and co-workers investigated the Venice Lagoon, a fragile estuarine ecosystem dominated by diversified anthropogenic activities, suspected to be a hot spot of plastic debris contamination [53]. Plastic debris of ≤1 mm or less was investigated in sediments collected from 10 sites chosen in shallow areas. Total abundances of plastic fragments varied from 2175 to 672 items/kg with higher concentrations generally found in the inner parts of the Lagoon. PE, PP, ethylene propylene (PEP), polyester (PEst), polyacrylonitrile (PAN), PS, alkyd resin (Alkyd), PVC, polyvinyl alcohol (PVOH) and NyL were identified. PE and PP were the most recurring polymer in the investigated samples which accounted for more than 82% of the total detected plastic debris in the whole sampling area. Among all classified shapes, irregular fragments accounted of the 87% of the total while films (2%) and pellets/granules (1%) were only occasionally recognized [54].
#### **4.4 Levels of microliter in biota**
The first report on the harmful effects of plastic debris ingestion on marine species in the Adriatic Sea was published in 1999 [55]. A dead dolphin *S. coeruleoalba* with the stomach occluded by different kinds of plastic materials was found near the island Krk, in the North Adriatic Sea. A following study on the logger head sea turtles, *C. caretta*, revealed a percentage of 35.2% of turtles sampled in the eastern Adriatic Sea were affected by plastic debris [55]. Occurrence of MPs in the gastrointestinal tract and gills of pelagic and demersal fish and marine mammals have been reported [56]. Few plastic debris accumulation studies have been performed in
*From Macroplastic to Microplastic Litter: Occurrence, Composition, Source Identification… DOI: http://dx.doi.org/10.5772/intechopen.81534*
the Adriatic Sea. Pellini et al. [57] aimed at characterizing the occurrence, amount, typology of microplastic litter in the gastrointestinal tract of a benthic fish, *S. solea*, in the northern and central Adriatic Sea. The digestive tract contents of over 500 individuals were collected from 60 sampling sites and examined for microplastics. These were recorded in 95% of sampled fish, with more than one microplastic item found in around 80% of the examined specimens. The most commonly found polymers were PVC, PP, PE, polyester (PES) and PA. In details, 72% of the total classified plastic debris were fragments and 28% were identified as fibers. The mean number of ingested microplastics was 1.6–1.7 items/fish. PVC and PA showed the highest densities in the northern Adriatic Sea, both inshore and off-shore while PE, PP and PET were more concentrated in coastal areas with the highest values offshore from the port of Rimini. These results confirm previous observations of Avio and co-workers [13] in various fish species collected along the Adriatic Sea. FTIR analyses indicated PE as the predominant polymer (65%) in the stomach of fish. More than 100 fish representatives of five commercial species like *S. pilchardus, S. acanthias, M. merlucius, M. barbatus C. lucernus* were collected from the Central and North Adriatic Sea. The mean number of ingested microplastics was 1.0–1.7 items/fish. In details, the shape of the plastic debris observed in the stomachs of the investigated samples was mostly fragments and line followed by film and pellet. The 18% of extracted microplastics exhibited the larger size class (from 5 to 1 mm), 43% was between 1 and 0.5 mm, 23% between 0.5 and 0.1 mm, and the 16% lower than 0.1 mm. The chemical characterization pointed out that approximately 65% of analyzed plastic fragments were PE, followed by PET, PS, PVC, Nylon and PP. These early findings suggest the possible accumulation of plastic debris through the food web. Despite of some recent findings point out that at the bottom of the food pyramid, filter feeders, such as mussels can ingest and incorporate MPs in their tissues [58], more research is needed to unveil the abundance, distribution and polymeric composition of plastic debris in marine organisms at different levels ecological web in areas like the Adriatic Sea were multiple anthropogenic activities coexist.
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**5. Conclusions**
The few available studies in the area prove the ubiquity of plastic pollution in the Adriatic Sea. The peculiar oceanographic conditions as well as the high levels of plastic debris recorded in all investigated matrices tend to classify such enclosed area as a hot spot of plastic contamination. Despite the distribution and circulation models appear to accurately estimate fluxes and final fate of marine plastic debris, sinks, sources, fate and residence times of different polymers at sea are the knowledge gaps that need to be addressed in the future to provide concrete info to support concrete actions toward plastic contamination reduction and remediation solutions.
## **Acknowledgements**
The authors wish to thank The International Research Institute of Stavanger and the National Research Council of Italy- Institute of Marine Science for technical assistance and financial support to publish this work.
### **Conflict of interest**
The authors declare no conflict of interest.
*Plastics in the Environment*
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**Author details**
Alessio Gomiero1,2\*, Pierluigi Strafella2 and Gianna Fabi<sup>2</sup>
1 NORCE Environment, Randaberg, Norway
2 National Research Council-Institute of Marine Sciences (CNR-ISMAR), Ancona, Italy
\*Address all correspondence to: [email protected]
© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
*From Macroplastic to Microplastic Litter: Occurrence, Composition, Source Identification… DOI: http://dx.doi.org/10.5772/intechopen.81534*
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[41] Pasquini G, Ronchi F, Strafella P, Scarcella G, Fortibuoni T. Seabed litter composition, distribution and sources in the Northern and Central Adriatic Sea (Mediterranean). Waste Management. 2016;**58**:41-51
[42] Blašković A, Fastelli P, Čižmek H, Guerranti C, Renzi M. Plastic litter in sediments from the Croatian marine protected area of the natural park of Telaščica bay (Adriatic Sea). Marine Pollution Bulletin. 2017;**114**(1):583-586
[43] Laglbauer BJ, Franco-Santos RM, Andreu-Cazenave M, Brunelli L, Papadatou M, Palatinus A, et al. Macrodebris and microplastics from beaches in Slovenia. Marine Pollution Bulletin. 2014;**89**(1-2):356-366
[44] Munari C, Scoponi M, Mistri M. Plastic debris in the Mediterranean Sea: Types, occurrence and distribution along Adriatic shorelines. Waste Management. 2017;**67**:385-391
[45] Conservacy O. A rising tide of ocean debris. 2009 Report. Washington, DC, USA; 2010
[46] PNUE/PAM/MEDPOL. Results of the assessment of the status of marine litter in the Mediterranean. In: Meeting of MED POL Focal Points No. 334; 2009. 91 pp
[47] Vlachogianni T, Fortibuoni T, Ronchi F, Zeri C, Mazziotti C, Tutman P, et al. Marine litter on the beaches of the Adriatic and Ionian Seas: An assessment of their abundance, composition and sources. Marine Pollution Bulletin. 2018;**131**:745-756
[48] Carlson DF, Suaria G, Aliani S, Fredj E, Fortibuoni T, Griffa A, et al. Combining litter observations with a regional ocean model to identify sources and sinks of floating debris in a semi-enclosed basin: The Adriatic Sea. Frontiers in Marine Science. 2017;**4**:78
[49] Strafella P, Fabi G, Spagnolo A, Grati F, Polidori P, Punzo E, et al. Spatial pattern and weight of seabed marine litter in the northern and Central Adriatic Sea. Marine Pollution Bulletin. 2015;**91**:120-127
[50] Melli V, Angiolillo M, Ronchi F, Canese S, Giovanardi O, Querin S, et al. The first assessment of marine debris in a site of community importance in the North-Western Adriatic Sea (Mediterranean Sea). Marine Pollution Bulletin. 2017;**114**:821-830
[51] Galgani F, Barnes DKA, Deudero S, Fossi MC, Ghiglione JF, Hema T, et al. Marine litter in the Mediterranean and black seas. In: Executive Summary. CIESM Work. Monogr. 46. 2014. pp. 7-20
[52] Guerranti C, Cannas S, Scopetani C, Fastelli P, Cincinelli A, Renzi M. Plastic litter in aquatic environments of Maremma Regional Park (Tyrrhenian Sea, Italy): Contribution by the Ombrone river and levels in marine sediments. Marine Pollution Bulletin. 2017;**117**:366-370
[53] Vianello A, Boldrin A, Guerriero P, Moschino V, Rella R, Sturaro A, et al. Microplastic particles in
sediments of lagoon of Venice, Italy: First observations on occurrence, spatial patterns and identification. Estuarine, Coastal and Shelf Science. 2013;**130**:54-61
[54] Vlachogianni TH, Anastasopoulou A, Fortibuoni T, Ronchi F, Zeri CH. Marine Litter Assessment in the Adriatic and Ionian Seas. IPA-Adriatic DeFishGear Project, MIO-ECSDE, HCMR and ISPRA. 2017. p. 168. ISBN: 978-960-6793-25-7
[55] Lazar B, Gračan R. Ingestion of marine debris by loggerhead sea turtles, *Caretta caretta*, in the Adriatic Sea. Marine Pollution Bulletin. 2011;**62**(1):43-47
[56] Dantas DV, Barletta M, Da Costa MF. The seasonal and spatial patterns of ingestion of polyfilament nylon fragments by estuarine drums (Sciaenidae). Environmental Science and Pollution Research. 2012;**19**(2):600-606
[57] Pellini G, Gomiero A, Fortibuoni T, Ferrà C, Grati F, Tassetti AN, et al. Characterization of microplastic litter in the gastrointestinal tract of *Solea solea* from the Adriatic Sea. Environmental Pollution. 2018;**234**:943-952
[58] Gomiero A, Strafella P, Maes T, Øysæd K-B, Fabi G. First record of the occurrence, composition of microplastic particles and fibers in native mussels collected from coastal and marine areas of the Northern and Central Adriatic Sea. In peer review
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ffcbf675-2a25-491a-a3fe-16b210576719.25
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Technological Approaches for the Reduction of Microplastic Pollution in Seawater Desalination Plants and for Sea Salt Extraction
*Katrin Schuhen, Michael Toni Sturm and Adrian Frank Herbort*
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ffcbf675-2a25-491a-a3fe-16b210576719.26
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**Abstract**
An increasingly serious and widespread problem is the introduction of plastics into the water cycle. The poor degradability leads to the plastic waste remaining in water for a long time and over time it fragments into smaller and smaller plastic particles. Both the visible plastic parts and in particular their decomposition products and functionalized plastic particles are an enormous burden. Seawater desalination and sea salt extraction are highly dependent on the quality of the seawater in terms of process utilization and cost structures, i.e., on the level of pollution. Especially microparticles represent a significant potential for blocking the microfiltration membranes (pore size > 100 nm) in the pretreatment and the very costly reverse osmosis (RO) membranes (pore size > 5 nm). An innovative approach for the removal of microplastics from industrially used seawater combines a chemically induced agglomeration and a new technological implementation step. The particular challenge in removing the synthetic impurities is not only their small size but also their inert properties against most of the physical and chemical additives for flocculation. With an easy implementation to existing systems, an economic aspect and a strong impact on the maritime ecological balance will be expected.
**Keywords:** microplastics, desalination, sea salt extraction, reverse osmosis, filtration, agglomeration, add-on technology
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**1. Introduction**
The oceans perform a vital function as a regulator of the climate and absorb 25% of the CO2 generated [1]. Through the production and consumption of foodstuffs, industrial and consumer goods, humankind produces large quantities of waste, whereof a considerable proportion ends up in the ocean sooner or later. Three-quarters of the waste in the ocean consists of plastic. This plastic is a steadily growing problem, costs the lives of ten thousands of animals every year and can also endanger humans.
Over 80% of the plastic material entering the ocean annually originates from landbased sources, which correlates with the fact that half of the world population lives in coastal regions [2, 3]. Large plastic waste represents the main contribution, including everyday objects like drink bottles and other types of plastic packaging. An estimated 4.8–12.7 million tons enter here annually [2]. The remaining input comes from plastic, which is released at sea, mainly from fishing—for example, due to lost and discarded fishing gear, which is estimated at 0.6 tons a year [4]. About 94% of the large plastic parts, which end up in the sea, sink with time to the ocean floor. Today, an average of 70 kg of plastic can be found on every square kilometer of the ocean floor.
Approximately, 350–400 years can pass before the plastic is completely degraded. As it moves through the seas, the plastic changes. Through weather conditions and waves, for example, it disintegrates into smaller and smaller fragments and from the macroplastic, the so-called secondary microplastics are formed [5]. If it directly enters the environment, it is designated as primary microplastics.
By definition, microplastics are small, solid, and water-insoluble plastic particles under 5 mm in size. In the meantime, these particles can be found in all bodies of water. They could even be detected in the Arctic [6].
The input of primary microplastics is estimated at 0.8–2.5 tons a year [7]. This enters mainly through tire abrasion and textile fibers, which enter the wastewater through washing clothes and thus end up in the environment. But also the dust from the wear on road paint, microplastics used in personal care products, marine coatings, and lost plastic pellets are important entry routes (**Figure 1**).
In general, a strong correlation can be observed between the population density and the microplastics concentration [8, 9] .The proximity to densely populated regions and poor waste management lead to particularly high levels of contamination [10, 11]. Sewage treatment plants or plastics manufacturing respectively processing companies are also important point sources and can release high volumes of plastic and microplastics locally (**Figure 2**) [12, 13].
**Figure 1.** *Global release of primary microplastics to the world oceans [7].*
*Technological Approaches for the Reduction of Microplastic Pollution in Seawater… DOI: http://dx.doi.org/10.5772/intechopen.81180*
#### **Figure 2.**
*Microplastic inputs and transport paths into the ocean [7, 15].*
Ports and industrial areas are especially contaminated with microplastic particles [14]. The majority of the microplastics remain near the shore [10]. In the Arabian Gulf along the coast, 4.38 × 104 –1.46 × 106 microplastic particles/km2 could be detected in the surface water [15].
Off the coast of South Africa, there were 257.9 ± 53.36 to 1215 ± 276.7 microplastic particles/m3 of water [12]. At the mouth of the Yangtze in the East China Sea, 4137.3 ± 2461.5 microplastic particles/m3 could be detected, whereby the concentration in the open sea was only 0.167 ± 0.138 microplastic particles/m3 [13].
The most common polymer types occurring in seawater are, in addition to polyethylene, polypropylene, and polystyrene, also polyamide, polyester, polymethylmethacrylate, polyvinyl chloride, polyoxymethylene, polyvinyl alcohol, polymethylacrylate, polyethylene terephthalate, alkyd resins, and polyurethane [16]. They are found in descending prevalence in the seawater.
The quantification of the inputs into the environment is, as a rule, based on a loss rate, which is calculated against the produced quantity of preproduced plastic [17]. The categorization of the industries, which manufacture preproduced plastic, is organized in producers (manufacture plastic material from raw materials), intermediaries, converters (convert preproduced plastic into products, or individual components), external waste disposers, and shipping companies (transport the material). By means of the difference between the respectively processed plastic quantities and the loss rate, the plastic quantities, which are released into the environment by the plastics industry alone, are revealed.
Forecasts assume an increasing plastic production volume in the future, which will lead to an increasing entry quantity of plastic and microplastics in the environment and seas [2]. In addition, microplastics arise continuously through the constant fragmentation of plastic already in the environment [5]. This leads to an ever higher contamination of the marine environment with microplastics.
Since the current analytical methods to detect microplastic in the aquatic environment have numerous shortcomings [18], the contamination of the marine environment can only be estimated. Particularly problematic here is that small plastic particles cannot be captured in most monitoring cases. The lower detection limit in the marine environment is usually 300 μm. With increased efforts, the lower detection limit can be reduced to 20–10 μm [16]. However, this is seldom practiced. Particles below the detection limit are not captured. Additionally, studies are not readily comparable, since there is no standardized monitoring procedure [18].
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**2. Seawater utilization**
Seawater utilization can be divided in three primary use areas: the use for agriculture, for the home, and for industry [19]. While the principal portion is used for agriculture in developing countries, a significantly increased proportion for use in households and industry can be seen in industrialized countries. Advancing industrialization also causes increased water consumption. Over the last century, the world population has quadrupled, while the water demand has increased sevenfold.
In general, usable water comes from surface water, groundwater, or fossil aquifers. In order to increase the supply of usable water, processes such as the desalination of seawater have been moving into focus for some time [20]. Seawater has most commonly been used as a coolant in energy generation and in industrial processes so far. It is also used in mining to extract minerals as well as for the hydraulic fracturing of gas and oil. It is additionally applied in production processes, such as sea salt extraction, aquaculture, algal cultivation as well as food manufacturing [21]. It is also used for temperature moderation in buildings and areas as well as for cold water fishery (**Figure 3**).
Two characteristic economic seawater application sectors are seawater desalination and sea salt extraction. For sea salt extraction, seawater is diverted into large basins. Over time, the water evaporates by the heat of the sun and wind and the previously dissolved salt remains [22]. The media currently reports again and again about the contamination of sea salt with high quantities of microplastics. Thus, 50–280 microplastic particles/kg of salt were detected in Spanish sea salt and in Chinese sea salt 550–681 microplastic particles/kg of salt [23, 24].
Seawater desalination is the production of drinking water and process water for industrial facilities or power plants from seawater through the reduction of the salt content. The desalination can be based on various processes, which remove the salts and minerals from the water. To some extent, usable ancillary products like table salt accrue. In addition to the already present burden from anthropogenic stressors, the chemicals added to the seawater against fouling and scaling as well as the metals dissolved by corrosion threaten marine ecosystems, e.g., in the Red Sea. An
**Figure 3.**
*Overview of the different seawater applications [21].*
*Technological Approaches for the Reduction of Microplastic Pollution in Seawater… DOI: http://dx.doi.org/10.5772/intechopen.81180*
investigation of 21 plants, which together produce 1.5 million m3 /day, accounted for 2.7 kg of chlorides, 3.6 kg of copper, and 9.5 kg antiscaling agents piped into the sea per day of seawater desalination [25].
Seawater desalination is practiced in numerous coastal semiarid regions [20]. In addition to small plants in areas with insufficient infrastructure with only a few hundred cubic meters of water a day, there are also large desalination plants, e.g., in southern Europe (Barcelona Seawater, 200,000 m3 /day), the USA (Claude Bud Lewis Carlsbad, 204,000 m3 /day), Israel (Sorek, 624,000 m3 /day), Australia (Kurnell, 250,000 m3 /day), and the Unites Arab Emirates (Dschabal Ali Block M, >2,000,000 m3 /day). Significant expansion of production capacities for seawater desalination is, for example, planned in the Persian Gulf. Due to the development of the steel, petrochemical, cement, aluminum, and energy industry, there will be a demand of roughly 940,000 m3 /day only for use in these sectors until 2030 in Iran alone [26].
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**3. Impact of increasing microplastic burden on seawater utilization and the water use cycle**
An increasing microplastic burden not only has implications for the costs and efficiency of the sea water utilization process, but also for the marine ecosystem and, as a consequence, also for humankind (**Figure 4**).
Bonded in microplastics, pollutants like, for example, dichlorodiphenyltrichloroethane (DDT), dioxins, or heavy metals can be transported and accumulated in organisms via ingested food [28]. Due to the manufacturing process, most polymer blends also contain harmful substances like softening agents or monomers, which in return can be released upon ingestion of the particles via food and exert a direct influence on the organism, since these substances are mostly classified as potentially harmful and/or carcinogenic [29]. It has already been shown in laboratory experiments that microplastics smaller than 150 μm can, after ingestion via food, enter the surrounding tissues, the bloodstream and, through these, the internal organs and also the brain [30]. There is then the risk of the formation of lesions and inflammations. Furthermore, oxidative stress, necrosis, and damage to DNA can be triggered, which again increases the risk of cancer. Neurological behavioral disorders are also possible [31]. Thus, there is a potential risk to human health from microplastics.
**Figure 4.** *Microplastics in the marine water cycle [27].*
#### *Plastics in the Environment*
In addition to the decrease in improper disposal, the search for replacement substances and the prohibition of microplastics as a product addition for everyday products, ensuring that the aquiferous processes are free of microplastics also represents a chance to reduce the degree of contamination due to microplastics in the water cycle. Besides the implementation of new technologies for the purification of wastewater in sewage treatment plants, this also includes the conceptual and technical development of new add-on technologies in seawater utilization processes in order to filter microplastics out of the inflowing seawater and eliminate it prior to the seawater utilization processes.
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**4. Ecological-chemical approach to the reduction of the microplastic burden in seawater-based processes**
At this time, there is no economical possibility yet to remove microplastics simply and cost-effectively from seawater. A promising research approach based on the adaptation of a concept by Herbort and Schuhen for freshwater systems and the simultaneous development of add-on technology for static (e.g., waterside plants) and mobile (e.g., ships) seawater utilization processes [32].
In the process developed by Herbort and Schuhen, silane-based microplastic agglomerates are formed according to the cloud point principle through the application of special organosilane-based precursors, which, via Van der Waals forces, have a high affinity to unreactive microplastics (IOCS, inert organicchemical macromolecules) and, at the same time, a high reactivity in water. [32–35]. A Video shows the fixation process in a batch reactor for use in wastewater treatment [36].
Organosilanes are hybrid compounds of inorganic silanes and organic hydrocarbons [37]. Through the selection of the functional groups in the organic unit (functional design), it is possible to exploit an adaptable system for the respective application (e.g., removal of reactive and/or inert organic-chemical compounds). By means of the substituent pattern within the organic unit and also directly on the silicon atom, the affinity of the organosilanes can be adapted to various polymer types and, simultaneously, the reactivity respectively the stability can be controlled.
Organosilanes with corresponding reactivity can react to organic-inorganic hybrid silica gels in the sol-gel process [38, 39]. In the first step, disposal groups are split off and reactive silanol groups generated through hydrolysis. The silanol groups subsequently form silanone bonds in a condensation reaction and link the organosilanes via a bridging unit (**Figure 5**).
A three-dimensional networked hybrid silica gel forms, which is stabilized via the respective bridging unit. Through the selection of the disposal groups and the organic groups, the properties and reactivity of the organosilanes can be specifically controlled. The selection of the disposal groups is decisive for the hydrolyzability of the organosilanes [40, 41].
The organic groups influence the water solubility, the stability of the resulting silanols, and the affinity of the organosilanes respectively the ability of the resulting silanols to fix microplastic particles. By choosing specific organic groups, the affinity to different polymer types can be controlled. Groups with low polarity can be used to attach to polymers with lower polarity like polyethylene or polypropylene. More polar polymers containing, e.g., heteroatoms, like polyester or polyamide, can be fixed by using organic groups with a similar chemical structure and polarity.
*Technological Approaches for the Reduction of Microplastic Pollution in Seawater… DOI: http://dx.doi.org/10.5772/intechopen.81180*
**Figure 5.** *Localization and agglomeration of microplastic particles [42].*
The interaction of the disposal groups and organic basis units must be so coordinated that the highest possible affinity to microplastics and optimal reaction kinetics are achieved [33]. The reaction kinetics takes on a decisive function within the research approach. It is influenced, among other things, by the water matrix and the temperature. Knowing that an increase in the water temperature accelerates the speed of the sol-gel process and that the temperature of seawater is subject to seasonal fluctuations and spatial variations, challenges are presented for the implementation of the concept in the seawater environment [39]. In addition, the factors of salinity, temperature, UV radiation, and pH value play a determining role and are also to be considered (**Figure 6**) [43].
Dissolved salts influence reaction speed and the reaction mechanism, for example, through the electrolyte effect [44, 45]. It results in the stabilization of the intermediate stages with higher ionic strength as well as catalytic effects or competitive influences of dissolved ions. The pH value influences reaction kinetics through the catalytic effect of hydroxide and oxonium ions [39]. Temperature differences directly influence the reaction speed [46, 47]. Thus, a temperature increase of 10°C is accompanied, as a rule, by approximately double the reaction speed. UV light can also facilitate the formation of reactive intermediate stages via a photocatalytic effect and thus accelerate the reaction [46]. However, it can also break down the precursors used or already linked molecules.
Initial experiments at a laboratory scale show that the salinity has a slowing influence on the fixation process and the entire process of agglomeration formation also works in an artificial salt water matrix. To produce the salt water (3.5% salt by mass), 27.5 g NaCl, 5 g MgCl2, 2 g MgSO4, 1 g KCl, and 0.5 g CaCl2 were dissolved in distilled water. This results in a mass concentration of 58.8% chloride (Cl<sup>−</sup>), 29.6% sodium (Na<sup>+</sup> ), 4.7% sulfate (SO4 <sup>2</sup>−), 4.9% magnesium (Mg2+), 1.5% calcium (Ca2+), and 0.5% potassium (K<sup>+</sup> ).
Subsequently, 0.1 g polyethylene powder (PE) (average particle size 350 μm), 0.1 g polypropylene powder (PP) (average particle size 350 μm), and 0.1 g of a 50:50 mixture of PE and PP were stirred in a beaker with 1 l of salt water respectively distilled water at room temperature. After 24 h, 0.15 ml agglomeration reagent was added and the mixture was stirred for an additional 24 h. The formed
#### **Figure 6.**
*Influence on the reaction kinetics of the fixation process in the marine environment (percentage by mass) [39, 43–47].*
aggregates were removed and dried at 60°C for 24 h. ESEM images were taken using a FEI Quanta 250 ESEM (FEI Company, Hillsboro, USA) equipped with a large field detector (LFD). The chamber pressures were between 60 and 80 Pa and the acceleration voltage between 7 and 20 kV. The remaining water was filtered using a paper filter (Rotilabo 111A, 12–15 μm pore size). The tare weight of the filter was noted before filtration using a AX105DR (Mettler Toledo, Switzerland). Afterward, the filter was dried at 105° C for 24 h and weighted again, to check if there is remaining microplastic in the water (accuracy ±0.2 mg).
In distilled water, an aggregation of the microplastic particles begins 15 s after the addition of the agglomeration reagent. After 2–3 min, the agglomeration is completed and an aggregate is present, which contains all of the microplastics. In the artificially produced salt water samples, the agglomeration process begins after 10 min and is concluded after 15 min. It is, therefore, significantly slower, but nevertheless fixes all the microplastics. This shows that salt water has a stabilizing effect on the reactive intermediate stages and thus slows down the sol-gel process, whereby the agglomeration starts later and also takes longer. **Figure 7** shows ESEM images of the mircoplastic particles used and their aggregates formed during the fixation process respectively fragments of the aggregate prepared for the images. How the microplastic particles are linked and embedded by the agglomeration reagent can be observed, where a considerable increase in size results. As the agglomeration reagent reacts to a solid hybrid silica, which will be removed within the aggregates from the water, residues in the water will be avoided. To ensure a complete removal of the agglomeration reagent even in the trace substance range, further TOC analysis and particle analysis will be proceeded [48, 49].
*Technological Approaches for the Reduction of Microplastic Pollution in Seawater… DOI: http://dx.doi.org/10.5772/intechopen.81180*
#### **Figure 7.**
*ESEM images of the microplastic blanks (a, b = PE; c, d = PP) and of the agglomerates formed during the fixation process (e = PE; f = PP; g = PE/PP (50:50)).*
## **5. Procedural implementation of microplastic elimination in seawater utilization processes**
The procedural implementation pursues the goal of increasing the service life of the existing desalination plants by already holding back the microplastic particles (0.1–5 μm) initially in pretreatment, which cause blockages of the membranes (pore size 0.002–0.1 μm). On the one hand, the service life of the microporous membranes can be thus extended and the operational expenses reduced (without the addition of suspect additives, such as, e.g., antiscalants) as well as, on the other hand, sustainably eliminating the much-criticized microplastic particles from the water cycle. As the removal is based on a physicochemical agglomeration process, it is not limited by particles size or shape like a filtration process. The agglomeration reagent can bind to fibers, films, and fragments in all size classes and fix them in big agglomerates. As a consequence, the quality of the water on the removal side near the coast/surface as well as on the output side will be improved through the reduction of the microplastic particle load.
In the application in sea salt extraction, contamination of the resulting sea salt and thus the transmission to people will be effectively avoided through the removal of microplastics from the seawater flowing into the evaporation basins.
This is possible through the combined development of a pretreatment stage of a series of stirred tanks and the inorganic-organic functional material. In addition, a high-performance cascade process is developed as add-on technology in order to
facilitate the material reaction and to make a throughput of >600 m3 a day possible. Continuous operation is sought by connecting several cascades. The adapted concept as well as the related technological implementation strategy provides for the first time the opportunity to remove plastic particles with a particle size of <5 μm effectively and sustainably from salt water in an upstream, modular pretreatment step. The diagram of the process is presented in **Figure 8**.
The process is divided in the following stages:
The feed stream, that is the extracted seawater with unwanted microplastic load, is piped to the first partial reactor via an existing suction pipe. This should take place by means of upstream, abrasion, and corrosion-resistant pumps and use the suction pipes present in the existing seawater desalination plants. Depending on the load of the feed stream, a defined amount of the organosilanes is simultaneously added by means of the dosing device and blended with the salt water already in the first partial reactor through a mechanical mixing concept.
In accordance with the concept, the addition of the material takes place in several steps within the process. In this way, the required initialization period of the material and the reaction time can be responded to through the variable design, dimensioning, and number of dosing stages, mixing installations as well as partial reactors. For this purpose, können reactors of different sizes respectively different volumes can be used so that, for example, the reaction starts in the first partial reactor and continues to react optimally in the subsequent reactors. The organosilane is mixed with the salt water according to the required concentration in the respective partial reactors, at which time it is successively bonded to the microplastics to be removed. Through injection or chemical interaction, the concentration of free,
**Figure 8.** *Process flow chart.*
*Technological Approaches for the Reduction of Microplastic Pollution in Seawater… DOI: http://dx.doi.org/10.5772/intechopen.81180*
#### **Table 1.**
*Technical target criteria.*
nonbonded microplastic particles should continuously decrease along the length of the reactor respectively the residence time. A consistent residence time of the material in the reactor is needed so that a complete conversion can be surmised. This is essential for the process, since an incomplete conversion would mean additional burdens through further foreign substances/particles.
In order to prevent the disintegration of the agglomerates, the reactor has a strict and clearly defined residence time distribution (RTD) so that ideally all particles can pass through the reactor in the same ideal time frame and grow and that the disintegration of the agglomerates can be avoided. The implementation of microplastic elimination in seawater utilization plants pursues the technical target criteria presented in **Table 1**.
## **6. Conclusions**
The innovative add-on technology for the removal of microplastics from industrial seawater utilization plants pursues the first problem solution regarding the risk of blocking from the immense microplastic particle load in the sea, among other things, in industrial, membrane-based seawater desalination plants (especially RO plants). Due to the significant reduction of the microplastic load in the pretreatment stage, alleviation results for the downstream RO membranes. The service life of the RO membranes will be significantly increased and the membranes can, thanks to the improved performance, be operated on a smaller scale and more cost-effectively.
Through the application of new add-on technology in sea salt extraction, the entry of potentially harmful microplastics in sea salt is reduced and thus, at the same time, also the contamination of everyday food. Based on the future increasing contamination of the oceans with microplastics, this technology helps to ensure the sustainable use of seawater.
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**Acknowledgements**
The research projects of Wasser 3.0 (www.wasserdreinull.de) are conducted by means of the financial support by the German Federal Ministry for Economic Affairs and Energy through the provision of ZIM (Central Innovation Program for SME) project funds. The enterprise abcr GmbH (www.abcr.de) from Karlsruhe (Germany) and Zahnen Technik GmbH (www.zahnen-technik.de) from Arzfeld (Germany) are directly involved in the project as industrial partners for the material science scale-up as well as for plant construction and engineering. Michael Sturm thanks the German Federal Environmental Foundation (DBU) for the support with a PhD scholarships (Aktenzeichen: 80018/174). The authors thank Carolin Hiller, Lukas Pohl, and Laura O'Connell for their contributions.
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**Conflict of interest**
We have no conflict of interest to declare. This research has not been submitted for publication nor has it been published in whole or in part elsewhere. We attest to the fact that all authors listed on the title page have contributed significantly to the work, have read the manuscript, attest to the validity and legitimacy of the data and its interpretation, and agree to its submission to the book.
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**Author details**
Katrin Schuhen1,2\*, Michael Toni Sturm1,3,4 and Adrian Frank Herbort1,3
1 Wasser 3.0/abcr GmbH, Karlsruhe, Germany
2 Wasser 3.0/Zahnen Technik, Arzfeld, Germany
3 Institute for Environmental Sciences Landau, University of Koblenz–Landau, Germany
4 Water Chemistry and Water Technology, Karlsruhe Institute of Technology (KIT), Engler-Bunte-Institute, Karlsruhe, Germany
\*Address all correspondence to: [email protected]
© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
*Technological Approaches for the Reduction of Microplastic Pollution in Seawater… DOI: http://dx.doi.org/10.5772/intechopen.81180*
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Elemental Analyzer/Isotope Ratio Mass Spectrometry (EA/ IRMS) as a Tool to Characterize Plastic Polymers in a Marine Environment
*Daniela Berto, Federico Rampazzo, Claudia Gion, Seta Noventa, Malgorzata Formalewicz, Francesca Ronchi, Umberto Traldi and Giordano Giorgi*
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**Abstract**
In the last 60 years, plastic has become a widely used material due to its versatility and wide range of applications. This characteristic, together with its persistence, makes plastic waste a growing environmental problem, particularly in the marine ecosystems. The production of plant-derived biodegradable plastic polymers is assuming increasing importance. Here, we report the results of a first preliminary characterization of carbon stable isotopes (δ 13C) of different plastic polymers (petroleum- and plant-derived) and a first experimental study aimed to determine carbon isotopic shift due to polymer degradation in an aquatic environment. The results showed that the δ 13C values determined in different packaging for food uses reflect the plant origin for "BIO" materials and the petroleum-derived source for plastic materials. Considering degradation, δ 13C values of both bio bags and HDPE bags showed a gradual decrease toward less negative values when kept immersed in seawater, recording a δ 13C variation (Δδ13C) of 1.15 and 1.78‰, respectively. With respect to other analytical methods, the characterization of the plastic polymer composition by isotope ratio mass spectrometry is advantageous due to low cost and rapidity of analysis, small amount of sample required, high sensitivity, and the possibility of analyzing colored samples.
**Keywords:** carbon isotopes, plastic polymers, EA/IRMS, plastic degradation, plastic pollution
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"section_idx": 37
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**1.1 Plastic debris in the ocean: a global environmental issue of the twenty-first century**
Since 1950, the production and use of plastics has been constantly increased reaching a global production of 280 million tons in 2016 (i.e., as thermoplastics and polyurethanes), with China as the major producer (29%) [1].
Plastics represent a group counting hundreds of different materials derived from fossil sources (e.g., oil and gas) among which the most produced are polypropylene (PP), high- and low-density polyethylene (HDPE, LDPE), polyvinyl chloride (PVC), polyurethane (PUR), polyethylene terephthalate (PET), and polystyrene (PS). Due to their high versatility, durability, low weight, and low cost, plastic materials find applications in almost any market sector, but primarily in packaging (39.9%) and building industries (19.7%) [1].
In recent years, the growing evidence about the massive presence of plastic litter in the ocean, its pressure on the marine environment and wildlife, and its impact on marine-related human activities (such as fishery, shipping, and tourism) has raised lot of attention in the scientific, regulatory, and civil communities (**Figure 1**).
Oceanographic surveys have recorded the presence of plastics in any geographical regions, including remote polar areas, and at any depth, from the sea surface to the seafloor of the oceans (**Figures 2–4**).
The amount of plastic debris in the sea is still unknown due to the large variability of its distribution as regards both spatial and temporal scale, which prevents accurate estimates. However, modeling studies have recently approximated that 5–13 million tons of plastics (i.e., equivalent to 1.5–4% of global plastic production) end up in the oceans every year [2].
The slow degradation rates of plastics under environmental conditions provide additional complexity to this global issue, by contributing to their accumulation in all terrestrial and aquatic environments. It has been estimated that, once in the ocean, the majority of manufactured polymers persist for decades and probably for centuries due to their low degradability (**Figure 5**) [3, 4].
In both terrestrial and marine environments, degradation of petroleum-derived plastics occurs through abiotic and biotic processes (i.e., UV degradation, hydrolysis, and decomposition by microorganisms), leading to their fragmentation into increasingly smaller pieces. Thus, plastic particles dispersed in the environment are commonly divided into three main classes based on their size: macro: >25 mm,
*Elemental Analyzer/Isotope Ratio Mass Spectrometry (EA/IRMS) as a Tool to Characterize... DOI: http://dx.doi.org/10.5772/intechopen.81485*
meso: 5–25 mm, and microplastics: <5 mm. The smaller-size class, which includes both primary microplastics (i.e., particles produced as such, e.g., plastic pellets, exfoliating cosmetics, or synthetic clothing fibers) and secondary particles (i.e., particles derived from the breakdown of larger plastic debris), is likely to be the most abundant in the ocean today [5].
**Figure 2.** *Marine litter on the beach (photo by Tomaso Fortibuoni).*
**Figure 3.** *Seabird nesting on plastic nets (public domain).*
**Figure 4.** *Tangle of fishing nets on a beach (photo by Francesca Ronchi).*
#### **Figure 5.**
*Estimated decomposition times of different types of garbage dispersed in the marine environment (illustration by Davide Zanella).*
The concern about the heavy contamination of the marine environment by plastics is related to the potential of plastic debris to cause harm to the inhabiting organisms via different mechanisms. Among the most alarming issues, there is an uptake and a bioaccumulation of plastic debris by marine organisms at almost all levels of the food web and the consequent trophic transfer. Recent studies have reported that micro and nanoplastics can easily be taken up and ingested by marine organisms (i.e., zooplankton, worms, bivalves, crustaceans, demersal and pelagic fishes, seabirds, reptiles, and mammals), resulting in a significant impact on the aquatic wildlife and possibly on human health via seafood consumption [6]. Furthermore, due to the large surface to volume ratio, microplastic fragments can potentially adsorb many kinds of common marine contaminants on their surface, in particular hydrophobic organic substances such as polychlorinated biphenyls, polyaromatic hydrocarbons, and organochlorine pesticides [7, 8]. This can promote their transport in the environment and induce toxic effects following ingestion and desorption (e.g., endocrine disruption, mutation, and cancer). Moreover, another source of concern is the possible release of additives commonly present in plastic formulations (i.e., bisphenol A, phthalates, and flame retardants) [8, 9], and although the leaching rates of these common additives in seawater are poorly known, their potential for toxicity is considered to be very high.
Several actions have currently been undertaking at national and international levels to tackle the contamination of marine environments by plastics. Their main aim is to achieve a general reduction of plastic use (in particular packaging and disposable items), recycling of plastic items at the end of their lifetime, and replacement of the use of plastics with more sustainable materials and biopolymers (e.g., plant-derived polymers [10]), which are more prone to degradation by microorganisms and show a shorter persistence once dispersed in the environment.
#### **1.2 Experimental approaches to assess plastic debris in environmental samples**
With the growing evidence of the severe impact caused by plastics on the wildlife, the assessment of the presence, behavior, and fate of plastics in the marine environments has become a fundamental research issue, highly advocated to the scope of putting in place more effective policies. However, especially for the smallest particles (i.e.,
#### *Elemental Analyzer/Isotope Ratio Mass Spectrometry (EA/IRMS) as a Tool to Characterize... DOI: http://dx.doi.org/10.5772/intechopen.81485*
microplastics), their efficient identification to the scope of assessing the plastic load in the environmental compartments (e.g., seawater, sediments, and biota) is a serious challenge for scientists. Many analytical techniques have been used to identify plastic debris in environmental samples, as largely reviewed in the literature [11, 12]. Among the most used approaches, there are electron scanning microscopy coupled with energy dispersive X-ray spectroscopy (SEM-EDS, ESEM-EDS), Raman spectroscopy, Fourier transform infrared spectroscopy (FT-IR) [13], and thermal analysis (pyro-GC/MS). Other analytical methods used to identify plastic materials are near infrared spectroscopy (NIRS), differential scanning calorimetry [14], and UV-VIS spectroscopy [15, 16].
Stable isotope analysis, which is an analytical technique that measures the relative abundance of stable isotopes yielding an isotope ratio that can be used as a research tool, is finding application in a growing number of different research fields and practical case studies. For instance, it is widely used to trace the origin of organic matter in various environments [17, 18], to track fraud in the food industries [19] and to identify microtraces of drugs, flammable liquids, and explosives in forensic cases [20]. This technique has been only rarely applied to assess the presence of microplastics in environmental samples [21]. Its potential for detecting plastic debris in environmental samples relies on different isotopic signatures of carbon in (i) petroleum-derived materials, (ii) C4 plants used in the synthesis of bioplastics, and (iii) marine samples' matrices (e.g., particulate organic matter, plankton, tissues of marine organisms, algae, and marine plants).
#### **1.3 Stable isotope analysis: principles of the method**
The term isotopes (from the Greek iso, same and topos, place) identifies atoms of the same chemical element, that is, the same place in the periodic table of the elements, that has the same atomic number but different atomic mass number. In other words, isotopes are atoms having the same number of protons and electrons (equal chemical properties) and a different number of neutrons (different physical properties). Each element has known isotopic forms, and in total, there are 275 isotopes of the 81 stable elements, in addition to over 800 radioactive isotopes (**Figure 6**).
Isotopes of a single element possess almost identical properties. They are commonly classified as natural or artificial, stable, or unstable. The quantification of the ratio between two isotopes allows to determine if two chemically similar environmental samples have different origins, related to the difference of the original sources. The isotopic distribution characterizing the sources may be influenced by phenomena of a different nature, which in turn may cause significant variations in the final products.
**Figure 6.** *Stable isotopes have a proton/neutron ratio lower than 1.5.*
Depending on the chemical element, variations in the relative mass abundance of its isotopes can be detected through the analysis of stable isotopes. Technological advances in isotope analysis have led to the development of scientific instruments able to measure very small variations in the abundance of stable isotopes with high precision and accuracy (mass spectrometry). Therefore, stable isotope analysis can be applied considering different elements, thus giving nowadays applications in different fields of science.
For a given chemically stable element, its isotopic composition in a sample (R) is equal to the ratio between the abundance of the heavy isotope with respect to the light one (e.g.,13C/12C), and it is expressed as deviation, in parts per thousand, from an international reference standard material (δ‰), according to the equation (Eq. (1)) given below:
$$\text{\{\{\%\}} \, = \, \left[ \{ \text{R}\_{\text{sample}} - \text{R}\_{\text{standard}} \} / \text{R}\_{\text{standard}} \right] \times \text{1000} \tag{1}$$
where Rsample is the mass ratio of the heavy isotope to the light isotope measured in a sample and Rstandard is the isotopic ratio defined for the standard. The standard reference material that is commonly used for carbon is Vienna Pee Dee Belemnite. Thus, positive δ values indicate that the heavy isotope is enriched in the sample compared to the standard, while negative δ values indicate that the heavy isotope is depleted in the sample.
The possibility of distinguishing two samples on the basis of their relative abundance of two isotopes bases on the phenomenon of isotopic fractionation, which can be enacted by a wide range of chemical (e.g., nitrification and ammonification), physical (e.g., evaporation and condensation), and biological (e.g., photosynthesis, assimilation, and excretion) processes. In fact, many natural (and anthropic) processes can alter the isotopic signature of a chemical element in a matrix by causing an imbalance of the isotope distribution that leads to a variation of its original isotopic signature [22]. Thus, as the extent of fractionation of many chemical elements have been proved to be sensitive to specific processes/variables, it can be used as a tool to investigate the involved process/variable itself. In general, two mechanisms of isotopic fractionation can be distinguished:
- heavy isotopes accumulate in oxidized products;
- • the isotopic fractionation is favored at low temperatures, since at high temperatures, the differences between the isotopes are attenuated;
- • the process is not relevant in the case of chemical reactions of gaseous substances and biological reactions.
- the preferential breaking of the bonds formed by light isotopes;
- • the preferential distribution of light isotopes in products and of the heavy ones in the reagents.
*Elemental Analyzer/Isotope Ratio Mass Spectrometry (EA/IRMS) as a Tool to Characterize... DOI: http://dx.doi.org/10.5772/intechopen.81485*
Given a chemical substance AB characterized by the presence of a certain isotopic distribution of element X, we can calculate the fractionation factor by dividing the ratio of the number of isotopes X in product A with the ratio of the number of isotopes X in product B (Eqs. (2) and (3)).
$$\begin{array}{rcl} \mathfrak{\alpha}\_{AB} &=& \frac{R\_A}{R\_B} = & \mathbf{1} + \left[ \frac{(\boldsymbol{\delta}\_A - \boldsymbol{\delta}\_B)}{1000} \right] \end{array} \tag{2}$$
where
$$R\_{\perp} = \frac{X\_h[\text{atoms of the heavier isotope (pure)}]}{X\_l[\text{atoms of the lighter isotopes}] \,\text{atm} \,\text{atm} \,\text{(abaudant)}} \tag{3}$$
However, the fractionation factor (α) is normally replaced by the isotopic enrichment factor (ε), which is defined as (α − 1) × 1000.
#### **1.4 Carbon isotope ratio as a tool in environmental studies**
Carbon and nitrogen isotope analysis is used to investigate the trophic web and the matter flows among the main components of an ecosystem (e.g., organic matter, producers, primary and secondary consumers); it can be used to understand chemical and biological processes occurring at both ecosystem and organism levels. Stable isotope analysis can also be a useful tool for assessing the origin of water, atmospheric, and soil pollution.
The two main carbon reserves in nature are represented by organic and inorganic carbon, which are characterized by different isotopic fingerprints due to the different processes in which they are involved (**Figure 7**). The inorganic carbon (carbonate) is involved in the exchange equilibrium among (i) atmospheric carbon dioxide, (ii) dissolved bicarbonate, and (iii) solid carbonate. The exchange reactions among these three forms lead to an enrichment of the heavy isotope in the
**Figure 7.** *Isotopic fingerprint of naturally occurring carbon.*
solid carbonate form (δ13C equal to 0‰). In contrast, the kinetic reactions which mainly involve the organic carbon (i.e., photosynthetic process) determine a concentration of the lightest isotope in the synthesized organic material (δ13C equal to about −25‰) [17].
The fractionation of organic carbon is mainly linked to the specific photosynthetic pathway featuring each plant. The terrestrial plants, classified as C3 and C4, can follow two different photosynthetic pathways. Both types synthesize organic matter characterized by δ13C values more negative than that of carbon dioxide (~−7‰), because during the photosynthesis, the produced organic substance accumulates the light isotope compared to the heavy one. The C3 plants, typical of temperate climates, produce the 3-phosphoglyceric acid, a compound with three carbon atoms (Calvin cycle) with an average value of δ13C of about −26.5‰. The C4 plants generate oxaloacetate, a compound with four carbon atoms (Hatch-Slack cycle) characterized by a value of δ13C around −12.5‰.
The chemical composition of animal tissues is related to the food sources they assimilate, and therefore, it reflects the isotopic composition of the diet [23, 24]. The enrichment between primary producers and consumers (herbivores) has been estimated to be approximately +5‰, whereas at the successive trophic levels, the enrichment is less marked (+1‰) [25]. Thus, the isotopic value detected in the tissues of an organism can be potentially used as an indicator of its trophic position. However, since the variation of the δ13C values due to trophic passages is relatively modest, δ13C is mainly used to trace the primary carbon source used [26].
Through the analysis of the stable carbon isotopes, it is also possible to differentiate terrestrial and marine trophic webs. The "marine" carbon derives from the dissolved inorganic carbon (dissolved bicarbonate) characterized by an isotopic value equal to about 0‰, while the "terrestrial" carbon derives from the atmospheric carbon dioxide which has a lower δ13C value (approximately −7‰). This difference is maintained at every trophic level both in the marine and terrestrial trophic chain (**Figure 8**).
#### **Figure 8.**
*Variations of δ13 carbon and δ15 nitrogen (‰) isotopes in different organisms of the terrestrial and marine food chain.*
*Elemental Analyzer/Isotope Ratio Mass Spectrometry (EA/IRMS) as a Tool to Characterize... DOI: http://dx.doi.org/10.5772/intechopen.81485*
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**2. Preliminary study to characterize plastic polymers using elemental analyzer/isotope ratio mass spectrometry (EA/IRMS)**
In 2016, Berto and collaborators carried out a preliminary study aimed at evaluating the potentials of stable isotope analysis to discriminate a wider range of plastic and bioplastic materials (including those highly used in packaging, such as shopping bags and plastic bottles for drinking water) (**Table 1**) to the scope of using this analytical technique for the identification of plastic debris in marine samples in future field surveys [27].
Furthermore, considering the lack of knowledge on possible changes in the carbon isotopic signature of plastics due to degradation processes in the marine environment, this study also investigated the variation of δ13C values of petroleum- and plant-derived polymers of packaging materials subjected to biotic and abiotic degradation. The study was carried out by using an isotope ratio mass spectrometer Delta V Advantage (Thermo Fisher Scientific, Bremen, Germany) coupled with an elemental analyzer Flash 2000 (Thermo Fisher Scientific, Bremen, Germany). The accuracy of the isotopic data was evaluated by the analysis of the certified polyethylene foil (−31.8 ± 0.2‰, IAEA-CH-7, International Atomic Energy Agency, Austria). The analytical precision of measurements was 0.2‰ for C.
**Table 1.**
*Plastic (petroleum and plant-derived polymers) and natural matrices analyzed in this study.*
*Elemental Analyzer/Isotope Ratio Mass Spectrometry (EA/IRMS) as a Tool to Characterize... DOI: http://dx.doi.org/10.5772/intechopen.81485*
This chapter gives a review of the main insights obtained and critically discusses the potentials of the carbon isotope ratio analysis to study the behavior and fate of plastics in the aquatic environment.
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**2.1 δ13C as a possible tool to investigate plastic polymers**
δ13C values recorded in this study for the most used petroleum-derived plastic polymers, plant-derived polymers, some commercial items made by petroleumand natural-derived polymers, which are largely found in the marine litter worldwide (i.e., food packaging items), and natural matrices are reported in **Figure 9**.
Due to their high stability and durability [28], in the last decades, petroleumderived plastic materials have largely replaced paper and other cellulose-based products with a continuously increasing trend. At the moment, a wide variety of petroleum-based synthetic polymers are produced worldwide (approximately a total of 335 million tons in 2016), and significant quantities of these polymers end up into natural ecosystems as waste products [1].
The δ13C values of the majority of the analyzed petroleum-derived plastic polymers ranged over a wide interval, that is, between −33.97 and –25.41‰. Only a few polymers, such as PTFE, silicon, and ABS showed more negative δ 13C values (−40.70 ± 1.17, −39.37 ± 0.27, and − 35.17 ± 0.98‰, respectively), possibly due to fractionation processes during their synthesis.
With the exclusion of PTFE, ABS, and silicon, the recorded δ13C range results are comparable to that reported for crude petroleum [29]. Petroleum is constituted by a complex mixture of organic substances, with a predominance of hydrocarbons, whose exact composition depends on the site of extraction. Petroleum usually shows negative values of δ13C, ranging between −34 and −18‰ depending on the specific extraction field. In fact, as reported by Stahl [29], petroleum could be originated from the lipid fraction of organic matter. In particular, the carbon
#### **Figure 9.**
*δ13C values determined in various petroleum- and plant-derived polymers, as well as in natural matrices analyzed in this study.*
isotopic value of petroleum can vary in relation to the marine vs. terrestrial origin of the source, with an enrichment of 12C with respect to 13C in the marine environment compared to the terrestrial one [30].
Interestingly, different δ13C values were recorded for some polymers as pure material and once in packaging commercial items. For instance, a significant (p < 0.05) more negative δ 13C value was determined in the HDPE shopping bag for food use with respect to the original HDPE polymer. This could be related to the addition of some organic additives (i.e., stabilizers) in the final materials used for food packaging. In fact, depending on the commercial use, plastic formulations can be enriched with monomeric ingredients to improve their processing, end-use performance, and appearance (e.g., colorants, photostabilizers, etc.). Among these possible additives, our preliminary data excluded colorants as the main cause of isotopic variation in the investigated samples. These results were confirmed by the lack of significant difference among polymers of different colors (p > 0.001). The independence of the δ13C value from the plastic color could provide an important analytical advantage to the isotopic approach over some of the other analytical methods used for plastic characterization. In particular, the spectroscopic methods have been proved to be limited by the color of the plastic samples, because of the occurrence of interferences due to a decrease of the diffuse reflection intensity in dark color samples [31]. Further investigation and larger analytical data set are required in order to strength these results.
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**2.2 δ13C as a possible tool to distinguish petroleum-derived plastics from plant-derived plastics**
To reduce the impact of plastic debris in the environment, recyclable and more biodegradable polymers (i.e., plant-derived polymers) have been introduced increasingly into the market [10]. Plant-derived plastic polymers used for food packaging, such as bags and bottles for drinking water, showed a significant difference in isotopic values with respect to the petroleum-derived plastic products. In fact, petroleum-derived packaging materials for food use, such as shopping bags for fruits and vegetables (HD PE) and plastic bottles for drinking water (PET), were characterized by the δ 13C mean values of −33.97 ± 1.15 and − 27.84 ± 1.71‰, respectively, whereas plant-derived supermarket envelopes ("BIO" bags) and bottles (PLA, a biodegradable polyester derived from the fermentation of starch and condensation of lactic acid) recorded the δ13C mean values of −25.30 ± 0.70 and −13.87 ± 2.18‰, respectively. As regards to the results obtained for "BIO" bags, values reflected those of C3 plants, while for PLA, the analyses highlighted δ13C values similar to those of C4 plants, suggesting their specific origin.
This difference suggests that stable isotope analysis could be a useful method to discriminate between petroleum-and plant-derived plastic debris [21, 27]. The most used biopolymers are in fact produced starting from C3 (rice, potatoes, cotton, and cellulose) and C4 (corn and sugarcane) plants, species which differ for photosynthetic pathways and, consequently, for the carbon fingerprint. C3 plants recorded more negative δ13C values (ranging from −30 to −25‰) than C4 plants (ranging from −13 to −11‰), in agreement with Suzuki et al. and authors therein [21]. Considering the isotopic signature of the "BIO" bags, a common and widespread biodegradable product used for many commercial purposes, δ13C values are generally comparable with those reported for C3 plants. Regarding "recycled" polymers, LD PE recycled envelopes showed a δ13C mean value of −27.75‰. The presence of a low quantity of other polymers as impurities or different recycle processes could explain the less negative average value with respect to the row LD PE (−30.19‰) given by an 13C enrichment or depletion (fractionation).
*Elemental Analyzer/Isotope Ratio Mass Spectrometry (EA/IRMS) as a Tool to Characterize... DOI: http://dx.doi.org/10.5772/intechopen.81485*
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"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
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"title": "Plastics in the Environment",
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**2.3 δ13C as a possible tool to study degradation processes in the marine environment**
The influence of natural degradation processes on the fractionation of carbon in plastic materials under marine conditions, according to a preliminary field study carried out by Berto et al. [27] in Venice lagoon, is showed in **Figure 10**. Over a 60-day period and under variable conditions of temperature and salinity (i.e., 24–35°C and 7.8–8.1, respectively), the δ 13C values of both "BIO" bags and HD PE bags showed a gradual decrease toward less negative values, recording a δ13C variation (Δδ13C) of 1.15 and 1.78‰, respectively. This shift could be reasonable due to physical, chemical, and/or biological degradation, even if the latter is a controversial matter.
The degradation of plastic polymers in the environment involves many factors (photodegradation, thermooxidation, hydrolysis, and biodegradation by microorganisms) [32], and it proceeds according to the rates highly dependent on the environmental conditions. For instance, several authors have reported that degradation processes and the rate of hydrolysis of most plastic polymers become insignificant in the ocean when the temperature and the concentration of oxygen are reduced [32, 33].
However, the physical/mechanical degradation occurring in the marine environment can alter the plastic polymers at the surface layer and favor the starting of microbial deterioration processes. By considering that, in many biochemical reactions, such as autotrophic fixation of CO2 by plants [34] and microbial degradation processes, the lightest isotope (12C) are preferentially used as a substrate over the heaviest isotopes, and the different isotopic values recorded by Berto et al. [27] for "pristine" and "aged" plastic materials sampled from the marine environment suggested the occurrence of degradation processes. Further studies are needed to evaluate the pathway and the time featuring this process.
In fact, some researchers are confident in thinking that biopolymer (such as cellulose in plants) plastics are not generally biodegradable. Bacteria and fungi coevolved with natural materials, while plastics have only been around for about 70 years. So microorganisms simply have not had much time to evolve the necessary biochemical tool kit to latch onto the plastic fibers, break them up into the constituent parts, and then use the resulting chemicals as a source of energy and carbon that they need to grow [35]. However, in 2016 a team of researchers from Kyoto Institute of Technology and Keio University, after collecting environmental samples
containing PET debris, observed a novel bacterium (*Ideonella sakaiensis* 201-F6) which is able to use PET plastic for carbon growth. This bacterium produces two distinct enzymes hydrolizing PET plastics into terephthalic acid and ethylene glycol. This discovery has potential importance for the recycling process of PET [36].
A large number of tests (respirometric, loss of weight, tensile strength, spectroscopic) have been conducted to evaluate the extent of degradation of polymers, either alone or in blended forms, mainly under terrestrial environmental conditions.
It is worth noting that most recalcitrant polymers can be degraded to some extent in the appropriate environment at the right concentration. A screening program to study the ability of organisms and enzymes in degrading plastic polymers in a marine environment is required, considering the increasing importance of biodegradable plastics in the last few years.
Considering the new data presented in this study, it is possible to hypothesize the new paths for stable isotope research applied to the plastic polymers in the environment.
## **3. Conclusive remarks**
In this chapter, we focused on plastic polymers, both petroleum- and plantderived, commonly used in commercial packaging products for food use, giving preliminary overview of their δ 13C values. The low difference of δ 13C values among polymers suggested that the different chemical pathways used for their synthesis did not induce fractionation of carbon stable isotopes, yielding to δ 13C values meaningful of the row material (i.e., petroleum and terrestrial plants). Thus, this technique showed interesting perspective for its application in discriminating petroleum- and plant-derived polymers in marine samples.
Furthermore, the method showed to be unaffected by additional variables, such as color, and thus, it seems a valuable alternative to the spectroscopy methods for the characterization of plastic polymers in marine samples, which in contrast found the analytical limitation especially with dark colored plastic samples.
Finally, an important potential of the isotope mass spectrometry is its application to the study of the degradation processes (abiotic and biotic) of plastic waste released in the marine environment and the assessment of the degradation rates. In particular, this technique could be applied for analysis of suspended plastic debris, after filtration of both marine and fresh water samples collected along the water column. In this regard, however, further studies are needed to discriminate the isotopic values of suspended organic matter from those of plastic polymers, with major concern for micro and nanoplastics. Such possible application is of particular interest for the estimation of the fate of plastics in the marine environment and the evaluation of the effectiveness of the policies developed to reduce the environmental impact of marine litter.
### **Acknowledgements**
The authors are grateful to Davide Zanella for graphical support and to Guido Giazzi and Luca Simonotti of Thermo Fischer Scientific for analytical support.
### **Conflict of interest**
No potential conflict of interest was reported by the authors.
*Elemental Analyzer/Isotope Ratio Mass Spectrometry (EA/IRMS) as a Tool to Characterize... DOI: http://dx.doi.org/10.5772/intechopen.81485*
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{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
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"title": "Plastics in the Environment",
"publisher": "IntechOpen",
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"section_idx": 43
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**Author details**
Daniela Berto1 \*, Federico Rampazzo1 , Claudia Gion1 , Seta Noventa1 , Malgorzata Formalewicz1 , Francesca Ronchi1 , Umberto Traldi2 and Giordano Giorgi3
1 Italian National Institute for Environmental Protection and Research (ISPRA), Chioggia (VE), Italy
2 Thermo Fisher Scientific, Rodano (MI), Italy
3 Italian National Institute for Environmental Protection and Research (ISPRA), Rome, Italy
\*Address all correspondence to: [email protected]
© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"book_id": "ffcbf675-2a25-491a-a3fe-16b210576719",
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"author": "",
"title": "Plastics in the Environment",
"publisher": "IntechOpen",
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"section_idx": 44
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Study of the Technical Feasibility of the Use of Polypropylene Residue in Composites for Automotive Industry
*Denis R. Dias, Maria José O. C. Guimarães, Christine R. Nascimento, Celio A. Costa, Giovanio L. de Oliveira, Mônica C. de Andrade, Ana Maria F. de Sousa, Ana Lúcia N. da Silva and Elen B. A. Vasques Pacheco*
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{
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"author": "",
"title": "Plastics in the Environment",
"publisher": "IntechOpen",
"isbn": "9781838804930",
"section_idx": 47
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ffcbf675-2a25-491a-a3fe-16b210576719.48
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**Abstract**
Polypropylene (PP) is widely used in short-term use artifacts, rapidly discarded and should partially replace neat PP. In addition, it is one of the polymers most used in the automobile industry. This study shows the technical feasibility of partially substituting neat PP for a post-consumer counterpart (PPr), as well as adding ground glass (GP), used as filler in the polymer matrix. Mechanical and thermal properties of the recycled blends (PP/PPr) and composites (PP/PPr/GP) were evaluated. The results demonstrated that the blend with the highest PPr content obtained a statistically significant decline in elastic modulus, but adding 5 wt% of GP to this blend increased this property, achieving a similar value in relation to neat PP. The composite developed may be a promising tailor-made product with properties resembling those of the virgin plastic. Thus, the automotive industry seems to be a good option for the use of PPr and GP composites and blends, without increasing product requirements.
**Keywords:** recycling, polypropylene, automotive, industry, PP, glass powder, composite
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{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"book_id": "ffcbf675-2a25-491a-a3fe-16b210576719",
"url": "https://mts.intechopen.com/storage/books/7479/authors_book/authors_book.pdf",
"author": "",
"title": "Plastics in the Environment",
"publisher": "IntechOpen",
"isbn": "9781838804930",
"section_idx": 48
}
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**1. Introduction**
The circular economy promotes sustainability by combining the hierarchy of reduction, reuse and recycling, considering economic and environmental issues [1]. The use of post-consumer materials to manufacture new ones for a new production cycle to minimize the waste of natural resources is the goal of the circular economy [2–6]. The same can be said of plastic, an important class of materials that meets many of society's needs.
Plastics are present in our daily life under different forms and applications, such as office supplies, toys, footwear, civil construction, electrical and electronic components, aerospace, food, the medical and textile industries, packaging, paint and varnish, as well as the automotive industry, among others. This array of applications is due to their desirable properties in a range of sectors [7, 8].
However, not all plastics return to the production cycle after use. In 2015, only 9% of plastics produced worldwide were recycled, 12% incinerated and the rest buried in landfills [9]. Recycling causes less environmental impact, as reported by Bernardo et al. [10], who assessed recycling in terms of global warming and total energy use, concluding that plastic materials generally display environmental and economic advantages over conventional materials throughout their life cycle, from raw material extraction to synthesis, transformation, transport, use, recovery and destination. Duval and Maclean [11] also found a decline in greenhouse gas emissions and energy required during recycling [12]. Mechanical recycling involves the addition of virgin or recycled material to maintain properties [13].
Plastics can be recycled in different ways, including mechanically, chemically and energetically. Chemical recycling involves physical processes, such as remolding [14–16], and the final product is a monomer or oligomer that can be used in the synthesis of other products. In energy recycling, the energy released from the burning of waste material is reused [14].
Closed-loop recycling occurs when the recycled material replaces the virgin material in the same production cycle as the original product [17, 18]. Open-loop recycling is when the recycled product is used in a different production cycle, that is, the product to be recycled is used to manufacture a product different from the original [17, 19, 20].
A crucial point to stimulate recycling is the search for a different market for the recycled material and more environmentally sustainable processes. Traditionally, recycled products compete with virgin material, which may hinder their market entry. Scientific studies that focus on recycling should also seek to obtain more economically feasible and technically useful recycled products.
The most widely used and manufactured plastics are high-density polyethylene (HDPE), polypropylene (PP), low-density polyethylene (LDPE), poly(ethylene terephthalate) (PET), poly(vinyl chloride) (PVC) and polystyrene (PS) [9]. In the automotive industry, PP is used to manufacture the following items: car trunk lids; battery trays and boxes; heater boxes; tool boxes; seat belt buckling boxes; rear view mirror boxes; electric junction boxes; hubcaps; carpets; battery guards (protection against short circuit); steering wheel covers; shock absorber covers; vacuum hoses; air hoses; consoles; bumpers; glove boxes, among several other uses [21–24].
To comply with the main technical demands of automobile manufacturers, PP compounds must exhibit a suitable balance between stiffness and tenacity, with good thermal resistance, as well as fewer imported raw materials, thereby achieving more competitive prices. In addition to these properties, PP shows good processability [24–26].
An important supplier of materials to the automotive industry is the industrial sector responsible for manufacturing laminated and tempered glass used in motor vehicle windows (laminated glass for windshields and tempered for the other windows). However, in the tempering and laminating processes an industrial residue consisting of glass powder is generated and disposed of in landfills, with no specific use for this material [27]. In addition to the origin of glass powder in laminating and tempering processes [28], the windows that are removed from automobiles are also discarded when they cannot be reused. In such cases, these parts can be collected and recovered, then submitted to separation processes (polymer protection film) and grinding. The glass powder produced can be incorporated into polymer materials, resulting in composites with different properties [29].
*Study of the Technical Feasibility of the Use of Polypropylene Residue in Composites… DOI: http://dx.doi.org/10.5772/intechopen.81147*
Incorporating mineral loads into PP has been the object of studies on the production of materials with different properties [30, 31]. Improving the properties of the final product depends on the type of load, particle size of the mineral load being used and degree of dispersion of these particles in the polymer matrix. The most widely used commercial mineral loads are talcum and calcium carbonate [32–34].
This study describes the addition of glass powder to a PP matrix in order to obtain reinforcement properties and compare them with those of conventional composites. The aim is to acquire different properties in the polypropylene composites and reuse a residue (in this case, glass powder). We also assessed the effect of adding recycled polypropylene on the final properties of composites in order to reuse both industrial (glass powder) and urban residue (PP recycled from packaging).
### **2. The use of polypropylene in a composite or mixture**
Polymers have been increasingly used in a number of applications as a substitute for traditional materials such as metal and ceramic, as homopolymers; formulated with additives, in the form of mixtures and polymer composites; or simply for their different properties, such as lightness, low transformation cost, resistance to corrosion, optimal thermal and electric insulation and easy conformation into complex shapes [33].
In general, the mechanical properties of polymers are not suitable in a number of applications owing to their lower resistance compared to metals and ceramics. However, the thermoplastic industry is growing due to ecological issues, in addition to the promising potential of these materials as mixtures or a composite matrix [35].
Compound systems formed by the combination of two polymer materials (mixtures) or a polymer material and a load (composites) are of significant technological interest due to the cost–benefit ratio. In both cases, the material consists of a continuous (matrix) and disperse phase, whose properties depend on good interaction between them [35].
The properties of interest for the automotive industry can be modified with studies on improving the polymer matrix, load, and polymer-load interface, among others. The interface is a link between the surface of the load and the matrix, and since the matrix receives the reinforcement, there is close contact between them, and there may or may not be adhesion. For a same combination of materials, different adhesion mechanisms can occur, such as mechanical, chemical, and electrostatic adhesion and by interdiffusion. The degree of reinforcement or improvement in mechanical behavior depends on a strong matrix-particle interface bond [36, 37].
The stress–strain behavior of many reinforced polymers or plastics can be changed by adhesion promoters and interfacial coupling agents (such as maleic anhydride) that alter adhesion and the nature of the matrix-load interface [38].
Polypropylene (PP) is a recyclable thermoplastic, that is, it melts when heated and hardens again when cooled, in a reversible process. Moreover, PP is easily mixed, primarily with organic reinforcing loads such as natural or inorganic fibers, including calcium carbonate, clay and talcum, and is widely used in structural applications [39–41].
The use of modified PP, especially for applications in the automotive industry, requires a suitable balance between stiffness and tenacity. In this scenario, the process of incorporating elastomeric materials, as well as mineral loads such as talcum and calcium carbonate (CaCO3), into the PP matrix has been widely used to achieve different properties [42, 43].
Nanofillers, such as silica and calcium carbonate nanoparticles, have been added to improve the final properties of the PP matrix [44, 45].
#### **2.1 Use of glass as an additive to the composite**
The use of glass in a polypropylene matrix has been extensively studied and employed its glass fiber form in materials in which mechanical properties such as tensile strength and resistance to impact are important [46, 47].
There are several groups of glass, including silica, oxynitride and phosphate, but the first is the most important raw material used in composites [48]. Short E-glass fibers, obtained from a mixture of Si, Al, B, Ca and Mg oxides, are normally used as reinforcement for thermoplastics due to their low cost when compared to aramid and carbon [49], in addition to better impact strength and stiffness [50].
The interfacial interaction of glass composites with a thermoplastic matrix is often very weak. Particularly with polyolefin polymers such as polypropylene, there is little or no chemical reaction between the glass and the matrix. The interest in polypropylene for applications as a matrix in composites has been growing and the adhesion of this nonpolar polymer to the glass surface, which is also nonpolar, is a daunting challenge [51, 52].
In addition to the use of glass fiber, there are also glass microsphere applications [53]; however, residual glass powder remains a poorly explored load as reinforcement.
### **2.2 Environmental justification for polypropylene and window glass, materials contained in automobiles**
Initiatives to develop more sustainable technological innovations and ecologically responsible management programs have been driven by a growth in environmental awareness and increasingly rigid legislation. The accumulation of plastic waste caused by the increase in per capital consumption of thermoplastic resins has prompted enormous research and efforts to substitute traditional thermoplastics [54].
To improve the production process, it is necessary to diagnose the flowchart of the process and manage inputs (water, energy, raw materials, etc.) and outputs (products, residues, effluents, atmospheric emissions, etc.). In general, inputs are natural resources that often cause environmental impact, such as ecosystem destruction, atmospheric pollution, etc. Outputs are environmental liabilities created by activities and residual materials (solid, liquid or gas) that, if not suitably managed, may cause permanent environmental impacts [55].
A sustainable production process contains a circular flow, where outputs are reintegrated into the process, which reduces impacts and costs in the generation of inputs and the destination and treatment of outputs. Recycling is an example of this type of sustainable production strategy and is therefore an attempt to reuse the material, natural resources and entropy expenditures in the production of a solid residue, reintroducing it into a new production process, thereby transforming the output of a process into the input of the same or another process [55].
The automotive industry is attempting to transform the car into a more sustainable and efficient product, not only in terms of the environment, but also from the consumer's financial standpoint. As such, the automotive industry has been working within the so-called DFE (Design for the Environment), that is, designing for the environment and introducing environmental variables in all the production strategies of the factory, such as product design (automobiles and parts), the process (manufacture of parts and assembly) and associated technologies (material treatment, painting, etc.) [28].
It is important to underscore that all participants in the life cycle of a product have shared responsibility. Thus, manufacturers, importers, distributors,
*Study of the Technical Feasibility of the Use of Polypropylene Residue in Composites… DOI: http://dx.doi.org/10.5772/intechopen.81147*
merchants, consumers and public cleaning concessionaires should promote the reuse of solid residues, transfer them to the production chain, reduce residue generation and encourage the development of products derived from recycled materials. The automotive industry, like all companies, is responsible for the entire process, from acquiring raw materials to discarding components, such as bumpers. Moreover, polypropylene is present in many automobile components.
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**3. Materials and methods**
#### **3.1 Materials**
The grade of the Virgin polypropylene (PP) was H 605. The values for the PP properties presented in **Table 1** were provided by the Braskem Company.
The post-industrial polypropylene (PPr) was a washed and ground material supplied by the Poli Injet Company (Brazil), while the residue came from the packaging industry. The properties of PPr are: melt temperature (Tm) of 161°C, crystallinity degree (χc) of 32%, and melt flow index (MFI) of 4.83 g/10 min (230°C/2.16 Kg). The methodologies to evaluate these properties are described below.
The glass powder (GP) used in this study was supplied by the Massfix Company (Brazil) and is ground from windshield scraps. **Figure 1** shows the morphology of GP samples examined under a scanning electron microscope (SEM). The GP sample is composed of irregular-shaped particles with a broad size distribution.
Polypropylene modified with maleic-anhydride (PPMA), and Polybond 3200 with 1 wt% of maleic anhydride (MA) were supplied by the Chemtura Company (USA). The melt flow rate specified by the supplier is 115 g/10 min at 190°C under 2.16 Kg.
#### **3.2 Blend and composite preparation**
**Table 2** shows the compositions of the blends and composites. GP, PP, PPr, and PPMA were dried in an oven at 60°C for 24 h before extrusion. Next, each composition was processed in a twin-screw extruder (TeckTril, L/D = 36, screw diameter = 20 mm) at a screw speed of 400 RPM and temperature profile of 90/12 0/150/160/185/200/220/240/260/260°C. The materials underwent injection molding to produce appropriate specimens for stress and impact strength tests, which were performed according to ASTM D638 and ASTM D256 standards, respectively. Injection molding was carried out in an Arburg 270 S injection machine, using a
#### **Table 1.** *Polypropylene H605 properties.*
#### **Figure 1.** *SEM micrographs of glass powder (GP) particles.*
#### **Table 2.**
*PP, PPr, glass powder and PPMA blends and (or) composites composition.*
temperature profile of 210/215/220/230/230°C. The injection and molding pressures were 1000 and 180 Bar, respectively. The mold temperature was 30°C, with a cooling time of 30 seconds.
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"section_idx": 50
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**3.3 Material characterizations**
Tensile properties were measured using a universal testing machine (EMIC, DL3000) based on ASTM D-638. Izod pendulum impact resistance was determined using a CEAST Resil Impactor tester based on ASTM D256. The data related to all the mechanical properties were based on the average of eight tested specimens.
Melting temperature (Tm), melt enthalpy (ΔHm) and crystallinity degree (χc) of the materials were determined using a Differential Scanning Calorimeter (STA 6000, Perkin Elmer) during the second heating scan. Samples weighing between 25 and 30 mg were heated from room temperature to 300°C at a heating rate of 10°C/min (first heating scan). The temperature was then lowered to 30°C at a heating rate of 10°C/min, and the samples submitted to a second heating scan under the same conditions as the first. Crystallinity degree was calculated using Eq. (1).
$$\% \text{ Cristalinidade} = \frac{\Delta \text{H}\_{\text{f}}}{\Delta \text{H}\_{100\%}} \ge 100 \tag{1}$$
where ΔHm is the endothermic enthalpy, ΔH100% the theoretical melting enthalpy of 100% crystalline PP (209 J/g) [56], and wt% the amount of PP in the blend or composite.
*Study of the Technical Feasibility of the Use of Polypropylene Residue in Composites… DOI: http://dx.doi.org/10.5772/intechopen.81147*
The cryogenic-fractured surface morphology of the materials was examined under a scanning electron microscope (SEM, FEI, Quanta 400, accelerating voltage at 25 kV, 800X). The fractured samples were coated with gold.
### **3.4 Statistical analysis**
The statistical analysis of the results was performed using STATISTICA 6 software. Analysis of variance (ANOVA) was applied to test for significant differences between the means. Residual normality and homogeneity of variances (Cochran C and Bartlett methods) were determined before univariate tests of significance and Fisher's least significant difference (LSD) test, using a significance level of α = 0.05.
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**4. Results and discussion**
#### **4.1 Morphological and thermal characterization**
**Figure 2** shows the cryogenic fractured specimen for comparison between PP, PPr1/GP, and PPr3/GP. Poor interaction can be observed between the polymer matrix and glass particles due to the presence of small voids, gaps, and unattached particles.
The addition of PPMA in GP/PPr composites resulted in the smoothing of fractured surfaces (**Figure 3**). Furthermore, unattached particles, micro-voids and the gap between the matrix and the filler were slightly reduced. As such, filler-matrix interaction improved due to the addition of the coupling agent. **Table 3** shows the thermal properties of composites containing GP.
The melting temperatures (Tm) of all samples were similar to those of neat PP, except for the presence of a small endothermic peak in some composites at 127°C. Based on literature data [57], the small peak at 127°C can be attributed to the polyethylene and contamination in PPr, which is very common due to the difficulty in separating PP from PE during the recycling process.
In general, the composites showed similar crystallinity degrees (χc) to those of neat PP (**Table 3**). The results are noteworthy because they suggest that the presence of post-consumer materials (PPr and GP) did not disturb the crystal formation of the final composite, which leads to the assumption that the final properties of the composite are maintained, even with the addition of postconsumer materials.
#### **Figure 3.**
*SEM micrographs in two sizes (100 and 50 μm) of PP/PPr1/GP/PPMA (75/10/5/10%w/w/w/w) and PPr3/GP (55/30/5/10%w/w/w/w) composites.*
#### **Table 3.**
*Thermal properties of PP and PP/PPr/GP-base composites.*
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**4.2 Mechanical properties**
Material tensile properties are important for both engineering and packaging applications, since they represent the ability of materials to withstand the load transferred in the longitudinal direction. Impact strength is essential to engineering applications, due to the need to bear high loads for very short periods of time. Thus, tensile properties are vital in evaluating bags or mooring ropes, while impact strength is a critical feature in recycled plastic fencing, furniture and automobile parts. Tensile properties, mean values and confidence intervals (Fisher's LSD test) are shown in **Figures 4**–**7**.
With respect to elastic modulus (**Figure 4**), a comparison between PPr1, PPr3, and PP showed that elastic modulus, which is related to composite rigidity, tends to decrease with the addition of PPr, but this effect is only significant for the composite with 30%wt of PPr. However, when recycled filler (GP) was added to PP/ PPr blends, the modulus rose until the mean composite modulus values were equal to those of virgin PP. This finding suggests that the previous decline in stiffness observed in PP/PPr blends can be solved by adding glass powder.
*Study of the Technical Feasibility of the Use of Polypropylene Residue in Composites… DOI: http://dx.doi.org/10.5772/intechopen.81147*
**Figure 4.** *Elastic modulus of PP and PP/PPr/GP-base composites.*
#### **Figure 5.**
*Stress x strain curves of composites samples with: (a) 0% and (b) 10% of PPMA.*
By contrast, the improvement in elastic modulus was reversed when PPMA was added to PP/PPr/GP composites. This is significant because good cohesion between the matrix and GP, as shown in **Figure 3**, was expected to improve the
**Figure 7.** *Impact strength of PP and PP/PPr/GP-base composites.*
tmodulus of the composite. One reason that may explain the previous undesirable result is the high amount of PPMA used as plasticizer. In other words, the amount of PPMA exceeded what was needed to coat the particle surfaces, diffusing in the
*Study of the Technical Feasibility of the Use of Polypropylene Residue in Composites… DOI: http://dx.doi.org/10.5772/intechopen.81147*
polymer matrix and influencing plasticizer properties or those of a third polymer component.
Yield stress is the maximum stress at which the material begins to exhibit permanent deformation. As the elastic limit shifts, the material does not return to its original dimensions after the applied stress is removed. This property is particularly important for automotive applications. The yield stress of composites is illustrated on median stress x strain curves (**Figure 5**), showing the PPMA effect.
Yield properties (**Figure 6**) are generally in accordance with the modulus trend, that is, the higher the stiffness, the greater the stress and lower the strain at the yield point of the material. This behavior became evident when virgin PP was compared to PPr/GP composites. For products whose performance is highly dependent on tensile properties, the PPr3/GP sample remains the best option, considering elastic modulus, yield and environmental aspects, since the properties are very similar to those of neat PP even with the addition of 30% PPr and 5%wt of GP.
With respect to the automotive applications of plastic materials, acceptable impact strength is one of the requirements and, in the case of composites, this property is highly sensitive to particle/matrix debonding during mechanical energy dissipation. **Figure 7** shows the mean values and confidence intervals of impact strength.
The PPr1/GP/PPMA sample showed somewhat better results compared to virgin PP, with a P-value of 0.0838, indicating no significant difference between results. The use of PPMA as a coupling agent for GP could be optimized to improve the impact strength of composites.
In the present study, the PPMA grade used exhibited a maleic anhydride level of 1.0% by weight, but other grades with higher levels and greater affinity to GP could be tested in future research.
Compared to virgin resin, blends of PP with PPr demonstrated poor impact properties (**Figure 7**). In addition to higher impurity levels, PPr is expected to show lower molecular weight than PP, and both factors can contribute to failure in recycled material blends. Fukuhara et al. [58] evaluated isotactic polypropylene with different molecular weight and observed less Izod impact strength in samples with lower molecular weights. Furthermore, any structural particularity in PPr able to influence PP crystalline morphology can modify mechanical properties. Xu et al. [59] studied the relationship between spherulite size and crystallinity in the impact strength of PP samples with several different nucleating agents. The authors reported that impact strength was primarily controlled by spherulite size for samples with low crystallinity. For high crystalline samples, crystallinity itself is the decisive factor in strength. The authors also observed that impact strength is greater in PP samples with small spherulites and lower crystallinity. Nevertheless, no clear relationship between the degree of crystallinity and impact strength of samples was observed in the present study (**Table 3**, **Figure 7**). As such, other factors, such as impurity content, may exert the greatest influence on impact results. Given that products such as furniture and automobile parts require high impact strength, suitable coupling agents should be added to recycled composites in order to enhance their properties.
### **5. Conclusions**
The greatest challenge to plastics in the automotive industry is in recycling. Some automotive manufacturers, such as Ford and Toyota, are recycling their vehicles plastics and reusing in the new vehicles, for example, old or damaged bumpers are recycled and reused in bumper reinforcement cores [60].
According to the results presented, it can be concluded that the properties did not vary significantly as a function of composition. Since the objective was to produce lower cost composites (incorporating recycled PP and glass powder) and more sustainable materials without significant loss of properties, this result is within the parameters established. In other words, it was possible to recover post-consumer materials, replacing the virgin resin without significant loss of mechanical integrity in the final product. The addition of maleated polypropylene (PPMA) was shown to significantly improve the toughness of the material.
In conclusion, based on the properties analyzed and the sustainable appeal of the new products, the powder-based composites displayed potential for use in various applications in the automotive industry, replacing conventional materials.
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**Acknowledgements**
The authors would like to thank Massfix company for donating the glass powder (GP), Poli Injet company for supplied the post-industrial polypropylene (PPr) and Braskem company for donating the virgin polypropylene, and PIBIC/UFRJ for the scholarship grants.
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**Conflict of interest**
The authors declare that there is no conflict of interest regarding the publication of this paper.
*Study of the Technical Feasibility of the Use of Polypropylene Residue in Composites… DOI: http://dx.doi.org/10.5772/intechopen.81147*
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**Author details**
Denis R. Dias1 , Maria José O. C. Guimarães1 , Christine R. Nascimento2 , Celio A. Costa 2 , Giovanio L. de Oliveira2 , Mônica C. de Andrade3 , Ana Maria F. de Sousa4 , Ana Lúcia N. da Silva5,6 and Elen B. A. Vasques Pacheco5,6\*
1 Departamento de Processos Orgânicos, Escola de Química, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil
2 Programa de Engenharia Metalúrgica e de Materiais, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil
3 Instituto Politécnico do Rio de Janeiro, Universidade do Estado do Rio de Janeiro, Nova Friburgo, Brazil
4 Instituto de Química, Universidade do Estado do Rio de Janeiro, Rio de Janeiro, Brazil
5 Instituto de Macromoléculas Professora Eloisa Mano, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil
6 Programa de Engenharia Ambiental, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil
\*Address all correspondence to: [email protected]
© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Biological Degradation of Polymers in the Environment
*John A. Glaser*
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**Abstract**
Polymers present to modern society remarkable performance characteristics desired by a wide range of consumers but the fate of polymers in the environment has become a massive management problem. Polymer applications offer molecular structures attractive to product engineers desirous of prolonged lifetime properties. These characteristics also figure prominently in the environmental lifetimes of polymers or plastics. Recently, reports of microbial degradation of polymeric materials offer new emerging technological opportunities to modify the enormous pollution threat incurred through use of polymers/plastics. A significant literature exists from which developmental directions for possible biological technologies can be discerned. Each report of microbial mediated degradation of polymers must be characterized in detail to provide the database from which a new technology developed. Part of the development must address the kinetics of the degradation process and find new approaches to enhance the rate of degradation. The understanding of the interaction of biotic and abiotic degradation is implicit to the technology development effort.
**Keywords:** polymers, plastics, degradation, microbial degradation, biofilms, extent of degradation
## **1. Introduction**
In 1869, the first synthetic polymer was invented in response to a commercial \$10,000 prize to provide a suitable replacement to ivory. A continuous string of discoveries and inventions contributed new polymers to meet the various requirements of society. Polymers are constructed of long chains of atoms, organized in repeating components or units often exceeding those found in nature. Plastic can refer to matter that is pliable and easily shaped. Recent usage finds it to be a name for materials called polymers. High molecular weight organic polymers derived from various hydrocarbon and petroleum materials are now referred to as plastics [1].
Synthetic polymers are constructed of long chains of smaller molecules connected by strong chemical bonds and arranged in repeating units which provide desirable properties. The chain length of the polymers and patterns of polymeric assembly provide properties such as strength, flexibility, and a lightweight feature that identify them as plastics. The properties have demonstrated the general utility of polymers and their manipulation for construction of a multitude of widely useful items leading to a world saturation and recognition of their unattractive properties too. A major trend of ever increasing consumption of plastics has been seen in the
#### **Table 1.**
*Selected features of major commercial thermoplastic polymers [7].*
areas of industrial and domestic applications. Much of this polymer production is composed of plastic materials that are generally non-biodegradable. This widespread use of plastics raises a significant threat to the environment due to the lack of proper waste management and a until recently cavalier community behavior to maintain proper control of this waste stream. Response to these conditions has elicited an effort to devise innovative strategies for plastic waste management, invention of biodegradable polymers, and education to promote proper disposal. Technologies available for current polymer degradation strategies are chemical, thermal, photo, and biological techniques [2–6]. The physical properties displayed in **Table 1** show little differences in density but remarkable differences in crystallinity and lifespan. Crystallinity has been shown to play a very directing role in certain biodegradation processes on select polymers.
Polymers are generally carbon-based commercialized polymeric materials that have been found to have desirable physical and chemical properties in a wide range of applications. A recent assessment attests to the broad range of commercial materials that entered to global economy since 1950 as plastics. The mass production of virgin polymers has been assessed to be 8300 million metric tons for the period of 1950 through 2015 [8]. Globally consumed at a pace of some 311 million tons per year with 90% having a petroleum origin, plastic materials have become a major worldwide solid waste problem. Plastic composition of solid waste has increased for less than 1% in 1960 to greater than 10% in 2005 which was attributed largely to packaging. Packaging plastics are recycled in remarkably low quantities. Should current production and waste management trends continue, landfill plastic waste and that in the natural environment could exceed 12,000 Mt of plastic waste by 2050 [9].
### **2. Polymer structures and features**
A polymer is easily recognized as a valuable chemical made of many repeating units [10]. The basic repeating unit of a polymer is referred to as the "-mer" with "poly-mer" denoting a chemical composed of many repeating units. Polymers can be chemically synthesized in a variety of ways depending on the chemical characteristics of the monomers thus forming a desired product. Nature affords many examples of polymers which can be used directly or transformed to form materials required by society serving specific needs. The polymers of concern are generally composed of carbon and hydrogen with extension to oxygen, nitrogen and chlorine functionalities (see **Figure 1** for examples). Chemical resistance, thermal and electrical insulation, strong and light-weight, and myriad applications where no alternative exists are polymer characteristics that continue to make polymers attractive. Significant polymer application can be found in the automotive, building and construction, and packaging industries [12].
*Biological Degradation of Polymers in the Environment DOI: http://dx.doi.org/10.5772/intechopen.85124*
**Figure 1.** *Structures of major commercial thermoplastic polymers [11].*
The environmental behavior of polymers can be only discerned through an understanding of the interaction between polymers and environment under ambient conditions. This interaction can be observed from surface properties changes that lead to new chemical functionality formation in the polymer matrix. New functional groups contribute to continued deterioration of the polymeric structure in conditions such as weathering. Discoloration and mechanical stiffness of the polymeric mass are often hallmarks of the degradative cycle in which heat, mechanical energy, radiation, and ozone are contributing factors [13].
Polyolefins (PO) are the front-runners of the global industrial polymer market where a broad range of commercial products contribute to our daily lives in the form o packaging, bottles, automobile parts and piping. The PO class family is comprised of saturated hydrocarbon polymers such as high-density polyethylene (HDPE), low-density polyethylene (LDPE) and linear low-density polyethylene (LLDPE), propylene and higher terminal olefins or monomer combinations as copolymers. The sources of these polymers are low-cost petrochemicals and natural gas with monomers production dependent on cracking or refining of petroleum. This class of polymers has a unique advantage derived from their basic composition of carbon and hydrogen in contrast to other available polymers such as polyurethanes, poly(vinyl chloride) and polyamides [14].
The copolymers of ethylene and propylene are produced in quantities that exceed 40% of plastics produced per annum with no production leveling in sight. This continuous increase suggests that as material use broadens yearly, the amount of waste will also increase and present waste disposal problems. Polyolefin biological and chemical inertness continues to be recognized as an advantage. However, this remarkable stability found at many environmental conditions and the degradation resistance leads to environmental accumulation and an obvious increase to visible pollution and ancillary contributing problems. Desired environmental properties impact the polyolefin market on the production side as well as product recyclability [15].
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**3. Biological degradation**
Biodegradation utilizes the functions of microbial species to convert organic substrates (polymers) to small molecular weight fragments that can be further degraded to carbon dioxide and water [16–21]. The physical and chemical properties of a polymer are important to biodegradation. Biodegradation efficiency
achieved by the microorganisms is directly related to the key properties such as molecular weight and crystallinity of the polymers. Enzymes engaged in polymer degradation initially are outside the cell and are referred to as exo-enzymes having a wide reactivity ranging from oxidative to hydrolytic functionality. Their action on the polymer can be generally described as depolymerization. The exo-enzymes generally degrade complex polymer structure to smaller, simple units that can take in the microbial cell to complete the process of degradation.
#### **3.1 Requirements to assay polymer biodegradation**
Polymer degradation proceeds to form new products during the degradation path leading to mineralization which results in the formation of process endproducts such as, e.g., CO2, H2O or CH4 [22]. Oxygen is the required terminal electron acceptor for the aerobic degradation process. Aerobic conditions lead to the formation of CO2 and H2O in addition to the cellular biomass of microorganisms during the degradation of the plastic forms. Where sulfidogenic conditions are found, polymer biodegradation leads to the formation of CO2 and H2O. Polymer degradation accomplished under anaerobic conditions produces organic acids, H2O, CO2, and CH4. Contrasting aerobic degradation with anaerobic conditions, the aerobic process is found to be more efficient. When considering energy production the anaerobic process produces less energy due to the absence of O2, serving the electron acceptor which is more efficient in comparison to CO2 and SO4 −2 [23].
As solid materials, plastics encounter the effects of biodegradation at the exposed surface. In the unweathered polymeric structure, the surface is affected by biodegradation whereas the inner part is generally unavailable to the effects of biodegradation. Weathering may mechanically affect the structural integrity of the plastic to permit intrusion of bacteria or fungal hyphae to initiate biodegradation at inner loci of the plastic. The rate of biodegradation is functionally dependent on the surface area of the plastic. As the microbial-colonized surface area increases, a faster biodegradation rate will be observed assuming all other environmental conditions to be equal [24].
Microorganisms can break organic chemicals into simpler chemical forms through biochemical transformation. Polymer biodegradation is a process in which any change in the polymer structure occurs as a result of polymer properties alteration resulting from the transformative action of microbial enzymes, molecular weight reduction, and changes to mechanical strength and surface properties attributable to microbial action. The biodegradation reaction for a carbon-based polymer under aerobic conditions can be formulated as follows:
$$\text{C}\_{\text{Polymer}} + \text{O}\_2 + \text{Biomass} \xrightarrow{\text{} \longrightarrow} \text{CO}\_2 + \text{H}\_2\text{O} + \text{C}\_{\text{biomass}} \tag{1}$$
Assimilation of the carbon comprising the polymer (Cpolymer) by microorganisms results in conversion to CO2 and H2O with production of more microbial biomass (Cbiomass). In turn, Cbiomass is mineralized across time by the microbial community or held in reserve as storage polymers [25].
The following set of equations is a more complete description of the aerobic plastic biodegradation process:
$$\begin{array}{c} \text{C}\_{\text{Polymer}} + \text{O}\_{2} \xrightarrow{\text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{H}\_{2}\text{O} + \text{C}\_{\text{biomass}} \\\\ \text{-} \phantom{\text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{-} \text{$$
#### *Biological Degradation of Polymers in the Environment DOI: http://dx.doi.org/10.5772/intechopen.85124*
where Cpolymer and newly formed oligomers are converted into Cbiomass but Cbiomass converts to CO2 under a different kinetics scheme. The conversion to CO2 is referred to as microbial mineralization. Each oligomeric fragment is expected to proceed through of sequential steps in which the chemical and physical properties are altered leading to the desired benign result. A technology for monitoring aerobic biodegradation has been developed and optimized for small organic pollutants using oxygen respirometry where the pollutant degrades at a sufficiently rapid rate for respirometry to provide expected rates of biodegradation. When polymers are considered, a variety of analytical approaches relating to physical and chemical changes are employed such as differential scanning calorimetry, scanning electron microscopy, thermal gravimetric analysis, Fourier transform infrared spectrometry, gas chromatograph-mass spectrometry, and atomic force microscopy [26].
Since most polymer disposal occurs in our oxygen atmosphere, it is important to recognize that aerobic biodegradation will be our focus but environmental anaerobic conditions do exist that may be useful to polymer degradation. The distinction between aerobic and anaerobic degradation is quite important since it has been observed that anaerobic conditions support slower biodegradation kinetics. Anaerobic biodegradation can occur in the environment in a variety of situations. Burial of polymeric materials initiates a complex series of chemical and biological reactions. Oxygen entrained in the buried materials is initially depleted by aerobic bacteria. The following oxygen depleted conditions provide conditions for the initiation of anaerobic biodegradation. The buried strata are generally covered by 3-m-thick layers which prevent oxygen replenishment. The alternate electron acceptors such as nitrate, sulfate, or methanogenic conditions enable the initiation of anaerobic biodegradation. Any introduction of oxygen will halt an established anaerobic degradation process.
#### **3.2 Formulation of newer biodegradation schema**
This formulation for the aerobic biodegradation of polymers can be improved due to the complexity of the processes involved in polymer biodegradation [27]. Biodegradation, defined as a decomposition of substances by the action of microorganisms, leading to mineralization and the formation of new biomass is not conveniently summarized. A new analysis is necessary to assist the formulation of comparative protocols to estimate biodegradability. In this context, polymer biodegradation is defined as a complex process composed of the stages of biodeterioration, biofragmentation, and assimilation [28].
The biological activity inferred in the term biodegradation is predominantly composed of, biological effects but within nature biotic and abiotic features act synergistically in the organic matter degradation process. Degradation modifying mechanical, physical and chemical properties of a material is generally referred to as deterioration. Abiotic and biotic effects combine to exert changes to these properties. This biological action occurs from the growth of microorganisms on the polymer surface or inside polymer material. Mechanical, chemical, and enzymatic means are exerted by microorganisms, thereby modifying the gross polymer material properties. Environmental conditions such as atmospheric pollutants, humidity, and weather strongly contribute to the overall process. The adsorbed pollutants can assist the material colonization by microbial species. A diverse collection of bacteria, protozoa, algae, and fungi are expected participants involved in biodeterioration. The development of different biota can increase biodeterioration by facilitating the production of simple molecules.
Fragmentation is a material breaking phenomenon required to meet the constraints for the subsequent event called assimilation. Polymeric material has a high
molecular weight which is restricted by its size in its transit across the cell wall or cytoplasmic membrane. Reduction of polymeric molecule size is indispensable to this process. Changes to molecular size can occur through the involvement of abiotic and biotic processes which are expected to reduce molecular weight and size. The utility of enzymes derived from the microbial biomass could provide the required molecular weight reductions. Mixtures of oligomers and/or monomers are the expected products of the biological fragmentation.
Assimilation describes the integration of atoms from fragments of polymeric materials inside microbial cells. The microorganisms benefit from the input of energy, electrons and elements (i.e., carbon, nitrogen, oxygen, phosphorus, sulfur and so forth) required for the cell growth. Assimilated substrates are expected to be derived from biodeterioration and biofragmentation effects. Non-assimilated materials, impermeable to cellular membranes, are subject to biotransformation reactions yielding products that may be assimilated. Molecules transported across the cell membrane can be oxidized through catabolic pathways for energy storage and structural cell elements. Assimilation supports microbial growth and reproduction as nutrient substrates (e.g., polymeric materials) are consumed from the environment.
#### **3.3 Factors affecting biodegradability**
The polymer substrate properties are highly important to any colonization of the surface by either bacteria or fungi [29]. The topology of the surface may also be important to the colonization process. The polymer properties of molecular weight, shape, size and additives are each unique features which can limit biodegradability. The molecular weight of a polymer can be very limiting since the microbial colonization depends on surface features that enable the microorganisms to establish a locus from which to expand growth. Polymer crystallinity can play a strong role since it has been observed that microbial attachment to the polymer surface occurs and utilizes polymer material in amorphous sections of the polymer surface. Polymer additives are generally low molecular weight organic chemicals that can provide a starting point for microbial colonization due to their ease of biodegradation (**Figure 2**).
Weather is responsible for the deterioration of most exposed materials. Abiotic contributors to these conditions are moisture in its variety of forms, non-ionizing radiation, and atmospheric temperature. When combined with wind effects, pollution, and atmospheric gases, the overall process of deterioration can be quite formable. The ultraviolet (UV) component of the solar spectrum contributes ionizing radiation which plays a significant role in initiating weathering effects. Visible and near-infrared radiation can also contribute to the weathering process. Other factors
#### **Figure 2.** *Factors controlling polymer biodegradation [30].*
#### *Biological Degradation of Polymers in the Environment DOI: http://dx.doi.org/10.5772/intechopen.85124*
couple with solar radiation synergistically to significantly influence the weathering processes. The quality and quantity of solar radiation, geographic location changes, time of day and year, and climatological conditions contribute to the overall effects. Effects of ozone and atmospheric pollutants are also important since each can interact with atmospheric radiation to result in mechanical stress such as stiffening and cracking. Moisture when combined with temperature effects can assist microbial colonization. The biotic contributors can strongly assist the colonization by providing the necessary nutrients for microbial growth. Hydrophilic surfaces may provide a more suitable place for colonization to ensue. Readily available exoenzymes from the colonized area can initiate the degradation process.
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**3.4 Biofilms**
Communities of microorganisms attached to a surface are referred to as biofilms [31]. The microorganisms forming a biofilm undergo remarkable changes during the transition from planktonic (free-swimming) biota to components of a complex, surface-attached community (**Figure 3**). The process is quite simple with planktonic microorganism encountering a surface where some adsorb followed by surface release to final attachment by the secretion of exopolysaccharides which act as an adhesive for the growing biofilm [33]. New phenotypic characteristics are exhibited by the bacteria of a biofilm in response to environmental signals. Initial cell-polymer surface interactions, biofilm maturation, and the return to planktonic mode of growth have regulatory circuits and genetic elements controlling these diverse functions. Studies have been conducted to explore the genetic basis of biofilm development with the development of new insights. Compositionally, these films have been found to be a single microbial species or multiple microbial species with attachment to a range of biotic and abiotic surfaces [34, 35]. Mixed-species biofilms are generally encountered in most environments. Under the proper nutrient and carbon substrate supply, biofilms can grow to massive sizes. With growth, the biofilm can achieve large film structures that may be sensitive to physical forces such as agitation. Under such energy regimes, the biofilm can detach. An example of biofilm attachment and utility can be found in the waste water treatment sector where large polypropylene disks are rotated through industrial or agriculture waste water and then exposed to the atmosphere to treat pollutants through the intermediacy of cultured biofilms attached to the rotating polypropylene disk.
Biofilm formation and activity to polymer biodegradation are complex and dynamic [36]. The physical attachment offers a unique scenario for the attached microorganism and its participation in the biodegradation. After attachment as a biofilm component, individual microorganisms can excrete exoenzymes which can provide a range of functions. Due to the mixed-species composition found in most
**Figure 3.**
*Microbial attachment processes to a polymer surface [32].*
**Figure 4.** *Biofilm formation and processes [34].*
environments, a broad spectrum of enzymatic activity is generally possible with wide functionalities. Biofilm formation can be assisted by the presence of pollutant chemical available at the polymer surface. The converse is also possible where surfaces contaminated with certain chemicals can prohibit biofilm formation. Biofilms continue to grow with the input of fresh nutrients, but when nutrients are deprived, the films will detach from the surface and return to a planktonic mode of growth. Overall hydrophobicity of the polymer surface and the surface charge of a bacterium may provide a reasonable prediction of surfaces to which a microorganism might colonize [37]. These initial cell-surface and cell-cell interactions are very useful to biofilm formation but incomplete (**Figure 4**). Microbial surfaces are heterogeneous, and can change widely in response to environmental changes. Five stages of biofilm development: have been identified as (1) initial attachment, (2) irreversible attachment, (3) maturation I, (4) maturation II, and (5) dispersion. Further research is required to provide the understanding of microbial components involved in biofilm development and regulation of their production to assemble to various facets of this complex microbial phenomenon [38].
The activities envisioned in this scenario (depicted in **Figure 4**) are the reversible adsorption of bacteria occurring at the later time scale, irreversible attachment of bacteria occurring at the second-minute time scale, growth and division of bacteria in hours-days, exopolymer production and biofilm formation in hours-days, and attachment and other organisms to biofilm in days-months.
#### **3.5 Standardized testing methods**
The evaluation of the extent of polymer biodegradation is made difficult by the dependence on polymer surface and the departure of degradation kinetics from the techniques available for small pollutant molecule techniques [39]. For applications for polymer biodegradation a variety of techniques have been applied. Visual observations, weight loss measurements, molar mass and mechanical properties, carbon dioxide evolution and/or oxygen consumption, radiolabeling, clear-zone formation, enzymatic degradation, and compost test under controlled conditions have been cited for their utility [27]. The testing regime must be explicitly described within a protocol of steps that can be collected for various polymers and compared on an equal basis. National and international efforts have developed such protocols to enable the desired comparisons using rigorous data collecting techniques and interpretation [40].
### **4. Environmental biodegradation of polymers**
The conventional polymers such as (PE), (PP), (PS), (PUR), and (PET) are recognized for their persistence in the environment [41]. Each of these polymers is subject
to very slow fragmentation to form small particles in a process expected to require centuries of exposure to photo-, physical, and biological degradation processes. Until recently, the commercial polymers were not expected to biodegrade. The current perspective supports polymer biodegradation with hopeful expectation that these newly encountered biodegradation processes can be transformed into technologies capable of providing major assistance to the ongoing task of waste polymer management.
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**4.1 Polyolefins**
The polyolefins such as polyethylene (PE) have been recognized as a polymer remarkably resistant to degradation [42]. Products made with PE are very diverse and a testament to its chemical and biological inertness. The biodegradation of the polyolefins is complex and incompletely understood. Pure strains elicited from the environment have been used to investigate metabolic pathways or to gain a better understanding of the effect that environmental conditions have on polyolefin degradation. This strategy ignores the importance of different microbial species that could participate in a cooperative process. Treatment of the complex environments associated with polymeric solid waste could be difficult with information based on pure strain analysis. Mixed and complex microbial communities have been used and encountered in different bioremediation environments [43].
A variety of common PE types, low-density PE (LDPE), high-density PE (HDPE), linear low-density PE (LLDPE) and cross-linked PE (XLPE), differ in their density, degree of branching and availability of functional groups at the surface. The type of polymer used as the substrate can strongly influence the microbial community structure colonizing PE surface. A significant number of microbial strains have been identified for the deterioration caused by their interaction with the polymer surface [44]. Microorganisms have been categorized for their involvement in PE colonization and biodegradation or the combination. Some research studies did not conduct all the tests required to verify PE biodegradation. A more inclusive approach to assessing community composition, including the non-culturable fraction of microorganisms invisible by traditional microbiology methods is required in future assessments. The diversity of microorganisms capable of degrading PE extends beyond 17 genera of bacteria and nine genera of fungi [45]. These numbers are expected to increase with the use of more sensitive isolation and characterization techniques using rDNA sequencing. Polymer additives can affect the kinds of microorganisms colonizing the surfaces of these polymers. The ability of microorganisms to colonize the PE surfaces exhibits a variety of effects on polymer properties. Seven different characteristics have been identified and are used to monitor the extent of polymer surface change resulting from biodegradation of the polymer. The characteristics are hydrophobicity/hydrophilicity, crystallinity, surface topography, functional groups on the surface, mechanical properties, and molecular weight distribution. The use of surfactants has become important to PE biodegradation. Complete solubilization of PE in water by a *Pseudomonas fluorescens* treated for a month followed by biosurfactant treatment for a subsequent month in the second month and finally a 10% sodium dodecyl sulfate treatment at 60°C for a third month led to complete polymer degradation. A combination of *P. fluorescens*, surfactant and biosurfactant treatments as a single treatment significantly exhibited polymer oxidation and biodegradation [46]. The metabolically diverse genus *Pseudomonas* has been investigated for its capabilities to degrade and metabolize synthetic plastics. *Pseudomonas* species found in environmental matrices have been identified to degrade a variety of polymers including PE, and PP [47]. The unique capabilities of *Pseudomonas* species related to degradation and metabolism of synthetic polymers requires a focus on: the interactions controlling cell surface
attachment of biofilms to polymer surfaces, extracellular polymer oxidation and/ or hydrolytic enzyme activity, metabolic pathways mediating polymer uptake and degradation of polymer fragments within the microbial cell through catabolism, and the importance of development of the implementation of enhancing factors such as pretreatments, microbial consortia and nutrient availability while minimizing the effects of constraining factors such as alternative carbon sources and inhibitory by-products. In an ancillary study, thermophilic consortia of *Brevibacillus* sps. and *Aneurinibacillus* sp. from waste management landfills and sewage treatment plants exhibited enhanced PE and PP degradation [48].
The larval stage of two waxworm species, *Galleria mellonella* and *Plodia interpunctella*, has been observed to degrade LDPE without pretreatment [49, 50]. The worms could macerate PE as thin film shopping bags and metabolize the film to ethylene glycol which in turn biodegrades rapidly. The remarkable ability to digest a polymer considered non-edible may parallel the worm's ability utilize beeswax as a food source. From the guts of *Plodia interpunctella* waxworms two strains of bacteria, *Enterobacter asburiae* YP1 and *Bacillus* sp. YP1, were isolated and found to degrade PE in laboratory conditions. The two strains of bacteria were shown to reduce the polymer film hydrophobicity during a 28-day incubation. Changes to the film surface as cavities and pits were observed using scanning electron microscopy and atomic-force microscopy. Simple contact of ~100 *Galleria mellonella* worms with a commercial PE shopping bag for 12 hours resulted in a mass loss of 92 mg. The waxworm research has been scrutinized and found to be lacking the necessary information to support the claims of the original *Galleria mellonella* report [51].
Polypropylene (PP) is very similar to PE, in solution behavior and electrical properties. Mechanical properties and thermal resistance are improved with the addition of the methyl group but chemical resistance decreases. There are three forms of propylene selectively formed from the monomer isotactic, syndiotactic, and atactic due to the different geometric relationships achievable through polymerization technology. PP properties are strongly directed by tacticity or the methyl group orientation as related the methyl groups in neighboring monomer units. Isotactic PP has a greater degree of crystallinity than atactic and syndiotactic PP and therefore more difficult to biodegrade. The high molar mass of PP prohibits permeation through the microbial cell membrane which thwarts metabolism by living organisms. It is generally recognized that abiotic degradation provides a foothold for microorganisms to form a biofilm. With partial destruction of the polymer surface by abiotic effects the microbes can then start breaking the damaged polymer chains [52].
#### **4.2 Polystyrene**
PS is a sturdy thermoplastic commonly used in short-lifetime items that contribute broadly to the mass of poorly controlled polymers [53]. Various forms of PS such as general purpose (GPPS)/oriented polystyrene (OPS), polystyrene foam, and expanded polystyrene (EPS) foam are available for different commercial leading to a broad solid waste composition. PS has been thought to be non-biodegradable. The rate of biodegradation encountered in the environment is very slow leading to prolonged persistence as solid waste. In the past, PS was recycled through mechanical, chemical, and thermal technologies yielding gaseous and liquid daughter products [54]. A rather large collection of studies has shown that PS is subject to biodegradation but at a very slow rate in the environment. A sheet of PS buried for 32 years. in soil showed no indication of biotic or abiotic degradation [55]. The hydrophobicity of the polymer surface, a function of molecular structure and composition, detracts from the effectiveness of microbial attachment [56, 57]. The general lack of water solubility of PS prohibits the transport into microbial cells for metabolism.
#### *Biological Degradation of Polymers in the Environment DOI: http://dx.doi.org/10.5772/intechopen.85124*
A narrow range of microorganisms have been elicited for the environment and found to degrade PS [53]. *Bacillus* and *Pseudomonas* strains isolated from soil samples have been shown to degrade brominated high impact PS. The activity was seen in weight loss and surface changes to the PS film. Soil invertebrates such as the larvae of the mealworm (*Tenebrio molitor* Linnaeus) have been shown to chew and eat Styrofoam [57]. Samples of the larvae were fed Styrofoam as the sole diet for 30 days and compared with worms fed a conventional diet. The worms feeding Styrofoam survived for 1 month after which they stopped eating as they entered the pupae stage and emerged as adults after a subsequent 2 weeks. It appears that Styrofoam feeding did not lead to any lethality for the mealworms. The ingested PS mass was efficiently depolymerized within the larval gut during the retention time of 24 hours and converted to CO2 [51]. This remarkable behavior by the mealworm can be considered the action of an efficient bioreactor. The mealworm can provide all the necessary components for PS treatment starting with chewing, ingesting, mixing, reacting with gut contents, and microbial degradation by gut microbial consortia. A PS-degrading bacterial strain *Exiguobacterium* sp. strain YT2 was isolated from the gut of mealworms and found to degrade PS films outside the mealworm gut. Superworms (*Zophobas morio*) were found to exhibit similar activity toward Styrofoam. Brominated high impact polystyrene (blend of polystyrene and polybutadiene) has been found to be degraded by *Pseudomonas* and *Bacillus* strains [58]. In a complementary study, four non-pathogenic cultures (*Enterobacter* sp., *Citrobacter sedlakii*, *Alcaligenes* sp. and *Brevundimonas diminuta*) were isolated from partially degraded polymer samples from a rural market setting and each were found to degrade high impact polystyrene [59].
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**4.3 Polyvinyl chloride**
PVC is manufactured in two forms rigid and flexible. The rigid form can be found in the construction industry as pipe or in structural applications. The soft and flexible form can be made through the incorporation of plasticizers such as phthalates. Credit cards, bottles, and non-food packaging are notable products with a PVC composition. PVC has been known from its inception as a polymer with remarkable resistance to degradation [60]. Thermal and photodegradation processes are widely recognized for their role in the weathering processes found with PVC [61, 62]. The recalcitrant feature of polyvinyl chloride resistance to biodegradation becomes a matter of environmental concern across the all processes extending from manufacturing to waste disposal. Few reports are available relating the extent of PVC biodegradation. Early studies investigated the biodegradation of low-molecular weight PVC by white rot fungi [63]. Plasticized PVC was found to be degraded by fungi such as *As. fumigatus*, *Phanerochaete chrysosporium*, *Lentinus tigrinus*, *As. niger*, and *Aspergillus sydowii* [64].
Modifying the PVC film composition with adjuvants such as cellulose and starch provided a substrate that fungi could also degrade [65]. Several investigations of soil bacteria for the ability to degrade PVC from enrichment cultures were conducted on different locations [66]. Mixed cultures containing bacteria and fungi were isolated and found to grow on plasticized PVC [67]. Significant differences were observed for the colonization by the various components of the mixed isolates during very long exposure times [68]. Significant drift in isolate activity was averted through the use of talc. Consortia composed of a combination of different bacterial strains of *Pseudomonas otitidis*, *Bacillus cereus*, and *Acanthopleurobacter pedis* have the ability to degrade PVC in the environment [64]. These results offer the opportunity to optimization conditions for consortia growth in PVC and use as a treatment technology to degrade large collections of PVC. PVC film blends were shown to degrade by partnering biodegradable polymers with PVC [69].
#### **4.4 Polyurethane**
PUR encompass a broad field of polymer synthesis where a di- or polyisocyanate is chemically linked through carbamate (urethane) formation. These thermosetting and thermoplastic polymers have been utilized to form microcellular foams, high performance adhesives, synthetic fibers, surface coatings, and automobile parts along with a myriad of other applications. The carbamate linkage can be severed by chemical and biological processes [70].
Aromatic esters and the extent of the crystalline fraction of the polymer have been identified as important factors affecting the biodegradation of PUR [71, 72]. Acid and base hydrolysis strategies can sever the carbamate bond of the polymer. Microbial ureases, esterases and proteases can enable the hydrolysis the carbamate and ester bonds of a PUR polymer [71, 73, 74]. Bacteria have been found to be good sources for enzymes capable of degrading PUR polymers [75–82]. Fungi are also quite capable of degrading PUR polymers [83–85]. Each of the enzyme systems has their preferential targets: ureases attack the urea linkages [86–88] with esterases and proteases hydrolyzing the ester bonds of the polyester PUR as a major mechanism for its enzymatic depolymerization [89–92]. PUR polymers appear to be more amenable to enzymatic depolymerization or degradation but further searches and inquiry into hitherto unrecognized microbial PUR degrading activities is expected to offer significant PUR degrading activities.
#### **4.5 Polyethylene terephthalate**
PET is a polyester commonly marketed as a thermoplastic polymer resin finding use as synthetic fibers in clothing and carpeting, food and liquid containers, manufactured objects made through thermoforming, and engineering resins with glass fiber. Composed of terephthalic acid and ethylene glycol through the formation of ester bonds, PET has found a substantial role in packaging materials, beverage bottles and the textile industry. Characterized as a recalcitrant polymer of remarkable durability, the polymer's properties are reflective of its aromatic units in its backbone and a limited polymer chain mobility [91]. In many of its commercial forms, PET is semicrystalline having crystalline and amorphous phases which has a major effect on PET biodegradability. The environmental accumulation of PET is a testament of its versatility and the apparent lack of chemical/physical mechanisms capable of attacking its structural integrity show it to be a major environmental pollution problem.
The durability and the resulting low biodegradability of PET are due to the presence of repeating aromatic terephthalate units in its backbone and the corresponding limited mobility of the polymer chains [92]. The semicrystalline PET polymer also contains both amorphous and crystalline fractions with a strong effect on its biodegradability. Crystallinity exceeding 30% in PET beverage bottles and fibers having even higher crystalline compositions presents major hurdles to enzyme-induced degradation [93, 94]. At higher temperatures, the amorphous fraction of PET becomes more flexible and available to enzymatic degradation [95, 96]. The hydrolysis of PET by enzymes has been identified as a surface erosion process [97–100]. The hydrophobic surface significantly limits biodegradation due to the limited ability for microbial attachment. The hydrophobic nature of PET poses a significant barrier to microbial colonization of the polymer surface thus attenuating effective adsorption and access by hydrolytic enzymes to accomplish the polymer degradation [101].
A wide array of hydrolytic enzymes including hydrolases, lipases, esterases, and cutinases has been shown to have the ability to hydrolyze amorphous PET polymers
#### *Biological Degradation of Polymers in the Environment DOI: http://dx.doi.org/10.5772/intechopen.85124*
and modify PET film surfaces. Microbes from a vast collection of waste sites and dumping situations have been studied for their ability to degrade PET. A subunit of PET, diethylene glycol phthalate has been found to be a source of carbon and energy necessary to the sustenance of microbial life. Enzyme modification may be effectively employed to improve the efficiency and specificity of the polyester degrading enzymes acknowledged to be active degraders of PET [102]. Significant efforts have been extended to developing an understanding of the enzymatic activity of high-performing candidate enzymes through selection processes, mechanistic probes, and enzyme engineering. In addition to hydrolytic enzymes already identified, enzymes found in thermophilic anaerobic sludge were found to degrade PET copolymers formed into beverage bottles [103].
Recently, the discovery of microbial activity capable of complete degradation of widely used beverage bottle plastic expands the range of technology options available for PET treatment. A microorganism isolated from the area adjacent to a plastic bottle-recycling facility was shown to aerobically degrade PET to small molecular daughter products and eventually to CO2 and H2O. This new research shows that a newly isolated microbial species, *Ideonella sakaiensis* 201-F6, degrades PET through hydrolytic transformations by the action of two enzymes, which are extracellular and intracellular hydrolases. A primary hydrolysis reaction intermediate, mono (hydroxy-2-ethyl) terephthalate is formed and can be subsequently degraded to ethylene glycol and terephthalic acid which can be utilized by the microorganism for growth [104–109].
This discovery could be a candidate as a single vessel system that could competently accomplish PET hydrolysis as an enzyme reactor. This may be the beginning of viable technology development applicable to the solution of the global plastic problem recognized for its terrestrial component as well as the water contamination problem found in the sea. These remarkable discoveries offer a new perspective on
**Figure 5.** *Microbial depolymerization of poly(ethylene terephthalate).*
the recalcitrant nature of PET and how future environmental management of PET waste may be conducted using the power of enzymes. The recognition of current limiting steps in the biological depolymerization of PET are expected to enable the design of a enzymes-based process to reutilized the natural assets contained in scrap PET [110] (**Figure 5**).
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**5. Conclusions**
The major commercial polymers have been shown to be biodegradable in a variety of circumstances despite a strong predisposition suggesting that many of these polymers were recalcitrant to the effects of biodegradation. The question of whether bioremediation can play a significant role in the necessary management of polymer waste remains to be determined. Treatment technology for massive waste polymer treatment must be sufficiently robust to be reliable at large scale use and adaptable to conditions throughout the environment where this treatment is required. The status of information relating to the application of biodegradation treatment to existing and future polymer solid waste is at early stages of development for several waste polymers. The discovery of that invertebrate species (insect larvae) can reduce the size of the waste polymer by ingesting and degradation in the gut via enzymes which aid or complete degradation is rather amazing and requires additional scrutiny. There is an outside change that a polymer recycling technology based on these findings is a future possibility.
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**Disclaimer**
The views expressed in this book chapter are those of the author and do not necessarily represent the views or policies of the U.S. Environmental Protection Agency.
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**Conflict of interest**
No "conflict of interest" is known or expected.
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**Author details**
John A. Glaser U.S. Environmental Protection Agency, Office of Research and Development, National Risk Management Research Laboratory, Cincinnati, Ohio, USA
\*Address all correspondence to: [email protected]
© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
*Biological Degradation of Polymers in the Environment DOI: http://dx.doi.org/10.5772/intechopen.85124*
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*Edited by Alessio Gomiero*
Plastics in the Environment is a collection of reviewed and relevant research chapters, ofering a comprehensive overview of recent developments in the feld of plastic pollution and how it is afecting the environment. Te book comprises single chapters authored by various researchers and edited by an expert active in the research area. All chapters are complete in themselves but united under a common research study topic. Tis publication aims at providing a thorough overview of the latest research eforts by international authors on the trending topic of plastics in the environment and opens new possible research paths for further novel developments.
Published in London, UK © 2019 IntechOpen © redstallion / iStock
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*Edited by Hassan Abid Yasser*
Linear algebra occupies a central place in modern mathematics. Also, it is a beautiful and mature field of mathematics, and mathematicians have developed highly effective methods for solving its problems. It is a subject well worth studying for its own sake. This book contains selected topics in linear algebra, which represent the recent contributions in the most famous and widely problems. It includes a wide range of theorems and applications in different branches of linear algebra, such as linear systems, matrices, operators, inequalities, etc. It continues to be a definitive resource for researchers, scientists and graduate students.
ISBN 978-953-51-0669-2
Linear Algebra - Theorems and Applications
Photo by Selim Dönmez / iStock
## Linear Algebra Theorems and Applications
*Edited by Hassan Abid Yasser*
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**LINEAR ALGEBRA – THEOREMS AND APPLICATIONS**
Edited by **Hassan Abid Yasser**
http://dx.doi.org/10.5772/3107 Edited by Hassan Abid Yasser
#### **Contributors**
Pattrawut Chansangiam, Abdulhadi Aminu, Mohammad Poursina, Imad M Khan, Kurt Anderson, Pedro C. Lozano, Luis Gonzalez, Matsuo Sato, Taro Kimura, Ricardo Soto, Raymond De Callafon, Francesco A. Costabile, Elisabetta Longo, Jadranka Micic Hot, Josip Pecaric, Agurtzane Amparan, Silvia Marcaida, Ion Zaballa
#### **© The Editor(s) and the Author(s) 2012**
The moral rights of the and the author(s) have been asserted.
All rights to the book as a whole are reserved by INTECH. The book as a whole (compilation) cannot be reproduced, distributed or used for commercial or non-commercial purposes without INTECH's written permission. Enquiries concerning the use of the book should be directed to INTECH rights and permissions department ([email protected]).
Violations are liable to prosecution under the governing Copyright Law.
Individual chapters of this publication are distributed under the terms of the Creative Commons Attribution 3.0 Unported License which permits commercial use, distribution and reproduction of the individual chapters, provided the original author(s) and source publication are appropriately acknowledged. If so indicated, certain images may not be included under the Creative Commons license. In such cases users will need to obtain permission from the license holder to reproduce the material. More details and guidelines concerning content reuse and adaptation can be foundat http://www.intechopen.com/copyright-policy.html.
#### **Notice**
Statements and opinions expressed in the chapters are these of the individual contributors and not necessarily those of the editors or publisher. No responsibility is accepted for the accuracy of information contained in the published chapters. The publisher assumes no responsibility for any damage or injury to persons or property arising out of the use of any materials, instructions, methods or ideas contained in the book.
First published in Croatia, 2012 by INTECH d.o.o. eBook (PDF) Published by IN TECH d.o.o. Place and year of publication of eBook (PDF): Rijeka, 2019. IntechOpen is the global imprint of IN TECH d.o.o. Printed in Croatia
Legal deposit, Croatia: National and University Library in Zagreb
Additional hard and PDF copies can be obtained from [email protected]
Linear Algebra - Theorems and Applications Edited by Hassan Abid Yasser p. cm. ISBN 978-953-51-0669-2 eBook (PDF) ISBN 978-953-51-5004-6
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"author": "",
"title": "Linear Algebra",
"publisher": "IntechOpen",
"isbn": "9789535106692",
"section_idx": 1
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ffcfaa45-0a39-4ca8-825b-7ff2f8f01119.2
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doab
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2025-04-07T04:13:03.954185
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20-4-2021 17:35
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{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"book_id": "ffcfaa45-0a39-4ca8-825b-7ff2f8f01119",
"url": "https://mts.intechopen.com/storage/books/2738/authors_book/authors_book.pdf",
"author": "",
"title": "Linear Algebra",
"publisher": "IntechOpen",
"isbn": "9789535106692",
"section_idx": 2
}
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ffcfaa45-0a39-4ca8-825b-7ff2f8f01119.3
|
**Meet the editor**
Dr Hassan A. Yasser received his B.Sc. and M.Sc. from the college of science, Baghdad University, Baghdad, Iraq in 1989 and 1995, respectively. He received his Ph.D. from the college of science, Basrah University, Basrah, Iraq in 2005. His current research interests include digital image processing, mathematical physics, and optical communication. He is currently a professor of
mathematical physics at the College of Science, Thi-Qar University, Iraq.
## Contents
#### **Preface XI**
#### Chapter 11 **Efficient Model Transition in Adaptive Multi-Resolution Modeling of Biopolymers 237** Mohammad Poursina, Imad M. Khan and Kurt S. Anderson
## Preface
The core of linear algebra is essential to every mathematician, and we not only treat this core, but add material that is essential to mathematicians in specific fields. This book is for advanced researchers. We presume you are already familiar with elementary linear algebra and that you know how to multiply matrices, solve linear systems, etc. We do not treat elementary material here, though we occasionally return to elementary material from a more advanced standpoint to show you what it really means. We have written a book that we hope will be broadly useful. In a few places we have succumbed to temptation and included material that is not quite so well known, but which, in our opinion, should be. We hope that you will be enlightened not only by the specific material in the book but also by its style of argument. We also hope this book will serve as a valuable reference throughout your mathematical career.
Chapter 1 reviews the metric Hermitian 3-algebra, which has been playing important roles recently in sting theory. It is classified by using a correspondence to a class of the super Lie algebra. It also reviews the Lie and Hermitian 3-algebra models of M-theory. Chapter 2 deals with algebraic analysis of Appell polynomials. It presents the determinantal approaches of Appell polynomials and the related topics, where many classical and non-classical examples are presented. Chapter 3 reviews a universal relation between combinatorics and the matrix model, and discusses its relation to the gauge theory. Chapter 4 covers the nonnegative matrices that have been a source of interesting and challenging mathematical problems. They arise in many applications such as: communications systems, biological systems, economics, ecology, computer sciences, machine learning, and many other engineering systems. Chapter 5 presents the central theory behind realization-based system identication and connects the theory to many tools in linear algebra, including the QR-decomposition, the singular value decomposition, and linear least-squares problems. Chapter 6 presents a novel iterative-recursive algorithm for computing GI for block matrices in the context of wireless MIMO communication systems within RFC. Chapter 7 deals with the development of the theory of operator means. It setups basic notations and states some background about operator monotone functions which play important roles in the theory of operator means. Chapter 8 studies a general formulation of Jensen's operator inequality for a continuous eld of self-adjoint operators and a eld of positive linear mappings. The aim of chapter 9 is to present a system of linear equation and inequalities in max-algebra. Max-algebra is an analogue of linear algebra developed on a pair of operations extended to matrices and vectors. Chapter 10 covers an efficient algorithm for the coarse to fine scale transition in multi-flexible-body systems with application to biomolecular systems that are modeled as articulated bodies and undergo discontinuous changes in the model definition. Finally, chapter 11 studies the structure of matrices dened over arbitrary elds whose elements are rational functions with no poles at innity and prescribed nite poles. Complete systems of invariants are provided for each one of these equivalence relations and the relationship between both systems of invariants is claried. This result can be seen as an extension of the classical theorem on pole assignment by Rosenbrock.
**Dr. Hassan Abid Yasser**
College of Science University of Thi-Qar, Thi-Qar Iraq
**Chapter 0 Chapter 1**
## **3-Algebras in String Theory**
Matsuo Sato
Additional information is available at the end of the chapter
http://dx.doi.org/10.5772/46480
## **1. Introduction**
In this chapter, we review 3-algebras that appear as fundamental properties of string theory. 3-algebra is a generalization of Lie algebra; it is defined by a tri-linear bracket instead of by a bi-linear bracket, and satisfies fundamental identity, which is a generalization of Jacobi identity [1–3]. We consider 3-algebras equipped with invariant metrics in order to apply them to physics.
It has been expected that there exists M-theory, which unifies string theories. In M-theory, some structures of 3-algebras were found recently. First, it was found that by using *u*(*N*) ⊕ *u*(*N*) Hermitian 3-algebra, we can describe a low energy effective action of N coincident supermembranes [4–8], which are fundamental objects in M-theory.
With this as motivation, 3-algebras with invariant metrics were classified [9–22]. Lie 3-algebras are defined in real vector spaces and tri-linear brackets of them are totally anti-symmetric in all the three entries. Lie 3-algebras with invariant metrics are classified into A<sup>4</sup> algebra, and Lorentzian Lie 3-algebras, which have metrics with indefinite signatures. On the other hand, Hermitian 3-algebras are defined in Hermitian vector spaces and their tri-linear brackets are complex linear and anti-symmetric in the first two entries, whereas complex anti-linear in the third entry. Hermitian 3-algebras with invariant metrics are classified into *u*(*N*) ⊕ *u*(*M*) and *sp*(2*N*) ⊕ *u*(1) Hermitian 3-algebras.
Moreover, recent studies have indicated that there also exist structures of 3-algebras in the Green-Schwartz supermembrane action, which defines full perturbative dynamics of a supermembrane. It had not been clear whether the total supermembrane action including fermions has structures of 3-algebras, whereas the bosonic part of the action can be described by using a tri-linear bracket, called Nambu bracket [23, 24], which is a generalization of Poisson bracket. If we fix to a light-cone gauge, the total action can be described by using Poisson bracket, that is, only structures of Lie algebra are left in this gauge [25]. However, it was shown under an approximation that the total action can be described by Nambu bracket if we fix to a semi-light-cone gauge [26]. In this gauge, the eleven dimensional space-time of M-theory is manifest in the supermembrane action, whereas only ten dimensional part is manifest in the light-cone gauge.
©2012 Sato, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ©2012 Sato, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The BFSS matrix theory is conjectured to describe an infinite momentum frame (IMF) limit of M-theory [27] and many evidences were found. The action of the BFSS matrix theory can be obtained by replacing Poisson bracket with a finite dimensional Lie algebra's bracket in the supermembrane action in the light-cone gauge. Because of this structure, only variables that represent the ten dimensional part of the eleven-dimensional space-time are manifest in the BFSS matrix theory. Recently, 3-algebra models of M-theory were proposed [26, 28, 29], by replacing Nambu bracket with finite dimensional 3-algebras' brackets in an action that is shown, by using an approximation, to be equivalent to the semi-light-cone supermembrane action. All the variables that represent the eleven dimensional space-time are manifest in these models. It was shown that if the DLCQ limit of the 3-algebra models of M-theory is taken, they reduce to the BFSS matrix theory [26, 28], as they should [30–35].
### **2. Definition and classification of metric Hermitian 3-algebra**
In this section, we will define and classify the Hermitian 3-algebras equipped with invariant metrics.
#### **2.1. General structure of metric Hermitian 3-algebra**
The metric Hermitian 3-algebra is a map *V* × *V* × *V* → *V* defined by (*x*, *y*, *z*) �→ [*x*, *y*; *z*], where the 3-bracket is complex linear in the first two entries, whereas complex anti-linear in the last entry, equipped with a metric < *x*, *y* >, satisfying the following properties: the fundamental identity
$$[[\mathbf{x}, \mathbf{y}; \mathbf{z}], \mathbf{v}; \mathbf{w}] = [[\mathbf{x}, \mathbf{v}; \mathbf{w}], \mathbf{y}; \mathbf{z}] + [\mathbf{x}, [\mathbf{y}, \mathbf{v}; \mathbf{w}]; \mathbf{z}] - [\mathbf{x}, \mathbf{y}; [\mathbf{z}, \mathbf{w}; \mathbf{v}]] \tag{1}$$
the metric invariance
$$<\langle x, v; w \rangle\_{\prime} y> - <\ge, [y, w; v]> = 0 \tag{2}$$
and the anti-symmetry
$$[\mathbf{x}, \mathbf{y}; \mathbf{z}] = -[\mathbf{y}, \mathbf{x}; \mathbf{z}] \tag{3}$$
for
$$
\langle x, y, z, v, w \in V \rangle \tag{4}
$$
The Hermitian 3-algebra generates a symmetry, whose generators *D*(*x*, *y*) are defined by
$$D(\mathbf{x}, \mathbf{y})z := [z, \mathbf{x}; \mathbf{y}] \tag{5}$$
From (1), one can show that *D*(*x*, *y*) form a Lie algebra,
$$D[D(\mathbf{x}, y), D(\mathbf{v}, w)] = D(D(\mathbf{x}, y)\mathbf{v}, \mathbf{w}) - D(\mathbf{v}, D(y, \mathbf{x})w) \tag{6}$$
There is an one-to-one correspondence between the metric Hermitian 3-algebra and a class of metric complex super Lie algebras [19]. Such a class satisfies the following conditions among complex super Lie algebras *S* = *S*<sup>0</sup> ⊕ *S*1, where *S*<sup>0</sup> and *S*<sup>1</sup> are even and odd parts, respectively. *<sup>S</sup>*<sup>1</sup> is decomposed as *<sup>S</sup>*<sup>1</sup> <sup>=</sup> *<sup>V</sup>* <sup>⊕</sup> *<sup>V</sup>*¯ , where *<sup>V</sup>* is an unitary representation of *<sup>S</sup>*0: for *<sup>a</sup>* <sup>∈</sup> *<sup>S</sup>*0, *u*, *v* ∈ *V*,
$$[a, u] \in V \tag{7}$$
and
2 Will-be-set-by-IN-TECH
The BFSS matrix theory is conjectured to describe an infinite momentum frame (IMF) limit of M-theory [27] and many evidences were found. The action of the BFSS matrix theory can be obtained by replacing Poisson bracket with a finite dimensional Lie algebra's bracket in the supermembrane action in the light-cone gauge. Because of this structure, only variables that represent the ten dimensional part of the eleven-dimensional space-time are manifest in the BFSS matrix theory. Recently, 3-algebra models of M-theory were proposed [26, 28, 29], by replacing Nambu bracket with finite dimensional 3-algebras' brackets in an action that is shown, by using an approximation, to be equivalent to the semi-light-cone supermembrane action. All the variables that represent the eleven dimensional space-time are manifest in these models. It was shown that if the DLCQ limit of the 3-algebra models of M-theory is taken, they
In this section, we will define and classify the Hermitian 3-algebras equipped with invariant
The metric Hermitian 3-algebra is a map *V* × *V* × *V* → *V* defined by (*x*, *y*, *z*) �→ [*x*, *y*; *z*], where the 3-bracket is complex linear in the first two entries, whereas complex anti-linear in the last
The Hermitian 3-algebra generates a symmetry, whose generators *D*(*x*, *y*) are defined by
There is an one-to-one correspondence between the metric Hermitian 3-algebra and a class of metric complex super Lie algebras [19]. Such a class satisfies the following conditions among complex super Lie algebras *S* = *S*<sup>0</sup> ⊕ *S*1, where *S*<sup>0</sup> and *S*<sup>1</sup> are even and odd parts, respectively. *<sup>S</sup>*<sup>1</sup> is decomposed as *<sup>S</sup>*<sup>1</sup> <sup>=</sup> *<sup>V</sup>* <sup>⊕</sup> *<sup>V</sup>*¯ , where *<sup>V</sup>* is an unitary representation of *<sup>S</sup>*0: for *<sup>a</sup>* <sup>∈</sup> *<sup>S</sup>*0,
[[*x*, *y*; *z*], *v*; *w*] = [[*x*, *v*; *w*], *y*; *z*]+[*x*, [*y*, *v*; *w*]; *z*] − [*x*, *y*; [*z*, *w*; *v*]] (1)
[*D*(*x*, *y*), *D*(*v*, *w*)] = *D*(*D*(*x*, *y*)*v*, *w*) − *D*(*v*, *D*(*y*, *x*)*w*) (6)
< [*x*, *v*; *w*], *y* > − < *x*, [*y*, *w*; *v*] >= 0 (2)
[*x*, *y*; *z*] = −[*y*, *x*; *z*] (3)
*x*, *y*, *z*, *v*, *w* ∈ *V* (4)
*D*(*x*, *y*)*z* := [*z*, *x*; *y*] (5)
[*a*, *u*] ∈ *V* (7)
reduce to the BFSS matrix theory [26, 28], as they should [30–35].
**2.1. General structure of metric Hermitian 3-algebra**
From (1), one can show that *D*(*x*, *y*) form a Lie algebra,
metrics.
for
*u*, *v* ∈ *V*,
the fundamental identity
the metric invariance
and the anti-symmetry
**2. Definition and classification of metric Hermitian 3-algebra**
entry, equipped with a metric < *x*, *y* >, satisfying the following properties:
$$<\langle a, u \rangle, v> + \langle a^\*, v \rangle> = 0 \tag{8}$$
*<sup>v</sup>*¯ <sup>∈</sup> *<sup>V</sup>*¯ is defined by
$$<\mathfrak{v}>\tag{9}$$
The super Lie bracket satisfies
$$[V,V] = 0, \quad [\bar{V}, \bar{V}] = 0 \tag{10}$$
From the metric Hermitian 3-algebra, we obtain the class of the metric complex super Lie algebra in the following way. The elements in *S*0, *V*, and *V*¯ are defined by (5), (4), and (9), respectively. The algebra is defined by (6) and
$$\begin{aligned} [D(x, y), z] &:= D(x, y)z = [z, x; y] \\ [D(x, y), \bar{z}] &:= -D(\bar{y'}, x)z = -[z, \bar{y}; x] \\ [x, \bar{y}] &:= D(x, y) \\ [x, y] &:= 0 \\ [\bar{x}, \bar{y}] &:= 0 \end{aligned} \tag{11}$$
One can show that this algebra satisfies the super Jacobi identity and (7)-(10) as in [19].
Inversely, from the class of the metric complex super Lie algebra, we obtain the metric Hermitian 3-algebra by
$$[\mathbf{x}, \mathbf{y}; \mathbf{z}] := \mathbf{a}[[\mathbf{y}, \mathbf{\bar{z}}], \mathbf{x}] \tag{12}$$
where *α* is an arbitrary constant. One can also show that this algebra satisfies (1)-(3) for (4) as in [19].
#### **2.2. Classification of metric Hermitian 3-algebra**
The classical Lie super algebras satisfying (7)-(10) are *A*(*m* − 1, *n* − 1) and *C*(*n* + 1). The even parts of *A*(*m* −1, *n* −1) and *C*(*n* +1) are *u*(*m*) ⊕ *u*(*n*) and *sp*(2*n*) ⊕ *u*(1), respectively. Because the metric Hermitian 3-algebra one-to-one corresponds to this class of the super Lie algebra, the metric Hermitian 3-algebras are classified into *u*(*m*) ⊕ *u*(*n*) and *sp*(2*n*) ⊕ *u*(1) Hermitian 3-algebras.
First, we will construct the *u*(*m*) ⊕ *u*(*n*) Hermitian 3-algebra from *A*(*m* − 1, *n* − 1), according to the relation in the previous subsection. *A*(*m* −1, *n* −1) is simple and is obtained by dividing *sl*(*m*, *n*) by its ideal. That is, *A*(*m* − 1, *n* − 1) = *sl*(*m*, *n*) when *m* �= *n* and *A*(*n* − 1, *n* − 1) = *sl*(*n*, *n*)/*λ*12*n*.
Real *sl*(*m*, *n*) is defined by
$$
\begin{pmatrix} h\_1 & c \\ ic^\dagger \ h\_2 \end{pmatrix} \tag{13}
$$
where *h*<sup>1</sup> and *h*<sup>2</sup> are *m* × *m* and *n* × *n* anti-Hermite matrices and *c* is an *n* × *m* arbitrary complex matrix. Complex *sl*(*m*, *n*) is a complexification of real *sl*(*m*, *n*), given by
$$
\begin{pmatrix} \alpha \ \beta \\ \gamma \ \delta \end{pmatrix} \tag{14}
$$
#### 4 Will-be-set-by-IN-TECH 4 Linear Algebra – Theorems and Applications
where *α*, *β*, *γ*, and *δ* are *m* × *m*, *n* × *m*, *m* × *n*, and *n* × *n* complex matrices that satisfy
$$\text{tr}\mathfrak{a} = \text{tr}\delta\tag{15}$$
Complex *<sup>A</sup>*(*<sup>m</sup>* <sup>−</sup> 1, *<sup>n</sup>* <sup>−</sup> <sup>1</sup>) is decomposed as *<sup>A</sup>*(*<sup>m</sup>* <sup>−</sup> 1, *<sup>n</sup>* <sup>−</sup> <sup>1</sup>) = *<sup>S</sup>*<sup>0</sup> <sup>⊕</sup> *<sup>V</sup>* <sup>⊕</sup> *<sup>V</sup>*¯ , where
$$\begin{aligned} \begin{pmatrix} \alpha & 0\\ 0 & \delta \end{pmatrix} & \in \mathcal{S}\_0\\ \begin{pmatrix} 0 & \beta\\ 0 & 0 \end{pmatrix} & \in \mathcal{V} \\ \begin{pmatrix} 0 & 0\\ \gamma & 0 \end{pmatrix} & \in \bar{\mathcal{V}} \end{aligned} \tag{16}$$
(9) is rewritten as *<sup>V</sup>* <sup>→</sup> *<sup>V</sup>*¯ defined by
$$B = \begin{pmatrix} 0 \ \beta \\ 0 \ 0 \end{pmatrix} \mapsto B^\dagger = \begin{pmatrix} 0 & 0 \\ \beta^\dagger \ 0 \end{pmatrix} \tag{17}$$
where *<sup>B</sup>* <sup>∈</sup> *<sup>V</sup>* and *<sup>B</sup>*† <sup>∈</sup> *<sup>V</sup>*¯ . (12) is rewritten as
$$\mathbb{E}\left[X,Y;Z\right] = \mathfrak{a}\left[\left[Y,Z^{\dagger}\right],X\right] = \mathfrak{a}\begin{pmatrix} 0 \ yz^{\dagger}x - xz^{\dagger}y\\ 0 & 0 \end{pmatrix} \tag{18}$$
for
$$\begin{aligned} X &= \begin{pmatrix} 0 \ x \\ 0 \ 0 \end{pmatrix} \in V \\ Y &= \begin{pmatrix} 0 \ y \\ 0 \ 0 \end{pmatrix} \in V \\ Z &= \begin{pmatrix} 0 \ z \\ 0 \ 0 \end{pmatrix} \in V \end{aligned} \tag{19}$$
As a result, we obtain the *u*(*m*) ⊕ *u*(*n*) Hermitian 3-algebra,
$$
\pi[\mathbf{x}, y; z] = \mathfrak{a}(yz^\dagger \mathbf{x} - \mathbf{x}z^\dagger y) \tag{20}
$$
where *x*, *y*, and *z* are arbitrary *n* × *m* complex matrices. This algebra was originally constructed in [8].
Inversely, from (20), we can construct complex *A*(*m* − 1, *n* − 1). (5) is rewritten as
$$D(\mathbf{x}, y) = (\mathbf{x}y^\dagger, y^\dagger \mathbf{x}) \in \mathbb{S}\_0 \tag{21}$$
(6) and (11) are rewritten as
$$\begin{aligned} [(xy^\dagger, y^\dagger x), (x'y^\dagger, y^\dagger x')] &= ([xy^\dagger, x'y^\dagger], [y^\dagger x, y'^\dagger x']) \\ [(xy^\dagger, y^\dagger x), z] &= xy^\dagger z - zy^\dagger x \\ [(xy^\dagger, y^\dagger x), w^\dagger] &= y^\dagger xw^\dagger - w^\dagger xy^\dagger \\ [x, y^\dagger] &= (xy^\dagger, y^\dagger x) \\ [x, y] &= 0 \\ [x^\dagger, y^\dagger] &= 0 \end{aligned} \tag{22}$$
This algebra is summarized as
4 Will-be-set-by-IN-TECH
tr*α* = tr*δ* (15)
<sup>∈</sup> *<sup>V</sup>*¯ (16)
(17)
(18)
(19)
where *α*, *β*, *γ*, and *δ* are *m* × *m*, *n* × *m*, *m* × *n*, and *n* × *n* complex matrices that satisfy
0 *β* 0 0 ∈ *V*
0 0 *γ* 0
�→ *<sup>B</sup>*† <sup>=</sup>
0 0 *β*† 0
<sup>0</sup> *yz*†*<sup>x</sup>* <sup>−</sup> *xz*†*<sup>y</sup>* 0 0
[*x*, *<sup>y</sup>*; *<sup>z</sup>*] = *<sup>α</sup>*(*yz*†*<sup>x</sup>* <sup>−</sup> *xz*†*y*) (20)
*<sup>D</sup>*(*x*, *<sup>y</sup>*)=(*xy*†, *<sup>y</sup>*†*x*) <sup>∈</sup> *<sup>S</sup>*<sup>0</sup> (21)
*y*�†], [*y*†*x*, *y*�†*x*�
])
*B* =
0 *β* 0 0
[*X*,*Y*; *Z*] = *α*[[*Y*, *Z*†], *X*] = *α*
*X* =
*Y* =
*Z* =
Inversely, from (20), we can construct complex *A*(*m* − 1, *n* − 1). (5) is rewritten as
*y*�†, *y*�†*x*�
[(*xy*†, *<sup>y</sup>*†*x*), *<sup>z</sup>*] = *xy*†*<sup>z</sup>* <sup>−</sup> *zy*†*<sup>x</sup>* [(*xy*†, *<sup>y</sup>*†*x*), *<sup>w</sup>*†] = *<sup>y</sup>*†*xw*† <sup>−</sup> *<sup>w</sup>*†*xy*†
As a result, we obtain the *u*(*m*) ⊕ *u*(*n*) Hermitian 3-algebra,
[(*xy*†, *y*†*x*),(*x*�
[*x*, *y*†]=(*xy*†, *y*†*x*)
[*x*, *y*] = 0
0 *x* 0 0 ∈ *V*
0 *y* 0 0 ∈ *V*
0 *z* 0 0 ∈ *V*
where *x*, *y*, and *z* are arbitrary *n* × *m* complex matrices. This algebra was originally
)] = ([*xy*†, *x*�
[*x*†, *y*†] = 0 (22)
(9) is rewritten as *<sup>V</sup>* <sup>→</sup> *<sup>V</sup>*¯ defined by
for
constructed in [8].
(6) and (11) are rewritten as
where *<sup>B</sup>* <sup>∈</sup> *<sup>V</sup>* and *<sup>B</sup>*† <sup>∈</sup> *<sup>V</sup>*¯ . (12) is rewritten as
Complex *<sup>A</sup>*(*<sup>m</sup>* <sup>−</sup> 1, *<sup>n</sup>* <sup>−</sup> <sup>1</sup>) is decomposed as *<sup>A</sup>*(*<sup>m</sup>* <sup>−</sup> 1, *<sup>n</sup>* <sup>−</sup> <sup>1</sup>) = *<sup>S</sup>*<sup>0</sup> <sup>⊕</sup> *<sup>V</sup>* <sup>⊕</sup> *<sup>V</sup>*¯ , where *α* 0 0 *δ* ∈ *S*<sup>0</sup>
$$
\begin{bmatrix}
\begin{pmatrix}
\mathbf{x}y^{\dagger} & z \\
\mathbf{w}^{\dagger} & y^{\dagger}\mathbf{x}
\end{pmatrix}
\end{bmatrix}
\begin{pmatrix}
\mathbf{x}'y^{\dagger} & z' \\
\mathbf{w}'^{\dagger} & y'^{\dagger}\mathbf{x}'
\end{pmatrix}
\begin{bmatrix}
\end{bmatrix}
\tag{23}
$$
which forms complex *A*(*m* − 1, *n* − 1).
Next, we will construct the *sp*(2*n*) ⊕ *u*(1) Hermitian 3-algebra from *C*(*n* + 1). Complex *C*(*n* + <sup>1</sup>) is decomposed as *<sup>C</sup>*(*<sup>n</sup>* <sup>+</sup> <sup>1</sup>) = *<sup>S</sup>*<sup>0</sup> <sup>⊕</sup> *<sup>V</sup>* <sup>⊕</sup> *<sup>V</sup>*¯ . The elements are given by
$$
\begin{pmatrix} a & 0 & 0 & 0 \\ 0 & -a & 0 & 0 \\ 0 & 0 & a & b \\ 0 & 0 & c & -a^T \end{pmatrix} \in \mathbb{S}\_0
$$
$$
\begin{pmatrix} 0 & 0 & x\_1 & x\_2 \\ 0 & 0 & 0 & 0 \\ 0 & x\_2^T & 0 & 0 \\ 0 & -x\_1^T & 0 & 0 \end{pmatrix} \in V
$$
$$
\begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & y\_1 & y\_2 \\ y\_2^T & 0 & 0 & 0 \\ -y\_1^T & 0 & 0 & 0 \end{pmatrix} \in V \tag{24}
$$
where *α* is a complex number, *a* is an arbitrary *n* × *n* complex matrix, *b* and *c* are *n* × *n* complex symmetric matrices, and *<sup>x</sup>*1, *<sup>x</sup>*2, *<sup>y</sup>*<sup>1</sup> and *<sup>y</sup>*<sup>2</sup> are *<sup>n</sup>* <sup>×</sup>1 complex matrices. (9) is rewritten as *<sup>V</sup>* <sup>→</sup> *<sup>V</sup>*¯ defined by *<sup>B</sup>* �→ *<sup>B</sup>*¯ <sup>=</sup> *UB*∗*U*−1, where *<sup>B</sup>* <sup>∈</sup> *<sup>V</sup>*, *<sup>B</sup>*¯ <sup>∈</sup> *<sup>V</sup>*¯ and
$$M = \begin{pmatrix} 0 \ 1 & 0 & 0 \\ 1 \ 0 & 0 & 0 \\ 0 \ 0 & 0 & \mathbf{1} \\ 0 \ 0 & -\mathbf{1} & 0 \end{pmatrix} \tag{25}$$
Explicitly,
$$B = \begin{pmatrix} 0 & 0 & \mathbf{x}\_1 \ \mathbf{x}\_2 \\ 0 & 0 & 0 & 0 \\ 0 & \mathbf{x}\_2^T & 0 & 0 \\ 0 & -\mathbf{x}\_1^T & 0 & 0 \end{pmatrix} \mapsto \bar{B} = \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 \ \mathbf{x}\_2^\* & -\mathbf{x}\_1^\* \\ -\mathbf{x}\_1^\dagger & 0 & 0 & 0 \\ -\mathbf{x}\_2^\dagger & 0 & 0 & 0 \end{pmatrix} \tag{26}$$
(12) is rewritten as
$$\begin{aligned} [X, Y; Z] &:= a[[Y, \bar{Z}], X] \\ &= a \left[ \left[ \begin{pmatrix} 0 & 0 & y\_1 \ y\_2 \\ 0 & 0 & 0 \\ 0 & y\_2^T & 0 & 0 \\ 0 & -y\_1^T & 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & z\_2^\* - z\_1^\* \\ -z\_1^\dagger & 0 & 0 & 0 \\ -z\_2^\dagger & 0 & 0 & 0 \end{pmatrix} \right], \begin{pmatrix} 0 & 0 & x\_1 \ x\_2 \\ 0 & 0 & 0 & 0 \\ 0 & x\_2^T & 0 & 0 \\ 0 & -x\_1^T & 0 & 0 \end{pmatrix} \right] \\ &= a \begin{pmatrix} 0 & 0 & w\_1 \ w\_2 \\ 0 & 0 & 0 & 0 \\ 0 & w\_2^T & 0 & 0 \\ 0 & -w\_1^T & 0 & 0 \end{pmatrix} \end{aligned} \tag{27}$$
#### 6 Will-be-set-by-IN-TECH 6 Linear Algebra – Theorems and Applications
for
$$\begin{aligned} X &= \begin{pmatrix} 0 & 0 & x\_1 \ x\_2 \\ 0 & 0 & 0 & 0 \\ 0 & x\_2^T & 0 & 0 \\ 0 & -x\_1^T & 0 & 0 \end{pmatrix} \in V \\ Y &= \begin{pmatrix} 0 & 0 & y\_1 \ y\_2 \\ 0 & 0 & 0 & 0 \\ 0 & y\_2^T & 0 & 0 \\ 0 & -y\_1^T & 0 & 0 \end{pmatrix} \in V \\ Z &= \begin{pmatrix} 0 & 0 & z\_1 \ z\_2 \\ 0 & 0 & 0 & 0 \\ 0 & z\_2^T & 0 & 0 \\ 0 & -z\_1^T & 0 & 0 \end{pmatrix} \in V \end{aligned} \tag{28}$$
where *w*<sup>1</sup> and *w*<sup>2</sup> are given by
$$(w\_1, w\_2) = -(y\_1 z\_1^\dagger + y\_2 z\_2^\dagger)(x\_1, x\_2) + (x\_1 z\_1^\dagger + x\_2 z\_2^\dagger)(y\_1, y\_2) + (x\_2 y\_1^T - x\_1 y\_2^T)(z\_{2\prime}^\* - z\_1^\*)\tag{29}$$
As a result, we obtain the *sp*(2*n*) ⊕ *u*(1) Hermitian 3-algebra,
$$\mathbf{a}\left[\mathbf{x},\mathbf{y};\mathbf{z}\right] = \mathbf{a}\left((\mathbf{y}\odot\mathbf{\bar{z}})\mathbf{x} + (\mathbf{z}\odot\mathbf{x})\mathbf{y} - (\mathbf{x}\odot\mathbf{y})\mathbf{\bar{z}}\right) \tag{30}$$
for *x* = (*x*1, *x*2), *y* = (*y*1, *y*2), *z* = (*z*1, *z*2), where *x*1, *x*2, *y*1, *y*2, *z*1, and *z*<sup>2</sup> are n-vectors and
$$\begin{aligned} \tilde{z} &= (z\_{2'}^\* - z\_1^\*) \\ a \odot b &= a\_1 \cdot b\_2 - a\_2 \cdot b\_1 \end{aligned} \tag{31}$$
#### **3. 3-algebra model of M-theory**
In this section, we review the fact that the supermembrane action in a semi-light-cone gauge can be described by Nambu bracket, where structures of 3-algebra are manifest. The 3-algebra Models of M-theory are defined based on the semi-light-cone supermembrane action. We also review that the models reduce to the BFSS matrix theory in the DLCQ limit.
#### **3.1. Supermembrane and 3-algebra model of M-theory**
The fundamental degrees of freedom in M-theory are supermembranes. The action of the covariant supermembrane action in M-theory [36] is given by
$$\begin{split} S\_{M2} = \int d^3 \sigma \left( \sqrt{-\mathcal{G}} + \frac{i}{4} \epsilon^{\mu \beta \gamma} \bar{\Psi} \Gamma\_{MN} \partial\_{\bar{n}} \Psi (\Pi\_{\beta}{}^{M} \Pi\_{\gamma}{}^{N} + \frac{i}{2} \Pi\_{\beta}{}^{M} \bar{\Psi} \Gamma^{N} \partial\_{\gamma} \Psi \\ - \frac{1}{12} \bar{\Psi} \Gamma^{M} \partial\_{\beta} \Psi \bar{\Psi} \Gamma^{N} \partial\_{\gamma} \Psi \rangle \right) \tag{32}$$
where *<sup>M</sup>*, *<sup>N</sup>* <sup>=</sup> 0, ··· , 10, *<sup>α</sup>*, *<sup>β</sup>*, *<sup>γ</sup>* <sup>=</sup> 0, 1, 2, *<sup>G</sup>αβ* <sup>=</sup> <sup>Π</sup> *<sup>M</sup> <sup>α</sup>* Π*β<sup>M</sup>* and Π *<sup>M</sup> <sup>α</sup>* <sup>=</sup> *∂αX<sup>M</sup>* <sup>−</sup> *<sup>i</sup>* <sup>2</sup>ΨΓ¯ *<sup>M</sup>∂α*Ψ. <sup>Ψ</sup> is a *SO*(1, 10) Majorana fermion.
This action is invariant under dynamical supertransformations,
$$
\delta \Psi = \epsilon
$$
$$
\delta X^M = -i \bar{\Psi} \Gamma^M \epsilon
$$
These transformations form the N = 1 supersymmetry algebra in eleven dimensions,
$$
\begin{bmatrix}
\delta\_1 \,\delta\_2\right] \mathbf{X}^M = -2i\epsilon\_1 \Gamma^M \epsilon\_2
$$
$$
[\delta\_1, \delta\_2] \mathbf{Y} = \mathbf{0} \tag{34}
$$
The action is also invariant under the *κ*-symmetry transformations,
$$
\begin{aligned}
\delta \Psi &= (1 + \Gamma) \kappa(\sigma) \\
\delta X^M &= i \bar{\Psi} \Gamma^M (1 + \Gamma) \kappa(\sigma)
\end{aligned}
\tag{35}
$$
where
6 Will-be-set-by-IN-TECH
0 0 *x*<sup>1</sup> *x*<sup>2</sup> 0 0 00 0 *x<sup>T</sup>*
0 0 *y*<sup>1</sup> *y*<sup>2</sup> 0 0 00 0 *y<sup>T</sup>*
0 0 *z*<sup>1</sup> *z*<sup>2</sup> 0 0 00 0 *z<sup>T</sup>*
<sup>0</sup> <sup>−</sup>*x<sup>T</sup>*
<sup>0</sup> <sup>−</sup>*y<sup>T</sup>*
<sup>0</sup> <sup>−</sup>*z<sup>T</sup>*
for *x* = (*x*1, *x*2), *y* = (*y*1, *y*2), *z* = (*z*1, *z*2), where *x*1, *x*2, *y*1, *y*2, *z*1, and *z*<sup>2</sup> are n-vectors and
*z*˜ = (*z*∗
review that the models reduce to the BFSS matrix theory in the DLCQ limit.
*i* 4
**3.1. Supermembrane and 3-algebra model of M-theory**
covariant supermembrane action in M-theory [36] is given by
�√−*<sup>G</sup>* <sup>+</sup>
where *<sup>M</sup>*, *<sup>N</sup>* <sup>=</sup> 0, ··· , 10, *<sup>α</sup>*, *<sup>β</sup>*, *<sup>γ</sup>* <sup>=</sup> 0, 1, 2, *<sup>G</sup>αβ* <sup>=</sup> <sup>Π</sup> *<sup>M</sup>*
<sup>2</sup> 0 0
⎞
⎟⎟⎠ ∈ *V*
⎞
⎟⎟⎠ ∈ *V*
⎞
⎟⎟⎠
<sup>2</sup>)(*y*1, *<sup>y</sup>*2)+(*x*2*y<sup>T</sup>*
[*x*, *y*; *z*] = *α*((*y* � *z*˜)*x* + (*z*˜ � *x*)*y* − (*x* � *y*)*z*˜) (30)
*<sup>β</sup>* <sup>Π</sup> *<sup>N</sup> <sup>γ</sup>* + *i* 2 Π *<sup>M</sup>*
*<sup>α</sup>* Π*β<sup>M</sup>* and Π *<sup>M</sup>*
<sup>12</sup>ΨΓ¯ *<sup>M</sup>∂β*ΨΨΓ¯ *<sup>N</sup>∂γ*Ψ)
− 1
*a* � *b* = *a*<sup>1</sup> · *b*<sup>2</sup> − *a*<sup>2</sup> · *b*<sup>1</sup> (31)
∈ *V* (28)
<sup>1</sup> <sup>−</sup> *<sup>x</sup>*1*y<sup>T</sup>*
<sup>2</sup> )(*z*<sup>∗</sup> 2, −*z*<sup>∗</sup>
*<sup>β</sup>* ΨΓ¯ *<sup>N</sup>∂γ*<sup>Ψ</sup>
*<sup>α</sup>* <sup>=</sup> *∂αX<sup>M</sup>* <sup>−</sup> *<sup>i</sup>*
�
(32)
<sup>2</sup>ΨΓ¯ *<sup>M</sup>∂α*Ψ. <sup>Ψ</sup>
<sup>1</sup> ) (29)
<sup>1</sup> 0 0
<sup>2</sup> 0 0
<sup>1</sup> 0 0
<sup>2</sup> 0 0
<sup>1</sup> 0 0
<sup>1</sup> <sup>+</sup> *<sup>x</sup>*2*z*†
<sup>2</sup>, −*z*<sup>∗</sup> 1 )
In this section, we review the fact that the supermembrane action in a semi-light-cone gauge can be described by Nambu bracket, where structures of 3-algebra are manifest. The 3-algebra Models of M-theory are defined based on the semi-light-cone supermembrane action. We also
The fundamental degrees of freedom in M-theory are supermembranes. The action of the
*�αβγ*ΨΓ¯ *MN∂α*Ψ(Π *<sup>M</sup>*
*X* =
*Y* =
*Z* =
<sup>2</sup>)(*x*1, *<sup>x</sup>*2)+(*x*1*z*†
As a result, we obtain the *sp*(2*n*) ⊕ *u*(1) Hermitian 3-algebra,
where *w*<sup>1</sup> and *w*<sup>2</sup> are given by
<sup>1</sup> <sup>+</sup> *<sup>y</sup>*2*z*†
**3. 3-algebra model of M-theory**
*SM*<sup>2</sup> =
is a *SO*(1, 10) Majorana fermion.
� *d*3*σ*
(*w*1, *<sup>w</sup>*2) = <sup>−</sup>(*y*1*z*†
⎛
⎜⎜⎝
⎛
⎜⎜⎝
⎛
⎜⎜⎝
for
$$\Gamma = \frac{1}{3!\sqrt{-G}} \epsilon^{a\beta\gamma} \Pi\_a^L \Pi\_\beta^M \Pi\_\gamma^N \Gamma\_{LMN} \tag{36}$$
If we fix the *κ*-symmetry (35) of the action by taking a semi-light-cone gauge [26]<sup>1</sup>
$$
\Gamma^{012}\Psi = -\Psi\tag{37}
$$
we obtain a semi-light-cone supermembrane action,
$$S\_{M\Sigma} = \int d^3 \sigma \left( \sqrt{-G} + \frac{i}{4} \epsilon^{\mu \beta \gamma} \left( \bar{\Psi} \Gamma\_{\mu \nu} \partial\_{\mathbf{a}} \Psi (\Pi\_{\beta}^{\ \mu} \Pi\_{\gamma}{}^{\nu} + \frac{i}{2} \Pi\_{\beta}{}^{\mu} \bar{\Psi} \Gamma^{\nu} \partial\_{\gamma} \Psi - \frac{1}{12} \bar{\Psi} \Gamma^{\mu} \partial\_{\beta} \Psi \bar{\Psi} \Gamma^{\nu} \partial\_{\gamma} \Psi \right)$$
$$+ \Psi \Gamma\_{I\bar{I}} \partial\_{\mathbf{a}} \Psi \partial\_{\beta} X^{I} \partial\_{\gamma} X^{J}) \right) \tag{38}$$
where *<sup>G</sup>αβ* <sup>=</sup> *<sup>h</sup>αβ* <sup>+</sup> <sup>Π</sup> *<sup>μ</sup> <sup>α</sup>* <sup>Π</sup>*βμ*, <sup>Π</sup> *<sup>μ</sup> <sup>α</sup>* <sup>=</sup> *∂αX<sup>μ</sup>* <sup>−</sup> *<sup>i</sup>* <sup>2</sup>ΨΓ¯ *μ∂α*Ψ, and *<sup>h</sup>αβ* <sup>=</sup> *∂αX<sup>I</sup>∂βXI*.
In [26], it is shown under an approximation up to the quadratic order in *∂αX<sup>μ</sup>* and *∂α*Ψ but exactly in *X<sup>I</sup>* , that this action is equivalent to the continuum action of the 3-algebra model of M-theory,
$$\begin{split} S\_{cl} &= \int d^3 \sigma \sqrt{-g} \Big( -\frac{1}{12} \{X^I, X^I, X^K\}^2 - \frac{1}{2} (A\_{\mu ab} \{\varphi^a, \varphi^b, X^I\})^2 \\ &- \frac{1}{3} F^{\mu \nu \lambda} A\_{\mu ab} A\_{\nu cd} A\_{\lambda cf} \{\varphi^a, \varphi^c, \varphi^d\} \{\varphi^b, \varphi^c, \varphi^f\} + \frac{1}{2} \Lambda \\ &- \frac{i}{2} \bar{\Psi} \Gamma^{\mu} A\_{\mu ab} \{\varphi^a, \varphi^b, \Psi\} + \frac{i}{4} \bar{\Psi} \Gamma\_{I\{}} \{X^I, X^I, \Psi\} \Big) \tag{39} \end{split} \tag{30}$$
where *<sup>I</sup>*, *<sup>J</sup>*, *<sup>K</sup>* <sup>=</sup> 3, ··· , 10 and {*ϕa*, *<sup>ϕ</sup>b*, *<sup>ϕ</sup>c*} <sup>=</sup> *�αβγ∂αϕ<sup>a</sup>∂βϕ<sup>b</sup>∂γ <sup>ϕ</sup><sup>c</sup>* is the Nambu-Poisson bracket. An invariant symmetric bilinear form is defined by *d*3*σ* √−*<sup>g</sup>ϕaϕ<sup>b</sup>* for complete basis *ϕ<sup>a</sup>* in three dimensions. Thus, this action is manifestly VPD covariant even when the world-volume metric is flat. *<sup>X</sup><sup>I</sup>* is a scalar and <sup>Ψ</sup> is a *SO*(1, 2) <sup>×</sup> *SO*(8) Majorana-Weyl fermion
<sup>1</sup> Advantages of a semi-light-cone gauges against a light-cone gauge are shown in [37–39]
#### 8 Will-be-set-by-IN-TECH 8 Linear Algebra – Theorems and Applications
satisfying (37). *Eμνλ* is a Levi-Civita symbol in three dimensions and Λ is a cosmological constant.
The continuum action of 3-algebra model of M-theory (39) is invariant under 16 dynamical supersymmetry transformations,
$$\begin{aligned} \delta X^I &= i\varepsilon \Gamma^I \Psi\\ \delta A\_\mu(\sigma, \sigma') &= \frac{i}{2} \varepsilon \Gamma\_\mu \Gamma\_I (X^I(\sigma) \Psi(\sigma') - X^I(\sigma') \Psi(\sigma)),\\ \delta \Psi &= -A\_{\mu ab} \{q^a, q^b, X^I\} \Gamma^\mu \Gamma\_I \varepsilon - \frac{1}{6} \{X^I, X^I, X^K\} \Gamma\_{IJK} \varepsilon \end{aligned} \tag{40}$$
where Γ012*�* = −*�*. These supersymmetries close into gauge transformations on-shell,
$$\begin{aligned} [\delta\_1, \delta\_2]X^I &= \Lambda\_{cd}\{\boldsymbol{\varphi^c}, \boldsymbol{\varphi^d}, X^I\} \\ [\delta\_1, \delta\_2]A\_{\mu ab}\{\boldsymbol{\varphi^a}, \boldsymbol{\varphi^b}, \quad \} &= \Lambda\_{ab}\{\boldsymbol{\varphi^a}, \boldsymbol{\varphi^b}, A\_{\mu cd}\{\boldsymbol{\varphi^c}, \boldsymbol{\varphi^d}, \quad \} \\ &- A\_{\mu ab}\{\boldsymbol{\varphi^a}, \boldsymbol{\varphi^b}, \Lambda\_{cd}\{\boldsymbol{\varphi^c}, \boldsymbol{\varphi^d}, \quad \} \} + 2i\bar{\varepsilon}\_2 \Gamma^{\boldsymbol{\nu}} \varepsilon\_1 O^A\_{\mu \nu} \\ [\delta\_1, \delta\_2]\Psi &= \Lambda\_{cd}\{\boldsymbol{\varphi^c}, \boldsymbol{\varphi^d}, \Psi\} + (i\bar{\varepsilon}\_2 \Gamma^{\boldsymbol{\mu}} \varepsilon\_1 \Gamma\_{\mu} - \frac{i}{4} \bar{\varepsilon}\_2 \Gamma^{KL} \varepsilon\_1 \Gamma\_{\boldsymbol{KL}}) \mathcal{O}^{\Psi} \end{aligned} \tag{41}$$
where gauge parameters are given by <sup>Λ</sup>*ab* <sup>=</sup> <sup>2</sup>*i�*¯2Γ*μ�*1*Aμab* <sup>−</sup> *<sup>i</sup>�*¯2Γ*JK�*1*X<sup>J</sup> aX<sup>K</sup> <sup>b</sup>* . *<sup>O</sup><sup>A</sup> μν* = 0 and *O*<sup>Ψ</sup> = 0 are equations of motions of *Aμν* and Ψ, respectively, where
$$\mathcal{O}^{A}\_{\mu\nu} = A\_{\mu ab} \{ \emptyset^{a}, \emptyset^{b}, A\_{\nu cd} \{ \emptyset^{c}, \emptyset^{d}, \quad \} \} - A\_{\nu ab} \{ \emptyset^{a}, \emptyset^{b}, A\_{\mu cd} \{ \emptyset^{c}, \emptyset^{d}, \quad \} \}$$
$$+ E\_{\mu\nu\lambda} (-\{ X^{I}, A\_{ab}^{\lambda} \{ \emptyset^{a}, \emptyset^{b}, X\_{I} \}, \quad \} + \frac{i}{2} \{ \Psi, \Gamma^{\lambda} \Psi, \quad \})$$
$$\mathcal{O}^{\Psi} = -\Gamma^{\mu} A\_{\mu ab} \{ \emptyset^{a}, \emptyset^{b}, \Psi \} + \frac{1}{2} \Gamma\_{II} \{ X^{I}, X^{I}, \Psi \} \tag{42}$$
(41) implies that a commutation relation between the dynamical supersymmetry transformations is
$$
\delta\_2 \delta\_1 - \delta\_1 \delta\_2 = 0 \tag{43}
$$
up to the equations of motions and the gauge transformations.
This action is invariant under a translation,
$$
\delta X^I(\sigma) = \eta^I, \qquad \delta A^\mu(\sigma, \sigma') = \eta^\mu(\sigma) - \eta^\mu(\sigma') \tag{44}
$$
where *η<sup>I</sup>* are constants.
The action is also invariant under 16 kinematical supersymmetry transformations
$$
\delta \Psi = \mathfrak{E} \tag{45}
$$
and the other fields are not transformed. *�*˜ is a constant and satisfy Γ012*�*˜ = *�*˜. *�*˜ and *�* should come from sixteen components of thirty-two N = 1 supersymmetry parameters in eleven dimensions, corresponding to eigen values ±1 of Γ012, respectively. This N = 1 supersymmetry consists of remaining 16 target-space supersymmetries and transmuted 16 *κ*-symmetries in the semi-light-cone gauge [25, 26, 40].
A commutation relation between the kinematical supersymmetry transformations is given by
$$
\tilde{\delta}\_2 \tilde{\delta}\_1 - \tilde{\delta}\_1 \tilde{\delta}\_2 = 0 \tag{46}
$$
A commutator of dynamical supersymmetry transformations and kinematical ones acts as
$$\begin{aligned} (\tilde{\delta}\_2 \delta\_1 - \delta\_1 \tilde{\delta}\_2) X^I(\sigma) &= i \tilde{\varepsilon}\_1 \Gamma^I \tilde{\varepsilon}\_2 \equiv \eta\_0^I \\ (\tilde{\delta}\_2 \delta\_1 - \delta\_1 \tilde{\delta}\_2) A^\mu(\sigma, \sigma') &= \frac{i}{2} \tilde{\varepsilon}\_1 \Gamma^\mu \Gamma\_I (X^I(\sigma) - X^I(\sigma')) \tilde{\varepsilon}\_2 \equiv \eta\_0^\mu(\sigma) - \eta\_0^\mu(\sigma') \end{aligned} \tag{47}$$
where the commutator that acts on the other fields vanishes. Thus, the commutation relation is given by
$$
\delta\_2 \delta\_1 - \delta\_1 \tilde{\delta}\_2 = \delta\_\eta \tag{48}
$$
where *δη* is a translation.
If we change a basis of the supersymmetry transformations as
$$\begin{aligned} \delta' &= \delta + \tilde{\delta} \\ \tilde{\delta}' &= \dot{\imath}(\delta - \tilde{\delta}) \end{aligned} \tag{49}$$
we obtain
8 Will-be-set-by-IN-TECH
satisfying (37). *Eμνλ* is a Levi-Civita symbol in three dimensions and Λ is a cosmological
The continuum action of 3-algebra model of M-theory (39) is invariant under 16 dynamical
(*σ*)Ψ(*σ*�
}Γ*μ*Γ*I�* <sup>−</sup> <sup>1</sup>
, *<sup>ϕ</sup>d*, <sup>Ψ</sup>} + (*i�*¯2Γ*μ�*1Γ*<sup>μ</sup>* <sup>−</sup> *<sup>i</sup>*
, *<sup>ϕ</sup>d*, }} − *<sup>A</sup>νab*{*ϕ<sup>a</sup>*
, *X<sup>J</sup>*
, *<sup>ϕ</sup>b*, *XI*}, } <sup>+</sup>
(41) implies that a commutation relation between the dynamical supersymmetry
1 2 <sup>Γ</sup>*I J*{*X<sup>I</sup>*
, *δAμ*(*σ*, *σ*�
˜
and the other fields are not transformed. *�*˜ is a constant and satisfy Γ012*�*˜ = *�*˜. *�*˜ and *�* should come from sixteen components of thirty-two N = 1 supersymmetry parameters in eleven dimensions, corresponding to eigen values ±1 of Γ012, respectively. This N = 1 supersymmetry consists of remaining 16 target-space supersymmetries and transmuted 16
The action is also invariant under 16 kinematical supersymmetry transformations
) <sup>−</sup> *<sup>X</sup><sup>I</sup>* (*σ*�
, *<sup>ϕ</sup>b*, *<sup>A</sup>μcd*{*ϕ<sup>c</sup>*
*i* 2
, *<sup>ϕ</sup>d*, }} <sup>+</sup> <sup>2</sup>*i�*¯2Γ*ν�*1*O<sup>A</sup>*
4
6 {*X<sup>I</sup>* , *X<sup>J</sup>*
)Ψ(*σ*)),
, *<sup>ϕ</sup>d*, }}
, *<sup>ϕ</sup>b*, *<sup>A</sup>μcd*{*ϕ<sup>c</sup>*
*δ*2*δ*<sup>1</sup> − *δ*1*δ*<sup>2</sup> = 0 (43)
*δ*Ψ = *�*˜ (45)
) = *<sup>η</sup>μ*(*σ*) <sup>−</sup> *<sup>η</sup>μ*(*σ*�
{Ψ¯ , <sup>Γ</sup>*λ*Ψ, })
*μν*
, *<sup>X</sup>K*}Γ*IJK�* (40)
*�*¯2Γ*KL�*1Γ*KL*)*O*<sup>Ψ</sup> (41)
, *<sup>ϕ</sup>d*, }}
) (44)
, Ψ} (42)
*μν* = 0 and
*aX<sup>K</sup> <sup>b</sup>* . *<sup>O</sup><sup>A</sup>*
constant.
supersymmetry transformations,
*O<sup>A</sup>*
transformations is
where *η<sup>I</sup>* are constants.
*μν* <sup>=</sup> *<sup>A</sup>μab*{*ϕ<sup>a</sup>*
*<sup>O</sup>*<sup>Ψ</sup> <sup>=</sup> <sup>−</sup>Γ*μAμab*{*ϕ<sup>a</sup>*
This action is invariant under a translation,
*δX<sup>I</sup>*
*κ*-symmetries in the semi-light-cone gauge [25, 26, 40].
*δX<sup>I</sup>* = *i�*¯Γ*<sup>I</sup>*
*δAμ*(*σ*, *σ*�
*<sup>δ</sup>*<sup>Ψ</sup> <sup>=</sup> <sup>−</sup>*Aμab*{*ϕ<sup>a</sup>*
[*δ*1, *<sup>δ</sup>*2]*X<sup>I</sup>* <sup>=</sup> <sup>Λ</sup>*cd*{*ϕ<sup>c</sup>*
[*δ*1, *<sup>δ</sup>*2]<sup>Ψ</sup> <sup>=</sup> <sup>Λ</sup>*cd*{*ϕ<sup>c</sup>*
<sup>+</sup>*Eμνλ*(−{*X<sup>I</sup>*
<sup>−</sup>*Aμab*{*ϕ<sup>a</sup>*
[*δ*1, *<sup>δ</sup>*2]*Aμab*{*ϕ<sup>a</sup>*
Ψ
) = *<sup>i</sup>* 2
*�*¯Γ*μ*Γ*I*(*X<sup>I</sup>*
, *ϕb*, *X<sup>I</sup>*
, *ϕd*, *X<sup>I</sup>* }
where gauge parameters are given by <sup>Λ</sup>*ab* <sup>=</sup> <sup>2</sup>*i�*¯2Γ*μ�*1*Aμab* <sup>−</sup> *<sup>i</sup>�*¯2Γ*JK�*1*X<sup>J</sup>*
*O*<sup>Ψ</sup> = 0 are equations of motions of *Aμν* and Ψ, respectively, where
, *<sup>ϕ</sup>b*, *<sup>A</sup>νcd*{*ϕ<sup>c</sup>*
, *A<sup>λ</sup> ab*{*ϕ<sup>a</sup>*
up to the equations of motions and the gauge transformations.
(*σ*) = *η<sup>I</sup>*
, *<sup>ϕ</sup>b*, <sup>Ψ</sup>} <sup>+</sup>
, *<sup>ϕ</sup>b*, } <sup>=</sup> <sup>Λ</sup>*ab*{*ϕ<sup>a</sup>*
, *<sup>ϕ</sup>b*, <sup>Λ</sup>*cd*{*ϕ<sup>c</sup>*
where Γ012*�* = −*�*. These supersymmetries close into gauge transformations on-shell,
$$\begin{aligned} \delta\_2' \delta\_1' - \delta\_1' \delta\_2' &= \delta\_\eta \\ \delta\_2' \delta\_1' - \tilde{\delta}\_1' \tilde{\delta}\_2' &= \delta\_\eta \\ \delta\_2' \delta\_1' - \delta\_1' \tilde{\delta}\_2' &= 0 \end{aligned} \tag{50}$$
These thirty-two supersymmetry transformations are summarised as Δ = (*δ*� , ˜ *δ*� ) and (50) implies the N = 1 supersymmetry algebra in eleven dimensions,
$$
\Delta\_2 \Delta\_1 - \Delta\_1 \Delta\_2 = \delta\_\eta \tag{51}
$$
#### **3.2. Lie 3-algebra models of M-theory**
In this and next subsection, we perform the second quantization on the continuum action of the 3-algebra model of M-theory: By replacing the Nambu-Poisson bracket in the action (39) with brackets of finite-dimensional 3-algebras, Lie and Hermitian 3-algebras, we obtain the Lie and Hermitian 3-algebra models of M-theory [26, 28], respectively. In this section, we review the Lie 3-algebra model.
If we replace the Nambu-Poisson bracket in the action (39) with a completely antisymmetric real 3-algebra's bracket [21, 22],
$$\begin{aligned} \int d^3 \sigma \sqrt{-g} &\rightarrow \left< \begin{array}{c} \\ \\ \end{array} \right> \\ \{\varphi^a, \varphi^b, \varphi^c\} &\rightarrow [T^a, T^b, T^c] \end{aligned} \tag{52}$$
we obtain the Lie 3-algebra model of M-theory [26, 28],
$$\begin{split} S\_0 &= \left\langle -\frac{1}{12} [X^I, X^I, X^K]^2 - \frac{1}{2} (A\_{\mu ab} [T^a, T^b, X^I])^2 \right. \\ &\left. -\frac{1}{3} E^{\mu \nu \lambda} A\_{\mu ab} A\_{\nu cd} A\_{\lambda ef} [T^a, T^c, T^d] [T^b, T^c, T^f] \right. \\ &\left. -\frac{i}{2} \bar{\Psi} \Gamma^{\mu} A\_{\mu ab} [T^a, T^b, \Psi] + \frac{i}{4} \bar{\Psi} \Gamma\_{II} [X^I, X^J, \Psi] \right\rangle \end{split} \tag{53}$$
We have deleted the cosmological constant Λ, which corresponds to an operator ordering ambiguity, as usual as in the case of other matrix models [27, 41].
This model can be obtained formally by a dimensional reduction of the N = 8 BLG model [4–6],
$$\begin{split} S\_{N=8BLG} &= \int d^3x \Big\langle -\frac{1}{12} [\mathbf{X}^I, \mathbf{X}^I, \mathbf{X}^K]^2 - \frac{1}{2} (D\_\mu \mathbf{X}^I)^2 - E^{\mu\nu\lambda} \Big( \frac{1}{2} A\_{\mu ab} \partial\_\nu A\_{\lambda cd} T^d [T^b, T^c, T^d] \\ &+ \frac{1}{3} A\_{\mu ab} A\_{\nu cd} A\_{\lambda cf} [T^d, T^c, T^d] [T^b, T^c, T^f] \Big) \\ &+ \frac{i}{2} \bar{\Psi} \Gamma^\mu D\_\mu \Psi + \frac{i}{4} \bar{\Psi} \Gamma\_{II} [X^I, X^I, \Psi] \Big) \end{split} \tag{54}$$
The formal relations between the Lie (Hermitian) 3-algebra models of M-theory and the N = 8 (N = 6) BLG models are analogous to the relation among the N = 4 super Yang-Mills in four dimensions, the BFSS matrix theory [27], and the IIB matrix model [41]. They are completely different theories although they are related to each others by dimensional reductions. In the same way, the 3-algebra models of M-theory and the BLG models are completely different theories.
The fields in the action (53) are spanned by the Lie 3-algebra *T<sup>a</sup>* as *X<sup>I</sup>* = *X<sup>I</sup> aT<sup>a</sup>*, Ψ = Ψ*aT<sup>a</sup>* and *A<sup>μ</sup>* = *A<sup>μ</sup> abT<sup>a</sup>* <sup>⊗</sup> *<sup>T</sup>b*, where *<sup>I</sup>* <sup>=</sup> 3, ··· , 10 and *<sup>μ</sup>* <sup>=</sup> 0, 1, 2. <> represents a metric for the 3-algebra. Ψ is a Majorana spinor of SO(1,10) that satisfies Γ012Ψ = Ψ. *Eμνλ* is a Levi-Civita symbol in three-dimensions.
Finite dimensional Lie 3-algebras with an invariant metric is classified into four-dimensional Euclidean A<sup>4</sup> algebra and the Lie 3-algebras with indefinite metrics in [9–11, 21, 22]. We do not choose A<sup>4</sup> algebra because its degrees of freedom are just four. We need an algebra with arbitrary dimensions N, which is taken to infinity to define M-theory. Here we choose the most simple indefinite metric Lie 3-algebra, so called the Lorentzian Lie 3-algebra associated with *u*(*N*) Lie algebra,
$$\begin{aligned} [T^{-1}, T^i, T^\emptyset] &= 0\\ [T^0, T^i, T^j] &= [T^i, T^j] = f^{ij} \, {}\_k T^k\\ [T^i, T^j, T^k] &= f^{ijk} T^{-1} \end{aligned} \tag{55}$$
where *<sup>a</sup>* <sup>=</sup> <sup>−</sup>1, 0, *<sup>i</sup>* (*<sup>i</sup>* <sup>=</sup> 1, ··· , *<sup>N</sup>*2). *<sup>T</sup><sup>i</sup>* are generators of *<sup>u</sup>*(*N*). A metric is defined by a symmetric bilinear form,
$$ = -1\tag{56}$$
$$ = \hbar^{ij} \tag{57}$$
and the other components are 0. The action is decomposed as
$$S = \text{Tr}(-\frac{1}{4}(\mathbf{x}\_0^K)^2[\mathbf{x}^I, \mathbf{x}^J]^2 + \frac{1}{2}(\mathbf{x}\_0^I[\mathbf{x}\_I, \mathbf{x}^I])^2 - \frac{1}{2}(\mathbf{x}\_0^I b\_\mu + [a\_\mu, \mathbf{x}^I])^2 - \frac{1}{2}E^{\mu\nu\lambda}b\_\mu[a\_\nu, a\_\lambda]$$
$$+ i\bar{\psi}\_0 \Gamma^\mu b\_\mu \psi - \frac{i}{2}\bar{\psi}\Gamma^\mu [a\_\mu, \psi] + \frac{i}{2}\mathbf{x}\_0^I \bar{\psi}\Gamma\_{II}[\mathbf{x}^I, \psi] - \frac{i}{2}\bar{\psi}b\_0 \Gamma\_{II}[\mathbf{x}^I, \mathbf{x}^I] \psi \tag{58}$$
where we have renamed *X<sup>I</sup>* <sup>0</sup> <sup>→</sup> *<sup>x</sup><sup>I</sup>* <sup>0</sup>, *<sup>X</sup><sup>I</sup> <sup>i</sup> <sup>T</sup><sup>i</sup>* <sup>→</sup> *<sup>x</sup><sup>I</sup>* , <sup>Ψ</sup><sup>0</sup> <sup>→</sup> *<sup>ψ</sup>*0, <sup>Ψ</sup>*iT<sup>i</sup>* <sup>→</sup> *<sup>ψ</sup>*, 2*Aμ*<sup>0</sup>*iT<sup>i</sup>* <sup>→</sup> *<sup>a</sup>μ*, and *Aμij*[*T<sup>i</sup>* , *T<sup>j</sup>* ] <sup>→</sup> *<sup>b</sup>μ*. *<sup>a</sup><sup>μ</sup>* correspond to the target coordinate matrices *<sup>X</sup>μ*, whereas *<sup>b</sup><sup>μ</sup>* are auxiliary fields.
In this action, *T*−<sup>1</sup> mode; *X<sup>I</sup>* <sup>−</sup>1, <sup>Ψ</sup>−<sup>1</sup> or *<sup>A</sup><sup>μ</sup>* <sup>−</sup>1*<sup>a</sup>* does not appear, that is they are unphysical modes. Therefore, the indefinite part of the metric (56) does not exist in the action and the Lie 3-algebra model of M-theory is ghost-free like a model in [42]. This action can be obtained by a dimensional reduction of the three-dimensional N = 8 BLG model [4–6] with the same 3-algebra. The BLG model possesses a ghost mode because of its kinetic terms with indefinite signature. On the other hand, the Lie 3-algebra model of M-theory does not possess a kinetic term because it is defined as a zero-dimensional field theory like the IIB matrix model [41].
This action is invariant under the translation
10 Will-be-set-by-IN-TECH
We have deleted the cosmological constant Λ, which corresponds to an operator ordering
This model can be obtained formally by a dimensional reduction of the N = 8 BLG model
, *T<sup>c</sup>*
ΨΓ¯ *I J*[*X<sup>I</sup>*
The formal relations between the Lie (Hermitian) 3-algebra models of M-theory and the N = 8 (N = 6) BLG models are analogous to the relation among the N = 4 super Yang-Mills in four dimensions, the BFSS matrix theory [27], and the IIB matrix model [41]. They are completely different theories although they are related to each others by dimensional reductions. In the same way, the 3-algebra models of M-theory and the BLG models are completely different
3-algebra. Ψ is a Majorana spinor of SO(1,10) that satisfies Γ012Ψ = Ψ. *Eμνλ* is a Levi-Civita
Finite dimensional Lie 3-algebras with an invariant metric is classified into four-dimensional Euclidean A<sup>4</sup> algebra and the Lie 3-algebras with indefinite metrics in [9–11, 21, 22]. We do not choose A<sup>4</sup> algebra because its degrees of freedom are just four. We need an algebra with arbitrary dimensions N, which is taken to infinity to define M-theory. Here we choose the most simple indefinite metric Lie 3-algebra, so called the Lorentzian Lie 3-algebra associated
, *Tb*] = 0
]=[*T<sup>i</sup>*
where *<sup>a</sup>* <sup>=</sup> <sup>−</sup>1, 0, *<sup>i</sup>* (*<sup>i</sup>* <sup>=</sup> 1, ··· , *<sup>N</sup>*2). *<sup>T</sup><sup>i</sup>* are generators of *<sup>u</sup>*(*N*). A metric is defined by a
])<sup>2</sup> <sup>−</sup> <sup>1</sup> 2 (*x<sup>I</sup>*
*<sup>ψ</sup>*¯Γ*μ*[*aμ*, *<sup>ψ</sup>*] + *<sup>i</sup>*
2 *xI* 0*ψ*¯Γ*I J*[*x<sup>J</sup>*
, *T<sup>j</sup>* ] = *f ij kTk*
, *Tk*] = *f ijkT*−<sup>1</sup> (55)
<sup>&</sup>lt; *<sup>T</sup>*<sup>−</sup>1, *<sup>T</sup>*<sup>0</sup> <sup>&</sup>gt; <sup>=</sup> <sup>−</sup><sup>1</sup> (56)
<sup>0</sup>*b<sup>μ</sup>* + [*aμ*, *<sup>x</sup><sup>I</sup>*
, *T<sup>j</sup>* > = *hij* (57)
])<sup>2</sup> <sup>−</sup> <sup>1</sup> 2
, *<sup>ψ</sup>*] <sup>−</sup> *<sup>i</sup>* 2 *Eμνλbμ*[*aν*, *aλ*]
, *x<sup>J</sup>*
]*ψ*) (58)
*ψ*¯0Γ*I J*[*x<sup>I</sup>*
)<sup>2</sup> <sup>−</sup> *<sup>E</sup>μνλ* <sup>1</sup>
, *T<sup>f</sup>* ]
, *Td*][*Tb*, *T<sup>e</sup>*
, *X<sup>J</sup>* , Ψ]
*abT<sup>a</sup>* <sup>⊗</sup> *<sup>T</sup>b*, where *<sup>I</sup>* <sup>=</sup> 3, ··· , 10 and *<sup>μ</sup>* <sup>=</sup> 0, 1, 2. <> represents a metric for the
<sup>2</sup> *<sup>A</sup>μab∂νAλcdT<sup>a</sup>*[*Tb*, *<sup>T</sup><sup>c</sup>*
, *Td*]
*aT<sup>a</sup>*, Ψ = Ψ*aT<sup>a</sup>*
(54)
<sup>2</sup> <sup>−</sup> <sup>1</sup> 2 (*DμX<sup>I</sup>*
*i* 4
The fields in the action (53) are spanned by the Lie 3-algebra *T<sup>a</sup>* as *X<sup>I</sup>* = *X<sup>I</sup>*
[*T*<sup>−</sup>1, *T<sup>a</sup>*
, *T<sup>j</sup>*
< *T<sup>i</sup>*
2
[*T*0, *T<sup>i</sup>*
[*Ti* , *T<sup>j</sup>*
and the other components are 0. The action is decomposed as
<sup>+</sup>*iψ*¯0Γ*μbμψ* <sup>−</sup> *<sup>i</sup>*
ambiguity, as usual as in the case of other matrix models [27, 41].
, *X<sup>J</sup>* , *XK*]
<sup>3</sup> *<sup>A</sup>μabAνcdAλe f* [*T<sup>a</sup>*
ΨΓ¯ *<sup>μ</sup>Dμ*Ψ +
+ 1
+ *i* 2
[4–6],
theories.
and *A<sup>μ</sup>* = *A<sup>μ</sup>*
symbol in three-dimensions.
with *u*(*N*) Lie algebra,
symmetric bilinear form,
*<sup>S</sup>* <sup>=</sup> Tr(−<sup>1</sup>
4 (*x<sup>K</sup>* <sup>0</sup> )2[*x<sup>I</sup>* , *x<sup>J</sup>* ] <sup>2</sup> + 1 2 (*x<sup>I</sup>* <sup>0</sup>[*xI*, *<sup>x</sup><sup>J</sup>*
*<sup>S</sup>*<sup>N</sup> <sup>=</sup>8*BLG* =
*d*3*x* − 1 <sup>12</sup> [*X<sup>I</sup>*
$$
\delta \mathbf{x}^I = \eta^I, \qquad \delta a^\mu = \eta^\mu \tag{59}
$$
where *η<sup>I</sup>* and *η<sup>μ</sup>* belong to *u*(1). This implies that eigen values of *x<sup>I</sup>* and *a<sup>μ</sup>* represent an eleven-dimensional space-time.
The action is also invariant under 16 kinematical supersymmetry transformations
$$
\delta\psi = \mathfrak{E} \tag{60}
$$
and the other fields are not transformed. *�*˜ belong to *u*(1) and satisfy Γ012*�*˜ = *�*˜. *�*˜ and *�* should come from sixteen components of thirty-two N = 1 supersymmetry parameters in eleven dimensions, corresponding to eigen values ±1 of Γ012, respectively, as in the previous subsection.
A commutation relation between the kinematical supersymmetry transformations is given by
$$
\delta\_2 \vec{\delta}\_1 - \vec{\delta}\_1 \vec{\delta}\_2 = 0 \tag{61}
$$
The action is invariant under 16 dynamical supersymmetry transformations,
$$\begin{aligned} \delta X^I &= i\varepsilon \Gamma^I \Psi\\ \delta A\_{\mu ab}[T^a, T^b, \quad ] &= i\varepsilon \Gamma\_{\mu} \Gamma\_I[X^I, \Psi, \quad ]\\ \delta \Psi &= -A\_{\mu ab}[T^a, T^b, X^I] \Gamma^\mu \Gamma\_I \varepsilon - \frac{1}{6}[X^I, X^J, X^K] \Gamma\_{IJK} \varepsilon \end{aligned} \tag{62}$$
where Γ012*�* = −*�*. These supersymmetries close into gauge transformations on-shell,
$$\begin{aligned} [\delta\_1, \delta\_2]X^I &= \Lambda\_{cd}[T^c, T^d, X^I] \\\\ [\delta\_1, \delta\_2]A\_{\mu ab}[T^a, T^b, \quad ] &= \Lambda\_{ab}[T^a, T^b, A\_{\mu cd}[T^c, T^d, \quad ]] \\\\ &- A\_{\mu ab}[T^d, T^b, \Lambda\_{cd}[T^c, T^d, \quad ]] + 2i\bar{\varepsilon}\_2 \Gamma^\nu \varepsilon\_1 O^A\_{\mu \nu} \end{aligned}$$
$$[\delta\_1, \delta\_2]\Psi = \Lambda\_{cd}[T^c, T^d, \Psi] + (i\bar{\varepsilon}\_2 \Gamma^\mu \varepsilon\_1 \Gamma\_\mu - \frac{i}{4}\bar{\varepsilon}\_2 \Gamma^{\text{KL}}\varepsilon\_1 \Gamma\_{\text{KL}})O^\Psi \tag{63}$$
#### 12 Will-be-set-by-IN-TECH 12 Linear Algebra – Theorems and Applications
where gauge parameters are given by <sup>Λ</sup>*ab* <sup>=</sup> <sup>2</sup>*i�*¯2Γ*μ�*1*Aμab* <sup>−</sup> *<sup>i</sup>�*¯2Γ*JK�*1*X<sup>J</sup> aX<sup>K</sup> <sup>b</sup>* . *<sup>O</sup><sup>A</sup> μν* = 0 and *O*<sup>Ψ</sup> = 0 are equations of motions of *Aμν* and Ψ, respectively, where
$$\mathbf{O}^{A}\_{\mu\nu} = A\_{\mu ab} [T^a, T^b, A\_{\nu cd} [T^c, T^d, \quad ]] - A\_{\nu ab} [T^a, T^b, A\_{\mu cd} [T^c, T^d, \quad ]]$$
$$+ E\_{\mu\nu\lambda} (-[X^I, A^\lambda\_{ab} [T^a, T^b, X\_I], \quad ] + \frac{i}{2} [\Psi, \Gamma^\lambda \Psi, \quad ])$$
$$\mathbf{O}^{\Psi} = -\Gamma^\mu A\_{\mu ab} [T^a, T^b, \Psi] + \frac{1}{2} \Gamma\_{II} [X^I, X^I, \Psi] \tag{64}$$
(63) implies that a commutation relation between the dynamical supersymmetry transformations is
$$
\delta\_2 \delta\_1 - \delta\_1 \delta\_2 = 0 \tag{65}
$$
up to the equations of motions and the gauge transformations.
The 16 dynamical supersymmetry transformations (62) are decomposed as
$$\begin{aligned} \delta \mathbf{x}^I &= i \bar{\epsilon} \Gamma^I \psi\\ \delta \mathbf{x}^I\_0 &= i \bar{\epsilon} \Gamma^I \psi\_0\\ \delta \mathbf{x}^I\_{-1} &= i \bar{\epsilon} \Gamma^I \psi\_{-1} \end{aligned}$$
$$\begin{aligned} \delta \psi &= -(b\_{\mu} \mathbf{x}^0\_0 + [a\_{\mu \nu} \mathbf{x}^I]) \Gamma^\mu \Gamma\_I \epsilon - \frac{1}{2} \mathbf{x}^I\_0 [\mathbf{x}^I, \mathbf{x}^K] \Gamma\_{IIK} \epsilon\\ \delta \psi\_0 &= 0 \end{aligned}$$
$$\begin{aligned} \delta \psi\_{-1} &= -\text{Tr}(b\_{\mu} \mathbf{x}^I) \Gamma^\mu \Gamma\_I \epsilon - \frac{1}{6} \text{Tr}([\mathbf{x}^I, \mathbf{x}^I] \mathbf{x}^K) \Gamma\_{IIK} \epsilon \\\ \delta a\_{\mu} &= i \bar{\epsilon} \Gamma\_{\mu} \Gamma\_I (\mathbf{x}^I, \psi \mathbf{I}) \\\ \delta b\_{\mu} &= i \bar{\epsilon} \Gamma\_{\mu} \Gamma\_I [\mathbf{x}^I, \psi \mathbf{I}] \end{aligned}$$
$$\begin{aligned} \delta A\_{\mu - 1i} &= i \bar{\epsilon} \Gamma\_{\mu} \Gamma\_I \frac{1}{2} (\mathbf{x}^I\_{-1} \Psi\_{\bar{\imath}} - \Psi\_{-1} \mathbf{x}^I\_i) \\\ \delta A\_{\mu - 10} &= i \bar{\epsilon} \Gamma\_{\mu} \Gamma\_I \frac{1}{2} (\mathbf{x}^I\_{-1} \Psi\_0 - \Psi\_{-1} \mathbf{x}^I\_0) \end{aligned} \tag{66}$$
and thus a commutator of dynamical supersymmetry transformations and kinematical ones acts as
$$(\delta\_2 \delta\_1 - \delta\_1 \tilde{\delta}\_2) \mathbf{x}^I = i \vec{\varepsilon}\_1 \boldsymbol{\Gamma}^I \vec{\varepsilon}\_2 \equiv \eta^I$$
$$(\delta\_2 \delta\_1 - \delta\_1 \tilde{\delta}\_2) a^\mu = i \vec{\varepsilon}\_1 \boldsymbol{\Gamma}^\mu \boldsymbol{\Gamma}\_I \mathbf{x}\_0^I \vec{\varepsilon}\_2 \equiv \eta^\mu$$
$$(\delta\_2 \delta\_1 - \delta\_1 \tilde{\delta}\_2) A^\mu\_{-1i} T^i = \frac{1}{2} i \vec{\varepsilon}\_1 \boldsymbol{\Gamma}^\mu \boldsymbol{\Gamma}\_I \mathbf{x}\_{-1}^I \vec{\varepsilon}\_2 \tag{67}$$
where the commutator that acts on the other fields vanishes. Thus, the commutation relation for physical modes is given by
$$
\delta\_2 \delta\_1 - \delta\_1 \tilde{\delta}\_2 = \delta\_\eta \tag{68}
$$
where *δη* is a translation.
(61), (65), and (68) imply the N = 1 supersymmetry algebra in eleven dimensions as in the previous subsection.
#### **3.3. Hermitian 3-algebra model of M-theory**
12 Will-be-set-by-IN-TECH
, *<sup>T</sup>d*, ]] <sup>−</sup> *<sup>A</sup>νab*[*T<sup>a</sup>*
, *<sup>T</sup>b*, *XI*], ] + *<sup>i</sup>*
, *X<sup>J</sup>*
])Γ*μ*Γ*I�* <sup>−</sup> <sup>1</sup>
6 Tr([*x<sup>I</sup>* , *x<sup>J</sup>*
)
<sup>−</sup>1*ψ<sup>i</sup>* <sup>−</sup> *<sup>ψ</sup>*−1*x<sup>I</sup>*
<sup>−</sup>1*ψ*<sup>0</sup> <sup>−</sup> *<sup>ψ</sup>*−1*x<sup>I</sup>*
and thus a commutator of dynamical supersymmetry transformations and kinematical ones
*δ*2)*a<sup>μ</sup>* = *i�*¯1Γ*μ*Γ*<sup>I</sup> x<sup>I</sup>*
where the commutator that acts on the other fields vanishes. Thus, the commutation relation
(61), (65), and (68) imply the N = 1 supersymmetry algebra in eleven dimensions as in the
*<sup>T</sup><sup>i</sup>* <sup>=</sup> <sup>1</sup> 2
*δ*2)*x<sup>I</sup>* = *i�*¯1Γ*<sup>I</sup>*
)Γ*μ*Γ*I�* <sup>−</sup> <sup>1</sup>
<sup>0</sup>*<sup>ψ</sup>* <sup>−</sup> *<sup>ψ</sup>*0*x<sup>I</sup>*
, *ψ*]
1 2 (*x<sup>I</sup>*
1 2 (*x<sup>I</sup>*
*<sup>δ</sup>*2)*A<sup>μ</sup>* −1*i*
˜ *<sup>δ</sup>*2*δ*<sup>1</sup> <sup>−</sup> *<sup>δ</sup>*<sup>1</sup> ˜ 2 *xI* 0[*x<sup>J</sup>*
*i* )
*�*˜2 <sup>≡</sup> *<sup>η</sup><sup>I</sup>*
<sup>0</sup>*�*˜2 <sup>≡</sup> *<sup>η</sup><sup>μ</sup>*
*i�*¯1Γ*μ*Γ*<sup>I</sup> x<sup>I</sup>*
2
*aX<sup>K</sup> <sup>b</sup>* . *<sup>O</sup><sup>A</sup>*
, *Td*, ]]
, Ψ] (64)
, *<sup>T</sup>b*, *<sup>A</sup>μcd*[*T<sup>c</sup>*
*δ*2*δ*<sup>1</sup> − *δ*1*δ*<sup>2</sup> = 0 (65)
, *xK*]Γ*IJK�*
]*xK*)Γ*IJK�*
<sup>0</sup>) (66)
<sup>−</sup>1*�*˜2 (67)
*δ*<sup>2</sup> = *δη* (68)
[Ψ¯ , Γ*λ*Ψ, ])
*μν* = 0 and
where gauge parameters are given by <sup>Λ</sup>*ab* <sup>=</sup> <sup>2</sup>*i�*¯2Γ*μ�*1*Aμab* <sup>−</sup> *<sup>i</sup>�*¯2Γ*JK�*1*X<sup>J</sup>*
*O*<sup>Ψ</sup> = 0 are equations of motions of *Aμν* and Ψ, respectively, where
, *<sup>T</sup>b*, *<sup>A</sup>νcd*[*T<sup>c</sup>*
, *A<sup>λ</sup> ab*[*T<sup>a</sup>*
up to the equations of motions and the gauge transformations.
*δx<sup>I</sup>* = *i�*¯Γ*<sup>I</sup>*
<sup>0</sup> = *<sup>i</sup>�*¯Γ*<sup>I</sup>*
<sup>−</sup><sup>1</sup> <sup>=</sup> *<sup>i</sup>�*¯Γ*<sup>I</sup>*
*δψ* <sup>=</sup> <sup>−</sup>(*bμx<sup>I</sup>*
*δψ*−<sup>1</sup> <sup>=</sup> <sup>−</sup>Tr(*bμx<sup>I</sup>*
*δa<sup>μ</sup>* = *i�*¯Γ*μ*Γ*I*(*x<sup>I</sup>*
*δb<sup>μ</sup>* = *i�*¯Γ*μ*Γ*I*[*x<sup>I</sup>*
*<sup>δ</sup>Aμ*−1*<sup>i</sup>* = *<sup>i</sup>�*¯Γ*μ*Γ*<sup>I</sup>*
*<sup>δ</sup>Aμ*−<sup>10</sup> = *<sup>i</sup>�*¯Γ*μ*Γ*<sup>I</sup>*
(˜
(˜
(˜
for physical modes is given by
where *δη* is a translation.
previous subsection.
*<sup>δ</sup>*2*δ*<sup>1</sup> <sup>−</sup> *<sup>δ</sup>*<sup>1</sup> ˜
*<sup>δ</sup>*2*δ*<sup>1</sup> <sup>−</sup> *<sup>δ</sup>*<sup>1</sup> ˜
*<sup>δ</sup>*2*δ*<sup>1</sup> <sup>−</sup> *<sup>δ</sup>*<sup>1</sup> ˜
*δψ*<sup>0</sup> = 0
*δx<sup>I</sup>*
*δx<sup>I</sup>*
, *<sup>T</sup>b*, <sup>Ψ</sup>] + <sup>1</sup>
The 16 dynamical supersymmetry transformations (62) are decomposed as
*ψ*
*ψ*0
*ψ*−<sup>1</sup>
<sup>0</sup> + [*aμ*, *<sup>x</sup><sup>I</sup>*
2 Γ*I J*[*X<sup>I</sup>*
(63) implies that a commutation relation between the dynamical supersymmetry
*O<sup>A</sup>*
transformations is
acts as
*μν* = *<sup>A</sup>μab*[*T<sup>a</sup>*
*<sup>O</sup>*<sup>Ψ</sup> <sup>=</sup> <sup>−</sup>Γ*μAμab*[*T<sup>a</sup>*
<sup>+</sup>*Eμνλ*(−[*X<sup>I</sup>*
In this subsection, we study the Hermitian 3-algebra models of M-theory [26]. Especially, we study mostly the model with the *u*(*N*) ⊕ *u*(*N*) Hermitian 3-algebra (20).
The continuum action (39) can be rewritten by using the triality of *SO*(8) and the *SU*(4) ×*U*(1) decomposition [8, 43, 44] as
$$\begin{split} S\_{cl} &= \int d^3 \sigma \sqrt{-g} \Big( -V - A\_{\mu \mu \iota} \{ Z^A, T^a, T^b \} A\_{dc}^{\mu} \{ Z\_A, T^c, T^d \} \\ &+ \frac{i}{3} E^{\mu \nu \lambda} A\_{\mu \mu \iota} A\_{\nu \ell \epsilon} \{ T^d, T^c, T^d \} \{ T^b, T^f, T^e \} \\ &+ i \bar{\psi}^A \Gamma^\mu A\_{\mu \mu \iota} \{ \psi\_A, T^a, T^b \} + \frac{i}{2} E\_{ABCD} \bar{\psi}^A \{ Z^C, Z^D, \psi^B \} - \frac{i}{2} E^{ABCD} Z\_D \{ \bar{\psi}\_A, \psi\_B, Z\_C \} \\ &- i \bar{\psi}^A \{ \psi\_A, Z^B, Z\_B \} + 2i \bar{\psi}^A \{ \psi\_B, Z^B, Z\_A \} \Big) \end{split} \tag{69}$$
where fields with a raised *A* index transform in the 4 of SU(4), whereas those with lowered one transform in the 4.¯ *<sup>A</sup>μba* (*<sup>μ</sup>* = 0, 1, 2) is an anti-Hermitian gauge field, *<sup>Z</sup><sup>A</sup>* and *ZA* are a complex scalar field and its complex conjugate, respectively. *ψ<sup>A</sup>* is a fermion field that satisfies
$$
\Gamma^{012}\psi\_A = -\psi\_A \tag{70}
$$
and *ψ<sup>A</sup>* is its complex conjugate. *Eμνλ* and *EABCD* are Levi-Civita symbols in three dimensions and four dimensions, respectively. The potential terms are given by
$$\begin{aligned} V &= \frac{2}{3} \mathbf{Y}\_B^{\mathbb{C}D} \mathbf{Y}\_{\mathbb{C}D}^{B} \\ \mathbf{Y}\_B^{\mathbb{C}D} &= \{ \mathbf{Z}^{\mathbb{C}}, \mathbf{Z}^{D}, \mathbf{Z}\_B \} - \frac{1}{2} \delta\_{\mathbb{B}}^{\mathbb{C}} \{ \mathbf{Z}^{\mathbb{E}}, \mathbf{Z}^{D}, \mathbf{Z}\_{\mathbb{E}} \} + \frac{1}{2} \delta\_{\mathbb{B}}^{D} \{ \mathbf{Z}^{\mathbb{E}}, \mathbf{Z}^{\mathbb{C}}, \mathbf{Z}\_{\mathbb{E}} \} \end{aligned} \tag{71}$$
If we replace the Nambu-Poisson bracket with a Hermitian 3-algebra's bracket [19, 20],
$$\begin{aligned} \int d^3 \sigma \sqrt{-g} &\rightarrow \left< \quad \right>\\ \{\varphi^a, \varphi^b, \varphi^c\} &\rightarrow [T^a, T^b; \bar{T}^c] \end{aligned} \tag{72}$$
we obtain the Hermitian 3-algebra model of M-theory [26],
$$\begin{split} S &= \left\langle -V - A\_{\mu\bar{b}a}[\boldsymbol{Z}^{A}, \boldsymbol{T}^{a}; \boldsymbol{\bar{T}}^{b}] \overline{A\_{\bar{d}c}^{\mu}[\boldsymbol{Z}\_{A}, \boldsymbol{T}^{c}; \boldsymbol{\bar{T}}^{d}]} + \frac{1}{3} \boldsymbol{E}^{\mu\nu\lambda} A\_{\mu\bar{b}a} A\_{\nu\bar{d}c} A\_{\lambda\bar{f}c} [\boldsymbol{T}^{a}, \boldsymbol{T}^{c}; \boldsymbol{\bar{T}}^{d}] \overline{[\boldsymbol{T}^{b}, \boldsymbol{T}^{f}; \boldsymbol{\bar{T}}^{p}]} \\ &+ i \boldsymbol{\bar{\psi}}^{A} \boldsymbol{\Gamma}^{\mu} A\_{\mu\bar{b}a} [\psi\_{A}, \boldsymbol{T}^{a}; \boldsymbol{\bar{T}}^{b}] + \frac{i}{2} \boldsymbol{E}\_{ABCD} \boldsymbol{\bar{\psi}}^{A} [\boldsymbol{Z}^{C}, \boldsymbol{Z}^{D}; \boldsymbol{\bar{\psi}}^{B}] - \frac{i}{2} \boldsymbol{E}^{\text{ABCD}} \boldsymbol{\mathcal{Z}}\_{D} [\bar{\psi}\_{A}, \psi\_{B}; \boldsymbol{\bar{\mathcal{Z}}\_{C}}] \\ &- i \boldsymbol{\bar{\psi}}^{A} [\psi\_{A}, \boldsymbol{\mathcal{Z}}^{B}; \boldsymbol{\bar{\mathcal{Z}}\_{B}}] + 2i \boldsymbol{i} \boldsymbol{\bar{\psi}}^{A} [\psi\_{B}, \boldsymbol{\mathcal{Z}}^{B}; \boldsymbol{\bar{\mathcal{Z}}\_{A}}] \end{split} \tag{73}$$
where the cosmological constant has been deleted for the same reason as before. The potential terms are given by
$$\begin{aligned} V &= \frac{2}{3} \mathbf{Y}\_B^{CD} \mathbf{\bar{Y}}\_{CD}^B\\ \mathbf{Y}\_B^{CD} &= [Z^\mathbb{C}, Z^D; \bar{Z}\_B] - \frac{1}{2} \delta\_B^\mathbb{C} [Z^E, Z^D; \bar{Z}\_E] + \frac{1}{2} \delta\_B^D [Z^E, Z^\mathbb{C}; \bar{Z}\_E] \end{aligned} \tag{74}$$
#### 14 Will-be-set-by-IN-TECH 14 Linear Algebra – Theorems and Applications
This matrix model can be obtained formally by a dimensional reduction of the N = 6 BLG action [8], which is equivalent to ABJ(M) action [7, 45]2,
$$\begin{split} S\_{N \leftarrow 6BLG} &= \int d^3x \Big\langle -V - D\_{\mu}Z^{A}\overline{D^{\mu}Z\_{A}} + E^{\mu\nu\lambda} \Big( \frac{1}{2} A\_{\mu\nu b} \partial\_{\nu}A\_{\lambda d\bar{a}} \, \bar{T}^{\bar{d}}[T^{a}, T^{b}; \bar{T}^{c}] \\ &+ \frac{1}{3} A\_{\mu\bar{b}a} A\_{\nu\bar{d}c} A\_{\lambda\bar{f}\epsilon} [T^{a}, T^{c}; \bar{T}^{d}] \overline{[T^{b}, T^{f}; \bar{T}^{c}]} \Big) \\ &- i\bar{\psi}^{A} \Gamma^{\mu} D\_{\mu} \psi\_{A} + \frac{i}{2} E\_{ABCD} \bar{\psi}^{A} [Z^{C}, Z^{D}; \psi^{B}] - \frac{i}{2} E^{ABCD} \bar{Z}\_{D} [\bar{\psi}\_{A}, \psi\_{B}; \bar{Z}\_{C}] \\ &- i\bar{\psi}^{A} \left[ \psi\_{A}, Z^{B}; \bar{Z}\_{B} \right] + 2i\bar{\psi}^{A} \{ \psi\_{B}, Z^{B}; \bar{Z}\_{A} \} \Big) \end{split} \tag{75}$$
The Hermitian 3-algebra models of M-theory are classified into the models with *u*(*m*) ⊕ *u*(*n*) Hermitian 3-algebra (20) and *sp*(2*n*) ⊕ *u*(1) Hermitian 3-algebra (30). In the following, we study the *u*(*N*) ⊕ *u*(*N*) Hermitian 3-algebra model. By substituting the *u*(*N*) ⊕ *u*(*N*) Hermitian 3-algebra (20) to the action (73), we obtain
$$\begin{split} S &= \text{Tr}\left(-\frac{(2\pi)^{2}}{k^{2}}V - (Z^{A}A^{R}\_{\mu} - A^{L}\_{\mu}Z^{A})(Z^{A}A^{R\mu} - A^{L\mu}Z^{A})^{\dagger} - \frac{k}{2\pi}\frac{i}{3}E^{\mu\lambda}(A^{R}\_{\mu}A^{R}\_{\nu}A^{R}\_{\lambda} - A^{L}\_{\mu}A^{L}\_{\nu}A^{L}\_{\lambda})\right) \\ &- \bar{\psi}^{A}\Gamma^{\mu}(\psi\_{A}A^{R}\_{\mu} - A^{L}\_{\mu}\psi\_{A}) + \frac{2\pi}{k}\left(i\text{E}\_{ABCD}\bar{\psi}^{A}Z^{C}\psi^{\dagger B}Z^{D} - i\text{E}^{ABCD}Z^{\dagger}\_{\alpha}\bar{\psi}^{\dagger}{}\_{A}Z^{\dagger}\_{\zeta}\psi\_{B} \\ &- i\bar{\psi}^{A}\psi\_{A}Z^{\dagger}\_{B}Z^{B} + i\bar{\psi}^{A}Z^{B}Z^{\dagger}\_{B}\psi\_{A} + 2i\bar{\psi}^{A}\psi\_{B}Z^{\dagger}\_{A}Z^{B} - 2i\bar{\psi}^{A}Z^{B}Z^{\dagger}\_{A}\psi\_{B})\right) \end{split} \tag{76}$$
where *A<sup>R</sup> <sup>μ</sup>* ≡ − *<sup>k</sup>* <sup>2</sup>*<sup>π</sup> iAμ*¯ *baT*†¯ *bTa* and *A<sup>L</sup> <sup>μ</sup>* ≡ − *<sup>k</sup>* <sup>2</sup>*<sup>π</sup> iAμ*¯ *baTaT*†¯ *<sup>b</sup>* are *<sup>N</sup>* <sup>×</sup> *<sup>N</sup>* Hermitian matrices. In the algebra, we have set *α* = <sup>2</sup>*<sup>π</sup> <sup>k</sup>* , where *k* is an integer representing the Chern-Simons level. We choose *k* = 1 in order to obtain 16 dynamical supersymmetries. *V* is given by
$$\begin{split} V &= + \frac{1}{3} \mathbf{Z}\_A^\dagger \mathbf{Z}^A \mathbf{Z}\_B^\dagger \mathbf{Z}^B \mathbf{Z}\_C^\dagger \mathbf{Z}^C + \frac{1}{3} \mathbf{Z}^A \mathbf{Z}\_A^\dagger \mathbf{Z}^B \mathbf{Z}\_B^\dagger \mathbf{Z}^C \mathbf{Z}\_C^\dagger + \frac{4}{3} \mathbf{Z}\_A^\dagger \mathbf{Z}^B \mathbf{Z}\_C^\dagger \mathbf{Z}^A \mathbf{Z}\_B^\dagger \mathbf{Z}^C \\ &- \mathbf{Z}\_A^\dagger \mathbf{Z}^A \mathbf{Z}\_B^\dagger \mathbf{Z}^C \mathbf{Z}\_C^\dagger \mathbf{Z}^B - \mathbf{Z}^A \mathbf{Z}\_A^\dagger \mathbf{Z}^B \mathbf{Z}\_C^\dagger \mathbf{Z}^C \mathbf{Z}\_B^\dagger \end{split} \tag{77}$$
By redefining fields as
$$Z^A \rightarrow \left(\frac{k}{2\pi}\right)^{\frac{1}{3}} Z^A$$
$$A^\mu \rightarrow \left(\frac{2\pi}{k}\right)^{\frac{1}{3}} A^\mu$$
$$\psi^A \rightarrow \left(\frac{k}{2\pi}\right)^{\frac{1}{6}} \psi^A \tag{78}$$
we obtain an action that is independent of Chern-Simons level:
$$\begin{split} S &= \text{Tr}\Big(-V - (Z^A A^R\_\mu - A^L\_\mu Z^A)(Z^A A^{R\mu} - A^{L\mu} Z^A)^\dagger - \frac{i}{3} E^{\mu\nu\lambda} (A^R\_\mu A^R\_\nu A^R\_\lambda - A^L\_\mu A^L\_\nu A^L\_\lambda) \\ &- \bar{\psi}^A \Gamma^\mu (\psi\_A A^R\_\mu - A^L\_\mu \psi\_A) + i E\_{ABCD} \bar{\psi}^A Z^C \psi^{\dagger B} Z^D - i E^{ABCD} Z^\dagger\_D \bar{\psi}^\dagger{}\_A Z^\dagger\_C \psi\_B \\ &- i \bar{\psi}^A \psi\_A Z^\dagger\_B Z^B + i \bar{\psi}^A Z^B Z^\dagger\_B \psi\_A + 2i \bar{\psi}^A \psi\_B Z^\dagger\_A Z^B - 2i \bar{\psi}^A Z^B Z^\dagger\_A \psi\_B \Big) \end{split} \tag{79}$$
<sup>2</sup> The authors of [46–49] studied matrix models that can be obtained by a dimensional reduction of the ABJM and ABJ gauge theories on *S*3. They showed that the models reproduce the original gauge theories on *S*<sup>3</sup> in planar limits.
as opposed to three-dimensional Chern-Simons actions.
14 Will-be-set-by-IN-TECH
This matrix model can be obtained formally by a dimensional reduction of the N = 6 BLG
The Hermitian 3-algebra models of M-theory are classified into the models with *u*(*m*) ⊕ *u*(*n*) Hermitian 3-algebra (20) and *sp*(2*n*) ⊕ *u*(1) Hermitian 3-algebra (30). In the following, we study the *u*(*N*) ⊕ *u*(*N*) Hermitian 3-algebra model. By substituting the *u*(*N*) ⊕ *u*(*N*)
*<sup>μ</sup>ZA*)(*Z<sup>A</sup> <sup>A</sup>Rμ*−*ALμZA*)†<sup>−</sup> *<sup>k</sup>*
<sup>2</sup>*<sup>π</sup> iAμ*¯
*AZBZ*†
*k* 2*π*
2*π k*
*k* 2*π*
*<sup>μ</sup>ZA*)(*Z<sup>A</sup> <sup>A</sup>R<sup>μ</sup>* <sup>−</sup> *<sup>A</sup>LμZA*)† <sup>−</sup> *<sup>i</sup>*
*<sup>B</sup>ψ<sup>A</sup>* <sup>+</sup> <sup>2</sup>*iψ*¯ *<sup>A</sup>ψBZ*†
*μψA*) + *iEABCDψ*¯ *AZCψ*†*BZD* <sup>−</sup> *iEABCDZ*†
<sup>2</sup> The authors of [46–49] studied matrix models that can be obtained by a dimensional reduction of the ABJM and ABJ gauge theories on *S*3. They showed that the models reproduce the original gauge theories on *S*<sup>3</sup> in planar limits.
*AZBZ*†
1 3 *Z<sup>A</sup>*
1 3 *Aμ*
1 6
*CZCZ*†
*<sup>k</sup>* (*iEABCDψ*¯ *AZCψ*†*BZD* <sup>−</sup> *iEABCDZ*†
*baTaT*†¯
*<sup>A</sup>Z<sup>B</sup>* <sup>−</sup> <sup>2</sup>*iψ*¯ *AZBZ*†
*BZCZ*† *<sup>C</sup>* + 4 3 *Z*† *AZBZ*†
<sup>2</sup> *<sup>A</sup>μcb*¯ *∂νA<sup>λ</sup>* ¯*daT*¯ ¯*d*[*T<sup>a</sup>*
]
2*π i* 3
*<sup>k</sup>* , where *k* is an integer representing the Chern-Simons level.
3
*<sup>A</sup>Z<sup>B</sup>* <sup>−</sup> <sup>2</sup>*iψ*¯ *AZBZ*†
*Eμνλ*(*A<sup>R</sup>*
2
*Eμνλ*(*A<sup>R</sup>*
*<sup>A</sup>ψB*) *<sup>μ</sup> <sup>A</sup><sup>R</sup> <sup>ν</sup> <sup>A</sup><sup>R</sup> <sup>λ</sup>* <sup>−</sup> *<sup>A</sup><sup>L</sup> μA<sup>L</sup> <sup>ν</sup> <sup>A</sup><sup>L</sup> λ*)
*<sup>b</sup>* are *<sup>N</sup>* <sup>×</sup> *<sup>N</sup>* Hermitian matrices. In
*CZAZ*† *BZ<sup>C</sup>*
*<sup>B</sup>* (77)
*ψ<sup>A</sup>* (78)
*<sup>μ</sup> <sup>A</sup><sup>R</sup> <sup>ν</sup> <sup>A</sup><sup>R</sup> <sup>λ</sup>* <sup>−</sup> *<sup>A</sup><sup>L</sup> μA<sup>L</sup> <sup>ν</sup> <sup>A</sup><sup>L</sup> λ*)
*<sup>D</sup>* ¯*ψ*† *AZ*† *<sup>C</sup>ψ<sup>B</sup>*
*<sup>A</sup>ψ<sup>B</sup>*
*<sup>D</sup>* ¯*ψ*† *AZ*† *<sup>C</sup>ψ<sup>B</sup>*
; *<sup>T</sup>*¯ ¯*d*][*Tb*, *<sup>T</sup><sup>f</sup>* ; *<sup>T</sup>*¯*e*¯
*EABCDψ*¯ *<sup>A</sup>*[*ZC*, *<sup>Z</sup>D*; *<sup>ψ</sup>B*] <sup>−</sup> *<sup>i</sup>*
, *Tb*; *T*¯ *<sup>c</sup>*¯ ]
*EABCDZ*¯ *<sup>D</sup>*[*ψ*¯*A*, *ψB*; *Z*¯*C*]
(75)
(76)
(79)
<sup>−</sup>*<sup>V</sup>* <sup>−</sup> *<sup>D</sup>μZAD<sup>μ</sup>ZA* <sup>+</sup> *<sup>E</sup>μνλ* <sup>1</sup>
*f e*[*T<sup>a</sup>* , *T<sup>c</sup>*
> *i* 2
<sup>−</sup>*iψ*¯ *<sup>A</sup>*[*ψA*, *<sup>Z</sup>B*; *<sup>Z</sup>*¯ *<sup>B</sup>*] + <sup>2</sup>*iψ*¯ *<sup>A</sup>*[*ψB*, *<sup>Z</sup>B*; *<sup>Z</sup>*¯ *<sup>A</sup>*]
*<sup>B</sup>ψ<sup>A</sup>* <sup>+</sup> <sup>2</sup>*iψ*¯ *<sup>A</sup>ψBZ*†
*<sup>μ</sup>* ≡ − *<sup>k</sup>*
We choose *k* = 1 in order to obtain 16 dynamical supersymmetries. *V* is given by
1 3 *ZAZ*†
*<sup>Z</sup><sup>A</sup>* <sup>→</sup>
*<sup>A</sup><sup>μ</sup>* <sup>→</sup>
*<sup>ψ</sup><sup>A</sup>* <sup>→</sup>
*<sup>C</sup>Z<sup>C</sup>* <sup>+</sup>
*<sup>C</sup>Z<sup>B</sup>* <sup>−</sup> *<sup>Z</sup>AZ*†
*baA<sup>ν</sup>* ¯*dcA<sup>λ</sup>* ¯
<sup>−</sup>*iψ*¯ *<sup>A</sup>*Γ*μDμψ<sup>A</sup>* <sup>+</sup>
action [8], which is equivalent to ABJ(M) action [7, 45]2,
+ 1 <sup>3</sup> *<sup>A</sup>μ*¯
Hermitian 3-algebra (20) to the action (73), we obtain
*<sup>μ</sup>* <sup>−</sup>*A<sup>L</sup>*
*μψA*) + <sup>2</sup>*<sup>π</sup>*
*bTa* and *A<sup>L</sup>*
*BZBZ*†
we obtain an action that is independent of Chern-Simons level:
*<sup>μ</sup>* <sup>−</sup> *<sup>A</sup><sup>L</sup>*
*<sup>μ</sup>* <sup>−</sup> *<sup>A</sup><sup>L</sup>*
*<sup>B</sup>Z<sup>B</sup>* <sup>+</sup> *<sup>i</sup>ψ*¯ *AZBZ*†
*BZCZ*†
*<sup>k</sup>*<sup>2</sup> *<sup>V</sup>*−(*Z<sup>A</sup> <sup>A</sup><sup>R</sup>*
<sup>2</sup>*<sup>π</sup> iAμ*¯
<sup>−</sup>*Z*†
<sup>−</sup>*<sup>V</sup>* <sup>−</sup> (*Z<sup>A</sup> <sup>A</sup><sup>R</sup>*
<sup>−</sup>*ψ*¯ *<sup>A</sup>*Γ*μ*(*ψ<sup>A</sup> <sup>A</sup><sup>R</sup>*
<sup>−</sup>*iψ*¯ *<sup>A</sup>ψAZ*†
*<sup>μ</sup>* <sup>−</sup> *<sup>A</sup><sup>L</sup>*
*<sup>B</sup>Z<sup>B</sup>* <sup>+</sup> *<sup>i</sup>ψ*¯ *AZBZ*†
*baT*†¯
*AZAZ*†
<sup>−</sup>*ψ*¯ *<sup>A</sup>*Γ*μ*(*ψ<sup>A</sup> <sup>A</sup><sup>R</sup>*
*<sup>μ</sup>* ≡ − *<sup>k</sup>*
the algebra, we have set *α* = <sup>2</sup>*<sup>π</sup>*
*V* = + 1 3 *Z*† *AZAZ*†
By redefining fields as
*S* = Tr
<sup>−</sup>*iψ*¯ *<sup>A</sup>ψAZ*†
*<sup>S</sup>*<sup>N</sup> <sup>=</sup>6*BLG* =
*S* = Tr −(2*π*)<sup>2</sup>
where *A<sup>R</sup>*
*d*3*x* If we rewrite the gauge fields in the action as *A<sup>L</sup> <sup>μ</sup>* = *A<sup>μ</sup>* + *b<sup>μ</sup>* and *A<sup>R</sup> <sup>μ</sup>* = *A<sup>μ</sup>* − *bμ*, we obtain
$$\begin{split} S &= \text{Tr}\Big(-V + ([A\_{\mu}, Z^{A}] + \{b\_{\mu}, Z^{A}\})([A^{\mu}, Z\_{A}] - \{b^{\mu}, Z\_{A}\}) + iE^{\mu\nu}(\frac{2}{3}b\_{\mu}b\_{\nu}b\_{\lambda} + 2A\_{\mu}A\_{\nu}b\_{\lambda}) \\ &+ \bar{\psi}^{A}\Gamma^{\mu}([A\_{\mu}, \psi\_{A}] + \{b\_{\mu}, \psi\_{A}\}) + iE\_{ABCD}\bar{\psi}^{A}Z^{C}\psi^{\dagger B}Z^{D} - iE^{\text{ABCD}}Z\_{D}^{\dagger}\bar{\psi}^{\dagger}{}\_{A}Z\_{C}^{\dagger}\psi\_{B} \\ &- i\bar{\psi}^{A}\psi\_{A}Z\_{B}^{\dagger}Z^{B} + i\bar{\psi}^{A}Z^{B}Z\_{B}^{\dagger}\psi\_{A} + 2i\bar{\psi}^{A}\psi\_{B}Z\_{A}^{\dagger}Z^{B} - 2i\bar{\psi}^{A}Z^{B}Z\_{A}^{\dagger}\psi\_{B} \end{split} \tag{80}$$
where [ , ] and { , } are the ordinary commutator and anticommutator, respectively. The *u*(1) parts of *A<sup>μ</sup>* decouple because *A<sup>μ</sup>* appear only in commutators in the action. *b<sup>μ</sup>* can be regarded as auxiliary fields, and thus *A<sup>μ</sup>* correspond to matrices *X<sup>μ</sup>* that represents three space-time coordinates in M-theory. Among *<sup>N</sup>* <sup>×</sup> *<sup>N</sup>* arbitrary complex matrices *<sup>Z</sup>A*, we need to identify matrices *<sup>X</sup><sup>I</sup>* (*<sup>I</sup>* <sup>=</sup> 3, ··· 10) representing the other space coordinates in M-theory, because the model possesses not *SO*(8) but *SU*(4) × *U*(1) symmetry. Our identification is
$$\begin{aligned} \mathbf{Z}^A &= \mathbf{i}X^{A+2} - \mathbf{X}^{A+6}, \\ \mathbf{X}^I &= \hat{\mathbf{X}}^I - \mathbf{i}\mathbf{x}^I \mathbf{1} \end{aligned} \tag{81}$$
where *X*ˆ *<sup>I</sup>* and *x<sup>I</sup>* are *su*(*N*) Hermitian matrices and real scalars, respectively. This is analogous to the identification when we compactify ABJM action, which describes N M2 branes, and obtain the action of N D2 branes [7, 50, 51]. We will see that this identification works also in our case. We should note that while the *su*(*N*) part is Hermitian, the *u*(1) part is anti-Hermitian. That is, an eigen-value distribution of *Xμ*, *ZA*, and not *X<sup>I</sup>* determine the spacetime in the Hermitian model. In order to define light-cone coordinates, we need to perform Wick rotation: *<sup>a</sup>*<sup>0</sup> → −*ia*0. After the Wick rotation, we obtain
$$A^0 = \hat{A}^0 - ia^0 \mathbf{1} \tag{82}$$
where *A*ˆ0 is a *su*(*N*) Hermitian matrix.
#### **3.4. DLCQ Limit of 3-algebra model of M-theory**
It was shown that M-theory in a DLCQ limit reduces to the BFSS matrix theory with matrices of finite size [30–35]. This fact is a strong criterion for a model of M-theory. In [26, 28], it was shown that the Lie and Hermitian 3-algebra models of M-theory reduce to the BFSS matrix theory with matrices of finite size in the DLCQ limit. In this subsection, we show an outline of the mechanism.
DLCQ limit of M-theory consists of a light-cone compactification, *x*<sup>−</sup> ≈ *x*<sup>−</sup> + 2*πR*, where *x*± = √ 1 2 (*x*<sup>10</sup> <sup>±</sup> *<sup>x</sup>*0), and Lorentz boost in *<sup>x</sup>*<sup>10</sup> direction with an infinite momentum. After appropriate scalings of fields [26, 28], we define light-cone coordinate matrices as
$$X^0 = \frac{1}{\sqrt{2}}(X^+ - X^-)$$
$$X^{10} = \frac{1}{\sqrt{2}}(X^+ + X^-) \tag{83}$$
We integrate out *b<sup>μ</sup>* by using their equations of motion.
#### 16 Will-be-set-by-IN-TECH 16 Linear Algebra – Theorems and Applications
A matrix compactification [52] on a circle with a radius R imposes the following conditions on *X*− and the other matrices *Y*:
$$\begin{cases} X^- - (2\pi R)1 = \mathcal{U}^\dagger X^- \mathcal{U} \\ Y = \mathcal{U}^\dagger Y \mathcal{U} \end{cases} \tag{84}$$
where *U* is a unitary matrix. In order to obtain a solution to (84), we need to take *N* → ∞ and consider matrices of infinite size [52]. A solution to (84) is given by *X*<sup>−</sup> = *X*¯ <sup>−</sup> + *X*˜ <sup>−</sup>, *Y* = *Y*˜ and
$$\mathcal{U} = \begin{pmatrix} \ddots \ \ddots \ \ddots \\ & 0 \ 1 & 0 \\ & & 0 \ 1 \\ & & & 0 \ 1 \\ & & & & 0 \ \ddots \\ & & & & & \ddots \end{pmatrix} \otimes \mathbf{1}\_{n \times n} \in \mathcal{U}(N) \tag{85}$$
Backgrounds *X*¯ <sup>−</sup> are
$$\mathcal{X}^- = -T^3 \mathfrak{x}\_0^- T^0 - (2\pi \mathbb{R}) \text{diag}(\cdots, s-1, s, s+1, \cdots) \otimes \mathbf{1}\_{\mathbb{R} \times \mathbb{R}} \tag{86}$$
in the Lie 3-algebra case, whereas
$$\bar{X}^- = -i(T^3\bar{x}^-)\mathbf{1} - i(2\pi R)\text{diag}(\cdot\cdot\cdot, s-1, s, s+1, \cdot\cdot) \otimes \mathbf{1}\_{\mathbb{R}\times\mathbb{N}}\tag{87}$$
in the Hermitian 3-algebra case. A fluctuation *x*˜ that represents *u*(*N*) parts of *X*˜ <sup>−</sup> and *Y*˜ is
$$
\begin{pmatrix}
\ddots & \ddots & \ddots & & & & & \\
\ddots & \check{\mathbf{x}}(0) & \check{\mathbf{x}}(1) & \check{\mathbf{x}}(2) & & & \ddots & \\
\vdots & \ddots & \check{\mathbf{x}}(0) & \check{\mathbf{x}}(1) & \check{\mathbf{x}}(2) & & \\
& \ddots & \check{\mathbf{x}}(-1) & \check{\mathbf{x}}(0) & \check{\mathbf{x}}(1) & \check{\mathbf{x}}(2) & \\
& & \check{\mathbf{x}}(-2) & \check{\mathbf{x}}(-1) & \check{\mathbf{x}}(0) & \check{\mathbf{x}}(1) & \check{\mathbf{x}}(2) & \\
& & & \check{\mathbf{x}}(-2) & \check{\mathbf{x}}(-1) & \check{\mathbf{x}}(0) & \check{\mathbf{x}}(1) & \check{\mathbf{x}}(2) \\
& & & & & \ddots & \\
& & & & & \check{\mathbf{x}}(-2) & \check{\mathbf{x}}(-1) & \check{\mathbf{x}}(0) & \overset{\cdot}{\cdot} \\
& & & & & \ddots & \ddots & \ddots \\
& & & & & & \ddots & \ddots & \ddots
\end{pmatrix}
\tag{88}
$$
Each *x*˜(*s*) is a *n* × *n* matrix, where *s* is an integer. That is, the (s, t)-th block is given by *x*˜*s*,*<sup>t</sup>* = *x*˜(*s* − *t*).
We make a Fourier transformation,
$$\mathfrak{X}(\mathbf{s}) = \frac{1}{2\pi\tilde{\mathbf{R}}} \int\_0^{2\pi\tilde{\mathbf{R}}} d\tau \mathbf{x}(\tau) e^{i\mathbf{s}\cdot\frac{\mathbf{r}}{R}} \tag{89}$$
where *<sup>x</sup>*(*τ*) is a *<sup>n</sup>* <sup>×</sup> *<sup>n</sup>* matrix in one-dimension and *RR*˜ <sup>=</sup> <sup>2</sup>*π*. From (86)-(89), the following identities hold:
$$\sum\_{t} \ddot{\mathbf{x}}\_{s,t} \ddot{\mathbf{x}}^{\prime}\_{t,\mu} = \frac{1}{2\pi\bar{\mathcal{R}}} \int\_{0}^{2\pi\bar{\mathcal{R}}} d\tau \, \mathbf{x}(\tau) \mathbf{x}^{\prime}(\tau) e^{i(s-u)\frac{\tau}{\bar{\mathcal{R}}}}$$
$$\text{tr}(\sum\_{s,t} \ddot{\mathbf{x}}\_{s,t} \ddot{\mathbf{x}}^{\prime}\_{t,s}) = V \frac{1}{2\pi\bar{\mathcal{R}}} \int\_{0}^{2\pi\bar{\mathcal{R}}} d\tau \, \text{tr}(\mathbf{x}(\tau) \mathbf{x}^{\prime}(\tau))$$
$$[\ddot{\mathbf{x}}^{\prime}, \ddot{\mathbf{x}}]\_{s,t} = \frac{1}{2\pi\bar{\mathcal{R}}} \int\_{0}^{2\pi\bar{\mathcal{R}}} d\tau \, \partial\_{\tau} \mathbf{x}(\tau) e^{i(s-t)\frac{\tau}{\bar{\mathcal{R}}}}\tag{90}$$
where tr is a trace over *n* × *n* matrices and *V* = ∑*<sup>s</sup>* 1. Next, we boost the system in *x*<sup>10</sup> direction:
$$
\tilde{X}^{\prime +} = \frac{1}{T} \tilde{X}^{+}
$$
$$
\tilde{X}^{\prime -} = T \tilde{X}^{-} \tag{91}
$$
The DLCQ limit is achieved when *T* → ∞, where the "novel Higgs mechanism" [51] is realized. In *T* → ∞, the actions of the 3-algebra models of M-theory reduce to that of the BFSS matrix theory [27] with matrices of finite size,
$$S = \frac{1}{\mathcal{S}^2} \int\_{-\infty}^{\infty} d\tau \text{tr}(\frac{1}{2} (D\_0 \mathbf{x}^P)^2 - \frac{1}{4} [\mathbf{x}^P, \mathbf{x}^Q]^2 + \frac{1}{2} \bar{\psi} \Gamma^0 D\_0 \psi - \frac{i}{2} \bar{\psi} \Gamma^P [\mathbf{x}\_P, \psi]) \tag{92}$$
where *P*, *Q* = 1, 2, ··· , 9.
#### **3.5. Supersymmetric deformation of Lie 3-algebra model of M-theory**
A supersymmetric deformation of the Lie 3-algebra Model of M-theory was studied in [53] (see also [54–56]). If we add mass terms and a flux term,
$$S\_{\mathfrak{M}} = \left\langle -\frac{1}{2}\mu^2 (\mathbf{X}^I)^2 - \frac{i}{2}\mu \bar{\Psi} \Gamma\_{3456} \Psi + H\_{IJKL} [\mathbf{X}^I, \mathbf{X}^I, \mathbf{X}^K] \mathbf{X}^L \right\rangle \tag{93}$$
such that
16 Will-be-set-by-IN-TECH
A matrix compactification [52] on a circle with a radius R imposes the following conditions on
*<sup>X</sup>*<sup>−</sup> <sup>−</sup> (2*πR*)<sup>1</sup> <sup>=</sup> *<sup>U</sup>*†*X*−*<sup>U</sup>*
where *U* is a unitary matrix. In order to obtain a solution to (84), we need to take *N* → ∞ and consider matrices of infinite size [52]. A solution to (84) is given by *X*<sup>−</sup> = *X*¯ <sup>−</sup> + *X*˜ <sup>−</sup>, *Y* = *Y*˜
⎞
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
...
in the Hermitian 3-algebra case. A fluctuation *x*˜ that represents *u*(*N*) parts of *X*˜ <sup>−</sup> and *Y*˜ is
... *<sup>x</sup>*˜(0) *<sup>x</sup>*˜(1) *<sup>x</sup>*˜(2) ...
*x*˜(−2) *x*˜(−1) *x*˜(0) *x*˜(1) *x*˜(2)
*x*˜(−2) *x*˜(−1) *x*˜(0) *x*˜(1) *x*˜(2)
... *<sup>x</sup>*˜(−2) *<sup>x</sup>*˜(−1) *<sup>x</sup>*˜(0) ...
Each *x*˜(*s*) is a *n* × *n* matrix, where *s* is an integer. That is, the (s, t)-th block is given by
� <sup>2</sup>*πR*˜ 0
*<sup>x</sup>*˜(−2) *<sup>x</sup>*˜(−1) *<sup>x</sup>*˜(0) *<sup>x</sup>*˜(1) ...
*dτx*(*τ*)*e*
*is <sup>τ</sup>*
... ... ...
... *<sup>x</sup>*˜(−1) *<sup>x</sup>*˜(0) *<sup>x</sup>*˜(1) *<sup>x</sup>*˜(2)
*<sup>x</sup>*˜(*s*) = <sup>1</sup>
2*πR*˜
*Y* = *U*†*YU* (84)
<sup>0</sup> *<sup>T</sup>*<sup>0</sup> <sup>−</sup> (2*πR*)diag(··· ,*<sup>s</sup>* <sup>−</sup> 1,*s*,*<sup>s</sup>* <sup>+</sup> 1, ···) <sup>⊗</sup> **<sup>1</sup>***n*×*<sup>n</sup>* (86)
<sup>−</sup>)**1** − *i*(2*πR*)diag(··· ,*s* − 1,*s*,*s* + 1, ···) ⊗ **1***n*×*<sup>n</sup>* (87)
⊗ 1*n*×*<sup>n</sup>* ∈ *U*(*N*) (85)
⎞
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
*<sup>R</sup>*˜ (89)
(88)
*X*− and the other matrices *Y*:
Backgrounds *X*¯ <sup>−</sup> are
*x*˜*s*,*<sup>t</sup>* = *x*˜(*s* − *t*).
We make a Fourier transformation,
*U* =
−
... ... ...
*<sup>X</sup>*¯ <sup>−</sup> <sup>=</sup> <sup>−</sup>*T*3*x*¯
*<sup>X</sup>*¯ <sup>−</sup> <sup>=</sup> <sup>−</sup>*i*(*T*3*x*¯
⎛
⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
in the Lie 3-algebra case, whereas
⎛
... ...
⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
and
$$\mathbf{H}\_{\rm IJKL} = \begin{cases} -\frac{\mu}{6} \mathbf{e}\_{\rm IJKL} \begin{pmatrix} \mathbf{I}\_{\prime} \mathbf{J}\_{\prime} \mathbf{K}\_{\prime} \mathbf{L} = \mathbf{3}, 4, 5, 6 \text{ or } \mathbf{7}, 8, 9, 10 \end{pmatrix} \\ \text{(} \mathbf{0}^{\prime} \text{)} \text{ (} \mathbf{0}^{\prime} \text{)} \text{ } \end{cases} \tag{94}$$
to the action (53), the total action *S*<sup>0</sup> + *Sm* is invariant under dynamical 16 supersymmetries,
$$\begin{aligned} \delta X^{I} &= i\varepsilon \Gamma^{I} \Psi\\ \delta A\_{\mu ab}[T^{a}, T^{b}, \quad ] &= i\varepsilon \Gamma\_{\mu} \Gamma\_{I}[X^{I}, \Psi, \quad ]\\ \delta \Psi &= -\frac{1}{6}[X^{I}, X^{J}, X^{K}] \Gamma\_{IJK}\varepsilon - A\_{\mu ab}[T^{a}, T^{b}, X^{I}] \Gamma^{\mu} \Gamma\_{I} \varepsilon + \mu \Gamma\_{3456} X^{I} \Gamma\_{I} \varepsilon \end{aligned} \tag{95}$$
From this action, we obtain various interesting solutions, including fuzzy sphere solutions [53].
18 Will-be-set-by-IN-TECH 18 Linear Algebra – Theorems and Applications
## **4. Conclusion**
The metric Hermitian 3-algebra corresponds to a class of the super Lie algebra. By using this relation, the metric Hermitian 3-algebras are classified into *u*(*m*) ⊕ *u*(*n*) and *sp*(2*n*) ⊕ *u*(1) Hermitian 3-algebras.
The Lie and Hermitian 3-algebra models of M-theory are obtained by second quantizations of the supermembrane action in a semi-light-cone gauge. The Lie 3-algebra model possesses manifest N = 1 supersymmetry in eleven dimensions. In the DLCQ limit, both the models reduce to the BFSS matrix theory with matrices of finite size as they should.
## **Acknowledgements**
We would like to thank T. Asakawa, K. Hashimoto, N. Kamiya, H. Kunitomo, T. Matsuo, S. Moriyama, K. Murakami, J. Nishimura, S. Sasa, F. Sugino, T. Tada, S. Terashima, S. Watamura, K. Yoshida, and especially H. Kawai and A. Tsuchiya for valuable discussions.
## **Author details**
Matsuo Sato *Hirosaki University, Japan*
### **5. References**
18 Will-be-set-by-IN-TECH
The metric Hermitian 3-algebra corresponds to a class of the super Lie algebra. By using this relation, the metric Hermitian 3-algebras are classified into *u*(*m*) ⊕ *u*(*n*) and *sp*(2*n*) ⊕ *u*(1)
The Lie and Hermitian 3-algebra models of M-theory are obtained by second quantizations of the supermembrane action in a semi-light-cone gauge. The Lie 3-algebra model possesses manifest N = 1 supersymmetry in eleven dimensions. In the DLCQ limit, both the models
We would like to thank T. Asakawa, K. Hashimoto, N. Kamiya, H. Kunitomo, T. Matsuo, S. Moriyama, K. Murakami, J. Nishimura, S. Sasa, F. Sugino, T. Tada, S. Terashima, S. Watamura,
[2] N. Kamiya, A structure theory of Freudenthal-Kantor triple systems, J. Algebra 110 (1987)
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[10] G. Papadopoulos, M2-branes, 3-Lie Algebras and Plucker relations, JHEP 0805 (2008)
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Theories and M2-branes on Orbifolds, JHEP 0809 (2008) 002.
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reduce to the BFSS matrix theory with matrices of finite size as they should.
K. Yoshida, and especially H. Kawai and A. Tsuchiya for valuable discussions.
[1] V. T. Filippov, n-Lie algebras, Sib. Mat. Zh. 26, No. 6, (1985) 126140.
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**4. Conclusion**
Hermitian 3-algebras.
**Acknowledgements**
**Author details**
**5. References**
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054.
*Hirosaki University, Japan*
Matsuo Sato
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## **Algebraic Theory of Appell Polynomials with Application to General Linear Interpolation Problem**
Francesco Aldo Costabile and Elisabetta Longo
Additional information is available at the end of the chapter
http://dx.doi.org/10.5772/46482
## **1. Introduction**
20 Will-be-set-by-IN-TECH
[43] H. Nishino, S. Rajpoot, Triality and Bagger-Lambert Theory, Phys. Lett. B671 (2009) 415. [44] A. Gustavsson, S-J. Rey, Enhanced N=8 Supersymmetry of ABJM Theory on R(8) and
[47] G. Ishiki, S. Shimasaki, A. Tsuchiya, Large N reduction for Chern-Simons theory on *S*3,
[48] H. Kawai, S. Shimasaki, A. Tsuchiya, Large N reduction on group manifolds,
[49] G. Ishiki, S. Shimasaki, A. Tsuchiya, A Novel Large-N Reduction on *S*3: Demonstration
[53] J. DeBellis, C. Saemann, R. J. Szabo, Quantized Nambu-Poisson Manifolds in a 3-Lie
[54] M. M. Sheikh-Jabbari, Tiny Graviton Matrix Theory: DLCQ of IIB Plane-Wave String
[55] J. Gomis, A. J. Salim, F. Passerini, Matrix Theory of Type IIB Plane Wave from
[56] K. Hosomichi, K. Lee, S. Lee, Mass-Deformed Bagger-Lambert Theory and its BPS
[45] O. Aharony, O. Bergman, D. L. Jafferis, Fractional M2-branes, JHEP 0811 (2008) 043. [46] M. Hanada, L. Mannelli, Y. Matsuo, Large-N reduced models of supersymmetric quiver,
Chern-Simons gauge theories and ABJM, arXiv:0907.4937 [hep-th].
[50] Y. Pang, T. Wang, From N M2's to N D2's, Phys. Rev. D78 (2008) 125007.
[52] W. Taylor, D-brane field theory on compact spaces, Phys. Lett. B394 (1997) 283.
in Chern-Simons Theory, arXiv:1001.4917 [hep-th].
Algebra Reduced Model, JHEP 1104 (2011) 075.
Theory, A Conjecture , JHEP 0409 (2004) 017.
Membranes, JHEP 0808 (2008) 002.
Objects, Phys.Rev. D78 (2008) 066015.
[51] S. Mukhi, C. Papageorgakis, M2 to D2, JHEP 0805 (2008) 085.
R(8)/Z(2), arXiv:0906.3568 [hep-th].
Phys. Rev. D80 (2009) 086004.
arXiv:0912.1456 [hep-th].
20 Linear Algebra – Theorems and Applications
In 1880 P. E. Appell ([1]) introduced and widely studied sequences of *n*-degree polynomials
$$A\_n\left(\mathbf{x}\right), n = 0, 1, \ldots \tag{1}$$
satisfying the differential relation
$$DA\_{\mathfrak{n}}\left(\mathbf{x}\right) = \mathfrak{n}A\_{\mathfrak{n}-1}(\mathbf{x}), \mathfrak{n} = 1, 2, \dots \tag{2}$$
Sequences of polynomials, verifying the (2), nowadays called Appell polynomials, have been well studied because of their remarkable applications not only in different branches of mathematics ([2, 3]) but also in theoretical physics and chemistry ([4, 5]). In 1936 an initial bibliography was provided by Davis ([6, p. 25]). In 1939 Sheffer ([7]) introduced a new class of polynomials which extends the class of Appell polynomials; he called these polynomials of type zero, but nowadays they are called Sheffer polynomials. Sheffer also noticed the similarities between Appell polynomials and the umbral calculus, introduced in the second half of the 19th century with the work of such mathematicians as Sylvester, Cayley and Blissard (for examples, see [8]). The Sheffer theory is mainly based on formal power series. In 1941 Steffensen ([9]) published a theory on Sheffer polynomials based on formal power series too. However, these theories were not suitable as they did not provide sufficient computational tools. Afterwards Mullin, Roman and Rota ([10–12]), using operators method, gave a beautiful theory of umbral calculus, including Sheffer polynomials. Recently, Di Bucchianico and Loeb ([13]) summarized and documented more than five hundred old and new findings related to Appell polynomial sequences. In last years attention has centered on finding a novel representation of Appell polynomials. For instance, Lehemer ([14]) illustrated six different approaches to representing the sequence of Bernoulli polynomials, which is a
©2012 Costabile and Longo, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ©2012 Costabile and Longo, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
#### 2 Will-be-set-by-IN-TECH 22 Linear Algebra – Theorems and Applications
special case of Appell polynomial sequences. Costabile ([15, 16]) also gave a new form of Bernoulli polynomials, called determinantal form, and later these ideas have been extended to Appell polynomial sequences. In fact, in 2010, Costabile and Longo ([17]) proposed an algebraic and elementary approach to Appell polynomial sequences. At the same time, Yang and Youn ([18]) also gave an algebraic approach, but with different methods. The approach to Appell polynomial sequences via linear algebra is an easily comprehensible mathematical tool, specially for non-specialists; that is very good because many polynomials arise in physics, chemistry and engineering. The present work concerns with these topics and it is organized as follows: in Section 2 we mention the Appell method ([1]); in Section 3 we provide the determinantal approach ([17]) and prove the equivalence with other definitions; in Section 4 classical and non-classical examples are given; in Section 5, by using elementary tools of linear algebra, general properties of Appell polynomials are provided; in Section 6 we mention Appell polynomials of second kind ([19, 20]) and, in Section 7 two classical examples are given; in Section 8 we provide an application to general linear interpolation problem([21]), giving, in Section 9, some examples; in Section 10 the Yang and Youn approach ([18]) is sketched; finally, in Section 11 conclusions close the work.
### **2. The Appell approach**
Let {*An*(*x*)}*<sup>n</sup>* be a sequence of *n*-degree polynomials satisfying the differential relation (2). Then we have
**Remark 1.** *There is a one-to-one correspondence of the set of such sequences* {*An*(*x*)}*<sup>n</sup> and the set of numerical sequences* {*αn*}*<sup>n</sup>* , *α*<sup>0</sup> � 0 *given by the explicit representation*
$$A\_n(\mathbf{x}) = a\_n + \binom{n}{1} a\_{n-1} \mathbf{x} + \binom{n}{2} a\_{n-2} \mathbf{x}^2 + \dots + a\_0 \mathbf{x}^n, \ n = 0, 1, \dots \tag{3}$$
Equation (3), in particular, shows explicitly that for each *n* ≥ 1 the polynomial *An* (*x*) is completely determined by *An*−<sup>1</sup> (*x*) and by the choice of the constant of integration *αn*.
**Remark 2.** *Given the formal power series*
$$a'(h) = a\_0 + \frac{h}{1!}a\_1 + \frac{h^2}{2!}a\_2 + \dots + \frac{h^n}{n!}a\_n + \dotsb, \quad a\_0 \neq 0,\tag{4}$$
*with α<sup>i</sup> i* = 0, 1, ... *real coefficients, the sequence of polynomials, An*(*x*)*, determined by the power series expansion of the product a* (*h*)*ehx, i.e.*
$$a\left(h\right)e^{h\mathbf{x}} = A\_0\left(\mathbf{x}\right) + \frac{h}{1!}A\_1\left(\mathbf{x}\right) + \frac{h^2}{2!}A\_2\left(\mathbf{x}\right) + \dots + \frac{h^n}{n!}A\_n\left(\mathbf{x}\right) + \dots \tag{5}$$
*satisfies (2).*
The function *a* (*h*) is said, by Appell, 'generating function' of the sequence {*An*(*x*)}*n*.
Appell also noticed various examples of sequences of polynomials verifying (2).
He also considered ([1]) an application of these polynomial sequences to linear differential equations, which is out of this context.
#### **3. The determinantal approach**
Let be *β<sup>i</sup>* ∈ **R**, *i* = 0, 1, ..., with *β*<sup>0</sup> �= 0.
We give the following
2 Will-be-set-by-IN-TECH
special case of Appell polynomial sequences. Costabile ([15, 16]) also gave a new form of Bernoulli polynomials, called determinantal form, and later these ideas have been extended to Appell polynomial sequences. In fact, in 2010, Costabile and Longo ([17]) proposed an algebraic and elementary approach to Appell polynomial sequences. At the same time, Yang and Youn ([18]) also gave an algebraic approach, but with different methods. The approach to Appell polynomial sequences via linear algebra is an easily comprehensible mathematical tool, specially for non-specialists; that is very good because many polynomials arise in physics, chemistry and engineering. The present work concerns with these topics and it is organized as follows: in Section 2 we mention the Appell method ([1]); in Section 3 we provide the determinantal approach ([17]) and prove the equivalence with other definitions; in Section 4 classical and non-classical examples are given; in Section 5, by using elementary tools of linear algebra, general properties of Appell polynomials are provided; in Section 6 we mention Appell polynomials of second kind ([19, 20]) and, in Section 7 two classical examples are given; in Section 8 we provide an application to general linear interpolation problem([21]), giving, in Section 9, some examples; in Section 10 the Yang and Youn approach ([18]) is sketched; finally,
Let {*An*(*x*)}*<sup>n</sup>* be a sequence of *n*-degree polynomials satisfying the differential relation (2).
**Remark 1.** *There is a one-to-one correspondence of the set of such sequences* {*An*(*x*)}*<sup>n</sup> and the set of*
Equation (3), in particular, shows explicitly that for each *n* ≥ 1 the polynomial *An* (*x*) is completely determined by *An*−<sup>1</sup> (*x*) and by the choice of the constant of integration *αn*.
2! *<sup>α</sup>*<sup>2</sup> <sup>+</sup> ··· <sup>+</sup>
*with α<sup>i</sup> i* = 0, 1, ... *real coefficients, the sequence of polynomials, An*(*x*)*, determined by the power series*
*h*2
He also considered ([1]) an application of these polynomial sequences to linear differential
*hn n*!
2! *<sup>A</sup>*<sup>2</sup> (*x*) <sup>+</sup> ··· <sup>+</sup>
*hn n*!
*<sup>α</sup>n*−2*x*<sup>2</sup> <sup>+</sup> ··· <sup>+</sup> *<sup>α</sup>*0*xn*, *<sup>n</sup>* <sup>=</sup> 0, 1, ... (3)
*α<sup>n</sup>* + ··· , *α*<sup>0</sup> � 0, (4)
*An* (*x*) + ··· , (5)
*n* 2
*numerical sequences* {*αn*}*<sup>n</sup>* , *α*<sup>0</sup> � 0 *given by the explicit representation*
*h* 1! *<sup>α</sup>*<sup>1</sup> <sup>+</sup>
*h*
*<sup>α</sup>n*−1*<sup>x</sup>* +
*h*2
1! *<sup>A</sup>*<sup>1</sup> (*x*) <sup>+</sup>
The function *a* (*h*) is said, by Appell, 'generating function' of the sequence {*An*(*x*)}*n*. Appell also noticed various examples of sequences of polynomials verifying (2).
*n* 1
in Section 11 conclusions close the work.
*An* (*x*) = *α<sup>n</sup>* +
**Remark 2.** *Given the formal power series*
*expansion of the product a* (*h*)*ehx, i.e.*
equations, which is out of this context.
*a* (*h*) = *α*<sup>0</sup> +
*a* (*h*)*ehx* = *A*<sup>0</sup> (*x*) +
**2. The Appell approach**
Then we have
*satisfies (2).*
**Definition 1.** *The polynomial sequence defined by*
$$\begin{cases} A\_{0}\left(\mathbf{x}\right) = \frac{1}{\beta\_{0}},\\ \begin{bmatrix} 1 & \mathbf{x} & \mathbf{x}^{2} & \cdots & \cdots & \mathbf{x}^{n-1} & \mathbf{x}^{n} \\ \beta\_{0}\left\beta\_{1} & \beta\_{2} & \cdots & \cdots & \beta\_{n-1} & \beta\_{n} \\ 0 & \beta\_{0}\left(\frac{1}{2}\right)\beta\_{1} & \cdots & \cdots & \binom{n-1}{1}\beta\_{n-2}\left(\frac{n}{2}\right)\beta\_{n-1} \\ 0 & 0 & \beta\_{0} & \cdots & \cdots & \binom{n-1}{2}\beta\_{n-3}\left(\frac{n}{2}\right)\beta\_{n-2} \\ \vdots & \ddots & \vdots & \vdots \\ \vdots & & \ddots & \vdots & \vdots \\ 0 & \cdots & \cdots & \cdots & 0 & \beta\_{0} & \binom{n}{n-1}\beta\_{1} \end{bmatrix}, n = 1, 2, \ldots \end{cases} (6)$$
*is called Appell polynomial sequence for βi*.
Then we have
**Theorem 1.** *If An* (*x*) *is the Appell polynomial sequence for β<sup>i</sup> the differential relation (2) holds.*
*Proof.* Using the properties of linearity we can differentiate the determinant (6), expand the resulting determinant with respect to the first column and recognize the factor *An*−<sup>1</sup> (*x*) after multiplication of the *<sup>i</sup>*-th row by *<sup>i</sup>* <sup>−</sup> 1, *<sup>i</sup>* <sup>=</sup> 2, ..., *<sup>n</sup>* and *<sup>j</sup>*-th column by <sup>1</sup> *<sup>j</sup>* , *j* = 1, ..., *n*.
**Theorem 2.** *If An* (*x*) *is the Appell polynomial sequence for β<sup>i</sup> we have the equality (3) with*
$$\begin{aligned} \mathbf{a}\_{0} &= \frac{1}{\beta\_{0}}, \end{aligned} \tag{7}$$
$$\mathbf{a}\_{i} = \frac{1}{(\beta\_{0})^{i+1}} \begin{vmatrix} \beta\_{1} & \beta\_{2} & \cdots & \cdots & \beta\_{i-1} & \beta\_{i} \\ & \beta\_{0} \binom{i}{2}\beta\_{1} & \cdots & \cdots & \binom{i-1}{2}\beta\_{i-2} & \binom{i}{1}\beta\_{i-1} \\ 0 & \beta\_{0} & \cdots & \cdots & \binom{i-1}{2}\beta\_{i-3} & \binom{i}{2}\beta\_{i-2} \\ \vdots & & \ddots & & \vdots & \vdots \\ \vdots & & \ddots & & \vdots \\ 0 & \cdots & \cdots & 0 & \beta\_{0} & \binom{i}{i-1}\beta\_{1} \end{vmatrix} = $$
$$= -\frac{1}{\beta\_{0}} \sum\_{k=0}^{i-1} \binom{i}{k} \beta\_{i-k} \mathbf{a}\_{k'} \qquad i = 1, 2, \dots, n. \tag{8}$$
#### 4 Will-be-set-by-IN-TECH 24 Linear Algebra – Theorems and Applications
*Proof.* From (6), by expanding the determinant *An* (*x*) with respect to the first row, we obtain the (3) with *α<sup>i</sup>* given by (7) and the determinantal form in (8); this is a determinant of an upper Hessenberg matrix of order *<sup>i</sup>* ([16]), then setting *<sup>α</sup><sup>i</sup>* = (−1)*<sup>i</sup>* (*β*0) *<sup>i</sup>*+<sup>1</sup> *<sup>α</sup><sup>i</sup>* for *<sup>i</sup>* <sup>=</sup> 1, 2, ..., *<sup>n</sup>*, we have
$$
\overline{\mathfrak{a}}\_{i} = \sum\_{k=0}^{i-1} (-1)^{i-k-1} h\_{k+1,i} q\_k \left( i \right) \overline{\mathfrak{a}}\_{k'} \tag{9}
$$
where:
$$h\_{l,m} = \begin{cases} \beta\_m & \text{for } l = 1, \\ \binom{m}{l-1} \beta\_{m-l+1} & \text{for } 1 < l \le m+1, \\ 0 & \text{for } l > m+1, \end{cases} \quad l, m = 1, 2, \dots, \mathbf{i}, \tag{10}$$
$$q\_k\left(i\right) = \prod\_{j=k+2}^{l} h\_{j,j-1} = \left(\beta\_0\right)^{i-k-1}, \quad k = 0, 1, \ldots, i-2,\tag{11}$$
$$q\_{i-1}\left(i\right) = 1.\tag{12}$$
By virtue of the previous setting, (9) implies
$$\begin{aligned} \overline{\mathfrak{a}}\_{i} &= \sum\_{k=0}^{i-2} (-1)^{i-k-1} \binom{i}{k} \beta\_{i-k} \left(\beta\_{0}\right)^{i-k-1} \overline{\mathfrak{a}}\_{k} + \binom{i}{i-1} \beta\_{1} \overline{\mathfrak{a}}\_{i-1} = \\ &= (-1)^{i} \left(\beta\_{0}\right)^{i+1} \left(-\frac{1}{\beta\_{0}} \sum\_{k=0}^{i-1} \binom{i}{k} \beta\_{i-k} \mathfrak{a}\_{k}\right). \end{aligned}$$
and the proof is concluded.
**Remark 3.** *We note that (7) and (8) are equivalent to*
$$\sum\_{k=0}^{i} \binom{i}{k} \beta\_{i-k} \alpha\_k = \begin{cases} 1 & i = 0 \\ 0 & i > 0 \end{cases} \tag{13}$$
*and that for each sequence of Appell polynomials there exist two sequences of numbers α<sup>i</sup> and β<sup>i</sup> related by (13).*
**Corollary 1.** *If An* (*x*) *is the Appell polynomial sequence for β<sup>i</sup> we have*
$$A\_{\boldsymbol{n}}\left(\mathbf{x}\right) = \sum\_{j=0}^{n} \binom{n}{j} A\_{\boldsymbol{n}-j}\left(\mathbf{0}\right) \mathbf{x}^{j}\,,\,\boldsymbol{n} = \mathbf{0}\text{,}\,\mathbf{1}\,,\tag{14}$$
*Proof.* Follows from Theorem 2 being
$$A\_{\dot{i}}\left(0\right) = a\_{\dot{i}\prime} \quad \dot{\iota} = 0, 1, \ldots, n. \tag{15}$$
**Remark 4.** *For computation we can observe that αn is a n-order determinant of a particular upper Hessenberg form and it's known that the algorithm of Gaussian elimination without pivoting for computing the determinant of an upper Hessenberg matrix is stable ([22, p. 27]).*
**Theorem 3.** *If a*(*h*) *is the function defined in (4) and An* (*x*) *is the polynomial sequence defined by (5), setting*
$$\begin{cases} \beta\_0 = \frac{1}{a\_0},\\ \beta\_n = -\frac{1}{a\_0} \left( \sum\_{k=1}^n \binom{n}{k} a\_k \beta\_{n-k} \right), \quad n = 1, 2, \dots \end{cases} \tag{16}$$
*we have that An*(*x*) *satisfies the (6), i.e. An*(*x*) *is the Appell polynomial sequence for βi.*
*Proof.* Let be
4 Will-be-set-by-IN-TECH
*Proof.* From (6), by expanding the determinant *An* (*x*) with respect to the first row, we obtain the (3) with *α<sup>i</sup>* given by (7) and the determinantal form in (8); this is a determinant of an upper
*<sup>i</sup>*+<sup>1</sup> *<sup>α</sup><sup>i</sup>* for *<sup>i</sup>* <sup>=</sup> 1, 2, ..., *<sup>n</sup>*, we
*l*, *m* = 1, 2, ..., *i*, (10)
*<sup>i</sup>*−*k*−<sup>1</sup> *hk*<sup>+</sup>1,*iqk* (*i*) *<sup>α</sup>k*, (9)
*<sup>i</sup>*−*k*−<sup>1</sup> , *<sup>k</sup>* <sup>=</sup> 0, 1, ..., *<sup>i</sup>* <sup>−</sup> 2, (11)
*<sup>β</sup>*1*αi*−<sup>1</sup> =
<sup>0</sup> *<sup>i</sup>* <sup>&</sup>gt; <sup>0</sup> (13)
, *n* = 0, 1, ... (14)
*Ai* (0) = *αi*, *i* = 0, 1, ..., *n*. (15)
*qi*−<sup>1</sup> (*i*) = 1. (12)
� , � *i i* − 1
�
*<sup>i</sup>*−*k*−<sup>1</sup> *<sup>α</sup><sup>k</sup>* <sup>+</sup>
*βi*−*kα<sup>k</sup>*
� 1 *i* = 0
Hessenberg matrix of order *<sup>i</sup>* ([16]), then setting *<sup>α</sup><sup>i</sup>* = (−1)*<sup>i</sup>* (*β*0)
*i*−1 ∑ *k*=0
*β<sup>m</sup>* for *l* = 1,
(−1)
*<sup>l</sup>*−1)*βm*−*l*+<sup>1</sup> for 1 <sup>&</sup>lt; *<sup>l</sup>* <sup>≤</sup> *<sup>m</sup>* <sup>+</sup> 1, 0 for *l* > *m* + 1,
*hj*,*j*−<sup>1</sup> = (*β*0)
*<sup>β</sup>i*−*<sup>k</sup>* (*β*0)
*i*−1 ∑ *k*=0 �*i k* �
*<sup>β</sup>i*−*kα<sup>k</sup>* =
*and that for each sequence of Appell polynomials there exist two sequences of numbers α<sup>i</sup> and β<sup>i</sup> related*
**Remark 4.** *For computation we can observe that αn is a n-order determinant of a particular upper Hessenberg form and it's known that the algorithm of Gaussian elimination without pivoting for*
*An*−*<sup>j</sup>* (0) *<sup>x</sup><sup>j</sup>*
*α<sup>i</sup>* =
*i* ∏ *j*=*k*+2
*i*−*k*−1 �*i k* �
> *i* ∑ *k*=0
**Corollary 1.** *If An* (*x*) *is the Appell polynomial sequence for β<sup>i</sup> we have*
*An* (*x*) =
�*i k* �
*n* ∑ *j*=0
*computing the determinant of an upper Hessenberg matrix is stable ([22, p. 27]).*
�*n j* �
*hl*,*<sup>m</sup>* =
⎧ ⎨ ⎩
*qk* (*i*) =
By virtue of the previous setting, (9) implies
*i*−2 ∑ *k*=0
= (−1) *i* (*β*0) *i*+1 � − 1 *β*0
**Remark 3.** *We note that (7) and (8) are equivalent to*
(−1)
*α<sup>i</sup>* =
*Proof.* Follows from Theorem 2 being
and the proof is concluded.
*by (13).*
( *<sup>m</sup>*
have
where:
$$\mathfrak{b}(h) = \mathfrak{f}\_0 + \frac{h}{1!} \mathfrak{f}\_1 + \frac{h^2}{2!} \mathfrak{f}\_2 + \dots + \frac{h^n}{n!} \mathfrak{f}\_n + \dotsb \tag{17}$$
with *β<sup>n</sup>* as in (16). Then we have *a* (*h*) *b* (*h*) = 1, where the product is intended in the Cauchy sense, i.e.:
$$a\left(h\right)b\left(h\right) = \sum\_{n=0}^{\infty} \sum\_{k=0}^{n} \binom{n}{k} \alpha\_k \beta\_{n-k} \frac{h^n}{n!}$$
Let us multiply both hand sides of equation
$$a(h)e^{h\mathbf{x}} = \sum\_{n=0}^{\infty} A\_{\mathbb{N}}\left(\mathbf{x}\right) \frac{h^n}{n!} \tag{18}$$
for <sup>1</sup> *<sup>a</sup>* (*h*) and, in the same equation, replace functions *<sup>e</sup>hx* and <sup>1</sup> *<sup>a</sup>* (*h*) by their Taylor series expansion at the origin; then (18) becomes
$$\sum\_{n=0}^{\infty} \frac{x^n h^n}{n!} = \sum\_{n=0}^{\infty} A\_n \left( x \right) \frac{h^n}{n!} \sum\_{n=0}^{\infty} \frac{h^n}{n!} \beta\_n. \tag{19}$$
By multiplying the series on the left hand side of (19) according to the Cauchy-product rules, previous equality leads to the following system of infinite equations in the unknown *An* (*x*), *n* = 0, 1, ...
$$\begin{cases} A\_0\left(\mathbf{x}\right)\boldsymbol{\beta}\_0 = 1, \\ A\_0\left(\mathbf{x}\right)\boldsymbol{\beta}\_1 + A\_1\left(\mathbf{x}\right)\boldsymbol{\beta}\_0 = \mathbf{x}, \\ A\_0\left(\mathbf{x}\right)\boldsymbol{\beta}\_2 + \binom{2}{1} A\_1\left(\mathbf{x}\right)\boldsymbol{\beta}\_1 + A\_2\left(\mathbf{x}\right)\boldsymbol{\beta}\_0 = \mathbf{x}^2, \\ \vdots \\ A\_0\left(\mathbf{x}\right)\boldsymbol{\beta}\_{\boldsymbol{n}} + \binom{n}{1} A\_1\left(\mathbf{x}\right)\boldsymbol{\beta}\_{\boldsymbol{n}-1} + \dots + A\_n\left(\mathbf{x}\right)\boldsymbol{\beta}\_0 = \mathbf{x}^n, \\ \vdots \end{cases} \tag{20}$$
From the first one of (20) we obtain the first one of (6). Moreover, the special form of the previous system (lower triangular) allows us to work out the unknown *An* (*x*) operating with the first *n* + 1 equations, only by applying the Cramer rule:
6 Will-be-set-by-IN-TECH 26 Linear Algebra – Theorems and Applications
$$A\_{\boldsymbol{n}}\left(\mathbf{x}\right) = \frac{1}{\left(\boldsymbol{\beta}\_{0}\right)^{\boldsymbol{n}+1}} \begin{vmatrix} \boldsymbol{\beta}\_{0} & \boldsymbol{0} & \boldsymbol{0} & \cdots & \boldsymbol{0} & \boldsymbol{1} \\ \boldsymbol{\beta}\_{1} & \boldsymbol{\beta}\_{0} & \boldsymbol{0} & \cdots & \boldsymbol{0} & \boldsymbol{x} \\ \boldsymbol{\beta}\_{2} & \binom{2}{1}\boldsymbol{\beta}\_{1} & \boldsymbol{\beta}\_{0} & \cdots & \boldsymbol{0} & \boldsymbol{x}^{2} \\ \vdots & & \ddots & & \vdots \\ \boldsymbol{\beta}\_{n-1} & \binom{n-1}{1}\boldsymbol{\beta}\_{n-2} & \cdots & \cdots & \boldsymbol{\beta}\_{0} & \boldsymbol{x}^{n-1} \\ \boldsymbol{\beta}\_{n} & \binom{n}{1}\boldsymbol{\beta}\_{n-1} & \cdots & \cdots & \binom{n}{n-1}\boldsymbol{\beta}\_{1} & \boldsymbol{x}^{n} \\ \end{vmatrix}$$
By transposition of the previous, we have
$$A\_n(\mathbf{x}) = \frac{1}{\left(\beta\_0\right)^{n+1}} \begin{vmatrix} \beta\_0 \ \beta\_1 & \beta\_2 & \cdots & \beta\_{n-1} & \beta\_n \\ 0 & \beta\_0 \ \binom{n}{1} \beta\_1 & \cdots & \binom{n-1}{1} \beta\_{n-2} & \binom{n}{1} \beta\_{n-1} \\ 0 & 0 & \beta\_0 & & \vdots \\ \vdots & & \ddots & & \vdots \\ 0 & 0 & 0 & \cdots & \beta\_0 & \binom{n}{n-1} \beta\_1 \\ 1 & x & x^2 & \cdots & x^{n-1} & x^n \end{vmatrix}, \quad n = 1, 2, \ldots, \tag{21}$$
.
that is exactly the second one of (6) after *n* circular row exchanges: more precisely, the *i*-th row moves to the (*i* + 1)-th position for *i* = 1, . . . , *n* − 1, the *n*-th row goes to the first position.
**Definition 2.** *The function a* (*h*)*ehx, as in (4) and (5), is said 'generating function' of the Appell polynomial sequence An* (*x*) *for βi.*
Theorems 1, 2, 3 concur to assert the validity of following
**Theorem 4** (Circular)**.** *If An* (*x*) *is the Appell polynomial sequence for β<sup>i</sup> we have*
$$(6) \Rightarrow (2) \Rightarrow (3) \Rightarrow (5) \Rightarrow (6).$$
*Proof.*
**(6)**⇒**(2):** Follows from Theorem 1.
**Remark 5.** *In virtue of the Theorem 4, any of the relations (2), (3), (5), (6) can be assumed as definition of Appell polynomial sequences.*
#### **4. Examples of Appell polynomial sequences**
The following are classical examples of Appell polynomial sequences.
**a)** Bernoulli polynomials ([17, 23]):
6 Will-be-set-by-IN-TECH
*β*<sup>0</sup> 0 0 ··· 0 1 *β*<sup>1</sup> *β*<sup>0</sup> 0 ··· 0 *x*
. ... .
<sup>1</sup>)*βn*−<sup>1</sup> ··· ··· ( *<sup>n</sup>*
<sup>1</sup>)*β*<sup>1</sup> *<sup>β</sup>*<sup>0</sup> ··· <sup>0</sup> *<sup>x</sup>*<sup>2</sup>
<sup>1</sup> )*βn*−<sup>2</sup> ··· ··· *<sup>β</sup>*<sup>0</sup> *<sup>x</sup>n*−<sup>1</sup>
*n* <sup>1</sup>)*βn*−<sup>1</sup>
. . .
. .
*<sup>n</sup>*−1)*β*<sup>1</sup>
. .
.
, *n* = 1, 2, ..., (21)
*<sup>n</sup>*−1)*β*<sup>1</sup> *<sup>x</sup><sup>n</sup>*
*An* (*x*) = <sup>1</sup>
By transposition of the previous, we have
*An* (*x*) <sup>=</sup> <sup>1</sup>
*polynomial sequence An* (*x*) *for βi.*
**(6)**⇒**(2):** Follows from Theorem 1.
*n*! . **(5)**⇒**(6):** Follows from Theorem 3.
*Proof.*
equation (2).
coefficients of *<sup>h</sup><sup>n</sup>*
*of Appell polynomial sequences.*
(*β*0) *n*+1
(*β*0) *n*+1
>
. .
Theorems 1, 2, 3 concur to assert the validity of following
0 *β*<sup>0</sup> ( 2 <sup>1</sup>)*β*<sup>1</sup> ··· (
0 0 *β*<sup>0</sup>
**Theorem 4** (Circular)**.** *If An* (*x*) *is the Appell polynomial sequence for β<sup>i</sup> we have*
*β*<sup>2</sup> ( 2
*n*−1
*β*<sup>0</sup> *β*<sup>1</sup> *β*<sup>2</sup> ··· *βn*−<sup>1</sup> *β<sup>n</sup>*
. ... .
00 0 ··· *<sup>β</sup>*<sup>0</sup> ( *<sup>n</sup>*
that is exactly the second one of (6) after *n* circular row exchanges: more precisely, the *i*-th row moves to the (*i* + 1)-th position for *i* = 1, . . . , *n* − 1, the *n*-th row goes to the first position.
**Definition 2.** *The function a* (*h*)*ehx, as in (4) and (5), is said 'generating function' of the Appell*
(6) ⇒ (2) ⇒ (3) ⇒ (5) ⇒ (6).
**(2)**⇒**(3):** Follows from Theorem 2, or more simply by direct integration of the differential
**(3 )**⇒**(5):** Follows ordering the Cauchy product of the developments *<sup>a</sup>*(*h*) and *<sup>e</sup>hx* with respect to the powers of *h* and recognizing polynomials *An*(*x*), expressed in form (3), as
**Remark 5.** *In virtue of the Theorem 4, any of the relations (2), (3), (5), (6) can be assumed as definition*
<sup>1</sup> *x x*<sup>2</sup> ··· *<sup>x</sup>n*−<sup>1</sup> *<sup>x</sup><sup>n</sup>*
*n*−1 <sup>1</sup> )*βn*−<sup>2</sup> (
. .
*βn*−<sup>1</sup> (
*β<sup>n</sup>* ( *n*
$$\beta\_i = \frac{1}{i+1}, \quad i = 0, 1, \ldots \tag{22}$$
$$a(h) = \frac{h}{e^h - 1};\tag{23}$$
**b)** Euler polynomials ([17, 23]):
$$
\beta\_0 = 1, \quad \beta\_i = \frac{1}{2}, \quad i = 1, 2, \dots \tag{24}
$$
$$a(h) = \frac{2}{e^h + 1};\tag{25}$$
**c)** Normalized Hermite polynomials ([17, 24]):
$$\beta\_{\dot{l}} = \frac{1}{\sqrt{\pi}} \int\_{-\infty}^{+\infty} e^{-\mathbf{x}^2} \mathbf{x}^{\dot{l}} d\mathbf{x} = \begin{cases} 0 & \text{for } i \text{ odd} \\ \frac{(i-1)(i-3)\cdots 3\cdot 1}{2^{\frac{\dot{l}}{2}}} & \text{for } i \text{ even} \end{cases} \quad \dot{\imath} = 0, 1, \ldots \tag{26}$$
$$a(h) = e^{-\frac{h^2}{4}};$$
$$a(h) = e^{-\frac{h^2}{4}};\tag{27}$$
**d)** Laguerre polynomials ([17, 24]):
$$\beta\_{\bar{i}} = \int\_0^{+\infty} e^{-\mathbf{x}} \mathbf{x}^{\bar{i}} d\mathbf{x} = \Gamma \left( \bar{i} + 1 \right) = \bar{i}!, \quad \bar{i} = 0, 1, \dots \tag{28}$$
$$a(h) = 1 - h;\tag{29}$$
The following are non-classical examples of Appell polynomial sequences.
#### **e)** Generalized Bernoulli polynomials
• with Jacobi weight ([17]):
$$\beta\_i = \int\_0^1 (1-x)^a x^\beta x^i dx = \frac{\Gamma(a+1)\,\Gamma\left(\beta+i+1\right)}{\Gamma\left(a+\beta+i+2\right)}, \quad a, \beta > -1, \quad i = 0, 1, \ldots \tag{30}$$
$$a(h) = \frac{1}{\int\_0^1 (1-x)^a x^\beta e^{hx} dx};\tag{31}$$
• of order *k* ([11]):
$$\beta\_i = \left(\frac{1}{i+1}\right)^k \text{ , } k \text{ integer}, \quad i = 0, 1, \dots \tag{32}$$
$$a(h) = \left(\frac{h}{e^h - 1}\right)^k;\tag{33}$$
#### 8 Will-be-set-by-IN-TECH 28 Linear Algebra – Theorems and Applications
**f)** Central Bernoulli polynomials ([25]):
$$
\beta\_{2i} = \frac{1}{i+1},
$$
$$
\beta\_{2i+1} = 0, \quad i = 0, 1, \dots \tag{34}
$$
$$a(h) = \frac{h}{\sinh(h)};$$
**g)** Generalized Euler polynomials ([17]):
$$\beta\_0 = 1,$$
$$\beta\_i = \frac{w\_1}{w\_1 + w\_2}, \quad w\_{1\prime} w\_2 > 0, \quad i = 1, 2, \dots \tag{36}$$
$$a(h) = \frac{w\_1 + w\_2}{w\_1 e^h + w\_2};\tag{37}$$
**h)** Generalized Hermite polynomials ([17]):
$$\begin{split} \beta\_{i} &= \frac{1}{\sqrt{\pi}} \int\_{-\infty}^{+\infty} e^{-|x|^{\
u}} x^{i} dx \\ &= \begin{cases} 0 & \text{for } i \text{ odd} \\ \frac{2}{a\sqrt{\pi}} \Gamma\left(\frac{i+1}{a}\right) & \text{for } i \text{ even} \end{cases} \quad \begin{split} i = 0, 1, ..., \\ \alpha > 0, \end{split} \end{split} \tag{38}$$
$$a(h) = \frac{\sqrt{\pi}}{\int\_{-\infty}^{\infty} e^{-|\mathbf{x}|^{k}} e^{h\mathbf{x}} d\mathbf{x}};\tag{39}$$
**i)** Generalized Laguerre polynomials ([17]):
$$\begin{split} \beta\_{i} &= \int\_{0}^{+\infty} e^{-ax} x^{i} dx \\ &= \frac{\Gamma\left(i+1\right)}{a^{i+1}} = \frac{i!}{a^{i+1}}, \quad a > 0, \quad i = 0, 1, \dots \end{split} \tag{40}$$
$$a(h) = a - h.\tag{41}$$
#### **5. General properties of Appell polynomials**
By elementary tools of linear algebra we can prove the general properties of Appell polynomials.
Let *An* (*x*), *n* = 0, 1, ..., be a polynomial sequence and *β<sup>i</sup>* ∈ **R**, *i* = 0, 1, ..., with *β*<sup>0</sup> �= 0.
**Theorem 5** (Recurrence)**.** *An* (*x*) *is the Appell polynomial sequence for β<sup>i</sup> if and only if*
$$A\_{\mathfrak{n}}(\mathbf{x}) = \frac{1}{\beta\_0} \left( \mathbf{x}^n - \sum\_{k=0}^{n-1} \binom{n}{k} \beta\_{n-k} A\_k(\mathbf{x}) \right), \quad n = 1, 2, \ldots \tag{42}$$
*Proof.* Follows observing that the following holds:
8 Will-be-set-by-IN-TECH
*β*2*i*+<sup>1</sup> = 0, *i* = 0, 1, ..., (34)
, *w*1, *w*<sup>2</sup> > 0, *i* = 1, 2, ..., (36)
; (37)
*<sup>e</sup>hxdx* ; (39)
*<sup>α</sup>i*+<sup>1</sup> , *<sup>α</sup>* <sup>&</sup>gt; 0, *<sup>i</sup>* <sup>=</sup> 0, 1, ..., (40)
*<sup>α</sup>* <sup>&</sup>gt; 0, (38)
, *n* = 1, 2, ... (42)
for *<sup>i</sup>* even , *<sup>i</sup>* <sup>=</sup> 0, 1, ...,
*a*(*h*) = *α* − *h*. (41)
; (35)
*<sup>β</sup>*2*<sup>i</sup>* <sup>=</sup> <sup>1</sup> *i* + 1 ,
*<sup>a</sup>*(*h*) = *<sup>h</sup>*
sinh(*h*)
**f)** Central Bernoulli polynomials ([25]):
**g)** Generalized Euler polynomials ([17]):
**h)** Generalized Hermite polynomials ([17]):
**i)** Generalized Laguerre polynomials ([17]):
polynomials.
*β*<sup>0</sup> = 1,
*<sup>β</sup><sup>i</sup>* <sup>=</sup> <sup>1</sup> <sup>√</sup>*<sup>π</sup>*
=
*a*(*h*) =
*β<sup>i</sup>* =
**5. General properties of Appell polynomials**
*An*(*x*) = <sup>1</sup>
*β*0
*<sup>x</sup><sup>n</sup>* <sup>−</sup>
*<sup>β</sup><sup>i</sup>* <sup>=</sup> *<sup>w</sup>*<sup>1</sup>
*<sup>a</sup>*(*h*) = *<sup>w</sup>*<sup>1</sup> <sup>+</sup> *<sup>w</sup>*<sup>2</sup>
*w*<sup>1</sup> + *w*<sup>2</sup>
*w*1*e<sup>h</sup>* + *w*<sup>2</sup>
+∞ −∞ *e* −|*x*| *α xi dx*
2 *α* <sup>√</sup>*<sup>π</sup>* <sup>Γ</sup>
<sup>∞</sup> <sup>−</sup><sup>∞</sup> *<sup>e</sup>*−|*x*<sup>|</sup> *α*
+∞ 0
<sup>=</sup> <sup>Γ</sup> (*<sup>i</sup>* <sup>+</sup> <sup>1</sup>)
*e* <sup>−</sup>*α<sup>x</sup> x<sup>i</sup> dx*
*<sup>α</sup>i*+<sup>1</sup> <sup>=</sup> *<sup>i</sup>*!
By elementary tools of linear algebra we can prove the general properties of Appell
Let *An* (*x*), *n* = 0, 1, ..., be a polynomial sequence and *β<sup>i</sup>* ∈ **R**, *i* = 0, 1, ..., with *β*<sup>0</sup> �= 0. **Theorem 5** (Recurrence)**.** *An* (*x*) *is the Appell polynomial sequence for β<sup>i</sup> if and only if*
> *n k*
*<sup>β</sup>n*−*kAk* (*x*)
*n*−1 ∑ *k*=0
0 for *i* odd
*<sup>i</sup>*+<sup>1</sup> *α*
<sup>√</sup>*<sup>π</sup>*
$$A\_n(\mathbf{x}) = \frac{\begin{vmatrix} 1 & x & x^2 & \cdots & \cdots & x^{n-1} & x^n \\ \beta\_0 \ \beta\_1 & \beta\_2 & \cdots & \cdots & \beta\_{n-1} & \beta\_n \\ 0 & \beta\_0 \ \binom{2}{1} \beta\_1 & \cdots & \cdots & \binom{n-1}{1} \beta\_{n-2} & \binom{n}{1} \beta\_{n-1} \\ 0 & 0 & \beta\_0 & \cdots & \cdots & \binom{n-1}{2} \beta\_{n-3} & \binom{n}{2} \beta\_{n-2} \\ \vdots & & & \ddots & & \vdots \\ \vdots & & & \ddots & \vdots & \vdots \\ \vdots & & & \ddots & \vdots & \vdots \\ 0 & \cdots & \cdots & \cdots & 0 & \beta\_0 & \binom{n}{n-1} \beta\_1 \end{vmatrix}}{\begin{vmatrix} \vdots \\ 0 \end{vmatrix}} + \dots \quad \begin{vmatrix} \beta\_0 \\ \beta\_1 \end{vmatrix} \begin{vmatrix} \beta\_0 \\ \beta\_1 \end{vmatrix} \tag{4.18}$$
$$= \frac{1}{\beta\_0} \left( x^n - \sum\_{k=0}^{n-1} \binom{n}{k} \beta\_{n-k} A\_k \left(x\right) \right), \quad n = 1, 2, \dots \tag{4.3}$$
$$\text{is the Aussul polynomial sequences for } \beta\_0 \text{ from (6), we can observe that } A\_n(\mathbf{x}) \text{ (4.1)}$$
In fact, if *An* (*x*) is the Appell polynomial sequence for *βi*, from (6), we can observe that *An*(*x*) is a determinant of an upper Hessenberg matrix of order *n* + 1 ([16]) and, proceeding as in Theorem 2, we can obtain the (43).
**Corollary 2.** *If An* (*x*) *is the Appell polynomial sequence for β<sup>i</sup> then*
$$\mathbf{x}^{n} = \sum\_{k=0}^{n} \binom{n}{k} \beta\_{n-k} A\_{k} \left( \mathbf{x} \right), \quad n = 0, 1, \ldots \tag{44}$$
*Proof.* Follows from (42).
**Corollary 3.** *Let* P*<sup>n</sup> be the space of polynomials of degree* ≤ *n and* {*An*(*x*)}*<sup>n</sup> be an Appell polynomial sequence, then* {*An*(*x*)}*<sup>n</sup> is a basis for* P*n.*
*Proof.* If we have
$$P\_n(\mathbf{x}) = \sum\_{k=0}^n a\_{n,k} \mathbf{x}^k \quad a\_{n,k} \in \mathbb{R}\_\prime \tag{45}$$
then, by Corollary 2, we get
$$P\_n(\mathbf{x}) = \sum\_{k=0}^n a\_{n,k} \sum\_{j=0}^k \binom{k}{j} \beta\_{k-j} A\_j(\mathbf{x}) = \sum\_{k=0}^n c\_{n,k} A\_k(\mathbf{x}),$$
where
$$c\_{n,k} = \sum\_{j=0}^{n-k} \binom{k+j}{k} a\_{k+j} \beta\_j. \tag{46}$$
**Remark 6.** *An alternative recurrence relation can be determined from (5) after differentiation with respect to h ([18, 26]).*
Let be *βi*, *γ<sup>i</sup>* ∈ **R**, *i* = 0, 1, ..., with *β*0, *γ*<sup>0</sup> �= 0.
Let us consider the Appell polynomial sequences *An* (*x*) and *Bn* (*x*), *n* = 0, 1, ..., for *β<sup>i</sup>* and *γi*, respectively, and indicate with (*AB*)*<sup>n</sup>* (*x*) the polynomial that is obtained replacing in *An* (*x*) the powers *x*0, *x*1, ..., *xn*, respectively, with the polynomials *B*<sup>0</sup> (*x*), *B*<sup>1</sup> (*x*), ..., *Bn* (*x*). Then we have
**Theorem 6.** *The sequences*
*are sequences of Appell polynomials again.*
*Proof. i*) Follows from the property of linearity of determinant.
*ii*) Expanding the determinant (*AB*)*<sup>n</sup>* (*x*) with respect to the first row we obtain
$$(AB)\_n\left(\mathbf{x}\right) = \frac{(-1)^n}{\left(\beta\_0\right)^{n+1}} \sum\_{j=0}^n (-1)^j \left(\beta\_0\right)^j \binom{n}{j} \overline{\pi}\_{n-j} B\_j\left(\mathbf{x}\right) =$$
$$= \sum\_{j=0}^n \frac{(-1)^{n-j}}{\left(\beta\_0\right)^{n-j+1}} \binom{n}{j} \overline{\pi}\_{n-j} B\_j\left(\mathbf{x}\right) \,,\tag{47}$$
where
$$
\overline{\alpha}\_{i} = \begin{vmatrix}
\overline{\alpha}\_{0} = 1, \\
& \begin{vmatrix}
\beta\_{1} & \beta\_{2} & \cdots & \cdots & \beta\_{i-1} & \beta\_{i} \\
\beta\_{0} \binom{i}{1}\beta\_{1} & \cdots & \cdots & \binom{i-1}{1}\beta\_{i-2} \binom{i}{1}\beta\_{i-1} \\
0 & \beta\_{0} & \cdots & \cdots & \binom{i-1}{2}\beta\_{i-3} \binom{i}{2}\beta\_{i-2} \\
\vdots & & \ddots & \vdots & \vdots \\
\vdots & & \ddots & \vdots & \vdots \\
\vdots & & \ddots & \vdots & \vdots \\
0 & \cdots & \cdots & 0 & \beta\_{0} & \binom{i}{i-1}\beta\_{1}
\end{vmatrix}, \quad i = 1, 2, \dots, n.
$$
We observe that
$$A\_i \begin{pmatrix} 0 \end{pmatrix} = \frac{(-1)^i}{\left(\beta\_0\right)^{i+1}} \overline{a}\_{i\prime} \quad i = 1, 2, \dots, n.$$
and hence (47) becomes
$$(AB)\_n \begin{pmatrix} \mathbf{x} \end{pmatrix} = \sum\_{j=0}^n \binom{n}{j} A\_{n-j} \begin{pmatrix} \mathbf{0} \end{pmatrix} B\_j \begin{pmatrix} \mathbf{x} \end{pmatrix}.\tag{48}$$
Differentiating both hand sides of (48) and since *Bj* (*x*) is a sequence of Appell polynomials, we deduce
$$\left(\left(AB\right)\_{\mathfrak{n}}\left(\mathbf{x}\right)\right)' = \mathfrak{n}\left(AB\right)\_{\mathfrak{n}-1}\left(\mathbf{x}\right).\tag{49}$$
Let us, now, introduce the Appell vector.
**Definition 3.** *If An* (*x*) *is the Appell polynomial sequence for β<sup>i</sup> the vector of functions An* (*x*) = [*A*0(*x*), ..., *An*(*x*)]*<sup>T</sup> is called Appell vector for βi.*
Then we have
10 Will-be-set-by-IN-TECH
Let us consider the Appell polynomial sequences *An* (*x*) and *Bn* (*x*), *n* = 0, 1, ..., for *β<sup>i</sup>* and *γi*, respectively, and indicate with (*AB*)*<sup>n</sup>* (*x*) the polynomial that is obtained replacing in *An* (*x*) the powers *x*0, *x*1, ..., *xn*, respectively, with the polynomials *B*<sup>0</sup> (*x*), *B*<sup>1</sup> (*x*), ..., *Bn* (*x*). Then we
Let be *βi*, *γ<sup>i</sup>* ∈ **R**, *i* = 0, 1, ..., with *β*0, *γ*<sup>0</sup> �= 0.
have
*ii*) (*AB*)*<sup>n</sup>* (*x*)
where
We observe that
we deduce
and hence (47) becomes
**Theorem 6.** *The sequences*
*i*) *λAn* (*x*) + *μBn* (*x*), *λ*, *μ* ∈ **R**,
*are sequences of Appell polynomials again.*
*α*<sup>0</sup> = 1,
*β*<sup>0</sup> ( 2
> . .
> . .
*α<sup>i</sup>* =
*Proof. i*) Follows from the property of linearity of determinant.
(*AB*)*<sup>n</sup>* (*x*) <sup>=</sup> (−1)
= *n* ∑ *j*=0
*ii*) Expanding the determinant (*AB*)*<sup>n</sup>* (*x*) with respect to the first row we obtain
*n*
(−1) *n*−*j*
*β*<sup>1</sup> *β*<sup>2</sup> ··· ··· *βi*−<sup>1</sup> *β<sup>i</sup>*
<sup>0</sup> ··· ··· <sup>0</sup> *<sup>β</sup>*<sup>0</sup> ( *<sup>i</sup>*
(*β*0)
*n* ∑ *j*=0 *n j*
Differentiating both hand sides of (48) and since *Bj* (*x*) is a sequence of Appell polynomials,
*Ai* (0) <sup>=</sup> (−1)
(*AB*)*<sup>n</sup>* (*x*) =
*i*−1 <sup>1</sup> )*βi*−<sup>2</sup> (*<sup>i</sup>*
*i*−1 <sup>2</sup> )*βi*−<sup>3</sup> (*<sup>i</sup>*
. . . . .
. . . . .
*i*
<sup>1</sup>)*β*<sup>1</sup> ··· ··· (
. ... .
. ... .
0 *β*<sup>0</sup> ··· ··· (
(*β*0)
*n* ∑ *j*=0
*n*−*j*+1
(−1) *j* (*β*0) *j n j*
> *n j*
> > <sup>1</sup>)*βi*−<sup>1</sup>
<sup>2</sup>)*βi*−<sup>2</sup>
*<sup>i</sup>*−1)*β*<sup>1</sup>
*<sup>i</sup>*+<sup>1</sup> *<sup>α</sup>i*, *<sup>i</sup>* = 1, 2, ..., *<sup>n</sup>*
*<sup>α</sup>n*−*jBj* (*x*) =
, *i* = 1, 2, ..., *n*.
*An*−*<sup>j</sup>* (0) *Bj* (*x*). (48)
((*AB*)*<sup>n</sup>* (*x*))� <sup>=</sup> *<sup>n</sup>* (*AB*)*n*−<sup>1</sup> (*x*). (49)
*<sup>α</sup>n*−*jBj* (*x*), (47)
(*β*0) *n*+1 **Theorem 7** (Matrix form)**.** *Let An* (*x*) *be a vector of polynomial functions. Then An* (*x*) *is the Appell vector for β<sup>i</sup> if and only if, putting*
$$(M)\_{i,j} = \begin{cases} \binom{i}{j} \beta\_{i-j} & i \ge j \\ 0 & \text{otherwise} \end{cases}, \qquad i, j = 0, \ldots, n \tag{50}$$
*and X*(*x*) = [1, *x*, ..., *xn*] *<sup>T</sup> the following relation holds*
$$X(\mathfrak{x}) = M \overline{A}\_{\mathfrak{n}}\left(\mathfrak{x}\right) \tag{51}$$
*or, equivalently,*
$$\overline{A}\_{\mathfrak{n}}\left(\mathbf{x}\right) = \left(M^{-1}\right)X(\mathbf{x})\_{\mathfrak{n}}\tag{52}$$
*being M*−<sup>1</sup> *the inverse matrix of M.*
*Proof.* If *An* (*x*) is the Appell vector for *β<sup>i</sup>* the result easily follows from Corollary 2.
Vice versa, observing that the matrix *M* defined by (50) is invertible, setting
$$\left(M^{-1}\right)\_{i,j} = \begin{cases} \binom{i}{j} \mathfrak{a}\_{i-j} & i \ge j \\ 0 & \text{otherwise} \end{cases}, \qquad i, j = 0, \ldots, n. \tag{53}$$
we have the (52) and therefore the (3) and, being the coefficients *α<sup>k</sup>* and *β<sup>k</sup>* related by (13), we have that *An*(*x*) is the Appell polynomial sequence for *βi*.
**Theorem 8** (Connection constants)**.** *Let An*(*x*) *and Bn*(*x*) *be the Appell vectors for β<sup>i</sup> and γi, respectively. Then*
$$
\overline{A}\_{\mathfrak{n}}(\mathfrak{x}) = \mathbb{C}\overline{B}\_{\mathfrak{n}}(\mathfrak{x}),
\tag{54}
$$
*where*
$$(\mathbf{C})\_{i,j} = \begin{cases} \binom{i}{j} c\_{i-j} & i \ge j \\ 0 & \text{otherwise} \end{cases}, \qquad i, j = 0, \ldots, n. \tag{55}$$
*with*
$$\mathcal{L}\_n = \sum\_{k=0}^n \binom{n}{k} \mathfrak{a}\_{n-k} \gamma\_k. \tag{56}$$
*Proof.* From Theorem 7 we have
$$X(\mathfrak{x}) = M\overline{A}\_n(\mathfrak{x})$$
with *M* as in (50) or, equivalently,
$$
\overline{A}\_n(\mathfrak{x}) = \left(M^{-1}\right)X(\mathfrak{x}),
$$
#### 12 Will-be-set-by-IN-TECH 32 Linear Algebra – Theorems and Applications
with *M*−<sup>1</sup> as in (53).
Always from Theorem 7 we get
with
$$(N)\_{i,j} = \begin{cases} \binom{i}{j} \gamma\_{i-j} & i \ge j \\ 0 & \text{otherwise} \end{cases}, \qquad i, j = 0, \ldots, n.$$
*X*(*x*) = *NBn* (*x*)
Then
$$
\overline{A}\_n\left(\mathbf{x}\right) = M^{-1}N\overline{B}\_n\left(\mathbf{x}\right),
$$
from which, setting *C* = *M*−1*N*, we have the thesis.
**Theorem 9** (Inverse relations)**.** *Let An* (*x*) *be the Appell polynomial sequence for β<sup>i</sup> then the following are inverse relations:*
$$\begin{cases} y\_n = \sum\_{k=0}^n \binom{n}{k} \beta\_{n-k} \mathbf{x}\_k \\ \mathbf{x}\_n = \sum\_{k=0}^n \binom{n}{k} A\_{n-k}(0) y\_k. \end{cases} \tag{57}$$
*Proof.* Let us remember that
*Ak*(0) = *αk*,
where the coefficients *α<sup>k</sup>* and *β<sup>k</sup>* are related by (13).
Moreover, setting *yn* = [*y*0, ..., *yn*] *<sup>T</sup>* and *xn* = [*x*0, ..., *xn*] *<sup>T</sup>*, from (57) we have
$$\begin{cases} \overline{y}\_n = M\_1 \overline{x}\_n \\ \overline{x}\_n = M\_2 \overline{y}\_n \end{cases}$$
with
$$(M\_1)\_{i,j} = \begin{cases} \binom{i}{j} \beta\_{i-j} & i \ge j \\ 0 & \text{otherwise} \end{cases}, \qquad i, j = 0, \ldots, n,\tag{58}$$
$$(M\_2)\_{i,j} = \begin{cases} \binom{i}{j} a\_{i-j} & i \ge j \\ 0 & \text{otherwise} \end{cases}, \qquad i, j = 0, \ldots, n,\tag{59}$$
and, from (13) we get
$$M\_1 M\_2 = I\_{n+1\prime}$$
i.e. (57) are inverse relations.
**Theorem 10** (Inverse relation between two Appell polynomial sequences)**.** *Let An*(*x*) *and Bn*(*x*) *be the Appell vectors for β<sup>i</sup> and γi, respectively. Then the following are inverse relations:*
$$\begin{cases} \overline{A}\_{\mathcal{U}}(\mathbf{x}) = \mathsf{C} \overline{B}\_{\mathcal{U}}(\mathbf{x}) \\ \overline{B}\_{\mathcal{U}}(\mathbf{x}) = \widetilde{\mathsf{C}A}\_{\mathcal{U}}(\mathbf{x}) \end{cases} \tag{60}$$
*with*
12 Will-be-set-by-IN-TECH
*X*(*x*) = *NBn* (*x*)
*An* (*x*) = *M*−1*NBn* (*x*),
**Theorem 9** (Inverse relations)**.** *Let An* (*x*) *be the Appell polynomial sequence for β<sup>i</sup> then the*
�*n k* �
�*n k* �
*Ak*(0) = *αk*,
� *yn* <sup>=</sup> *<sup>M</sup>*1*xn xn* = *M*2*yn*
*<sup>T</sup>* and *xn* = [*x*0, ..., *xn*]
)*βi*−*<sup>j</sup> <sup>i</sup>* ≥ *<sup>j</sup>*
)*αi*−*<sup>j</sup> <sup>i</sup>* ≥ *<sup>j</sup>*
*M*1*M*<sup>2</sup> = *In*+1,
**Theorem 10** (Inverse relation between two Appell polynomial sequences)**.** *Let An*(*x*) *and Bn*(*x*) *be the Appell vectors for β<sup>i</sup> and γi, respectively. Then the following are inverse relations:*
� *An*(*x*) = *CBn*(*x*)
<sup>0</sup> *otherwise* , *<sup>i</sup>*, *<sup>j</sup>* <sup>=</sup> 0, ..., *<sup>n</sup>*.
*<sup>β</sup>n*−*kxk*
*An*−*k*(0)*yk*.
*<sup>T</sup>*, from (57) we have
<sup>0</sup> *otherwise* , *<sup>i</sup>*, *<sup>j</sup>* <sup>=</sup> 0, ..., *<sup>n</sup>*, (58)
<sup>0</sup> *otherwise* , *<sup>i</sup>*, *<sup>j</sup>* <sup>=</sup> 0, ..., *<sup>n</sup>*, (59)
*Bn*(*x*) = *<sup>C</sup>*�*An*(*x*) (60)
(57)
)*γi*−*<sup>j</sup> <sup>i</sup>* ≥ *<sup>j</sup>*
with *M*−<sup>1</sup> as in (53).
with
Then
with
and, from (13) we get
i.e. (57) are inverse relations.
Always from Theorem 7 we get
*following are inverse relations:*
*Proof.* Let us remember that
Moreover, setting *yn* = [*y*0, ..., *yn*]
(*N*)*i*,*<sup>j</sup>* =
from which, setting *C* = *M*−1*N*, we have the thesis.
where the coefficients *α<sup>k</sup>* and *β<sup>k</sup>* are related by (13).
(*M*1)*i*,*<sup>j</sup>* =
(*M*2)*i*,*<sup>j</sup>* =
� ( *i j*
� ( *i j*
� ( *i j*
> ⎧ ⎪⎪⎪⎨
*yn* =
*xn* =
*n* ∑ *k*=0
*n* ∑ *k*=0
⎪⎪⎪⎩
$$(\mathbf{C})\_{i,j} = \begin{cases} \binom{i}{j} \mathbf{c}\_{i-j} & \mathbf{i} \ge j \\ 0 & \text{otherwise} \end{cases}, \qquad \left(\tilde{\mathbf{C}}\right)\_{i,j} = \begin{cases} \binom{i}{j} \tilde{\mathbf{c}}\_{i-j} & \mathbf{i} \ge j \\ 0 & \text{otherwise} \end{cases}, \quad \mathbf{i}, j = \mathbf{0}, \ldots, n,\tag{61}$$
$$\mathbf{c}\_{\mathrm{ll}} = \sum\_{k=0}^{n} \binom{n}{k} A\_{\mathrm{ll}-k}(0) \gamma\_{k\prime} \quad \tilde{\mathbf{c}}\_{\mathrm{ll}} = \sum\_{k=0}^{n} \binom{n}{k} B\_{\mathrm{ll}-k}(0) \beta\_{k\prime} \tag{62}$$
*Proof.* Follows from Theorem 8, after observing that
$$\sum\_{k=0}^{n} \binom{n}{k} c\_{n-k} \tilde{c}\_k = \begin{cases} 1 & n = 0 \\ 0 & n > 0 \end{cases} \tag{63}$$
and therefore
$$
\mathbf{C}\widetilde{\mathbf{C}} = I\_{n+1}.\tag{7}
$$
**Theorem 11** (Binomial identity)**.** *If An* (*x*) *is the Appell polynomial sequence for β<sup>i</sup> we have*
$$A\_{\boldsymbol{n}}\left(\mathbf{x}+\boldsymbol{y}\right) = \sum\_{i=0}^{n} \binom{n}{i} A\_{i}\left(\mathbf{x}\right) \boldsymbol{y}^{n-i}, \quad \boldsymbol{n} = \mathbf{0}, 1, \ldots \tag{64}$$
*Proof.* Starting by the Definition 1 and using the identity
$$(\mathbf{x} + \mathbf{y})^{\dot{i}} = \sum\_{k=0}^{\dot{i}} \binom{\dot{i}}{k} \mathbf{y}^{k} \mathbf{x}^{\dot{i}-k},\tag{65}$$
we infer
$$A\_{n}\left(\mathbf{x}+\mathbf{y}\right)=\frac{\left(-1\right)^{n}}{\left(\beta\_{0}\right)^{n+1}}\begin{vmatrix}1&(\mathbf{x}+\mathbf{y})&\cdots&(\mathbf{x}+\mathbf{y})^{n-1}&(\mathbf{x}+\mathbf{y})^{n}\\\beta\_{0}&\beta\_{1}&\cdots&\beta\_{n-1}&\beta\_{n}\\0&\ddots&\ddots&\vdots\\\vdots&\ddots&\vdots\\0&\cdots&\cdots&\beta\_{0}&\beta\_{1}\binom{n}{n-1}\\\vdots&\vdots&\vdots&\vdots\\\beta\_{0}&\beta\_{1}\left(\mathbf{i}^{i}\_{1}\right)\mathbf{x}^{1}&\left(\mathbf{i}^{i}\_{2}\right)\mathbf{x}^{2}&\cdots&\left(\mathbf{i}^{n}\_{i}\right)\mathbf{x}^{n-i-1}&\binom{n}{i}\mathbf{x}^{n-i}\\\\beta\_{0}&\beta\_{1}\left(\mathbf{i}^{i+1}\_{1}\right)\beta\_{2}\left(\mathbf{i}^{i+2}\_{2}\right)\cdots&\beta\_{n-i-1}\left(\mathbf{i}^{n}\_{i}\right)&\beta\_{n-i}\left(\mathbf{i}^{i}\_{i}\right)\\0&\beta\_{0}&\beta\_{1}\left(\mathbf{i}^{i+2}\_{i+1}\right)\cdots&\beta\_{n-i-1}\left(\mathbf{i}^{n-1}\_{i}\right)\beta\_{n-i-1}\left(\mathbf{i}^{n}\_{i+1}\right)\\\vdots&\vdots&\vdots&\vdots\\\beta\_{0}&\cdots&\beta\_{0}&\cdots&\beta\_{1}\\\vdots&\vdots&\vdots&\vdots\\0&\cdots&\cdots&0&\beta\_{0}&\beta\_{1}\left(\mathbf{i}^{n}\_{n-1}\right)\end{vmatrix}$$
#### 14 Will-be-set-by-IN-TECH 34 Linear Algebra – Theorems and Applications
We divide, now, each *j*−th column, *j* = 2, ..., *n* − *i* + 1, for ( *i*+*j*−1 *<sup>i</sup>* ) and multiply each *h*−th row, *h* = 3, ..., *n* − *i* + 1, for ( *i*+*h*−2 *<sup>i</sup>* ). Thus we finally obtain
$$\begin{aligned} A\_n(x+y) &= \\ &= \sum\_{i=0}^n \frac{\binom{i+1}{i} \cdots \binom{n}{i}}{\binom{i+1}{i} \cdots \binom{n-1}{i}} y^i \frac{(-1)^{n-i}}{(\beta\_0)^{n-i+1}} \\ &\qquad \left| \begin{array}{cccc} 1 & \mathbf{x}^1 & \mathbf{x}^2 & \cdots & \mathbf{x}^{n-i-1} & \mathbf{x}^{n-i} \\ \beta\_0 \ \beta\_1 & \beta\_2 & \cdots & \beta\_{n-i-1} & \beta\_{n-i} \\ 0 & \beta\_0 \ \beta\_1 (^1\_1)^{\cdot} \cdot \cdots \beta\_{n-i-2} (^{n-1}\_{-1}) & \beta\_{n-i-1} (^{n-i}\_1) \\ \vdots & & & \vdots \\ \beta\_0 & & & \vdots \\ \vdots & & \ddots & & \vdots \\ 0 & \cdots & \cdots & 0 & \beta\_0 \end{array} \right| \\ &= \sum\_{i=0}^n \binom{n}{i} A\_{n-i}(x) y^i = \sum\_{i=0}^n \binom{n}{i} A\_i(x) y^{n-i}. \end{aligned}$$
**Theorem 12** (Generalized Appell identity)**.** *Let An*(*x*) *and Bn*(*x*) *be the Appell polynomial sequences for β<sup>i</sup> and γi*, *respectively. Then, if Cn*(*x*) *is the Appell polynomial sequence for δ<sup>i</sup> with*
$$\begin{cases} \delta\_0 = \frac{1}{\mathbb{C}\_0(0)} \prime \\ \delta\_i = -\frac{1}{\mathbb{C}\_0(0)} \sum\_{k=1}^i \binom{i}{k} \delta\_{i-k} \mathbb{C}\_k(0) \prime \ i = 1, \ldots \end{cases} \tag{66}$$
*and*
$$\mathbf{C}\_{i}(\mathbf{0}) = \sum\_{j=0}^{i} \binom{i}{j} B\_{i-j}(\mathbf{0}) A\_{j}(\mathbf{0})\_{\prime} \tag{67}$$
*where Ai*(0) *and Bi*(0) *are related to β<sup>i</sup> and γi, respectively, by relations similar to (66), we have*
$$\mathcal{C}\_{\mathfrak{n}}(y+z) = \sum\_{k=0}^{n} \binom{n}{k} A\_{k}(y) B\_{n-k}(z). \tag{68}$$
*Proof.* Starting from (3) we have
$$\mathbb{C}\_{n}(y+z) = \sum\_{k=0}^{n} \binom{n}{k} \mathbb{C}\_{n-k}(0)(y+z)^{k}.\tag{69}$$
Then, applying (67) and the well-known classical binomial identity, after some calculation, we obtain the thesis.
**Theorem 13** (Combinatorial identities)**.** *Let An*(*x*) *and Bn*(*x*) *be the Appell polynomial sequences for β<sup>i</sup> and γi*, *respectively. Then the following relations holds:*
34 Linear Algebra – Theorems and Applications Algebraic Theory of Appell Polynomials with Application to General Linear Interpolation Problem <sup>15</sup> Algebraic Theory of Appell Polynomials with Application to General Linear Interpolation Problem 35
$$\sum\_{k=0}^{n} \binom{n}{k} A\_k(\mathbf{x}) B\_{n-k}(-\mathbf{x}) = \sum\_{k=0}^{n} \binom{n}{k} A\_k(\mathbf{0}) B\_{n-k}(\mathbf{0}),\tag{70}$$
$$\sum\_{k=0}^{n} \binom{n}{k} A\_k(\mathbf{x}) B\_{n-k}(z) = \sum\_{k=0}^{n} \binom{n}{k} A\_k(\mathbf{x} + z) B\_{n-k}(0). \tag{71}$$
*Proof.* If *Cn*(*x*) is the Appell polynomial sequence for *δ<sup>i</sup>* defined as in (66), from the generalized Appell identity, we have
$$\sum\_{k=0}^{n} \binom{n}{k} A\_k(\mathfrak{x}) B\_{n-k}(-\mathfrak{x}) = \mathbf{C}\_n(0) = \sum\_{k=0}^{n} \binom{n}{k} A\_k(0) B\_{n-k}(0)$$
and *<sup>n</sup>*
14 Will-be-set-by-IN-TECH
*n*−1 *i* )
*An*−*<sup>i</sup>* (*x*) *<sup>y</sup><sup>i</sup>* <sup>=</sup>
*i* ∑ *k*=1 (*i*
*i* ∑ *j*=0 �*i j* �
*where Ai*(0) *and Bi*(0) *are related to β<sup>i</sup> and γi, respectively, by relations similar to (66), we have*
�*n k* �
*n* ∑ *k*=0
*n* ∑ *k*=0 �*n k* �
Then, applying (67) and the well-known classical binomial identity, after some calculation, we
**Theorem 13** (Combinatorial identities)**.** *Let An*(*x*) *and Bn*(*x*) *be the Appell polynomial sequences*
**Theorem 12** (Generalized Appell identity)**.** *Let An*(*x*) *and Bn*(*x*) *be the Appell polynomial sequences for β<sup>i</sup> and γi*, *respectively. Then, if Cn*(*x*) *is the Appell polynomial sequence for δ<sup>i</sup> with*
*<sup>y</sup><sup>i</sup>* (−1)
(*β*0)
<sup>1</sup> *<sup>x</sup>*<sup>1</sup> *<sup>x</sup>*<sup>2</sup> ··· *<sup>x</sup>n*−*i*−<sup>1</sup> *<sup>x</sup>n*−*<sup>i</sup> β*<sup>0</sup> *β*<sup>1</sup> *β*<sup>2</sup> ··· *βn*−*i*−<sup>1</sup> *βn*−*<sup>i</sup>*
. ... .
0 ... ... 0 *β*<sup>0</sup> *β*1( *<sup>n</sup>*−*<sup>i</sup>*
*n* ∑ *i*=0 �*n i* �
<sup>1</sup>) ··· *<sup>β</sup>n*−*i*−2(
*n*−*i*
*n*−*i*+1
*n*−*i*−1
<sup>1</sup> ) *<sup>β</sup>n*−*i*−1(
*n*−*i* 1 ) � � � � � � � � � � � � � � �
=
. . .
. .
*Ai* (*x*) *yn*−*<sup>i</sup>*
*<sup>n</sup>*−*i*−1)
.
*<sup>k</sup>*)*δi*−*kCk*(0), *<sup>i</sup>* <sup>=</sup> 1, ..., (66)
*Bi*−*j*(0)*Aj*(0), (67)
*Ak*(*y*)*Bn*−*k*(*z*). (68)
*Cn*−*k*(0)(*<sup>y</sup>* <sup>+</sup> *<sup>z</sup>*)*k*. (69)
*<sup>i</sup>* ). Thus we finally obtain
( *i*+1 *<sup>i</sup>* )···( *n i*)
0 *β*<sup>0</sup> *β*1(
2
( *i*+1 *<sup>i</sup>* )···( *i*+*j*−1
*<sup>i</sup>* ) and multiply each *h*−th row,
We divide, now, each *j*−th column, *j* = 2, ..., *n* − *i* + 1, for (
*i*+*h*−2
= *n* ∑ *i*=0
= *n* ∑ *i*=0
⎧ ⎪⎨
⎪⎩
*Proof.* Starting from (3) we have
obtain the thesis.
� � � � � � � � � � � � � � �
. . . *β*<sup>0</sup>
. .
> �*n i* �
*δ*<sup>0</sup> = <sup>1</sup> *<sup>C</sup>*0(0),
*<sup>δ</sup><sup>i</sup>* <sup>=</sup> <sup>−</sup> <sup>1</sup> *C*0(0)
*Ci*(0) =
*Cn*(*y* + *z*) =
*Cn*(*y* + *z*) =
*for β<sup>i</sup> and γi*, *respectively. Then the following relations holds:*
*An* (*x* + *y*) =
*h* = 3, ..., *n* − *i* + 1, for (
*and*
$$\sum\_{k=0}^{n} \binom{n}{k} A\_k(\mathbf{x}) B\_{n-k}(\mathbf{z}) = \mathbb{C}\_{\mathbb{R}}(\mathbf{x} + \mathbf{z}) = \sum\_{k=0}^{n} \binom{n}{k} A\_k(\mathbf{x} + \mathbf{z}) B\_{n-k}(\mathbf{0}).$$
$$\square$$
**Theorem 14** (Forward difference)**.** *If An* (*x*) *is the Appell polynomial sequence for β<sup>i</sup> we have*
$$\Delta A\_{\rm ll} \left( \mathbf{x} \right) \equiv A\_{\rm ll} \left( \mathbf{x} + \mathbf{1} \right) - A\_{\rm ll} \left( \mathbf{x} \right) = \sum\_{i=0}^{n-1} \binom{n}{i} A\_{i} \left( \mathbf{x} \right), \quad n = 0, 1, \ldots \tag{72}$$
*Proof.* The desired result follows from (64) with *y* = 1.
**Theorem 15** (Multiplication Theorem)**.** *Let An*(*x*) *be the Appell vector for βi.*
*The following identities hold:*
$$
\overline{A}\_{\mathbb{H}}\left(m\mathbf{x}\right) = B\left(\mathbf{x}\right)\overline{A}\_{\mathbb{H}}\left(\mathbf{x}\right) \qquad n = 0, 1, \ldots \qquad m = 1, 2, \ldots. \tag{73}
$$
$$
\overline{A}\_{\text{ll}}(m\mathbf{x}) = M^{-1}DX(\mathbf{x}) \qquad n = 0, 1, \ldots \qquad m = 1, 2, \ldots \tag{74}
$$
*where*
$$(B(\mathbf{x}))\_{i,j} = \begin{cases} \binom{i}{j} (m-1)^{i-j} \mathbf{x}^{i-j} & \mathbf{i} \ge j \\ 0 & \text{otherwise} \end{cases}, \qquad \mathbf{i}, \mathbf{j} = \mathbf{0}, \dots, n,\tag{75}$$
*D* = *diag*[1, *m*, ..., *mn*] *and M*−<sup>1</sup> *defined as in (53).*
*Proof.* The (73) follows from (64) setting *y* = *x* (*m* − 1). In fact we get
$$A\_{\boldsymbol{n}}\left(m\mathbf{x}\right) = \sum\_{i=0}^{n} \binom{n}{i} A\_{\boldsymbol{i}}\left(\mathbf{x}\right) \left(m-1\right)^{n-i} \mathbf{x}^{n-i}.\tag{76}$$
The (74) follows from Theorem 7. In fact we get
$$
\overline{A}\_{\mathfrak{N}}(m\mathbf{x}) = M^{-1}X(m\mathbf{x}) = M^{-1}DX(\mathbf{x}),\tag{77}
$$
#### 16 Will-be-set-by-IN-TECH 36 Linear Algebra – Theorems and Applications
and
$$A\_{\mathfrak{n}}\left(m\mathbf{x}\right) = \sum\_{i=0}^{n} \binom{n}{i} \mathfrak{a}\_{n-i} m^i \mathfrak{x}^i. \tag{78}$$
**Theorem 16** (Differential equation)**.** *If An* (*x*) *is the Appell polynomial sequence for β<sup>i</sup> then An* (*x*) *satisfies the linear differential equation:*
$$\frac{\beta\_n}{n!}y^{(n)}(\mathbf{x}) + \frac{\beta\_{n-1}}{(n-1)!}y^{(n-1)}(\mathbf{x}) + \dots + \frac{\beta\_2}{2!}y^{(2)}(\mathbf{x}) + \beta\_1 y^{(1)}(\mathbf{x}) + \beta\_0 y(\mathbf{x}) = \mathbf{x}^n \tag{79}$$
*Proof.* From Theorem 5 we have
$$A\_{n+1}(\mathbf{x}) = \frac{1}{\beta\_0} \left( \mathbf{x}^{n+1} - \sum\_{k=0}^{n} \binom{n+1}{k+1} \beta\_{k+1} A\_{n-k}(\mathbf{x}) \right). \tag{80}$$
From Theorem 1 we find that
$$A\_{n+1}'(\mathbf{x}) = (n+1)A\_{\mathbb{R}}(\mathbf{x}), \quad \text{and} \quad A\_{n-k}(\mathbf{x}) = \frac{A\_{\mathbb{R}}^{(k)}(\mathbf{x})}{n(n-1)...(n-k+1)},\tag{81}$$
and replacing *An*−*k*(*x*) in the (80) we obtain
$$A\_{n+1}(\mathbf{x}) = \frac{1}{\beta\_0} \left( \mathbf{x}^{n+1} - (n+1) \sum\_{k=0}^n \beta\_{k+1} \frac{A\_n^{(k)}(\mathbf{x})}{(k+1)!} \right). \tag{82}$$
Differentiating both hand sides of the last one and replacing *A*� *<sup>n</sup>*+1(*x*) with (*n* + 1)*An*(*x*), after some calculation we obtain the thesis.
**Remark 7.** *An alternative differential equation for Appell polynomial sequences can be determined by the recurrence relation referred to in Remark 6 ([18, 26]).*
#### **6. Appell polynomial sequences of second kind**
Let *f* : *I* ⊂ **R** → **R** and Δ be the finite difference operator ([23]), i.e.:
$$
\Delta[f](\mathbf{x}) = f(\mathbf{x} + \mathbf{1}) - f(\mathbf{x}) \tag{83}
$$
we define the finite difference operator of order *i*, with *i* ∈ **N**, as
$$
\Delta^i[f](\mathbf{x}) = \Delta(\Delta^{i-1}[f](\mathbf{x})) = \sum\_{j=0}^i (-1)^{i-j} \binom{i}{j} f(\mathbf{x} + j), \tag{84}
$$
meaning Δ<sup>0</sup> = *I* and Δ<sup>1</sup> = Δ, where *I* is the identity operator. Let the sequence of falling factorial defined by
$$\begin{cases} \left(\mathbf{x}\right)\_0 = 1, \\ \left(\mathbf{x}\right)\_n = \mathbf{x}\left(\mathbf{x} - 1\right)\left(\mathbf{x} - 2\right)\cdots\left(\mathbf{x} - n + 1\right), n = 1, 2, \ldots \end{cases} \tag{85}$$
we give the following
16 Will-be-set-by-IN-TECH
**Theorem 16** (Differential equation)**.** *If An* (*x*) *is the Appell polynomial sequence for β<sup>i</sup> then An* (*x*)
*n* ∑ *k*=0
2! *<sup>y</sup>*(2)
*n* + 1 *k* + 1
*n* ∑ *k*=0
*βk*+<sup>1</sup>
(*x*) + ... <sup>+</sup> *<sup>β</sup>*<sup>2</sup>
*n i*
*<sup>α</sup>n*−*im<sup>i</sup> xi*
(*x*) + *β*1*y*(1)
*<sup>β</sup>k*+1*An*−*k*(*x*)
*A*(*k*) *<sup>n</sup>* (*x*) (*k* + 1)!
Δ[ *f* ](*x*) = *f*(*x* + 1) − *f*(*x*), (83)
*i j* *<sup>n</sup>* (*x*) *n*(*n* − 1)...(*n* − *k* + 1)
. (78)
(*x*) + *β*0*y*(*x*) = *x<sup>n</sup>* (79)
. (80)
. (82)
*<sup>n</sup>*+1(*x*) with (*n* + 1)*An*(*x*), after
*f*(*x* + *j*), (84)
, (81)
*n* ∑ *i*=0
*An* (*mx*) =
*y*(*n*−1)
*β*0
*β*0
Differentiating both hand sides of the last one and replacing *A*�
**6. Appell polynomial sequences of second kind**
Let *f* : *I* ⊂ **R** → **R** and Δ be the finite difference operator ([23]), i.e.:
we define the finite difference operator of order *i*, with *i* ∈ **N**, as
meaning Δ<sup>0</sup> = *I* and Δ<sup>1</sup> = Δ, where *I* is the identity operator.
[ *f* ](*x*) = Δ(Δ*i*−1[ *f* ](*x*)) =
*<sup>x</sup>n*+<sup>1</sup> <sup>−</sup>
*<sup>n</sup>*+1(*x*)=(*<sup>n</sup>* <sup>+</sup> <sup>1</sup>)*An*(*x*), and *An*−*k*(*x*) = *<sup>A</sup>*(*k*)
*<sup>x</sup>n*+<sup>1</sup> <sup>−</sup> (*<sup>n</sup>* <sup>+</sup> <sup>1</sup>)
**Remark 7.** *An alternative differential equation for Appell polynomial sequences can be determined by*
*i* ∑ *j*=0
(−1)*i*−*<sup>j</sup>*
(*x*)*<sup>n</sup>* <sup>=</sup> *<sup>x</sup>* (*<sup>x</sup>* <sup>−</sup> <sup>1</sup>) (*<sup>x</sup>* <sup>−</sup> <sup>2</sup>)···(*<sup>x</sup>* <sup>−</sup> *<sup>n</sup>* <sup>+</sup> <sup>1</sup>), *<sup>n</sup>* <sup>=</sup> 1, 2, ..., (85)
and
*satisfies the linear differential equation:*
*Proof.* From Theorem 5 we have
From Theorem 1 we find that
*A*�
(*x*) + *<sup>β</sup>n*−<sup>1</sup> (*n* − 1)!
and replacing *An*−*k*(*x*) in the (80) we obtain
some calculation we obtain the thesis.
*An*+1(*x*) = <sup>1</sup>
*An*+1(*x*) = <sup>1</sup>
*the recurrence relation referred to in Remark 6 ([18, 26]).*
Δ*i*
Let the sequence of falling factorial defined by
(*x*)<sup>0</sup> <sup>=</sup> 1,
*βn n*! *y*(*n*) **Definition 4.** *Let ßi* ∈ **R***, i* = 0, 1, ..., *with ß*<sup>0</sup> �= 0*. The polynomial sequence*
$$\begin{cases} \mathcal{A}\_{0}\left(\mathbf{x}\right) = \frac{1}{\beta\_{0}},\\ \begin{bmatrix} \mathcal{A}\_{0}\left(\mathbf{x}\right) = \frac{1}{\beta\_{0}}\\ \mathcal{A}\_{0} \begin{bmatrix} \mathcal{A}\_{1} & \mathcal{A}\_{2} & \cdots & \cdots & \mathcal{A}\_{n-1} & \langle \mathbf{x} \rangle\_{n}\\ \mathcal{A}\_{0} & \beta\_{1} & \beta\_{2} & \cdots & \cdots & \beta\_{n-1} & \beta\_{n}\\ 0 & \beta\_{0} & \binom{n}{1}\mathcal{A}\_{1} & \cdots & \cdots & \binom{n-1}{1}\mathcal{A}\_{n-2} & \binom{n}{1}\mathcal{A}\_{n-1}\\ 0 & 0 & \beta\_{0} & \cdots & \cdots & \binom{n-1}{2}\mathcal{A}\_{n-3} & \binom{n}{2}\mathcal{A}\_{n-2}\\ \vdots & & \ddots & & \vdots & \vdots\\ \vdots & & \ddots & \vdots & \vdots\\ 0 & \cdots & \cdots & \cdots & 0 & \beta\_{0} & \binom{n}{n-1}\mathcal{A}\_{1} \end{bmatrix}, n = 1, 2, \dots \end{cases} \tag{86}$$
*is called Appell polynomial sequence of second kind.*
Then, we have
**Theorem 17.** *For Appell polynomial sequences of second kind we get*
$$
\Delta \mathcal{A}\_{\mathbb{H}} \left( \mathbf{x} \right) = n \mathcal{A}\_{\mathbb{H}-1} \left( \mathbf{x} \right) \quad n = 1, 2, \dots \tag{87}
$$
*Proof.* By the well-known relation ([23])
$$
\Delta\left(\mathbf{x}\right)\_n = n \left(\mathbf{x}\right)\_{n-1}, \quad n = 1, 2, \ldots,\tag{88}
$$
applying the operator Δ to the definition (86) and using the properties of linearity of Δ we have
$$\Delta \mathcal{A}\_{\mathbb{R}}(\mathbf{x}) = \frac{(-1)^{n}}{\left(\boldsymbol{\beta}\_{0}\right)^{n+1}} \begin{vmatrix} \Delta \mathbf{1} \ \Delta \left(\mathbf{x}\right)\_{1} \ \Delta \left(\mathbf{x}\right)\_{2} & \cdots & \cdots & \Delta \left(\mathbf{x}\right)\_{n-1} & \Delta \left(\mathbf{x}\right)\_{n} \\ \boldsymbol{\beta}\_{0} & \boldsymbol{\beta}\_{1} & \boldsymbol{\beta}\_{2} & \cdots & \cdots & \boldsymbol{\beta}\_{n-1} & \boldsymbol{\beta}\_{n} \\ 0 & \boldsymbol{\beta}\_{0} & \binom{2}{1} \boldsymbol{\beta}\_{1} & \cdots & \cdots & \binom{n-1}{1} \boldsymbol{\beta}\_{n-2} & \binom{n}{1} \boldsymbol{\beta}\_{n-1} \\ 0 & 0 & \boldsymbol{\beta}\_{0} & \cdots & \cdots & \binom{n-1}{2} \boldsymbol{\beta}\_{n-3} & \binom{n}{2} \boldsymbol{\beta}\_{n-2} \\ \vdots & & & \ddots & \vdots & \vdots \\ \vdots & & & \ddots & \vdots & \vdots \\ \vdots & & & \ddots & \vdots & \vdots \\ 0 & \cdots & \cdots & \cdots & 0 & \beta\_{0} & \binom{n}{n-1} \boldsymbol{\beta}\_{1} \end{vmatrix}, n = 1, 2, \ldots \tag{89}$$
We can expand the determinant in (89) with respect to the first column and, after multiplying the *<sup>i</sup>*-th row by *<sup>i</sup>* <sup>−</sup> 1, *<sup>i</sup>* <sup>=</sup> 2, ..., *<sup>n</sup>* and the *<sup>j</sup>*-th column by <sup>1</sup> *<sup>j</sup>* , *j* = 1, ..., *n*, we can recognize the factor A*n*−<sup>1</sup> (*x*).
We can observe that the structure of the determinant in (86) is similar to that one of the determinant in (6). In virtue of this it is possible to obtain a dual theory of Appell polynomials of first kind, in the sense that similar properties can be proven ([19]).
For example, the generating function is
$$H(\mathbf{x}, h) = a(h)(1 + h)^{\mathbf{x}},\tag{90}$$
where *a*(*h*) is an invertible formal series of power.
#### **7. Examples of Appell polynomial sequences of second kind**
The following are classical examples of Appell polynomial sequences of second kind.
**a)** Bernoulli polynomials of second kind ([19, 23]):
$$\pounds\_{i} = \frac{(-1)^{l}}{i+1} \text{i!} , \; i = 0, 1, \dots \tag{91}$$
$$H(\mathbf{x}, h) = \frac{h(1+h)^{\mathbf{x}}}{\ln(1+h)};\tag{92}$$
**b)** Boole polynomials ([19, 23]):
$$\begin{array}{c} \{\}\_{i} = \begin{cases} 1, \ i = 0 \\ \frac{1}{2}, \ i = 1 \\ 0, \ i = 2, \ldots \end{cases} \end{array} \tag{93}$$
$$H(\mathbf{x}, h) = \frac{\mathbf{2}(1+h)^{\mathbf{x}}}{\mathbf{2} + h}. \tag{94}$$
#### **8. An application to general linear interpolation problem**
Let *X* be the linear space of real functions defined in the interval [0, 1] continuous and with continuous derivatives of all necessary orders. Let *L* be a linear functional on *X* such that *L*(1) �= 0. If in (6) and respectively in (86) we set
$$\beta\_{\dot{i}} = L(\mathbf{x}^{\dot{i}}), \qquad \beta\_{\dot{i}} = L((\mathbf{x})\_{\dot{i}}), \quad \dot{i} = 0, 1, \dots \tag{95}$$
*An*(*x*) and A*n*(*x*) will be said Appell polynomial sequences of first or of second kind related to the functional *L* and denoted by *AL*,*n*(*x*) and A*L*,*n*(*x*), respectively.
**Remark 8.** *The generating function of the sequence AL*,*n*(*x*) *is*
$$G(\mathbf{x}, h) = \frac{e^{\mathbf{x}h}}{L\_{\mathbf{x}}(e^{\mathbf{x}h})},\tag{96}$$
*and for* A*L*,*n*(*x*) *is*
$$H(\mathbf{x}, h) = \frac{(1 + h)^{\mathbf{x}}}{L\_{\mathbf{x}}((1 + h)^{\mathbf{x}})} \prime \tag{97}$$
*where Lx means that the functional L is applied to the argument as a function of x.*
*Proof.* For *AL*,*n*(*x*) if *<sup>G</sup>*(*x*, *<sup>h</sup>*) = *<sup>a</sup>*(*h*)*exh* with <sup>1</sup> *<sup>a</sup>*(*h*) <sup>=</sup> <sup>∞</sup> ∑ *i*=0 *βi hi <sup>i</sup>*! we have
$$G(\mathbf{x},t) = \frac{e^{\mathbf{x}\mathbf{h}}}{\frac{1}{a(\hbar)}} = \frac{e^{\mathbf{x}\mathbf{h}}}{\sum\_{i=0}^{\infty} \beta\_i \frac{\mathbf{h}^i}{l!}} = \frac{e^{\mathbf{x}\mathbf{h}}}{\sum\_{i=0}^{\infty} L(\mathbf{x}^i) \frac{\mathbf{h}^i}{l!}} = \frac{e^{\mathbf{x}\mathbf{h}}}{L\left(\sum\_{i=0}^{\infty} \mathbf{x}^i \frac{\mathbf{h}^i}{l!}\right)} = \frac{e^{\mathbf{x}\mathbf{h}}}{L\_{\mathbf{x}}(e^{\mathbf{x}\mathbf{h}})}.$$
For A*L*,*n*(*x*), the proof similarly follows.
Then, we have
18 Will-be-set-by-IN-TECH
**7. Examples of Appell polynomial sequences of second kind**
The following are classical examples of Appell polynomial sequences of second kind.
*ßi* <sup>=</sup> (−1)
*ßi* =
**8. An application to general linear interpolation problem**
⎧ ⎨ ⎩ *i*
*<sup>H</sup>*(*x*, *<sup>h</sup>*) = *<sup>h</sup>*(<sup>1</sup> <sup>+</sup> *<sup>h</sup>*)*<sup>x</sup>*
*<sup>H</sup>*(*x*, *<sup>h</sup>*) = <sup>2</sup>(<sup>1</sup> <sup>+</sup> *<sup>h</sup>*)*<sup>x</sup>*
Let *X* be the linear space of real functions defined in the interval [0, 1] continuous and with continuous derivatives of all necessary orders. Let *L* be a linear functional on *X* such that
*An*(*x*) and A*n*(*x*) will be said Appell polynomial sequences of first or of second kind related
*<sup>G</sup>*(*x*, *<sup>h</sup>*) = *<sup>e</sup>xh*
*<sup>H</sup>*(*x*, *<sup>h</sup>*) = (<sup>1</sup> <sup>+</sup> *<sup>h</sup>*)*<sup>x</sup>*
*where Lx means that the functional L is applied to the argument as a function of x.*
*Lx*(*exh*)
*Lx*((1 + *h*)*x*)
*<sup>a</sup>*(*h*) <sup>=</sup> <sup>∞</sup> ∑ *i*=0 *βi hi*
*ln*(1 + *h*)
1, *i* = 0 1 <sup>2</sup> , *i* = 1 0, *i* = 2, ...
*H*(*x*, *h*) = *a*(*h*)(1 + *h*)*x*, (90)
*<sup>i</sup>* <sup>+</sup> <sup>1</sup> *<sup>i</sup>*!, *<sup>i</sup>* <sup>=</sup> 0, 1, ..., (91)
; (92)
<sup>2</sup> <sup>+</sup> *<sup>h</sup>* . (94)
, (96)
, (97)
), *ßi* = *L*((*x*)*i*), *i* = 0, 1, ..., (95)
*<sup>i</sup>*! we have
(93)
For example, the generating function is
**b)** Boole polynomials ([19, 23]):
*and for* A*L*,*n*(*x*) *is*
where *a*(*h*) is an invertible formal series of power.
**a)** Bernoulli polynomials of second kind ([19, 23]):
*L*(1) �= 0. If in (6) and respectively in (86) we set
*Proof.* For *AL*,*n*(*x*) if *<sup>G</sup>*(*x*, *<sup>h</sup>*) = *<sup>a</sup>*(*h*)*exh* with <sup>1</sup>
*β<sup>i</sup>* = *L*(*x<sup>i</sup>*
**Remark 8.** *The generating function of the sequence AL*,*n*(*x*) *is*
to the functional *L* and denoted by *AL*,*n*(*x*) and A*L*,*n*(*x*), respectively.
**Theorem 18.** *Let ω<sup>i</sup>* ∈ **R**, *i* = 0, ..., *n*, *the polynomials*
$$P\_n(\mathbf{x}) = \sum\_{i=0}^n \frac{\omega\_i}{i!} A\_{L,i}(\mathbf{x})\_\prime \tag{98}$$
$$P\_n^\*(\mathbf{x}) = \sum\_{i=0}^n \frac{\omega\_i}{i!} \mathcal{A}\_{L,i}(\mathbf{x}) \tag{99}$$
*are the unique polynomials of degree less than or equal to n*, *such that*
$$L(P\_n^{(i)}) = i!\omega\_{i\prime} \quad i = 0, \ldots, n,\tag{100}$$
$$L(\Delta^l P\_n^\*) = \mathbf{i}!\omega\_{\mathbf{i}\prime} \quad \mathbf{i} = \mathbf{0}, \ldots, n. \tag{101}$$
*Proof.* The proof follows observing that, by the hypothesis on functional *L* there exists a unique polynomial of degree ≤ *n* verifying (100) and , respectively, (101); moreover from the properties of *AL*,*i*(*x*) and A*L*,*i*(*x*), we have
$$L(A\_{L,i}^{(j)}(\\\mathbf{x})) = i(i-1)...(i-j+1)L(A\_{L,i-j}(\\\mathbf{x})) = j! \binom{i}{j} \delta\_{ij} \tag{102}$$
$$L(\Delta^i \mathcal{A}\_{L,i}(\mathbf{x})) = i(i-1)...(i-j+1)L(\mathcal{A}\_{L,i-j}(\mathbf{x})) = j! \binom{i}{j} \delta\_{ij\prime} \tag{103}$$
where *δij* is the Kronecker symbol.
From (102) and (103) it is easy to prove that the polynomials (98) and (99) verify (100) and (101), respectively.
**Remark 9.** *For every linear functional L on X,* {*AL*,*i*(*x*)}, {A*L*,*i*(*x*)}, *i* = 0, ..., *n*, *are basis for* P*<sup>n</sup> and,* ∀*Pn*(*x*) ∈ P*n, we have*
$$P\_n(\mathbf{x}) = \sum\_{i=0}^n \frac{L(P\_n^{(i)})}{i!} \ A\_{L,i}(\mathbf{x}),\tag{104}$$
$$P\_n(\mathbf{x}) = \sum\_{i=0}^n \frac{L(\Delta^i P\_n)}{i!} \, \mathcal{A}\_{L,i}(\mathbf{x}). \tag{105}$$
Let us consider a function *f* ∈ *X*. Then we have the following
#### 20 Will-be-set-by-IN-TECH 40 Linear Algebra – Theorems and Applications
**Theorem 19.** *The polynomials*
$$P\_{L,n}[f](\mathbf{x}) = \sum\_{i=0}^{n} \frac{L(f^{(i)})}{i!} \, A\_{L,i}(\mathbf{x}),\tag{106}$$
$$P\_{L,n}^\*[f](\mathbf{x}) = \sum\_{i=0}^n \frac{L(\Delta^i f)}{i!} \, \mathcal{A}\_{L,i}(\mathbf{x}) \tag{107}$$
*are the unique polynomial of degree* ≤ *n such that*
$$L(P\_{L,n}[f]^{(i)}) = L(f^{(i)}), \ i = 0, \ldots, n,$$
$$L(\Delta^i P\_{L,n}^\*[f]) = L(\Delta^i f), \ i = 0, \ldots, n.$$
*Proof.* Setting *<sup>ω</sup><sup>i</sup>* <sup>=</sup> *<sup>L</sup>*(*<sup>f</sup>* (*i*)) *<sup>i</sup>*! , and respectively, *<sup>ω</sup><sup>i</sup>* <sup>=</sup> *<sup>L</sup>*(Δ*<sup>i</sup> <sup>f</sup>*) *<sup>i</sup>*! , *i* = 0, ..., *n*, the result follows from Theorem 18.
**Definition 5.** *The polynomials (106) and (107) are called Appell interpolation polynomial for f of first and of second kind, respectively.*
Now it is interesting to consider the estimation of the remainders
$$R\_{L, \mathbb{N}}[f](\mathbf{x}) = f(\mathbf{x}) - P\_{L, \mathbb{N}}[f](\mathbf{x}), \ \forall \mathbf{x} \in [0, 1], \tag{108}$$
$$R\_{L, \mathbf{u}}^{\*}[f](\mathbf{x}) = f(\mathbf{x}) - P\_{L, \mathbf{u}}^{\*}[f](\mathbf{x}), \ \forall \mathbf{x} \in [0, 1]. \tag{109}$$
**Remark 10.** *For any f* ∈ P*<sup>n</sup>*
$$R\_{L, \mathbb{N}}[f](\mathbf{x}) = \mathbf{0}, \quad R\_{L, \mathbb{N}}[\mathbf{x}^{n+1}] \neq \mathbf{0}, \text{ } \forall \mathbf{x} \in [\mathbf{0}, \mathbf{1}], \tag{110}$$
$$R\_{L,n}^\*[f](\mathbf{x}) = \mathbf{0}, \quad R\_{L,n}^\*[(\mathbf{x})\_{n+1}] \neq \mathbf{0}, \ \forall \mathbf{x} \in [0,1], \tag{111}$$
*i. e. the polynomial operators (106) and (107) are exact on* P*n.*
For a fixed *x* we may consider the remainder *RL*,*n*[ *f* ](*x*) and *R*<sup>∗</sup> *<sup>L</sup>*,*n*[ *f* ](*x*) as linear functionals which act on *f* and annihilate all elements of P*n*. From Peano's Theorem ([27, p. 69]) if a linear functional has this property, then it must also have a simple representation in terms of *f*(*n*+1). Therefore we have
**Theorem 20.** *Let f* <sup>∈</sup> *<sup>C</sup>n*+<sup>1</sup> [*a*, *<sup>b</sup>*] , *the following relations hold*
$$R\_{L, \mathbb{n}}(f, \mathbf{x}) = \frac{1}{n!} \int\_0^1 K\_{\mathbb{n}}(\mathbf{x}, t) f^{(n+1)}\left(t\right) dt, \quad \forall \mathbf{x} \in \left[0, 1\right], \tag{112}$$
$$R\_{L,n}^\*(f, \mathbf{x}) = \frac{1}{n!} \int\_0^1 K\_n^\*(\mathbf{x}, t) f^{(n+1)}\left(t\right) dt, \quad \forall \mathbf{x} \in \left[0, 1\right], \tag{113}$$
*where*
$$K\_{\mathfrak{n}}(\mathbf{x},t) = \mathcal{R}\_{L,\mathfrak{n}}\left[ (\mathbf{x}-t)\_{+}^{\mathfrak{n}} \right] = (\mathbf{x}-t)\_{+}^{\mathfrak{n}} - \sum\_{i=0}^{\mathfrak{n}} \binom{\mathfrak{n}}{i} L\left( (\mathbf{x}-t)\_{+}^{\mathfrak{n}-i} \right) A\_{L,i}(\mathbf{x})\_{i} \tag{114}$$
40 Linear Algebra – Theorems and Applications Algebraic Theory of Appell Polynomials with Application to General Linear Interpolation Problem <sup>21</sup> Algebraic Theory of Appell Polynomials with Application to General Linear Interpolation Problem 41
$$K\_n^\*(\mathbf{x}, t) = R\_{L, n}^\* \left[ (\mathbf{x} - t)\_+^n \right] = (\mathbf{x} - t)\_+^n - \sum\_{i=0}^n \frac{L \left( \Delta^i (\mathbf{x} - t)\_+^n \right)}{i!} \mathcal{A}\_{L, i}(\mathbf{x}). \tag{115}$$
*Proof.* After some calculation, the results follow by Remark 10 and Peano's Theorem.
**Remark 11** (Bounds)**.** *If f*(*n*+1) ∈ L*p*[0, 1] *and Kn*(*x*, *<sup>t</sup>*), *<sup>K</sup>*<sup>∗</sup> *<sup>n</sup>*(*x*, *<sup>t</sup>*) ∈ L*q*[0, 1] *with* <sup>1</sup> *<sup>p</sup>* <sup>+</sup> <sup>1</sup> *<sup>q</sup>* = 1 *then we apply the Hölder's inequality so that*
$$\begin{aligned} \left| \mathcal{R}\_{L,n}[f](\mathbf{x}) \right| &\leq \frac{1}{n!} \left( \int\_0^1 |\mathcal{K}\_n(\mathbf{x},t)|^q \, dt \right)^{\frac{1}{q}} \left( \int\_0^1 \left| f^{(n+1)} \begin{pmatrix} t \end{pmatrix}^p \, dt \right)^{\frac{1}{p}} \right) \\\ \left| \mathcal{R}\_{L,n}^\*[f](\mathbf{x}) \right| &\leq \frac{1}{n!} \left( \int\_0^1 |\mathcal{K}\_n^\*(\mathbf{x},t)|^q \, dt \right)^{\frac{1}{q}} \left( \int\_0^1 \left| f^{(n+1)} \begin{pmatrix} t \end{pmatrix}^p \, dt \right)^{\frac{1}{p}} .\end{aligned}$$
The two most important cases are *p* = *q* = 2 and *q* = 1, *p* = ∞ :
**i)** for *p* = *q* = 2 we have the estimates
$$|R\_{L,n}[f](\mathbf{x})| \le \sigma\_n \left| ||f|| \right| \, \mathsf{//} \, \_{\prime} \left| R\_{L,n}^\*[f](\mathbf{x}) \right| \le \sigma\_n^\* \left| ||f|| \right| \, \_{\prime} \tag{116}$$
where
20 Will-be-set-by-IN-TECH
*n* ∑ *i*=0
*n* ∑ *i*=0
) = *L*(*f* (*i*)
**Definition 5.** *The polynomials (106) and (107) are called Appell interpolation polynomial for f of first*
which act on *f* and annihilate all elements of P*n*. From Peano's Theorem ([27, p. 69]) if a linear functional has this property, then it must also have a simple representation in terms of *f*(*n*+1).
*L*(*f*(*i*))
*L*(Δ*<sup>i</sup> f*)
*<sup>L</sup>*,*n*[ *<sup>f</sup>* ]) = *<sup>L</sup>*(Δ*<sup>i</sup> <sup>f</sup>*), *<sup>i</sup>* <sup>=</sup> 0, ..., *<sup>n</sup>*.
), *i* = 0, ..., *n*,
*RL*,*n*[ *f* ](*x*) = *f*(*x*) − *PL*,*n*[ *f* ](*x*), ∀*x* ∈ [0, 1], (108)
*RL*,*n*[ *<sup>f</sup>* ](*x*) = 0, *RL*,*n*[*xn*+1] �<sup>=</sup> 0, <sup>∀</sup>*<sup>x</sup>* <sup>∈</sup> [0, 1], (110)
*<sup>i</sup>*! *AL*,*i*(*x*), (106)
*<sup>i</sup>*! <sup>A</sup>*L*,*i*(*x*) (107)
*<sup>i</sup>*! , *i* = 0, ..., *n*, the result follows from
*<sup>L</sup>*,*n*[ *f* ](*x*), ∀*x* ∈ [0, 1]. (109)
*<sup>L</sup>*,*n*[(*x*)*n*+1] �= 0, ∀*x* ∈ [0, 1], (111)
*Kn*(*x*, *<sup>t</sup>*)*f*(*n*+1) (*t*) *dt*, <sup>∀</sup>*<sup>x</sup>* <sup>∈</sup> [0, 1] , (112)
*<sup>n</sup>*(*x*, *<sup>t</sup>*)*f*(*n*+1) (*t*) *dt*, <sup>∀</sup>*<sup>x</sup>* <sup>∈</sup> [0, 1] , (113)
(*<sup>x</sup>* <sup>−</sup> *<sup>t</sup>*)*n*−*<sup>i</sup>* +
*<sup>L</sup>*,*n*[ *f* ](*x*) as linear functionals
*AL*,*i*(*x*), (114)
*PL*,*n*[ *f* ](*x*) =
*<sup>L</sup>*,*n*[ *f* ](*x*) =
(*i*)
*<sup>i</sup>*! , and respectively, *<sup>ω</sup><sup>i</sup>* <sup>=</sup> *<sup>L</sup>*(Δ*<sup>i</sup> <sup>f</sup>*)
*P*∗
*L*(*PL*,*n*[ *f* ]
*L*(Δ*<sup>i</sup> P*∗
Now it is interesting to consider the estimation of the remainders
*<sup>L</sup>*,*n*[ *f* ](*x*) = *f*(*x*) − *P*<sup>∗</sup>
*<sup>L</sup>*,*n*[ *f* ](*x*) = 0, *R*<sup>∗</sup>
For a fixed *x* we may consider the remainder *RL*,*n*[ *f* ](*x*) and *R*<sup>∗</sup>
*n*! 1 0
*n*! 1 0 *K*∗
= (*x* − *t*)
*n* + −
*n* ∑ *i*=0 *n i L*
*R*∗
*R*∗
*i. e. the polynomial operators (106) and (107) are exact on* P*n.*
**Theorem 20.** *Let f* <sup>∈</sup> *<sup>C</sup>n*+<sup>1</sup> [*a*, *<sup>b</sup>*] , *the following relations hold*
*RL*,*n*(*<sup>f</sup>* , *<sup>x</sup>*) = <sup>1</sup>
*<sup>L</sup>*,*n*(*<sup>f</sup>* , *<sup>x</sup>*) = <sup>1</sup>
(*x* − *t*) *n* +
*R*∗
*Kn*(*x*, *t*) = *RL*,*<sup>n</sup>*
*are the unique polynomial of degree* ≤ *n such that*
**Theorem 19.** *The polynomials*
*Proof.* Setting *<sup>ω</sup><sup>i</sup>* <sup>=</sup> *<sup>L</sup>*(*<sup>f</sup>* (*i*))
*and of second kind, respectively.*
**Remark 10.** *For any f* ∈ P*<sup>n</sup>*
Therefore we have
*where*
Theorem 18.
$$(\sigma\_n)^2 = \left(\frac{1}{n!}\right)^2 \int\_0^1 \left(K\_n(\mathbf{x}, t)\right)^2 dt, \quad (\sigma\_n^\*)^2 = \left(\frac{1}{n!}\right)^2 \int\_0^1 \left(K\_n^\*(\mathbf{x}, t)\right)^2 dt,\tag{117}$$
and
$$|||f|||^2 = \int\_0^1 \left( f^{(n+1)}\left(t\right) \right)^2 dt;\tag{118}$$
**ii)** for *q* = 1, *p* = ∞ we have that
$$\left| \left| R\_{L, \mathbb{n}}[f](\mathbf{x}) \right| \leq \frac{1}{n!} M\_{\mathbb{n} + 1} \int\_0^1 \left| K\_{\mathbb{n}}(\mathbf{x}, t) \right| dt, \quad \left| R\_{L, \mathbb{n}}^\*[f](\mathbf{x}) \right| \leq \frac{1}{n!} M\_{\mathbb{n} + 1} \int\_0^1 \left| K\_{\mathbb{n}}^\*(\mathbf{x}, t) \right| dt,\tag{119}$$
where
$$M\_{\mathbb{N}+1} = \sup\_{a \le x \le b} \left| f^{(\mathbb{n}+1)} \left( \mathbf{x} \right) \right|. \tag{120}$$
A further polynomial operator can be determined as follows: for any fixed *z* ∈ [0, 1] we consider the polynomial
$$\overline{P}\_{\text{L,n}}[f](\mathbf{x}) \equiv f(\mathbf{z}) + P\_{\text{L,n}}[f](\mathbf{x}) - P\_{\text{L,n}}[f](\mathbf{z}) = f(\mathbf{z}) + \sum\_{i=1}^{n} \frac{L(f^{(i)})}{i!} \left(A\_{\text{L,i}}(\mathbf{x}) - A\_{\text{L,i}}(\mathbf{z})\right), \tag{121}$$
and, respectively,
$$\left[\overline{P}\_{\mathrm{L},\mathrm{u}}^{\*}[f](\mathbf{x}) \equiv f(\mathbf{z}) + P\_{\mathrm{L},\mathrm{u}}^{\*}[f](\mathbf{x}) - P\_{\mathrm{L},\mathrm{u}}^{\*}[f](\mathbf{z}) = f(\mathbf{z}) + \sum\_{i=1}^{n} \frac{L(\Delta^{i}f)}{i!} \left(\mathcal{A}\_{\mathrm{L},i}(\mathbf{x}) - \mathcal{A}\_{\mathrm{L},i}(\mathbf{z})\right). \tag{122}$$
#### Then we have the following
**Theorem 21.** *The polynomials PL*,*n*[ *f* ](*x*)*, P*<sup>∗</sup> *<sup>L</sup>*,*n*[ *f* ](*x*) *are approximating polynomials of degree n for f*(*x*)*, i.e.:*
$$\forall \mathbf{x} \in [0, 1], \quad f(\mathbf{x}) = \overline{P}\_{L, \mathbb{H}}[f](\mathbf{x}) + \overline{R}\_{L, \mathbb{H}}[f](\mathbf{x}), \tag{123}$$
$$f(\mathbf{x}) = \overline{P}\_{L,n}^\*[f](\mathbf{x}) + \overline{R}\_{L,n}^\*[f](\mathbf{x}),\tag{124}$$
*where*
$$
\overline{R}\_{L,n}[f](\mathbf{x}) = \mathcal{R}\_{L,n}[f](\mathbf{x}) - \mathcal{R}\_{L,n}[f](\mathbf{z}),\tag{125}
$$
$$\overline{R}\_{L,n}^\*[f](\mathbf{x}) = R\_{L,n}^\*[f](\mathbf{x}) - R\_{L,n}^\*[f](\mathbf{z}),\tag{126}$$
*with*
$$
\overline{R}\_{L,\mathbb{M}}[\mathbf{x}^i] = \mathbf{0}, \quad i = \mathbf{0}, \dots \\
\mathbf{n}, \quad \overline{R}\_{L,\mathbb{M}}[\mathbf{x}^{n+1}] \neq \mathbf{0}, \tag{127}
$$
$$
\overline{\mathcal{R}}\_{\mathbf{L},\mathbb{M}}^{\*}[(\mathbf{x})\_{i}] = \mathbf{0}, \quad i = \mathbf{0}, \ldots, n, \quad \overline{\mathcal{R}}\_{\mathbf{L},\mathbb{M}}^{\*}[(\mathbf{x})\_{n+1}] \neq \mathbf{0}. \tag{128}
$$
*Proof.* ∀*x* ∈ [0, 1] and for any fixed *z* ∈ [0, 1], from (108), we have
$$f(\mathbf{x}) - f(z) = P\_{\mathcal{L}, \mathfrak{n}}[f](\mathbf{x}) - P\_{\mathcal{L}, \mathfrak{n}}[f](z) + \mathcal{R}\_{\mathcal{L}, \mathfrak{n}}[f](\mathbf{x}) - \mathcal{R}\_{\mathcal{L}, \mathfrak{n}}[f](z),$$
from which we get (123) and (125). The exactness of the polynomial *PL*,*n*[ *f* ](*x*) follows from the exactness of the polynomial *PL*,*n*[ *f* ](*x*).
Proceeding in the same manner we can prove the result for the polynomial *P*<sup>∗</sup> *<sup>L</sup>*,*n*[ *f* ](*x*).
**Remark 12.** *The polynomials PL*,*n*[ *f* ](*x*)*, P*<sup>∗</sup> *<sup>L</sup>*,*n*[ *f* ](*x*) *satisfy the interpolation conditions*
$$
\overline{P}\_{L,n}[f](z) = f(z), \quad L(\overline{P}\_{L,n}^{(i)}[f]) = L(f^{(i)}), \text{ i } i = 1, \dots, n,\tag{129}
$$
$$
\overline{P}\_{L,n}^\*[f](z) = f(z), \quad L(\Delta^i \overline{P}\_{L,n}^\*[f]) = L(\Delta^i f), \; i = 1, \ldots, n. \tag{130}
$$
#### **9. Examples of Appell interpolation polynomials**
**a)** Taylor interpolation and classical interpolation on equidistant points: Assuming
$$L(f) = f(\mathbf{x}\_0), \qquad \mathbf{x}\_0 \in [0, 1], \tag{131}$$
the polynomials *PL*,*n*[ *f* ](*x*) and *P*<sup>∗</sup> *<sup>L</sup>*,*n*[ *f* ](*x*) are, respectively, the Taylor interpolation polynomial and the classical interpolation polynomial on equidistant points;
- Bernoulli interpolation of first kind ([15, 21]): Assuming
$$L(f) = \int\_0^1 f(\mathbf{x})d\mathbf{x},\tag{132}$$
the interpolation polynomials *PL*,*n*[ *f* ](*x*) and *PL*,*n*[ *f* ](*x*) become
$$P\_{\rm L,n}[f](\mathbf{x}) = \int\_0^1 f(\mathbf{x})d\mathbf{x} + \sum\_{i=1}^n \frac{f^{(i-1)}(1) - f^{(i-1)}(0)}{i!} B\_{\mathbf{i}}(\mathbf{x}) \, \, \, \, \tag{133}$$
42 Linear Algebra – Theorems and Applications Algebraic Theory of Appell Polynomials with Application to General Linear Interpolation Problem <sup>23</sup> Algebraic Theory of Appell Polynomials with Application to General Linear Interpolation Problem 43
$$\overline{P}\_{L,\mathbb{II}}[f](\mathbf{x}) = f(\mathbf{0}) + \sum\_{i=1}^{n} \frac{f^{(i-1)}(\mathbf{1}) - f^{(i-1)}(\mathbf{0})}{i!} \left(\mathcal{B}\_{\mathbf{i}}(\mathbf{x}) - \mathcal{B}\_{\mathbf{i}}(\mathbf{0})\right),\tag{134}$$
where *Bi*(*x*) are the classical Bernoulli polynomials ([17, 23]);
• Bernoulli interpolation of second kind ([19]): Assuming
$$L(f) = \left[D\Delta^{-1}f\right]\_{x=0},\tag{135}$$
where Δ−<sup>1</sup> denote the indefinite summation operator and is defined as the linear operator inverse of the finite difference operator Δ, the interpolation polynomials *P*∗ *<sup>L</sup>*,*n*[ *<sup>f</sup>* ](*x*) and *<sup>P</sup>*<sup>∗</sup> *<sup>L</sup>*,*n*[ *f* ](*x*) become
$$P\_{L,n}^\*[f](\mathbf{x}) = [\Delta^{-1}Df]\_{\mathbf{x}=0} + \sum\_{i=0}^{n-1} f'(i)\mathcal{B}\_{n,i}^{II}(\mathbf{x})\,. \tag{136}$$
$$\overline{P}\_{\mathcal{L},\mathfrak{u}}^{\*}[f](\mathbf{x}) = f(\mathbf{0}) + \sum\_{i=0}^{n-1} f'(i) \left( \mathcal{B}\_{n,i}^{II}(\mathbf{x}) - \mathcal{B}\_{n,i}^{II}(\mathbf{0}) \right), \tag{137}$$
where
22 Will-be-set-by-IN-TECH
*f*(*x*) = *P*<sup>∗</sup>
*<sup>L</sup>*,*n*[ *f* ](*x*) *are approximating polynomials of degree n for*
*<sup>L</sup>*,*n*[ *f* ](*x*), (124)
*<sup>L</sup>*,*n*[ *f* ](*z*), (126)
*<sup>L</sup>*,*n*[(*x*)*n*+1] � 0. (128)
*<sup>L</sup>*,*n*[ *f* ](*x*).
), *i* = 1, ..., *n*, (129)
*<sup>L</sup>*,*n*[ *<sup>f</sup>* ]) = *<sup>L</sup>*(Δ*<sup>i</sup> <sup>f</sup>*), *<sup>i</sup>* = 1, ..., *<sup>n</sup>*. (130)
∀*x* ∈ [0, 1] , *f*(*x*) = *PL*,*n*[ *f* ](*x*) + *RL*,*n*[ *f* ](*x*), (123)
*RL*,*n*[ *f* ](*x*) = *RL*,*n*[ *f* ](*x*) − *RL*,*n*[ *f* ](*z*), (125)
] = 0, *<sup>i</sup>* <sup>=</sup> 0, .., *<sup>n</sup>*, *RL*,*n*[*xn*+1] � 0, (127)
*<sup>L</sup>*,*n*[ *f* ](*x*) *satisfy the interpolation conditions*
*L*(*f*) = *f*(*x*0), *x*<sup>0</sup> ∈ [0, 1], (131)
*<sup>f</sup>*(*i*−<sup>1</sup>)(1) <sup>−</sup> *<sup>f</sup>* (*i*−<sup>1</sup>)(0)
*<sup>L</sup>*,*n*[ *f* ](*x*) are, respectively, the Taylor interpolation
*f*(*x*)*dx*, (132)
*<sup>i</sup>*! *Bi* (*x*), (133)
*<sup>L</sup>*,*n*[ *f* ](*x*) + *R*<sup>∗</sup>
*<sup>L</sup>*,*n*[ *f* ](*x*) − *R*<sup>∗</sup>
*f*(*x*) − *f*(*z*) = *PL*,*n*[ *f* ](*x*) − *PL*,*n*[ *f* ](*z*) + *RL*,*n*[ *f* ](*x*) − *RL*,*n*[ *f* ](*z*),
from which we get (123) and (125). The exactness of the polynomial *PL*,*n*[ *f* ](*x*) follows from
*P*∗
*<sup>L</sup>*,*n*[ *<sup>f</sup>* ]) = *<sup>L</sup>*(*f*(*i*)
Then we have the following
*f*(*x*)*, i.e.:*
*where*
*with*
**Theorem 21.** *The polynomials PL*,*n*[ *f* ](*x*)*, P*<sup>∗</sup>
*R*∗
*RL*,*n*[*x<sup>i</sup>*
*Proof.* ∀*x* ∈ [0, 1] and for any fixed *z* ∈ [0, 1], from (108), we have
*PL*,*n*[ *<sup>f</sup>* ](*z*) = *<sup>f</sup>*(*z*), *<sup>L</sup>*(*P*(*i*)
*<sup>L</sup>*,*n*[ *<sup>f</sup>* ](*z*) = *<sup>f</sup>*(*z*), *<sup>L</sup>*(Δ*<sup>i</sup>*
**a)** Taylor interpolation and classical interpolation on equidistant points:
polynomial and the classical interpolation polynomial on equidistant points;
*L*(*f*) =
*f*(*x*)*dx* +
the interpolation polynomials *PL*,*n*[ *f* ](*x*) and *PL*,*n*[ *f* ](*x*) become
1 0
1 0
*n* ∑ *i*=1
**9. Examples of Appell interpolation polynomials**
*R*∗
the exactness of the polynomial *PL*,*n*[ *f* ](*x*).
**Remark 12.** *The polynomials PL*,*n*[ *f* ](*x*)*, P*<sup>∗</sup>
*P*∗
the polynomials *PL*,*n*[ *f* ](*x*) and *P*<sup>∗</sup>
**b)** Bernoulli interpolation of first and of second kind: • Bernoulli interpolation of first kind ([15, 21]):
*PL*,*n*[ *f* ](*x*) =
Assuming
Assuming
*<sup>L</sup>*,*n*[ *f* ](*x*) = *R*<sup>∗</sup>
*<sup>L</sup>*,*n*[(*x*)*i*] = 0, *i* = 0, .., *n*, *R*<sup>∗</sup>
Proceeding in the same manner we can prove the result for the polynomial *P*<sup>∗</sup>
$$\mathcal{B}\_{n,i}^{II}(\mathbf{x}) = \sum\_{j=i}^{n-1} \binom{j}{i} \frac{(-1)^{j-i}}{(j+1)!} \mathcal{B}\_{j+1}^{II}(\mathbf{x}) \, , \tag{138}$$
and *BI I <sup>j</sup>* (*x*) are the Bernoulli polynomials of second kind ([19]);
- Euler interpolation ([21]): Assuming
$$L(f) = \frac{f(0) + f(1)}{2},\tag{139}$$
the interpolation polynomials *PL*,*n*[ *f* ](*x*) and *PL*,*n*[ *f* ](*x*) become
$$P\_{\mathbf{L},n}[f](\mathbf{x}) = \frac{f(0) + f(1)}{2} + \sum\_{i=1}^{n} \frac{f^{(i)}(0) + f^{(i)}(1)}{2i!} E\_i(\mathbf{x}) \, , \tag{140}$$
$$\overline{P}\_{\mathbf{L},n}[f](\mathbf{x}) = f(\mathbf{0}) + \sum\_{i=1}^{n} \frac{f^{(i)}(\mathbf{0}) + f^{(i)}(\mathbf{1})}{2i!} \left( E\_i \left( \mathbf{x} \right) - E\_i \left( \mathbf{0} \right) \right);\tag{141}$$
• Boole interpolation ([19]): Assuming
$$L(f) = [Mf]\_{\mathbf{x}=\mathbf{0}} \, \prime \tag{142}$$
where *M f* is defined by
$$Mf(\mathbf{x}) = \frac{f(\mathbf{x}) + f(\mathbf{x} + 1)}{2},\tag{143}$$
the interpolation polynomials *P*∗ *<sup>L</sup>*,*n*[ *<sup>f</sup>* ](*x*) and *<sup>P</sup>*<sup>∗</sup> *<sup>L</sup>*,*n*[ *f* ](*x*) become
$$P\_{L, \text{II}}^{\*}[f](\mathbf{x}) = \frac{f(0) + f(1)}{2} \mathcal{E}\_{\mathbf{n}, 0}^{II}(\mathbf{x}) + \sum\_{i=1}^{n} \frac{f(i) + f(i+1)}{2} \mathcal{E}\_{\mathbf{n}, i}^{II}(\mathbf{x}),\tag{144}$$
24 Will-be-set-by-IN-TECH 44 Linear Algebra – Theorems and Applications
$$\left[\overline{P}\_{L,n}^\*[f](\mathbf{x}) = f(0) + \sum\_{i=1}^n \frac{f(i) + f(i+1)}{2} \left(\mathcal{E}\_{n,i}^{II}(\mathbf{x}) - \mathcal{E}\_{n,i}^{II}(0)\right),\tag{145}$$
where
$$\mathcal{E}\_{n,i}^{II}(\mathbf{x}) = \sum\_{j=i}^{n} \binom{j}{i} \frac{(-1)^{j-i}}{j!} E\_j^{II}(\mathbf{x}) \, , \tag{146}$$
and *EI I <sup>j</sup>* (*x*) are the Boole polynomials ([19]).
#### **10. The algebraic approach of Yang and Youn**
Yang and Youn ([18]) also proposed an algebraic approach to Appell polynomial sequences but with different methods. In fact, they referred the Appell sequence, *sn*(*x*), to an invertible analytic function g(t):
$$s\_{\hbar}(\mathbf{x}) = \left[ \frac{d^n}{dt} \left( \frac{1}{\mathbf{g}(t)} e^{\mathbf{x}t} \right) \right]\_{t=0} \text{ \,\, \text{\,} \,\tag{147}}$$
and called Appell vector for *g*(*t*) the vector
$$\overline{S}\_{\mathfrak{n}}(\mathfrak{x}) = \left[ \mathbf{s}\_{0}(\mathfrak{x}), \dots, \mathbf{s}\_{\mathfrak{n}}(\mathfrak{x}) \right]^{T}. \tag{148}$$
Then, they proved that
$$\overline{S}\_{\boldsymbol{\eta}}(\boldsymbol{x}) = P\_{\boldsymbol{\eta}} \left[ \frac{1}{g(t)} \right]\_{t=0} \mathcal{W}\_{\boldsymbol{\eta}} \left[ \boldsymbol{e}^{\mathbf{x}t} \right]\_{t=0} = \mathcal{W}\_{\boldsymbol{\eta}} \left[ \frac{1}{g(t)} \boldsymbol{e}^{\mathbf{x}t} \right]\_{t=0} \tag{149}$$
being *Wn* [ *f*(*t*)] = � *f*(*t*), *f* � (*t*), ..., *f*(*n*)(*t*) �*T* and *Pn*[ *f*(*t*)] the generalized Pascal functional matrix of *f*(*t*) ([28]) defined by
$$(P\_{\boldsymbol{\eta}}[f(t)])\_{i,j} = \begin{cases} \binom{i}{j} f^{(i-j)}(t) & i \ge j \\ 0 & \text{otherwise} \end{cases}, \qquad i, j = 0, \ldots, n. \tag{150}$$
Expressing the (149) in matrix form we have
$$
\overline{S}\_{\mathfrak{n}}(\mathfrak{x}) = SX(\mathfrak{x}),
\tag{151}
$$
with
$$S = \begin{bmatrix} s\_{00} & 0 & 0 & \cdots & 0 \\ s\_{10} & s\_{11} & 0 & \cdots & 0 \\ s\_{20} & s\_{21} & s\_{22} & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ s\_{n0} & s\_{n1} & s\_{n2} & \cdots & s\_{nn} \end{bmatrix}, \quad X(x) = \begin{bmatrix} 1, x, \dots, x^n \end{bmatrix}^T,\tag{152}$$
where
$$s\_{i,j} = \binom{i}{j} \left[ \left( \frac{1}{g(t)} \right)^{(i-j)} \right]\_{t=0}, \qquad i = 0, \ldots, n, \quad j = 0, \ldots, i. \tag{153}$$
It is easy to see that the matrix *S* coincides with the matrix *M*−<sup>1</sup> introduced in Section 5, Theorem 7.
#### **11. Conclusions**
24 Will-be-set-by-IN-TECH
*f*(*i*) + *f*(*i* + 1) 2
> � (−1)*j*−*<sup>i</sup> <sup>j</sup>*! *<sup>E</sup>I I*
� E *I I n*,*i*
*t*=0
� 1 *g*(*t*) *ext* �
*<sup>t</sup>*=<sup>0</sup> = *Wn*
(*x*) − E *I I n*,*i* (0) �
*<sup>j</sup>* (*x*), (146)
, (147)
*<sup>T</sup>* . (148)
, (149)
*<sup>T</sup>* , (152)
*t*=0
and *Pn*[ *f*(*t*)] the generalized Pascal functional
<sup>0</sup> *otherwise* , *<sup>i</sup>*, *<sup>j</sup>* <sup>=</sup> 0, ..., *<sup>n</sup>*. (150)
*Sn*(*x*) = *SX*(*x*), (151)
, *i* = 0, ..., *n*, *j* = 0, ..., *i*. (153)
, *X*(*x*) = [1, *x*, ..., *xn*]
, (145)
*n* ∑ *i*=1
> *n* ∑ *j*=*i*
�*j i*
Yang and Youn ([18]) also proposed an algebraic approach to Appell polynomial sequences but with different methods. In fact, they referred the Appell sequence, *sn*(*x*), to an invertible
> � 1 *g*(*t*) *ext* ��
*Sn*(*x*) = [*s*0(*x*), ...,*sn*(*x*)]
*P*∗
and called Appell vector for *g*(*t*) the vector
� *f*(*t*), *f* �
matrix of *f*(*t*) ([28]) defined by
*Sn*(*x*) = *Pn*
(*Pn*[ *f*(*t*)])*i*,*<sup>j</sup>* =
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
. . . . . . . . . ... . . .
� �� <sup>1</sup>
*g*(*t*)
Expressing the (149) in matrix form we have
*S* =
�*i j*
*si*,*<sup>j</sup>* =
where
and *EI I*
analytic function g(t):
Then, they proved that
being *Wn* [ *f*(*t*)] =
with
where
Theorem 7.
*<sup>L</sup>*,*n*[ *f* ](*x*) = *f*(0) +
*<sup>j</sup>* (*x*) are the Boole polynomials ([19]).
**10. The algebraic approach of Yang and Youn**
E *I I n*,*i* (*x*) =
*sn*(*x*) =
� 1 *g*(*t*) �
(*t*), ..., *f*(*n*)(*t*)
� ( *i j*
*s*<sup>00</sup> 0 0 ··· 0 *s*<sup>10</sup> *s*<sup>11</sup> 0 ··· 0 *s*<sup>20</sup> *s*<sup>21</sup> *s*<sup>22</sup> ··· 0
*sn*<sup>0</sup> *sn*<sup>1</sup> *sn*<sup>2</sup> ··· *snn*
�(*i*−*j*) �
� *dn dt*
*t*=0 *Wn* � *ext*�
�*T*
)*<sup>f</sup>* (*i*−*<sup>j</sup>*)(*t*) *<sup>i</sup>* <sup>≥</sup> *<sup>j</sup>*
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
*t*=0
It is easy to see that the matrix *S* coincides with the matrix *M*−<sup>1</sup> introduced in Section 5,
We have presented an elementary algebraic approach to the theory of Appell polynomials. Given a sequence of real numbers *βi*, *i* = 0, 1, ..., *β*<sup>0</sup> � 0, a polynomial sequence on determinantal form, called of Appell, has been built. The equivalence of this approach with others existing was proven and, almost always using elementary tools of linear algebra, most important properties od Appell polynomials were proven too. A dual theory referred to the finite difference operator Δ has been proposed. This theory has provided a class of polynomials called Appell polynomials of second kind. Finally, given a linear functional *L*, with *L*(1) � 0, and defined
$$L(\mathbf{x}^i) = \beta\_{i\prime} \quad \left(L((\mathbf{x})\_i) = \beta\_i\right), \tag{154}$$
the linear interpolation problem
$$L(P\_n^{(i)}) = i!\omega\_{i\prime} \qquad \left(L(\Delta^i P\_n) = i!\omega\_i\right), \quad P\_n \in \mathcal{P}\_n \quad \omega\_i \in \mathbb{R},\tag{155}$$
has been considered and its solution has been expressed by the basis of Appell polynomials related to the functional *L* by (154). This problem can be extended to appropriate real functions, providing a new approximating polynomial, the remainder of which can be estimated too. This theory is susceptible of extension to the more general class of Sheffer polynomials and to the bi-dimensional case.
#### **Author details**
Costabile Francesco Aldo and Longo Elisabetta *Department of Mathematics, University of Calabria, Rende, CS, Italy.*
#### **12. References**
## **An Interpretation of Rosenbrock's Theorem via Local Rings**
A. Amparan, S. Marcaida and I. Zaballa
Additional information is available at the end of the chapter
http://dx.doi.org/10.5772/81051
**1. Introduction**
26 Will-be-set-by-IN-TECH
[10] Mullin R, Rota G.C (1970) On the foundations of combinatorial theory III. Theory of binomial enumeration. B. Harris (Ed.) Graph Theory and its Applications, Academic
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[20] Fort T (1942) Generalizations of the Bernoulli polynomials and numbers and
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[27] Davis P.J (1975) Interpolation & Approximation. Dover Publication, Inc. New York. [28] Yang Y, Micek C (2007) Generalized Pascal functional matrix and its applications. Linear
polynomials. Rendiconti di matematica e delle sue applicazioni 26: 1-12.
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[23] Jordan C (1965) Calculus of finite difference. Chealsea Pub. Co. New York.
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46 Linear Algebra – Theorems and Applications
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Consider a linear time invariant system
$$
\dot{\mathbf{x}}(t) = A\mathbf{x}(t) + Bu(t) \tag{1}
$$
to be identified with the pair of matrices (*A*, *<sup>B</sup>*) where *<sup>A</sup>* <sup>∈</sup> **<sup>F</sup>***n*×*n*, *<sup>B</sup>* <sup>∈</sup> **<sup>F</sup>***n*×*<sup>m</sup>* and **<sup>F</sup>** <sup>=</sup> **<sup>R</sup>** or **C** the fields of the real or complex numbers. If state-feedback *u*(*t*) = *Fx*(*t*) + *v*(*t*) is applied to system (1), Rosenbrock's Theorem on pole assignment (see [14]) characterizes for the closed-loop system
$$
\dot{\mathbf{x}}(t) = (A + BF)\mathbf{x}(t) + Bv(t), \tag{2}
$$
the invariant factors of its state-space matrix *A* + *BF*. This result can be seen as the solution of an inverse problem; that of finding a non-singular polynomial matrix with prescribed invariant factors and left Wiener–Hopf factorization indices at infinity. To see this we recall that the invariant factors form a complete system of invariants for the finite equivalence of polynomial matrices (this equivalence relation will be revisited in Section 2) and it will be seen in Section 4 that any polynomial matrix is left Wiener–Hopf equivalent at infinity to a diagonal matrix Diag(*sk*<sup>1</sup> ,...,*skm* ), where the non-negative integers *k*1,..., *km* (that can be assumed in non-increasing order) form a complete system of invariants for the left Wiener–Hopf equivalence at infinity. Consider now the transfer function matrix *<sup>G</sup>*(*s*)=(*sI* <sup>−</sup> (*<sup>A</sup>* <sup>+</sup> *BF*))−1*<sup>B</sup>* of (2). This is a rational matrix that can be written as an irreducible matrix fraction description *G*(*s*) = *N*(*s*)*P*(*s*)−1, where *N*(*s*) and *P*(*s*) are right coprime polynomial matrices. In the terminology of [18], *P*(*s*) is a polynomial matrix representation of (2), concept that is closely related to that of polynomial model introduced by Fuhrmann (see for example [8] and the references therein). It turns out that all polynomial matrix representations of a system are right equivalent (see [8, 18]), that is, if *P*1(*s*) and *P*2(*s*) are polynomial matrix representations of the same system there exists a unimodular matrix *U*(*s*) such that *P*2(*s*) = *P*1(*s*)*U*(*s*). Therefore all polynomial matrix representations of (2) have the same invariant factors, which are the invariant factors of *sIn* − (*A* + *BF*) except for some trivial ones. Furthermore, all polynomial
©2012 Amparan et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ©2012 Amparan et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
#### 2 Will-be-set-by-IN-TECH 48 Linear Algebra – Theorems and Applications
matrix representations also have the same left Wiener– Hopf factorization indices at infinity, which are equal to the controllability indices of (2) and (1), because the controllability indices are invariant under feedback. With all this in mind it is not hard to see that Rosenbrock's Theorem on pole assignment is equivalent to finding necessary and sufficient conditions for the existence of a non-singular polynomial matrix with prescribed invariant factors and left Wiener–Hopf factorization indices at infinity. This result will be precisely stated in Section 5 once all the elements that appear are properly defined. In addition, there is a similar result to Rosenbrock's Theorem on pole assignment but involving the infinite structure (see [1]).
Our goal is to generalize both results (the finite and infinite versions of Rosenbrock's Theorem) for rational matrices defined on arbitrary fields via local rings. This will be done in Section 5 and an extension to arbitrary fields of the concept of Wiener–Hopf equivalence will be needed. This concept is very well established for complex valued rational matrix functions (see for example [6, 10]). Originally it requires a closed contour, *γ*, that divides the extended complex plane (**C** ∪ {∞}) into two parts: the inner domain (Ω+) and the region outside *γ* (Ω−), which contains the point at infinity. Then two non-singular *m* × *m* complex rational matrices *T*1(*s*) and *T*2(*s*), with no poles and no zeros in *γ*, are said to be left Wiener–Hopf equivalent with respect to *γ* if there are *m* × *m* matrices *U*−(*s*) and *U*+(*s*) with no poles and no zeros in Ω<sup>−</sup> ∪ *γ* and Ω<sup>+</sup> ∪ *γ*, respectively, such that
$$T\_2(s) = \mathcal{U}\_-(s)T\_1(s)\mathcal{U}\_+(s). \tag{3}$$
It can be seen, then, that any non-singular *m* × *m* complex rational matrix *T*(*s*) is left Wiener–Hopf equivalent with respect to *γ* to a diagonal matrix
$$\text{Diag}\left(\left(\text{s}-\text{z}\_{0}\right)^{k\_{1}}, \dots, \left(\text{s}-\text{z}\_{0}\right)^{k\_{m}}\right) \tag{4}$$
where *z*<sup>0</sup> is any complex number in Ω<sup>+</sup> and *k*<sup>1</sup> ≥··· ≥ *km* are integers uniquely determined by *T*(*s*). They are called the left Wiener–Hopf factorization indices of *T*(*s*) with respect to *γ* (see again [6, 10]). The generalization to arbitrary fields relies on the following idea: We can identify Ω<sup>+</sup> ∪ *γ* and (Ω<sup>−</sup> ∪ *γ*) \ {∞} with two sets *M* and *M*� , respectively, of maximal ideals of **C**[*s*]. In fact, to each *z*<sup>0</sup> ∈ **C** we associate the ideal generated by *s* − *z*0, which is a maximal ideal of **C**[*s*]. Notice that *s* − *z*<sup>0</sup> is also a prime polynomial of **C**[*s*] but *M* and *M*� , as defined, cannot contain the zero ideal, which is prime. Thus we are led to consider the set Specm(**C**[*s*]) of maximal ideals of **C**[*s*]. By using this identification we define the left Wiener–Hopf equivalence of rational matrices over an arbitrary field **F** with respect to a subset *M* of Specm(**F**[*s*]), the set of all maximal ideals of **F**[*s*]. In this study local rings play a fundamental role. They will be introduced in Section 2. Localization techniques have been used previously in the algebraic theory of linear systems (see, for example, [7]). In Section 3 the algebraic structure of the rings of proper rational functions with prescribed finite poles is studied (i.e., for a fixed *M* ⊆ Specm(**F**[*s*]) the ring of proper rational functions *p*(*s*) *<sup>q</sup>*(*s*) with gcd(*g*(*s*), *π*(*s*)) = 1 for all (*π*(*s*)) ∈ *M*). It will be shown that if there is an ideal generated by a linear polynomial outside *M* then the set of proper rational functions with no poles in *M* is an Euclidean domain and all rational matrices can be classified according to their Smith–McMillan invariants. In this case, two types of invariants live together for any non-singular rational matrix and any set *M* ⊆ Specm(**F**[*s*]): its Smith–McMillan and left Wiener–Hopf invariants. In Section 5 we show that a Rosenbrock-like Theorem holds true that completely characterizes the relationship between these two types of invariants.
#### **2. Preliminaries**
2 Will-be-set-by-IN-TECH
matrix representations also have the same left Wiener– Hopf factorization indices at infinity, which are equal to the controllability indices of (2) and (1), because the controllability indices are invariant under feedback. With all this in mind it is not hard to see that Rosenbrock's Theorem on pole assignment is equivalent to finding necessary and sufficient conditions for the existence of a non-singular polynomial matrix with prescribed invariant factors and left Wiener–Hopf factorization indices at infinity. This result will be precisely stated in Section 5 once all the elements that appear are properly defined. In addition, there is a similar result to Rosenbrock's Theorem on pole assignment but involving the infinite structure (see [1]).
Our goal is to generalize both results (the finite and infinite versions of Rosenbrock's Theorem) for rational matrices defined on arbitrary fields via local rings. This will be done in Section 5 and an extension to arbitrary fields of the concept of Wiener–Hopf equivalence will be needed. This concept is very well established for complex valued rational matrix functions (see for example [6, 10]). Originally it requires a closed contour, *γ*, that divides the extended complex plane (**C** ∪ {∞}) into two parts: the inner domain (Ω+) and the region outside *γ* (Ω−), which contains the point at infinity. Then two non-singular *m* × *m* complex rational matrices *T*1(*s*) and *T*2(*s*), with no poles and no zeros in *γ*, are said to be left Wiener–Hopf equivalent with respect to *γ* if there are *m* × *m* matrices *U*−(*s*) and *U*+(*s*) with no poles and no zeros in Ω<sup>−</sup> ∪ *γ*
It can be seen, then, that any non-singular *m* × *m* complex rational matrix *T*(*s*) is left
where *z*<sup>0</sup> is any complex number in Ω<sup>+</sup> and *k*<sup>1</sup> ≥··· ≥ *km* are integers uniquely determined by *T*(*s*). They are called the left Wiener–Hopf factorization indices of *T*(*s*) with respect to *γ* (see again [6, 10]). The generalization to arbitrary fields relies on the following idea: We
ideals of **C**[*s*]. In fact, to each *z*<sup>0</sup> ∈ **C** we associate the ideal generated by *s* − *z*0, which is a maximal ideal of **C**[*s*]. Notice that *s* − *z*<sup>0</sup> is also a prime polynomial of **C**[*s*] but *M* and
*<sup>q</sup>*(*s*) with gcd(*g*(*s*), *π*(*s*)) = 1 for all (*π*(*s*)) ∈ *M*). It will be shown that if there is an ideal generated by a linear polynomial outside *M* then the set of proper rational functions with no poles in *M* is an Euclidean domain and all rational matrices can be classified according to their Smith–McMillan invariants. In this case, two types of invariants live together for any non-singular rational matrix and any set *M* ⊆ Specm(**F**[*s*]): its Smith–McMillan and left Wiener–Hopf invariants. In Section 5 we show that a Rosenbrock-like Theorem holds true that
completely characterizes the relationship between these two types of invariants.
, as defined, cannot contain the zero ideal, which is prime. Thus we are led to consider the set Specm(**C**[*s*]) of maximal ideals of **C**[*s*]. By using this identification we define the left Wiener–Hopf equivalence of rational matrices over an arbitrary field **F** with respect to a subset *M* of Specm(**F**[*s*]), the set of all maximal ideals of **F**[*s*]. In this study local rings play a fundamental role. They will be introduced in Section 2. Localization techniques have been used previously in the algebraic theory of linear systems (see, for example, [7]). In Section 3 the algebraic structure of the rings of proper rational functions with prescribed finite poles is studied (i.e., for a fixed *M* ⊆ Specm(**F**[*s*]) the ring of proper rational functions
(*<sup>s</sup>* <sup>−</sup> *<sup>z</sup>*0)*k*<sup>1</sup> ,...,(*<sup>s</sup>* <sup>−</sup> *<sup>z</sup>*0)*km*
Wiener–Hopf equivalent with respect to *γ* to a diagonal matrix
Diag
can identify Ω<sup>+</sup> ∪ *γ* and (Ω<sup>−</sup> ∪ *γ*) \ {∞} with two sets *M* and *M*�
*T*2(*s*) = *U*−(*s*)*T*1(*s*)*U*+(*s*). (3)
(4)
, respectively, of maximal
and Ω<sup>+</sup> ∪ *γ*, respectively, such that
*M*�
*p*(*s*)
In the sequel **F**[*s*] will denote the ring of polynomials with coefficients in an arbitrary field **F** and Specm(**F**[*s*]) the set of all maximal ideals of **F**[*s*], that is,
$$\text{Spec}(\mathbb{F}[s]) = \left\{ (\pi(s)) : \pi(s) \in \mathbb{F}[s], \text{ irreducible, monic, different from } 1 \right\}. \tag{5}$$
Let *π*(*s*) ∈ **F**[*s*] be a monic irreducible non-constant polynomial. Let *S* = **F**[*s*] \ (*π*(*s*)) be the multiplicative subset of **F**[*s*] whose elements are coprime with *π*(*s*). We denote by **F***π*(*s*) the quotient ring of **F**[*s*] by *S*; i.e., *S*−1**F**[*s*]:
$$\mathcal{F}\_{\pi}(s) = \left\{ \frac{p(s)}{q(s)} : p(s), q(s) \in \mathbb{F}[s], \gcd(q(s), \pi(s)) = 1 \right\}.\tag{6}$$
This is the localization of **F**[*s*] at (*π*(*s*)) (see [5]). The units of **F***π*(*s*) are the rational functions *u*(*s*) = *<sup>p</sup>*(*s*) *<sup>q</sup>*(*s*) such that gcd(*p*(*s*), *π*(*s*)) = 1 and gcd(*q*(*s*), *π*(*s*)) = 1. Consequentially,
$$\mathcal{F}\_{\pi}(s) = \left\{ \mathfrak{u}(s)\pi(s)^d : \mathfrak{u}(s) \text{ is a unit and } d \ge 0 \right\} \cup \{ 0 \}. \tag{7}$$
For any *M* ⊆ Specm(**F**[*s*]), let
$$\begin{array}{lcl} \mathbb{F}\_{M}(\mathbf{s}) = \bigcap\_{(\pi(\mathbf{s})) \in M} \mathbb{F}\_{\pi}(\mathbf{s})\\ = \left\{ \frac{p(\mathbf{s})}{q(\mathbf{s})} : p(\mathbf{s}), q(\mathbf{s}) \in \mathbb{F}[\mathbf{s}], \ \gcd(q(\mathbf{s}), \pi(\mathbf{s})) = 1 \,\forall \,(\pi(\mathbf{s})) \in M \right\}. \end{array} \tag{8}$$
This is a ring whose units are the rational functions *u*(*s*) = *<sup>p</sup>*(*s*) *<sup>q</sup>*(*s*) such that for all ideals (*π*(*s*)) ∈ *M*, gcd(*p*(*s*), *π*(*s*)) = 1 and gcd(*q*(*s*), *π*(*s*)) = 1. Notice that, in particular, if *M* = Specm(**F**[*s*]) then **F***M*(*s*) = **F**[*s*] and if *M* = ∅ then **F***M*(*s*) = **F**(*s*), the field of rational functions.
Moreover, if *α*(*s*) ∈ **F**[*s*] is a non-constant polynomial whose prime factorization, *α*(*s*) = *<sup>k</sup>α*1(*s*)*d*<sup>1</sup> ··· *<sup>α</sup>m*(*s*)*dm* , satisfies the condition that (*αi*(*s*)) <sup>∈</sup> *<sup>M</sup>* for all *<sup>i</sup>*, we will say that *<sup>α</sup>*(*s*) factorizes in *M* or *α*(*s*) has all its zeros in *M*. We will consider that the only polynomials that factorize in *M* = ∅ are the constants. We say that a non-zero rational function factorizes in *M* if both its numerator and denominator factorize in *M*. In this case we will say that the rational function has all its zeros and poles in *M*. Similarly, we will say that *<sup>p</sup>*(*s*) *<sup>q</sup>*(*s*) has no poles in *M* if *p*(*s*) �= 0 and gcd(*q*(*s*), *π*(*s*)) = 1 for all ideals (*π*(*s*)) ∈ *M*. And it has no zeros in *M* if gcd(*p*(*s*), *<sup>π</sup>*(*s*)) = 1 for all ideals (*π*(*s*)) <sup>∈</sup> *<sup>M</sup>*. In other words, it is equivalent that *<sup>p</sup>*(*s*) *<sup>q</sup>*(*s*) has no poles and no zeros in *M* and that *<sup>p</sup>*(*s*) *<sup>q</sup>*(*s*) is a unit of **F***M*(*s*). So, a non-zero rational function factorizes in *<sup>M</sup>* if and only if it is a unit in **<sup>F</sup>**Specm(**F**[*s*])\*M*(*s*).
Let **<sup>F</sup>***M*(*s*)*m*×*<sup>m</sup>* denote the set of *<sup>m</sup>* <sup>×</sup> *<sup>m</sup>* matrices with elements in **<sup>F</sup>***M*(*s*). A matrix is invertible in **F***M*(*s*)*m*×*<sup>m</sup>* if all its elements are in **F***M*(*s*) and its determinant is a unit in **F***M*(*s*). We denote by Gl*m*(**F***M*(*s*)) the group of units of **F***M*(*s*)*m*×*m*.
**Remark 1.** Let *M*1, *M*<sup>2</sup> ⊆ Specm(**F**[*s*]). Notice that
1. If *M*<sup>1</sup> ⊆ *M*<sup>2</sup> then **F***M*<sup>1</sup> (*s*) ⊇ **F***M*<sup>2</sup> (*s*) and Gl*m*(**F***M*<sup>1</sup> (*s*)) ⊇ Gl*m*(**F***M*<sup>2</sup> (*s*)).
2. **<sup>F</sup>***M*1∪*M*<sup>2</sup> (*s*) = **<sup>F</sup>***M*<sup>1</sup> (*s*) ∩ **<sup>F</sup>***M*<sup>2</sup> (*s*) and Gl*m*(**F***M*1∪*M*<sup>2</sup> (*s*)) = Gl*m*(**F***M*<sup>1</sup> (*s*)) ∩ Gl*m*(**F***M*<sup>2</sup> (*s*)).
#### 4 Will-be-set-by-IN-TECH 50 Linear Algebra – Theorems and Applications
For any *M* ⊆ Specm(**F**[*s*]) the ring **F***M*(*s*) is a principal ideal domain (see [3]) and its field of fractions is **<sup>F</sup>**(*s*). Two matrices *<sup>T</sup>*1(*s*), *<sup>T</sup>*2(*s*) <sup>∈</sup> **<sup>F</sup>**(*s*)*m*×*<sup>m</sup>* are equivalent with respect to *<sup>M</sup>* if there exist matrices *U*(*s*), *V*(*s*) ∈ Gl*m*(**F***M*(*s*)) such that *T*2(*s*) = *U*(*s*)*T*1(*s*)*V*(*s*). Since **F***M*(*s*) is a principal ideal domain, for all non-singular *<sup>G</sup>*(*s*) <sup>∈</sup> **<sup>F</sup>***M*(*s*)*m*×*<sup>m</sup>* (see [13]) there exist matrices *U*(*s*), *V*(*s*) ∈ Gl*m*(**F***M*(*s*)) such that
$$G(s) = \mathcal{U}(s) \operatorname{Diag}(\mathfrak{a}\_1(s), \dots, \mathfrak{a}\_{\mathfrak{m}}(s)) V(s) \tag{9}$$
with *α*1(*s*) | ··· | *αm*(*s*) ("|" stands for divisibility) monic polynomials factorizing in *M*, unique up to multiplication by units of **F***M*(*s*). The diagonal matrix is the Smith normal form of *G*(*s*) with respect to *M* and *α*1(*s*),..., *αm*(*s*) are called the invariant factors of *G*(*s*) with respect to *M*. Now we introduce the Smith–McMillan form with respect to *M*. Assume that *<sup>T</sup>*(*s*) <sup>∈</sup> **<sup>F</sup>**(*s*)*m*×*<sup>m</sup>* is a non-singular rational matrix. Then *<sup>T</sup>*(*s*) = *<sup>G</sup>*(*s*) *<sup>d</sup>*(*s*) with *<sup>G</sup>*(*s*) <sup>∈</sup> **<sup>F</sup>***M*(*s*)*m*×*<sup>m</sup>* and *d*(*s*) ∈ **F**[*s*] monic, factorizing in *M*. Let *G*(*s*) = *U*(*s*) Diag(*α*1(*s*),..., *αm*(*s*))*V*(*s*) be the Smith normal form with respect to *M* of *G*(*s*), i.e., *U*(*s*), *V*(*s*) invertible in **F***M*(*s*)*m*×*<sup>m</sup>* and *α*1(*s*) |···| *αm*(*s*) monic polynomials factorizing in *M*. Then
$$T(s) = \mathcal{U}(s)\operatorname{Diag}\left(\frac{\mathfrak{e}\_1(s)}{\psi\_1(s)}, \dots, \frac{\mathfrak{e}\_m(s)}{\psi\_m(s)}\right)V(s) \tag{10}$$
where *�i*(*s*) *<sup>ψ</sup>i*(*s*) are irreducible rational functions, which are the result of dividing *<sup>α</sup>i*(*s*) by *<sup>d</sup>*(*s*) and canceling the common factors. They satisfy that *�*1(*s*) | ··· | *�m*(*s*), *ψm*(*s*) | ··· | *ψ*1(*s*) are monic polynomials factorizing in *M*. The diagonal matrix in (10) is the Smith–McMillan form with respect to *M*. The rational functions *�i*(*s*) *<sup>ψ</sup>i*(*s*), *<sup>i</sup>* = 1, . . . , *<sup>m</sup>*, are called the invariant rational functions of *T*(*s*) with respect to *M* and constitute a complete system of invariants of the equivalence with respect to *M* for rational matrices.
In particular, if *M* = Specm(**F**[*s*]) then **F**Specm(**F**[*s*])(*s*) = **F**[*s*], the matrices *U*(*s*), *V*(*s*) ∈ Gl*m*(**F**[*s*]) are unimodular matrices, (10) is the global Smith–McMillan form of a rational matrix (see [15] or [14] when **F** = **R** or **C**) and *�i*(*s*) *<sup>ψ</sup>i*(*s*) are the global invariant rational functions of *T*(*s*).
From now on rational matrices will be assumed to be non-singular unless the opposite is specified. Given any *M* ⊆ Specm(**F**[*s*]) we say that an *m* × *m* non-singular rational matrix has no zeros and no poles in *M* if its global invariant rational functions are units of **F***M*(*s*). If its global invariant rational functions factorize in *M*, the matrix has its global finite structure localized in *M* and we say that the matrix has all zeros and poles in *M*. The former means that *<sup>T</sup>*(*s*) <sup>∈</sup> Gl*m*(**F***M*(*s*)) and the latter that *<sup>T</sup>*(*s*) <sup>∈</sup> Gl*m*(**F**Specm(**F**[*s*])\*M*(*s*)) because det *T*(*s*) = det *U*(*s*) det *V*(*s*) *�*1(*s*)···*�m*(*s*) *<sup>ψ</sup>*1(*s*)···*ψm*(*s*) and det *<sup>U</sup>*(*s*), det *<sup>V</sup>*(*s*) are non-zero constants. The following result clarifies the relationship between the global finite structure of any rational matrix and its local structure with respect to any *M* ⊆ Specm(**F**[*s*]).
**Proposition 2.** *Let M* <sup>⊆</sup> Specm(**F**[*s*])*. Let T*(*s*) <sup>∈</sup> **<sup>F</sup>**(*s*)*m*×*<sup>m</sup> be non-singular with <sup>α</sup>*1(*s*) *<sup>β</sup>*1(*s*),..., *<sup>α</sup>m*(*s*) *βm*(*s*) *its global invariant rational functions and let �*1(*s*) *<sup>ψ</sup>*1(*s*),..., *�m*(*s*) *<sup>ψ</sup>m*(*s*) *be irreducible rational functions such that �*1(*s*) | ··· | *�m*(*s*)*, ψm*(*s*) | ··· | *ψ*1(*s*) *are monic polynomials factorizing in M. The following properties are equivalent:*
$$T(s) = \mathcal{U}\_1(s) \operatorname{Diag} \left( \frac{\epsilon\_1(s)}{\psi\_1(s)}, \dots, \frac{\epsilon\_m(s)}{\psi\_m(s)} \right) \mathcal{U}\_2(s), \tag{11}$$
*i.e., �*1(*s*) *<sup>ψ</sup>*1(*s*),..., *�m*(*s*) *<sup>ψ</sup>m*(*s*) *are the invariant rational functions of T*(*s*) *with respect to M. 3. αi*(*s*) = *�i*(*s*)*�*� *i* (*s*) *and βi*(*s*) = *ψi*(*s*)*ψ*� *i* (*s*) *with �*� *i* (*s*), *ψ*� *i* (*s*) ∈ **F**[*s*] *units of* **F***M*(*s*)*, for i* = 1, . . . , *m.*
**Proof**.- 1 <sup>⇒</sup> 2. Since the global invariant rational functions of *TL*(*s*) are *�*1(*s*) *<sup>ψ</sup>*1(*s*),..., *�m*(*s*) *<sup>ψ</sup>m*(*s*), there exist *<sup>W</sup>*1(*s*), *<sup>W</sup>*2(*s*) <sup>∈</sup> Gl*m*(**F**[*s*]) such that *TL*(*s*) = *<sup>W</sup>*1(*s*) Diag *�*1(*s*) *<sup>ψ</sup>*1(*s*),..., *�m*(*s*) *ψm*(*s*) *W*2(*s*). As **F**Specm(**F**[*s*])(*s*) = **F**[*s*], by Remark 1.1, *W*1(*s*), *W*2(*s*) ∈ Gl*m*(**F***M*(*s*)). Therefore, putting *U*1(*s*) = *W*1(*s*) and *U*2(*s*) = *W*2(*s*)*TR*(*s*) it follows that *U*1(*s*) and *U*2(*s*) are invertible in **<sup>F</sup>***M*(*s*)*m*×*<sup>m</sup>* and *<sup>T</sup>*(*s*) = *<sup>U</sup>*1(*s*) Diag *�*1(*s*) *<sup>ψ</sup>*1(*s*),..., *�m*(*s*) *ψm*(*s*) *U*2(*s*).
2 ⇒ 3. There exist unimodular matrices *V*1(*s*), *V*2(*s*) ∈ **F**[*s*] *<sup>m</sup>*×*<sup>m</sup>* such that
$$T(s) = V\_1(s) \operatorname{Diag} \left( \frac{\mathfrak{a}\_1(s)}{\mathfrak{F}\_1(s)}, \dots, \frac{\mathfrak{a}\_m(s)}{\mathfrak{F}\_m(s)} \right) V\_2(s) \tag{12}$$
with *<sup>α</sup>i*(*s*) *<sup>β</sup>i*(*s*) irreducible rational functions such that *<sup>α</sup>*1(*s*) |···| *<sup>α</sup>m*(*s*) and *<sup>β</sup>m*(*s*) |···| *<sup>β</sup>*1(*s*) are monic polynomials. Write *<sup>α</sup>i*(*s*) *<sup>β</sup>i*(*s*) <sup>=</sup> *pi*(*s*)*p*� *i* (*s*) *qi*(*s*)*q*� *i* (*s*) such that *pi*(*s*), *qi*(*s*) factorize in *M* and *p*� *i* (*s*), *q*� *i* (*s*) factorize in Specm(**F**[*s*]) \ *M*. Then
$$T(s) = V\_1(s) \operatorname{Diag}\left(\frac{p\_1(s)}{q\_1(s)}, \dots, \frac{p\_m(s)}{q\_m(s)}\right) \operatorname{Diag}\left(\frac{p\_1'(s)}{q\_1'(s)}, \dots, \frac{p\_m'(s)}{q\_m'(s)}\right) V\_2(s) \tag{13}$$
with *<sup>V</sup>*1(*s*) and Diag *<sup>p</sup>*� <sup>1</sup>(*s*) *q*� <sup>1</sup>(*s*) ,..., *<sup>p</sup>*� *<sup>m</sup>*(*s*) *q*� *<sup>m</sup>*(*s*) *V*2(*s*) invertible in **F***M*(*s*)*m*×*m*. Since the Smith–McMillan form with respect to *M* is unique we get that *pi*(*s*) *qi*(*s*) <sup>=</sup> *�i*(*s*) *<sup>ψ</sup>i*(*s*).
3 ⇒ 1. Write (12) as
4 Will-be-set-by-IN-TECH
For any *M* ⊆ Specm(**F**[*s*]) the ring **F***M*(*s*) is a principal ideal domain (see [3]) and its field of fractions is **<sup>F</sup>**(*s*). Two matrices *<sup>T</sup>*1(*s*), *<sup>T</sup>*2(*s*) <sup>∈</sup> **<sup>F</sup>**(*s*)*m*×*<sup>m</sup>* are equivalent with respect to *<sup>M</sup>* if there exist matrices *U*(*s*), *V*(*s*) ∈ Gl*m*(**F***M*(*s*)) such that *T*2(*s*) = *U*(*s*)*T*1(*s*)*V*(*s*). Since **F***M*(*s*) is a principal ideal domain, for all non-singular *<sup>G</sup>*(*s*) <sup>∈</sup> **<sup>F</sup>***M*(*s*)*m*×*<sup>m</sup>* (see [13]) there exist matrices
with *α*1(*s*) | ··· | *αm*(*s*) ("|" stands for divisibility) monic polynomials factorizing in *M*, unique up to multiplication by units of **F***M*(*s*). The diagonal matrix is the Smith normal form of *G*(*s*) with respect to *M* and *α*1(*s*),..., *αm*(*s*) are called the invariant factors of *G*(*s*) with respect to *M*. Now we introduce the Smith–McMillan form with respect to *M*. Assume that
and *d*(*s*) ∈ **F**[*s*] monic, factorizing in *M*. Let *G*(*s*) = *U*(*s*) Diag(*α*1(*s*),..., *αm*(*s*))*V*(*s*) be the Smith normal form with respect to *M* of *G*(*s*), i.e., *U*(*s*), *V*(*s*) invertible in **F***M*(*s*)*m*×*<sup>m</sup>* and
> *�*1(*s*) *ψ*1(*s*)
and canceling the common factors. They satisfy that *�*1(*s*) | ··· | *�m*(*s*), *ψm*(*s*) | ··· | *ψ*1(*s*) are monic polynomials factorizing in *M*. The diagonal matrix in (10) is the Smith–McMillan
rational functions of *T*(*s*) with respect to *M* and constitute a complete system of invariants of
In particular, if *M* = Specm(**F**[*s*]) then **F**Specm(**F**[*s*])(*s*) = **F**[*s*], the matrices *U*(*s*), *V*(*s*) ∈ Gl*m*(**F**[*s*]) are unimodular matrices, (10) is the global Smith–McMillan form of a rational
From now on rational matrices will be assumed to be non-singular unless the opposite is specified. Given any *M* ⊆ Specm(**F**[*s*]) we say that an *m* × *m* non-singular rational matrix has no zeros and no poles in *M* if its global invariant rational functions are units of **F***M*(*s*). If its global invariant rational functions factorize in *M*, the matrix has its global finite structure localized in *M* and we say that the matrix has all zeros and poles in *M*. The former means that *<sup>T</sup>*(*s*) <sup>∈</sup> Gl*m*(**F***M*(*s*)) and the latter that *<sup>T</sup>*(*s*) <sup>∈</sup> Gl*m*(**F**Specm(**F**[*s*])\*M*(*s*)) because
following result clarifies the relationship between the global finite structure of any rational
*that �*1(*s*) | ··· | *�m*(*s*)*, ψm*(*s*) | ··· | *ψ*1(*s*) *are monic polynomials factorizing in M. The following*
*<sup>ψ</sup>*1(*s*),..., *�m*(*s*)
**Proposition 2.** *Let M* <sup>⊆</sup> Specm(**F**[*s*])*. Let T*(*s*) <sup>∈</sup> **<sup>F</sup>**(*s*)*m*×*<sup>m</sup> be non-singular with <sup>α</sup>*1(*s*)
matrix and its local structure with respect to any *M* ⊆ Specm(**F**[*s*]).
*<sup>ψ</sup>i*(*s*) are irreducible rational functions, which are the result of dividing *<sup>α</sup>i*(*s*) by *<sup>d</sup>*(*s*)
,..., *�m*(*s*) *ψm*(*s*) *<sup>ψ</sup>*1(*s*)···*ψm*(*s*) and det *<sup>U</sup>*(*s*), det *<sup>V</sup>*(*s*) are non-zero constants. The
*<sup>T</sup>*(*s*) <sup>∈</sup> **<sup>F</sup>**(*s*)*m*×*<sup>m</sup>* is a non-singular rational matrix. Then *<sup>T</sup>*(*s*) = *<sup>G</sup>*(*s*)
*T*(*s*) = *U*(*s*) Diag
*α*1(*s*) |···| *αm*(*s*) monic polynomials factorizing in *M*. Then
form with respect to *M*. The rational functions *�i*(*s*)
the equivalence with respect to *M* for rational matrices.
matrix (see [15] or [14] when **F** = **R** or **C**) and *�i*(*s*)
det *T*(*s*) = det *U*(*s*) det *V*(*s*) *�*1(*s*)···*�m*(*s*)
*properties are equivalent:*
*its global invariant rational functions and let �*1(*s*)
*G*(*s*) = *U*(*s*) Diag(*α*1(*s*),..., *αm*(*s*))*V*(*s*) (9)
*<sup>d</sup>*(*s*) with *<sup>G</sup>*(*s*) <sup>∈</sup> **<sup>F</sup>***M*(*s*)*m*×*<sup>m</sup>*
*V*(*s*) (10)
*<sup>β</sup>*1(*s*),..., *<sup>α</sup>m*(*s*)
*βm*(*s*)
*<sup>ψ</sup>i*(*s*), *<sup>i</sup>* = 1, . . . , *<sup>m</sup>*, are called the invariant
*<sup>ψ</sup>i*(*s*) are the global invariant rational functions
*<sup>ψ</sup>m*(*s*) *be irreducible rational functions such*
*U*(*s*), *V*(*s*) ∈ Gl*m*(**F***M*(*s*)) such that
where *�i*(*s*)
of *T*(*s*).
$$T(s) = V\_1(s) \operatorname{Diag}\left(\frac{\epsilon\_1(s)}{\psi\_1(s)}, \dots, \frac{\epsilon\_m(s)}{\psi\_m(s)}\right) \operatorname{Diag}\left(\frac{\epsilon'\_1(s)}{\psi'\_1(s)}, \dots, \frac{\epsilon'\_m(s)}{\psi'\_m(s)}\right) V\_2(s) . \tag{14}$$
It follows that *<sup>T</sup>*(*s*) = *TL*(*s*)*TR*(*s*) with *TL*(*s*) = *<sup>V</sup>*1(*s*) Diag *�*1(*s*) *<sup>ψ</sup>*1(*s*),..., *�m*(*s*) *ψm*(*s*) and *TR*(*s*) = Diag *�*� <sup>1</sup>(*s*) *ψ*� 1(*s*),..., *�*� *<sup>m</sup>*(*s*) *ψ*� *<sup>m</sup>*(*s*) *V*2(*s*) ∈ Gl*m*(**F***M*(*s*)).
**Corollary 3.** *Let T*(*s*) <sup>∈</sup> **<sup>F</sup>**(*s*)*m*×*<sup>m</sup> be non-singular and M*1, *<sup>M</sup>*<sup>2</sup> <sup>⊆</sup> Specm(**F**[*s*]) *such that M*<sup>1</sup> <sup>∩</sup> *<sup>M</sup>*<sup>2</sup> <sup>=</sup> <sup>∅</sup>*. If �<sup>i</sup>* <sup>1</sup>(*s*) *ψi* <sup>1</sup>(*s*) ,..., *�<sup>i</sup> <sup>m</sup>*(*s*) *ψi <sup>m</sup>*(*s*) *are the invariant rational functions of T*(*s*) *with respect to Mi, i* <sup>=</sup> 1, 2*, then �*<sup>1</sup> <sup>1</sup> (*s*)*�*<sup>2</sup> <sup>1</sup> (*s*) *ψ*1 <sup>1</sup> (*s*)*ψ*<sup>2</sup> <sup>1</sup> (*s*) ,..., *�*<sup>1</sup> *m*(*s*)*�*<sup>2</sup> *<sup>m</sup>*(*s*) *ψ*1 *m*(*s*)*ψ*<sup>2</sup> *<sup>m</sup>*(*s*) *are the invariant rational functions of T*(*s*) *with respect to M*<sup>1</sup> <sup>∪</sup> *<sup>M</sup>*2*.*
#### 6 Will-be-set-by-IN-TECH 52 Linear Algebra – Theorems and Applications
**Proof**.- Let *<sup>α</sup>*1(*s*) *<sup>β</sup>*1(*s*),..., *<sup>α</sup>m*(*s*) *<sup>β</sup>m*(*s*) be the global invariant rational functions of *<sup>T</sup>*(*s*). By Proposition 2, *αi*(*s*) = *�*<sup>1</sup> *<sup>i</sup>* (*s*)*n*<sup>1</sup> *<sup>i</sup>* (*s*), *<sup>β</sup>i*(*s*) = *<sup>ψ</sup>*<sup>1</sup> *<sup>i</sup>* (*s*)*d*<sup>1</sup> *<sup>i</sup>* (*s*), with *<sup>n</sup>*<sup>1</sup> *<sup>i</sup>* (*s*), *<sup>d</sup>*<sup>1</sup> *<sup>i</sup>* (*s*) ∈ **F**[*s*] units of **F***M*<sup>1</sup> (*s*). On the other hand *αi*(*s*) = *�*<sup>2</sup> *<sup>i</sup>* (*s*)*n*<sup>2</sup> *<sup>i</sup>* (*s*), *<sup>β</sup>i*(*s*) = *<sup>ψ</sup>*<sup>2</sup> *<sup>i</sup>* (*s*)*d*<sup>2</sup> *<sup>i</sup>* (*s*), with *<sup>n</sup>*<sup>2</sup> *<sup>i</sup>* (*s*), *<sup>d</sup>*<sup>2</sup> *<sup>i</sup>* (*s*) ∈ **F**[*s*] units of **F***M*<sup>2</sup> (*s*). So, *�*1 *<sup>i</sup>* (*s*)*n*<sup>1</sup> *<sup>i</sup>* (*s*) = *�*<sup>2</sup> *<sup>i</sup>* (*s*)*n*<sup>2</sup> *<sup>i</sup>* (*s*) or equivalently *<sup>n</sup>*<sup>1</sup> *<sup>i</sup>* (*s*) = *�*<sup>2</sup> *<sup>i</sup>* (*s*)*n*<sup>2</sup> *<sup>i</sup>* (*s*) *�*1 *<sup>i</sup>* (*s*) , *<sup>n</sup>*<sup>2</sup> *<sup>i</sup>* (*s*) = *�*<sup>1</sup> *<sup>i</sup>* (*s*)*n*<sup>1</sup> *<sup>i</sup>* (*s*) *�*2 *<sup>i</sup>* (*s*) . The polynomials *�*1 *<sup>i</sup>* (*s*), *�*<sup>2</sup> *<sup>i</sup>* (*s*) are coprime because *�*<sup>1</sup> *<sup>i</sup>* (*s*) factorizes in *<sup>M</sup>*1, *�*<sup>2</sup> *<sup>i</sup>* (*s*) factorizes in *M*<sup>2</sup> and *M*<sup>1</sup> ∩ *M*<sup>2</sup> = ∅. In consequence *�*<sup>1</sup> *<sup>i</sup>* (*s*) <sup>|</sup> *<sup>n</sup>*<sup>2</sup> *<sup>i</sup>* (*s*) and *�*<sup>2</sup> *<sup>i</sup>* (*s*) <sup>|</sup> *<sup>n</sup>*<sup>1</sup> *<sup>i</sup>* (*s*). Therefore, there exist polynomials *a*(*s*), unit of **F***M*<sup>2</sup> (*s*), and *a*� (*s*), unit of **F***M*<sup>1</sup> (*s*), such that *n*<sup>2</sup> *<sup>i</sup>* (*s*) = *�*<sup>1</sup> *<sup>i</sup>* (*s*)*a*(*s*), *<sup>n</sup>*<sup>1</sup> *<sup>i</sup>* (*s*) = *�*<sup>2</sup> *<sup>i</sup>* (*s*)*a*� (*s*). Since *αi*(*s*) = *�*<sup>1</sup> *<sup>i</sup>* (*s*)*n*<sup>1</sup> *<sup>i</sup>* (*s*) = *�*<sup>1</sup> *<sup>i</sup>* (*s*)*�*<sup>2</sup> *<sup>i</sup>* (*s*)*a*� (*s*) and *αi*(*s*) = *�*<sup>2</sup> *<sup>i</sup>* (*s*)*n*<sup>2</sup> *<sup>i</sup>* (*s*) = *�*<sup>2</sup> *<sup>i</sup>* (*s*)*�*<sup>1</sup> *<sup>i</sup>* (*s*)*a*(*s*). This implies that *a*(*s*) = *a*� (*s*) unit of **<sup>F</sup>***M*<sup>1</sup> (*s*) ∩ **<sup>F</sup>***M*<sup>2</sup> (*s*) = **<sup>F</sup>***M*1∪*M*<sup>2</sup> (*s*). Following the same ideas we can prove that *βi*(*s*) = *ψ*<sup>1</sup> *<sup>i</sup>* (*s*)*ψ*<sup>2</sup> *<sup>i</sup>* (*s*)*b*(*s*) with *<sup>b</sup>*(*s*) a unit of **<sup>F</sup>***M*1∪*M*<sup>2</sup> (*s*). By Proposition 2 *�*1 <sup>1</sup> (*s*)*�*<sup>2</sup> <sup>1</sup> (*s*) *ψ*1 <sup>1</sup> (*s*)*ψ*<sup>2</sup> <sup>1</sup> (*s*) ,..., *�*<sup>1</sup> *m*(*s*)*�*<sup>2</sup> *<sup>m</sup>*(*s*) *ψ*1 *m*(*s*)*ψ*<sup>2</sup> *<sup>m</sup>*(*s*) are the invariant rational functions of *<sup>T</sup>*(*s*) with respect to *<sup>M</sup>*<sup>1</sup> <sup>∪</sup> *<sup>M</sup>*2.
**Corollary 4.** *Let M*1, *M*<sup>2</sup> ⊆ Specm(**F**[*s*])*. Two non-singular matrices are equivalent with respect to M*<sup>1</sup> ∪ *M*<sup>2</sup> *if and only if they are equivalent with respect to M*<sup>1</sup> *and with respect to M*2*.*
**Proof**.- Notice that by Remark 1.2 two matrices *<sup>T</sup>*1(*s*), *<sup>T</sup>*2(*s*) <sup>∈</sup> **<sup>F</sup>**(*s*)*m*×*<sup>m</sup>* are equivalent with respect to *<sup>M</sup>*<sup>1</sup> <sup>∪</sup> *<sup>M</sup>*<sup>2</sup> if and only if there exist *<sup>U</sup>*1(*s*), *<sup>U</sup>*2(*s*) invertible in **<sup>F</sup>***M*<sup>1</sup> (*s*)*m*×*<sup>m</sup>* <sup>∩</sup>**F***M*<sup>2</sup> (*s*)*m*×*<sup>m</sup>* such that *T*2(*s*) = *U*1(*s*)*T*1(*s*)*U*2(*s*). Since *U*1(*s*) and *U*2(*s*) are invertible in both **F***M*<sup>1</sup> (*s*)*m*×*<sup>m</sup>* and **F***M*<sup>2</sup> (*s*)*m*×*<sup>m</sup>* then *T*1(*s*) and *T*2(*s*) are equivalent with respect to *M*<sup>1</sup> and with respect to *M*2.
Conversely, if *T*1(*s*) and *T*2(*s*) are equivalent with respect to *M*<sup>1</sup> and with respect to *M*<sup>2</sup> then, by the necessity of this result, they are equivalent with respect to *M*<sup>1</sup> \ (*M*<sup>1</sup> ∩ *M*2), with respect to *<sup>M</sup>*<sup>2</sup> \ (*M*<sup>1</sup> <sup>∩</sup> *<sup>M</sup>*2) and with respect to *<sup>M</sup>*<sup>1</sup> <sup>∩</sup> *<sup>M</sup>*2. Let *�*<sup>1</sup> <sup>1</sup> (*s*) *ψ*1 <sup>1</sup> (*s*) ,..., *�*<sup>1</sup> *<sup>m</sup>*(*s*) *ψ*1 *<sup>m</sup>*(*s*) be the invariant rational functions of *<sup>T</sup>*1(*s*) and *<sup>T</sup>*2(*s*) with respect to *<sup>M</sup>*<sup>1</sup> \ (*M*<sup>1</sup> <sup>∩</sup> *<sup>M</sup>*2), *�*<sup>2</sup> <sup>1</sup> (*s*) *ψ*2 <sup>1</sup> (*s*) ,..., *�*<sup>2</sup> *<sup>m</sup>*(*s*) *ψ*2 *<sup>m</sup>*(*s*) be the invariant rational functions of *<sup>T</sup>*1(*s*) and *<sup>T</sup>*2(*s*) with respect to *<sup>M</sup>*<sup>2</sup> \ (*M*<sup>1</sup> <sup>∩</sup> *<sup>M</sup>*2) and *�*<sup>3</sup> <sup>1</sup> (*s*) *ψ*3 <sup>1</sup> (*s*) ,..., *�*<sup>3</sup> *<sup>m</sup>*(*s*) *ψ*3 *<sup>m</sup>*(*s*) be the invariant rational functions of *T*1(*s*) and *T*2(*s*) with respect to *M*<sup>1</sup> ∩ *M*2. By Corollary 3 *�*1 <sup>1</sup> (*s*) *ψ*1 <sup>1</sup> (*s*) *�*2 <sup>1</sup> (*s*) *ψ*2 <sup>1</sup> (*s*) *�*3 <sup>1</sup> (*s*) *ψ*3 <sup>1</sup> (*s*) ,..., *�*<sup>1</sup> *<sup>m</sup>*(*s*) *ψ*1 *<sup>m</sup>*(*s*) *�*2 *<sup>m</sup>*(*s*) *ψ*2 *<sup>m</sup>*(*s*) *�*3 *<sup>m</sup>*(*s*) *ψ*3 *<sup>m</sup>*(*s*) must be the invariant rational functions of *<sup>T</sup>*1(*s*) and *<sup>T</sup>*2(*s*) with respect to *M*<sup>1</sup> ∪ *M*2. Therefore, *T*1(*s*) and *T*2(*s*) are equivalent with respect to *M*<sup>1</sup> ∪ *M*2.
Let **F***pr*(*s*) be the ring of proper rational functions, that is, rational functions with the degree of the numerator at most the degree of the denominator. The units in this ring are the rational functions whose numerators and denominators have the same degree. They are called biproper rational functions. A matrix *<sup>B</sup>*(*s*) <sup>∈</sup> **<sup>F</sup>***pr*(*s*)*m*×*<sup>m</sup>* is said to be biproper if it is a unit in **F***pr*(*s*)*m*×*<sup>m</sup>* or, what is the same, if its determinant is a biproper rational function.
Recall that a rational function *t*(*s*) has a pole (zero) at ∞ if *t* <sup>1</sup> *s* has a pole (zero) at 0. Following this idea, we can define the local ring at ∞ as the set of rational functions, *t*(*s*), such that *t* <sup>1</sup> *s* does not have 0 as a pole, that is, **<sup>F</sup>**∞(*s*) = *t*(*s*) ∈ **F**(*s*) : *t* <sup>1</sup> *s* ∈ **F***s*(*s*) . If *t*(*s*) = *<sup>p</sup>*(*s*) *<sup>q</sup>*(*s*) with *<sup>p</sup>*(*s*) = *ats<sup>t</sup>* <sup>+</sup> *at*+1*st*+<sup>1</sup> <sup>+</sup> ··· <sup>+</sup> *apsp*, *ap* �<sup>=</sup> 0, *<sup>q</sup>*(*s*) = *brs<sup>r</sup>* <sup>+</sup> *br*+1*sr*+<sup>1</sup> <sup>+</sup> ··· <sup>+</sup> *bqsq*, *bq* � 0, *<sup>p</sup>* <sup>=</sup> *<sup>d</sup>*(*p*(*s*)), *<sup>q</sup>* <sup>=</sup> *<sup>d</sup>*(*q*(*s*)), where *<sup>d</sup>*(·) stands for "degree of", then
$$t\left(\frac{1}{s}\right) = \frac{\frac{a\_l}{s^l} + \frac{a\_{l+1}}{s^{l+1}} + \dots + \frac{a\_l}{s^l}}{\frac{b\_r}{s^r} + \frac{b\_{r+1}}{s^{r+1}} + \dots + \frac{b\_l}{s^l}} = \frac{a\_l s^{p-t} + a\_{l+1} s^{p-t-1} + \dots + a\_p}{b\_r s^{q-r} + b\_{r+1} s^{q-r-1} + \dots + b\_q} s^{q-p} = \frac{f(s)}{g(s)} s^{q-p}.\tag{15}$$
$$\text{As } \mathbb{F}\_{\mathbf{s}}(\mathbf{s}) = \left\{ \frac{f(s)}{g(s)} s^d : f(0) \neq 0, g(0) \neq 0 \text{ and } d \geq 0 \right\} \cup \{0\}, \text{ then }$$
$$\mathbb{F}\_{\mathbf{so}}(s) = \left\{ \frac{p(s)}{q(s)} \in \mathbb{F}(s) : d(q(s)) \geq d(p(s)) \right\}. \tag{16}$$
Thus, this set is the ring of proper rational functions, **F***pr*(*s*).
6 Will-be-set-by-IN-TECH
*<sup>i</sup>* (*s*), *<sup>d</sup>*<sup>1</sup>
*<sup>i</sup>* (*s*)*n*<sup>2</sup> *<sup>i</sup>* (*s*) *�*1 *<sup>i</sup>* (*s*) , *<sup>n</sup>*<sup>2</sup>
(*s*) and *αi*(*s*) = *�*<sup>2</sup>
*<sup>i</sup>* (*s*), with *<sup>n</sup>*<sup>2</sup>
*<sup>i</sup>* (*s*) = *�*<sup>2</sup>
*<sup>i</sup>* (*s*) factorizes in *<sup>M</sup>*1, *�*<sup>2</sup>
**Corollary 4.** *Let M*1, *M*<sup>2</sup> ⊆ Specm(**F**[*s*])*. Two non-singular matrices are equivalent with respect to*
**Proof**.- Notice that by Remark 1.2 two matrices *<sup>T</sup>*1(*s*), *<sup>T</sup>*2(*s*) <sup>∈</sup> **<sup>F</sup>**(*s*)*m*×*<sup>m</sup>* are equivalent with respect to *<sup>M</sup>*<sup>1</sup> <sup>∪</sup> *<sup>M</sup>*<sup>2</sup> if and only if there exist *<sup>U</sup>*1(*s*), *<sup>U</sup>*2(*s*) invertible in **<sup>F</sup>***M*<sup>1</sup> (*s*)*m*×*<sup>m</sup>* <sup>∩</sup>**F***M*<sup>2</sup> (*s*)*m*×*<sup>m</sup>* such that *T*2(*s*) = *U*1(*s*)*T*1(*s*)*U*2(*s*). Since *U*1(*s*) and *U*2(*s*) are invertible in both **F***M*<sup>1</sup> (*s*)*m*×*<sup>m</sup>* and **F***M*<sup>2</sup> (*s*)*m*×*<sup>m</sup>* then *T*1(*s*) and *T*2(*s*) are equivalent with respect to *M*<sup>1</sup> and with respect to
Conversely, if *T*1(*s*) and *T*2(*s*) are equivalent with respect to *M*<sup>1</sup> and with respect to *M*<sup>2</sup> then, by the necessity of this result, they are equivalent with respect to *M*<sup>1</sup> \ (*M*<sup>1</sup> ∩ *M*2), with respect
the invariant rational functions of *T*1(*s*) and *T*2(*s*) with respect to *M*<sup>1</sup> ∩ *M*2. By Corollary 3
with respect to *M*<sup>1</sup> ∪ *M*2. Therefore, *T*1(*s*) and *T*2(*s*) are equivalent with respect to *M*<sup>1</sup> ∪ *M*2. Let **F***pr*(*s*) be the ring of proper rational functions, that is, rational functions with the degree of the numerator at most the degree of the denominator. The units in this ring are the rational functions whose numerators and denominators have the same degree. They are called biproper rational functions. A matrix *<sup>B</sup>*(*s*) <sup>∈</sup> **<sup>F</sup>***pr*(*s*)*m*×*<sup>m</sup>* is said to be biproper if it is a unit in
Following this idea, we can define the local ring at ∞ as the set of rational functions, *t*(*s*),
*<sup>q</sup>*(*s*) with *<sup>p</sup>*(*s*) = *ats<sup>t</sup>* <sup>+</sup> *at*+1*st*+<sup>1</sup> <sup>+</sup> ··· <sup>+</sup> *apsp*, *ap* �<sup>=</sup> 0, *<sup>q</sup>*(*s*) = *brs<sup>r</sup>* <sup>+</sup> *br*+1*sr*+<sup>1</sup> <sup>+</sup> ··· <sup>+</sup>
*<sup>i</sup>* (*s*) <sup>|</sup> *<sup>n</sup>*<sup>1</sup>
*<sup>i</sup>* (*s*), with *<sup>n</sup>*<sup>1</sup>
*<sup>i</sup>* (*s*)*d*<sup>2</sup>
(*s*), unit of **F***M*<sup>1</sup> (*s*), such that *n*<sup>2</sup>
*M*<sup>1</sup> ∪ *M*<sup>2</sup> *if and only if they are equivalent with respect to M*<sup>1</sup> *and with respect to M*2*.*
*<sup>i</sup>* (*s*)*a*�
*<sup>β</sup>m*(*s*) be the global invariant rational functions of *<sup>T</sup>*(*s*). By Proposition 2,
*<sup>i</sup>* (*s*), *<sup>d</sup>*<sup>2</sup>
*<sup>i</sup>* (*s*) = *�*<sup>1</sup>
*<sup>i</sup>* (*s*) = *�*<sup>1</sup>
(*s*) unit of **<sup>F</sup>***M*<sup>1</sup> (*s*) ∩ **<sup>F</sup>***M*<sup>2</sup> (*s*) = **<sup>F</sup>***M*1∪*M*<sup>2</sup> (*s*). Following the same ideas
*<sup>m</sup>*(*s*) are the invariant rational functions of *<sup>T</sup>*(*s*) with respect to *<sup>M</sup>*<sup>1</sup> <sup>∪</sup> *<sup>M</sup>*2.
<sup>1</sup> (*s*) *ψ*1 <sup>1</sup> (*s*) ,..., *�*<sup>1</sup> *<sup>m</sup>*(*s*) *ψ*1
*<sup>m</sup>*(*s*) must be the invariant rational functions of *<sup>T</sup>*1(*s*) and *<sup>T</sup>*2(*s*)
<sup>1</sup> *s*
*t*(*s*) ∈ **F**(*s*) : *t*
<sup>1</sup> (*s*) *ψ*2 <sup>1</sup> (*s*) ,..., *�*<sup>2</sup> *<sup>m</sup>*(*s*) *ψ*2
*<sup>i</sup>* (*s*)*n*<sup>2</sup>
*<sup>i</sup>* (*s*)*b*(*s*) with *<sup>b</sup>*(*s*) a unit of **<sup>F</sup>***M*1∪*M*<sup>2</sup> (*s*). By Proposition 2
*<sup>i</sup>* (*s*) ∈ **F**[*s*] units of **F***M*<sup>1</sup> (*s*). On the other
*<sup>i</sup>* (*s*)*n*<sup>1</sup> *<sup>i</sup>* (*s*) *�*2
*<sup>i</sup>* (*s*). Therefore, there exist polynomials *a*(*s*),
*<sup>i</sup>* (*s*)*a*(*s*), *<sup>n</sup>*<sup>1</sup>
*<sup>i</sup>* (*s*) = *�*<sup>2</sup>
*<sup>i</sup>* (*s*) ∈ **F**[*s*] units of **F***M*<sup>2</sup> (*s*). So,
*<sup>i</sup>* (*s*) factorizes in *M*<sup>2</sup> and *M*<sup>1</sup> ∩ *M*<sup>2</sup> =
*<sup>i</sup>* (*s*) . The polynomials
*<sup>i</sup>* (*s*) = *�*<sup>2</sup>
*<sup>m</sup>*(*s*) be the invariant rational
<sup>1</sup> (*s*) *ψ*3 <sup>1</sup> (*s*)
*<sup>m</sup>*(*s*) be the invariant
,..., *�*<sup>3</sup> *<sup>m</sup>*(*s*) *ψ*3 *<sup>m</sup>*(*s*) be
has a pole (zero) at 0.
∈ **F***s*(*s*)
. If
<sup>1</sup> *s*
*<sup>i</sup>* (*s*)*�*<sup>1</sup>
*<sup>i</sup>* (*s*)*a*� (*s*).
*<sup>i</sup>* (*s*)*a*(*s*). This
**Proof**.- Let *<sup>α</sup>*1(*s*)
hand *αi*(*s*) = *�*<sup>2</sup>
*<sup>i</sup>* (*s*)*n*<sup>1</sup>
*<sup>i</sup>* (*s*) = *�*<sup>2</sup>
∅. In consequence *�*<sup>1</sup>
unit of **F***M*<sup>2</sup> (*s*), and *a*�
implies that *a*(*s*) = *a*�
,..., *�*<sup>1</sup>
we can prove that *βi*(*s*) = *ψ*<sup>1</sup>
*ψ*1 *m*(*s*)*ψ*<sup>2</sup>
*m*(*s*)*�*<sup>2</sup> *<sup>m</sup>*(*s*)
,..., *�*<sup>1</sup> *<sup>m</sup>*(*s*) *ψ*1 *<sup>m</sup>*(*s*) *�*2 *<sup>m</sup>*(*s*) *ψ*2 *<sup>m</sup>*(*s*) *�*3 *<sup>m</sup>*(*s*) *ψ*3
Since *αi*(*s*) = *�*<sup>1</sup>
*αi*(*s*) = *�*<sup>1</sup>
*�*1 *<sup>i</sup>* (*s*)*n*<sup>1</sup>
*�*1 *<sup>i</sup>* (*s*), *�*<sup>2</sup>
*�*1 <sup>1</sup> (*s*)*�*<sup>2</sup> <sup>1</sup> (*s*)
*ψ*1 <sup>1</sup> (*s*)*ψ*<sup>2</sup> <sup>1</sup> (*s*)
*M*2.
*�*1 <sup>1</sup> (*s*) *ψ*1 <sup>1</sup> (*s*) *�*2 <sup>1</sup> (*s*) *ψ*2 <sup>1</sup> (*s*) *�*3 <sup>1</sup> (*s*) *ψ*3 <sup>1</sup> (*s*)
such that *t*
*t*(*s*) = *<sup>p</sup>*(*s*)
<sup>1</sup> *s*
*<sup>β</sup>*1(*s*),..., *<sup>α</sup>m*(*s*)
*<sup>i</sup>* (*s*)*n*<sup>2</sup>
*<sup>i</sup>* (*s*) are coprime because *�*<sup>1</sup>
*<sup>i</sup>* (*s*)*n*<sup>1</sup>
*<sup>i</sup>* (*s*) <sup>|</sup> *<sup>n</sup>*<sup>2</sup>
*<sup>i</sup>* (*s*) = *�*<sup>1</sup>
to *<sup>M</sup>*<sup>2</sup> \ (*M*<sup>1</sup> <sup>∩</sup> *<sup>M</sup>*2) and with respect to *<sup>M</sup>*<sup>1</sup> <sup>∩</sup> *<sup>M</sup>*2. Let *�*<sup>1</sup>
functions of *<sup>T</sup>*1(*s*) and *<sup>T</sup>*2(*s*) with respect to *<sup>M</sup>*<sup>1</sup> \ (*M*<sup>1</sup> <sup>∩</sup> *<sup>M</sup>*2), *�*<sup>2</sup>
Recall that a rational function *t*(*s*) has a pole (zero) at ∞ if *t*
rational functions of *<sup>T</sup>*1(*s*) and *<sup>T</sup>*2(*s*) with respect to *<sup>M</sup>*<sup>2</sup> \ (*M*<sup>1</sup> <sup>∩</sup> *<sup>M</sup>*2) and *�*<sup>3</sup>
**F***pr*(*s*)*m*×*<sup>m</sup>* or, what is the same, if its determinant is a biproper rational function.
does not have 0 as a pole, that is, **F**∞(*s*) =
*<sup>i</sup>* (*s*)*n*<sup>2</sup>
*<sup>i</sup>* (*s*), *<sup>β</sup>i*(*s*) = *<sup>ψ</sup>*<sup>1</sup>
*<sup>i</sup>* (*s*)*d*<sup>1</sup>
*<sup>i</sup>* (*s*) or equivalently *<sup>n</sup>*<sup>1</sup>
*<sup>i</sup>* (*s*) and *�*<sup>2</sup>
*<sup>i</sup>* (*s*)*�*<sup>2</sup>
*<sup>i</sup>* (*s*)*ψ*<sup>2</sup>
*<sup>i</sup>* (*s*), *<sup>β</sup>i*(*s*) = *<sup>ψ</sup>*<sup>2</sup>
Two rational matrices *<sup>T</sup>*1(*s*), *<sup>T</sup>*2(*s*) <sup>∈</sup> **<sup>F</sup>**(*s*)*m*×*<sup>m</sup>* are equivalent at infinity if there exist biproper matrices *B*1(*s*), *B*2(*s*) ∈ Gl*m*(**F***pr*(*s*)) such that *T*2(*s*) = *B*1(*s*)*T*1(*s*)*B*2(*s*). Given a non-singular rational matrix *<sup>T</sup>*(*s*) <sup>∈</sup> **<sup>F</sup>**(*s*)*m*×*<sup>m</sup>* (see [15]) there always exist *<sup>B</sup>*1(*s*), *<sup>B</sup>*2(*s*) <sup>∈</sup> Gl*m*(**F***pr*(*s*)) such that
$$T(\mathbf{s}) = B\_1(\mathbf{s}) \operatorname{Diag}(\mathbf{s}^{q\_1}, \dots, \mathbf{s}^{q\_m}) B\_2(\mathbf{s}) \tag{17}$$
where *q*<sup>1</sup> ≥ ··· ≥ *qm* are integers. They are called the invariant orders of *T*(*s*) at infinity and the rational functions *sq*<sup>1</sup> ,...,*sqm* are called the invariant rational functions of *T*(*s*) at infinity.
## **3. Structure of the ring of proper rational functions with prescribed finite poles**
Let *<sup>M</sup>*� <sup>⊆</sup> Specm(**F**[*s*]). Any non-zero rational function *<sup>t</sup>*(*s*) can be uniquely written as *<sup>t</sup>*(*s*) = *<sup>n</sup>*(*s*) *d*(*s*) *n*� (*s*) *<sup>d</sup>*�(*s*) where *<sup>n</sup>*(*s*) *<sup>d</sup>*(*s*) is an irreducible rational function factorizing in *<sup>M</sup>*� and *<sup>n</sup>*� (*s*) *<sup>d</sup>*�(*s*) is a unit of **F***M*�(*s*). Define the following function over **F**(*s*) \ {0} (see [15], [16]):
$$\begin{array}{ccc} \delta: \mathbb{F}(s) \mid \{0\} \to & \mathbb{Z} \\ t(s) & \mapsto d(d'(s)) - d(n'(s)). \end{array} \tag{18}$$
This mapping is not a discrete valuation of **F**(*s*) if *M*� � ∅: Given two non-zero elements *t*1(*s*), *t*2(*s*) ∈ **F**(*s*) it is clear that *δ*(*t*1(*s*)*t*2(*s*)) = *δ*(*t*1(*s*)) + *δ*(*t*2(*s*)); but it may not satisfy that *δ*(*t*1(*s*) + *t*2(*s*)) ≥ min(*δ*(*t*1(*s*)), *δ*(*t*2(*s*))). For example, let *M*� = {(*s* − *a*) ∈ Specm(**R**[*s*]) : *a* ∈/ [−2, <sup>−</sup>1]}. Put *<sup>t</sup>*1(*s*) = *<sup>s</sup>*+0.5 *<sup>s</sup>*+1.5 and *<sup>t</sup>*2(*s*) = *<sup>s</sup>*+2.5 *<sup>s</sup>*+1.5 . We have that *<sup>δ</sup>*(*t*1(*s*)) = *<sup>d</sup>*(*<sup>s</sup>* <sup>+</sup> 1.5) <sup>−</sup> *<sup>d</sup>*(1) = 1, *δ*(*t*2(*s*)) = *d*(*s* + 1.5) − *d*(1) = 1 but *δ*(*t*1(*s*) + *t*2(*s*)) = *δ*(2) = 0.
However, if *M*� = ∅ and *t*(*s*) = *<sup>n</sup>*(*s*) *<sup>d</sup>*(*s*) ∈ **F**(*s*) where *n*(*s*), *d*(*s*) ∈ **F**[*s*], *d*(*s*) � 0, the map
$$\delta\_{\infty} \colon \mathbb{F}(s) \to \mathbb{Z} \cup \{+\infty\} \tag{19}$$
defined via *δ*∞(*t*(*s*)) = *d*(*d*(*s*)) − *d*(*n*(*s*)) if *t*(*s*) � 0 and *δ*∞(*t*(*s*)) = +∞ if *t*(*s*) = 0 is a discrete valuation of **F**(*s*).
Consider the subset of **F**(*s*), **F***M*�(*s*) ∩ **F***pr*(*s*), consisting of all proper rational functions with poles in Specm(**F**[*s*]) \ *M*� , that is, the elements of **F***M*�(*s*) ∩ **F***pr*(*s*) are proper rational functions whose denominators are coprime with all the polynomials *π*(*s*) such that (*π*(*s*)) ∈ *M*� . Notice that *<sup>g</sup>*(*s*) <sup>∈</sup> **<sup>F</sup>***M*�(*s*) <sup>∩</sup> **<sup>F</sup>***pr*(*s*) if and only if *<sup>g</sup>*(*s*) = *<sup>n</sup>*(*s*) *<sup>n</sup>*� (*s*) *<sup>d</sup>*�(*s*) where:
(a) *n*(*s*) ∈ **F**[*s*] is a polynomial factorizing in *M*� ,
(b) *<sup>n</sup>*� (*s*) *<sup>d</sup>*�(*s*) is an irreducible rational function and a unit of **F***M*�(*s*),
(c) *δ*(*g*(*s*)) − *d*(*n*(*s*)) ≥ 0 or equivalently *δ*∞(*g*(*s*)) ≥ 0.
In particular (*c*) implies that *<sup>n</sup>*� (*s*) *<sup>d</sup>*�(*s*) ∈ **F***pr*(*s*). The units in **F***M*�(*s*) ∩ **F***pr*(*s*) are biproper rational functions *<sup>n</sup>*� (*s*) *<sup>d</sup>*�(*s*) , that is *d*(*n*� (*s*)) = *d*(*d*� (*s*)), with *n*� (*s*), *d*� (*s*) factorizing in Specm(**F**[*s*]) \ *M*� . Furthermore, **F***M*�(*s*) ∩ **F***pr*(*s*) is an integral domain whose field of fractions is **F**(*s*) provided that *M*� �= Specm(**F**[*s*])(see, for example, [15, Prop.5.22]). Notice that for *M*� = Specm(**F**[*s*]), **F***M*�(*s*) ∩ **F***pr*(*s*) = **F**[*s*] ∩ **F***pr*(*s*) = **F**.
Assume that there are ideals in Specm(**F**[*s*]) \ *M*� generated by linear polynomials and let (*s* − *<sup>a</sup>*) be any of them. The elements of **<sup>F</sup>***M*�(*s*) <sup>∩</sup> **<sup>F</sup>***pr*(*s*) can be written as *<sup>g</sup>*(*s*) = *<sup>n</sup>*(*s*)*u*(*s*) <sup>1</sup> (*s*−*a*)*<sup>d</sup>* where *n*(*s*) ∈ **F**[*s*] factorizes in *M*� , *u*(*s*) is a unit in **F***M*�(*s*) ∩ **F***pr*(*s*) and *d* = *δ*(*g*(*s*)) ≥ *d*(*n*(*s*)). If **F** is algebraically closed, for example **F** = **C**, and *M*� �= Specm(**F**[*s*]) the previous condition is always fulfilled.
The divisibility in **F***M*�(*s*) ∩ **F***pr*(*s*) is characterized in the following lemma.
**Lemma 5.** *Let M*� ⊆ Specm(**F**[*s*])*. Let g*1(*s*), *g*2(*s*) ∈ **F***M*�(*s*) ∩ **F***pr*(*s*) *be such that g*1(*s*) = *<sup>n</sup>*1(*s*) *<sup>n</sup>*� <sup>1</sup>(*s*) *d*� <sup>1</sup>(*s*) *and g*2(*s*) = *<sup>n</sup>*2(*s*) *<sup>n</sup>*� <sup>2</sup>(*s*) *d*� <sup>2</sup>(*s*) *with n*1(*s*), *<sup>n</sup>*2(*s*) <sup>∈</sup> **<sup>F</sup>**[*s*] *factorizing in M*� *and <sup>n</sup>*� <sup>1</sup>(*s*) *d*� <sup>1</sup>(*s*) , *<sup>n</sup>*� <sup>2</sup>(*s*) *d*� <sup>2</sup>(*s*) *irreducible rational functions, units of* **F***M*�(*s*)*. Then g*1(*s*) *divides g*2(*s*) *in* **F***M*�(*s*) ∩ **F***pr*(*s*) *if and only if*
$$n\_1(\mathbf{s}) \mid n\_2(\mathbf{s}) \text{ in } \mathbb{F}[\mathbf{s}] \tag{20}$$
$$
\delta(\mathcal{g}\_1(\mathbf{s})) - d(n\_1(\mathbf{s})) \le \delta(\mathcal{g}\_2(\mathbf{s})) - d(n\_2(\mathbf{s})).\tag{21}
$$
**Proof**.- If *<sup>g</sup>*1(*s*) <sup>|</sup> *<sup>g</sup>*2(*s*) then there exists *<sup>g</sup>*(*s*) = *<sup>n</sup>*(*s*) *<sup>n</sup>*� (*s*) *<sup>d</sup>*�(*s*) ∈ **F***M*�(*s*) ∩ **F***pr*(*s*), with *n*(*s*) ∈ **F**[*s*] factorizing in *M*� and *n*� (*s*), *d*� (*s*) ∈ **F**[*s*] coprime, factorizing in Specm(**F**[*s*]) \ *M*� , such that *<sup>g</sup>*2(*s*) = *<sup>g</sup>*(*s*)*g*1(*s*). Equivalently, *<sup>n</sup>*2(*s*) *<sup>n</sup>*� <sup>2</sup>(*s*) *d*� <sup>2</sup>(*s*) <sup>=</sup> *<sup>n</sup>*(*s*) *<sup>n</sup>*� (*s*) *<sup>d</sup>*�(*s*) *<sup>n</sup>*1(*s*) *<sup>n</sup>*� <sup>1</sup>(*s*) *d*� <sup>1</sup>(*s*) <sup>=</sup> *<sup>n</sup>*(*s*)*n*1(*s*) *<sup>n</sup>*� (*s*)*n*� <sup>1</sup>(*s*) *d*�(*s*)*d*� <sup>1</sup>(*s*) . So *n*2(*s*) = *n*(*s*)*n*1(*s*) and *δ*(*g*2(*s*)) − *d*(*n*2(*s*)) = *δ*(*g*(*s*)) − *d*(*n*(*s*)) + *δ*(*g*1(*s*)) − *d*(*n*1(*s*)). Moreover, as *g*(*s*) is a proper rational function, *δ*(*g*(*s*)) − *d*(*n*(*s*)) ≥ 0 and *δ*(*g*2(*s*)) − *d*(*n*2(*s*)) ≥ *δ*(*g*1(*s*)) − *d*(*n*1(*s*)).
Conversely, if *n*1(*s*) | *n*2(*s*) then there is *n*(*s*) ∈ **F**[*s*], factorizing in *M*� , such that *n*2(*s*) = *<sup>n</sup>*(*s*)*n*1(*s*). Write *<sup>g</sup>*(*s*) = *<sup>n</sup>*(*s*) *<sup>n</sup>*� (*s*) *<sup>d</sup>*�(*s*) where *<sup>n</sup>*� (*s*) *<sup>d</sup>*�(*s*) is an irreducible fraction representation of *n*� 2(*s*)*d*� <sup>1</sup>(*s*) *d*� 2(*s*)*n*� 1(*s*), i.e., *<sup>n</sup>*� (*s*) *<sup>d</sup>*�(*s*) <sup>=</sup> *<sup>n</sup>*� 2(*s*)*d*� <sup>1</sup>(*s*) *d*� 2(*s*)*n*� <sup>1</sup>(*s*) after canceling possible common factors. Thus *<sup>n</sup>*� <sup>2</sup>(*s*) *d*� <sup>2</sup>(*s*) <sup>=</sup> *<sup>n</sup>*� (*s*) *d*�(*s*) *n*� <sup>1</sup>(*s*) *d*� <sup>1</sup>(*s*) and
$$\begin{array}{l} \delta(\mathcal{g}(\mathbf{s})) - d(n(\mathbf{s})) = d(d'(\mathbf{s})) - d(n'(\mathbf{s})) - d(n(\mathbf{s})) \\ = d(d\_2'(\mathbf{s})) + d(n\_1'(\mathbf{s})) - d(n\_2'(\mathbf{s})) - d(d\_1'(\mathbf{s})) - d(n\_2(\mathbf{s})) + d(n\_1(\mathbf{s})) \\ = \delta(\mathcal{g}\_2(\mathbf{s})) - d(n\_2(\mathbf{s})) - (\delta(\mathcal{g}\_1(\mathbf{s})) - d(n\_1(\mathbf{s}))) \ge 0. \end{array} \tag{22}$$
Then *g*(*s*) ∈ **F***M*�(*s*) ∩ **F***pr*(*s*) and *g*2(*s*) = *g*(*s*)*g*1(*s*).
Notice that condition (20) means that *g*1(*s*) | *g*2(*s*) in **F***M*�(*s*) and condition (21) means that *g*1(*s*) | *g*2(*s*) in **F***pr*(*s*). So, *g*1(*s*) | *g*2(*s*) in **F***M*�(*s*) ∩ **F***pr*(*s*) if and only if *g*1(*s*) | *g*2(*s*) simultaneously in **F***M*�(*s*) and **F***pr*(*s*).
8 Will-be-set-by-IN-TECH
(*s*)), with *n*�
Furthermore, **F***M*�(*s*) ∩ **F***pr*(*s*) is an integral domain whose field of fractions is **F**(*s*) provided that *M*� �= Specm(**F**[*s*])(see, for example, [15, Prop.5.22]). Notice that for *M*� = Specm(**F**[*s*]),
Assume that there are ideals in Specm(**F**[*s*]) \ *M*� generated by linear polynomials and let (*s* − *<sup>a</sup>*) be any of them. The elements of **<sup>F</sup>***M*�(*s*) <sup>∩</sup> **<sup>F</sup>***pr*(*s*) can be written as *<sup>g</sup>*(*s*) = *<sup>n</sup>*(*s*)*u*(*s*) <sup>1</sup>
If **F** is algebraically closed, for example **F** = **C**, and *M*� �= Specm(**F**[*s*]) the previous condition
**Lemma 5.** *Let M*� ⊆ Specm(**F**[*s*])*. Let g*1(*s*), *g*2(*s*) ∈ **F***M*�(*s*) ∩ **F***pr*(*s*) *be such that g*1(*s*) =
*irreducible rational functions, units of* **F***M*�(*s*)*. Then g*1(*s*) *divides g*2(*s*) *in* **F***M*�(*s*) ∩ **F***pr*(*s*) *if and*
<sup>2</sup>(*s*) *d*�
So *n*2(*s*) = *n*(*s*)*n*1(*s*) and *δ*(*g*2(*s*)) − *d*(*n*2(*s*)) = *δ*(*g*(*s*)) − *d*(*n*(*s*)) + *δ*(*g*1(*s*)) − *d*(*n*1(*s*)). Moreover, as *g*(*s*) is a proper rational function, *δ*(*g*(*s*)) − *d*(*n*(*s*)) ≥ 0 and *δ*(*g*2(*s*)) −
(*s*)
(*s*)) − *d*(*n*(*s*))
= *δ*(*g*2(*s*)) − *d*(*n*2(*s*)) − (*δ*(*g*1(*s*)) − *d*(*n*1(*s*))) ≥ 0.
<sup>1</sup>(*s*)) − *d*(*n*�
The divisibility in **F***M*�(*s*) ∩ **F***pr*(*s*) is characterized in the following lemma.
Conversely, if *n*1(*s*) | *n*2(*s*) then there is *n*(*s*) ∈ **F**[*s*], factorizing in *M*�
(*s*) *<sup>d</sup>*�(*s*) where *<sup>n</sup>*�
(*s*)) − *d*(*n*�
<sup>2</sup>(*s*)) + *d*(*n*�
<sup>2</sup>(*s*) *d*�
**Proof**.- If *<sup>g</sup>*1(*s*) <sup>|</sup> *<sup>g</sup>*2(*s*) then there exists *<sup>g</sup>*(*s*) = *<sup>n</sup>*(*s*) *<sup>n</sup>*�
that *<sup>g</sup>*2(*s*) = *<sup>g</sup>*(*s*)*g*1(*s*). Equivalently, *<sup>n</sup>*2(*s*) *<sup>n</sup>*�
2(*s*)*d*� <sup>1</sup>(*s*)
= *d*(*d*�
Then *g*(*s*) ∈ **F***M*�(*s*) ∩ **F***pr*(*s*) and *g*2(*s*) = *g*(*s*)*g*1(*s*).
*d*� 2(*s*)*n*�
(*s*), *d*�
,
*<sup>d</sup>*�(*s*) ∈ **F***pr*(*s*). The units in **F***M*�(*s*) ∩ **F***pr*(*s*) are biproper rational
, *u*(*s*) is a unit in **F***M*�(*s*) ∩ **F***pr*(*s*) and *d* = *δ*(*g*(*s*)) ≥ *d*(*n*(*s*)).
*n*1(*s*) | *n*2(*s*) *in* **F**[*s*] (20)
*<sup>d</sup>*�(*s*) ∈ **F***M*�(*s*) ∩ **F***pr*(*s*), with *n*(*s*) ∈ **F**[*s*]
<sup>1</sup>(*s*) <sup>=</sup> *<sup>n</sup>*(*s*)*n*1(*s*) *<sup>n</sup>*�
<sup>1</sup>(*s*) *d*�
*<sup>d</sup>*�(*s*) is an irreducible fraction representation of
<sup>1</sup>(*s*)) − *d*(*n*2(*s*)) + *d*(*n*1(*s*))
<sup>2</sup>(*s*) *with n*1(*s*), *<sup>n</sup>*2(*s*) <sup>∈</sup> **<sup>F</sup>**[*s*] *factorizing in M*� *and <sup>n</sup>*�
*δ*(*g*1(*s*)) − *d*(*n*1(*s*)) ≤ *δ*(*g*2(*s*)) − *d*(*n*2(*s*)). (21)
(*s*) ∈ **F**[*s*] coprime, factorizing in Specm(**F**[*s*]) \ *M*�
(*s*) *<sup>d</sup>*�(*s*) *<sup>n</sup>*1(*s*) *<sup>n</sup>*�
(*s*)
<sup>2</sup>(*s*) <sup>=</sup> *<sup>n</sup>*(*s*) *<sup>n</sup>*�
<sup>1</sup>(*s*) after canceling possible common factors. Thus *<sup>n</sup>*�
<sup>2</sup>(*s*)) − *d*(*d*�
(*s*) factorizing in Specm(**F**[*s*]) \ *M*�
.
(*s*−*a*)*<sup>d</sup>*
, such
(*s*)*n*� <sup>1</sup>(*s*) *d*�(*s*)*d*� <sup>1</sup>(*s*) .
(*s*) *d*�(*s*) *n*� <sup>1</sup>(*s*) *d*� <sup>1</sup>(*s*)
(22)
, such that *n*2(*s*) =
<sup>2</sup>(*s*) *d*� <sup>2</sup>(*s*) <sup>=</sup> *<sup>n</sup>*�
<sup>1</sup>(*s*) *d*� <sup>1</sup>(*s*) , *<sup>n</sup>*� <sup>2</sup>(*s*) *d*� <sup>2</sup>(*s*)
(*s*), *d*�
(a) *n*(*s*) ∈ **F**[*s*] is a polynomial factorizing in *M*�
In particular (*c*) implies that *<sup>n</sup>*�
*<sup>d</sup>*�(*s*) , that is *d*(*n*�
**F***M*�(*s*) ∩ **F***pr*(*s*) = **F**[*s*] ∩ **F***pr*(*s*) = **F**.
where *n*(*s*) ∈ **F**[*s*] factorizes in *M*�
<sup>1</sup>(*s*) *and g*2(*s*) = *<sup>n</sup>*2(*s*) *<sup>n</sup>*�
factorizing in *M*� and *n*�
*d*(*n*2(*s*)) ≥ *δ*(*g*1(*s*)) − *d*(*n*1(*s*)).
*<sup>n</sup>*(*s*)*n*1(*s*). Write *<sup>g</sup>*(*s*) = *<sup>n</sup>*(*s*) *<sup>n</sup>*�
(*s*) *<sup>d</sup>*�(*s*) <sup>=</sup> *<sup>n</sup>*�
*δ*(*g*(*s*)) − *d*(*n*(*s*)) = *d*(*d*�
1(*s*), i.e., *<sup>n</sup>*�
(*s*)
(c) *δ*(*g*(*s*)) − *d*(*n*(*s*)) ≥ 0 or equivalently *δ*∞(*g*(*s*)) ≥ 0.
*<sup>d</sup>*�(*s*) is an irreducible rational function and a unit of **F***M*�(*s*),
(*s*)
(*s*)) = *d*(*d*�
(b) *<sup>n</sup>*� (*s*)
functions *<sup>n</sup>*�
is always fulfilled.
*<sup>n</sup>*1(*s*) *<sup>n</sup>*� <sup>1</sup>(*s*) *d*�
*only if*
*n*� 2(*s*)*d*� <sup>1</sup>(*s*)
*d*� 2(*s*)*n*�
and
**Lemma 6.** *Let M*� ⊆ Specm(**F**[*s*])*. Let g*1(*s*), *g*2(*s*) ∈ **F***M*�(*s*) ∩ **F***pr*(*s*) *be such that g*1(*s*) = *<sup>n</sup>*1(*s*) *<sup>n</sup>*� <sup>1</sup>(*s*) *d*� <sup>1</sup>(*s*) *and g*2(*s*) = *<sup>n</sup>*2(*s*) *<sup>n</sup>*� <sup>2</sup>(*s*) *d*� <sup>2</sup>(*s*) *as in Lemma 5. If n*1(*s*) *and n*2(*s*) *are coprime in* **<sup>F</sup>**[*s*] *and either δ*(*g*1(*s*)) = *d*(*n*1(*s*)) *or δ*(*g*2(*s*)) = *d*(*n*2(*s*)) *then g*1(*s*) *and g*2(*s*) *are coprime in* **F***M*�(*s*) ∩ **F***pr*(*s*)*.*
**Proof**.- Suppose that *g*1(*s*) and *g*2(*s*) are not coprime. Then there exists a non-unit *g*(*s*) = *n*(*s*) *<sup>n</sup>*� (*s*) *<sup>d</sup>*�(*s*) ∈ **F***M*�(*s*) ∩ **F***pr*(*s*) such that *g*(*s*) | *g*1(*s*) and *g*(*s*) | *g*2(*s*). As *g*(*s*) is not a unit, *n*(*s*) is not a constant or *δ*(*g*(*s*)) > 0. If *n*(*s*) is not a constant then *n*(*s*) | *n*1(*s*) and *n*(*s*) | *n*2(*s*) which is impossible because *n*1(*s*) and *n*2(*s*) are coprime. Otherwise, if *n*(*s*) is a constant then *δ*(*g*(*s*)) > 0 and we have that *δ*(*g*(*s*)) ≤ *δ*(*g*1(*s*)) − *d*(*n*1(*s*)) and *δ*(*g*(*s*)) ≤ *δ*(*g*2(*s*)) − *d*(*n*2(*s*)). But this is again impossible.
It follows from this Lemma that if *g*1(*s*), *g*2(*s*) are coprime in both rings **F***M*�(*s*) and **F***pr*(*s*) then *g*1(*s*), *g*2(*s*) are coprime in **F***M*�(*s*) ∩ **F***pr*(*s*). The following example shows that the converse is not true in general.
**Example 7.** Suppose that **<sup>F</sup>** <sup>=</sup> **<sup>R</sup>** and *<sup>M</sup>*� <sup>=</sup> Specm(**R**[*s*]) \ {(*s*<sup>2</sup> <sup>+</sup> <sup>1</sup>)}. It is not difficult to prove that *<sup>g</sup>*1(*s*) = *<sup>s</sup>*<sup>2</sup> *<sup>s</sup>*<sup>2</sup>+<sup>1</sup> and *<sup>g</sup>*2(*s*) = *<sup>s</sup> <sup>s</sup>*<sup>2</sup>+<sup>1</sup> are coprime elements in **R***M*�(*s*) ∩ **R***pr*(*s*). Assume that there exists a non-unit *g*(*s*) = *n*(*s*) *<sup>n</sup>*� (*s*) *<sup>d</sup>*�(*s*) ∈ **R***M*�(*s*) ∩ **R***pr*(*s*) such that *g*(*s*) | *g*1(*s*) and *<sup>g</sup>*(*s*) <sup>|</sup> *<sup>g</sup>*2(*s*). Then *<sup>n</sup>*(*s*) <sup>|</sup> *<sup>s</sup>*2, *<sup>n</sup>*(*s*) <sup>|</sup> *<sup>s</sup>* and *<sup>δ</sup>*(*g*(*s*)) <sup>−</sup> *<sup>d</sup>*(*n*(*s*)) = 0. Since *<sup>g</sup>*(*s*) is not a unit, *<sup>n</sup>*(*s*) cannot be a constant. Hence, *n*(*s*) = *cs*, *c* �= 0, and *δ*(*g*(*s*)) = 1, but this is impossible because *d*� (*s*) and *n*� (*s*) are powers of *s*<sup>2</sup> + 1. Therefore *g*1(*s*) and *g*2(*s*) must be coprime. However *n*1(*s*) = *s*<sup>2</sup> and *n*2(*s*) = *s* are not coprime.
Now, we have the following property when there are ideals in Specm(**F**[*s*]) \ *M*� , *M*� ⊆ Specm(**F**[*s*]), generated by linear polynomials.
**Lemma 8.** *Let M*� ⊆ Specm(**F**[*s*])*. Assume that there are ideals in* Specm(**F**[*s*]) \ *M*� *generated by linear polynomials and let* (*s* − *a*) *be any of them. Let g*1(*s*), *g*2(*s*) ∈ **F***M*�(*s*) ∩ **F***pr*(*s*) *be such that g*1(*s*) = *n*1(*s*)*u*1(*s*) <sup>1</sup> (*s*−*a*)*<sup>d</sup>*<sup>1</sup> *and g*2(*s*) = *<sup>n</sup>*2(*s*)*u*2(*s*) <sup>1</sup> (*s*−*a*)*<sup>d</sup>*<sup>2</sup> *. If g*1(*s*) *and g*2(*s*) *are coprime in* **F***M*�(*s*)∩**F***pr*(*s*) *then n*1(*s*) *and n*2(*s*) *are coprime in* **F**[*s*] *and either d*<sup>1</sup> = *d*(*n*1(*s*)) *or d*<sup>2</sup> = *d*(*n*2(*s*))*.*
**Proof**.- Suppose that *n*1(*s*) and *n*2(*s*) are not coprime in **F**[*s*]. Then there exists a non-constant *<sup>n</sup>*(*s*) <sup>∈</sup> **<sup>F</sup>**[*s*] such that *<sup>n</sup>*(*s*) <sup>|</sup> *<sup>n</sup>*1(*s*) and *<sup>n</sup>*(*s*) <sup>|</sup> *<sup>n</sup>*2(*s*). Let *<sup>d</sup>* <sup>=</sup> *<sup>d</sup>*(*n*(*s*)). Then *<sup>g</sup>*(*s*) = *<sup>n</sup>*(*s*) <sup>1</sup> (*s*−*a*)*<sup>d</sup>* is not a unit in **F***M*�(*s*) ∩ **F***pr*(*s*) and divides *g*1(*s*) and *g*2(*s*) because 0 = *d* − *d*(*n*(*s*)) ≤ *d*<sup>1</sup> − *d*(*n*1(*s*)) and 0 = *d* − *d*(*n*(*s*)) ≤ *d*<sup>2</sup> − *d*(*n*2(*s*)). This is impossible, so *n*1(*s*) and *n*2(*s*) must be coprime.
Now suppose that *d*<sup>1</sup> > *d*(*n*1(*s*)) and *d*<sup>2</sup> > *d*(*n*2(*s*)). Let *d* = min{*d*<sup>1</sup> − *d*(*n*1(*s*)), *d*<sup>2</sup> − *<sup>d</sup>*(*n*2(*s*))}. We have that *<sup>d</sup>* <sup>&</sup>gt; 0. Thus *<sup>g</sup>*(*s*) = <sup>1</sup> (*s*−*a*)*<sup>d</sup>* is not a unit in **<sup>F</sup>***M*�(*s*) <sup>∩</sup> **<sup>F</sup>***pr*(*s*) and divides *g*1(*s*) and *g*2(*s*) because *d* ≤ *d*<sup>1</sup> − *d*(*n*1(*s*)) and *d* ≤ *d*<sup>2</sup> − *d*(*n*2(*s*)). This is again impossible and either *d*<sup>1</sup> = *d*(*n*1(*s*)) or *d*<sup>2</sup> = *d*(*n*2(*s*)).
The above lemmas yield a characterization of coprimeness of elements in **F***M*�(*s*) ∩ **F***pr*(*s*) when *M*� excludes at least one ideal generated by a linear polynomial.
Following the same steps as in [16, p. 11] and [15, p. 271] we get the following result.
**Lemma 9.** *Let M*� ⊆ Specm(**F**[*s*]) *and assume that there is at least an ideal in* Specm(**F**[*s*]) \ *M*� *generated by a linear polynomial. Then* **F***M*�(*s*) ∩ **F***pr*(*s*) *is a Euclidean domain.*
The following examples show that if all ideals generated by polynomials of degree one are in *M*� , the ring **F***M*�(*s*) ∩ **F***pr*(*s*) may not be a Bezout domain. Thus, it may not be a Euclidean domain. Even more, it may not be a greatest common divisor domain.
**Example 10.** Let **<sup>F</sup>** <sup>=</sup> **<sup>R</sup>** and *<sup>M</sup>*� <sup>=</sup> Specm(**R**[*s*]) \ {(*s*<sup>2</sup> <sup>+</sup> <sup>1</sup>)}. Let *<sup>g</sup>*1(*s*) = *<sup>s</sup>*<sup>2</sup> *<sup>s</sup>*<sup>2</sup>+<sup>1</sup> , *<sup>g</sup>*2(*s*) = *<sup>s</sup> s*<sup>2</sup>+1 ∈ **R***M*�(*s*) ∩ **R***pr*(*s*). We have seen, in the previous example, that *g*1(*s*), *g*2(*s*) are coprime. We show now that the Bezout identity is not fulfilled, that is, there are not *a*(*s*), *b*(*s*) ∈ **R***M*�(*s*) ∩ **R***pr*(*s*) such that *a*(*s*)*g*1(*s*) + *b*(*s*)*g*2(*s*) = *u*(*s*), with *u*(*s*) a unit in **R***M*�(*s*) ∩ **R***pr*(*s*). Elements in **<sup>R</sup>***M*�(*s*) <sup>∩</sup> **<sup>R</sup>***pr*(*s*) are of the form *<sup>n</sup>*(*s*) (*s*<sup>2</sup>+1)*<sup>d</sup>* with *<sup>n</sup>*(*s*) relatively prime with *<sup>s</sup>*<sup>2</sup> <sup>+</sup> 1 and 2*<sup>d</sup>* <sup>≥</sup> *d*(*n*(*s*)) and the units in **R***M*�(*s*) ∩ **R***pr*(*s*) are non-zero constants. We will see that there are not elements *a*(*s*) = *<sup>n</sup>*(*s*) (*s*<sup>2</sup>+1)*<sup>d</sup>* , *<sup>b</sup>*(*s*) = *<sup>n</sup>*� (*s*) (*s*<sup>2</sup>+1)*<sup>d</sup>*� with *<sup>n</sup>*(*s*) and *<sup>n</sup>*� (*s*) coprime with *<sup>s</sup>*<sup>2</sup> <sup>+</sup> 1, 2*<sup>d</sup>* <sup>≥</sup> *<sup>d</sup>*(*n*(*s*)) and 2*d*� ≥ *d*(*n*� (*s*)) such that *a*(*s*)*g*1(*s*) + *b*(*s*)*g*2(*s*) = *c*, with *c* non-zero constant. Assume that *n*(*s*) (*s*<sup>2</sup>+1)*<sup>d</sup> s*2 *<sup>s</sup>*<sup>2</sup>+<sup>1</sup> <sup>+</sup> *<sup>n</sup>*� (*s*) (*s*<sup>2</sup>+1)*<sup>d</sup>*� *<sup>s</sup> <sup>s</sup>*<sup>2</sup>+<sup>1</sup> <sup>=</sup> *<sup>c</sup>*. We conclude that *<sup>c</sup>*(*s*<sup>2</sup> <sup>+</sup> <sup>1</sup>)*d*+<sup>1</sup> or *<sup>c</sup>*(*s*<sup>2</sup> <sup>+</sup> <sup>1</sup>)*d*� <sup>+</sup><sup>1</sup> is a multiple of *s*, which is impossible.
**Example 11.** Let **<sup>F</sup>** <sup>=</sup> **<sup>R</sup>** and *<sup>M</sup>*� <sup>=</sup> Specm(**R**[*s*]) \ {(*s*<sup>2</sup> <sup>+</sup> <sup>1</sup>)}. A fraction *<sup>g</sup>*(*s*) = *<sup>n</sup>*(*s*) (*s*<sup>2</sup>+1)*<sup>d</sup>* ∈ **<sup>R</sup>***M*�(*s*) <sup>∩</sup> **<sup>R</sup>***pr*(*s*) if and only if 2*<sup>d</sup>* <sup>−</sup> *<sup>d</sup>*(*n*(*s*)) <sup>≥</sup> 0. Let *<sup>g</sup>*1(*s*) = *<sup>s</sup>*<sup>2</sup> (*s*<sup>2</sup>+1)<sup>3</sup> , *<sup>g</sup>*2(*s*) = *<sup>s</sup>*(*s*+1) (*s*<sup>2</sup>+1)<sup>4</sup> ∈ **R***M*�(*s*) ∩ **R***pr*(*s*). By Lemma 5:
If *<sup>n</sup>*(*s*) <sup>|</sup> *<sup>s</sup>*<sup>2</sup> and *<sup>n</sup>*(*s*) <sup>|</sup> *<sup>s</sup>*(*<sup>s</sup>* <sup>+</sup> <sup>1</sup>) then *<sup>n</sup>*(*s*) = *<sup>c</sup>* or *<sup>n</sup>*(*s*) = *cs* with *<sup>c</sup>* a non-zero constant. Then *g*(*s*) | *g*1(*s*) and *g*(*s*) | *g*2(*s*) if and only if *n*(*s*) = *c* and *d* ≤ 2 or *n*(*s*) = *cs* and 2*d* ≤ 5. So, the list of common divisors of *g*1(*s*) and *g*2(*s*) is:
$$\left\{c, \frac{c}{s^2 + 1}, \frac{c}{(s^2 + 1)^2}, \frac{cs}{s^2 + 1}, \frac{cs}{(s^2 + 1)^2} : c \in \mathbb{F}, c \neq 0\right\}.\tag{23}$$
If there would be a greatest common divisor, say *<sup>n</sup>*(*s*) (*s*<sup>2</sup>+1)*<sup>d</sup>* , then *<sup>n</sup>*(*s*) = *cs* because *<sup>n</sup>*(*s*) must be a multiple of *c* and *cs*. Thus such a greatest common divisor should be either *cs <sup>s</sup>*<sup>2</sup>+<sup>1</sup> or *cs* (*s*<sup>2</sup>+1)<sup>2</sup> , but *<sup>c</sup>* (*s*<sup>2</sup>+1)<sup>2</sup> does not divide neither of them because
$$A = \delta\left(\frac{c}{(s^2+1)^2}\right) - d(c) > \max\left\{\delta\left(\frac{cs}{s^2+1}\right) - d(cs), \delta\left(\frac{cs}{(s^2+1)^2}\right) - d(cs)\right\} = 3. \tag{24}$$
Thus, *g*1(*s*) and *g*2(*s*) do not have greatest common divisor.
#### **3.1. Smith–McMillan form**
10 Will-be-set-by-IN-TECH
The above lemmas yield a characterization of coprimeness of elements in **F***M*�(*s*) ∩ **F***pr*(*s*)
**Lemma 9.** *Let M*� ⊆ Specm(**F**[*s*]) *and assume that there is at least an ideal in* Specm(**F**[*s*]) \ *M*�
The following examples show that if all ideals generated by polynomials of degree one are in
**R***M*�(*s*) ∩ **R***pr*(*s*). We have seen, in the previous example, that *g*1(*s*), *g*2(*s*) are coprime. We show now that the Bezout identity is not fulfilled, that is, there are not *a*(*s*), *b*(*s*) ∈ **R***M*�(*s*) ∩ **R***pr*(*s*) such that *a*(*s*)*g*1(*s*) + *b*(*s*)*g*2(*s*) = *u*(*s*), with *u*(*s*) a unit in **R***M*�(*s*) ∩ **R***pr*(*s*). Elements
*d*(*n*(*s*)) and the units in **R***M*�(*s*) ∩ **R***pr*(*s*) are non-zero constants. We will see that there are not
*<sup>s</sup>*<sup>2</sup>+<sup>1</sup> <sup>=</sup> *<sup>c</sup>*. We conclude that *<sup>c</sup>*(*s*<sup>2</sup> <sup>+</sup> <sup>1</sup>)*d*+<sup>1</sup> or *<sup>c</sup>*(*s*<sup>2</sup> <sup>+</sup> <sup>1</sup>)*d*�
(*s*)) such that *a*(*s*)*g*1(*s*) + *b*(*s*)*g*2(*s*) = *c*, with *c* non-zero constant. Assume that
(*s*<sup>2</sup>+1)*<sup>d</sup>*� with *<sup>n</sup>*(*s*) and *<sup>n</sup>*�
**Example 11.** Let **<sup>F</sup>** <sup>=</sup> **<sup>R</sup>** and *<sup>M</sup>*� <sup>=</sup> Specm(**R**[*s*]) \ {(*s*<sup>2</sup> <sup>+</sup> <sup>1</sup>)}. A fraction *<sup>g</sup>*(*s*) = *<sup>n</sup>*(*s*)
If *<sup>n</sup>*(*s*) <sup>|</sup> *<sup>s</sup>*<sup>2</sup> and *<sup>n</sup>*(*s*) <sup>|</sup> *<sup>s</sup>*(*<sup>s</sup>* <sup>+</sup> <sup>1</sup>) then *<sup>n</sup>*(*s*) = *<sup>c</sup>* or *<sup>n</sup>*(*s*) = *cs* with *<sup>c</sup>* a non-zero constant. Then *g*(*s*) | *g*1(*s*) and *g*(*s*) | *g*2(*s*) if and only if *n*(*s*) = *c* and *d* ≤ 2 or *n*(*s*) = *cs* and 2*d* ≤ 5. So, the
, *cs*
− *d*(*cs*), *δ*
(*s*<sup>2</sup> <sup>+</sup> <sup>1</sup>)<sup>2</sup> : *<sup>c</sup>* <sup>∈</sup> **<sup>F</sup>**, *<sup>c</sup>* �<sup>=</sup> <sup>0</sup>
*cs* (*s*<sup>2</sup> + 1)<sup>2</sup>
*s*<sup>2</sup> + 1
, the ring **F***M*�(*s*) ∩ **F***pr*(*s*) may not be a Bezout domain. Thus, it may not be a Euclidean
*<sup>s</sup>*<sup>2</sup>+<sup>1</sup> , *<sup>g</sup>*2(*s*) = *<sup>s</sup>*
<sup>+</sup><sup>1</sup> is a multiple of
. (23)
*<sup>s</sup>*<sup>2</sup>+<sup>1</sup> or *cs*
(*s*<sup>2</sup>+1)<sup>2</sup> ,
= 3. (24)
(*s*<sup>2</sup>+1)*<sup>d</sup>* ∈
(*s*<sup>2</sup>+1)<sup>4</sup> ∈
(*s*) coprime with *<sup>s</sup>*<sup>2</sup> <sup>+</sup> 1, 2*<sup>d</sup>* <sup>≥</sup> *<sup>d</sup>*(*n*(*s*))
(*s*<sup>2</sup>+1)<sup>3</sup> , *<sup>g</sup>*2(*s*) = *<sup>s</sup>*(*s*+1)
(*s*<sup>2</sup>+1)*<sup>d</sup>* , then *<sup>n</sup>*(*s*) = *cs* because *<sup>n</sup>*(*s*) must be
− *d*(*cs*)
(*s*<sup>2</sup>+1)*<sup>d</sup>* with *<sup>n</sup>*(*s*) relatively prime with *<sup>s</sup>*<sup>2</sup> <sup>+</sup> 1 and 2*<sup>d</sup>* <sup>≥</sup>
*s*<sup>2</sup>+1 ∈
Following the same steps as in [16, p. 11] and [15, p. 271] we get the following result.
when *M*� excludes at least one ideal generated by a linear polynomial.
*generated by a linear polynomial. Then* **F***M*�(*s*) ∩ **F***pr*(*s*) *is a Euclidean domain.*
domain. Even more, it may not be a greatest common divisor domain.
in **<sup>R</sup>***M*�(*s*) <sup>∩</sup> **<sup>R</sup>***pr*(*s*) are of the form *<sup>n</sup>*(*s*)
(*s*) (*s*<sup>2</sup>+1)*<sup>d</sup>*� *<sup>s</sup>*
**R***M*�(*s*) ∩ **R***pr*(*s*). By Lemma 5:
(*s*<sup>2</sup>+1)*<sup>d</sup>* , *<sup>b</sup>*(*s*) = *<sup>n</sup>*�
elements *a*(*s*) = *<sup>n</sup>*(*s*)
*s*, which is impossible.
and 2*d*� ≥ *d*(*n*�
*s*2 *<sup>s</sup>*<sup>2</sup>+<sup>1</sup> <sup>+</sup> *<sup>n</sup>*�
*n*(*s*) (*s*<sup>2</sup>+1)*<sup>d</sup>*
but *<sup>c</sup>*
4 = *δ*
*c* (*s*<sup>2</sup> + 1)<sup>2</sup>
**Example 10.** Let **<sup>F</sup>** <sup>=</sup> **<sup>R</sup>** and *<sup>M</sup>*� <sup>=</sup> Specm(**R**[*s*]) \ {(*s*<sup>2</sup> <sup>+</sup> <sup>1</sup>)}. Let *<sup>g</sup>*1(*s*) = *<sup>s</sup>*<sup>2</sup>
(*s*)
**<sup>R</sup>***M*�(*s*) <sup>∩</sup> **<sup>R</sup>***pr*(*s*) if and only if 2*<sup>d</sup>* <sup>−</sup> *<sup>d</sup>*(*n*(*s*)) <sup>≥</sup> 0. Let *<sup>g</sup>*1(*s*) = *<sup>s</sup>*<sup>2</sup>
• *<sup>g</sup>*(*s*) <sup>|</sup> *<sup>g</sup>*1(*s*) <sup>⇔</sup> *<sup>n</sup>*(*s*) <sup>|</sup> *<sup>s</sup>*<sup>2</sup> and 0 <sup>≤</sup> <sup>2</sup>*<sup>d</sup>* <sup>−</sup> *<sup>d</sup>*(*n*(*s*)) <sup>≤</sup> <sup>6</sup> <sup>−</sup> <sup>2</sup> <sup>=</sup> <sup>4</sup>
list of common divisors of *g*1(*s*) and *g*2(*s*) is:
*<sup>c</sup>*, *<sup>c</sup> s*<sup>2</sup> + 1
• *g*(*s*) | *g*2(*s*) ⇔ *n*(*s*) | *s*(*s* + 1) and 0 ≤ 2*d* − *d*(*n*(*s*)) ≤ 8 − 2 = 6.
, *<sup>c</sup>*
If there would be a greatest common divisor, say *<sup>n</sup>*(*s*)
(*s*<sup>2</sup>+1)<sup>2</sup> does not divide neither of them because
− *d*(*c*) > *max*
Thus, *g*1(*s*) and *g*2(*s*) do not have greatest common divisor.
(*s*<sup>2</sup> <sup>+</sup> <sup>1</sup>)<sup>2</sup> , *cs*
*δ cs s*<sup>2</sup> + 1
a multiple of *c* and *cs*. Thus such a greatest common divisor should be either *cs*
*M*�
A matrix *<sup>U</sup>*(*s*) is invertible in **<sup>F</sup>***M*�(*s*)*m*×*<sup>m</sup>* <sup>∩</sup> **<sup>F</sup>***pr*(*s*)*m*×*<sup>m</sup>* if *<sup>U</sup>*(*s*) <sup>∈</sup> **<sup>F</sup>***M*�(*s*)*m*×*<sup>m</sup>* <sup>∩</sup> **<sup>F</sup>***pr*(*s*)*m*×*<sup>m</sup>* and its determinant is a unit in both rings, **F***M*�(*s*) and **F***pr*(*s*), i.e., *U*(*s*) ∈ Gl*m*(**F***M*�(*s*) ∩ **F***pr*(*s*)) if and only if *U*(*s*) ∈ Gl*m*(**F***M*�(*s*)) ∩ Gl*m*(**F***pr*(*s*)).
Two matrices *<sup>G</sup>*1(*s*), *<sup>G</sup>*2(*s*) <sup>∈</sup> **<sup>F</sup>***M*�(*s*)*m*×*<sup>m</sup>* <sup>∩</sup> **<sup>F</sup>***pr*(*s*)*m*×*<sup>m</sup>* are equivalent in **<sup>F</sup>***M*�(*s*) <sup>∩</sup> **<sup>F</sup>***pr*(*s*) if there exist *<sup>U</sup>*1(*s*), *<sup>U</sup>*2(*s*) invertible in **<sup>F</sup>***M*�(*s*)*m*×*<sup>m</sup>* <sup>∩</sup> **<sup>F</sup>***pr*(*s*)*m*×*<sup>m</sup>* such that
$$\mathbf{G\_2(s)} = \mathcal{U}\_1(s)\mathcal{G}\_1(s)\mathcal{U}\_2(s). \tag{25}$$
If there are ideals in Specm(**F**[*s*]) \ *M*� generated by linear polynomials then **F***M*�(*s*) ∩ **F***pr*(*s*) is an Euclidean ring and any matrix with elements in **F***M*�(*s*) ∩ **F***pr*(*s*) admits a Smith normal form (see [13], [15] or [16]). Bearing in mind the characterization of divisibility in **F***M*�(*s*) ∩ **F***pr*(*s*) given in Lemma 5 we have
**Theorem 12.** *(Smith normal form in* **F***M*�(*s*) ∩ **F***pr*(*s*)*) Let M*� ⊆ Specm(**F**[*s*])*. Assume that there are ideals in* Specm(**F**[*s*]) \ *M*� *generated by linear polynomials and let* (*s* − *a*) *be one of them. Let G*(*s*) <sup>∈</sup> **<sup>F</sup>***M*�(*s*)*m*×*<sup>m</sup>* <sup>∩</sup> **<sup>F</sup>***pr*(*s*)*m*×*<sup>m</sup> be non-singular. Then there exist U*1(*s*), *<sup>U</sup>*2(*s*) *invertible in* **<sup>F</sup>***M*�(*s*)*m*×*<sup>m</sup>* <sup>∩</sup> **<sup>F</sup>***pr*(*s*)*m*×*<sup>m</sup> such that*
$$G(s) = \mathcal{U}\_1(s) \operatorname{Diag} \left( n\_1(s) \frac{1}{(s-a)^{d\_1}}, \dots, n\_m(s) \frac{1}{(s-a)^{d\_m}} \right) \mathcal{U}\_2(s) \tag{26}$$
*with n*1(*s*)|···|*nm*(*s*) *monic polynomials factorizing in M*� *and d*1,..., *dm integers such that* 0 ≤ *d*<sup>1</sup> − *d*(*n*1(*s*)) ≤···≤ *dm* − *d*(*nm*(*s*))*.*
Under the hypothesis of the last theorem *n*1(*s*) <sup>1</sup> (*s*−*a*)*<sup>d</sup>*<sup>1</sup> ,..., *nm*(*s*) <sup>1</sup> (*s*−*a*)*dm* form a complete system of invariants for the equivalence in **F***M*�(*s*)∩**F***pr*(*s*) and are called the invariant rational functions of *G*(*s*) in **F***M*�(*s*) ∩ **F***pr*(*s*). Notice that 0 ≤ *d*<sup>1</sup> ≤ ··· ≤ *dm* because *ni*(*s*) divides *ni*<sup>+</sup>1(*s*).
Recall that the field of fractions of **F***M*�(*s*) ∩ **F***pr*(*s*) is **F**(*s*) when *M*� �= Specm(**F**[*s*]). Thus we can talk about equivalence of matrix rational functions. Two rational matrices *T*1(*s*), *T*2(*s*) ∈ **<sup>F</sup>**(*s*)*m*×*<sup>m</sup>* are equivalent in **<sup>F</sup>***M*�(*s*) <sup>∩</sup> **<sup>F</sup>***pr*(*s*) if there are *<sup>U</sup>*1(*s*), *<sup>U</sup>*2(*s*) invertible in **<sup>F</sup>***M*�(*s*)*m*×*<sup>m</sup>* <sup>∩</sup> **F***pr*(*s*)*m*×*<sup>m</sup>* such that
$$T\_2(s) = \mathcal{U}\_1(s)T\_1(s)\mathcal{U}\_2(s). \tag{27}$$
When all ideals generated by linear polynomials are not in *M*� , each rational matrix admits a reduction to Smith–McMillan form with respect to **F***M*�(*s*) ∩ **F***pr*(*s*).
**Theorem 13.** *(Smith–McMillan form in* **F***M*�(*s*) ∩ **F***pr*(*s*)*) Let M*� ⊆ Specm(**F**[*s*])*. Assume that there are ideals in* Specm(**F**[*s*]) \ *M*� *generated by linear polynomials and let* (*s* − *a*) *be any of them. Let T*(*s*) <sup>∈</sup> **<sup>F</sup>**(*s*)*m*×*<sup>m</sup> be a non-singular matrix. Then there exist U*1(*s*), *<sup>U</sup>*2(*s*) *invertible in* **<sup>F</sup>***M*�(*s*)*m*×*<sup>m</sup>* <sup>∩</sup> **<sup>F</sup>***pr*(*s*)*m*×*<sup>m</sup> such that*
$$T(s) = \mathcal{U}\_1(s) \text{Diag}\left(\frac{\frac{\varepsilon\_1(s)}{(s-a)^{n\_1}}}{\frac{\psi\_1(s)}{(s-a)^{d\_1}}}, \dots, \frac{\frac{\varepsilon\_m(s)}{(s-a)^{n\_m}}}{\frac{\psi\_m(s)}{(s-a)^{d\_m}}}\right) \mathcal{U}\_2(s) \tag{28}$$
#### 12 Will-be-set-by-IN-TECH 58 Linear Algebra – Theorems and Applications
*with �i*(*s*) (*s*−*a*)*ni* , *<sup>ψ</sup>i*(*s*) (*s*−*a*)*di* <sup>∈</sup> **<sup>F</sup>***M*�(*s*) <sup>∩</sup> **<sup>F</sup>***pr*(*s*) *coprime for all i such that �i*(*s*)*, <sup>ψ</sup>i*(*s*) *are monic polynomials factorizing in M*� *, �i*(*s*) (*s*−*a*)*ni divides �i*+1(*s*) (*s*−*a*) *ni*+<sup>1</sup> *for i* <sup>=</sup> 1, . . . , *<sup>m</sup>* <sup>−</sup> <sup>1</sup> *while <sup>ψ</sup>i*(*s*) (*s*−*a*)*di divides <sup>ψ</sup>i*−<sup>1</sup>(*s*) (*s*−*a*) *di*−<sup>1</sup> *for i* = 2, . . . , *m.*
The elements *�i*(*s*) (*s*−*a*) *ni ψi*(*s*) (*s*−*a*) *di* of the diagonal matrix, satisfying the conditions of the previous theorem,
constitute a complete system of invariant for the equivalence in **F***M*�(*s*) ∩ **F***pr*(*s*) of rational matrices. However, this system of invariants is not minimal. A smaller one can be obtained by substituting each pair of positive integers (*ni*, *di*) by its difference *li* = *ni* − *di*.
**Theorem 14.** *Under the conditions of Theorem 13, �i*(*s*) *ψi*(*s*) 1 (*s*−*a*)*<sup>l</sup> <sup>i</sup> with �i*(*s*)*, ψi*(*s*) *monic and coprime polynomials factorizing in M*� *, �i*(*s*) | *�i*+1(*s*) *while <sup>ψ</sup>i*(*s*) | *<sup>ψ</sup>i*−1(*s*) *and l*1,..., *lm integers such that l*<sup>1</sup> + *d*(*ψ*1(*s*)) − *d*(*�*1(*s*)) ≤ ··· ≤ *lm* + *d*(*ψm*(*s*)) − *d*(*�m*(*s*)) *also constitute a complete system of invariants for the equivalence in* **F***M*�(*s*) ∩ **F***pr*(*s*)*.*
**Proof**.- We only have to show that from the system *�i*(*s*) *ψi*(*s*) 1 (*s*−*a*)*<sup>l</sup> i* , *i* = 1, . . . , *m*, satisfying the conditions of Theorem 14, the system *�i*(*s*) (*s*−*a*) *ni ψi*(*s*) (*s*−*a*) *di* , *i* = 1, . . . , *n*, can be constructed satisfying the conditions of Theorem 13.
Suppose that *�i*(*s*), *ψi*(*s*) are monic and coprime polynomials factorizing in *M*� such that *�i*(*s*) | *�i*+1(*s*) and *<sup>ψ</sup>i*(*s*) | *<sup>ψ</sup>i*−1(*s*). And suppose also that *<sup>l</sup>*1,..., *lm* are integers such that *l*<sup>1</sup> + *d*(*ψ*1(*s*)) − *d*(*�*1(*s*)) ≤··· ≤ *lm* + *d*(*ψm*(*s*)) − *d*(*�m*(*s*)). If *li* + *d*(*ψi*(*s*)) − *d*(*�i*(*s*)) ≤ 0 for all *i*, we define non-negative integers *ni* = *d*(*�i*(*s*)) and *di* = *d*(*�i*(*s*)) − *li* for *i* = 1, . . . , *m*. If *li* + *d*(*ψi*(*s*)) − *d*(*�i*(*s*)) > 0 for all *i*, we define *ni* = *li* + *d*(*ψi*(*s*)) and *di* = *d*(*ψi*(*s*)). Otherwise there is an index *k* ∈ {2, . . . , *m*} such that
$$d\_{k-1} + d(\psi\_{k-1}(s)) - d(\varepsilon\_{k-1}(s)) \le 0 < l\_k + d(\psi\_k(s)) - d(\varepsilon\_k(s)).\tag{29}$$
Define now the non-negative integers *ni*, *di* as follows:
$$m\_i = \begin{cases} d(\varepsilon\_i(s)) & \text{if } i < k \\ l\_i + d(\psi\_i(s)) \text{ if } i \ge k \end{cases} \quad d\_i = \begin{cases} d(\varepsilon\_i(s)) - l\_i \text{ if } i < k \\ d(\psi\_i(s)) & \text{if } i \ge k \end{cases} \tag{30}$$
Notice that *li* = *ni* − *di*. Moreover,
$$m\_i - d(\mathfrak{e}\_i(s)) = \begin{cases} 0 & \text{if } i < k \\ l\_i + d(\psi\_i(s)) - d(\mathfrak{e}\_i(s)) \text{ if } i \ge k \end{cases} \tag{31}$$
$$d\_i - d(\psi\_i(s)) = \begin{cases} -l\_i - d(\psi\_i(s)) + d(\varepsilon\_i(s)) \text{ if } i < k \\ 0 & \text{if } i \ge k \end{cases} \tag{32}$$
and using (29), (30)
$$n\_1 - d(\varepsilon\_1(s)) = \dots = n\_{k-1} - d(\varepsilon\_{k-1}(s)) = 0 < n\_k - d(\varepsilon\_k(s)) \le \dots \le n\_m - d(\varepsilon\_m(s)) \tag{33}$$
$$d\_1 - d(\psi\_1(s)) \ge \dots \ge d\_{k-1} - d(\psi\_{k-1}(s)) \ge 0 = d\_k - d(\psi\_k(s)) = \dots = d\_m - d(\psi\_m(s)).\tag{34}$$
In any case *�i*(*s*) (*s*−*a*)*ni* and *<sup>ψ</sup>i*(*s*) (*s*−*a*)*di* are elements of **<sup>F</sup>***M*�(*s*) <sup>∩</sup> **<sup>F</sup>***pr*(*s*). Now, on the one hand *�i*(*s*), *ψi*(*s*) are coprime and *ni* − *d*(*�i*(*s*)) = 0 or *di* − *d*(*ψi*(*s*)) = 0. This means (Lemma 6) that *�i*(*s*) (*s*−*a*)*ni* , *<sup>ψ</sup>i*(*s*) (*s*−*a*)*di* are coprime for all *<sup>i</sup>*. On the other hand *�i*(*s*) <sup>|</sup> *�i*+1(*s*) and 0 <sup>≤</sup> *ni* <sup>−</sup> *<sup>d</sup>*(*�i*(*s*)) <sup>≤</sup> *ni*<sup>+</sup><sup>1</sup> <sup>−</sup> *<sup>d</sup>*(*�i*+1(*s*)). Then (Lemma 5) *�i*(*s*) (*s*−*a*)*ni* divides *�i*+1(*s*) (*s*−*a*) *ni*+<sup>1</sup> . Similarly, since *<sup>ψ</sup>i*(*s*) | *<sup>ψ</sup>i*−1(*s*) and 0 <sup>≤</sup> *di* <sup>−</sup> *<sup>d</sup>*(*ψi*(*s*)) <sup>≤</sup> *di*−<sup>1</sup> <sup>−</sup> *<sup>d</sup>*(*ψi*−1(*s*)), it follows that *<sup>ψ</sup>i*(*s*) (*s*−*a*)*di* divides *<sup>ψ</sup>i*−<sup>1</sup>(*s*) (*s*−*a*) *di*−<sup>1</sup> .
We call *�i*(*s*) *ψi*(*s*) 1 (*s*−*a*)*<sup>l</sup> i* , *i* = 1, . . . , *m*, the invariant rational functions of *T*(*s*) in **F***M*�(*s*) ∩ **F***pr*(*s*).
There is a particular case worth considering: If *M*� = ∅ then **F**∅(*s*) ∩ **F***pr*(*s*) = **F***pr*(*s*) and (*s*) ∈ Specm(**F**[*s*]) \ *M*� = Specm(**F**[*s*]). In this case, we obtain the invariant rational functions of *T*(*s*) at infinity (recall (17)).
#### **4. Wiener–Hopf equivalence**
12 Will-be-set-by-IN-TECH
constitute a complete system of invariant for the equivalence in **F***M*�(*s*) ∩ **F***pr*(*s*) of rational matrices. However, this system of invariants is not minimal. A smaller one can be obtained
*l*<sup>1</sup> + *d*(*ψ*1(*s*)) − *d*(*�*1(*s*)) ≤ ··· ≤ *lm* + *d*(*ψm*(*s*)) − *d*(*�m*(*s*)) *also constitute a complete system of*
Suppose that *�i*(*s*), *ψi*(*s*) are monic and coprime polynomials factorizing in *M*� such that *�i*(*s*) | *�i*+1(*s*) and *<sup>ψ</sup>i*(*s*) | *<sup>ψ</sup>i*−1(*s*). And suppose also that *<sup>l</sup>*1,..., *lm* are integers such that *l*<sup>1</sup> + *d*(*ψ*1(*s*)) − *d*(*�*1(*s*)) ≤··· ≤ *lm* + *d*(*ψm*(*s*)) − *d*(*�m*(*s*)). If *li* + *d*(*ψi*(*s*)) − *d*(*�i*(*s*)) ≤ 0 for all *i*, we define non-negative integers *ni* = *d*(*�i*(*s*)) and *di* = *d*(*�i*(*s*)) − *li* for *i* = 1, . . . , *m*. If *li* + *d*(*ψi*(*s*)) − *d*(*�i*(*s*)) > 0 for all *i*, we define *ni* = *li* + *d*(*ψi*(*s*)) and *di* = *d*(*ψi*(*s*)). Otherwise
*ψi*(*s*)
by substituting each pair of positive integers (*ni*, *di*) by its difference *li* = *ni* − *di*.
*�i*(*s*) (*s*−*a*) *ni ψi*(*s*) (*s*−*a*) *di*
(*s*−*a*)*di* <sup>∈</sup> **<sup>F</sup>***M*�(*s*) <sup>∩</sup> **<sup>F</sup>***pr*(*s*) *coprime for all i such that �i*(*s*)*, <sup>ψ</sup>i*(*s*) *are monic polynomials*
*ni*+<sup>1</sup> *for i* <sup>=</sup> 1, . . . , *<sup>m</sup>* <sup>−</sup> <sup>1</sup> *while <sup>ψ</sup>i*(*s*)
of the diagonal matrix, satisfying the conditions of the previous theorem,
1 (*s*−*a*)*<sup>l</sup>*
*ψi*(*s*)
*lk*−<sup>1</sup> + *<sup>d</sup>*(*ψk*−1(*s*)) − *<sup>d</sup>*(*�k*−1(*s*)) ≤ <sup>0</sup> < *lk* + *<sup>d</sup>*(*ψk*(*s*)) − *<sup>d</sup>*(*�k*(*s*)). (29)
0 if *i* < *k*
<sup>−</sup>*li* <sup>−</sup> *<sup>d</sup>*(*ψi*(*s*)) + *<sup>d</sup>*(*�i*(*s*)) if *<sup>i</sup>* <sup>&</sup>lt; *<sup>k</sup>*
*<sup>n</sup>*<sup>1</sup> − *<sup>d</sup>*(*�*1(*s*)) = ··· = *nk*−<sup>1</sup> − *<sup>d</sup>*(*�k*−1(*s*)) = <sup>0</sup> < *nk* − *<sup>d</sup>*(*�k*(*s*)) ≤···≤ *nm* − *<sup>d</sup>*(*�m*(*s*)) (33)
*<sup>d</sup>*<sup>1</sup> − *<sup>d</sup>*(*ψ*1(*s*)) ≥···≥ *dk*−<sup>1</sup> − *<sup>d</sup>*(*ψk*−1(*s*)) ≥ <sup>0</sup> = *dk* − *<sup>d</sup>*(*ψk*(*s*)) = ··· = *dm* − *<sup>d</sup>*(*ψm*(*s*)). (34)
*<sup>d</sup>*(*�i*(*s*)) <sup>−</sup> *li* if *<sup>i</sup>* <sup>&</sup>lt; *<sup>k</sup>*
*li* <sup>+</sup> *<sup>d</sup>*(*ψi*(*s*)) <sup>−</sup> *<sup>d</sup>*(*�i*(*s*)) if *<sup>i</sup>* <sup>≥</sup> *<sup>k</sup>* (31)
<sup>0</sup> if *<sup>i</sup>* <sup>≥</sup> *<sup>k</sup>* (32)
*<sup>d</sup>*(*ψi*(*s*)) if *<sup>i</sup>* <sup>≥</sup> *<sup>k</sup>* (30)
*, �i*(*s*) | *�i*+1(*s*) *while <sup>ψ</sup>i*(*s*) | *<sup>ψ</sup>i*−1(*s*) *and l*1,..., *lm integers such that*
1 (*s*−*a*)*<sup>l</sup> i*
, *i* = 1, . . . , *n*, can be constructed satisfying the
(*s*−*a*)*di divides <sup>ψ</sup>i*−<sup>1</sup>(*s*)
*<sup>i</sup> with �i*(*s*)*, ψi*(*s*) *monic and coprime*
, *i* = 1, . . . , *m*, satisfying the
(*s*−*a*)
*di*−<sup>1</sup> *for*
*with �i*(*s*)
(*s*−*a*)*ni* , *<sup>ψ</sup>i*(*s*)
*, �i*(*s*)
*�i*(*s*) (*s*−*a*) *ni ψi*(*s*) (*s*−*a*) *di*
*polynomials factorizing in M*�
conditions of Theorem 13.
(*s*−*a*)*ni divides �i*+1(*s*)
**Theorem 14.** *Under the conditions of Theorem 13, �i*(*s*)
**Proof**.- We only have to show that from the system *�i*(*s*)
*invariants for the equivalence in* **F***M*�(*s*) ∩ **F***pr*(*s*)*.*
conditions of Theorem 14, the system
there is an index *k* ∈ {2, . . . , *m*} such that
*ni* =
Notice that *li* = *ni* − *di*. Moreover,
and using (29), (30)
Define now the non-negative integers *ni*, *di* as follows:
*ni* − *d*(*�i*(*s*)) =
*di* − *d*(*ψi*(*s*)) =
*d*(*�i*(*s*)) if *i* < *k*
*li* <sup>+</sup> *<sup>d</sup>*(*ψi*(*s*)) if *<sup>i</sup>* <sup>≥</sup> *<sup>k</sup> di* <sup>=</sup>
(*s*−*a*)
*factorizing in M*�
*i* = 2, . . . , *m.*
The elements
The left Wiener–Hopf equivalence of rational matrices with respect to a closed contour in the complex plane has been extensively studied ([6] or [10]). Now we present the generalization to arbitrary fields ([4]).
**Definition 15.** *Let M and M*� *be subsets of* Specm(**F**[*s*]) *such that M* ∪ *M*� = Specm(**F**[*s*])*. Let <sup>T</sup>*1(*s*), *<sup>T</sup>*2(*s*) <sup>∈</sup> **<sup>F</sup>**(*s*)*m*×*<sup>m</sup> be two non-singular rational matrices with no zeros and no poles in M* <sup>∩</sup> *<sup>M</sup>*� *. The matrices T*1(*s*), *T*2(*s*) *are said to be left Wiener–Hopf equivalent with respect to* (*M*, *M*� ) *if there exist both U*1(*s*) *invertible in* **<sup>F</sup>***M*�(*s*)*m*×*<sup>m</sup>* <sup>∩</sup> **<sup>F</sup>***pr*(*s*)*m*×*<sup>m</sup> and U*2(*s*) *invertible in* **<sup>F</sup>***M*(*s*)*m*×*<sup>m</sup> such that*
$$T\_2(\mathbf{s}) = \mathcal{U}\_1(\mathbf{s}) T\_1(\mathbf{s}) \mathcal{U}\_2(\mathbf{s}).\tag{35}$$
This is, in fact, an equivalence relation as it is easily seen. It would be an equivalence relation even if no condition about the union and intersection of *M* and *M*� were imposed. It will be seen later on that these conditions are natural assumptions for the existence of unique diagonal representatives in each class.
The right Wiener–Hopf equivalence with respect to (*M*, *M*� ) is defined in a similar manner: There are invertible matrices *<sup>U</sup>*1(*s*) in **<sup>F</sup>***M*�(*s*)*m*×*<sup>m</sup>* <sup>∩</sup> **<sup>F</sup>***pr*(*s*)*m*×*<sup>m</sup>* and *<sup>U</sup>*2(*s*) in **<sup>F</sup>***M*(*s*)*m*×*<sup>m</sup>* such that
$$T\_2(s) = \mathcal{U}\_2(s)T\_1(s)\mathcal{U}\_1(s). \tag{36}$$
In the following only the left Wiener–Hopf equivalence will be considered, but, by transposition, all results hold for the right Wiener–Hopf equivalence as well.
The aim of this section is to obtain a complete system of invariants for the Wiener–Hopf equivalence with respect to (*M*, *M*� ) of rational matrices, and to obtain, if possible, a canonical form.
There is a particular case that is worth-considering: If *M* = Specm(**F**[*s*]) and *M*� = ∅, the invertible matrices in **<sup>F</sup>**∅(*s*)*m*×*<sup>m</sup>* <sup>∩</sup> **<sup>F</sup>***pr*(*s*)*m*×*<sup>m</sup>* are the biproper matrices and the invertible matrices in **<sup>F</sup>**Specm(**F**[*s*])(*s*)*m*×*<sup>m</sup>* are the unimodular matrices. In this case, the left Wiener–Hopf equivalence with respect to (*M*, *M*� )=(Specm(**F**[*s*]), ∅) is the so-called left Wiener–Hopf equivalence at infinity (see [9]). It is known that any non-singular rational matrix is left Wiener–Hopf equivalent at infinity to a diagonal matrix Diag(*sg*<sup>1</sup> ,...,*sgm* ) where *g*1,..., *gm*
are integers, that is, for any non-singular *<sup>T</sup>*(*s*) <sup>∈</sup> **<sup>F</sup>**(*s*)*m*×*<sup>m</sup>* there exist both a biproper matrix *B*(*s*) ∈ Gl*m*(**F***pr*(*s*)) and a unimodular matrix *U*(*s*) ∈ Gl*m*(**F**[*s*]) such that
$$T(\mathbf{s}) = B(\mathbf{s}) \operatorname{Diag}(\mathbf{s}^{\mathcal{G}\_1}, \dots, \mathbf{s}^{\mathcal{G}\_m}) \mathcal{U}(\mathbf{s}) \tag{37}$$
where *g*<sup>1</sup> ≥ ··· ≥ *gm* are integers uniquely determined by *T*(*s*). They are called the left Wiener–Hopf factorization indices at infinity and form a complete system of invariants for the left Wiener–Hopf equivalence at infinity. These are the basic objects that will produce the complete system of invariants for the left Wiener–Hopf equivalence with respect to (*M*, *M*� ).
For polynomial matrices, their left Wiener–Hopf factorization indices at infinity are the column degrees of any right equivalent (by a unimodular matrix) column proper matrix. Namely, a polynomial matrix is column proper if it can be written as *Pc* Diag(*sg*<sup>1</sup> ,...,*sgm* ) + *<sup>L</sup>*(*s*) with *Pc* <sup>∈</sup> **<sup>F</sup>***m*×*<sup>m</sup>* non-singular, *<sup>g</sup>*1,..., *gm* non-negative integers and *<sup>L</sup>*(*s*) a polynomial matrix such that the degree of the *i*th column of *L*(*s*) smaller than *gi*, 1 ≤ *i* ≤ *m*. Let *P*(*s*) ∈ **F**[*s*] *<sup>m</sup>*×*<sup>m</sup>* be non-singular polynomial. There exists a unimodular matrix *<sup>V</sup>*(*s*) <sup>∈</sup> **<sup>F</sup>**[*s*] *<sup>m</sup>*×*<sup>m</sup>* such that *P*(*s*)*V*(*s*) is column proper. The column degrees of *P*(*s*)*V*(*s*) are uniquely determined by *P*(*s*), although *V*(*s*) is not (see [9], [12, p. 388], [17]). Since *P*(*s*)*V*(*s*) is column proper, it can be written as *P*(*s*)*V*(*s*) = *PcD*(*s*) + *L*(*s*) with *Pc* non-singular, *D*(*s*) = Diag(*sg*<sup>1</sup> ,...,*sgm* ) and the degree of the *i*th column of *L*(*s*) smaller than *gi*, 1 ≤ *i* ≤ *m*. Then *P*(*s*)*V*(*s*) = (*Pc* + *L*(*s*)*D*(*s*)−1)*D*(*s*). Put *B*(*s*) = *Pc* + *L*(*s*)*D*(*s*)−1. Since *Pc* is non-singular and *L*(*s*)*D*(*s*)−<sup>1</sup> is a strictly proper matrix, *B*(*s*) is biproper, and *P*(*s*) = *B*(*s*)*D*(*s*)*U*(*s*) where *U*(*s*) = *V*(*s*)−1.
The left Wiener–Hopf factorization indices at infinity can be used to associate a sequence of integers with every non-singular rational matrix and every *M* ⊆ Specm(**F**[*s*]). This is done as follows: If *<sup>T</sup>*(*s*) <sup>∈</sup> **<sup>F</sup>**(*s*)*m*×*<sup>m</sup>* then it can always be written as *<sup>T</sup>*(*s*) = *TL*(*s*)*TR*(*s*) such that the global invariant rational functions of *TL*(*s*) factorize in *M* and *TR*(*s*) ∈ Gl*m*(**F***M*(*s*)) or, equivalently, the global invariant rational functions of *TR*(*s*) factorize in Specm(**F**[*s*]) \ *M* (see Proposition 2). There may be many factorizations of this type, but it turns out (see [1, Proposition 3.2] for the polynomial case) that the left factors in all of them are right equivalent. This means that if *T*(*s*) = *TL*1(*s*)*TR*1(*s*) = *TL*2(*s*)*TR*2(*s*) with the global invariant rational functions of *TL*1(*s*) and *TL*2(*s*) factorizing in *M* and the global invariant rational functions of *TR*1(*s*) and *TR*2(*s*) factorizing in Specm(**F**[*s*]) \ *M* then there is a unimodular matrix *U*(*s*) such that *TL*1(*s*) = *TL*2(*s*)*U*(*s*). In particular, *TL*1(*s*) and *TL*2(*s*) have the same left Wiener–Hopf factorization indices at infinity. Thus the following definition makes sense:
**Definition 16.** *Let T*(*s*) <sup>∈</sup> **<sup>F</sup>**(*s*)*m*×*<sup>m</sup> be a non-singular rational matrix and M* <sup>⊆</sup> Specm(**F**[*s*])*. Let TL*(*s*), *TR*(*s*) <sup>∈</sup> **<sup>F</sup>**(*s*)*m*×*<sup>m</sup> such that*
*Then the left Wiener–Hopf factorization indices of T*(*s*) *with respect to M are defined to be the left Wiener–Hopf factorization indices of TL*(*s*) *at infinity.*
In the particular case that *M* = Specm(**F**[*s*]), we can put *TL*(*s*) = *T*(*s*) and *TR*(*s*) = *Im*. Therefore, the left Wiener–Hopf factorization indices of *T*(*s*) with respect to Specm(**F**[*s*]) are the left Wiener–Hopf factorization indices of *T*(*s*) at infinity.
We prove now that the left Wiener–Hopf equivalence with respect to (*M*, *M*� ) can be characterized through the left Wiener–Hopf factorization indices with respect to *M*.
**Theorem 17.** *Let M*, *M*� ⊆ Specm(**F**[*s*]) *be such that M* ∪ *M*� = Specm(**F**[*s*])*. Let T*1(*s*)*, T*2(*s*) ∈ **<sup>F</sup>**(*s*)*m*×*<sup>m</sup> be two non-singular rational matrices with no zeros and no poles in M* <sup>∩</sup> *<sup>M</sup>*� *. The matrices T*1(*s*) *and T*2(*s*) *are left Wiener–Hopf equivalent with respect to* (*M*, *M*� ) *if and only if T*1(*s*) *and T*2(*s*) *have the same left Wiener–Hopf factorization indices with respect to M.*
**Proof.-** By Proposition 2 we can write *T*1(*s*) = *TL*1(*s*)*TR*1(*s*), *T*2(*s*) = *TL*2(*s*)*TR*2(*s*) with the global invariant rational functions of *TL*1(*s*) and of *TL*2(*s*) factorizing in *M* \ *M*� (recall that *T*1(*s*) and *T*2(*s*) have no zeros and no poles in *M* ∩ *M*� ) and the global invariant rational functions of *TR*1(*s*) and of *TR*2(*s*) factorizing in *M*� \ *M*.
Assume that *T*1(*s*), *T*2(*s*) have the same left Wiener–Hopf factorization indices with respect to *M*. By definition, *T*1(*s*) and *T*2(*s*) have the same left Wiener–Hopf factorization indices with respect to *M* if *TL*1(*s*) and *TL*2(*s*) have the same left Wiener–Hopf factorization indices at infinity. This means that there exist matrices *B*(*s*) ∈ Gl*m*(**F***pr*(*s*)) and *U*(*s*) ∈ Gl*m*(**F**[*s*]) such that *TL*2(*s*) = *B*(*s*)*TL*1(*s*)*U*(*s*). We have that *T*2(*s*) = *TL*2(*s*)*TR*2(*s*) = *B*(*s*)*TL*1(*s*)*U*(*s*)*TR*2(*s*) = *B*(*s*)*T*1(*s*)(*TR*1(*s*)−1*U*(*s*)*TR*2(*s*)). We aim to prove that *B*(*s*) = *TL*2(*s*)*U*(*s*)−<sup>1</sup>*TL*1(*s*)−<sup>1</sup> is invertible in **<sup>F</sup>***M*�(*s*)*m*×*<sup>m</sup>* and *TR*1(*s*)−1*U*(*s*)*TR*2(*s*) <sup>∈</sup> *Glm*(**F***M*(*s*)). Since the global invariant rational functions of *TL*2(*s*) and *TL*1(*s*) factorize in *M* \ *M*� , *TL*2(*s*), *TL*1(*s*) <sup>∈</sup> **<sup>F</sup>***M*�(*s*)*m*×*<sup>m</sup>* and *<sup>B</sup>*(*s*) <sup>∈</sup> **<sup>F</sup>***M*�(*s*)*m*×*m*. Moreover, det *<sup>B</sup>*(*s*) is a unit in **<sup>F</sup>***M*�(*s*)*m*×*<sup>m</sup>* as desired. Now, *TR*1(*s*)−1*U*(*s*)*TR*2(*s*) <sup>∈</sup> *Glm*(**F***M*(*s*)) because *TR*1(*s*), *TR*2(*s*) <sup>∈</sup> **<sup>F</sup>***M*(*s*)*m*×*<sup>m</sup>* and det *TR*1(*s*) and det *TR*2(*s*) factorize in *M*� \ *M*. Therefore *T*1(*s*) and *T*2(*s*) are left Wiener–Hopf equivalent with respect to (*M*, *M*� ).
Conversely, let *U*1(*s*) ∈ Gl*m*(**F***M*�(*s*)) ∩ Gl*m*(**F***pr*(*s*)) and *U*2(*s*) ∈ Gl*m*(**F***M*(*s*)) such that *T*1(*s*) = *U*1(*s*)*T*2(*s*)*U*2(*s*). Hence, *T*1(*s*) = *TL*1(*s*)*TR*1(*s*) = *U*1(*s*)*TL*2(*s*)*TR*2(*s*)*U*2(*s*). Put *TL*2(*s*) = *U*1(*s*)*TL*2(*s*) and *TR*2(*s*) = *TR*2(*s*)*U*2(*s*). Therefore,
$$\text{(i)}\ T\_1(\mathbf{s}) = T\_{L1}(\mathbf{s})T\_{R1}(\mathbf{s}) = \overline{T}\_{L2}(\mathbf{s})\overline{T}\_{R2}(\mathbf{s})\_{\text{A}}$$
14 Will-be-set-by-IN-TECH
are integers, that is, for any non-singular *<sup>T</sup>*(*s*) <sup>∈</sup> **<sup>F</sup>**(*s*)*m*×*<sup>m</sup>* there exist both a biproper matrix
where *g*<sup>1</sup> ≥ ··· ≥ *gm* are integers uniquely determined by *T*(*s*). They are called the left Wiener–Hopf factorization indices at infinity and form a complete system of invariants for the left Wiener–Hopf equivalence at infinity. These are the basic objects that will produce the complete system of invariants for the left Wiener–Hopf equivalence with respect to (*M*, *M*�
For polynomial matrices, their left Wiener–Hopf factorization indices at infinity are the column degrees of any right equivalent (by a unimodular matrix) column proper matrix. Namely, a polynomial matrix is column proper if it can be written as *Pc* Diag(*sg*<sup>1</sup> ,...,*sgm* ) + *<sup>L</sup>*(*s*) with *Pc* <sup>∈</sup> **<sup>F</sup>***m*×*<sup>m</sup>* non-singular, *<sup>g</sup>*1,..., *gm* non-negative integers and *<sup>L</sup>*(*s*) a polynomial matrix such that the degree of the *i*th column of *L*(*s*) smaller than *gi*, 1 ≤ *i* ≤ *m*. Let *P*(*s*) ∈
*<sup>m</sup>*×*<sup>m</sup>* be non-singular polynomial. There exists a unimodular matrix *<sup>V</sup>*(*s*) <sup>∈</sup> **<sup>F</sup>**[*s*]
factorization indices at infinity. Thus the following definition makes sense:
*iii) the global invariant rational functions of TR*(*s*) *factorize in* Specm(**F**[*s*]) \ *M.*
*ii) the global invariant rational functions of TL*(*s*) *factorize in M, and*
the left Wiener–Hopf factorization indices of *T*(*s*) at infinity.
*Wiener–Hopf factorization indices of TL*(*s*) *at infinity.*
*TL*(*s*), *TR*(*s*) <sup>∈</sup> **<sup>F</sup>**(*s*)*m*×*<sup>m</sup> such that*
*i) T*(*s*) = *TL*(*s*)*TR*(*s*)*,*
**Definition 16.** *Let T*(*s*) <sup>∈</sup> **<sup>F</sup>**(*s*)*m*×*<sup>m</sup> be a non-singular rational matrix and M* <sup>⊆</sup> Specm(**F**[*s*])*. Let*
*Then the left Wiener–Hopf factorization indices of T*(*s*) *with respect to M are defined to be the left*
In the particular case that *M* = Specm(**F**[*s*]), we can put *TL*(*s*) = *T*(*s*) and *TR*(*s*) = *Im*. Therefore, the left Wiener–Hopf factorization indices of *T*(*s*) with respect to Specm(**F**[*s*]) are
that *P*(*s*)*V*(*s*) is column proper. The column degrees of *P*(*s*)*V*(*s*) are uniquely determined by *P*(*s*), although *V*(*s*) is not (see [9], [12, p. 388], [17]). Since *P*(*s*)*V*(*s*) is column proper, it can be written as *P*(*s*)*V*(*s*) = *PcD*(*s*) + *L*(*s*) with *Pc* non-singular, *D*(*s*) = Diag(*sg*<sup>1</sup> ,...,*sgm* ) and the degree of the *i*th column of *L*(*s*) smaller than *gi*, 1 ≤ *i* ≤ *m*. Then *P*(*s*)*V*(*s*) = (*Pc* + *L*(*s*)*D*(*s*)−1)*D*(*s*). Put *B*(*s*) = *Pc* + *L*(*s*)*D*(*s*)−1. Since *Pc* is non-singular and *L*(*s*)*D*(*s*)−<sup>1</sup> is a strictly proper matrix, *B*(*s*) is biproper, and *P*(*s*) = *B*(*s*)*D*(*s*)*U*(*s*) where *U*(*s*) = *V*(*s*)−1. The left Wiener–Hopf factorization indices at infinity can be used to associate a sequence of integers with every non-singular rational matrix and every *M* ⊆ Specm(**F**[*s*]). This is done as follows: If *<sup>T</sup>*(*s*) <sup>∈</sup> **<sup>F</sup>**(*s*)*m*×*<sup>m</sup>* then it can always be written as *<sup>T</sup>*(*s*) = *TL*(*s*)*TR*(*s*) such that the global invariant rational functions of *TL*(*s*) factorize in *M* and *TR*(*s*) ∈ Gl*m*(**F***M*(*s*)) or, equivalently, the global invariant rational functions of *TR*(*s*) factorize in Specm(**F**[*s*]) \ *M* (see Proposition 2). There may be many factorizations of this type, but it turns out (see [1, Proposition 3.2] for the polynomial case) that the left factors in all of them are right equivalent. This means that if *T*(*s*) = *TL*1(*s*)*TR*1(*s*) = *TL*2(*s*)*TR*2(*s*) with the global invariant rational functions of *TL*1(*s*) and *TL*2(*s*) factorizing in *M* and the global invariant rational functions of *TR*1(*s*) and *TR*2(*s*) factorizing in Specm(**F**[*s*]) \ *M* then there is a unimodular matrix *U*(*s*) such that *TL*1(*s*) = *TL*2(*s*)*U*(*s*). In particular, *TL*1(*s*) and *TL*2(*s*) have the same left Wiener–Hopf
*T*(*s*) = *B*(*s*) Diag(*sg*<sup>1</sup> ,...,*sgm* )*U*(*s*) (37)
).
*<sup>m</sup>*×*<sup>m</sup>* such
*B*(*s*) ∈ Gl*m*(**F***pr*(*s*)) and a unimodular matrix *U*(*s*) ∈ Gl*m*(**F**[*s*]) such that
**F**[*s*]
(ii) the global invariant rational functions of *TL*1(*s*) and of *TL*2(*s*) factorize in *M*, and (iii)the global invariant rational functions of *TR*1(*s*) and of *TR*2(*s*) factorize in Specm(**F**[*s*]) \ *M*.
Then *TL*1(*s*) and *TL*2(*s*) are right equivalent (see the remark previous to Definition 16). So, there exists *U*(*s*) ∈ Gl*m*(**F**[*s*]) such that *TL*1(*s*) = *TL*2(*s*)*U*(*s*). Thus, *TL*1(*s*) = *U*1(*s*)*TL*2(*s*)*U*(*s*). Since *U*1(*s*) is biproper and *U*(*s*) is unimodular *TL*1(*s*), *TL*2(*s*) have the same left Wiener–Hopf factorization indices at infinity. Consequentially, *T*1(*s*) and *T*2(*s*) have the same left Wiener–Hopf factorization indices with respect to *M*.
In conclusion, for non-singular rational matrices with no zeros and no poles in *M* ∩ *M*� the left Wiener–Hopf factorization indices with respect to *M* form a complete system of invariants for the left Wiener–Hopf equivalence with respect to (*M*, *M*� ) with *M* ∪ *M*� = Specm(**F**[*s*]).
A straightforward consequence of the above theorem is the following Corollary
**Corollary 18.** *Let M*, *M*� ⊆ Specm(**F**[*s*]) *be such that M* ∪ *M*� = Specm(**F**[*s*])*. Let T*1(*s*)*, <sup>T</sup>*2(*s*) <sup>∈</sup> **<sup>F</sup>**(*s*)*m*×*<sup>m</sup> be non-singular with no zeros and no poles in M* <sup>∩</sup> *<sup>M</sup>*� *. Then T*1(*s*) *and T*2(*s*) *are left Wiener–Hopf equivalent with respect to* (*M*, *M*� ) *if and only if for any factorizations T*1(*s*) = *TL*1(*s*)*TR*1(*s*) *and T*2(*s*) = *TL*2(*s*)*TR*2(*s*) *satisfying the conditions (i)–(iii) of Definition 16, TL*1(*s*) *and TL*2(*s*) *are left Wiener–Hopf equivalent at infinity.*
Next we deal with the problem of factorizing or reducing a rational matrix to diagonal form by Wiener–Hopf equivalence. It will be shown that if there exists in *M* an ideal generated by a monic irreducible polynomial of degree equal to 1 which is not in *M*� , then any non-singular rational matrix, with no zeros and no poles in *M* ∩ *M*� admits a factorization with respect to (*M*, *M*� ). Afterwards, some examples will be given in which these conditions on *M* and *M*� are removed and factorization fails to exist.
**Theorem 19.** *Let M*, *M*� ⊆ Specm(**F**[*s*]) *be such that M* ∪ *M*� = Specm(**F**[*s*])*. Assume that there are ideals in M* \ *<sup>M</sup>*� *generated by linear polynomials. Let* (*<sup>s</sup>* <sup>−</sup> *<sup>a</sup>*) *be any of them and T*(*s*) <sup>∈</sup> **<sup>F</sup>**(*s*)*m*×*<sup>m</sup> a non-singular matrix with no zeros and no poles in M* ∩ *M*� *. There exist both U*1(*s*) *invertible in* **<sup>F</sup>***M*�(*s*)*m*×*<sup>m</sup>* <sup>∩</sup> **<sup>F</sup>***pr*(*s*)*m*×*<sup>m</sup> and U*2(*s*) *invertible in* **<sup>F</sup>***M*(*s*)*m*×*<sup>m</sup> such that*
$$T(s) = \mathcal{U}\_1(s) \operatorname{Diag}((s-a)^{k\_1}, \dots, (s-a)^{k\_w}) \mathcal{U}\_2(s), \tag{38}$$
*where k*<sup>1</sup> ≥ ··· ≥ *km are integers uniquely determined by T*(*s*)*. Moreover, they are the left Wiener–Hopf factorization indices of T*(*s*) *with respect to M.*
**Proof.-** The matrix *T*(*s*) can be written (see Proposition 2) as *T*(*s*) = *TL*(*s*)*TR*(*s*) with the global invariant rational functions of *TL*(*s*) factorizing in *M* \ *M*� and the global invariant rational functions of *TR*(*s*) factorizing in Specm(**F**[*s*]) \ *M* = *M*� \ *M*. As *k*1,..., *km* are the left Wiener–Hopf factorization indices of *TL*(*s*) at infinity, there exist matrices *U*(*s*) ∈ Gl*m*(**F**[*s*]) and *<sup>B</sup>*(*s*) <sup>∈</sup> Gl*m*(**F***pr*(*s*)) such that *TL*(*s*) = *<sup>B</sup>*(*s*)*D*1(*s*)*U*(*s*) with *<sup>D</sup>*1(*s*) = Diag(*sk*<sup>1</sup> ,...,*skm* ). Put *<sup>D</sup>*(*s*) = Diag((*<sup>s</sup>* <sup>−</sup> *<sup>a</sup>*)*k*<sup>1</sup> ,...,(*<sup>s</sup>* <sup>−</sup> *<sup>a</sup>*)*km* ) and *<sup>U</sup>*1(*s*) = *<sup>B</sup>*(*s*) Diag *<sup>s</sup>k*<sup>1</sup> (*s*−*a*)*<sup>k</sup>*<sup>1</sup> ,..., *<sup>s</sup>km* (*s*−*a*)*km* . Then *TL*(*s*) = *U*1(*s*)*D*(*s*)*U*(*s*). If *U*2(*s*) = *U*(*s*)*TR*(*s*) then this matrix is invertible in **F***M*(*s*)*m*×*<sup>m</sup>* and *<sup>T</sup>*(*s*) = *<sup>U</sup>*1(*s*) Diag((*<sup>s</sup>* <sup>−</sup> *<sup>a</sup>*)*k*<sup>1</sup> ,...,(*<sup>s</sup>* <sup>−</sup> *<sup>a</sup>*)*km* )*U*2(*s*). We only have to prove that *<sup>U</sup>*1(*s*) is invertible in **<sup>F</sup>***M*�(*s*)*m*×*<sup>m</sup>* <sup>∩</sup> **<sup>F</sup>***pr*(*s*)*m*×*m*. It is clear that *<sup>U</sup>*1(*s*) is in **<sup>F</sup>***pr*(*s*)*m*×*<sup>m</sup>* and biproper. Moreover, the global invariant rational functions of *TL*(*s*) *U*1(*s*) = *TL*(*s*)(*D*(*s*)*U*(*s*))−<sup>1</sup> factorize in *M* \ *M*� . Therefore, *U*1(*s*) is invertible in **F***M*�(*s*)*m*×*m*.
We prove now the uniqueness of the factorization. Assume that *T*(*s*) also factorizes as
$$T(s) = \tilde{\mathcal{U}}\_1(s) \operatorname{Diag}((s-a)^{\tilde{k}\_1}, \dots, (s-a)^{\tilde{k}\_w}) \tilde{\mathcal{U}}\_2(s), \tag{39}$$
with ˜ *<sup>k</sup>*<sup>1</sup> ≥···≥ ˜ *km* integers. Then,
$$\text{Diag}((s-a)^{\tilde{k}\_1}, \dots, (s-a)^{\tilde{k}\_m}) = \tilde{\text{U}}\_1(s)^{-1} \text{U}\_1(s) \text{Diag}((s-a)^{\tilde{k}\_1}, \dots, (s-a)^{\tilde{k}\_m}) \text{U}\_2(s) \tilde{\text{U}}\_2(s)^{-1}. \tag{40}$$
The diagonal matrices have no zeros and no poles in *M* ∩ *M*� (because (*s* − *a*) ∈ *M* \ *M*� ) and they are left Wiener–Hopf equivalent with respect to (*M*, *M*� ). By Theorem 17, they have the same left Wiener–Hopf factorization indices with respect to *M*. Thus, ˜ *ki* = *ki* for all *i* = 1, . . . , *m*.
Following [6] we could call left Wiener–Hopf factorization indices with respect to (*M*, *M*� ) the exponents *k*<sup>1</sup> ≥···≥ *km* appearing in the diagonal matrix of Theorem 19. They are, actually, the left Wiener–Hopf factorization indices with respect to *M*.
Several examples follow that exhibit some remarkable features about the results that have been proved so far. The first two examples show that if no assumption is made on the intersection and/or union of *M* and *M*� then existence and/or uniqueness of diagonal factorization may fail to exist.
**Example 20.** If *P*(*s*) is a polynomial matrix with zeros in *M* ∩ *M*� then the existence of invertible matrices *U*1(*s*) ∈ Gl*m*(**F***M*�(*s*)) ∩ Gl*m*(**F***pr*(*s*)) and *U*2(*s*) ∈ Gl*m*(**F***M*(*s*)) such that *<sup>P</sup>*(*s*) = *<sup>U</sup>*1(*s*) Diag((*<sup>s</sup>* <sup>−</sup> *<sup>a</sup>*)*k*<sup>1</sup> ,...,(*<sup>s</sup>* <sup>−</sup> *<sup>a</sup>*)*km* )*U*2(*s*) with (*<sup>s</sup>* <sup>−</sup> *<sup>a</sup>*) <sup>∈</sup> *<sup>M</sup>* \ *<sup>M</sup>*� may fail. In fact, suppose that *M* = {(*s*),(*s* + 1)}, *M*� = Specm **F**[*s*] \ {(*s*)}. Therefore, *M* ∩ *M*� = {(*s* + 1)} and (*s*) ∈ *M* \ *M*� . Consider *p*1(*s*) = *s* + 1. Assume that *s* + 1 = *u*1(*s*)*sku*2(*s*) with *u*1(*s*) a unit in **F***M*�(*s*) ∩ **F***pr*(*s*) and *u*2(*s*) a unit in **F***M*(*s*). Thus, *u*1(*s*) = *c* a nonzero constant and *u*2(*s*) = <sup>1</sup> *c s*+1 *sk* which is not a unit in **F***M*(*s*).
16 Will-be-set-by-IN-TECH
Next we deal with the problem of factorizing or reducing a rational matrix to diagonal form by Wiener–Hopf equivalence. It will be shown that if there exists in *M* an ideal generated by
rational matrix, with no zeros and no poles in *M* ∩ *M*� admits a factorization with respect to
**Theorem 19.** *Let M*, *M*� ⊆ Specm(**F**[*s*]) *be such that M* ∪ *M*� = Specm(**F**[*s*])*. Assume that there are ideals in M* \ *<sup>M</sup>*� *generated by linear polynomials. Let* (*<sup>s</sup>* <sup>−</sup> *<sup>a</sup>*) *be any of them and T*(*s*) <sup>∈</sup> **<sup>F</sup>**(*s*)*m*×*<sup>m</sup>*
*where k*<sup>1</sup> ≥ ··· ≥ *km are integers uniquely determined by T*(*s*)*. Moreover, they are the left*
**Proof.-** The matrix *T*(*s*) can be written (see Proposition 2) as *T*(*s*) = *TL*(*s*)*TR*(*s*) with the global invariant rational functions of *TL*(*s*) factorizing in *M* \ *M*� and the global invariant rational functions of *TR*(*s*) factorizing in Specm(**F**[*s*]) \ *M* = *M*� \ *M*. As *k*1,..., *km* are the left Wiener–Hopf factorization indices of *TL*(*s*) at infinity, there exist matrices *U*(*s*) ∈ Gl*m*(**F**[*s*]) and *<sup>B</sup>*(*s*) <sup>∈</sup> Gl*m*(**F***pr*(*s*)) such that *TL*(*s*) = *<sup>B</sup>*(*s*)*D*1(*s*)*U*(*s*) with *<sup>D</sup>*1(*s*) = Diag(*sk*<sup>1</sup> ,...,*skm* ).
*TL*(*s*) = *U*1(*s*)*D*(*s*)*U*(*s*). If *U*2(*s*) = *U*(*s*)*TR*(*s*) then this matrix is invertible in **F***M*(*s*)*m*×*<sup>m</sup>* and *<sup>T</sup>*(*s*) = *<sup>U</sup>*1(*s*) Diag((*<sup>s</sup>* <sup>−</sup> *<sup>a</sup>*)*k*<sup>1</sup> ,...,(*<sup>s</sup>* <sup>−</sup> *<sup>a</sup>*)*km* )*U*2(*s*). We only have to prove that *<sup>U</sup>*1(*s*) is invertible in **<sup>F</sup>***M*�(*s*)*m*×*<sup>m</sup>* <sup>∩</sup> **<sup>F</sup>***pr*(*s*)*m*×*m*. It is clear that *<sup>U</sup>*1(*s*) is in **<sup>F</sup>***pr*(*s*)*m*×*<sup>m</sup>* and biproper. Moreover, the global invariant rational functions of *TL*(*s*) *U*1(*s*) = *TL*(*s*)(*D*(*s*)*U*(*s*))−<sup>1</sup>
˜
The diagonal matrices have no zeros and no poles in *M* ∩ *M*� (because (*s* − *a*) ∈ *M* \ *M*�
Following [6] we could call left Wiener–Hopf factorization indices with respect to (*M*, *M*�
exponents *k*<sup>1</sup> ≥···≥ *km* appearing in the diagonal matrix of Theorem 19. They are, actually,
Several examples follow that exhibit some remarkable features about the results that have been proved so far. The first two examples show that if no assumption is made on the intersection and/or union of *M* and *M*� then existence and/or uniqueness of diagonal
*<sup>k</sup>*<sup>1</sup> ,...,(*<sup>s</sup>* <sup>−</sup> *<sup>a</sup>*)
˜
*km* ) = *<sup>U</sup>*˜ <sup>1</sup>(*s*)−1*U*1(*s*) Diag((*<sup>s</sup>* <sup>−</sup> *<sup>a</sup>*)*k*<sup>1</sup> ,...,(*<sup>s</sup>* <sup>−</sup> *<sup>a</sup>*)*km* )*U*2(*s*)*U*˜ <sup>2</sup>(*s*)<sup>−</sup>1.
. Therefore, *U*1(*s*) is invertible in **F***M*�(*s*)*m*×*m*. We prove now the uniqueness of the factorization. Assume that *T*(*s*) also factorizes as
). Afterwards, some examples will be given in which these conditions on *M* and *M*�
*<sup>T</sup>*(*s*) = *<sup>U</sup>*1(*s*) Diag((*<sup>s</sup>* <sup>−</sup> *<sup>a</sup>*)*k*<sup>1</sup> ,...,(*<sup>s</sup>* <sup>−</sup> *<sup>a</sup>*)*km* )*U*2(*s*), (38)
, then any non-singular
*. There exist both U*1(*s*) *invertible in*
*<sup>s</sup>k*<sup>1</sup>
(*s*−*a*)*<sup>k</sup>*<sup>1</sup> ,..., *<sup>s</sup>km*
*km* )*U*˜ <sup>2</sup>(*s*), (39)
). By Theorem 17, they have
*ki* = *ki* for all *i* =
(*s*−*a*)*km*
. Then
(40)
) the
)
a monic irreducible polynomial of degree equal to 1 which is not in *M*�
are removed and factorization fails to exist.
*a non-singular matrix with no zeros and no poles in M* ∩ *M*�
*Wiener–Hopf factorization indices of T*(*s*) *with respect to M.*
**<sup>F</sup>***M*�(*s*)*m*×*<sup>m</sup>* <sup>∩</sup> **<sup>F</sup>***pr*(*s*)*m*×*<sup>m</sup> and U*2(*s*) *invertible in* **<sup>F</sup>***M*(*s*)*m*×*<sup>m</sup> such that*
Put *<sup>D</sup>*(*s*) = Diag((*<sup>s</sup>* <sup>−</sup> *<sup>a</sup>*)*k*<sup>1</sup> ,...,(*<sup>s</sup>* <sup>−</sup> *<sup>a</sup>*)*km* ) and *<sup>U</sup>*1(*s*) = *<sup>B</sup>*(*s*) Diag
*<sup>T</sup>*(*s*) = *<sup>U</sup>*˜ <sup>1</sup>(*s*) Diag((*<sup>s</sup>* <sup>−</sup> *<sup>a</sup>*)
*km* integers. Then,
˜
and they are left Wiener–Hopf equivalent with respect to (*M*, *M*�
the left Wiener–Hopf factorization indices with respect to *M*.
the same left Wiener–Hopf factorization indices with respect to *M*. Thus, ˜
*<sup>k</sup>*<sup>1</sup> ,...,(*<sup>s</sup>* <sup>−</sup> *<sup>a</sup>*)
(*M*, *M*�
factorize in *M* \ *M*�
*<sup>k</sup>*<sup>1</sup> ≥···≥ ˜
˜
factorization may fail to exist.
Diag((*s* − *a*)
with ˜
1, . . . , *m*.
**Example 21.** If *M* ∪ *M*� �= Specm **F**[*s*] then the factorization indices with respect to (*M*, *M*� ) may be not unique. Suppose that (*β*(*s*)) ∈/ *M* ∪ *M*� , (*π*(*s*)) ∈ *M* \ *M*� with *d*(*π*(*s*)) = 1 and *<sup>p</sup>*(*s*) = *<sup>u</sup>*1(*s*)*π*(*s*)*ku*2(*s*), with *<sup>u</sup>*1(*s*) a unit in **<sup>F</sup>***M*�(*s*) <sup>∩</sup> **<sup>F</sup>***pr*(*s*) and *<sup>u</sup>*2(*s*) a unit in **<sup>F</sup>***M*(*s*). Then *<sup>p</sup>*(*s*) can also be factorized as *<sup>p</sup>*(*s*) = *<sup>u</sup>*˜1(*s*)*π*(*s*)*k*−*d*(*β*(*s*))*u*˜2(*s*) with *<sup>u</sup>*˜1(*s*) = *<sup>u</sup>*1(*s*) *<sup>π</sup>*(*s*)*<sup>d</sup>*(*β*(*s*)) *<sup>β</sup>*(*s*) a unit in **F***M*�(*s*) ∩ **F***pr*(*s*) and *u*˜2(*s*) = *β*(*s*)*u*2(*s*) a unit in **F***M*(*s*).
The following example shows that if all ideals generated by polynomials of degree equal to one are in *M*� \ *M* then a factorization as in Theorem 19 may not exist.
**Example 22.** Suppose that **<sup>F</sup>** <sup>=</sup> **<sup>R</sup>**. Consider *<sup>M</sup>* <sup>=</sup> {(*s*<sup>2</sup> <sup>+</sup> <sup>1</sup>)} ⊆ Specm(**R**[*s*]) and *<sup>M</sup>*� <sup>=</sup> Specm(**R**[*s*]) \ {(*s*<sup>2</sup> <sup>+</sup> <sup>1</sup>)}. Let
$$P(s) = \begin{bmatrix} s & 0 \\ -s^2 \ (s^2+1)^2 \end{bmatrix}.\tag{41}$$
Notice that *P*(*s*) has no zeros and no poles in *M* ∩ *M*� = ∅. We will see that it is not possible to find invertible matrices *<sup>U</sup>*1(*s*) <sup>∈</sup> **<sup>R</sup>***M*�(*s*)2×<sup>2</sup> <sup>∩</sup> **<sup>R</sup>***pr*(*s*)2×<sup>2</sup> and *<sup>U</sup>*2(*s*) <sup>∈</sup> **<sup>R</sup>***M*(*s*)2×<sup>2</sup> such that
$$\mathcal{U}\_1(s)P(s)\mathcal{U}\_2(s) = \text{Diag}((p(s)/q(s))^{c\_1}, (p(s)/q(s))^{c\_2}).\tag{42}$$
We can write *<sup>p</sup>*(*s*) *<sup>q</sup>*(*s*) <sup>=</sup> *<sup>u</sup>*(*s*)(*s*<sup>2</sup> <sup>+</sup> <sup>1</sup>)*<sup>a</sup>* with *<sup>u</sup>*(*s*) a unit in **<sup>R</sup>***M*(*s*) and *<sup>a</sup>* <sup>∈</sup> **<sup>Z</sup>**. Therefore,
$$\text{Diag}((p(s)/q(s))^{\varepsilon\_1}, (p(s)/q(s))^{\varepsilon\_2}) = \text{Diag}((s^2+1)^{ac\_1}, (s^2+1)^{ac\_2}) \cdot \text{Diag}(u(s)^{\varepsilon\_1}, u(s)^{\varepsilon\_2}) . \tag{43}$$
Diag(*u*(*s*)*c*<sup>1</sup> , *u*(*s*)*c*<sup>2</sup> ) is invertible in **R***M*(*s*)2×<sup>2</sup> and *P*(*s*) is also left Wiener–Hopf equivalent with respect to (*M*, *M*� ) to the diagonal matrix Diag((*s*<sup>2</sup> + 1)*ac*<sup>1</sup> ,(*s*<sup>2</sup> + 1)*ac*<sup>2</sup> ).
Assume that there exist invertible matrices *<sup>U</sup>*1(*s*) <sup>∈</sup> **<sup>R</sup>***M*�(*s*)2×<sup>2</sup> <sup>∩</sup> **<sup>R</sup>***pr*(*s*)2×<sup>2</sup> and *<sup>U</sup>*2(*s*) <sup>∈</sup> **<sup>R</sup>***M*(*s*)2×<sup>2</sup> such that *<sup>U</sup>*1(*s*)*P*(*s*)*U*2(*s*) = Diag((*s*<sup>2</sup> <sup>+</sup> <sup>1</sup>)*d*<sup>1</sup> ,(*s*<sup>2</sup> <sup>+</sup> <sup>1</sup>)*d*<sup>2</sup> ), with *<sup>d</sup>*<sup>1</sup> <sup>≥</sup> *<sup>d</sup>*<sup>2</sup> integers. Notice first that det *U*1(*s*) is a nonzero constant and since det *P*(*s*) = *s*(*s*<sup>2</sup> + 1)<sup>2</sup> and det *U*2(*s*) is a rational function with numerator and denominator relatively prime with *s*<sup>2</sup> + 1, it follows that *cs*(*s*<sup>2</sup> + 1)<sup>2</sup> det *U*2(*s*)=(*s*<sup>2</sup> + 1)*d*1+*d*<sup>2</sup> . Thus, *d*<sup>1</sup> + *d*<sup>2</sup> = 2. Let
$$\mathcal{U}\_{1}(s)^{-1} = \begin{bmatrix} b\_{11}(s) \ b\_{12}(s) \\ b\_{21}(s) \ b\_{22}(s) \end{bmatrix}, \quad \mathcal{U}\_{2}(s) = \begin{bmatrix} \mu\_{11}(s) \ \mu\_{12}(s) \\ \mu\_{21}(s) \ \mu\_{22}(s) \end{bmatrix}. \tag{44}$$
From *P*(*s*)*U*2(*s*) = *U*1(*s*)−<sup>1</sup> Diag((*s*<sup>2</sup> + 1)*d*<sup>1</sup> ,(*s*<sup>2</sup> + 1)*d*<sup>2</sup> ) we get
*su*11(*s*) = *b*11(*s*)(*s* <sup>2</sup> + 1)*d*<sup>1</sup> , (45)
$$s - s^2 u\_{11}(s) + (s^2 + 1)^2 u\_{21}(s) = b\_{21}(s)(s^2 + 1)^{d\_1} \,\,\,\,\tag{46}$$
$$
\mathfrak{su}\_{12}(\mathbf{s}) = b\_{12}(\mathbf{s})(\mathbf{s}^2 + 1)^{d\_2}, \tag{47}
$$
$$-s^2\mu\_{12}(s) + (s^2+1)^2\mu\_{22}(s) = b\_{22}(s)(s^2+1)^{d\_2}.\tag{48}$$
As *<sup>u</sup>*11(*s*) <sup>∈</sup> **<sup>R</sup>***M*(*s*) and *<sup>b</sup>*11(*s*) <sup>∈</sup> **<sup>R</sup>***M*�(*s*) <sup>∩</sup> **<sup>R</sup>***pr*(*s*), we can write *<sup>u</sup>*11(*s*) = *<sup>f</sup>*1(*s*) *<sup>g</sup>*1(*s*) and *<sup>b</sup>*11(*s*) = *h*1(*s*) (*s*<sup>2</sup>+1)*<sup>q</sup>*<sup>1</sup> with *<sup>f</sup>*1(*s*), *<sup>g</sup>*1(*s*), *<sup>h</sup>*1(*s*) <sup>∈</sup> **<sup>R</sup>**[*s*], gcd(*g*1(*s*),*s*<sup>2</sup> <sup>+</sup> <sup>1</sup>) = 1 and *<sup>d</sup>*(*h*1(*s*)) <sup>≤</sup> <sup>2</sup>*q*1. Therefore, by (45), *s <sup>f</sup>*1(*s*) *<sup>g</sup>*1(*s*) <sup>=</sup> *<sup>h</sup>*1(*s*) (*s*<sup>2</sup>+1)*<sup>q</sup>*<sup>1</sup> (*s*<sup>2</sup> <sup>+</sup> <sup>1</sup>)*d*<sup>1</sup> . Hence, *<sup>u</sup>*11(*s*) = *<sup>f</sup>*1(*s*) or *<sup>u</sup>*11(*s*) = *<sup>f</sup>*1(*s*) *<sup>s</sup>* . In the same way and using (47), *<sup>u</sup>*12(*s*) = *<sup>f</sup>*2(*s*) or *<sup>u</sup>*12(*s*) = *<sup>f</sup>*2(*s*) *<sup>s</sup>* with *f*2(*s*) a polynomial. Moreover, by (47), *d*<sup>2</sup> must be non-negative. Hence, *d*<sup>1</sup> ≥ *d*<sup>2</sup> ≥ 0. Using now (46) and (48) and bearing in mind again that *u*21(*s*), *u*22(*s*) ∈ **R***M*(*s*) and *b*21(*s*), *b*22(*s*) ∈ **R***M*�(*s*) ∩ **R***pr*(*s*), we conclude that *u*21(*s*) and *u*22(*s*) are polynomials.
We can distinguish two cases: *d*<sup>1</sup> = 2, *d*<sup>2</sup> = 0 and *d*<sup>1</sup> = *d*<sup>2</sup> = 1. If *d*<sup>1</sup> = 2 and *d*<sup>2</sup> = 0, by (47), *b*12(*s*) is a polynomial and since *b*12(*s*) is proper, it is constant: *b*12(*s*) = *c*1. Thus *u*12(*s*) = *<sup>c</sup>*<sup>1</sup> *s* . By (48), *<sup>b</sup>*22(*s*) = <sup>−</sup>*c*1*<sup>s</sup>* + (*s*<sup>2</sup> <sup>+</sup> <sup>1</sup>)2*u*22(*s*). Since *<sup>u</sup>*22(*s*) is polynomial and *<sup>b</sup>*22(*s*) is proper, *<sup>b</sup>*22(*s*) is also constant and then *u*22(*s*) = 0 and *c*<sup>1</sup> = 0. Consequentially, *b*22(*s*) = 0, and *b*12(*s*) = 0. This is impossible because *U*1(*s*) is invertible.
If *d*<sup>1</sup> = *d*<sup>2</sup> = 1 then , using (46),
$$\begin{split} b\_{21}(s) &= \frac{-s^2 \mu\_{11}(s) + (s^2+1)^2 \mu\_{21}(s)}{s^2+1} = \frac{-s^2 \frac{h\_{11}(s)}{s}(s^2+1) + (s^2+1)^2 \mu\_{21}(s)}{s^2+1} \\ &= -s b\_{11}(s) + (s^2+1) \mu\_{21}(s) = -s \frac{h\_{1}(s)}{(s^2+1)^{\eta\_1}} + (s^2+1) \mu\_{21}(s) \\ &= \frac{-s h\_{1}(s) + (s^2+1)^{\eta\_1+1} \mu\_{21}(s)}{(s^2+1)^{\phi\_1}}. \end{split} \tag{49}$$
Notice that *<sup>d</sup>*(−*sh*1(*s*)) <sup>≤</sup> <sup>1</sup> <sup>+</sup> <sup>2</sup>*q*<sup>1</sup> and *<sup>d</sup>*((*s*<sup>2</sup> <sup>+</sup> <sup>1</sup>)*q*1+1*u*21(*s*)) = <sup>2</sup>(*q*<sup>1</sup> <sup>+</sup> <sup>1</sup>) + *<sup>d</sup>*(*u*21(*s*)) <sup>≥</sup> <sup>2</sup>*q*<sup>1</sup> <sup>+</sup> <sup>2</sup> unless *<sup>u</sup>*21(*s*) = 0. Hence, if *<sup>u</sup>*21(*s*) �<sup>=</sup> 0, *<sup>d</sup>*(−*sh*1(*s*)+(*s*<sup>2</sup> <sup>+</sup> <sup>1</sup>)*q*1+1*u*21(*s*)) <sup>≥</sup> <sup>2</sup>*q*<sup>1</sup> <sup>+</sup> 2 which is greater than *d*((*s*<sup>2</sup> + 1)*q*<sup>1</sup> ) = 2*q*1. This cannot happen because *b*21(*s*) is proper. Thus, *u*21(*s*) = 0. In the same way and reasoning with (48) we get that *u*22(*s*) is also zero. This is again impossible because *U*2(*s*) is invertible. Therefore no left Wiener–Hopf factorization of *P*(*s*) with respect to (*M*, *M*� ) exits.
We end this section with an example where the left Wiener–Hopf factorization indices of the matrix polynomial in the previous example are computed. Then an ideal generated by a polynomial of degree 1 is added to *M* and the Wiener–Hopf factorization indices of the same matrix are obtained in two different cases.
**Example 23.** Let **<sup>F</sup>** <sup>=</sup> **<sup>R</sup>** and *<sup>M</sup>* <sup>=</sup> {(*s*<sup>2</sup> <sup>+</sup> <sup>1</sup>)}. Consider the matrix
$$P(s) = \begin{bmatrix} s & 0 \\ -s^2 \ (s^2+1)^2 \end{bmatrix} \tag{50}$$
which has a zero at 0. It can be written as *P*(*s*) = *P*1(*s*)*P*2(*s*) with
$$P\_1(s) = \begin{bmatrix} 1 & 0 \\ -s \ (s^2 + 1)^2 \end{bmatrix}, \quad P\_2(s) = \begin{bmatrix} s \ 0 \\ 0 \ 1 \end{bmatrix}, \tag{51}$$
where the global invariant factors of *P*1(*s*) are powers of *s*<sup>2</sup> + 1 and the global invariant factors of *P*2(*s*) are relatively prime with *s*<sup>2</sup> + 1. Moreover, the left Wiener–Hopf factorization indices of *P*1(*s*) at infinity are 3, 1 (add the first column multiplied by *s*<sup>3</sup> + 2*s* to the second column; the result is a column proper matrix with column degrees 1 and 3). Therefore, the left Wiener–Hopf factorization indices of *P*(*s*) with respect to *M* are 3, 1.
Consider now *<sup>M</sup>*˜ <sup>=</sup> {(*s*<sup>2</sup> <sup>+</sup> <sup>1</sup>),(*s*)} and *<sup>M</sup>*˜ � <sup>=</sup> Specm(**R**[*s*]) \ *<sup>M</sup>*˜ . There is a unimodular matrix *<sup>U</sup>*(*s*) = 1 *s*<sup>2</sup> + 2 0 1 , invertible in **<sup>R</sup>***M*˜ (*s*)2×2, such that *<sup>P</sup>*(*s*)*U*(*s*) = *s s*<sup>3</sup> <sup>+</sup> <sup>2</sup>*<sup>s</sup>* <sup>−</sup>*s*<sup>2</sup> <sup>1</sup> is column proper with column degrees 3 and 2. We can write
$$P(s)\mathcal{U}(s) = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \begin{bmatrix} s^2 & 0 \\ 0 & s^3 \end{bmatrix} + \begin{bmatrix} s \ 2s \\ 0 \ 1 \end{bmatrix} = \mathcal{B}(s) \begin{bmatrix} s^2 & 0 \\ 0 \ s^3 \end{bmatrix} \tag{52}$$
where *B*(*s*) is the following biproper matrix
18 Will-be-set-by-IN-TECH
(*s*<sup>2</sup>+1)*<sup>q</sup>*<sup>1</sup> with *<sup>f</sup>*1(*s*), *<sup>g</sup>*1(*s*), *<sup>h</sup>*1(*s*) <sup>∈</sup> **<sup>R</sup>**[*s*], gcd(*g*1(*s*),*s*<sup>2</sup> <sup>+</sup> <sup>1</sup>) = 1 and *<sup>d</sup>*(*h*1(*s*)) <sup>≤</sup> <sup>2</sup>*q*1. Therefore,
(47), *d*<sup>2</sup> must be non-negative. Hence, *d*<sup>1</sup> ≥ *d*<sup>2</sup> ≥ 0. Using now (46) and (48) and bearing in mind again that *u*21(*s*), *u*22(*s*) ∈ **R***M*(*s*) and *b*21(*s*), *b*22(*s*) ∈ **R***M*�(*s*) ∩ **R***pr*(*s*), we conclude
We can distinguish two cases: *d*<sup>1</sup> = 2, *d*<sup>2</sup> = 0 and *d*<sup>1</sup> = *d*<sup>2</sup> = 1. If *d*<sup>1</sup> = 2 and *d*<sup>2</sup> = 0, by (47), *b*12(*s*) is a polynomial and since *b*12(*s*) is proper, it is constant: *b*12(*s*) = *c*1. Thus *u*12(*s*) = *<sup>c</sup>*<sup>1</sup>
By (48), *<sup>b</sup>*22(*s*) = <sup>−</sup>*c*1*<sup>s</sup>* + (*s*<sup>2</sup> <sup>+</sup> <sup>1</sup>)2*u*22(*s*). Since *<sup>u</sup>*22(*s*) is polynomial and *<sup>b</sup>*22(*s*) is proper, *<sup>b</sup>*22(*s*) is also constant and then *u*22(*s*) = 0 and *c*<sup>1</sup> = 0. Consequentially, *b*22(*s*) = 0, and *b*12(*s*) = 0.
Notice that *<sup>d</sup>*(−*sh*1(*s*)) <sup>≤</sup> <sup>1</sup> <sup>+</sup> <sup>2</sup>*q*<sup>1</sup> and *<sup>d</sup>*((*s*<sup>2</sup> <sup>+</sup> <sup>1</sup>)*q*1+1*u*21(*s*)) = <sup>2</sup>(*q*<sup>1</sup> <sup>+</sup> <sup>1</sup>) + *<sup>d</sup>*(*u*21(*s*)) <sup>≥</sup> <sup>2</sup>*q*<sup>1</sup> <sup>+</sup> <sup>2</sup> unless *<sup>u</sup>*21(*s*) = 0. Hence, if *<sup>u</sup>*21(*s*) �<sup>=</sup> 0, *<sup>d</sup>*(−*sh*1(*s*)+(*s*<sup>2</sup> <sup>+</sup> <sup>1</sup>)*q*1+1*u*21(*s*)) <sup>≥</sup> <sup>2</sup>*q*<sup>1</sup> <sup>+</sup> 2 which is greater than *d*((*s*<sup>2</sup> + 1)*q*<sup>1</sup> ) = 2*q*1. This cannot happen because *b*21(*s*) is proper. Thus, *u*21(*s*) = 0. In the same way and reasoning with (48) we get that *u*22(*s*) is also zero. This is again impossible because *U*2(*s*) is invertible. Therefore no left Wiener–Hopf factorization of *P*(*s*)
We end this section with an example where the left Wiener–Hopf factorization indices of the matrix polynomial in the previous example are computed. Then an ideal generated by a polynomial of degree 1 is added to *M* and the Wiener–Hopf factorization indices of the same
> *s* 0 <sup>−</sup>*s*<sup>2</sup> (*s*<sup>2</sup> <sup>+</sup> <sup>1</sup>)<sup>2</sup>
where the global invariant factors of *P*1(*s*) are powers of *s*<sup>2</sup> + 1 and the global invariant factors of *P*2(*s*) are relatively prime with *s*<sup>2</sup> + 1. Moreover, the left Wiener–Hopf factorization
*s* 0 0 1
, *P*2(*s*) =
*<sup>s</sup>*<sup>2</sup>+<sup>1</sup> <sup>=</sup> <sup>−</sup>*s*<sup>2</sup> *<sup>b</sup>*11(*s*)
<sup>=</sup> <sup>−</sup>*sb*11(*s*)+(*s*<sup>2</sup> <sup>+</sup> <sup>1</sup>)*u*21(*s*) = <sup>−</sup>*<sup>s</sup>*
(*s*<sup>2</sup>+1)*<sup>q</sup>*<sup>1</sup> .
(*s*<sup>2</sup>+1)*<sup>q</sup>*<sup>1</sup> (*s*<sup>2</sup> <sup>+</sup> <sup>1</sup>)*d*<sup>1</sup> . Hence, *<sup>u</sup>*11(*s*) = *<sup>f</sup>*1(*s*) or *<sup>u</sup>*11(*s*) = *<sup>f</sup>*1(*s*)
<sup>2</sup> + 1)2*u*22(*s*) = *b*22(*s*)(*s*
<sup>2</sup> + 1)*d*<sup>2</sup> , (47)
*<sup>s</sup>* with *f*2(*s*) a polynomial. Moreover, by
*<sup>s</sup>* (*s*<sup>2</sup>+1)+(*s*<sup>2</sup>+1)<sup>2</sup>*u*21(*s*) *s*<sup>2</sup>+1
(*s*<sup>2</sup>+1)*<sup>q</sup>*<sup>1</sup> + (*s*<sup>2</sup> <sup>+</sup> <sup>1</sup>)*u*21(*s*)
, (50)
, (51)
*h*1(*s*)
<sup>2</sup> + 1)*d*<sup>2</sup> . (48)
*<sup>g</sup>*1(*s*) and *<sup>b</sup>*11(*s*) =
*<sup>s</sup>* . In the same
*s* .
(49)
*su*12(*s*) = *b*12(*s*)(*s*
As *<sup>u</sup>*11(*s*) <sup>∈</sup> **<sup>R</sup>***M*(*s*) and *<sup>b</sup>*11(*s*) <sup>∈</sup> **<sup>R</sup>***M*�(*s*) <sup>∩</sup> **<sup>R</sup>***pr*(*s*), we can write *<sup>u</sup>*11(*s*) = *<sup>f</sup>*1(*s*)
− *s*
way and using (47), *<sup>u</sup>*12(*s*) = *<sup>f</sup>*2(*s*) or *<sup>u</sup>*12(*s*) = *<sup>f</sup>*2(*s*)
*h*1(*s*)
by (45), *s <sup>f</sup>*1(*s*)
*<sup>g</sup>*1(*s*) <sup>=</sup> *<sup>h</sup>*1(*s*)
that *u*21(*s*) and *u*22(*s*) are polynomials.
If *d*<sup>1</sup> = *d*<sup>2</sup> = 1 then , using (46),
with respect to (*M*, *M*�
This is impossible because *U*1(*s*) is invertible.
*<sup>b</sup>*21(*s*) = <sup>−</sup>*s*2*u*11(*s*)+(*s*<sup>2</sup>+1)<sup>2</sup>*u*21(*s*)
) exits.
matrix are obtained in two different cases.
<sup>=</sup> <sup>−</sup>*sh*1(*s*)+(*s*<sup>2</sup>+1)*<sup>q</sup>*1+1*u*21(*s*)
**Example 23.** Let **<sup>F</sup>** <sup>=</sup> **<sup>R</sup>** and *<sup>M</sup>* <sup>=</sup> {(*s*<sup>2</sup> <sup>+</sup> <sup>1</sup>)}. Consider the matrix
which has a zero at 0. It can be written as *P*(*s*) = *P*1(*s*)*P*2(*s*) with
*P*1(*s*) =
*P*(*s*) =
1 0 <sup>−</sup>*<sup>s</sup>* (*s*<sup>2</sup> <sup>+</sup> <sup>1</sup>)<sup>2</sup>
<sup>2</sup>*u*12(*s*)+(*s*
$$B(s) = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} + \begin{bmatrix} s \ 2s \\ 0 \ 1 \end{bmatrix} \begin{bmatrix} s^{-2} & 0 \\ 0 & s^{-3} \end{bmatrix} = \begin{bmatrix} \frac{1}{s} & \frac{s^2 + 2}{s^2} \\ -1 & \frac{1}{s^3} \end{bmatrix}.\tag{53}$$
Moreover, the denominators of its entries are powers of *<sup>s</sup>* and det *<sup>B</sup>*(*s*) = (*s*<sup>2</sup>+1)<sup>2</sup> *<sup>s</sup>*<sup>4</sup> . Therefore, *<sup>B</sup>*(*s*) is invertible in **<sup>R</sup>***M*˜ �(*s*)2×<sup>2</sup> <sup>∩</sup> **<sup>R</sup>***pr*(*s*)2×2. Since *<sup>B</sup>*(*s*)−1*P*(*s*)*U*(*s*) = Diag(*s*2,*s*3), the left Wiener–Hopf factorization indices of *P*(*s*) with respect to *M*˜ are 3, 2.
If *<sup>M</sup>*˜ <sup>=</sup> {(*s*<sup>2</sup> <sup>+</sup> <sup>1</sup>),(*<sup>s</sup>* <sup>−</sup> <sup>1</sup>)}, for example, a similar procedure shows that *<sup>P</sup>*(*s*) has 3, 1 as left Wiener–Hopf factorization indices with respect to *M*˜ ; the same indices as with respect to *M*. The reason is that *s* − 1 is not a divisor of det *P*(*s*) and so *P*(*s*) = *P*1(*s*)*P*2(*s*) with *P*1(*s*) and *P*2(*s*) as in (51) and *P*1(*s*) factorizing in *M*˜ .
**Remark 24.** It must be noticed that a procedure has been given to compute, at least theoretically, the left Wiener–Hopf factorization indices of any rational matrix with respect to any subset *M* of Specm(**F**[*s*]). In fact, given a rational matrix *T*(*s*) and *M*, write *T*(*s*) = *TL*(*s*)*TR*(*s*) with the global invariant rational functions of *TL*(*s*) factorizing in *M*, and the global invariant rational functions of *TR*(*s*) factorizing in Specm(**F**[*s*]) \ *M* (for example, using the global Smith–McMillan form of *T*(*s*)). We need to compute the left Wiener–Hopf factorization indices at infinity of the rational matrix *TL*(*s*). The idea is as follows: Let *d*(*s*) be the monic least common denominator of all the elements of *TL*(*s*). The matrix *TL*(*s*) can be written as *TL*(*s*) = *<sup>P</sup>*(*s*) *<sup>d</sup>*(*s*) , with *P*(*s*) polynomial. The left Wiener–Hopf factorization indices of *P*(*s*) at infinity are the column degrees of any column proper matrix right equivalent to *P*(*s*). If *k*1,..., *km* are the left Wiener–Hopf factorization indices at infinity of *P*(*s*) then *k*<sup>1</sup> + *d*,..., *km* + *d* are the left Wiener–Hopf factorization indices of *TL*(*s*), where *d* = *d*(*d*(*s*)) (see [1]). Free and commercial software exists that compute such column degrees.
#### **5. Rosenbrock's Theorem via local rings**
As said in the Introduction, Rosenbrock's Theorem ([14]) on pole assignment by state feedback provides, in its polynomial formulation, a complete characterization of the relationship between the invariant factors and the left Wiener–Hopf factorization indices at infinity of any non-singular matrix polynomial. The precise statement of this result is the following theorem: **Theorem 25.** *Let g*<sup>1</sup> ≥ ··· ≥ *gm and α*1(*s*) | ··· | *αm*(*s*) *be non-negative integers and monic polynomials, respectively. Then there exists a non-singular matrix P*(*s*) ∈ **F**[*s*] *<sup>m</sup>*×*<sup>m</sup> with α*1(*s*),..., *αm*(*s*) *as invariant factors and g*1,..., *gm as left Wiener–Hopf factorization indices at infinity if and only if the following relation holds:*
$$(g\_1, \ldots, g\_m) \prec (d(\mathfrak{a}\_m(\mathbf{s})), \ldots, d(\mathfrak{a}\_1(\mathbf{s}))).\tag{54}$$
Symbol ≺ appearing in (54) is the majorization symbol (see [11]) and it is defined as follows: If (*a*1,..., *am*) and (*b*1,..., *bm*) are two finite sequences of real numbers and *a*[1] ≥···≥ *a*[*m*] and *b*[1] ≥···≥ *b*[*m*] are the given sequences arranged in non-increasing order then (*a*1,..., *am*) ≺ (*b*1,..., *bm*) if
$$\sum\_{i=1}^{l} a\_{[i]} \le \sum\_{i=1}^{l} b\_{[i]}, \quad 1 \le j \le m-1 \tag{55}$$
with equality for *j* = *m*.
The above Theorem 25 can be extended to cover rational matrix functions. Any rational matrix *T*(*s*) can be written as *<sup>N</sup>*(*s*) *<sup>d</sup>*(*s*) where *d*(*s*) is the monic least common denominator of all the elements of *T*(*s*) and *N*(*s*) is polynomial. It turns out that the invariant rational functions of *T*(*s*) are the invariant factors of *N*(*s*) divided by *d*(*s*) after canceling common factors. We also have the following characterization of the left Wiener– Hopf factorization indices at infinity of *T*(*s*): these are those of *N*(*s*) plus the degree of *d*(*s*) (see [1]). Bearing all this in mind one can easily prove (see [1])
**Theorem 26.** *Let g*<sup>1</sup> ≥ ··· ≥ *gm be integers and <sup>α</sup>*1(*s*) *<sup>β</sup>*1(*s*),..., *<sup>α</sup>m*(*s*) *<sup>β</sup>m*(*s*) *irreducible rational functions, where αi*(*s*), *βi*(*s*) ∈ **F**[*s*] *are monic such that α*1(*s*) | ··· | *αm*(*s*) *while βm*(*s*) | ··· | *β*1(*s*)*. Then there exists a non-singular rational matrix T*(*s*) <sup>∈</sup> **<sup>F</sup>**(*s*)*m*×*<sup>m</sup> with g*1,..., *gm as left Wiener–Hopf factorization indices at infinity and <sup>α</sup>*1(*s*) *<sup>β</sup>*1(*s*),..., *<sup>α</sup>m*(*s*) *<sup>β</sup>m*(*s*) *as global invariant rational functions if and only if*
$$d(g\_1, \ldots, g\_{\mathfrak{m}}) \prec (d(a\_{\mathfrak{m}}(\mathbf{s})) - d(\beta\_{\mathfrak{m}}(\mathbf{s})), \ldots, d(a\_1(\mathbf{s})) - d(\beta\_1(\mathbf{s}))).\tag{56}$$
Recall that for *M* ⊆ Specm(**F**[*s*]) any rational matrix *T*(*s*) can be factorized into two matrices (see Proposition 2) such that the global invariant rational functions and the left Wiener–Hopf factorization indices at infinity of the left factor of *T*(*s*) give the invariant rational functions and the left Wiener–Hopf factorization indices of *T*(*s*) with respect to *M*. Using Theorem 26 on the left factor of *T*(*s*) we get:
**Theorem 27.** *Let M* <sup>⊆</sup> Specm(**F**[*s*])*. Let k*<sup>1</sup> ≥ ··· ≥ *km be integers and �*1(*s*) *<sup>ψ</sup>*1(*s*),..., *�m*(*s*) *<sup>ψ</sup>m*(*s*) *be irreducible rational functions such that �*1(*s*) |···| *�m*(*s*)*, ψm*(*s*) |···| *ψ*1(*s*) *are monic polynomials factorizing in M. Then there exists a non-singular matrix T*(*s*) <sup>∈</sup> **<sup>F</sup>**(*s*)*m*×*<sup>m</sup> with �*1(*s*) *<sup>ψ</sup>*1(*s*),..., *�m*(*s*) *<sup>ψ</sup>m*(*s*) *as invariant rational functions with respect to M and k*1,..., *km as left Wiener–Hopf factorization indices with respect to M if and only if*
$$d(k\_1, \ldots, k\_m) \prec (d(\varepsilon\_m(s)) - d(\psi\_m(s)), \ldots, d(\varepsilon\_1(s)) - d(\psi\_1(s))).\tag{57}$$
Theorem 27 relates the left Wiener–Hopf factorization indices with respect to *M* and the finite structure inside *M*. Our last result will relate the left Wiener–Hopf factorization indices with respect to *M* and the structure outside *M*, including that at infinity. The next Theorem is an extension of Rosenbrock's Theorem to the point at infinity, which was proved in [1]:
**Theorem 28.** *Let g*<sup>1</sup> ≥ ··· ≥ *gm and q*<sup>1</sup> ≥ ··· ≥ *qm be integers. Then there exists a non-singular matrix T*(*s*) <sup>∈</sup> **<sup>F</sup>**(*s*)*m*×*<sup>m</sup> with g*1,..., *gm as left Wiener–Hopf factorization indices at infinity and sq*<sup>1</sup> ,...,*sqm as invariant rational functions at infinity if and only if*
$$(\mathbf{g}\_1, \dots, \mathbf{g}\_m) \prec (q\_{1'}, \dots, q\_m). \tag{58}$$
Notice that Theorem 26 can be obtained from Theorem 27 when *M* = Specm(**F**[*s*]). In the same way, taking into account that the equivalence at infinity is a particular case of the equivalence in **F***M*�(*s*) ∩ **F***pr*(*s*) when *M*� = ∅, we can give a more general result than that of Theorem 28. Specifically, necessary and sufficient conditions can be provided for the existence of a non-singular rational matrix with prescribed left Wiener–Hopf factorization indices with respect to *M* and invariant rational functions in **F***M*�(*s*) ∩ **F***pr*(*s*).
**Theorem 29.** *Let M*, *M*� ⊆ Specm(**F**[*s*]) *be such that M* ∪ *M*� = Specm(**F**[*s*])*. Assume that there are ideals in M* \ *M*� *generated by linear polynomials and let* (*s* − *a*) *be any of them. Let <sup>k</sup>*<sup>1</sup> ≥···≥ *km be integers, �*1(*s*) *<sup>ψ</sup>*1(*s*),..., *�m*(*s*) *<sup>ψ</sup>m*(*s*) *irreducible rational functions such that �*1(*s*)|···|*�m*(*s*)*, ψm*(*s*)|···|*ψ*1(*s*) *are monic polynomials factorizing in M*� \ *M and l*1,..., *lm integers such that l*<sup>1</sup> + *d*(*ψ*1(*s*)) − *d*(*�*1(*s*)) ≤ ··· ≤ *lm* + *d*(*ψm*(*s*)) − *d*(*�m*(*s*))*. Then there exists a non-singular matrix T*(*s*) <sup>∈</sup> **<sup>F</sup>**(*s*)*m*×*<sup>m</sup> with no zeros and no poles in M* <sup>∩</sup> *<sup>M</sup>*� *with k*1,..., *km as left Wiener–Hopf factorization indices with respect to M and �*1(*s*) *ψ*1(*s*) 1 (*s*−*a*)*<sup>l</sup>* <sup>1</sup> ,..., *�m*(*s*) *ψm*(*s*) 1 (*s*−*a*)*lm as invariant rational functions in* **F***M*�(*s*) ∩ **F***pr*(*s*) *if and only if the following condition holds:*
$$(k\_1, \ldots, k\_m) \prec (-l\_1, \ldots, -l\_m). \tag{59}$$
The proof of this theorem will be given along the following two subsections. We will use several auxiliary results that will be stated and proved when needed.
#### **5.1. Necessity**
20 Will-be-set-by-IN-TECH
**Theorem 25.** *Let g*<sup>1</sup> ≥ ··· ≥ *gm and α*1(*s*) | ··· | *αm*(*s*) *be non-negative integers and*
*α*1(*s*),..., *αm*(*s*) *as invariant factors and g*1,..., *gm as left Wiener–Hopf factorization indices at*
Symbol ≺ appearing in (54) is the majorization symbol (see [11]) and it is defined as follows: If (*a*1,..., *am*) and (*b*1,..., *bm*) are two finite sequences of real numbers and *a*[1] ≥···≥ *a*[*m*] and *b*[1] ≥···≥ *b*[*m*] are the given sequences arranged in non-increasing order then (*a*1,..., *am*) ≺
The above Theorem 25 can be extended to cover rational matrix functions. Any rational matrix
elements of *T*(*s*) and *N*(*s*) is polynomial. It turns out that the invariant rational functions of *T*(*s*) are the invariant factors of *N*(*s*) divided by *d*(*s*) after canceling common factors. We also have the following characterization of the left Wiener– Hopf factorization indices at infinity of *T*(*s*): these are those of *N*(*s*) plus the degree of *d*(*s*) (see [1]). Bearing all this in mind one can
*where αi*(*s*), *βi*(*s*) ∈ **F**[*s*] *are monic such that α*1(*s*) | ··· | *αm*(*s*) *while βm*(*s*) | ··· | *β*1(*s*)*. Then there exists a non-singular rational matrix T*(*s*) <sup>∈</sup> **<sup>F</sup>**(*s*)*m*×*<sup>m</sup> with g*1,..., *gm as left Wiener–Hopf*
Recall that for *M* ⊆ Specm(**F**[*s*]) any rational matrix *T*(*s*) can be factorized into two matrices (see Proposition 2) such that the global invariant rational functions and the left Wiener–Hopf factorization indices at infinity of the left factor of *T*(*s*) give the invariant rational functions and the left Wiener–Hopf factorization indices of *T*(*s*) with respect to *M*. Using Theorem 26
*irreducible rational functions such that �*1(*s*) |···| *�m*(*s*)*, ψm*(*s*) |···| *ψ*1(*s*) *are monic polynomials*
*invariant rational functions with respect to M and k*1,..., *km as left Wiener–Hopf factorization indices*
*<sup>β</sup>*1(*s*),..., *<sup>α</sup>m*(*s*)
**Theorem 27.** *Let M* <sup>⊆</sup> Specm(**F**[*s*])*. Let k*<sup>1</sup> ≥ ··· ≥ *km be integers and �*1(*s*)
*factorizing in M. Then there exists a non-singular matrix T*(*s*) <sup>∈</sup> **<sup>F</sup>**(*s*)*m*×*<sup>m</sup> with �*1(*s*)
(*g*1,..., *gm*) ≺ (*d*(*αm*(*s*)),..., *d*(*α*1(*s*))). (54)
*<sup>d</sup>*(*s*) where *d*(*s*) is the monic least common denominator of all the
*<sup>β</sup>*1(*s*),..., *<sup>α</sup>m*(*s*)
(*g*1,..., *gm*) ≺ (*d*(*αm*(*s*)) − *d*(*βm*(*s*)),..., *d*(*α*1(*s*)) − *d*(*β*1(*s*))). (56)
(*k*1,..., *km*) ≺ (*d*(*�m*(*s*)) − *d*(*ψm*(*s*)),..., *d*(*�*1(*s*)) − *d*(*ψ*1(*s*))). (57)
*b*[*i*], 1 ≤ *j* ≤ *m* − 1 (55)
*<sup>β</sup>m*(*s*) *as global invariant rational functions if and only*
*<sup>β</sup>m*(*s*) *irreducible rational functions,*
*<sup>ψ</sup>*1(*s*),..., *�m*(*s*)
*<sup>ψ</sup>*1(*s*),..., *�m*(*s*)
*<sup>ψ</sup>m*(*s*) *be*
*<sup>ψ</sup>m*(*s*) *as*
*<sup>m</sup>*×*<sup>m</sup> with*
*monic polynomials, respectively. Then there exists a non-singular matrix P*(*s*) ∈ **F**[*s*]
*infinity if and only if the following relation holds:*
*j* ∑ *i*=1
**Theorem 26.** *Let g*<sup>1</sup> ≥ ··· ≥ *gm be integers and <sup>α</sup>*1(*s*)
*factorization indices at infinity and <sup>α</sup>*1(*s*)
on the left factor of *T*(*s*) we get:
*with respect to M if and only if*
*a*[*i*] ≤
*j* ∑ *i*=1
(*b*1,..., *bm*) if
with equality for *j* = *m*.
easily prove (see [1])
*if*
*T*(*s*) can be written as *<sup>N</sup>*(*s*)
We can give the following result for rational matrices using a similar result given in Lemma 4.2 in [2] for matrix polynomials.
**Lemma 30.** *Let M*, *<sup>M</sup>*� <sup>⊆</sup> Specm(**F**[*s*]) *be such that M* <sup>∪</sup> *<sup>M</sup>*� <sup>=</sup> Specm(**F**[*s*])*. Let T*(*s*) <sup>∈</sup> **<sup>F</sup>**(*s*)*m*×*<sup>m</sup> be a non-singular matrix with no zeros and no poles in M* ∩ *M*� *with g*<sup>1</sup> ≥ ··· ≥ *gm as left Wiener–Hopf factorization indices at infinity and k*<sup>1</sup> ≥ ··· ≥ *km as left Wiener–Hopf factorization indices with respect to M. If �*1(*s*) *<sup>ψ</sup>*1(*s*),..., *�m*(*s*) *<sup>ψ</sup>m*(*s*) *are the invariant rational functions of T*(*s*) *with respect to M*� *then*
$$(g\_1 - k\_1, \dots, g\_{\mathfrak{m}} - k\_{\mathfrak{m}}) \prec (d(\varepsilon\_{\mathfrak{m}}(\mathbf{s})) - d(\psi\_{\mathfrak{m}}(\mathbf{s})), \dots, d(\varepsilon\_1(\mathbf{s})) - d(\psi\_1(\mathbf{s}))).\tag{60}$$
It must be pointed out that (*g*<sup>1</sup> − *k*1,..., *gm* − *km*) may be an unordered *m*-tuple.
**Proof**.- By Proposition 2 there exist unimodular matrices *U*(*s*), *V*(*s*) ∈ **F**[*s*] *<sup>m</sup>*×*<sup>m</sup>* such that
$$T(s) = \mathcal{U}(s)\operatorname{Diag}\left(\frac{a\_1(s)}{\beta\_1(s)}, \dots, \frac{a\_m(s)}{\beta\_m(s)}\right)\operatorname{Diag}\left(\frac{\varepsilon\_1(s)}{\psi\_1(s)}, \dots, \frac{\varepsilon\_m(s)}{\psi\_m(s)}\right)V(s) \tag{61}$$
with *<sup>α</sup>i*(*s*) | *<sup>α</sup>i*+1(*s*), *<sup>β</sup>i*(*s*) | *<sup>β</sup>i*−1(*s*), *�i*(*s*) | *�i*+1(*s*), *<sup>ψ</sup>i*(*s*) | *<sup>ψ</sup>i*−1(*s*), *<sup>α</sup>i*(*s*), *<sup>β</sup>i*(*s*) units in **<sup>F</sup>***M*�\*M*(*s*) and *�i*(*s*), *<sup>ψ</sup>i*(*s*) factorizing in *<sup>M</sup>*� \ *<sup>M</sup>* because *<sup>T</sup>*(*s*) has no poles and no zeros in *<sup>M</sup>* <sup>∩</sup> *M*� . Therefore *<sup>T</sup>*(*s*) = *TL*(*s*)*TR*(*s*), where *TL*(*s*) = *<sup>U</sup>*(*s*) Diag *<sup>α</sup>*1(*s*) *<sup>β</sup>*1(*s*),..., *<sup>α</sup>m*(*s*) *βm*(*s*) has *k*1,..., *km* as left Wiener–Hopf factorization indices at infinity and *TR*(*s*) = Diag *�*1(*s*) *<sup>ψ</sup>*1(*s*),..., *�m*(*s*) *ψm*(*s*) *V*(*s*) has *�*1(*s*) *<sup>ψ</sup>*1(*s*),..., *�m*(*s*) *<sup>ψ</sup>m*(*s*) as global invariant rational functions. Let *<sup>d</sup>*(*s*) = *<sup>β</sup>*1(*s*)*ψ*1(*s*). Hence,
$$d(s)T(s) = \mathcal{U}(s)\operatorname{Diag}(\overline{\mathfrak{a}}\_1(s), \dots, \overline{\mathfrak{a}}\_{\mathfrak{m}}(s))\operatorname{Diag}(\overline{\mathfrak{e}}\_1(s), \dots, \overline{\mathfrak{e}}\_{\mathfrak{m}}(s))V(s) \tag{62}$$
with *<sup>α</sup>i*(*s*) = *<sup>α</sup>i*(*s*) *<sup>β</sup>i*(*s*) *<sup>β</sup>*1(*s*) units in **<sup>F</sup>***M*�\*M*(*s*) and *�i*(*s*) = *�i*(*s*) *<sup>ψ</sup>i*(*s*)*ψ*1(*s*) factorizing in *<sup>M</sup>*� \ *M*. Put *P*(*s*) = *d*(*s*)*T*(*s*). Its left Wiener–Hopf factorization indices at infinity are *g*<sup>1</sup> + *d*(*d*(*s*)),..., *gm* + *d*(*d*(*s*)) [1, Lemma 2.3]. The matrix *P*1(*s*) = *U*(*s*) Diag(*α*1(*s*),..., *αm*(*s*)) = *β*1(*s*)*TL*(*s*) has *k*<sup>1</sup> + *d*(*β*1(*s*)),..., *km* + *d*(*β*1(*s*)) as left Wiener–Hopf factorization indices at infinity. Now if *P*2(*s*) = Diag(*�*1(*s*),..., *�m*(*s*))*V*(*s*) = *ψ*1(*s*)*TR*(*s*) then its invariant factors are *�*1(*s*),..., *�m*(*s*), *P*(*s*) = *P*1(*s*)*P*2(*s*) and, by [2, Lemma 4.2],
$$(g\_1 + d(d(s)) - k\_1 - d(\beta\_1(s)), \dots, g\_{\mathfrak{M}} + d(d(s)) - k\_{\mathfrak{M}} - d(\beta\_1(s))) \prec (d(\overline{\varepsilon}\_{\mathfrak{m}}(s)), \dots, d(\overline{\varepsilon}\_1(s))).\tag{63}$$
Therefore, (60) follows.
#### *5.1.1. Proof of Theorem 29: Necessity*
If *�*1(*s*) *ψ*1(*s*) 1 (*s*−*a*)*<sup>l</sup>* <sup>1</sup> ,..., *�m*(*s*) *ψm*(*s*) 1 (*s*−*a*)*lm* are the invariant rational functions of *<sup>T</sup>*(*s*) in **<sup>F</sup>***M*�(*s*) <sup>∩</sup> **<sup>F</sup>***pr*(*s*) then there exist matrices *<sup>U</sup>*1(*s*), *<sup>U</sup>*2(*s*) invertible in **<sup>F</sup>***M*�(*s*)*m*×*<sup>m</sup>* <sup>∩</sup> **<sup>F</sup>***pr*(*s*)*m*×*<sup>m</sup>* such that
$$T(s) = \mathcal{U}\_1(s) \operatorname{Diag}\left(\frac{\varepsilon\_1(s)}{\psi\_1(s)} \frac{1}{(s-a)^{l\_1}}, \dots, \frac{\varepsilon\_m(s)}{\psi\_m(s)} \frac{1}{(s-a)^{l\_n}}\right) \mathcal{U}\_2(s). \tag{64}$$
We analyze first the finite structure of *T*(*s*) with respect to *M*� . If *<sup>D</sup>*1(*s*) = Diag((*<sup>s</sup>* <sup>−</sup> *<sup>a</sup>*)−*l*<sup>1</sup> , ...,(*<sup>s</sup>* <sup>−</sup> *<sup>a</sup>*)−*lm* ) <sup>∈</sup> **<sup>F</sup>***M*�(*s*)*m*×*m*, we can write *<sup>T</sup>*(*s*) as follows:
$$T(s) = \mathcal{U}\_1(s) \operatorname{Diag} \left( \frac{\mathfrak{e}\_1(s)}{\psi\_1(s)}, \dots, \frac{\mathfrak{e}\_m(s)}{\psi\_m(s)} \right) D\_1(s) \mathcal{U}\_2(s), \tag{65}$$
with *<sup>U</sup>*1(*s*) and *<sup>D</sup>*1(*s*)*U*2(*s*) invertible matrices in **<sup>F</sup>***M*�(*s*)*m*×*m*. Thus *�*1(*s*) *<sup>ψ</sup>*1(*s*),..., *�m*(*s*) *<sup>ψ</sup>m*(*s*) are the invariant rational functions of *T*(*s*) with respect to *M*� . Let *g*<sup>1</sup> ≥ ··· ≥ *gm* be the left Wiener–Hopf factorization indices of *T*(*s*) at infinity. By Lemma 30 we have
$$(\mathfrak{g}\_1 - k\_1, \dots, \mathfrak{g}\_m - k\_m) \prec (d(\varepsilon\_m(s)) - d(\mathfrak{\psi}\_m(s)), \dots, d(\varepsilon\_1(s)) - d(\mathfrak{\psi}\_1(s))).\tag{66}$$
As far as the structure of *T*(*s*) at infinity is concerned, let
$$D\_2(s) = \text{Diag}\left(\frac{\mathfrak{c}\_1(s)}{\psi\_1(s)} \frac{s^{l\_1 + d(\psi\_1(s)) - d(\mathfrak{e}\_1(s))}}{(s - a)^{l\_1}}, \dots, \frac{\mathfrak{e}\_m(s)}{\psi\_m(s)} \frac{s^{l\_m + d(\psi\_m(s)) - d(\mathfrak{e}\_m(s))}}{(s - a)^{l\_m}}\right). \tag{67}$$
Then *D*2(*s*) ∈ *Gl*(**F***pr*(*s*)) and
22 Will-be-set-by-IN-TECH
with *<sup>α</sup>i*(*s*) | *<sup>α</sup>i*+1(*s*), *<sup>β</sup>i*(*s*) | *<sup>β</sup>i*−1(*s*), *�i*(*s*) | *�i*+1(*s*), *<sup>ψ</sup>i*(*s*) | *<sup>ψ</sup>i*−1(*s*), *<sup>α</sup>i*(*s*), *<sup>β</sup>i*(*s*) units in **<sup>F</sup>***M*�\*M*(*s*) and *�i*(*s*), *<sup>ψ</sup>i*(*s*) factorizing in *<sup>M</sup>*� \ *<sup>M</sup>* because *<sup>T</sup>*(*s*) has no poles and no zeros in *<sup>M</sup>* <sup>∩</sup>
*M*. Put *P*(*s*) = *d*(*s*)*T*(*s*). Its left Wiener–Hopf factorization indices at infinity are *g*<sup>1</sup> + *d*(*d*(*s*)),..., *gm* + *d*(*d*(*s*)) [1, Lemma 2.3]. The matrix *P*1(*s*) = *U*(*s*) Diag(*α*1(*s*),..., *αm*(*s*)) = *β*1(*s*)*TL*(*s*) has *k*<sup>1</sup> + *d*(*β*1(*s*)),..., *km* + *d*(*β*1(*s*)) as left Wiener–Hopf factorization indices at infinity. Now if *P*2(*s*) = Diag(*�*1(*s*),..., *�m*(*s*))*V*(*s*) = *ψ*1(*s*)*TR*(*s*) then its invariant factors
(*g*<sup>1</sup> + *d*(*d*(*s*)) − *k*<sup>1</sup> − *d*(*β*1(*s*)),..., *gm* + *d*(*d*(*s*)) − *km* − *d*(*β*1(*s*))) ≺ (*d*(*�m*(*s*)),..., *d*(*�*1(*s*))).
then there exist matrices *<sup>U</sup>*1(*s*), *<sup>U</sup>*2(*s*) invertible in **<sup>F</sup>***M*�(*s*)*m*×*<sup>m</sup>* <sup>∩</sup> **<sup>F</sup>***pr*(*s*)*m*×*<sup>m</sup>* such that
*�*1(*s*) *ψ*1(*s*)
1 (*<sup>s</sup>* − *<sup>a</sup>*)*l*<sup>1</sup>
*�*1(*s*) *ψ*1(*s*)
with *<sup>U</sup>*1(*s*) and *<sup>D</sup>*1(*s*)*U*2(*s*) invertible matrices in **<sup>F</sup>***M*�(*s*)*m*×*m*. Thus *�*1(*s*)
Wiener–Hopf factorization indices of *T*(*s*) at infinity. By Lemma 30 we have
*sl*1+*d*(*ψ*1(*s*))−*d*(*�*1(*s*)) (*<sup>s</sup>* − *<sup>a</sup>*)*l*<sup>1</sup>
We analyze first the finite structure of *T*(*s*) with respect to *M*�
...,(*<sup>s</sup>* <sup>−</sup> *<sup>a</sup>*)−*lm* ) <sup>∈</sup> **<sup>F</sup>***M*�(*s*)*m*×*m*, we can write *<sup>T</sup>*(*s*) as follows:
*T*(*s*) = *U*1(*s*) Diag
invariant rational functions of *T*(*s*) with respect to *M*�
As far as the structure of *T*(*s*) at infinity is concerned, let
*�*1(*s*) *ψ*1(*s*) Diag
*<sup>ψ</sup>m*(*s*) as global invariant rational functions. Let *<sup>d</sup>*(*s*) = *<sup>β</sup>*1(*s*)*ψ*1(*s*). Hence,
*d*(*s*)*T*(*s*) = *U*(*s*) Diag(*α*1(*s*),..., *αm*(*s*)) Diag(*�*1(*s*),..., *�m*(*s*))*V*(*s*) (62)
(*s*−*a*)*lm* are the invariant rational functions of *<sup>T</sup>*(*s*) in **<sup>F</sup>***M*�(*s*) <sup>∩</sup> **<sup>F</sup>***pr*(*s*)
1 (*<sup>s</sup>* − *<sup>a</sup>*)*lm* *U*2(*s*). (64)
*<sup>ψ</sup>m*(*s*) are the
. (67)
. If *<sup>D</sup>*1(*s*) = Diag((*<sup>s</sup>* <sup>−</sup> *<sup>a</sup>*)−*l*<sup>1</sup> ,
*D*1(*s*)*U*2(*s*), (65)
*<sup>ψ</sup>*1(*s*),..., *�m*(*s*)
. Let *g*<sup>1</sup> ≥ ··· ≥ *gm* be the left
*slm*+*d*(*ψm*(*s*))−*d*(*�m*(*s*)) (*<sup>s</sup>* − *<sup>a</sup>*)*lm*
,..., *�m*(*s*) *ψm*(*s*)
,..., *�m*(*s*) *ψm*(*s*)
(*g*<sup>1</sup> − *k*1,..., *gm* − *km*) ≺ (*d*(*�m*(*s*)) − *d*(*ψm*(*s*)),..., *d*(*�*1(*s*)) − *d*(*ψ*1(*s*))). (66)
,..., *�m*(*s*) *ψm*(*s*)
*�*1(*s*) *ψ*1(*s*) ,..., *�m*(*s*) *ψm*(*s*)
*<sup>β</sup>*1(*s*),..., *<sup>α</sup>m*(*s*)
*α*1(*s*)
,..., *<sup>α</sup>m*(*s*) *βm*(*s*)
*<sup>β</sup>i*(*s*) *<sup>β</sup>*1(*s*) units in **<sup>F</sup>***M*�\*M*(*s*) and *�i*(*s*) = *�i*(*s*)
*<sup>m</sup>*×*<sup>m</sup>* such that
*V*(*s*) (61)
has *k*1,..., *km*
(63)
*ψm*(*s*) *V*(*s*)
*βm*(*s*)
*<sup>ψ</sup>*1(*s*),..., *�m*(*s*)
*<sup>ψ</sup>i*(*s*)*ψ*1(*s*) factorizing in *<sup>M</sup>*� \
*�*1(*s*)
**Proof**.- By Proposition 2 there exist unimodular matrices *U*(*s*), *V*(*s*) ∈ **F**[*s*]
*α*1(*s*) *β*1(*s*)
. Therefore *T*(*s*) = *TL*(*s*)*TR*(*s*), where *TL*(*s*) = *U*(*s*) Diag
are *�*1(*s*),..., *�m*(*s*), *P*(*s*) = *P*1(*s*)*P*2(*s*) and, by [2, Lemma 4.2],
1
*T*(*s*) = *U*1(*s*) Diag
as left Wiener–Hopf factorization indices at infinity and *TR*(*s*) = Diag
*T*(*s*) = *U*(*s*) Diag
*M*�
has *�*1(*s*)
If *�*1(*s*) *ψ*1(*s*)
*<sup>ψ</sup>*1(*s*),..., *�m*(*s*)
with *<sup>α</sup>i*(*s*) = *<sup>α</sup>i*(*s*)
Therefore, (60) follows.
1 (*s*−*a*)*<sup>l</sup>*
*5.1.1. Proof of Theorem 29: Necessity*
<sup>1</sup> ,..., *�m*(*s*) *ψm*(*s*)
*D*2(*s*) = Diag
$$T(\mathbf{s}) = \mathcal{U}\_1(\mathbf{s}) \operatorname{Diag} \left( \mathbf{s}^{-l\_1 - d(\psi\_1(\mathbf{s})) + d(\varepsilon\_1(\mathbf{s}))}, \dots, \mathbf{s}^{-l\_n - d(\psi\_n(\mathbf{s})) + d(\varepsilon\_n(\mathbf{s}))} \right) \mathcal{D}\_2(\mathbf{s}) \mathcal{U}\_2(\mathbf{s}) \tag{68}$$
where *<sup>U</sup>*1(*s*) <sup>∈</sup> **<sup>F</sup>***pr*(*s*)*m*×*<sup>m</sup>* and *<sup>D</sup>*2(*s*)*U*2(*s*) <sup>∈</sup> **<sup>F</sup>***pr*(*s*)*m*×*<sup>m</sup>* are biproper matrices. Therefore *s*−*l*1−*d*(*ψ*1(*s*))+*d*(*�*1(*s*)), ..., *s*−*lm*−*d*(*ψm*(*s*))+*d*(*�m*(*s*)) are the invariant rational functions of *T*(*s*) at infinity. By Theorem 28
$$(g\_1, \ldots, g\_{\mathfrak{m}}) \prec (-l\_1 - d(\psi\_1(\mathbf{s})) + d(\varepsilon\_1(\mathbf{s})), \ldots, -l\_{\mathfrak{m}} - d(\psi\_{\mathfrak{m}}(\mathbf{s})) + d(\varepsilon\_{\mathfrak{m}}(\mathbf{s}))).\tag{69}$$
Let *σ* ∈ Σ*<sup>m</sup>* (the symmetric group of order *m*) be a permutation such that *gσ*(1) − *kσ*(1) ≥···≥ *gσ*(*m*) − *kσ*(*m*) and define *ci* = *gσ*(*i*) − *kσ*(*i*), *i* = 1, . . . , *m*. Using (66) and (69) we obtain
$$\begin{array}{ll}\sum\_{j=1}^{r}k\_{j} + \sum\_{j=1}^{r} \left(d(\varepsilon\_{j}(s)) - d(\psi\_{j}(s))\right) \leq \sum\_{j=1}^{r}k\_{j} + \sum\_{j=m-r+1}^{m} c\_{j} \\ \leq \sum\_{j=1}^{r}k\_{j} + \sum\_{j=1}^{r} (\mathfrak{g}\_{j} - k\_{j}) = \sum\_{j=1}^{r} \mathfrak{g}\_{j} \\ \leq \sum\_{j=1}^{r} -l\_{j} + \sum\_{j=1}^{r} \left(d(\varepsilon\_{j}(s)) - d(\psi\_{j}(s))\right) \end{array} \tag{70}$$
for *r* = 1, . . . , *m* − 1. When *r* = *m* the previous inequalities are all equalities and condition (59) is satisfied.
**Remark 31.** It has been seen in the above proof that if a matrix has *�*1(*s*) *ψ*1(*s*) 1 (*s*−*a*)*<sup>l</sup>* <sup>1</sup> ,..., *�m*(*s*) *ψm*(*s*) 1 (*s*−*a*)*lm* as invariant rational functions in **<sup>F</sup>***M*�(*s*) <sup>∩</sup> **<sup>F</sup>***pr*(*s*) then *�*1(*s*) *<sup>ψ</sup>*1(*s*),..., *�m*(*s*) *<sup>ψ</sup>m*(*s*) are its invariant rational functions with respect to *<sup>M</sup>*� and *<sup>s</sup>*−*l*1−*d*(*ψ*1(*s*))+*d*(*�*1(*s*)), ..., *s*−*lm*−*d*(*ψm*(*s*))+*d*(*�m*(*s*)) are its invariant rational functions at infinity.
#### **5.2. Sufficiency**
Let *a*, *b* ∈ **F** be arbitrary elements such that *ab* �= 1. Consider the changes of indeterminate
$$f(s) = a + \frac{1}{s - b'} \quad \vec{f}(s) = b + \frac{1}{s - a} \tag{71}$$
and notice that *f*( ˜ *f*(*s*)) = ˜ *f*(*f*(*s*)) = *s*. For *α*(*s*) ∈ **F**[*s*], let **F**[*s*] \ (*α*(*s*)) denote the multiplicative subset of **F**[*s*] whose elements are coprime with *α*(*s*). For *a*, *b* ∈ **F** as above define
$$\begin{array}{lcl} t\_{a,b}: \mathbb{F}[s] \to \mathbb{F}[s] \backslash (s-b) \\ \pi(s) \mapsto (s-b)^{d(\pi(s))} \, \pi\left(a + \frac{1}{s-b}\right) = (s-b)^{d(\pi(s))} \, \pi(f(s)) \, . \end{array} \tag{72}$$
In words, if *<sup>π</sup>*(*s*) = *pd*(*<sup>s</sup>* <sup>−</sup> *<sup>a</sup>*)*<sup>d</sup>* <sup>+</sup> *pd*−1(*<sup>s</sup>* <sup>−</sup> *<sup>a</sup>*)*d*−<sup>1</sup> <sup>+</sup> ··· <sup>+</sup> *<sup>p</sup>*1(*<sup>s</sup>* <sup>−</sup> *<sup>a</sup>*) + *<sup>p</sup>*<sup>0</sup> (*pd* �<sup>=</sup> 0) then
$$t\_{a,b}(\pi(s)) = p\_0(s-b)^d + p\_1(s-b)^{d-1} + \dots + p\_{d-1}(s-b) + p\_d. \tag{73}$$
In general *d*(*ta*,*b*(*π*(*s*))) ≤ *d*(*π*(*s*)) with equality if and only if *π*(*s*) ∈ **F**[*s*] \ (*s* − *a*). This shows that the restriction *ha*,*<sup>b</sup>* : **F**[*s*] \ (*s* − *a*) → **F**[*s*] \ (*s* − *b*) of *ta*,*<sup>b</sup>* to **F**[*s*] \ (*s* − *a*) is a bijection. In addition *h*−<sup>1</sup> *<sup>a</sup>*,*<sup>b</sup>* is the restriction of *tb*,*<sup>a</sup>* to **F**[*s*] \ (*s* − *b*); i.e.,
$$\begin{aligned} \mathcal{H}\_{a,b}^{-1}: \mathbb{F}[s] \backslash (s-b) &\to \mathbb{F}[s] \backslash (s-a) \\ a(s) &\mapsto (s-a)^{d(a(s))} a \left(b + \frac{1}{s-a}\right) = (s-a)^{d(a(s))} a(\tilde{f}(s)) \end{aligned} \tag{74}$$
#### 24 Will-be-set-by-IN-TECH 70 Linear Algebra – Theorems and Applications
or *h*−<sup>1</sup> *<sup>a</sup>*,*<sup>b</sup>* = *hb*,*a*.
In what follows we will think of *a*, *b* as given elements of **F** and the subindices of *ta*,*b*, *ha*,*<sup>b</sup>* and *h*−<sup>1</sup> *<sup>a</sup>*,*<sup>b</sup>* will be removed. The following are properties of *<sup>h</sup>* (and *<sup>h</sup>*−1) that can be easily proved.
**Lemma 32.** *Let π*1(*s*), *π*2(*s*) ∈ **F**[*s*] \ (*s* − *a*)*. The following properties hold:*
As a consequence the map
$$\begin{array}{ccc} H: \mathsf{Spec}\, \mathsf{m}\left(\mathsf{F}[s]\right) \backslash \{ (s-a) \} \to \mathsf{Spec}\, \mathsf{m}\left(\mathsf{F}[s]\right) \backslash \{ (s-b) \} \\\ (\pi(s)) & \mapsto & \left(\frac{1}{p\_0}h(\pi(s))\right) \end{array} \tag{75}$$
with *p*<sup>0</sup> = *π*(*a*), is a bijection whose inverse is
$$\begin{array}{ccc} H^{-1}: \operatorname{Spec}\left(\mathbb{F}[s]\right) \backslash \{ (s-b) \} \to \operatorname{Spec}\left(\mathbb{F}[s]\right) \backslash \{ (s-a) \} \\\ (a(s)) & \mapsto & \left(\frac{1}{a\_0}h^{-1}(a(s))\right) \end{array} \tag{76}$$
where *<sup>a</sup>*<sup>0</sup> <sup>=</sup> *<sup>α</sup>*(*b*). In particular, if *<sup>M</sup>*� <sup>⊆</sup> Specm(**F**[*s*]) \ {(*<sup>s</sup>* <sup>−</sup> *<sup>a</sup>*)} and *<sup>M</sup>*˜ <sup>=</sup> Specm(**F**[*s*]) \ (*M*� <sup>∪</sup> {(*s* − *a*)}) (i.e. the complementary subset of *M*� in Specm (**F**[*s*]) \ {(*s* − *a*)}) then
$$H(\tilde{M}) = \text{Spec}\, \text{cm}\left(\mathbb{F}[s]\right) \backslash \left(H(M') \cup \{(s-b)\}\right). \tag{77}$$
In what follows and for notational simplicity we will assume *b* = 0.
**Lemma 33.** *Let M*� ⊆ Specm (**F**[*s*]) \ {(*s* − *a*)} *where a* ∈ **F** *is an arbitrary element of* **F***.*
**Proof.-** 1. Let *<sup>π</sup>*(*s*) = *<sup>c</sup>π*1(*s*)*g*<sup>1</sup> ··· *<sup>π</sup>m*(*s*)*gm* with *<sup>c</sup>* �<sup>=</sup> 0 constant, (*πi*(*s*)) <sup>∈</sup> *<sup>M</sup>*� and *gi* <sup>≥</sup> 1. Then *<sup>h</sup>*(*π*(*s*)) = *<sup>c</sup>*(*h*(*π*1(*s*)))*g*<sup>1</sup> ···(*h*(*πm*(*s*)))*gm* . By Lemma 32 *<sup>h</sup>*(*πi*(*s*)) is an irreducible polynomial (that may not be monic). If *ci* is the leading coefficient of *h*(*πi*(*s*)) then <sup>1</sup> *ci h*(*πi*(*s*)) is monic, irreducible and ( <sup>1</sup> *ci h*(*πi*(*s*))) ∈ *H*(*M*� ). Hence *h*(*π*(*s*)) factorizes in *H*(*M*� ).
2. If *<sup>π</sup>*(*s*) <sup>∈</sup> **<sup>F</sup>**[*s*] is a unit of **<sup>F</sup>***M*�(*s*) then it can be written as *<sup>π</sup>*(*s*)=(*<sup>s</sup>* <sup>−</sup> *<sup>a</sup>*)*gπ*1(*s*) where *<sup>g</sup>* <sup>≥</sup> 0 and *<sup>π</sup>*1(*s*) is a unit of **<sup>F</sup>***M*�∪{(*s*−*a*)}(*s*). Therefore *<sup>π</sup>*1(*s*) factorizes in Specm(**F**[*s*]) \ (*M*� <sup>∪</sup> {(*s* − *a*)}). Since *t*(*π*(*s*)) = *h*(*π*1(*s*)), it factorizes in (recall that we are assuming *b* = 0) *H*(Specm(**F**[*s*]) \ (*M*� ∪ {(*s* − *a*)}) = Specm(**F**[*s*]) \ (*H*(*M*� ) ∪ {(*s*)}). So, *t*(*π*(*s*)) is a unit of **F***H*(*M*�)(*s*).
**Lemma 34.** *Let a* ∈ **F** *be an arbitrary element. Then*
$$\text{1. } \text{If } M' \subseteq \text{Spec}\left(\mathbb{F}[s]\right) \\
\text{ } \{\left(s-a\right)\} \text{ and } \mathcal{U}(s) \in \text{Gl}\_{\mathfrak{m}}(\mathbb{F}\_{M'}(s)) \text{ then } \mathcal{U}(f(s)) \in \text{Gl}\_{\mathfrak{m}}(\mathbb{F}\_{H(M')}(s)).$$
**Proof.-** Let *<sup>p</sup>*(*s*) *<sup>q</sup>*(*s*) with *p*(*s*), *q*(*s*) ∈ **F**[*s*].
24 Will-be-set-by-IN-TECH
In what follows we will think of *a*, *b* as given elements of **F** and the subindices of *ta*,*b*, *ha*,*<sup>b</sup>* and
*<sup>a</sup>*,*<sup>b</sup>* will be removed. The following are properties of *<sup>h</sup>* (and *<sup>h</sup>*−1) that can be easily proved.
**Lemma 32.** *Let π*1(*s*), *π*2(*s*) ∈ **F**[*s*] \ (*s* − *a*)*. The following properties hold:*
*3. If π*1(*s*) *is an irreducible polynomial then h*(*π*1(*s*)) *is an irreducible polynomial.*
*4. If π*1(*s*), *π*2(*s*) *are coprime polynomials then h*(*π*1(*s*))*, h*(*π*2(*s*)) *are coprime polynomials.*
*H* : Specm (**F**[*s*]) \ {(*s* − *a*)} → Specm (**F**[*s*]) \ {(*s* − *b*)} (*π*(*s*)) �→ ( <sup>1</sup>
*<sup>H</sup>*−<sup>1</sup> : Specm (**F**[*s*]) \ {(*<sup>s</sup>* <sup>−</sup> *<sup>b</sup>*)} <sup>→</sup> Specm (**F**[*s*]) \ {(*<sup>s</sup>* <sup>−</sup> *<sup>a</sup>*)} (*α*(*s*)) �→ ( <sup>1</sup>
where *<sup>a</sup>*<sup>0</sup> <sup>=</sup> *<sup>α</sup>*(*b*). In particular, if *<sup>M</sup>*� <sup>⊆</sup> Specm(**F**[*s*]) \ {(*<sup>s</sup>* <sup>−</sup> *<sup>a</sup>*)} and *<sup>M</sup>*˜ <sup>=</sup> Specm(**F**[*s*]) \ (*M*� <sup>∪</sup>
**F**[*s*]) \ (*H*(*M*�
**Proof.-** 1. Let *<sup>π</sup>*(*s*) = *<sup>c</sup>π*1(*s*)*g*<sup>1</sup> ··· *<sup>π</sup>m*(*s*)*gm* with *<sup>c</sup>* �<sup>=</sup> 0 constant, (*πi*(*s*)) <sup>∈</sup> *<sup>M</sup>*� and *gi* <sup>≥</sup> 1. Then *<sup>h</sup>*(*π*(*s*)) = *<sup>c</sup>*(*h*(*π*1(*s*)))*g*<sup>1</sup> ···(*h*(*πm*(*s*)))*gm* . By Lemma 32 *<sup>h</sup>*(*πi*(*s*)) is an irreducible polynomial
2. If *<sup>π</sup>*(*s*) <sup>∈</sup> **<sup>F</sup>**[*s*] is a unit of **<sup>F</sup>***M*�(*s*) then it can be written as *<sup>π</sup>*(*s*)=(*<sup>s</sup>* <sup>−</sup> *<sup>a</sup>*)*gπ*1(*s*) where *<sup>g</sup>* <sup>≥</sup> 0 and *<sup>π</sup>*1(*s*) is a unit of **<sup>F</sup>***M*�∪{(*s*−*a*)}(*s*). Therefore *<sup>π</sup>*1(*s*) factorizes in Specm(**F**[*s*]) \ (*M*� <sup>∪</sup> {(*s* − *a*)}). Since *t*(*π*(*s*)) = *h*(*π*1(*s*)), it factorizes in (recall that we are assuming *b* = 0)
*1. If M*� ⊆ Specm (**F**[*s*]) \ {(*s* − *a*)} *and U*(*s*) ∈ Gl*m*(**F***M*�(*s*)) *then U*(*f*(*s*)) ∈ Gl*m*(**F***H*(*M*�)(*s*))*.*
). Hence *h*(*π*(*s*)) factorizes in *H*(*M*�
) ∪ {(*s* − *b*)
)*.*
{(*s* − *a*)}) (i.e. the complementary subset of *M*� in Specm (**F**[*s*]) \ {(*s* − *a*)}) then
**Lemma 33.** *Let M*� ⊆ Specm (**F**[*s*]) \ {(*s* − *a*)} *where a* ∈ **F** *is an arbitrary element of* **F***.*
*H*(*M*˜ ) = Specm
*1. If π*(*s*) ∈ **F**[*s*] *factorizes in M*� *then h*(*π*(*s*)) *factorizes in H*(*M*�
*h*(*πi*(*s*))) ∈ *H*(*M*�
*H*(Specm(**F**[*s*]) \ (*M*� ∪ {(*s* − *a*)}) = Specm(**F**[*s*]) \ (*H*(*M*�
**Lemma 34.** *Let a* ∈ **F** *be an arbitrary element. Then*
In what follows and for notational simplicity we will assume *b* = 0.
*2. If π*(*s*) ∈ **F**[*s*] *is a unit of* **F***M*�(*s*) *then t*(*π*(*s*)) *is a unit of* **F***H*(*M*�)(*s*)*.*
(that may not be monic). If *ci* is the leading coefficient of *h*(*πi*(*s*)) then <sup>1</sup>
*<sup>p</sup>*<sup>0</sup> *<sup>h</sup>*(*π*(*s*))) (75)
*<sup>a</sup>*<sup>0</sup> *<sup>h</sup>*−1(*α*(*s*))) (76)
*ci*
).
) ∪ {(*s*)}). So, *t*(*π*(*s*)) is a unit of
). (77)
*h*(*πi*(*s*)) is monic,
or *h*−<sup>1</sup>
*h*−<sup>1</sup>
*<sup>a</sup>*,*<sup>b</sup>* = *hb*,*a*.
*1. h*(*π*1(*s*)*π*2(*s*)) = *h*(*π*1(*s*))*h*(*π*2(*s*))*. 2. If π*1(*s*) | *π*2(*s*) *then h*(*π*1(*s*)) | *h*(*π*2(*s*))*.*
with *p*<sup>0</sup> = *π*(*a*), is a bijection whose inverse is
As a consequence the map
irreducible and ( <sup>1</sup>
**F***H*(*M*�)(*s*).
*ci*
$$\frac{p(f(s))}{q(f(s))} = \frac{s^{d(p(s))} p(f(s))}{s^{d(q(s))} q(f(s))} s^{d(q(s)) - d(p(s))} = \frac{t(p(s))}{t(q(s))} s^{d(q(s)) - d(p(s))}.\tag{78}$$
1. Assume that *<sup>U</sup>*(*s*) <sup>∈</sup> Gl*m*(**F***M*�(*s*)) and let *<sup>p</sup>*(*s*) *<sup>q</sup>*(*s*) be any element of *U*(*s*). Therefore *q*(*s*) is a unit of **F***M*�(*s*) and, by Lemma 33.2, *t*(*q*(*s*)) is a unit of **F***H*(*M*�)(*s*). Moreover, *s* is also a unit of **<sup>F</sup>***H*(*M*�)(*s*). Hence, *<sup>p</sup>*(*f*(*s*)) *<sup>q</sup>*(*f*(*s*)) <sup>∈</sup> **<sup>F</sup>***H*(*M*�)(*s*). Furthermore, if det *<sup>U</sup>*(*s*) = *<sup>p</sup>*˜(*s*) *<sup>q</sup>*˜(*s*) , it is a unit of **F***M*�(*s*) and det *U*(*f*(*s*)) = *<sup>p</sup>*˜(*f*(*s*)) *<sup>q</sup>*˜(*f*(*s*)) is a unit of **F***H*(*M*�)(*s*).
2. If *<sup>p</sup>*(*s*) *<sup>q</sup>*(*s*) is any element of *U*(*s*) ∈ Gl*m*(**F***s*−*a*(*s*)) then *q*(*s*) ∈ **F**[*s*] \ (*s* − *a*) and so *d*(*h*(*q*(*s*))) = *d*(*q*(*s*)). Since *s* − *a* may divide *p*(*s*) we have that *d*(*t*(*p*(*s*))) ≤ *d*(*p*(*s*)). Hence, *<sup>d</sup>*(*h*(*q*(*s*))) <sup>−</sup> *<sup>d</sup>*(*q*(*s*)) <sup>≥</sup> *<sup>d</sup>*(*t*(*p*(*s*)) <sup>−</sup> *<sup>d</sup>*(*p*(*s*)) and *<sup>p</sup>*(*f*(*s*)) *<sup>q</sup>*(*f*(*s*)) <sup>=</sup> *<sup>t</sup>*(*p*(*s*)) *<sup>h</sup>*(*q*(*s*))*sd*(*q*(*s*))−*d*(*p*(*s*)) <sup>∈</sup> **<sup>F</sup>***pr*(*s*). Moreover if det *U*(*s*) = *<sup>p</sup>*˜(*s*) *<sup>q</sup>*˜(*s*) then *p*˜(*s*), *q*˜(*s*) ∈ **F**[*s*] \ (*s* − *a*), *d*(*h*(*p*˜(*s*))) = *d*(*p*˜(*s*)) and *d*(*h*(*q*˜(*s*))) = *d*(*q*˜(*s*)). Thus, det *U*(*f*(*s*)) = *<sup>h</sup>*(*p*˜(*s*)) *<sup>h</sup>*(*q*˜(*s*))*sd*(*q*˜(*s*))−*d*(*p*˜(*s*)) is a biproper rational function, i.e., a unit of **F***pr*(*s*).
3. If *<sup>U</sup>*(*s*) <sup>∈</sup> Gl*m*(**F***pr*(*s*)) and *<sup>p</sup>*(*s*) *<sup>q</sup>*(*s*) is any element of *U*(*s*) then *d*(*q*(*s*)) ≥ *d*(*p*(*s*)). Since *p*(*f*(*s*)) *<sup>q</sup>*(*f*(*s*)) <sup>=</sup> *<sup>t</sup>*(*p*(*s*)) *<sup>t</sup>*(*q*(*s*))*sd*(*q*(*s*))−*d*(*p*(*s*)) and *<sup>t</sup>*(*p*(*s*)), *<sup>t</sup>*(*q*(*s*)) <sup>∈</sup> **<sup>F</sup>**[*s*] \ (*s*) we obtain that *<sup>U</sup>*(*f*(*s*)) <sup>∈</sup> **<sup>F</sup>***s*(*s*)*m*×*m*. In addition, if det *<sup>U</sup>*(*s*) = *<sup>p</sup>*˜(*s*) *<sup>q</sup>*˜(*s*) , which is a unit of **F***pr*(*s*), then *d*(*q*˜(*s*)) = *d*(*p*˜(*s*)) and since *<sup>t</sup>*(*p*˜(*s*)), *<sup>t</sup>*(*q*˜(*s*)) <sup>∈</sup> **<sup>F</sup>**[*s*] \ (*s*) we conclude that det *<sup>U</sup>*(*f*(*s*)) = *<sup>t</sup>*(*p*˜(*s*)) *<sup>t</sup>*(*q*˜(*s*)) is a unit of **F***s*(*s*).
4. It is a consequence of 1., 2. and Remark 1.2.
**Proposition 35.** *Let M* <sup>⊆</sup> Specm(**F**[*s*]) *and* (*<sup>s</sup>* <sup>−</sup> *<sup>a</sup>*) <sup>∈</sup> *M. If T*(*s*) <sup>∈</sup> **<sup>F</sup>**(*s*)*m*×*<sup>m</sup> is non-singular with ni*(*s*) *di*(*s*) = (*<sup>s</sup>* <sup>−</sup> *<sup>a</sup>*)*gi �i*(*s*) *<sup>ψ</sup>i*(*s*) (*�i*(*s*), *<sup>ψ</sup>i*(*s*) ∈ **<sup>F</sup>**[*s*] \ (*<sup>s</sup>* − *<sup>a</sup>*)) *as invariant rational functions with respect to M then T*(*f*(*s*))*<sup>T</sup>* <sup>∈</sup> **<sup>F</sup>**(*s*)*m*×*<sup>m</sup> is a non-singular matrix with* <sup>1</sup> *ci h*(*�i*(*s*)) *<sup>h</sup>*(*ψi*(*s*))*s*−*gi*+*d*(*ψi*(*s*))−*d*(*�i*(*s*)) *as invariant rational functions in* **<sup>F</sup>***H*(*M*\{(*s*−*a*)})(*s*)*m*×*<sup>m</sup>* <sup>∩</sup> **<sup>F</sup>***pr*(*s*)*m*×*<sup>m</sup> where ci* <sup>=</sup> *�i*(*a*) *<sup>ψ</sup>i*(*a*)*.*
**Proof.-** Since (*<sup>s</sup>* <sup>−</sup> *<sup>a</sup>*)*gi �i*(*s*) *<sup>ψ</sup>*(*s*) are the invariant rational functions of *T*(*s*) with respect to *M*, there are *U*1(*s*), *U*2(*s*) ∈ *Glm*(**F***M*(*s*)) such that
$$T(s) = \mathcal{U}\_1(s) \operatorname{Diag} \left( (s-a)^{\mathcal{G}\_1} \frac{\varepsilon\_1(s)}{\psi\_1(s)}, \dots, (s-a)^{\mathcal{G}\_m} \frac{\varepsilon\_m(s)}{\psi\_m(s)} \right) \mathcal{U}\_2(s). \tag{79}$$
#### 26 Will-be-set-by-IN-TECH 72 Linear Algebra – Theorems and Applications
Notice that (*f*(*s*) − *a*) *gi �i*(*f*(*s*)) *<sup>ψ</sup>i*(*f*(*s*)) <sup>=</sup> *<sup>h</sup>*(*�i*(*s*)) *<sup>h</sup>*(*ψi*(*s*))*s*−*gi*+*d*(*ψi*(*s*))−*d*(*�i*(*s*)). Let *ci* <sup>=</sup> *�i*(*a*) *<sup>ψ</sup>i*(*a*), which is a non-zero constant, and put *D* = Diag (*c*1,..., *cm*). Hence,
$$\left(T(f(s))^T = \mathcal{U}\_2(f(s))^T \mathcal{D}L(s)\mathcal{U}\_1(f(s))^T\right.\tag{80}$$
with
$$L(s) = \text{Diag}\left(\frac{1}{c\_1} \frac{h(\varepsilon\_1(s))}{h(\psi\_1(s))} s^{-\mathcal{G}+d(\psi\_1(s))-d(\varepsilon\_1(s))}, \dots, \frac{1}{c\_m} \frac{h(\varepsilon\_m(s))}{h(\psi\_m(s))} s^{-\mathcal{g}\_m+d(\psi\_n(s))-d(\varepsilon\_n(s))}\right). \tag{81}$$
By 4 of Lemma 34 matrices *<sup>U</sup>*1(*f*(*s*))*T*, *<sup>U</sup>*2(*f*(*s*))*<sup>T</sup>* <sup>∈</sup> Gl*m*(**F***H*(*M*\{(*s*−*a*)})(*s*)) <sup>∩</sup> Gl*m*(**F***pr*(*s*)) and the Proposition follows.
**Proposition 36.** *Let M*, *M*� ⊆ Specm(**F**[*s*]) *such that M* ∪ *M*� = Specm(**F**[*s*])*. Assume that there are ideals in M* \ *M*� *generated by linear polynomials and let* (*s* − *a*) *be any of them. If T*(*s*) ∈ **<sup>F</sup>**(*s*)*m*×*<sup>m</sup> is a non-singular rational matrix with no poles and no zeros in M* <sup>∩</sup> *<sup>M</sup>*� *and k*1,..., *km as left Wiener–Hopf factorization indices with respect to M then T*(*f*(*s*))*<sup>T</sup>* <sup>∈</sup> **<sup>F</sup>**(*s*)*m*×*<sup>m</sup> is a non-singular rational matrix with no poles and no zeros in H*(*M* ∩ *M*� ) *and* −*km*,..., −*k*<sup>1</sup> *as left Wiener–Hopf factorization indices with respect to H*(*M*� ) ∪ {(*s*)}*.*
**Proof.-** By Theorem 19 there are matrices *<sup>U</sup>*1(*s*) invertible in **<sup>F</sup>***M*�(*s*)*m*×*<sup>m</sup>* <sup>∩</sup> **<sup>F</sup>***pr*(*s*)*m*×*<sup>m</sup>* and *<sup>U</sup>*2(*s*) invertible in **<sup>F</sup>***M*(*s*)*m*×*<sup>m</sup>* such that *<sup>T</sup>*(*s*) = *<sup>U</sup>*1(*s*) Diag (*s* − *a*) *<sup>k</sup>*<sup>1</sup> ,...,(*<sup>s</sup>* <sup>−</sup> *<sup>a</sup>*) *km U*2(*s*). By Lemma 34 *<sup>U</sup>*2(*f*(*s*))*<sup>T</sup>* is invertible in **<sup>F</sup>***H*(*M*\{(*s*−*a*)})(*s*)*m*×*<sup>m</sup>* <sup>∩</sup> **<sup>F</sup>***pr*(*s*)*m*×*<sup>m</sup>* and *<sup>U</sup>*1(*f*(*s*))*<sup>T</sup>* is invertible in **<sup>F</sup>***H*(*M*�)(*s*)*m*×*<sup>m</sup>* <sup>∩</sup> **<sup>F</sup>***s*(*s*)*m*×*<sup>m</sup>* <sup>=</sup> **<sup>F</sup>***H*(*M*�)∪{(*s*)}(*s*)*m*×*m*. Moreover, *<sup>H</sup>*(*<sup>M</sup>* \ {(*<sup>s</sup>* <sup>−</sup> *<sup>a</sup>*)}) <sup>∪</sup> *H*(*M*� ) ∪ {(*s*)} = Specm(**F**[*s*]) and *H*(*M* \ {(*s* − *a*)}) ∩ (*H*(*M*� ) ∪ {(*s*)}) = *H*(*M* ∩ *M*� ). Thus, *<sup>T</sup>*(*f*(*s*))*<sup>T</sup>* <sup>=</sup> *<sup>U</sup>*2(*f*(*s*))*<sup>T</sup>* Diag *s*−*k*<sup>1</sup> ,...,*s*−*km <sup>U</sup>*1(*f*(*s*))*<sup>T</sup>* has no poles and no zeros in *<sup>H</sup>*(*<sup>M</sup>* <sup>∩</sup> *M*� ) and −*km*,..., −*k*<sup>1</sup> are its left Wiener–Hopf factorization indices with respect to *H*(*M*� ) ∪ {(*s*)}.
#### *5.2.1. Proof of Theorem 29: Sufficiency*
Let *<sup>k</sup>*<sup>1</sup> ≥ ··· ≥ *km* be integers, *�*1(*s*) *<sup>ψ</sup>*1(*s*),..., *�m*(*s*) *<sup>ψ</sup>m*(*s*) irreducible rational functions such that *�*1(*s*) | ··· | *�m*(*s*), *ψm*(*s*) | ··· | *ψ*1(*s*) are monic polynomials factorizing in *M*� \ *M* and *l*1,..., *lm* integers such that *l*<sup>1</sup> + *d*(*ψ*1(*s*)) − *d*(*�*1(*s*)) ≤ ··· ≤ *lm* + *d*(*ψm*(*s*)) − *d*(*�m*(*s*)) and satisfying (59).
Since *�i*(*s*) and *ψi*(*s*) are coprime polynomials that factorize in *M*� \ *M* and (*s* − *a*) ∈ *M* \ *M*� , by Lemmas 32 and 33, *<sup>h</sup>*(*�*1(*s*)) *<sup>h</sup>*(*ψ*1(*s*))*sl*1+*d*(*ψ*1(*s*))−*d*(*�*1(*s*)),..., *<sup>h</sup>*(*�m*(*s*)) *<sup>h</sup>*(*ψm*(*s*))*slm*+*d*(*ψm*(*s*))−*d*(*�m*(*s*)) are irreducible rational functions with numerators and denominators polynomials factorizing in *H*(*M*� ) ∪ {(*s*)} (actually, in *H*(*M*� \ *M*) ∪ {(*s*)}) and such that each numerator divides the next one and each denominator divides the previous one.
By (59) and Theorem 27 there is a matrix *<sup>G</sup>*(*s*) <sup>∈</sup> **<sup>F</sup>**(*s*)*m*×*<sup>m</sup>* with <sup>−</sup>*km*,..., <sup>−</sup>*k*<sup>1</sup> as left Wiener–Hopf factorization indices with respect to *H*(*M*� ) ∪ {(*s*)} and 1 *c*1 *h*(*�*1(*s*)) *<sup>h</sup>*(*ψ*1(*s*))*sl*1+*d*(*ψ*1(*s*))−*d*(*�*1(*s*)),..., <sup>1</sup> *cm h*(*�m*(*s*)) *<sup>h</sup>*(*ψm*(*s*))*slm*+*d*(*ψm*(*s*))−*d*(*�m*(*s*)) as invariant rational functions with respect to *H*(*M*� ) ∪ {(*s*)} where *ci* <sup>=</sup> *�i*(*a*) *<sup>ψ</sup>i*(*a*), *<sup>i</sup>* = 1, . . . , *<sup>m</sup>*. Notice that *<sup>G</sup>*(*s*) has no zeros and poles in *H*(*M* ∩ *M*� ) because the numerator and denominator of each rational function *h*(*�i*(*s*)) *<sup>h</sup>*(*ψi*(*s*))*sli*+*d*(*ψi*(*s*))−*d*(*�i*(*s*)) factorizes in *<sup>H</sup>*(*M*� \ *<sup>M</sup>*) ∪ {(*s*)} and so it is a unit of **<sup>F</sup>***H*(*M*∩*M*�)(*s*).
Put *M* = *H*(*M*� ) ∪ {(*s*)} and *M* � = *H*(*M* \ {(*s* − *a*)}). As remarked in the proof of Proposition 36, *M* ∪ *M* � = Specm(**F**[*s*]) and *M* ∩ *M* � = *H*(*M* ∩ *M*� ). Now (*s*) ∈ *M* so that we can apply Proposition 35 to *G*(*s*) with the change of indeterminate *f* (*s*) = <sup>1</sup> *<sup>s</sup>*−*<sup>a</sup>* . Thus the invariant rational functions of *G*(*f* (*s*))*<sup>T</sup>* in **<sup>F</sup>***M*�(*s*) <sup>∩</sup> **<sup>F</sup>***pr*(*s*) are *�*1(*s*) *ψ*1(*s*) 1 (*s*−*a*)*<sup>l</sup>* <sup>1</sup> ,..., *�m*(*s*) *ψm*(*s*) 1 (*s*−*a*)*lm* .
On the other hand *M* � = *H*(*M* \ {(*s* − *a*)}) ⊆ Specm(**F**[*s*]) \ {(*s*)} and so (*s*) ∈ *M* \ *M* � . Then we can apply Proposition 36 to *G*(*s*) with *f* (*s*) = <sup>1</sup> *<sup>s</sup>*−*<sup>a</sup>* so that *<sup>G</sup>*(*<sup>f</sup>* (*s*))*<sup>T</sup>* is a non-singular matrix with no poles and no zeros in *<sup>H</sup>*−1(*<sup>M</sup>* <sup>∩</sup> *<sup>M</sup>* � ) = *<sup>H</sup>*−1(*H*(*<sup>M</sup>* <sup>∩</sup> *<sup>M</sup>*� )) = *M* ∩ *M*� and *<sup>k</sup>*1,..., *km* as left Wiener–Hopf factorization indices with respect to *<sup>H</sup>*−1(*<sup>M</sup>* � ) ∪ {(*s* − *a*)} = (*M* \ {(*s* − *a*)}) ∪ {(*s* − *a*)} = *M*. The theorem follows by letting *T*(*s*) = *G*(*f* (*s*))*T*.
**Remark 37.** Notice that when *M*� = ∅ and *M* = Specm(**F**[*s*]) in Theorem 29 we obtain Theorem 28 (*qi* = −*li*).
#### **Author details**
26 Will-be-set-by-IN-TECH
<sup>−</sup>*g*1+*d*(*ψ*1(*s*))−*d*(*�*1(*s*)),..., <sup>1</sup>
By 4 of Lemma 34 matrices *<sup>U</sup>*1(*f*(*s*))*T*, *<sup>U</sup>*2(*f*(*s*))*<sup>T</sup>* <sup>∈</sup> Gl*m*(**F***H*(*M*\{(*s*−*a*)})(*s*)) <sup>∩</sup> Gl*m*(**F***pr*(*s*)) and
**Proposition 36.** *Let M*, *M*� ⊆ Specm(**F**[*s*]) *such that M* ∪ *M*� = Specm(**F**[*s*])*. Assume that there are ideals in M* \ *M*� *generated by linear polynomials and let* (*s* − *a*) *be any of them. If T*(*s*) ∈ **<sup>F</sup>**(*s*)*m*×*<sup>m</sup> is a non-singular rational matrix with no poles and no zeros in M* <sup>∩</sup> *<sup>M</sup>*� *and k*1,..., *km as left Wiener–Hopf factorization indices with respect to M then T*(*f*(*s*))*<sup>T</sup>* <sup>∈</sup> **<sup>F</sup>**(*s*)*m*×*<sup>m</sup> is a non-singular*
**Proof.-** By Theorem 19 there are matrices *<sup>U</sup>*1(*s*) invertible in **<sup>F</sup>***M*�(*s*)*m*×*<sup>m</sup>* <sup>∩</sup> **<sup>F</sup>***pr*(*s*)*m*×*<sup>m</sup>* and
By Lemma 34 *<sup>U</sup>*2(*f*(*s*))*<sup>T</sup>* is invertible in **<sup>F</sup>***H*(*M*\{(*s*−*a*)})(*s*)*m*×*<sup>m</sup>* <sup>∩</sup> **<sup>F</sup>***pr*(*s*)*m*×*<sup>m</sup>* and *<sup>U</sup>*1(*f*(*s*))*<sup>T</sup>* is invertible in **<sup>F</sup>***H*(*M*�)(*s*)*m*×*<sup>m</sup>* <sup>∩</sup> **<sup>F</sup>***s*(*s*)*m*×*<sup>m</sup>* <sup>=</sup> **<sup>F</sup>***H*(*M*�)∪{(*s*)}(*s*)*m*×*m*. Moreover, *<sup>H</sup>*(*<sup>M</sup>* \ {(*<sup>s</sup>* <sup>−</sup> *<sup>a</sup>*)}) <sup>∪</sup>
) and −*km*,..., −*k*<sup>1</sup> are its left Wiener–Hopf factorization indices with respect to *H*(*M*�
··· | *�m*(*s*), *ψm*(*s*) | ··· | *ψ*1(*s*) are monic polynomials factorizing in *M*� \ *M* and *l*1,..., *lm* integers such that *l*<sup>1</sup> + *d*(*ψ*1(*s*)) − *d*(*�*1(*s*)) ≤ ··· ≤ *lm* + *d*(*ψm*(*s*)) − *d*(*�m*(*s*)) and satisfying
Since *�i*(*s*) and *ψi*(*s*) are coprime polynomials that factorize in *M*� \ *M* and (*s* − *a*) ∈
irreducible rational functions with numerators and denominators polynomials factorizing in
By (59) and Theorem 27 there is a matrix *<sup>G</sup>*(*s*) <sup>∈</sup> **<sup>F</sup>**(*s*)*m*×*<sup>m</sup>* with <sup>−</sup>*km*,..., <sup>−</sup>*k*<sup>1</sup>
*<sup>h</sup>*(*ψ*1(*s*))*sl*1+*d*(*ψ*1(*s*))−*d*(*�*1(*s*)),..., *<sup>h</sup>*(*�m*(*s*))
) ∪ {(*s*)} (actually, in *H*(*M*� \ *M*) ∪ {(*s*)}) and such that each numerator divides the next
) ∪ {(*s*)}*.*
*<sup>h</sup>*(*ψi*(*s*))*s*−*gi*+*d*(*ψi*(*s*))−*d*(*�i*(*s*)). Let *ci* <sup>=</sup> *�i*(*a*)
*cm*
*T*(*f*(*s*))*<sup>T</sup>* = *U*2(*f*(*s*))*TDL*(*s*)*U*1(*f*(*s*))*<sup>T</sup>* (80)
*h*(*�m*(*s*)) *<sup>h</sup>*(*ψm*(*s*))*<sup>s</sup>*
> (*s* − *a*)
*<sup>ψ</sup>i*(*a*), which is a
. (81)
−*gm*+*d*(*ψm*(*s*))−*d*(*�m*(*s*))
) *and* −*km*,..., −*k*<sup>1</sup> *as left Wiener–Hopf*
*<sup>k</sup>*<sup>1</sup> ,...,(*<sup>s</sup>* <sup>−</sup> *<sup>a</sup>*)
*<sup>h</sup>*(*ψm*(*s*))*slm*+*d*(*ψm*(*s*))−*d*(*�m*(*s*)) are
) ∪ {(*s*)} and
) ∪ {(*s*)}) = *H*(*M* ∩ *M*�
*<sup>U</sup>*1(*f*(*s*))*<sup>T</sup>* has no poles and no zeros in *<sup>H</sup>*(*<sup>M</sup>* <sup>∩</sup>
*<sup>ψ</sup>m*(*s*) irreducible rational functions such that *�*1(*s*) |
*<sup>h</sup>*(*ψm*(*s*))*slm*+*d*(*ψm*(*s*))−*d*(*�m*(*s*)) as invariant rational functions
*<sup>ψ</sup>i*(*a*), *<sup>i</sup>* = 1, . . . , *<sup>m</sup>*. Notice that *<sup>G</sup>*(*s*) has no zeros
*km U*2(*s*).
). Thus,
) ∪
Notice that (*f*(*s*) − *a*)
*L*(*s*) = Diag
the Proposition follows.
1 *c*1
with
*H*(*M*�
{(*s*)}.
(59).
*M* \ *M*�
*H*(*M*�
1 *c*1 *h*(*�*1(*s*))
with respect to *H*(*M*�
*M*�
*gi �i*(*f*(*s*))
*h*(*�*1(*s*)) *<sup>h</sup>*(*ψ*1(*s*))*<sup>s</sup>*
*rational matrix with no poles and no zeros in H*(*M* ∩ *M*�
*U*2(*s*) invertible in **F***M*(*s*)*m*×*<sup>m</sup>* such that *T*(*s*) = *U*1(*s*) Diag
) ∪ {(*s*)} = Specm(**F**[*s*]) and *H*(*M* \ {(*s* − *a*)}) ∩ (*H*(*M*�
*s*−*k*<sup>1</sup> ,...,*s*−*km*
*<sup>ψ</sup>*1(*s*),..., *�m*(*s*)
as left Wiener–Hopf factorization indices with respect to *H*(*M*�
*h*(*�m*(*s*))
*cm*
) ∪ {(*s*)} where *ci* <sup>=</sup> *�i*(*a*)
*factorization indices with respect to H*(*M*�
*T*(*f*(*s*))*<sup>T</sup>* = *U*2(*f*(*s*))*<sup>T</sup>* Diag
*5.2.1. Proof of Theorem 29: Sufficiency*
Let *<sup>k</sup>*<sup>1</sup> ≥ ··· ≥ *km* be integers, *�*1(*s*)
*<sup>h</sup>*(*ψ*1(*s*))*sl*1+*d*(*ψ*1(*s*))−*d*(*�*1(*s*)),..., <sup>1</sup>
, by Lemmas 32 and 33, *<sup>h</sup>*(*�*1(*s*))
one and each denominator divides the previous one.
non-zero constant, and put *D* = Diag (*c*1,..., *cm*). Hence,
*<sup>ψ</sup>i*(*f*(*s*)) <sup>=</sup> *<sup>h</sup>*(*�i*(*s*))
A. Amparan, S. Marcaida, I. Zaballa *Universidad del País Vasco/Euskal Herriko Unibertsitatea UPV/EHU, Spain*
#### **6. References**
- [12] Kailath, T. [1980]. *Linear systems*, Prentice Hall, New Jersey.
- [13] Newman, M. [1972]. *Integral matrices*, Academic Press, New York and London.
- [14] Rosenbrock, H. H. [1970]. *State-space and multivariable theory*, Thomas Nelson and Sons, London.
- [15] Vardulakis, A. I. G. [1991]. *Linear multivariable control*, John Wiley and Sons, New York.
- [16] Vidyasagar, M. [1985]. *Control system synthesis. A factorization approach*, The MIT Press, New York.
- [17] Wolovich, W. A. [1974]. *Linear multivariable systems*, Springer-Verlag, New York.
- [18] Zaballa, I. [1997]. Controllability and hermite indices of matrix pairs, *Int. J. Control* 68(1): 61–86.
## **Gauge Theory, Combinatorics, and Matrix Models**
Taro Kimura
28 Will-be-set-by-IN-TECH
[14] Rosenbrock, H. H. [1970]. *State-space and multivariable theory*, Thomas Nelson and Sons,
[15] Vardulakis, A. I. G. [1991]. *Linear multivariable control*, John Wiley and Sons, New York. [16] Vidyasagar, M. [1985]. *Control system synthesis. A factorization approach*, The MIT Press,
[18] Zaballa, I. [1997]. Controllability and hermite indices of matrix pairs, *Int. J. Control*
[13] Newman, M. [1972]. *Integral matrices*, Academic Press, New York and London.
[17] Wolovich, W. A. [1974]. *Linear multivariable systems*, Springer-Verlag, New York.
[12] Kailath, T. [1980]. *Linear systems*, Prentice Hall, New Jersey.
London.
New York.
68(1): 61–86.
Additional information is available at the end of the chapter
http://dx.doi.org/10.5772/46481
**1. Introduction**
Quantum field theory is the most universal method in physics, applied to all the area from condensed-matter physics to high-energy physics. The standard tool to deal with quantum field theory is the perturbation method, which is quite useful if we know the vacuum of the system, namely the starting point of our analysis. On the other hand, sometimes the vacuum itself is not obvious due to the quantum nature of the system. In that case, since the perturbative method is not available any longer, we have to treat the theory in a non-perturbative way.
Supersymmetric gauge theory plays an important role in study on the non-perturbative aspects of quantum field theory. The milestone paper by Seiberg and Witten proposed a *solution* to N = 2 supersymmetric gauge theory [48, 49], which completely describes the low energy effective behavior of the theory. Their solution can be written down by an auxiliary complex curve, called *Seiberg-Witten curve*, but its meaning was not yet clear and the origin was still mysterious. Since the establishment of Seiberg-Witten theory, tremendous number of works are devoted to understand the Seiberg-Witten's solution, not only by physicists but also mathematicians. In this sense the solution was not a *solution* at that time, but just a *starting point* of the exploration.
One of the most remarkable progress in N = 2 theories referring to Seiberg-Witten theory is then the exact derivation of the gauge theory partition function by performing the integral over the instanton moduli space [43]. The partition function is written down by multiple partitions, thus we can discuss it in a combinatorial way. It was mathematically proved that the partition function correctly reproduces the Seiberg-Witten solution. This means Seiberg-Witten theory was mathematically established at that time.
The recent progress on the four dimensional N = 2 supersymmetric gauge theory has revealed a remarkable relation to the two dimensional conformal field theory [1]. This relation provides the explicit interpretation for the partition function of the four dimensional gauge theory as the conformal block of the two dimensional Liouville field theory. It is naturally regarded as a consequence of the M-brane compactifications [23, 60], and also reproduces
#### 2 Linear Algebra 76 Linear Algebra – Theorems and Applications
the results of Seiberg-Witten theory. It shows how Seiberg-Witten curve characterizes the corresponding four dimensional gauge theory, and thus we can obtain a novel viewpoint of Seiberg-Witten theory.
Based on the connection between the two and four dimensional theories, established results on the two dimensional side can be reconsidered from the viewpoint of the four dimensional theory, and vice versa. One of the useful applications is the matrix model description of the supersymmetric gauge theory [12, 16, 17, 47]. This is based on the fact that the conformal block on the sphere can be also regarded as the matrix integral, which is called Dotsenko-Fateev integral representation [14, 15]. In this direction some extensions of the matrix model description are performed by starting with the two dimensional conformal field theory.
Another type of the matrix model is also investigated so far [27, 28, 30, 52, 53]. This is apparently different from the Dotsenko-Fateev type matrix models, but both of them correctly reproduce the results of the four dimensional gauge theory, e.g. Seiberg-Witten curve. While these studies mainly focus on rederiving the gauge theory results, the present author reveals the new kind of Seiberg-Witten curve by studying the corresponding new matrix model [27, 28]. Such a matrix models is directly derived from the combinatorial representation of the partition function by considering its asymptotic behavior. This treatment is quite analogous to the matrix integral representation of the combinatorial object, for example, the longest increasing subsequences in random permutations [3], the non-equilibrium stochastic model, so-called TASEP [26], and so on (see also [46]). Their remarkable connection to the Tracy-Widom distribution [56] can be understood from the viewpoint of the random matrix theory through the Robinson-Schensted-Knuth (RSK) correspondence (see e.g. [51]).
In this article we review such a universal relation between combinatorics and the matrix model, and discuss its relation to the gauge theory. The gauge theory consequence can be naturally extacted from such a matrix model description. Actually the spectral curve of the matrix model can be interpreted as Seiberg-Witten curve for N = 2 supersymmetric gauge theory. This identification suggests some aspects of the gauge theory are also described by the significant universality of the matrix model.
This article is organized as follows. In section 2 we introduce statistical models defined in a combinaorial manner. These models are based on the Plancherel measure on a combinatorial object, and its origin from the gauge theory perspective is also discussed. In section 3 it is shown that the matrix model is derived from the combinatorial model by considering its asymptotic limit. There are various matrix integral representations, corresponding to some deformations of the combinatorial model. In section 4 we investigate the large matrix size limit of the matrix model. It is pointed out that the algebraic curve is quite useful to study one-point function. Its relation to Seiberg-Witten theory is also discussed. Section 5 is devoted to conclusion.
## **2. Combinatorial partition function**
In this section we introduce several kinds of combinatorial models. Their partition functions are defined as summation over partitions with a certain weight function, which is called *Plancherel measure*. It is also shown that such a combinatorial partition function is obtained by performing the path integral for supersymmetric gauge theories.
**Figure 1.** Graphical representation of a partition *λ* = (5, 4, 3, 1, 1) and its transposed partition *λ*ˇ = (5, 3, 2, 2, 1) by the associated Young diagrams. There are 5 non-zero entries in both of them, �(*λ*) = *λ*ˇ <sup>1</sup> = 5 and �(*λ*ˇ) = *λ*<sup>1</sup> = 5.
#### **2.1. Random partition model**
2 Linear Algebra
the results of Seiberg-Witten theory. It shows how Seiberg-Witten curve characterizes the corresponding four dimensional gauge theory, and thus we can obtain a novel viewpoint of
Based on the connection between the two and four dimensional theories, established results on the two dimensional side can be reconsidered from the viewpoint of the four dimensional theory, and vice versa. One of the useful applications is the matrix model description of the supersymmetric gauge theory [12, 16, 17, 47]. This is based on the fact that the conformal block on the sphere can be also regarded as the matrix integral, which is called Dotsenko-Fateev integral representation [14, 15]. In this direction some extensions of the matrix model description are performed by starting with the two dimensional conformal field theory.
Another type of the matrix model is also investigated so far [27, 28, 30, 52, 53]. This is apparently different from the Dotsenko-Fateev type matrix models, but both of them correctly reproduce the results of the four dimensional gauge theory, e.g. Seiberg-Witten curve. While these studies mainly focus on rederiving the gauge theory results, the present author reveals the new kind of Seiberg-Witten curve by studying the corresponding new matrix model [27, 28]. Such a matrix models is directly derived from the combinatorial representation of the partition function by considering its asymptotic behavior. This treatment is quite analogous to the matrix integral representation of the combinatorial object, for example, the longest increasing subsequences in random permutations [3], the non-equilibrium stochastic model, so-called TASEP [26], and so on (see also [46]). Their remarkable connection to the Tracy-Widom distribution [56] can be understood from the viewpoint of the random matrix
theory through the Robinson-Schensted-Knuth (RSK) correspondence (see e.g. [51]).
significant universality of the matrix model.
**2. Combinatorial partition function**
by performing the path integral for supersymmetric gauge theories.
to conclusion.
In this article we review such a universal relation between combinatorics and the matrix model, and discuss its relation to the gauge theory. The gauge theory consequence can be naturally extacted from such a matrix model description. Actually the spectral curve of the matrix model can be interpreted as Seiberg-Witten curve for N = 2 supersymmetric gauge theory. This identification suggests some aspects of the gauge theory are also described by the
This article is organized as follows. In section 2 we introduce statistical models defined in a combinaorial manner. These models are based on the Plancherel measure on a combinatorial object, and its origin from the gauge theory perspective is also discussed. In section 3 it is shown that the matrix model is derived from the combinatorial model by considering its asymptotic limit. There are various matrix integral representations, corresponding to some deformations of the combinatorial model. In section 4 we investigate the large matrix size limit of the matrix model. It is pointed out that the algebraic curve is quite useful to study one-point function. Its relation to Seiberg-Witten theory is also discussed. Section 5 is devoted
In this section we introduce several kinds of combinatorial models. Their partition functions are defined as summation over partitions with a certain weight function, which is called *Plancherel measure*. It is also shown that such a combinatorial partition function is obtained
Seiberg-Witten theory.
Let us first recall a partition of a positive integer *n*: it is a way of writing *n* as a sum of positive integers
$$
\lambda = (\lambda\_1, \lambda\_2, \dots, \lambda\_{\ell(\lambda)}) \tag{1}
$$
satisfying the following conditions,
$$m = \sum\_{i=1}^{\ell(\lambda)} \lambda\_i \equiv |\lambda|\_{\prime} \qquad \lambda\_1 \ge \lambda\_2 \ge \cdots \cdot \lambda\_{\ell(\lambda)} > 0 \tag{2}$$
Here �(*λ*) is the number of non-zero entries in *λ*. Now it is convenient to define *λ<sup>i</sup>* = 0 for *i* > �(*λ*). Fig. 2 shows *Young diagram*, which graphically describes a partition *λ* = (5, 4, 2, 1, 1) with �(*λ*) = 5.
It is known that the partition is quite usefull for representation theory. We can obtain an irreducible representation of symmetric group S*n*, which is in one-to-one correspondence with a partition *λ* with |*λ*| = *n*. For such a finite group, one can define a natural measure, which is called *Plancherel measure*,
$$
\mu\_{\mathfrak{n}}(\lambda) = \frac{(\dim \lambda)^2}{n!} \tag{3}
$$
This measure is normalized as
$$\sum\_{\substack{\lambda \text{ s.t. } |\lambda|=n}} \mu\_n(\lambda) = 1 \tag{4}$$
It is also interpreted as Fourier transform of Haar measure on the group. This measure has another useful representation, which is described in a combinatorial way,
$$\mu\_{\mathfrak{n}}(\lambda) = n! \prod\_{(i,j)\in\lambda} \frac{1}{h(i,j)^2} \tag{5}$$
This *h*(*i*, *j*) is called *hook length*, which is defined with *arm length* and *leg length*,
$$\begin{aligned} h(i,j) &= a(i,j) + l(i,j) + 1, \\ a(i,j) &= \lambda\_i - j, \\ l(i,j) &= \lambda\_j - i \end{aligned} \tag{6}$$
Here *λ*ˇ stands for the transposed partition. Thus the height of a partition *λ* can be explicitly written as �(*λ*) = *λ*ˇ 1.
**Figure 2.** Combinatorics of Young diagram. Definitions of hook, arm and leg lengths are shown in (6). For the shaded box in this figure, *a*(2, 3) = 4, *l*(2, 3) = 3, and *h*(2, 3) = 8.
With this combinatorial measure, we now introduce the following partition function,
$$Z\_{\mathbf{U}(1)} = \sum\_{\lambda} \left(\frac{\Lambda}{\hbar}\right)^{2|\lambda|} \prod\_{(i,j)\in\lambda} \frac{1}{\hbar (i,j)^2} \tag{7}$$
This model is often called *random partition model*. Here Λ is regarded as a parameter like a *chemical potential*, or a *fugacity*, and ¯*h* stands for the size of boxes.
Note that a deformed model, which includes higher Casimir potentials, is also investigated in detail [19],
$$Z\_{\text{higher}} = \sum\_{\lambda} \prod\_{(i,j) \in \lambda} \frac{1}{h(i,j)^2} \prod\_{k=1} e^{-\mathcal{G}k\mathcal{C}\_k(\lambda)} \tag{8}$$
In this case the chemical potential term is absorbed by the linear potential term. There is an interesting interpretation of this deformation in terms of topological string, gauge theory and so on [18, 38].
In order to compute the U(1) partition function it is useful to rewrite it in a "canonical form" instead of the "grand canonical form" which is originally shown in (7),
$$Z\_{\mathbf{U}(1)} = \sum\_{n=0} \sum\_{\substack{\lambda \text{ s.t. } |\lambda|=n}} \left(\frac{\Lambda}{\hbar}\right)^{2n} \prod\_{(i,j) \in \lambda} \frac{1}{\hbar (i,j)^2} \tag{9}$$
Due to the normalization condition (4), this partition function can be computed as
$$Z\_{\mathbf{U}(1)} = \exp\left(\frac{\Lambda}{\hbar}\right)^2\tag{10}$$
Although this is explicitly solvable, its universal property and explicit connections to other models are not yet obvious. We will show, in section 3 and section 4, the matrix model description plays an important role in discussing such an interesting aspect of the combinatorial model.
Now let us remark one interesting observation, which is partially related to the following discussion. The combinatorial partition function (7) has another field theoretical representation using the free boson field [44]. We now consider the following coherent state,
$$|\psi\rangle = \exp\left(\frac{\Lambda}{\hbar}a\_{-1}\right)|0\rangle \tag{11}$$
Here we introduce Heisenberg algebra, satisfying the commutation relation, [*an*, *am*] = *nδn*+*m*,0, and the vacuum |0� annihilated by any positive modes, *an*|0� = 0 for *n* > 0. Then it is easy to show the norm of this state gives rise to the partition function,
$$Z\_{\mathbf{U}(1)} = \langle \psi | \psi \rangle \tag{12}$$
Similar kinds of observation is also performed for generalized combinatorial models introduced in section 2.2 [22, 44, 55].
Let us then introduce some generalizations of the U(1) model. First is what we call *β-deformed model* including an arbitrary parameter *β* ∈ **R**,
$$Z\_{\mathbf{U}(1)}^{(\beta)} = \sum\_{\lambda} \left(\frac{\Lambda}{\hbar}\right)^{2|\lambda|} \prod\_{(i,j)\in\lambda} \frac{1}{h\_{\beta}(i,j)h^{\beta}(i,j)}\tag{13}$$
Here we involve the deformed hook lengths,
4 Linear Algebra
**Figure 2.** Combinatorics of Young diagram. Definitions of hook, arm and leg lengths are shown in (6).
<sup>2</sup>|*λ*|
This model is often called *random partition model*. Here Λ is regarded as a parameter like a
Note that a deformed model, which includes higher Casimir potentials, is also investigated in
In this case the chemical potential term is absorbed by the linear potential term. There is an interesting interpretation of this deformation in terms of topological string, gauge theory and
In order to compute the U(1) partition function it is useful to rewrite it in a "canonical form"
Λ *h*¯
> Λ *h*¯ <sup>2</sup>
<sup>2</sup>*<sup>n</sup>*
∏ (*i*,*j*)∈*λ* 1
1 *<sup>h</sup>*(*i*, *<sup>j</sup>*)<sup>2</sup> ∏ *k*=1 *e*
∏ (*i*,*j*)∈*λ* 1
*<sup>h</sup>*(*i*, *<sup>j</sup>*)<sup>2</sup> (7)
<sup>−</sup>*gkCk* (*λ*) (8)
*<sup>h</sup>*(*i*, *<sup>j</sup>*)<sup>2</sup> (9)
(10)
With this combinatorial measure, we now introduce the following partition function,
Λ *h*¯
∏ (*i*,*j*)∈*λ*
∑ *λ s*.*t*. |*λ*|=*n*
Due to the normalization condition (4), this partition function can be computed as
*Z*U(1) = exp
Although this is explicitly solvable, its universal property and explicit connections to other models are not yet obvious. We will show, in section 3 and section 4, the matrix
For the shaded box in this figure, *a*(2, 3) = 4, *l*(2, 3) = 3, and *h*(2, 3) = 8.
*chemical potential*, or a *fugacity*, and ¯*h* stands for the size of boxes.
*Z*higher = ∑
instead of the "grand canonical form" which is originally shown in (7),
*n*=0
*Z*U(1) = ∑
detail [19],
so on [18, 38].
*Z*U(1) = ∑
*λ*
*λ*
$$h\_{\beta}(\mathbf{i}, \mathbf{j}) = a(\mathbf{i}, \mathbf{j}) + \beta l(\mathbf{i}, \mathbf{j}) + 1, \qquad h^{\beta}(\mathbf{i}, \mathbf{j}) = a(\mathbf{i}, \mathbf{j}) + \beta l(\mathbf{i}, \mathbf{j}) + \beta \tag{14}$$
This generalized model corresponds to Jack polynomial, which is a kind of symmetric polynomial obtained by introducing a free parameter to Schur polynomial [34]. This Jack polynomial is applied to several physical theories: quantum integrable model called Calogero-Sutherland model [10, 54], quantum Hall effect [4–6] and so on.
Second is a further generalized model involving two free parameters,
$$Z\_{\mathbf{U}(1)}^{(q,t)} = \sum\_{\lambda} \left(\frac{\Lambda}{\hbar}\right)^{2|\lambda|} \prod\_{(i,j)\in\lambda} \frac{(1-q)(1-q^{-1})}{(1-q^{a(i,j)+1}t^{l(i,j)})(1-q^{-a(i,j)}t^{-l(i,j)-1})} \tag{15}$$
This is just a *q*-analog of the previous combinatorial model. One can see this is reduced to the *<sup>β</sup>*-deformed model (13) in the limit of *<sup>q</sup>* <sup>→</sup> 1 with fixing *<sup>t</sup>* <sup>=</sup> *<sup>q</sup>β*. This generalization is also related to the symmetric polynomial, which is called *Macdonald polynomial* [34]. This symmetric polynomial is used to study Ruijsenaars-Schneider model [45], and the stochastic process based on this function has been recently proposed [8].
Next is **Z***r*-generalization of the model, which is defined as
$$Z\_{\text{orbifold},\mathcal{U}(1)} = \sum\_{\lambda} \left(\frac{\Lambda}{\hbar}\right)^{2|\lambda|} \prod\_{\Gamma:\text{inv}\subset\lambda} \frac{1}{h(i,j)^2} \tag{16}$$
**Figure 3.** Γ-invariant sector for U(1) theory with *λ* = (8, 5, 5, 4, 2, 2, 2, 1). Numbers in boxes stand for their hook lengths *<sup>h</sup>*(*i*, *<sup>j</sup>*) = *<sup>λ</sup><sup>i</sup>* <sup>−</sup> *<sup>j</sup>* <sup>+</sup> *<sup>λ</sup>*<sup>ˇ</sup> *<sup>j</sup>* <sup>−</sup> *<sup>i</sup>* <sup>+</sup> 1. Shaded boxes are invariant under the action of <sup>Γ</sup> <sup>=</sup> **<sup>Z</sup>**3.
Here the product is taken only for the Γ-invariant sector as shown in Fig. 3,
$$h(i,j) = a(i,j) + l(i,j) + 1 \equiv 0 \pmod{r} \tag{17}$$
This restriction is considered in order to study the four dimensional supersymmetric gauge theory on orbifold **R**4/**Z***<sup>r</sup>* ∼= **C**2/**Z***<sup>r</sup>* [11, 20, 27], thus we call this *orbifold partition function*. This also corresponds to a certain symmetric polynomial [57] (see also [32]), which is related to the Calogero-Sutherland model involving spin degrees of freedom. We can further generalize this model (16) to the *β*- or the *q*-deformed **Z***r*-orbifold model, and the generic toric orbifold model [28].
Let us comment on a relation between the orbifold partition function and the *q*-deformed model. Taking the limit of *q* → 1, the latter is reduced to the U(1) model because the *q*-integer is just replaced by the usual integer in such a limit,
$$[\mathfrak{x}]\_q \equiv \frac{1 - q^{-\mathfrak{x}}}{1 - q^{-1}} \longrightarrow \mathfrak{x} \tag{18}$$
This can be easily shown by l'Hopital's rule and so on. On the other hand, parametrizing *q* → *ωrq* with *ω<sup>r</sup>* = exp(2*πi*/*r*) being the primitive *r*-th root of unity, we have
$$\frac{1 - (\omega\_r q)^{-\chi}}{1 - (\omega\_r q)^{-1}} \xrightarrow{q \to 1} \begin{cases} \text{x } (\mathfrak{x} \equiv 0, \text{ mod } r) \\ 1 \ (\mathfrak{x} \not\equiv 0, \text{ mod } r) \end{cases} \tag{19}$$
Therefore the orbifold partition function (16) is derived from the *q*-deformed one (15) by taking this root of unity limit. This prescription is useful to study its asymptotic behavior.
#### **2.2. Gauge theory partition function**
The path integral in quantum field theory involves some kinds of divergence, which are due to infinite degrees of freedom in the theory. On the other hand, we can exactly perform the path integral for several highly supersymmetric theories. We now show that the gauge theory partition function can be described in a combinatorial way, and yields some extended versions of the model we have introduced in section 2.1.
The main part of the gauge theory path integral is just evaluation of the moduli space volume for a topological excitation, for example, a vortex in two dimensional theory and an instanton in four dimensional theory. Here we concentrate on the four dimentional case. See [13, 21, 50] for the two dimensional vortex partition function. The most usuful method to deal with the instanton is ADHM construction [2]. According to this, the instanton moduli space for *k*-instanton in SU(*n*) gauge theory on **R**4, is written as a kind of hyper-Kähler quotient,
6 Linear Algebra
**Figure 3.** Γ-invariant sector for U(1) theory with *λ* = (8, 5, 5, 4, 2, 2, 2, 1). Numbers in boxes stand for their hook lengths *<sup>h</sup>*(*i*, *<sup>j</sup>*) = *<sup>λ</sup><sup>i</sup>* <sup>−</sup> *<sup>j</sup>* <sup>+</sup> *<sup>λ</sup>*<sup>ˇ</sup> *<sup>j</sup>* <sup>−</sup> *<sup>i</sup>* <sup>+</sup> 1. Shaded boxes are invariant under the action of <sup>Γ</sup> <sup>=</sup> **<sup>Z</sup>**3.
This restriction is considered in order to study the four dimensional supersymmetric gauge theory on orbifold **R**4/**Z***<sup>r</sup>* ∼= **C**2/**Z***<sup>r</sup>* [11, 20, 27], thus we call this *orbifold partition function*. This also corresponds to a certain symmetric polynomial [57] (see also [32]), which is related to the Calogero-Sutherland model involving spin degrees of freedom. We can further generalize this model (16) to the *β*- or the *q*-deformed **Z***r*-orbifold model, and the generic toric orbifold model
Let us comment on a relation between the orbifold partition function and the *q*-deformed model. Taking the limit of *q* → 1, the latter is reduced to the U(1) model because the *q*-integer
This can be easily shown by l'Hopital's rule and so on. On the other hand, parametrizing
Therefore the orbifold partition function (16) is derived from the *q*-deformed one (15) by taking this root of unity limit. This prescription is useful to study its asymptotic behavior.
The path integral in quantum field theory involves some kinds of divergence, which are due to infinite degrees of freedom in the theory. On the other hand, we can exactly perform the path integral for several highly supersymmetric theories. We now show that the gauge theory partition function can be described in a combinatorial way, and yields some extended versions
*<sup>x</sup>* (*<sup>x</sup>* <sup>≡</sup> 0, mod *<sup>r</sup>*)
[*x*]*<sup>q</sup>* <sup>≡</sup> <sup>1</sup> <sup>−</sup> *<sup>q</sup>*−*<sup>x</sup>*
*q*→1 −→
*q* → *ωrq* with *ω<sup>r</sup>* = exp(2*πi*/*r*) being the primitive *r*-th root of unity, we have
<sup>1</sup> <sup>−</sup> (*ωrq*)−*<sup>x</sup>* <sup>1</sup> − (*ωrq*)−<sup>1</sup>
*h*(*i*, *j*) = *a*(*i*, *j*) + *l*(*i*, *j*) + 1 ≡ 0 (mod *r*) (17)
<sup>1</sup> <sup>−</sup> *<sup>q</sup>*−<sup>1</sup> −→ *<sup>x</sup>* (18)
<sup>1</sup> (*<sup>x</sup>* �≡ 0, mod *<sup>r</sup>*) (19)
-
-
Here the product is taken only for the Γ-invariant sector as shown in Fig. 3,
is just replaced by the usual integer in such a limit,
**2.2. Gauge theory partition function**
of the model we have introduced in section 2.1.
[28].
$$\mathcal{M}\_{\mathfrak{n},k} = \left\{ (B\_1, B\_2, I\_\prime J) \,|\,\mu\_\mathbb{R} = 0, \mu\_\mathbb{C} = 0 \right\} / \,\mathsf{U}(k) \tag{20}$$
$$B\_{1,2} \in \text{Hom}(\mathbb{C}^k, \mathbb{C}^k), \quad I \in \text{Hom}(\mathbb{C}^n, \mathbb{C}^k), \quad I \in \text{Hom}(\mathbb{C}^k, \mathbb{C}^n) \tag{21}$$
$$
\mu\_{\mathbb{R}} = [B\_1 \, B\_1^\dagger] + [B\_2 \, B\_2^\dagger] + II^\dagger - I^\dagger I,\tag{22}
$$
$$
\mu\_{\mathbb{C}} = [B\_1, B\_2] + II \tag{23}
$$
The *k* × *k* matrix condition *μ***<sup>R</sup>** = *μ***<sup>C</sup>** = 0, and parameters (*B*1, *B*2, *I*, *J*) satisfying this condition are called ADHM equation and ADHM data. Note that they are identified under the following U(*k*) transformation,
$$(\mathbf{g}\_1, \mathbf{g}\_2, \mathbf{l}\_\prime \mathbf{l}\_\prime) \sim (\mathbf{g} \mathbf{g}\_1 \mathbf{g}^{-1}, \mathbf{g} \mathbf{g}\_2 \mathbf{g}^{-1}, \mathbf{g} \mathbf{l}\_\prime \mathbf{l}\_\prime \mathbf{g}^{-1}), \qquad \mathbf{g} \in \mathbf{U}(k) \tag{24}$$
Thus all we have to do is to estimate the volume of this parameter space. However it is well known that there are some singularities in this moduli space, so that one has to regularize it in order to obtain a meaningful result. Its regularized volume had been derived by applying the localization formula to the moduli space integral [41], and it was then shown that the partition function correctly reproduces Seiberg-Witten theory [43].
We then consider the action of isometries on **C**<sup>2</sup> ∼= **R**<sup>4</sup> for the ADHM data. If we assign (*z*1, *<sup>z</sup>*2) <sup>→</sup> (*ei�*<sup>1</sup> *<sup>z</sup>*1,*ei�*<sup>2</sup> *<sup>z</sup>*2) for the spatial coordinate of **<sup>C</sup>**2, and U(1)*n*−<sup>1</sup> rotation coming from the gauge symmetry SU(*n*), ADHM data transform as
$$(B\_1, B\_2, I, I) \quad \longrightarrow \quad \left(T\_1 B\_{1\prime} T\_2 B\_{2\prime} I T\_a^{-1}, T\_1 T\_2 T\_a I\right) \tag{25}$$
where we define the torus actions as *Ta* <sup>=</sup> diag(*eia*<sup>1</sup> , ··· ,*eian* ) <sup>∈</sup> <sup>U</sup>(1)*n*−1, *<sup>T</sup><sup>α</sup>* <sup>=</sup> *<sup>e</sup>i�α* <sup>∈</sup> <sup>U</sup>(1)2. Note that these toric actions are based on the maximal torus of the gauge theory symmetry, <sup>U</sup>(1)<sup>2</sup> <sup>×</sup> <sup>U</sup>(1)*n*−<sup>1</sup> <sup>⊂</sup> SO(4) <sup>×</sup> SU(*n*). We have to consider the fixed point of these isometries up to gauge transformation *g* ∈ U(*k*) to perform the localization formula.
The localization formula in the instanton moduli space is based on the vector field *ξ*∗, which is associated with *<sup>ξ</sup>* <sup>∈</sup> <sup>U</sup>(1)<sup>2</sup> <sup>×</sup> <sup>U</sup>(1)*n*−1. It generates the one-parameter flow *<sup>e</sup>t<sup>ξ</sup>* on the moduli space M, corresponding to the isometries. The vector field is represented by the element of the maximal torus of the gauge theory symmetry under the Ω-background deformation. The gauge theory action is invariant under the deformed BRST transformation, whose generator satisfies *ξ*<sup>∗</sup> = {*Q*∗, *Q*∗}/2. Thus this generator can be interpreted as the equivariant derivative *d<sup>ξ</sup>* = *d* + *iξ*<sup>∗</sup> where *iξ*<sup>∗</sup> stands for the contraction with the vector field *ξ*∗. The localization formula is given by
$$\int\_{\mathcal{M}} \mathfrak{a}(\boldsymbol{\xi}) = (-2\pi)^{n/2} \sum\_{\mathbf{x}\_0} \frac{\mathfrak{a}\_0(\boldsymbol{\xi})(\mathbf{x}\_0)}{\mathbf{det}^{1/2} \mathcal{L}\_{\mathbf{x}\_0}} \tag{26}$$
#### 8 Linear Algebra 82 Linear Algebra – Theorems and Applications
where *α*(*ξ*) is an equivariant form, which is related to the gauge theory action. *α*0(*ξ*) is zero degree part and L*x*<sup>0</sup> : *Tx*0M → *Tx*0M is the map generated by the vector field *ξ*<sup>∗</sup> at the fixed points *x*0. These fixed points are defined as *ξ*∗(*x*0) = 0 up to U(*k*) transformation of the instanton moduli space.
Let us then study the fixed point in the moduli space. The fixed point condition for them are obtained from the infinitesimal version of (24) and (25) as
$$(\phi\_{\dot{l}} - \phi\_{\dot{l}} + \varepsilon\_a)B\_{\text{a,ij}} = 0, \qquad (\phi\_{\dot{l}} - a\_{\dot{l}})I\_{\text{il}} = 0, \qquad (-\phi\_{\dot{l}} + a\_{\dot{l}} + \varepsilon)I\_{\text{li}} = 0 \tag{27}$$
where the element of U(*k*) gauge transformation is diagonalized as *<sup>e</sup>i<sup>φ</sup>* <sup>=</sup> diag(*eiφ*<sup>1</sup> , ··· ,*eiφ<sup>k</sup>* ) <sup>∈</sup> U(*k*) with *�* = *�*<sup>1</sup> + *�*2. We can show that an eigenvalue of *φ* turns out to be
$$a\_l + (j-1)\varepsilon\_1 + (i-1)\varepsilon\_2 \tag{28}$$
and the corresponding eigenvector is given by
$$B\_1^{j-1} B\_2^{i-1} I\_l \tag{29}$$
Since *φ* is a finite dimensional matrix, we can obtain *kl* independent vectors from (29) with *k*<sup>1</sup> + ··· + *kn* = *k*. This means that the solution of this condition can be characterized by *n*-tuple Young diagrams, or partitions *<sup>λ</sup>* = (*λ*(1), ··· , *<sup>λ</sup>*(*n*)) [42]. Thus the characters of the vector spaces are yielding
$$V = \sum\_{l=1}^{n} \sum\_{\substack{(i,j) \in \lambda^{(l)}}} T\_{a\_l} T\_1^{-j+1} T\_2^{-i+1} \, \prime \qquad W = \sum\_{l=1}^{n} T\_{a\_l} \tag{30}$$
and that of the tangent space at the fixed point under the isometries can be represented in terms of the *n*-tuple partition as
$$\begin{split} \chi\_{\vec{\lambda}} &= -V^\* V (1 - T\_1) (1 - T\_2) + W^\* V + V^\* W T\_1 T\_2 \\ &= \sum\_{l,m}^n \sum\_{(i,j) \in \lambda^{(l)}} \left( T\_{a\_{ml}} T\_1^{-\lambda\_j^{(l)} + i} T\_2^{\lambda\_i^{(m)} - j + 1} + T\_{a\_{lm}} T\_1^{\lambda\_j^{(l)} - i + 1} T\_2^{-\lambda\_i^{(m)} + j} \right) \end{split} \tag{31}$$
Here *λ*ˇ is a conjugated partition. Therefore the instanton partition function is obtained by reading the weight function from the character [43, 44],
$$\mathbf{Z\_{SU(n)}} = \sum\_{\vec{\lambda}} \Lambda^{2\mathbf{n}[\vec{\lambda}]} Z\_{\vec{\lambda}} \tag{32}$$
$$Z\_{\vec{\lambda}} = \prod\_{i=1}^{n} \prod\_{\substack{\vec{\lambda} \ \vec{\lambda} \ \vec{\lambda} \ \vdots \ \vec{\lambda} \ \vec{\lambda} \ \vdots \ \vec{\lambda} \ \vdots \ \vec{\lambda} \ \vdots \ \vec{\lambda} \ \vdots \ \vec{\lambda} \ \vdots \ \vec{\lambda} \end{pmatrix}} \frac{1}{1 + \prod\_{\substack{\vec{\lambda} \ \vec{\lambda} \ \vec{\lambda} \ \vdots \ \vec{\lambda} \ \vdots \ \vec{\lambda} \ \vdots \ \vec{\lambda} \ \vdots \ \vdots \ \vdots \ \vdots}} $$
$$Z\_{\vec{\lambda}} = \prod\_{l,m} \prod\_{\substack{(i,j) \in \lambda^{(l)} \ a\_{ml}}} \overline{a\_{ml} + \varepsilon\_2(\lambda\_i^{(m)} - j + 1) - \varepsilon\_1(\vec{\lambda}\_j^{(l)} - i)} \overline{a\_{lm} - \varepsilon\_2(\lambda\_i^{(m)} - j) + \varepsilon\_1(\vec{\lambda}\_j^{(l)} - i + 1)} \tag{33}$$
This is regarded as a generalized model of (7) or (13). Furthermore by lifting it to the five dimensional theory on **<sup>R</sup>**<sup>4</sup> <sup>×</sup> *<sup>S</sup>*1, one can obtain a generalized version of the *<sup>q</sup>*-deformed partition function (15). Actually it is easy to see these SU(*n*) models are reduced to the U(1) models in the case of *n* = 1. Note, if we take into account other matter contributions in addition to the vector multiplet, this partition function involves the associated combinatorial factors. We can extract various properties of the gauge theory from these partition functions, especially its asymptotic behavior.
#### **3. Matrix model description**
In this section we discuss the matrix model description of the combinatorial partition function. The matrix integral representation can be treated in a standard manner, which is developed in the random matrix theory [40].
#### **3.1. Matrix integral**
8 Linear Algebra
where *α*(*ξ*) is an equivariant form, which is related to the gauge theory action. *α*0(*ξ*) is zero degree part and L*x*<sup>0</sup> : *Tx*0M → *Tx*0M is the map generated by the vector field *ξ*<sup>∗</sup> at the fixed points *x*0. These fixed points are defined as *ξ*∗(*x*0) = 0 up to U(*k*) transformation of the
Let us then study the fixed point in the moduli space. The fixed point condition for them are
where the element of U(*k*) gauge transformation is diagonalized as *<sup>e</sup>i<sup>φ</sup>* <sup>=</sup> diag(*eiφ*<sup>1</sup> , ··· ,*eiφ<sup>k</sup>* ) <sup>∈</sup>
*Bj*−<sup>1</sup> <sup>1</sup> *<sup>B</sup>i*−<sup>1</sup>
*Tal <sup>T</sup>*−*j*+<sup>1</sup>
*<sup>λ</sup>* = −*V*∗*V*(1 − *T*1)(1 − *T*2) + *W*∗*V* + *V*∗*WT*1*T*<sup>2</sup>
<sup>−</sup>*λ*<sup>ˇ</sup> (*l*) *<sup>j</sup>* +*i* <sup>1</sup> *<sup>T</sup>λ*(*m*)
1
*<sup>i</sup>* <sup>−</sup> *<sup>j</sup>* <sup>+</sup> <sup>1</sup>) <sup>−</sup> *�*1(*λ*<sup>ˇ</sup> (*l*)
Since *φ* is a finite dimensional matrix, we can obtain *kl* independent vectors from (29) with *k*<sup>1</sup> + ··· + *kn* = *k*. This means that the solution of this condition can be characterized by
<sup>1</sup> *<sup>T</sup>*−*i*+<sup>1</sup>
and that of the tangent space at the fixed point under the isometries can be represented in
Here *λ*ˇ is a conjugated partition. Therefore the instanton partition function is obtained by
This is regarded as a generalized model of (7) or (13). Furthermore by lifting it to the five dimensional theory on **<sup>R</sup>**<sup>4</sup> <sup>×</sup> *<sup>S</sup>*1, one can obtain a generalized version of the *<sup>q</sup>*-deformed partition function (15). Actually it is easy to see these SU(*n*) models are reduced to the U(1) models in the case of *n* = 1. Note, if we take into account other matter contributions in
*<sup>i</sup>* −*j*+1 <sup>2</sup> + *Talm T*
<sup>2</sup> , *W* =
*<sup>λ</sup>* (32)
*<sup>j</sup>* − *i*)
U(*k*) with *�* = *�*<sup>1</sup> + *�*2. We can show that an eigenvalue of *φ* turns out to be
(*φ<sup>i</sup>* − *φ<sup>j</sup>* + *�α*)*Bα*,*ij* = 0, (*φ<sup>i</sup>* − *al*)*Iil* = 0, (−*φ<sup>i</sup>* + *al* + *�*)*Jli* = 0 (27)
*al* + (*j* − 1)*�*<sup>1</sup> + (*i* − 1)*�*<sup>2</sup> (28)
<sup>2</sup> *Il* (29)
*Tal* (30)
*<sup>i</sup>* +*j*
1
*<sup>i</sup>* <sup>−</sup> *<sup>j</sup>*) + *�*1(*λ*<sup>ˇ</sup> (*l*)
(31)
*<sup>j</sup>* − *i* + 1)
(33)
2
*<sup>λ</sup>* = (*λ*(1), ··· , *<sup>λ</sup>*(*n*)) [42]. Thus the characters of the
*n* ∑ *l*=1
*λ*ˇ (*l*) *<sup>j</sup>* −*i*+1 <sup>1</sup> *<sup>T</sup>*−*λ*(*m*)
*alm* <sup>−</sup> *�*2(*λ*(*m*)
instanton moduli space.
obtained from the infinitesimal version of (24) and (25) as
and the corresponding eigenvector is given by
*n*-tuple Young diagrams, or partitions
*V* =
∑ (*i*,*j*)∈*λ*(*l*)
reading the weight function from the character [43, 44],
*aml* <sup>+</sup> *�*2(*λ*(*m*)
*Taml T*
*n* ∑ *l*=1
∑ (*i*,*j*)∈*λ*(*l*)
vector spaces are yielding
terms of the *n*-tuple partition as
*χ*
Λ2*n*<sup>|</sup> *λ*| *Z*
∏ (*i*,*j*)∈*λ*(*l*)
*Z*SU(*n*) = ∑
*Z <sup>λ</sup>* = *λ*
*n* ∏ *l*,*m*
= *n* ∑ *l*,*m* Let us consider the following *N* × *N* matrix integral,
$$Z\_{\text{matrix}} = \int \mathcal{D}X \, e^{-\frac{1}{\hbar} \text{Tr} \, V(X)} \tag{34}$$
Here *X* is an hermitian matrix, and D*X* is the associated matrix measure. This matrix can be diagonalized by a unitary transformation, *gXg*−<sup>1</sup> <sup>=</sup> diag(*x*1, ··· , *xN*) with *<sup>g</sup>* <sup>∈</sup> <sup>U</sup>(*N*), and the integrand is invariant under this transformation, Tr *V*(*X*) = Tr *V*(*gXg*−1) = ∑*<sup>N</sup> <sup>i</sup>*=<sup>1</sup> *V*(*xi*). On the other hand, we have to take care of the matrix measure in (34): the non-trivial Jacobian is arising from the matrix diagonalization (see, e.g. [40]),
$$\mathcal{D}X = \mathcal{D}x \mathcal{D}U \Delta(x)^2 \tag{35}$$
The Jacobian part is called *Vandermonde determinant*, which is written as
$$\Delta(\mathbf{x}) = \prod\_{i$$
and D*U* is the Haar measure, which is invariant under unitary transformation, D(*gU*) = <sup>D</sup>*U*. The diagonal part is simply given by <sup>D</sup>*<sup>x</sup>* <sup>≡</sup> <sup>∏</sup>*<sup>N</sup> <sup>i</sup>*=<sup>1</sup> *dxi*. Therefore, by integrating out the off-diagonal part, the matrix integral (34) is reduced to the integral over the matrix eigenvalues,
$$Z\_{\text{matrix}} = \int \mathcal{D}\mathbf{x} \,\Delta(\mathbf{x})^2 \, e^{-\frac{1}{\hbar} \sum\_{i=1}^{N} V(\mathbf{x}\_i)} \tag{37}$$
This expression is up to a constant factor, associated with the volume of the unitary group, vol(U(*N*)), coming from the off-diagonal integral.
When we consider a real symmetric or a quaternionic self-dual matrix, it can be diagonalized by orthogonal/symplectic transformation. In these cases, the Jacobian part is slightly modified,
$$Z\_{\text{matrix}} = \int \mathcal{D}\mathbf{x} \,\Delta(\mathbf{x})^{2\mathcal{G}} \, e^{-\frac{1}{\hbar} \sum\_{i=1}^{N} V(\mathbf{x}\_{i})} \tag{38}$$
The power of the Vandermonde determinant is given by *β* = <sup>1</sup> <sup>2</sup> , 1, 2 for symmetric, hermitian and self-dual, respcecively.1 They correspond to orthogonal, unitary, symplectic ensembles in random matrix theory, and the model with a generic *β* ∈ **R** is called *β-ensemble matrix model*.
<sup>1</sup> This notation is different from the standard one: 2*<sup>β</sup>* <sup>→</sup> *<sup>β</sup>* <sup>=</sup> 1, 2, 4 for symmetric, hermitian and self-dual matrices.
**Figure 4.** Shape of Young diagram can be represented by introducing one-dimensional exclusive particles. Positions of particles would be interpreted as eigenvalues of the matrix.
#### **3.2.** U(1) **partition function**
We would like to show an essential connection between the combinatorial partition function and the matrix model. By considering the thermodynamical limit of the partition function, it can be represented as a matrix integral discussed above.
Let us start with the most fundamental partition function (7). The main part of its partition function is the product all over the boxes in the partition *λ*. After some calculations, we can show this combinatorial factor is rewritten as
$$\prod\_{(i,j)\in\lambda} \frac{1}{h(i,j)} = \prod\_{i$$
where *N* is an arbitrary integer satisfying *N* > �(*λ*). This can be also represented in an infinite product form,
$$\prod\_{(i,j)\in\lambda} \frac{1}{h(i,j)} = \prod\_{i$$
These expressions correspond to an embedding of the finite dimensional symmetric group S*<sup>N</sup>* into the infinite dimensional one S∞.
By introducing a new set of variables *ξ<sup>i</sup>* = *λ<sup>i</sup>* + *N* − *i* + 1, we have another representation of the partition function,
$$Z\_{\mathsf{U}(1)} = \sum\_{\lambda} \left(\frac{\Lambda}{\hbar}\right)^{2\sum\_{i=1}^{N} \xi\_i - N(N+1)} \prod\_{i$$
These new variables satisfy *ξ<sup>i</sup>* > *ξ*<sup>2</sup> > ··· > *ξ*�(*λ*) while the original ones satisfy *λ*<sup>1</sup> ≥ *λ*<sup>2</sup> ≥ ··· ≥ *λ*�(*λ*). This means {*ξi*} and {*λi*} are interpreted as fermionic and bosonic degrees of freedom. Fig. 4 shows the correspondence between the bosinic and fermionic variables. The bosonic excitation is regarded as density fluctuation of the fermionic particles around the Fermi energy. This is just the bosonization method, which is often used to study quantum one-dimensional systems (For example, see [24]). Especially we concentrate only on either of the Fermi points. Thus it yields the chiral conformal field theory.
We would like to show that the matrix integral form is obtained from the expression (41). First we rewrite the summation over partitions as
$$\sum\_{\lambda} = \sum\_{\lambda\_1 \ge \cdots \ge \lambda\_N} = \sum\_{\substack{\mathfrak{F}\_1 > \cdots > \mathfrak{F}\_N \\ \mathfrak{F}\_1 < \cdots < \mathfrak{F}\_N}} = \frac{1}{N!} \sum\_{\substack{\mathfrak{F}\_1 \cdots \mathfrak{F}\_N \\ \mathfrak{F}\_1 < \cdots < \mathfrak{F}\_N}} \tag{42}$$
Then, introducing another variable defined as *xi* = *h*¯ *ξi*, it can be regarded as a continuous variable in the large *N* limit,
$$N \longrightarrow \infty, \qquad \hbar \longrightarrow 0, \qquad \hbar N = \mathcal{O}(1) \tag{43}$$
This is called 't Hooft limit. The measure for this variable is given by
10 Linear Algebra
**Figure 4.** Shape of Young diagram can be represented by introducing one-dimensional exclusive
We would like to show an essential connection between the combinatorial partition function and the matrix model. By considering the thermodynamical limit of the partition function, it
Let us start with the most fundamental partition function (7). The main part of its partition function is the product all over the boxes in the partition *λ*. After some calculations, we can
> *N* ∏ *i*=1
*λ<sup>i</sup>* − *λ<sup>j</sup>* + *j* − *i*
1
<sup>Γ</sup>(*λ<sup>i</sup>* <sup>+</sup> *<sup>N</sup>* <sup>−</sup> *<sup>i</sup>* <sup>+</sup> <sup>1</sup>) (39)
*<sup>j</sup>* <sup>−</sup> *<sup>i</sup>* (40)
(*λ<sup>i</sup>* − *λ<sup>j</sup>* + *j* − *i*)
where *N* is an arbitrary integer satisfying *N* > �(*λ*). This can be also represented in an infinite
∞ ∏ *i*<*j*
These expressions correspond to an embedding of the finite dimensional symmetric group
By introducing a new set of variables *ξ<sup>i</sup>* = *λ<sup>i</sup>* + *N* − *i* + 1, we have another representation of
*<sup>i</sup>*=<sup>1</sup> *ξi*−*N*(*N*+1) *N*
These new variables satisfy *ξ<sup>i</sup>* > *ξ*<sup>2</sup> > ··· > *ξ*�(*λ*) while the original ones satisfy *λ*<sup>1</sup> ≥ *λ*<sup>2</sup> ≥ ··· ≥ *λ*�(*λ*). This means {*ξi*} and {*λi*} are interpreted as fermionic and bosonic degrees of freedom. Fig. 4 shows the correspondence between the bosinic and fermionic variables. The bosonic excitation is regarded as density fluctuation of the fermionic particles around the Fermi energy. This is just the bosonization method, which is often used to study quantum one-dimensional systems (For example, see [24]). Especially we concentrate only on either of
∏ *i*<*j*
(*ξ<sup>i</sup>* <sup>−</sup> *<sup>ξ</sup>j*)<sup>2</sup>
*N* ∏ *i*=1
1
<sup>Γ</sup>(*ξi*)<sup>2</sup> (41)
particles. Positions of particles would be interpreted as eigenvalues of the matrix.
*N* ∏ *i*<*j*
∏ (*i*,*j*)∈*λ*
1 *<sup>h</sup>*(*i*, *<sup>j</sup>*) <sup>=</sup>
can be represented as a matrix integral discussed above.
1 *<sup>h</sup>*(*i*, *<sup>j</sup>*) <sup>=</sup>
show this combinatorial factor is rewritten as
∏ (*i*,*j*)∈*λ*
S*<sup>N</sup>* into the infinite dimensional one S∞.
*Z*U(1) = ∑
*λ*
Λ *h*¯
the Fermi points. Thus it yields the chiral conformal field theory.
<sup>2</sup> <sup>∑</sup>*<sup>N</sup>*
**3.2.** U(1) **partition function**
product form,
the partition function,
$$d\mathbf{x}\_i \approx \hbar \sim \frac{1}{N} \tag{44}$$
Therefore the partition function (41) is rewritten as the following matrix integral,
$$Z\_{\mathbb{U}(1)} \approx \int \mathcal{D}\mathbf{x} \,\Delta(\mathbf{x})^2 \, e^{-\frac{1}{\hbar} \sum\_{l=1}^{N} V(\mathbf{x}\_l)} \tag{45}$$
Here the matrix potential is derived from the asymptotic behavior of the Γ-function,
$$
\hbar \log \Gamma(\mathbf{x}/\hbar) \longrightarrow \mathbf{x} \log \mathbf{x} - \mathbf{x}, \qquad \hbar \longrightarrow \mathbf{0} \tag{46}
$$
Since this variable can take a negative value, the potential term should be simply extended to the region of *x* < 0. Thus, taking into account the fugacity parameter Λ, the matrix potential is given by
$$V(\mathbf{x}) = 2\left[\mathbf{x}\log\left|\frac{\mathbf{x}}{\Lambda}\right| - \mathbf{x}\right] \tag{47}$$
This is the simplest version of the **CP**<sup>1</sup> matrix model [18]. If we start with the partition function including the higher Casimir operators (8), the associated integral expression just yields the **CP**<sup>1</sup> matrix model.
Let us comment on other possibilities to obtain the matrix model. It is shown that the matrix integral form can be derived without taking the large *N* limit [19]. Anyway one can see that it is reduced to the model we discussed above in the large *N* limit. There is another kind of the matrix model derived from the combinatorial partition function by *poissonizing* the probability measure. In this case, only the linear potential is arising in the matrix potential term. Such a matrix model is called Bessel-type matrix model, where its short range fluctuation is described by the Bessel kernel.
Next we shall derive the matrix model corresponding to the *β*-deformed U(1) model (13). The combinatorial part of the partition function is similarly given by
$$\prod\_{\{\boldsymbol{i},\boldsymbol{j}\}\in\lambda} \frac{1}{h\_{\boldsymbol{\beta}}(\boldsymbol{i},\boldsymbol{j})h^{\boldsymbol{\beta}}(\boldsymbol{i},\boldsymbol{j})} = \Gamma(\boldsymbol{\beta})^{N} \prod\_{i
$$\times \prod\_{i=1}^{N} \frac{1}{\Gamma(\lambda\_{i}+\beta(N-i)+\beta)} \frac{1}{\Gamma(\lambda\_{i}+\beta(N-i)+1)}\tag{48}$$
$$
#### 12 Linear Algebra 86 Linear Algebra – Theorems and Applications
In this case we shall introduce the following variables, *ξ* (*β*) *<sup>i</sup>* = *λ<sup>i</sup>* + *β*(*N* − *i*) + 1 or *ξ* (*β*) *<sup>i</sup>* = *λ<sup>i</sup>* + *β*(*N* − *i*) + *β*, satisfying *ξ* (*β*) *<sup>i</sup>* − *ξ* (*β*) *<sup>i</sup>*+<sup>1</sup> ≥ *β*. This means the parameter *β* characterizes how they are exclusive. They satisfy the generalized fractional exclusive statistics for *β* � 1 [25] (see also [32]). They are reduced to fermions and bosons for *β* = 1 and *β* = 0, respectively. Then, rescaling the variables, *xi* = *h*¯ *ξ* (*β*) *<sup>i</sup>* , the combinatorial part (48) in the 't Hooft limit yields
$$\prod\_{(i,j)\in\lambda} \frac{1}{h\_{\beta}(i,j)h^{\beta}(i,j)} \longrightarrow \Delta(\mathbf{x})^{2\beta} \ e^{-\frac{1}{\hbar} \sum\_{i=1}^{N} V(\mathbf{x}\_i)}\tag{49}$$
Here we use <sup>Γ</sup>(*<sup>α</sup>* <sup>+</sup> *<sup>β</sup>*)/Γ(*α*) <sup>∼</sup> *<sup>α</sup><sup>β</sup>* with *<sup>α</sup>* <sup>→</sup> <sup>∞</sup>. The matrix potential obtained here is the same as (47). Therefore the matrix model associated with the *β*-deformed partition function is given by
$$Z\_{\mathbf{U}(1)}^{(\beta)} \approx \int \mathcal{D}\mathbf{x} \,\Delta(\mathbf{x})^{2\beta} \, e^{-\frac{1}{\hbar} \sum\_{l=1}^{N} V(\mathbf{x}\_l)} \tag{50}$$
This is just the *β*-ensemble matrix model shown in (38).
We can consider the matrix model description of the (*q*, *t*)-deformed partition function. In this case the combinatorial part of (15) is written as
$$\prod\_{(i,j)\in\lambda} \frac{1-q}{1-q^{a(i,j)+1}t^{l(i,j)}} = (1-q)^{|\lambda|} \prod\_{i
$$\prod\_{i=1}^{N} \frac{1-q^{-1}}{1-q^{-1}|\lambda|} \prod\_{j=1}^{N} (q^{-\lambda\_i+\lambda\_j+1}t^{-j+i-1};q)\_{\infty} \prod\_{i=1}^{N} \qquad (qt^{-1};q)\_{\infty}$$
$$
$$\prod\_{(i,j)\in\lambda} \frac{1-q^{-1}}{1-q^{-a(i,j)}t^{-l(i,j)-1}} = (1-q^{-1})^{|\lambda|} \prod\_{i$$
Here (*x*; *<sup>q</sup>*)*<sup>n</sup>* <sup>=</sup> <sup>∏</sup>*n*−<sup>1</sup> *<sup>m</sup>*=0(<sup>1</sup> <sup>−</sup> *xqm*) is the *<sup>q</sup>*-Pochhammer symbol. When we parametrize *<sup>q</sup>* <sup>=</sup> *<sup>e</sup>*−*hR*¯ and *<sup>t</sup>* <sup>=</sup> *<sup>q</sup>β*, a set of the variables {*<sup>ξ</sup>* (*β*) *<sup>i</sup>* } plays an important role in considering the large *N* limit as well as the *β*-deformed model. Thus, rescaling these as *xi* = *h*¯ *ξ* (*β*) *<sup>i</sup>* and taking the 't Hooft limit, we obtain the integral expression of the *q*-deformed partition function,
$$Z\_{\mathbf{U}(1)}^{(q,t)} \approx \int \mathcal{D}\mathbf{x} \ (\Delta\_{\mathbf{R}}(\mathbf{x}))^{2\beta} \ e^{-\frac{1}{\hbar} \sum\_{i=1}^{N} V\_{\mathbf{R}}(\mathbf{x}\_i)} \tag{53}$$
The matrix measure and potential are given by
$$\Delta\_{\mathbb{R}}(\mathbf{x}) = \prod\_{i$$
$$V\_R(\mathbf{x}) = -\frac{1}{R} \left[ \text{Li}\_2 \left( e^{R\mathbf{x}} \right) - \text{Li}\_2 \left( e^{-R\mathbf{x}} \right) \right] \tag{55}$$
We will discuss how to obtain these expressions below. We can see they are reduced to the standard ones in the limit of *R* → 0,
$$
\Delta\_R(\mathbf{x}) \longrightarrow \Delta(\mathbf{x}), \qquad V\_R(\mathbf{x}) \longrightarrow V(\mathbf{x}) \tag{56}
$$
Note that this hyperbolic-type matrix measure is also investigated in the Chern-Simons matrix model [35], which is extensively involved with the recent progress on the three dimensional supersymmetric gauge theory via the localization method [36].
Let us comment on useful formulas to derive the integral expression (53). The measure part is relevant to the asymptotic form of the following function,
$$\frac{(\mathbf{x};q)\_{\infty}}{(\mathbf{t}\mathbf{x};q)\_{\infty}} \longrightarrow \frac{(\mathbf{x};q)\_{\infty}}{(\mathbf{t}\mathbf{x};q)\_{\infty}}\bigg|\_{q\to 1} = (1-\mathbf{x})^{\beta}, \qquad \mathbf{x} \longrightarrow \infty \tag{57}$$
This essentially corresponds to the *<sup>q</sup>* <sup>→</sup> 1 limit of the *<sup>q</sup>*-Vandermonde determinant2,
$$\Delta\_{q,t}^2(\mathbf{x}) = \prod\_{i \neq j}^N \frac{(\mathbf{x}\_i/\mathbf{x}\_j; q)\_{\infty}}{(t\mathbf{x}\_i/\mathbf{x}\_j; q)\_{\infty}} \tag{58}$$
Then, to investigate the matrix potential term, we now introduce the quantum dilogarithm function,
$$g(\mathbf{x};q) = \prod\_{n=1}^{\infty} \left(1 - \frac{1}{\mathbf{x}} q^n\right) \tag{59}$$
Its asymptotic expansion is given by (see, e.g. [19])
$$\log g(\mathbf{x}; q = e^{-\hbar R}) = -\frac{1}{\hbar R} \sum\_{m=0}^{\infty} \text{Li}\_{2-m} \left( \mathbf{x}^{-1} \right) \frac{B\_{m}}{m!} (\hbar R)^{m} \tag{60}$$
where *Bm* is the *m*-th Bernouilli number, and Li*m*(*x*) = ∑<sup>∞</sup> *<sup>k</sup>*=<sup>1</sup> *<sup>x</sup>k*/*k<sup>m</sup>* is the polylogarithm function. The potential term is coming from the leading term of this expression.
#### **3.3.** SU(*n*) **partition function**
12 Linear Algebra
they are exclusive. They satisfy the generalized fractional exclusive statistics for *β* � 1 [25] (see also [32]). They are reduced to fermions and bosons for *β* = 1 and *β* = 0, respectively.
*<sup>h</sup>β*(*i*, *<sup>j</sup>*)*hβ*(*i*, *<sup>j</sup>*) −→ <sup>Δ</sup>(*x*)2*<sup>β</sup> <sup>e</sup>*
Here we use <sup>Γ</sup>(*<sup>α</sup>* <sup>+</sup> *<sup>β</sup>*)/Γ(*α*) <sup>∼</sup> *<sup>α</sup><sup>β</sup>* with *<sup>α</sup>* <sup>→</sup> <sup>∞</sup>. The matrix potential obtained here is the same as (47). Therefore the matrix model associated with the *β*-deformed partition function is given
<sup>D</sup>*<sup>x</sup>* <sup>Δ</sup>(*x*)2*<sup>β</sup> <sup>e</sup>*
We can consider the matrix model description of the (*q*, *t*)-deformed partition function. In this
Here (*x*; *<sup>q</sup>*)*<sup>n</sup>* <sup>=</sup> <sup>∏</sup>*n*−<sup>1</sup> *<sup>m</sup>*=0(<sup>1</sup> <sup>−</sup> *xqm*) is the *<sup>q</sup>*-Pochhammer symbol. When we parametrize *<sup>q</sup>* <sup>=</sup> *<sup>e</sup>*−*hR*¯
− 1 *<sup>h</sup>*¯ <sup>∑</sup>*<sup>N</sup>*
(*qλi*−*λj*+1*tj*−*i*−1; *q*)<sup>∞</sup> (*qλi*−*λj*+1*tj*−*<sup>i</sup>*
(*q*−*λi*+*λj*+1*t*
(*q*−*λi*+*λj*+1*t*−*j*+*<sup>i</sup>*
<sup>2</sup>*<sup>β</sup> e* − 1 *<sup>h</sup>*¯ <sup>∑</sup>*<sup>N</sup>*
> − Li2 *e* −*Rx*
Δ*R*(*x*) −→ Δ(*x*), *VR*(*x*) −→ *V*(*x*) (56)
; *q*)<sup>∞</sup>
*N* ∏ *i*=1
<sup>−</sup>*j*+*i*−1; *q*)<sup>∞</sup>
; *q*)<sup>∞</sup>
*<sup>i</sup>* } plays an important role in considering the large *N*
(*β*)
− 1 *<sup>h</sup>*¯ <sup>∑</sup>*<sup>N</sup>*
*<sup>i</sup>*+<sup>1</sup> ≥ *β*. This means the parameter *β* characterizes how
*<sup>i</sup>* , the combinatorial part (48) in the 't Hooft limit yields
*<sup>i</sup>* = *λ<sup>i</sup>* + *β*(*N* − *i*) + 1 or *ξ*
*<sup>i</sup>*=<sup>1</sup> *<sup>V</sup>*(*xi*) (49)
*<sup>i</sup>*=<sup>1</sup> *<sup>V</sup>*(*xi*) (50)
(*qλi*+1*tN*−*<sup>i</sup>*
*N* ∏ *i*=1
(*q*; *q*)<sup>∞</sup>
(*β*)
*<sup>i</sup>*=<sup>1</sup> *VR*(*xi*) (53)
<sup>2</sup> (*xi* <sup>−</sup> *xj*) (54)
; *q*)<sup>∞</sup>
(*qt*−1; *q*)<sup>∞</sup> (*q*−*λi*+1*t*−*N*+*i*−1; *q*)<sup>∞</sup>
*<sup>i</sup>* and taking the 't
(51)
(52)
(55)
(*β*) *<sup>i</sup>* =
In this case we shall introduce the following variables, *ξ*
∏ (*i*,*j*)∈*λ*
This is just the *β*-ensemble matrix model shown in (38).
case the combinatorial part of (15) is written as
<sup>1</sup> <sup>−</sup> *<sup>q</sup>a*(*i*,*j*)+1*tl*(*i*,*j*) = (<sup>1</sup> <sup>−</sup> *<sup>q</sup>*)|*λ*<sup>|</sup>
<sup>1</sup> <sup>−</sup> *<sup>q</sup>*−*a*(*i*,*j*)*t*−*l*(*i*,*j*)−<sup>1</sup> = (<sup>1</sup> <sup>−</sup> *<sup>q</sup>*−1)|*λ*<sup>|</sup>
The matrix measure and potential are given by
1 − *q*
<sup>1</sup> <sup>−</sup> *<sup>q</sup>*−<sup>1</sup>
and *<sup>t</sup>* <sup>=</sup> *<sup>q</sup>β*, a set of the variables {*<sup>ξ</sup>*
standard ones in the limit of *R* → 0,
(*β*) *<sup>i</sup>* − *ξ*
*Z*(*β*) U(1) ≈
(*β*)
(*β*)
1
*N* ∏ *i*<*j*
(*β*)
Hooft limit, we obtain the integral expression of the *q*-deformed partition function,
*N* ∏ *i*<*j*
*R* Li2 *eRx*
D*x* (Δ*R*(*x*))
2 *<sup>R</sup>* sinh *<sup>R</sup>*
We will discuss how to obtain these expressions below. We can see they are reduced to the
limit as well as the *β*-deformed model. Thus, rescaling these as *xi* = *h*¯ *ξ*
Δ*R*(*x*) =
*VR*(*x*) = <sup>−</sup> <sup>1</sup>
*Z*(*q*,*t*) U(1) ≈
*N* ∏ *i*<*j*
*λ<sup>i</sup>* + *β*(*N* − *i*) + *β*, satisfying *ξ*
by
∏ (*i*,*j*)∈*λ*
∏ (*i*,*j*)∈*λ*
Then, rescaling the variables, *xi* = *h*¯ *ξ*
Generalizing the result shown in section 3.2, we deal with the combinatorial partition function for SU(*n*) gauge theory (32). Its matrix model description is evolved in [30].
The combinatorial factor of the SU(*n*) partition function (33) can be represented as
$$Z\_{\vec{\lambda}} = \frac{1}{\mathfrak{e}\_2^{2\mathfrak{n}[\vec{\lambda}]}} \prod\_{(l,i)\neq(m,j)} \frac{\Gamma(\lambda\_i^{(l)} - \lambda\_j^{(m)} + \beta(j-i) + b\_{lm} + \beta)}{\Gamma(\lambda\_i^{(l)} - \lambda\_j^{(m)} + \beta(j-i) + b\_{lm})} \frac{\Gamma(\beta(j-i) + b\_{lk})}{\Gamma(\beta(j-i) + b\_{lk} + \beta)}\tag{61}$$
where we define parameters as *β* = −*�*1/*�*2, *blm* = *alm*/*�*2. This is an infinite product expression of the partition function. Anyway in this case one can see it is useful to introduce *n* kinds of fermionic variables, corresponding to the *n*-tupe partition,
$$\mathfrak{F}\_{i}^{(l)} = \lambda\_{i}^{(l)} + \beta(N - i) + 1 + b\_{l} \tag{62}$$
<sup>2</sup> This expression is up to logarithmic term, which can be regarded as the zero mode contribution of the free boson field. See [28, 29] for details.
**Figure 5.** The decomposition of the partition for **Z***r*=3. First suppose the standard correspondence between the one-dimensional particles and the original partition, and then rearrange them with respect to mod *r*.
Then, assuming *blm* � 1, let us introduce a set of variables,
$$(\mathfrak{J}\_1, \mathfrak{J}\_2, \dots, \mathfrak{J}\_{nN}) = (\mathfrak{J}\_1^{(n)}, \dots, \mathfrak{J}\_N^{(n)}, \mathfrak{J}\_1^{(n-1)}, \dots, \dots, \mathfrak{J}\_N^{(2)}, \mathfrak{J}\_1^{(1)}, \dots, \mathfrak{J}\_N^{(1)}) \tag{63}$$
satisfying *ζ*<sup>1</sup> > *ζ*<sup>2</sup> > ··· > *ζnN*. The combinatorial factor (61) is rewritten with these variables as
$$Z\_{\widetilde{\lambda}} = \frac{1}{\mathfrak{e}\_{\mathbf{2}}^{2n|\widetilde{\lambda}|}} \prod\_{i$$
From this expression we can obtain the matrix model description for SU(*n*) gauge theory partition function, by rescaling *xi* = *h*¯ *ζ<sup>i</sup>* with reparametrizing ¯*h* = *�*2,
$$Z\_{\rm SU(n)} \approx \int \mathcal{D}\mathbf{x} \,\Delta(\mathbf{x})^{2\mathcal{B}} \, e^{-\frac{1}{\hbar} \sum\_{l=1}^{nN} V\_{\rm SU(n)}(\mathbf{x}\_l)} \,\tag{65}$$
In this case the matrix potential is given by
$$V\_{\rm SU(n)}(\mathbf{x}) = 2 \sum\_{l=1}^{n} \left[ (\mathbf{x} - a\_l) \log \left| \frac{\mathbf{x} - a\_l}{\Lambda} \right| - (\mathbf{x} - a\_l) \right] \tag{66}$$
Note that this matrix model is regarded as the U(1) matrix model with external fields *al*. We will discuss how to extract the gauge theory consequences from this matrix model in section 4.
#### **3.4. Orbifold partition function**
The matrix model description for the random partition model is also possible for the orbifold theory. We would like to derive another kind of the matrix model from the combinatorial orbifold partition function (16). We now concentrate on the U(1) orbifold partition function for simplicity. See [27, 28] for details of the SU(*n*) theory.
To obtain the matrix integral representation of the combinatorial partition function, we have to find the associated one-dimensional particle description of the combinatorial factor. In this case, although the combinatorial weight itself is the same as the standard U(1) model, there is restriction on its product region. Thus it is useful to introduce another basis obtained by dividing the partition as follows,
$$\left\{ r\left(\lambda\_i^{(u)} + N^{(u)} - i\right) + u \Big| i = 1, \dots, N^{(u)}, u = 0, \dots, r - 1 \right\} = \{\lambda\_i + N - i | i = 1, \dots, N\} \tag{67}$$
Fig.5 shows the meaning of this procedure graphically. We now assume *N*(*u*) = *N* for all *u*. With these one-dimensional particles, we now utilize the relation between the orbifold partition function and the *q*-deformed model as discussed in section 2.1. Its calculation is quite straightforward, but a little bit complicated. See [27, 28] for details.
After some computations, we finally obtain the matrix model for the *β*-deformed orbifold partition function,
$$Z\_{\text{orbifold},\mathcal{U}(1)}^{(\mathcal{G})} \approx \int \mathcal{D}\vec{\mathcal{X}} \left(\Delta\_{\text{orb}}^{(\mathcal{G})}(\mathbf{x})\right)^2 e^{-\frac{1}{\hbar}\sum\_{a=0}^{r-1}\sum\_{l=1}^{N} V(\mathbf{x}\_l^{(a)})} \tag{68}$$
In this case, we have a multi-matrix integral representation, since we introduce *r* kinds of partitions from the original partition. The matrix measure and the matrix potential are given as follows,
$$\mathcal{D}\vec{\mathcal{X}} = \prod\_{u=0}^{r-1} \prod\_{i=1}^{N} d\boldsymbol{x}\_i^{(u)} \tag{69}$$
$$\left(\Delta\_{\text{orb}}^{(\beta)}(\mathbf{x})\right)^{2} = \prod\_{u=0}^{r-1} \prod\_{i$$
$$V(\mathbf{x}) = \frac{2}{r} \left[ \mathbf{x} \log \left| \frac{\mathbf{x}}{\Lambda} \right| - \mathbf{x} \right] \tag{71}$$
The matrix measure consists of two parts, interaction between eigenvalues from the same matrix and that between eigenvalues from different matrices. Note that in the case of *β* = 1, because the interaction part in the matrix measure beteen different matrices is vanishing, this multi-matrix model is simply reduced to the one-matrix model.
#### **4. Large** *N* **analysis**
14 Linear Algebra
**Figure 5.** The decomposition of the partition for **Z***r*=3. First suppose the standard correspondence between the one-dimensional particles and the original partition, and then rearrange them with respect
> (*n*) *<sup>N</sup>* , *ξ*
Γ(*ζ<sup>i</sup>* − *ζ<sup>j</sup>* + *β*) Γ(*ζ<sup>i</sup>* − *ζj*)
satisfying *ζ*<sup>1</sup> > *ζ*<sup>2</sup> > ··· > *ζnN*. The combinatorial factor (61) is rewritten with these variables
From this expression we can obtain the matrix model description for SU(*n*) gauge theory
<sup>D</sup>*<sup>x</sup>* <sup>Δ</sup>(*x*)2*<sup>β</sup> <sup>e</sup>*
(*x* − *al*)log
Note that this matrix model is regarded as the U(1) matrix model with external fields *al*. We will discuss how to extract the gauge theory consequences from this matrix model in section 4.
The matrix model description for the random partition model is also possible for the orbifold theory. We would like to derive another kind of the matrix model from the combinatorial orbifold partition function (16). We now concentrate on the U(1) orbifold partition function
(*n*−1)
*nN* ∏ *i*=1
> − 1 *<sup>h</sup>*¯ <sup>∑</sup>*nN*
> >
*x* − *al* Λ
<sup>−</sup> (*<sup>x</sup>* <sup>−</sup> *al*)
*n* ∏ *l*=1
<sup>1</sup> , ······ , *ξ*
(2) *<sup>N</sup>* , *ξ* (1) <sup>1</sup> , ··· , *ξ*
Γ(−*ζ<sup>i</sup>* + *bl* + 1)
(1)
<sup>Γ</sup>(*ζ<sup>i</sup>* <sup>−</sup> *bl* <sup>−</sup> <sup>1</sup> <sup>+</sup> *<sup>β</sup>*) (64)
*<sup>i</sup>*=<sup>1</sup> *<sup>V</sup>*SU(*n*)(*xi*) (65)
*<sup>N</sup>* ) (63)
(66)
Then, assuming *blm* � 1, let us introduce a set of variables,
(*n*) <sup>1</sup> , ··· , *ξ*
*nN* ∏ *i*<*j*
partition function, by rescaling *xi* = *h*¯ *ζ<sup>i</sup>* with reparametrizing ¯*h* = *�*2,
*n* ∑ *l*=1 *Z*SU(*n*) ≈
*V*SU(*n*)(*x*) = 2
for simplicity. See [27, 28] for details of the SU(*n*) theory.
(*ζ*1, *ζ*2, ··· , *ζnN*)=(*ξ*
*Z*� *<sup>λ</sup>* <sup>=</sup> <sup>1</sup> *�* 2*n*| � *λ*| 2
In this case the matrix potential is given by
**3.4. Orbifold partition function**
to mod *r*.
as
One of the most important aspects of the matrix model is universality arising in the large *N* limit. The universality class described by the matrix model covers huge kinds of the statistical models, in particular its characteristic fluctuation rather than the eigenvalue density function. In the large *N* limit, which is regarded as a justification to apply a kind of the mean field approximation, anslysis of the matrix model is extremely reduced to the saddle point equation and a simple fluctuation around it.
#### **4.1. Saddle point equation and spectral curve**
Let us first define the prepotential, which is also interpreted as the effective action for the eigenvalues, from the matrix integral representation
#### 16 Linear Algebra 90 Linear Algebra – Theorems and Applications
(37),
$$-\frac{1}{\hbar^2}\mathcal{F}(\{\mathbf{x}\_i\}) = -\frac{1}{\hbar}\sum\_{i=1}^N V(\mathbf{x}\_i) + 2\sum\_{i$$
This is essentially the genus zero part of the prepotential. In the large *N* limit, in particular 't Hooft limit (43) with *Nh*¯ ≡ *t*, we shall investigate the saddle point equation for the matrix integral. We can obtain the condition for criticality by differentiating the prepotential,
$$V'(\mathbf{x}\_i) = 2\hbar \sum\_{j(\neq i)}^{N} \frac{1}{\mathbf{x}\_j - \mathbf{x}\_i}, \qquad \text{for all } i \tag{73}$$
This is also given by the extremal condition of the effective potential defined as
$$V\_{\rm eff}(\mathbf{x}\_{i}) = V(\mathbf{x}\_{i}) - 2\hbar \sum\_{j(\neq i)}^{N} \log(\mathbf{x}\_{i} - \mathbf{x}\_{j}) \tag{74}$$
This potential involves a logarithmic Coulomb repulsion between eigenvalues. If the 't Hooft coupling is small, the potential term dominates the Coulomb interaction and eigenvalues concentrate on extrema of the potential *V*� (*x*) = 0. On the other hand, as the coupling gets bigger, the eigenvalue distribution is extended.
To deal with such a situation, we now define the density of eigenvalues,
$$\rho(\mathbf{x}) = \frac{1}{N} \sum\_{i=1}^{N} \delta(\mathbf{x} - \mathbf{x}\_i) \tag{75}$$
where *xi* is the solution of the criticality condition (73). In the large *N* limit, it is natural to think this eigenvalue distribution is smeared, and becomes a continuous function. Furthermore, we assume the eigenvalues are distributed around the critical points of the potential *V*(*x*) as linear segments. Thus we generically denote the *l*-th segment for *ρ*(*x*) as C*l*, and the total number of eigenvalues *N* splits into *n* integers for these segments,
$$N = \sum\_{l=1}^{n} N\_l \tag{76}$$
where *Nl* is the number of eigenvalues in the interval C*l*. The density of eigenvalues *ρ*(*x*) takes non-zero value only on the segment C*l*, and is normalized as
$$\int\_{\mathcal{C}\_l} d\mathbf{x} \,\rho(\mathbf{x}) = \frac{N\_l}{N} \equiv \nu\_l \tag{77}$$
where we call it *filling fraction*. According to these fractions, we can introduce the partial 't Hooft parameters, *tl* = *Nlh*¯. Note there are *n* 't Hooft couplings and filling fractions, but only *<sup>n</sup>* <sup>−</sup> 1 fractions are independent since they have to satisfy <sup>∑</sup>*<sup>n</sup> <sup>l</sup>*=<sup>1</sup> *ν<sup>l</sup>* = 1 while all the 't Hooft couplings are independent.
We then introduce the resolvent for this model as an auxiliary function, a kind of Green function. By taking the large *N* limit, it can be given by the integral representation,
#### 90 Linear Algebra – Theorems and Applications Gauge Theory, Combinatorics, and Matrix Models <sup>17</sup> Gauge Theory, Combinatorics, and Matrix Models 91
$$
\omega(\mathbf{x}) = t \int dy \, \frac{\rho(y)}{\mathbf{x} - y} \tag{78}
$$
This means that the density of states is regarded as the Hilbert transformation of this resolvent function. Indeed the density of states is associated with the discontinuities of the resolvent,
$$\rho(\mathbf{x}) = -\frac{1}{2\pi it} \left( \omega(\mathbf{x} + i\boldsymbol{\epsilon}) - \omega(\mathbf{x} - i\boldsymbol{\epsilon}) \right) \tag{79}$$
Thus all we have to do is to determine the resolvent instead of the density of states with satisfying the asymptotic behavior,
$$
\omega(\mathbf{x}) \longrightarrow \frac{1}{\mathbf{x}'} \qquad \mathbf{x} \longrightarrow \infty \tag{80}
$$
Writing down the prepotential with the density of states,
$$\mathcal{F}(\{\mathbf{x}\_{i}\}) = t \int d\mathbf{x} \,\rho(\mathbf{x}) V(\mathbf{x}) - t^{2} \mathbf{P} \int d\mathbf{x} dy \,\rho(\mathbf{x}) \rho(y) \log(\mathbf{x} - y) \tag{81}$$
the criticality condition is given by
$$\frac{1}{2t}V'(\mathbf{x}) = \mathbf{P} \int dy \, \frac{\rho(y)}{\mathbf{x} - y} \tag{82}$$
Here P stands for the principal value. Thus this saddle point equation can be also written in the following convenient form to discuss its analytic property,
$$V'(\mathbf{x}) = \omega(\mathbf{x} + i\varepsilon) + \omega(\mathbf{x} - i\varepsilon) \tag{83}$$
On the other hand, we have another convenient form to treat the saddle point equation, which is called *loop equation*, given by
$$\left(y^2(\mathbf{x}) - V'(\mathbf{x})^2 + \mathcal{R}(\mathbf{x}) = 0\right) \tag{84}$$
where we denote
16 Linear Algebra
This is essentially the genus zero part of the prepotential. In the large *N* limit, in particular 't Hooft limit (43) with *Nh*¯ ≡ *t*, we shall investigate the saddle point equation for the matrix integral. We can obtain the condition for criticality by differentiating the prepotential,
> 1 *xj* − *xi*
> > *N* ∑ *j*(�=*i*)
This potential involves a logarithmic Coulomb repulsion between eigenvalues. If the 't Hooft coupling is small, the potential term dominates the Coulomb interaction and eigenvalues
> *N* ∑ *i*=1
where *xi* is the solution of the criticality condition (73). In the large *N* limit, it is natural to think this eigenvalue distribution is smeared, and becomes a continuous function. Furthermore, we assume the eigenvalues are distributed around the critical points of the potential *V*(*x*) as linear segments. Thus we generically denote the *l*-th segment for *ρ*(*x*) as C*l*, and the total number of
> *n* ∑ *l*=1
where *Nl* is the number of eigenvalues in the interval C*l*. The density of eigenvalues *ρ*(*x*) takes
*dx <sup>ρ</sup>*(*x*) = *Nl*
where we call it *filling fraction*. According to these fractions, we can introduce the partial 't Hooft parameters, *tl* = *Nlh*¯. Note there are *n* 't Hooft couplings and filling fractions, but only
We then introduce the resolvent for this model as an auxiliary function, a kind of Green
function. By taking the large *N* limit, it can be given by the integral representation,
*N* =
*V*(*xi*) + 2
*N* ∑ *i*<*j*
log(*xi* − *xj*) (72)
, for all *i* (73)
log(*xi* − *xj*) (74)
(*x*) = 0. On the other hand, as the coupling gets
*δ*(*x* − *xi*) (75)
*Nl* (76)
*<sup>N</sup>* <sup>≡</sup> *<sup>ν</sup><sup>l</sup>* (77)
*<sup>l</sup>*=<sup>1</sup> *ν<sup>l</sup>* = 1 while all the 't Hooft
*h*¯
*N* ∑ *j*(�=*i*)
This is also given by the extremal condition of the effective potential defined as
*V*eff(*xi*) = *V*(*xi*) − 2¯*h*
To deal with such a situation, we now define the density of eigenvalues,
*<sup>ρ</sup>*(*x*) = <sup>1</sup> *N*
*N* ∑ *i*=1
(37),
− 1
concentrate on extrema of the potential *V*�
bigger, the eigenvalue distribution is extended.
eigenvalues *N* splits into *n* integers for these segments,
non-zero value only on the segment C*l*, and is normalized as
*<sup>n</sup>* <sup>−</sup> 1 fractions are independent since they have to satisfy <sup>∑</sup>*<sup>n</sup>*
couplings are independent.
C*l*
*<sup>h</sup>*¯ <sup>2</sup> <sup>F</sup>({*xi*}) = <sup>−</sup><sup>1</sup>
(*xi*) = 2¯*h*
*V*�
$$y(\mathbf{x}) = V'(\mathbf{x}) - 2\omega(\mathbf{x}) = -2\omega\_{\text{sing}}(\mathbf{x})\tag{85}$$
$$R(\mathbf{x}) = \frac{4t}{N} \sum\_{i=1}^{N} \frac{V'(\mathbf{x}) - V'(\mathbf{x}\_i)}{\mathbf{x} - \mathbf{x}\_i} \tag{86}$$
It is obtained from the saddle point equation by multiplying 1/(*x* − *xi*) and taking their summation and the large *N* limit. This representation (84) is more appropriate to reveal its geometric meaning. Indeed this algebraic curve is interpreted as the hyperelliptic curve which is given by resolving the singular form,
$$\left(y^2(\mathbf{x}) - V'(\mathbf{x})\right)^2 = 0\tag{87}$$
The genus of the Riemann surface is directly related to the number of cuts of the corresponding resolvent. The filling fraction, or the partial 't Hooft coupling, is simply given by the contour
#### 18 Linear Algebra 92 Linear Algebra – Theorems and Applications
integral on the hyperelliptic curve
$$\text{tr}\_l = \frac{1}{2\pi i} \oint\_{\mathcal{C}\_l} d\mathbf{x} \,\omega\_{\text{sing}}(\mathbf{x}) = -\frac{1}{4\pi i} \oint\_{\mathcal{C}\_l} d\mathbf{x} \, y(\mathbf{x})\tag{88}$$
#### **4.2. Relation to Seiberg-Witten theory**
We now discuss the relation between Seiberg-Witten curve and the matrix model. In the first place, the matrix model captures the asymptotic behavior of the combinatorial representation of the partition function. The energy functional, which is derived from the asymptotics of the partition function [44], in terms of the profile function
$$\mathcal{E}\_{\Lambda}(f) = \frac{1}{4} \text{P} \int\_{y<\infty} d\mathbf{x} dy \, f^{\prime\prime}(\mathbf{x}) f^{\prime\prime}(y) (\mathbf{x} - y)^2 \left( \log \left( \frac{\mathbf{x} - y}{\Lambda} \right) - \frac{\mathbf{3}}{2} \right) \tag{89}$$
can be rewritten as
$$\mathcal{E}\_{\Lambda}(\varrho) = -\mathbb{P} \int\_{\mathbf{x} \neq \mathbf{y}} d\mathbf{x} dy \, \frac{\varrho(\mathbf{x})\varrho(\mathbf{y})}{(\mathbf{x} - \mathbf{y})^2} - 2 \int d\mathbf{x} \, \varrho(\mathbf{x}) \log \prod\_{l=1}^{N} \left(\frac{\mathbf{x} - a\_l}{\Lambda}\right) \tag{90}$$
up to the perturbative contribution
$$\frac{1}{2} \sum\_{l,m} (a\_l - a\_m)^2 \log\left(\frac{a\_l - a\_m}{\Lambda}\right) \tag{91}$$
by identifying
$$f(\mathbf{x}) - \sum\_{l=1}^{n} |\mathbf{x} - a\_l| = \varrho(\mathbf{x}) \tag{92}$$
Then integrating (90) by parts, we have
$$\mathcal{E}\_{\Lambda}(\varrho) = -\mathbb{P} \int\_{\mathbf{x} \neq \mathbf{y}} d\mathbf{x} dy \,\varrho'(\mathbf{x}) \varrho'(\mathbf{y}) \log(\mathbf{x} - \mathbf{y}) + 2 \int d\mathbf{x} \,\varrho'(\mathbf{x}) \sum\_{l=1}^{n} \left[ (\mathbf{x} - a\_{l}) \log\left(\frac{\mathbf{x} - a\_{l}}{\Lambda}\right) - (\mathbf{x} - a\_{l}) \right] \tag{93}$$
This is just the matrix model discussed in section 3.3 if we identify *�*� (*x*) = *ρ*(*x*). Therefore analysis of this matrix model is equivalent to that of [**?** ]. But in this section we reconsider the result of the gauge theory from the viewpoint of the matrix model.
We can introduce a regular function on the complex plane, except at the infinity,
$$P\_n(\mathbf{x}) = \Lambda^n \left( e^{y/2} + e^{-y/2} \right) \equiv \Lambda^n \left( w + \frac{1}{w} \right) \tag{94}$$
It is because the saddle point equation (83) yields the following equation,
$$e^{y(\mathbf{x}+i\varepsilon)/2} + e^{-y(\mathbf{x}+i\varepsilon)/2} = e^{y(\mathbf{x}-i\varepsilon)/2} + e^{-y(\mathbf{x}-i\varepsilon)/2} \tag{95}$$
This entire function turns out to be a monic polynomial *Pn*(*x*) = *<sup>x</sup><sup>n</sup>* <sup>+</sup> ··· , because it is an analytic function with the following asymptotic behavior,
$$
\Lambda^n e^{y/2} = \Lambda^n e^{-\omega(\mathbf{x})} \prod\_{l=1}^n \left( \frac{\mathbf{x} - a\_l}{\Lambda} \right) \longrightarrow \mathbf{x}^n, \qquad \mathbf{x} \longrightarrow \infty \tag{96}
$$
Here *w* should be the smaller root with the boundary condition as
$$w \longrightarrow \frac{\Lambda^n}{\mathfrak{x}^n} \, \Big|\qquad \mathfrak{x} \longrightarrow \infty \tag{97}$$
thus we now identify
18 Linear Algebra
*dx <sup>ω</sup>*sing(*x*) = <sup>−</sup> <sup>1</sup>
We now discuss the relation between Seiberg-Witten curve and the matrix model. In the first place, the matrix model captures the asymptotic behavior of the combinatorial representation of the partition function. The energy functional, which is derived from the asymptotics of the
*dxdy f* ��(*x*)*<sup>f</sup>* ��(*y*)(*<sup>x</sup>* <sup>−</sup> *<sup>y</sup>*)<sup>2</sup>
(*<sup>x</sup>* <sup>−</sup> *<sup>y</sup>*)<sup>2</sup> <sup>−</sup> <sup>2</sup>
(*al* <sup>−</sup> *am*)<sup>2</sup> log
*n* ∑ *l*=1
> *dx �*� (*x*) *n* ∑ *l*=1
analysis of this matrix model is equivalent to that of [**?** ]. But in this section we reconsider the
−*y*/2
<sup>≡</sup> <sup>Λ</sup>*<sup>n</sup> w* + 1 *w*
*<sup>y</sup>*(*x*−*i�*)/2 + *e*
*al* <sup>−</sup> *am* Λ
*dxdy �*(*x*)*�*(*y*)
*f*(*x*) −
(*y*)log(*x* − *y*) +2
We can introduce a regular function on the complex plane, except at the infinity,
<sup>−</sup>*y*(*x*+*i�*)/2 = *e*
This entire function turns out to be a monic polynomial *Pn*(*x*) = *<sup>x</sup><sup>n</sup>* <sup>+</sup> ··· , because it is an
*e <sup>y</sup>*/2 + *e*
It is because the saddle point equation (83) yields the following equation,
This is just the matrix model discussed in section 3.3 if we identify *�*�
result of the gauge theory from the viewpoint of the matrix model.
*Pn*(*x*) = Λ*<sup>n</sup>*
*<sup>y</sup>*(*x*+*i�*)/2 + *e*
analytic function with the following asymptotic behavior,
4*πi* C*l*
> log
*dx �*(*x*)log
*<sup>x</sup>* <sup>−</sup> *<sup>y</sup>* Λ
> *N* ∏ *l*=1
(*x* − *al*)log
− 3 2
*<sup>x</sup>* <sup>−</sup> *al* Λ
*<sup>x</sup>* <sup>−</sup> *al* Λ
<sup>−</sup>*y*(*x*−*i�*)/2 (95)
(*x*) = *ρ*(*x*). Therefore
*dx y*(*x*) (88)
(89)
(90)
(91)
− (*x* − *al*)
(93)
(94)
integral on the hyperelliptic curve
**4.2. Relation to Seiberg-Witten theory**
<sup>E</sup>Λ(*f*) = <sup>1</sup>
EΛ(*�*) = −P
up to the perturbative contribution
Then integrating (90) by parts, we have
*dxdy �*�
*e*
(*x*)*�*�
*x*�=*y*
can be rewritten as
by identifying
EΛ(*�*) = −P
*tl* <sup>=</sup> <sup>1</sup> 2*πi* C*l*
partition function [44], in terms of the profile function
*x*�=*y*
> 1 <sup>2</sup> ∑ *l*,*m*
4 P *y*<*x*
$$w = e^{-y/2} \tag{98}$$
Therefore from the hyperelliptic curve (94) we can relate Seiberg-Witten curve to the spectral curve of the matrix model,
$$\begin{split} dS &= \frac{1}{2\pi i} \mathbf{x} \frac{dw}{w} \\ &= -\frac{1}{2\pi i} \log w \, dx \\ &= \frac{1}{4\pi i} y(x) dz \end{split} \tag{99}$$
Note that it is shown in [37, 38] we have to take the vanishing fraction limit to obtain the Coulomb moduli from the matrix model contour integral. This is the essential difference between the profile function method and the matrix model description.
#### **4.3. Eigenvalue distribution**
We now demonstrate that the eigenvalue distribution function is indeed derived from the spectral curve of the matrix model. The spectral curve (94) in the case of *n* = 1 with setting Λ = 1 and *Pn*=1(*x*) = *x* is written as
$$x = w + \frac{1}{w} \tag{100}$$
From this relation the singular part of the resolvent can be extracted as
$$
\omega\_{\text{sing}}(\mathbf{x}) = \arccos \left( \frac{\mathbf{x}}{2} \right) \tag{101}
$$
This has a branch cut only on *x* ∈ [−2, 2], namely a one-cut solution. Thus the eigenvalue distribution function is witten as follows at least on *x* ∈ [−2, 2],
$$\rho(\mathbf{x}) = \frac{1}{\pi} \arccos\left(\frac{\mathbf{x}}{2}\right) \tag{102}$$
Note that this function has a non-zero value at the left boundary of the cut, *ρ*(−2) = 1, while at the right boundary we have *ρ*(2) = 0. Equivalently we now choose the cut of arccos function in this way. This seems a little bit strange because the eigenvalue density has to vanish except for on the cut. On the other hand, recalling the meaning of the eigenvalues, i.e. positions of one-dimensional particles, as shown in Fig. 4, this situation is quite reasonable. The region below the Fermi level is filled of the particles, and thus the density has to be a non-zero constant in such a region. This is just a property of the Fermi distribution function. (1/*N* correction could be interpreted as a finite temperature effect.) Therefore the total eigenvalue
**Figure 6.** The eigenvalue distribution function for the U(1) model.
distribution function is given by
$$\rho(\mathbf{x}) = \begin{cases} 1 & \mathbf{x} < -2 \\ \frac{1}{\pi} \arccos\left(\frac{\mathbf{x}}{2}\right) & |\mathbf{x}| < 2 \\ 0 & \mathbf{x} > 2 \end{cases} \tag{103}$$
Remark the eigenvalue density (103) is quite similar to the Wigner's semi-circle distribution function, especially its behavior around the edge,
$$\rho\_{\text{circ}}(\mathbf{x}) = \frac{1}{\pi} \sqrt{1 - \left(\frac{\mathbf{x}}{2}\right)^2} \longrightarrow \frac{1}{\pi} \sqrt{2 - \mathbf{x}}, \qquad \mathbf{x} \longrightarrow \mathbf{2} \tag{104}$$
The fluctuation at the spectral edge of the random matrix obeys Tracy-Widom distribution [56], thus it is natural that the edge fluctuation of the combinatorial model is also described by Tracy-Widom distribution. This remarkable fact was actually shown by [9]. Evolving such a similarity to the gaussian random matrix theory, the kernel of this model is also given by the following sine kernel,
$$K(\mathbf{x}, y) = \frac{\sin \rho\_0 \pi (\mathbf{x} - y)}{\pi (\mathbf{x} - y)} \tag{105}$$
where *ρ*<sup>0</sup> is the averaged density of eigenvalues. This means the U(1) combinatorial model belongs to the GUE random matrix universal class [40]. Then all the correlation functions can be written as a determinant of this kernel,
$$\rho(\mathbf{x}\_1, \dots, \mathbf{x}\_k) = \det\left[\mathcal{K}(\mathbf{x}\_i, \mathbf{x}\_j)\right]\_{1 \le i, j \le k} \tag{106}$$
Let us then remark a relation to the profile function of the Young diagram. It was shown that the shape of the Young diagram goes to the following form in the thermodynamical limit [33, 58, 59],
$$\Omega(\mathbf{x}) = \begin{cases} \frac{2}{\pi} \left( \mathbf{x} \arcsin \frac{\mathbf{x}}{2} + \sqrt{4 - \mathbf{x}^2} \right) & |\mathbf{x}| < 2 \\\ & |\mathbf{x}| \end{cases} \tag{107}$$
Rather than this profile function itself, the derivative of this function is more relevant to our study,
$$\Omega'(\mathbf{x}) = \begin{cases} -1 & \mathbf{x} < -2 \\ \frac{2}{\pi} \arcsin\left(\frac{\mathbf{x}}{2}\right) & |\mathbf{x}| < 2 \\ 1 & \mathbf{x} > 2 \end{cases} \tag{108}$$
One can see the eigenvalue density (103) is directly related to this derivative function (108) as
$$\rho(\mathbf{x}) = \frac{1 - \Omega'(\mathbf{x})}{2} \tag{109}$$
This relation is easily obtained from the correspondence between the Young diagram and the one-dimensional particle as shown in Fig. 4.
#### **5. Conclusion**
20 Linear Algebra
**Figure 6.** The eigenvalue distribution function for the U(1) model.
function, especially its behavior around the edge,
be written as a determinant of this kernel,
Ω(*x*) =
Ω� (*x*) =
*<sup>ρ</sup>*circ(*x*) = <sup>1</sup>
*ρ*(*x*) =
*π* � 1 − � *x* 2 �2 −→ 1 *π*
⎧ ⎨ ⎩
1
*<sup>π</sup>* arccos � *<sup>x</sup>*
Remark the eigenvalue density (103) is quite similar to the Wigner's semi-circle distribution
The fluctuation at the spectral edge of the random matrix obeys Tracy-Widom distribution [56], thus it is natural that the edge fluctuation of the combinatorial model is also described by Tracy-Widom distribution. This remarkable fact was actually shown by [9]. Evolving such a similarity to the gaussian random matrix theory, the kernel of this model is also given by the
*<sup>K</sup>*(*x*, *<sup>y</sup>*) = sin *<sup>ρ</sup>*0*π*(*<sup>x</sup>* <sup>−</sup> *<sup>y</sup>*)
where *ρ*<sup>0</sup> is the averaged density of eigenvalues. This means the U(1) combinatorial model belongs to the GUE random matrix universal class [40]. Then all the correlation functions can
�
Let us then remark a relation to the profile function of the Young diagram. It was shown that the shape of the Young diagram goes to the following form in the thermodynamical limit
Rather than this profile function itself, the derivative of this function is more relevant to our
<sup>2</sup> <sup>+</sup> <sup>√</sup>
*x* arcsin *<sup>x</sup>*
⎧ ⎨ ⎩
2 *<sup>π</sup>* arcsin � *<sup>x</sup>*
*K*(*xi*, *xj*)
<sup>4</sup> − *<sup>x</sup>*<sup>2</sup> �
−1 *x* < −2
� <sup>|</sup>*x*<sup>|</sup> <sup>&</sup>lt; <sup>2</sup> 1 *x* > 2
2
<sup>|</sup>*x*| |*x*<sup>|</sup> <sup>&</sup>gt; <sup>2</sup> (107)
�
*ρ*(*x*1, ··· , *xk*) = det
� 2 *π* � 1 *x* < −2
� <sup>|</sup>*x*<sup>|</sup> <sup>&</sup>lt; <sup>2</sup> 0 *x* > 2
<sup>√</sup><sup>2</sup> <sup>−</sup> *<sup>x</sup>*, *<sup>x</sup>* −→ <sup>2</sup> (104)
*<sup>π</sup>*(*<sup>x</sup>* <sup>−</sup> *<sup>y</sup>*) (105)
<sup>1</sup>≤*i*,*j*,≤*<sup>k</sup>* (106)
(103)
(108)
2
distribution function is given by
following sine kernel,
[33, 58, 59],
study,
In this article we have investigated the combinatorial statistical model through its matrix model description. Starting from the U(1) model, which is motivated by representation theory, we have dealt with its *β*-deformation and *q*-deformation. We have shown that its non-Abelian generalization, including external field parameters, is obtained as the four dimensional supersymmetric gauge theory partition function. We have also referred to the orbifold partition function, and its relation to the *q*-deformed model through the root of unity limit.
We have then shown the matrix integral representation is derived from such a combinatorial partition function by considering its asymptotic behavior in the large *N* limit. Due to variety of the combinatorial model, we can obtain the *β*-ensemble matrix model, the hyperbolic matrix model, and those with external fields. Furthermore from the orbifold partition function the multi-matrix model is derived.
Based on the matrix model description, we have study the asymptotic behavior of the combinatorial models in the large *N* limit. In this limit we can extract various important properties of the matrix model by analysing the saddle point equation. Introducing the resolvent as an auxiliary function, we have obtained the algebraic curve for the matrix model, which is called the spectral curve. We have shown it can be interpreted as Seiberg-Witten curve, and then the eigenvalue distribution function is also obtained from this algebraic curve.
Let us comment on some possibilities of generalization and perspective. As discussed in this article we can obtain various interesting results from Macdonald polynomial by taking the corresponding limit. It is interesting to research its matrix model consequence from the exotic limit of Macdonald polynomial. For example, the *q* → 0 limit of Macdonald polynomial, which is called Hall-Littlewood polynomial, is not investigated with respect to its connection with the matrix model. We also would like to study properties of the *BC*-type polynomial [31], which is associated with the corresponding root system. Recalling the meaning of the *q*-deformation in terms of the gauge theory, namely lifting up to the five dimensional theory **<sup>R</sup>**<sup>4</sup> <sup>×</sup> *<sup>S</sup>*<sup>1</sup> by taking into account all the Kaluza-Klein modes, it seems interesting to study the six dimensional theory on **<sup>R</sup>**<sup>4</sup> <sup>×</sup> *<sup>T</sup>*2. In this case it is natural to obtain the elliptic generalization of the matrix model. It can not be interpreted as matrix integral representation any longer, however the large *N* analysis could be anyway performed in the standard manner. We would like to expect further develpopment beyond this work.
#### **Author details**
Taro Kimura *Mathematical Physics Laboratory, RIKEN Nishina Center, Japan*
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## **Nonnegative Inverse Eigenvalue Problem**
Ricardo L. Soto
24 Linear Algebra
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Additional information is available at the end of the chapter
http://dx.doi.org/10.5772/48279
## **1. Introduction**
Nonnegative matrices have long been a sorce of interesting and challenging mathematical problems. They are real matrices with all their entries being nonnegative and arise in a number of important application areas: communications systems, biological systems, economics, ecology, computer sciences, machine learning, and many other engineering systems. Inverse eigenvalue problems constitute an important subclass of inverse problems that arise in the context of mathematical modeling and parameter identification. A simple application of such problems is the construction of Leontief models in economics [1]-[3].
The *nonnegative inverse eigenvalue problem* (*NIEP*) is the problem of characterizing those lists Λ = {*λ*1, *λ*2, ..., *λn*} of complex numbers which can be the spectra of *n* × *n* entrywise nonnegative matrices. If there exists a nonnegative matrix *A* with spectrum Λ we say that Λ is realized by *A* and that *A* is the realizing matrix. A set *K* of conditions is said to be a *realizability criterion* if any list Λ = {*λ*1, *λ*2, ..., *λn*}, real or complex, satisfying conditions *K* is realizable. The *NIEP is an open problem.* A full solution is unlikely in the near future. The problem has only been solved for *n* = 3 by Loewy and London ([4], 1978) and for *n* = 4 by Meehan ([5], 1998) and Torre-Mayo et al.([6], 2007). The case *n* = 5 has been solved for matrices of trace zero in ([7], 1999). Other results, mostly in terms of sufficient conditions for the problem to have a solution (in the case of a complex list Λ), have been obtained, in chronological order, in [8]-[13].
Two main subproblems of the *NIEP* are of great interest: the *real nonnegative inverse eigenvalue problem* (*RNIEP*), in which Λ is a list of real numbers, and the *symmetric nonnegative inverse eigenvalue problem* (*SNIEP*), in which the realizing matrix must be symmetric. Both problems, *RNIEP* and *SNIEP* are equivalent for *n* ≤ 4 (see [14]), but they are different otherwise (see [15]). Moreover, both problems remains unsolved for *n* ≥ 5. The *NIEP* is also of interest for nonnegative matrices with a particular structure, like stochastic and doubly stochastic, circulant, persymmetric, centrosymmetric, Hermitian, Toeplitz, etc.
The first sufficient conditions for the existence of a nonnegative matrix with a given real spectrum (*RNIEP*) were obtained by Suleimanova ([16], 1949) and Perfect ([17, 18], 1953 and
1955). Other sufficient conditions have also been obtained, in chronological order in [19]-[26], (see also [27, 28], and references therein for a comprehensive survey).
The first sufficient conditions for the *SNIEP* were obtained by Fiedler ([29], 1974). Other results for symmetric realizability have been obtained in [8, 30] and [31]-[33]. Recently, new sufficient conditions for the *SNIEP* have been given in [34]-[37].
#### **1.1. Necessary conditions**
Let *A* be a nonnegative matrix with spectrum Λ = {*λ*1, *λ*2, ..., *λn*}. Then, from the Perron Frobenius theory we have the following basic necessary conditions
$$\begin{array}{l} \text{(1)} \ \overline{\Lambda} = \{ \overline{\lambda\_1}, \dots, \overline{\lambda\_n} \} = \Lambda \\ \text{(2)} \ \max\_{\overline{j}} \{ \left| \lambda\_{\overline{j}} \right| \} \in \Lambda \\ \text{(3)} \ s\_m(\Lambda) = \sum\_{j=1}^n \lambda\_{\overline{j}}^m \ge 0, \ m = 1, 2, \dots, \end{array} \tag{1}$$
where Λ = Λ means that Λ is closed under complex comjugation.
Moreover, we have
$$\begin{array}{ll}(4) \ (s\_k(\Lambda))^m \le n^{m-1} s\_{km}(\Lambda), \ k, m = 1, 2, \dots \\ (5) \ (s\_2(\Lambda))^2 \le (n-1) s\_4(\Lambda), \ n \text{ odd}, \ tr(A) = 0. \end{array} \tag{2}$$
Necessary condition (4) is due to Loewy and London [4]. Necessary condition (5), which is a refinement of (4), is due to Laffey and Meehan [38]. The list Λ = {5, 4, −3, −3, −3} for instance, satisfies all above necessary conditions, except condition (5). Therefore Λ is not a realizable list. In [39] it was obtained a new necessary condition, which is independent of the previous ones. This result is based on the Newton's inequalities associated to the normalized coefficients of the characteristic polynomial of an M-matrix or an inverse M-matrix.
The chapter is organized as follows: In section 2 we introduce two important matrix results, due to Brauer and Rado, which have allowed to obtain many of the most general sufficient conditions for the *RNIEP*, the *SNIEP* and the complex case. In section 3 we consider the real case and we introduce, without proof (we indicate where the the proofs can be found), two sufficient conditions with illustrative examples. We consider, in section 4, the symmetric case. Here we introduce a symmetric version of the Rado result, Theorem 2, and we set, without proof (see the appropriate references), three sufficient conditions, which are, as far as we know, the most general sufficient conditions for the *SNIEP*. In section 5, we discuss the complex (non real) case. Here we present several results with illustrative examples. Section 6 is devoted to discuss some Fiedler results and Guo results, which are very related with the problem and have been employed with success to derive sufficient conditions. Finally, in section 7, we introduce some open questions.
### **2. Brauer and Rado Theorems**
A real matrix *A* = (*aij*)*<sup>n</sup> <sup>i</sup>*=<sup>1</sup> is said to have *constant row sums* if all its rows sum up to the same constant, say, *α*, that is, *n* ∑ *j*=1 *aij* = *α*, *i* = 1, . . . , *n*. The set of all real matrices with constant row sums equal to *α* is denoted by CS*α*. It is clear that any matrix in CS*<sup>α</sup>* has eigenvector **<sup>e</sup>** = (1, 1, . . . , 1)*<sup>T</sup>* corresponding to the eigenvalue *<sup>α</sup>*. Denote by **<sup>e</sup>***<sup>k</sup>* the *<sup>n</sup>*−dimensional vector with one in the *k* − *th* position and zeros elsewhere.
It is well known that the problem of finding a nonnegative matrix with spectrum Λ = {*λ*1,..., *λn*} is equivalent to the problem of finding a nonnegative matrix in CS*λ*<sup>1</sup> with spectrum Λ (see [40]). This will allow us to exploit the advantages of two important theorems, Brauer Theorem and Rado Theorem, which will be introduced in this section.
The spectra of circulant nonnegative matrices have been characterized in [9], while in [10], a simple complex generalization of Suleimanova result has been proved, and efficient and general sufficient conditions for the realizability of partitioned spectra, with the partition allowing some of its pieces to be nonrealizable, provided there are other pieces, which are realizable and, in certain way, compensate the nonnrealizability of the former, have been obtained. This is the procedure which we call *negativity compensation*. This strategy, based in the use of the following two perturbation results, together with the properties of real matrices with constant row sums, has proved to be successful.
**Theorem 1.** *Brauer [41] Let A be an n* × *n arbitrary matrix with eigenvalues λ*1,..., *λn*. *Let* **v** = (*v*1, ..., *vn*)*<sup>T</sup> an eigenvector of A associated with the eigenvalue λ<sup>k</sup> and let* **q** = (*q*1, ..., *qn*)*<sup>T</sup> be any n-dimensional vector. Then the matrix A* <sup>+</sup> **vq***<sup>T</sup> has eigenvalues <sup>λ</sup>*1,..., *<sup>λ</sup>k*−1, *<sup>λ</sup><sup>k</sup>* <sup>+</sup> *vTq*, *λk*<sup>+</sup>1,..., *λn*.
**Proof.** Let *U* be an *n* × *n* nonsingular matrix such that
2 Will-be-set-by-IN-TECH
1955). Other sufficient conditions have also been obtained, in chronological order in [19]-[26],
The first sufficient conditions for the *SNIEP* were obtained by Fiedler ([29], 1974). Other results for symmetric realizability have been obtained in [8, 30] and [31]-[33]. Recently, new sufficient
Let *A* be a nonnegative matrix with spectrum Λ = {*λ*1, *λ*2, ..., *λn*}. Then, from the Perron
*<sup>j</sup>* ≥ 0, *m* = 1, 2, . . . ,
(5) (*s*2(Λ))<sup>2</sup> <sup>≤</sup> (*<sup>n</sup>* <sup>−</sup> <sup>1</sup>)*s*4(Λ), *<sup>n</sup>* odd, *tr*(*A*) = 0. (2)
*<sup>i</sup>*=<sup>1</sup> is said to have *constant row sums* if all its rows sum up to the same
*aij* = *α*, *i* = 1, . . . , *n*. The set of all real matrices with constant
(1)
(see also [27, 28], and references therein for a comprehensive survey).
Frobenius theory we have the following basic necessary conditions
*λj* } ∈ <sup>Λ</sup>
(1) Λ = {*λ*1,..., *λn*} = Λ
*<sup>j</sup>*=<sup>1</sup> *<sup>λ</sup><sup>m</sup>*
(4) (*sk*(Λ))*<sup>m</sup>* <sup>≤</sup> *<sup>n</sup>m*−<sup>1</sup>*skm*(Λ), *<sup>k</sup>*, *<sup>m</sup>* <sup>=</sup> 1, 2, . . .
coefficients of the characteristic polynomial of an M-matrix or an inverse M-matrix.
Necessary condition (4) is due to Loewy and London [4]. Necessary condition (5), which is a refinement of (4), is due to Laffey and Meehan [38]. The list Λ = {5, 4, −3, −3, −3} for instance, satisfies all above necessary conditions, except condition (5). Therefore Λ is not a realizable list. In [39] it was obtained a new necessary condition, which is independent of the previous ones. This result is based on the Newton's inequalities associated to the normalized
The chapter is organized as follows: In section 2 we introduce two important matrix results, due to Brauer and Rado, which have allowed to obtain many of the most general sufficient conditions for the *RNIEP*, the *SNIEP* and the complex case. In section 3 we consider the real case and we introduce, without proof (we indicate where the the proofs can be found), two sufficient conditions with illustrative examples. We consider, in section 4, the symmetric case. Here we introduce a symmetric version of the Rado result, Theorem 2, and we set, without proof (see the appropriate references), three sufficient conditions, which are, as far as we know, the most general sufficient conditions for the *SNIEP*. In section 5, we discuss the complex (non real) case. Here we present several results with illustrative examples. Section 6 is devoted to discuss some Fiedler results and Guo results, which are very related with the problem and have been employed with success to derive sufficient conditions. Finally, in section 7, we
row sums equal to *α* is denoted by CS*α*. It is clear that any matrix in CS*<sup>α</sup>* has eigenvector
conditions for the *SNIEP* have been given in [34]-[37].
(2) max*j*{
(3) *sm*(Λ) = ∑*<sup>n</sup>*
where Λ = Λ means that Λ is closed under complex comjugation.
**1.1. Necessary conditions**
Moreover, we have
introduce some open questions.
A real matrix *A* = (*aij*)*<sup>n</sup>*
constant, say, *α*, that is,
**2. Brauer and Rado Theorems**
*n* ∑ *j*=1
$$U^{-1}AU = \begin{bmatrix} \lambda\_1 \ \* \ \* \ \cdots \ \* \\\\ \lambda\_2 \ \* \ \vdots \\\\ \cdot \ \* \ \* \\\\ \lambda\_n \end{bmatrix}$$
is an upper triangular matrix, where we choose the first column of *U* as **v** (*U* there exists from a well known result of Schur). Then,
$$(\mathbf{U}^{-1}(A+\mathbf{v}\mathbf{q}^{T})\mathbf{U} = \mathbf{U}^{-1}A\mathbf{U} + \begin{bmatrix} q\_{1} \ q\_{2} \cdot \cdots \ q\_{n} \\\\\\\\\\\end{bmatrix} \mathbf{U} = \begin{bmatrix} \lambda\_{1} + \mathbf{q}^{T}\mathbf{v} & \* & \cdots & \* \\\\ & \lambda\_{2} & \ddots & \vdots \\\\&& & \ddots & \* \\\\ & & & \lambda\_{n} \end{bmatrix} .$$
and the result follows. This proof is due to Reams [42].
**Theorem 2.** *Rado [18] Let A be an n* × *n arbitrary matrix with eigenvalues λ*1,..., *λ<sup>n</sup> and let* Ω = *diag*{*λ*1,..., *λr*} *for some r* ≤ *n*. *Let X be an n* × *r matrix with rank r such that its columns x*1, *x*2,..., *xr satisfy Axi* = *λixi*, *i* = 1, . . . ,*r*. *Let C be an r* × *n arbitrary matrix. Then the matrix A* + *XC has eigenvalues μ*1,..., *μr*, *λr*+1,..., *λn*, *where μ*1,..., *μ<sup>r</sup> are eigenvalues of the matrix* Ω + *CX*.
**Proof.** Let *<sup>S</sup>* <sup>=</sup> [*<sup>X</sup>* <sup>|</sup> *<sup>Y</sup>*] a nonsingular matrix with *<sup>S</sup>*−<sup>1</sup> <sup>=</sup> [ *U <sup>V</sup>*]. Then *UX* = *Ir*, *VY* = *In*−*<sup>r</sup>* and *VX* = 0, *UY* = 0. Let *C* = [*C*<sup>1</sup> | *C*2] , *X* = [ *X*<sup>1</sup> *X*<sup>2</sup> ], *Y* = [ *Y*1 *Y*2 ]. Then, since *AX* = *X*Ω,
$$S^{-1}AS = \begin{bmatrix} \mathcal{U} \\ V \end{bmatrix} \begin{bmatrix} X\Omega \mid AY \end{bmatrix} = \begin{bmatrix} \Omega \downarrow \mathcal{U}AY \\ \mathbf{0} \ \mathcal{V}AY \end{bmatrix}$$
and
$$\mathbf{S}^{-1}\mathbf{X}\mathbf{C}\mathbf{S} = \begin{bmatrix} I\_r \\ 0 \end{bmatrix} \begin{bmatrix} \mathbf{C}\_1 \ \mathbf{C}\_2 \end{bmatrix} \mathbf{S} = \begin{bmatrix} \mathbf{C}\_1 \ \mathbf{C}\_2 \\ 0 \ 0 \end{bmatrix} \begin{bmatrix} X\_1 \ Y\_1 \\ X\_2 \ Y\_2 \end{bmatrix} = \begin{bmatrix} \mathbf{C}\mathbf{X} \ \mathbf{C}\mathbf{Y} \\ 0 \ 0 \end{bmatrix}.$$
Thus,
$$\mathbf{S}^{-1}(A+\mathbf{XC})\mathbf{S} = \mathbf{S}^{-1}A\mathbf{S} + \mathbf{S}^{-1}\mathbf{XCS} = \begin{bmatrix} \Omega + \mathbf{C}X \ UAAY + \mathbf{C}Y\\ \mathbf{0} & \forall AY \end{bmatrix}\mathbf{A}$$
and we have *σ*(*A* + *XC*) = *σ*(Ω + *CX*) + *σ*(*A*) − *σ*(Ω).
#### **3. Real nonnegative inverse eigenvalue problem.**
Regarding the *RNIEP,* by applying Brauer Theorem and Rado Theorem, efficient and general sufficient conditions have been obtained in [18, 22, 24, 36].
**Theorem 3.** *[24] Let* Λ = {*λ*1, *λ*2, ..., *λn*} *be a given list of real numbers. Suppose that: i*) *There exists a partition* Λ = Λ<sup>1</sup> ∪ ... ∪ Λ*t*, *where*
Λ*<sup>k</sup>* = {*λk*1, *λk*2,... *λkpk* }, *λ*<sup>11</sup> = *λ*1, *λk*<sup>1</sup> ≥···≥ *λkpk* , *λk*<sup>1</sup> ≥ 0,
*k* = 1, . . . , *t*, *such that for each sublist* Λ*<sup>k</sup> we associate a corresponding list*
$$\Gamma\_k = \{\omega\_{k'}\lambda\_{k2'}...\lambda\_{kp\_k}\}, \ 0 \le \omega\_k \le \lambda\_{1\nu}$$
*which is realizable by a nonnegative matrix Ak* ∈ CS*ω<sup>k</sup> of order pk*. *ii*) *There exists a nonnegative matrix B* ∈ CS*λ*<sup>1</sup> *with eigenvalues λ*1, *λ*21, ..., *λt*<sup>1</sup> *(the first elements of the lists* Λ*k) and diagonal entries ω*1, *ω*2,..., *ω<sup>t</sup> (the first elements of the lists* Γ*k). Then* Λ *is realizable by a nonnegative matrix A* ∈ CS*λ*<sup>1</sup> .
Perfect [18] gave conditions under which *λ*1, *λ*2, ..., *λ<sup>t</sup>* and *ω*1, *ω*2,..., *ω<sup>t</sup>* are the eigenvalues and the diagonal entries, respectively, of a *t* × *t* nonnegative matrix *B* ∈ *CSλ*<sup>1</sup> . For *t* = 2 it is necessary and sufficient that *λ*<sup>1</sup> + *λ*<sup>2</sup> = *ω*<sup>1</sup> + *ω*2, with 0 ≤ *ω<sup>i</sup>* ≤ *λ*1. For *t* = 3 Perfect gave the following result:
**Theorem 4.** *[18] The real numbers λ*1, *λ*2, *λ*<sup>3</sup> *and ω*1, *ω*2, *ω*<sup>3</sup> *are the eigenvalues and the diagonal entries, respectively, of a* 3 × 3 *nonnegative matrix B* ∈ CS*λ*<sup>1</sup> , *if and only if:*
$$\begin{array}{ll} i) & 0 \le \omega\_i \le \lambda\_1, \quad i = 1, 2, 3\\ ii) & \lambda\_1 + \lambda\_2 + \lambda\_3 = \omega\_1 + \omega\_2 + \omega\_3\\ iii) & \lambda\_1\lambda\_2 + \lambda\_1\lambda\_3 + \lambda\_2\lambda\_3 \le \omega\_1\omega\_2 + \omega\_1\omega\_3 + \omega\_2\omega\_3\\ iv) & \max\_k \omega\_k \ge \lambda\_2 \end{array} \tag{3}$$
*Then, an appropriate* 3 × 3 *nonnegative matrix B is*
$$B = \begin{bmatrix} \omega\_1 & 0 & \lambda\_1 - \omega\_1 \\ \lambda\_1 - \omega\_2 - p & \omega\_2 & p \\ 0 & \lambda\_1 - \omega\_3 & \omega\_3 \end{bmatrix} \tag{4}$$
*where*
$$p = \frac{1}{\lambda\_1 - \omega\_3} (\omega\_1 \omega\_2 + \omega\_1 \omega\_3 + \omega\_2 \omega\_3 - \lambda\_1 \lambda\_2 + \lambda\_1 \lambda\_3 + \lambda\_2 \lambda\_3).$$
For *t* ≥ 4, we only have a sufficient condition:
4 Will-be-set-by-IN-TECH
� *C*<sup>1</sup> *C*<sup>2</sup> 0 0
Regarding the *RNIEP,* by applying Brauer Theorem and Rado Theorem, efficient and general
Λ*<sup>k</sup>* = {*λk*1, *λk*2,... *λkpk* }, *λ*<sup>11</sup> = *λ*1, *λk*<sup>1</sup> ≥···≥ *λkpk* , *λk*<sup>1</sup> ≥ 0,
Γ*<sup>k</sup>* = {*ωk*, *λk*2, ..., *λkpk* }, 0 ≤ *ω<sup>k</sup>* ≤ *λ*1,
*ii*) *There exists a nonnegative matrix B* ∈ CS*λ*<sup>1</sup> *with eigenvalues λ*1, *λ*21, ..., *λt*<sup>1</sup> *(the first elements of*
Perfect [18] gave conditions under which *λ*1, *λ*2, ..., *λ<sup>t</sup>* and *ω*1, *ω*2,..., *ω<sup>t</sup>* are the eigenvalues and the diagonal entries, respectively, of a *t* × *t* nonnegative matrix *B* ∈ *CSλ*<sup>1</sup> . For *t* = 2 it is necessary and sufficient that *λ*<sup>1</sup> + *λ*<sup>2</sup> = *ω*<sup>1</sup> + *ω*2, with 0 ≤ *ω<sup>i</sup>* ≤ *λ*1. For *t* = 3 Perfect gave the
**Theorem 4.** *[18] The real numbers λ*1, *λ*2, *λ*<sup>3</sup> *and ω*1, *ω*2, *ω*<sup>3</sup> *are the eigenvalues and the diagonal*
*iii*) *λ*1*λ*<sup>2</sup> + *λ*1*λ*<sup>3</sup> + *λ*2*λ*<sup>3</sup> ≤ *ω*1*ω*<sup>2</sup> + *ω*1*ω*<sup>3</sup> + *ω*2*ω*<sup>3</sup>
*λ*<sup>1</sup> − *ω*<sup>2</sup> − *p ω*<sup>2</sup> *p* 0 *λ*<sup>1</sup> − *ω*<sup>3</sup> *ω*<sup>3</sup>
*ω*<sup>1</sup> 0 *λ*<sup>1</sup> − *ω*<sup>1</sup>
(*ω*1*ω*<sup>2</sup> + *ω*1*ω*<sup>3</sup> + *ω*2*ω*<sup>3</sup> − *λ*1*λ*<sup>2</sup> + *λ*1*λ*<sup>3</sup> + *λ*2*λ*3).
⎤
⎦ , (4)
**Theorem 3.** *[24] Let* Λ = {*λ*1, *λ*2, ..., *λn*} *be a given list of real numbers. Suppose that:*
� � *X*<sup>1</sup> *Y*<sup>1</sup> *X*<sup>2</sup> *Y*<sup>2</sup>
�
� = � *CX CY* 0 0
Ω + *CX UAY* + *CY* 0 *VAY*
� .
> � ,
> > (3)
[*C*<sup>1</sup> | *C*2] *S* =
*S*−1(*A* + *XC*)*S* = *S*−1*AS* + *S*−1*XCS* =
*k* = 1, . . . , *t*, *such that for each sublist* Λ*<sup>k</sup> we associate a corresponding list*
*the lists* Λ*k) and diagonal entries ω*1, *ω*2,..., *ω<sup>t</sup> (the first elements of the lists* Γ*k).*
*entries, respectively, of a* 3 × 3 *nonnegative matrix B* ∈ CS*λ*<sup>1</sup> , *if and only if:*
*iv*) *maxkω<sup>k</sup>* ≥ *λ*<sup>2</sup>
⎡ ⎣
*B* =
*Then, an appropriate* 3 × 3 *nonnegative matrix B is*
*<sup>p</sup>* <sup>=</sup> <sup>1</sup>
*λ*<sup>1</sup> − *ω*<sup>3</sup>
*i*) 0 ≤ *ω<sup>i</sup>* ≤ *λ*1, *i* = 1, 2, 3 *ii*) *λ*<sup>1</sup> + *λ*<sup>2</sup> + *λ*<sup>3</sup> = *ω*<sup>1</sup> + *ω*<sup>2</sup> + *ω*<sup>3</sup>
*which is realizable by a nonnegative matrix Ak* ∈ CS*ω<sup>k</sup> of order pk*.
*Then* Λ *is realizable by a nonnegative matrix A* ∈ CS*λ*<sup>1</sup> .
and
Thus,
following result:
*where*
*S*−1*XCS* =
� *Ir* 0 �
and we have *σ*(*A* + *XC*) = *σ*(Ω + *CX*) + *σ*(*A*) − *σ*(Ω).
**3. Real nonnegative inverse eigenvalue problem.**
sufficient conditions have been obtained in [18, 22, 24, 36].
*i*) *There exists a partition* Λ = Λ<sup>1</sup> ∪ ... ∪ Λ*t*, *where*
$$\begin{array}{ll} i) & 0 \le \omega\_k \le \lambda\_{1\prime} \ k = 1, 2, \dots, t\_{\prime} \\ ii) & \omega\_1 + \omega\_2 \cdot \dots + \omega\_t = \lambda\_1 + \lambda\_2 \cdot \dots + \lambda\_{t\prime} \\ iii) & \omega\_k \ge \lambda\_{k\prime} \ \omega\_1 \ge \lambda\_{k\prime} \ k = 2, 3, \dots, t\_{\prime} \end{array} \tag{5}$$
with the following matrix *B* ∈ *CSλ*<sup>1</sup> having eigenvalues and diagonal entries *λ*1, *λ*2,..., *λ<sup>t</sup>* and *ω*1, *ω*2,..., *ωt*, respectively:
$$B = \begin{bmatrix} \omega\_1 & \omega\_2 - \lambda\_2 \cdot \cdots \cdot \omega\_r - \lambda\_t \\ \omega\_1 - \lambda\_2 & \omega\_2 & \cdots \cdot \omega\_r - \lambda\_t \\ \vdots & \vdots & \ddots & \vdots \\ \omega\_1 - \lambda\_t \cdot \omega\_2 - \lambda\_2 \cdot \cdots \cdot \omega\_t \end{bmatrix} \tag{6}$$
**Example 1.** *Let us consider the list* Λ = {6, 1, 1, −4, −4} *with the partition*
$$
\Lambda\_1 = \{6, -4\}, \ \Lambda\_2 = \{1, -4\}, \ \Lambda\_3 = \{1\}
$$
*and the realizable associated lists*
$$
\Gamma\_1 = \{4, -4\}, \; \Gamma\_2 = \{4, -4\}, \; \Gamma\_3 = \{0\}.
$$
*From (4) we compute the* 3 × 3 *nonnegative matrix*
$$B = \begin{bmatrix} 4 \ 0 \ 2 \\ \frac{3}{2} \ 4 \ \frac{1}{2} \\ 0 \ 6 \ 0 \end{bmatrix}$$
*with eigenvalues* 6, 1, 1, *and diagonal entries* 4, 4, 0. *Then*
$$\begin{aligned} A &= \begin{bmatrix} 0 \ 4 \ 0 \ 0 \ 0 \\ 4 \ 0 \ 0 \ 0 \\ 0 \ 0 \ 0 \ 4 \ 0 \\ 0 \ 0 \ 4 \ 0 \\ 0 \ 0 \ 0 \ 0 \\ 0 \end{bmatrix} + \begin{bmatrix} 1 \ 0 \ 0 \\ 1 \ 0 \ 0 \\ 0 \ 1 \ 0 \\ 0 \ 0 \ 1 \\ 0 \ 0 \ 1 \end{bmatrix} \begin{bmatrix} 0 \ 0 \ 0 \ 0 \ 2 \\ \frac{3}{2} \ 0 \ 0 \ 0 \ \frac{1}{2} \\ 0 \ 0 \ 6 \ 0 \ 0 \end{bmatrix} \\ &= \begin{bmatrix} 0 \ 4 \ 0 \ 0 \ 2 \\ 4 \ 0 \ 0 \ 0 \ 2 \\ \frac{3}{2} \ 0 \ 0 \ 4 \ \frac{1}{2} \\ \frac{3}{2} \ 0 \ 4 \ 0 \ \frac{1}{2} \\ 0 \ 0 \ 6 \ 0 \ 0 \end{bmatrix} \end{aligned}$$
*is nonnegative with spectrum* Λ.
A map of sufficient conditions for the *RNIEP* it was constructed in [28]*,* There, the sufficient conditions were compared to establish inclusion or independence relations between them. It is also shown in [28] that the criterion given by Theorem 3 contains all realizability criteria for lists of real numbers studied therein. In [36], from a new special partition, Theorem 3 is extended. Now, the first element *λk*<sup>1</sup> of the sublist Λ*<sup>k</sup>* need not to be nonnegative and the realizable auxiliar list Γ*<sup>k</sup>* = {*ωk*, *λk*1, ..., *λkpk* } contains one more element. Moreover, the number of lists of the partition depend on the number of elements of the first list Λ1, and some lists Λ*<sup>k</sup>* can be empty.
**Theorem 5.** *[36] Let* Λ = {*λ*1, *λ*2,..., *λn*} *be a list of real numbers and let the partition* Λ = Λ<sup>1</sup> ∪···∪ Λ*p*1+<sup>1</sup> *be such that*
$$\Lambda\_k = \{\lambda\_{k1}, \lambda\_{k2}, \dots \lambda\_{kp\_k}\}, \ \lambda\_{11} = \lambda\_{1\prime} \ \lambda\_{k1} \ge \lambda\_{k2} \ge \dots \ge \lambda\_{kp\_{k^{\prime}}}$$
*k* = 1, . . . , *p*<sup>1</sup> + 1, *where p*<sup>1</sup> *is the number of elements of the list* Λ<sup>1</sup> *and some of the lists* Λ*<sup>k</sup> can be empty. Let ω*2,..., *ωp*1+<sup>1</sup> *be real numbers satisfying* 0 ≤ *ω<sup>k</sup>* ≤ *λ*1, *k* = 2, . . . , *p*<sup>1</sup> + 1. *Suppose that the following conditions hold:*
*i*) *For each k* = 2, . . . , *p*<sup>1</sup> + 1, *there exists a nonnegative matrix Ak* ∈ CS*ω<sup>k</sup> with spectrum* Γ*<sup>k</sup>* = {*ωk*, *λk*1, ..., *λkpk* },
*ii*) *There exists a p*<sup>1</sup> × *p*<sup>1</sup> *nonnegative matrix B* ∈ CS*λ*1, *with spectrum* Λ<sup>1</sup> *and with diagonal entries ω*2,..., *ωp*1+1.
*Then* Λ *is realizable by a nonnegative matrix A* ∈ CS*λ*<sup>1</sup> .
**Example 2.** *With this extension, the authors show for instance, that the list*
$$\{5, 4, 0, -3, -3, -3\}$$
*is realizable, which can not be done from the criterion given by Theorem 3. In fact, let the partition*
$$\begin{aligned} \Lambda\_1 &= \{5, 4, 0, -3\}, \,\Lambda\_2 = \{-3\}, \,\Lambda\_3 = \{-3\} \text{ with} \\ \Gamma\_2 &= \{3, -3\}, \,\Gamma\_3 = \{3, -3\}, \,\Gamma\_4 = \Gamma\_5 = \{0\}. \end{aligned}$$
*The nonnegative matrix*
$$B = \begin{bmatrix} \mathbf{3} \ \mathbf{0} \ \mathbf{2} \ \mathbf{0} \\ \mathbf{0} \ \mathbf{3} \ \mathbf{0} \ \mathbf{2} \\ \mathbf{3} \ \mathbf{0} \ \mathbf{0} \ \mathbf{2} \\ \mathbf{0} \ \mathbf{3} \ \mathbf{2} \ \mathbf{0} \end{bmatrix}$$
*has spectrum* Λ<sup>1</sup> *and diagonal entries* 3, 3, 0, 0. *It is clear that*
$$A\_2 = A\_3 = \begin{bmatrix} 0 \ 3 \\ 3 \ 0 \end{bmatrix} \text{ realizes } \Gamma\_2 = \Gamma\_3.$$
*Then*
$$A = \begin{bmatrix} A\_2 \\ A\_3 \\ 0 \\ 0 \end{bmatrix} + \begin{bmatrix} 1 \ 0 \ 0 \ 0 \\ 1 \ 0 \ 0 \ 0 \\ 0 \ 1 \ 0 \ 0 \\ 0 \ 1 \ 0 \ 0 \\ 0 \ 0 \ 1 \ 0 \\ 0 \ 0 \ 0 \ 1 \end{bmatrix} \begin{bmatrix} 0 \ 0 \ 0 \ 0 \ 2 \ 0 \\ 0 \ 0 \ 0 \ 0 \ 0 \ 2 \\ 3 \ 0 \ 0 \ 0 \ 0 \ 2 \\ 0 \ 0 \ 3 \ 0 \ 2 \ 0 \\ 0 \ 0 \ 3 \ 0 \ 2 \ 0 \end{bmatrix} = \begin{bmatrix} 0 \ 3 \ 0 \ 0 \ 0 \ 2 \ 0 \\ 3 \ 0 \ 0 \ 0 \ 2 \ 0 \\ 0 \ 0 \ 0 \ 3 \ 0 \ 2 \\ 0 \ 0 \ 3 \ 0 \ 0 \ 2 \\ 3 \ 0 \ 0 \ 0 \ 0 \ 2 \\ 0 \ 0 \ 3 \ 0 \ 2 \ 0 \end{bmatrix}$$
*has the desired spectrum* {5, 4, 0, −3, −3, −3}.
#### **4. Symmetric nonnegative inverse eigenvalue problem**
Several realizability criteria which were first obtained for the *RNIEP* have later been shown to be symmetric realizability criteria as well. For example, Kellogg criterion [19] was showed by Fiedler [29] to imply symmetric realizability. It was proved by Radwan [8] that Borobia's criterion [21] is also a symmetric realizability criterion, and it was proved in [33] that Soto's criterion for the *RNIEP* is also a criterion for the *SNIEP.* In this section we shall consider the most general and efficient symmetric realizability criteria for the *SNIEP* (as far as we know they are). We start by introducing a symmetric version of the Rado Theorem:.
**Theorem 6.** *[34] Let A be an n* × *n symmetric matrix with spectrum* Λ = {*λ*1, *λ*2,..., *λn*} *and for some r* ≤ *n*, *let* {**x**1, **x**2,..., **x***r*} *be an orthonormal set of eigenvectors of A spanning the invariant subspace associated with λ*1, *λ*2,..., *λr*. *Let X be the n* × *r matrix with i* − *th column* **x***i*, *let* Ω = *diag*{*λ*1,..., *<sup>λ</sup>r*}, *and let C be any r* <sup>×</sup> *r symmetric matrix. Then the symmetric matrix A* <sup>+</sup> *XCX<sup>T</sup> has eigenvalues μ*1, *μ*2,..., *μr*, *λr*+1,..., *λn*, *where μ*1, *μ*2,..., *μ<sup>r</sup> are the eigenvalues of the matrix* Ω + *C*.
**Proof.** Since the columns of *X* are an orthonormal set, we may complete *X* to an orthogonal matrix *<sup>W</sup>* = [*X Y*], that is, *<sup>X</sup>TX* <sup>=</sup> *Ir*, *<sup>Y</sup>TY* <sup>=</sup> *In*−*r*, *<sup>X</sup>TY* <sup>=</sup> 0, *<sup>Y</sup>TX* <sup>=</sup> 0. Then
$$W^{-1}AW = \begin{bmatrix} X^T \\ Y^T \end{bmatrix} A \begin{bmatrix} X & Y \end{bmatrix} = \begin{bmatrix} \Omega \ X^T A Y \\ 0 \ Y^T A Y \end{bmatrix}.$$
$$W^{-1}(\mathbf{X} \mathbf{C} \mathbf{X}^T)W = \begin{bmatrix} I\_r \\ \mathbf{0} \end{bmatrix} \mathbf{C} \begin{bmatrix} I\_r & \mathbf{0} \end{bmatrix} = \begin{bmatrix} \mathbf{C} \ \mathbf{0} \\ \mathbf{0} \ \mathbf{0} \end{bmatrix}.$$
Therefore,
6 Will-be-set-by-IN-TECH
**Theorem 5.** *[36] Let* Λ = {*λ*1, *λ*2,..., *λn*} *be a list of real numbers and let the partition* Λ =
Λ*<sup>k</sup>* = {*λk*1, *λk*2,... *λkpk* }, *λ*<sup>11</sup> = *λ*1, *λk*<sup>1</sup> ≥ *λk*<sup>2</sup> ≥···≥ *λkpk* , *k* = 1, . . . , *p*<sup>1</sup> + 1, *where p*<sup>1</sup> *is the number of elements of the list* Λ<sup>1</sup> *and some of the lists* Λ*<sup>k</sup> can be empty. Let ω*2,..., *ωp*1+<sup>1</sup> *be real numbers satisfying* 0 ≤ *ω<sup>k</sup>* ≤ *λ*1, *k* = 2, . . . , *p*<sup>1</sup> + 1. *Suppose that*
*i*) *For each k* = 2, . . . , *p*<sup>1</sup> + 1, *there exists a nonnegative matrix Ak* ∈ CS*ω<sup>k</sup> with spectrum* Γ*<sup>k</sup>* =
*ii*) *There exists a p*<sup>1</sup> × *p*<sup>1</sup> *nonnegative matrix B* ∈ CS*λ*1, *with spectrum* Λ<sup>1</sup> *and with diagonal entries*
{5, 4, 0, −3, −3, −3} *is realizable, which can not be done from the criterion given by Theorem 3. In fact, let the partition*
> Λ<sup>1</sup> = {5, 4, 0, −3}, Λ<sup>2</sup> = {−3}, Λ<sup>3</sup> = {−3} *with* Γ<sup>2</sup> = {3, −3}, Γ<sup>3</sup> = {3, −3}, Γ<sup>4</sup> = Γ<sup>5</sup> = {0}.
⎤ ⎥ ⎥ ⎦
*realizes* Γ<sup>2</sup> = Γ3.
⎤ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
⎡ ⎢ ⎢ ⎣
*B* =
� 0 3 3 0 �
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
⎡ ⎢ ⎢ ⎣
Several realizability criteria which were first obtained for the *RNIEP* have later been shown to be symmetric realizability criteria as well. For example, Kellogg criterion [19] was showed by Fiedler [29] to imply symmetric realizability. It was proved by Radwan [8] that Borobia's criterion [21] is also a symmetric realizability criterion, and it was proved in [33] that Soto's criterion for the *RNIEP* is also a criterion for the *SNIEP.* In this section we shall consider the most general and efficient symmetric realizability criteria for the *SNIEP* (as far as we know
Λ<sup>1</sup> ∪···∪ Λ*p*1+<sup>1</sup> *be such that*
*the following conditions hold:*
*Then* Λ *is realizable by a nonnegative matrix A* ∈ CS*λ*<sup>1</sup> .
*has spectrum* Λ<sup>1</sup> *and diagonal entries* 3, 3, 0, 0. *It is clear that*
*A*<sup>2</sup> = *A*<sup>3</sup> =
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
**4. Symmetric nonnegative inverse eigenvalue problem**
they are). We start by introducing a symmetric version of the Rado Theorem:.
⎤ ⎥ ⎥ ⎦ +
**Example 2.** *With this extension, the authors show for instance, that the list*
{*ωk*, *λk*1, ..., *λkpk* },
*The nonnegative matrix*
*A* =
⎡ ⎢ ⎢ ⎣
*A*2 *A*3 0 0
*has the desired spectrum* {5, 4, 0, −3, −3, −3}.
*Then*
*ω*2,..., *ωp*1+1.
$$W^{-1}(A + XCX^T)W = \begin{bmatrix} \Omega + \mathcal{C} \ X^T A Y\\ 0 & Y^T A Y \end{bmatrix}$$
and *<sup>A</sup>* + *XCX<sup>T</sup>* is symmetric with eigenvalues *<sup>μ</sup>*1,..., *<sup>μ</sup>r*, *<sup>λ</sup>r*+1,..., *<sup>λ</sup>n*.
By using Theorem 6, the following sufficient condition was proved in [34]:
**Theorem 7.** *[34] Let* Λ = {*λ*1, *λ*2,..., *λn*} *be a list of real numbers with λ*<sup>1</sup> ≥ *λ*<sup>2</sup> ≥···≥ *λ<sup>n</sup> and, for some t* ≤ *n*, *let ω*1,..., *ω<sup>t</sup> be real numbers satisfying* 0 ≤ *ω<sup>k</sup>* ≤ *λ*1, *k* = 1, . . . , *t*. *Suppose there exists:*
*i*) *a partition* Λ = Λ<sup>1</sup> ∪···∪ Λ*<sup>t</sup> with*
$$
\Lambda\_{\mathbf{k}} = \{\lambda\_{\mathbf{k}1\prime}\lambda\_{\mathbf{k}2\prime}\dots\lambda\_{\mathbf{k}p\_{\mathbf{k}}}\}, \ \lambda\_{11} = \lambda\_{1\prime} \quad \lambda\_{\mathbf{k}1} \ge 0, \ \lambda\_{\mathbf{k}1} \ge \lambda\_{\mathbf{k}2} \ge \dots \ge \lambda\_{\mathbf{k}p\_{\mathbf{k}}}.
$$
*such that for each k* = 1, . . . , *t*, *the list* Γ*<sup>k</sup>* = {*ωk*, *λk*2, ..., *λkpk* } *is realizable by a symmetric nonnegative matrix Ak of order pk*, *and*
*ii*) *a t* × *t symmetric nonnegative matrix B with eigenvalues λ*11, *λ*21, ..., *λt*1} *and with diagonal entries ω*1, *ω*2,..., *ωt*.
*Then* Λ *is realizable by a symmetric nonnegative matrix.*
**Proof.** Since *Ak* is a *pk* × *pk* symmetric nonnegative matrix realizing Γ*k*, then *A* = *diag*{*A*1, *A*2,..., *At*} is symmetric nonnegative with spectrum Γ<sup>1</sup> ∪ Γ<sup>2</sup> ∪ ··· ∪ Γ*t*. Let {**x**1,..., **x***t*} be an orthonormal set of eigenvectors of *A* associated with *ω*1,..., *ωt*, respectively. Then the *n* × *t* matrix *X* with *i* − *th* column **x***<sup>i</sup>* satisfies *AX* = *X*Ω for Ω = *dig*{*ω*1,..., *ωt*}. Moreover, *X* is entrywise nonnegative, since each **x***<sup>i</sup>* contains the Perron eigenvector of *Ai* and zeros. Now, if we set *C* = *B* − Ω, the matrix *C* is symmetric nonnegative and Ω + *C* has eigenvalues *λ*1,..., *λt*. Therefore, by Theorem 6 the symmetric matrix *A* + *XCX<sup>T</sup>* has spectrum Λ. Besides, it is nonnegative since all the entries of *A*, *X*, and *C* are nonnegative.
Theorem 7 not only ensures the existence of a realizing matrix, but it also allows to construct the realizing matrix. Of course, the key is to know under which conditions does there exists a *t* × *t* symmetrix nonnegative matrix *B* with eigenvalues *λ*1,..., *λ<sup>t</sup>* and diagonal entries *ω*1,..., *ωt*.
#### 8 Will-be-set-by-IN-TECH 106 Linear Algebra – Theorems and Applications
The following conditions for the existence of a real symmetric matrix, not necessarily nonnegative, with prescribed eigenvalues and diagonal entries are due to Horn [43]: *There exists a real symmetric matrix with eigenvalues λ*<sup>1</sup> ≥ *λ*<sup>2</sup> ≥ ··· ≥ *λ<sup>t</sup> and diagonal entries ω*<sup>1</sup> ≥ *ω*<sup>2</sup> ≥···≥ *ω<sup>t</sup> if and only if*
$$\begin{aligned} \sum\_{i=1}^{k} \lambda\_i &\ge \sum\_{i=1}^{k} \omega\_{i\star} \ k = 1, \dots, t-1 \\ \sum\_{i=1}^{t} \lambda\_i &= \sum\_{i=1}^{t} \omega\_i \end{aligned} \tag{7}$$
For *t* = 2, the conditions (7) become
$$
\lambda\_1 \ge \omega\_{1\prime} \quad \lambda\_1 + \lambda\_2 = \omega\_1 + \omega\_{2\prime}
$$
and they are also sufficient for the existence of a 2 × 2 symmetric nonnegative matrix *B* with eigenvalues *λ*<sup>1</sup> ≥ *λ*<sup>2</sup> and diagonal entries *ω*<sup>1</sup> ≥ *ω*<sup>2</sup> ≥ 0, namely,
$$B = \left[ \frac{\omega\_1}{\sqrt{(\lambda\_1 - \omega\_1)(\lambda\_1 - \omega\_2)}} \sqrt{\frac{(\lambda\_1 - \omega\_1)(\lambda\_1 - \omega\_2)}{\omega\_2}} \right].$$
For *t* = 3, we have the following conditions:
**Lemma 1.** *[29] The conditions*
$$\begin{array}{c} \lambda\_1 \ge \omega\_1\\ \lambda\_1 + \lambda\_2 \ge \omega\_1 + \omega\_2\\ \lambda\_1 + \lambda\_2 + \lambda\_3 = \omega\_1 + \omega\_2 + \omega\_3\\ \omega\_1 \ge \lambda\_2 \end{array} \tag{8}$$
*are necessary and sufficient for the existence of a* 3 × 3 *symmetric nonnegative matrix B with eigenvalues λ*<sup>1</sup> ≥ *λ*<sup>2</sup> ≥ *λ*<sup>3</sup> *and diagonal entries ω*<sup>1</sup> ≥ *ω*<sup>2</sup> ≥ *ω*<sup>3</sup> ≥ 0.
In [34], the following symmetric nonnegative matrix *B*, satisfying conditions (8), it was constructed:
$$B = \begin{bmatrix} \omega\_1 & \sqrt{\frac{\mu - \omega\_3}{2\mu - \omega\_2 - \omega\_3}} s & \sqrt{\frac{\mu - \omega\_2}{2\mu - \omega\_2 - \omega\_3}} s\\ \sqrt{\frac{\mu - \omega\_2}{2\mu - \omega\_2 - \omega\_3}} s & \omega\_2 & \sqrt{(\mu - \omega\_2)(\mu - \omega\_3)}\\ \sqrt{\frac{\mu - \omega\_2}{2\mu - \omega\_2 - \omega\_3}} s & \sqrt{(\mu - \omega\_2)(\mu - \omega\_3)} & \omega\_3 \end{bmatrix},\tag{9}$$
where *<sup>μ</sup>* <sup>=</sup> *<sup>λ</sup>*<sup>1</sup> <sup>+</sup> *<sup>λ</sup>*<sup>2</sup> <sup>−</sup> *<sup>ω</sup>*1; *<sup>s</sup>* <sup>=</sup> �(*λ*<sup>1</sup> <sup>−</sup> *<sup>μ</sup>*)(*λ*<sup>1</sup> <sup>−</sup> *<sup>ω</sup>*1).
For *t* ≥ 4 we have only a sufficient condition:
**Theorem 8.** *Fiedler [29] If λ*<sup>1</sup> ≥···≥ *λ<sup>t</sup> and ω*<sup>1</sup> ≥···≥ *ω<sup>t</sup> satisfy*
$$\begin{aligned} \text{i)} \quad & \sum\_{i=1}^{s} \lambda\_i \ge \sum\_{i=1}^{s} \omega\_{i\prime} \text{ s} = 1, \dots, t-1 \\ \text{ii)} \quad & \sum\_{i=1}^{t} \lambda\_i = \sum\_{i=1}^{t} \omega\_i \\ \text{iii)} \, & \omega\_{k-1} \ge \lambda\_{k\prime} \text{ } k = 2, \dots, t-1 \end{aligned} \tag{10}$$
*then there exists a t* × *t symmetric nonnegative matrix with eigenvalues λ*1,..., *λ<sup>t</sup> and diagonal entries ω*1,..., *ωt*.
Observe that
8 Will-be-set-by-IN-TECH
The following conditions for the existence of a real symmetric matrix, not necessarily nonnegative, with prescribed eigenvalues and diagonal entries are due to Horn [43]: *There exists a real symmetric matrix with eigenvalues λ*<sup>1</sup> ≥ *λ*<sup>2</sup> ≥ ··· ≥ *λ<sup>t</sup> and diagonal entries*
*λ*<sup>1</sup> ≥ *ω*1, *λ*<sup>1</sup> + *λ*<sup>2</sup> = *ω*<sup>1</sup> + *ω*2, and they are also sufficient for the existence of a 2 × 2 symmetric nonnegative matrix *B* with
(*λ*<sup>1</sup> − *ω*1)(*λ*<sup>1</sup> − *ω*2) *ω*<sup>2</sup>
*λ*<sup>1</sup> ≥ *ω*<sup>1</sup> *λ*<sup>1</sup> + *λ*<sup>2</sup> ≥ *ω*<sup>1</sup> + *ω*<sup>2</sup> *λ*<sup>1</sup> + *λ*<sup>2</sup> + *λ*<sup>3</sup> = *ω*<sup>1</sup> + *ω*<sup>2</sup> + *ω*<sup>3</sup> *ω*<sup>1</sup> ≥ *λ*<sup>2</sup>
*are necessary and sufficient for the existence of a* 3 × 3 *symmetric nonnegative matrix B with*
In [34], the following symmetric nonnegative matrix *B*, satisfying conditions (8), it was
�(*<sup>μ</sup>* <sup>−</sup> *<sup>ω</sup>*2)(*<sup>μ</sup>* <sup>−</sup> *<sup>ω</sup>*3) *<sup>ω</sup>*<sup>3</sup>
*ωi*, *s* = 1, . . . , *t* − 1
� *<sup>μ</sup>*−*ω*<sup>3</sup> <sup>2</sup>*μ*−*ω*2−*ω*<sup>3</sup> *<sup>s</sup>*
> *s* ∑ *i*=1
*t* ∑ *i*=1 *ωi*
*iii*) *<sup>ω</sup>k*−<sup>1</sup> ≥ *<sup>λ</sup>k*, *<sup>k</sup>* = 2, . . . , *<sup>t</sup>* − <sup>1</sup>
*λ<sup>i</sup>* =
�
*ωi*, *k* = 1, . . . , *t* − 1
⎫ ⎪⎪⎪⎬
⎪⎪⎪⎭
(*λ*<sup>1</sup> − *ω*1)(*λ*<sup>1</sup> − *ω*2)
⎫ ⎪⎪⎬
⎪⎪⎭
� *<sup>μ</sup>*−*ω*<sup>2</sup> <sup>2</sup>*μ*−*ω*2−*ω*<sup>3</sup> *<sup>s</sup>*
�(*<sup>μ</sup>* <sup>−</sup> *<sup>ω</sup>*2)(*<sup>μ</sup>* <sup>−</sup> *<sup>ω</sup>*3)
⎫ ⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎭
⎤ ⎥ ⎥ ⎥
, (10)
<sup>⎦</sup> , (9)
� . (7)
(8)
*ω*<sup>1</sup> ≥ *ω*<sup>2</sup> ≥···≥ *ω<sup>t</sup> if and only if*
For *t* = 2, the conditions (7) become
*B* =
For *t* = 3, we have the following conditions:
**Lemma 1.** *[29] The conditions*
*B* =
⎡ ⎢ ⎢ ⎢ ⎣
constructed:
*k* ∑ *i*=1 *λ<sup>i</sup>* ≥
eigenvalues *λ*<sup>1</sup> ≥ *λ*<sup>2</sup> and diagonal entries *ω*<sup>1</sup> ≥ *ω*<sup>2</sup> ≥ 0, namely,
� *ω*<sup>1</sup>
*eigenvalues λ*<sup>1</sup> ≥ *λ*<sup>2</sup> ≥ *λ*<sup>3</sup> *and diagonal entries ω*<sup>1</sup> ≥ *ω*<sup>2</sup> ≥ *ω*<sup>3</sup> ≥ 0.
<sup>2</sup>*μ*−*ω*2−*ω*<sup>3</sup> *<sup>s</sup> <sup>ω</sup>*<sup>2</sup>
**Theorem 8.** *Fiedler [29] If λ*<sup>1</sup> ≥···≥ *λ<sup>t</sup> and ω*<sup>1</sup> ≥···≥ *ω<sup>t</sup> satisfy*
*s* ∑ *i*=1 *λ<sup>i</sup>* ≥
*i*)
*ii*) *t* ∑ *i*=1
*ω*1
� *<sup>μ</sup>*−*ω*<sup>3</sup>
� *<sup>μ</sup>*−*ω*<sup>2</sup> <sup>2</sup>*μ*−*ω*2−*ω*<sup>3</sup> *<sup>s</sup>*
where *<sup>μ</sup>* <sup>=</sup> *<sup>λ</sup>*<sup>1</sup> <sup>+</sup> *<sup>λ</sup>*<sup>2</sup> <sup>−</sup> *<sup>ω</sup>*1; *<sup>s</sup>* <sup>=</sup> �(*λ*<sup>1</sup> <sup>−</sup> *<sup>μ</sup>*)(*λ*<sup>1</sup> <sup>−</sup> *<sup>ω</sup>*1).
For *t* ≥ 4 we have only a sufficient condition:
�
*t* ∑ *i*=1
*k* ∑ *i*=1
*t* ∑ *i*=1 *ωi*
*λ<sup>i</sup>* =
$$B = \begin{bmatrix} 5 & 2 & \frac{1}{2} & \frac{1}{2} \\ 2 & 5 & \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} & 5 & 2 \\ \frac{1}{2} & \frac{1}{2} & 2 & 5 \end{bmatrix}$$
has eigenvalues 8, 6, 3, 3, but *λ*<sup>2</sup> = 6 > 5 = *ω*1.
**Example 3.** *Let us consider the list* Λ = {7, 5, 1, −3, −4, −6} *with the partition*
$$\begin{aligned} \Lambda\_1 &= \{7, -6\}, & \Lambda\_2 &= \{5, -4\}, & \Lambda\_3 &= \{1, -3\} \text{ with} \\ \Gamma\_1 &= \{6, -6\}, & \Gamma\_2 &= \{4, -4\}, & \Gamma\_3 &= \{3, -3\}. \end{aligned}$$
*We look for a symmetric nonnegative matrix B with eigenvalues* 7, 5, 1 *and diagonal entries* 6, 4, 3. *Then conditions (8) are satisfied and from (9) we compute*
$$B = \begin{bmatrix} 6 & \sqrt{\frac{3}{5}} \sqrt{\frac{2}{5}} \\ \sqrt{\frac{3}{5}} & 4 & \sqrt{6} \\ \sqrt{\frac{2}{5}} & \sqrt{6} & 3 \end{bmatrix} \text{ and } \mathbb{C} = B - \Omega\_{\prime}$$
*where* Ω = *diag*{6, 4, 3}. *The symmetric matrices*
$$A\_1 = \begin{bmatrix} 0 \ 6 \\ 6 \ 0 \end{bmatrix}, \ A\_2 = \begin{bmatrix} 0 \ 4 \\ 4 \ 0 \end{bmatrix}, \ A\_3 = \begin{bmatrix} 0 \ 3 \\ 3 \ 0 \end{bmatrix}$$
*realize* Γ1, Γ2, Γ3. *Then*
$$A = \begin{bmatrix} A\_1 \\ & A\_2 \\ & & A\_3 \end{bmatrix} + \text{XCX}^T \text{ \\_where \ X = \begin{bmatrix} \frac{\sqrt{2}}{2} & 0 & 0 \\ \frac{\sqrt{2}}{2} & 0 & 0 \\ 0 & \frac{\sqrt{2}}{2} & 0 \\ 0 & \frac{\sqrt{2}}{2} & 0 \\ 0 & 0 & \frac{\sqrt{2}}{2} \\ 0 & 0 & \frac{\sqrt{2}}{2} \end{bmatrix}.$$
*is symmetric nonnegative with spectrum* Λ.
In the same way as Theorem 3 was extended to Theorem 5 (in the real case), Theorem 7 was also extended to the following result:
**Theorem 9.** *[36] Let* Λ = {*λ*1, *λ*2,..., *λn*} *be a list of real numbers and let the partition* Λ = Λ<sup>1</sup> ∪···∪ Λ*p*1+<sup>1</sup> *be such that*
$$\Lambda\_k = \{\lambda\_{k1}, \lambda\_{k2}, \dots \lambda\_{kp\_k}\}, \ \lambda\_{11} = \lambda\_{1\prime} \ \lambda\_{k1} \ge \lambda\_{k2} \ge \dots \ge \lambda\_{kp\_{k^{\prime}}}$$
*k* = 1, . . . , *p*<sup>1</sup> + 1, *where* Λ<sup>1</sup> *is symmetrically realizable, p*<sup>1</sup> *is the number of elements of* Λ<sup>1</sup> *and some lists* Λ*<sup>k</sup> can be empty. Let ω*2,..., *ωp*1+<sup>1</sup> *be real numbers satisfying* 0 ≤ *ω<sup>k</sup>* ≤ *λ*1, *k* = 2, . . . , *p*<sup>1</sup> + 1. *Suppose that the following conditions hold:*
*i*) *For each k* = 2, . . . , *p*<sup>1</sup> + 1, *there exists a symmetric nonnegative matrix Ak with spectrum* Γ*<sup>k</sup>* = {*ωk*, *λk*1, ..., *λkpk* },
*ii*) *There exists a p*<sup>1</sup> × *p*<sup>1</sup> *symmetric nonnegative matrix B with spectrum* Λ<sup>1</sup> *and with diagonal entries ω*2,..., *ωp*1+1.
*Then* Λ *is symmetrically realizable.*
**Example 4.** *Now, from Theorem 9, we can see that there exists a symmetric nonnegative matrix with spectrum* Λ = {5, 4, 0, −3, −3, −3}*, which can not be seen from Theorem 7. Moreover, we can compute a realizing matrix. In fact, let the partition*
$$\begin{aligned} \Lambda\_1 &= \{5, 4, 0, -3\}, \,\Lambda\_2 = \{-3\}, \,\Lambda\_3 = \{-3\} \text{ with} \\ \Gamma\_2 &= \{3, -3\}, \,\Gamma\_3 = \{3, -3\}, \,\Gamma\_4 = \Gamma\_5 = \{0\}. \end{aligned}$$
*The symmetric nonnegative matrix*
$$B = \begin{bmatrix} 3 & 0 & \sqrt{6} & 0 \\ 0 & 3 & 0 & \sqrt{6} \\ \sqrt{6} & 0 & 0 & 2 \\ 0 & \sqrt{6} & 2 & 0 \end{bmatrix}$$
*has spectrum* Λ<sup>1</sup> *and diagonal entries* 3, 3, 0, 0. *Let* Ω = *diag*{3, 3, 0, 0} *and*
$$X = \begin{bmatrix} \frac{\sqrt{2}}{2} & 0 & 0 \ 0\\ \frac{\sqrt{2}}{2} & 0 & 0 \ 0\\ 0 & \frac{\sqrt{2}}{2} & 0 \ 0\\ 0 & \frac{\sqrt{2}}{2} & 0 \ 0\\ 0 & 0 & 1 \ 0\\ 0 & 0 & 0 \ 1 \end{bmatrix}, \ A\_2 = A\_3 = \begin{bmatrix} 0 \ 3\\ 3 \ 0 \end{bmatrix}, \ \mathbf{C} = \mathbf{B} - \boldsymbol{\Omega}.$$
*Then, from Theorem 6 we obtain*
$$A = \begin{bmatrix} A\_2 \\ & A\_3 \\ & & 0 \\ & & 0 \end{bmatrix} + \mathbf{X} \mathbf{C} \mathbf{X}^T \mathbf{A}$$
*which is symmetric nonnegative with spectrum* Λ.
The following result, although is not written in the fashion of a sufficient condition, is indeed a very general and efficient sufficient condition for the *SNIEP*.
**Theorem 10.** *[35] Let A be an n* × *n irreducible symmetric nonnegative matrix with spectrum* Λ = {*λ*1, *λ*2,..., *λn*}, *Perron eigenvalue λ*<sup>1</sup> *and a diagonal element c*. *Let B be an m* × *m symmetric nonnegative matrix with spectrum* Γ = {*μ*1, *μ*2,..., *μm*} *and Perron eigenvalue μ*1.
*i*) *If μ*<sup>1</sup> ≤ *c*, *then there exists a symmetric nonnegative matrix C*, *of order* (*n* + *m* − 1), *with spectrum* {*λ*1,..., *λn*, *μ*2,..., *μm*}.
*ii*) *If μ*<sup>1</sup> ≥ *c*, *then there exists a symmetric nonnegative matrix C*, *of order* (*n* + *m* − 1), *with spectrum* {*λ*<sup>1</sup> + *μ*<sup>1</sup> − *c*, *λ*2,..., *λn*, *μ*2,..., *μm*}.
**Example 5.** *The following example, given in [35], shows that* {7, 5, 0, −4, −4, −4} *with the partition*
$$\Lambda = \{7, 5, 0, -4\}, \ \Gamma = \{4, -4\}.$$
*satisfies conditions of Theorem 10, where*
10 Will-be-set-by-IN-TECH
*i*) *For each k* = 2, . . . , *p*<sup>1</sup> + 1, *there exists a symmetric nonnegative matrix Ak with spectrum* Γ*<sup>k</sup>* =
*ii*) *There exists a p*<sup>1</sup> × *p*<sup>1</sup> *symmetric nonnegative matrix B with spectrum* Λ<sup>1</sup> *and with diagonal entries*
**Example 4.** *Now, from Theorem 9, we can see that there exists a symmetric nonnegative matrix with spectrum* Λ = {5, 4, 0, −3, −3, −3}*, which can not be seen from Theorem 7. Moreover, we can*
> Λ<sup>1</sup> = {5, 4, 0, −3}, Λ<sup>2</sup> = {−3}, Λ<sup>3</sup> = {−3} *with* Γ<sup>2</sup> = {3, −3}, Γ<sup>3</sup> = {3, −3}, Γ<sup>4</sup> = Γ<sup>5</sup> = {0}.
> > 3 0 <sup>√</sup>6 0 <sup>030</sup> <sup>√</sup><sup>6</sup> <sup>√</sup>60 0 2 <sup>0</sup> <sup>√</sup>62 0
, *A*<sup>2</sup> = *A*<sup>3</sup> =
⎤ ⎥ ⎥ ⎦
The following result, although is not written in the fashion of a sufficient condition, is indeed
**Theorem 10.** *[35] Let A be an n* × *n irreducible symmetric nonnegative matrix with spectrum* Λ = {*λ*1, *λ*2,..., *λn*}, *Perron eigenvalue λ*<sup>1</sup> *and a diagonal element c*. *Let B be an m* × *m symmetric*
*i*) *If μ*<sup>1</sup> ≤ *c*, *then there exists a symmetric nonnegative matrix C*, *of order* (*n* + *m* − 1), *with spectrum*
*ii*) *If μ*<sup>1</sup> ≥ *c*, *then there exists a symmetric nonnegative matrix C*, *of order* (*n* + *m* − 1), *with spectrum*
+ *XCXT*,
�
, *C* = *B* − Ω.
⎤ ⎥ ⎥ ⎦
*B* =
*has spectrum* Λ<sup>1</sup> *and diagonal entries* 3, 3, 0, 0. *Let* Ω = *diag*{3, 3, 0, 0} *and*
*A* =
a very general and efficient sufficient condition for the *SNIEP*.
⎡ ⎢ ⎢ ⎣
*nonnegative matrix with spectrum* Γ = {*μ*1, *μ*2,..., *μm*} *and Perron eigenvalue μ*1.
*A*2 *A*3 0 0
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎣
*Suppose that the following conditions hold:*
*compute a realizing matrix. In fact, let the partition*
*X* =
*which is symmetric nonnegative with spectrum* Λ.
*Then, from Theorem 6 we obtain*
{*λ*1,..., *λn*, *μ*2,..., *μm*}.
{*λ*<sup>1</sup> + *μ*<sup>1</sup> − *c*, *λ*2,..., *λn*, *μ*2,..., *μm*}.
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ √2 <sup>2</sup> 0 00 √2 <sup>2</sup> 0 00
0 √2 <sup>2</sup> 0 0
*Then* Λ *is symmetrically realizable.*
*The symmetric nonnegative matrix*
{*ωk*, *λk*1, ..., *λkpk* },
*ω*2,..., *ωp*1+1.
$$A = \begin{bmatrix} 4 \ 0 \ 0 \ \dot{b} \ 0 \\ 0 \ 4 \ 0 \ \dot{d} \\ b \ 0 \ 0 \ \sqrt{6} \\ 0 \ d \ \sqrt{6} \ 0 \end{bmatrix} \text{ with } b^2 + d^2 = 23, \text{ } bd = 4\sqrt{6}.$$
*is symmetric nonnegative with spectrum* Λ. *Then there exists a symmetric nonnegative matrix C with spectrum* {7, 5, 0, −4, −4} *and a diagonal element* 4. *By applying again Theorem 10 to the lists* {7, 5, 0, −4, −4} *and* {4, −4}, *we obtain the desired symmetric nonnegative matrix.*
It is not hard to show that both results, Theorem 9 and Theorem 10, are equivalent (see [44]). Thus, the list in the Example 4 is also realizable from Theorem 10, while the list in the example 5 is also realizable from Theorem 9.
#### **5. List of complex numbers**
In this section we consider lists of complex nonreal numbers. We start with a complex generalization of a well known result of Suleimanova, usually considered as one of the important results in the *RNIEP (see [16]): The list λ*<sup>1</sup> > 0 > *λ*<sup>2</sup> ≥···≥ *λ<sup>n</sup>* is the spectrum of a nonnegative matrix if and only if *λ*<sup>1</sup> + *λ*<sup>2</sup> + ··· + *λ<sup>n</sup>* ≥ 0.
**Theorem 11.** *[10] Let* Λ = {*λ*0, *λ*1,..., *λn*} *be a list of complex numbers closed under complex conjugation, with*
$$\Lambda' = \{\lambda\_1, \dots, \lambda\_n\} \subset \{z \in \mathbb{C} : \operatorname{Re} z \le 0; \ |\operatorname{Re} z| \ge |\operatorname{Im} z|\}.$$
*Then* Λ *is realizable if and only if n* ∑ *i*=0 *λ<sup>i</sup>* ≥ 0.
**Proof.** Suppose that the elements of Λ� are ordered in such a way that *λ*2*p*+1,..., *λ<sup>n</sup>* are real and *λ*1,..., *λ*2*<sup>p</sup>* are complex nonreal, with
$$\lambda x\_k = \text{Re}\,\lambda\_{2k-1} = \text{Re}\,\lambda\_{2k} \text{ and } \, y\_k = \text{Im}\,\lambda\_{2k-1} = \text{Im}\,\lambda\_{2k}$$
for *k* = 1, . . . , *p*. Consider the matrix
$$B = \begin{bmatrix} 0 & 0 & 0 & \cdot \\ -\mathbf{x}\_1 + y\_1 & \mathbf{x}\_1 - y\_1 & \cdot \\ -\mathbf{x}\_1 - y\_1 & y\_1 & \mathbf{x}\_1 & \cdot \\ \vdots & \vdots & \vdots & \ddots \\ -\mathbf{x}\_p + y\_p & 0 & 0 & \cdot & \mathbf{x}\_p - y\_p \\ -\mathbf{x}\_p - y\_p & 0 & 0 & \cdot & y\_p & \mathbf{x}\_p \\ -\lambda\_{2p+1} & 0 & 0 & \cdot & & \lambda\_{2p+1} \\ \vdots & \vdots & \vdots & \ddots & & \ddots \\ -\lambda\_n & 0 & \cdot & & & \lambda\_n \end{bmatrix}.$$
It is clear that *B* ∈ CS<sup>0</sup> with spectrum {0, *λ*1,..., *λn*} and all the entries on its first column are nonnegative. Define **q** = (*q*0, *q*1,..., *qn*)*<sup>T</sup>* with *q*<sup>0</sup> = *λ*<sup>0</sup> + *n* ∑ *i*=1 *λ<sup>i</sup>* and
*qk* = − Re *λ<sup>k</sup>* for *k* = 1, . . . , 2*p* and *qk* = −*λ<sup>k</sup>* for *k* = 2*p* + 1, . . . , *n*.
Then, from the Brauer Theorem 1 *A* = *B* + **eq***<sup>T</sup>* is nonnegative with spectrum Λ.
In the case when all numbers in the given list, except one (the Perron eigenvalue), have real parts smaller than or equal to zero, remarkably simple necessary and sufficient conditions were obtained in [11].
**Theorem 12.** *[11] Let λ*2, *λ*3,..., *λ<sup>n</sup> be nonzero complex numbers with real parts less than or equal to zero and let λ*<sup>1</sup> *be a positive real number. Then the list* Λ = {*λ*1, *λ*2,..., *λn*} *is the nonzero spectrum of a nonnegative matrix if the following conditions are satisfied:*
$$\begin{aligned} \text{i)} \quad &\Lambda = \Lambda\\ \text{ii)} \ s\_1 = \sum\_{i=1}^n \lambda\_i \ge 0\\ \text{iii)} \ s\_2 = \sum\_{i=1}^n \lambda\_i^2 \ge 0 \end{aligned} \tag{11}$$
*The minimal number of zeros that need to be added to* Λ *to make it realizable is the smallest nonnegative integer N for which the following inequality is satisfied:*
$$s\_1^2 \le (n+N)s\_2.$$
*Furthermore, the list* {*λ*1, *λ*2,..., *λn*, 0, . . . , 0} *can be realized by C* + *αI*, *where C is a nonnegative companion matrix with trace zero, α is a nonnegative scalar and I is the n* × *n identity matrix.*
**Corollary 1.** *[11] Let λ*2, *λ*3,..., *λ<sup>n</sup> be complex numbers with real parts less than or equal to zero and let λ*<sup>1</sup> *be a positive real number. Then the list* Λ = {*λ*1, *λ*2,..., *λn*} *is the spectrum of a nonnegative matrix if and only if the following conditions are satisfied:*
$$\begin{aligned} \begin{array}{l} i) \quad \overline{\Lambda} = \Lambda\\ ii) \ s\_1 = \sum\_{\substack{i=1\\i=1\\i\neq 1}}^n \lambda\_i \ge 0\\ iii) \ s\_2 = \sum\_{i=1}^n \lambda\_i^2 \ge 0\\ iv) \quad s\_1^2 \le ns\_2 \end{array} \tag{12}$$
**Example 6.** *The list* Λ = {8, −1 + 3*i*, −1 − 3*i*, −2 + 5*i*, −2 − 5*i*} *satisfies conditions (12). Then* Λ *is the spectrum of the nonnegative companion matrix*
$$\mathbf{C} = \begin{bmatrix} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 2320 & 494 & 278 & 1 & 2 \end{bmatrix}.$$
*Observe that Theorem 11 gives no information about the realizability of* Λ.
*The list* {19, −1 + 11*i*, −1 − 11*i*, −3 + 8*i*, −3 − 8*i*} *was given in [11]. It does not satisfy conditions (12): s*<sup>1</sup> = 11, *s*<sup>2</sup> = 11 *and s*<sup>2</sup> <sup>1</sup> *ns*2. *The inequality* <sup>11</sup><sup>2</sup> <sup>≤</sup> (<sup>5</sup> <sup>+</sup> *<sup>N</sup>*)<sup>11</sup> *is satisfied for N* <sup>≥</sup> 6. *Then we need to add* 6 *zeros to the list to make it realizable.*
Theorem 3 (in section 3), can also be extended to the complex case:
**Theorem 13.** *[13] Let* Λ = {*λ*2, *λ*3,..., *λn*} *be a list of complex numbers such that* Λ = Λ, *λ*<sup>1</sup> ≥ max*<sup>i</sup>* |*λi*| , *i* = 2, . . . , *n*, *and n* ∑ *i*=1 *λ<sup>i</sup>* ≥ 0. *Suppose that: i*) *there exists a partition* Λ = Λ<sup>1</sup> ∪···∪ Λ*<sup>t</sup> with*
$$\Lambda\_k = \{\lambda\_{k1}, \lambda\_{k2}, \dots, \lambda\_{kp}\}, \ \lambda\_{11} = \lambda\_{1\nu}$$
*k* = 1, . . . , *t*, *such that* Γ*<sup>k</sup>* = {*ωk*, *λk*2, ..., *λkpk* } *is realizable by a nonnegative matrix Ak* ∈ CS*ω<sup>k</sup>* , *and*
*ii*) *there exists a t* × *t nonnegative matrix B* ∈ CS*λ*<sup>1</sup> , *with eigenvalues λ*1, *λ*21,..., *λt*<sup>1</sup> *(the first elements of the lists* Λ*k*) *and with diagonal entries ω*1, *ω*2,..., *ω<sup>t</sup> (the Perron eigenvalues of the lists* Γ*k*).
*Then* Λ *is realizable.*
**Example 7.** *Let* Λ = {7, 1, −2, −2, −2 + 4*i*, −2 − 4*i*}. *Consider the partition*
$$\begin{aligned} \Lambda\_1 &= \{7, 1, -2, -2\}, \,\,\Lambda\_2 = \{-2 + 4i\}, \,\,\Lambda\_3 = \{-2 - 4i\} \text{ with} \\ \Gamma\_1 &= \{3, 1, -2, -2\}, \,\,\Gamma\_2 = \{0\}, \,\,\Gamma\_3 = \{0\}. \end{aligned}$$
*We look for a nonnegative matrix B* ∈ CS<sup>7</sup> *with eigenvalues* 7, −2 + 4*i*, −2 − 4*i and diagonal entries* 3, 0, 0, *and a nonnegative matrix A*<sup>1</sup> *realizing* Γ1. *They are*
$$B = \begin{bmatrix} 3 & 0 \ 4 \\ \frac{41}{7} & 0 \ \frac{8}{7} \\ 0 & 7 \ 0 \end{bmatrix} \quad \text{and} \quad A\_1 = \begin{bmatrix} 0 \ 2 \ 0 \ 1 \\ 2 \ 0 \ 0 \ 1 \\ 0 \ 1 \ 0 \ 2 \\ 0 \ 1 \ 2 \ 0 \end{bmatrix}.$$
*Then*
12 Will-be-set-by-IN-TECH
It is clear that *B* ∈ CS<sup>0</sup> with spectrum {0, *λ*1,..., *λn*} and all the entries on its first column are
*qk* = − Re *λ<sup>k</sup>* for *k* = 1, . . . , 2*p* and *qk* = −*λ<sup>k</sup>* for *k* = 2*p* + 1, . . . , *n*.
In the case when all numbers in the given list, except one (the Perron eigenvalue), have real parts smaller than or equal to zero, remarkably simple necessary and sufficient conditions
**Theorem 12.** *[11] Let λ*2, *λ*3,..., *λ<sup>n</sup> be nonzero complex numbers with real parts less than or equal to zero and let λ*<sup>1</sup> *be a positive real number. Then the list* Λ = {*λ*1, *λ*2,..., *λn*} *is the nonzero spectrum*
> *n* ∑ *i*=1
*n* ∑ *i*=1 *λ*2 *<sup>i</sup>* ≥ 0
*The minimal number of zeros that need to be added to* Λ *to make it realizable is the smallest nonnegative*
<sup>1</sup> ≤ (*n* + *N*)*s*2.
*Furthermore, the list* {*λ*1, *λ*2,..., *λn*, 0, . . . , 0} *can be realized by C* + *αI*, *where C is a nonnegative companion matrix with trace zero, α is a nonnegative scalar and I is the n* × *n identity matrix.*
**Corollary 1.** *[11] Let λ*2, *λ*3,..., *λ<sup>n</sup> be complex numbers with real parts less than or equal to zero and let λ*<sup>1</sup> *be a positive real number. Then the list* Λ = {*λ*1, *λ*2,..., *λn*} *is the spectrum of a nonnegative*
> *n* ∑ *i*=1
*n* ∑ *i*=1 *λ*2 *<sup>i</sup>* ≥ 0
<sup>1</sup> ≤ *ns*<sup>2</sup>
0 1 0 00 0 0 1 00 0 0 0 10 0 0 0 01 2320 494 278 1 2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ .
**Example 6.** *The list* Λ = {8, −1 + 3*i*, −1 − 3*i*, −2 + 5*i*, −2 − 5*i*} *satisfies conditions (12). Then* Λ
*λ<sup>i</sup>* ≥ 0
*i*) Λ = Λ
*ii*) *s*<sup>1</sup> =
*iii*) *s*<sup>2</sup> =
*iv*) *s*<sup>2</sup>
*C* =
⎡ ⎢ ⎢ ⎢ ⎢ ⎣ *λ<sup>i</sup>* ≥ 0
*i*) Λ = Λ
*ii*) *s*<sup>1</sup> =
*iii*) *s*<sup>2</sup> =
*s* 2
Then, from the Brauer Theorem 1 *A* = *B* + **eq***<sup>T</sup>* is nonnegative with spectrum Λ.
*n* ∑ *i*=1
*λ<sup>i</sup>* and
(11)
(12)
nonnegative. Define **q** = (*q*0, *q*1,..., *qn*)*<sup>T</sup>* with *q*<sup>0</sup> = *λ*<sup>0</sup> +
*of a nonnegative matrix if the following conditions are satisfied:*
*integer N for which the following inequality is satisfied:*
*matrix if and only if the following conditions are satisfied:*
*is the spectrum of the nonnegative companion matrix*
were obtained in [11].
$$A = \begin{bmatrix} A\_1 \\ 0 \\ 0 \end{bmatrix} + \begin{bmatrix} 1 \ 0 \ 0 \\ 1 \ 0 \ 0 \\ 1 \ 0 \ 0 \\ 1 \ 0 \ 0 \\ 0 \ 1 \ 0 \\ 0 \ 0 \ 1 \end{bmatrix} \begin{bmatrix} 0 & 0 \ 0 & 0 \ 0 & 4 \\ \frac{41}{7} & 0 \ 0 & 0 \ 0 & 8 \\ 0 & 0 \ 0 & 0 & 7 \end{bmatrix} = \begin{bmatrix} 0 & 2 \ 0 \ 1 \ 0 \ 4 \\ 2 \ 0 \ 0 \ 1 \ 0 \ 4 \\ 0 \ 1 \ 0 \ 2 \ 0 \ 4 \\ 0 \ 1 \ 2 \ 0 \ 0 \ 4 \\ \frac{41}{7} & 0 \ 0 \ 0 \ 0 \ 8 \\ 0 \ 0 \ 0 \ 0 \ 7 & 0 \end{bmatrix}$$
*has the spectrum* Λ.
#### **6. Fiedler and Guo results**
One of the most important works about the *SNIEP* is due to Fiedler [29]. In [29] Fiedler showed, as it was said before, that Kellogg sufficient conditions for the *RNIEP* are also sufficient for the *SNIEP*. Three important and very useful results of Fiedler are:
**Lemma 2.** *[29] Let A be a symmetric m* × *m matrix with eigenvalues α*1,..., *αm*, *A***u** = *α*1**u**, �**u**� = 1. *Let B be a symmetric n* × *n matrix with eigenvalues β*1,..., *βn*, *B***v** = *β*1**v**, �**v**� = 1. *Then for any ρ*, *the matrix*
$$\mathbf{C} = \begin{bmatrix} A & \rho \mathbf{u} \mathbf{v}^T \\ \rho \mathbf{v} \mathbf{u}^T & B \end{bmatrix}$$
*has eigenvalues α*2,..., *αm*, *β*2,..., *βn*, *γ*1, *γ*2, *where γ*1, *γ*<sup>2</sup> *are eigenvalues of the matrix*
$$
\tilde{\mathcal{C}} = \begin{bmatrix} \alpha\_1 & \rho \\ \rho & \beta\_1 \end{bmatrix}.
$$
**Lemma 3.** *[29] If* {*α*1,..., *αm*} *and* {*β*1,..., *βn*} *are lists symmetrically realizable and α*<sup>1</sup> ≥ *β*1*, then for any t* ≥ 0, *the list*
$$\{\alpha\_1 + t\_\prime \beta\_1 - t\_\prime \alpha\_2, \dots, \alpha\_m, \beta\_{2'}, \dots, \beta\_n\}$$
*is also symmetrically realizable.*
**Lemma 4.** *[29] If* Λ = {*λ*1, *λ*2,..., *λn*} *is symmetrically realizable by a nonnegative matrix and if t* > 0, *then*
$$\Lambda\_{\mathbf{f}} = \{\lambda\_1 + t\_\prime \lambda\_{2\prime}, \dots, \lambda\_n\},$$
*is symmetrically realizable by a positive matrix.*
**Remark 1.** *It is not hard to see that Lemma 2 can be obtained from Theorem 6. In fact, it is enough to consider*
$$\begin{aligned} \mathbf{C} &= \begin{bmatrix} A \\ & B \end{bmatrix} + \begin{bmatrix} \mathbf{u} \ \mathbf{0} \\ \mathbf{0} \ \mathbf{v} \end{bmatrix} \begin{bmatrix} 0 \ \rho \\ \rho \ \mathbf{0} \end{bmatrix} \begin{bmatrix} \mathbf{u}^T \ \mathbf{0}^T \\ \mathbf{0}^T \ \mathbf{v}^T \end{bmatrix} \\ &= \begin{bmatrix} A & \rho \mathbf{u} \mathbf{v}^T \\ \rho \mathbf{v} \mathbf{u}^T & B \end{bmatrix} \end{aligned}$$
*which is symmetric with eigenvalues γ*1, *γ*2, *α*2,..., *αm*, *β*2,..., *βn*, *where γ*1, *γ*<sup>2</sup> *are eigenvalues of*
$$B = \begin{bmatrix} \alpha\_1 & \rho \\ \rho & \beta\_1 \end{bmatrix}.$$
Now we consider a relevant result due to Guo [45]:
**Theorem 14.** *[45] If the list of complex numbers* Λ = {*λ*1, *λ*2,..., *λn*} *is realizable, where λ*<sup>1</sup> *is the Perron eigenvalue and λ*<sup>2</sup> ∈ **R**, *then for any t* ≥ 0 *the list* Λ*<sup>t</sup>* = {*λ*<sup>1</sup> + *t*, *λ*<sup>2</sup> ± *t*, *λ*3,..., *λn*} *is also realizable.*
**Corollary 2.** *[45] If the list of real numbers* Λ = {*λ*1, *λ*2,..., *λn*} *is realizable and t*<sup>1</sup> = *n* ∑ *i*=2 |*ti*| *with ti* ∈ **R**, *i* = 2, . . . , *n*, *then the list* Λ*<sup>t</sup>* = {*λ*<sup>1</sup> + *t*1, *λ*<sup>2</sup> + *t*2,..., *λ<sup>n</sup>* + *tn*} *is also realizable.*
**Example 8.** *Let* Λ = {8, 6, 3, 3, −5, −5, −5, −5} *be a given list. Since the lists* Λ<sup>1</sup> = Λ<sup>2</sup> = {7, 3, −5, −5} *are both realizable (see [22] to apply a simple criterion, which shows the realizability of* Λ<sup>1</sup> = Λ2*), then*
$$
\Lambda\_1 \cup \Lambda\_2 = \{7, 7, 3, 3, -5, -5, -5, -5\}
$$
*is also realizable. Now, from Theorem 14, with t* = 1, Λ *is realizable.*
Guo also sets the following two questions:
Question 1: Do complex eigenvalues of nonnegative matrices have a property similar to Theorem 14?
Question 2: If the list Λ = {*λ*1, *λ*2,..., *λn*} is symmetrically realizable, and *t* > 0, is the list Λ*<sup>t</sup>* = {*λ*<sup>1</sup> + *t*, *λ*<sup>2</sup> ± *t*, *λ*3,..., *λn*} symmetrically realizable?.
It was shown in [12] and also in [46] that Question 1 has an affirmative answer.
**Theorem 15.** *[12] Let* Λ = {*λ*1, *a* + *bi*, *a* − *bi*, *λ*4,..., *λn*} *be a realizable list of complex numbers. Then for all t* ≥ 0, *the perturbed list*
$$\Lambda\_t = \{\lambda\_1 + 2t, a - t + bi, a - t - bi, \lambda\_{4\prime}, \dots, \lambda\_n\}$$
*is also realizable.*
14 Will-be-set-by-IN-TECH
**Lemma 2.** *[29] Let A be a symmetric m* × *m matrix with eigenvalues α*1,..., *αm*, *A***u** = *α*1**u**, �**u**� = 1. *Let B be a symmetric n* × *n matrix with eigenvalues β*1,..., *βn*, *B***v** = *β*1**v**, �**v**� = 1. *Then for any*
> *A ρ***uv***<sup>T</sup> ρ***vu***<sup>T</sup> B*
**Lemma 3.** *[29] If* {*α*1,..., *αm*} *and* {*β*1,..., *βn*} *are lists symmetrically realizable and α*<sup>1</sup> ≥ *β*1*,*
{*α*<sup>1</sup> + *t*, *β*<sup>1</sup> − *t*, *α*2,..., *αm*, *β*2,..., *βn*}
**Lemma 4.** *[29] If* Λ = {*λ*1, *λ*2,..., *λn*} *is symmetrically realizable by a nonnegative matrix and if*
Λ*<sup>t</sup>* = {*λ*<sup>1</sup> + *t*, *λ*2,..., *λn*}
**Remark 1.** *It is not hard to see that Lemma 2 can be obtained from Theorem 6. In fact, it is enough to*
,
*which is symmetric with eigenvalues γ*1, *γ*2, *α*2,..., *αm*, *β*2,..., *βn*, *where γ*1, *γ*<sup>2</sup> *are eigenvalues of*
**Theorem 14.** *[45] If the list of complex numbers* Λ = {*λ*1, *λ*2,..., *λn*} *is realizable, where λ*<sup>1</sup> *is the Perron eigenvalue and λ*<sup>2</sup> ∈ **R**, *then for any t* ≥ 0 *the list* Λ*<sup>t</sup>* = {*λ*<sup>1</sup> + *t*, *λ*<sup>2</sup> ± *t*, *λ*3,..., *λn*} *is also*
**Example 8.** *Let* Λ = {8, 6, 3, 3, −5, −5, −5, −5} *be a given list. Since the lists* Λ<sup>1</sup> = Λ<sup>2</sup> = {7, 3, −5, −5} *are both realizable (see [22] to apply a simple criterion, which shows the realizability*
Λ<sup>1</sup> ∪ Λ<sup>2</sup> = {7, 7, 3, 3, −5, −5, −5, −5}
*B* = *α*1 *ρ ρ β*<sup>1</sup>
**Corollary 2.** *[45] If the list of real numbers* Λ = {*λ*1, *λ*2,..., *λn*} *is realizable and t*<sup>1</sup> =
*ti* ∈ **R**, *i* = 2, . . . , *n*, *then the list* Λ*<sup>t</sup>* = {*λ*<sup>1</sup> + *t*1, *λ*<sup>2</sup> + *t*2,..., *λ<sup>n</sup>* + *tn*} *is also realizable.*
0 *ρ ρ* 0
> .
**u***<sup>T</sup>* **0***<sup>T</sup>* **0***<sup>T</sup>* **v***<sup>T</sup>* *n* ∑ *i*=2
.
*C* =
*has eigenvalues α*2,..., *αm*, *β*2,..., *βn*, *γ*1, *γ*2, *where γ*1, *γ*<sup>2</sup> *are eigenvalues of the matrix*
*C* = *α*1 *ρ ρ β*<sup>1</sup>
*ρ*, *the matrix*
*t* > 0, *then*
*consider*
*realizable.*
*of* Λ<sup>1</sup> = Λ2*), then*
*then for any t* ≥ 0, *the list*
*is also symmetrically realizable.*
*is symmetrically realizable by a positive matrix.*
*C* = *A B* + **u 0 0 v**
=
*is also realizable. Now, from Theorem 14, with t* = 1, Λ *is realizable.*
Now we consider a relevant result due to Guo [45]:
*A ρ***uv***<sup>T</sup> ρ***vu***<sup>T</sup> B*
Question 2, however, remains open. An affirmative answer to Question 2, in the case that the symmetric realizing matrix is a nonnegative circulant matrix or it is a nonnegative left circulant matrix, it was given in [47]. The use of circulant matrices has been shown to be very useful for the *NIEP* [9, 24]. In [24] it was given a necessary and sufficient condition for a list of 5 real numbers, which corresponds to a even-conjugate vector, to be the spectrum of 5 × 5 symmetric nonnegative circulant matrix:
**Lemma 5.** *[24] Let λ* = (*λ*1, *λ*2, *λ*3, *λ*3, *λ*2)*<sup>T</sup> be a vector of real numbers (even-conjugate) such that*
$$\begin{aligned} \lambda\_1 &\ge \left| \lambda\_j \right|, \ j = 2, 3\\ \lambda\_1 &\ge \lambda\_2 \ge \lambda\_3\\ \lambda\_1 + 2\lambda\_2 + 2\lambda\_3 &\ge 0 \end{aligned} \tag{13}$$
*A necessary and sufficient condition for* {*λ*1, *λ*2, *λ*3, *λ*3, *λ*2} *to be the spectrum of a symmetric nonnegative circulant matrix is*
$$
\lambda\_1 + (\lambda\_3 - \lambda\_2) \frac{\sqrt{5} - 1}{2} - \lambda\_2 \ge 0. \tag{14}
$$
**Example 9.** *From Lemma 5 we may know, for instance, that the list* {6, 1, 1, −4, −4} *is the spectrum of a symmetric nonnegative circulant matrix.*
#### **7. Some open questions**
We finish this chapter by setting two open questions:
Question 1: *If the list of real numbers* Λ = {*λ*1, *λ*2,..., *λn*} *is symmetrically realizable, and t* > 0, *is the list* Λ*<sup>t</sup>* = {*λ*<sup>1</sup> + *t*, *λ*<sup>2</sup> ± *t*, *λ*3,..., *λn*} also symmetrically realizable?
Some progress has been done about this question. In [47], it was given an affirmative answer to Question 1, in the case that the realizing matrix is symmetric nonnegative circulant matrix or it is nonnegative left circulant matrix. In [48] it was shown that if 1 > *λ*<sup>2</sup> ≥ ··· ≥ *λ<sup>n</sup>* ≥ 0, then Theorem 14 holds for positive stochastic, positive doubly stochastic and positive symmetric matrices.
Question 2: *How adding one or more zeros to a list can lead to its symmetric realizability by different symmetric patterned matrices?*
The famous Boyle-Handelman Theorem [49] gives a nonconstructive proof of the fact that if *sk* = *λ<sup>k</sup>* <sup>1</sup> <sup>+</sup> *<sup>λ</sup><sup>k</sup>* <sup>2</sup> <sup>+</sup> ··· <sup>+</sup> *<sup>λ</sup><sup>k</sup> <sup>n</sup>* > 0, for *k* = 1, 2, . . . , then there exists a nonnegative number *N* for which the list {*λ*1,..., *λn*, 0, . . . , 0}, with *N* zeros added, is realizable. In [11] Laffey and Šmigoc completely solve the *NIEP* for lists of complex numbers Λ = {*λ*1,..., *λn*}, closed under conjugation, with *λ*2,..., *λ<sup>n</sup>* having real parts smaller than or equal to zero. They show the existence of *N* ≥ 0 for which Λ with *N* zeros added is realizable and show how to compute the least such *N*. The situation for symmetrically realizable spectra is different and even less is known.
## **8. Conclusion**
The *nonnegative* inverse eigenvalue problem is an open and difficult problem. A full solution is unlikely in the near future. A number of partial results are known in the literature about the problem, most of them in terms of sufficient conditions. Some matrix results, like Brauer Theorem (Theorem 1), Rado Theorem (Theorem 2), and its symmetric version (Theorem 6) have been shown to be very useful to derive good sufficient conditions. This way, however, seems to be quite narrow and may be other techniques should be explored and applied.
## **Author details**
Ricardo L. Soto
*Department of Mathematics, Universidad Católica del Norte, Casilla 1280, Antofagasta, Chile.*
## **9. References**
[11] T. J. Laffey, H. Šmigoc (2006) Nonnegative realization of spectra having negative real parts. In: Linear Algebra Appl. 416 148-159.
16 Will-be-set-by-IN-TECH
The famous Boyle-Handelman Theorem [49] gives a nonconstructive proof of the fact that if
for which the list {*λ*1,..., *λn*, 0, . . . , 0}, with *N* zeros added, is realizable. In [11] Laffey and Šmigoc completely solve the *NIEP* for lists of complex numbers Λ = {*λ*1,..., *λn*}, closed under conjugation, with *λ*2,..., *λ<sup>n</sup>* having real parts smaller than or equal to zero. They show the existence of *N* ≥ 0 for which Λ with *N* zeros added is realizable and show how to compute the least such *N*. The situation for symmetrically realizable spectra is different and even less
The *nonnegative* inverse eigenvalue problem is an open and difficult problem. A full solution is unlikely in the near future. A number of partial results are known in the literature about the problem, most of them in terms of sufficient conditions. Some matrix results, like Brauer Theorem (Theorem 1), Rado Theorem (Theorem 2), and its symmetric version (Theorem 6) have been shown to be very useful to derive good sufficient conditions. This way, however, seems to be quite narrow and may be other techniques should be explored and applied.
*Department of Mathematics, Universidad Católica del Norte, Casilla 1280, Antofagasta, Chile.*
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**8. Conclusion**
**Author details** Ricardo L. Soto
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## **Identification of Linear, Discrete-Time Filters via Realization**
Daniel N. Miller and Raymond A. de Callafon
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problem. In: International Mathematical Forum 6 N◦ 50, 2447-2460.
Algebra 3 119-128.
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133 (3) (2004) 711-717.
The realization of a discrete-time, linear, time-invariant (LTI) filter from its impulse response provides insight into the role of linear algebra in the analysis of both dynamical systems and rational functions. For an LTI filter, a sequence of output data measured over some finite period of time may be expressed as the linear combination of the past input and the input measured over that same period. For a finite-dimensional LTI filter, the mapping from past input to future output is a finite-rank linear operator, and the effect of past input, that is, the memory of the system, may be represented as a finite-dimensional vector. This vector is the *state* of the system.
The central idea of realization theory is to first identify the mapping from past input to future output and to then factor it into two parts: a map from the input to the state and another from the state to the output. This factorization guarantees that the resulting system representation is both casual and finite-dimensional; thus it can be physically constructed, or *realized*.
*System identification* is the science of constructing dynamic models from experimentally measured data. Realization-based identification methods construct models by estimating the mapping from past input to future output based on this measured data. The non-deterministic nature of the estimation process causes this mapping to have an arbitrarily large rank, and so a rank-reduction step is required to factor the mapping into a suitable state-space model. Both these steps must be carefully considered to guarantee unbiased estimates of dynamic systems.
The foundations of realization theory are primarily due to Kalman and first appear in the landmark paper of [1], though the problem is not defined explicitly until [2], which also coins the term "realization" as being the a state-space model of a linear system constructed from an experimentally measured impulse response. It was [3] that introduced the structured-matrix approach now synonymous with the term "realization theory" by re-interpreting a theorem originally due to [4] in a state-space LTI system framework.
Although Kalman's original definition of "realization" implied an identification problem, it was not until [5] proposed rank-reduction by means of the singular-value decomposition
#### 2 Will-be-set-by-IN-TECH 118 Linear Algebra – Theorems and Applications
that Ho's method became feasible for use with non-deterministic data sets. The combination of Ho's method and the singular-value decomposition was finally generalized to use with experimentally measured data by Kung in [6].
With the arrival of Kung's method came the birth of what is now known as the field of *subspace identification* methods. These methods use structured matrices of arbitrary input and output data to estimate a state-sequence from the system. The system is then identified from the propagation of the state over time. While many subspace methods exist, the most popular are the Multivariable Output-Error State Space (MOESP) family, due to [7], and the Numerical Algorithms for Subspace State-Space System Identification (N4SID) family, due to [8]. Related to subspace methods is the Eigensystem Realization Algorithm [9], which applies Kung's algorithm to impulse-response estimates, which are typically estimated through an Observer/Kalman Filter Identification (OKID) algorithm [10].
This chapter presents the central theory behind realization-based system identification in a chronological context, beginning with Kronecker's theorem, proceeding through the work of Kalman and Kung, and presenting a generalization of the procedure to arbitrary sets of data. This journey provides an interesting perspective on the original role of linear algebra in the analysis of rational functions and highlights the similarities of the different representations of LTI filters. Realization theory is a diverse field that connects many tools of linear algebra, including structured matrices, the QR-decomposition, the singular-value decomposition, and linear least-squares problems.
## **2. Transfer-function representations**
We begin by reviewing some properties of discrete-time linear filters, focusing on the role of infinite series expansions in analyzing the properties of rational functions. The reconstruction of a transfer function from an infinite impulse response is equivalent to the reconstruction of a rational function from its Laurent series expansion. The reconstruction problem is introduced and solved by forming structured matrices of impulse-response coefficients.
### **2.1. Difference equations and transfer functions**
Discrete-time linear filters are most frequently encountered in the form of difference equations that relate an input signal *uk* to an output signal *yk*. A simple example is an output *yk* determined by a weighted sum of the inputs from *uk* to *uk*−*m*,
$$y\_k = b\_m u\_k + b\_{m-1} u\_{k-1} + \cdots + b\_0 u\_{k-m}.\tag{1}$$
More commonly, the output *yk* also contains a weighted sum of previous outputs, such as a weighted sum of samples from *yk*−<sup>1</sup> to *yk*−*n*,
$$y\_k = b\_m u\_k + b\_{m-1} u\_{k-1} + \dots + b\_0 u\_{k-m} - a\_{n-1} y\_{k-1} - a\_{n-2} y\_{k-2} - \dots + a\_0 y\_{k-n} \tag{2}$$
The impulse response of a filter is the output sequence *gk* = *yk* generated from an input
$$\mu\_k = \begin{cases} 1 & k = 0, \\ 0 & k \neq 0. \end{cases} \tag{3}$$
The parameters *gk* are the impulse-response coefficients, and they completely describe the behavior of an LTI filter through the convolution
2 Will-be-set-by-IN-TECH
that Ho's method became feasible for use with non-deterministic data sets. The combination of Ho's method and the singular-value decomposition was finally generalized to use with
With the arrival of Kung's method came the birth of what is now known as the field of *subspace identification* methods. These methods use structured matrices of arbitrary input and output data to estimate a state-sequence from the system. The system is then identified from the propagation of the state over time. While many subspace methods exist, the most popular are the Multivariable Output-Error State Space (MOESP) family, due to [7], and the Numerical Algorithms for Subspace State-Space System Identification (N4SID) family, due to [8]. Related to subspace methods is the Eigensystem Realization Algorithm [9], which applies Kung's algorithm to impulse-response estimates, which are typically estimated through an
This chapter presents the central theory behind realization-based system identification in a chronological context, beginning with Kronecker's theorem, proceeding through the work of Kalman and Kung, and presenting a generalization of the procedure to arbitrary sets of data. This journey provides an interesting perspective on the original role of linear algebra in the analysis of rational functions and highlights the similarities of the different representations of LTI filters. Realization theory is a diverse field that connects many tools of linear algebra, including structured matrices, the QR-decomposition, the singular-value decomposition, and
We begin by reviewing some properties of discrete-time linear filters, focusing on the role of infinite series expansions in analyzing the properties of rational functions. The reconstruction of a transfer function from an infinite impulse response is equivalent to the reconstruction of a rational function from its Laurent series expansion. The reconstruction problem is introduced
Discrete-time linear filters are most frequently encountered in the form of difference equations that relate an input signal *uk* to an output signal *yk*. A simple example is an output *yk*
More commonly, the output *yk* also contains a weighted sum of previous outputs, such as a
The impulse response of a filter is the output sequence *gk* = *yk* generated from an input
*uk* =
*yk* = *bmuk* + *bm*−<sup>1</sup>*uk*−<sup>1</sup> + ··· + *<sup>b</sup>*0*uk*−*<sup>m</sup>* − *an*−<sup>1</sup>*yk*−<sup>1</sup> − *an*−<sup>2</sup>*yk*−<sup>2</sup> −··· + *<sup>a</sup>*0*yk*−*n*. (2)
1 *k* = 0,
*yk* = *bmuk* + *bm*−<sup>1</sup>*uk*−<sup>1</sup> + ··· + *<sup>b</sup>*0*uk*−*m*. (1)
<sup>0</sup> *<sup>k</sup>* �<sup>=</sup> 0. (3)
and solved by forming structured matrices of impulse-response coefficients.
experimentally measured data by Kung in [6].
linear least-squares problems.
**2. Transfer-function representations**
weighted sum of samples from *yk*−<sup>1</sup> to *yk*−*n*,
**2.1. Difference equations and transfer functions**
determined by a weighted sum of the inputs from *uk* to *uk*−*m*,
Observer/Kalman Filter Identification (OKID) algorithm [10].
$$y\_k = \sum\_{j=0}^{\infty} g\_j u\_{k-j}.\tag{4}$$
Filters of type (1) are called finite-impulse response (FIR) filters because *gk* is a finite-length sequence that settles to 0 once *k* > *m*. Filters of type (2) are called infinite impulse response (IIR) filters since generally the impulse response will never completely settle to 0.
A system is stable if a bounded *uk* results in a bounded *yk*. Because the output of LTI filters is a linear combination of the input and previous output, any input-output sequence can be formed from a linear superposition of other input-output sequences. Hence proving that the system has a bounded output for a single input sequence is necessary and sufficient to prove the stability of an LTI filter. The simplest input to consider is an impulse, and so a suitable definition of system stability is that the absolute sum of the impulse response is bounded,
$$\sum\_{k=0}^{\infty} |g\_k| < \infty. \tag{5}$$
Though the impulse response completely describes the behavior of an LTI filter, it does so with an infinite number of parameters. For this reason, discrete-time LTI filters are often written as transfer functions of a complex variable *z*. This enables analysis of filter stability and computation of the filter's frequency response in a finite number of calculations, and it simplifies convolution operations into basic polynomial algebra.
The transfer function is found by grouping output and input terms together and taking the *Z*-transform of both signals. Let *Y*(*z*) = ∑<sup>∞</sup> *<sup>k</sup>*=−<sup>∞</sup> *ykz*−*<sup>k</sup>* be the *<sup>Z</sup>*-transform of *yk* and *<sup>U</sup>*(*z*) be the *Z*-transform of *uk*. From the property
$$\mathcal{Z}\left[y\_{k-1}\right] = \mathcal{Y}(z)z^{-1}$$
the relationship between *Y*(*z*) and *U*(*z*) may be expressed in polynomials of *z* as
$$a(z)\mathcal{Y}(z) = b(z)\mathcal{U}(z).$$
The ratio of these two polynomials is the filter's transfer function
$$G(z) = \frac{b(z)}{a(z)} = \frac{b\_m z^m + b\_{m-1} z^{m-1} + \dots + b\_1 z + b\_0}{z^n + a\_{n-1} z^{n-1} + \dots + a\_1 z + a\_0}.\tag{6}$$
When *n* ≥ *m*, *G*(*z*) is *proper*. If the transfer function is not proper, then the difference equations will have *yk* dependent on future input samples such as *uk*+1. Proper transfer functions are required for causality, and thus all physical systems have proper transfer function representations. When *n* > *m*, the system is *strictly proper*. Filters with strictly proper transfer functions have no feed-through terms; the output *yk* does not depend on *uk*, only the preceding input *uk*−1, *uk*−2, . . . . In this chapter, we assume all systems are causal and all transfer functions proper.
If *a*(*z*) and *b*(*z*) have no common roots, then the rational function *G*(*z*) is *coprime*, and the order *n* of *G*(*z*) cannot be reduced. Fractional representations are not limited to
#### 4 Will-be-set-by-IN-TECH 120 Linear Algebra – Theorems and Applications
single-input-single-output systems. For vector-valued input signals *uk* <sup>∈</sup> **<sup>R</sup>***nu* and output signals *yk* <sup>∈</sup> **<sup>R</sup>***ny* , an LTI filter may be represented as an *ny* <sup>×</sup> *nu* matrix of rational functions *Gij*(*z*), and the system will have matrix-valued impulse-response coefficients. For simplicity, we will assume that transfer function representations are single-input-single-output, though all results presented here generalize to the multi-input-multi-output case.
#### **2.2. Stability of transfer function representations**
Because the effect of *b*(*z*) is equivalent to a finite-impulse response filter, the only requirement for *b*(*z*) to produce a stable system is that its coefficients be bounded, which we may safely assume is always the case. Thus the stability of a transfer function *G*(*z*) is determined entirely by *a*(*z*), or more precisely, the roots of *a*(*z*). To see this, suppose *a*(*z*) is factored into its roots, which are the poles *pi* of *G*(*z*),
$$G(z) = \frac{b(z)}{\prod\_{i=1}^{n} (z - p\_i)}.\tag{7}$$
To guarantee a bounded *yk*, it is sufficient to study a single pole, which we will denote simply as *p*. Thus we wish to determine necessary and sufficient conditions for stability of the system
$$G'(z) = \frac{1}{z - p}.\tag{8}$$
Note that *p* may be complex. Assume that |*z*| > |*p*|. *G*� (*z*) then has the Laurent-series expansion
$$G'(z) = z^{-1} \left(\frac{1}{1 - pz^{-1}}\right) = z^{-1} \sum\_{k=0}^{\infty} p^k z^{-k} = \sum\_{k=1}^{\infty} p^{k-1} z^{-k}.\tag{9}$$
From the time-shift property of the *z*-transform, it is immediately clear that the sequence
$$\mathbf{g}'\_{k} = \begin{cases} 0 & k = 1, \\ p^{k-1} & k > 1, \end{cases} \tag{10}$$
is the impulse response of *G*� (*z*). If we require that (9) is absolutely summable and let |*z*| = 1, the result is the original stability requirement (5), which may be written in terms of *p* as
$$\sum\_{k=1}^{\infty} \left| p^{k-1} \right| < \infty.$$
This is true if and only if |*p*| < 1, and thus *G*� (*z*) is stable if and only if |*p*| < 1. Finally, from (7) we may deduce that a system is stable if and only if all the poles of *G*(*z*) satisfy the property |*pi*| < 1.
#### **2.3. Construction of transfer functions from impulse responses**
Transfer functions are a convenient way of representing complex system dynamics in a finite number of parameters, but the coefficients of *a*(*z*) and *b*(*z*) cannot be measured directly. The impulse response of a system can be found experimentally by either direct measurement or from other means such as taking the inverse Fourier transform of a measured frequency response [11]. It cannot, however, be represented in a finite number of parameters. Thus the conversion between transfer functions and impulse responses is an extremely useful tool.
For a single-pole system such as (8), the expansion (9) provides an obvious means of reconstructing a transfer function from a measured impulse response: given any 2 sequential impulse-response coefficients *gk* and *gk*+1, the pole of *G*� (*z*) may be found from
4 Will-be-set-by-IN-TECH
single-input-single-output systems. For vector-valued input signals *uk* <sup>∈</sup> **<sup>R</sup>***nu* and output signals *yk* <sup>∈</sup> **<sup>R</sup>***ny* , an LTI filter may be represented as an *ny* <sup>×</sup> *nu* matrix of rational functions *Gij*(*z*), and the system will have matrix-valued impulse-response coefficients. For simplicity, we will assume that transfer function representations are single-input-single-output, though
Because the effect of *b*(*z*) is equivalent to a finite-impulse response filter, the only requirement for *b*(*z*) to produce a stable system is that its coefficients be bounded, which we may safely assume is always the case. Thus the stability of a transfer function *G*(*z*) is determined entirely by *a*(*z*), or more precisely, the roots of *a*(*z*). To see this, suppose *a*(*z*) is factored into its roots,
> *<sup>G</sup>*(*z*) = *<sup>b</sup>*(*z*) ∏*<sup>n</sup>*
> > *G*�
From the time-shift property of the *z*-transform, it is immediately clear that the sequence
the result is the original stability requirement (5), which may be written in terms of *p* as
we may deduce that a system is stable if and only if all the poles of *G*(*z*) satisfy the property
Transfer functions are a convenient way of representing complex system dynamics in a finite number of parameters, but the coefficients of *a*(*z*) and *b*(*z*) cannot be measured directly. The impulse response of a system can be found experimentally by either direct measurement or from other means such as taking the inverse Fourier transform of a measured frequency response [11]. It cannot, however, be represented in a finite number of parameters. Thus the conversion between transfer functions and impulse responses is an extremely useful tool.
∞ ∑ *k*=1 *<sup>p</sup>k*−<sup>1</sup> <sup>&</sup>lt; <sup>∞</sup>.
To guarantee a bounded *yk*, it is sufficient to study a single pole, which we will denote simply as *p*. Thus we wish to determine necessary and sufficient conditions for stability of the system
(*z*) = <sup>1</sup>
*<sup>i</sup>*=1(*z* − *pi*)
*z* − *p*
*pkz*−*<sup>k</sup>* =
(*z*). If we require that (9) is absolutely summable and let |*z*| = 1,
∞ ∑ *k*=1
*<sup>p</sup>k*−<sup>1</sup> *<sup>k</sup>* <sup>&</sup>gt; 1, (10)
(*z*) is stable if and only if |*p*| < 1. Finally, from (7)
<sup>=</sup> *<sup>z</sup>*−<sup>1</sup> <sup>∞</sup> ∑ *k*=0
0 *k* = 1,
. (7)
. (8)
(*z*) then has the Laurent-series
*pk*−1*z*−*k*. (9)
all results presented here generalize to the multi-input-multi-output case.
**2.2. Stability of transfer function representations**
Note that *p* may be complex. Assume that |*z*| > |*p*|. *G*�
1 <sup>1</sup> − *pz*−<sup>1</sup>
> *g*� *<sup>k</sup>* =
**2.3. Construction of transfer functions from impulse responses**
which are the poles *pi* of *G*(*z*),
*G*�
This is true if and only if |*p*| < 1, and thus *G*�
is the impulse response of *G*�
(*z*) = *z*−<sup>1</sup>
expansion
$$p = \mathcal{g}\_k^{-1} \mathcal{g}\_{k+1}.\tag{11}$$
Notice that this is true for any *k*, and the impulse response can be said to have a *shift-invariant* property in this respect.
Less clear is the case when an impulse response is generated by a system with higher-order *a*(*z*) and *b*(*z*). In fact, there is no guarantee that an arbitrary impulse response is the result of a linear system of difference equations at all. For an LTI filter, however, the coefficients of the impulse response exhibit a linear dependence which may be used to not only verify the linearity of the system, but to construct a transfer function representation as well. The exact nature of this linear dependence may be found by forming a structured matrix of impulse response coefficients and examining its behavior when the indices of the coefficients are shifted forward by a single increment, similar to the single-pole case in (11). The result is stated in the following theorem, originally due to Kronecker [4] and adopted from the English translation of [12].
**Theorem 1** (Kronecker's Theorem)**.** *Suppose G*(*z*) : **C** → **C** *is an infinite series of descending powers of z, starting with z*−1*,*
$$G(z) = g\_1 z^{-1} + g\_2 z^{-2} + g\_3 z^{-3} + \cdots = \sum\_{k=1}^{\infty} g\_k z^{-k}.\tag{12}$$
*Assume G(z) is analytic (the series converges) for all* |*z*| > 1*. Let H be an infinitely large matrix of the form*
$$H = \begin{bmatrix} \\$1 & \\$2 & \\$3 & \cdots \\ \\$2 & \\$3 & \\$4 & \cdots \\ \\$3 & \\$4 & \\$5 & \cdots \\ \vdots & \vdots & \vdots \end{bmatrix} \tag{13}$$
*Then H has finite rank n if and only if G*(*z*) *is a strictly proper, coprime, rational function of degree n with poles inside the unit circle. That is, G*(*z*) *has an alternative representation*
$$G(z) = \frac{b(z)}{a(z)} = \frac{b\_m z^m + b\_{m-1} z^{m-1} + \dots + b\_1 z + b\_0}{z^n + a\_{n-1} z^{n-1} + \dots + a\_1 z + a\_0},\tag{14}$$
*in which m* < *n, all roots of a*(*z*) *satisfy* |*z*| < 1*, a*(*z*) *and b*(*z*) *have no common roots, and we have assumed without loss of generality that a*(*z*) *is monic.*
To prove Theorem 1, we first prove that for *k* > *n*, *gk* must be linearly dependent on the previous *n* terms of the series for *H* to have finite rank.
**Theorem 2.** *The infinitely large matrix H is of finite rank n if and only if there exists a finite sequence α*1, *α*2, ··· , *α<sup>n</sup> such that for k* ≥ *n,*
$$\mathbf{g}\_{k+1} = \sum\_{j=1}^{n} a\_j \mathbf{g}\_{k-j+1\nu} \tag{15}$$
*and n is the smallest number with this property.*
*Proof.* Let *hk* be the row of *H* beginning with *gk*. If *H* has rank *n*, then the first *n* + 1 rows of *H* are linearly dependent. This implies that for some 1 ≤ *p* ≤ *n*, *hp*+<sup>1</sup> is a linear combination of *h*1,..., *hp*, and thus there exists some sequence *α<sup>k</sup>* such that
$$h\_{p+1} = \sum\_{j=1}^{p} \alpha\_j h\_{p-j+1}.\tag{16}$$
The structure and infinite size of *H* imply that such a relationship must hold for all following rows of *H*, so that for *q* ≥ 0
$$h\_{q+p+1} = \sum\_{j=1}^{p} \alpha\_j h\_{q+p-j+1}.$$
Hence any row *hk*, *k* > *p*, can be expressed as a linear combination of the previous *p* rows. Since *H* has at least *n* linearly independent rows, *p* = *n*, and since this applies element-wise, rank(*H*) = *n* implies (15).
Alternatively, (15) implies a relationship of the form (16) exists, and hence rank(*H*) = *p*. Since *n* is the smallest possible *p*, this implies rank(*H*) = *n*.
We now prove Theorem 1.
*Proof.* Suppose *G*(*z*) is a coprime rational function of the form (14) with series expansion (12), which we know exists, since *G*(*z*) is analytic for |*z*| < 1. Without loss of generality, let *m* = *n* − 1, since we may always let *bk* = 0 for some *k*. Hence
$$\frac{b\_{n-1}z^{n-1} + b\_{n-2}z^{n-2} + \cdots + b\_1z + b\_0}{z^n + a\_{n-1}z^{n-1} + \cdots + a\_1z + a\_0} = g\_1z^{-1} + g\_2z^{-2} + g\_3z^{-3} + \cdots + z^{-n}$$
Multiplying both sides by the denominator of the left,
$$\begin{aligned} (b\_{n-1}z^{n-1} + b\_{n-2}z^{n-2} + \cdots + b\_1z + b\_0 \\ &= g\_1 z^{n-1} + (g\_2 + g\_1 a\_{n-1})z^{n-2} + (g\_3 + g\_2 a\_{n-1} + g\_1 a\_{n-2})z^{n-3} + \cdots \end{aligned}$$
and equating powers of *z*, we find
$$\begin{aligned} b\_{n-1} &= g\_1 \\ b\_{n-2} &= g\_2 + g\_1 a\_{n-1} \\ b\_{n-3} &= g\_3 + g\_2 a\_{n-1} + g\_1 a\_{n-2} \\ &\vdots \\ b\_1 &= g\_{n-1} + g\_{n-2} a\_{n-1} + \dots + g\_1 a\_2 \\ b\_0 &= g\_n + g\_{n-1} a\_{n-1} + \dots + g\_1 a\_1 \\ 0 &= g\_{k+1} + g\_k a\_{n-1} + \dots + g\_{k-n+1} a\_0 \qquad k \ge n. \end{aligned} \tag{17}$$
From this, we have, for *k* ≥ *n*,
$$\mathcal{g}\_{k+1} = \sum\_{j=1}^{n} -a\_j \mathcal{g}\_{k-j+1}$$
which not only shows that (15) holds, but also shows that *α<sup>j</sup>* = −*aj*. Hence by Theorem 2, *H* must have finite rank.
Conversely, suppose *H* has finite rank. Then (15) holds, and we may construct *a*(*z*) from *α<sup>k</sup>* and *b*(*z*) from (17) to create a rational function. This function must be coprime since its order *n* is the smallest possible.
The construction in Theorem 1 is simple to extend to the case in which *G*(*z*) is only proper and not strictly proper; the additional coefficient *bn* is simply the feed-through term in the impulse response, that is, *g*0.
A result of Theorem 2 is that given finite-dimensional, full-rank matrices
$$H\_k = \begin{bmatrix} \mathcal{S}k & \mathcal{S}k+1 & \cdots & \mathcal{S}k+n-1\\ \mathcal{S}k+1 & \mathcal{S}k+2 & \cdots & \mathcal{S}k+n\\ \vdots & \vdots & & \vdots\\ \mathcal{S}k+n-1 & \mathcal{S}k+n & \cdots & \mathcal{S}k+2n-2 \end{bmatrix} \tag{18}$$
and
6 Will-be-set-by-IN-TECH
*Proof.* Let *hk* be the row of *H* beginning with *gk*. If *H* has rank *n*, then the first *n* + 1 rows of *H* are linearly dependent. This implies that for some 1 ≤ *p* ≤ *n*, *hp*+<sup>1</sup> is a linear combination
> *p* ∑ *j*=1
The structure and infinite size of *H* imply that such a relationship must hold for all following
*p* ∑ *j*=1
Hence any row *hk*, *k* > *p*, can be expressed as a linear combination of the previous *p* rows. Since *H* has at least *n* linearly independent rows, *p* = *n*, and since this applies element-wise,
Alternatively, (15) implies a relationship of the form (16) exists, and hence rank(*H*) = *p*. Since
*Proof.* Suppose *G*(*z*) is a coprime rational function of the form (14) with series expansion (12), which we know exists, since *G*(*z*) is analytic for |*z*| < 1. Without loss of generality, let
*<sup>α</sup>jhq*<sup>+</sup>*p*−*j*<sup>+</sup>1.
*<sup>α</sup>jhp*−*j*<sup>+</sup>1. (16)
<sup>=</sup> *<sup>g</sup>*1*z*−<sup>1</sup> <sup>+</sup> *<sup>g</sup>*2*z*−<sup>2</sup> <sup>+</sup> *<sup>g</sup>*3*z*−<sup>3</sup> <sup>+</sup> ···
(17)
<sup>=</sup> *<sup>g</sup>*1*zn*−<sup>1</sup> + (*g*<sup>2</sup> <sup>+</sup> *<sup>g</sup>*1*an*−1)*zn*−<sup>2</sup> + (*g*<sup>3</sup> <sup>+</sup> *<sup>g</sup>*2*an*−<sup>1</sup> <sup>+</sup> *<sup>g</sup>*1*an*−2)*zn*−<sup>3</sup> <sup>+</sup> ··· ,
*hp*+<sup>1</sup> =
*hq*+*p*+<sup>1</sup> =
*and n is the smallest number with this property.*
rows of *H*, so that for *q* ≥ 0
rank(*H*) = *n* implies (15).
We now prove Theorem 1.
of *h*1,..., *hp*, and thus there exists some sequence *α<sup>k</sup>* such that
*n* is the smallest possible *p*, this implies rank(*H*) = *n*.
*m* = *n* − 1, since we may always let *bk* = 0 for some *k*. Hence
*bn*−1*zn*−<sup>1</sup> <sup>+</sup> *bn*−2*zn*−<sup>2</sup> <sup>+</sup> ··· <sup>+</sup> *<sup>b</sup>*1*<sup>z</sup>* <sup>+</sup> *<sup>b</sup>*<sup>0</sup> *<sup>z</sup><sup>n</sup>* <sup>+</sup> *an*−1*zn*−<sup>1</sup> <sup>+</sup> ··· <sup>+</sup> *<sup>a</sup>*1*<sup>z</sup>* <sup>+</sup> *<sup>a</sup>*<sup>0</sup>
Multiplying both sides by the denominator of the left,
*bn*−<sup>1</sup> = *<sup>g</sup>*<sup>1</sup>
. . .
*bn*−<sup>2</sup> = *<sup>g</sup>*<sup>2</sup> + *<sup>g</sup>*1*an*−<sup>1</sup>
*bn*−<sup>3</sup> = *<sup>g</sup>*<sup>3</sup> + *<sup>g</sup>*2*an*−<sup>1</sup> + *<sup>g</sup>*1*an*−<sup>2</sup>
*<sup>b</sup>*<sup>1</sup> = *gn*−<sup>1</sup> + *gn*−<sup>2</sup>*an*−<sup>1</sup> + ··· + *<sup>g</sup>*1*a*<sup>2</sup> *<sup>b</sup>*<sup>0</sup> = *gn* + *gn*−<sup>1</sup>*an*−<sup>1</sup> + ··· + *<sup>g</sup>*1*a*<sup>1</sup>
<sup>0</sup> = *gk*<sup>+</sup><sup>1</sup> + *gk an*−<sup>1</sup> + ··· + *gk*−*n*+1*a*<sup>0</sup> *<sup>k</sup>* ≥ *<sup>n</sup>*.
*bn*−1*zn*−<sup>1</sup> <sup>+</sup> *bn*−2*zn*−<sup>2</sup> <sup>+</sup> ··· <sup>+</sup> *<sup>b</sup>*1*<sup>z</sup>* <sup>+</sup> *<sup>b</sup>*<sup>0</sup>
and equating powers of *z*, we find
$$H\_{k+1} = \begin{bmatrix} \mathcal{g}k+1 & \mathcal{g}k+2 & \cdots & \mathcal{g}k+n\\ \mathcal{g}k+2 & \mathcal{g}k+3 & \cdots & \mathcal{g}k+n+1\\ \vdots & \vdots & & \vdots\\ \mathcal{g}k+n & \mathcal{g}k+n+1 & \cdots & \mathcal{g}k+2n-1 \end{bmatrix}'\tag{19}$$
the coefficients of *a*(*z*) may be calculated as
$$\begin{bmatrix} 0 \ 0 \ \cdots \ 0 \ \ -a\_0 \\ 1 \ 0 \ \cdots \ 0 \ \ -a\_1 \\ 0 \ 1 \ \cdots \ 0 \ \ -a\_2 \\ \vdots \ \vdots \ \ddots \ \vdots \ \vdots \\ 0 \ 0 \ \cdots \ 1 \ -a\_{n-1} \end{bmatrix} = H\_k^{-1} H\_{k+1}.\tag{20}$$
Notice that (11) is in fact a special case of (20). Thus we need only know the first 2*n* + 1 impulse-response coefficients to reconstruct the transfer function *G*(*z*): 2*n* to form the matrices *Hk* and *Hk*<sup>+</sup><sup>1</sup> from (18) and (19), respectively, and possibly the initial coefficient *g*<sup>0</sup> in case of an *n*th-order *b*(*z*).
Matrices with the structure of *H* are useful enough to have a special name. A *Hankel matrix H* is a matrix constructed from a sequence {*hk*} so that each element *H*(*j*,*k*) = *hj*<sup>+</sup>*k*. For the Hankel matrix in (13), *hk* = *gk*−1. *Hk* also has an interesting property implied by (20): its row space is invariant under shifting of the index *k*. Because its symmetric, this is also true for its column space. Thus this matrix is also often referred to as being *shift-invariant*.
While (20) provides a potential method of identifying a system from a measured impulse response, this is not a reliable method to use with measured impulse response coefficients that are corrupted by noise. The exact linear dependence of the coefficients will not be identical for all *k*, and the structure of (20) will not be preserved. Inverting *Hk* will also invert any noise on *gk*, potentially amplifying high-frequency noise content. Finally, the system order *n* is required to be known beforehand, which is usually not the case if only an impulse response is available. Fortunately, these difficulties may all be overcome by reinterpreting the results Kronecker's theorem in a state-space framework. First, however, we more carefully examine the role of the Hankel matrix in the behavior of LTI filters.
#### **2.4. Hankel and Toeplitz operators**
The Hankel matrix of impulse response coefficients (13) is more than a tool for computing the transfer function representation of a system from its impulse response. It also defines the mapping of past input signals to future output signals. To define exactly what this means, we write the convolution of (4) around sample *k* = 0 in matrix form as
where the vectors and matrix have been partitioned into sections for *k* < 0 and *k* ≥ 0. The output for *k* ≥ 0 may then be split into two parts:
$$
\underbrace{\begin{bmatrix} y\_0 \\ y\_1 \\ y\_2 \\ \vdots \\ y\_f \end{bmatrix}}\_{\mathcal{Y}f} = \underbrace{\begin{bmatrix} \mathfrak{g}\_1 \ \mathfrak{g}\_2 \ \mathfrak{g}\_3 \cdots \cdots \\ \mathfrak{g}\_2 \ \mathfrak{g}\_3 \ \mathfrak{g}\_4 \ \cdots \\ \mathfrak{g}\_3 \ \mathfrak{g}\_4 \ \mathfrak{g}\_5 \cdots \\ \vdots \\ \mathfrak{i} \end{bmatrix}}\_{\mathcal{H}} \begin{bmatrix} u\_{-1} \\ u\_{-2} \\ u\_{-3} \\ \vdots \\ \vdots \\ u\_p \end{bmatrix} + \underbrace{\begin{bmatrix} \mathfrak{g}\_0 & \cdots & 0 \\ \mathfrak{g}\_1 \ \mathfrak{g}\_0 & \vdots \\ \mathfrak{g}\_1 \ \mathfrak{g}\_0 & \mathfrak{i} \\ \vdots & \vdots & \vdots \\ \mathfrak{i} & \mathfrak{i} & \mathfrak{i} \end{bmatrix}}\_{\mathcal{U}} \begin{bmatrix} u\_0 \\ u\_1 \\ u\_2 \\ \vdots \\ \vdots \end{bmatrix}' \tag{21}
$$
,
where the subscripts *p* and *f* denote "past" and "future," respectively. The system Hankel matrix *H* has returned to describe the effects of the past input *up* on the future output *yf* . Also present is the matrix *T*, which represents the convolution of future input *uf* with the impulse response. Matrices such as *T* with constant diagonals are called *Toeplitz* matrices.
From (21), it can be seen that *H* defines the effects of past input on future output. One interpretation of this is that *H* represents the "memory" of the system. Because *H* is a linear mapping from *up* to *yf* , the induced matrix 2-norm of *H*, ||*H*||2, can be considered a function norm, and in a sense, ||*H*||<sup>2</sup> is a measure of the "gain" of the system. ||*H*||<sup>2</sup> is often called the *Hankel-norm* of a system, and it plays an important role in model reduction and in the analysis of anti-causal systems. More information on this aspect of linear systems can be found in the literature of robust control, for instance, [13].
#### **3. State-space representations**
8 Will-be-set-by-IN-TECH
for all *k*, and the structure of (20) will not be preserved. Inverting *Hk* will also invert any noise on *gk*, potentially amplifying high-frequency noise content. Finally, the system order *n* is required to be known beforehand, which is usually not the case if only an impulse response is available. Fortunately, these difficulties may all be overcome by reinterpreting the results Kronecker's theorem in a state-space framework. First, however, we more carefully examine
The Hankel matrix of impulse response coefficients (13) is more than a tool for computing the transfer function representation of a system from its impulse response. It also defines the mapping of past input signals to future output signals. To define exactly what this means, we
... ··· <sup>0</sup>
where the vectors and matrix have been partitioned into sections for *k* < 0 and *k* ≥ 0. The
*u*−<sup>1</sup> *u*−<sup>2</sup> *u*−<sup>3</sup> . . .
� �� � *up*
where the subscripts *p* and *f* denote "past" and "future," respectively. The system Hankel matrix *H* has returned to describe the effects of the past input *up* on the future output *yf* . Also present is the matrix *T*, which represents the convolution of future input *uf* with the impulse
From (21), it can be seen that *H* defines the effects of past input on future output. One interpretation of this is that *H* represents the "memory" of the system. Because *H* is a linear mapping from *up* to *yf* , the induced matrix 2-norm of *H*, ||*H*||2, can be considered a function norm, and in a sense, ||*H*||<sup>2</sup> is a measure of the "gain" of the system. ||*H*||<sup>2</sup> is often called the *Hankel-norm* of a system, and it plays an important role in model reduction and in the analysis
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
+
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
*g*<sup>0</sup> ··· 0
. . . ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
*u*0 *u*1 *u*2 . . .
� �� � *uf*
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
, (21)
*g*<sup>1</sup> *g*<sup>0</sup>
*g*<sup>2</sup> *g*<sup>1</sup> *g*<sup>0</sup> . . . . . . . . . ...
� �� � *T*
. . . ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
. . . *u*−<sup>3</sup> *u*−<sup>2</sup> *u*−<sup>1</sup> *u*0 *u*1 *u*2 . . .
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
,
the role of the Hankel matrix in the behavior of LTI filters.
write the convolution of (4) around sample *k* = 0 in matrix form as
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
··· *g*<sup>0</sup>
··· *g*<sup>1</sup> *g*<sup>0</sup> ··· *g*<sup>2</sup> *g*<sup>1</sup> *g*<sup>0</sup> ··· *g*<sup>3</sup> *g*<sup>2</sup> *g*<sup>1</sup> *g*<sup>0</sup> ··· *g*<sup>4</sup> *g*<sup>3</sup> *g*<sup>2</sup> *g*<sup>1</sup> *g*<sup>0</sup> ··· *g*<sup>5</sup> *g*<sup>4</sup> *g*<sup>3</sup> *g*<sup>2</sup> *g*<sup>1</sup> *g*<sup>0</sup>
. . . . . . . . . . . . . . . . . . ...
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
response. Matrices such as *T* with constant diagonals are called *Toeplitz* matrices.
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
**2.4. Hankel and Toeplitz operators**
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
output for *k* ≥ 0 may then be split into two parts:
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
. . . . . . . . .
*g*<sup>1</sup> *g*<sup>2</sup> *g*<sup>3</sup> ··· *g*<sup>2</sup> *g*<sup>3</sup> *g*<sup>4</sup> ··· *g*<sup>3</sup> *g*<sup>4</sup> *g*<sup>5</sup> ···
� �� � *H*
=
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
*y*0 *y*1 *y*2 . . .
� �� � *yf*
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
. . . *y*−<sup>3</sup> *y*−<sup>2</sup> *y*−<sup>1</sup> *y*0 *y*1 *y*2 . . .
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
=
Although transfer functions define system behavior completely with a finite number of parameters and simplify frequency-response calculations, they are cumbersome to manipulate when the input or output is multi-dimensional or when initial conditions must be considered. The other common representation of LTI filters is the state-space form
$$\begin{aligned} \mathbf{x}\_{k+1} &= A\mathbf{x}\_k + Bu\_k\\ y\_k &= \mathbf{C}\mathbf{x}\_k + Du\_{k\prime} \end{aligned} \tag{22}$$
in which *xk* <sup>∈</sup> **<sup>R</sup>***<sup>n</sup>* is the system state. The matrices *<sup>A</sup>* <sup>∈</sup> **<sup>R</sup>***n*×*n*, *<sup>B</sup>* <sup>∈</sup> **<sup>R</sup>***n*×*nu* , *<sup>C</sup>* <sup>∈</sup> **<sup>R</sup>***ny*×*n*, and *<sup>D</sup>* <sup>∈</sup> **<sup>R</sup>***ny*×*nu* completely parameterize the system. Only *<sup>D</sup>* uniquely defines the input-output behavior; any nonsingular matrix *T*� may be used to change the state basis via the relationships
$$\mathbf{x}' = T'\mathbf{x} \qquad A' = T'AT'^{-1} \qquad B' = T'B \qquad C' = CT'^{-1}.$$
The *Z*-transform may also be applied to the state-space equations (22) to find
$$\begin{array}{llll} \mathcal{Z}[\mathbf{x}\_{k+1}] = A \mathcal{Z}[\mathbf{x}\_{k}] + B \mathcal{Z}[\mathbf{u}\_{k}] & \Rightarrow & X(z)z = AX(z) + BI(z) \\ \mathcal{Z}[y\_{k}] = \mathcal{C} \mathcal{Z}[\mathbf{x}\_{k}] + D \mathcal{Z}[\mathbf{u}\_{k}] & \Rightarrow & Y(z) = \mathcal{C}X(z) + DI(z) \end{array}$$
$$\frac{Y(z)}{U(z)} = G(z) \qquad \qquad G(z) = \mathcal{C} \left(zI - A\right)^{-1}B + D,\tag{23}$$
and thus, if (22) is the state-space representation of the single-variable system (6), then *a*(*z*) is the characteristic polynomial of *A*, det(*zI* − *A*).
Besides clarifying the effect of initial conditions on the output, state-space representations are inherently causal, and (23) will always result in a proper system (strictly proper if *D* = 0). For this reason, state-space representations are often called *realizable* descriptions; while the forward-time-shift of *z* is an inherently non-causal operation, state-space systems may always be constructed in reality.
#### **3.1. Stability, controllability, and observability of state-space representations**
The system impulse response is simple to formulate in terms of the state-space parameters by calculation of the output to a unit impulse with *x*<sup>0</sup> = 0:
$$\mathcal{g}\_k = \begin{cases} D & k = 0, \\ CA^{k-1}B & k > 0 \end{cases} \tag{24}$$
Notice the similarity of (10) and (24). In fact, from the eigenvalue decomposition of *A*,
$$A = V\Lambda V^{-1}\prime$$
we find <sup>∞</sup>
$$\sum\_{k=1}^{\infty} |\mathcal{g}\_k| = \sum\_{k=1}^{\infty} \left| \mathcal{C} A^{k-1} B \right| = \sum\_{k=1}^{\infty} |\mathcal{C} V| \left( |\Lambda^{k-1}| \right) \left| V^{-1} B \right|.$$
The term <sup>|</sup>Λ*k*−1<sup>|</sup> will only converge if the largest eigenvalue of *<sup>A</sup>* is within the unit circle, and thus the condition that all eigenvalues *λ<sup>i</sup>* of *A* satisfy |*λi*| < 1 is a necessary and sufficient condition for stability.
For state-space representations, there is the possibility that a combination of *A* and *B* will result in a system for which *xk* cannot be entirely controlled by the input *uk*. Expressing *xk* in a matrix-form similar to (21) as
$$\mathbf{x}\_{k} = \mathcal{C} \begin{bmatrix} u\_{k-1} \\ u\_{k-2} \\ u\_{k-3} \\ \vdots \end{bmatrix}, \qquad \mathcal{C} = \begin{bmatrix} B \ AB \ A^2 B \ \cdots \end{bmatrix} \tag{25}$$
demonstrates that *xk* is in subspace **<sup>R</sup>***<sup>n</sup>* if and only if <sup>C</sup> has rank *<sup>n</sup>*. <sup>C</sup> is the *controllability matrix* and the system is *controllable* if it has full row rank.
Similarly, the state *xk* may not uniquely determine the output for some combinations of *A* and *C*. Expressing the evolution of the output as a function of the state in matrix-form as
$$\begin{bmatrix} y\_k \\ y\_{k+1} \\ y\_{k+2} \\ \vdots \end{bmatrix} = \mathcal{O} \mathbf{x}\_{k\prime} \qquad \mathcal{O} = \begin{bmatrix} \mathbf{C} \\ \mathbf{C}A \\ \mathbf{C}A^2 \\ \vdots \end{bmatrix}$$
demonstrates that there is no nontrivial null space in the mapping from *xk* to *yk* if and only if O has rank *n*. O is the *observability matrix* and the system is *observable* if it has full column rank.
Systems that are both controllable and observable are called *minimal*, and for minimal systems, the dimension *n* of the state variable cannot be reduced. In the next section we show that minimal state-space system representations convert to coprime transfer functions that are found through (23).
#### **3.2. Construction of state-space representations from impulse responses**
The fact that the denominator of *G*(*z*) is the characteristic polynomial of *A* not only allows for the calculation of a transfer function from a state-space representation, but provides an alternative version of Kronecker's theorem for state-space systems, known as the Ho-Kalman Algorithm [3]. From the Caley-Hamilton theorem, if *a*(*z*) is the characteristic polynomial of *A*, then *a*(*A*) = 0, and
$$CA^k a(A)B = CA^k \left( A^n + a\_{n-1}A^{n-1} + \dots + a\_1A + a\_0 \right) B$$
$$= CA^{k+n}B + \sum\_{j=0}^{n-1} a\_j \mathbf{C} A^{k+j} B\_\prime$$
which implies
$$
\mathbb{C}A^{k+n}B = -\sum\_{j=0}^{n-1} a\_j \mathbb{C}A^{k+j}B.\tag{26}
$$
Indeed, substitution of (24) into (26) and rearrangement of the indices leads to (15). Additionally, substitution of (24) into the product of O and C shows that
$$\mathcal{OC} = \begin{bmatrix} \mathcal{C}B & \mathcal{C}AB \ \mathcal{C}A^2B & \cdots \\ \mathcal{C}AB \ \mathcal{C}A^2B \ \mathcal{C}A^3B & \cdots \\ \mathcal{C}A^2B \ \mathcal{C}A^3B \ \mathcal{C}A^4B & \cdots \\ \vdots & \vdots & \vdots \end{bmatrix} = \begin{bmatrix} \mathcal{g}\_1 \ \mathcal{g}\_2 \ \mathcal{g}\_3 \ \cdots \\ \mathcal{g}\_2 \ \mathcal{g}\_3 \ \mathcal{g}\_4 \ \cdots \\ \mathcal{g}\_3 \ \mathcal{g}\_4 \ \mathcal{g}\_5 \ \cdots \\ \vdots & \vdots \end{bmatrix} = H\_{\mathcal{A}}$$
which confirms our previous statement that *H* effectively represents the memory of the system. Because
$$\text{rank}(H) = \min\{\text{rank}(\mathcal{O}), \text{rank}(\mathcal{C})\}\_{\mathcal{M}}$$
we see that rank(*H*) = *n* implies the state-space system (22) is minimal.
If the entries of *H* are shifted forward by one index to form
$$
\overline{H} = \begin{bmatrix}
\\$2 \text{ }\\$3 \text{ }\\$4 & \cdots \\
\\$3 \text{ }\\$4 \text{ }\\$5 & \cdots \\
\\$4 \text{ }\\$5 \text{ }\\$6 & \cdots \\
\vdots & \vdots & \vdots
\end{bmatrix},
$$
then once again substituting (24) reveals
10 Will-be-set-by-IN-TECH
The term <sup>|</sup>Λ*k*−1<sup>|</sup> will only converge if the largest eigenvalue of *<sup>A</sup>* is within the unit circle, and thus the condition that all eigenvalues *λ<sup>i</sup>* of *A* satisfy |*λi*| < 1 is a necessary and sufficient
For state-space representations, there is the possibility that a combination of *A* and *B* will result in a system for which *xk* cannot be entirely controlled by the input *uk*. Expressing *xk* in
<sup>⎦</sup> , <sup>C</sup> <sup>=</sup> �
demonstrates that *xk* is in subspace **<sup>R</sup>***<sup>n</sup>* if and only if <sup>C</sup> has rank *<sup>n</sup>*. <sup>C</sup> is the *controllability matrix*
Similarly, the state *xk* may not uniquely determine the output for some combinations of *A* and
⎡ ⎢ ⎢ ⎢ ⎣ *C CA CA*<sup>2</sup> . . .
⎤ ⎥ ⎥ ⎥ ⎦
<sup>⎦</sup> <sup>=</sup> <sup>O</sup>*xk*, <sup>O</sup> <sup>=</sup>
demonstrates that there is no nontrivial null space in the mapping from *xk* to *yk* if and only if O has rank *n*. O is the *observability matrix* and the system is *observable* if it has full column
Systems that are both controllable and observable are called *minimal*, and for minimal systems, the dimension *n* of the state variable cannot be reduced. In the next section we show that minimal state-space system representations convert to coprime transfer functions that are
The fact that the denominator of *G*(*z*) is the characteristic polynomial of *A* not only allows for the calculation of a transfer function from a state-space representation, but provides an alternative version of Kronecker's theorem for state-space systems, known as the Ho-Kalman Algorithm [3]. From the Caley-Hamilton theorem, if *a*(*z*) is the characteristic polynomial of
> *n*−1 ∑ *j*=0
*<sup>A</sup><sup>n</sup>* <sup>+</sup> *an*−1*An*−<sup>1</sup> <sup>+</sup> ··· <sup>+</sup> *<sup>a</sup>*1*<sup>A</sup>* <sup>+</sup> *<sup>a</sup>*<sup>0</sup>
*ajCA<sup>k</sup>*+*<sup>j</sup>*
*B*,
*ajCA<sup>k</sup>*+*<sup>j</sup>*
*n*−1 ∑ *j*=0
**3.2. Construction of state-space representations from impulse responses**
�
= *CAk*<sup>+</sup>*nB* +
*CAk*<sup>+</sup>*nB* <sup>=</sup> <sup>−</sup>
*C*. Expressing the evolution of the output as a function of the state in matrix-form as
*B AB A*2*<sup>B</sup>* ···� (25)
� *B*
*B*. (26)
condition for stability.
rank.
found through (23).
*A*, then *a*(*A*) = 0, and
which implies
a matrix-form similar to (21) as
*xk* = C
⎡ ⎢ ⎢ ⎢ ⎣
*CA<sup>k</sup> a*(*A*)*B* = *CA<sup>k</sup>*
*yk yk*<sup>+</sup><sup>1</sup> *yk*<sup>+</sup><sup>2</sup> . . .
⎤ ⎥ ⎥ ⎥
and the system is *controllable* if it has full row rank.
⎡ ⎢ ⎢ ⎢ ⎣ *uk*−<sup>1</sup> *uk*−<sup>2</sup> *uk*−<sup>3</sup> . . .
⎤ ⎥ ⎥ ⎥
$$
\overline{H} = \mathcal{O}A\mathcal{C}.\tag{27}
$$
Thus the row space and column space of *H* are invariant under a forward-shift of the indices, implying the same shift-invariant structure seen in (20).
The appearance of *A* in (27) hints at a method for constructing a state-space realization from an impulse response. Suppose the impulse response is known exactly, and let *Hr* be a finite slice of *H* with *r* block rows and *L* columns,
$$H\_r = \begin{bmatrix} \mathcal{S}1 & \mathcal{S}2 & \mathcal{S}3 & \cdots & \mathcal{S}L \\ \mathcal{S}2 & \mathcal{S}3 & \mathcal{S}4 & \cdots & \mathcal{S}L+1 \\ \mathcal{S}3 & \mathcal{S}4 & \mathcal{S}5 & \cdots & \mathcal{S}L+2 \\ \vdots & \vdots & \vdots & & \vdots \\ \mathcal{S}r-1 & \mathcal{S}r & \mathcal{S}r+1 & \cdots & \mathcal{S}r+L-1 \end{bmatrix}.$$
Then any appropriately dimensioned factorization
$$\begin{aligned} \mathcal{H}\_{\mathcal{I}} = \mathcal{O}\_{\mathcal{I}} \mathcal{C}\_{L} = \begin{bmatrix} \mathsf{C} \\ \mathsf{C}A \\ \mathsf{C}A^{2} \\ \vdots \\ \mathsf{C}A^{r-1} \end{bmatrix} \begin{bmatrix} \mathcal{B} \ AB \ A^{2} \mathcal{B} \ \cdots \ A^{L-1} \mathcal{B} \end{bmatrix} \tag{28}$$
may be used to find *A* for some arbitrary state basis as
$$A = \left(\mathcal{O}\_r\right)^\dagger \overline{H}\_r \left(\mathcal{C}\_L\right)^\dagger \tag{29}$$
where *Hr* is *Hr* with the indices shifted forward once and (·)† is the Moore-Penrose pseudoinverse. *C* taken from the first block row of O*r*, *B* taken from the first block column of C*L*, and *D* taken from *g*<sup>0</sup> then provides a complete and minimal state-space realization from an impulse response. Because *Hr* has rank *n* and det(*zI* − *A*) has degree *n*, we know from Kronecker's theorem that *G*(*z*) taken from (23) will be coprime.
However, as mentioned before, the impulse response of the system is rarely known exactly. In this case only an estimate *H*ˆ*<sup>r</sup>* with a non-deterministic error term is available:
$$
\hat{H}\_r = H\_r + E.
$$
Because *E* is non-deterministic, *H*ˆ will always have full rank, regardless of the number of rows *r*. Thus *n* cannot be determined from examining the rank of *H*, and even if *n* is known beforehand, a factorization (28) for *r* > *n* will not exist. Thus we must find a way of reducing the rank of *H*ˆ*<sup>r</sup>* in order to find a state-space realization.
#### **3.3. Rank-reduction of the Hankel matrix estimate**
If *H*ˆ*<sup>r</sup>* has full rank, or if *n* is unknown, its rank must be reduced prior to factorization. The obvious tool for reducing the rank of matrices is the *singular-value decomposition* (SVD). Assume for now that *n* is known. The SVD of *H*ˆ*<sup>r</sup>* is
$$
\hat{H}\_r = l\Sigma V^T
$$
where *U* and *V<sup>T</sup>* are orthogonal matrices and Σ is a diagonal matrix containing the nonnegative *singular values σ<sup>i</sup>* ordered from largest to smallest. The SVD for a matrix is unique and guaranteed to exist, and the number of nonzero singular values of a matrix is equal to its rank [14].
Because *U* and *V<sup>T</sup>* are orthogonal, the SVD satisfies
$$\hat{H}\_{\mathbf{r}} = \left| \left| \mathbf{J} \Sigma \mathbf{V}^{T} \right| \right|\_{2} = ||\Sigma||\_{2} = \sigma\_{1} \tag{30}$$
where ||·||<sup>2</sup> is the induced matrix 2-norm, and
$$\hat{H}\_{l} = \left\| \left| U \Sigma V^{T} \right| \right\|\_{F} = ||\Sigma||\_{F} = \left( \sum\_{i}^{l} \sigma\_{i}^{2} \right)^{1/2} \tag{31}$$
where ||·||*<sup>F</sup>* is the Frobenius norm. Equation (30) also shows that the Hankel norm of a system is the maximum singular value of *Hr*. From (30) and (31), we can directly see that if the SVD of *Hr* is partitioned into
$$
\hat{H}\_r = \begin{bmatrix} \mathcal{U}\_n \ \mathcal{U}\_s \end{bmatrix} \begin{bmatrix} \Sigma\_n & 0 \\ 0 & \Sigma\_s \end{bmatrix} \begin{bmatrix} V\_n^T \\ V\_s^T \end{bmatrix} \mathcal{I}
$$
where *Un* is the first *<sup>n</sup>* columns of *<sup>U</sup>*, <sup>Σ</sup>*<sup>n</sup>* is the upper-left *<sup>n</sup>* <sup>×</sup> *<sup>n</sup>* block of <sup>Σ</sup>, and *<sup>V</sup><sup>T</sup> <sup>n</sup>* is the first *n* rows of *VT*, the solution to the rank-reduction problem is [14]
$$Q = \underset{\text{rank}(Q) = n}{\text{arg min}} \left||Q - \hat{H}\_r||\_2 = \underset{\text{rank}(Q) = n}{\text{arg min}} \left||Q - \hat{H}\_r||\_F = \mathcal{U}\_n \Sigma\_n V\_n^T.$$
Additionally, the error resulting from the rank reduction is
$$\mathcal{e} = \left\| \left| \mathcal{Q} - \hat{H}\_{\prime} \right\| \right\|\_{2} = \sigma\_{n+1\prime}$$
which suggests that if the rank of *Hr* is not known beforehand, it can be determined by examining the nonzero singular values in the deterministic case or by searching for a significant drop-off in singular values if only a noise-corrupted estimate is available.
#### **3.4. Identifying the state-space realization**
From a rank-reduced *H*ˆ*r*, any factorization
$$
\hat{H}\_r = \hat{\mathcal{O}}\_r \hat{\mathcal{C}}\_L
$$
can be used to estimate O*<sup>r</sup>* and C*L*. The error in the state-space realization, however, will depend on the chosen state basis. Generally we would like to have a state variable with a norm ||*xk*||<sup>2</sup> in between ||*uk*||<sup>2</sup> and ||*yk*||2. As first proposed in [5], choosing the factorization
$$\mathcal{O}\_r = \mathsf{U}\_n \Sigma\_n^{1/2} \qquad \text{and} \qquad \mathcal{C}\_L = \Sigma\_n^{1/2} V\_n^T \tag{32}$$
results in
12 Will-be-set-by-IN-TECH
where *Hr* is *Hr* with the indices shifted forward once and (·)† is the Moore-Penrose pseudoinverse. *C* taken from the first block row of O*r*, *B* taken from the first block column of C*L*, and *D* taken from *g*<sup>0</sup> then provides a complete and minimal state-space realization from an impulse response. Because *Hr* has rank *n* and det(*zI* − *A*) has degree *n*, we know from
However, as mentioned before, the impulse response of the system is rarely known exactly. In
*H*ˆ*<sup>r</sup>* = *Hr* + *E*. Because *E* is non-deterministic, *H*ˆ will always have full rank, regardless of the number of rows *r*. Thus *n* cannot be determined from examining the rank of *H*, and even if *n* is known beforehand, a factorization (28) for *r* > *n* will not exist. Thus we must find a way of reducing
If *H*ˆ*<sup>r</sup>* has full rank, or if *n* is unknown, its rank must be reduced prior to factorization. The obvious tool for reducing the rank of matrices is the *singular-value decomposition* (SVD).
*H*ˆ*<sup>r</sup>* = *U*Σ*V<sup>T</sup>* where *U* and *V<sup>T</sup>* are orthogonal matrices and Σ is a diagonal matrix containing the nonnegative *singular values σ<sup>i</sup>* ordered from largest to smallest. The SVD for a matrix is unique and guaranteed to exist, and the number of nonzero singular values of a matrix is equal to its
*<sup>F</sup>* <sup>=</sup> ||Σ||*<sup>F</sup>* <sup>=</sup>
where ||·||*<sup>F</sup>* is the Frobenius norm. Equation (30) also shows that the Hankel norm of a system is the maximum singular value of *Hr*. From (30) and (31), we can directly see that if the SVD
> <sup>2</sup> = arg min rank(*Q*)=*n*
*l* ∑ *i σ*2 *i*
*V<sup>T</sup> n V<sup>T</sup> s* ,
*<sup>Q</sup>* <sup>−</sup> *<sup>H</sup>*ˆ*<sup>r</sup>*
<sup>2</sup> <sup>=</sup> ||Σ||<sup>2</sup> <sup>=</sup> *<sup>σ</sup>*<sup>1</sup> (30)
*<sup>F</sup>* <sup>=</sup> *Un*Σ*nV<sup>T</sup>*
*n* .
(31)
*<sup>n</sup>* is the first
1/2
Kronecker's theorem that *G*(*z*) taken from (23) will be coprime.
the rank of *H*ˆ*<sup>r</sup>* in order to find a state-space realization.
**3.3. Rank-reduction of the Hankel matrix estimate**
Assume for now that *n* is known. The SVD of *H*ˆ*<sup>r</sup>* is
Because *U* and *V<sup>T</sup>* are orthogonal, the SVD satisfies
*H*ˆ*<sup>r</sup>* = *U*Σ*V<sup>T</sup>*
where ||·||<sup>2</sup> is the induced matrix 2-norm, and
*Q* = arg min rank(*Q*)=*n* *H*ˆ*<sup>r</sup>* = *U*Σ*V<sup>T</sup>*
*H*ˆ*<sup>r</sup>* =
*n* rows of *VT*, the solution to the rank-reduction problem is [14]
*<sup>Q</sup>* <sup>−</sup> *<sup>H</sup>*ˆ*<sup>r</sup>*
*Un Us* Σ*<sup>n</sup>* 0 0 Σ*<sup>s</sup>*
where *Un* is the first *<sup>n</sup>* columns of *<sup>U</sup>*, <sup>Σ</sup>*<sup>n</sup>* is the upper-left *<sup>n</sup>* <sup>×</sup> *<sup>n</sup>* block of <sup>Σ</sup>, and *<sup>V</sup><sup>T</sup>*
rank [14].
of *Hr* is partitioned into
this case only an estimate *H*ˆ*<sup>r</sup>* with a non-deterministic error term is available:
$$|||\mathcal{O}\_r|||\_2 = ||\mathcal{C}\_L||\_2 = \sqrt{||\hat{H}\_r||\_2} \tag{33}$$
and thus, from a functional perspective, the mappings from input to state and state to output will have equal magnitudes, and each entry of the state vector *xk* will have similar magnitudes. State-space realizations that satisfy (33) are sometimes called *internally balanced* realizations [11]. (Alternative definitions of a "balanced" realization exist, however, and it is generally wise to verify the definition in each context.)
Choosing the factorization (32) also simplifies computation of the estimate *A*ˆ, since
$$\begin{aligned} \hat{A} &= \left(\mathcal{O}\_r\right)^\dagger \overleftarrow{H}\_r \left(\mathcal{C}\_L\right)^\dagger \\ &= \Sigma\_n^{-1/2} \mathcal{U}\_n^T \overleftarrow{H}\_r V\_n \Sigma\_n^{-1/2} . \end{aligned}$$
By estimating *<sup>B</sup>*<sup>ˆ</sup> as the first block column of <sup>C</sup><sup>ˆ</sup> *<sup>L</sup>*, *<sup>C</sup>*<sup>ˆ</sup> as the first block row of <sup>O</sup><sup>ˆ</sup> *<sup>L</sup>*, and *<sup>D</sup>*<sup>ˆ</sup> as *<sup>g</sup>*0, a complete state-space realization (*A*ˆ, *B*ˆ, *C*ˆ, *D*ˆ ) is identified from this method.
#### **3.5. Pitfalls of direct realization from an impulse response**
Even though the rank-reduction process allows for realization from a noise-corrupted estimate of an impulse response, identification methods that generate a system estimate from a Hankel matrix constructed from an estimated impulse response have numerous difficulties when applied to noisy measurements. Measuring an impulse response directly is often infeasible; high-frequency damping may result in a measurement that has a very brief response before the signal-to-noise ratio becomes prohibitively small, and a unit pulse will often excite high-frequency nonlinearities that degrade the quality of the resulting estimate.
Taking the inverse Fourier transform of the frequency response guarantees that the estimates of the Markov parameters will converge as the dataset grows only so long as the input is broadband. Generally input signals decay in magnitude at higher frequencies, and calculation of the frequency response by inversion of the input will amplify high-frequency noise. We would prefer an identification method that is guaranteed to provide a system estimate that converges to the true system as the amount of data measured increases and that avoids inverting the input. Fortunately, the relationship between input and output data in (21) may be used to formulate just such an identification procedure.
#### **4. Realization from input-output data**
To avoid the difficulties in constructing a system realization from an estimated impulse response, we will form a realization-based identification procedure applicable to measured input-output data. To sufficiently account for non-deterministic effects in measured data, we add a noise term *vk* <sup>∈</sup> **<sup>R</sup>***ny* to the output to form the noise-perturbed state-space equations
$$\begin{aligned} \mathbf{x}\_{k+1} &= A\mathbf{x}\_k + B\mathbf{u}\_k\\ \mathbf{y}\_k &= \mathbf{C}\mathbf{x}\_k + D\mathbf{u}\_k + \mathbf{v}\_k. \end{aligned} \tag{34}$$
We assume that the noise signal *vk* is generated by a stationary stochastic process, which may be either white or colored. This includes the case in which the state is disturbed by process noise, so that the noise process may have the same poles as the deterministic system. (See [15] for a thorough discussion of representations of noise in the identification context.)
#### **4.1. Data-matrix equations**
The goal is to construct a state-space realization using the relationships in (21), but doing so requires a complete characterization of the row space of *Hr*. To this end, we expand a finite-slice of the future output vector to form a block-Hankel matrix of output data with *r* block rows,
$$Y = \begin{bmatrix} \begin{array}{cccc} \mathcal{Y}\_0 & \mathcal{Y}\_1 & \mathcal{Y}\_2 & \cdots & \mathcal{Y}\_L \\ \mathcal{Y}\_1 & \mathcal{Y}\_2 & \mathcal{Y}\_3 & \cdots & \mathcal{Y}\_{L+1} \\ \mathcal{Y}\_2 & \mathcal{Y}\_3 & \mathcal{Y}\_4 & \cdots & \mathcal{Y}\_{L+2} \\ \vdots & \vdots & \vdots & & \vdots \\ \vdots & \vdots & \vdots & & \vdots \\ \mathcal{Y}\_{r-1} & \mathcal{Y}\_r & \mathcal{Y}\_{r+1} & \cdots & \mathcal{Y}\_{r+L-1} \end{array} \end{bmatrix}.$$
This matrix is related to a block-Hankel matrix of future input data
$$\mathcal{U}\_f = \begin{bmatrix} u\_0 & u\_1 & u\_2 & \cdots & u\_L \\ u\_1 & u\_2 & u\_3 & \cdots & u\_{L+1} \\ u\_2 & u\_3 & u\_4 & \cdots & u\_{L+2} \\ \vdots & \vdots & \vdots & & \vdots \\ u\_{r-1} & u\_r & u\_{r+1} & \cdots & u\_{r+L-1} \end{bmatrix} \prime$$
a block-Toeplitz matrix of past input data
$$\mathcal{U}\_p = \begin{bmatrix} \mu\_{-1} & \mu\_0 & \mu\_1 & \cdots & \mu\_{L-1} \\ \mu\_{-2} & \mu\_{-1} & \mu\_0 & \cdots & \mu\_{L-2} \\ \mu\_{-3} & \mu\_{-2} & \mu\_{-1} & \cdots & \mu\_{L-3} \\ \vdots & \vdots & \vdots & & \vdots \end{bmatrix} \prime$$
a finite-dimensional block-Toeplitz matrix
14 Will-be-set-by-IN-TECH
of the frequency response by inversion of the input will amplify high-frequency noise. We would prefer an identification method that is guaranteed to provide a system estimate that converges to the true system as the amount of data measured increases and that avoids inverting the input. Fortunately, the relationship between input and output data in (21) may
To avoid the difficulties in constructing a system realization from an estimated impulse response, we will form a realization-based identification procedure applicable to measured input-output data. To sufficiently account for non-deterministic effects in measured data, we add a noise term *vk* <sup>∈</sup> **<sup>R</sup>***ny* to the output to form the noise-perturbed state-space equations
We assume that the noise signal *vk* is generated by a stationary stochastic process, which may be either white or colored. This includes the case in which the state is disturbed by process noise, so that the noise process may have the same poles as the deterministic system. (See [15]
The goal is to construct a state-space realization using the relationships in (21), but doing so requires a complete characterization of the row space of *Hr*. To this end, we expand a finite-slice of the future output vector to form a block-Hankel matrix of output data with *r*
> *y*<sup>0</sup> *y*<sup>1</sup> *y*<sup>2</sup> ··· *yL y*<sup>1</sup> *y*<sup>2</sup> *y*<sup>3</sup> ··· *yL*+<sup>1</sup> *y*<sup>2</sup> *y*<sup>3</sup> *y*<sup>4</sup> ··· *yL*+<sup>2</sup>
*yr*−<sup>1</sup> *yr yr*+<sup>1</sup> ··· *yr*+*L*−<sup>1</sup>
*u*<sup>0</sup> *u*<sup>1</sup> *u*<sup>2</sup> ··· *uL u*<sup>1</sup> *u*<sup>2</sup> *u*<sup>3</sup> ··· *uL*+<sup>1</sup> *u*<sup>2</sup> *u*<sup>3</sup> *u*<sup>4</sup> ··· *uL*+<sup>2</sup>
*ur*−<sup>1</sup> *ur ur*+<sup>1</sup> ··· *ur*+*L*−<sup>1</sup>
*u*−<sup>1</sup> *u*<sup>0</sup> *u*<sup>1</sup> ··· *uL*−<sup>1</sup> *u*−<sup>2</sup> *u*−<sup>1</sup> *u*<sup>0</sup> ··· *uL*−<sup>2</sup> *u*−<sup>3</sup> *u*−<sup>2</sup> *u*−<sup>1</sup> ··· *uL*−<sup>3</sup>
. .
. .
. .
. . ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ .
> ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ,
⎤ ⎥ ⎥ ⎥ ⎦ ,
*yk* <sup>=</sup> *Cxk* <sup>+</sup> *Duk* <sup>+</sup> *vk*. (34)
*xk*<sup>+</sup><sup>1</sup> = *Axk* + *Buk*
for a thorough discussion of representations of noise in the identification context.)
*Y* =
This matrix is related to a block-Hankel matrix of future input data
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
*Uf* =
*Up* =
a block-Toeplitz matrix of past input data
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
. . . . . . . .
. . . . . . . .
⎡ ⎢ ⎢ ⎢ ⎣
. . . . . . . . . . . .
be used to formulate just such an identification procedure.
**4. Realization from input-output data**
**4.1. Data-matrix equations**
block rows,
$$T = \begin{bmatrix} \ \mathfrak{g}\_0 & \cdots & 0 \\ \mathfrak{g}\_1 & \mathfrak{g}\_0 & & \vdots \\ \mathfrak{g}\_2 & \mathfrak{g}\_1 & \mathfrak{g}\_0 \\ \vdots & \vdots & \vdots & \ddots \\ \mathfrak{g}\_{r-1} & \mathfrak{g}\_{r-2} & \mathfrak{g}\_{r-3} & \cdots & \mathfrak{g}\_0 \end{bmatrix} \cdot \mathbf{n}$$
the system Hankel matrix *H*, and a block-Hankel matrix *V* formed from noise data *vk* with the same indices as *Y* by the equation
$$Y = H\mathcal{U}\_p + T\mathcal{U}\_f + V.\tag{35}$$
If the entries of *Yf* are shifted forward by one index to form
$$
\overline{Y} = \begin{bmatrix}
\mathcal{Y}\_1 & \mathcal{Y}\_2 & \mathcal{Y}\_3 & \cdots & \mathcal{Y}\_{L+1} \\
\mathcal{Y}\_2 & \mathcal{Y}\_3 & \mathcal{Y}\_4 & \cdots & \mathcal{Y}\_{L+2} \\
\mathcal{Y}\_3 & \mathcal{Y}\_4 & \mathcal{Y}\_5 & \cdots & \mathcal{Y}\_{L+3} \\
\vdots & \vdots & \vdots & & \vdots \\
\vdots & \vdots & \vdots & & \vdots \\
\mathcal{Y}\_r \ \mathcal{Y}\_{r+1} \ \mathcal{Y}\_{r+2} & \cdots & \mathcal{Y}\_{r+L}
\end{bmatrix}
$$
then *Yf* is related to the shifted system Hankel matrix *H*, the past input data *Up*, *T* with a block column appended to the left, and *Uf* with a block row appended to the bottom,
$$
\overline{T} = \begin{bmatrix} \mathcal{S}\_1 \\ \mathcal{S}\_2 \\ \mathcal{S}\_3 \\ \vdots \\ \mathcal{S}\_r \end{bmatrix}, \qquad \qquad \overline{\mathcal{U}}\_f = \begin{bmatrix} \mathcal{U}\_f \\ \frac{\mathcal{U}\_r \ \mathcal{U}\_{r+1} \ \mathcal{U}\_{r+2} \ \cdots \ \mathcal{U}\_{r+L}} \end{bmatrix}.
$$
and a block-Hankel matrix *V* of noise data *vk* with the same indices as *Y* by the equation
$$
\overline{Y} = \overline{H}\underline{U}\_p + \overline{T}\,\overline{U}\_f + \overline{V}.\tag{36}
$$
From (25), the state is equal to the column vectors of *Up* multiplied by the entries of the controllability matrix C, which we may represent as the block-matrix
$$X = \begin{bmatrix} \mathfrak{x}\_0 \ \mathfrak{x}\_1 \ \mathfrak{x}\_2 \ \cdots \ \mathfrak{x}\_L \end{bmatrix} = \mathcal{C} \mathcal{U}\_{p\nu}$$
which is an alternative means of representing the memory of the system at sample 0, 1, .... The two data matrix equations (35) and (36) may then be written as
$$Y = \mathcal{O}\_r X + T\mathcal{U}\_f + V\_r \tag{37}$$
$$
\overline{Y} = \mathcal{O}\_{\overline{r}} A X + \overline{T} \, \overline{U}\_f + \overline{V}. \tag{38}
$$
Equation (37) is basis for the field of system identification methods known as subspace methods. Subspace identification methods typically fall into one of two categories. First, because a shifted observability matrix
$$
\overline{\mathcal{O}} = \begin{bmatrix} \mathcal{C}A \\ \mathcal{C}A^2 \\ \mathcal{C}A^3 \\ \vdots \end{bmatrix}'
$$
satisfies
$$\text{im}(\mathcal{O}) = \text{im}(\overline{\mathcal{O}})\_\prime$$
where im(·) of denotes the row space (often called the "image"), the row-space of O is shift-invariant, and *A* may be identified from estimates O*<sup>r</sup>* and O*<sup>r</sup>* as
$$
\vec{A} = \mathcal{O}\_r^\dagger \vec{\mathcal{O}}\_r.
$$
Alternatively, because a forward-propagated sequence of states
$$
\overline{X} = AX
$$
satisfies
$$\text{im}(X^T) = \text{im}(\overline{X}^T)\_\prime$$
the column-space of *<sup>X</sup>* is shift-invariant, and *<sup>A</sup>* may be identified from estimates *<sup>X</sup>*<sup>ˆ</sup> and <sup>ˆ</sup> *X* as
$$
\hat{A} = \mathring{\overline{X}} \hat{X}^{\dagger}.
$$
In both instances, the system dynamics are estimated by propagating the indices forward by one step and examining a propagation of linear dynamics, not unlike (20) from Kronecker's theorem. Details of these methods may be found in [16] and [17]. In the next section we present a system identification method that constructs system estimates from the shift-invariant structure of *Y* itself.
#### **4.2. Identification from shift-invariance of output measurements**
Equations (37) and (38) still contain the effects of the future input in *Uf* . To remove these effects from the output, we must first add some assumptions about *Uf* . First, we assume that *Uf* has full row rank. This is true for any *Uf* with a smooth frequency response or if *Uf* is generated from some pseudo-random sequence. Next, we assume that the initial conditions in *X* do not somehow cancel out the effects of future input. A sufficient condition for this is to require
$$\text{rank}\left(\left[\frac{X}{\overline{U}\_f}\right]\right) = n + rn\_{\mu}$$
to have full row rank. Although these assumptions might appear restrictive at first, since it is impossible to verify without knowledge of *X*, it is generally true with the exception of some pathological cases.
Next, we form the null-space projector matrix
$$
\Pi = I\_{L+1} - \overline{\mathcal{U}}\_f^T \left( \overline{\mathcal{U}}\_f \overline{\mathcal{U}}\_f^T \right)^{-1} \overline{\mathcal{U}}\_{f'} \tag{39}
$$
which has the property
16 Will-be-set-by-IN-TECH
⎡ ⎢ ⎢ ⎢ ⎣
im(O) = im(O), where im(·) of denotes the row space (often called the "image"), the row-space of O is
> *<sup>A</sup>*<sup>ˆ</sup> <sup>=</sup> <sup>O</sup><sup>ˆ</sup> † *r* ˆ O*r*.
> > *X* = *AX*
im(*XT*) = im(*X<sup>T</sup>*
the column-space of *<sup>X</sup>* is shift-invariant, and *<sup>A</sup>* may be identified from estimates *<sup>X</sup>*<sup>ˆ</sup> and <sup>ˆ</sup>
*<sup>A</sup>*<sup>ˆ</sup> = <sup>ˆ</sup> *XX*ˆ †.
**4.2. Identification from shift-invariance of output measurements**
rank
<sup>Π</sup> <sup>=</sup> *IL*+<sup>1</sup> <sup>−</sup> *<sup>U</sup><sup>T</sup>*
�� *X Uf* ��
In both instances, the system dynamics are estimated by propagating the indices forward by one step and examining a propagation of linear dynamics, not unlike (20) from Kronecker's theorem. Details of these methods may be found in [16] and [17]. In the next section we present a system identification method that constructs system estimates from the
Equations (37) and (38) still contain the effects of the future input in *Uf* . To remove these effects from the output, we must first add some assumptions about *Uf* . First, we assume that *Uf* has full row rank. This is true for any *Uf* with a smooth frequency response or if *Uf* is generated from some pseudo-random sequence. Next, we assume that the initial conditions in *X* do not somehow cancel out the effects of future input. A sufficient condition for this is to
to have full row rank. Although these assumptions might appear restrictive at first, since it is impossible to verify without knowledge of *X*, it is generally true with the exception of some
> *f* � *Uf <sup>U</sup><sup>T</sup> f* �−<sup>1</sup>
= *n* + *rnu*
*Uf* , (39)
),
*X* as
*CA CA*<sup>2</sup> *CA*<sup>3</sup> . . .
⎤ ⎥ ⎥ ⎥ ⎦ ,
O =
shift-invariant, and *A* may be identified from estimates O*<sup>r</sup>* and O*<sup>r</sup>* as
Alternatively, because a forward-propagated sequence of states
because a shifted observability matrix
shift-invariant structure of *Y* itself.
satisfies
satisfies
require
pathological cases.
Next, we form the null-space projector matrix
$$
\overline{\mathcal{U}}\_f \Pi = 0.
$$
We know the inverse of (*Uf <sup>U</sup><sup>T</sup> <sup>f</sup>* ) exists, since we assume *Uf* has full row rank. Projector matrices such as (39) have many interesting properties. Their eigenvalues are all 0 or 1, and if they are symmetric, they separate the subspace of real vectors — in this case, vectors in **R***L*+<sup>1</sup> — into a subspace and its orthogonal complement. In fact, it is simple to verify that the null space of *Uf* contains the null space of *Uf* as a subspace, since
$$
\overline{\mathcal{U}}\_f \Pi = \left[ \frac{\mathcal{U}\_f}{\cdots} \right] \Pi = 0.
$$
Thus multiplication of (37) and (38) on the right by Π results in
$$\text{YTI} = \mathcal{O}\_r \text{XTI} + \text{VII}\_r \tag{40}$$
$$
\overline{Y}\overline{\Pi} = \mathcal{O}\_I A X \Pi + \overline{V} \Pi. \tag{41}
$$
It is also unnecessary to compute the projected products *Y*Π and *Y*Π directly, since from the QR-decomposition
$$
\begin{bmatrix} \overline{\boldsymbol{\mathcal{U}}}^T \boldsymbol{Y}^T \end{bmatrix} = \begin{bmatrix} \boldsymbol{Q}\_1 \ \boldsymbol{Q}\_2 \end{bmatrix} \begin{bmatrix} \boldsymbol{R}\_{11} \ \boldsymbol{R}\_{12} \\ \boldsymbol{0} \ \boldsymbol{R}\_{22} \end{bmatrix} \prime$$
we have
$$Y = R\_{12}^T Q\_1^T + R\_{22}^T Q\_2^T \tag{42}$$
and *U* = *R<sup>T</sup>* 11*Q<sup>T</sup>* <sup>1</sup> . Substitution into (39) reveals
$$
\Pi = I - Q\_1 Q\_1^T. \tag{43}
$$
Because the columns of *Q*<sup>1</sup> and *Q*<sup>2</sup> are orthogonal, multiplication of (42) on the right by (43) results in
$$\mathcal{Y}\Pi = \mathcal{R}\_{22}^{\top}\mathcal{Q}\_{2}^{\top}.$$
A similar result holds for *Y*Π. Taking the QR-decomposition of the data can alternatively be thought of as using the principle of superposition to construct new sequences of input-output data through a Gram-Schmidt-type orthogonalization process. A detailed discussion of this interpretation can be found in [18].
Thus we have successfully removed the effects of future input on the output while retaining the effects of the past, which is the foundation of the realization process. We still must account for non-deterministic effects in *V* and *V*. To do so, we look for some matrix *Z* such that
$$\begin{array}{c} \mathsf{V}\Pi\!\!\!\!Z^{\mathsf{T}} \to 0, \\\\ \overline{\mathsf{V}}\Pi\!\!\!\!Z^{\mathsf{T}} \to 0. \end{array}$$
This requires the content of *Z* to be statistically independent of the process that generates *vk*. The input *uk* is just such a signal, so long as the filter output is not a function of the input that is, the data was measured in open-loop operation,. If we begin measuring input before *k* = 0 at some sample *k* = −*ζ* and construct *Z* as a block-Hankel matrix of past input data,
$$Z = \frac{1}{L} \begin{bmatrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \end{matrix} \end{matrix} \end{matrix} & \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \end{matrix} \end{matrix} \end{matrix} \end{bmatrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \end{matrix} \end{matrix} \end{matrix} \end{bmatrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \end{matrix} \end{matrix} \end{matrix} \end{bmatrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \end{matrix} \end{matrix} \end{matrix} \end{bmatrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \end{matrix} \end{matrix} \end{matrix} \end{bmatrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \end{matrix} \end{matrix} \end{matrix} \end{bmatrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \end{matrix} \end{matrix} \end{matrix} \end{pmatrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \end{matrix} \end{matrix} \end{matrix} \end{bmatrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{bmatrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \end{matrix} \end{matrix} \end{matrix} \end{$$
then multiplication of (40) and (41) on the right by *Z<sup>T</sup>* results in
$$\text{YITZ}^{T} \rightarrow \mathcal{O}\_{\text{I}} \text{XIIZ}^{T},\tag{44}$$
$$
\overline{\mathcal{Y}}\Pi\mathcal{Z}^{\mathsf{T}} \to \mathcal{O}\_{\mathsf{T}}A\Pi\mathcal{Z}^{\mathsf{T}},\tag{45}
$$
as *<sup>L</sup>* <sup>→</sup> <sup>∞</sup>. Note the term <sup>1</sup> *<sup>L</sup>* in *Z* is necessary to keep (44) and (45) bounded.
Finally we are able to perform our rank-reduction technique directly on measured data without needing to first estimate the impulse response. From the SVD
$$\mathbf{Y}\Pi\mathbf{Z}^{\mathsf{T}} = \mathcal{U}\Sigma V^{\mathsf{T}}{}\_{\mathsf{A}}$$
we may estimate the order *n* by looking for a sudden decrease in singular values. From the partitioning
$$\begin{aligned} \,^j\text{T}\Pi Z^T &= \begin{bmatrix} \mathcal{U}\_{\boldsymbol{n}} \ \mathcal{U}\_{\boldsymbol{s}} \end{bmatrix} \begin{bmatrix} \Sigma\_{\boldsymbol{n}} & \mathbf{0} \\ \mathbf{0} & \Sigma\_{\boldsymbol{s}} \end{bmatrix} \begin{bmatrix} V\_{\boldsymbol{n}}^T \\ V\_{\boldsymbol{s}}^T \end{bmatrix} \,^j\text{T} \end{aligned} $$
we may estimate <sup>O</sup>*<sup>r</sup>* and *<sup>X</sup>*Π*Z<sup>T</sup>* from the factorization
$$
\mathcal{O}\_r = \mathsf{U}\_n \Sigma\_n^{1/2} \qquad \text{and} \qquad \hat{\mathsf{X}} \Pi \mathbb{Z}^T = \Sigma\_n^{1/2} V\_n^T.
$$
*A* may then be estimated as
$$\begin{split} \boldsymbol{\hat{A}} &= \left(\boldsymbol{\mathcal{O}}\_{r}\right)^{\dagger} \overline{\boldsymbol{\Upsilon}} \boldsymbol{\Pi} \boldsymbol{Z}^{T} \left(\boldsymbol{\hat{X}} \boldsymbol{\Pi} \boldsymbol{Z}^{T}\right)^{\dagger} = \boldsymbol{\Sigma}\_{n}^{-1/2} \boldsymbol{\mathcal{U}}\_{n}^{T} \overline{\boldsymbol{\Upsilon}} \boldsymbol{\Pi} \boldsymbol{Z}^{T} \boldsymbol{V}\_{n} \boldsymbol{\Sigma}\_{n}^{-1/2} \\ &\approx \left(\boldsymbol{\mathcal{O}}\_{r}\right)^{\dagger} \left(\overline{\boldsymbol{\mathcal{H}}} \boldsymbol{\mathcal{U}}\_{p} \boldsymbol{\Pi}\right) \left(\boldsymbol{\mathcal{C}}\_{L} \boldsymbol{\mathcal{U}}\_{p} \boldsymbol{\Pi}\right)^{\dagger} \approx \left(\boldsymbol{\mathcal{O}}\_{r}\right)^{\dagger} \overline{\boldsymbol{\mathcal{H}}} \left(\boldsymbol{\mathcal{C}}\_{L}\right)^{\dagger} .\end{split}$$
And so we have returned to our original relationship (29).
While *<sup>C</sup>* may be estimated from the top block row of <sup>O</sup>ˆ*r*, our projection has lost the column space of *Hr* that we previously used to estimate *B*, and initial conditions in *X* prevent us from estimating *D* directly. Fortunately, if *A* and *C* are known, then the remaining terms *B*, *D*, and an initial condition *x*<sup>0</sup> are linear in the input output data, and may be estimated by solving a linear-least-squares problem.
#### **4.3. Estimation of** *B***,** *D***, and** *x*0
The input-to-state terms *B* and *D* may be estimated by examining the convolution with the state-space form of the impulse response. Expanding (24) with the input and including an initial condition *x*<sup>0</sup> results in
$$y\_k = \mathbb{C}A^k x\_0 + \sum\_{j=0}^{k-1} \mathbb{C}A^{k-j-1}Bu\_j + Du\_k + v\_k. \tag{46}$$
Factoring out *B* and *D* on the right provides
18 Will-be-set-by-IN-TECH
*u*−*<sup>ζ</sup> u*−*ζ*+<sup>1</sup> *u*−*ζ*+<sup>2</sup> ··· *u*−*ζ*+*<sup>L</sup> u*−*ζ*+<sup>1</sup> *u*−*ζ*+<sup>2</sup> *u*−*ζ*+<sup>3</sup> ··· *u*−*ζ*+*L*+<sup>1</sup> *u*−*ζ*+<sup>2</sup> *u*−*ζ*+<sup>3</sup> *u*−*ζ*+<sup>4</sup> ··· *u*−*ζ*+*L*+<sup>2</sup>
*u*−<sup>1</sup> *u*<sup>0</sup> *u*<sup>1</sup> ··· *ur*+*L*−<sup>2</sup>
*<sup>L</sup>* in *Z* is necessary to keep (44) and (45) bounded.
*<sup>n</sup>* and *<sup>X</sup>*<sup>ˆ</sup> <sup>Π</sup>*Z<sup>T</sup>* = <sup>Σ</sup>1/2
= Σ−1/2 *<sup>n</sup> <sup>U</sup><sup>T</sup>*
Finally we are able to perform our rank-reduction technique directly on measured data
*Y*Π*Z<sup>T</sup>* = *U*Σ*VT*, we may estimate the order *n* by looking for a sudden decrease in singular values. From the
> *Un Us* � � Σ*<sup>n</sup>* 0 0 Σ*<sup>s</sup>*
*X*ˆ Π*Z<sup>T</sup>* �†
*HUp*Π� �C*LUp*Π�† <sup>≈</sup> (O*r*)
While *<sup>C</sup>* may be estimated from the top block row of <sup>O</sup>ˆ*r*, our projection has lost the column space of *Hr* that we previously used to estimate *B*, and initial conditions in *X* prevent us from estimating *D* directly. Fortunately, if *A* and *C* are known, then the remaining terms *B*, *D*, and an initial condition *x*<sup>0</sup> are linear in the input output data, and may be estimated by solving a
The input-to-state terms *B* and *D* may be estimated by examining the convolution with the state-space form of the impulse response. Expanding (24) with the input and including an
> *k*−1 ∑ *j*=0
. .
� �*V<sup>T</sup> n V<sup>T</sup> s* � ,
. .
*<sup>Y</sup>*Π*Z<sup>T</sup>* → O*rX*Π*ZT*, (44) *<sup>Y</sup>*Π*Z<sup>T</sup>* → O*rA*Π*ZT*, (45)
> *<sup>n</sup> <sup>V</sup><sup>T</sup> n* .
*<sup>n</sup> <sup>Y</sup>*Π*ZTVn*Σ−1/2 *n*
† .
*CAk*−*j*−<sup>1</sup>*Buj* + *Duk* + *vk*. (46)
† *<sup>H</sup>* (C*L*)
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ,
*k* = 0 at some sample *k* = −*ζ* and construct *Z* as a block-Hankel matrix of past input data,
*<sup>Z</sup>* <sup>=</sup> <sup>1</sup> *L*
as *<sup>L</sup>* <sup>→</sup> <sup>∞</sup>. Note the term <sup>1</sup>
*A* may then be estimated as
linear-least-squares problem.
initial condition *x*<sup>0</sup> results in
**4.3. Estimation of** *B***,** *D***, and** *x*0
partitioning
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
then multiplication of (40) and (41) on the right by *Z<sup>T</sup>* results in
. . . . . . . .
without needing to first estimate the impulse response. From the SVD
*Y*Π*Z<sup>T</sup>* = �
we may estimate <sup>O</sup>*<sup>r</sup>* and *<sup>X</sup>*Π*Z<sup>T</sup>* from the factorization
*A*ˆ = � Oˆ*r*
≈ (O*r*)
<sup>O</sup>ˆ*<sup>r</sup>* <sup>=</sup> *Un*Σ1/2
�† *<sup>Y</sup>*Π*Z<sup>T</sup>* �
*yk* = *CAkx*<sup>0</sup> +
† �
And so we have returned to our original relationship (29).
$$y\_k = \mathbb{C}A^k x\_0 + \left(\sum\_{j=0}^{k-1} \mathfrak{u}\_k^T \otimes \mathbb{C}A^{k-j-1}\right) \text{vec}(B) + \left(\mathfrak{u}\_k^T \otimes I\_{n\_y}\right) \text{vec}(D) + v\_{k'}$$
in which vec(·) is the operation that stacks the columns of a matrix on one another, ⊗ is the (coincidentally named) Kronecker product, and we have made use of the identity
$$\text{vec}(AXB) = (B^T \otimes A)\text{vec}(X).$$
Grouping the unknown terms together results in
$$y\_k = \left[ \mathbb{C}A^k \, \sum\_{j=0}^{k-1} u\_k^T \otimes \mathbb{C}A^{k-j-1} \, u\_k^T \otimes I\_{\mathbb{N}\_y} \right] \begin{bmatrix} \varkappa\_0\\ \mathrm{vec}(B) \\ \mathrm{vec}(D) \end{bmatrix} + v\_k.$$
Thus by forming the regressor
$$\phi\_k = \left[ \triangle \hat{A}^k \; \Sigma\_{j=0}^{k-1} \; \boldsymbol{\mu}\_k^T \otimes \triangle \hat{A}^{k-j-1} \; \boldsymbol{\mu}\_k^T \otimes \boldsymbol{I}\_{\boldsymbol{\eta}\_k} \right],$$
from the estimates *A*ˆ and *C*ˆ, estimates of *B* and *D* may be found from the least-squares solution of the linear system of *N* equations
$$
\begin{bmatrix} y\_0 \\ y\_1 \\ y\_2 \\ \vdots \\ y\_N \end{bmatrix} = \begin{bmatrix} \phi\_0 \\ \phi\_1 \\ \phi\_2 \\ \vdots \\ \phi\_N \end{bmatrix} \begin{bmatrix} \hat{x}\_0 \\ \text{vec}(\hat{B}) \\ \text{vec}(\hat{D}) \end{bmatrix}.
$$
Note that *N* is arbitrary and does not need to be related in any way to the indices of the data matrix equations. This can be useful, since for large-dimensional systems, the regressor *φ<sup>k</sup>* may become very computationally expensive to compute.
#### **5. Conclusion**
Beginning with the construction of a transfer function from an impulse response, we have constructed a method for identification of state-space realizations of linear filters from measured input-output data, introducing the fundamental concepts of realization theory of linear systems along the way. Computing a state-space realization from measured input-output data requires many tools of linear algebra: projections and the QR-decomposition, rank reduction and the singular-value decomposition, and linear least squares. The principles of realization theory provide insight into the different representations of linear systems, as well as the role of rational functions and series expansions in linear algebra.
#### **Author details**
Daniel N. Miller and Raymond A. de Callafon *University of California, San Diego, USA*
### **6. References**
## **Partition-Matrix Theory Applied to the Computation of Generalized-Inverses for MIMO Systems in Rayleigh Fading Channels**
P. Cervantes, L. F. González, F. J. Ortiz and A. D. García
Additional information is available at the end of the chapter
http://dx.doi.org/10.5772/48198
## **1. Introduction**
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[14] G.H. Golub and C.F. Van Loan. *Matrix Computations*. The Johns Hopkins University
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[17] Peter Van Overschee and Bart De Moor. *Subspace Identification for Linear Systems: Theory,*
[18] Tohru Katayama. Role of LQ Decomposition in Subspace Identification Methods. In Alessandro Chiuso, Stefano Pinzoni, and Augusto Ferrante, editors, *Modeling, Estimation*
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Partition-Matrix Theory and Generalized-Inverses are interesting topics explored in linear algebra and matrix computation. Partition-Matrix Theory is associated with the problem of properly partitioning a matrix into block matrices (i.e. an array of matrices), and is a matrix computation tool widely employed in several scientific-technological application areas. For instance, blockwise Toeplitz-based covariance matrices are used to model structural properties for space-time multivariate adaptive processing in radar applications [1], Jacobian response matrices are partitioned into several block-matrix instances in order to enhance medical images for Electrical-Impedance-Tomography [2], design of stateregulators and partial-observers for non-controllable/non-observable linear continuous systems contemplates matrix blocks for controllable/non-controllable and observable/nonobservable eigenvalues [3]. The Generalized-Inverse is a common and natural problem found in a vast of applications. In control robotics, non-collocated partial linearization is applied to underactuated mechanical systems through inertia-decoupling regulators which employ a pseudoinverse as part of a modified input control law [4]. At sliding-mode control structures, a Right-Pseudoinverse is incorporated into a state-feedback control law in order to stabilize electromechanical non-linear systems [5]. Under the topic of system identification, definition of a Left-Pseudoinverse is present in auto-regressive movingaverage models (ARMA) for matching dynamical properties of unknown systems [6]. An interesting approach arises whenever Partition-Matrix Theory and Generalized-Inverse are combined together yielding attractive solutions for solving the problem of block matrix inversion [7-10]. Nevertheless, several assumptions and restrictions regarding numerical stability and structural properties are considered for these alternatives. For example, an attractive pivot-free block matrix inversion algorithm is proposed in [7], which
unfortunately exhibits an overhead in matrix multiplications that are required in order to guarantee full-rank properties for particular blocks within it. For circumventing the expense in rank deficiency, [8] offers block-matrix completion strategies in order to find the Generalized-Inverse of any non-singular block matrix (irrespective of the singularity of their constituting sub-blocks). However, the existence of intermediate matrix inverses and pseudoinverses throughout this algorithm still rely on full-rank assumptions, as well as introducing more hardness to the problem. The proposals exposed in [9-10] avoid completion strategies and contemplate all possible scenarios for avoiding any rank deficiency among each matrix sub-block, yet demanding full-rank assumptions for each scenario. In this chapter, an iterative-recursive algorithm for computing a Left-Pseudoinverse (LPI) of a MIMO channel matrix is developed by combining Partition-Matrix Theory and Generalized-Inverse concepts. For this approach, no matrix-operations' overhead nor any particular block matrix full-rank assumptions are needed because of structural attributes of the MIMO channel matrix, which models dynamical properties of a Rayleigh fading channel (RFC) within wireless MIMO communication systems.
The content of this work is outlined as follows. Section 2 provides a description of the MIMO communication link, pointing out its principal physical effects and the mathematical model considered for RFC-based environments. Section 3 defines formally the problem of computing the Left-Pseudoinverse as the Generalized-Inverse for the MIMO channel matrix applying Partition-Matrix Theory concepts. Section 4 presents linear algebra and matrix computation concepts and tools needed for tracking a solution for the aforementioned problem. Section 5 analyzes important properties of the MIMO channel matrix derived from a Rayleigh fading channel scenario. Section 6 explains the proposed novel algorithm. Section 7 presents a brief analysis of VLSI (Very Large Scale of Integration) aspects towards implementation of arithmetic operations presented in this algorithm. Section 8 concludes the chapter. Due to the vast literature about MIMO systems, and to the best of the authors' knowledge, this chapter provides a nice and strategic list of references in order to easily correlate essential concepts between matrix theory and MIMO systems. For instance, [11-16] describe and analyze information and system aspects about MIMO communication systems, as well as studying MIMO channel matrix behavior under RFC-based environments; [17-18] contain all useful linear algebra and matrix computation theoretical concepts around the mathematical background immersed in MIMO systems; [19-21] provide practical guidelines and examples for MIMO channel matrix realizations comprising RFC scenarios; [22] treats the formulation and development of the algorithm presented in this chapter; [23-27] detail a splendid survey on architectural aspects for implementing several arithmetic operations.
### **2. MIMO systems**
In the context of wireless communication systems, MIMO (Multiple-Input Multiple-Output) is an extension of the classical SISO (Single-Input Single-Output) communication paradigm, where instead of having a communication link composed of a single transmitter-end and a receiver-end element (or antenna), wireless MIMO communication systems (or just MIMO systems) consist of an array of multiple elements at both the transmission and reception parts [11-16,19-21]. Generally speaking, the MIMO communication link contains *<sup>T</sup> <sup>n</sup>* transmitter-end and *<sup>R</sup> <sup>n</sup>* receiver-end antennas sendingand-receiving information through a wireless channel. Extensive studies on MIMO systems and commercial devices already employing them reveal that these communication systems offer promising results in terms of: a) spectral efficiency and channel capacity enhancements (many user-end applications supporting high-data rates at limited available bandwidth); b) improvements on Bit-Error-Rate (BER) performance; and c) practical feasability already seen in several wireless communication standards. The conceptualization of this paradigm is illustrated in figure 1, where Tx is the transmitterend, Rx the receiver-end, and Chx the channel.
138 Linear Algebra – Theorems and Applications
**2. MIMO systems**
unfortunately exhibits an overhead in matrix multiplications that are required in order to guarantee full-rank properties for particular blocks within it. For circumventing the expense in rank deficiency, [8] offers block-matrix completion strategies in order to find the Generalized-Inverse of any non-singular block matrix (irrespective of the singularity of their constituting sub-blocks). However, the existence of intermediate matrix inverses and pseudoinverses throughout this algorithm still rely on full-rank assumptions, as well as introducing more hardness to the problem. The proposals exposed in [9-10] avoid completion strategies and contemplate all possible scenarios for avoiding any rank deficiency among each matrix sub-block, yet demanding full-rank assumptions for each scenario. In this chapter, an iterative-recursive algorithm for computing a Left-Pseudoinverse (LPI) of a MIMO channel matrix is developed by combining Partition-Matrix Theory and Generalized-Inverse concepts. For this approach, no matrix-operations' overhead nor any particular block matrix full-rank assumptions are needed because of structural attributes of the MIMO channel matrix, which models dynamical properties of a
Rayleigh fading channel (RFC) within wireless MIMO communication systems.
The content of this work is outlined as follows. Section 2 provides a description of the MIMO communication link, pointing out its principal physical effects and the mathematical model considered for RFC-based environments. Section 3 defines formally the problem of computing the Left-Pseudoinverse as the Generalized-Inverse for the MIMO channel matrix applying Partition-Matrix Theory concepts. Section 4 presents linear algebra and matrix computation concepts and tools needed for tracking a solution for the aforementioned problem. Section 5 analyzes important properties of the MIMO channel matrix derived from a Rayleigh fading channel scenario. Section 6 explains the proposed novel algorithm. Section 7 presents a brief analysis of VLSI (Very Large Scale of Integration) aspects towards implementation of arithmetic operations presented in this algorithm. Section 8 concludes the chapter. Due to the vast literature about MIMO systems, and to the best of the authors' knowledge, this chapter provides a nice and strategic list of references in order to easily correlate essential concepts between matrix theory and MIMO systems. For instance, [11-16] describe and analyze information and system aspects about MIMO communication systems, as well as studying MIMO channel matrix behavior under RFC-based environments; [17-18] contain all useful linear algebra and matrix computation theoretical concepts around the mathematical background immersed in MIMO systems; [19-21] provide practical guidelines and examples for MIMO channel matrix realizations comprising RFC scenarios; [22] treats the formulation and development of the algorithm presented in this chapter; [23-27] detail a splendid survey on architectural aspects for implementing several arithmetic operations.
In the context of wireless communication systems, MIMO (Multiple-Input Multiple-Output) is an extension of the classical SISO (Single-Input Single-Output) communication paradigm, where instead of having a communication link composed of a single transmitter-end and a receiver-end element (or antenna), wireless MIMO communication systems (or just MIMO systems) consist of an array of multiple elements at both the
**Figure 1.** The MIMO system: conceptualization for the MIMO communication paradigm.
Notice that information sent from the trasnmission part (Tx label on figure 1) will suffer from several degradative and distorional effects inherent in the channel (Chx label on figure 1), forcing the reception part (Rx label on figure 1) to decode information properly. Information at Rx will suffer from degradations caused by time, frequency, and spatial characteristics of the MIMO communication link [11-12,14]. These issues are directly related to: i) the presence of physical obstacles obstructing the Line-of-Sight (LOS) between Tx and Rx (existance of non-LOS); ii) time delays between received and transmitted information signals due to Tx and Rx dynamical properties (time-selectivity of Chx); iii) frequency distortion and interference among signal carriers through Chx (frequencyselectivity of Chx); iv) correlation of information between receiver-end elements. Fading (or fading mutlipath) and noise are the most common destructive phenomena that significantly affect information at Rx [11-16]. Fading is a combination of time-frequency replicas of the trasnmitted information as a consequence of the MIMO system phenomena i)-iv) exposed before, whereas noise affects information at every receiver-end element under an additve or multiplicative way. As a consequence, degradation of signal information rests mainly upon magnitude attenuation and time-frequency shiftings. The simplest treatable MIMO communication link has a slow-flat quasi-static fading channel (proper of a non-LOS indoor environment). For this type of scenario, a well-known dynamical-stochastic model considers a Rayleigh fading channel (RFC) [13,15-16,19-21], which gives a quantitative clue of how information has been degradated by means of Chx. Moreover, this type of channels allows to: a) distiguish among each information block tranmitted from the *<sup>T</sup> <sup>n</sup>* elements at every Chx realization (i.e. the time during which the channel's properties remain unvariant); and b) implement easily symbol decoding tasks related to channel equalization (CE) techniques. Likewise, noise is commonly assumed to have additive effects over Rx. Once again, all of these assumptions provide a treatable information-decoding problem (refered as MIMO demodulation [12]), and the mathematical model that suits the aforementioned MIMO communication link characteristics will be represented by
$$y = Hx + \eta \tag{1}$$
where: 1 1 [ ] *T T n n <sup>j</sup> <sup>x</sup>* is a complex-valued *<sup>T</sup> <sup>n</sup>* dimensional transmitted vector with entries drawn from a Gaussian-integer finite-lattice constellation (digital modulators, such as: *q*-QAM, QPSK); <sup>1</sup> *Rn <sup>y</sup>* is a complex-valued *<sup>R</sup> <sup>n</sup>* dimensional received vector; 1 *Rn* is a *<sup>R</sup> <sup>n</sup>* dimensional independent-identically-distributed (idd) complexcircularly-symmetric (ccs) Additive White Gaussian Noise (AWGN) vector; and *R T n n <sup>H</sup>* is the *R T n n* dimensional MIMO channel matrix whose entries model: a) the RFC-based environment behavior according to a Gaussian probabilistic density function with zero-mean and 0.5-variance statistics; and b) the time-invariant transfer function (which measures the degradation of the signal information) between the i-th receiver-end and the j-th trasnmitter-end antennas [11-16,19-21]. Figure 2 gives a representation of (1). As shown therein, the MIMO communication link model stated in (1) can be also expressed as
$$
\begin{bmatrix} y\_1 \\ \vdots \\ y\_{n\_R} \end{bmatrix} = \begin{bmatrix} h\_{11} & \cdots & h\_{1n\_T} \\ \vdots & & \vdots \\ h\_{n\_R 1} & \cdots & h\_{n\_R n\_T} \end{bmatrix} \begin{bmatrix} \mathbf{x}\_1 \\ \vdots \\ \mathbf{x}\_{n\_T} \end{bmatrix} + \begin{bmatrix} \eta\_1 \\ \vdots \\ \eta\_{n\_R} \end{bmatrix} \tag{2}
$$
Notice from (1-2) that an important requisite for CE purposes within RFC scenarios is that *H* is provided somehow to the Rx. This MIMO system requirement is classically known as Channel State Information (CSI) [11-16]. In the sequel of this work, symbol-decoding efforts will consider the problem of finding *x* from *y* regarding CSI at the Rx part within a slowflat quasi-static RFC-based environment as modeled in (1-2). In simpler words, Rx must find *x* from degradated information *y* through calculating an inversion over *H* . Moreover, *R T n n* is commonly assumed for MIMO demodulation tasks [13-14] because it guarantees linear independency between row-entries of matrix *H* in (2), yielding a nonhomogeneous overdetermined system of linear equations.
**Figure 2.** Representation for the MIMO communication link model according to *y Hx* . Here, each dotted arrow represents an entry ij *h* in *H* which determines channel degradation between the j-th transmitter and the i-th receiver elements. AWGN appears additively in each receiver-end antenna.
#### **3. Problem definition**
140 Linear Algebra – Theorems and Applications
characteristics will be represented by
*T T n n*
overdetermined system of linear equations.
where: 1 1 [ ]
1 *Rn*
expressed as
information rests mainly upon magnitude attenuation and time-frequency shiftings. The simplest treatable MIMO communication link has a slow-flat quasi-static fading channel (proper of a non-LOS indoor environment). For this type of scenario, a well-known dynamical-stochastic model considers a Rayleigh fading channel (RFC) [13,15-16,19-21], which gives a quantitative clue of how information has been degradated by means of Chx. Moreover, this type of channels allows to: a) distiguish among each information block tranmitted from the *<sup>T</sup> <sup>n</sup>* elements at every Chx realization (i.e. the time during which the channel's properties remain unvariant); and b) implement easily symbol decoding tasks related to channel equalization (CE) techniques. Likewise, noise is commonly assumed to have additive effects over Rx. Once again, all of these assumptions provide a treatable information-decoding problem (refered as MIMO demodulation [12]), and the mathematical model that suits the aforementioned MIMO communication link
> *y Hx*
entries drawn from a Gaussian-integer finite-lattice constellation (digital modulators, such as: *q*-QAM, QPSK); <sup>1</sup> *Rn <sup>y</sup>* is a complex-valued *<sup>R</sup> <sup>n</sup>* dimensional received vector;
is a *<sup>R</sup> <sup>n</sup>* dimensional independent-identically-distributed (idd) complexcircularly-symmetric (ccs) Additive White Gaussian Noise (AWGN) vector; and *R T n n <sup>H</sup>* is the *R T n n* dimensional MIMO channel matrix whose entries model: a) the RFC-based environment behavior according to a Gaussian probabilistic density function with zero-mean and 0.5-variance statistics; and b) the time-invariant transfer function (which measures the degradation of the signal information) between the i-th receiver-end and the j-th trasnmitter-end antennas [11-16,19-21]. Figure 2 gives a representation of (1). As shown therein, the MIMO communication link model stated in (1) can be also
11 1 1 1 1
*R R RT T R*
*n n nn n n*
Notice from (1-2) that an important requisite for CE purposes within RFC scenarios is that *H* is provided somehow to the Rx. This MIMO system requirement is classically known as Channel State Information (CSI) [11-16]. In the sequel of this work, symbol-decoding efforts will consider the problem of finding *x* from *y* regarding CSI at the Rx part within a slowflat quasi-static RFC-based environment as modeled in (1-2). In simpler words, Rx must find *x* from degradated information *y* through calculating an inversion over *H* . Moreover, *R T n n* is commonly assumed for MIMO demodulation tasks [13-14] because it guarantees linear independency between row-entries of matrix *H* in (2), yielding a nonhomogeneous
*T*
*n*
1
*y x h h*
*y h hx*
*<sup>j</sup> <sup>x</sup>* is a complex-valued *<sup>T</sup> <sup>n</sup>* dimensional transmitted vector with
(1)
(2)
Recall for the moment the mathematical model provided in (1). Consider <sup>r</sup> and <sup>i</sup> to be the real and imaginary parts of a complex-valued matrix (vector) , that is, r i *<sup>j</sup>* . Then, Equation (1) can be expanded as follows:
$$\mathbf{y}^{\mathbf{r}} + j\mathbf{y}^{\mathbf{i}} = \left( H^{\mathbf{r}}\mathbf{x}^{\mathbf{r}} - H^{\mathbf{i}}\mathbf{x}^{\mathbf{i}} + \boldsymbol{\eta}^{\mathbf{r}} \right) + j\left( H^{\mathbf{i}}\mathbf{x}^{\mathbf{r}} + H^{\mathbf{i}}\mathbf{x}^{\mathbf{i}} + \boldsymbol{\eta}^{\mathbf{i}} \right) \tag{3}$$
It can be noticed from Equation (3) that: r i <sup>1</sup> , *Tn x x* ; r i <sup>1</sup> , *Rn y y* ; r i <sup>1</sup> , *Rn* ; and r i , *R T n n H H* . An alternative representation for the MIMO communication link model in (2) can be expressed as
$$
\begin{bmatrix} y^{\text{r}} \\ y^{\text{i}} \end{bmatrix} = \begin{bmatrix} H^{\text{r}} & -H^{\text{i}} \\ H^{\text{i}} & H^{\text{r}} \end{bmatrix} \begin{bmatrix} \mathbf{x}^{\text{r}} \\ \mathbf{x}^{\text{i}} \end{bmatrix} + \begin{bmatrix} \boldsymbol{\eta}^{\text{r}} \\ \boldsymbol{\eta}^{\text{i}} \end{bmatrix} \tag{4}
$$
where r i 2 1 <sup>Y</sup> *Rn <sup>y</sup> y* , r i 2 2 i r <sup>h</sup> *R T H H n n H H* , <sup>r</sup> 2 1 i <sup>X</sup> *<sup>T</sup> <sup>x</sup> <sup>n</sup> x* , and <sup>r</sup> 2 1 i N . *<sup>R</sup> n*
CSI is still needed for MIMO demodulation purposes involving (4). Moreover, if 2 *N n r R* and 2 *N n t T* , then *N N r t* . Obviously, while seeking for a solution of signal vector X from (4), the reception part Rx will provide also the solution for signal vector *x* , and thus MIMO demodulation tasks will be fulfilled. This problem can be defined formally into the following manner:
*Definition 1.* Given parameters 2 *N n r R* and <sup>2</sup> *N n t T* , and a block-matrix <sup>h</sup> *N N r t* , there exists an operator 1 1 : *N NN N r rt t* which solves the matrixblock equation Y=hX+N so that Y,h X . ■
From Definition 1, the following affirmations hold: i) CSI over h is a necessary condition as an input argument for the operator ; and ii) can be naïvely defined as a Generalized-Inverse of the block-matrix h . In simpler terms, † X=h Y 1 is associated with Y,h and † <sup>h</sup> *N N t r* stands for the Generalized-Inverse of the block-matrix h , where †T T <sup>1</sup> h hh h [17-18]. Clearly, <sup>1</sup> and <sup>T</sup> represent the inverse and transpose matrix operations over real-valued matrices. As a concluding remark, computing the Generalized-Inverse † h can be separated into two operations: 1) a block-matrix inversion <sup>T</sup> <sup>1</sup> h h 2; 2) a typical matrix multiplication T T <sup>1</sup> hh h . For these tasks, Partition-Matrix Theory will be employed in order to find a novel algorithm for computing a Generalized-Inverse related to (4).
### **4. Mathematical background**
#### **4.1. Partition-matrix theory**
Partition-Matrix Theory embraces structures related to block matrices (or partition matrices: an array of matrices) [17-18]. Furthermore, a block-matrix *L* with *nq mp* dimension can be constructed (or partitioned) consistently according to matrix sub-blocks *A* , *B* , *C* , and *D* of *n m* , *n p* , *q m* , and *q p* dimensions, respectively, yielding
$$L = \begin{bmatrix} A & B \\ C & D \end{bmatrix} \tag{5}$$
An interesting operation to be performed for these structures given in (5) is the inversion, i.e. a blockwise inversion <sup>1</sup> *L* . For instance, let *nm nm <sup>L</sup>* be a full-rank real-valued block matrix (the subsequent treatment is also valid for complex-valued entities, i.e.
2 Notice that <sup>T</sup> <sup>h</sup> *N N t r* and <sup>T</sup> <sup>1</sup> h h *N N t t* .
<sup>1</sup> In the context of MIMO systems, this matrix operation is commonly found in Babai estimators for symbol-decoding purposes at the Rx part [12,13]. For the reader's interest, refer to [11-16] for other MIMO demodulation techniques.
*nm nm <sup>L</sup>* ). An alternative partition can be performed with *n n <sup>A</sup>* , *n m <sup>B</sup>* , *m n <sup>C</sup>* , and *m m <sup>D</sup>* . Assume also *A* and *D* to be full-rank matrices. Then,
$$L^{-1} = \begin{bmatrix} \left(A - BD^{-1}\mathbb{C}\right)^{-1} & -\left(A - BD^{-1}\mathbb{C}\right)^{-1} BD^{-1} \\\\ -\left(D - CA^{-1}\mathcal{B}\right)^{-1}CA^{-1} & \left(D - CA^{-1}\mathcal{B}\right)^{-1} \end{bmatrix} \tag{6}$$
This strategy (to be proved in the next part) requires additonally and mandatorily full-rank over matrices <sup>1</sup> *A BD C* and <sup>1</sup> *D CA B* . The simple case is defined for *L a b c d* (indistinctly for 2 2 or 2 2 ). Once again, assuming det *<sup>L</sup>* 0 , *<sup>a</sup>* <sup>0</sup> , and *<sup>d</sup>* <sup>0</sup> (related to full-rank restictions within block-matrix *L* ):
$$L^{-1} = \begin{bmatrix} \left(a - bd^{-1}c\right)^{-1} & -\left(a - bd^{-1}c\right)^{-1}bd^{-1} \\ -\left(d - ca^{-1}b\right)^{-1}ca^{-1} & \left(d - ca^{-1}b\right)^{-1} \end{bmatrix} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \end{bmatrix}$$
where evidently *ad bc* <sup>0</sup> , ( )( ) *nm nm* <sup>1</sup> *a bd <sup>c</sup>* <sup>0</sup> , and <sup>1</sup> *d ca <sup>b</sup>* <sup>0</sup> .
#### **4.2. Matrix Inversion Lemma**
142 Linear Algebra – Theorems and Applications
*Definition 1.* Given parameters 2 *N n r R*
block equation Y=hX+N so that Y,h X . ■
and <sup>T</sup>
hh h
**4. Mathematical background**
**4.1. Partition-matrix theory**
i.e. a blockwise inversion <sup>1</sup> *L*
Notice that <sup>T</sup> <sup>h</sup> *N N t r* and <sup>T</sup> <sup>1</sup>
1
2
<sup>h</sup> *N N r t* , there exists an operator 1 1
be separated into two operations: 1) a block-matrix inversion <sup>T</sup> <sup>1</sup>
and *D* of *n m* , *n p* , *q m* , and *q p* dimensions, respectively, yielding
h h *N N t t* .
order to find a novel algorithm for computing a Generalized-Inverse related to (4).
:
following manner:
[17-18]. Clearly, <sup>1</sup>
multiplication T T <sup>1</sup>
(4), the reception part Rx will provide also the solution for signal vector *x* , and thus MIMO demodulation tasks will be fulfilled. This problem can be defined formally into the
From Definition 1, the following affirmations hold: i) CSI over h is a necessary condition as an input argument for the operator ; and ii) can be naïvely defined as a Generalized-Inverse of the block-matrix h . In simpler terms, † X=h Y 1 is associated with Y,h and
† <sup>h</sup> *N N t r* stands for the Generalized-Inverse of the block-matrix h , where †T T <sup>1</sup>
real-valued matrices. As a concluding remark, computing the Generalized-Inverse † h can
Partition-Matrix Theory embraces structures related to block matrices (or partition matrices: an array of matrices) [17-18]. Furthermore, a block-matrix *L* with *nq mp* dimension can be constructed (or partitioned) consistently according to matrix sub-blocks *A* , *B* , *C* ,
> *A B <sup>L</sup> C D*
An interesting operation to be performed for these structures given in (5) is the inversion,
block matrix (the subsequent treatment is also valid for complex-valued entities, i.e.
In the context of MIMO systems, this matrix operation is commonly found in Babai estimators for symbol-decoding purposes at the Rx part [12,13]. For the reader's interest, refer to [11-16] for other MIMO demodulation techniques.
and <sup>2</sup> *N n t T*
, and a block-matrix
h hh h
h h 2; 2) a typical matrix
(5)
*N NN N r rt t* which solves the matrix-
represent the inverse and transpose matrix operations over
. For these tasks, Partition-Matrix Theory will be employed in
. For instance, let *nm nm <sup>L</sup>* be a full-rank real-valued
The Matrix Inversion Lemma is an indirect consequence of inverting non-singular block matrices [17-18], either real-valued or complex-valued, e.g., under certain restrictions 3. Lemma 1 states this result.
*Lemma 1*. Let *r r* , *r s* , *s s* , and *s r* be real-valued or complexvalued matrices. Assume these matrices to be non-singular: , , , and 1 1 . Then,
$$\left(\Psi + \Sigma\mathbf{1}\mathbf{\Xi}\right)^{-1} = \Psi^{-1} - \Psi^{-1}\Sigma\left(\mathbf{I}^{-1} + \Xi\Psi^{-1}\Sigma\right)^{-1}\Xi\Psi^{-1} \tag{7}$$
**Proof**. The validation of (7) must satisfy
$$\begin{aligned} \text{i.i.} \qquad & \left(\Psi + \Sigma\boldsymbol{\Sigma}\boldsymbol{\Xi}\right) \cdot \left(\Psi^{-1} - \Psi^{-1}\boldsymbol{\Sigma}\left(\boldsymbol{\chi}^{-1} + \boldsymbol{\Xi}\Psi^{-1}\boldsymbol{\Sigma}\right)^{-1}\boldsymbol{\Xi}\Psi^{-1}\right) = \boldsymbol{I}\_{r^{-}} \text{ and} \\\\ & \left(\Psi^{-1} - \Psi^{-1}\boldsymbol{\Sigma}\left(\boldsymbol{\chi}^{-1} + \boldsymbol{\Xi}\Psi^{-1}\boldsymbol{\Sigma}\right)^{-1}\boldsymbol{\Xi}\Psi^{-1}\right) \cdot \left(\Psi + \Sigma\boldsymbol{\Upsilon}\boldsymbol{\Xi}\right) = \boldsymbol{I}\_{r^{-}} \text{, where } \boldsymbol{I}\_{r} \text{ represents the } r \times r \text{ identity} \\\\ & \text{matrix. Notice the existence of matrices } \;\Psi^{-1}, \;\ \operatorname{\boldsymbol{\chi}}^{-1}, \; \begin{pmatrix} \Psi + \Sigma\boldsymbol{\chi}\boldsymbol{\Xi} \end{pmatrix}^{-1} \text{ and } \begin{pmatrix} \operatorname{\boldsymbol{\chi}}^{-1} + \boldsymbol{\Xi}\Psi^{-1}\boldsymbol{\Sigma} \end{pmatrix}^{-1}. \end{aligned}$$
Manipulating i) shows:
<sup>3</sup> Refer to [3,7-10,17,18] to review lemmata exposed for these issues and related results.
$$
\left(\Psi + \Sigma\Upsilon\Xi\right) \cdot \left(\Psi^{-1} - \Psi^{-1}\Sigma\left(\Upsilon^{-1} + \Xi\Psi^{-1}\Sigma\right)^{-1}\Xi\Psi^{-1}\right)
$$
$$
= I\_r - \Sigma\left(\Upsilon^{-1} + \Xi\Psi^{-1}\Sigma\right)^{-1}\Xi\Psi^{-1} + \Sigma\Upsilon\Xi\Psi^{-1} - \Sigma\Upsilon\Xi\Psi^{-1}\Sigma\left(\Upsilon^{-1} + \Xi\Psi^{-1}\Sigma\right)^{-1}\Xi\Psi^{-1}
$$
$$
= I\_r + \Sigma\Upsilon\Xi\Psi^{-1} - \Sigma\Upsilon\left(\Upsilon^{-1} + \Xi\Psi^{-1}\Sigma\right)\left(\Upsilon^{-1} + \Xi\Psi^{-1}\Sigma\right)^{-1}\Xi\Psi^{-1}
$$
$$
= I\_r + \Sigma\Upsilon\Xi\Psi^{-1} - \Sigma\Upsilon\Xi\Psi^{-1} = I\_r.
$$
Likewise for ii):
$$\begin{aligned} \left(\Psi^{-1} - \Psi^{-1}\Sigma\left(\mathbf{I}^{-1} + \Xi\Psi^{-1}\Sigma\right)^{-1}\Xi\Psi^{-1}\right)\cdot\left(\Psi + \Sigma\Upsilon\Xi\right) \\\\ = \left(\Psi^{-1} + \Psi^{-1}\Sigma\Upsilon\Xi - \Psi^{-1}\Sigma\left(\mathbf{I}^{-1} + \Xi\Psi^{-1}\Sigma\right)^{-1}\Xi - \Psi^{-1}\Sigma\left(\mathbf{I}^{-1} + \Xi\Psi^{-1}\Sigma\right)^{-1}\Xi\Psi^{-1}\Sigma\Upsilon\Xi\right) \\\\ = I\_r + \Psi^{-1}\Sigma\Upsilon\Xi - \Psi^{-1}\Sigma\left(\mathbf{I}^{-1} + \Xi\Psi^{-1}\Sigma\right)^{-1}\left(\mathbf{I}^{-1} + \Xi\Psi^{-1}\Sigma\right)\mathbf{I}\Xi \\\\ = I\_r + \Psi^{-1}\Sigma\Upsilon\Xi - \Psi^{-1}\Sigma\Upsilon\Xi = I\_r. \ \blacksquare$$
Now it is pertinent to demonstrate (6) with the aid of Lemma 1. It must be verified that both <sup>1</sup> *LL* and 1 *L L* must be equal to the *nm nm* identity block matrix ( ) 0 0 *n nm mn m n m I <sup>I</sup> <sup>I</sup>* , with consistent-dimensional identity and zero sub-blocks: *nI* , *mI* ; 0*n m* , 0*m n* , respectively. We start by calulating
$$LL^{-1} = \begin{bmatrix} A & B \\ C & D \end{bmatrix} \begin{bmatrix} \left(A - BD^{-1} \mathbb{C}\right)^{-1} & -\left(A - BD^{-1} \mathbb{C}\right)^{-1} BD^{-1} \\\\ -\left(D - CA^{-1}\mathbb{B}\right)^{-1}CA^{-1} & \left(D - CA^{-1}\mathbb{B}\right)^{-1} \end{bmatrix} \tag{8}$$
and
$$L^{-1}L = \begin{bmatrix} \left(A - BD^{-1}\mathbb{C}\right)^{-1} & -\left(A - BD^{-1}\mathbb{C}\right)^{-1}BD^{-1} \\\\ -\left(D - CA^{-1}B\right)^{-1}CA^{-1} & \left(D - CA^{-1}B\right)^{-1} \end{bmatrix} \begin{bmatrix} A & B \\ \mathbb{C} & D \end{bmatrix} \tag{9}$$
by applying (7) in Lemma 1 to both matrices <sup>1</sup> <sup>1</sup> *n n A BD C* and <sup>1</sup> <sup>1</sup> *m m D CA B* , which are present in (8) and (9), and recalling full-rank conditions not only over those matrices but also for *A* and *D* , yields the relations
Partition-Matrix Theory Applied to the Computation of Generalized-Inverses for MIMO Systems in Rayleigh Fading Channels 145
$$\left(A - BD^{-1}\mathbb{C}\right)^{-1} = A^{-1} + A^{-1}B\left(D - CA^{-1}B\right)^{-1}CA^{-1} \tag{10}$$
$$\left(D - \mathbb{C}A^{-1}B\right)^{-1} = D^{-1} + D^{-1}\mathbb{C}\left(A - \mathbb{B}D^{-1}\mathbb{C}\right)^{-1}BD^{-1} \tag{11}$$
Using (10-11) in (8-9), the following results arise:
144 Linear Algebra – Theorems and Applications
Likewise for ii):
<sup>1</sup> *LL* and 1 *L L*
0
*I <sup>I</sup> <sup>I</sup>*
0
0*n m* , 0*m n* , respectively. We start by calulating
*A B LL C D*
1
1
*n nm mn m*
( )
and
*n m*
<sup>1</sup> 1 11 1 1
*rI*
1 1 . *r r I I*
<sup>1</sup> 1 11 1 1
1 1 1 11 1 11 1 1 *<sup>r</sup> <sup>I</sup>*
> <sup>1</sup> 1 11 1 1 1 *<sup>r</sup> <sup>I</sup>*
> > 1 1 . *r r I I* ■
Now it is pertinent to demonstrate (6) with the aid of Lemma 1. It must be verified that both
must be equal to the *nm nm* identity block matrix
1 1 <sup>1</sup> 1 1
1 1 11 1
*A BD C A BD C BD*
by applying (7) in Lemma 1 to both matrices <sup>1</sup> <sup>1</sup> *n n A BD C* and
<sup>1</sup> <sup>1</sup> *m m D CA B* , which are present in (8) and (9), and recalling full-rank conditions
*D CA B CA D CA B*
not only over those matrices but also for *A* and *D* , yields the relations
*D CA B CA D CA B A B L L*
, with consistent-dimensional identity and zero sub-blocks: *nI* , *mI* ;
1 1 1 1 <sup>1</sup>
*C D*
(8)
(9)
1 1 11 1
*A BD C A BD C BD*
<sup>1</sup> 1 1 11 1 1
1 1 1 1 1 1 11 1 1 *rI*
a. for operations involved in sub-blocks of <sup>1</sup> *LL* :
$$A\left(A-BD^{-1}C\right)^{-1} - B\left(D-CA^{-1}B\right)^{-1}CA^{-1}$$
$$= A\left[A^{-1} + A^{-1}B\left(D-CA^{-1}B\right)^{-1}CA^{-1}\right] - B\left(D-CA^{-1}B\right)^{-1}CA^{-1}$$
$$= I\_n + B\left(D-CA^{-1}B\right)^{-1}CA^{-1} - B\left(D-CA^{-1}B\right)^{-1}CA^{-1} = I\_n + I\_n$$
$$\quad - A\left(A-BD^{-1}C\right)^{-1}BD^{-1} + B\left(D-CA^{-1}B\right)^{-1}CA^{-1} = I\_n + I\_n$$
$$= -A\left[A^{-1} + A^{-1}B\left(D-CA^{-1}B\right)^{-1}CA^{-1}\right]BD^{-1} + B\left(D-CA^{-1}B\right)^{-1}$$
$$= -BD^{-1} - B\left(D-CA^{-1}B\right)^{-1}CA^{-1}BD^{-1} + B\left(D-CA^{-1}B\right)^{-1}$$
$$= -BD^{-1} - B\left(D-CA^{-1}B\right)^{-1}CA^{-1}BD^{-1} + B\left(D-CA^{-1}B\right)^{-1}$$
$$= -BD^{-1} - B\left(D-CA^{-1}B\right)^{-1}\left(-CA^{-1}B\right)D^{-1} = 0\_{n \times n};$$
$$C\left(A-BD^{-1}C\right)^{-1} = D\left(D-CA^{-1}B\right)^{-1}CA^{-1}$$
$$= C\left(A-BD^{-1}C\right)^{-1} - D\left[D^{-1} + D^{-1}C\left(A-BD^{-1}C\right)^{-1}BD^{-1}\right]AC^{-1}$$
$$= C\left(A-BD^{-1}C\right)^{-1} - CA^{-1} - C\left(A-BD^{-1}C\right)^{-1}BD^{-1}CA^{-1}$$
$$\mathcal{L} = -\mathbb{C}\left(A - BD^{-1}\mathbb{C}\right)^{-1}BD^{-1} + I\_m + \mathbb{C}\left(A - BD^{-1}\mathbb{C}\right)^{-1}BD^{-1} = I\_m;$$
thus, <sup>1</sup> ( ) *n m LL I* .
b. for operations involved in sub-blocks of <sup>1</sup> *L L* :
$$\begin{aligned} \left(A - BD^{-1}\mathbb{C}\right)^{-1}A - \left(A - BD^{-1}\mathbb{C}\right)^{-1}BD^{-1}\mathbb{C} \\\\ = \left(A - BD^{-1}\mathbb{C}\right)^{-1}\left[A - BD^{-1}\mathbb{C}\right] = I\_n; \\\\ \left(A - BD^{-1}\mathbb{C}\right)^{-1}B - \left(A - BD^{-1}\mathbb{C}\right)^{-1}BD^{-1}D = 0\_{n \times n}; \\\\ -\left(D - CA^{-1}B\right)^{-1}CA^{-1}A + \left(D - CA^{-1}B\right)^{-1}\mathbb{C} = 0\_{n \times n}; \\\\ -\left(D - CA^{-1}B\right)^{-1}CA^{-1}B + \left(D - CA^{-1}B\right)^{-1}D \\\\ = -\left(D - CA^{-1}B\right)^{-1}\left[-CA^{-1}B + D\right] = I\_m; \end{aligned}$$
thus, <sup>1</sup> ( ) *n m LL I* .
#### **4.3. Generalized-Inverse**
The concept of Generalized-Inverse is an extension of a matrix inversion operations applied to non-singular rectangular matrices [17-18]. For notation purposes and without loss of generalization, *G* and *G*<sup>T</sup> denote the rank of a rectangular matrix M*m n G* , and <sup>H</sup> *G G*<sup>T</sup> is the transpose-conjugate of *G* (when M= *m n <sup>G</sup>* ) or <sup>T</sup> *G G*<sup>T</sup> is the transpose of *<sup>G</sup>* (when M= *m n <sup>G</sup>* ), respectively.
*Definition 2*. Let M*m n G* and 0 min , *G mn* . Then, there exists a matrix † M*n m G* (identified as the Generalized-Inverse), such that it satisfies several conditions for the following cases:
**case i**: if *m n* and 0 min , *G mn <sup>G</sup> n* , then there exists a unique matrix † <sup>M</sup>*n m G G* (identified as Left-Pseudoinverse: LPI) such that *GG In* , satisfying: a) *GG G G* , and b) *G GG G* . Therefore, the LPI matrix is proposed as <sup>1</sup> *G GG G* T T .
**case ii**: if *m n* and det 0 *G G <sup>n</sup>* , then there exists a unique matrix † 1 M*n n G G* (identified as Inverse) such that 1 1 *G G GG In* .
**case iii**: if *m n* and 0 min , *G mn <sup>G</sup> m* , then there exists a unique matrix † <sup>M</sup>*n m G G* (identified as Right-Pseudoinverse: RPI) such that *GG Im* , satisfying: a) *GG G G* , and b) *G GG G* . Therefore, the RPI matrix is proposed as <sup>1</sup> *G G GG* T T . ■
Given the mathematical structure for † *G* provided in Definition 2, it can be easily validated that: 1) For a LPI matrix stipulated in case i, † *GG G G* and †† † *G GG G* with <sup>1</sup> † *G GG G* T T ; 2) For a RPI matrix stipulated in case iii, † *GG G G* and †† † *G GG G* with <sup>1</sup> † *G G GG* T T ; iii) For the Inverse in case ii, 1 1 *G G G G G GG G* T TT T . For a uniqueness test for all cases, assume the existance of matrices † <sup>1</sup> <sup>M</sup>*n m <sup>G</sup>* and † <sup>2</sup> <sup>M</sup>*n m <sup>G</sup>* such that † *GG I* <sup>1</sup> *<sup>n</sup>* and † *GG I* <sup>2</sup> *<sup>n</sup>* (for case i), and † *GG I* <sup>1</sup> *<sup>m</sup>* and † *GG I* <sup>2</sup> *<sup>m</sup>* (for case iii). Notice immediately, † † 1 2 0 *G GG <sup>n</sup>* (for case i) and † † 1 2 <sup>0</sup> *GG G <sup>m</sup>* (for case iii), which obligates † † *G G* 1 2 for both cases, because of full-rank properties over *G* . Clearly, case ii is a particular consequence of cases i and iii.
### **5. The MIMO channel matrix**
146 Linear Algebra – Theorems and Applications
b. for operations involved in sub-blocks of <sup>1</sup> *L L* :
thus, <sup>1</sup>
thus, <sup>1</sup>
generalization,
following cases:
( ) *n m LL I*
.
**4.3. Generalized-Inverse**
(when M= *m n <sup>G</sup>* ), respectively.
**case i**: if *m n* and 0 min ,
**case ii**: if *m n* and det 0 *G G*
(identified as Inverse) such that 1 1 *G G GG In*
**case iii**: if *m n* and 0 min ,
*Definition 2*. Let M*m n G* and 0 min ,
( ) *n m LL I* . 1 1 1 1 1 1 ; *m m C A BD C BD C A BD C BD I I*
1 1 1 1 <sup>1</sup> *A BD C A A BD C BD <sup>C</sup>*
<sup>1</sup> 1 1 ; *<sup>n</sup> A BD C A BD C I*
1 1 <sup>1</sup> 1 1 <sup>0</sup> ; *A BD C B A BD C n m BD D*
1 1 11 1 <sup>0</sup> ; *D CA B CA A D CA B <sup>C</sup> m n*
1 1 11 1 *D CA B CA B D CA B <sup>D</sup>*
<sup>1</sup> 1 1 ; *D CA B CA B D mI*
The concept of Generalized-Inverse is an extension of a matrix inversion operations applied to non-singular rectangular matrices [17-18]. For notation purposes and without loss of
is the transpose-conjugate of *G* (when M= *m n <sup>G</sup>* ) or <sup>T</sup> *G G*<sup>T</sup> is the transpose of *<sup>G</sup>*
(identified as the Generalized-Inverse), such that it satisfies several conditions for the
*G mn*
† <sup>M</sup>*n m G G* (identified as Left-Pseudoinverse: LPI) such that *GG In*
*GG G G* , and b) *G GG G* . Therefore, the LPI matrix is proposed as <sup>1</sup>
.
*G mn*
matrix † <sup>M</sup>*n m G G* (identified as Right-Pseudoinverse: RPI) such that *GG Im*
*G* and *G*<sup>T</sup> denote the rank of a rectangular matrix M*m n G* , and <sup>H</sup> *G G*<sup>T</sup>
*G mn* . Then, there exists a matrix † M*n m G*
*<sup>n</sup>* , then there exists a unique matrix † 1 M*n n G G*
*<sup>G</sup> n* , then there exists a unique matrix
*<sup>G</sup> m* , then there exists a unique
, satisfying: a)
,
*G GG G* T T .
The MIMO channel matrix is the mathematical representation for modeling the degradation phenomena presented in the RFC scenario presented in (2). The elements ij *<sup>h</sup>* in *R T n n <sup>H</sup>* represent a time-invariant transfer function (possesing spectral information about magnitude and phase profiles) between a j-th transmitter and an i-th receiver antenna. Once again, dynamical properties of physical phenomena 4 such as path-loss, shadowing, multipath, Doppler spreading, coherence time, absorption, reflection, scattering, diffraction, basestation-user motion, antenna's physical properties-dimensions, information correlation, associated with a slow-flat quasi-static RFC scenario (proper of a non-LOS indoor wireless environments) are highlighted into a statistical model represented by matrix *H* . For † *<sup>H</sup>* purposes, CSI is a necessary feature required at the reception part in (2), as well as the *R T n n* condition. Table 1 provides several *R T n n* MIMO channel matrix realizations for RFC-based environments [19-21]. On table 1: a) MIMO , *R T n n* : refers to the MIMO communication link configuration, i.e. amount of receiver-end and transmitter-end elements; b) *H*<sup>m</sup> : refers to a MIMO channel matrix realization; c) *<sup>H</sup>*<sup>m</sup> : refers to the corresponding LPI, computed as <sup>1</sup> H H mm m † *H HH H* <sup>m</sup> ; d) h : blockwise matrix version for *<sup>H</sup>*<sup>m</sup> ; e) <sup>+</sup> <sup>h</sup> : refers to the corresponding LPI, computed as <sup>1</sup> †T T h hh h . As an additional point of analysis, full-rank properties over *H* and h (and thus the existance of matrices *H* , <sup>1</sup> *H* , <sup>+</sup> h , and <sup>1</sup> h ) are validated and corroborated through a MATLAB simulation-driven model regarding frequency-selective and time-invariant properties for several RFC-based scenarios at different MIMO configurations. Experimental data were generated upon <sup>6</sup> 10 MIMO channel matrix realizations. As illustrated in figure 3, a common pattern is found regarding the statistical evolution for full-rank properties of *H* and h with *R T n n* at several typical MIMO configurations, for instance, MIMO 2, 2 , MIMO 4, 2 , and MIMO 4, 4 . It is plotted therein REAL(H,h) against IMAG(H,h), where each axis label denote respectively the real and imaginary parts of: a) det( ) *<sup>H</sup>* and det(h) when *R T n n* , and b) <sup>H</sup> det *H H* and <sup>T</sup> det h h when . Blue crosses indicate the behavior of ( ) *H* related to det( ) *H* and <sup>H</sup> det *H H*
<sup>4</sup> We suggest the reader consulting references [11-16] for a detail and clear explanation on these narrowband and wideband physical phenomena presented in wireless MIMO communication systems.
(det(H) legend on top-left margin), while red crosses indicate the behavior of (h) related to det(h) and <sup>T</sup> det h h (det(h) legend on top-left margin). The black-circled zone intersected with black-dotted lines locates the 0 0 *j* value. As depicted on figures (4)-(5), a closer glance at this statistical behavior reveals a prevalence on full-rank properties of *H* and h , meaning that non of the determinants det( ) *H* , det(h) , <sup>H</sup> det *H H* and <sup>T</sup> det h h is equal to zero (behavior enclosed by the light-blue region and delimited by blue/red-dotted lines).
**Figure 3.** MIMO channel matrix rank-determinant behavior for several realizations for *H* and h . This statistical evolution is a common pattern found for several MIMO configurations involving slow-flat quasi-static RFC-based environments with *R T n n* .
(det(H) legend on top-left margin), while red crosses indicate the behavior of
enclosed by the light-blue region and delimited by blue/red-dotted lines).
det(h) and <sup>T</sup> det h h (det(h) legend on top-left margin). The black-circled zone intersected with black-dotted lines locates the 0 0 *j* value. As depicted on figures (4)-(5), a closer glance at this statistical behavior reveals a prevalence on full-rank properties of *H* and h , meaning that non of the determinants det( ) *H* , det(h) , <sup>H</sup> det *H H* and <sup>T</sup> det h h is equal to zero (behavior
**Figure 3.** MIMO channel matrix rank-determinant behavior for several realizations for *H* and h . This statistical evolution is a common pattern found for several MIMO configurations involving slow-flat
quasi-static RFC-based environments with *R T n n* .
(h) related to
**Table 1.** MIMO channel matrix realizations for several MIMO communication link configurations at slow-flat quasi-static RFC scenarios.
**Figure 4.** MIMO channel matrix rank-determinant behavior for several realizations for *H* . Full-rank properties for *H* and <sup>H</sup> *H H* preveal for RFC-based environments (light-blue region delimited by bluedotted lines).
**Figure 5.** MIMO channel matrix rank-determinant behavior for several realizations for h . Full-rank properties for h and Th h preveal for RFC-based environments (light-blue region delimited by reddotted line).
#### **6. Proposed algorithm**
150 Linear Algebra – Theorems and Applications
properties for *H* and <sup>H</sup>
dotted lines).
dotted line).
**Figure 4.** MIMO channel matrix rank-determinant behavior for several realizations for *H* . Full-rank
**Figure 5.** MIMO channel matrix rank-determinant behavior for several realizations for h . Full-rank properties for h and Th h preveal for RFC-based environments (light-blue region delimited by red-
*H H* preveal for RFC-based environments (light-blue region delimited by blue-
The proposal for a novel algorithm for computing a LPI matrix <sup>+</sup> 2 2 <sup>h</sup> *T R n n* (with *R T n n* ) is based on the block-matrix structure of h as exhibited in (4). This idea is an extension of the approach presented in [22]. The existence for this Generalized-Inverse matrix is supported on the statistical properties of the slow-flat quasi-static RFC scenario which impact directly on the singularity of *H* at every MIMO channel matrix realization. Keeping in mind that other approaches attempting to solve the block-matrix inversion problem [7-10] requires several constraints and conditions, the subsequent proposal does not require any restriction at all mainly due to the aforementioned properties of *H* . From (4), it is suggested
$$\text{that}\\
\begin{bmatrix} \mathbf{x}^{\mathsf{t}}\\\mathbf{x}^{\mathsf{t}} \end{bmatrix} \text{ is some}\\
\text{show related to}\\
\begin{bmatrix} \Re\left\{\mathbf{H}^{+}\right\} & -\Im\left\{\mathbf{H}^{+}\right\} \\\\ \Im\left\{\mathbf{H}^{+}\right\} & \Re\left\{\mathbf{H}^{+}\right\} \end{bmatrix} \cdot \mathbf{y} \\
\text{ :} \text{hence, calculating } \mathbf{h}^{+} \text{ will lead to this}\\
\begin{bmatrix} \mathbf{h}^{+} \\\\ \end{bmatrix} \text{ is a vector of } \mathbf{h}$$
solution. Let r *A H* and i *B H* . It is kwon a priori that *<sup>A</sup> <sup>T</sup> jB n* . Then h *A B B A*
with h 2 *<sup>T</sup> <sup>t</sup> n N* . Define the matrix as <sup>T</sup> h h *N N t t* , where *<sup>M</sup> <sup>L</sup> L M* with
T T *T T n n M AA BB* , <sup>T</sup> T T *T T n n L AB AB* , and *Nt* as a direct consequence from 2 2 *R T r t n n N N* . It can be seen that
$$\mathbf{h}^{+} = \tilde{\mathbf{\Omega}}^{-1} \mathbf{h}^{\mathbf{T}} \in \mathbb{R}^{N\_{t} \times N\_{r}} \tag{12}$$
For simplicity, matrix operations involved in (12) require classic multiply-and-accumulate operations between row-entries of 1 *N N t t* and column-entries of T <sup>h</sup> *N N t r* . Notice immediately that the critical and essential task of computing <sup>+</sup> h relies on finding the block matrix inverse <sup>1</sup> 5. The strategy to be followed in order to solve <sup>1</sup> in (12) will consist of the following steps: 1) the proposition of partitioning without any restriction on rankdefficiency over inner matrix sub-blocks; 2) the definition of iterative multiply-andaccumulate operations within sub-blocks comprised in ; 3) the recursive definition for compacting the overall blockwise matrix inversion. Keep in mind that matrix can be also
viewed as 1,1 1, ,1 , *t t t t N N N N* . The symmetry presented in *<sup>M</sup> <sup>L</sup> L M* will motivate
the development for the pertinent LPI-based algorithm. From (12) and by the use of Lemma 1 it can be concluded that <sup>1</sup> *Q P P Q* , where <sup>1</sup> <sup>1</sup> *T T n n Q M LM L* , *T T n n P QX* ,
<sup>5</sup> Notice that + 1 T T *A jB M jL A jB* . Moreover, <sup>1</sup> *M jL T T n n j* , where 1 1 <sup>1</sup> 1 1 *M LM L L ML M ML* and 1 1 <sup>1</sup> 1 1 *M L M LM L L ML M* .
and <sup>1</sup> *T T n n X LM* . Interesting enough, full-rank is identified at each matrix sub-block in the main diagonal of (besides *Q nT* ). This structural behavior serves as the leitmotiv for the construction of an algorithm for computing the blockwise inverse <sup>1</sup> . Basically speaking and concerning step 1) of this strategy, the matrix partition procedure obeys the assignments (13-16) defined as:
$$\mathcal{W}\_{k} = \begin{bmatrix} \tilde{o}\_{N\_{l}-\{2k+1\}, N\_{l}-\{2k+1\}} & \tilde{o}\_{N\_{l}-\{2k+1\}, N\_{l}-2k} \\ \tilde{o}\_{N\_{l}-2k, N\_{l}-\{2k+1\}} & \tilde{o}\_{N\_{l}-2k, N\_{l}-2k} \end{bmatrix} \in \mathbb{R}^{2 \times 2} \tag{13}$$
$$X\_k = \begin{bmatrix} \tilde{o}\tilde{o}\_{N\_t - \{2k+1\}, N\_t - \{2k-1\}} & \cdots & \tilde{o}\_{N\_t - \{2k+1\}, N\_t} \\ \tilde{o}\tilde{o}\_{N\_t - 2k, N\_t - \{2k-1\}} & \cdots & \tilde{o}\_{N\_t - 2k, N\_t} \end{bmatrix} \in \mathbb{R}^{2 \times 2k} \tag{14}$$
$$Y\_k = \begin{bmatrix} \tilde{o}\_{N\_l - \{2k - 1\}, N\_l - \{2k + 1\}} & \tilde{o}\_{N\_l - \{2k - 1\}, N\_l - 2k} \\ \vdots & \vdots \\ \tilde{o}\_{N\_l, N\_l - \{2k + 1\}} & \tilde{o}\_{N\_l, N\_l - 2k} \end{bmatrix} \in \mathbb{R}^{2k \times 2} \tag{15}$$
$$Z\_0 = \begin{bmatrix} \tilde{o}\_{N\_t - 1, N\_t - 1} & \tilde{o}\_{N\_T - 1, N\_t} \\ \tilde{o}\_{N\_t, N\_t - 1} & \tilde{o}\_{N\_t, N\_t} \end{bmatrix} \in \mathbb{R}^{2 \times 2} \tag{16}$$
The matrix partition over obeys the index *<sup>k</sup>* 1:1:*Nt* 2 1 . Because of the evenrectangular dimensions of , matirx owns exactly an amount of <sup>2</sup> *N n t T* sub-block matrices of 2 2 dimension along its main diagonal. Interesting enough, due to RFC-based environment characteristics studied in (1) and (4), it is found that:
$$
\rho\left(\mathcal{W}\_k\right) = \rho\left(Z\_0\right) = 2 \tag{17}
$$
After performing these structural characteristics for , and with the use of (13-16), step 2) of the strategy consists of the following iterative operations also indexed by *<sup>k</sup>* 1:1:*Nt* 2 1 , in the sense of performing:
$$
\phi\_k = \mathcal{W}\_k - X\_k Z\_{k-1}^{-1} Y\_k \tag{18}
$$
$$
\alpha\_k = \phi\_k^{-1} X\_k Z\_{k-1}^{-1} \tag{19}
$$
$$
\theta\_k = Z\_{k-1}^{-1} + Z\_{k-1}^{-1} Y\_k a\_k \tag{20}
$$
Here: 1 22 1 *k k Zk* , 2 2 *k* , 2 2*<sup>k</sup> k* , and 2 2 *k k k* . Steps stated in (18-20) help to construct intermediate sub-blocks as
Partition-Matrix Theory Applied to the Computation of Generalized-Inverses for MIMO Systems in Rayleigh Fading Channels 153
$$\begin{array}{c} \widetilde{\boldsymbol{\Omega}}\_{k} = \begin{bmatrix} \underline{\boldsymbol{W}\_{k}} & \underline{\boldsymbol{X}\_{k}} \\ \underline{\boldsymbol{z}}\_{2\times 2} & \underline{\boldsymbol{z}}\_{2\times 2k} \\ \underline{\boldsymbol{Y}\_{k}} & \underline{\boldsymbol{Z}\_{k-1}} \end{bmatrix} \rightarrow \begin{array}{c} \underline{\boldsymbol{\Theta}\_{k}^{-1}} & \underline{\boldsymbol{\Theta}\_{k}^{-1}} \\ \underline{\boldsymbol{Y}\_{k}} & \underline{\boldsymbol{Z}\_{k-1}} \end{array} \rightarrow \begin{array}{c} \underline{\boldsymbol{\Theta}\_{k}^{-1}} & \underline{\boldsymbol{\Theta}\_{k}} \\ \underline{\boldsymbol{\Theta}\_{k}} & \underline{\boldsymbol{Z}\_{k}} \end{array} \tag{21}$$
152 Linear Algebra – Theorems and Applications
assignments (13-16) defined as:
the main diagonal of (besides *Q nT*
*k*
0
environment characteristics studied in (1) and (4), it is found that:
*Z*
*W*
*k*
*k*
*<sup>k</sup>* 1:1:*Nt* 2 1 , in the sense of performing:
, 2 2 *k*
construct intermediate sub-blocks as
Here: 1 22 1 *k k Zk*
*Y*
*X*
and <sup>1</sup> *T T n n X LM* . Interesting enough, full-rank is identified at each matrix sub-block in
for the construction of an algorithm for computing the blockwise inverse <sup>1</sup> . Basically speaking and concerning step 1) of this strategy, the matrix partition procedure obeys the
*N k N k N k Nk*
*N kN k N kN k*
2, 2 1 2 , *t t t t t t t t*
2 1, 2 1 2 1, 2
, 21 , 2
,1 , *tt Tt t t t t NN N N NN NN*
The matrix partition over obeys the index *<sup>k</sup>* 1:1:*Nt* 2 1 . Because of the evenrectangular dimensions of , matirx owns exactly an amount of <sup>2</sup> *N n t T* sub-block matrices of 2 2 dimension along its main diagonal. Interesting enough, due to RFC-based
<sup>0</sup> <sup>2</sup> *W Z <sup>k</sup>*
After performing these structural characteristics for , and with the use of (13-16), step 2) of the strategy consists of the following iterative operations also indexed by
*k k kk k W XZ Y*<sup>1</sup>
1 1 *k k k kk Z ZY* 1 1
, and 2 2 *k k k*
1 1
1
>
*NN k NN k*
*t t t t*
*N kN k N kN*
*tt tt*
*N k N k N k Nk*
2, 2 1 2, 2 *tt tt t t t t*
>
2 1, 2 1 2 1, 2 2 2
2 1, 2 1 2 1, 2 2
*N kN k N kN k*
1, 1 1, 2 2
, 2 2*<sup>k</sup> k*
*k k kk X Z* <sup>1</sup>
). This structural behavior serves as the leitmotiv
(13)
(14)
2 2
(17)
(18)
(19)
(20)
. Steps stated in (18-20) help to
(15)
*k*
(16)
The dimensions of each real-valued sub-block in (21) are indicated consistently 6. For step 3) of the strategy, a recursion step 1 1 <sup>1</sup> ( ) *Z Z k k* is provided in terms of the assignment 1 1 2( 1) 2( 1) *k k Zk k* . Clearly, only inversions of *Wk* , *Z*<sup>0</sup> , and *<sup>k</sup>* (which are 2 2 matrices, yielding correspondingly <sup>1</sup> *Wk* , <sup>1</sup> *<sup>Z</sup>*<sup>0</sup> , and <sup>1</sup> *k* ) are required to be performed throughout this iterative-recursive process, unlike the operation linked to <sup>1</sup> *Zk* <sup>1</sup> , which comes from a previous updating step associated with the recursion belonging to <sup>1</sup> *Zk* . Although *Nt* assures the existance of <sup>1</sup> , full-rank requirements outlined in (17) and non-zero determinants for (18) are strongly needed for this iterative-recursive algorithm to work accordingly. Also, full-rank is expected for every recursive outcome related to 1 1 <sup>1</sup> ( ) *Z Z k k* . Again, thank to the characteristics of the slow-flat quasi-static RFC-based environment in which these operations are involved among every MIMO channel matrix realization, conditions in (17) and full-rank of (18) are always satisfied. These issues are corroborated with the aid of the same MATLAB-based simulation framework used to validate full-rank properties over *H* and h . The statistical evolution for the determinants for *Wk* , *Z*<sup>0</sup> , and *<sup>k</sup>* , and the behavior of singularity within the 1 1 <sup>1</sup> ( ) *Z Z k k* recursion are respectively illustrated in figures (6)-(8). MIMO 2, 2 , MIMO 4, 2 , and MIMO 4, 4 were the MIMO communication link configurations considered for these tests. These simulation-driven outcomes provide supportive evidence for the proper functionality of the proposed iterative-recursive algorithm for computing <sup>1</sup> involving matrix sub-block inversions. On each figure, the statistical evolution for the determinants associated with *Z*<sup>0</sup> , *Wk* , *<sup>k</sup>* , and 1 1 <sup>1</sup> ( ) *Z Z k k* are respectively indicated by labels det(Zo), det(Wk), det(Fik), and det(iZk,iZkm1), while the light-blue zone at bottom delimited by a red-dotted line exhibits the gap which marks the avoidance in rankdeficincy over the involved matrices. The zero-determinant value is marked with a black circle.
The next point of analysis for the behavior of the + h LPI-based iterative-recursive algorithm is complexity, which in essence will consist of a demand in matrix partitions (amount of matrix sub-blocks: PART) and arithmetic operations (amount of additions-subtractions: ADD-SUB; multiplications: MULT; and divisions: DIV). Let PART-mtx and ARITH-ops be the nomenclature for complexity cost related to matrix partitions and arithmetic operations, respectively. Without loss of generalization, define *C* as the complexity in terms of the
6 Matrix structure given in (21) is directly derived from applying Equation (6), and by the use of Lemma 1 as 1 1 1 11 1 1 *k k k k k k k k kk k kk* <sup>1</sup> 1 1 1 1 *Z YW X Z Z Y W X Z Y X Z* . See that this expansion is preferable instead of 1 1 1 11 1 1 *W X Z Y W W X Z YW X YW k kk k k k k k k k k k k* 1 1 , which is undesirable due to an unnecessary matrix operation overhead related to computing *<sup>k</sup>* <sup>1</sup> *<sup>Z</sup>* , e.g. inverting <sup>1</sup> *<sup>k</sup>* <sup>1</sup> *<sup>Z</sup>* , which comes preferably from the 1 1 <sup>1</sup> ( ) *k k Z Z* recursion.
costs PART-mtx and ARITH-ops belonging to operations involved in . Henceforth, 1 1T *CC C* h h denotes the cost of computing + h as the sum of the costs of inverting and multiplying <sup>1</sup> by T <sup>h</sup> . It is evident that: a) 1 T *<sup>C</sup>* <sup>h</sup> implies PART=0 and ARITH-ops itemized into MULT= <sup>2</sup> <sup>8</sup> *R T n n* , ADD-SUB= 4 21 *RT T nn n* , and DIV=0; b) 1T T -1 *CC C* h h (h h) . Clearly, T *<sup>C</sup>* h h demands no partitions at all, but with a ARITH-ops cost of MULT= <sup>2</sup> <sup>8</sup> *R T n n* , and ADD-SUB= <sup>2</sup> 42 1 *R T n n* . However, the principal complexity relies critically on T -1 *<sup>C</sup>* (h h) , which is the backbone for + <sup>h</sup> , as presented in [22]. Table 2 summerizes these complexity results. For this treatment, T -1 *<sup>C</sup>* (h h) consists of 3 2 *<sup>T</sup> <sup>n</sup>* partitions, MULT = 1 1 6 *T n I k k C* , ADD-SUB = 1 1 1 *T n II k k C* , and DIV = 1 1 1 *T n III k k C* . The ARITH-ops cost depends on *<sup>I</sup> <sup>k</sup> <sup>C</sup>* , *II <sup>k</sup> <sup>C</sup>* , and *III <sup>k</sup> <sup>C</sup>* ; the constant factors for each one of these items are proper of the complexity presented in <sup>1</sup> <sup>0</sup> *C Z* . The remain of the complexities, i.e. *I <sup>k</sup> <sup>C</sup>* , *II <sup>k</sup> <sup>C</sup>* , and *III <sup>k</sup> <sup>C</sup>* , are calculated according to the iterative stpes defined in (18-20) and (21), particularly expressed in terms of
$$\mathbb{C}\left[\boldsymbol{\phi}\_{k}^{-1}\right] + \mathbb{C}\left[-\boldsymbol{\alpha}\_{k}\right] + \mathbb{C}\left[-\theta\_{k}\boldsymbol{Y}\_{k}\boldsymbol{W}\_{k}^{-1}\right] + \mathbb{C}\left[\boldsymbol{\theta}\_{k}\right] \tag{22}$$
It can be checked out that: a) no PART-mtx cost is required; b) the ARITH-ops cost employs (22) for each item, yielding: 2 40 24 12 *<sup>I</sup> <sup>k</sup> Ckk* (for MULT), 2 40 2 *II <sup>k</sup> C k* (for ADD\_SUB), and 2 *III <sup>k</sup> <sup>C</sup>* (for DIV).
An illustrative application example is given next. It considers a MIMO channel matrix realization obeying statistical behavior according to (1) and a MIMO 4, 4 configuration:
0.3059 0.7543 0.8107 0.2082 0.2314 0.4892 0.416 1.0189 1.1777 0.0419 0.8421 0.9448 0.1235 0.6067 1.5437 0.4039 0.0886 0.0676 0.8409 0.5051 0.132 0.8867 0.0964 0.2828 0.2034 0.5886 0.0266 1.1 *j jj j jjjj <sup>H</sup> j jj j j j* 4 4 48 0.5132 1.1269 0.0806 0.4879 *j j* with *H* 4 . As a consequence, 2.4516 1.2671 0.1362 2.7028 0 1.9448 0.6022 0.2002 1.2671 4.5832 1.7292 1.3776 1.9448 0 1.229 2.4168 0.1362 1.7292 3.0132 0.0913 0.6022 1.229 0 0.862 2.7028 1.3776 0.0913 4.0913 0.2002 2.4168 0.862 0 0 1.9448 0.6022 0.2 002 2.4516 1.2671 0.1362 2.7028 1.9448 0 1.229 2.4168 1.2671 4.5832 1.7292 1.3776 0.6022 1.229 0 0.862 0.1362 1.7292 3.0132 0.0913 0.2002 2.4168 0.862 0 2.7028 1.3776 0.0913 4.0913 8 8 with 8 .
1T T -1 *CC C* h h (h h)
3 2 *<sup>T</sup> <sup>n</sup>* partitions, MULT =
*<sup>k</sup> <sup>C</sup>* , and *III*
*<sup>k</sup> <sup>C</sup>* (for DIV).
*I <sup>k</sup> <sup>C</sup>* , *II*
and 2 *III*
consequence,
ARITH-ops cost depends on *<sup>I</sup>*
particularly expressed in terms of
0.2034 0.5886 0.0266 1.1
0 1.9448 0.6022 0.2
*j j*
costs PART-mtx and ARITH-ops belonging to operations involved in . Henceforth, 1 1T *CC C* h h denotes the cost of computing + h as the sum of the costs of
inverting and multiplying <sup>1</sup> by T <sup>h</sup> . It is evident that: a) 1 T *<sup>C</sup>* <sup>h</sup> implies PART=0
and ARITH-ops itemized into MULT= <sup>2</sup> <sup>8</sup> *R T n n* , ADD-SUB= 4 21 *RT T nn n* , and DIV=0; b)
ARITH-ops cost of MULT= <sup>2</sup> <sup>8</sup> *R T n n* , and ADD-SUB= <sup>2</sup> 42 1 *R T n n* . However, the principal complexity relies critically on T -1 *<sup>C</sup>* (h h) , which is the backbone for + <sup>h</sup> , as presented in [22].
Table 2 summerizes these complexity results. For this treatment, T -1 *<sup>C</sup>* (h h) consists of
1
*T n II k k C*
*<sup>k</sup> <sup>C</sup>* , are calculated according to the iterative stpes defined in (18-20) and (21),
*<sup>k</sup> Ckk* (for MULT), 2 40 2 *II*
1
(22)
, and DIV =
*<sup>k</sup> <sup>C</sup>* ; the constant factors for each one of these
<sup>0</sup> *C Z* . The remain of the complexities, i.e.
4 4
with
1
*III k k C*
*T n*
1
. The
1
*<sup>k</sup> C k* (for ADD\_SUB),
8 8 with
8 .
*H* 4 . As a
1
1
*T n I k k C*
6
, ADD-SUB =
*<sup>k</sup> <sup>C</sup>* , and *III*
1 1 *<sup>k</sup> <sup>k</sup> kk k k C C C YW C*
It can be checked out that: a) no PART-mtx cost is required; b) the ARITH-ops cost employs
An illustrative application example is given next. It considers a MIMO channel matrix realization obeying statistical behavior according to (1) and a MIMO 4, 4 configuration:
48 0.5132 1.1269 0.0806 0.4879 *j j*
002 2.4516 1.2671 0.1362 2.7028
1
0.3059 0.7543 0.8107 0.2082 0.2314 0.4892 0.416 1.0189 1.1777 0.0419 0.8421 0.9448 0.1235 0.6067 1.5437 0.4039 0.0886 0.0676 0.8409 0.5051 0.132 0.8867 0.0964 0.2828
*jjjj <sup>H</sup>*
*j jj j*
*j jj j*
2.4516 1.2671 0.1362 2.7028 0 1.9448 0.6022 0.2002 1.2671 4.5832 1.7292 1.3776 1.9448 0 1.229 2.4168 0.1362 1.7292 3.0132 0.0913 0.6022 1.229 0 0.862 2.7028 1.3776 0.0913 4.0913 0.2002 2.4168 0.862 0
1.9448 0 1.229 2.4168 1.2671 4.5832 1.7292 1.3776 0.6022 1.229 0 0.862 0.1362 1.7292 3.0132 0.0913 0.2002 2.4168 0.862 0 2.7028 1.3776 0.0913 4.0913
*<sup>k</sup> <sup>C</sup>* , *II*
items are proper of the complexity presented in <sup>1</sup>
(22) for each item, yielding: 2 40 24 12 *<sup>I</sup>*
. Clearly, T *<sup>C</sup>* h h demands no partitions at all, but with a
**Figure 6.** Statistical evolution of the rank-determinant behaviour concerning <sup>0</sup> *Z* , *Wk* , *<sup>k</sup>* , and 1 1 <sup>1</sup> ( ) *k k Z Z* for a MIMO 2, 2 configuration.
**Figure 7.** Statistical evolution of the rank-determinant behaviour concerning <sup>0</sup> *Z* , *Wk* , *<sup>k</sup>* , and 1 1 <sup>1</sup> ( ) *k k Z Z* for a MIMO 4, 2 configuration.
**Figure 7.** Statistical evolution of the rank-determinant behaviour concerning <sup>0</sup> *Z* , *Wk* , *<sup>k</sup>*
for a MIMO 4, 2 configuration.
, and 1 1 <sup>1</sup> ( ) *k k Z Z*
**Figure 8.** Statistical evolution of the rank-determinant behaviour concerning <sup>0</sup> *Z* , *Wk* , *<sup>k</sup>* , and 1 1 <sup>1</sup> ( ) *k k Z Z* for a MIMO 4, 4 configuration.
**Table 2.** Complexity cost results of the LPI-based iterative-recursive algorithm for <sup>+</sup> <sup>h</sup> .
Applying partition criteria (13-16) and given *k* 1:1:3 , the following matrix sub-blocks are generated:
$$W\_1 = \begin{bmatrix} 2.4516 & -1.2671 \\ -1.2671 & 4.5832 \end{bmatrix}^t$$
$$X\_1 = \begin{bmatrix} 0.1362 & -2.7028 \\ -1.7292 & 1.3776 \end{bmatrix}^t \quad Y\_1 = \begin{bmatrix} 0.1362 & -1.7292 \\ -2.7028 & 1.3776 \end{bmatrix} \quad Z\_{\
u} = \begin{bmatrix} 3.0132 & 0.0913 \\ 0.0913 & 4.0913 \end{bmatrix}^t,$$
$$W\_2 = \begin{bmatrix} 3.0132 & 0.0913 \\ 0.0913 & 4.0913 \end{bmatrix}^t,$$
$$X\_2 = \begin{bmatrix} -0.6022 & 1.2280 & 0 & 0.862 \\ 0.2002 & 2.4168 & -0.862 & 0 \end{bmatrix}^t \quad Y\_2 = \begin{bmatrix} -0.6022 & 0.2002 \\ 1.229 & 2.4168 \\ 0 & -0.862 \\ 0.862 & 0 \end{bmatrix}^t$$
$$\begin{aligned} W\_{3} &= \begin{bmatrix} 2.4516 & -1.2671 \\ -1.2671 & 4.5832 \end{bmatrix}, Y\_{3} = \begin{bmatrix} 0.1382 & -2.7028 & 0 & -1.9448 & 0.6022 & -0.202 \\ -1.7292 & 1.3776 & 1.9448 & 0 & -1.229 & -2.418 \end{bmatrix}, \\\\ \mathbf{and} \quad Y\_{j} &= \begin{bmatrix} 0.1362 & -1.7292 \\ -2.7028 & 1.3776 \\ 0 & 1.9448 \\ -1.9448 & 0 \\ 0.6022 & -1.229 \\ -0.2002 & -2.418 \end{bmatrix}. \end{aligned}$$
generated:
1
2
0.1362 2.7028 1.7292 1.3776 *<sup>X</sup>* ,
<sup>1</sup>
**Table 2.** Complexity cost results of the LPI-based iterative-recursive algorithm for <sup>+</sup> <sup>h</sup> .
1
2
0.6022 1.2290 0 0.862 0.2002 2.4168 0.862 0 *<sup>X</sup>* , <sup>2</sup>
Applying partition criteria (13-16) and given *k* 1:1:3 , the following matrix sub-blocks are
2.4516 1.2671 1.2671 4.5832 *<sup>W</sup>* ,
0.1362 1.7292 2.7028 1.3776 *<sup>Y</sup>* ,
3.0132 0.0913 0.0913 4.0913 *<sup>W</sup>*
<sup>0</sup>
,
3.0132 0.0913 0.0913 4.0913 *<sup>Z</sup>* ,
0.6022 0.2002 1.229 2.4168 0 0.862 0.862 0
*Y* ,
$$
\phi\_1 = \mathcal{W}\_1 - X\_1 Z\_0^{-1} Y\_1, \quad \alpha\_1 = \phi\_1^{-1} X\_1 Z\_0^{-1}, \quad \theta\_1 = Z\_0^{-1} + Z\_0^{-1} Y\_1 \alpha\_1 \tag{23}
$$
$$
\boldsymbol{\phi}\_{2} = \boldsymbol{W}\_{2} - \boldsymbol{X}\_{2}\boldsymbol{Z}\_{1}^{-1}\boldsymbol{Y}\_{2}, \quad \boldsymbol{\alpha}\_{2} = \boldsymbol{\phi}\_{2}^{-1}\boldsymbol{X}\_{2}\boldsymbol{Z}\_{1}^{-1}, \quad \boldsymbol{\theta}\_{2} = \boldsymbol{Z}\_{1}^{-1} + \boldsymbol{Z}\_{1}^{-1}\boldsymbol{Y}\_{2}\boldsymbol{\alpha}\_{2} \tag{24}
$$
$$
\boldsymbol{\phi\_3} = \boldsymbol{W\_3} - \boldsymbol{X\_3} \boldsymbol{Z\_2}^{-1} \boldsymbol{Y\_{3'}} \quad \boldsymbol{\alpha\_3} = \boldsymbol{\phi\_3^{-1}} \boldsymbol{X\_3} \boldsymbol{Z\_2}^{-1} \; \prime \quad \boldsymbol{\theta\_3} = \boldsymbol{Z\_2}^{-1} + \boldsymbol{Z\_2}^{-1} \boldsymbol{Y\_3} \boldsymbol{\alpha\_3} \tag{25}
$$
From (21), the matrix assignments related to recursion 1 1 <sup>1</sup> ( ) *Z Z k k* produces the following intermediate blockwise matrix results:
$$\begin{aligned} Z\_1^{-1} \left( Z\_0^{-1} \right) = \tilde{\Omega}\_1^{-1} = \begin{bmatrix} \theta\_1^{-1} & -\alpha\_1 \\ -\theta\_1 Y\_1 W\_1^{-1} & \theta\_1 \end{bmatrix} = \begin{bmatrix} 1.5765 & 0.1235 & -0.0307 & 1.0005 \\ 0.1235 & 0.3332 & 0.1867 & -0.0348 \\ -0.0307 & 0.1867 & 0.4432 & -0.093 \\ 1.0005 & -0.0348 & -0.093 & 0.9191 \end{bmatrix} \prime \\\ Z\_2^{-1} \left( Z\_1^{-1} \right) = \tilde{\Omega}\_2^{-1} = \begin{bmatrix} \theta\_2^{-1} & -\alpha\_2 \\ \theta\_2^{-1} \theta\_2 W\_2^{-1} & \theta\_2 \end{bmatrix} \end{aligned}$$
from 1 1 <sup>1</sup> ( ) *Z Z k k* corresponds to <sup>1</sup> , and is further used for calculating + 8 1 T <sup>8</sup> h h . Moreover, notice that full-rank properties are always presented in matrices 0 *Z* , *W*<sup>1</sup> , *W*<sup>2</sup> , *W*<sup>3</sup> , 1 , <sup>2</sup> , <sup>3</sup> , <sup>1</sup> <sup>1</sup> *<sup>Z</sup>* , <sup>1</sup> <sup>2</sup> *<sup>Z</sup>* , and <sup>1</sup> <sup>3</sup> *<sup>Z</sup>* .
### **7. VLSI implementation aspects**
The arithmetic operations presented in the algorithm for computing <sup>+</sup> h can be implemented under a modular-iterative fashion towards a VLSI (Very Large Scale of Integration) design. The partition strategy comprised in (13-16) provides modularity, while (18-20) is naturally associated with iterativeness; recursion is just used for constructing matrix-blocks in (21). Several well-studied aspects aid to implement a further VLSI architecture [23-27] given the nature of the mathematical structure of the algorithm. For instance, systolic arrays [25-27] are a suitable choice for efficient, parallel-processing architectures concerning matrix multiplications-additions. Bidimensional processing arrays are typical architectural outcomes, whose design consist basically in interconnecting processing elements (PE) among different array layers. The configuration of each PE comes from projection or linear mapping techniques [25-27] derived from multiplications and additions presented in (18-20). Also, systolic arrays tend to concurrently perform arithmetic operations dealing with the matrix concatenated multiplications <sup>1</sup> *XZ Y kk k* <sup>1</sup> , 1 1 *k kk X Z* <sup>1</sup> , <sup>1</sup> *Z Y k kk* <sup>1</sup> , and <sup>1</sup> *kk k Y W* presented in (18-20). Consecutive additions inside every PE can be favourably implemented via Carry-Save-Adder (CSA) architectures [23-24], while multiplications may recur to Booth multipliers [23-24] in order to reduce latencies caused by adding acummulated partial products. Divisions presented in <sup>1</sup> *Wk* , <sup>1</sup> *<sup>Z</sup>*<sup>0</sup> , and <sup>1</sup> *k* can be built through regular shift-and-subtract modules or classic serial-parallel subtractors [23-24]; in fact, CORDIC (Coordinate Rotate Digital Computer) processors [23] are also employed and configured in order to solve numerical divisions. The aforementioned architectural aspects offer an attractive and alternative framework for consolidating an ultimate VLSI design for implementing the <sup>+</sup> h algorithm without compromising the overall system data throughput (intrinsicly related to operation frequencies) for it.
### **8. Conclusions**
This chapter presented the development of a novel iterative-recursive algorithm for computing a Left-Pseudoinverse (LPI) as a Generalized-Inverse for a MIMO channel matrix within a Rayleigh fading channel (RFC). The formulation of this algorithm consisted in the following step: i) first, structural properties for the MIMO channel matrix acquired permanent full-rank due to statistical properties of the RFC scenario; ii) second, Partition-Matrix Theory was applied allowing the generation of a block-matrix version of the MIMO channel matrix; iii) third, iterative addition-multiplication operations were applied at these matrix sub-blocks in order to construct blockwise sub-matrix inverses, and recursively reusing them for obtaining the LPI. For accomplishing this purpose, required mathematical background and MIMO systems concepts were provided for consolidating a solid scientific framework to understand the context of the problem this algorithm was attempting to solve. Proper functionality for this approach was validated through simulation-driven experiments, as well as providing an example of this operation. As an additional remark, some VLSI aspects and architectures were outlined for basically implementing arithmetic operations within the proposed LPI-based algorithm.
## **Author details**
160 Linear Algebra – Theorems and Applications
<sup>2</sup> *<sup>Z</sup>* , and <sup>1</sup>
**7. VLSI implementation aspects**
<sup>3</sup> *<sup>Z</sup>* .
1 , <sup>2</sup> , <sup>3</sup> , <sup>1</sup> <sup>1</sup> *<sup>Z</sup>* , <sup>1</sup>
<sup>1</sup> *Z Y k kk* <sup>1</sup>
, and <sup>1</sup>
**8. Conclusions**
Moreover, notice that full-rank properties are always presented in matrices 0 *Z* , *W*<sup>1</sup> , *W*<sup>2</sup> , *W*<sup>3</sup> ,
The arithmetic operations presented in the algorithm for computing <sup>+</sup> h can be implemented under a modular-iterative fashion towards a VLSI (Very Large Scale of Integration) design. The partition strategy comprised in (13-16) provides modularity, while (18-20) is naturally associated with iterativeness; recursion is just used for constructing matrix-blocks in (21). Several well-studied aspects aid to implement a further VLSI architecture [23-27] given the nature of the mathematical structure of the algorithm. For instance, systolic arrays [25-27] are a suitable choice for efficient, parallel-processing architectures concerning matrix multiplications-additions. Bidimensional processing arrays are typical architectural outcomes, whose design consist basically in interconnecting processing elements (PE) among different array layers. The configuration of each PE comes from projection or linear mapping techniques [25-27] derived from multiplications and additions presented in (18-20). Also, systolic arrays tend to concurrently perform arithmetic
operations dealing with the matrix concatenated multiplications <sup>1</sup> *XZ Y kk k* <sup>1</sup>
adding acummulated partial products. Divisions presented in <sup>1</sup> *Wk*
throughput (intrinsicly related to operation frequencies) for it.
favourably implemented via Carry-Save-Adder (CSA) architectures [23-24], while multiplications may recur to Booth multipliers [23-24] in order to reduce latencies caused by
built through regular shift-and-subtract modules or classic serial-parallel subtractors [23-24]; in fact, CORDIC (Coordinate Rotate Digital Computer) processors [23] are also employed and configured in order to solve numerical divisions. The aforementioned architectural aspects offer an attractive and alternative framework for consolidating an ultimate VLSI design for implementing the <sup>+</sup> h algorithm without compromising the overall system data
This chapter presented the development of a novel iterative-recursive algorithm for computing a Left-Pseudoinverse (LPI) as a Generalized-Inverse for a MIMO channel matrix within a Rayleigh fading channel (RFC). The formulation of this algorithm consisted in the following step: i) first, structural properties for the MIMO channel matrix acquired permanent full-rank due to statistical properties of the RFC scenario; ii) second, Partition-Matrix Theory was applied allowing the generation of a block-matrix version of the MIMO channel matrix; iii) third, iterative addition-multiplication operations were applied at these matrix sub-blocks in order to construct blockwise sub-matrix inverses, and recursively reusing them for obtaining the LPI. For accomplishing this purpose, required mathematical background and MIMO systems concepts were provided for consolidating a solid scientific framework to understand the context of the problem this algorithm was attempting to solve.
*kk k Y W* presented in (18-20). Consecutive additions inside every PE can be
, and <sup>1</sup> *k*
, <sup>1</sup> *<sup>Z</sup>*<sup>0</sup>
, 1 1 *k kk X Z* <sup>1</sup> ,
can be
P. Cervantes and L. F. González *Instituto Tecnológico y de Estudios Superiores de Monterrey, Campus Guadalajara, ITESM University, Mexico*
F. J. Ortiz and A. D. García *Instituto Tecnológico y de Estudios Superiores de Monterrey, Campus Estado de México, ITESM University, Mexico*
## **Acknowledgement**
This work was supported by CONACYT (National Council of Science and Technology) under the supervision, revision, and sponsorship of ITESM University (Instituto Tecnológico y de Estudios Superiores de Monterrey).
### **9. References**
- [9] Choi Y (2009) New Form of Block Matrix Inversion. International Conference on Advanced Intelligent Mechatronics. July 2009: 1952-1957.
- [10] Choi Y, and Cheong J. (2009) New Expressions of 2X2 Block Matrix Inversion and Their Application. IEEE Transactions on Automatic Control, vol. 54, no. 11. November 2009: 2648-2653.
- [11] Fontán FP, and Espiñera PM (2008) Modeling the Wireless Propagation Channel. Wiley. 268 p.
- [12] El-Hajjar M, and Hanzo L (2010) Multifunctional MIMO Systems: A Combined Diversity and Multiplexing Design Perspective. IEEE Wireless Communications. April 2010: 73-79.
- [13] Biglieri E, *et al* (2007) MIMO Wireless Communications. Cambridge University Press: United Kingdom. 344 p.
- [14] Jankiraman M (2004) Space-Time Codes and MIMO Systems. Artech House: United States. 327 p.
- [15] Biglieri E, Proakis J, and Shamai S (1998) Fading Channels: Information-Theoretic and Communications Aspects. IEEE Transactions on Information Theory, vol. 44, no. 6. October 1998: 2619-2692.
- [16] Almers P, Bonek E, Burr A, *et al* (2007) Survey of Channel and Radio Propagation Models for Wireless MIMO Systems. EURASIP Journal on Wireless Communications and Networking, vol. 2011, issue 1. January 2007: 19 p.
- [17] Golub GH, and Van Loan CF (1996) Matrix Computations. The Johns Hopkins University Press. 694 p.
- [18] Serre D (2001) Matrices: Theory and Applications. Springer Verlag. 202 p.
- [19] R&S®. Rohde & Schwarz GmbH & Co. KG. WLAN 802.11n: From SISO to MIMO. Application Note: 1MA179\_9E. Available: www.rohde-schwarz.com: 59 p.
- [20] © Agilent Technologies, Inc. (2008) Agilent MIMO Wireless LAN PHY Layer [RF] : Operation & Measurement: Application Note: 1509. Available: www.agilent.com: 48 p.
- [21] Paul T, and Ogunfunmi T (2008) Wireless LAN Comes of Age : Understanding the IEEE 802.11n Amendment. IEEE Circuits and Systems Magazine. First Quarter 2008: 28-54.
- [22] Cervantes P, González VM, and Mejía PA (2009) Left-Pseudoinverse MIMO Channel Matrix Computation. 19th International Conference on Electronics, Communications, and Computers (CONIELECOMP 2009). July 2009: 134-138.
- [23] Milos E, and Tomas L (2004) Digital Arithmetic. Morgan Kauffmann Publishers. 709 p.
- [24] Parhi KK (1999) VLSI Digital Signal Processing Systems: Design and Implementation. John Wiley & Sons. 784 p.
- [25] Song SW (1994) Systolic Algorithms: Concepts, Synthesis, and Evolution. Institute of Mathematics, University of Sao Paulo, Brazil. Available: http://www.ime.usp.br/~song/ papers/cimpa.pdf. DOI number: 10.1.1.160.4057: 40 p.
- [26] Kung SY (1985) VLSI Array Processors. IEEE ASSP Magazine. July 1985: 4-22.
- [27] Jagadish HV, Rao SK, and Kailath T (1987) Array Architectures for Iterative Algorithms. Proceedings of the IEEE, vol. 75, no. 9. September 1987: 1304-1321.
## **Operator Means and Applications**
### Pattrawut Chansangiam
162 Linear Algebra – Theorems and Applications
2648-2653.
2010: 73-79.
States. 327 p.
United Kingdom. 344 p.
October 1998: 2619-2692.
University Press. 694 p.
John Wiley & Sons. 784 p.
268 p.
[9] Choi Y (2009) New Form of Block Matrix Inversion. International Conference on
[10] Choi Y, and Cheong J. (2009) New Expressions of 2X2 Block Matrix Inversion and Their Application. IEEE Transactions on Automatic Control, vol. 54, no. 11. November 2009:
[11] Fontán FP, and Espiñera PM (2008) Modeling the Wireless Propagation Channel. Wiley.
[12] El-Hajjar M, and Hanzo L (2010) Multifunctional MIMO Systems: A Combined Diversity and Multiplexing Design Perspective. IEEE Wireless Communications. April
[13] Biglieri E, *et al* (2007) MIMO Wireless Communications. Cambridge University Press:
[14] Jankiraman M (2004) Space-Time Codes and MIMO Systems. Artech House: United
[15] Biglieri E, Proakis J, and Shamai S (1998) Fading Channels: Information-Theoretic and Communications Aspects. IEEE Transactions on Information Theory, vol. 44, no. 6.
[16] Almers P, Bonek E, Burr A, *et al* (2007) Survey of Channel and Radio Propagation Models for Wireless MIMO Systems. EURASIP Journal on Wireless Communications
[17] Golub GH, and Van Loan CF (1996) Matrix Computations. The Johns Hopkins
[19] R&S®. Rohde & Schwarz GmbH & Co. KG. WLAN 802.11n: From SISO to MIMO.
[20] © Agilent Technologies, Inc. (2008) Agilent MIMO Wireless LAN PHY Layer [RF] : Operation & Measurement: Application Note: 1509. Available: www.agilent.com: 48 p. [21] Paul T, and Ogunfunmi T (2008) Wireless LAN Comes of Age : Understanding the IEEE 802.11n Amendment. IEEE Circuits and Systems Magazine. First Quarter 2008: 28-54. [22] Cervantes P, González VM, and Mejía PA (2009) Left-Pseudoinverse MIMO Channel Matrix Computation. 19th International Conference on Electronics, Communications,
[23] Milos E, and Tomas L (2004) Digital Arithmetic. Morgan Kauffmann Publishers. 709 p. [24] Parhi KK (1999) VLSI Digital Signal Processing Systems: Design and Implementation.
[25] Song SW (1994) Systolic Algorithms: Concepts, Synthesis, and Evolution. Institute of Mathematics, University of Sao Paulo, Brazil. Available: http://www.ime.usp.br/~song/
[27] Jagadish HV, Rao SK, and Kailath T (1987) Array Architectures for Iterative Algorithms.
[26] Kung SY (1985) VLSI Array Processors. IEEE ASSP Magazine. July 1985: 4-22.
Proceedings of the IEEE, vol. 75, no. 9. September 1987: 1304-1321.
[18] Serre D (2001) Matrices: Theory and Applications. Springer Verlag. 202 p.
Application Note: 1MA179\_9E. Available: www.rohde-schwarz.com: 59 p.
Advanced Intelligent Mechatronics. July 2009: 1952-1957.
and Networking, vol. 2011, issue 1. January 2007: 19 p.
and Computers (CONIELECOMP 2009). July 2009: 134-138.
papers/cimpa.pdf. DOI number: 10.1.1.160.4057: 40 p.
Additional information is available at the end of the chapter
http://dx.doi.org/10.5772/46479
### **1. Introduction**
The theory of scalar means was developed since the ancient Greek by the Pythagoreans until the last century by many famous mathematicians. See the development of this subject in a survey article [24]. In Pythagorean school, various means are defined via the method of proportions (in fact, they are solutions of certain algebraic equations). The theory of matrix and operator means started from the presence of the notion of parallel sum as a tool for analyzing multi-port electrical networks in engineering; see [1]. Three classical means, namely, arithmetic mean, harmonic mean and geometric mean for matrices and operators are then considered, e.g., in [3, 4, 11, 12, 23]. These means play crucial roles in matrix and operator theory as tools for studying monotonicity and concavity of many interesting maps between algebras of operators; see the original idea in [3]. Another important mean in mathematics, namely the power mean, is considered in [6]. The parallel sum is characterized by certain properties in [22]. The parallel sum and these means share some common properties. This leads naturally to the definitions of the so-called connection and mean in a seminal paper [17]. This class of means cover many in-practice operator means. A major result of Kubo-Ando states that there are one-to-one correspondences between connections, operator monotone functions on the non-negative reals and finite Borel measures on the extended half-line. The mean theoretic approach has many applications in operator inequalities (see more information in Section 8), matrix and operator equations (see e.g. [2, 19]) and operator entropy. The concept of operator entropy plays an important role in mathematical physics. The *relative operator entropy* is defined in [13] for invertible positive operators *A*, *B* by
$$S(A|B) = A^{1/2} \log(A^{-1/2}BA^{-1/2})A^{1/2}.\tag{1}$$
In fact, this formula comes from the Kubo-Ando theory–*S*(·|·) is the connection corresponds to the operator monotone function *t* �→ log *t*. See more information in [7, Chapter IV] and its references.
In this chapter, we treat the theory of operator means by weakening the original definition of connection in such a way that the same theory is obtained. Moreover, there is a one-to-one correspondence between connections and finite Borel measures on the unit interval. Each connection can be regarded as a weighed series of weighed harmonic means. Hence, every mean in Kubo-Ando's sense corresponds to a probability Borel measure on the unit interval.
©2012 Chansangiam, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ©2012 Chansangiam, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Various characterizations of means are obtained; one of them is a usual property of scalar mean, namely, the betweenness property. We provide some new properties of abstract operator connections, involving operator monotonicity and concavity, which include specific operator means as special cases.
For benefits of readers, we provide the development of the theory of operator means. In Section 2, we setup basic notations and state some background about operator monotone functions which play important roles in the theory of operator means. In Section 3, we consider the parallel sum together with its physical interpretation in electrical circuits. The arithmetic mean, the geometric mean and the harmonic mean of positive operators are investigated and characterized in Section 4. The original definition of connection is improved in Section 5 in such a way that the same theory is obtained. In Section 6, several characterizations and examples of Kubo-Ando means are given. We provide some new properties of general operator connections, related to operator monotonicity and concavity, in Section 7. Many operator versions of classical inequalities are obtained via the mean-theoretic approach in Section 8.
## **2. Preliminaries**
Throughout, let *B*(H) be the von Neumann algebra of bounded linear operators acting on a Hilbert space <sup>H</sup>. Let *<sup>B</sup>*(H)*sa* be the real vector space of self-adjoint operators on <sup>H</sup>. Equip *<sup>B</sup>*(H) with a natural partial order as follows. For *<sup>A</sup>*, *<sup>B</sup>* <sup>∈</sup> *<sup>B</sup>*(H)*sa*, we write *<sup>A</sup> <sup>B</sup>* if *<sup>B</sup>* <sup>−</sup> *<sup>A</sup>* is a positive operator. The notation *<sup>T</sup>* <sup>∈</sup> *<sup>B</sup>*(H)<sup>+</sup> or *<sup>T</sup>* 0 means that *<sup>T</sup>* is a positive operator. The case that *<sup>T</sup>* 0 and *<sup>T</sup>* is invertible is denoted by *<sup>T</sup>* <sup>&</sup>gt; 0 or *<sup>T</sup>* <sup>∈</sup> *<sup>B</sup>*(H)++. Unless otherwise stated, every limit in *B*(H) is taken in the strong-operator topology. Write *An* → *A* to indicate that *An* converges strongly to *<sup>A</sup>*. If *An* is a sequence in *<sup>B</sup>*(H)*sa*, the expression *An* <sup>↓</sup> *<sup>A</sup>* means that *An* is a decreasing sequence and *An* → *A*. Similarly, *An* ↑ *A* tells us that *An* is increasing and *An* → *A*. We always reserve *A*, *B*, *C*, *D* for positive operators. The set of non-negative real numbers is denoted by **R**+.
**Remark 0.1.** It is important to note that if *An* is a decreasing sequence in *<sup>B</sup>*(H)*sa* such that *An A*, then *An* → *A* if and only if �*Anx*, *x*�→�*Ax*, *x*� for all *x* ∈ H. Note first that this sequence is convergent by the order completeness of *B*(H). For the sufficiency, if *x* ∈ H, then
$$\|(A\_{\mathfrak{n}} - A)^{1/2}\mathbf{x}\|^2 = \langle (A\_{\mathfrak{n}} - A)^{1/2}\mathbf{x}, (A\_{\mathfrak{n}} - A)^{1/2}\mathbf{x} \rangle = \langle (A\_{\mathfrak{n}} - A)\mathbf{x}, \mathbf{x} \rangle \to 0$$
and hence �(*An* − *A*)*x*� → 0.
The spectrum of *T* ∈ *B*(H) is defined by
Sp(*T*) = {*λ* ∈ **C** : *T* − *λI* is not invertible}.
Then Sp(*T*) is a nonempty compact Hausdorff space. Denote by *C*(Sp(*T*)) the C∗-algebra of continuous functions from Sp(*T*) to **C**. Let *T* ∈ *B*(H) be a normal operator and *z* : Sp(*T*) → **C** the inclusion. Then there exists a unique unital ∗-homomorphism *φ* : *C*(Sp(*T*)) → *B*(H) such that *φ*(*z*) = *T*, i.e.,
• *φ*( ¯ *f*)=(*φ*(*f*))<sup>∗</sup> for all *f* ∈ *C*(Sp(*T*))
$$\bullet \quad \phi(1) = I.$$
2 Will-be-set-by-IN-TECH
Various characterizations of means are obtained; one of them is a usual property of scalar mean, namely, the betweenness property. We provide some new properties of abstract operator connections, involving operator monotonicity and concavity, which include specific
For benefits of readers, we provide the development of the theory of operator means. In Section 2, we setup basic notations and state some background about operator monotone functions which play important roles in the theory of operator means. In Section 3, we consider the parallel sum together with its physical interpretation in electrical circuits. The arithmetic mean, the geometric mean and the harmonic mean of positive operators are investigated and characterized in Section 4. The original definition of connection is improved in Section 5 in such a way that the same theory is obtained. In Section 6, several characterizations and examples of Kubo-Ando means are given. We provide some new properties of general operator connections, related to operator monotonicity and concavity, in Section 7. Many operator versions of classical inequalities are obtained via the mean-theoretic
Throughout, let *B*(H) be the von Neumann algebra of bounded linear operators acting on a Hilbert space <sup>H</sup>. Let *<sup>B</sup>*(H)*sa* be the real vector space of self-adjoint operators on <sup>H</sup>. Equip *<sup>B</sup>*(H) with a natural partial order as follows. For *<sup>A</sup>*, *<sup>B</sup>* <sup>∈</sup> *<sup>B</sup>*(H)*sa*, we write *<sup>A</sup> <sup>B</sup>* if *<sup>B</sup>* <sup>−</sup> *<sup>A</sup>* is a positive operator. The notation *<sup>T</sup>* <sup>∈</sup> *<sup>B</sup>*(H)<sup>+</sup> or *<sup>T</sup>* 0 means that *<sup>T</sup>* is a positive operator. The case that *<sup>T</sup>* 0 and *<sup>T</sup>* is invertible is denoted by *<sup>T</sup>* <sup>&</sup>gt; 0 or *<sup>T</sup>* <sup>∈</sup> *<sup>B</sup>*(H)++. Unless otherwise stated, every limit in *B*(H) is taken in the strong-operator topology. Write *An* → *A* to indicate that *An* converges strongly to *<sup>A</sup>*. If *An* is a sequence in *<sup>B</sup>*(H)*sa*, the expression *An* <sup>↓</sup> *<sup>A</sup>* means that *An* is a decreasing sequence and *An* → *A*. Similarly, *An* ↑ *A* tells us that *An* is increasing and *An* → *A*. We always reserve *A*, *B*, *C*, *D* for positive operators. The set of non-negative
**Remark 0.1.** It is important to note that if *An* is a decreasing sequence in *<sup>B</sup>*(H)*sa* such that *An A*, then *An* → *A* if and only if �*Anx*, *x*�→�*Ax*, *x*� for all *x* ∈ H. Note first that this sequence is convergent by the order completeness of *B*(H). For the sufficiency, if *x* ∈ H, then
�(*An* <sup>−</sup> *<sup>A</sup>*)1/2*x*�<sup>2</sup> <sup>=</sup> �(*An* <sup>−</sup> *<sup>A</sup>*)1/2*x*,(*An* <sup>−</sup> *<sup>A</sup>*)1/2*x*� <sup>=</sup> �(*An* <sup>−</sup> *<sup>A</sup>*)*x*, *<sup>x</sup>*� → <sup>0</sup>
Sp(*T*) = {*λ* ∈ **C** : *T* − *λI* is not invertible}. Then Sp(*T*) is a nonempty compact Hausdorff space. Denote by *C*(Sp(*T*)) the C∗-algebra of continuous functions from Sp(*T*) to **C**. Let *T* ∈ *B*(H) be a normal operator and *z* : Sp(*T*) → **C** the inclusion. Then there exists a unique unital ∗-homomorphism *φ* : *C*(Sp(*T*)) → *B*(H) such
operator means as special cases.
approach in Section 8.
**2. Preliminaries**
real numbers is denoted by **R**+.
and hence �(*An* − *A*)*x*� → 0.
that *φ*(*z*) = *T*, i.e.,
• *φ* is linear
The spectrum of *T* ∈ *B*(H) is defined by
• *φ*(*f g*) = *φ*(*f*)*φ*(*g*) for all *f* , *g* ∈ *C*(Sp(*T*))
Moreover, *φ* is isometric. We call the unique isometric ∗-homomorphism which sends *f* ∈ *C*(Sp(*T*)) to *φ*(*f*) ∈ *B*(H) the *continuous functional calculus* of *T*. We write *f*(*T*) for *φ*(*f*).
A continuous real-valued function *f* on an interval *I* is called an *operator monotone function* if one of the following equivalent conditions holds:
This concept is introduced in [20]; see also [7, 10, 15, 16]. Every operator monotone function is always continuously differentiable and monotone increasing. Here are examples of operator monotone functions:
The next result is called the Löwner-Heinz's inequality [20].
**Theorem 0.3.** *For A*, *<sup>B</sup>* <sup>∈</sup> *<sup>B</sup>*(H)<sup>+</sup> *and r* <sup>∈</sup> [0, 1]*, if A B, then A<sup>r</sup> <sup>B</sup>r. That is the map t* �→ *<sup>t</sup> <sup>r</sup> is an operator monotone function on* **<sup>R</sup>**<sup>+</sup> *for any r* <sup>∈</sup> [0, 1]*.*
A key result about operator monotone functions is that there is a one-to-one correspondence between nonnegative operator monotone functions on **R**<sup>+</sup> and finite Borel measures on [0, ∞] via integral representations. We give a variation of this result in the next proposition.
**Proposition 0.4.** *A continuous function f* : **<sup>R</sup>**<sup>+</sup> <sup>→</sup> **<sup>R</sup>**<sup>+</sup> *is operator monotone if and only if there exists a finite Borel measure μ on* [0, 1] *such that*
$$f(\mathbf{x}) = \int\_{[0,1]} \mathbf{1} \mathbf{1}\_t \mathbf{x} \, d\mu(t), \quad \mathbf{x} \in \mathbb{R}^+. \tag{2}$$
*Here, the weighed harmonic mean* !*<sup>t</sup> is defined for a*, *b* > 0 *by*
$$a \, !\_{t} b = [(1 - t)a^{-1} + tb^{-1}]^{-1} \tag{3}$$
*and extended to a*, *b* 0 *by continuity. Moreover, the measure μ is unique. Hence, there is a one-to-one correspondence between operator monotone functions on the non-negative reals and finite Borel measures on the unit interval.*
*Proof.* Recall that a continuous function *<sup>f</sup>* : **<sup>R</sup>**<sup>+</sup> <sup>→</sup> **<sup>R</sup>**<sup>+</sup> is operator monotone if and only if there exists a unique finite Borel measure *ν* on [0, ∞] such that
$$f(\mathbf{x}) = \int\_{[0,\infty]} \phi\_{\mathbf{x}}(\lambda) \, d\nu(\lambda), \quad \mathbf{x} \in \mathbb{R}^+$$
where
$$\phi\_{\mathfrak{X}}(\lambda) = \frac{\mathfrak{x}(\lambda + 1)}{\mathfrak{x} + \lambda} \text{ for } \lambda > 0, \quad \phi\_{\mathfrak{X}}(0) = 1, \quad \phi\_{\mathfrak{X}}(\infty) = \infty.$$
Consider the Borel measurable function *<sup>ψ</sup>* : [0, 1] <sup>→</sup> [0, <sup>∞</sup>], *<sup>t</sup>* �→ *<sup>t</sup>* <sup>1</sup>−*<sup>t</sup>* . Then, for each *<sup>x</sup>* <sup>∈</sup> **<sup>R</sup>**+,
$$\begin{aligned} \int\_{[0,\infty]} \phi\_{\mathfrak{x}}(\lambda) \, d\nu(\lambda) &= \int\_{[0,1]} \phi\_{\mathfrak{x}} \circ \psi(t) \, d\nu \psi(t) \\ &= \int\_{[0,1]} \frac{\mathfrak{x}}{\mathfrak{x} - \mathfrak{x}t + t} \, d\nu \psi(t) \\ &= \int\_{[0,1]} 1 \, !\_{\mathfrak{f}} \, d\nu \psi(t) .\end{aligned}$$
Now, set *μ* = *νψ*. Since *ψ* is bijective, there is a one-to-one corresponsence between the finite Borel measures on [0, ∞] of the form *ν* and the finite Borel measures on [0, 1] of the form *νψ*. The map *f* �→ *μ* is clearly well-defined and bijective.
#### **3. Parallel sum: A notion from electrical networks**
In connections with electrical engineering, Anderson and Duffin [1] defined the *parallel sum* of two positive definite matrices *A* and *B* by
$$A:B=(A^{-1}+B^{-1})^{-1}.\tag{4}$$
The impedance of an electrical network can be represented by a positive (semi)definite matrix. If *A* and *B* are impedance matrices of multi-port networks, then the parallel sum *A* : *B* indicates the total impedance of two electrical networks connected in parallel. This notion plays a crucial role for analyzing multi-port electrical networks because many physical interpretations of electrical circuits can be viewed in a form involving parallel sums. This is a starting point of the study of matrix and operator means. This notion can be extended to invertible positive operators by the same formula.
**Lemma 0.5.** *Let A*, *<sup>B</sup>*, *<sup>C</sup>*, *<sup>D</sup>*, *An*, *Bn* <sup>∈</sup> *<sup>B</sup>*(H)++ *for all n* <sup>∈</sup> **<sup>N</sup>***.*
*(1) If An* <sup>↓</sup> *A, then A*−<sup>1</sup> *<sup>n</sup>* <sup>↑</sup> *<sup>A</sup>*−1*. If An* <sup>↑</sup> *A, then A*−<sup>1</sup> *<sup>n</sup>* <sup>↓</sup> *<sup>A</sup>*−1*.*
*Proof.* (1) Assume *An* <sup>↓</sup> *<sup>A</sup>*. Then *<sup>A</sup>*−<sup>1</sup> *<sup>n</sup>* is increasing and, for each *<sup>x</sup>* ∈ H,
$$\langle (A\_{\mathfrak{n}}^{-1} - A^{-1})\mathbf{x}, \mathbf{x} \rangle = \langle (A - A\_{\mathfrak{n}})A^{-1}\mathbf{x}, A\_{\mathfrak{n}}^{-1}\mathbf{x} \rangle \lesssim \|(A - A\_{\mathfrak{n}})A^{-1}\mathbf{x}\| \|A\_{\mathfrak{n}}^{-1}\| \|\mathbf{x}\| \to 0.$$
(2) Follow from (1).
4 Will-be-set-by-IN-TECH
*<sup>a</sup>* !*<sup>t</sup> <sup>b</sup>* = [(<sup>1</sup> <sup>−</sup> *<sup>t</sup>*)*a*−<sup>1</sup> <sup>+</sup> *tb*−1]
*and extended to a*, *b* 0 *by continuity. Moreover, the measure μ is unique. Hence, there is a one-to-one correspondence between operator monotone functions on the non-negative reals and finite*
*Proof.* Recall that a continuous function *<sup>f</sup>* : **<sup>R</sup>**<sup>+</sup> <sup>→</sup> **<sup>R</sup>**<sup>+</sup> is operator monotone if and only if
[0,1]
= [0,1]
= [0,1]
Now, set *μ* = *νψ*. Since *ψ* is bijective, there is a one-to-one corresponsence between the finite Borel measures on [0, ∞] of the form *ν* and the finite Borel measures on [0, 1] of the form *νψ*.
In connections with electrical engineering, Anderson and Duffin [1] defined the *parallel sum* of
The impedance of an electrical network can be represented by a positive (semi)definite matrix. If *A* and *B* are impedance matrices of multi-port networks, then the parallel sum *A* : *B* indicates the total impedance of two electrical networks connected in parallel. This notion plays a crucial role for analyzing multi-port electrical networks because many physical interpretations of electrical circuits can be viewed in a form involving parallel sums. This is a starting point of the study of matrix and operator means. This notion can be extended to
*<sup>φ</sup>x*(*λ*) *<sup>d</sup>ν*(*λ*), *<sup>x</sup>* <sup>∈</sup> **<sup>R</sup>**<sup>+</sup>
*<sup>x</sup>* <sup>+</sup> *<sup>λ</sup>* for *<sup>λ</sup>* <sup>&</sup>gt; 0, *<sup>φ</sup>x*(0) = 1, *<sup>φ</sup>x*(∞) = *<sup>x</sup>*.
*φ<sup>x</sup>* ◦ *ψ*(*t*) *dνψ*(*t*)
*dνψ*(*t*)
*A* : *B* = (*A*−<sup>1</sup> + *B*−1)<sup>−</sup>1. (4)
*x x* − *xt* + *t*
1 !*<sup>t</sup> x dνψ*(*t*).
<sup>−</sup><sup>1</sup> (3)
<sup>1</sup>−*<sup>t</sup>* . Then, for each *<sup>x</sup>* <sup>∈</sup> **<sup>R</sup>**+,
*Here, the weighed harmonic mean* !*<sup>t</sup> is defined for a*, *b* > 0 *by*
there exists a unique finite Borel measure *ν* on [0, ∞] such that
*<sup>φ</sup>x*(*λ*) = *<sup>x</sup>*(*<sup>λ</sup>* <sup>+</sup> <sup>1</sup>)
[0,∞]
The map *f* �→ *μ* is clearly well-defined and bijective.
invertible positive operators by the same formula.
**Lemma 0.5.** *Let A*, *<sup>B</sup>*, *<sup>C</sup>*, *<sup>D</sup>*, *An*, *Bn* <sup>∈</sup> *<sup>B</sup>*(H)++ *for all n* <sup>∈</sup> **<sup>N</sup>***.*
*(1) If An* <sup>↓</sup> *A, then A*−<sup>1</sup> *<sup>n</sup>* <sup>↑</sup> *<sup>A</sup>*−1*. If An* <sup>↑</sup> *A, then A*−<sup>1</sup> *<sup>n</sup>* <sup>↓</sup> *<sup>A</sup>*−1*.*
two positive definite matrices *A* and *B* by
**3. Parallel sum: A notion from electrical networks**
*f*(*x*) =
Consider the Borel measurable function *<sup>ψ</sup>* : [0, 1] <sup>→</sup> [0, <sup>∞</sup>], *<sup>t</sup>* �→ *<sup>t</sup>*
[0,∞]
*φx*(*λ*) *dν*(*λ*) =
*Borel measures on the unit interval.*
where
(3) Let *An*, *Bn* <sup>∈</sup> *<sup>B</sup>*(H)++ be such that *An* <sup>↓</sup> *<sup>A</sup>* and *Bn* <sup>↓</sup> *<sup>A</sup>* where *<sup>A</sup>*, *<sup>B</sup>* <sup>&</sup>gt; 0. Then *<sup>A</sup>*−<sup>1</sup> *<sup>n</sup>* <sup>↑</sup> *<sup>A</sup>*−<sup>1</sup> and *<sup>B</sup>*−<sup>1</sup> *<sup>n</sup>* <sup>↑</sup> *<sup>B</sup>*−1. So, *<sup>A</sup>*−<sup>1</sup> *<sup>n</sup>* <sup>+</sup> *<sup>B</sup>*−<sup>1</sup> *<sup>n</sup>* is an increasing sequence in *<sup>B</sup>*(H)<sup>+</sup> such that
$$A\_n^{-1} + B\_n^{-1} \to A^{-1} + B^{-1}\prime$$
i.e. *<sup>A</sup>*−<sup>1</sup> *<sup>n</sup>* <sup>+</sup> *<sup>B</sup>*−<sup>1</sup> *<sup>n</sup>* <sup>↑</sup> *<sup>A</sup>*−<sup>1</sup> <sup>+</sup> *<sup>B</sup>*−1. By (1), we thus have (*A*−<sup>1</sup> *<sup>n</sup>* <sup>+</sup> *<sup>B</sup>*−<sup>1</sup> *<sup>n</sup>* )−<sup>1</sup> <sup>↓</sup> (*A*−<sup>1</sup> <sup>+</sup> *<sup>B</sup>*−1)−1.
(4) Let *An*, *Bn* <sup>∈</sup> *<sup>B</sup>*(H)++ be such that *An* <sup>↓</sup> *<sup>A</sup>* and *Bn* <sup>↓</sup> *<sup>B</sup>*. Then, by (2), *An* : *Bn* is a decreasing sequence of positive operators. The order completeness of *B*(H) guaruntees the existence of the strong limit of *An* : *Bn*. Let *A*� *<sup>n</sup>* and *B*� *<sup>n</sup>* be another sequences such that *A*� *<sup>n</sup>* ↓ *A* and *B*� *<sup>n</sup>* ↓ *B*. Note that for each *n*, *m* ∈ **N**, we have *An An* + *A*� *<sup>m</sup>* − *A* and *Bn Bn* + *B*� *<sup>m</sup>* − *B*. Then
$$A\_{\mathfrak{n}} : B\_{\mathfrak{n}} \leqslant \left( A\_{\mathfrak{n}} + A\_{\mathfrak{m}}' - A \right) : (B\_{\mathfrak{n}} + B\_{\mathfrak{m}}' - B).$$
Note that as *n* → ∞, *An* + *A*� *<sup>m</sup>* − *A* → *A*� *<sup>m</sup>* and *Bn* + *B*� *<sup>m</sup>* − *B* → *B*� *<sup>m</sup>*. We have that as *n* → ∞,
> (*An* + *A*� *<sup>m</sup>* − *A*) : (*Bn* + *B*� *<sup>m</sup>* − *B*) → *A*� *<sup>m</sup>* : *B*� *m*.
Hence, lim*n*→<sup>∞</sup> *An* : *Bn A*� *<sup>m</sup>* : *B*� *<sup>m</sup>* and lim*n*→<sup>∞</sup> *An* : *Bn* lim*m*→<sup>∞</sup> *A*� *<sup>m</sup>* : *B*� *<sup>m</sup>*. By symmetry, lim*n*→<sup>∞</sup> *An* : *Bn* lim*m*→<sup>∞</sup> *A*� *<sup>m</sup>* : *B*� *m*.
We define the *parallel sum* of *A*, *B* 0 to be
$$A:B = \lim\_{\epsilon \downarrow 0} (A + \epsilon I):(B + \epsilon I) \tag{5}$$
where the limit is taken in the strong-operator topology.
**Lemma 0.6.** *For each x* ∈ H*,*
$$<\langle (A:B)\mathbf{x}, \mathbf{x} \rangle = \inf \{ \langle Ay, y \rangle + \langle Bz, z \rangle : y, z \in \mathcal{H}, y + z = \mathbf{x} \}. \tag{6}$$
*Proof.* First, assume that *A*, *B* are invertible. Then for all *x*, *y* ∈ H,
$$\begin{aligned} &\langle Ay, y \rangle + \langle B(x - y), x - y \rangle - \langle (A : B)x, x \rangle \\ &= \langle Ay, y \rangle + \langle Bx, x \rangle - 2\text{Re}\langle Bx, y \rangle + \langle By, y \rangle - \langle (B - B(A + B)^{-1}B)x, x \rangle \\ &= \langle (A + B)y, y \rangle - 2\text{Re}\langle Bx, y \rangle + \langle (A + B)^{-1}Bx, Bx \rangle \\ &= \|(A + B)^{1/2}y\|^2 - 2\text{Re}\langle Bx, y \rangle + \|(A + B)^{-1/2}Bx\|^2 \\ &\ge 0. \end{aligned}$$
With *y* = (*A* + *B*)−1*Bx*, we have
�*Ay*, *y*� + �*B*(*x* − *y*), *x* − *y*�−�(*A* : *B*)*x*, *x*� = 0.
Hence, we have the claim for *A*, *B* > 0. For *A*, *B* 0, consider *A* + *I* and *B* + *I* where ↓ 0.
**Remark 0.7.** This lemma has a physical interpretation, called the *Maxwell's minimum power principle*. Recall that a positive operator represents the impedance of a electrical network while the power dissipation of network with impedance *A* and current *x* is the inner product �*Ax*, *x*�. Consider two electrical networks connected in parallel. For a given current input *x*, the current will divide *x* = *y* + *z*, where *y* and *z* are currents of each network, in such a way that the power dissipation is minimum.
**Theorem 0.8.** *The parallel sum satisfies*
*Proof.* (1) The monotonicity follows from the formula (5) and Lemma 0.5(2).
(2) For each *x*, *y*, *z* ∈ H such that *x* = *y* + *z*, by the previous lemma,
$$
\begin{aligned}
\langle \mathcal{S}^\*(A:B)Sx, x \rangle &= \langle (A:B)Sx, Sx \rangle \\
&\le \langle ASy, Sy \rangle + \langle S^\*BSz, z \rangle \\
&= \langle \mathcal{S}^\*ASy, y \rangle + \langle \mathcal{S}^\*BSz, z \rangle.
\end{aligned}
$$
Again, the previous lemma assures *S*∗(*A* : *B*)*S* (*S*∗*AS*) : (*S*∗*BS*).
(3) Let *An* and *Bn* be decreasing sequences in *<sup>B</sup>*(H)<sup>+</sup> such that *An* <sup>↓</sup> *<sup>A</sup>* and *Bn* <sup>↓</sup> *<sup>B</sup>*. Then *An* : *Bn* is decreasing and *A* : *B An* : *Bn* for all *n* ∈ **N**. We have that, by the joint monotonicity of parallel sum, for all > 0
$$A\_n: B\_n \leqslant (A\_n + \epsilon I): (B\_n + \epsilon I).$$
Since *An* + *I* ↓ *A* + *I* and *Bn* + *I* ↓ *B* + *I*, by Lemma 3.1.4(3) we have *An* : *Bn* ↓ *A* : *B*.
**Remark 0.9.** The positive operator *S*∗*AS* represents the impedance of a network connected to a transformer. The transformer inequality means that the impedance of parallel connection with transformer first is greater than that with transformer last.
**Proposition 0.10.** *The set of positive operators on* H *is a partially ordered commutative semigroup with respect to the parallel sum.*
*Proof.* For *A*, *B*, *C* > 0, we have (*A* : *B*) : *C* = *A* : (*B* : *C*) and *A* : *B* = *B* : *A*. The continuity from above in Theorem 0.8 implies that (*A* : *B*) : *C* = *A* : (*B* : *C*) and *A* : *B* = *B* : *A* for all *A*, *B*, *C* 0. The monotonicity of the parallel sum means that the positive operators form a partially ordered semigroup.
**Theorem 0.11.** *For A*, *B*, *C*, *D* 0*, we have the series-parallel inequality*
$$(A+B):(\mathbb{C}+D)\geqslant A:\mathbb{C}+B:D.\tag{7}$$
*In other words, the parallel sum is concave.*
6 Will-be-set-by-IN-TECH
�*Ay*, *y*� + �*B*(*x* − *y*), *x* − *y*�−�(*A* : *B*)*x*, *x*� = 0. Hence, we have the claim for *A*, *B* > 0. For *A*, *B* 0, consider *A* + *I* and *B* + *I* where
**Remark 0.7.** This lemma has a physical interpretation, called the *Maxwell's minimum power principle*. Recall that a positive operator represents the impedance of a electrical network while the power dissipation of network with impedance *A* and current *x* is the inner product �*Ax*, *x*�. Consider two electrical networks connected in parallel. For a given current input *x*, the current will divide *x* = *y* + *z*, where *y* and *z* are currents of each network, in such a way that the power
With *y* = (*A* + *B*)−1*Bx*, we have
dissipation is minimum.
**Theorem 0.8.** *The parallel sum satisfies*
monotonicity of parallel sum, for all > 0
*with respect to the parallel sum.*
partially ordered semigroup.
*(1) monotonicity: A*<sup>1</sup> *A*2, *B*<sup>1</sup> *B*<sup>2</sup> ⇒ *A*<sup>1</sup> : *B*<sup>1</sup> *A*<sup>2</sup> : *B*2*.*
*(2) transformer inequality: S*∗(*A* : *B*)*S* (*S*∗*AS*) : (*S*∗*BS*) *for every S* ∈ *B*(H)*.*
*Proof.* (1) The monotonicity follows from the formula (5) and Lemma 0.5(2).
�*S*∗(*A* : *B*)*Sx*, *x*� = �(*A* : *B*)*Sx*, *Sx*�
(3) Let *An* and *Bn* be decreasing sequences in *<sup>B</sup>*(H)<sup>+</sup> such that *An* <sup>↓</sup> *<sup>A</sup>* and *Bn* <sup>↓</sup> *<sup>B</sup>*. Then *An* : *Bn* is decreasing and *A* : *B An* : *Bn* for all *n* ∈ **N**. We have that, by the joint
*An* : *Bn* (*An* + *I*) : (*Bn* + *I*).
**Remark 0.9.** The positive operator *S*∗*AS* represents the impedance of a network connected to a transformer. The transformer inequality means that the impedance of parallel connection
**Proposition 0.10.** *The set of positive operators on* H *is a partially ordered commutative semigroup*
*Proof.* For *A*, *B*, *C* > 0, we have (*A* : *B*) : *C* = *A* : (*B* : *C*) and *A* : *B* = *B* : *A*. The continuity from above in Theorem 0.8 implies that (*A* : *B*) : *C* = *A* : (*B* : *C*) and *A* : *B* = *B* : *A* for all *A*, *B*, *C* 0. The monotonicity of the parallel sum means that the positive operators form a
Since *An* + *I* ↓ *A* + *I* and *Bn* + *I* ↓ *B* + *I*, by Lemma 3.1.4(3) we have *An* : *Bn* ↓ *A* : *B*.
�*ASy*, *Sy*� + �*S*∗*BSz*, *z*� = �*S*∗*ASy*, *y*� + �*S*∗*BSz*, *z*�.
*(3) continuity from above: if An* ↓ *A and Bn* ↓ *B, then An* : *Bn* ↓ *A* : *B.*
(2) For each *x*, *y*, *z* ∈ H such that *x* = *y* + *z*, by the previous lemma,
Again, the previous lemma assures *S*∗(*A* : *B*)*S* (*S*∗*AS*) : (*S*∗*BS*).
with transformer first is greater than that with transformer last.
↓ 0.
*Proof.* For each *x*, *y*, *z* ∈ H such that *x* = *y* + *z*, we have by the previous lemma that
$$
\begin{aligned}
\langle (A:\mathbb{C} + B:D)\mathbf{x}, \mathbf{x} \rangle &= \langle (A:\mathbb{C})\mathbf{x}, \mathbf{x} \rangle + \langle (B:D)\mathbf{x}, \mathbf{x} \rangle \\
&\leqslant \langle Ay, y \rangle + \langle \mathbb{C}z, z \rangle + \langle By, y \rangle + \langle Dz, z \rangle \\
&= \langle (A + B)y, y \rangle + \langle (\mathbb{C} + D)z, z \rangle.
\end{aligned}
$$
Applying the previous lemma yields (*A* + *B*) : (*C* + *D*) *A* : *C* + *B* : *D*.
**Remark 0.12.** The ordinary sum of operators represents the total impedance of two networks with series connection while the parallel sum indicates the total impedance of two networks with parallel connection. So, the series-parallel inequality means that the impedance of a series-parallel connection is greater than that of a parallel-series connection.
#### **4. Classical means: arithmetic, harmonic and geometric means**
Some desired properties of any object that is called a "mean" *<sup>M</sup>* on *<sup>B</sup>*(H)<sup>+</sup> should have are given here.
(A4). *transformer inequality*: *X*∗*M*(*A*, *B*)*X M*(*X*∗*AX*, *X*∗*BX*) for *X* ∈ *B*(H);
In order to study matrix or operator means in general, the first step is to consider three classical means in mathematics, namely, arithmetic, geometric and harmonic means.
The *arithmetic mean* of *<sup>A</sup>*, *<sup>B</sup>* <sup>∈</sup> *<sup>B</sup>*(H)<sup>+</sup> is defined by
$$A \sqcap B = \frac{1}{2}(A+B). \tag{8}$$
Then the arithmetic mean satisfies the properties (A1)–(A9). In fact, the properties (A5) and (A6) can be replaced by a stronger condition:
*X*∗*M*(*A*, *B*)*X* = *M*(*X*∗*AX*, *X*∗*BX*) for all *X* ∈ *B*(H).
Moreover, the arithmetic mean satisfies
*affinity*: *<sup>M</sup>*(*kA* <sup>+</sup> *<sup>C</sup>*, *kB* <sup>+</sup> *<sup>C</sup>*) = *kM*(*A*, *<sup>B</sup>*) + *<sup>C</sup>* for *<sup>k</sup>* <sup>∈</sup> **<sup>R</sup>**+.
Define the *harmonic mean* of positive operators *<sup>A</sup>*, *<sup>B</sup>* <sup>∈</sup> *<sup>B</sup>*(H)<sup>+</sup> by
$$A \, !B = 2(A : B) = \lim\_{\epsilon \downarrow 0} 2(A\_{\epsilon}^{-1} + B\_{\epsilon}^{-1})^{-1} \tag{9}$$
where *A* ≡ *A* + *I* and *B* ≡ *B* + *I*. Then the harmonic mean satisfies the properties (A1)–(A9).
The geometric mean of matrices is defined in [23] and studied in details in [3]. A usage of congruence transformations for treating geometric means is given in [18]. For a given invertible operator *C* ∈ *B*(H), define
$$
\Gamma\_{\mathbb{C}} : B(\mathcal{H})^{sa} \to B(\mathcal{H})^{sa}, A \mapsto \mathbb{C}^\* A \mathbb{C}.
$$
Then each Γ*<sup>C</sup>* is a linear isomorphism with inverse Γ*C*−<sup>1</sup> and is called a *congruence transformation*. The set of congruence transformations is a group under multiplication. Each congruence transformation preserves positivity, invertibility and, hence, strictly positivity. In fact, <sup>Γ</sup>*<sup>C</sup>* maps *<sup>B</sup>*(H)<sup>+</sup> and *<sup>B</sup>*(H)++ onto themselves. Note also that <sup>Γ</sup>*<sup>C</sup>* is order-preserving.
Define the *geometric mean* of *A*, *B* > 0 by
$$A \# B = A^{1/2} (A^{-1/2} B A^{-1/2})^{1/2} A^{1/2} = \Gamma\_{A^{1/2}} \circ \Gamma\_{A^{-1/2}}^{1/2} (B). \tag{10}$$
Then *A* # *B* > 0 for *A*, *B* > 0. This formula comes from two natural requirements: This definition should coincide with the usual geometric mean in **R**+: *A* # *B* = (*AB*)1/2 provided that *AB* = *BA*. The second condition is that, for any invertible *T* ∈ *B*(H),
$$T^\*(A \# B)T = (T^\*AT) \# (T^\*BT). \tag{11}$$
The next theorem characterizes the geometric mean of *A* and *B* in term of the solution of a certain operator equation.
**Theorem 0.13.** *For each A*, *B* > 0*, the Riccati equation* Γ*X*(*A*−1) := *XA*−1*X* = *B has a unique positive solution, namely, X* = *A* # *B.*
*Proof.* The direct computation shows that (*A* # *B*)*A*−1(*A* # *B*) = *B*. Suppose there is another positive solution *Y* 0. Then
$$(A^{-1/2}XA^{-1/2})^2 = A^{-1/2}XA^{-1}XA^{-1/2} = A^{-1/2}YA^{-1}YA^{-1/2} = (A^{-1/2}YA^{-1/2})^2.$$
The uniqueness of positive square roots implies that *A*−1/2*XA*−1/2 = *A*−1/2*YA*−1/2, i.e., *X* = *Y*.
**Theorem 0.14** (Maximum property of geometric mean)**.** *For A*, *B* > 0*,*
$$A \# B = \max \{ X \geqslant 0 \, : \, XA^{-1}X \leqslant B \} \tag{12}$$
*where the maximum is taken with respect to the positive semidefinite ordering.*
*Proof.* If *XA*−1*X B*, then
8 Will-be-set-by-IN-TECH
↓0
where *A* ≡ *A* + *I* and *B* ≡ *B* + *I*. Then the harmonic mean satisfies the properties
The geometric mean of matrices is defined in [23] and studied in details in [3]. A usage of congruence transformations for treating geometric means is given in [18]. For a given
<sup>Γ</sup>*<sup>C</sup>* : *<sup>B</sup>*(H)*sa* <sup>→</sup> *<sup>B</sup>*(H)*sa*, *<sup>A</sup>* �→ *<sup>C</sup>*∗*AC*.
Then each Γ*<sup>C</sup>* is a linear isomorphism with inverse Γ*C*−<sup>1</sup> and is called a *congruence transformation*. The set of congruence transformations is a group under multiplication. Each congruence transformation preserves positivity, invertibility and, hence, strictly positivity. In fact, <sup>Γ</sup>*<sup>C</sup>* maps *<sup>B</sup>*(H)<sup>+</sup> and *<sup>B</sup>*(H)++ onto themselves. Note also that <sup>Γ</sup>*<sup>C</sup>* is order-preserving.
*<sup>A</sup>* # *<sup>B</sup>* <sup>=</sup> *<sup>A</sup>*1/2(*A*−1/2*BA*−1/2)1/2*A*1/2 <sup>=</sup> <sup>Γ</sup>*A*1/2 ◦ <sup>Γ</sup>1/2
that *AB* = *BA*. The second condition is that, for any invertible *T* ∈ *B*(H),
**Theorem 0.14** (Maximum property of geometric mean)**.** *For A*, *B* > 0*,*
*where the maximum is taken with respect to the positive semidefinite ordering.*
Then *A* # *B* > 0 for *A*, *B* > 0. This formula comes from two natural requirements: This definition should coincide with the usual geometric mean in **R**+: *A* # *B* = (*AB*)1/2 provided
The next theorem characterizes the geometric mean of *A* and *B* in term of the solution of a
**Theorem 0.13.** *For each A*, *B* > 0*, the Riccati equation* Γ*X*(*A*−1) := *XA*−1*X* = *B has a unique*
*Proof.* The direct computation shows that (*A* # *B*)*A*−1(*A* # *B*) = *B*. Suppose there is another
(*A*−1/2*XA*−1/2)<sup>2</sup> = *A*−1/2*XA*−1*XA*−1/2 = *A*−1/2*YA*−1*YA*−1/2 = (*A*−1/2*YA*−1/2)2.
The uniqueness of positive square roots implies that *A*−1/2*XA*−1/2 = *A*−1/2*YA*−1/2, i.e.,
*T*∗(*A* # *B*)*T* = (*T*∗*AT*) # (*T*∗*BT*). (11)
*<sup>A</sup>* # *<sup>B</sup>* <sup>=</sup> max{*<sup>X</sup>* 0 : *XA*−1*<sup>X</sup> <sup>B</sup>*} (12)
2(*A*−<sup>1</sup>
+ *<sup>B</sup>*−<sup>1</sup> )−<sup>1</sup> (9)
*<sup>A</sup>*−1/2 (*B*). (10)
Moreover, the arithmetic mean satisfies
invertible operator *C* ∈ *B*(H), define
Define the *geometric mean* of *A*, *B* > 0 by
certain operator equation.
*positive solution, namely, X* = *A* # *B.*
positive solution *Y* 0. Then
*X* = *Y*.
(A1)–(A9).
*affinity*: *<sup>M</sup>*(*kA* <sup>+</sup> *<sup>C</sup>*, *kB* <sup>+</sup> *<sup>C</sup>*) = *kM*(*A*, *<sup>B</sup>*) + *<sup>C</sup>* for *<sup>k</sup>* <sup>∈</sup> **<sup>R</sup>**+.
Define the *harmonic mean* of positive operators *<sup>A</sup>*, *<sup>B</sup>* <sup>∈</sup> *<sup>B</sup>*(H)<sup>+</sup> by
*A* ! *B* = 2(*A* : *B*) = lim
$$(A^{-1/2}XA^{-1/2})^2 = A^{-1/2}XA^{-1}XA^{-1/2} \lesssim A^{-1/2}BA^{-1/2}$$
and *A*−1/2*XA*−1/2 (*A*−1/2*BA*−1/2)1/2 i.e. *X A* # *B* by Theorem 0.3.
Recall the fact that if *f* : [*a*, *b*] → **C** is continuous and *An* → *A* with Sp(*An*) ⊆ [*a*, *b*] for all *n* ∈ **N**, then Sp(*A*) ⊆ [*a*, *b*] and *f*(*An*) → *f*(*A*).
**Lemma 0.15.** *Let A*, *<sup>B</sup>*, *<sup>C</sup>*, *<sup>D</sup>*, *An*, *Bn* <sup>∈</sup> *<sup>B</sup>*(H)++ *for all n* <sup>∈</sup> **<sup>N</sup>***.*
*Proof.* (1) The extremal characterization allows us to prove only that (*A* # *B*)*C*−1(*A* # *B*) *D*. Indeed,
$$\begin{aligned} (A \# B) \mathbb{C}^{-1} (A \# B) &= A^{1/2} (A^{-1/2} B A^{-1/2})^{1/2} A^{1/2} \mathbb{C}^{-1} A^{1/2} (A^{-1/2} B A^{-1/2})^{1/2} A^{1/2} \\ &\leqslant A^{1/2} (A^{-1/2} B A^{-1/2})^{1/2} A^{1/2} A^{-1} A^{1/2} (A^{-1/2} B A^{-1/2})^{1/2} A^{1/2} \\ &= \mathcal{B} \\ &\leqslant D. \end{aligned}$$
(2) Assume *An* ↓ *A* and *Bn* ↓ *B*. Then *An* # *Bn* is a decreasing sequence of strictly positive operators which is bounded below by 0. The order completeness of *B*(H) implies that this sequence converges strongly to a positive operator. Since *A*−<sup>1</sup> *<sup>n</sup> A*−1, the Löwner-Heinz's inequality assures that *<sup>A</sup>*−1/2 *<sup>n</sup> <sup>A</sup>*−1/2 and hence �*A*−1/2 *<sup>n</sup>* � �*A*−1/2� for all *<sup>n</sup>* <sup>∈</sup> **<sup>N</sup>**. Note also that �*Bn*� �*B*1� for all *n* ∈ **N**. Recall that the multiplication is jointly continuous in the strong-operator topology if the first variable is bounded in norm. So, *<sup>A</sup>*−1/2 *<sup>n</sup> BnA*−1/2 *<sup>n</sup>* converges strongly to *A*−1/2*BA*−1/2. It follows that
$$(A\_n^{-1/2}B\_nA\_n^{-1/2})^{1/2} \to (A^{-1/2}BA^{-1/2})^{1/2}.$$
Since *A*1/2 *<sup>n</sup>* is norm-bounded by �*A*1/2� by Löwner-Heinz's inequality, we conclude that
$$A\_n^{1/2} (A\_n^{-1/2} \mathcal{B}\_n A\_n^{-1/2})^{1/2} A\_n^{1/2} \to A^{1/2} (A^{-1/2} \mathcal{B} A^{-1/2})^{1/2} A^{1/2}.$$
The proof of (3) is just the same as the case of harmonic mean.
We define the *geometric mean* of *A*, *B* 0 by
$$A \# B = \lim\_{\epsilon \downarrow 0} (A + \epsilon I) \# (B + \epsilon I). \tag{13}$$
Then *A* # *B* 0 for any *A*, *B* 0.
**Theorem 0.16.** *The geometric mean enjoys the following properties*
*Proof.* (1) Use the formula (13) and Lemma 0.15 (1).
(2) Follows from Lemma 0.15 and the definition of the geometric mean.
(3) The unique positive solution to the equation *XA*−1*X* = *A* is *X* = *A*.
(4) The unique positive solution to the equation *X*−1*A*−1*X*−<sup>1</sup> = *B* is *X*−<sup>1</sup> = *A* # *B*. But this equstion is equivalent to *XAX* = *B*−1. So, *A*−<sup>1</sup> # *B*−<sup>1</sup> = *X* = (*A* # *B*)−1.
(5) The equation *XA*−1*X* = *B* has the same solution to the equation *XB*−1*X* = *A* by taking inverse in both sides.
(6) We have
$$\begin{aligned} \Gamma\_{\mathbb{C}}(A \# B)(\Gamma\_{\mathbb{C}}(A))^{-1} \Gamma\_{\mathbb{C}}(A \# B) &= \Gamma\_{\mathbb{C}}(A \# B) \Gamma\_{\mathbb{C}^{-1}}(A^{-1}) \Gamma\_{\mathbb{C}}(A \# B) \\ &= \Gamma\_{\mathbb{C}}((A \# B)A^{-1}(A \# B)) \\ &= \Gamma\_{\mathbb{C}}(B). \end{aligned}$$
Then apply Theorem 0.13.
The congruence invariance asserts that <sup>Γ</sup>*<sup>C</sup>* is an isomorphism on *<sup>B</sup>*(H)++ with respect to the operation of taking the geometric mean.
**Lemma 0.17.** *For A* > 0 *and B* 0*, the operator*
$$
\begin{pmatrix} A & \mathbf{C} \\ \mathbf{C}^\* \ \mathbf{B} \end{pmatrix}
$$
*is positive if and only if B* <sup>−</sup> *<sup>C</sup>*∗*A*−1*C is positive, i.e., B <sup>C</sup>*∗*A*−1*C.*
*Proof.* By setting
$$X = \begin{pmatrix} I & -A^{-1}C \\ 0 & I \end{pmatrix} \cdot$$
we compute
$$\begin{aligned} \Gamma\_X \begin{pmatrix} A & \mathbb{C} \\ \mathbb{C}^\* \end{pmatrix} &= \begin{pmatrix} I & 0 \\ -\mathbb{C}^\* A^{-1} & I \end{pmatrix} \begin{pmatrix} A & \mathbb{C} \\ \mathbb{C}^\* B \end{pmatrix} \begin{pmatrix} I - A^{-1} \mathbb{C} \\ 0 & I \end{pmatrix} \\ &= \begin{pmatrix} A & 0 \\ 0 \ B - \mathbb{C}^\* A^{-1} \mathbb{C} \end{pmatrix} .\end{aligned}$$
Since Γ*<sup>G</sup>* preserves positivity, we obtain the desired result.
**Theorem 0.18.** *The geometric mean A* # *B of A*, *<sup>B</sup>* <sup>∈</sup> *<sup>B</sup>*(H)<sup>+</sup> *is the largest operator X* <sup>∈</sup> *<sup>B</sup>*(H)*sa for which the operator*
$$
\begin{pmatrix} A & X \\ X^\* & B \end{pmatrix} \tag{14}
$$
*is positive.*
10 Will-be-set-by-IN-TECH
(4) The unique positive solution to the equation *X*−1*A*−1*X*−<sup>1</sup> = *B* is *X*−<sup>1</sup> = *A* # *B*. But this
(5) The equation *XA*−1*X* = *B* has the same solution to the equation *XB*−1*X* = *A* by taking
<sup>Γ</sup>*C*(*<sup>A</sup>* # *<sup>B</sup>*)(Γ*C*(*A*))−1Γ*C*(*<sup>A</sup>* # *<sup>B</sup>*) = <sup>Γ</sup>*C*(*<sup>A</sup>* # *<sup>B</sup>*)Γ*C*−<sup>1</sup> (*A*−1)Γ*C*(*<sup>A</sup>* # *<sup>B</sup>*)
The congruence invariance asserts that <sup>Γ</sup>*<sup>C</sup>* is an isomorphism on *<sup>B</sup>*(H)++ with respect to the
*A C C*∗ *B*
*<sup>I</sup>* <sup>−</sup>*A*−1*<sup>C</sup>* 0 *I*
,
*A C C*∗ *B*
> .
*<sup>I</sup>* <sup>−</sup>*A*−1*<sup>C</sup>* 0 *I*
= Γ*C*(*B*).
= Γ*C*((*A* # *B*)*A*−1(*A* # *B*))
**Theorem 0.16.** *The geometric mean enjoys the following properties*
*(6) congruence invariance:* Γ*C*(*A*) # Γ*C*(*B*) = Γ*C*(*A* # *B*) *for all invertible C.*
(2) Follows from Lemma 0.15 and the definition of the geometric mean. (3) The unique positive solution to the equation *XA*−1*X* = *A* is *X* = *A*.
equstion is equivalent to *XAX* = *B*−1. So, *A*−<sup>1</sup> # *B*−<sup>1</sup> = *X* = (*A* # *B*)−1.
*(1) monotonicity: A*<sup>1</sup> *A*2, *B*<sup>1</sup> *B*<sup>2</sup> ⇒ *A*<sup>1</sup> # *B*<sup>1</sup> *A*<sup>2</sup> # *B*2*. (2) continuity from above: An* ↓ *A*, *Bn* ↓ *B* ⇒ *An* # *Bn* ↓ *A* # *B.*
*Proof.* (1) Use the formula (13) and Lemma 0.15 (1).
*(3) fixed point property: A* # *A* = *A. (4) self-duality:* (*A* # *B*)−<sup>1</sup> = *A*−<sup>1</sup> # *B*−1*.*
*(5) symmetry: A* # *B* = *B* # *A.*
inverse in both sides.
Then apply Theorem 0.13.
*Proof.* By setting
we compute
operation of taking the geometric mean.
Γ*X*
*A C C*∗ *B*
**Lemma 0.17.** *For A* > 0 *and B* 0*, the operator*
*is positive if and only if B* <sup>−</sup> *<sup>C</sup>*∗*A*−1*C is positive, i.e., B <sup>C</sup>*∗*A*−1*C.*
=
Since Γ*<sup>G</sup>* preserves positivity, we obtain the desired result.
=
*X* =
*I* 0 <sup>−</sup>*C*∗*A*−<sup>1</sup> *<sup>I</sup>*
*A* 0
<sup>0</sup> *<sup>B</sup>* <sup>−</sup> *<sup>C</sup>*∗*A*−1*<sup>C</sup>*
(6) We have
*Proof.* By continuity argumeny, we may assume that *A*, *B* > 0. If *X* = *A* # *B*, then the operator (14) is positive by Lemma 0.17. Let *<sup>X</sup>* <sup>∈</sup> *<sup>B</sup>*(H)*sa* be such that the operator (14) is positive. Then Lemma 0.17 again implies that *XA*−1*X B* and
$$(A^{-1/2}XA^{-1/2})^2 = A^{-1/2}XA^{-1}XA^{-1/2} \lesssim A^{-1/2}BA^{-1/2}A$$
The Löwner-Heinz's inequality forces *A*−1/2*XA*−1/2 (*A*−1/2*BA*−1/2)1/2. Now, applying Γ*A*1/2 yields *X A* # *B*.
**Remark 0.19.** The arithmetric mean and the harmonic mean can be easily defined for multivariable positive operators. The case of geometric mean is not easy, even for the case of matrices. Many authors tried to defined geometric means for multivariable positive semidefinite matrices but there is no satisfactory definition until 2004 in [5].
#### **5. Operator connections**
We see that the arithmetic, harmonic and geometric means share the properties (A1)–(A9) in common. A mean in general should have algebraic, order and topological properties. Kubo and Ando [17] proposed the following definition:
**Definition 0.20.** <sup>A</sup> *connection* on *<sup>B</sup>*(H)<sup>+</sup> is a binary operation *<sup>σ</sup>* on *<sup>B</sup>*(H)<sup>+</sup> satisfying the following axioms for all *A*, *A*� , *B*, *B*� , *<sup>C</sup>* <sup>∈</sup> *<sup>B</sup>*(H)+:
The term "connection" comes from the study of electrical network connections.
**Example 0.21.** The following are examples of connections:
From now on, assume dim H = ∞. Consider the following property:
(M3') *separate continuity from above*: if *An*, *Bn* <sup>∈</sup> *<sup>B</sup>*(H)<sup>+</sup> satisfy *An* <sup>↓</sup> *<sup>A</sup>* and *Bn* <sup>↓</sup> *<sup>B</sup>*, then *An σ B* ↓ *A σ B* and *A σ Bn* ↓ *A σ B*.
The condition (M3') is clearly weaker than (M3). The next theorem asserts that we can improve the definition of Kubo-Ando by replacing (M3) with (M3') and still get the same theory. This theorem also provides an easier way for checking a binary opertion to be a connection.
**Theorem 0.22.** *If a binary operation <sup>σ</sup> on B*(H)<sup>+</sup> *satisfies (M1), (M2) and (M3'), then <sup>σ</sup> satisfies (M3), that is, σ is a connection.*
Denote by *OM*(**R**+) the set of operator monotone functions from **R**<sup>+</sup> to **R**+. If a binary operation *σ* has a property (A), we write *σ* ∈ *BO*(*A*). The following properties for a binary operation *<sup>σ</sup>* and a function *<sup>f</sup>* : **<sup>R</sup>**<sup>+</sup> <sup>→</sup> **<sup>R</sup>**<sup>+</sup> play important roles:
(P) : If a projection *<sup>P</sup>* <sup>∈</sup> *<sup>B</sup>*(H)<sup>+</sup> commutes with *<sup>A</sup>*, *<sup>B</sup>* <sup>∈</sup> *<sup>B</sup>*(H)+, then
*P*(*A σ B*)=(*PA*) *σ* (*PB*)=(*A σ B*)*P*;
(F) : *<sup>f</sup>*(*t*)*<sup>I</sup>* <sup>=</sup> *<sup>I</sup> <sup>σ</sup>* (*tI*) for any *<sup>t</sup>* <sup>∈</sup> **<sup>R</sup>**+.
**Proposition 0.23.** *The transformer inequality (M2) implies*
*Proof.* For *A*, *B* 0 and *C* > 0, we have
$$\mathbb{C}^{-1}[(\mathbb{C}\mathcal{A}\mathbb{C})\,\sigma\,(\mathbb{C}\mathcal{B}\mathbb{C})]\mathbb{C}^{-1}\leqslant (\mathbb{C}^{-1}\mathbb{C}\mathcal{A}\mathbb{C}\mathbb{C}^{-1})\,\sigma\,(\mathbb{C}^{-1}\mathbb{C}\mathcal{B}\mathbb{C}\mathbb{C}^{-1})=A\,\sigma\,B$$
and hence (*CAC*) *σ* (*CBC*) *C*(*A σ B*)*C*. The positive homogeneity comes from the congruence invariance by setting *<sup>C</sup>* <sup>=</sup> <sup>√</sup>*αI*.
**Lemma 0.24.** *Let f* : **<sup>R</sup>**<sup>+</sup> <sup>→</sup> **<sup>R</sup>**<sup>+</sup> *be an increasing function. If <sup>σ</sup> satisfies the positive homogeneity, (M3') and (F), then f is continuous.*
*Proof.* To show that *<sup>f</sup>* is right continuous at each *<sup>t</sup>* <sup>∈</sup> **<sup>R</sup>**+, consider a sequence *tn* in **<sup>R</sup>**<sup>+</sup> such that *tn* ↓ *t*. Then by (M3')
$$f(t\_n)I = I\sigma \, t\_n I \downarrow I\sigma \, tI = f(t)I\prime$$
i.e. *f*(*tn*) ↓ *f*(*t*). To show that *f* is left continuous at each *t* > 0, consider a sequence *tn* > 0 such that *tn* ↑ *t*. Then *t* <sup>−</sup><sup>1</sup> *<sup>n</sup>* <sup>↓</sup> *<sup>t</sup>* <sup>−</sup><sup>1</sup> and
$$\begin{aligned} \lim t\_n^{-1} f(t\_n) I &= \lim t\_n^{-1} (I \, \sigma \, t\_n I) = \lim (t\_n^{-1} I) \, \sigma I = (t^{-1} I) \, \sigma I \\ &= t^{-1} (I \, \sigma \, tI) = t^{-1} f(t) I \end{aligned}$$
Since *f* is increasing, *t* <sup>−</sup><sup>1</sup> *<sup>n</sup> <sup>f</sup>*(*tn*) is decreasing. So, *<sup>t</sup>* �→ *<sup>t</sup>* <sup>−</sup><sup>1</sup> *f*(*t*) and *f* are left continuous. **Lemma 0.25.** *Let <sup>σ</sup> be a binary operation on B*(H)<sup>+</sup> *satisfying (M3') and (P). If f* : **<sup>R</sup>**<sup>+</sup> <sup>→</sup> **<sup>R</sup>**<sup>+</sup> *is an increasing continuous function such that <sup>σ</sup> and f satisfy (F), then f*(*A*) = *<sup>I</sup> <sup>σ</sup> A for any A* <sup>∈</sup> *<sup>B</sup>*(H)+*.*
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(M3') *separate continuity from above*: if *An*, *Bn* <sup>∈</sup> *<sup>B</sup>*(H)<sup>+</sup> satisfy *An* <sup>↓</sup> *<sup>A</sup>* and *Bn* <sup>↓</sup> *<sup>B</sup>*, then
The condition (M3') is clearly weaker than (M3). The next theorem asserts that we can improve the definition of Kubo-Ando by replacing (M3) with (M3') and still get the same theory. This theorem also provides an easier way for checking a binary opertion to be a connection.
**Theorem 0.22.** *If a binary operation <sup>σ</sup> on B*(H)<sup>+</sup> *satisfies (M1), (M2) and (M3'), then <sup>σ</sup> satisfies*
Denote by *OM*(**R**+) the set of operator monotone functions from **R**<sup>+</sup> to **R**+. If a binary operation *σ* has a property (A), we write *σ* ∈ *BO*(*A*). The following properties for a binary
*P*(*A σ B*)=(*PA*) *σ* (*PB*)=(*A σ B*)*P*;
*C*−1[(*CAC*) *σ* (*CBC*)]*C*−<sup>1</sup> (*C*−1*CACC*−1) *σ* (*C*−1*CBCC*−1) = *A σ B*
and hence (*CAC*) *σ* (*CBC*) *C*(*A σ B*)*C*. The positive homogeneity comes from the
**Lemma 0.24.** *Let f* : **<sup>R</sup>**<sup>+</sup> <sup>→</sup> **<sup>R</sup>**<sup>+</sup> *be an increasing function. If <sup>σ</sup> satisfies the positive homogeneity,*
*Proof.* To show that *<sup>f</sup>* is right continuous at each *<sup>t</sup>* <sup>∈</sup> **<sup>R</sup>**+, consider a sequence *tn* in **<sup>R</sup>**<sup>+</sup> such
*f*(*tn*)*I* = *I σ tn I* ↓ *I σ tI* = *f*(*t*)*I*,
i.e. *f*(*tn*) ↓ *f*(*t*). To show that *f* is left continuous at each *t* > 0, consider a sequence *tn* > 0
*<sup>n</sup>* (*I σ tn I*) = lim(*t*
<sup>−</sup><sup>1</sup> *f*(*t*)*I*
−1
*<sup>n</sup> I*) *σ I* = (*t*
<sup>−</sup><sup>1</sup> *I*) *σ I*
<sup>−</sup><sup>1</sup> *f*(*t*) and *f* are left continuous.
From now on, assume dim H = ∞. Consider the following property:
operation *<sup>σ</sup>* and a function *<sup>f</sup>* : **<sup>R</sup>**<sup>+</sup> <sup>→</sup> **<sup>R</sup>**<sup>+</sup> play important roles:
**Proposition 0.23.** *The transformer inequality (M2) implies*
(P) : If a projection *<sup>P</sup>* <sup>∈</sup> *<sup>B</sup>*(H)<sup>+</sup> commutes with *<sup>A</sup>*, *<sup>B</sup>* <sup>∈</sup> *<sup>B</sup>*(H)+, then
*•* Congruence invariance*: For A*, *B* 0 *and C* > 0*, C*(*AσB*)*C* = (*CAC*) *σ* (*CBC*)*; •* Positive homogeneity*: For A*, *B* 0 *and α* ∈ (0, ∞)*, α*(*A σ B*)=(*αA*) *σ* (*αB*)*.*
*An σ B* ↓ *A σ B* and *A σ Bn* ↓ *A σ B*.
*(M3), that is, σ is a connection.*
(F) : *<sup>f</sup>*(*t*)*<sup>I</sup>* <sup>=</sup> *<sup>I</sup> <sup>σ</sup>* (*tI*) for any *<sup>t</sup>* <sup>∈</sup> **<sup>R</sup>**+.
*Proof.* For *A*, *B* 0 and *C* > 0, we have
congruence invariance by setting *<sup>C</sup>* <sup>=</sup> <sup>√</sup>*αI*.
<sup>−</sup><sup>1</sup> *<sup>n</sup>* <sup>↓</sup> *<sup>t</sup>*
lim *t* −1 <sup>−</sup><sup>1</sup> and
= *t*
−1
<sup>−</sup><sup>1</sup> *<sup>n</sup> <sup>f</sup>*(*tn*) is decreasing. So, *<sup>t</sup>* �→ *<sup>t</sup>*
<sup>−</sup>1(*I σ tI*) = *t*
*<sup>n</sup> f*(*tn*)*I* = lim *t*
*(M3') and (F), then f is continuous.*
that *tn* ↓ *t*. Then by (M3')
such that *tn* ↑ *t*. Then *t*
Since *f* is increasing, *t*
*Proof.* First consider *<sup>A</sup>* <sup>∈</sup> *<sup>B</sup>*(H)<sup>+</sup> in the form <sup>∑</sup>*<sup>m</sup> <sup>i</sup>*=<sup>1</sup> *<sup>λ</sup>iPi* where {*Pi*}*<sup>m</sup> <sup>i</sup>*=<sup>1</sup> is an orthogonal family of projections with sum *I* and *λ<sup>i</sup>* > 0 for all *i* = 1, . . . , *m*. Since each *Pi* commutes with *A*, we have by the property (P) that
$$\begin{aligned} I\,\sigma\,A &= \sum P\_{\bar{l}}(I\,\sigma\,A) = \sum P\_{\bar{l}}\,\sigma\,P\_{\bar{l}}A = \sum P\_{\bar{l}}\,\sigma\,\lambda\_{\bar{l}}P\_{\bar{l}} \\ &= \sum P\_{\bar{l}}(I\,\sigma\,\lambda\_{\bar{l}}I) = \sum f(\lambda\_{\bar{l}})P\_{\bar{l}} = f(A). \end{aligned}$$
Now, consider *<sup>A</sup>* <sup>∈</sup> *<sup>B</sup>*(H)+. Then there is a sequence *An* of strictly positive operators in the above form such that *An* ↓ *A*. Then *I σ An* ↓ *I σ A* and *f*(*An*) converges strongly to *f*(*A*). Hence, *I σ A* = lim *I σ An* = lim *f*(*An*) = *f*(*A*).
*Proof of Theorem 0.22:* Let *σ* ∈ *BO*(*M*1, *M*2, *M*3� ). As in [17], the conditions (M1) and (M2) imply that *<sup>σ</sup>* satisfies (P) and there is a function *<sup>f</sup>* : **<sup>R</sup>**<sup>+</sup> <sup>→</sup> **<sup>R</sup>**<sup>+</sup> subject to (F). If 0 *<sup>t</sup>*<sup>1</sup> *<sup>t</sup>*2, then by (M1)
$$f(t\_1)I = I\sigma\left(t\_1I\right) \leqslant I\sigma\left(t\_2I\right) = f(t\_2)I\nu$$
i.e. *f*(*t*1) *f*(*t*2). The assumption (M3') is enough to guarantee that *f* is continuous by Lemma 0.24. Then Lemma 0.25 results in *f*(*A*) = *IσA* for all *A* 0. Now, (M1) and the fact that dim <sup>H</sup> <sup>=</sup> <sup>∞</sup> yield that *<sup>f</sup>* is operator monotone. If there is another *<sup>g</sup>* <sup>∈</sup> *OM*(**R**+) satisfying (F), then *f*(*t*)*I* = *I σ tI* = *g*(*t*)*I* for each *t* 0, i.e. *f* = *g*. Thus, we establish a well-defined map *σ* ∈ *BO*(*M*1, *M*2, *M*3� ) �→ *<sup>f</sup>* <sup>∈</sup> *OM*(**R**+) such that *<sup>σ</sup>* and *<sup>f</sup>* satisfy (F).
Now, given *<sup>f</sup>* <sup>∈</sup> *OM*(**R**+), we construct *<sup>σ</sup>* from the integral representation (2) in Proposition 0.4. Define a binary operation *<sup>σ</sup>* : *<sup>B</sup>*(H)<sup>+</sup> <sup>×</sup> *<sup>B</sup>*(H)<sup>+</sup> <sup>→</sup> *<sup>B</sup>*(H)<sup>+</sup> by
$$A \,\sigma \, B = \int\_{\left[0,1\right]} A \, !\_{l} \, B \, d\mu(t) \tag{15}$$
where the integral is taken in the sense of Bochner. Consider *<sup>A</sup>*, *<sup>B</sup>* <sup>∈</sup> *<sup>B</sup>*(H)<sup>+</sup> and set *Ft* <sup>=</sup> *<sup>A</sup>* !*<sup>t</sup> <sup>B</sup>* for each *t* ∈ [0, 1]. Since *A* �*A*�*I* and *B* �*B*�*I*, we get
$$A \, !\_t B \leqslant \|A\| \|I\| \, !\_t \, \|B\| \|I\| \quad = \quad \frac{\|A\| \|\|B\|\|}{t \|\|A\|\| + (1 - t) \|\|B\|\|} I.$$
By Banach-Steinhaus' theorem, there is an *M* > 0 such that �*Ft*� *M* for all *t* ∈ [0, 1]. Hence,
$$\int\_{[0,1]} \|F\_t\| \, d\mu(t) \lesssim \int\_{[0,1]} M \, d\mu(t) < \infty.$$
So, *Ft* is Bochner integrable. Since *Ft* 0 for all *t* ∈ [0, 1], [0,1] *Ft dμ*(*t*) 0. Thus, *A σ B* is a well-defined element in *<sup>B</sup>*(H)+. The monotonicity (M1) and the transformer inequality (M2) come from passing the monotonicity and the transformer inequality of the weighed harmonic mean through the Bochner integral. To show (M3'), let *An* ↓ *A* and *Bn* ↓ *B*. Then *An* !*<sup>t</sup> B* ↓ *A* !*<sup>t</sup> B* for *t* ∈ [0, 1] by the monotonicity and the separate continuity from above of the weighed harmonic mean. Let *ξ* ∈ *H*. Define a bounded linear map Φ : *B*(H) → **C** by Φ(*T*) = �*Tξ*, *ξ*�.
#### 14 Will-be-set-by-IN-TECH 176 Linear Algebra – Theorems and Applications
For each *n* ∈ **N**, set *Tn*(*t*) = *An* !*<sup>t</sup> B* and put *T*∞(*t*) = *A* !*<sup>t</sup> B*. Then for each *n* ∈ **N** ∪ {∞}, Φ ◦ *Tn* is Bochner integrable and
$$\langle \int T\_{\mathfrak{n}}(t) \, d\mu(t) \xi\_t \, \xi \rangle = \Phi(\int T\_{\mathfrak{n}}(t) \, d\mu(t)) = \int \Phi \circ T\_{\mathfrak{n}}(t) \, d\mu(t).$$
Since *Tn*(*t*) ↓ *T*∞(*t*), we have that �*Tn*(*t*)*ξ*, *ξ*�→�*T*∞(*t*)*ξ*, *ξ*� as *n* → ∞ for each *t* ∈ [0, 1]. We obtain from the dominated convergence theorem that
$$\lim\_{n \to \infty} \langle (A\_n \,\sigma \, B)\tilde{\xi}, \tilde{\xi} \rangle = \lim\_{n \to \infty} \langle \int T\_n(t) \, d\mu(t) \tilde{\xi}, \tilde{\xi} \rangle$$
$$= \lim\_{n \to \infty} \int \langle T\_n(t) \tilde{\xi}, \tilde{\xi} \rangle \, d\mu(t)$$
$$= \int \langle T\_{\infty}(t) \tilde{\xi}, \tilde{\xi} \rangle \, d\mu(t)$$
$$= \langle \int T\_{\infty}(t) d\mu(t) \tilde{\xi}, \tilde{\xi} \rangle$$
$$= \langle (A \,\sigma \, B)\tilde{\xi}, \tilde{\xi} \rangle.$$
So, *An σ B* ↓ *A σ B*. Similarly, *A σ Bn* ↓ *A σ B*. Thus, *σ* satisfies (M3'). It is easy to see that *f*(*t*)*I* = *I σ* (*tI*) for *t* 0. This shows that the map *σ* �→ *f* is surjective.
To show the injectivity of this map, let *σ*1, *σ*<sup>2</sup> ∈ *BO*(*M*1, *M*2, *M*3� ) be such that *σ<sup>i</sup>* �→ *f* where, for each *t* 0, *I σ<sup>i</sup>* (*tI*) = *f*(*t*)*I*, *i* = 1, 2. Since *σ<sup>i</sup>* satisfies the property (P), we have *I σ<sup>i</sup> A* = *f*(*A*) for *A* 0 by Lemma 0.25. Since *σ<sup>i</sup>* satisfies the congruence invariance, we have that for *A* > 0 and *B* 0,
$$A\,\sigma\_{\mathrm{i}}\,\mathrm{B} = A^{1/2}(\mathrm{I}\,\sigma\_{\mathrm{i}}\,\mathrm{A}^{-1/2}\mathrm{BA}^{-1/2})A^{1/2} = A^{1/2}f(\mathrm{A}^{-1/2}\mathrm{BA}^{-1/2})A^{1/2}, \quad \mathrm{i} = 1,2.$$
For each *A*, *B* 0, we obtain by (M3') that
$$\begin{aligned} A \,\sigma\_1 \,\, &B = \lim\_{\epsilon \downarrow 0} A\_{\epsilon} \,\, \sigma\_1 \,\, \, \\ &= \lim\_{\epsilon \downarrow 0} A\_{\epsilon}^{1/2} (I \,\sigma\_1 \, A\_{\epsilon}^{-1/2} B A\_{\epsilon}^{-1/2}) A\_{\epsilon}^{1/2} \\ &= \lim\_{\epsilon \downarrow 0} A\_{\epsilon}^{1/2} f(A\_{\epsilon}^{-1/2} B A\_{\epsilon}^{-1/2}) A\_{\epsilon}^{1/2} \\ &= \lim\_{\epsilon \downarrow 0} A\_{\epsilon}^{1/2} (I \,\sigma\_2 \, A\_{\epsilon}^{-1/2} B A\_{\epsilon}^{-1/2}) A\_{\epsilon}^{1/2} \\ &= \lim\_{\epsilon \downarrow 0} A\_{\epsilon} \,\, \sigma\_2 \, \, \, \, \\ &= A \,\, \sigma\_2 \, B\_{\epsilon} \end{aligned}$$
where *<sup>A</sup>�* <sup>≡</sup> *<sup>A</sup>* <sup>+</sup> *�I*. That is *<sup>σ</sup>*<sup>1</sup> <sup>=</sup> *<sup>σ</sup>*2. Therefore, there is a bijection between *OM*(**R**+) and *BO*(*M*1, *M*2, *M*3� ). Every element in *BO*(*M*1, *M*2, *M*3� ) admits an integral representation (15). Since the weighed harmonic mean possesses the joint continuity (M3), so is any element in *BO*(*M*1, *M*2, *M*3� ).
The next theorem is a fundamental result of [17].
**Theorem 0.26.** *There is a one-to-one correspondence between connections σ and operator monotone functions f on the non-negative reals satisfying*
$$f(t)I = I\sigma\left(tI\right), \quad t \in \mathbb{R}^+.\tag{16}$$
*There is a one-to-one correspondence between connections σ and finite Borel measures ν on* [0, ∞] *satisfying*
$$A\,\sigma\,B = \int\_{\left[0,\infty\right]} \frac{t+1}{t} (tA\,:\,B) \,d\nu(t), \quad A, B \geqslant 0. \tag{17}$$
*Moreover, the map σ* �→ *f is an affine order-isomorphism between connections and non-negative operator monotone functions on* **<sup>R</sup>**+*. Here, the order-isomorphism means that when <sup>σ</sup> <sup>i</sup>* �→ *fi for <sup>i</sup>* <sup>=</sup> 1, 2*, A <sup>σ</sup>* <sup>1</sup>*<sup>B</sup> <sup>A</sup> <sup>σ</sup>* <sup>2</sup>*B for all A*, *<sup>B</sup>* <sup>∈</sup> *<sup>B</sup>*(H)<sup>+</sup> *if and only if f*<sup>1</sup> *<sup>f</sup>*2*.*
Each connection *<sup>σ</sup>* on *<sup>B</sup>*(H)<sup>+</sup> produces a unique scalar function on **<sup>R</sup>**+, denoted by the same notation, satisfying
$$(s\,\sigma\,t)I = (sI)\,\sigma\,(tI), \quad s, t \in \mathbb{R}^+.\tag{18}$$
Let *<sup>s</sup>*, *<sup>t</sup>* <sup>∈</sup> **<sup>R</sup>**+. If *<sup>s</sup>* <sup>&</sup>gt; 0, then *<sup>s</sup> <sup>σ</sup> <sup>t</sup>* <sup>=</sup> *s f*(*t*/*s*). If *<sup>t</sup>* <sup>&</sup>gt; 0, then *<sup>s</sup> <sup>σ</sup> <sup>t</sup>* <sup>=</sup> *t f*(*s*/*t*).
14 Will-be-set-by-IN-TECH
For each *n* ∈ **N**, set *Tn*(*t*) = *An* !*<sup>t</sup> B* and put *T*∞(*t*) = *A* !*<sup>t</sup> B*. Then for each *n* ∈ **N** ∪ {∞},
Since *Tn*(*t*) ↓ *T*∞(*t*), we have that �*Tn*(*t*)*ξ*, *ξ*�→�*T*∞(*t*)*ξ*, *ξ*� as *n* → ∞ for each *t* ∈ [0, 1]. We
<sup>=</sup> lim*n*→<sup>∞</sup>
=
= �
So, *An σ B* ↓ *A σ B*. Similarly, *A σ Bn* ↓ *A σ B*. Thus, *σ* satisfies (M3'). It is easy to see that
for each *t* 0, *I σ<sup>i</sup>* (*tI*) = *f*(*t*)*I*, *i* = 1, 2. Since *σ<sup>i</sup>* satisfies the property (P), we have *I σ<sup>i</sup> A* = *f*(*A*) for *A* 0 by Lemma 0.25. Since *σ<sup>i</sup>* satisfies the congruence invariance, we have that for
*A σ<sup>i</sup> B* = *A*1/2(*I σ<sup>i</sup> A*−1/2*BA*−1/2)*A*1/2 = *A*1/2 *f*(*A*−1/2*BA*−1/2)*A*1/2, *i* = 1, 2.
*�* (*<sup>I</sup> <sup>σ</sup>*<sup>1</sup> *<sup>A</sup>*−1/2
*�* (*<sup>I</sup> <sup>σ</sup>*<sup>2</sup> *<sup>A</sup>*−1/2
where *<sup>A</sup>�* <sup>≡</sup> *<sup>A</sup>* <sup>+</sup> *�I*. That is *<sup>σ</sup>*<sup>1</sup> <sup>=</sup> *<sup>σ</sup>*2. Therefore, there is a bijection between *OM*(**R**+) and
Since the weighed harmonic mean possesses the joint continuity (M3), so is any element in
*� <sup>f</sup>*(*A*−1/2
*� BA*−1/2
*� BA*−1/2
).
*� BA*−1/2
*�* )*A*1/2 *�*
*�* )*A*1/2 *�*
) admits an integral representation (15).
*�* )*A*1/2 *�*
*Tn*(*t*) *dμ*(*t*)) =
= �(*A σ B*)*ξ*, *ξ*�.
*Tn*(*t*) *dμ*(*t*)*ξ*, *ξ*�
�*Tn*(*t*)*ξ*, *ξ*� *dμ*(*t*)
�*T*∞(*t*)*ξ*, *ξ*� *dμ*(*t*)
*T*∞(*t*)*dμ*(*t*)*ξ*, *ξ*�
Φ ◦ *Tn*(*t*) *dμ*(*t*).
) be such that *σ<sup>i</sup>* �→ *f* where,
lim*n*→∞�(*An <sup>σ</sup> <sup>B</sup>*)*ξ*, *<sup>ξ</sup>*� <sup>=</sup> lim*n*→∞�
*f*(*t*)*I* = *I σ* (*tI*) for *t* 0. This shows that the map *σ* �→ *f* is surjective.
To show the injectivity of this map, let *σ*1, *σ*<sup>2</sup> ∈ *BO*(*M*1, *M*2, *M*3�
*A σ*<sup>1</sup> *B* = lim
*�*↓0
= lim *�*↓0
= lim *�*↓0
= lim *�*↓0
= lim *�*↓0
= *A σ*<sup>2</sup> *B*,
). Every element in *BO*(*M*1, *M*2, *M*3�
The next theorem is a fundamental result of [17].
*A� σ*<sup>1</sup> *B*
*A*1/2
*A*1/2
*A*1/2
*A� σ*<sup>2</sup> *B*
Φ ◦ *Tn* is Bochner integrable and
�
*A* > 0 and *B* 0,
*BO*(*M*1, *M*2, *M*3�
*BO*(*M*1, *M*2, *M*3�
For each *A*, *B* 0, we obtain by (M3') that
*Tn*(*t*) *dμ*(*t*)*ξ*, *ξ*� = Φ(
obtain from the dominated convergence theorem that
**Theorem 0.27.** *There is a one-to-one correspondence between connections and finite Borel measures on the unit interval. In fact, every connection takes the form*
$$A\,\sigma\,B = \int\_{[0,1]} A\,!\_{\text{f}}\, B\,d\mu(t), \quad A\,\,B \gtrless 0 \tag{19}$$
*for some finite Borel measure μ on* [0, 1]*. Moreover, the map μ* �→ *σ is affine and order-preserving. Here, the order-presering means that when μ<sup>i</sup>* �→ *σ<sup>i</sup> (i=1,2), if μ*1(*E*) *μ*2(*E*) *for all Borel sets E in* [0, 1]*, then A <sup>σ</sup>*<sup>1</sup> *<sup>B</sup> <sup>A</sup> <sup>σ</sup>*<sup>2</sup> *B for all A*, *<sup>B</sup>* <sup>∈</sup> *<sup>B</sup>*(H)+*.*
*Proof.* The proof of the first part is contained in the proof of Theorem 0.22. This map is affine because of the linearity of the map *<sup>μ</sup>* �→ *f d<sup>μ</sup>* on the set of finite positive measures and the bijective correspondence between connections and Borel measures. It is straight forward to show that this map is order-preserving.
**Remark 0.28.** Let us consider operator connections from electrical circuit viewpoint. A general connection represents a formulation of making a new impedance from two given impedances. The integral representation (19) shows that such a formulation can be described as a weighed series connection of (infinite) weighed harmonic means. From this point of view, the theory of operator connections can be regarded as a mathematical theory of electrical circuits.
**Definition 0.29.** Let *σ* be a connection. The operator monotone function *f* in (16) is called the *representing function* of *σ*. If *μ* is the measure corresponds to *σ* in Theorem 0.27, the measure *μψ*−<sup>1</sup> that takes a Borel set *E* in [0, ∞] to *μ*(*ψ*−1(*E*)) is called the *representing measure* of *σ* in the Kubo-Ando's theory. Here, *ψ* : [0, 1] → [0, ∞] is a homeomorphism *t* �→ *t*/(1 − *t*).
Since every connection *σ* has an integral representation (19), properties of weighed harmonic means reflect properties of a general connection. Hence, every connection *σ* satisfies the following properties for all *A*, *B* 0, *T* ∈ *B*(H) and invertible *X* ∈ *B*(H):
16 Will-be-set-by-IN-TECH 178 Linear Algebra – Theorems and Applications
Moreover, if *A*, *B* > 0,
$$A\,\sigma\,\mathbf{B} = A^{1/2}f(A^{-1/2}\mathbf{B}A^{-1/2})A^{1/2} \tag{20}$$
and, in general, for each *A*, *B* 0,
$$A \,\sigma \, B = \lim\_{\epsilon \downarrow 0} A\_{\epsilon} \,\sigma \, B\_{\epsilon} \tag{21}$$
where *A�* ≡ *A* + *�I* and *B�* ≡ *B* + *�I*. These properties are useful tools for deriving operator inequalities involving connections. The formulas (20) and (21) give a way for computing the formula of connection from its representing function.
**Remark 0.31.** The map *σ* �→ *μ*, where *μ* is the representing measure of *σ*, is not order-preserving in general. Indeed, the representing measure of is given by *μ* = (*δ*<sup>0</sup> + *δ*∞)/2 while the representing measure of ! is given by *δ*1. We have ! but *δ*<sup>1</sup> *μ*.
#### **6. Operator means**
According to [24], a (scalar) mean is a binary operation *M* on (0, ∞) such that *M*(*s*, *t*) lies between *s* and *t* for any *s*, *t* > 0. For a connection, this property is equivalent to various properties in the next theorem.
**Theorem 0.32.** *The following are equivalent for a connection <sup>σ</sup> on B*(H)+*:*
*Proof.* Clearly, (i) ⇒ (iii) ⇒ (iv). The implication (iii) ⇒ (ii) follows from the congruence invariance and the continuity from above of *σ*. The monotonicity of *σ* is used to prove (ii) ⇒ (i). Since
$$I\sigma I = \int\_{[0,1]} I \mathbf{1}\_t \, I \, d\mu(t) = \mu([0,1])I\nu$$
we obtain that (iv) ⇒ (v) ⇒ (iii).
16 Will-be-set-by-IN-TECH
*A σ B* = lim *�*↓0
where *A�* ≡ *A* + *�I* and *B�* ≡ *B* + *�I*. These properties are useful tools for deriving operator inequalities involving connections. The formulas (20) and (21) give a way for computing the
**Example 0.30.** 1. The left- and the right-trivial means have representing functions given by *t* �→ 1 and *t* �→ *t*, respectively. The representing measures of the left- and the right-trivial means are given respectively by *δ*<sup>0</sup> and *δ*<sup>∞</sup> where *δ<sup>x</sup>* is the Dirac measure at *x*. So, the *α*-weighed arithmetic mean has the representing function *t* �→ (1 − *α*) + *αt* and it has
3. The harmonic mean has the representing function *t* �→ 2*t*/(1 + *t*) while *t* �→ *t*/(1 + *t*)
**Remark 0.31.** The map *σ* �→ *μ*, where *μ* is the representing measure of *σ*, is not order-preserving in general. Indeed, the representing measure of is given by *μ* = (*δ*<sup>0</sup> +
According to [24], a (scalar) mean is a binary operation *M* on (0, ∞) such that *M*(*s*, *t*) lies between *s* and *t* for any *s*, *t* > 0. For a connection, this property is equivalent to various
*Proof.* Clearly, (i) ⇒ (iii) ⇒ (iv). The implication (iii) ⇒ (ii) follows from the congruence invariance and the continuity from above of *σ*. The monotonicity of *σ* is used to prove (ii) ⇒
*I* !*<sup>t</sup> I dμ*(*t*) = *μ*([0, 1])*I*,
*δ*∞)/2 while the representing measure of ! is given by *δ*1. We have ! but *δ*<sup>1</sup> *μ*.
**Theorem 0.32.** *The following are equivalent for a connection <sup>σ</sup> on B*(H)+*:*
*(i) σ satisfies the* betweenness property*, i.e. A B* ⇒ *A A σ B B. (ii) <sup>σ</sup> satisfies the* fixed point property*, i.e. A <sup>σ</sup> <sup>A</sup>* <sup>=</sup> *A for all A* <sup>∈</sup> *<sup>B</sup>*(H)+*.*
*(v) the representing measure μ of σ is normalized, i.e. μ is a probability measure.*
[0,1]
*(iv) the representing function f of σ is normalized, i.e. f*(1) = 1*.*
*I σ I* =
) *t*(*A σ A*�
)+(1 − *t*)(*B σ B*�
*A� σ B�* (21)
*A σ B* = *A*1/2 *f*(*A*−1/2*BA*−1/2)*A*1/2 (20)
1/2.
) for *t* ∈ [0, 1].
• *transformer inequality*: *T*∗(*A σ B*)*T* (*T*∗*AT*) *σ* (*T*∗*BT*); • *congruence invariance*: *X*∗(*A σ B*)*X* = (*X*∗*AX*) *σ* (*X*∗*BX*);
• *concavity*: (*tA* + (1 − *t*)*B*) *σ* (*tA*� + (1 − *t*)*B*�
formula of connection from its representing function.
(1 − *α*)*δ*<sup>0</sup> + *αδ*<sup>∞</sup> as the representing measure.
corrsponds to the parallel sum.
**6. Operator means**
properties in the next theorem.
*(iii) σ is normalized, i.e. I σ I* = *I.*
(i). Since
2. The geometric mean has the representing function *t* �→ *t*
Moreover, if *A*, *B* > 0,
and, in general, for each *A*, *B* 0,
**Definition 0.33.** A *mean* is a connection satisfying one, and thus all, of the properties in the previous theorem.
Hence, every mean in Kubo-Ando's sense satisfies the desired properties (A1)–(A9) in Section 3. As a consequence of Theorem 0.32, a convex combination of means is a mean.
**Theorem 0.34.** *Given a Hilbert space* H*, there exist affine bijections between any pair of the following objects:*
*Moreover, these correspondences between (i) and (ii) are order isomorphic. Hence, there exists an affine order isomorphism between the means on the positive operators acting on different Hilbert spaces.*
*Proof.* Follow from Theorems 0.27 and 0.32.
**Example 0.35.** The left- and right-trivial means, weighed arithmetic means, the geometric mean and the harmonic mean are means. The parallel sum is not a mean since its representing function is not normalized.
**Example 0.36.** The function *t* �→ *t <sup>α</sup>* is an operator monotone function on **<sup>R</sup>**<sup>+</sup> for each *<sup>α</sup>* <sup>∈</sup> [0, 1] by the Löwner-Heinz's inequality. So it produces a mean, denoted by #*α*, on *<sup>B</sup>*(H)+. By the direct computation,
$$s \#\_{\mathfrak{a}} t = s^{1-\mathfrak{a}} t^{\mathfrak{a}} \, , \tag{22}$$
i.e. #*<sup>α</sup>* is the *α*-weighed geometric mean on **R**+. So the *α*-weighed geometric mean on **R**<sup>+</sup> is really a Kubo-Ando mean. The *<sup>α</sup>-weighed geometric mean* on *<sup>B</sup>*(H)<sup>+</sup> is defined to be the mean corresponding to that mean on **R**+. Since *t <sup>α</sup>* has an integral expression
$$t^{\mathfrak{a}} = \frac{\sin \mathfrak{a} \pi}{\pi} \int\_0^\infty \frac{t \lambda^{\mathfrak{a}-1}}{t+\lambda} \, dm(\lambda) \tag{23}$$
(see [7]) where *m* denotes the Lebesgue measure, the representing measure of #*<sup>α</sup>* is given by
$$d\mu(\lambda) = \frac{\sin \alpha \pi}{\pi} \frac{\lambda^{\alpha - 1}}{\lambda + 1} \, dm(\lambda). \tag{24}$$
**Example 0.37.** Consider the operator monotone function
$$t \mapsto \frac{t}{(1-\alpha)t + \alpha'} \quad \text{ } t \gg 0 \text{, } \alpha \in [0, 1].$$
The direct computation shows that
$$s \restriction\_{\mathfrak{A}} t = \begin{cases} ((1-a)s^{-1} + at^{-1})^{-1} \text{, s} \, t > 0; \\ 0, & \text{otherwise,} \end{cases} \tag{25}$$
which is the *<sup>α</sup>*-weighed harmonic mean. We define the *<sup>α</sup>*-*weighed harmonic mean* on *<sup>B</sup>*(H)<sup>+</sup> to be the mean corresponding to this operator monotone function.
**Example 0.38.** Consider the operator monotone function *f*(*t*)=(*t* − 1)/ log *t* for *t* > 0, *t* �= 1, *<sup>f</sup>*(0) <sup>≡</sup> 0 and *<sup>f</sup>*(1) <sup>≡</sup> 1. Then it gives rise to a mean, denoted by *<sup>λ</sup>*, on *<sup>B</sup>*(H)+. By the direct computation,
$$s \wedge t = \begin{cases} \frac{s - t}{\log s - \log t}, s > 0, t > 0, s \neq t;\\ s\_\prime & s = t \\ 0, & \text{otherwise} \end{cases} \tag{26}$$
i.e. *λ* is the logarithmic mean on **R**+. So the logarithmic mean on **R**<sup>+</sup> is really a mean in Kubo-Ando's sense. The *logarithmic mean* on *<sup>B</sup>*(H)<sup>+</sup> is defined to be the mean corresponding to this operator monotone function.
**Example 0.39.** The map *t* �→ (*t <sup>r</sup>* + *t* <sup>1</sup>−*r*)/2 is operator monotone for any *<sup>r</sup>* <sup>∈</sup> [0, 1]. This function produces a mean on *<sup>B</sup>*(H)+. The computation shows that
$$(\mathbf{s}, t) \mapsto \frac{s^r t^{1-r} + s^{1-r} t^r}{2}.$$
However, the corresponding mean on *<sup>B</sup>*(H)<sup>+</sup> is not given by the formula
$$(A, B) \mapsto \frac{A^r B^{1-r} + A^{1-r} B^r}{2} \tag{27}$$
since it is not a binary operation on *<sup>B</sup>*(H)+. In fact, the formula (27) is considered in [8], called the *Heinz mean* of *A* and *B*.
**Example 0.40.** For each *p* ∈ [−1, 1] and *α* ∈ [0, 1], the map
$$t \mapsto [(1 - \mathfrak{a}) + \mathfrak{a}t^p]^{1/p}$$
is an operator monotone function on **R**+. Here, the case *p* = 0 is understood that we take limit as *p* → 0. Then
$$s \#\_{p, \mathfrak{a}} t = [(1 - \mathfrak{a})s^p + at^p]^{1/p}. \tag{28}$$
The corresponding mean on *<sup>B</sup>*(H)<sup>+</sup> is called the *quasi-arithmetic power mean* with parameter (*p*, *α*), defined for *A* > 0 and *B* 0 by
$$A \#\_{p\mu} B = A^{1/2} [(1 - \mathfrak{a})I + \mathfrak{a} (A^{-1/2} \mathfrak{B} A^{-1/2})^p]^{1/p} A^{1/2}. \tag{29}$$
The class of quasi-arithmetic power means contain many kinds of means: The mean #1,*<sup>α</sup>* is the *α*-weighed arithmetic mean. The case #0,*<sup>α</sup>* is the *α*-weighed geometric mean. The case #−1,*<sup>α</sup>* is the *α*-weighed harmonic mean. The mean #*p*,1/2 is the *power mean* or *binomial mean* of order *p*. These means satisfy the property that
$$A \#\_{p,a} B = B \#\_{p,1-a} A. \tag{30}$$
Moreover, they are interpolated in the sense that for all *p*, *q*, *α* ∈ [0, 1],
$$(A\#\_{r,p}B)\#\_{r,\mathfrak{a}}(A\#\_{r,\mathfrak{g}}B) \;=\; A\#\_{r,(1-\mathfrak{a})p+\mathfrak{a}\mathfrak{g}}B.\tag{31}$$
**Example 0.41.** If *σ*1, *σ*<sup>2</sup> are means such that *σ*<sup>1</sup> *σ*2, then there is a family of means that interpolates between *σ*<sup>1</sup> and *σ*2, namely, (1 − *α*)*σ*<sup>1</sup> + *ασ*<sup>2</sup> for all *α* ∈ [0, 1]. Note that the map *α* �→ (1 − *α*)*σ*<sup>1</sup> + *ασ*<sup>2</sup> is increasing. For instance, the *Heron mean* with weight *α* ∈ [0, 1] is defined to be *h<sup>α</sup>* = (1 − *α*) # + *α* . This family is the linear interpolations between the geometric mean and the arithmetic mean. The representing function of *h<sup>α</sup>* is given by
$$t \mapsto (1 - \alpha)t^{1/2} + \frac{\alpha}{2}(1 + t).$$
The case *α* = 2/3 is called the *Heronian mean* in the literature.
18 Will-be-set-by-IN-TECH
which is the *<sup>α</sup>*-weighed harmonic mean. We define the *<sup>α</sup>*-*weighed harmonic mean* on *<sup>B</sup>*(H)<sup>+</sup> to
**Example 0.38.** Consider the operator monotone function *f*(*t*)=(*t* − 1)/ log *t* for *t* > 0, *t* �= 1, *<sup>f</sup>*(0) <sup>≡</sup> 0 and *<sup>f</sup>*(1) <sup>≡</sup> 1. Then it gives rise to a mean, denoted by *<sup>λ</sup>*, on *<sup>B</sup>*(H)+. By the direct
log *<sup>s</sup>*−log *<sup>t</sup>* , *<sup>s</sup>* <sup>&</sup>gt; 0, *<sup>t</sup>* <sup>&</sup>gt; 0,*<sup>s</sup>* �<sup>=</sup> *<sup>t</sup>*;
<sup>1</sup>−*<sup>r</sup>* + *s*1−*rt*
*ArB*<sup>1</sup>−*<sup>r</sup>* + *A*1−*rBr*
*p*] 1/*p*
*p*]
*<sup>A</sup>* #*p*,*<sup>α</sup> <sup>B</sup>* = *<sup>B</sup>* #*p*,1−*<sup>α</sup> <sup>A</sup>*. (30)
(*<sup>A</sup>* #*r*,*<sup>p</sup> <sup>B</sup>*) #*r*,*<sup>α</sup>* (*<sup>A</sup>* #*r*,*<sup>q</sup> <sup>B</sup>*) = *<sup>A</sup>* #*r*,(1−*α*)*p*+*α<sup>q</sup> <sup>B</sup>*. (31)
*r* <sup>2</sup> .
<sup>1</sup>−*r*)/2 is operator monotone for any *<sup>r</sup>* <sup>∈</sup> [0, 1]. This
<sup>2</sup> (27)
1/*p*. (28)
1/*pA*1/2. (29)
(26)
be the mean corresponding to this operator monotone function.
*s λ t* =
⎧ ⎨ ⎩
*<sup>r</sup>* + *t*
(*s*, *t*) �→
However, the corresponding mean on *<sup>B</sup>*(H)<sup>+</sup> is not given by the formula
(*A*, *B*) �→
**Example 0.40.** For each *p* ∈ [−1, 1] and *α* ∈ [0, 1], the map
function produces a mean on *<sup>B</sup>*(H)+. The computation shows that
*s*−*t*
*s*, *s* = *t* 0, otherwise,
*srt*
since it is not a binary operation on *<sup>B</sup>*(H)+. In fact, the formula (27) is considered in [8], called
*t* �→ [(1 − *α*) + *αt*
*<sup>s</sup>* #*p*,*<sup>α</sup> <sup>t</sup>* = [(<sup>1</sup> <sup>−</sup> *<sup>α</sup>*)*s<sup>p</sup>* <sup>+</sup> *<sup>α</sup><sup>t</sup>*
*<sup>A</sup>* #*p*,*<sup>α</sup> <sup>B</sup>* <sup>=</sup> *<sup>A</sup>*1/2[(<sup>1</sup> <sup>−</sup> *<sup>α</sup>*)*<sup>I</sup>* <sup>+</sup> *<sup>α</sup>*(*A*−1/2*BA*−1/2)*p*]
Moreover, they are interpolated in the sense that for all *p*, *q*, *α* ∈ [0, 1],
is an operator monotone function on **R**+. Here, the case *p* = 0 is understood that we take
The corresponding mean on *<sup>B</sup>*(H)<sup>+</sup> is called the *quasi-arithmetic power mean* with parameter
The class of quasi-arithmetic power means contain many kinds of means: The mean #1,*<sup>α</sup>* is the *α*-weighed arithmetic mean. The case #0,*<sup>α</sup>* is the *α*-weighed geometric mean. The case #−1,*<sup>α</sup>* is the *α*-weighed harmonic mean. The mean #*p*,1/2 is the *power mean* or *binomial mean* of order *p*.
i.e. *λ* is the logarithmic mean on **R**+. So the logarithmic mean on **R**<sup>+</sup> is really a mean in Kubo-Ando's sense. The *logarithmic mean* on *<sup>B</sup>*(H)<sup>+</sup> is defined to be the mean corresponding
computation,
to this operator monotone function.
**Example 0.39.** The map *t* �→ (*t*
the *Heinz mean* of *A* and *B*.
limit as *p* → 0. Then
(*p*, *α*), defined for *A* > 0 and *B* 0 by
These means satisfy the property that
#### **7. Applications to operator monotonicity and concavity**
In this section, we generalize the matrix and operator monotonicity and concavity in the literature (see e.g. [3, 9]) in such a way that the geometric mean, the harmonic mean or specific operator means are replaced by general connections. Recall the following terminology. A continuous function *f* : *I* → **R** is called an *operator concave function* if
$$f(tA + (1 - t)B) \geqslant tf(A) + (1 - t)f(B)$$
for any *t* ∈ [0, 1] and Hermitian operators *A*, *B* ∈ *B*(H) whose spectrums are contained in the interval *I* and for all Hilbert spaces H. A well-known result is that a continuous function *<sup>f</sup>* : **<sup>R</sup>**<sup>+</sup> <sup>→</sup> **<sup>R</sup>**<sup>+</sup> is operator monotone if and only if it is operator concave. Hence, the maps *t* �→ *t <sup>r</sup>* and *<sup>t</sup>* �→ log *<sup>t</sup>* are operator concave for *<sup>r</sup>* <sup>∈</sup> [0, 1]. Let <sup>H</sup> and <sup>K</sup> be Hilbert spaces. A map Φ : *B*(H) → *B*(K) is said to be *positive* if Φ(*A*) 0 whenever *A* 0. It is called *unital* if Φ(*I*) = *I*. We say that a positive map Φ is *strictly positive* if Φ(*A*) > 0 when *A* > 0. A map Ψ from a convex subset <sup>C</sup> of *<sup>B</sup>*(H)*sa* to *<sup>B</sup>*(K)*sa* is called *concave* if for each *<sup>A</sup>*, *<sup>B</sup>* ∈ C and *<sup>t</sup>* <sup>∈</sup> [0, 1],
$$
\Psi(tA + (1-t)B) \gtrsim t\Psi(A) + (1-t)\Psi(B).
$$
A map <sup>Ψ</sup> : *<sup>B</sup>*(H)*sa* <sup>→</sup> *<sup>B</sup>*(K)*sa* is called *monotone* if *<sup>A</sup> <sup>B</sup>* assures <sup>Ψ</sup>(*A*) <sup>Ψ</sup>(*B*). So, in particular, the map *<sup>A</sup>* �→ *<sup>A</sup><sup>r</sup>* is monotone and concave on *<sup>B</sup>*(H)<sup>+</sup> for each *<sup>r</sup>* <sup>∈</sup> [0, 1]. The map *<sup>A</sup>* �→ log *<sup>A</sup>* is monotone and concave on *<sup>B</sup>*(H)++.
Note first that, from the previous section, the quasi-arithmetic power mean (*A*, *B*) �→ *A* #*p*,*<sup>α</sup> B* is monotone and concave for any *p* ∈ [−1, 1] and *α* ∈ [0, 1]. In particular, the following are monotone and concave:
Recall the following lemma from [9].
**Lemma 0.42** (Choi's inequality)**.** *If* Φ : *B*(H) → *B*(K) *is linear, strictly positive and unital, then for every A* > 0*,* Φ(*A*)−<sup>1</sup> Φ(*A*−1)*.*
**Proposition 0.43.** *If* Φ : *B*(H) → *B*(K) *is linear and strictly positive, then for any A*, *B* > 0
$$
\Phi(A)\Phi(B)^{-1}\Phi(A) \lesssim \Phi(AB^{-1}A).\tag{32}
$$
*Proof.* For each *<sup>X</sup>* <sup>∈</sup> *<sup>B</sup>*(H), set <sup>Ψ</sup>(*X*) = <sup>Φ</sup>(*A*)−1/2Φ(*A*1/2*XA*1/2)Φ(*A*)−1/2. Then <sup>Ψ</sup> is a unital strictly positive linear map. So, by Choi's inequality, Ψ(*A*)−<sup>1</sup> Ψ(*A*−1) for all *A* > 0. For each *A*, *B* > 0, we have by Lemma 0.42 that
$$\Phi(A)^{1/2}\Phi(B)^{-1}\Phi(A)^{1/2} = \Psi(A^{-1/2}BA^{-1/2})^{-1}$$
$$\lesssim \Psi\left((A^{-1/2}BA^{-1/2})^{-1}\right)$$
$$= \Phi(A)^{-1/2}\Phi(AB^{-1}A)\Phi(A)^{-1/2}.$$
So, we have the claim.
**Theorem 0.44.** *If* Φ : *B*(H) → *B*(K) *is a positive linear map which is norm-continuous, then for any connection <sup>σ</sup> on B*(K)<sup>+</sup> *and for each A*, *<sup>B</sup>* <sup>&</sup>gt; <sup>0</sup>*,*
$$
\Phi(A \,\sigma \, B) \lessdot \Phi(A) \,\sigma \, \Phi(B). \tag{33}
$$
*If, addition,* Φ *is strongly continuous, then* (33) *holds for any A*, *B* 0*.*
*Proof.* First, consider *A*, *B* > 0. Assume that Φ is strictly positive. For each *X* ∈ *B*(H), set
$$\Psi(X) = \Phi(B)^{-1/2} \Phi(B^{1/2} X B^{1/2}) \Phi(B)^{-1/2}.$$
Then Ψ is a unital strictly positive linear map. So, by Choi's inequality, Ψ(*C*)−<sup>1</sup> Ψ(*C*−1) for all *<sup>C</sup>* <sup>&</sup>gt; 0. For each *<sup>t</sup>* <sup>∈</sup> [0, 1], put *Xt* <sup>=</sup> *<sup>B</sup>*−1/2(*<sup>A</sup>* !*<sup>t</sup> <sup>B</sup>*)*B*−1/2 <sup>&</sup>gt; 0. We obtain from the previous proposition that
$$\begin{aligned} \Phi(A\,^\circ\_t B) &= \Phi(B)^{1/2} \Psi(X\_t) \Phi(B)^{1/2} \\\\ &\leqslant \Phi(B)^{1/2} [\Psi(X\_t^{-1})]^{-1} \Phi(B)^{1/2} \\\\ &= \Phi(B) [\Phi(B((1-t)A^{-1} + tB^{-1})B)]^{-1} \Phi(B) \\\\ &= \Phi(B) [(1-t)\Phi(BA^{-1}B) + t\Phi(B)]^{-1} \Phi(B) \\\\ &\leqslant \Phi(B) [(1-t)\Phi(B)\Phi(A)^{-1}\Phi(B) + t\Phi(B)]^{-1} \Phi(B) \\\\ &= \Phi(A) \, ^\circ\_t \Phi(B) .\end{aligned}$$
For general case of Φ, consider the family Φ*�*(*A*) = Φ(*A*) + *�I* where *�* > 0. Since the map (*A*, *<sup>B</sup>*) �→ *<sup>A</sup>* !*<sup>t</sup> <sup>B</sup>* = [(<sup>1</sup> <sup>−</sup> *<sup>t</sup>*)*A*−<sup>1</sup> <sup>+</sup> *tB*−1] <sup>−</sup><sup>1</sup> is norm-continuous, we arrive at
$$
\Phi(A \restriction\_t B) \lesssim \Phi(A) \restriction\_t \Phi(B).
$$
For each connection *σ*, since Φ is a bounded linear operator, we have
$$\begin{aligned} \Phi(A \,\sigma \, B) &= \Phi(\int\_{[0,1]} A \, !\_t \, B \, d\mu(t)) = \int\_{[0,1]} \Phi(A \, !\_t \, B) \, d\mu(t) \\ &\leqslant \int\_{[0,1]} \Phi(A) \, !\_t \, \Phi(B) \, d\mu(t) = \Phi(A) \, \sigma \, \Phi(B) .\end{aligned}$$
Suppose further that Φ is strongly continuous. Then, for each *A*, *B* 0,
$$\begin{aligned} \Phi(A \,\, \sigma \,\, B) &= \Phi(\lim\_{\epsilon \downarrow 0} (A + \epsilon I) \,\, \sigma \,(B + \epsilon I)) = \lim\_{\epsilon \downarrow 0} \Phi((A + \epsilon I) \,\, \sigma \,(B + \epsilon I)) \\ &\leqslant \lim\_{\epsilon \downarrow 0} \Phi(A + \epsilon I) \,\, \sigma \, \Phi(B + \epsilon I) = \Phi(A) \,\, \sigma \, \Phi(B). \end{aligned}$$
The proof is complete.
20 Will-be-set-by-IN-TECH
*Proof.* For each *<sup>X</sup>* <sup>∈</sup> *<sup>B</sup>*(H), set <sup>Ψ</sup>(*X*) = <sup>Φ</sup>(*A*)−1/2Φ(*A*1/2*XA*1/2)Φ(*A*)−1/2. Then <sup>Ψ</sup> is a unital strictly positive linear map. So, by Choi's inequality, Ψ(*A*)−<sup>1</sup> Ψ(*A*−1) for all *A* > 0. For
Φ(*A*)Φ(*B*)−1Φ(*A*) Φ(*AB*−1*A*). (32)
(*A*−1/2*BA*−1/2)−<sup>1</sup>
= Φ(*A*)−1/2Φ(*AB*−1*A*)Φ(*A*)<sup>−</sup>1/2.
Φ(*A σ B*) Φ(*A*) *σ* Φ(*B*). (33)
**Proposition 0.43.** *If* Φ : *B*(H) → *B*(K) *is linear and strictly positive, then for any A*, *B* > 0
Φ(*A*)1/2Φ(*B*)−1Φ(*A*)1/2 = Ψ(*A*−1/2*BA*−1/2)−<sup>1</sup>
Ψ
**Theorem 0.44.** *If* Φ : *B*(H) → *B*(K) *is a positive linear map which is norm-continuous, then for*
*Proof.* First, consider *A*, *B* > 0. Assume that Φ is strictly positive. For each *X* ∈ *B*(H), set
Ψ(*X*) = Φ(*B*)−1/2Φ(*B*1/2*XB*1/2)Φ(*B*)<sup>−</sup>1/2.
Then Ψ is a unital strictly positive linear map. So, by Choi's inequality, Ψ(*C*)−<sup>1</sup> Ψ(*C*−1) for all *<sup>C</sup>* <sup>&</sup>gt; 0. For each *<sup>t</sup>* <sup>∈</sup> [0, 1], put *Xt* <sup>=</sup> *<sup>B</sup>*−1/2(*<sup>A</sup>* !*<sup>t</sup> <sup>B</sup>*)*B*−1/2 <sup>&</sup>gt; 0. We obtain from the previous
*<sup>t</sup>* )]−1Φ(*B*)1/2
<sup>=</sup> <sup>Φ</sup>(*B*)[Φ(*B*((<sup>1</sup> <sup>−</sup> *<sup>t</sup>*)*A*−<sup>1</sup> <sup>+</sup> *tB*−1)*B*)]−1Φ(*B*)
<sup>=</sup> <sup>Φ</sup>(*B*)[(<sup>1</sup> <sup>−</sup> *<sup>t</sup>*)Φ(*BA*−1*B*) + *<sup>t</sup>*Φ(*B*)]−1Φ(*B*)
For general case of Φ, consider the family Φ*�*(*A*) = Φ(*A*) + *�I* where *�* > 0. Since the map
Φ(*A* !*<sup>t</sup> B*) Φ(*A*)!*<sup>t</sup>* Φ(*B*).
<sup>Φ</sup>(*B*)[(<sup>1</sup> <sup>−</sup> *<sup>t</sup>*)Φ(*B*)Φ(*A*)−1Φ(*B*) + *<sup>t</sup>*Φ(*B*)]−1Φ(*B*)
<sup>−</sup><sup>1</sup> is norm-continuous, we arrive at
each *A*, *B* > 0, we have by Lemma 0.42 that
*any connection <sup>σ</sup> on B*(K)<sup>+</sup> *and for each A*, *<sup>B</sup>* <sup>&</sup>gt; <sup>0</sup>*,*
*If, addition,* Φ *is strongly continuous, then* (33) *holds for any A*, *B* 0*.*
Φ(*A* !*<sup>t</sup> B*) = Φ(*B*)1/2Ψ(*Xt*)Φ(*B*)1/2
Φ(*B*)1/2[Ψ(*X*−<sup>1</sup>
= Φ(*A*)!*<sup>t</sup>* Φ(*B*).
(*A*, *<sup>B</sup>*) �→ *<sup>A</sup>* !*<sup>t</sup> <sup>B</sup>* = [(<sup>1</sup> <sup>−</sup> *<sup>t</sup>*)*A*−<sup>1</sup> <sup>+</sup> *tB*−1]
So, we have the claim.
proposition that
As a special case, if Φ : *Mn*(**C**) → *Mn*(**C**) is a positive linear map, then for any connection *σ* and for any positive semidefinite matrices *A*, *B* ∈ *Mn*(**C**), we have
$$
\Phi(A\sigma B) \lessapprox \Phi(A)\,\sigma\,\Phi(B).
$$
In particular, Φ(*A*) #*p*,*<sup>α</sup>* Φ(*B*) Φ(*A*) #*p*,*<sup>α</sup>* Φ(*B*) for any *p* ∈ [−1, 1] and *α* ∈ [0, 1].
**Theorem 0.45.** *If* <sup>Φ</sup>1, <sup>Φ</sup><sup>2</sup> : *<sup>B</sup>*(H)<sup>+</sup> <sup>→</sup> *<sup>B</sup>*(K)<sup>+</sup> *are concave, then the map*
$$(A\_1, A\_2) \mapsto \Phi\_1(A\_1) \,\sigma \,\Phi\_2(A\_2) \tag{34}$$
*is concave for any connection <sup>σ</sup> on B*(K)+*.*
*Proof.* Let *A*1, *A*� <sup>1</sup>, *A*2, *A*� <sup>2</sup> 0 and *t* ∈ [0, 1]. The concavity of Φ<sup>1</sup> and Φ<sup>2</sup> means that for *i* = 1, 2
$$
\Phi\_i(tA\_i + (1-t)A\_i') \geqslant t\Phi\_i(A\_i) + (1-t)\Phi\_i(A\_i').
$$
It follows from the monotonicity and concavity of *σ* that
$$\begin{aligned} \Phi\_1(tA\_1 + (1-t)A\_1') \sigma \Phi\_2(tA\_2 + (1-t)A\_2')\\ &\geqslant \left[t\Phi\_1(A\_1) + (1-t)\Phi\_1(A\_1')\right] \sigma \left[t\Phi\_2(A\_2) + (1-t)\Phi\_2(A\_2')\right] \\ &\geqslant t\left[\Phi\_1(A\_1)\sigma\Phi\_2(A\_2)\right] + (1-t)\left[\Phi\_1(A\_1)\sigma\Phi\_2(A\_2)\right]. \end{aligned}$$
This shows the concavity of the map (*A*1, *A*2) �→ Φ1(*A*1) *σ* Φ2(*A*2) .
In particular, if Φ<sup>1</sup> and Φ<sup>2</sup> are concave, then so is (*A*, *B*) �→ Φ1(*A*) #*p*,*α*Φ2(*B*) for *p* ∈ [−1, 1] and *α* ∈ [0, 1].
**Corollary 0.46.** *Let <sup>σ</sup> be a connection. Then, for any operator monotone functions f* , *<sup>g</sup>* : **<sup>R</sup>**<sup>+</sup> <sup>→</sup> **<sup>R</sup>**+*, the map* (*A*, *B*) �→ *f*(*A*) *σ g*(*B*) *is concave. In particular,*
**Theorem 0.47.** *If* <sup>Φ</sup>1, <sup>Φ</sup><sup>2</sup> : *<sup>B</sup>*(H)<sup>+</sup> <sup>→</sup> *<sup>B</sup>*(K)<sup>+</sup> *are monotone, then the map*
$$(A\_1, A\_2) \mapsto \Phi\_1(A\_1) \,\sigma \,\Phi\_2(A\_2) \tag{35}$$
*is monotone for any connection <sup>σ</sup> on B*(K)+*.*
*Proof.* Let *A*<sup>1</sup> *A*� <sup>1</sup> and *A*<sup>2</sup> *A*� <sup>2</sup>. Then Φ1(*A*1) Φ1(*A*� <sup>1</sup>) and Φ2(*A*2) Φ2(*A*� <sup>2</sup>) by the monotonicity of Φ<sup>1</sup> and Φ2. Now, the monotonicity of *σ* forces Φ1(*A*1) *σ* Φ2(*A*2) Φ1(*A*� <sup>1</sup>) *σ* Φ2(*A*� 2).
In particular, if Φ<sup>1</sup> and Φ<sup>2</sup> are monotone, then so is (*A*, *B*) �→ Φ1(*A*) #*p*,*α*Φ2(*B*) for *p* ∈ [−1, 1] and *α* ∈ [0, 1].
**Corollary 0.48.** *Let <sup>σ</sup> be a connection. Then, for any operator monotone functions f* , *<sup>g</sup>* : **<sup>R</sup>**<sup>+</sup> <sup>→</sup> **<sup>R</sup>**+*, the map* (*A*, *B*) �→ *f*(*A*) *σ g*(*B*) *is monotone. In particular,*
*(1) the map* (*A*, *<sup>B</sup>*) �→ *<sup>A</sup><sup>r</sup> <sup>σ</sup> <sup>B</sup><sup>s</sup> is monotone on B*(H)<sup>+</sup> *for any r*,*<sup>s</sup>* <sup>∈</sup> [0, 1]*,*
*(2) the map* (*A*, *<sup>B</sup>*) �→ (log *<sup>A</sup>*) *<sup>σ</sup>* (log *<sup>B</sup>*) *is monotone on B*(H)++*.*
**Corollary 0.49.** *Let <sup>σ</sup> be a connection on B*(H)+*. If* <sup>Φ</sup>1, <sup>Φ</sup><sup>2</sup> : *<sup>B</sup>*(H)<sup>+</sup> <sup>→</sup> *<sup>B</sup>*(H)<sup>+</sup> *is monotone and strongly continuous, then the map*
$$(A, B) \mapsto \Phi\_1(A) \,\sigma \,\Phi\_2(B) \tag{36}$$
*is a connection on B*(H)+*. Hence, the map*
$$f(A, B) \mapsto f(A) \,\sigma \,\mathrm{g}(B) \tag{37}$$
*is a connection for any operator monotone functions f* , *<sup>g</sup>* : **<sup>R</sup>**<sup>+</sup> <sup>→</sup> **<sup>R</sup>**+*.*
*Proof.* The monotonicity of this map follows from the previous result. It is easy to see that this map satisfies the transformer inequality. Since Φ<sup>1</sup> and Φ<sup>2</sup> strongly continuous, this binary operation satisfies the (separate or joint) continuity from above. The last statement follows from the fact that if *An* ↓ *A*, then Sp(*An*) ⊆ [0, �*A*1�] for all *n* and hence *f*(*An*) → *f*(*A*).
#### **8. Applications to operator inequalities**
In this section, we apply Kubo-Ando's theory in order to get simple proofs of many classical inequalities in the context of operators.
**Theorem 0.50** (AM-LM-GM-HM inequalities)**.** *For A*, *B* 0*, we have*
$$A \,\!\!\/ B \leqslant A \,\#\!\/ B \leqslant A \,\lambda \,\!\/ B \leqslant A \,\!\/ \to \!\!\/ B. \tag{38}$$
*Proof.* It is easy to see that, for each *t* > 0, *t* �= 1,
$$\frac{2t}{1+t} \le t^{1/2} \le \frac{t-1}{\log t} \le \frac{1+t}{2}.$$
Now, we apply the order isomorphism which converts inequalities of operator monotone functions to inequalities of the associated operator connections.
**Theorem 0.51** (Weighed AM-GM-HM inequalities)**.** *For A*, *B* 0 *and α* ∈ [0, 1]*, we have*
$$A \upharpoonright\_{\mathfrak{A}} B \lesssim A \,\#\_{\mathfrak{A}} B \lesssim A \,\top\_{\mathfrak{A}} B. \tag{39}$$
*Proof.* Apply the order isomorphism to the following inequalities:
$$\frac{t}{(1-\alpha)t+\alpha} \leqslant t^{\alpha} \leqslant 1-\alpha+\alpha t, \quad t \geqslant 0.$$
The next two theorems are given in [21].
**Theorem 0.52.** *For each i* <sup>=</sup> 1, ··· , *n, let Ai*, *Bi* <sup>∈</sup> *<sup>B</sup>*(H)+*. Then for each connection <sup>σ</sup>*
$$\sum\_{i=1}^{n} (A\_i \,\sigma \, B\_i) \lessapprox \sum\_{i=1}^{n} A\_i \,\sigma \sum\_{i=1}^{n} B\_i. \tag{40}$$
*Proof.* Use the concavity of *σ* together with the induction.
By replacing *σ* with appropriate connections, we get some interesting inequalities. (1) Cauchy-Schwarz's inequality: For *Ai*, *Bi* <sup>∈</sup> *<sup>B</sup>*(H)*sa*,
$$\sum\_{i=1}^{n} A\_i^2 \# B\_i^2 \leqslant \sum\_{i=1}^{n} A\_i^2 \# \sum\_{i=1}^{n} B\_i^2. \tag{41}$$
(2) Hölder's inequality: For *Ai*, *Bi* <sup>∈</sup> *<sup>B</sup>*(H)<sup>+</sup> and *<sup>p</sup>*, *<sup>q</sup>* <sup>&</sup>gt; 0 such that 1/*<sup>p</sup>* <sup>+</sup> 1/*<sup>q</sup>* <sup>=</sup> 1,
$$\sum\_{i=1}^{n} A\_i^p \#\_{1/p} B\_i^q \lesssim \sum\_{i=1}^{n} A\_i^p \#\_{1/p} \sum\_{i=1}^{n} B\_i^q. \tag{42}$$
(3) Minkowski's inequality: For *Ai*, *Bi* <sup>∈</sup> *<sup>B</sup>*(H)++,
$$
\left(\sum\_{i=1}^{n} (A\_i + B\_i)^{-1}\right)^{-1} \geqslant \left(\sum\_{i=1}^{n} A\_i^{-1}\right)^{-1} + \left(\sum\_{i=1}^{n} B\_i^{-1}\right)^{-1}.\tag{43}
$$
**Theorem 0.53.** *Let Ai*, *Bi* <sup>∈</sup> *<sup>B</sup>*(H)+*, i* <sup>=</sup> 1, ··· , *n, be such that*
$$A\_1 - A\_2 - \dots - A\_n \geqslant 0 \quad \text{and} \quad B\_1 - B\_2 - \dots - B\_n \geqslant 0.$$
*Then*
22 Will-be-set-by-IN-TECH
<sup>2</sup>. Then Φ1(*A*1) Φ1(*A*�
the monotonicity of Φ<sup>1</sup> and Φ2. Now, the monotonicity of *σ* forces Φ1(*A*1) *σ* Φ2(*A*2)
In particular, if Φ<sup>1</sup> and Φ<sup>2</sup> are monotone, then so is (*A*, *B*) �→ Φ1(*A*) #*p*,*α*Φ2(*B*) for *p* ∈ [−1, 1]
**Corollary 0.48.** *Let <sup>σ</sup> be a connection. Then, for any operator monotone functions f* , *<sup>g</sup>* : **<sup>R</sup>**<sup>+</sup> <sup>→</sup> **<sup>R</sup>**+*,*
**Corollary 0.49.** *Let <sup>σ</sup> be a connection on B*(H)+*. If* <sup>Φ</sup>1, <sup>Φ</sup><sup>2</sup> : *<sup>B</sup>*(H)<sup>+</sup> <sup>→</sup> *<sup>B</sup>*(H)<sup>+</sup> *is monotone and*
*Proof.* The monotonicity of this map follows from the previous result. It is easy to see that this map satisfies the transformer inequality. Since Φ<sup>1</sup> and Φ<sup>2</sup> strongly continuous, this binary operation satisfies the (separate or joint) continuity from above. The last statement follows from the fact that if *An* ↓ *A*, then Sp(*An*) ⊆ [0, �*A*1�] for all *n* and hence *f*(*An*) → *f*(*A*).
In this section, we apply Kubo-Ando's theory in order to get simple proofs of many classical
1/2
Now, we apply the order isomorphism which converts inequalities of operator monotone
*t* − 1 log *<sup>t</sup>*
(*A*1, *A*2) �→ Φ1(*A*1) *σ* Φ2(*A*2) (35)
(*A*, *B*) �→ Φ1(*A*) *σ* Φ2(*B*) (36)
(*A*, *B*) �→ *f*(*A*) *σ g*(*B*) (37)
*A* ! *B A* # *B A λ B A B*. (38)
1 + *t* 2 . <sup>1</sup>) and Φ2(*A*2) Φ2(*A*�
<sup>2</sup>) by
**Theorem 0.47.** *If* <sup>Φ</sup>1, <sup>Φ</sup><sup>2</sup> : *<sup>B</sup>*(H)<sup>+</sup> <sup>→</sup> *<sup>B</sup>*(K)<sup>+</sup> *are monotone, then the map*
*is monotone for any connection <sup>σ</sup> on B*(K)+*.*
<sup>1</sup> and *A*<sup>2</sup> *A*�
*the map* (*A*, *B*) �→ *f*(*A*) *σ g*(*B*) *is monotone. In particular,*
*(1) the map* (*A*, *<sup>B</sup>*) �→ *<sup>A</sup><sup>r</sup> <sup>σ</sup> <sup>B</sup><sup>s</sup> is monotone on B*(H)<sup>+</sup> *for any r*,*<sup>s</sup>* <sup>∈</sup> [0, 1]*,*
*(2) the map* (*A*, *<sup>B</sup>*) �→ (log *<sup>A</sup>*) *<sup>σ</sup>* (log *<sup>B</sup>*) *is monotone on B*(H)++*.*
*is a connection for any operator monotone functions f* , *<sup>g</sup>* : **<sup>R</sup>**<sup>+</sup> <sup>→</sup> **<sup>R</sup>**+*.*
**Theorem 0.50** (AM-LM-GM-HM inequalities)**.** *For A*, *B* 0*, we have*
2*t* <sup>1</sup> <sup>+</sup> *<sup>t</sup> <sup>t</sup>*
functions to inequalities of the associated operator connections.
*Proof.* Let *A*<sup>1</sup> *A*�
<sup>1</sup>) *σ* Φ2(*A*�
2).
*strongly continuous, then the map*
*is a connection on B*(H)+*. Hence, the map*
**8. Applications to operator inequalities**
*Proof.* It is easy to see that, for each *t* > 0, *t* �= 1,
inequalities in the context of operators.
Φ1(*A*�
and *α* ∈ [0, 1].
$$A\_1 \sigma \, B\_1 - \sum\_{i=2}^n A\_i \, \sigma \, B\_i \gtrsim \left( A\_1 - \sum\_{i=2}^n A\_i \right) \sigma \left( B\_1 - \sum\_{i=2}^n B\_i \right). \tag{44}$$
*Proof.* Substitute *A*<sup>1</sup> to *A*<sup>1</sup> − *A*<sup>2</sup> −···− *An* and *B*<sup>1</sup> to *B*<sup>1</sup> − *B*<sup>2</sup> −···− *Bn* in (40).
Here are consequences.
(1) Aczél's inequality: For *Ai*, *Bi* <sup>∈</sup> *<sup>B</sup>*(H)*sa*, if
$$A\_1^2 - A\_2^2 - \dots - A\_n^2 \gg 0 \quad \text{and} \quad B\_1^2 - B\_2^2 - \dots - B\_n^2 \gg 0,$$
then
$$A\_1^2 \# B\_1^2 - \sum\_{i=2}^n A\_i^2 \# B\_i^2 \geqslant \left(A\_1^2 - \sum\_{i=2}^n A\_i^2\right) \# \left(B\_1^2 - \sum\_{i=2}^n B\_i^2\right). \tag{45}$$
(2) Popoviciu's inequality: For *Ai*, *Bi* <sup>∈</sup> *<sup>B</sup>*(H)<sup>+</sup> and *<sup>p</sup>*, *<sup>q</sup>* <sup>&</sup>gt; 0 such that 1/*<sup>p</sup>* <sup>+</sup> 1/*<sup>q</sup>* <sup>=</sup> 1, if *p*, *q* > 0 are such that 1/*p* + 1/*q* = 1 and
$$A\_1^p - A\_2^p - \dots - A\_n^p \gtrsim 0 \quad \text{and} \quad B\_1^q - B\_2^q - \dots - B\_n^q \gtrsim 0,$$
then
$$A\_1^p \#\_{1/p} B\_1^q - \sum\_{i=2}^n A\_i^p \#\_{1/p} B\_i^q \geqslant \left( A\_1^p - \sum\_{i=2}^n A\_i^p \right) \#\_{1/p} \left( B\_1^q - \sum\_{i=2}^n B\_i^q \right). \tag{46}$$
(3) Bellman's inequality: For *Ai*, *Bi* <sup>∈</sup> *<sup>B</sup>*(H)++, if
$$A\_1^{-1} - A\_2^{-1} - \dots - A\_n^{-1} > 0 \quad \text{and} \quad B\_1^{-1} - B\_2^{-1} - \dots - B\_n^{-1} > 0\_n$$
then
$$\left[ (A\_1^{-1} + B\_1^{-1}) - \sum\_{i=2}^n (A\_i + B\_i)^{-1} \right]^{-1} \leqslant \left( A\_1^{-1} - \sum\_{i=2}^n A\_i^{-1} \right)^{-1} + \left( B\_1^{-1} - \sum\_{i=2}^n B\_i^{-1} \right)^{-1}.\tag{47}$$
The mean-theoretic approach can be used to prove the famous Furuta's inequality as follows. We cite [14] for the proof.
**Theorem 0.54** (Furuta's inequality)**.** *For A B* 0*, we have*
$$(B^r A^p B^r)^{1/q} \gtrsim B^{(p+2r)/q} \tag{48}$$
$$A^{(p+2r)/q} \gtrless (A^r B^p A^r)^{1/q} \tag{49}$$
*where r* 0, *p* 0, *q* 1 *and* (1 + 2*r*)*q p* + 2*r.*
*Proof.* By the continuity argument, assume that *A*, *B* > 0. Note that (48) and (49) are equivalent. Indeed, if (48) holds, then (49) comes from applying (48) to *A*−<sup>1</sup> *B*−<sup>1</sup> and taking inverse on both sides. To prove (48), first consider the case 0 *p* 1. We have *Bp*+2*<sup>r</sup>* = *BrBpBr BrApBr* and the Löwner-Heinz's inequality (LH) implies the desired result. Now, consider the case *p* 1 and *q* = (*p* + 2*r*)/(1 + 2*r*), since (48) for *q* > (*p* + 2*r*)/(1 + 2*r*) can be obtained by (LH). Let *f*(*t*) = *t* 1/*<sup>q</sup>* and let *σ* be the associated connection (in fact, *σ* = #1/*q*). Must show that, for any *r* 0,
24 Will-be-set-by-IN-TECH
*<sup>n</sup>* 0 and *<sup>B</sup>*<sup>2</sup>
(2) Popoviciu's inequality: For *Ai*, *Bi* <sup>∈</sup> *<sup>B</sup>*(H)<sup>+</sup> and *<sup>p</sup>*, *<sup>q</sup>* <sup>&</sup>gt; 0 such that 1/*<sup>p</sup>* <sup>+</sup> 1/*<sup>q</sup>* <sup>=</sup> 1, if
*<sup>n</sup>* 0 and *<sup>B</sup><sup>q</sup>*
*<sup>n</sup>* > 0 and *<sup>B</sup>*−<sup>1</sup>
The mean-theoretic approach can be used to prove the famous Furuta's inequality as follows.
*A*(*p*+2*r*)/*<sup>q</sup>* (*A<sup>r</sup>*
*Proof.* By the continuity argument, assume that *A*, *B* > 0. Note that (48) and (49) are equivalent. Indeed, if (48) holds, then (49) comes from applying (48) to *A*−<sup>1</sup> *B*−<sup>1</sup> and taking inverse on both sides. To prove (48), first consider the case 0 *p* 1. We have *Bp*+2*<sup>r</sup>* = *BrBpBr BrApBr* and the Löwner-Heinz's inequality (LH) implies the desired result. Now, consider the case *p* 1 and *q* = (*p* + 2*r*)/(1 + 2*r*), since (48) for *q* > (*p* + 2*r*)/(1 + 2*r*)
*<sup>i</sup> Ap* 1 − *n* ∑ *i*=2 *Ap i* #1/*<sup>p</sup>*
<sup>1</sup> <sup>−</sup> *<sup>B</sup>*<sup>2</sup>
<sup>1</sup> <sup>−</sup> *<sup>B</sup><sup>q</sup>*
<sup>1</sup> <sup>−</sup> *<sup>B</sup>*−<sup>1</sup>
−<sup>1</sup> + *B*−<sup>1</sup> 1 −
*A*−<sup>1</sup> *i*
*BpAr*
*n* ∑ *i*=2 <sup>2</sup> −···− *<sup>B</sup>*<sup>2</sup>
<sup>2</sup> −···− *<sup>B</sup><sup>q</sup>*
*Bq* 1 − *n* ∑ *i*=2 *Bq i*
<sup>2</sup> −···− *<sup>B</sup>*−<sup>1</sup> *<sup>n</sup>* <sup>&</sup>gt; 0,
)1/*<sup>q</sup> B*(*p*+2*r*)/*<sup>q</sup>* (48)
1/*<sup>q</sup>* and let *σ* be the associated connection (in fact,
*n* ∑ *i*=2
)1/*<sup>q</sup>* (49)
*B*−<sup>1</sup> *i*
−<sup>1</sup>
. (47)
*<sup>n</sup>* 0,
*<sup>n</sup>* 0,
. (45)
. (46)
Here are consequences.
then
then
then (*A*−<sup>1</sup> <sup>1</sup> <sup>+</sup> *<sup>B</sup>*−<sup>1</sup>
(1) Aczél's inequality: For *Ai*, *Bi* <sup>∈</sup> *<sup>B</sup>*(H)*sa*, if
<sup>2</sup> −···− *<sup>A</sup>*<sup>2</sup>
<sup>2</sup> −···− *<sup>A</sup><sup>p</sup>*
<sup>2</sup> −···− *<sup>A</sup>*−<sup>1</sup>
(*Ai* + *Bi*)−<sup>1</sup>
**Theorem 0.54** (Furuta's inequality)**.** *For A B* 0*, we have*
−<sup>1</sup> *A*−<sup>1</sup> 1 −
(*BrApBr*
*A*2 <sup>1</sup> <sup>−</sup> *<sup>A</sup>*<sup>2</sup>
*A*2 <sup>1</sup> # *<sup>B</sup>*<sup>2</sup> 1 − *n* ∑ *i*=2 *A*2 *<sup>i</sup>* # *<sup>B</sup>*<sup>2</sup> *<sup>i</sup> A*2 1 − *n* ∑ *i*=2 *A*2 *i* # *B*2 1 − *n* ∑ *i*=2 *B*2 *i*
*p*, *q* > 0 are such that 1/*p* + 1/*q* = 1 and
1 − *n* ∑ *i*=2 *Ap <sup>i</sup>* #1/*<sup>p</sup> <sup>B</sup><sup>q</sup>*
(3) Bellman's inequality: For *Ai*, *Bi* <sup>∈</sup> *<sup>B</sup>*(H)++, if
*Ap* <sup>1</sup> <sup>−</sup> *<sup>A</sup><sup>p</sup>*
*Ap* <sup>1</sup> #1/*<sup>p</sup> <sup>B</sup><sup>q</sup>*
*A*−<sup>1</sup> <sup>1</sup> <sup>−</sup> *<sup>A</sup>*−<sup>1</sup>
<sup>1</sup> ) −
We cite [14] for the proof.
*n* ∑ *i*=2
*where r* 0, *p* 0, *q* 1 *and* (1 + 2*r*)*q p* + 2*r.*
can be obtained by (LH). Let *f*(*t*) = *t*
$$B^{-2r} \sigma A^p \gtrsim B. \tag{50}$$
For 0 *r* <sup>1</sup> <sup>2</sup> , we have by (LH) that *<sup>A</sup>*2*<sup>r</sup> <sup>B</sup>*2*<sup>r</sup>* and
$$B^{-2r}\sigma A^p \gtrsim A^{-2r}\sigma A^p = A^{-2r(1-1/q)}A^{p/q} = A \gtrsim B = B^{-2r}\sigma B^p.$$
Now, set *s* = 2*r* + <sup>1</sup> <sup>2</sup> and *q*<sup>1</sup> = (*p* + 2*s*)/(1 + 2*s*) 1. Let *f*1(*t*) = *t* 1/*q*<sup>1</sup> and consider the associated connection *σ*1. The previous step, the monotonicity and the congruence invariance of connections imply that
$$\begin{aligned} \mathcal{B}^{-2s} \sigma\_1 A^p &= \mathcal{B}^{-r} [\mathcal{B}^{-(2r+1)} \sigma\_1 \left( \mathcal{B}^r A^p B^r \right)] \mathcal{B}^{-r} \\ &\geqslant \mathcal{B}^{-r} [(\mathcal{B}^r A^p B^r)^{-1/q\_1} \sigma\_1 \left( \mathcal{B}^r A^p B^r \right)] \mathcal{B}^{-r} \\ &= \mathcal{B}^{-r} (\mathcal{B}^r A^p B^r)^{1/q} \mathcal{B}^{-r} \\ &\geqslant \mathcal{B}^{-r} \mathcal{B}^{1+2r} \mathcal{B}^{-r} \\ &= \mathcal{B}. \end{aligned}$$
Note that the above result holds for *A*, *B* 0 via the continuity of a connection. The desired equation (50) holds for all *r* 0 by repeating this process.
#### **Acknowledgement**
The author thanks referees for article processing.
#### **Author details**
Pattrawut Chansangiam *King Mongkut's Institute of Technology Ladkrabang, Thailand*
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## **Recent Research on Jensen's Inequality for OpÜrators**
Jadranka Mićić and Josip Pečarić
Additional information is available at the end of the chapter
http://dx.doi.org/10.5772/48468
26 Will-be-set-by-IN-TECH
[11] Fujii, J. (1978). Arithmetico-geometric mean of operators, *Mathematica Japonica*, Vol. 23,
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## **1. Introduction**
The self-adjoint operators on Hilbert spaces with their numerous applications play an important part in the operator theory. The bounds research for self-adjoint operators is a very useful area of this theory. There is no better inequality in bounds examination than Jensen's inequality. It is an extensively used inequality in various fields of mathematics.
Let *I* be a real interval of any type. A continuous function *f* : *I* → **R** is said to be operator convex if
$$f\left(\lambda\mathbf{x} + (1-\lambda)y\right) \le \lambda f(\mathbf{x}) + (1-\lambda)f(y) \tag{1}$$
holds for each *λ* ∈ [0, 1] and every pair of self-adjoint operators *x* and *y* (acting) on an infinite dimensional Hilbert space *H* with spectra in *I* (the ordering is defined by setting *x* ≤ *y* if *y* − *x* is positive semi-definite).
Let *f* be an operator convex function defined on an interval *I*. Ch. Davis [1] proved1 a Schwarz inequality
$$f\left(\phi(\mathbf{x})\right) \le \phi\left(f(\mathbf{x})\right) \tag{2}$$
where *φ*: A → *B*(*K*) is a unital completely positive linear mapping from a *C*∗-algebra A to linear operators on a Hilbert space *K*, and *x* is a self-adjoint element in A with spectrum in *I*. Subsequently M. D. Choi [2] noted that it is enough to assume that *φ* is unital and positive. In fact, the restriction of *φ* to the commutative *C*∗-algebra generated by *x* is automatically completely positive by a theorem of Stinespring.
F. Hansen and G. K. Pedersen [3] proved a Jensen type inequality
$$f\left(\sum\_{i=1}^{n}a\_i^\*\mathbf{x}\_ia\_i\right) \le \sum\_{i=1}^{n}a\_i^\*f(\mathbf{x}\_i)a\_i \tag{3}$$
<sup>1</sup> There is small typo in the proof. Davis states that *φ* by Stinespring's theorem can be written on the form *φ*(*x*) = *Pρ*(*x*)*P* where *ρ* is a ∗-homomorphism to *B*(*H*) and *P* is a projection on *H*. In fact, *H* may be embedded in a Hilbert space *K* on which *ρ* and *P* acts. The theorem then follows by the calculation *f*(*φ*(*x*)) = *f*(*Pρ*(*x*)*P*) ≤ *P f*(*ρ*(*x*))*P* = *Pρ*(*f*(*x*)*P* = *φ*(*f*(*x*)), where the pinching inequality, proved by Davis in the same paper, is applied.
©2012 Mi´ci´c and Peˇcari´c, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ©2012 Mićić and Pečarić, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
for operator convex functions *f* defined on an interval *I* = [0, *α*) (with *α* ≤ ∞ and *f*(0) ≤ 0) and self-adjoint operators *x*1,..., *xn* with spectra in *I* assuming that ∑*<sup>n</sup> <sup>i</sup>*=<sup>1</sup> *a*<sup>∗</sup> *<sup>i</sup> ai* = **1**. The restriction on the interval and the requirement *f*(0) ≤ 0 was subsequently removed by B. Mond and J. Peˇcari´c in [4], cf. also [5].
The inequality (3) is in fact just a reformulation of (2) although this was not noticed at the time. It is nevertheless important to note that the proof given in [3] and thus the statement of the theorem, when restricted to *n* × *n* matrices, holds for the much richer class of 2*n* × 2*n* matrix convex functions. Hansen and Pedersen used (3) to obtain elementary operations on functions, which leave invariant the class of operator monotone functions. These results then served as the basis for a new proof of Löwner's theorem applying convexity theory and Krein-Milman's theorem.
B. Mond and J. Peˇcari´c [6] proved the inequality
$$f\left(\sum\_{i=1}^{n} w\_i \phi\_i(\mathbf{x}\_i)\right) \le \sum\_{i=1}^{n} w\_i \phi\_i(f(\mathbf{x}\_i)) \tag{4}$$
for operator convex functions *f* defined on an interval *I*, where *φ<sup>i</sup>* : *B*(*H*) → *B*(*K*) are unital positive linear mappings, *x*1,..., *xn* are self-adjoint operators with spectra in *I* and *w*1,..., *wn* are are non-negative real numbers with sum one.
Also, B. Mond, J. Peˇcari´c, T. Furuta et al. [6–11] observed conversed of some special case of Jensen's inequality. So in [10] presented the following generalized converse of a Schwarz inequality (2)
$$\left[F\left[\phi\left(f(A)\right),\operatorname{g}\left(\phi(A)\right)\right]\right] \le \max\_{m \le t \le M} F\left[f(m) + \frac{f(M) - f(m)}{M - m} (t - m), \operatorname{g}(t)\right] \mathbf{1}\_{\tilde{n}} \tag{5}$$
for convex functions *f* defined on an interval [*m*, *M*], *m* < *M*, where *g* is a real valued continuous function on [*m*, *M*], *F*(*u*, *v*) is a real valued function defined on *U* × *V*, matrix non-decreasing in *u*, *U* ⊃ *f* [*m*, *M*], *V* ⊃ *g*[*m*, *M*], *φ* : *Hn* → *Hn*˜ is a unital positive linear mapping and *A* is a Hermitian matrix with spectrum contained in [*m*, *M*].
There are a lot of new research on the classical Jensen inequality (4) and its reverse inequalities. For example, J.I. Fujii et all. in [12, 13] expressed these inequalities by externally dividing points.
#### **2. Classic results**
In this section we present a form of Jensen's inequality which contains (2), (3) and (4) as special cases. Since the inequality in (4) was the motivating step for obtaining converses of Jensen's inequality using the so-called Mond-Peˇcari´c method, we also give some results pertaining to converse inequalities in the new formulation.
We recall some definitions. Let *T* be a locally compact Hausdorff space and let A be a *C*∗-algebra of operators on some Hilbert space *H*. We say that a field (*xt*)*t*∈*<sup>T</sup>* of operators in A is continuous if the function *t* �→ *xt* is norm continuous on *T*. If in addition *μ* is a Radon measure on *T* and the function *t* �→ �*xt*� is integrable, then we can form *the Bochner integral <sup>T</sup> xt dμ*(*t*), which is the unique element in A such that
$$\int \varphi\left(\int\_T \mathbf{x}\_t \, d\mu(t)\right) = \int\_T \varphi(\mathbf{x}\_t) \, d\mu(t).$$
for every linear functional *ϕ* in the norm dual A∗.
2 Will-be-set-by-IN-TECH
for operator convex functions *f* defined on an interval *I* = [0, *α*) (with *α* ≤ ∞ and *f*(0) ≤
restriction on the interval and the requirement *f*(0) ≤ 0 was subsequently removed by B.
The inequality (3) is in fact just a reformulation of (2) although this was not noticed at the time. It is nevertheless important to note that the proof given in [3] and thus the statement of the theorem, when restricted to *n* × *n* matrices, holds for the much richer class of 2*n* × 2*n* matrix convex functions. Hansen and Pedersen used (3) to obtain elementary operations on functions, which leave invariant the class of operator monotone functions. These results then served as the basis for a new proof of Löwner's theorem applying convexity theory and
*<sup>i</sup>*=<sup>1</sup> *a*<sup>∗</sup>
*wiφi*(*f*(*xi*)) (4)
*<sup>M</sup>* <sup>−</sup> *<sup>m</sup>* (*<sup>t</sup>* <sup>−</sup> *<sup>m</sup>*), *<sup>g</sup>*(*t*)
1*n*˜ (5)
*<sup>i</sup> ai* = **1**. The
0) and self-adjoint operators *x*1,..., *xn* with spectra in *I* assuming that ∑*<sup>n</sup>*
*wiφi*(*xi*)
*m*≤*t*≤*M*
mapping and *A* is a Hermitian matrix with spectrum contained in [*m*, *M*].
*F*
≤ *n* ∑ *i*=1
for operator convex functions *f* defined on an interval *I*, where *φ<sup>i</sup>* : *B*(*H*) → *B*(*K*) are unital positive linear mappings, *x*1,..., *xn* are self-adjoint operators with spectra in *I* and *w*1,..., *wn*
Also, B. Mond, J. Peˇcari´c, T. Furuta et al. [6–11] observed conversed of some special case of Jensen's inequality. So in [10] presented the following generalized converse of a Schwarz
for convex functions *f* defined on an interval [*m*, *M*], *m* < *M*, where *g* is a real valued continuous function on [*m*, *M*], *F*(*u*, *v*) is a real valued function defined on *U* × *V*, matrix non-decreasing in *u*, *U* ⊃ *f* [*m*, *M*], *V* ⊃ *g*[*m*, *M*], *φ* : *Hn* → *Hn*˜ is a unital positive linear
There are a lot of new research on the classical Jensen inequality (4) and its reverse inequalities. For example, J.I. Fujii et all. in [12, 13] expressed these inequalities by externally dividing
In this section we present a form of Jensen's inequality which contains (2), (3) and (4) as special cases. Since the inequality in (4) was the motivating step for obtaining converses of Jensen's inequality using the so-called Mond-Peˇcari´c method, we also give some results pertaining to
We recall some definitions. Let *T* be a locally compact Hausdorff space and let A be a *C*∗-algebra of operators on some Hilbert space *H*. We say that a field (*xt*)*t*∈*<sup>T</sup>* of operators in A is continuous if the function *t* �→ *xt* is norm continuous on *T*. If in addition *μ* is a Radon
*<sup>f</sup>*(*m*) + *<sup>f</sup>*(*M*) <sup>−</sup> *<sup>f</sup>*(*m*)
Mond and J. Peˇcari´c in [4], cf. also [5].
B. Mond and J. Peˇcari´c [6] proved the inequality
are are non-negative real numbers with sum one.
*F* [*φ* (*f*(*A*)), *g* (*φ*(*A*))] ≤ max
converse inequalities in the new formulation.
*f n* ∑ *i*=1
Krein-Milman's theorem.
inequality (2)
points.
**2. Classic results**
Assume furthermore that there is a field (*φt*)*t*∈*<sup>T</sup>* of positive linear mappings *φ<sup>t</sup>* : A→B from A to another C∗-algebra B of operators on a Hilbert space *K*. We recall that a linear mapping *φ<sup>t</sup>* : A→B is said to be a positive mapping if *φt*(*xt*) ≥ 0 for all *xt* ≥ 0. We say that such a field is continuous if the function *t* �→ *φt*(*x*) is continuous for every *x* ∈ A. Let the C∗-algebras include the identity operators and the function *t* �→ *φt*(1*H*) be integrable with *<sup>T</sup> <sup>φ</sup>t*(1*H*) *<sup>d</sup>μ*(*t*) = *<sup>k</sup>*1*<sup>K</sup>* for some positive scalar *<sup>k</sup>*. Specially, if *<sup>T</sup> φt*(1*H*) *dμ*(*t*) = 1*K*, we say that a *field* (*φt*)*t*∈*<sup>T</sup>* is *unital*.
Let *B*(*H*) be the *C*∗-algebra of all bounded linear operators on a Hilbert space *H*. We define bounds of an operator *x* ∈ *B*(*H*) by
$$m\_{\mathbf{x}} = \inf\_{\|\boldsymbol{\xi}\|=1} \langle \mathbf{x} \boldsymbol{\xi}, \boldsymbol{\xi} \rangle \quad \text{and} \quad M\_{\mathbf{x}} = \sup\_{\|\boldsymbol{\xi}\|=1} \langle \mathbf{x} \boldsymbol{\xi}, \boldsymbol{\xi} \rangle \tag{6}$$
for *ξ* ∈ *H*. If Sp(*x*) denotes the spectrum of *x*, then Sp(*x*) ⊆ [*mx*, *Mx*].
For an operator *<sup>x</sup>* <sup>∈</sup> *<sup>B</sup>*(*H*) we define operators <sup>|</sup>*x*|, *<sup>x</sup>*+, *<sup>x</sup>*<sup>−</sup> by
$$|\mathbf{x}| = (\mathbf{x}^\* \mathbf{x})^{1/2}, \qquad \mathbf{x}^+ = (|\mathbf{x}| + \mathbf{x})/2, \qquad \mathbf{x}^- = (|\mathbf{x}| - \mathbf{x})/2$$
Obviously, if *<sup>x</sup>* is self-adjoint, then <sup>|</sup>*x*<sup>|</sup> = (*x*2)1/2 and *<sup>x</sup>*+, *<sup>x</sup>*<sup>−</sup> <sup>≥</sup> 0 (called positive and negative parts of *<sup>x</sup>* <sup>=</sup> *<sup>x</sup>*<sup>+</sup> <sup>−</sup> *<sup>x</sup>*−).
#### **2.1. Jensen's inequality with operator convexity**
Firstly, we give a general formulation of Jensen's operator inequality for a unital field of positive linear mappings (see [14]).
**Theorem 1.** *Let f* : *I* → **R** *be an operator convex function defined on an interval I and let* A *and* B *be unital C*∗*-algebras acting on a Hilbert space H and K respectively. If* (*φt*)*t*∈*<sup>T</sup> is a unital field of positive linear mappings φ<sup>t</sup>* : A→B *defined on a locally compact Hausdorff space T with a bounded Radon measure μ*, *then the inequality*
$$f\left(\int\_{T} \phi\_{t}(\mathbf{x}\_{t}) \, d\mu(t)\right) \le \int\_{T} \phi\_{t}(f(\mathbf{x}\_{t})) \, d\mu(t) \tag{7}$$
*holds for every bounded continuous field* (*xt*)*t*∈*<sup>T</sup> of self-adjoint elements in* A *with spectra contained in I*.
*Proof.* We first note that the function *t* �→ *φt*(*xt*) ∈ B is continuous and bounded, hence integrable with respect to the bounded Radon measure *μ*. Furthermore, the integral is an element in the multiplier algebra *M*(B) acting on *K*. We may organize the set *CB*(*T*, A) of bounded continuous functions on *T* with values in A as a normed involutive algebra by applying the point-wise operations and setting
$$\left\|(y\_t)\_{t\in T}\right\| = \sup\_{t\in T} \left\|y\_t\right\| \qquad (y\_t)\_{t\in T} \in \mathcal{CB}(T, \mathcal{A}),$$
and it is not difficult to verify that the norm is already complete and satisfy the *C*∗-identity. In fact, this is a standard construction in *C*∗-algebra theory. It follows that *f*((*xt*)*t*∈*T*) = (*f*(*xt*))*t*∈*T*. We then consider the mapping
$$
\pi \colon \mathsf{CB}(T, \mathcal{A}) \to M(\mathcal{B}) \subseteq B(K).
$$
defined by setting
$$\pi\left(\left(\mathfrak{x}\_{t}\right)\_{t\in T}\right) = \int\_{T} \phi\_{t}\left(\mathfrak{x}\_{t}\right)d\mu(t)$$
and note that it is a unital positive linear map. Setting *x* = (*xt*)*t*∈*<sup>T</sup>* ∈ *CB*(*T*, A), we use inequality (2) to obtain
$$f\left(\pi\left((\mathbf{x}\_{l})\_{l\in T}\right)\right) = f\left(\pi(\mathbf{x})\right) \le \pi\left(f(\mathbf{x})\right) = \pi\left(f\left((\mathbf{x}\_{l})\_{l\in T}\right)\right) = \pi\left(\left(f(\mathbf{x}\_{l})\right)\_{l\in T}\right)$$
but this is just the statement of the theorem.
#### **2.2. Converses of Jensen's inequality**
In the present context we may obtain results of the Li-Mathias type cf. [15, Chapter 3] and [16, 17].
**Theorem 2.** *Let T be a locally compact Hausdorff space equipped with a bounded Radon measure μ. Let* (*xt*)*t*∈*<sup>T</sup> be a bounded continuous field of self-adjoint elements in a unital C*∗*-algebra* A *with spectra in* [*m*, *M*]*, m* < *M. Furthermore, let* (*φt*)*t*∈*<sup>T</sup> be a field of positive linear mappings φ<sup>t</sup>* : A → B *from* A *to another unital C*∗−*algebra* B*, such that the function t* �→ *φt*(1*H*) *is integrable with <sup>T</sup> φt*(1*H*) *dμ*(*t*) = *k*1*<sup>K</sup> for some positive scalar k. Let mx and Mx, mx* ≤ *Mx, be the bounds of the self-adjoint operator x* = *<sup>T</sup> φt*(*xt*) *dμ*(*t*) *and f* : [*m*, *M*] → **R***, g* : [*mx*, *Mx*] → **R***, F* : *U* × *V* → **R** *be functions such that* (*k f*)([*m*, *M*]) ⊂ *U*, *g* ([*mx*, *Mx*]) ⊂ *V and F is bounded. If F is operator monotone in the first variable, then*
$$\begin{split} \inf\_{\boldsymbol{\mu}\_{\boldsymbol{\mu}\_{\boldsymbol{\nu}}} \leq \boldsymbol{z} \leq M\_{\boldsymbol{\mu}}} & F \left[ k \cdot h\_{1} \left( \frac{1}{\tilde{k}} z \right), \boldsymbol{g}(\boldsymbol{z}) \right] \mathbf{1}\_{K} \leq F \left[ \int\_{T} \boldsymbol{\Phi}\_{l} \left( f(\mathbf{x}\_{l}) \right) d\boldsymbol{\mu}(t), \boldsymbol{g} \left( \int\_{T} \boldsymbol{\Phi}\_{l} (\mathbf{x}\_{l}) d\boldsymbol{\mu}(t) \right) \right] \\ & \leq \sup\_{\boldsymbol{m}\_{\boldsymbol{\nu}} \leq \boldsymbol{z} \leq M\_{\boldsymbol{\nu}}} F \left[ k \cdot h\_{2} \left( \frac{1}{\tilde{k}} z \right), \boldsymbol{g}(\boldsymbol{z}) \right] \mathbf{1}\_{K} \end{split} \tag{8}$$
*holds for every operator convex function h*<sup>1</sup> *on* [*m*, *M*] *such that h*<sup>1</sup> ≤ *f and for every operator concave function h*<sup>2</sup> *on* [*m*, *M*] *such that h*<sup>2</sup> ≥ *f .*
*Proof.* We prove only RHS of (8). Let *h*<sup>2</sup> be operator concave function on [*m*, *M*] such that *f*(*z*) ≤ *h*2(*z*) for every *z* ∈ [*m*, *M*]. By using the functional calculus, it follows that *f*(*xt*) ≤ *h*2(*xt*) for every *t* ∈ *T*. Applying the positive linear mappings *φ<sup>t</sup>* and integrating, we obtain
$$\int\_{T} \phi\_{t} \left( f(\mathbf{x}\_{t}) \right) d\mu(t) \le \int\_{T} \phi\_{t} \left( h\_{2}(\mathbf{x}\_{t}) \right) d\mu(t)$$
Furthermore, replacing *φ<sup>t</sup>* by <sup>1</sup> *<sup>k</sup> <sup>φ</sup><sup>t</sup>* in Theorem 1, we obtain <sup>1</sup> *k T φ<sup>t</sup>* (*h*2(*xt*)) *dμ*(*t*) ≤ *h*2 1 *k T φt*(*xt*) *dμ*(*t*) , which gives *T φ<sup>t</sup>* (*f*(*xt*)) *dμ*(*t*) ≤ *k* · *h*<sup>2</sup> 1 *k T φt*(*xt*) *dμ*(*t*) . Since *mx* <sup>1</sup>*<sup>K</sup>* <sup>≤</sup> *<sup>T</sup> φt*(*xt*)*dμ*(*t*) ≤ *Mx* 1*K*, then using operator monotonicity of *F*(·, *v*) we obtain
$$F\left[\int\_{T} \phi\_{l}\left(f(\mathbf{x}\_{l})\right)d\mu(t), \mathbf{g}\left(\int\_{T} \phi\_{l}(\mathbf{x}\_{l})d\mu(t)\right)\right] \tag{9}$$
$$0 \le \mathcal{F}\left[k \cdot h\_2\left(\frac{1}{k} \int\_T \phi\_l(\mathbf{x}\_l) \, d\mu(t)\right), \mathcal{g}\left(\int\_T \phi\_l(\mathbf{x}\_l) d\mu(t)\right)\right] \le \sup\_{m\_z \le z \le M\_x} \mathcal{F}\left[k \cdot h\_2\left(\frac{1}{k} z\right), \mathcal{g}(z)\right] \mathbf{1}\_K$$
Applying RHS of (8) for a convex function *f* (or LHS of (8) for a concave function *f*) we obtain the following generalization of (5).
**Theorem 3.** *Let* (*xt*)*t*∈*T, mx, Mx and* (*φt*)*t*∈*<sup>T</sup> be as in Theorem 2. Let f* : [*m*, *<sup>M</sup>*] → **<sup>R</sup>***, g* : [*mx*, *Mx*] → **R***, F* : *U* × *V* → **R** *be functions such that* (*k f*)([*m*, *M*]) ⊂ *U*, *g* ([*mx*, *Mx*]) ⊂ *V and F is bounded. If F is operator monotone in the first variable and f is convex on the interval* [*m*, *M*]*, then*
$$\begin{aligned} &F\left[\int\_{T} \phi\_{l}\left(f(\mathbf{x}\_{l})\right)d\mu(t), \mathbf{g}\left(\int\_{T} \phi\_{l}(\mathbf{x}\_{l})d\mu(t)\right)\right] \\ &\leq \sup\_{m\_{\boldsymbol{x}} \leq z \leq M\_{\boldsymbol{x}}} F\left[\frac{Mk-z}{M-m}f(m) + \frac{z-km}{M-m}f(M), \mathbf{g}(z)\right] \mathbf{1}\_{K} \end{aligned} \tag{10}$$
*In the dual case (when f is concave) the opposite inequalities hold in* (10) *with* inf *instead of* sup*.*
*Proof.* We prove only the convex case. For convex *<sup>f</sup>* the inequality *<sup>f</sup>*(*z*) <sup>≤</sup> *<sup>M</sup>*−*<sup>z</sup> <sup>M</sup>*−*<sup>m</sup> <sup>f</sup>*(*m*) + *<sup>z</sup>*−*<sup>m</sup> <sup>M</sup>*−*<sup>m</sup> <sup>f</sup>*(*M*) holds for every *<sup>z</sup>* <sup>∈</sup> [*m*, *<sup>M</sup>*]. Thus, by putting *<sup>h</sup>*2(*z*) = *<sup>M</sup>*−*<sup>z</sup> <sup>M</sup>*−*<sup>m</sup> <sup>f</sup>*(*m*) + *<sup>z</sup>*−*<sup>m</sup> <sup>M</sup>*−*<sup>m</sup> <sup>f</sup>*(*M*) in (9) we obtain (10).
Numerous applications of the previous theorem can be given (see [15]). Applying Theorem 3 for the function *F*(*u*, *v*) = *u* − *αv* and *k* = 1, we obtain the following generalization of [15, Theorem 2.4].
**Corollary 4.** *Let* (*xt*)*t*∈*T, mx, Mx be as in Theorem 2 and* (*φt*)*t*∈*<sup>T</sup> be a unital field of positive linear mappings φ<sup>t</sup>* : A→B*. If f* : [*m*, *M*] → **R** *is convex on the interval* [*m*, *M*]*, m* < *M, and g* : [*m*, *M*] → **R***, then for any α* ∈ **R**
$$\int\_{T} \phi\_{l} \left( f(\mathbf{x}\_{l}) \right) d\mu(t) \le \mathfrak{a} \left( \int\_{T} \phi\_{l}(\mathbf{x}\_{l}) d\mu(t) \right) + \mathfrak{C} \mathbf{1}\_{K} \tag{11}$$
*where*
4 Will-be-set-by-IN-TECH
bounded continuous functions on *T* with values in A as a normed involutive algebra by
and it is not difficult to verify that the norm is already complete and satisfy the *C*∗-identity. In fact, this is a standard construction in *C*∗-algebra theory. It follows that *f*((*xt*)*t*∈*T*) =
*π*: *CB*(*T*, A) → *M*(B) ⊆ *B*(*K*)
*T*
and note that it is a unital positive linear map. Setting *x* = (*xt*)*t*∈*<sup>T</sup>* ∈ *CB*(*T*, A), we use
In the present context we may obtain results of the Li-Mathias type cf. [15, Chapter 3] and
**Theorem 2.** *Let T be a locally compact Hausdorff space equipped with a bounded Radon measure μ. Let* (*xt*)*t*∈*<sup>T</sup> be a bounded continuous field of self-adjoint elements in a unital C*∗*-algebra* A *with spectra in* [*m*, *M*]*, m* < *M. Furthermore, let* (*φt*)*t*∈*<sup>T</sup> be a field of positive linear mappings φ<sup>t</sup>* : A → B *from* A *to another unital C*∗−*algebra* B*, such that the function t* �→ *φt*(1*H*) *is integrable with*
*<sup>T</sup> φt*(1*H*) *dμ*(*t*) = *k*1*<sup>K</sup> for some positive scalar k. Let mx and Mx, mx* ≤ *Mx, be the bounds of the*
*be functions such that* (*k f*)([*m*, *M*]) ⊂ *U*, *g* ([*mx*, *Mx*]) ⊂ *V and F is bounded. If F is operator*
*T*
*holds for every operator convex function h*<sup>1</sup> *on* [*m*, *M*] *such that h*<sup>1</sup> ≤ *f and for every operator concave*
*Proof.* We prove only RHS of (8). Let *h*<sup>2</sup> be operator concave function on [*m*, *M*] such that *f*(*z*) ≤ *h*2(*z*) for every *z* ∈ [*m*, *M*]. By using the functional calculus, it follows that *f*(*xt*) ≤ *h*2(*xt*) for every *t* ∈ *T*. Applying the positive linear mappings *φ<sup>t</sup>* and integrating, we obtain
> *T*
1 *k z* , *g*(*z*) 1*K*
1*<sup>K</sup>* ≤ *F*
*F k* · *h*<sup>2</sup>
*φ<sup>t</sup>* (*f*(*xt*)) *dμ*(*t*) ≤
<sup>≤</sup> sup *mx*≤*z*≤*Mx*
*<sup>T</sup> φt*(*xt*) *dμ*(*t*) *and f* : [*m*, *M*] → **R***, g* : [*mx*, *Mx*] → **R***, F* : *U* × *V* → **R**
*φ<sup>t</sup>* (*f*(*xt*)) *dμ*(*t*), *g*
*φ<sup>t</sup>* (*h*2(*xt*)) *dμ*(*t*)
*T*
*φt*(*xt*)*dμ*(*t*)
(8)
�*yt*� (*yt*)*t*∈*<sup>T</sup>* ∈ *CB*(*T*, A)
*φt*(*xt*) *dμ*(*t*)
*f* (*xt*)*t*∈*<sup>T</sup>* = *π*
*f*(*xt*) *t*∈*T*
applying the point-wise operations and setting
(*f*(*xt*))*t*∈*T*. We then consider the mapping
but this is just the statement of the theorem.
**2.2. Converses of Jensen's inequality**
defined by setting
[16, 17].
inequality (2) to obtain
*self-adjoint operator x* =
inf *mx*≤*z*≤*Mx*
*monotone in the first variable, then*
*F k* · *h*<sup>1</sup>
*function h*<sup>2</sup> *on* [*m*, *M*] *such that h*<sup>2</sup> ≥ *f .*
1 *k z* , *g*(*z*)
> *T*
�(*yt*)*t*∈*T*� = sup
*<sup>f</sup>* (*<sup>π</sup>* ((*xt*)*t*∈*T*)) <sup>=</sup> *<sup>f</sup>*(*π*(*x*)) <sup>≤</sup> *<sup>π</sup>*(*f*(*x*)) = *<sup>π</sup>*
*t*∈*T*
*π* ((*xt*)*t*∈*T*) =
$$\begin{aligned} \mathcal{C} &= \max\_{m\_{\boldsymbol{z}} \le \boldsymbol{z} \le M\_{\boldsymbol{x}}} \left\{ \frac{M-\boldsymbol{z}}{M-m} f(m) + \frac{\boldsymbol{z}-m}{M-m} f(M) - \mathfrak{a}g(\boldsymbol{z}) \right\} \\ &\le \max\_{m \le \boldsymbol{z} \le M} \left\{ \frac{M-\boldsymbol{z}}{M-m} f(m) + \frac{\boldsymbol{z}-m}{M-m} f(M) - \mathfrak{a}g(\boldsymbol{z}) \right\} \end{aligned}$$
*If furthermore αg is strictly convex differentiable, then the constant C* ≡ *C*(*m*, *M*, *f* , *g*, *α*) *can be written more precisely as*
$$\mathcal{C} = \frac{M - z\_0}{M - m} f(m) + \frac{z\_0 - m}{M - m} f(M) - \mathfrak{a} \mathfrak{g}(z\_0)$$
*where*
$$z\_0 = \begin{cases} \text{g}'^{-1}\left(\frac{f(M) - f(m)}{a(M-m)}\right) & \text{if} \quad \text{ag}'(m\_X) \le \frac{f(M) - f(m)}{M-m} \le \text{ag}'(M\_X) \\ m\_X & \text{if} \quad \text{ag}'(m\_X) \ge \frac{f(M) - f(m)}{M-m} \\ M\_X & \text{if} \quad \text{ag}'(M\_X) \le \frac{f(M) - f(m)}{M-m} \end{cases}$$
*In the dual case (when f is concave and αg is strictly concave differentiable) the opposite inequalities hold in* (11) *with* min *instead of* max *with the opposite condition while determining z*0*.*
#### **3. Inequalities with conditions on spectra**
In this section we present Jensens's operator inequality for real valued continuous convex functions with conditions on the spectra of the operators. A discrete version of this result is given in [18]. Also, we obtain generalized converses of Jensen's inequality under the same conditions.
Operator convexity plays an essential role in (2). In fact, the inequality (2) will be false if we replace an operator convex function by a general convex function. For example, M.D. Choi in [2, Remark 2.6] considered the function *f*(*t*) = *t* <sup>4</sup> which is convex but not operator convex. He demonstrated that it is sufficient to put dim*H* = 3, so we have the matrix case as follows. Let <sup>Φ</sup> : *<sup>M</sup>*3(**C**) → *<sup>M</sup>*2(**C**) be the contraction mapping <sup>Φ</sup>((*aij*)1≤*i*,*j*≤3)=(*aij*)1≤*i*,*j*≤2. If
$$A = \begin{pmatrix} 1 \ 0 \ 1 \\ 0 \ 0 \ 1 \\ 1 \ 1 \ 1 \end{pmatrix}, \text{ then } \Phi(A)^4 = \begin{pmatrix} 1 \ 0 \\ 0 \ 0 \end{pmatrix} \not\le \begin{pmatrix} 9 \ 5 \\ 5 \ 3 \end{pmatrix} = \Phi(A^4) \text{ and no relation between } \Phi(A)^4 \text{ and } \Phi(A)^4 \text{ is equal to } \Phi(A)$$
Φ(*A*4) under the operator order.
**Example 5.** *It appears that the inequality* (7) *will be false if we replace the operator convex function by a general convex function. We give a small example for the matrix cases and T* = {1, 2}*. We define mappings* <sup>Φ</sup>1, <sup>Φ</sup><sup>2</sup> : *<sup>M</sup>*3(**C**) <sup>→</sup> *<sup>M</sup>*2(**C**) *by* <sup>Φ</sup>1((*aij*)1≤*i*,*j*≤3) = <sup>1</sup> <sup>2</sup> (*aij*)1≤*i*,*j*≤2*,* <sup>Φ</sup><sup>2</sup> = <sup>Φ</sup>1*. Then* Φ1(*I*3) + Φ2(*I*3) = *I*2*.*
$$D \quad \mathcal{Y}$$
$$X\_1 = 2\begin{pmatrix} 1 \ 0 \ 1 \\ 0 \ 0 \ 1 \\ 1 \ 1 \ 1 \end{pmatrix} \quad \text{and} \quad X\_2 = 2\begin{pmatrix} 1 \ 0 \ 0 \\ 0 \ 0 \ 0 \\ 0 \ 0 \ 0 \end{pmatrix}$$
*then*
$$\left(\Phi\_1(X\_1) + \Phi\_2(X\_2)\right)^4 = \begin{pmatrix} 16 \ 0 \\ 0 \ 0 \end{pmatrix} \not\le \begin{pmatrix} 80 \ 40 \\ 40 \ 24 \end{pmatrix} = \Phi\_1\left(X\_1^4\right) + \Phi\_2\left(X\_2^4\right)$$
*Given the above, there is no relation between* (Φ1(*X*1) + Φ2(*X*2)) <sup>4</sup> *and* <sup>Φ</sup><sup>1</sup> � *X*4 1 � + Φ<sup>2</sup> � *X*4 2 � *under the operator order. We observe that in the above case the following stands X* = Φ1(*X*1) + <sup>Φ</sup>2(*X*2) = � 2 0 0 0� *and* [*mx*, *Mx*]=[0, 2]*,* [*m*1, *M*1] ⊂ [−1.60388, 4.49396]*,* [*m*2, *M*2]=[0, 2]*, i.e.*
$$(m\_{\mathbf{x}\prime}M\_{\mathbf{x}}) \subset [m\_{1\prime}M\_1] \cup [m\_{2\prime}M\_2],$$
*(see Fig. 1.a).*
6 Will-be-set-by-IN-TECH
*If furthermore αg is strictly convex differentiable, then the constant C* ≡ *C*(*m*, *M*, *f* , *g*, *α*) *can be*
*if αg*�
*In the dual case (when f is concave and αg is strictly concave differentiable) the opposite inequalities*
In this section we present Jensens's operator inequality for real valued continuous convex functions with conditions on the spectra of the operators. A discrete version of this result is given in [18]. Also, we obtain generalized converses of Jensen's inequality under the same
Operator convexity plays an essential role in (2). In fact, the inequality (2) will be false if we replace an operator convex function by a general convex function. For example, M.D.
convex. He demonstrated that it is sufficient to put dim*H* = 3, so we have the matrix case as follows. Let <sup>Φ</sup> : *<sup>M</sup>*3(**C**) → *<sup>M</sup>*2(**C**) be the contraction mapping <sup>Φ</sup>((*aij*)1≤*i*,*j*≤3)=(*aij*)1≤*i*,*j*≤2. If
**Example 5.** *It appears that the inequality* (7) *will be false if we replace the operator convex function by a general convex function. We give a small example for the matrix cases and T* = {1, 2}*. We*
⎠ *and X*<sup>2</sup> = 2
*under the operator order. We observe that in the above case the following stands X* = Φ1(*X*1) +
(*mx*, *Mx*) ⊂ [*m*1, *M*1] ∪ [*m*2, *M*2]
⎛ ⎝
� = Φ<sup>1</sup> � *X*4 1 � + Φ<sup>2</sup> � *X*4 2 �
*and* [*mx*, *Mx*]=[0, 2]*,* [*m*1, *M*1] ⊂ [−1.60388, 4.49396]*,* [*m*2, *M*2]=[0, 2]*,*
100 000 000
⎞ ⎠
<sup>4</sup> *and* <sup>Φ</sup><sup>1</sup>
� *X*4 1 � + Φ<sup>2</sup> � *X*4 2 �
*<sup>M</sup>* <sup>−</sup> *<sup>m</sup> <sup>f</sup>*(*M*) <sup>−</sup> *<sup>α</sup>g*(*z*0)
(*mx*) <sup>≤</sup> *<sup>f</sup>*(*M*)−*f*(*m*)
(*mx*) <sup>≥</sup> *<sup>f</sup>*(*M*)−*f*(*m*) *M*−*m*
(*Mx*) <sup>≤</sup> *<sup>f</sup>*(*M*)−*f*(*m*) *M*−*m*
*<sup>M</sup>*−*<sup>m</sup>* <sup>≤</sup> *<sup>α</sup>g*�
(*Mx*)
<sup>4</sup> which is convex but not operator
<sup>2</sup> (*aij*)1≤*i*,*j*≤2*,* <sup>Φ</sup><sup>2</sup> = <sup>Φ</sup>1*. Then*
= Φ(*A*4) and no relation between Φ(*A*)<sup>4</sup> and
*<sup>M</sup>* <sup>−</sup> *<sup>m</sup> <sup>f</sup>*(*m*) + *<sup>z</sup>*<sup>0</sup> <sup>−</sup> *<sup>m</sup>*
�
*hold in* (11) *with* min *instead of* max *with the opposite condition while determining z*0*.*
*<sup>C</sup>* <sup>=</sup> *<sup>M</sup>* <sup>−</sup> *<sup>z</sup>*<sup>0</sup>
� *<sup>f</sup>*(*M*)−*f*(*m*) *α*(*M*−*m*)
*mx if αg*�
*Mx if αg*�
*written more precisely as*
*z*<sup>0</sup> =
⎧ ⎪⎪⎨
*g*�−<sup>1</sup>
**3. Inequalities with conditions on spectra**
Choi in [2, Remark 2.6] considered the function *f*(*t*) = *t*
� 1 0 0 0 � �≤ � 9 5 5 3 �
*define mappings* <sup>Φ</sup>1, <sup>Φ</sup><sup>2</sup> : *<sup>M</sup>*3(**C**) <sup>→</sup> *<sup>M</sup>*2(**C**) *by* <sup>Φ</sup>1((*aij*)1≤*i*,*j*≤3) = <sup>1</sup>
⎛ ⎝
<sup>4</sup> = � 16 0 0 0 � �≤ � 80 40 40 24
*Given the above, there is no relation between* (Φ1(*X*1) + Φ2(*X*2))
101 001 111
⎞
*X*<sup>1</sup> = 2
(Φ1(*X*1) + Φ2(*X*2))
⎪⎪⎩
*where*
conditions.
⎛ ⎝
101 001 111
Φ1(*I*3) + Φ2(*I*3) = *I*2*.*
⎞
Φ(*A*4) under the operator order.
<sup>⎠</sup> , then <sup>Φ</sup>(*A*)<sup>4</sup> <sup>=</sup>
*A* =
*I) If*
*then*
Φ2(*X*2) =
*i.e.*
� 2 0 0 0 �
**Figure 1.** Spectral conditions for a convex function *f*
*II) If*
$$X\_1 = \begin{pmatrix} -14 & 0 & 1 \\ 0 & -2 & -1 \\ 1 & -1 & -1 \end{pmatrix} \quad \text{and} \quad X\_2 = \begin{pmatrix} 15 \ 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 15 \end{pmatrix}.$$
*then*
$$\left(\Phi\_1(X\_1) + \Phi\_2(X\_2)\right)^4 = \begin{pmatrix} \frac{1}{15} & 0\\ 0 & 0 \end{pmatrix} < \begin{pmatrix} 89660 & -247\\ -247 & 51 \end{pmatrix} = \Phi\_1\left(X\_1^4\right) + \Phi\_2\left(X\_2^4\right)$$
*So we have that an inequality of type* (7) *now is valid. In the above case the following stands <sup>X</sup>* <sup>=</sup> <sup>Φ</sup>1(*X*1) + <sup>Φ</sup>2(*X*2) = �<sup>1</sup> <sup>2</sup> 0 0 0� *and* [*mx*, *Mx*]=[0, 0.5]*,* [*m*1, *M*1] ⊂ [−14.077, −0.328566]*,* [*m*2, *M*2]=[2, 15]*, i.e.*
$$(m\_{\mathcal{X}}, M\_{\mathcal{X}}) \cap [m\_{\mathcal{Y}}, M\_1] = \bigcirc \quad \text{and} \quad (m\_{\mathcal{X}}, M\_{\mathcal{X}}) \cap [m\_{\mathcal{Z}}, M\_2] = \bigcirc$$
*(see Fig. 1.b).*
#### **3.1. Jensen's inequality without operator convexity**
It is no coincidence that the inequality (7) is valid in Example 18-II). In the following theorem we prove a general result when Jensen's operator inequality (7) holds for convex functions.
**Theorem 6.** *Let* (*xt*)*t*∈*<sup>T</sup> be a bounded continuous field of self-adjoint elements in a unital C*∗*-algebra* A *defined on a locally compact Hausdorff space T equipped with a bounded Radon measure μ. Let mt and Mt, mt* ≤ *Mt, be the bounds of xt, t* ∈ *T. Let* (*φt*)*t*∈*<sup>T</sup> be a unital field of positive linear mappings φ<sup>t</sup>* : A→B *from* A *to another unital C*∗−*algebra* B*. If*
$$(m\_{\mathcal{X}\prime}M\_{\mathcal{X}}) \cap [m\_{t\prime}M\_{t}] = \mathcal{Q}\_{\prime} \qquad t \in T$$
*where mx and Mx, mx* <sup>≤</sup> *Mx, are the bounds of the self-adjoint operator x* <sup>=</sup> � *<sup>T</sup> φt*(*xt*) *dμ*(*t*)*, then*
$$f\left(\int\_{T} \phi\_{l}(\mathbf{x}\_{l}) \, d\mu(t)\right) \le \int\_{T} \phi\_{l}(f(\mathbf{x}\_{l})) \, d\mu(t) \tag{12}$$
*holds for every continuous convex function f* : *I* → **R** *provided that the interval I contains all mt*, *Mt. If f* : *I* → **R** *is concave, then the reverse inequality is valid in* (12)*.*
#### 8 Will-be-set-by-IN-TECH 196 Linear Algebra – Theorems and Applications
*Proof.* We prove only the case when *f* is a convex function. If we denote *m* = inf *t*∈*T* {*mt*} and *M* = sup *t*∈*T* {*Mt*}, then [*m*, *M*] ⊆ *I* and *m*1*<sup>H</sup>* ≤ *At* ≤ *M*1*H*, *t* ∈ *T*. It follows *<sup>m</sup>*1*<sup>K</sup>* <sup>≤</sup> *<sup>T</sup> φt*(*xt*) *dμ*(*t*) ≤ *M*1*K*. Therefore [*mx*, *Mx*] ⊆ [*m*, *M*] ⊆ *I*. **a)** Let *mx* < *Mx*. Since *f* is convex on [*mx*, *Mx*], then
$$f(z) \le \frac{M\_{\text{X}} - z}{M\_{\text{X}} - m\_{\text{X}}} f(m\_{\text{X}}) + \frac{z - m\_{\text{X}}}{M\_{\text{X}} - m\_{\text{X}}} f(M\_{\text{X}}), \quad z \in [m\_{\text{X}}, M\_{\text{X}}] \tag{13}$$
but since *f* is convex on [*mt*, *Mt*] and since (*mx*, *Mx*) ∩ [*mt*, *Mt*] = ∅, then
$$f(z) \ge \frac{M\_{\mathbf{x}} - z}{M\_{\mathbf{x}} - m\_{\mathbf{x}}} f(m\_{\mathbf{x}}) + \frac{z - m\_{\mathbf{x}}}{M\_{\mathbf{x}} - m\_{\mathbf{x}}} f(M\_{\mathbf{x}}), \quad z \in [m\_{l}, M\_{l}], \quad t \in T \tag{14}$$
Since *mx*1*<sup>K</sup>* <sup>≤</sup> *<sup>T</sup> φt*(*xt*) *dμ*(*t*) ≤ *Mx*1*K*, then by using functional calculus, it follows from (13)
$$f\left(\int\_{T} \phi\_{l}(\mathbf{x}\_{l}) \, d\mu(t)\right) \le \frac{M\_{\mathbf{x}} \mathbf{1}\_{K} - \int\_{T} \phi\_{l}(\mathbf{x}\_{l}) \, d\mu(t)}{M\_{\mathbf{x}} - m\_{\mathbf{x}}} f(m\_{\mathbf{x}}) + \frac{\int\_{T} \phi\_{l}(\mathbf{x}\_{l}) \, d\mu(t) - m\_{\mathbf{x}} \mathbf{1}\_{K}}{M\_{\mathbf{x}} - m\_{\mathbf{x}}} f(M\_{\mathbf{x}}) \tag{15}$$
On the other hand, since *mt*1*<sup>H</sup>* ≤ *xt* ≤ *Mt*1*H*, *t* ∈ *T*, then by using functional calculus, it follows from (14)
$$f\left(\mathbf{x}\_{t}\right) \ge \frac{M\_{\mathbf{x}}\mathbf{1}\_{H} - \mathbf{x}\_{t}}{M\_{\mathbf{x}} - m\_{\mathbf{x}}} f(m\_{\mathbf{x}}) + \frac{\mathbf{x}\_{t} - m\_{\mathbf{x}}\mathbf{1}\_{H}}{M\_{\mathbf{x}} - m\_{\mathbf{x}}} f(M\_{\mathbf{x}}), \qquad t \in T$$
Applying a positive linear mapping *φ<sup>t</sup>* and summing, we obtain
$$\int\_{T} \phi\_{l} \left( f(\mathbf{x}\_{l}) \right) \, d\mu(t) \ge \frac{M\_{\mathbf{x}} \mathbf{1}\_{K} - \int\_{T} \phi\_{l}(\mathbf{x}\_{l}) \, d\mu(t)}{M\_{\mathbf{x}} - m\_{\mathbf{x}}} f(m\_{\mathbf{x}}) + \frac{\int\_{T} \phi\_{l}(\mathbf{x}\_{l}) \, d\mu(t) - m\_{\mathbf{x}} \mathbf{1}\_{K}}{M\_{\mathbf{x}} - m\_{\mathbf{x}}} f(M\_{\mathbf{x}}) \tag{16}$$
since *<sup>T</sup> φt*(1*H*) *dμ*(*t*) = 1*K*. Combining the two inequalities (15) and (16), we have the desired inequality (12).
**b)** Let *mx* = *Mx*. Since *f* is convex on [*m*, *M*], we have
$$f(z) \ge f(m\_{\ge}) + l(m\_{\ge})(z - m\_{\ge}) \quad \text{for every } z \in [m\_{\prime}M] \tag{17}$$
where *l* is the subdifferential of *f* . Since *m*1*<sup>H</sup>* ≤ *xt* ≤ *M*1*H*, *t* ∈ *T*, then by using functional calculus, applying a positive linear mapping *φ<sup>t</sup>* and summing, we obtain from (17)
$$\int\_{T} \phi\_{t} \left( f(\mathbf{x}\_{t}) \right) \, d\mu(\mathbf{t}) \ge f(m\_{\mathbf{x}}) \mathbf{1}\_{\mathcal{K}} + l(m\_{\mathbf{x}}) \left( \int\_{T} \phi\_{t}(\mathbf{x}\_{t}) \, d\mu(\mathbf{t}) - m\_{\mathbf{x}} \mathbf{1}\_{\mathcal{K}} \right).$$
Since *mx*1*<sup>K</sup>* = *<sup>T</sup> φt*(*xt*) *dμ*(*t*), it follows
$$\int\_{T} \phi\_{t} \left( f(\mathbf{x}\_{t}) \right) \, d\mu(t) \ge f(m\_{\mathbf{x}}) \mathbf{1}\_{K} = f \left( \int\_{T} \phi\_{t}(\mathbf{x}\_{t}) \, d\mu(t) \right).$$
which is the desired inequality (12).
Putting *φt*(*y*) = *aty* for every *y* ∈ A, where *at* ≥ 0 is a real number, we obtain the following obvious corollary of Theorem 6.
**Corollary 7.** *Let* (*xt*)*t*∈*<sup>T</sup> be a bounded continuous field of self-adjoint elements in a unital C*∗*-algebra* A *defined on a locally compact Hausdorff space T equipped with a bounded Radon measure μ. Let mt and Mt, mt* ≤ *Mt, be the bounds of xt, t* ∈ *T. Let* (*at*)*t*∈*<sup>T</sup> be a continuous field of nonnegative real numbers such that <sup>T</sup> at dμ*(*t*) = 1*. If*
$$(m\_{\mathcal{X}}, M\_{\mathcal{X}}) \cap [m\_{\mathcal{Y}}, M\_{\mathcal{t}}] = \bigotimes\_{\prime} \qquad t \in T^\*$$
*where mx and Mx, mx* <sup>≤</sup> *Mx, are the bounds of the self-adjoint operator x* <sup>=</sup> *<sup>T</sup> atxt dμ*(*t*)*, then*
$$f\left(\int\_{T} a\_t \mathbf{x}\_t \, d\mu(t)\right) \le \int\_{T} a\_t f(\mathbf{x}\_t) \, d\mu(t) \tag{18}$$
*holds for every continuous convex function f* : *I* → **R** *provided that the interval I contains all mt*, *Mt.*
#### **3.2. Converses of Jensen's inequality with conditions on spectra**
8 Will-be-set-by-IN-TECH
{*Mt*}, then [*m*, *M*] ⊆ *I* and *m*1*<sup>H</sup>* ≤ *At* ≤ *M*1*H*, *t* ∈ *T*. It follows
*<sup>T</sup> φt*(*xt*) *dμ*(*t*) ≤ *Mx*1*K*, then by using functional calculus, it follows from (13)
*f*(*mx*) +
*Mx* − *mx*
*f*(*mx*) +
*<sup>T</sup> φt*(1*H*) *dμ*(*t*) = 1*K*. Combining the two inequalities (15) and (16), we have the desired
where *l* is the subdifferential of *f* . Since *m*1*<sup>H</sup>* ≤ *xt* ≤ *M*1*H*, *t* ∈ *T*, then by using functional
Putting *φt*(*y*) = *aty* for every *y* ∈ A, where *at* ≥ 0 is a real number, we obtain the following
*f*(*z*) ≥ *f*(*mx*) + *l*(*mx*)(*z* − *mx*) for every *z* ∈ [*m*, *M*] (17)
*T*
> *T*
*t*∈*T* {*mt*}
*f*(*Mx*) (15)
*f*(*Mx*) (16)
*f*(*Mx*), *z* ∈ [*mx*, *Mx*] (13)
*f*(*Mx*), *z* ∈ [*mt*, *Mt*], *t* ∈ *T* (14)
*<sup>T</sup> φt*(*xt*) *dμ*(*t*) − *mx*1*<sup>K</sup> Mx* − *mx*
*f*(*Mx*), *t* ∈ *T*
*<sup>T</sup> φt*(*xt*) *dμ*(*t*) − *mx*1*<sup>K</sup> Mx* − *mx*
*φt*(*xt*) *dμ*(*t*) − *mx*1*<sup>K</sup>*
*φt*(*xt*) *dμ*(*t*)
*Proof.* We prove only the case when *f* is a convex function. If we denote *m* = inf
*<sup>f</sup>*(*mx*) + *<sup>z</sup>* <sup>−</sup> *mx*
*Mx* − *mx*
On the other hand, since *mt*1*<sup>H</sup>* ≤ *xt* ≤ *Mt*1*H*, *t* ∈ *T*, then by using functional calculus, it
*<sup>f</sup>*(*mx*) + *xt* <sup>−</sup> *mx*1*<sup>H</sup>*
*Mx* − *mx*
*<sup>T</sup> φt*(*xt*) *dμ*(*t*) ≤ *M*1*K*. Therefore [*mx*, *Mx*] ⊆ [*m*, *M*] ⊆ *I*.
but since *f* is convex on [*mt*, *Mt*] and since (*mx*, *Mx*) ∩ [*mt*, *Mt*] = ∅, then
*Mx* − *mx*
*<sup>f</sup>*(*mx*) + *<sup>z</sup>* <sup>−</sup> *mx*
*<sup>T</sup> φt*(*xt*) *dμ*(*t*)
*<sup>T</sup> φt*(*xt*) *dμ*(*t*)
calculus, applying a positive linear mapping *φ<sup>t</sup>* and summing, we obtain from (17)
*φ<sup>t</sup>* (*f*(*xt*)) *dμ*(*t*) ≥ *f*(*mx*)1*<sup>K</sup>* = *f*
*Mx* − *mx*
*φ<sup>t</sup>* (*f*(*xt*)) *dμ*(*t*) ≥ *f*(*mx*)1*<sup>K</sup>* + *l*(*mx*)
**a)** Let *mx* < *Mx*. Since *f* is convex on [*mx*, *Mx*], then
*<sup>f</sup>*(*z*) <sup>≤</sup> *Mx* <sup>−</sup> *<sup>z</sup>*
*Mx* − *mx*
*<sup>f</sup>*(*z*) <sup>≥</sup> *Mx* <sup>−</sup> *<sup>z</sup>*
≤ *Mx* − *mx*
*Mx*1*<sup>K</sup>* <sup>−</sup>
*<sup>f</sup>* (*xt*) <sup>≥</sup> *Mx*1*<sup>H</sup>* <sup>−</sup> *xt*
*Mx* − *mx*
Applying a positive linear mapping *φ<sup>t</sup>* and summing, we obtain
*Mx*1*<sup>K</sup>* <sup>−</sup>
**b)** Let *mx* = *Mx*. Since *f* is convex on [*m*, *M*], we have
*<sup>T</sup> φt*(*xt*) *dμ*(*t*), it follows
*T*
which is the desired inequality (12).
obvious corollary of Theorem 6.
and *M* = sup
Since *mx*1*<sup>K</sup>* <sup>≤</sup>
follows from (14)
*φt*(*xt*) *dμ*(*t*)
*φ<sup>t</sup>* (*f*(*xt*)) *dμ*(*t*) ≥
*T*
*f T*
*T*
since
inequality (12).
Since *mx*1*<sup>K</sup>* =
*<sup>m</sup>*1*<sup>K</sup>* <sup>≤</sup>
*t*∈*T*
Using the condition on spectra we obtain the following extension of Theorem 3.
**Theorem 8.** *Let* (*xt*)*t*∈*<sup>T</sup> be a bounded continuous field of self-adjoint elements in a unital C*∗*-algebra* A *defined on a locally compact Hausdorff space T equipped with a bounded Radon measure μ. Furthermore, let* (*φt*)*t*∈*<sup>T</sup> be a field of positive linear mappings φ<sup>t</sup>* : A→B *from* A *to another unital <sup>C</sup>*∗−*algebra* <sup>B</sup>*, such that the function t* �→ *<sup>φ</sup>t*(1*H*) *is integrable with <sup>T</sup> φt*(1*H*) *dμ*(*t*) = *k*1*<sup>K</sup> for some positive scalar k. Let mt and Mt, mt* ≤ *Mt, be the bounds of xt, t* ∈ *T, m* = inf *t*∈*T* {*mt*}*, M* = sup *t*∈*T* {*Mt*}*,*
*and mx and Mx, mx* < *Mx, be the bounds of x* = *<sup>T</sup> φt*(*xt*) *dμ*(*t*)*. If*
$$(m\_{\mathcal{X}}, M\_{\mathcal{X}}) \cap [m\_{\mathcal{Y}}, M\_{\mathcal{t}}] = \bigotimes\_{\mathsf{T}} \qquad t \in T$$
*and f* : [*m*, *M*] → **R***, g* : [*mx*, *Mx*] → **R***, F* : *U* × *V* → **R** *are functions such that* (*k f*)([*m*, *M*]) ⊂ *U*, *g* ([*mx*, *Mx*]) ⊂ *V, f is convex, F is bounded and operator monotone in the first variable, then*
$$\begin{split} \inf\_{\boldsymbol{m}\_{\boldsymbol{x}} \le \boldsymbol{z} \le M\_{\mathcal{X}}} & F \left[ \frac{M\_{\mathcal{X}}k - \boldsymbol{z}}{M\_{\mathcal{X}} - \boldsymbol{m}\_{\boldsymbol{x}}} f(\boldsymbol{m}\_{\boldsymbol{x}}) + \frac{\boldsymbol{z} - k\boldsymbol{m}\_{\boldsymbol{x}}}{M\_{\mathcal{X}} - \boldsymbol{m}\_{\boldsymbol{x}}} f(\boldsymbol{M}\_{\boldsymbol{x}}), \boldsymbol{g}(\boldsymbol{z}) \right] \mathbf{1}\_{K} \\ & F \left[ \int\_{T} \phi\_{l} \left( f(\boldsymbol{x}\_{t}) \right) d\mu(t), \boldsymbol{g} \left( \int\_{T} \phi\_{l}(\mathbf{x}\_{t}) d\mu(t) \right) \right] \\ \leq \sup\_{\boldsymbol{m}\_{\boldsymbol{x}} \le \boldsymbol{z} \le M\_{\mathcal{X}}} & F \left[ \frac{M\boldsymbol{k} - \boldsymbol{z}}{M - \boldsymbol{m}} f(\boldsymbol{m}) + \frac{\boldsymbol{z} - k\boldsymbol{m}}{M - \boldsymbol{m}} f(\boldsymbol{M}), \boldsymbol{g}(\boldsymbol{z}) \right] \mathbf{1}\_{K} \end{split} \tag{19}$$
*In the dual case (when f is concave) the opposite inequalities hold in* (19) *by replacing* inf *and* sup *with* sup *and* inf*, respectively.*
*Proof.* We prove only LHS of (19). It follows from (14) (compare it to (16))
$$\int\_{T} \phi\_{l} \left( f(\mathbf{x}\_{l}) \right) \, d\mu(t) \ge \frac{M\_{\mathbf{x}} k \mathbf{1}\_{K} - \int\_{T} \phi\_{l}(\mathbf{x}\_{l}) \, d\mu(t)}{M\_{\mathbf{x}} - m\_{\mathbf{x}}} f(m\_{\mathbf{x}}) + \frac{\int\_{T} \phi\_{l}(\mathbf{x}\_{l}) \, d\mu(t) - m\_{\mathbf{x}} k \mathbf{1}\_{K}}{M\_{\mathbf{x}} - m\_{\mathbf{x}}} f(M\_{\mathbf{x}})$$
since *<sup>T</sup> φt*(1*H*) *dμ*(*t*) = *k*1*K*. By using operator monotonicity of *F*(·, *v*) we obtain
$$\begin{aligned} &\quad \begin{bmatrix} F\left[\int\_{T} \phi\_{t}\left(f(\mathbf{x}\_{t})\right) d\mu(t), \operatorname{g}\left(\int\_{T} \phi\_{t}(\mathbf{x}\_{t}) \, d\mu(t)\right) \right] \\ \geq &F\left[\frac{M\_{i}k\_{\mathrm{I}} - \int\_{T} \phi\_{t}(\mathbf{x}\_{t}) \, d\mu(t)}{M\_{\mathrm{I}} - m\_{\mathrm{X}}} f(m\_{\mathrm{X}}) + \frac{\int\_{T} \phi\_{t}(\mathbf{x}\_{t}) \, d\mu(t) - m\_{\mathrm{i}}k \mathbf{1}\_{\mathrm{K}}}{M\_{\mathrm{I}} - m\_{\mathrm{X}}} f(M\_{\mathrm{X}}), \operatorname{g}\left(\int\_{T} \phi\_{t}(\mathbf{x}\_{t}) \, d\mu(t)\right) \right] \\ \geq &\inf\_{m\_{\mathrm{I}} \leq z \leq M\_{\mathrm{X}}} F\left[\frac{M\_{\mathrm{X}}k - z}{M\_{\mathrm{X}} - m\_{\mathrm{X}}} f(m\_{\mathrm{X}}) + \frac{z - km\_{\mathrm{X}}}{M\_{\mathrm{X}} - m\_{\mathrm{X}}} f(M\_{\mathrm{X}}), \operatorname{g}(z) \right] \mathbbm{1}\_{\mathrm{K}} \end{aligned}$$
Putting *<sup>F</sup>*(*u*, *<sup>v</sup>*) = *<sup>u</sup>* <sup>−</sup> *<sup>α</sup><sup>v</sup>* or *<sup>F</sup>*(*u*, *<sup>v</sup>*) = *<sup>v</sup>*−1/2*uv*−1/2 in Theorem 8, we obtain the next corollary. **Corollary 9.** *Let* (*xt*)*t*∈*T, mt, Mt, mx, Mx, m, M,* (*φt*)*t*∈*<sup>T</sup> be as in Theorem 8 and f* : [*m*, *<sup>M</sup>*] → **<sup>R</sup>***, g* : [*mx*, *Mx*] → **R** *be continuous functions. If*
$$(m\_{\mathcal{X}\prime}M\_{\mathcal{X}}) \cap [m\_{\mathcal{U}\prime}M\_{\mathcal{U}}] = \bigotimes\_{\prime} \qquad t \in T$$
*and f is convex, then for any α* ∈ **R**
$$\begin{split} \min\_{m\_{\boldsymbol{x}} \le z \le M\_{\boldsymbol{x}}} & \left\{ \frac{M\_{\boldsymbol{x}}k - z}{M\_{\boldsymbol{x}} - m\_{\boldsymbol{x}}} f(m\_{\boldsymbol{x}}) + \frac{z - km\_{\boldsymbol{x}}}{M\_{\boldsymbol{x}} - m\_{\boldsymbol{x}}} f(M\_{\boldsymbol{x}}) - \operatorname{g}(z) \right\} 1\_{\mathcal{K}} + \operatorname{ag} \left( \int\_{T} \phi\_{l}(\mathbf{x}\_{l}) d\mu(t) \right) \\ & \qquad \le \int\_{T} \phi\_{l} \left( f(\mathbf{x}\_{l}) \right) d\mu(t) \\ & \qquad \le \operatorname{ag} \left( \int\_{T} \phi\_{l}(\mathbf{x}\_{l}) d\mu(t) \right) + \max\_{m\_{\boldsymbol{x}} \le z \le M\_{\boldsymbol{x}}} \left\{ \frac{Mk - z}{M - m} f(m) + \frac{z - km}{M - m} f(M) - \operatorname{g}(z) \right\} 1\_{\mathcal{K}} \end{split} \tag{20}$$
*If additionally g* > 0 *on* [*mx*, *Mx*]*, then*
$$\begin{split} \min\_{m\_{\mathbf{x}} \le z \le M\_{\mathbf{x}}} & \left\{ \frac{\frac{M\_{\mathbf{x}}k - z}{M\_{\mathbf{r}} - m\_{\mathbf{x}}} f(m\_{\mathbf{x}}) + \frac{z - km\_{\mathbf{x}}}{M\_{\mathbf{x}} - m\_{\mathbf{x}}} f(M\_{\mathbf{x}})}{g(z)} \right\} \operatorname{g} \left( \int\_{T} \phi\_{l}(\mathbf{x}\_{l}) d\mu(t) \right) \\ \le \int\_{T} \phi\_{l} \left( f(\mathbf{x}\_{l}) \right) d\mu(t) \le \max\_{m\_{\mathbf{x}} \le z \le M\_{\mathbf{x}}} & \left\{ \frac{\frac{M\mathbf{k} - z}{M - m} f(m) + \frac{z - km}{M - m} f(M)}{g(z)} \right\} g\left( \int\_{T} \phi\_{l}(\mathbf{x}\_{l}) d\mu(t) \right) \end{split} \tag{21}$$
*In the dual case (when f is concave) the opposite inequalities hold in* (20) *by replacing* min *and* max *with* max *and* min*, respectively. If additionally g* > 0 *on* [*mx*, *Mx*]*, then the opposite inequalities also hold in* (21) *by replacing* min *and* max *with* max *and* min*, respectively.*
#### **4. Refined Jensen's inequality**
In this section we present a refinement of Jensen's inequality for real valued continuous convex functions given in Theorem 6. A discrete version of this result is given in [19].
To obtain our result we need the following two lemmas.
**Lemma 10.** *Let f be a convex function on an interval I, m*, *M* ∈ *I and p*1, *p*<sup>2</sup> ∈ [0, 1] *such that p*<sup>1</sup> + *p*<sup>2</sup> = 1*. Then*
$$\min\{p\_1, p\_2\} \left[ f(m) + f(M) - 2f\left(\frac{m+M}{2}\right) \right] \le p\_1 f(m) + p\_2 f(M) - f(p\_1 m + p\_2 M) \tag{22}$$
*Proof.* These results follows from [20, Theorem 1, p. 717].
**Lemma 11.** *Let x be a bounded self-adjoint elements in a unital C*∗*-algebra* A *of operators on some Hilbert space H. If the spectrum of x is in* [*m*, *M*]*, for some scalars m* < *M, then*
$$f\left(\mathbf{x}\right) \le \frac{M\mathbf{1}\_H - \mathbf{x}}{M - m} f(m) + \frac{\mathbf{x} - m\mathbf{1}\_H}{M - m} f(M) - \delta\_f \tilde{\mathbf{x}} \tag{23}$$
$$f(resp.\quad f\left(\mathbf{x}\right) \ge \frac{M\mathbf{1}\_H - \mathbf{x}}{M - m} f(m) + \frac{\mathbf{x} - m\mathbf{1}\_H}{M - m} f(M) + \delta\_f \tilde{\mathbf{x}} \quad (1)$$
*holds for every continuous convex* (*resp. concave*) *function f* : [*m*, *M*] → **R***, where*
$$\delta\_f = f(m) + f(M) - 2f\left(\frac{m+M}{2}\right) \quad \text{(resp. } \delta\_f = 2f\left(\frac{m+M}{2}\right) - f(m) - f(M)\text{)}$$
$$and \quad \tilde{\mathbf{x}} = \frac{1}{2}\mathbf{1}\_H - \frac{1}{M-m} \left| \mathbf{x} - \frac{m+M}{2}\mathbf{1}\_H \right| $$
*Proof.* We prove only the convex case. It follows from (22) that
$$\begin{split} f\left(p\_1 m + p\_2 M\right) &\leq p\_1 f(m) + p\_2 f(M) \\ &- \min\{p\_1, p\_2\} \left(f(m) + f(M) - 2f\left(\frac{m+M}{2}\right)\right) \end{split} \tag{24}$$
for every *p*1, *p*<sup>2</sup> ∈ [0, 1] such that *p*<sup>1</sup> + *p*<sup>2</sup> = 1 . For any *z* ∈ [*m*, *M*] we can write
$$f\left(z\right) = f\left(\frac{M-z}{M-m}m + \frac{z-m}{M-m}M\right).$$
Then by using (24) for *<sup>p</sup>*<sup>1</sup> <sup>=</sup> *<sup>M</sup>*−*<sup>z</sup> <sup>M</sup>*−*<sup>m</sup>* and *<sup>p</sup>*<sup>2</sup> <sup>=</sup> *<sup>z</sup>*−*<sup>m</sup> <sup>M</sup>*−*<sup>m</sup>* we obtain
$$\begin{split} f(z) &\leq \frac{M-z}{M-m} f(m) + \frac{z-m}{M-m} f(M) \\ &\quad - \left(\frac{1}{2} - \frac{1}{M-m} \left| z - \frac{m+M}{2} \right| \right) \left( f(m) + f(M) - 2f\left(\frac{m+M}{2}\right) \right) \end{split} \tag{25}$$
since
10 Will-be-set-by-IN-TECH
Putting *<sup>F</sup>*(*u*, *<sup>v</sup>*) = *<sup>u</sup>* <sup>−</sup> *<sup>α</sup><sup>v</sup>* or *<sup>F</sup>*(*u*, *<sup>v</sup>*) = *<sup>v</sup>*−1/2*uv*−1/2 in Theorem 8, we obtain the next corollary. **Corollary 9.** *Let* (*xt*)*t*∈*T, mt, Mt, mx, Mx, m, M,* (*φt*)*t*∈*<sup>T</sup> be as in Theorem 8 and f* : [*m*, *<sup>M</sup>*] → **<sup>R</sup>***,*
(*mx*, *Mx*) ∩ [*mt*, *Mt*] = ∅, *t* ∈ *T*
*f*(*Mx*) − *g*(*z*)
*φ<sup>t</sup>* (*f*(*xt*)) *dμ*(*t*)
*Mx*−*mx <sup>f</sup>*(*Mx*)
*Mk*−*<sup>z</sup> <sup>M</sup>*−*<sup>m</sup> <sup>f</sup>*(*m*) + *<sup>z</sup>*−*km*
*In the dual case (when f is concave) the opposite inequalities hold in* (20) *by replacing* min *and* max *with* max *and* min*, respectively. If additionally g* > 0 *on* [*mx*, *Mx*]*, then the opposite inequalities also*
In this section we present a refinement of Jensen's inequality for real valued continuous
**Lemma 10.** *Let f be a convex function on an interval I, m*, *M* ∈ *I and p*1, *p*<sup>2</sup> ∈ [0, 1] *such that*
**Lemma 11.** *Let x be a bounded self-adjoint elements in a unital C*∗*-algebra* A *of operators on some*
*<sup>M</sup>* <sup>−</sup> *<sup>m</sup> <sup>f</sup>*(*m*) + *<sup>x</sup>* <sup>−</sup> *<sup>m</sup>*1*<sup>H</sup>*
*<sup>M</sup>* <sup>−</sup> *<sup>m</sup> <sup>f</sup>*(*m*) + *<sup>x</sup>* <sup>−</sup> *<sup>m</sup>*1*<sup>H</sup>*
convex functions given in Theorem 6. A discrete version of this result is given in [19].
*m* + *M* 2
*Hilbert space H. If the spectrum of x is in* [*m*, *M*]*, for some scalars m* < *M, then*
*<sup>f</sup>* (*x*) <sup>≤</sup> *<sup>M</sup>*1*<sup>H</sup>* <sup>−</sup> *<sup>x</sup>*
(*resp. f* (*x*) <sup>≥</sup> *<sup>M</sup>*1*<sup>H</sup>* <sup>−</sup> *<sup>x</sup>*
*g*(*z*)
*Mk* <sup>−</sup> *<sup>z</sup>*
*<sup>M</sup>* <sup>−</sup> *<sup>m</sup> <sup>f</sup>*(*m*) + *<sup>z</sup>* <sup>−</sup> *km*
*g T*
*<sup>M</sup>*−*<sup>m</sup> <sup>f</sup>*(*M*)
1*<sup>K</sup>* + *αg*
*T*
*<sup>M</sup>* <sup>−</sup> *<sup>m</sup> <sup>f</sup>*(*M*) <sup>−</sup> *<sup>g</sup>*(*z*)
*φt*(*xt*)*dμ*(*t*)
≤ *p*<sup>1</sup> *f*(*m*) + *p*<sup>2</sup> *f*(*M*) − *f*(*p*1*m* + *p*2*M*) (22)
*<sup>M</sup>* <sup>−</sup> *<sup>m</sup> <sup>f</sup>*(*M*) <sup>−</sup> *<sup>δ</sup><sup>f</sup> <sup>x</sup>* (23)
*<sup>M</sup>* <sup>−</sup> *<sup>m</sup> <sup>f</sup>*(*M*) + *<sup>δ</sup><sup>f</sup> <sup>x</sup>* )
*g T* *φt*(*xt*)*dμ*(*t*)
*φt*(*xt*)*dμ*(*t*)
1*K* (20)
(21)
*<sup>f</sup>*(*mx*) + *<sup>z</sup>* <sup>−</sup> *kmx*
≤ *T*
<sup>+</sup> max *mx*≤*z*≤*Mx*
*Mx*−*mx <sup>f</sup>*(*mx*) + *<sup>z</sup>*−*kmx*
*hold in* (21) *by replacing* min *and* max *with* max *and* min*, respectively.*
To obtain our result we need the following two lemmas.
*f*(*m*) + *f*(*M*) − 2 *f*
*Proof.* These results follows from [20, Theorem 1, p. 717].
*g*(*z*)
*Mx* − *mx*
*g* : [*mx*, *Mx*] → **R** *be continuous functions. If*
*and f is convex, then for any α* ∈ **R**
*Mxk* <sup>−</sup> *<sup>z</sup> Mx* − *mx*
*If additionally g* > 0 *on* [*mx*, *Mx*]*, then*
min *mx*≤*z*≤*Mx*
**4. Refined Jensen's inequality**
*φt*(*xt*)*dμ*(*t*)
*<sup>φ</sup><sup>t</sup>* (*f*(*xt*)) *<sup>d</sup>μ*(*t*) <sup>≤</sup> max *mx*≤*z*≤*Mx*
*Mx k*−*z*
min *mx*≤*z*≤*Mx*
≤ *αg*
≤ *T*
*p*<sup>1</sup> + *p*<sup>2</sup> = 1*. Then*
min{*p*1, *p*2}
*T*
$$\min\left\{\frac{M-z}{M-m}, \frac{z-m}{M-m}\right\} = \frac{1}{2} - \frac{1}{M-m} \left| z - \frac{m+M}{2} \right|^2$$
Finally we use the continuous functional calculus for a self-adjoint operator *x*: *f* , *g* ∈ C(*I*), *Sp*(*x*) ⊆ *I* and *f* ≤ *g* on *I* implies *f*(*x*) ≤ *g*(*x*); and *h*(*z*) = |*z*| implies *h*(*x*) = |*x*|. Then by using (25) we obtain the desired inequality (23).
**Theorem 12.** *Let* (*xt*)*t*∈*<sup>T</sup> be a bounded continuous field of self-adjoint elements in a unital C*∗*-algebra* A *defined on a locally compact Hausdorff space T equipped with a bounded Radon measure μ. Let mt and Mt, mt* ≤ *Mt, be the bounds of xt, t* ∈ *T. Let* (*φt*)*t*∈*<sup>T</sup> be a unital field of positive linear mappings φ<sup>t</sup>* : A→B *from* A *to another unital C*∗−*algebra* B*. Let*
$$(m\_{\mathbf{x}\prime}M\_{\mathbf{x}}) \cap [m\_{\mathbf{l}\prime}M\_{\mathbf{l}}] = \bigcirc \,, \quad \mathbf{t} \in T\prime \qquad \text{and} \qquad m < M\_{\mathbf{l}\prime}$$
*where mx and Mx, mx* <sup>≤</sup> *Mx, be the bounds of the operator x* <sup>=</sup> *<sup>T</sup> φt*(*xt*) *dμ*(*t*) *and*
$$m = \sup\left\{M\_l \colon M\_l \le m\_{\ge \prime}t \in T\right\},\ M = \inf\left\{m\_l \colon m\_l \ge M\_{\ge \prime}t \in T\right\},$$
*If f* : *I* → **R** *is a continuous convex* (*resp. concave*) *function provided that the interval I contains all mt*, *Mt, then*
$$f\left(\int\_{T} \phi\_{t}(\mathbf{x}\_{t}) \, d\mu(t)\right) \le \int\_{T} \phi\_{t}(f(\mathbf{x}\_{t})) \, d\mu(t) - \delta\_{f} \tilde{\mathbf{x}} \le \int\_{T} \phi\_{t}(f(\mathbf{x}\_{t})) \, d\mu(t) \tag{26}$$
*resp.*
$$f\left(\int\_{T} \phi\_{t}(\mathbf{x}\_{t}) \, d\mu(\mathbf{t})\right) \ge \int\_{T} \phi\_{t}(f(\mathbf{x}\_{t})) \, d\mu(\mathbf{t}) - \delta\_{f} \tilde{\mathbf{x}} \ge \int\_{T} \phi\_{t}(f(\mathbf{x}\_{t})) \, d\mu(\mathbf{t}) \tag{27}$$
*holds, where*
$$\delta\_f \equiv \delta\_f(\vec{m}, \bar{M}) = f(\vec{m}) + f(\bar{M}) - 2f\left(\frac{\vec{m} + \bar{M}}{2}\right)$$
$$\text{ (resp. } \quad \delta\_f \equiv \delta\_f(\vec{m}, \bar{M}) = 2f\left(\frac{\vec{m} + \bar{M}}{2}\right) - f(\bar{m}) - f(\bar{M}) \text{ )}\tag{28}$$
$$\tilde{\mathbf{x}} \equiv \tilde{\mathbf{x}}\_{\mathbf{x}}(\vec{m}, \bar{M}) = \frac{1}{2}\mathbf{1}\_{\mathbf{K}} - \frac{1}{\bar{M} - \bar{m}} \left| \mathbf{x} - \frac{\vec{m} + \bar{M}}{2}\mathbf{1}\_{\mathbf{K}} \right|$$
$$\vdots \quad \vdots \quad \vdots \quad \vdots \quad \vdots \quad \vdots \quad \vdots$$
*and <sup>m</sup>*¯ <sup>∈</sup> [*m*, *mA*]*, <sup>M</sup>*¯ <sup>∈</sup> [*MA*, *<sup>M</sup>*]*, <sup>m</sup>*¯ <sup>&</sup>lt; *M, are arbitrary numbers.* ¯
*Proof.* We prove only the convex case. Since *x* = *<sup>T</sup> φt*(*xt*) *dμ*(*t*) ∈ B is the self-adjoint elements such that *<sup>m</sup>*¯ <sup>1</sup>*<sup>K</sup>* <sup>≤</sup> *mx*1*<sup>K</sup>* <sup>≤</sup> *<sup>T</sup> <sup>φ</sup>t*(*xt*) *<sup>d</sup>μ*(*t*) <sup>≤</sup> *Mx*1*<sup>K</sup>* <sup>≤</sup> *<sup>M</sup>*¯ <sup>1</sup>*<sup>K</sup>* and *<sup>f</sup>* is convex on [*m*¯ , *<sup>M</sup>*¯ ] <sup>⊆</sup> *<sup>I</sup>*, then by Lemma 11 we obtain
$$f\left(\int\_{T} \phi\_{l}(\mathbf{x}\_{l}) \, d\mu(t)\right) \leq \frac{\bar{M}\mathbf{1}\_{K} - \int\_{T} \phi\_{l}(\mathbf{x}\_{l}) \, d\mu(t)}{\bar{M} - \bar{m}} f(\bar{m}) + \frac{\int\_{T} \phi\_{l}(\mathbf{x}\_{l}) \, d\mu(t) - \bar{m}\mathbf{1}\_{K}}{\bar{M} - \bar{m}} f(\bar{M}) - \delta\_{f} \tilde{\mathbf{x}} \tag{29}$$
where *<sup>δ</sup><sup>f</sup>* and *<sup>x</sup>* are defined by (28).
But since *<sup>f</sup>* is convex on [*mt*, *Mt*] and (*mx*, *Mx*) <sup>∩</sup> [*mt*, *Mt*] = <sup>∅</sup> implies (*m*¯ , *<sup>M</sup>*¯ ) <sup>∩</sup> [*mt*, *Mt*] = <sup>∅</sup>, then
$$f\left(\mathbf{x}\_{t}\right) \geq \frac{\bar{M}\mathbf{1}\_{H} - \mathbf{x}\_{t}}{\bar{M} - \bar{m}} f(\bar{m}) + \frac{\mathbf{x}\_{t} - \bar{m}\mathbf{1}\_{H}}{\bar{M} - \bar{m}} f(\bar{M}), \quad t \in T$$
Applying a positive linear mapping *<sup>φ</sup>t*, integrating and adding <sup>−</sup>*δ<sup>f</sup> <sup>x</sup>*, we obtain
$$\int\_{T} \phi\_{l} \left( f(\mathbf{x}\_{l}) \right) \, d\mu(\mathbf{t}) - \delta\_{f} \tilde{\mathbf{x}} \ge \frac{\bar{M} \mathbf{1}\_{\mathcal{K}} - \int\_{T} \phi\_{l}(\mathbf{x}\_{l}) \, d\mu(\mathbf{t})}{\bar{M} - \bar{m}} f(\bar{m}) + \frac{\int\_{T} \phi\_{l}(\mathbf{x}\_{l}) \, d\mu(\mathbf{t}) - \bar{m} \mathbf{1}\_{\mathcal{K}}}{\bar{M} - \bar{m}} f(\bar{M}) - \delta\_{f} \tilde{\mathbf{x}} \tag{30}$$
since *<sup>T</sup> φt*(1*H*) *dμ*(*t*) = 1*K*. Combining the two inequalities (29) and (30), we have LHS of (26). Since *<sup>δ</sup><sup>f</sup>* <sup>≥</sup> 0 and *<sup>x</sup>* <sup>≥</sup> 0, then we have RHS of (26).
If *<sup>m</sup>* <sup>&</sup>lt; *<sup>M</sup>* and *mx* <sup>=</sup> *Mx*, then the inequality (26) holds, but *<sup>δ</sup>f*(*mx*, *Mx*) *<sup>x</sup>*(*mx*, *Mx*) is not defined (see Example 13 I) and II)).
**Example 13.** *We give examples for the matrix cases and T* = {1, 2}*. Then we have refined inequalities given in Fig. 2. We put f*(*t*) = *t* <sup>4</sup> *which is convex but not operator convex in* (26)*. Also, we define mappings* <sup>Φ</sup>1, <sup>Φ</sup><sup>2</sup> : *<sup>M</sup>*3(**C**) <sup>→</sup> *<sup>M</sup>*2(**C**) *as follows:* <sup>Φ</sup>1((*aij*)1≤*i*,*j*≤3) = <sup>1</sup> <sup>2</sup> (*aij*)1≤*i*,*j*≤2*,* <sup>Φ</sup><sup>2</sup> = <sup>Φ</sup><sup>1</sup> *(then* Φ1(*I*3) + Φ2(*I*3) = *I*2*).*
*I) First, we observe an example when δ<sup>f</sup> X is equal to the difference RHS and LHS of Jensen's inequality. If X*<sup>1</sup> = −3*I*<sup>3</sup> *and X*<sup>2</sup> = 2*I*3*, then X* = Φ1(*X*1) + Φ2(*X*2) = −0.5*I*2*, so m* = −3*, <sup>M</sup>* <sup>=</sup> <sup>2</sup>*. We also put <sup>m</sup>*¯ <sup>=</sup> <sup>−</sup><sup>3</sup> *and <sup>M</sup>*¯ <sup>=</sup> <sup>2</sup>*. We obtain*
$$\left(\Phi\_1(X\_1) + \Phi\_2(X\_2)\right)^4 = 0.0625I\_2 < 48.5I\_2 = \Phi\_1\left(X\_1^4\right) + \Phi\_2\left(X\_2^4\right)$$
$$\begin{aligned} &\text{if } \{\phi\_1(\mathbf{x}\_1)\} + \mathsf{f}\left\{\phi\_2(\mathbf{x}\_2)\right\} \leq \phi\_1(\{\mathsf{f}\left(\mathbf{x}\_1\right)\} + \phi\_2(\{\mathsf{f}\left(\mathbf{x}\_2\right)\} - \bar{\phi}\_1\overline{\mathbf{X}}, \mathsf{f}\})\\ &\text{where} \\ &8\_t = \mathsf{f}\left(\overline{\mathbf{m}}\right) \oplus \mathsf{f}\left(\mathbf{M}\right) - 2\mathsf{f}\left(\overline{\mathbf{M}} + \overline{\mathbf{m}}\right)/2\right) \\ &\overline{\mathbf{X}} = \frac{1}{2}\mathsf{f}\_\mathsf{K} - \frac{1}{\overline{\mathbf{M}} - \overline{\mathbf{m}}} \Big| \phi\_\mathsf{K}(\mathbf{x}\_1) + \phi\_2(\mathbf{x}\_2) - \frac{\overline{\mathbf{M}} + \overline{\mathbf{m}}}{2}\mathsf{f}\_\mathsf{K} \end{aligned}$$
**Figure 2.** Refinement for two operators and a convex function *f*
*and its improvement*
12 Will-be-set-by-IN-TECH
*<sup>δ</sup><sup>f</sup>* <sup>≡</sup> *<sup>δ</sup>f*(*m*¯ , *<sup>M</sup>*¯ ) = *<sup>f</sup>*(*m*¯) + *<sup>f</sup>*(*M*¯ ) <sup>−</sup> <sup>2</sup> *<sup>f</sup>*
*<sup>φ</sup>t*(*f*(*xt*)) *<sup>d</sup>μ*(*t*) <sup>−</sup> *<sup>δ</sup><sup>f</sup> <sup>x</sup>* <sup>≥</sup>
*<sup>m</sup>*¯ +*M*¯ 2
<sup>2</sup> <sup>1</sup>*<sup>K</sup>* <sup>−</sup> <sup>1</sup> *<sup>M</sup>*¯ <sup>−</sup>*m*¯
*T*
> *<sup>m</sup>*¯ +*M*¯ 2
<sup>−</sup> *<sup>f</sup>*(*m*¯) <sup>−</sup> *<sup>f</sup>*(*M*¯ ) )
*<sup>T</sup> <sup>φ</sup>t*(*xt*) *<sup>d</sup>μ*(*t*) <sup>≤</sup> *Mx*1*<sup>K</sup>* <sup>≤</sup> *<sup>M</sup>*¯ <sup>1</sup>*<sup>K</sup>* and *<sup>f</sup>* is convex on
*<sup>T</sup> φt*(*xt*) *dμ*(*t*) − *m*¯ 1*<sup>K</sup>*
*<sup>T</sup> φt*(*xt*) *dμ*(*t*) − *m*¯ 1*<sup>K</sup>*
*<sup>M</sup>*¯ <sup>−</sup> *<sup>m</sup>*¯ *<sup>f</sup>*(*M*¯ ), *<sup>t</sup>* <sup>∈</sup> *<sup>T</sup>*
<sup>4</sup> *which is convex but not operator convex in* (26)*. Also, we define*
*X*4 1 + Φ<sup>2</sup> *X*4 2
*<sup>x</sup>* <sup>−</sup> *<sup>m</sup>*¯ <sup>+</sup>*M*¯ <sup>2</sup> 1*<sup>K</sup>*
*φt*(*f*(*xt*)) *dμ*(*t*)
*<sup>T</sup> φt*(*xt*) *dμ*(*t*) ∈ B is the self-adjoint
*<sup>M</sup>*¯ <sup>−</sup> *<sup>m</sup>*¯ *<sup>f</sup>*(*M*¯ ) <sup>−</sup> *<sup>δ</sup><sup>f</sup> <sup>x</sup>* (29)
*<sup>M</sup>*¯ <sup>−</sup> *<sup>m</sup>*¯ *<sup>f</sup>*(*M*¯ ) <sup>−</sup> *<sup>δ</sup><sup>f</sup> <sup>x</sup>*
<sup>2</sup> (*aij*)1≤*i*,*j*≤2*,* <sup>Φ</sup><sup>2</sup> = <sup>Φ</sup><sup>1</sup> *(then*
(30)
(27)
(28)
*resp.*
> *f T*
then
*T*
since
*holds, where*
*f T*
*φt*(*xt*) *dμ*(*t*)
≥ *T*
(*resp. <sup>δ</sup><sup>f</sup>* <sup>≡</sup> *<sup>δ</sup>f*(*m*¯ , *<sup>M</sup>*¯ ) = <sup>2</sup> *<sup>f</sup>*
*and <sup>m</sup>*¯ <sup>∈</sup> [*m*, *mA*]*, <sup>M</sup>*¯ <sup>∈</sup> [*MA*, *<sup>M</sup>*]*, <sup>m</sup>*¯ <sup>&</sup>lt; *M, are arbitrary numbers.* ¯
*Proof.* We prove only the convex case. Since *x* =
*<sup>M</sup>*¯ <sup>1</sup>*<sup>K</sup>* <sup>−</sup>
*<sup>f</sup>* (*xt*) <sup>≥</sup> *<sup>M</sup>*¯ <sup>1</sup>*<sup>H</sup>* <sup>−</sup> *xt*
(26). Since *<sup>δ</sup><sup>f</sup>* <sup>≥</sup> 0 and *<sup>x</sup>* <sup>≥</sup> 0, then we have RHS of (26).
*<sup>M</sup>* <sup>=</sup> <sup>2</sup>*. We also put <sup>m</sup>*¯ <sup>=</sup> <sup>−</sup><sup>3</sup> *and <sup>M</sup>*¯ <sup>=</sup> <sup>2</sup>*. We obtain*
(Φ1(*X*1) + Φ2(*X*2))
*<sup>M</sup>*¯ <sup>1</sup>*<sup>K</sup>* <sup>−</sup>
*mappings* <sup>Φ</sup>1, <sup>Φ</sup><sup>2</sup> : *<sup>M</sup>*3(**C**) <sup>→</sup> *<sup>M</sup>*2(**C**) *as follows:* <sup>Φ</sup>1((*aij*)1≤*i*,*j*≤3) = <sup>1</sup>
elements such that *<sup>m</sup>*¯ <sup>1</sup>*<sup>K</sup>* <sup>≤</sup> *mx*1*<sup>K</sup>* <sup>≤</sup>
*φt*(*xt*) *dμ*(*t*)
[*m*¯ , *<sup>M</sup>*¯ ] <sup>⊆</sup> *<sup>I</sup>*, then by Lemma 11 we obtain
≤
where *<sup>δ</sup><sup>f</sup>* and *<sup>x</sup>* are defined by (28).
*<sup>φ</sup><sup>t</sup>* (*f*(*xt*)) *<sup>d</sup>μ*(*t*) <sup>−</sup> *<sup>δ</sup><sup>f</sup> <sup>x</sup>* <sup>≥</sup>
defined (see Example 13 I) and II)).
*given in Fig. 2. We put f*(*t*) = *t*
Φ1(*I*3) + Φ2(*I*3) = *I*2*).*
*<sup>x</sup>* <sup>≡</sup> *<sup>x</sup>x*(*m*¯ , *<sup>M</sup>*¯ ) = <sup>1</sup>
*<sup>T</sup> φt*(*xt*) *dμ*(*t*) *<sup>M</sup>*¯ <sup>−</sup> *<sup>m</sup>*¯ *<sup>f</sup>*(*m*¯) +
Applying a positive linear mapping *<sup>φ</sup>t*, integrating and adding <sup>−</sup>*δ<sup>f</sup> <sup>x</sup>*, we obtain
But since *<sup>f</sup>* is convex on [*mt*, *Mt*] and (*mx*, *Mx*) <sup>∩</sup> [*mt*, *Mt*] = <sup>∅</sup> implies (*m*¯ , *<sup>M</sup>*¯ ) <sup>∩</sup> [*mt*, *Mt*] = <sup>∅</sup>,
*<sup>M</sup>*¯ <sup>−</sup> *<sup>m</sup>*¯ *<sup>f</sup>*(*m*¯) + *xt* <sup>−</sup> *<sup>m</sup>*¯ <sup>1</sup>*<sup>H</sup>*
*<sup>T</sup> φt*(*xt*) *dμ*(*t*) *<sup>M</sup>*¯ <sup>−</sup> *<sup>m</sup>*¯ *<sup>f</sup>*(*m*¯) +
*<sup>T</sup> φt*(1*H*) *dμ*(*t*) = 1*K*. Combining the two inequalities (29) and (30), we have LHS of
If *<sup>m</sup>* <sup>&</sup>lt; *<sup>M</sup>* and *mx* <sup>=</sup> *Mx*, then the inequality (26) holds, but *<sup>δ</sup>f*(*mx*, *Mx*) *<sup>x</sup>*(*mx*, *Mx*) is not
**Example 13.** *We give examples for the matrix cases and T* = {1, 2}*. Then we have refined inequalities*
*I) First, we observe an example when δ<sup>f</sup> X is equal to the difference RHS and LHS of Jensen's*
*inequality. If X*<sup>1</sup> = −3*I*<sup>3</sup> *and X*<sup>2</sup> = 2*I*3*, then X* = Φ1(*X*1) + Φ2(*X*2) = −0.5*I*2*, so m* = −3*,*
<sup>4</sup> = 0.0625*I*<sup>2</sup> < 48.5*I*<sup>2</sup> = <sup>Φ</sup><sup>1</sup>
$$\left(\Phi\_1(X\_1) + \Phi\_2(X\_2)\right)^4 = 0.0625I\_2 = \Phi\_1\left(X\_1^4\right) + \Phi\_2\left(X\_2^4\right) - 48.4375I\_2$$
*since δ<sup>f</sup>* = 96.875*, X*� = 0.5*I*2. *We remark that in this case mx* = *Mx* = −1/2 *and X*�(*mx*, *Mx*) *is not defined.*
*II) Next, we observe an example when δ<sup>f</sup> X is not equal to the difference RHS and LHS of Jensen's* � *inequality and mx* = *Mx. If*
$$\begin{aligned} \mathbf{X}\_1 = \begin{pmatrix} -1 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & -1 \end{pmatrix}, \ \mathbf{X}\_2 = \begin{pmatrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 4 \end{pmatrix}, \ \text{then } \mathbf{X} = \frac{1}{2} \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \text{ and } \mathbf{m} = -1, \ \mathbf{M} = 2. \end{aligned}$$
*In this case <sup>x</sup>*�(*mx*, *Mx*) *is not defined, since mx* <sup>=</sup> *Mx* <sup>=</sup> 1/2*. We have*
$$\left(\Phi\_1(X\_1) + \Phi\_2(X\_2)\right)^4 = \frac{1}{16} \begin{pmatrix} 1 \ 0 \\ 0 \ 1 \end{pmatrix} < \begin{pmatrix} \frac{17}{2} & 0 \\ 0 & \frac{97}{2} \end{pmatrix} = \Phi\_1\left(X\_1^4\right) + \Phi\_2\left(X\_2^4\right)$$
*and putting <sup>m</sup>*¯ <sup>=</sup> <sup>−</sup>1*, <sup>M</sup>*¯ <sup>=</sup> <sup>2</sup> *we obtain <sup>δ</sup><sup>f</sup>* <sup>=</sup> 135/8*, <sup>X</sup>*� <sup>=</sup> *<sup>I</sup>*2/2 *which give the following improvement*
$$\left(\Phi\_1(X\_1) + \Phi\_2(X\_2)\right)^4 = \frac{1}{16} \begin{pmatrix} 1 \ 0 \\ 0 \ 1 \end{pmatrix} < \frac{1}{16} \begin{pmatrix} 1 & 0 \\ 0 \ 641 \end{pmatrix} = \Phi\_1 \begin{pmatrix} X\_1^4 \\ \end{pmatrix} + \Phi\_2 \begin{pmatrix} X\_2^4 \\ \end{pmatrix} - \frac{135}{16} \begin{pmatrix} 1 \ 0 \\ 0 \ 1 \end{pmatrix}$$
*III) Next, we observe an example with matrices that are not special. If*
$$X\_1 = \begin{pmatrix} -4 & 1 & 1 \\ 1 & -2 & -1 \\ 1 & -1 & -1 \end{pmatrix} \quad \text{and} \quad X\_2 = \begin{pmatrix} 5 & -1 & -1 \\ -1 & 2 & 1 \\ -1 & 1 & 3 \end{pmatrix}, \quad \text{then} \quad X = \frac{1}{2} \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}.$$
*so m*<sup>1</sup> = −4.8662*, M*<sup>1</sup> = −0.3446*, m*<sup>2</sup> = 1.3446*, M*<sup>2</sup> = 5.8662*, m* = −0.3446*, M* = 1.3446 *and we put m*¯ = *m, M*¯ = *M (rounded to four decimal places). We have*
$$\left(\Phi\_1(X\_1) + \Phi\_2(X\_2)\right)^4 = \frac{1}{16} \begin{pmatrix} 1 \ 0 \\ 0 \ 0 \end{pmatrix} < \begin{pmatrix} \frac{1283}{2} & -255 \\ -255 & \frac{237}{2} \end{pmatrix} = \Phi\_1\left(X\_1^4\right) + \Phi\_2\left(X\_2^4\right)$$
*and its improvement*
$$\begin{aligned} \left(\Phi\_1(X\_1) + \Phi\_2(X\_2)\right)^4 &= \frac{1}{16} \begin{pmatrix} 1 & 0\\ 0 & 0 \end{pmatrix} \\ &< \begin{pmatrix} 639.9213 & -255\\ -255 & 117.8559 \end{pmatrix} = \Phi\_1 \begin{pmatrix} X\_1^4\\ X\_1^4 \end{pmatrix} + \Phi\_2 \begin{pmatrix} X\_2^4\\ X\_2^4 \end{pmatrix} - \begin{pmatrix} 1.5787 & 0\\ 0 & 0.6441 \end{pmatrix} \end{aligned}$$
*(rounded to four decimal places), since δ<sup>f</sup>* = 3.1574*, X* = 0.5 0 0 0.2040 *. But, if we put m*¯ = *mx* = 0*, <sup>M</sup>*¯ <sup>=</sup> *Mx* <sup>=</sup> 0.5*, then <sup>X</sup>* <sup>=</sup> **<sup>0</sup>***, so we do not have an improvement of Jensen's inequality. Also, if we put <sup>m</sup>*¯ <sup>=</sup> <sup>0</sup>*, <sup>M</sup>*¯ <sup>=</sup> <sup>1</sup>*, then <sup>X</sup>* <sup>=</sup> 0.5 1 0 0 1 *, <sup>δ</sup><sup>f</sup>* <sup>=</sup> 7/8 *and <sup>δ</sup><sup>f</sup> <sup>X</sup>* <sup>=</sup> 0.4375 1 0 0 1 *, which is worse than the above improvement.*
Putting Φ*t*(*y*) = *aty* for every *y* ∈ A, where *at* ≥ 0 is a real number, we obtain the following obvious corollary of Theorem 12.
**Corollary 14.** *Let* (*xt*)*t*∈*<sup>T</sup> be a bounded continuous field of self-adjoint elements in a unital C*∗*-algebra* A *defined on a locally compact Hausdorff space T equipped with a bounded Radon measure μ. Let mt and Mt, mt* ≤ *Mt, be the bounds of xt, t* ∈ *T. Let* (*at*)*t*∈*<sup>T</sup> be a continuous field of nonnegative real numbers such that <sup>T</sup> at dμ*(*t*) = 1*. Let*
$$(m\_{\mathbf{x}\prime}M\_{\mathbf{x}}) \cap [m\_{\mathbf{t}\prime}M\_{\mathbf{t}}] = \bigcirc \mathbf{t} \in T, \qquad \text{and} \qquad m < M$$
*where mx and Mx, mx* <sup>≤</sup> *Mx, are the bounds of the operator x* <sup>=</sup> *<sup>T</sup> φt*(*xt*) *dμ*(*t*) *and*
*m* = sup {*Mt* : *Mt* ≤ *mx*, *t* ∈ *T*} , *M* = inf {*mt* : *mt* ≥ *Mx*, *t* ∈ *T*}
*If f* : *I* → **R** *is a continuous convex* (*resp. concave*) *function provided that the interval I contains all mt*, *Mt, then*
$$\begin{cases} \int\_{T} a\_{t} \mathbf{x}\_{t} \, d\mu(t) \\ \text{(resp. } \quad f \left( \int\_{T} a\_{t} \mathbf{x}\_{t} \, d\mu(t) \right) \ge \int\_{T} a\_{t} f(\mathbf{x}\_{t}) \, d\mu(t) \end{cases}$$
$$\text{(resp. } \quad f \left( \int\_{T} a\_{t} \mathbf{x}\_{t} \, d\mu(t) \right) \ge \int\_{T} a\_{t} f(\mathbf{x}\_{t}) \, d\mu(t) + \delta\_{f} \tilde{\mathbf{x}} \ge \int\_{T} a\_{t} f(\mathbf{x}\_{t}) \, d\mu(t) \text{ )$$
*holds, where δ<sup>f</sup> is defined by* (28)*,* ˜ *x*˜ = <sup>1</sup> <sup>2</sup> <sup>1</sup>*<sup>H</sup>* <sup>−</sup> <sup>1</sup> *<sup>M</sup>*¯ <sup>−</sup>*m*¯ *<sup>T</sup> atxt <sup>d</sup>μ*(*t*) <sup>−</sup> *<sup>m</sup>*¯ <sup>+</sup>*M*¯ <sup>2</sup> 1*<sup>H</sup> and m*¯ ∈ [*m*, *mA*]*, <sup>M</sup>*¯ <sup>∈</sup> [*MA*, *<sup>M</sup>*]*, <sup>m</sup>*¯ <sup>&</sup>lt; *M, are arbitrary numbers.* ¯
#### **5. Extension Jensen's inequality**
In this section we present an extension of Jensen's operator inequality for *n*−tuples of self-adjoint operators, unital *n*−tuples of positive linear mappings and real valued continuous convex functions with conditions on the spectra of the operators.
In a discrete version of Theorem 6 we prove that Jensen's operator inequality holds for every continuous convex function and for every *n*−tuple of self-adjoint operators (*A*1,..., *An*), for every *n*−tuple of positive linear mappings (Φ1,..., Φ*n*) in the case when the interval with bounds of the operator *A* = ∑*<sup>n</sup> <sup>i</sup>*=<sup>1</sup> Φ*i*(*Ai*) has no intersection points with the interval with bounds of the operator *Ai* for each *i* = 1, . . . , *n*, i.e. when (*mA*, *MA*) ∩ [*mi*, *Mi*] = ∅ for *i* = 1, . . . , *n*, where *mA* and *MA*, *mA* ≤ *MA*, are the bounds of *A*, and *mi* and *Mi*, *mi* ≤ *Mi*, are the bounds of *Ai*, *i* = 1, . . . , *n*. It is interesting to consider the case when (*mA*, *MA*) ∩ [*mi*, *Mi*] = ∅ is valid for several *i* ∈ {1, . . . , *n*}, but not for all *i* = 1, . . . , *n*. We study it in the following theorem (see [21]).
**Theorem 15.** *Let* (*A*1,..., *An*) *be an n*−*tuple of self-adjoint operators Ai* ∈ *B*(*H*) *with the bounds mi and Mi, mi* ≤ *Mi, i* = 1, . . . , *n. Let* (Φ1,..., Φ*n*) *be an n*−*tuple of positive linear mappings* Φ*<sup>i</sup>* : *<sup>B</sup>*(*H*) <sup>→</sup> *<sup>B</sup>*(*K*)*, such that* <sup>∑</sup>*<sup>n</sup> <sup>i</sup>*=<sup>1</sup> Φ*i*(1*H*) = 1*K. For* 1 ≤ *n*<sup>1</sup> < *n, we denote m* = min{*m*1,..., *mn*<sup>1</sup> }*, <sup>M</sup>* <sup>=</sup> max{*M*1,..., *Mn*<sup>1</sup> } *and* <sup>∑</sup>*n*<sup>1</sup> *<sup>i</sup>*=<sup>1</sup> <sup>Φ</sup>*i*(1*H*) = *<sup>α</sup>* <sup>1</sup>*K,* <sup>∑</sup>*<sup>n</sup> <sup>i</sup>*=*n*1+<sup>1</sup> Φ*i*(1*H*) = *β* 1*K, where α*, *β* > 0*, α* + *β* = 1*. If*
$$(m,M)\cap[m\_{i\cdot},M\_i]=\mathcal{Q}\_{\prime}\qquad i=n\_1+1,\ldots,n\_n$$
*and one of two equalities*
$$\frac{1}{\alpha} \sum\_{i=1}^{n\_1} \Phi\_i(A\_i) = \frac{1}{\beta} \sum\_{i=n\_1+1}^n \Phi\_i(A\_i) = \sum\_{i=1}^n \Phi\_i(A\_i)$$
*is valid, then*
14 Will-be-set-by-IN-TECH
*<sup>M</sup>*¯ <sup>=</sup> *Mx* <sup>=</sup> 0.5*, then <sup>X</sup>* <sup>=</sup> **<sup>0</sup>***, so we do not have an improvement of Jensen's inequality. Also, if we put*
Putting Φ*t*(*y*) = *aty* for every *y* ∈ A, where *at* ≥ 0 is a real number, we obtain the following
**Corollary 14.** *Let* (*xt*)*t*∈*<sup>T</sup> be a bounded continuous field of self-adjoint elements in a unital C*∗*-algebra* A *defined on a locally compact Hausdorff space T equipped with a bounded Radon measure μ. Let mt and Mt, mt* ≤ *Mt, be the bounds of xt, t* ∈ *T. Let* (*at*)*t*∈*<sup>T</sup> be a continuous field of nonnegative real*
(*mx*, *Mx*) ∩ [*mt*, *Mt*] = ∅, *t* ∈ *T*, *and m* < *M*
*m* = sup {*Mt* : *Mt* ≤ *mx*, *t* ∈ *T*} , *M* = inf {*mt* : *mt* ≥ *Mx*, *t* ∈ *T*}
*If f* : *I* → **R** *is a continuous convex* (*resp. concave*) *function provided that the interval I contains all*
*<sup>T</sup> at <sup>f</sup>*(*xt*) *<sup>d</sup>μ*(*t*) <sup>−</sup> *<sup>δ</sup><sup>f</sup>* ˜
*<sup>T</sup> at <sup>f</sup>*(*xt*) *<sup>d</sup>μ*(*t*) + *<sup>δ</sup><sup>f</sup>* ˜
*, δ<sup>f</sup>* = 7/8 *and δ<sup>f</sup> X* = 0.4375
0.5 0 0 0.2040 1.5787 0 0 0.6441
*<sup>T</sup> φt*(*xt*) *dμ*(*t*) *and*
*<sup>T</sup> at f*(*xt*) *dμ*(*t*)
*<sup>T</sup> at f*(*xt*) *dμ*(*t*) )
*and m*¯ ∈ [*m*, *mA*]*,*
<sup>2</sup> 1*<sup>H</sup>*
*<sup>x</sup>*˜ <sup>≤</sup>
*<sup>x</sup>*˜ <sup>≥</sup>
*<sup>i</sup>*=<sup>1</sup> Φ*i*(*Ai*) has no intersection points with the interval
*<sup>T</sup> atxt <sup>d</sup>μ*(*t*) <sup>−</sup> *<sup>m</sup>*¯ <sup>+</sup>*M*¯
1 0 0 1 *. But, if we put m*¯ = *mx* = 0*,*
*, which is worse than the*
<sup>4</sup> <sup>=</sup> <sup>1</sup> 16 1 0 0 0
= Φ<sup>1</sup> *X*4 1 + Φ<sup>2</sup> *X*4 2 −
*and its improvement*
<
*<sup>m</sup>*¯ <sup>=</sup> <sup>0</sup>*, <sup>M</sup>*¯ <sup>=</sup> <sup>1</sup>*, then <sup>X</sup>* <sup>=</sup> 0.5
obvious corollary of Theorem 12.
*f T*
*T*
*<sup>M</sup>*¯ <sup>∈</sup> [*MA*, *<sup>M</sup>*]*, <sup>m</sup>*¯ <sup>&</sup>lt; *M, are arbitrary numbers.* ¯
**5. Extension Jensen's inequality**
with bounds of the operator *A* = ∑*<sup>n</sup>*
(*resp. f*
*holds, where δ<sup>f</sup> is defined by* (28)*,* ˜
*above improvement.*
*numbers such that*
*mt*, *Mt, then*
(Φ1(*X*1) + Φ2(*X*2))
639.9213 −255 −255 117.8559
*(rounded to four decimal places), since δ<sup>f</sup>* = 3.1574*, X* =
1 0 0 1
*<sup>T</sup> at dμ*(*t*) = 1*. Let*
*where mx and Mx, mx* <sup>≤</sup> *Mx, are the bounds of the operator x* <sup>=</sup>
*atxt dμ*(*t*)
*atxt dμ*(*t*)
<sup>≤</sup>
<sup>≥</sup>
*x*˜ = <sup>1</sup>
convex functions with conditions on the spectra of the operators.
<sup>2</sup> <sup>1</sup>*<sup>H</sup>* <sup>−</sup> <sup>1</sup> *<sup>M</sup>*¯ <sup>−</sup>*m*¯
In this section we present an extension of Jensen's operator inequality for *n*−tuples of self-adjoint operators, unital *n*−tuples of positive linear mappings and real valued continuous
In a discrete version of Theorem 6 we prove that Jensen's operator inequality holds for every continuous convex function and for every *n*−tuple of self-adjoint operators (*A*1,..., *An*), for every *n*−tuple of positive linear mappings (Φ1,..., Φ*n*) in the case when the interval
with bounds of the operator *Ai* for each *i* = 1, . . . , *n*, i.e. when (*mA*, *MA*) ∩ [*mi*, *Mi*] = ∅
$$\frac{1}{\alpha} \sum\_{i=1}^{n\_1} \Phi\_i(f(A\_i)) \le \sum\_{i=1}^n \Phi\_i(f(A\_i)) \le \frac{1}{\beta} \sum\_{i=n\_1+1}^n \Phi\_i(f(A\_i)) \tag{31}$$
*holds for every continuous convex function f* : *I* → **R** *provided that the interval I contains all mi*, *Mi, i* = 1, . . . , *n. If f* : *I* → **R** *is concave, then the reverse inequality is valid in* (31)*.*
*Proof.* We prove only the case when *f* is a convex function. Let us denote
$$A = \frac{1}{\mathfrak{a}} \sum\_{i=1}^{n\_1} \Phi\_i(A\_i), \qquad B = \frac{1}{\beta} \sum\_{i=n\_1+1}^n \Phi\_i(A\_i), \qquad \mathsf{C} = \sum\_{i=1}^n \Phi\_i(A\_i).$$
It is easy to verify that *A* = *B* or *B* = *C* or *A* = *C* implies *A* = *B* = *C*. **a)** Let *m* < *M*. Since *f* is convex on [*m*, *M*] and [*mi*, *Mi*] ⊆ [*m*, *M*] for *i* = 1, . . . , *n*1, then
$$f(z) \le \frac{M-z}{M-m} f(m) + \frac{z-m}{M-m} f(M), \quad z \in [m\_i, M\_i] \text{ for } i = 1, \dots, n\_1 \tag{32}$$
but since *f* is convex on all [*mi*, *Mi*] and (*m*, *M*) ∩ [*mi*, *Mi*] = ∅ for *i* = *n*<sup>1</sup> + 1, . . . , *n*, then
$$f(z) \ge \frac{M-z}{M-m} f(m) + \frac{z-m}{M-m} f(M), \quad z \in [m\_i, M\_i] \text{ for } i = n\_1 + 1, \dots, n \tag{33}$$
Since *mi*1*<sup>H</sup>* ≤ *Ai* ≤ *Mi*1*H*, *i* = 1, . . . , *n*1, it follows from (32)
$$f\left(A\_{i}\right) \le \frac{M\mathbf{1}\_{H} - A\_{i}}{M - m} f(m) + \frac{A\_{i} - m\mathbf{1}\_{H}}{M - m} f(M), \qquad i = 1, \dots, n\_{1}$$
Applying a positive linear mapping Φ*<sup>i</sup>* and summing, we obtain
$$\sum\_{i=1}^{n\_1} \Phi\_i \left( f(A\_i) \right) \le \frac{M\mathfrak{a}\mathbf{1}\_K - \sum\_{i=1}^{n\_1} \Phi\_i(A\_i)}{M - m} f(m) + \frac{\sum\_{i=1}^{n\_1} \Phi\_i(A\_i) - m\mathfrak{a}\mathbf{1}\_K}{M - m} f(M)$$
#### 16 Will-be-set-by-IN-TECH 204 Linear Algebra – Theorems and Applications
since ∑*n*<sup>1</sup> *<sup>i</sup>*=<sup>1</sup> Φ*i*(1*H*) = *α*1*K*. It follows
$$\frac{1}{\alpha} \sum\_{i=1}^{n\_1} \Phi\_i \left( f(A\_i) \right) \le \frac{M \mathbf{1}\_K - A}{M - m} f(m) + \frac{A - m \mathbf{1}\_K}{M - m} f(M) \tag{34}$$
Similarly to (34) in the case *mi*1*<sup>H</sup>* ≤ *Ai* ≤ *Mi*1*H*, *i* = *n*<sup>1</sup> + 1, . . . , *n*, it follows from (33)
$$\frac{1}{\beta} \sum\_{i=n\_1+1}^n \Phi\_i \left( f(A\_i) \right) \ge \frac{M \mathbf{1}\_K - B}{M - m} f(m) + \frac{B - m \mathbf{1}\_K}{M - m} f(M) \tag{35}$$
Combining (34) and (35) and taking into account that *A* = *B*, we obtain
$$\frac{1}{\alpha} \sum\_{i=1}^{n\_1} \Phi\_{\bar{l}} \left( f(A\_{\bar{l}}) \right) \le \frac{1}{\beta} \sum\_{i=n\_1+1}^{n} \Phi\_{\bar{l}} \left( f(A\_{\bar{l}}) \right) \tag{36}$$
It follows
$$\begin{split} \frac{1}{\mathfrak{a}} \sum\_{i=1}^{n\_1} \Phi\_i(f(A\_i)) &= \sum\_{i=1}^{n\_1} \Phi\_i(f(A\_i)) + \frac{\beta}{\mathfrak{a}} \sum\_{i=1}^{n\_1} \Phi\_i(f(A\_i)) \\ &\leq \sum\_{i=1}^{n\_1} \Phi\_i(f(A\_i)) + \sum\_{i=n\_1+1}^{n} \Phi\_i(f(A\_i)) \end{split} \tag{\text{by } \mathfrak{a} + \beta = 1}$$
$$\begin{aligned} &= \sum\_{i=1}^{n} \Phi\_i(f(A\_i)) \\ &\leq \frac{\alpha}{\beta} \sum\_{i=n\_1+1}^{n} \Phi\_i(f(A\_i)) + \sum\_{i=n\_1+1}^{n} \Phi\_i(f(A\_i)) &\quad &(\text{by (36)}) \\ &= \frac{1}{\beta} \sum\_{i=n\_1+1}^{n} \Phi\_i(f(A\_i)) &\quad &(\text{by } \alpha + \beta = 1) \end{aligned}$$
which gives the desired double inequality (31).
**b)** Let *m* = *M*. Since [*mi*, *Mi*] ⊆ [*m*, *M*] for *i* = 1, . . . , *n*1, then *Ai* = *m*1*<sup>H</sup>* and *f*(*Ai*) = *f*(*m*)1*<sup>H</sup>* for *i* = 1, . . . , *n*1. It follows
$$\frac{1}{\mathfrak{a}}\sum\_{i=1}^{\eta\_1} \Phi\_{\bar{i}}(A\_i) = m \mathbf{1}\_K \qquad \text{and} \qquad \frac{1}{\mathfrak{a}}\sum\_{i=1}^{\eta\_1} \Phi\_{\bar{i}}\left(f(A\_i)\right) = f(m)\mathbf{1}\_K \tag{37}$$
On the other hand, since *f* is convex on *I*, we have
$$f(z) \ge f(m) + l(m)(z - m) \quad \text{for every } z \in I \tag{38}$$
where *l* is the subdifferential of *f* . Replacing *z* by *Ai* for *i* = *n*<sup>1</sup> + 1, . . . , *n*, applying Φ*<sup>i</sup>* and summing, we obtain from (38) and (37)
$$\begin{aligned} \frac{1}{\beta} \sum\_{i=n\_1+1}^n \Phi\_i\left(f(A\_i)\right) &\geq f(m)\mathbf{1}\_K + l(m) \left(\frac{1}{\beta} \sum\_{i=n\_1+1}^n \Phi\_i(A\_i) - m\mathbf{1}\_K\right) \\ &= f(m)\mathbf{1}\_K = \frac{1}{\alpha} \sum\_{i=1}^{n\_1} \Phi\_i\left(f(A\_i)\right) \end{aligned}$$
So (36) holds again. The remaining part of the proof is the same as in the case a).
**Remark 16.** *We obtain the equivalent inequality to the one in Theorem 15 in the case when* ∑*n <sup>i</sup>*=<sup>1</sup> Φ*i*(1*H*) = *γ* 1*K, for some positive scalar γ. If α* + *β* = *γ and one of two equalities*
$$\frac{1}{\alpha} \sum\_{i=1}^{m\_1} \Phi\_i(A\_i) = \frac{1}{\beta} \sum\_{i=n\_1+1}^n \Phi\_i(A\_i) = \frac{1}{\gamma} \sum\_{i=1}^n \Phi\_i(A\_i)$$
*is valid, then*
16 Will-be-set-by-IN-TECH
1 *β*
*n* ∑ *i*=*n*1+1
*n* ∑ *i*=*n*1+1
Φ*i*(*f*(*Ai*)) (by *α* + *β* = 1)
*f*(*z*) ≥ *f*(*m*) + *l*(*m*)(*z* − *m*) for every *z* ∈ *I* (38)
*n* ∑ *i*=*n*1+1
Φ*<sup>i</sup>* (*f*(*Ai*))
*<sup>M</sup>* <sup>−</sup> *<sup>m</sup> <sup>f</sup>*(*m*) + *<sup>A</sup>* <sup>−</sup> *<sup>m</sup>*1*<sup>K</sup>*
*<sup>M</sup>* <sup>−</sup> *<sup>m</sup> <sup>f</sup>*(*m*) + *<sup>B</sup>* <sup>−</sup> *<sup>m</sup>*1*<sup>K</sup>*
*<sup>M</sup>* <sup>−</sup> *<sup>m</sup> <sup>f</sup>*(*M*) (34)
*<sup>M</sup>* <sup>−</sup> *<sup>m</sup> <sup>f</sup>*(*M*) (35)
Φ*<sup>i</sup>* (*f*(*Ai*)) (36)
Φ*i*(*f*(*Ai*)) (by *α* + *β* = 1)
Φ*i*(*f*(*Ai*)) (by (36))
Φ*i*(*f*(*Ai*)) (by (36))
Φ*<sup>i</sup>* (*f*(*Ai*)) = *f*(*m*)1*<sup>K</sup>* (37)
Φ*i*(*Ai*) − *m*1*<sup>K</sup>*
<sup>Φ</sup>*<sup>i</sup>* (*f*(*Ai*)) <sup>≤</sup> *<sup>M</sup>*1*<sup>K</sup>* <sup>−</sup> *<sup>A</sup>*
Similarly to (34) in the case *mi*1*<sup>H</sup>* ≤ *Ai* ≤ *Mi*1*H*, *i* = *n*<sup>1</sup> + 1, . . . , *n*, it follows from (33)
<sup>Φ</sup>*<sup>i</sup>* (*f*(*Ai*)) <sup>≥</sup> *<sup>M</sup>*1*<sup>K</sup>* <sup>−</sup> *<sup>B</sup>*
Φ*<sup>i</sup>* (*f*(*Ai*)) ≤
*α*
Φ*i*(*f*(*Ai*)) +
*n*1 ∑ *i*=1
*n* ∑ *i*=*n*1+1
**b)** Let *m* = *M*. Since [*mi*, *Mi*] ⊆ [*m*, *M*] for *i* = 1, . . . , *n*1, then *Ai* = *m*1*<sup>H</sup>* and *f*(*Ai*) = *f*(*m*)1*<sup>H</sup>*
where *l* is the subdifferential of *f* . Replacing *z* by *Ai* for *i* = *n*<sup>1</sup> + 1, . . . , *n*, applying Φ*<sup>i</sup>* and
*α*
*α*
*n*1 ∑ *i*=1
1 *β*
*n*1 ∑ *i*=1
Combining (34) and (35) and taking into account that *A* = *B*, we obtain
<sup>Φ</sup>*i*(*f*(*Ai*)) + *<sup>β</sup>*
Φ*i*(*f*(*Ai*)) +
Φ*i*(*f*(*Ai*))
<sup>Φ</sup>*i*(*Ai*) = *<sup>m</sup>*1*<sup>K</sup>* and <sup>1</sup>
Φ*<sup>i</sup>* (*f*(*Ai*)) ≥ *f*(*m*)1*<sup>K</sup>* + *l*(*m*)
<sup>=</sup> *<sup>f</sup>*(*m*)1*<sup>K</sup>* <sup>=</sup> <sup>1</sup>
So (36) holds again. The remaining part of the proof is the same as in the case a).
*n* ∑ *i*=*n*1+1
*n* ∑ *i*=*n*1+1
since ∑*n*<sup>1</sup>
It follows
1 *α*
*n*1 ∑ *i*=1
*<sup>i</sup>*=<sup>1</sup> Φ*i*(1*H*) = *α*1*K*. It follows 1 *α*
> 1 *β*
Φ*i*(*f*(*Ai*)) =
*n*1 ∑ *i*=1
*n* ∑ *i*=*n*1+1
> 1 *α*
*n*1 ∑ *i*=1
≤ *n*1 ∑ *i*=1
= *n* ∑ *i*=1
≤ *α β*
<sup>=</sup> <sup>1</sup> *β*
On the other hand, since *f* is convex on *I*, we have
which gives the desired double inequality (31).
for *i* = 1, . . . , *n*1. It follows
1 *α*
*n*1 ∑ *i*=1
summing, we obtain from (38) and (37)
*n* ∑ *i*=*n*1+1
1 *β*
*n*1 ∑ *i*=1
$$\frac{1}{\alpha} \sum\_{i=1}^{n\_1} \Phi\_i(f(A\_i)) \le \frac{1}{\gamma} \sum\_{i=1}^n \Phi\_i(f(A\_i)) \le \frac{1}{\beta} \sum\_{i=n\_1+1}^n \Phi\_i(f(A\_i))$$
*holds for every continuous convex function f .*
**Remark 17.** *Let the assumptions of Theorem 15 be valid. 1. We observe that the following inequality*
$$f\left(\frac{1}{\beta}\sum\_{i=n\_1+1}^n \Phi\_i(A\_i)\right) \le \frac{1}{\beta} \sum\_{i=n\_1+1}^n \Phi\_i(f(A\_i))$$
*holds for every continuous convex function f* : *I* → **R***.*
*Indeed, by the assumptions of Theorem 15 we have*
$$\max \mathbf{1}\_H \le \sum\_{i=1}^{n\_1} \Phi\_i(f(A\_i)) \le M\alpha \mathbf{1}\_H \quad \text{and} \quad \frac{1}{\alpha} \sum\_{i=1}^{n\_1} \Phi\_i(A\_i) = \frac{1}{\beta} \sum\_{i=n\_1+1}^n \Phi\_i(A\_i)$$
*which implies*
$$m\mathbf{1}\_H \le \sum\_{i=n\_1+1}^n \frac{1}{\beta} \Phi\_i(f(A\_i)) \le M\mathbf{1}\_H.$$
*Also* (*m*, *<sup>M</sup>*) <sup>∩</sup> [*mi*, *Mi*] = <sup>∅</sup> *for i* <sup>=</sup> *<sup>n</sup>*<sup>1</sup> <sup>+</sup> 1, . . . , *n and* <sup>∑</sup>*<sup>n</sup> i*=*n*1+1 1 *<sup>β</sup>*Φ*i*(1*H*) = 1*<sup>K</sup> hold. So we can apply Theorem 6 on operators An*1+1,..., *An and mappings* <sup>1</sup> *<sup>β</sup>*Φ*<sup>i</sup> and obtain the desired inequality.*
*2. We denote by mC and MC the bounds of C* = ∑*<sup>n</sup> <sup>i</sup>*=<sup>1</sup> Φ*i*(*Ai*)*. If* (*mC*, *MC*) ∩ [*mi*, *Mi*] = ∅*, i* = 1, . . . , *n*<sup>1</sup> *or f is an operator convex function on* [*m*, *M*]*, then the double inequality* (31) *can be extended from the left side if we use Jensen's operator inequality (see [16, Theorem 2.1])*
$$\begin{aligned} f\left(\sum\_{i=1}^n \Phi\_i(A\_i)\right) &= f\left(\frac{1}{n} \sum\_{i=1}^{n\_1} \Phi\_i(A\_i)\right) \\ &\le \frac{1}{n} \sum\_{i=1}^{n\_1} \Phi\_i(f(A\_i)) \le \sum\_{i=1}^n \Phi\_i(f(A\_i)) \le \frac{1}{\beta} \sum\_{i=n\_1+1}^n \Phi\_i(f(A\_i)) \end{aligned}$$
**Example 18.** *If neither assumptions* (*mC*, *MC*) ∩ [*mi*, *Mi*] = ∅*, i* = 1, . . . , *n*1*, nor f is operator convex in Remark 17 - 2. is satisfied and if* 1 < *n*<sup>1</sup> < *n, then* (31) *can not be extended by Jensen's operator inequality, since it is not valid. Indeed, for n*<sup>1</sup> = 2 *we define mappings* Φ1, Φ<sup>2</sup> : *M*3(**C**) → *<sup>M</sup>*2(**C**) *by* <sup>Φ</sup>1((*aij*)1≤*i*,*j*≤3) = *<sup>α</sup>* <sup>2</sup> (*aij*)1≤*i*,*j*≤2*,* <sup>Φ</sup><sup>2</sup> = <sup>Φ</sup>1*. Then* <sup>Φ</sup>1(*I*3) + <sup>Φ</sup>2(*I*3) = *<sup>α</sup>I*2*. If*
$$A\_1 = 2\begin{pmatrix} 1 \ 0 \ 1 \\ 0 \ 0 \ 1 \\ 1 \ 1 \ 1 \end{pmatrix} \quad \text{and} \quad A\_2 = 2\begin{pmatrix} 1 \ 0 \ 0 \\ 0 \ 0 \ 0 \\ 0 \ 0 \ 0 \end{pmatrix}$$
*then*
$$\left(\frac{1}{a}\Phi\_1(A\_1) + \frac{1}{a}\Phi\_2(A\_2)\right)^4 = \frac{1}{a^4}\begin{pmatrix} 16 \ 0 \\ 0 \ 0 \end{pmatrix} \not\le \frac{1}{a}\begin{pmatrix} 80 \ 40 \\ 40 \ 24 \end{pmatrix} = \frac{1}{a}\Phi\_1\left(A\_1^4\right) + \frac{1}{a}\Phi\_2\left(A\_2^4\right)$$
*for every α* ∈ (0, 1)*. We observe that f*(*t*) = *t* <sup>4</sup> *is not operator convex and* (*mC*, *MC*) <sup>∩</sup> [*mi*, *Mi*] �<sup>=</sup> ∅, *since C* = *A* = <sup>1</sup> *<sup>α</sup>*Φ1(*A*1) + <sup>1</sup> *<sup>α</sup>*Φ2(*A*2) = <sup>1</sup> *α* 2 0 0 0 , [*mC*, *MC*]=[0, 2/*α*]*,* [*m*1, *M*1] ⊂ [−1.60388, 4.49396] *and* [*m*2, *M*2]=[0, 2]*.*
With respect to Remark 16, we obtain the following obvious corollary of Theorem 15.
**Corollary 19.** *Let* (*A*1,..., *An*) *be an n*−*tuple of self-adjoint operators Ai* ∈ *B*(*H*) *with the bounds mi and Mi, mi* ≤ *Mi, i* = 1, . . . , *n. For some* 1 ≤ *n*<sup>1</sup> < *n, we denote m* = min{*m*1,..., *mn*<sup>1</sup> }*, M* = max{*M*1,..., *Mn*<sup>1</sup> }*. Let* (*p*1,..., *pn*) *be an n*−*tuple of non-negative numbers, such that* 0 < ∑*n*<sup>1</sup> *<sup>i</sup>*=<sup>1</sup> *pi* <sup>=</sup> **pn1** <sup>&</sup>lt; **pn** <sup>=</sup> <sup>∑</sup>*<sup>n</sup> <sup>i</sup>*=<sup>1</sup> *pi. If*
$$(m,M)\cap[m\_{i\cdot},M\_i]=\mathcal{Q}\_{\prime}\qquad i=n\_1+1,\ldots,n\_n$$
*and one of two equalities*
$$\frac{1}{\mathbf{p\_{n\_1}}} \sum\_{i=1}^{n\_1} p\_i A\_i = \frac{1}{\mathbf{p\_{n\_1}}} \sum\_{i=1}^n p\_i A\_i = \frac{1}{\mathbf{p\_{n\_1}} - \mathbf{p\_{n\_1}}} \sum\_{i=n\_1+1}^n p\_i A\_i$$
*is valid, then*
$$\frac{1}{\mathbf{p}\_{\mathbf{n}\_{1}}}\sum\_{i=1}^{n\_{1}}p\_{i}f(A\_{i}) \le \frac{1}{\mathbf{p}\_{\mathbf{n}}}\sum\_{i=1}^{n}p\_{i}f(A\_{i}) \le \frac{1}{\mathbf{p}\_{\mathbf{n}} - \mathbf{p}\_{\mathbf{n}\_{1}}}\sum\_{i=n\_{1}+1}^{n}p\_{i}f(A\_{i})\tag{39}$$
*holds for every continuous convex function f* : *I* → **R** *provided that the interval I contains all mi*, *Mi, i* = 1, . . . , *n.*
*If f* : *I* → **R** *is concave, then the reverse inequality is valid in* (39)*.*
As a special case of Corollary 19 we can obtain a discrete version of Corollary 7 as follows.
**Corollary 20** (Discrete version of Corollary 7)**.** *Let* (*A*1,..., *An*) *be an n*−*tuple of self-adjoint operators Ai* ∈ *B*(*H*) *with the bounds mi and Mi, mi* ≤ *Mi, i* = 1, . . . , *n. Let* (*α*1,..., *αn*) *be an <sup>n</sup>*−*tuple of nonnegative real numbers such that* <sup>∑</sup>*<sup>n</sup> <sup>i</sup>*=<sup>1</sup> *α<sup>i</sup>* = 1*. If*
$$(m\_{A\prime}M\_A) \cap [m\_{\dot{1}\prime}M\_{\dot{1}}] = \bigcirc \qquad \quad \dot{\mathbf{i}} = \mathbf{1}, \ldots, \mathbf{n} \tag{40}$$
*where mA and MA, mA* <sup>≤</sup> *MA, are the bounds of A* <sup>=</sup> <sup>∑</sup>*<sup>n</sup> <sup>i</sup>*=<sup>1</sup> *αiAi, then*
$$f\left(\sum\_{i=1}^{n} \alpha\_i A\_i\right) \le \sum\_{i=1}^{n} \alpha\_i f(A\_i) \tag{41}$$
*holds for every continuous convex function f* : *I* → **R** *provided that the interval I contains all mi*, *Mi.*
*Proof.* We prove only the convex case. We define (*n* + 1)−tuple of operators (*B*1,..., *Bn*+1), *Bi* <sup>∈</sup> *<sup>B</sup>*(*H*), by *<sup>B</sup>*<sup>1</sup> <sup>=</sup> *<sup>A</sup>* <sup>=</sup> <sup>∑</sup>*<sup>n</sup> <sup>i</sup>*=<sup>1</sup> *<sup>α</sup>iAi* and *Bi* = *Ai*−1, *<sup>i</sup>* = 2, . . . , *<sup>n</sup>* + 1. Then *mB*<sup>1</sup> = *mA*, *MB*<sup>1</sup> = *MA* are the bounds of *<sup>B</sup>*<sup>1</sup> and *mBi* = *mi*−1, *MBi* = *Mi*−<sup>1</sup> are the ones of *Bi*, *<sup>i</sup>* = 2, . . . , *n* + 1. Also, we define (*n* + 1)−tuple of non-negative numbers (*p*1,..., *pn*+1) by *p*<sup>1</sup> = 1 and *pi* <sup>=</sup> *<sup>α</sup>i*−1, *<sup>i</sup>* <sup>=</sup> 2, . . . , *<sup>n</sup>* <sup>+</sup> 1. Then <sup>∑</sup>*n*+<sup>1</sup> *<sup>i</sup>*=<sup>1</sup> *pi* = 2 and by using (40) we have
$$(m\_{B\_{1'}}M\_{B\_1}) \cap [m\_{B\_{1'}}M\_{B\_1}] = \bigcirc, \qquad \mathbf{i} = \mathbf{2}, \ldots, n+1\tag{42}$$
Since *<sup>n</sup>*+<sup>1</sup>
18 Will-be-set-by-IN-TECH
*<sup>α</sup>*Φ2(*A*2) = <sup>1</sup>
With respect to Remark 16, we obtain the following obvious corollary of Theorem 15.
*α* 2 0 0 0
**Corollary 19.** *Let* (*A*1,..., *An*) *be an n*−*tuple of self-adjoint operators Ai* ∈ *B*(*H*) *with the bounds mi and Mi, mi* ≤ *Mi, i* = 1, . . . , *n. For some* 1 ≤ *n*<sup>1</sup> < *n, we denote m* = min{*m*1,..., *mn*<sup>1</sup> }*, M* = max{*M*1,..., *Mn*<sup>1</sup> }*. Let* (*p*1,..., *pn*) *be an n*−*tuple of non-negative numbers, such that* 0 <
(*m*, *M*) ∩ [*mi*, *Mi*] = ∅, *i* = *n*<sup>1</sup> + 1, . . . , *n*
*pi f*(*Ai*) ≤
*holds for every continuous convex function f* : *I* → **R** *provided that the interval I contains all mi*, *Mi,*
As a special case of Corollary 19 we can obtain a discrete version of Corollary 7 as follows.
**Corollary 20** (Discrete version of Corollary 7)**.** *Let* (*A*1,..., *An*) *be an n*−*tuple of self-adjoint operators Ai* ∈ *B*(*H*) *with the bounds mi and Mi, mi* ≤ *Mi, i* = 1, . . . , *n. Let* (*α*1,..., *αn*) *be an*
*holds for every continuous convex function f* : *I* → **R** *provided that the interval I contains all mi*, *Mi.*
*<sup>i</sup>*=<sup>1</sup> *α<sup>i</sup>* = 1*. If*
(*mA*, *MA*) ∩ [*mi*, *Mi*] = ∅, *i* = 1, . . . , *n* (40)
*<sup>i</sup>*=<sup>1</sup> *αiAi, then*
*α<sup>i</sup> f*(*Ai*) (41)
*piAi* <sup>=</sup> <sup>1</sup>
**pn** − **pn1**
1 **pn** − **pn1**
*n* ∑ *i*=*n*1+1
> *n* ∑ *i*=*n*1+1
*piAi*
*pi f*(*Ai*) (39)
<sup>=</sup> <sup>1</sup> *α* Φ<sup>1</sup> *A*4 1 + 1 *α* Φ<sup>2</sup> *A*4 2
<sup>4</sup> *is not operator convex and* (*mC*, *MC*) <sup>∩</sup> [*mi*, *Mi*] �<sup>=</sup>
, [*mC*, *MC*]=[0, 2/*α*]*,* [*m*1, *M*1] ⊂
*then*
∑*n*<sup>1</sup>
1 *α*
<sup>Φ</sup>1(*A*1) + <sup>1</sup>
∅, *since C* = *A* = <sup>1</sup>
*<sup>i</sup>*=<sup>1</sup> *pi* <sup>=</sup> **pn1** <sup>&</sup>lt; **pn** <sup>=</sup> <sup>∑</sup>*<sup>n</sup>*
*and one of two equalities*
*is valid, then*
*i* = 1, . . . , *n.*
*α*
*for every α* ∈ (0, 1)*. We observe that f*(*t*) = *t*
[−1.60388, 4.49396] *and* [*m*2, *M*2]=[0, 2]*.*
Φ2(*A*2)
<sup>4</sup> <sup>=</sup> <sup>1</sup> *α*4 16 0 0 0 �≤ 1 *α* 80 40 40 24
*<sup>α</sup>*Φ1(*A*1) + <sup>1</sup>
*<sup>i</sup>*=<sup>1</sup> *pi. If*
1 **pn1**
*n*1 ∑ *i*=1
*<sup>n</sup>*−*tuple of nonnegative real numbers such that* <sup>∑</sup>*<sup>n</sup>*
*where mA and MA, mA* <sup>≤</sup> *MA, are the bounds of A* <sup>=</sup> <sup>∑</sup>*<sup>n</sup>*
*f n* ∑ *i*=1 *αiAi* ≤ *n* ∑ *i*=1
1 **pn1**
*n*1 ∑ *i*=1
*pi f*(*Ai*) ≤
*If f* : *I* → **R** *is concave, then the reverse inequality is valid in* (39)*.*
*piAi* <sup>=</sup> <sup>1</sup>
**pn**
*n* ∑ *i*=1
1 **pn**
*n* ∑ *i*=1
$$\sum\_{i=1}^{n+1} p\_i B\_i = B\_1 + \sum\_{i=2}^{n+1} p\_i B\_i = \sum\_{i=1}^n \alpha\_i A\_i + \sum\_{i=1}^n \alpha\_i A\_i = 2B\_1$$
then
$$p\_1 B\_1 = \frac{1}{2} \sum\_{i=1}^{n+1} p\_i B\_i = \sum\_{i=2}^{n+1} p\_i B\_i \tag{43}$$
Taking into account (42) and (43), we can apply Corollary 19 for *n*<sup>1</sup> = 1 and *Bi*, *pi* as above, and we get
$$p\_1 f(\mathcal{B}\_1) \le \frac{1}{2} \sum\_{i=1}^{n+1} p\_i f(\mathcal{B}\_i) \le \sum\_{i=2}^{n+1} p\_i f(\mathcal{B}\_i)$$
which gives the desired inequality (41).
#### **6. Extension of the refined Jensen's inequality**
There is an extensive literature devoted to Jensen's inequality concerning different refinements and extensive results, see, for example [22–29].
In this section we present an extension of the refined Jensen's inequality obtained in Section 4 and a refinement of the same inequality obtained in Section 5.
**Theorem 21.** *Let* (*A*1,..., *An*) *be an n*−*tuple of self-adjoint operators Ai* ∈ *B*(*H*) *with the bounds mi and Mi, mi* ≤ *Mi, i* = 1, . . . , *n. Let* (Φ1,..., Φ*n*) *be an n*−*tuple of positive linear mappings* <sup>Φ</sup>*<sup>i</sup>* : *<sup>B</sup>*(*H*) <sup>→</sup> *<sup>B</sup>*(*K*)*, such that* <sup>∑</sup>*n*<sup>1</sup> *<sup>i</sup>*=<sup>1</sup> <sup>Φ</sup>*i*(1*H*) = *<sup>α</sup>* <sup>1</sup>*K,* <sup>∑</sup>*<sup>n</sup> <sup>i</sup>*=*n*1+<sup>1</sup> Φ*i*(1*H*) = *β* 1*K, where* 1 ≤ *n*<sup>1</sup> < *n, α*, *β* > 0 *and α* + *β* = 1*. Let mL* = min{*m*1,..., *mn*<sup>1</sup> }*, MR* = max{*M*1,..., *Mn*<sup>1</sup> } *and*
$$\begin{aligned} m &= \max\left\{ M\_{\bar{i}} \colon M\_{\bar{i}} \le m\_{L'} \, \middle| \, \bar{i} \in \{ n\_1 + 1, \dots, n \} \right\} \\ M &= \min\left\{ m\_{\bar{i}} \colon m\_{\bar{i}} \ge M\_{R'} \, \middle| \, \bar{i} \in \{ n\_1 + 1, \dots, n \} \right\} \end{aligned}$$
*If*
$$(\mathfrak{m}\_{L\prime}M\_R) \cap [\mathfrak{m}\_{\natural}M\_{\bar{i}}] = \bigcirc \,, \quad \mathfrak{i} = \mathfrak{n}\_1 + \mathfrak{1}, \ldots, \mathfrak{n}\_{\prime} \qquad \text{and} \qquad \mathfrak{m} < M$$
*and one of two equalities*
$$\frac{1}{\pi} \sum\_{i=1}^{n\_1} \Phi\_i(A\_i) = \sum\_{i=1}^n \Phi\_i(A\_i) = \frac{1}{\beta} \sum\_{i=n\_1+1}^n \Phi\_i(A\_i)$$
*is valid, then*
$$\begin{split} \frac{1}{n} \sum\_{i=1}^{n\_1} \Phi\_i(f(A\_i)) &\le \frac{1}{n} \sum\_{i=1}^{n\_1} \Phi\_i(f(A\_i)) + \beta \delta\_f \tilde{A} \le \sum\_{i=1}^n \Phi\_i(f(A\_i)) \\ &\le \frac{1}{\beta} \sum\_{i=n\_1+1}^n \Phi\_i(f(A\_i)) - a \delta\_f \tilde{A} \le \frac{1}{\beta} \sum\_{i=n\_1+1}^n \Phi\_i(f(A\_i)) \end{split} \tag{44}$$
*holds for every continuous convex function f* : *I* → **R** *provided that the interval I contains all mi*, *Mi, i* = 1, . . . , *n, where*
$$\begin{aligned} \delta\_f \equiv \delta\_f(\bar{m}, \bar{M}) &= f(\bar{m}) + f(\bar{M}) - 2f\left(\frac{\bar{m} + \bar{M}}{2}\right) \\ \tilde{A} \equiv \tilde{A}\_{A, \Phi, n\_1, a}(\bar{m}, \bar{M}) &= \frac{1}{2} \mathbf{1}\_K - \frac{1}{a(\bar{M} - \bar{m})} \sum\_{i=1}^{n\_1} \Phi\_i\left(\left|A\_i - \frac{\bar{m} + \bar{M}}{2} \mathbf{1}\_H\right|\right) \end{aligned} \tag{45}$$
*and <sup>m</sup>*¯ <sup>∈</sup> [*m*, *mL*]*, <sup>M</sup>*¯ <sup>∈</sup> [*MR*, *<sup>M</sup>*]*, <sup>m</sup>*¯ <sup>&</sup>lt; *M, are arbitrary numbers. If f* ¯ : *<sup>I</sup>* <sup>→</sup> **<sup>R</sup>** *is concave, then the reverse inequality is valid in* (44)*.*
*Proof.* We prove only the convex case. Let us denote
$$A = \frac{1}{\mathfrak{a}} \sum\_{i=1}^{n\_1} \Phi\_i(A\_i), \qquad B = \frac{1}{\mathfrak{d}} \sum\_{i=n\_1+1}^n \Phi\_i(A\_i), \qquad \mathsf{C} = \sum\_{i=1}^n \Phi\_i(A\_i).$$
It is easy to verify that *A* = *B* or *B* = *C* or *A* = *C* implies *A* = *B* = *C*.
Since *<sup>f</sup>* is convex on [*m*¯ , *<sup>M</sup>*¯ ] and Sp(*Ai*) <sup>⊆</sup> [*mi*, *Mi*] <sup>⊆</sup> [*m*¯ , *<sup>M</sup>*¯ ] for *<sup>i</sup>* <sup>=</sup> 1, . . . , *<sup>n</sup>*1, it follows from Lemma 11 that
$$f\left(A\_{i}\right) \leq \frac{\tilde{M}1\_{H} - A\_{i}}{\tilde{M} - \tilde{m}} f(\tilde{m}) + \frac{A\_{i} - \tilde{m}1\_{H}}{\tilde{M} - \tilde{m}} f(\tilde{M}) - \delta\_{f} \tilde{A}\_{i\prime} \qquad i = 1, \dots, n\_{1}$$
holds, where *<sup>δ</sup><sup>f</sup>* <sup>=</sup> *<sup>f</sup>*(*m*¯) + *<sup>f</sup>*(*M*¯ ) <sup>−</sup> <sup>2</sup> *<sup>f</sup> <sup>m</sup>*¯ +*M*¯ 2 and *<sup>A</sup><sup>i</sup>* <sup>=</sup> <sup>1</sup> <sup>2</sup> <sup>1</sup>*<sup>H</sup>* <sup>−</sup> <sup>1</sup> *<sup>M</sup>*¯ <sup>−</sup>*m*¯ *Ai* <sup>−</sup> *<sup>m</sup>*¯ <sup>+</sup>*M*¯ <sup>2</sup> 1*<sup>H</sup>* . Applying a positive linear mapping Φ*<sup>i</sup>* and summing, we obtain
$$\begin{aligned} \Sigma\_{i=1}^{\eta\_1} \Phi\_{\bar{l}} \left( f(A\_{\bar{l}}) \right) &\leq \frac{\bar{M} \mathfrak{a} \mathbf{1}\_K - \sum\_{i=1}^{\eta\_1} \Phi\_{\bar{l}}(A\_{\bar{l}})}{\bar{M} - \bar{m}} f(\bar{m}) + \frac{\sum\_{i=1}^{\eta\_1} \Phi\_{\bar{l}}(A\_{\bar{l}}) - \bar{m} \mathfrak{a} \mathbf{1}\_K}{\bar{M} - \bar{m}} f(\bar{M}) \\ &- \delta\_f \left( \frac{\underline{a}}{2} \mathbf{1}\_K - \frac{1}{\bar{M} - \bar{m}} \sum\_{i=1}^{\eta\_1} \Phi\_{\bar{l}} \left( \left| A\_{\bar{l}} - \frac{\bar{m} + \bar{M}}{2} \mathbf{1}\_H \right| \right) \right) \end{aligned}$$
since ∑*n*<sup>1</sup> *<sup>i</sup>*=<sup>1</sup> Φ*i*(1*H*) = *α*1*K*. It follows that
$$\frac{1}{\alpha} \sum\_{i=1}^{n\_1} \Phi\_i \left( f(A\_i) \right) \le \frac{\bar{M} \mathbf{1}\_K - A}{\bar{M} - \bar{m}} f(\bar{m}) + \frac{A - \bar{m} \mathbf{1}\_K}{\bar{M} - \bar{m}} f(\bar{M}) - \delta\_f \tilde{A} \tag{46}$$
where *<sup>A</sup>* <sup>=</sup> <sup>1</sup> <sup>2</sup> <sup>1</sup>*<sup>K</sup>* <sup>−</sup> <sup>1</sup> *<sup>α</sup>*(*M*¯ <sup>−</sup>*m*¯) <sup>∑</sup>*n*<sup>1</sup> *<sup>i</sup>*=<sup>1</sup> Φ*<sup>i</sup> Ai* <sup>−</sup> *<sup>m</sup>*¯ <sup>+</sup>*M*¯ <sup>2</sup> 1*<sup>H</sup>* .
Additionally, since *<sup>f</sup>* is convex on all [*mi*, *Mi*] and (*m*¯ , *<sup>M</sup>*¯ ) <sup>∩</sup> [*mi*, *Mi*] = <sup>∅</sup>, *<sup>i</sup>* <sup>=</sup> *<sup>n</sup>*<sup>1</sup> <sup>+</sup> 1, . . . , *<sup>n</sup>*, then
$$f(A\_i) \ge \frac{\bar{M}\mathbf{1}\_H - A\_i}{\bar{M} - \bar{m}} f(\bar{m}) + \frac{A\_i - \bar{m}\mathbf{1}\_H}{\bar{M} - \bar{m}} f(\bar{M}), \qquad i = n\_1 + 1, \dots, n$$
It follows
$$\frac{1}{\beta} \sum\_{i=n\_1+1}^n \Phi\_i \left( f(A\_i) \right) - \delta\_f \tilde{A} \ge \frac{\tilde{M} \mathbf{1}\_K - B}{\tilde{M} - \bar{m}} f(\bar{m}) + \frac{B - \bar{m} \mathbf{1}\_K}{\tilde{M} - \bar{m}} f(\bar{M}) - \delta\_f \tilde{A} \tag{47}$$
Combining (46) and (47) and taking into account that *A* = *B*, we obtain
$$\frac{1}{\alpha} \sum\_{i=1}^{n\_1} \Phi\_{\bar{l}} \left( f(A\_{\bar{l}}) \right) \le \frac{1}{\mathcal{P}} \sum\_{i=n\_1+1}^n \Phi\_{\bar{l}} \left( f(A\_{\bar{l}}) \right) - \delta\_{\bar{f}} \tilde{A} \tag{48}$$
Next, we obtain
20 Will-be-set-by-IN-TECH
*holds for every continuous convex function f* : *I* → **R** *provided that the interval I contains all mi*, *Mi,*
*<sup>α</sup>*(*M*¯ <sup>−</sup> *<sup>m</sup>*¯)
*and <sup>m</sup>*¯ <sup>∈</sup> [*m*, *mL*]*, <sup>M</sup>*¯ <sup>∈</sup> [*MR*, *<sup>M</sup>*]*, <sup>m</sup>*¯ <sup>&</sup>lt; *M, are arbitrary numbers. If f* ¯ : *<sup>I</sup>* <sup>→</sup> **<sup>R</sup>** *is concave, then the*
*n* ∑ *i*=*n*1+1
Since *<sup>f</sup>* is convex on [*m*¯ , *<sup>M</sup>*¯ ] and Sp(*Ai*) <sup>⊆</sup> [*mi*, *Mi*] <sup>⊆</sup> [*m*¯ , *<sup>M</sup>*¯ ] for *<sup>i</sup>* <sup>=</sup> 1, . . . , *<sup>n</sup>*1, it follows from
*<sup>m</sup>*¯ +*M*¯ 2
*<sup>i</sup>*=<sup>1</sup> Φ*i*(*Ai*) *<sup>M</sup>*¯ <sup>−</sup> *<sup>m</sup>*¯ *<sup>f</sup>*(*m*¯) + <sup>∑</sup>*n*<sup>1</sup>
*n*1 ∑ *i*=1 Φ*i*
*<sup>M</sup>*¯ <sup>−</sup> *<sup>m</sup>*¯ *<sup>f</sup>*(*m*¯) + *<sup>A</sup>* <sup>−</sup> *<sup>m</sup>*¯ <sup>1</sup>*<sup>K</sup>*
<sup>2</sup> 1*<sup>H</sup>* .
Additionally, since *<sup>f</sup>* is convex on all [*mi*, *Mi*] and (*m*¯ , *<sup>M</sup>*¯ ) <sup>∩</sup> [*mi*, *Mi*] = <sup>∅</sup>, *<sup>i</sup>* <sup>=</sup> *<sup>n</sup>*<sup>1</sup> <sup>+</sup> 1, . . . , *<sup>n</sup>*,
*<sup>M</sup>*¯ <sup>−</sup> *<sup>m</sup>*¯
*Ai* <sup>−</sup> *<sup>m</sup>*¯ <sup>+</sup>*M*¯
*n*1 ∑ *i*=1 Φ*i*
Φ*i*(*Ai*), *C* =
*<sup>M</sup>*¯ <sup>−</sup> *<sup>m</sup>*¯ *<sup>f</sup>*(*M*¯ ) <sup>−</sup> *<sup>δ</sup><sup>f</sup> <sup>A</sup><sup>i</sup>*, *<sup>i</sup>* <sup>=</sup> 1, . . . , *<sup>n</sup>*<sup>1</sup>
*Ai* <sup>−</sup> *<sup>m</sup>*¯ <sup>+</sup> *<sup>M</sup>*¯
*<sup>M</sup>*¯ <sup>−</sup> *<sup>m</sup>*¯ *<sup>f</sup>*(*M*¯ ), *<sup>i</sup>* <sup>=</sup> *<sup>n</sup>*<sup>1</sup> <sup>+</sup> 1, . . . , *<sup>n</sup>*
*<sup>M</sup>*¯ <sup>−</sup> *<sup>m</sup>*¯ *<sup>f</sup>*(*m*¯) + *<sup>B</sup>* <sup>−</sup> *<sup>m</sup>*¯ <sup>1</sup>*<sup>K</sup>*
and *<sup>A</sup><sup>i</sup>* <sup>=</sup> <sup>1</sup>
*m*¯ + *M*¯ 2
*Ai* <sup>−</sup> *<sup>m</sup>*¯ <sup>+</sup> *<sup>M</sup>*¯
*n* ∑ *i*=1
<sup>2</sup> <sup>1</sup>*<sup>H</sup>* <sup>−</sup> <sup>1</sup>
*<sup>i</sup>*=<sup>1</sup> Φ*i*(*Ai*) − *m*¯ *α*1*<sup>K</sup>*
<sup>2</sup> <sup>1</sup>*<sup>H</sup>*
*<sup>M</sup>*¯ <sup>−</sup>*m*¯
*<sup>M</sup>*¯ <sup>−</sup> *<sup>m</sup>*¯ *<sup>f</sup>*(*M*¯ )
*<sup>M</sup>*¯ <sup>−</sup> *<sup>m</sup>*¯ *<sup>f</sup>*(*M*¯ ) <sup>−</sup> *<sup>δ</sup><sup>f</sup> <sup>A</sup>* (46)
*<sup>M</sup>*¯ <sup>−</sup> *<sup>m</sup>*¯ *<sup>f</sup>*(*M*¯ ) <sup>−</sup> *<sup>δ</sup><sup>f</sup> <sup>A</sup>* (47)
*Ai* <sup>−</sup> *<sup>m</sup>*¯ <sup>+</sup>*M*¯
<sup>2</sup> 1*<sup>H</sup>* .
<sup>2</sup> <sup>1</sup>*<sup>H</sup>*
Φ*i*(*Ai*)
(45)
*<sup>δ</sup><sup>f</sup>* <sup>≡</sup> *<sup>δ</sup>f*(*m*¯ , *<sup>M</sup>*¯ ) = *<sup>f</sup>*(*m*¯) + *<sup>f</sup>*(*M*¯ ) <sup>−</sup> <sup>2</sup> *<sup>f</sup>*
<sup>1</sup>*<sup>K</sup>* <sup>−</sup> <sup>1</sup>
*β*
2
<sup>Φ</sup>*i*(*Ai*), *<sup>B</sup>* <sup>=</sup> <sup>1</sup>
It is easy to verify that *A* = *B* or *B* = *C* or *A* = *C* implies *A* = *B* = *C*.
*<sup>M</sup>*¯ <sup>−</sup> *<sup>m</sup>*¯ *<sup>f</sup>*(*m*¯) + *Ai* <sup>−</sup> *<sup>m</sup>*¯ <sup>1</sup>*<sup>H</sup>*
Applying a positive linear mapping Φ*<sup>i</sup>* and summing, we obtain
*α* 2
− *δ<sup>f</sup>*
*<sup>i</sup>*=<sup>1</sup> Φ*i*(1*H*) = *α*1*K*. It follows that
*<sup>α</sup>*(*M*¯ <sup>−</sup>*m*¯) <sup>∑</sup>*n*<sup>1</sup>
*<sup>f</sup>*(*Ai*) <sup>≥</sup> *<sup>M</sup>*¯ <sup>1</sup>*<sup>H</sup>* <sup>−</sup> *Ai*
*<sup>M</sup>*¯ *<sup>α</sup>*1*<sup>K</sup>* <sup>−</sup> <sup>∑</sup>*n*<sup>1</sup>
<sup>Φ</sup>*<sup>i</sup>* (*f*(*Ai*)) <sup>≤</sup> *<sup>M</sup>*¯ <sup>1</sup>*<sup>K</sup>* <sup>−</sup> *<sup>A</sup>*
*<sup>M</sup>*¯ <sup>−</sup> *<sup>m</sup>*¯ *<sup>f</sup>*(*m*¯) + *Ai* <sup>−</sup> *<sup>m</sup>*¯ <sup>1</sup>*<sup>H</sup>*
<sup>Φ</sup>*<sup>i</sup>* (*f*(*Ai*)) <sup>−</sup> *<sup>δ</sup><sup>f</sup> <sup>A</sup>* <sup>≥</sup> *<sup>M</sup>*¯ <sup>1</sup>*<sup>K</sup>* <sup>−</sup> *<sup>B</sup>*
*<sup>i</sup>*=<sup>1</sup> Φ*<sup>i</sup>*
<sup>1</sup>*<sup>K</sup>* <sup>−</sup> <sup>1</sup>
*<sup>A</sup>* <sup>≡</sup> *<sup>A</sup><sup>A</sup>*,Φ,*n*1,*α*(*m*¯ , *<sup>M</sup>*¯ ) = <sup>1</sup>
*Proof.* We prove only the convex case. Let us denote
*n*1 ∑ *i*=1
*<sup>f</sup>* (*Ai*) <sup>≤</sup> *<sup>M</sup>*¯ <sup>1</sup>*<sup>H</sup>* <sup>−</sup> *Ai*
holds, where *<sup>δ</sup><sup>f</sup>* <sup>=</sup> *<sup>f</sup>*(*m*¯) + *<sup>f</sup>*(*M*¯ ) <sup>−</sup> <sup>2</sup> *<sup>f</sup>*
*<sup>i</sup>*=<sup>1</sup> Φ*<sup>i</sup>* (*f*(*Ai*)) ≤
1 *α*
<sup>2</sup> <sup>1</sup>*<sup>K</sup>* <sup>−</sup> <sup>1</sup>
*n* ∑ *i*=*n*1+1
1 *β*
*n*1 ∑ *i*=1
*i* = 1, . . . , *n, where*
Lemma 11 that
∑*n*<sup>1</sup>
since ∑*n*<sup>1</sup>
where *<sup>A</sup>* <sup>=</sup> <sup>1</sup>
then
It follows
*reverse inequality is valid in* (44)*.*
*<sup>A</sup>* <sup>=</sup> <sup>1</sup> *α*
$$\begin{aligned} &\frac{1}{\alpha}\sum\_{i=1}^{n}\Phi\_{i}(f(A\_{i})) \\ &=\sum\_{i=1}^{n\_{1}}\Phi\_{i}(f(A\_{i})) + \frac{\beta}{\alpha}\sum\_{i=1}^{n\_{1}}\Phi\_{i}(f(A\_{i})) \quad (\text{by }\alpha+\beta=1) \\ &\leq\sum\_{i=1}^{n\_{1}}\Phi\_{i}(f(A\_{i})) + \sum\_{i=n\_{1}+1}^{n}\Phi\_{i}(f(A\_{i})) - \beta\delta\_{f}\tilde{A} \quad \quad (\text{by (48)}) \\ &\leq\frac{\alpha}{\beta}\sum\_{i=n\_{1}+1}^{n}\Phi\_{i}(f(A\_{i})) - \alpha\delta\_{f}\tilde{A} + \sum\_{i=n\_{1}+1}^{n}\Phi\_{i}(f(A\_{i})) - \beta\delta\_{f}\tilde{A} \quad \quad \text{(by (48))} \\ &=\frac{1}{\beta}\sum\_{i=n\_{1}+1}^{n}\Phi\_{i}(f(A\_{i})) - \delta\_{f}\tilde{A} \quad \text{(by }\alpha+\beta=1) \end{aligned}$$
which gives the following double inequality
$$\frac{1}{n}\sum\_{i=1}^{n\_1} \Phi\_i(f(A\_i)) \le \sum\_{i=1}^n \Phi\_i(f(A\_i)) - \beta \delta\_f \tilde{A} \le \frac{1}{\beta} \sum\_{i=n\_1+1}^n \Phi\_i(f(A\_i)) - \delta\_f \tilde{A}$$
Adding *βδ<sup>f</sup> A* in the above inequalities, we get
$$\frac{1}{n}\sum\_{i=1}^{n\_1} \Phi\_i(f(A\_i)) + \beta \delta\_f \tilde{A} \le \sum\_{i=1}^n \Phi\_i(f(A\_i)) \le \frac{1}{\beta} \sum\_{i=n\_1+1}^n \Phi\_i(f(A\_i)) - a \delta\_f \tilde{A} \tag{49}$$
Now, we remark that *<sup>δ</sup><sup>f</sup>* <sup>≥</sup> 0 and *<sup>A</sup>* <sup>≥</sup> 0. (Indeed, since *<sup>f</sup>* is convex, then *<sup>f</sup>* ((*m*¯ <sup>+</sup> *<sup>M</sup>*¯ )/2) <sup>≤</sup> (*f*(*m*¯) + *<sup>f</sup>*(*M*¯ ))/2, which implies that *<sup>δ</sup><sup>f</sup>* <sup>≥</sup> 0. Also, since
$$\mathsf{Sp}(A\_{i}) \subseteq [\vec{m}, \bar{M}] \quad \Rightarrow \quad \left| A\_{i} - \frac{\bar{M} + \bar{m}}{2} \mathbf{1}\_{H} \right| \leq \frac{\bar{M} - \bar{m}}{2} \mathbf{1}\_{H} \qquad i = 1, \ldots, n\_{1}$$
then *<sup>n</sup>*<sup>1</sup>
$$\sum\_{i=1}^{n\_1} \Phi\_i \left( \left| A\_i - \frac{\bar{M} + \tilde{m}}{2} \mathbf{1}\_H \right| \right) \le \frac{\bar{M} - \tilde{m}}{2} \mathfrak{a} \mathbf{1}\_K$$
which gives
$$0 \le \frac{1}{2} \mathbf{1}\_K - \frac{1}{\alpha (\tilde{M} - \bar{m})} \sum\_{i=1}^{n\_1} \Phi\_i \left( \left| A\_i - \frac{\bar{M} + \bar{m}}{2} \mathbf{1}\_H \right| \right) = \tilde{A} \ )\ .$$
#### 22 Will-be-set-by-IN-TECH 210 Linear Algebra – Theorems and Applications
Consequently, the following inequalities
$$\frac{1}{\alpha} \sum\_{i=1}^{n\_1} \Phi\_i(f(A\_i)) \le \frac{1}{\alpha} \sum\_{i=1}^{n\_1} \Phi\_i(f(A\_i)) + \beta \delta\_f \tilde{A}$$
$$\frac{1}{\beta} \sum\_{i=n\_1+1}^n \Phi\_i(f(A\_i)) - \alpha \delta\_f \tilde{A} \le \frac{1}{\beta} \sum\_{i=n\_1+1}^n \Phi\_i(f(A\_i))$$
hold, which with (49) proves the desired series inequalities (44).
**Example 22.** *We observe the matrix case of Theorem 21 for f*(*t*) = *t* <sup>4</sup>*, which is the convex function but not operator convex, n* = 4*, n*<sup>1</sup> = 2 *and the bounds of matrices as in Fig. 3. We show an example*
**Figure 3.** An example a convex function and the bounds of four operators
*such that*
$$\begin{aligned} \frac{1}{a} \left( \Phi\_1(A\_1^4) + \Phi\_2(A\_2^4) \right) &< \frac{1}{a} \left( \Phi\_1(A\_1^4) + \Phi\_2(A\_2^4) \right) + \beta \delta\_f \tilde{A} \\ &< \Phi\_1(A\_1^4) + \Phi\_2(A\_2^4) + \Phi\_3(A\_3^4) + \Phi\_4(A\_4^4) \\ &< \frac{1}{\tilde{\mathcal{P}}} \left( \Phi\_3(A\_3^4) + \Phi\_4(A\_4^4) \right) - a \delta\_f \tilde{A} < \frac{1}{\tilde{\mathcal{P}}} \left( \Phi\_3(A\_3^4) + \Phi\_4(A\_4^4) \right) \end{aligned} \tag{50}$$
*holds, where <sup>δ</sup><sup>f</sup>* <sup>=</sup> *<sup>M</sup>*¯ <sup>4</sup> <sup>+</sup> *<sup>m</sup>*¯ <sup>4</sup> <sup>−</sup> (*M*¯ <sup>+</sup> *<sup>m</sup>*¯)4/8 *and*
$$\tilde{A} = \frac{1}{2}I\_2 - \frac{1}{\mathfrak{a}(\bar{M} - \bar{m})} \left( \Phi\_1 \left( |A\_1 - \frac{\bar{M} + \tilde{m}}{2} I\_{\bar{h}}| \right) + \Phi\_2 \left( |A\_2 - \frac{\bar{M} + \tilde{m}}{2} I\_3| \right) \right)$$
*We define mappings* <sup>Φ</sup>*<sup>i</sup>* : *<sup>M</sup>*3(**C**) <sup>→</sup> *<sup>M</sup>*2(**C**) *as follows:* <sup>Φ</sup>*i*((*ajk*)1≤*j*,*k*≤3) = <sup>1</sup> <sup>4</sup> (*ajk*)1≤*j*,*k*≤2*, i* = 1, . . . , 4*. Then* ∑<sup>4</sup> *<sup>i</sup>*=<sup>1</sup> <sup>Φ</sup>*i*(*I*3) = *<sup>I</sup>*<sup>2</sup> *and <sup>α</sup>* <sup>=</sup> *<sup>β</sup>* <sup>=</sup> <sup>1</sup> 2 *.*
$$Let$$
$$A\_1 = 2\begin{pmatrix} 2 & 9/8 \ 1 \\ 9/8 & 2 & 0 \\ 1 & 0 & 3 \end{pmatrix}, A\_2 = 3\begin{pmatrix} 2 & 9/8 \ 0 \\ 9/8 & 1 & 0 \\ 0 & 0 & 2 \end{pmatrix}, A\_3 = -3\begin{pmatrix} 4 & 1/2 \ 1 \\ 1/2 & 4 & 0 \\ 1 & 0 & 2 \end{pmatrix}, A\_4 = 12\begin{pmatrix} 5/3 \ 1/2 \ 0 \\ 1/2 \ 3/2 \ 0 \\ 0 & 0 \end{pmatrix}$$
*Then m*<sup>1</sup> = 1.28607*, M*<sup>1</sup> = 7.70771*, m*<sup>2</sup> = 0.53777*, M*<sup>2</sup> = 5.46221*, m*<sup>3</sup> = −14.15050*, M*<sup>3</sup> = −4.71071*, m*<sup>4</sup> = 12.91724*, M*<sup>4</sup> = 36.*, so mL* = *m*2*, MR* = *M*1*, m* = *M*<sup>3</sup> *and M* = *m*<sup>4</sup> *(rounded to*
#### 210 Linear Algebra – Theorems and Applications Recent Research on Jensen's Inequality for Oparators <sup>23</sup> Recent Research on Jensen's Inequality for Operators 211
*five decimal places). Also,*
$$\frac{1}{\mathcal{A}}\left(\Phi\_1(A\_1) + \Phi\_2(A\_2)\right) = \frac{1}{\mathcal{B}}\left(\Phi\_3(A\_3) + \Phi\_4(A\_4)\right) = \begin{pmatrix} 4 & 9/4\\ 9/4 & 3 \end{pmatrix}.$$
*and*
22 Will-be-set-by-IN-TECH
1 *α*
Φ*i*(*f*(*Ai*)) − *αδ<sup>f</sup> A*� ≤
*n*1 ∑ *i*=1
*but not operator convex, n* = 4*, n*<sup>1</sup> = 2 *and the bounds of matrices as in Fig. 3. We show an example*
Φ*i*(*f*(*Ai*)) + *βδ<sup>f</sup> A*�
*n* ∑ *i*=*n*1+1
<sup>1</sup>) + <sup>Φ</sup>2(*A*<sup>4</sup>
<sup>3</sup>) + <sup>Φ</sup>4(*A*<sup>4</sup>
<sup>2</sup>) + <sup>Φ</sup>3(*A*<sup>4</sup>
<sup>2</sup> *Ih*<sup>|</sup>
*β* � Φ3(*A*<sup>4</sup>
> � + Φ<sup>2</sup> �
⎛ ⎝
4 1/2 1 1/2 4 0 1 02
⎞
⎠ , *A*<sup>4</sup> = 12
<sup>−</sup> *αδ<sup>f</sup> <sup>A</sup>*� <sup>&</sup>lt; <sup>1</sup>
<sup>|</sup>*A*<sup>1</sup> <sup>−</sup> *<sup>M</sup>*¯ <sup>+</sup> *<sup>m</sup>*¯
2 *.*
⎞
⎠ , *A*<sup>3</sup> = −3
*Then m*<sup>1</sup> = 1.28607*, M*<sup>1</sup> = 7.70771*, m*<sup>2</sup> = 0.53777*, M*<sup>2</sup> = 5.46221*, m*<sup>3</sup> = −14.15050*, M*<sup>3</sup> = −4.71071*, m*<sup>4</sup> = 12.91724*, M*<sup>4</sup> = 36.*, so mL* = *m*2*, MR* = *M*1*, m* = *M*<sup>3</sup> *and M* = *m*<sup>4</sup> *(rounded to*
2) �
+ *βδ<sup>f</sup> A*�
<sup>|</sup>*A*<sup>2</sup> <sup>−</sup> *<sup>M</sup>*¯ <sup>+</sup> *<sup>m</sup>*¯
4) �
<sup>3</sup>) + <sup>Φ</sup>4(*A*<sup>4</sup>
<sup>4</sup>) (50)
<sup>2</sup> *<sup>I</sup>*3<sup>|</sup>
��
⎛ ⎝
<sup>4</sup> (*ajk*)1≤*j*,*k*≤2*, i* =
5/3 1/2 0 1/2 3/2 0 0 03
⎞ ⎠
Φ*i*(*f*(*Ai*))
<sup>4</sup>*, which is the convex function*
1 *β*
Consequently, the following inequalities
1 *α* � Φ1(*A*<sup>4</sup>
< <sup>1</sup> *β* � Φ3(*A*<sup>4</sup>
*<sup>A</sup>*� <sup>=</sup> <sup>1</sup>
2 9/8 1 9/8 2 0 1 03
1, . . . , 4*. Then* ∑<sup>4</sup>
⎛ ⎝
*Let*
*A*<sup>1</sup> = 2
*holds, where <sup>δ</sup><sup>f</sup>* <sup>=</sup> *<sup>M</sup>*¯ <sup>4</sup> <sup>+</sup> *<sup>m</sup>*¯ <sup>4</sup> <sup>−</sup> (*M*¯ <sup>+</sup> *<sup>m</sup>*¯)4/8 *and*
*<sup>α</sup>*(*M*¯ <sup>−</sup> *<sup>m</sup>*¯)
<sup>2</sup> *<sup>I</sup>*<sup>2</sup> <sup>−</sup> <sup>1</sup>
⎞
⎠ , *A*<sup>2</sup> = 3
*such that*
1 *α*
1 *β*
*n*1 ∑ *i*=1
*n* ∑ *i*=*n*1+1
Φ*i*(*f*(*Ai*)) ≤
hold, which with (49) proves the desired series inequalities (44).
**Example 22.** *We observe the matrix case of Theorem 21 for f*(*t*) = *t*
**Figure 3.** An example a convex function and the bounds of four operators
<sup>1</sup>) + <sup>Φ</sup>2(*A*<sup>4</sup>
<sup>3</sup>) + <sup>Φ</sup>4(*A*<sup>4</sup>
� Φ<sup>1</sup> �
*<sup>i</sup>*=<sup>1</sup> <sup>Φ</sup>*i*(*I*3) = *<sup>I</sup>*<sup>2</sup> *and <sup>α</sup>* <sup>=</sup> *<sup>β</sup>* <sup>=</sup> <sup>1</sup>
⎛ ⎝
< Φ1(*A*<sup>4</sup>
2) � < <sup>1</sup> *α* � Φ1(*A*<sup>4</sup>
<sup>1</sup>) + <sup>Φ</sup>2(*A*<sup>4</sup>
4) �
*We define mappings* <sup>Φ</sup>*<sup>i</sup>* : *<sup>M</sup>*3(**C**) <sup>→</sup> *<sup>M</sup>*2(**C**) *as follows:* <sup>Φ</sup>*i*((*ajk*)1≤*j*,*k*≤3) = <sup>1</sup>
2 9/8 0 9/8 1 0 0 02
$$A\_f \equiv \frac{1}{\alpha} \left( \Phi\_1(A\_1^4) + \Phi\_2(A\_2^4) \right) = \begin{pmatrix} 989.00391 \ 663.46875 \\ 663.46875 \ 526.12891 \end{pmatrix}$$
$$\mathbf{C}\_f \equiv \Phi\_1(A\_1^4) + \Phi\_2(A\_2^4) + \Phi\_3(A\_3^4) + \Phi\_4(A\_4^4) = \begin{pmatrix} 68093.14258 \ 48477.98437 \\ 48477.98437 \ 51335.39258 \end{pmatrix}$$
$$B\_f \equiv \frac{1}{\beta} \left( \Phi\_3(A\_3^4) + \Phi\_4(A\_4^4) \right) = \begin{pmatrix} 135197.28125 & 96292.5 \\ 96292.5 & 102144.65625 \end{pmatrix}$$
*Then*
$$A\_f < \mathbb{C}\_f < \mathbb{B}\_f \tag{51}$$
*holds (which is consistent with* (31)*).*
*We will choose three pairs of numbers* (*m*¯ , *<sup>M</sup>*¯ )*, <sup>m</sup>*¯ <sup>∈</sup> [−4.71071, 0.53777]*, <sup>M</sup>*¯ <sup>∈</sup> [7.70771, 12.91724] *as follows*
$$\begin{aligned} \text{i) } \ \vec{m} = m\_{\perp} = 0.53777, \ \vec{M} = M\_{R} = 7.70771, \ \text{then} \\ \tilde{\Delta}\_{1} = \beta \delta\_{f} \tilde{A} = 0.5 \cdot 2951.69249 \cdot \begin{pmatrix} 0.15678 \ 0.09030 \\ 0.09030 \ 0.15943 \end{pmatrix} = \begin{pmatrix} 231.38908 \ 133.26139 \\ 133.26139 \ 235.29515 \end{pmatrix} \\ \text{ii) } \ \vec{m} = m = -4.71071, \ \vec{M} = M = 12.91724, \ \text{then} \\ \tilde{\Delta}\_{2} = \beta \delta\_{f} \tilde{A} = 0.5 \cdot 27766.07963 \cdot \begin{pmatrix} 0.36022 \ 0.03573 \\ 0.03573 \ 0.36155 \end{pmatrix} = \begin{pmatrix} 5000.89860 & 496.04498 \\ 496.04498 & 5019.50711 \end{pmatrix} \\ \text{iii) } \ \vec{m} = -1, \ \tilde{M} = 10, \text{then} \\ \tilde{\Delta}\_{3} = \beta \delta\_{f} \tilde{A} = 0.5 \cdot 9180.875 \cdot \begin{pmatrix} 0.28203 \ 0.08975 \\ 0.08975 \ 0.27557 \end{pmatrix} = \begin{pmatrix} 1294.66 \ 411.999 \\ 411.999 & 1265. \end{pmatrix} \end{aligned}$$
*New, we obtain the following improvement of* (51) *(see* (50)*)*
$$\begin{aligned} \text{i)} \quad A\_f &< A\_f + \tilde{\Lambda}\_1 = \begin{pmatrix} 1220.3929796.73014 \\ 796.73014 & 761.42406 \end{pmatrix} \\ &< \mathbb{C}\_f < \begin{pmatrix} 134965.89217 & 96159.23861 \\ 96159.23861 & 1109.903610 \end{pmatrix} = B\_f - \tilde{\Lambda}\_1 < B\_f \\ \text{ii)} \quad A\_f &< A\_f + \tilde{\Lambda}\_2 = \begin{pmatrix} 5989.90251 & 1159.51373 \\ 1159.51373 & 5545.63601 \end{pmatrix} \\ &< \mathbb{C}\_f < \begin{pmatrix} 130196.38265 & 95796.45502 \\ 95796.45502 & 97125.14914 \end{pmatrix} = B\_f - \tilde{\Lambda}\_2 < B\_f \\ \text{iii)} \quad A\_f &< A\_f + \tilde{\Lambda}\_3 = \begin{pmatrix} 2283.66362 & 1075.46746 \\ 1075.46746 & 1791.12874 \end{pmatrix} \\ &< \mathbb{C}\_f < \begin{pmatrix} 133902.62153 & 95890.50129 \\ 95880.50129 & 100879.65641 \end{pmatrix} = B\_f - \tilde{\Lambda}\_3 < B\_f \end{aligned}$$
#### 24 Will-be-set-by-IN-TECH 212 Linear Algebra – Theorems and Applications
Using Theorem 21 we get the following result.
**Corollary 23.** *Let the assumptions of Theorem 21 hold. Then*
$$\frac{1}{\alpha} \sum\_{i=1}^{n\_1} \Phi\_i(f(A\_i)) \le \frac{1}{\alpha} \sum\_{i=1}^{n\_1} \Phi\_i(f(A\_i)) + \gamma\_1 \delta\_f \tilde{A} \le \frac{1}{\beta} \sum\_{i=n\_1+1}^n \Phi\_i(f(A\_i)) \tag{52}$$
*and*
$$\frac{1}{n}\sum\_{i=1}^{n\_1} \Phi\_i(f(A\_i)) \le \frac{1}{\mathcal{\beta}}\sum\_{i=n\_1+1}^n \Phi\_i(f(A\_i)) - \gamma\_2 \delta\_f \tilde{A} \le \frac{1}{\mathcal{\beta}}\sum\_{i=n\_1+1}^n \Phi\_i(f(A\_i))\tag{53}$$
*holds for every γ*1, *γ*<sup>2</sup> *in the close interval joining α and β, where δ<sup>f</sup> and A are defined by* (45)*.*
*Proof.* Adding *αδ<sup>f</sup> A* in (44) and noticing *δ<sup>f</sup> A* ≥ 0, we obtain
$$\frac{1}{\alpha} \sum\_{i=1}^{n\_1} \Phi\_i(f(A\_i)) \le \frac{1}{\alpha} \sum\_{i=1}^{n\_1} \Phi\_i(f(A\_i)) + \alpha \delta\_f \tilde{A} \le \frac{1}{\beta} \sum\_{i=n\_1+1}^n \Phi\_i(f(A\_i))$$
Taking into account the above inequality and the left hand side of (44) we obtain (52).
Similarly, subtracting *βδ<sup>f</sup> A* in (44) we obtain (53).
**Remark 24.** *We can obtain extensions of inequalities which are given in Remark 16 and 17. Also, we can obtain a special case of Theorem 21 with the convex combination of operators Ai putting* Φ*i*(*B*) = *αiB, for i* = 1, . . . , *n, similarly as in Corollary 19. Finally, applying this result, we can give another proof of Corollary 14. The interested reader can see the details in [30].*
#### **Author details**
Jadranka Mi´ci´c
*Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, Ivana Luˇci´ca 5, 10000 Zagreb, Croatia*
Josip Peˇcari´c
*Faculty of Textile Technology, University of Zagreb, Prilaz baruna Filipovi´ca 30, 10000 Zagreb, Croatia*
#### **7. References**
[6] Mond B, Peˇcari´c J (1994) Converses of Jensen's inequality for several operators, Rev. Anal. Numér. Théor. Approx. 23: 179-183.
24 Will-be-set-by-IN-TECH
Φ*i*(*f*(*Ai*)) + *γ*1*δ<sup>f</sup> A* ≤
Φ*i*(*f*(*Ai*)) − *γ*2*δ<sup>f</sup> A* ≤
Φ*i*(*f*(*Ai*)) + *αδ<sup>f</sup> A* ≤
**Remark 24.** *We can obtain extensions of inequalities which are given in Remark 16 and 17. Also, we can obtain a special case of Theorem 21 with the convex combination of operators Ai putting* Φ*i*(*B*) = *αiB, for i* = 1, . . . , *n, similarly as in Corollary 19. Finally, applying this result, we can give another*
*Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, Ivana Luˇci´ca 5,*
*Faculty of Textile Technology, University of Zagreb, Prilaz baruna Filipovi´ca 30, 10000 Zagreb, Croatia*
[1] Davis C (1957) A Schwarz inequality for convex operator functions. Proc. Amer. Math.
[2] Choi M.D (1974) A Schwarz inequality for positive linear maps on *C*∗-algebras. Illinois
[3] Hansen F, Pedersen G.K (1982) Jensen's inequality for operators and Löwner's theorem.
[4] Mond B, Peˇcari´c J (1995) On Jensen's inequality for operator convex functions. Houston
[5] Hansen F, Pedersen G.K (2003) Jensen's operator inequality. Bull. London Math. Soc. 35:
1 *β*
> 1 *β*
1 *β*
*n* ∑ *i*=*n*1+1
*n* ∑ *i*=*n*1+1
> *n* ∑ *i*=*n*1+1
Φ*i*(*f*(*Ai*)) (52)
Φ*i*(*f*(*Ai*)) (53)
(45)*.*
Φ*i*(*f*(*Ai*))
Using Theorem 21 we get the following result.
1 *α*
1 *α*
*n*1 ∑ *i*=1
> 1 *α*
**Author details**
*10000 Zagreb, Croatia*
Jadranka Mi´ci´c
Josip Peˇcari´c
**7. References**
Soc. 8: 42-44.
553-564.
J. Math. 18: 565-574.
J. Math. 21: 739-754.
Math. Ann. 258: 229-241.
*n*1 ∑ *i*=1
*and*
*n*1 ∑ *i*=1
**Corollary 23.** *Let the assumptions of Theorem 21 hold. Then*
1 *α*
1 *β*
*Proof.* Adding *αδ<sup>f</sup> A* in (44) and noticing *δ<sup>f</sup> A* ≥ 0, we obtain
1 *α*
*proof of Corollary 14. The interested reader can see the details in [30].*
*n*1 ∑ *i*=1
Φ*i*(*f*(*Ai*)) ≤
Similarly, subtracting *βδ<sup>f</sup> A* in (44) we obtain (53).
*n*1 ∑ *i*=1
*n* ∑ *i*=*n*1+1
*holds for every γ*1, *γ*<sup>2</sup> *in the close interval joining α and β, where δ<sup>f</sup> and A are defined by*
Taking into account the above inequality and the left hand side of (44) we obtain (52).
Φ*i*(*f*(*Ai*)) ≤
Φ*i*(*f*(*Ai*)) ≤
- [28] Xiao Z.G, Srivastava H.M, Zhang Z.H (2010) Further refinements of the Jensen inequalities based upon samples with repetitions. Math. Comput. Modelling 51: 592-600.
- [29] Wang L.C, Ma X.F, Liu L.H (2009) A note on some new refinements of Jensen's inequality for convex functions. J. Inequal. Pure Appl. Math. 10. 2. Art. 48: 6 p.
- [30] Mi´ci´c J, Peˇcari´c J, Peri´c J (2012) Extension of the refined Jensen's operator inequality with condition on spectra. Ann. Funct. Anal. 3: 67-85.
## **A Linear System of Both Equations and Inequalities in Max-Algebra**
Abdulhadi Aminu
26 Will-be-set-by-IN-TECH
[28] Xiao Z.G, Srivastava H.M, Zhang Z.H (2010) Further refinements of the Jensen inequalities based upon samples with repetitions. Math. Comput. Modelling 51:
[29] Wang L.C, Ma X.F, Liu L.H (2009) A note on some new refinements of Jensen's inequality
[30] Mi´ci´c J, Peˇcari´c J, Peri´c J (2012) Extension of the refined Jensen's operator inequality with
for convex functions. J. Inequal. Pure Appl. Math. 10. 2. Art. 48: 6 p.
condition on spectra. Ann. Funct. Anal. 3: 67-85.
592-600.
Additional information is available at the end of the chapter
http://dx.doi.org/10.5772/48195
## **1. Introduction**
The aim of this chapter is to present a system of linear equation and inequalities in max-algebra. Max-algebra is an analogue of linear algebra developed on the pair of operations (⊕, ⊗) extended to matrices and vectors, where *a* ⊕ *b* = *max*(*a*, *b*) and *a* ⊗ *b* = *a* + *b* for *a*, *b* ∈ **R**. The system of equations *A* ⊗ *x* = *c* and inequalities *B* ⊗ *x* ≤ *d* have each been studied in the literature. We will present necessary and sufficient conditions for the solvability of a system consisting of these two systems and also develop a polynomial algorithm for solving max-linear program whose constraints are max-linear equations and inequalities. Moreover, some solvability concepts of an inteval system of linear equations and inequalities will also be presented.
Max-algebraic linear systems were investigated in the first publications which deal with the introduction of algebraic structures called (max,+) algebras. Systems of equations with variables only on one side were considered in these publications [1, 2] and [3]. Other systems with a special structure were investigated in the context of solving eigenvalue problems in correspondence with algebraic structures or synchronisation of discrete event systems, see [4] and also [1] for additional information. Given a matrix *A*, a vector *b* of an appropriate size, using the notation ⊕ = max, ⊗ = plus, the studied systems had one of the following forms: *A* ⊗ *x* = *b*, *A* ⊗ *x* = *x* or *A* ⊗ *x* = *x* ⊕ *b*. An infinite dimensional generalisation can be found in [5].
In [1] Cuninghame-Green showed that the problem *A* ⊗ *x* = *b* can be solved using residuation [6]. That is the equality in *A* ⊗ *x* = *b* be relaxed so that the set of its sub-solutions is studied. It was shown that the greatest solution of *A* ⊗ *x* ≤ *b* is given by *x*¯ where
$$\mathfrak{x}\_{\mathfrak{j}} = \min\_{i \in M} (b\_i \otimes a\_{ij}^{-1}) \text{ for all } j \in N$$
The equation *A* ⊗ *x* = *b* is also solved using the above result as follows: The equation *A* ⊗ *x* = *b* has solution if and only if *A* ⊗ *x*¯ = *b*. Also, Gaubert [7] proposed a method for solving the one-sided system *x* = *A* ⊗ *x* ⊕ *b* using rational calculus.
> ©2012 Aminu, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ©2012 Aminu, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Zimmermann [3] developed a method for solving *A* ⊗ *x* = *b* by set covering and also presented an algorithm for solving max-linear programs with one sided constraints. This method is proved to has a computational complexity of *O*(*mn*), where *m*, *n* are the number of rows and columns of input matrices respectively. Akian, Gaubert and Kolokoltsov [5] extended Zimmermann's solution method by set covering to the case of functional Galois connections.
Butkovic [8] developed a max-algebraic method for finding all solutions to a system of inequalities *xi* − *xj* > *bij*, *i*, *j* = 1, ..., *n* using *n* generators. Using this method Butkovic [8] developed a pseudopolynomial algorithm which either finds a bounded mixed-integer solution, or decides that no such solution exists. Summary of these results can be found in [9] and [10]
Cechlarova and Diko [11] proposed a method for resolving infeasibility of the system ´ *A* ⊗ *x* = *b* . The techniques presented in this method are to modify the right-hand side as little as possible or to omit some equations. It was shown that the problem of finding the minimum number of those equations is NP-complete.
## **2. Max-algebra and some basic definitions**
In this section we introduce max-algebra, give the essential definitions and show how the operations of max-algebra can be extended to matrices and vectors.
In max-algebra, we replace addition and multiplication, the binary operations in conventional linear algebra, by maximum and addition respectively. For any problem that involves adding numbers together and taking the maximum of numbers, it may be possible to describe it in max-algebra. A problem that is nonlinear when described in conventional terms may be converted to a max-algebraic problem that is linear with respect to (⊕, ⊗)=(max, +).
**Definition 1.** The max-plus semiring **R** is the set **R** ∪ {−∞}, equipped with the addition (*a*, *b*) �→ max(*a*, *b*) and multiplication (*a*, *b*) �→ *a* + *b* denoted by ⊕ and ⊗ respectively. That is *a* ⊕ *b* = max(*a*, *b*) and *a* ⊗ *b* = *a* + *b*. The identity element for the addition (or zero) is −∞, and the identity element for the multiplication (or unit) is 0.
**Definition 2.** The min-plus semiring **R**min is the set **R** ∪ {+∞}, equipped with the addition (*a*, *<sup>b</sup>*) �→ min(*a*, *<sup>b</sup>*) and multiplication (*a*, *<sup>b</sup>*) �→ *<sup>a</sup>* <sup>+</sup> *<sup>b</sup>* denoted by <sup>⊕</sup>� and <sup>⊗</sup>� respectively. The zero is +∞, and the unit is 0. The name tropical semiring is also used as a synonym of min-plus when the ground set is **N**.
The completed max-plus semiring **R***max* is the set **R** ∪ {±∞}, equipped with the addition (*a*, *b*) �→ max(*a*, *b*) and multiplication (*a*, *b*) �→ *a* + *b*, with the convention that −∞ + (+∞) = +∞ + (−∞) = −∞. The completed min-plus semiring **R**min is defined in the dual way.
**Proposition 1.** The following properties hold for all *a*, *b*, *c* ∈ **R**:
$$\begin{aligned} a \oplus b &= b \oplus a \\ a \otimes b &= b \otimes a \\ a \oplus (b \oplus c) &= (a \oplus b) \oplus c \\ a \otimes (b \otimes c) &= (a \otimes b) \otimes c \end{aligned}$$
216 Linear Algebra – Theorems and Applications A Linear System of Both Equations and Inequalities in Max-Algebra <sup>3</sup> A Linear System of Both Equations and Inequalities in Max-Algebra 217
$$a \otimes (b \oplus c) = a \otimes b \oplus a \otimes c$$
$$a \oplus (-\infty) = -\infty = (-\infty) \oplus a$$
$$a \otimes 0 = a = 0 \otimes a$$
$$a \otimes a^{-1} = 0, a, a^{-1} \in \mathbb{R}$$
*Proof.*
2 Will-be-set-by-IN-TECH
Zimmermann [3] developed a method for solving *A* ⊗ *x* = *b* by set covering and also presented an algorithm for solving max-linear programs with one sided constraints. This method is proved to has a computational complexity of *O*(*mn*), where *m*, *n* are the number of rows and columns of input matrices respectively. Akian, Gaubert and Kolokoltsov [5] extended Zimmermann's solution method by set covering to the case of functional Galois
Butkovic [8] developed a max-algebraic method for finding all solutions to a system of inequalities *xi* − *xj* > *bij*, *i*, *j* = 1, ..., *n* using *n* generators. Using this method Butkovic [8] developed a pseudopolynomial algorithm which either finds a bounded mixed-integer solution, or decides that no such solution exists. Summary of these results can be found in [9]
Cechlarova and Diko [11] proposed a method for resolving infeasibility of the system ´ *A* ⊗ *x* = *b* . The techniques presented in this method are to modify the right-hand side as little as possible or to omit some equations. It was shown that the problem of finding the minimum
In this section we introduce max-algebra, give the essential definitions and show how the
In max-algebra, we replace addition and multiplication, the binary operations in conventional linear algebra, by maximum and addition respectively. For any problem that involves adding numbers together and taking the maximum of numbers, it may be possible to describe it in max-algebra. A problem that is nonlinear when described in conventional terms may be converted to a max-algebraic problem that is linear with respect to (⊕, ⊗)=(max, +).
**Definition 1.** The max-plus semiring **R** is the set **R** ∪ {−∞}, equipped with the addition (*a*, *b*) �→ max(*a*, *b*) and multiplication (*a*, *b*) �→ *a* + *b* denoted by ⊕ and ⊗ respectively. That is *a* ⊕ *b* = max(*a*, *b*) and *a* ⊗ *b* = *a* + *b*. The identity element for the addition (or zero) is −∞,
**Definition 2.** The min-plus semiring **R**min is the set **R** ∪ {+∞}, equipped with the addition
zero is +∞, and the unit is 0. The name tropical semiring is also used as a synonym of min-plus
The completed max-plus semiring **R***max* is the set **R** ∪ {±∞}, equipped with the addition (*a*, *b*) �→ max(*a*, *b*) and multiplication (*a*, *b*) �→ *a* + *b*, with the convention that −∞ + (+∞) = +∞ + (−∞) = −∞. The completed min-plus semiring **R**min is defined in the dual way.
> *a* ⊕ *b* = *b* ⊕ *a a* ⊗ *b* = *b* ⊗ *a a* ⊕ (*b* ⊕ *c*)=(*a* ⊕ *b*) ⊕ *c a* ⊗ (*b* ⊗ *c*)=(*a* ⊗ *b*) ⊗ *c*
and <sup>⊗</sup>�
respectively. The
connections.
and [10]
number of those equations is NP-complete.
**2. Max-algebra and some basic definitions**
operations of max-algebra can be extended to matrices and vectors.
and the identity element for the multiplication (or unit) is 0.
when the ground set is **N**.
(*a*, *<sup>b</sup>*) �→ min(*a*, *<sup>b</sup>*) and multiplication (*a*, *<sup>b</sup>*) �→ *<sup>a</sup>* <sup>+</sup> *<sup>b</sup>* denoted by <sup>⊕</sup>�
**Proposition 1.** The following properties hold for all *a*, *b*, *c* ∈ **R**:
The statements follow from the definitions.
**Proposition 2.** For all *a*, *b*, *c* ∈ **R** the following properties hold:
$$\begin{array}{l} a \le b \implies a \oplus c \le b \oplus c \\\\ a \le b \iff a \otimes c \le b \otimes c, c \in \mathbb{R} \\\\ a \le b \iff a \oplus b = b \\\\ a > b \iff a \otimes c > b \otimes c, -\infty < c < +\infty \end{array}$$
*Proof.* The statements follow from definitions.
The pair of operations (⊕,⊗) is extended to matrices and vectors as in the conventional linear algebra as follows: For *A* = (*aij*), *B* = (*bij*) of compatible sizes and *α* ∈ **R** we have:
$$\begin{aligned} A \oplus B &= (a\_{ij} \oplus b\_{ij}) \\ A \otimes B &= \left(\sum\_{k} ^{\oplus} a\_{ik} \otimes b\_{kj}\right) \\ \mathfrak{a} \otimes A &= (\mathfrak{a} \otimes a\_{ij}) \end{aligned}$$
**Example 1.**
$$
\begin{pmatrix} 3 \ 1 \ 5 \\ 2 \ 1 \ 5 \end{pmatrix} \oplus \begin{pmatrix} -1 & 0 \ 2 \\ 6 \ -5 \ 4 \end{pmatrix} = \begin{pmatrix} 3 \ 1 \ 5 \\ 6 \ 1 \ 5 \end{pmatrix}
$$
**Example 2.**
$$\begin{pmatrix} -4 \ 1 \ -5 \\ 3 \ 0 \ 8 \end{pmatrix} \otimes \begin{pmatrix} -1 \ 2 \\ 1 \ 7 \\ 3 \ 1 \end{pmatrix}$$
$$\mathbf{J} = \begin{pmatrix} (-4 + (-1)) \oplus (1 + 1) \oplus (-5 + 3) & (-4 + 2) \oplus (1 + 7) \oplus (-5 + 1) \\ (3 + (-1)) \oplus (0 + 1) \oplus (8 + 3) & (3 + 2) \oplus (0 + 7) \oplus (8 + 1) \end{pmatrix} = \begin{pmatrix} 2 \ 8 \\ 11 \ 9 \end{pmatrix}$$
**Example 3.**
$$10 \otimes \begin{pmatrix} 7 \ -3 \ 2 \\ 6 \ -1 \ 0 \end{pmatrix} = \begin{pmatrix} 17 \ 7 \ 12 \\ 16 \ 11 \ 10 \end{pmatrix}$$
#### **Proposition 3.**
For *<sup>A</sup>*, *<sup>B</sup>*, *<sup>C</sup>* <sup>∈</sup> **<sup>R</sup>***m*×*<sup>n</sup>* of compatible sizes, the following properties hold:
$$A \oplus B = B \oplus A$$
$$A \oplus (B \oplus \mathbb{C}) = (A \oplus B) \oplus \mathbb{C}$$
$$A \otimes (B \otimes \mathbb{C}) = (A \otimes B) \otimes \mathbb{C}$$
$$A \otimes (B \oplus \mathbb{C}) = A \otimes B \oplus A \otimes \mathbb{C}$$
$$(A \oplus B) \otimes \mathbb{C} = A \otimes \mathbb{C} \oplus B \otimes \mathbb{C}$$
*Proof.*
The statements follow from the definitions.
#### **Proposition 4.**
The following hold for *A*, *B*, *C*, *a*, *b*, *c*, *x*, *y* of compatible sizes and *α*, *β* ∈ **R**:
$$\begin{aligned} A \otimes (\mathfrak{a} \otimes B) &= \mathfrak{a} \otimes (A \otimes B) \\ \mathfrak{a} \otimes (A \oplus B) &= \mathfrak{a} \otimes A \oplus \mathfrak{a} \otimes B \\ (\mathfrak{a} \oplus \beta) \otimes A &= \mathfrak{a} \otimes A \oplus \mathfrak{f} \otimes B \\ \mathfrak{x}^T \otimes \mathfrak{a} \otimes \mathfrak{y} &= \mathfrak{a} \otimes \mathfrak{x}^T \otimes \mathfrak{y} \\ \mathfrak{a} \le b &\Longrightarrow \mathfrak{c}^T \otimes \mathfrak{a} \leq \mathfrak{c}^T \otimes b \\ A \le B &\Longrightarrow A \oplus \mathbb{C} \leq B \oplus \mathbb{C} \\ A \le B &\Longrightarrow A \otimes \mathbb{C} \leq B \otimes \mathbb{C} \\ A \le B &\Longleftrightarrow A \oplus B = B \end{aligned}$$
*Proof.* The statements follow from the definition of the pair of operations (⊕,⊗).
**Definition 3.** Given real numbers *a*, *b*, *c*, . . . , a max-algebraic *diagonal matrix* is defined as:
$$\text{diag}(a, b, c, \dots) = \begin{pmatrix} a & & & & \\ & b & & -\infty & \\ & & c & & \\ & -\infty & \ddots & \\ & & & \ddots \\ & & & & \ddots \end{pmatrix}\_{\lambda\_{-}}$$
Given a vector *d* = (*d*1, *d*2,..., *dn*), the *diagonal of the vector d* is denoted as diag(*d*) = diag(*d*1, *d*2,..., *dn*).
**Definition 4.** Max-algebraic *identity matrix* is a diagonal matrix with all diagonal entries zero. We denote by *I* an identity matrix. Therefore, *identity matrix I* = diag(0, 0, 0, . . .).
It is obvious that *A* ⊗ *I* = *I* ⊗ *A* for any matrices *A* and *I* of compatible sizes.
**Definition 5.** Any matrix that can be obtained from the identity matrix, *I*, by permuting its rows and or columns is called a *permutation matrix*. A matrix arising as a product of a diagonal matrix and a permutation matrix is called a *generalised permutation matrix* [12].
**Definition 6.** A matrix *<sup>A</sup>* <sup>∈</sup> **<sup>R</sup>***n*×*<sup>n</sup>* is *invertible* if there exists a matrix *<sup>B</sup>* <sup>∈</sup> **<sup>R</sup>***n*×*<sup>n</sup>* , such that *A* ⊗ *B* = *B* ⊗ *A* = *I*. The matrix *B* is unique and will be called the *inverse* of *A*. We will henceforth denote *B* by *A*−1.
It has been shown in [1] that a matrix is *invertible* if and only if it is a generalised permutation matrix. If *x* = (*x*1,..., *xn*) we will denote *x*−<sup>1</sup> = (*x*−<sup>1</sup> <sup>1</sup> ,..., *<sup>x</sup>*−<sup>1</sup> *<sup>n</sup>* ), that is *<sup>x</sup>*−<sup>1</sup> <sup>=</sup> <sup>−</sup>*x*, in conventional notation.
#### **Example 4.**
4 Will-be-set-by-IN-TECH
*A* ⊕ *B* = *B* ⊕ *A A* ⊕ (*B* ⊕ *C*)=(*A* ⊕ *B*) ⊕ *C A* ⊗ (*B* ⊗ *C*)=(*A* ⊗ *B*) ⊗ *C A* ⊗ (*B* ⊕ *C*) = *A* ⊗ *B* ⊕ *A* ⊗ *C* (*A* ⊕ *B*) ⊗ *C* = *A* ⊗ *C* ⊕ *B* ⊗ *C*
For *<sup>A</sup>*, *<sup>B</sup>*, *<sup>C</sup>* <sup>∈</sup> **<sup>R</sup>***m*×*<sup>n</sup>* of compatible sizes, the following properties hold:
The following hold for *A*, *B*, *C*, *a*, *b*, *c*, *x*, *y* of compatible sizes and *α*, *β* ∈ **R**:
*A* ⊗ (*α* ⊗ *B*) = *α* ⊗ (*A* ⊗ *B*) *α* ⊗ (*A* ⊕ *B*) = *α* ⊗ *A* ⊕ *α* ⊗ *B* (*α* ⊕ *β*) ⊗ *A* = *α* ⊗ *A* ⊕ *β* ⊗ *B <sup>x</sup><sup>T</sup>* <sup>⊗</sup> *<sup>α</sup>* <sup>⊗</sup> *<sup>y</sup>* <sup>=</sup> *<sup>α</sup>* <sup>⊗</sup> *<sup>x</sup><sup>T</sup>* <sup>⊗</sup> *<sup>y</sup>*
*Proof.* The statements follow from the definition of the pair of operations (⊕,⊗).
diag(*a*, *b*, *c*,...) =
**Definition 3.** Given real numbers *a*, *b*, *c*, . . . , a max-algebraic *diagonal matrix* is defined as:
⎛
*a*
*b* −∞ *c* <sup>−</sup><sup>∞</sup> ...
...
⎞
⎟⎟⎟⎟⎟⎟⎠
⎜⎜⎜⎜⎜⎜⎝
Given a vector *d* = (*d*1, *d*2,..., *dn*), the *diagonal of the vector d* is denoted as diag(*d*) =
**Definition 4.** Max-algebraic *identity matrix* is a diagonal matrix with all diagonal entries zero.
**Definition 5.** Any matrix that can be obtained from the identity matrix, *I*, by permuting its rows and or columns is called a *permutation matrix*. A matrix arising as a product of a diagonal
We denote by *I* an identity matrix. Therefore, *identity matrix I* = diag(0, 0, 0, . . .).
It is obvious that *A* ⊗ *I* = *I* ⊗ *A* for any matrices *A* and *I* of compatible sizes.
matrix and a permutation matrix is called a *generalised permutation matrix* [12].
*<sup>a</sup>* <sup>≤</sup> *<sup>b</sup>* <sup>=</sup><sup>⇒</sup> *<sup>c</sup><sup>T</sup>* <sup>⊗</sup> *<sup>a</sup>* <sup>≤</sup> *<sup>c</sup><sup>T</sup>* <sup>⊗</sup> *<sup>b</sup> A* ≤ *B* =⇒ *A* ⊕ *C* ≤ *B* ⊕ *C A* ≤ *B* =⇒ *A* ⊗ *C* ≤ *B* ⊗ *C A* ≤ *B* ⇐⇒ *A* ⊕ *B* = *B*
The statements follow from the definitions.
**Proposition 3.**
*Proof.*
**Proposition 4.**
diag(*d*1, *d*2,..., *dn*).
Consider the following matrices
$$A = \begin{pmatrix} -\infty & -\infty & 3\\ 5 & -\infty & -\infty \\ -\infty & 8 & -\infty \end{pmatrix} \text{ and } B = \begin{pmatrix} -\infty & -5 & -\infty \\ -\infty & -\infty & -8 \\ -3 & -\infty & -\infty \end{pmatrix}.$$
The matrix *B* is an inverse of *A* because,
$$A \otimes B = \begin{pmatrix} -\infty & -\infty & 3 \\ 5 & -\infty & -\infty \\ -\infty & 8 & -\infty \end{pmatrix} \otimes \begin{pmatrix} -\infty & -5 & -\infty \\ -\infty & -\infty & -8 \\ 3 & -\infty & -\infty \end{pmatrix} = \begin{pmatrix} 0 & -\infty & -\infty \\ -\infty & 0 & -\infty \\ -\infty & -\infty & 0 \end{pmatrix}$$
Given a matrix *<sup>A</sup>* = (*aij*) <sup>∈</sup> **<sup>R</sup>**, the *transpose* of *<sup>A</sup>* will be denoted by *<sup>A</sup>T*, that is *<sup>A</sup><sup>T</sup>* = (*aji*). Structures of discrete-event dynamic systems may be represented by square matrices *A* over the semiring:
$$\overline{\mathbb{R}} = (\{ -\infty \} \cup \mathbb{R}, \oplus, \otimes) = (\{ -\infty \} \cup \mathbb{R}, \max, +)$$
The system $\mathbb{R}$ is embedded in the self-dual system:
$$\overline{\overline{\mathbb{R}}} = (\{-\infty\} \cup \mathbb{R}\{+\infty\}, \oplus, \otimes, \stackrel{\circ}{\oplus}, \stackrel{\circ}{\otimes}) = (\{-\infty\} \cup \mathbb{R}\{+\infty\}, \max, +, \min, +)$$
Basic algebraic properties for <sup>⊕</sup>� and <sup>⊗</sup>� are similar to those of ⊕ and ⊗ described earlier. These are obtained by swapping <sup>≤</sup> and <sup>≥</sup> . Extension of the pair (⊕� , <sup>⊗</sup>� ) to matrices and vectors is as follows:
Given *A*, *B* of compatible sizes and *α* ∈ **R**, we define the following:
$$\begin{aligned} A \oplus^{'} B &= (a\_{ij} \oplus^{'} b\_{ij}) \\ A \otimes^{'} B &= \left( \sum\_{k}^{\oplus^{'}} a\_{ik} \otimes^{'} b\_{kj} \right) = \min\_{k} (a\_{ik} + b\_{kj}) \\ \alpha \otimes^{'} A &= (\alpha \otimes^{'} a\_{ij}) \end{aligned}$$
Also, properties of matrices for the pair (⊕� , <sup>⊗</sup>� ) are similar to those of (⊕, ⊗), just swap ≤ and ≥. For any matrix *A* = [*aij*] over **R**, the *conjugate* matrix is *A*<sup>∗</sup> = [−*aji*] obtained by negation and transposition, that is *<sup>A</sup>* <sup>=</sup> <sup>−</sup>*AT*.
**Proposition 5.** The following relations hold for any matrices *U*, *V*, *W* over **R** .
$$(\mathcal{U}\otimes^{'}V)\otimes\mathcal{W}\leq\mathcal{U}\otimes^{'}(V\otimes\mathcal{W})\tag{1}$$
$$\mathcal{U}\ll(\mathcal{U}^\*\otimes \stackrel{\circ}{\mathcal{W}}\mathcal{W})\le\mathcal{W}\tag{2}$$
$$\mathcal{U}\lhd\left(\mathcal{U}^\*\otimes\ulcorner(\mathcal{U}\otimes\mathcal{W})\right)=\mathcal{U}\otimes\mathcal{W}\tag{3}$$
*Proof.* Follows from the definitions.
#### **3. The Multiprocessor Interactive System (MPIS): A practical application**
Linear equations and inequalities in max-algebra have a considerable number of applications, the model we present here is called the *multiprocessor interactive system (MPIS)* which is formulated as follows:
Products *P*1,..., *Pm* are prepared using *n* processors, every processor contributing to the completion of each product by producing a partial product. It is assumed that every processor can work on all products simultaneously and that all these actions on a processor start as soon as the processor is ready to work. Let *aij* be the duration of the work of the *j th* processor needed to complete the partial product for *Pi* (*i* = 1, . . . , *m*; *j* = 1, . . . , *n*). Let us denote by *xj* the starting time of the *j th* processor (*j* = 1, . . . , *n*). Then, all partial products for *Pi* (*i* = 1, . . . , *m*; *j* = 1, . . . , *n*) will be ready at time max(*ai*<sup>1</sup> + *x*1,..., *ain* + *xn*). If the completion times *b*1,..., *bm* are given for each product then the starting times have to satisfy the following system of equations:
$$\max(a\_{i1} + \mathbf{x}\_{1\prime} \dots \mathbf{a}\_{in} + \mathbf{x}\_n) = b\_i \text{ for all } i \in M$$
Using the notation *a* ⊕ *b* = *max*(*a*, *b*) and *a* ⊗ *b* = *a* + *b* for *a*, *b* ∈ **R** extended to matrices and vectors in the same way as in linear algebra, then this system can be written as
$$A \circledast \mathfrak{x} = b$$
Any system of the form (4) is called 'one-sided max-linear system'. Also, if the requirement is that each product is to be produced on or before the completion times *b*1,..., *bm*, then the starting times have to satisfy
$$\max(a\_{i1} + \mathbf{x}\_{1'} \dots \mathbf{a}\_{in} + \mathbf{x}\_n) \le b\_i \text{ for all } i \in M$$
which can also be written as
$$A \otimes \mathfrak{x} \le b \tag{5}$$
The system of inequalities (5) is called 'one-sided max-linear system of inequalities'.
#### **4. Linear equations and inequalities in max-algebra**
In this section we will present a system of linear equation and inequalities in max-algebra. Solvability conditions for linear system and inequalities will each be presented. A system consisting of max-linear equations and inequalities will also be discussed and necessary and sufficient conditions for the solvability of this system will be presented.
#### **4.1. System of equations**
In this section we present a solution method for the system *A* ⊗ *x* = *b* as given in [1, 3, 13] and also in the monograph [10]. Results concerning the existence and uniqueness of solution to the system will also be presented.
$$\text{Given } A = (a\_{ij}) \in \overline{\mathbb{R}}^{m \times n} \text{ and } b = (b\_1, \dots, b\_m)^T \in \overline{\mathbb{R}}^m \text{, a system of the form}$$
$$A \otimes \mathbf{x} = b \tag{6}$$
is called a *one-sided max-linear system*, some times we may omit 'max-linear' and say one-sided system. This system can be written using the conventional notation as follows
$$\max\_{\{j=1,\ldots,n\}} (a\_{l\bar{l}} + x\_{\bar{l}}) = b\_{\bar{l}\bar{\iota}} \; i \in M \tag{7}$$
The system in (7) can be written after subtracting the right-hand sides constants as
$$\max\_{j=1,\dots,n} (a\_{ij}\otimes b\_i^{-1} + \mathfrak{x}\_j) = \mathbf{0}, \; i \in M$$
A one-sided max-linear system whose all right hand side constants are zero is called *normalised max-linear system* or just *normalised* and the process of subtracting the right-hand side constants is called *normalisation*. Equivalently, *normalisation* is the process of multiplying the system (6) by the matrix *B* � from the left. That is
$$\mathcal{B}' \otimes \mathcal{A} \otimes \mathfrak{x} = \mathcal{B}' \otimes b = 0$$
where,
6 Will-be-set-by-IN-TECH
**3. The Multiprocessor Interactive System (MPIS): A practical application** Linear equations and inequalities in max-algebra have a considerable number of applications, the model we present here is called the *multiprocessor interactive system (MPIS)* which is
Products *P*1,..., *Pm* are prepared using *n* processors, every processor contributing to the completion of each product by producing a partial product. It is assumed that every processor can work on all products simultaneously and that all these actions on a processor start as soon
needed to complete the partial product for *Pi* (*i* = 1, . . . , *m*; *j* = 1, . . . , *n*). Let us denote
(*i* = 1, . . . , *m*; *j* = 1, . . . , *n*) will be ready at time max(*ai*<sup>1</sup> + *x*1,..., *ain* + *xn*). If the completion times *b*1,..., *bm* are given for each product then the starting times have to satisfy the following
max(*ai*<sup>1</sup> + *x*1,..., *ain* + *xn*) = *bi* for all *i* ∈ *M*
Using the notation *a* ⊕ *b* = *max*(*a*, *b*) and *a* ⊗ *b* = *a* + *b* for *a*, *b* ∈ **R** extended to matrices and
Any system of the form (4) is called 'one-sided max-linear system'. Also, if the requirement is that each product is to be produced on or before the completion times *b*1,..., *bm*, then the
max(*ai*<sup>1</sup> + *x*1,..., *ain* + *xn*) ≤ *bi* for all *i* ∈ *M*
In this section we will present a system of linear equation and inequalities in max-algebra. Solvability conditions for linear system and inequalities will each be presented. A system consisting of max-linear equations and inequalities will also be discussed and necessary and
In this section we present a solution method for the system *A* ⊗ *x* = *b* as given in [1, 3, 13] and also in the monograph [10]. Results concerning the existence and uniqueness of solution
The system of inequalities (5) is called 'one-sided max-linear system of inequalities'.
**4. Linear equations and inequalities in max-algebra**
sufficient conditions for the solvability of this system will be presented.
Given *<sup>A</sup>* = (*aij*) <sup>∈</sup> **<sup>R</sup>***m*×*<sup>n</sup>* and *<sup>b</sup>* = (*b*1,..., *bm*)*<sup>T</sup>* <sup>∈</sup> **<sup>R</sup>***m*, a system of the form
*th* processor (*j* = 1, . . . , *n*). Then, all partial products for *Pi*
*A* ⊗ *x* = *b* (4)
*A* ⊗ *x* ≤ *b* (5)
*A* ⊗ *x* = *b* (6)
*th* processor
as the processor is ready to work. Let *aij* be the duration of the work of the *j*
vectors in the same way as in linear algebra, then this system can be written as
formulated as follows:
system of equations:
by *xj* the starting time of the *j*
starting times have to satisfy
which can also be written as
**4.1. System of equations**
to the system will also be presented.
$$\boldsymbol{b}^{\prime} = \text{diag}(\boldsymbol{b}\_1^{-1}, \boldsymbol{b}\_2^{-1}, \dots, \boldsymbol{b}\_m^{-1}) = \text{diag}(\boldsymbol{b}^{-1})$$
For instance, consider the following one-sided system:
$$
\begin{pmatrix} -2 \ 1 \ 3 \\ 3 \ 0 \ 2 \\ 1 \ 2 \ 1 \end{pmatrix} \otimes \begin{pmatrix} x\_1 \\ x\_2 \\ x\_3 \end{pmatrix} = \begin{pmatrix} 5 \\ 6 \\ 3 \end{pmatrix} \tag{8}
$$
After normalisation, this system is equivalent to
$$
\begin{pmatrix} -7 \ -4 \ -2 \\ -3 \ -6 \ -4 \\ -2 \ -1 \ -2 \end{pmatrix} \otimes \begin{pmatrix} \mathbf{x}\_1 \\ \mathbf{x}\_2 \\ \mathbf{x}\_3 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}.
$$
That is after multiplying the system (8) by
$$
\begin{pmatrix}
\end{pmatrix},
$$
Consider the first equation of the normalised system above, that is *max*(*x*<sup>1</sup> − 7, *x*<sup>2</sup> − 4, *x*<sup>3</sup> − <sup>2</sup>) = 0. This means that if (*x*1, *<sup>x</sup>*2, *<sup>x</sup>*3)*<sup>T</sup>* is a solution to this system then *<sup>x</sup>*<sup>1</sup> <sup>≤</sup> 7,*x*<sup>2</sup> <sup>≤</sup> 4, *x*<sup>3</sup> ≤ 2 and at least one of these inequalities will be satisfied with equality. From the other equations of the system, we have for *x*<sup>1</sup> ≤ 3, *x*<sup>1</sup> ≤ 2, hence we have *x*<sup>1</sup> ≤ *min*(7, 3, 2) = −*max*(−7, −3, −2) = −*x*¯1 where −*x*¯1 is the column 1 maximum. It is clear that for all *j* then *xj* ≤ *x*¯*j*, where −*x*¯*<sup>j</sup>* is the column *j* maximum. At the same time equality must be attained in some of these inequalities so that in every row there is at least one column maximum which is attained by *xj*. This observation was made in [3].
**Definition 7.** A matrix *A* is called *doubly* **R***-astic* [14, 15], if it has at least one finite element on each row and on each column.
#### 8 Will-be-set-by-IN-TECH 222 Linear Algebra – Theorems and Applications
We introduce the following notations
$$\begin{aligned} S(A, b) &= \{ \mathbf{x} \in \overline{\mathbb{R}}^n ; A \otimes \mathbf{x} = b \} \\ M\_{\bar{j}} &= \{ k \in M ; b\_k \otimes a\_{k\bar{j}}^{-1} = \max\_{\bar{i}} (b\_{\bar{i}} \otimes a\_{i\bar{j}}^{-1}) \} \text{ for all } j \in \mathbb{N} \\ \bar{\mathbf{x}}(A, b)\_{\bar{j}} &= \min\_{\bar{i} \in M} (b\_{\bar{i}} \otimes a\_{i\bar{j}}^{-1}) \text{ for all } j \in \mathbb{N} \end{aligned}$$
We now consider the cases when *A* = −∞ and/or *b* = −∞. Suppose that *b* = −∞. Then *S*(*A*, *b*) can simply be written as
$$S(A, b) = \{ \mathbf{x} \in \mathbb{R}^n ; \mathbf{x}\_j = -\infty, \text{ if } A\_j \neq -\infty, \ j \in N \}$$
Therefore if *<sup>A</sup>* <sup>=</sup> <sup>−</sup><sup>∞</sup> we have *<sup>S</sup>*(*A*, *<sup>b</sup>*) = **<sup>R</sup>***<sup>n</sup>* . Now, if *<sup>A</sup>* <sup>=</sup> <sup>−</sup><sup>∞</sup> and *<sup>b</sup>* �<sup>=</sup> <sup>−</sup><sup>∞</sup> then *<sup>S</sup>*(*A*, *<sup>b</sup>*) = <sup>∅</sup>. Thus, we may assume in this section that *A* = −∞ and *b* �= −∞. If *bk* = −∞ for some *<sup>k</sup>* <sup>∈</sup> *<sup>M</sup>* then for any *<sup>x</sup>* <sup>∈</sup> *<sup>S</sup>*(*A*, *<sup>b</sup>*) we have *xj* <sup>=</sup> <sup>−</sup><sup>∞</sup> if *akj* �<sup>=</sup> <sup>−</sup>∞, *<sup>j</sup>* <sup>∈</sup> *<sup>N</sup>*, as a result the *<sup>k</sup>th* equation could be removed from the system together with every column *j* in the matrix *A* where *akj* �= −∞ (if any), and set the corresponding *xj* = −∞. Consequently, we may assume without loss of generality that *<sup>b</sup>* <sup>∈</sup> **<sup>R</sup>***m*.
Moreover, if *<sup>b</sup>* <sup>∈</sup> **<sup>R</sup>***<sup>m</sup>* and *<sup>A</sup>* has an <sup>−</sup><sup>∞</sup> row then *<sup>S</sup>*(*A*, *<sup>b</sup>*) = <sup>∅</sup>. If there is an <sup>−</sup><sup>∞</sup> column *<sup>j</sup>* in *<sup>A</sup>* then *xj* may take on any value in a solution *x*. Thus, in what follows we assume without loss of generality that *<sup>A</sup>* is doubly **<sup>R</sup>** <sup>−</sup> *astic* and *<sup>b</sup>* <sup>∈</sup> **<sup>R</sup>***m*.
**Theorem 1.** Let *<sup>A</sup>* = (*aij*) <sup>∈</sup> **<sup>R</sup>***m*×*<sup>n</sup>* be doubly **<sup>R</sup>** <sup>−</sup> *astic* and *<sup>b</sup>* <sup>∈</sup> **<sup>R</sup>***m*. Then *<sup>x</sup>* <sup>∈</sup> *<sup>S</sup>*(*A*, *<sup>b</sup>*) if and only if
$$\begin{aligned} \text{i) } &\ge \bar{\mathfrak{x}}(A, b) \text{ and} \\ \text{ii) } &\bigcup\_{j \in \mathcal{N}\_{\mathcal{X}}} M\_j = M \text{ where } \mathcal{N}\_{\mathcal{X}} = \{ j \in \mathcal{N}; \mathfrak{x}\_j = \mathfrak{x}(A, b)\_j \} \end{aligned}$$
*Proof.* Suppose *x* ∈ *S*(*A*, *b*). Thus we have,
$$\begin{aligned} A \otimes \mathfrak{x} &= b \\ \iff \max\_{j} (a\_{ij} + \mathfrak{x}\_{j}) &= b\_{i} \text{ for all } i \in M \\ \iff a\_{ij} + \mathfrak{x}\_{j} &= b\_{i} \text{ for some } j \in N \\ \iff \mathfrak{x}\_{j} &\le b\_{i} \otimes a\_{ij}^{-1} \text{ for all } i \in M \\ \iff \mathfrak{x}\_{j} &\le \min\_{i \in M} (b\_{i} \otimes a\_{ij}^{-1}) \text{ for all } j \in N \end{aligned}$$
Hence, *x* ≤ *x*¯ .
Now that *<sup>x</sup>* ∈ *<sup>S</sup>*(*A*, *<sup>b</sup>*). Since *Mj* ⊆ *<sup>M</sup>* we only need to show that *<sup>M</sup>* ⊆ *<sup>j</sup>*∈*Nx Mj*. Let *<sup>k</sup>* ∈ *<sup>M</sup>*. Since *bk* <sup>=</sup> *akj* <sup>⊗</sup> *xj* <sup>&</sup>gt; <sup>−</sup><sup>∞</sup> for some *<sup>j</sup>* <sup>∈</sup> *<sup>N</sup>* and *<sup>x</sup>*−<sup>1</sup> *<sup>j</sup>* ≥ *x*¯ −1 *<sup>j</sup>* <sup>≥</sup> *aij* <sup>⊗</sup> *<sup>b</sup>*−<sup>1</sup> *<sup>i</sup>* for every *i* ∈ *M* we have *x*−<sup>1</sup> *<sup>j</sup>* <sup>=</sup> *akj* <sup>⊗</sup> *<sup>b</sup>*−<sup>1</sup> *<sup>k</sup>* <sup>=</sup> max*i*∈*<sup>M</sup> aij* <sup>⊗</sup> *<sup>b</sup>*−<sup>1</sup> *<sup>i</sup>* . Hence *k* ∈ *Mj* and *xj* = *x*¯*j*.
Suppose that *<sup>x</sup>* ≤ *<sup>x</sup>*¯ and *<sup>j</sup>*∈*Nx Mj* = *<sup>M</sup>*. Let *<sup>k</sup>* ∈ *<sup>M</sup>*, *<sup>j</sup>* ∈ *<sup>N</sup>*. Then *akj* ⊗ *xj* ≤ *bk* if *akj* = −∞. If *akj* �= −∞ then
$$a\_{k\mathbf{j}} \otimes \mathbf{x}\_{\mathbf{j}} \le a\_{k\mathbf{j}} \otimes \mathbf{\bar{x}}\_{\mathbf{j}} \le a\_{k\mathbf{j}} \otimes b\_{\mathbf{k}} \otimes a\_{k\mathbf{j}}^{-1} = b\_{\mathbf{k}} \tag{9}$$
Therefore *A* ⊗ *x* ≤ *b*. At the same time *k* ∈ *Mj* for some *j* ∈ *N* satisfying *xj* = *x*¯*j*. For this *j* both inequalities in (9) are equalities and thus *A* ⊗ *x* = *b*.
The following is a summary of prerequisites proved in [1] and [12]:
**Theorem 2.** Let *<sup>A</sup>* = (*aij*) <sup>∈</sup> **<sup>R</sup>***m*×*<sup>n</sup>* be doubly **<sup>R</sup>** <sup>−</sup> *astic* and *<sup>b</sup>* <sup>∈</sup> **<sup>R</sup>***m*. The system *<sup>A</sup>* <sup>⊗</sup> *<sup>x</sup>* <sup>=</sup> *<sup>b</sup>* has a solution if and only if *x*¯(*A*, *b*) is a solution.
*Proof.* Follows from Theorem 1.
8 Will-be-set-by-IN-TECH
*kj* = max
*ij* ) for all *j* ∈ *N*
We now consider the cases when *A* = −∞ and/or *b* = −∞. Suppose that *b* = −∞. Then
Therefore if *<sup>A</sup>* <sup>=</sup> <sup>−</sup><sup>∞</sup> we have *<sup>S</sup>*(*A*, *<sup>b</sup>*) = **<sup>R</sup>***<sup>n</sup>* . Now, if *<sup>A</sup>* <sup>=</sup> <sup>−</sup><sup>∞</sup> and *<sup>b</sup>* �<sup>=</sup> <sup>−</sup><sup>∞</sup> then *<sup>S</sup>*(*A*, *<sup>b</sup>*) = <sup>∅</sup>. Thus, we may assume in this section that *A* = −∞ and *b* �= −∞. If *bk* = −∞ for some *<sup>k</sup>* <sup>∈</sup> *<sup>M</sup>* then for any *<sup>x</sup>* <sup>∈</sup> *<sup>S</sup>*(*A*, *<sup>b</sup>*) we have *xj* <sup>=</sup> <sup>−</sup><sup>∞</sup> if *akj* �<sup>=</sup> <sup>−</sup>∞, *<sup>j</sup>* <sup>∈</sup> *<sup>N</sup>*, as a result the *<sup>k</sup>th* equation could be removed from the system together with every column *j* in the matrix *A* where *akj* �= −∞ (if any), and set the corresponding *xj* = −∞. Consequently, we may assume
Moreover, if *<sup>b</sup>* <sup>∈</sup> **<sup>R</sup>***<sup>m</sup>* and *<sup>A</sup>* has an <sup>−</sup><sup>∞</sup> row then *<sup>S</sup>*(*A*, *<sup>b</sup>*) = <sup>∅</sup>. If there is an <sup>−</sup><sup>∞</sup> column *<sup>j</sup>* in *<sup>A</sup>* then *xj* may take on any value in a solution *x*. Thus, in what follows we assume without loss
**Theorem 1.** Let *<sup>A</sup>* = (*aij*) <sup>∈</sup> **<sup>R</sup>***m*×*<sup>n</sup>* be doubly **<sup>R</sup>** <sup>−</sup> *astic* and *<sup>b</sup>* <sup>∈</sup> **<sup>R</sup>***m*. Then *<sup>x</sup>* <sup>∈</sup> *<sup>S</sup>*(*A*, *<sup>b</sup>*) if and
*Mj* = *M* where *Nx* = {*j* ∈ *N*; *xj* = *x*¯(*A*, *b*)*j*}
*<sup>j</sup>* (*aij* <sup>+</sup> *xj*) = *bi* for all *<sup>i</sup>* <sup>∈</sup> *<sup>M</sup>*
*ij* for all *i* ∈ *M*
*<sup>j</sup>* ≥ *x*¯ −1
*<sup>i</sup>* . Hence *k* ∈ *Mj* and *xj* = *x*¯*j*.
*ij* )for all *j* ∈ *N*
*<sup>j</sup>* <sup>≥</sup> *aij* <sup>⊗</sup> *<sup>b</sup>*−<sup>1</sup>
*<sup>j</sup>*∈*Nx Mj* = *<sup>M</sup>*. Let *<sup>k</sup>* ∈ *<sup>M</sup>*, *<sup>j</sup>* ∈ *<sup>N</sup>*. Then *akj* ⊗ *xj* ≤ *bk* if *akj* = −∞. If
*<sup>j</sup>*∈*Nx Mj*. Let *<sup>k</sup>* ∈ *<sup>M</sup>*.
*<sup>i</sup>* for every *i* ∈ *M* we have
*kj* = *bk* (9)
⇐⇒ *aij* + *xj* = *bi* for some *j* ∈ *N*
*<sup>i</sup>*∈*M*(*bi* <sup>⊗</sup> *<sup>a</sup>*−<sup>1</sup>
*akj* <sup>⊗</sup> *xj* <sup>≤</sup> *akj* <sup>⊗</sup> *<sup>x</sup>*¯*<sup>j</sup>* <sup>≤</sup> *akj* <sup>⊗</sup> *bk* <sup>⊗</sup> *<sup>a</sup>*−<sup>1</sup>
*<sup>i</sup>* (*bi* <sup>⊗</sup> *<sup>a</sup>*−<sup>1</sup>
; *xj* = −∞, if *Aj* �= −∞, *j* ∈ *N*}
*ij* )} for all *j* ∈ *N*
; *A* ⊗ *x* = *b*}
We introduce the following notations
*S*(*A*, *b*) can simply be written as
without loss of generality that *<sup>b</sup>* <sup>∈</sup> **<sup>R</sup>***m*.
only if
Hence, *x* ≤ *x*¯ .
*<sup>j</sup>* <sup>=</sup> *akj* <sup>⊗</sup> *<sup>b</sup>*−<sup>1</sup>
*akj* �= −∞ then
Suppose that *<sup>x</sup>* ≤ *<sup>x</sup>*¯ and
*x*−<sup>1</sup>
of generality that *<sup>A</sup>* is doubly **<sup>R</sup>** <sup>−</sup> *astic* and *<sup>b</sup>* <sup>∈</sup> **<sup>R</sup>***m*.
ii) *j*∈*Nx*
Since *bk* <sup>=</sup> *akj* <sup>⊗</sup> *xj* <sup>&</sup>gt; <sup>−</sup><sup>∞</sup> for some *<sup>j</sup>* <sup>∈</sup> *<sup>N</sup>* and *<sup>x</sup>*−<sup>1</sup>
*<sup>k</sup>* <sup>=</sup> max*i*∈*<sup>M</sup> aij* <sup>⊗</sup> *<sup>b</sup>*−<sup>1</sup>
*Proof.* Suppose *x* ∈ *S*(*A*, *b*). Thus we have,
i) *x* ≤ *x*¯(*A*, *b*) and
*A* ⊗ *x* = *b* ⇐⇒ max
⇐⇒ *xj* <sup>≤</sup> *bi* <sup>⊗</sup> *<sup>a</sup>*−<sup>1</sup>
⇐⇒ *xj* ≤ min
Now that *<sup>x</sup>* ∈ *<sup>S</sup>*(*A*, *<sup>b</sup>*). Since *Mj* ⊆ *<sup>M</sup>* we only need to show that *<sup>M</sup>* ⊆
*<sup>S</sup>*(*A*, *<sup>b</sup>*) = {*<sup>x</sup>* <sup>∈</sup> **<sup>R</sup>***<sup>n</sup>*
*x*¯(*A*, *b*)*<sup>j</sup>* = min
*Mj* <sup>=</sup> {*<sup>k</sup>* <sup>∈</sup> *<sup>M</sup>*; *bk* <sup>⊗</sup> *<sup>a</sup>*−<sup>1</sup>
*<sup>S</sup>*(*A*, *<sup>b</sup>*) = {*<sup>x</sup>* <sup>∈</sup> **<sup>R</sup>***<sup>n</sup>*
*<sup>i</sup>*∈*M*(*bi* <sup>⊗</sup> *<sup>a</sup>*−<sup>1</sup>
Since *x*¯(*A*, *b*) has played an important role in the solution of *A* ⊗ *x* = *b*. This vector *x*¯ is called the *principal solution* to *A* ⊗ *x* = *b* [1], and we will call it likewise. The principal solution will also be used when studying the systems *A* ⊗ *x* ≤ *b* and also when solving the one-sided system containing both equations and inequalities. The one-sided systems containing both equations and inequalities have been studied in [16] and the result will be presented later in this chapter.
Note that the principal solution may not be a solution to the system *A* ⊗ *x* = *b*. More precisely, the following are observed in [12]:
**Corollary 1.** Let *<sup>A</sup>* = (*aij*) <sup>∈</sup> **<sup>R</sup>***m*×*<sup>n</sup>* be doubly **<sup>R</sup>** <sup>−</sup> *astic* and *<sup>b</sup>* <sup>∈</sup> **<sup>R</sup>***m*. Then the following three statements are equivalent:
$$\begin{aligned} \text{i) } S(A, b) &\neq \bigotimes \\ \text{ii) } \bar{x}(A, b) &\in S(A, b) \\ \text{iii) } \bigcup\_{j \in N} M\_j &= M \end{aligned}$$
*Proof.*
The statements follow from Theorems 1 and 2.
For the existence of a unique solution to the max-linear system *A* ⊗ *x* = *b* we have the following corollary:
**Corollary 2.** Let *<sup>A</sup>* = (*aij*) <sup>∈</sup> **<sup>R</sup>***m*×*<sup>n</sup>* be doubly **<sup>R</sup>** <sup>−</sup> *astic* and *<sup>b</sup>* <sup>∈</sup> **<sup>R</sup>***m*. Then *<sup>S</sup>*(*A*, *<sup>b</sup>*) = {*x*¯(*A*, *<sup>b</sup>*)} if and only if
$$\begin{aligned} \text{i) } & \bigcup\_{j \in N} M\_j = M \text{ and} \\ \text{ii) } & \bigcup\_{j \in N} M\_j \neq M \text{ for any } N' \subseteq N, N' \neq N \end{aligned}$$
*Proof.* Follows from Theorem 1.
The question of solvability and unique solvability of the system *A* ⊗ *x* = *b* was linked to the set covering and minimal set covering problem of combinatorics in [12].
#### **4.2. System of inequalities**
In this section we show how a solution to the one-sided system of inequalities can be obtained. Let *<sup>A</sup>* = (*aij*) <sup>∈</sup> **<sup>R</sup>***m*×*<sup>n</sup>* and *<sup>b</sup>* = (*b*1,..., *bm*)*<sup>T</sup>* <sup>∈</sup> **<sup>R</sup>**. A system of the form:
$$A \otimes \mathfrak{x} \le b$$
is called *one-sided max-linear system of inequalities* or just *one-sided system of inequalities*. The one-sided system of inequalities has received some attention in the past, see [1, 3] and [17] for more information. Here, we will only present a result which shows that the principal solution, *x*¯(*A*, *b*) is the greatest solution to (10). That is if (10) has a solution then *x*¯(*A*, *b*) is the greatest of all the solutions. We denote the solution set of (10) by *S*(*A*, *b*, ≤). That is
$$S(A, b \le) = \{ \mathfrak{x} \in \mathbb{R}^n ; A \otimes \mathfrak{x} \le b \}$$
**Theorem 3.** *x* ∈ *S*(*A*, *b*, ≤) if and only if *x* ≤ *x*¯(*A*, *b*).
*Proof.* Suppose *x* ∈ *S*(*A*, *b*, ≤). Then we have
$$\begin{aligned} A \otimes \mathfrak{x} &\leq b \\ \iff \max\_{j} (a\_{ij} + \mathfrak{x}\_{j}) &\leq b\_{i} \text{ for all } i \\ \iff a\_{ij} + \mathfrak{x}\_{j} &\leq b\_{i} \text{ for all } i, j \\ \iff \mathfrak{x}\_{j} &\leq b\_{i} \otimes a\_{ij}^{-1} \text{ for all } i, j \\ \iff \mathfrak{x}\_{j} &\leq \min\_{i} (b\_{i} \otimes a\_{ij}^{-1}) \text{ for all } j \\ \iff \mathfrak{x} &\leq \mathfrak{x}(A, b) \end{aligned}$$
and the proof is now complete.
The system of inequalities
$$\begin{aligned} A \otimes \mathfrak{x} &\le b \\ \mathfrak{C} \otimes \mathfrak{x} &\ge d \end{aligned} \tag{11}$$
was discussed in [18] where the following result was presented.
**Lemma 1.** A system of inequalities (11) has a solution if and only if *C* ⊗ *x*¯(*A*, *b*) ≥ *d*
#### **4.3. A system containing of both equations and inequalities**
In this section a system containing both equations and inequalities will be presented, the results were taken from [16]. Let *<sup>A</sup>* = (*aij*) <sup>∈</sup> **<sup>R</sup>***k*×*n*, *<sup>C</sup>* = (*cij*) <sup>∈</sup> **<sup>R</sup>***r*×*n*, *<sup>b</sup>* = (*b*1,..., *bk*)*<sup>T</sup>* <sup>∈</sup> **<sup>R</sup>***<sup>k</sup>* and *<sup>d</sup>* = (*d*1,..., *dr*)*<sup>T</sup>* <sup>∈</sup> **<sup>R</sup>***r*. A *one-sided max-linear system with both equations and inequalities* is of the form:
$$\begin{aligned} A \otimes \mathfrak{x} &= b \\ \mathfrak{C} \otimes \mathfrak{x} &\le d \end{aligned} \tag{12}$$
We shall use the following notation throughout this paper
$$R = \{1, 2, \ldots, r\}$$
$$S(A, C, b, d) = \{x \in \mathbb{R}^n; A \otimes x = b \text{ and } C \otimes x \le d\}$$
$$S(C, d, \le) = \{x \in \mathbb{R}^n; C \otimes x \le d\}$$
$$\mathfrak{F}\_j(\mathbb{C}, d) = \min\_{i \in K} (d\_i \otimes c\_{ij}^{-1}) \text{ for all } j \in N$$
$$K = \{1, \ldots, k\}$$
$$K\_j = \left\{k \in K; b\_k \otimes a\_{kj}^{-1} = \min\_{i \in K} \left(b\_i \otimes a\_{ij}^{-1}\right)\right\} \text{ for all } j \in N$$
$$\mathfrak{F}\_j(A, b) = \min\_{i \in K} (b\_i \otimes a\_{ij}^{-1}) \text{ for all } j \in N$$
$$\mathfrak{F} = (\mathfrak{F}\_1, \ldots, \mathfrak{F}\_n)^T$$
$$I = \{j \in N; \mathfrak{F}\_j(\mathbb{C}, d) \ge \mathfrak{F}\_j(A, b)\} \text{ and }$$
$$L = N \mid f$$
We also define the vector *x*ˆ = (*x*ˆ1, *x*ˆ2, ..., *x*ˆ*n*)*T*, where
$$\mathfrak{X}\_{\dot{f}}(A,\mathsf{C},b,d) \equiv \begin{cases} \mathfrak{X}\_{\dot{f}}(A,b) & \text{if } \dot{f} \in I \\ \mathfrak{X}\_{\dot{f}}(\mathsf{C},d) & \text{if } \dot{f} \in L \end{cases} \tag{13}$$
and *Nx*<sup>ˆ</sup> = {*j* ∈ *N*; *x*ˆ*<sup>j</sup>* = *x*¯*j*}.
10 Will-be-set-by-IN-TECH
In this section we show how a solution to the one-sided system of inequalities can be obtained.
is called *one-sided max-linear system of inequalities* or just *one-sided system of inequalities*. The one-sided system of inequalities has received some attention in the past, see [1, 3] and [17] for more information. Here, we will only present a result which shows that the principal solution, *x*¯(*A*, *b*) is the greatest solution to (10). That is if (10) has a solution then *x*¯(*A*, *b*) is the greatest
*<sup>S</sup>*(*A*, *<sup>b</sup>*, <sup>≤</sup>) = {*<sup>x</sup>* <sup>∈</sup> **<sup>R</sup>***n*; *<sup>A</sup>* <sup>⊗</sup> *<sup>x</sup>* <sup>≤</sup> *<sup>b</sup>*}
*<sup>j</sup>* (*aij* <sup>+</sup> *xj*) <sup>≤</sup> *bi* for all *<sup>i</sup>*
*<sup>i</sup>* (*bi* <sup>⊗</sup> *<sup>a</sup>*−<sup>1</sup>
*A* ⊗ *x* ≤ *b*
In this section a system containing both equations and inequalities will be presented, the results were taken from [16]. Let *<sup>A</sup>* = (*aij*) <sup>∈</sup> **<sup>R</sup>***k*×*n*, *<sup>C</sup>* = (*cij*) <sup>∈</sup> **<sup>R</sup>***r*×*n*, *<sup>b</sup>* = (*b*1,..., *bk*)*<sup>T</sup>* <sup>∈</sup> **<sup>R</sup>***<sup>k</sup>* and *<sup>d</sup>* = (*d*1,..., *dr*)*<sup>T</sup>* <sup>∈</sup> **<sup>R</sup>***r*. A *one-sided max-linear system with both equations and inequalities* is
**Lemma 1.** A system of inequalities (11) has a solution if and only if *C* ⊗ *x*¯(*A*, *b*) ≥ *d*
*ij* for all *i*, *j*
*ij* )for all *j*
*<sup>C</sup>* <sup>⊗</sup> *<sup>x</sup>* <sup>≥</sup> *<sup>d</sup>* (11)
⇐⇒ *aij* + *xj* ≤ *bi* for all *i*, *j*
⇐⇒ *xj* <sup>≤</sup> *bi* <sup>⊗</sup> *<sup>a</sup>*−<sup>1</sup>
⇐⇒ *xj* ≤ min
was discussed in [18] where the following result was presented.
**4.3. A system containing of both equations and inequalities**
⇐⇒ *x* ≤ *x*¯(*A*, *b*)
*A* ⊗ *x* ≤ *b* (10)
Let *<sup>A</sup>* = (*aij*) <sup>∈</sup> **<sup>R</sup>***m*×*<sup>n</sup>* and *<sup>b</sup>* = (*b*1,..., *bm*)*<sup>T</sup>* <sup>∈</sup> **<sup>R</sup>**. A system of the form:
of all the solutions. We denote the solution set of (10) by *S*(*A*, *b*, ≤). That is
*A* ⊗ *x* ≤ *b* ⇐⇒ max
**Theorem 3.** *x* ∈ *S*(*A*, *b*, ≤) if and only if *x* ≤ *x*¯(*A*, *b*).
*Proof.* Suppose *x* ∈ *S*(*A*, *b*, ≤). Then we have
and the proof is now complete.
The system of inequalities
of the form:
**4.2. System of inequalities**
**Theorem 4.** Let *<sup>A</sup>* = (*aij*) <sup>∈</sup> **<sup>R</sup>***k*×*n*, *<sup>C</sup>* = (*cij*) <sup>∈</sup> **<sup>R</sup>***r*×*n*, *<sup>b</sup>* = (*b*1,..., *bk*)*<sup>T</sup>* <sup>∈</sup> **<sup>R</sup>***<sup>k</sup>* and *<sup>d</sup>* <sup>=</sup> (*d*1,..., *dr*)*<sup>T</sup>* <sup>∈</sup> **<sup>R</sup>***r*. Then the following three statements are equivalent:
$$\begin{aligned} \text{(i)} \quad &S(A,\mathsf{C},b,d) \neq \bigotimes \\ \text{(ii)} \quad &\hat{\mathsf{x}}(A,\mathsf{C},b,d) \in S(A,\mathsf{C},b,d) \\ \text{(iii)} \quad &\bigcup\_{j \in I} K\_j = K \end{aligned}$$
*Proof.* (*i*) =⇒ (*ii*). Let *x* ∈ *S*(*A*, *C*, *b*, *d*), therefore *x* ∈ *S*(*A*, *b*) and *x* ∈ *S*(*C*, *d*, ≤). Since *x* ∈ *S*(*C*, *d*, ≤), it follows from Theorem 3 that *x* ≤ *x*¯(*C*, *d*). Now that *x* ∈ *S*(*A*, *b*) and also *x* ∈ *S*(*C*, *d*, ≤), we need to show that *x*¯*j*(*C*, *d*) ≥ *x*¯*j*(*A*, *b*) for all *j* ∈ *Nx* (that is *Nx* ⊆ *J*). Let *j* ∈ *Nx* then *xj* = *x*¯*j*(*A*, *b*). Since *x* ∈ *S*(*C*, *d*, ≤) we have *x* ≤ *x*¯(*C*, *d*) and therefore *<sup>x</sup>*¯*j*(*A*, *<sup>b</sup>*) ≤ *<sup>x</sup>*¯*j*(*C*, *<sup>d</sup>*) thus *<sup>j</sup>* ∈ *<sup>J</sup>*. Hence, *Nx* ⊆ *<sup>J</sup>* and by Theorem 1 *<sup>j</sup>*∈*<sup>J</sup> Kj* = *<sup>K</sup>*. This also proves (*i*) =⇒ (*iii*) (*iii*) =⇒ (*i*). Suppose *<sup>j</sup>*∈*<sup>J</sup> Kj* = *<sup>K</sup>*. Since *<sup>x</sup>*ˆ(*A*, *<sup>C</sup>*, *<sup>b</sup>*, *<sup>d</sup>*) ≤ *<sup>x</sup>*¯(*C*, *<sup>d</sup>*) we have *<sup>x</sup>*ˆ(*A*, *<sup>C</sup>*, *<sup>b</sup>*, *<sup>d</sup>*) ∈ *<sup>S</sup>*(*C*, *<sup>d</sup>*, ≤). Also *<sup>x</sup>*ˆ(*A*, *<sup>C</sup>*, *<sup>b</sup>*, *<sup>d</sup>*) ≤ *<sup>x</sup>*¯(*A*, *<sup>b</sup>*) and *Nx*<sup>ˆ</sup> ⊇ *<sup>J</sup>* gives *<sup>j</sup>*∈*Nx*<sup>ˆ</sup>(*A*,*C*,*b*,*d*) *Kj* <sup>⊇</sup> *<sup>j</sup>*∈*<sup>J</sup> Kj* = *<sup>K</sup>*. Hence *<sup>j</sup>*∈*Nx*<sup>ˆ</sup>(*A*,*C*,*b*,*d*) *Kj* = *<sup>K</sup>*, therefore *<sup>x</sup>*ˆ(*A*, *<sup>C</sup>*, *<sup>b</sup>*, *<sup>d</sup>*) ∈ *<sup>S</sup>*(*A*, *<sup>b</sup>*) and *<sup>x</sup>*ˆ(*A*, *<sup>C</sup>*, *<sup>b</sup>*, *<sup>d</sup>*) ∈ *<sup>S</sup>*(*C*, *<sup>d</sup>*, ≤). Hence *x*ˆ(*A*, *C*, *b*, *d*) ∈ *S*(*A*, *C*, *b*, *d*) (that is *S*(*A*, *C*, *b*, *d*) �= ∅) and this also proves (*iii*) =⇒ (*ii*).
**Theorem 5.** Let *<sup>A</sup>* = (*aij*) <sup>∈</sup> **<sup>R</sup>***k*×*n*, *<sup>C</sup>* = (*cij*) <sup>∈</sup> **<sup>R</sup>***r*×*n*, *<sup>b</sup>* = (*b*1,..., *bk*)*<sup>T</sup>* <sup>∈</sup> **<sup>R</sup>***<sup>k</sup>* and *<sup>d</sup>* <sup>=</sup> (*d*1,..., *dr*)*<sup>T</sup>* <sup>∈</sup> **<sup>R</sup>***r*. Then *<sup>x</sup>* <sup>∈</sup> *<sup>S</sup>*(*A*, *<sup>C</sup>*, *<sup>b</sup>*, *<sup>d</sup>*) if and only if
> (i) *x* ≤ *x*ˆ(*A*, *C*, *b*, *d*) and (ii) *j*∈*Nx Kj* = *K* where *Nx* = {*j* ∈ *N* ; *xj* = *x*¯*j*(*A*, *b*)}
*Proof.* (=⇒) Let *x* ∈ *S*(*A*, *C*, *b*, *d*), then *x* ≤ *x*¯(*A*, *b*) and *x* ≤ *x*¯(*C*, *d*). Since *x*ˆ(*A*, *C*, *b*, *d*) = *<sup>x</sup>*¯(*A*, *<sup>b</sup>*) <sup>⊕</sup>� *x*¯(*C*, *d*) we have *x* ≤ *x*ˆ(*A*, *C*, *b*, *d*). Also, *x* ∈ *S*(*A*, *C*, *b*, *d*) implies that *x* ∈ *S*(*C*, *d*, ≤). It follows from Theorem 1 that *<sup>j</sup>*∈*Nx Kj* = *<sup>K</sup>*.
(⇐=) Suppose that *<sup>x</sup>* <sup>≤</sup> *<sup>x</sup>*ˆ(*A*, *<sup>C</sup>*, *<sup>b</sup>*, *<sup>d</sup>*) = *<sup>x</sup>*¯(*A*, *<sup>b</sup>*) <sup>⊕</sup>� *x*¯(*C*, *d*) and *<sup>j</sup>*∈*Nx Kj* = *<sup>K</sup>*. It follows from Theorem 1 that *x* ∈ *S*(*A*, *b*), also by Theorem 3 *x* ∈ *S*(*C*, *d*, ≤). Thus *x* ∈ *S*(*A*, *b*) ∩ *S*(*C*, *d*, ≤) = *S*(*A*, *C*, *b*, *d*).
We introduce the symbol |*X*| which stands for the number of elements of the set *X*.
**Lemma 2.** Let *<sup>A</sup>* = (*aij*) <sup>∈</sup> **<sup>R</sup>***k*×*n*, *<sup>C</sup>* = (*cij*) <sup>∈</sup> **<sup>R</sup>***r*×*n*, *<sup>b</sup>* = (*b*1,..., *bk*)*<sup>T</sup>* <sup>∈</sup> **<sup>R</sup>***<sup>k</sup>* and *<sup>d</sup>* <sup>=</sup> (*d*1,..., *dr*)*<sup>T</sup>* <sup>∈</sup> **<sup>R</sup>***r*. If <sup>|</sup>*S*(*A*, *<sup>C</sup>*, *<sup>b</sup>*, *<sup>d</sup>*)<sup>|</sup> <sup>=</sup> 1 then <sup>|</sup>*S*(*A*, *<sup>b</sup>*)<sup>|</sup> <sup>=</sup> 1.
*Proof.* Suppose <sup>|</sup>*S*(*A*, *<sup>C</sup>*, *<sup>b</sup>*, *<sup>d</sup>*)<sup>|</sup> <sup>=</sup> 1, that is *<sup>S</sup>*(*A*, *<sup>C</sup>*, *<sup>b</sup>*, *<sup>d</sup>*) = {*x*} for an *<sup>x</sup>* <sup>∈</sup> **<sup>R</sup>***n*. Since *S*(*A*, *C*, *b*, *d*) = {*x*} we have *x* ∈ *S*(*A*, *b*) and thus *S*(*A*, *b*) �= ∅. For contradiction, suppose |*S*(*A*, *b*)| > 1. We need to check the following two cases: (i) *L* �= ∅ and (ii) *L* = ∅ where *L* = *N* \ *J*, and show in each case that |*S*(*A*, *C*, *b*, *d*)| > 1.
**Proof of Case (i)**, that is *L* �= ∅: Suppose that *L* contains only one element say *n* ∈ *N* i.e *L* = {*n*}. Since *x* ∈ *S*(*A*, *C*, *b*, *d*) it follows from Theorem 4that *x*ˆ(*A*, *C*, *b*, *d*) ∈ *S*(*A*, *C*, *b*, *d*). That is *x* = *x*ˆ(*A*, *C*, *b*, *d*) =
(*x*¯1(*A*, *<sup>b</sup>*), *<sup>x</sup>*¯2(*A*, *<sup>b</sup>*),..., *<sup>x</sup>*¯*n*−1(*A*, *<sup>b</sup>*), *<sup>x</sup>*¯*n*(*C*, *<sup>d</sup>*)) ∈ *<sup>S</sup>*(*A*, *<sup>C</sup>*, *<sup>b</sup>*, *<sup>d</sup>*). It can also be seen that, *<sup>x</sup>*¯(*C*, *<sup>d</sup>*)*<sup>n</sup>* < *x*¯*n*(*A*, *b*) and any vector of the form *z* =
(*x*¯1(*A*, *<sup>b</sup>*), *<sup>x</sup>*¯2(*A*, *<sup>b</sup>*),..., *<sup>x</sup>*¯*n*−1(*A*, *<sup>b</sup>*), *<sup>α</sup>*) ∈ *<sup>S</sup>*(*A*, *<sup>C</sup>*, *<sup>b</sup>*, *<sup>d</sup>*), where *<sup>α</sup>* ≤ *<sup>x</sup>*¯*n*(*C*, *<sup>d</sup>*). Hence |*S*(*A*, *C*, *b*, *d*)| > 1. If *L* contains more than one element, then the proof is done in a similar way.
**Proof of Case (ii)**, that is *L* = ∅ (*J* = *N*): Suppose that *J* = *N*. Then we have *x*ˆ(*A*, *C*, *b*, *d*) = *x*¯(*A*, *b*) ≤ *x*¯(*C*, *d*). Suppose without loss of generality that *x*, *x* � ∈ *S*(*A*, *b*) such that *x* �= *x* � . Then *x* ≤ *x*¯(*A*, *b*) ≤ *x*¯(*C*, *d*) and also *x* � ≤ *x*¯(*A*, *b*) ≤ *x*¯(*C*, *d*). Thus, *x*, *x* � ∈ *S*(*C*, *d*, ≤). Consequently, *x*, *x* � ∈ *S*(*A*, *C*, *b*, *d*) and *x* �= *x* � . Hence |*S*(*A*, *C*, *b*, *d*)| > 1.
**Theorem 6.** Let *<sup>A</sup>* = (*aij*) <sup>∈</sup> **<sup>R</sup>***k*×*n*, *<sup>C</sup>* = (*cij*) <sup>∈</sup> **<sup>R</sup>***r*×*n*, *<sup>b</sup>* = (*b*1,..., *bk*)*<sup>T</sup>* <sup>∈</sup> **<sup>R</sup>***<sup>k</sup>* and *<sup>d</sup>* <sup>=</sup> (*d*1,..., *dr*)*<sup>T</sup>* <sup>∈</sup> **<sup>R</sup>***r*. If <sup>|</sup>*S*(*A*, *<sup>C</sup>*, *<sup>b</sup>*, *<sup>d</sup>*)<sup>|</sup> <sup>=</sup> 1 then *<sup>J</sup>* <sup>=</sup> *<sup>N</sup>*.
*Proof.* Suppose |*S*(*A*, *<sup>C</sup>*, *<sup>b</sup>*, *<sup>d</sup>*)| = 1. It follows from Theorem 4 that *<sup>j</sup>*∈*<sup>J</sup> Kj* = *<sup>K</sup>*. Also, |*S*(*A*, *C*, *b*, *d*)| = 1 implies that |*S*(*A*, *b*)| = 1 (Lemma 2). Moreover, |*S*(*A*, *b*)| = 1 implies that *<sup>j</sup>*∈*<sup>N</sup> Kj* <sup>=</sup> *<sup>K</sup>* and *<sup>j</sup>*∈*N*� *Kj* �<sup>=</sup> *<sup>K</sup>*, *<sup>N</sup>*� <sup>⊆</sup> *<sup>N</sup>*, *<sup>N</sup>*� �= *<sup>N</sup>* (Theorem 2). Since *<sup>J</sup>* ⊆ *<sup>N</sup>* and *<sup>j</sup>*∈*<sup>J</sup> Kj* = *<sup>K</sup>*, we have *J* = *N*.
**Corollary 3.** Let *<sup>A</sup>* = (*aij*) <sup>∈</sup> **<sup>R</sup>***k*×*n*, *<sup>C</sup>* = (*cij*) <sup>∈</sup> **<sup>R</sup>***r*×*n*, *<sup>b</sup>* = (*b*1,..., *bk*)*<sup>T</sup>* <sup>∈</sup> **<sup>R</sup>***<sup>k</sup>* and *<sup>d</sup>* <sup>=</sup> (*d*1,..., *dr*)*<sup>T</sup>* <sup>∈</sup> **<sup>R</sup>***r*. If <sup>|</sup>*S*(*A*, *<sup>C</sup>*, *<sup>b</sup>*, *<sup>d</sup>*)<sup>|</sup> <sup>=</sup> 1 then *<sup>S</sup>*(*A*, *<sup>C</sup>*, *<sup>b</sup>*, *<sup>d</sup>*) = {*x*¯(*A*, *<sup>b</sup>*)}.
*Proof.* The statement follows from Theorem 6 and Lemma 2.
12 Will-be-set-by-IN-TECH
*<sup>j</sup>*∈*Nx*<sup>ˆ</sup>(*A*,*C*,*b*,*d*) *Kj* = *<sup>K</sup>*, therefore *<sup>x</sup>*ˆ(*A*, *<sup>C</sup>*, *<sup>b</sup>*, *<sup>d</sup>*) ∈ *<sup>S</sup>*(*A*, *<sup>b</sup>*) and *<sup>x</sup>*ˆ(*A*, *<sup>C</sup>*, *<sup>b</sup>*, *<sup>d</sup>*) ∈ *<sup>S</sup>*(*C*, *<sup>d</sup>*, ≤). Hence
**Theorem 5.** Let *<sup>A</sup>* = (*aij*) <sup>∈</sup> **<sup>R</sup>***k*×*n*, *<sup>C</sup>* = (*cij*) <sup>∈</sup> **<sup>R</sup>***r*×*n*, *<sup>b</sup>* = (*b*1,..., *bk*)*<sup>T</sup>* <sup>∈</sup> **<sup>R</sup>***<sup>k</sup>* and *<sup>d</sup>* <sup>=</sup>
*Proof.* (=⇒) Let *x* ∈ *S*(*A*, *C*, *b*, *d*), then *x* ≤ *x*¯(*A*, *b*) and *x* ≤ *x*¯(*C*, *d*). Since *x*ˆ(*A*, *C*, *b*, *d*) =
Theorem 1 that *x* ∈ *S*(*A*, *b*), also by Theorem 3 *x* ∈ *S*(*C*, *d*, ≤). Thus *x* ∈ *S*(*A*, *b*) ∩ *S*(*C*, *d*, ≤) =
**Lemma 2.** Let *<sup>A</sup>* = (*aij*) <sup>∈</sup> **<sup>R</sup>***k*×*n*, *<sup>C</sup>* = (*cij*) <sup>∈</sup> **<sup>R</sup>***r*×*n*, *<sup>b</sup>* = (*b*1,..., *bk*)*<sup>T</sup>* <sup>∈</sup> **<sup>R</sup>***<sup>k</sup>* and *<sup>d</sup>* <sup>=</sup>
*Proof.* Suppose <sup>|</sup>*S*(*A*, *<sup>C</sup>*, *<sup>b</sup>*, *<sup>d</sup>*)<sup>|</sup> <sup>=</sup> 1, that is *<sup>S</sup>*(*A*, *<sup>C</sup>*, *<sup>b</sup>*, *<sup>d</sup>*) = {*x*} for an *<sup>x</sup>* <sup>∈</sup> **<sup>R</sup>***n*. Since *S*(*A*, *C*, *b*, *d*) = {*x*} we have *x* ∈ *S*(*A*, *b*) and thus *S*(*A*, *b*) �= ∅. For contradiction, suppose |*S*(*A*, *b*)| > 1. We need to check the following two cases: (i) *L* �= ∅ and (ii) *L* = ∅ where
**Proof of Case (i)**, that is *L* �= ∅: Suppose that *L* contains only one element say *n* ∈ *N* i.e *L* = {*n*}. Since *x* ∈ *S*(*A*, *C*, *b*, *d*) it follows from Theorem 4that *x*ˆ(*A*, *C*, *b*, *d*) ∈ *S*(*A*, *C*, *b*, *d*).
(*x*¯1(*A*, *<sup>b</sup>*), *<sup>x</sup>*¯2(*A*, *<sup>b</sup>*),..., *<sup>x</sup>*¯*n*−1(*A*, *<sup>b</sup>*), *<sup>x</sup>*¯*n*(*C*, *<sup>d</sup>*)) ∈ *<sup>S</sup>*(*A*, *<sup>C</sup>*, *<sup>b</sup>*, *<sup>d</sup>*). It can also be seen that, *<sup>x</sup>*¯(*C*, *<sup>d</sup>*)*<sup>n</sup>* <
(*x*¯1(*A*, *<sup>b</sup>*), *<sup>x</sup>*¯2(*A*, *<sup>b</sup>*),..., *<sup>x</sup>*¯*n*−1(*A*, *<sup>b</sup>*), *<sup>α</sup>*) ∈ *<sup>S</sup>*(*A*, *<sup>C</sup>*, *<sup>b</sup>*, *<sup>d</sup>*), where *<sup>α</sup>* ≤ *<sup>x</sup>*¯*n*(*C*, *<sup>d</sup>*). Hence |*S*(*A*, *C*, *b*, *d*)| > 1. If *L* contains more than one element, then the proof is done in a similar
**Proof of Case (ii)**, that is *L* = ∅ (*J* = *N*): Suppose that *J* = *N*. Then we have *x*ˆ(*A*, *C*, *b*, *d*) =
**Theorem 6.** Let *<sup>A</sup>* = (*aij*) <sup>∈</sup> **<sup>R</sup>***k*×*n*, *<sup>C</sup>* = (*cij*) <sup>∈</sup> **<sup>R</sup>***r*×*n*, *<sup>b</sup>* = (*b*1,..., *bk*)*<sup>T</sup>* <sup>∈</sup> **<sup>R</sup>***<sup>k</sup>* and *<sup>d</sup>* <sup>=</sup>
�
�
�
≤ *x*¯(*A*, *b*) ≤ *x*¯(*C*, *d*). Thus, *x*, *x*
. Hence |*S*(*A*, *C*, *b*, *d*)| > 1.
∈ *S*(*A*, *b*) such that *x* �=
∈ *S*(*C*, *d*, ≤).
�
*<sup>j</sup>*∈*Nx Kj* = *<sup>K</sup>*.
We introduce the symbol |*X*| which stands for the number of elements of the set *X*.
*Kj* = *K* where *Nx* = {*j* ∈ *N* ; *xj* = *x*¯*j*(*A*, *b*)}
*x*¯(*C*, *d*) we have *x* ≤ *x*ˆ(*A*, *C*, *b*, *d*). Also, *x* ∈ *S*(*A*, *C*, *b*, *d*) implies that *x* ∈ *S*(*C*, *d*, ≤).
*x*¯(*C*, *d*) and
*x*ˆ(*A*, *C*, *b*, *d*) ∈ *S*(*A*, *C*, *b*, *d*) (that is *S*(*A*, *C*, *b*, *d*) �= ∅) and this also proves (*iii*) =⇒ (*ii*).
*<sup>j</sup>*∈*<sup>J</sup> Kj* = *<sup>K</sup>*. Since *<sup>x</sup>*ˆ(*A*, *<sup>C</sup>*, *<sup>b</sup>*, *<sup>d</sup>*) ≤ *<sup>x</sup>*¯(*C*, *<sup>d</sup>*) we have *<sup>x</sup>*ˆ(*A*, *<sup>C</sup>*, *<sup>b</sup>*, *<sup>d</sup>*) ∈
*<sup>j</sup>*∈*Nx*<sup>ˆ</sup>(*A*,*C*,*b*,*d*) *Kj* <sup>⊇</sup>
*<sup>j</sup>*∈*<sup>J</sup> Kj* = *<sup>K</sup>*. Hence
*<sup>j</sup>*∈*Nx Kj* = *<sup>K</sup>*. It follows from
(*i*) =⇒ (*iii*)
*<sup>x</sup>*¯(*A*, *<sup>b</sup>*) <sup>⊕</sup>�
*S*(*A*, *C*, *b*, *d*).
(*iii*) =⇒ (*i*). Suppose
*<sup>S</sup>*(*C*, *<sup>d</sup>*, ≤). Also *<sup>x</sup>*ˆ(*A*, *<sup>C</sup>*, *<sup>b</sup>*, *<sup>d</sup>*) ≤ *<sup>x</sup>*¯(*A*, *<sup>b</sup>*) and *Nx*<sup>ˆ</sup> ⊇ *<sup>J</sup>* gives
(*d*1,..., *dr*)*<sup>T</sup>* <sup>∈</sup> **<sup>R</sup>***r*. Then *<sup>x</sup>* <sup>∈</sup> *<sup>S</sup>*(*A*, *<sup>C</sup>*, *<sup>b</sup>*, *<sup>d</sup>*) if and only if
(ii) *j*∈*Nx*
(⇐=) Suppose that *<sup>x</sup>* <sup>≤</sup> *<sup>x</sup>*ˆ(*A*, *<sup>C</sup>*, *<sup>b</sup>*, *<sup>d</sup>*) = *<sup>x</sup>*¯(*A*, *<sup>b</sup>*) <sup>⊕</sup>�
(*d*1,..., *dr*)*<sup>T</sup>* <sup>∈</sup> **<sup>R</sup>***r*. If <sup>|</sup>*S*(*A*, *<sup>C</sup>*, *<sup>b</sup>*, *<sup>d</sup>*)<sup>|</sup> <sup>=</sup> 1 then <sup>|</sup>*S*(*A*, *<sup>b</sup>*)<sup>|</sup> <sup>=</sup> 1.
*L* = *N* \ *J*, and show in each case that |*S*(*A*, *C*, *b*, *d*)| > 1.
*x*¯(*A*, *b*) ≤ *x*¯(*C*, *d*). Suppose without loss of generality that *x*, *x*
∈ *S*(*A*, *C*, *b*, *d*) and *x* �= *x*
It follows from Theorem 1 that
That is *x* = *x*ˆ(*A*, *C*, *b*, *d*) =
way.
*x* �
Consequently, *x*, *x*
*x*¯*n*(*A*, *b*) and any vector of the form *z* =
. Then *x* ≤ *x*¯(*A*, *b*) ≤ *x*¯(*C*, *d*) and also *x*
(*d*1,..., *dr*)*<sup>T</sup>* <sup>∈</sup> **<sup>R</sup>***r*. If <sup>|</sup>*S*(*A*, *<sup>C</sup>*, *<sup>b</sup>*, *<sup>d</sup>*)<sup>|</sup> <sup>=</sup> 1 then *<sup>J</sup>* <sup>=</sup> *<sup>N</sup>*.
�
(i) *x* ≤ *x*ˆ(*A*, *C*, *b*, *d*) and
**Corollary 4.** Let *<sup>A</sup>* = (*aij*) <sup>∈</sup> **<sup>R</sup>***k*×*n*, *<sup>C</sup>* = (*cij*) <sup>∈</sup> **<sup>R</sup>***r*×*n*, *<sup>b</sup>* = (*b*1,..., *bk*)*<sup>T</sup>* <sup>∈</sup> **<sup>R</sup>***<sup>k</sup>* and *<sup>d</sup>* <sup>=</sup> (*d*1,..., *dr*)*<sup>T</sup>* <sup>∈</sup> **<sup>R</sup>***k*. Then, the following three statements are equivalent:
$$\begin{aligned} \text{(i)} \quad &|S(A, \mathbb{C}, b, d)| = 1\\ \text{(ii)} \quad &|S(A, b)| = 1 \text{ and } f = N\\ \text{(iii)} \quad &\bigcup\_{j \in J} K\_j = \text{Kand} \bigcup\_{j \in J'} K\_j \neq K, \text{ for every } f' \subseteq f, f' \neq f, \text{ and } f = N \end{aligned}$$
*Proof.* (*i*) =⇒ (*ii*) Follows from Lemma 2 and Theorem 6. (*ii*) =⇒ (*i*) Let *J* = *N*, therefore *x*¯ ≤ *x*¯(*C*, *d*) and thus *S*(*A*, *b*) ⊆ *S*(*C*, *d*, ≤). Therefore we have *S*(*A*, *C*, *b*, *d*) = *S*(*A*, *b*) ∩ *S*(*C*, *d*, ≤) = *S*(*A*, *b*). Hence |*S*(*A*, *C*, *b*, *d*)| = 1. (*ii*) =⇒ (*iii*) Suppose that *S*(*A*, *b*) = {*x*} and *J* = *N*. It follows from Theorem 2 that *<sup>j</sup>*∈*<sup>N</sup> Kj* <sup>=</sup> *<sup>K</sup>* and *<sup>j</sup>*∈*N*� *Kj* �<sup>=</sup> *<sup>K</sup>*, *<sup>N</sup>*� <sup>⊆</sup> *<sup>N</sup>*, *<sup>N</sup>*� �= *N*. Since *J* = *N* the statement now follows from Theorem 2.
(*iii*) =⇒ (*ii*) It is immediate that *J* = *N* and the statement now follows from Theorem 2.
**Theorem 7.** Let *<sup>A</sup>* = (*aij*) <sup>∈</sup> **<sup>R</sup>***k*×*n*, *<sup>C</sup>* = (*cij*) <sup>∈</sup> **<sup>R</sup>***r*×*n*, *<sup>b</sup>* = (*b*1,..., *bk*)*<sup>T</sup>* <sup>∈</sup> **<sup>R</sup>***<sup>k</sup>* and *<sup>d</sup>* <sup>=</sup> (*d*1,..., *dr*)*<sup>T</sup>* <sup>∈</sup> **<sup>R</sup>***k*. If <sup>|</sup>*S*(*A*, *<sup>C</sup>*, *<sup>b</sup>*, *<sup>d</sup>*)<sup>|</sup> <sup>&</sup>gt; 1 then <sup>|</sup>*S*(*A*, *<sup>C</sup>*, *<sup>b</sup>*, *<sup>d</sup>*)<sup>|</sup> is infinite .
*Proof.* Suppose |*S*(*A*, *<sup>C</sup>*, *<sup>b</sup>*, *<sup>d</sup>*)| > 1. By Corollary 4 we have *<sup>j</sup>*∈*<sup>J</sup> Kj* = *<sup>K</sup>*, for some *<sup>J</sup>* ⊆ *<sup>N</sup>*, *<sup>J</sup>* �= *<sup>N</sup>*(that is ∃*<sup>j</sup>* ∈ *<sup>N</sup>* such that *<sup>x</sup>*¯*j*(*A*, *<sup>b</sup>*) > *<sup>x</sup>*¯*j*(*C*, *<sup>d</sup>*)). Now *<sup>J</sup>* ⊆ *<sup>N</sup>* and *<sup>j</sup>*∈*<sup>J</sup> Kj* = *<sup>K</sup>*, Theorem 5 implies that any vector *x* = (*x*1, *x*2, ..., *xn*)*<sup>T</sup>* of the form
$$\mathfrak{x}\_{\mathfrak{j}} \equiv \begin{cases} \mathfrak{x}\_{\mathfrak{j}}(A, b) & \text{if } j \in J \\ y \le \mathfrak{x}\_{\mathfrak{j}}(C, d) & \text{if } j \in L \end{cases}$$
is in *S*(*A*, *C*, *b*, *d*), and the statement follows.
**Remark 1.** From Theorem 7 we can say that the number of solutions to the one-sided system containing both equations and inequalities can only be 0, 1, or ∞.
The vector *x*ˆ(*A*, *C*, *b*, *d*) plays an important role in the solution of the one-sided system containing both equations and inequalities. This role is the same as that of the principal solution *x*¯(*A*, *b*) to the one-sided max-linear system *A* ⊗ *x* = *b*, see [19] for more details.
#### **5. Max-linear program with equation and inequality constraints**
Suppose that the vector *<sup>f</sup>* = (*f*1, *<sup>f</sup>*2, ..., *fn*)*<sup>T</sup>* <sup>∈</sup> **<sup>R</sup>***<sup>n</sup>* is given. The task of minimizing [maximizing]the function *<sup>f</sup>*(*x*) = *<sup>f</sup> <sup>T</sup>* <sup>⊗</sup> *<sup>x</sup>* <sup>=</sup> max(*f*<sup>1</sup> <sup>+</sup> *<sup>x</sup>*1, *<sup>f</sup>*<sup>1</sup> <sup>+</sup> *<sup>x</sup>*2..., *fn* <sup>+</sup> *xn*) subject to (12) is called max-linear program with one-sided equations and inequalities and will be denoted by MLPmin <sup>≤</sup> and [MLPmax <sup>≤</sup> ]. We denote the sets of optimal solutions by *<sup>S</sup>*min(*A*, *<sup>C</sup>*, *<sup>b</sup>*, *<sup>d</sup>*) and *S*max(*A*, *C*, *b*, *d*), respectively.
**Lemma 3.** Suppose *<sup>f</sup>* <sup>∈</sup> **<sup>R</sup>***<sup>n</sup>* and let *<sup>f</sup>*(*x*) = *<sup>f</sup> <sup>T</sup>* <sup>⊗</sup> *<sup>x</sup>* be defined on **<sup>R</sup>***n*. Then, (i) *<sup>f</sup>*(*x*) is max-linear, i.e. *<sup>f</sup>*(*<sup>λ</sup>* <sup>⊗</sup> *<sup>x</sup>* <sup>⊕</sup> *<sup>μ</sup>* <sup>⊗</sup> *<sup>y</sup>*) = *<sup>λ</sup>* <sup>⊗</sup> *<sup>f</sup>*(*x*) <sup>⊕</sup> *<sup>μ</sup>* <sup>⊗</sup> *<sup>f</sup>*(*y*) for every *<sup>x</sup>*, *<sup>y</sup>* <sup>∈</sup> **<sup>R</sup>***n*. (ii) *<sup>f</sup>*(*x*) is isotone, i.e. *<sup>f</sup>*(*x*) <sup>≤</sup> *<sup>f</sup>*(*y*) for every *<sup>x</sup>*, *<sup>y</sup>* <sup>∈</sup> **<sup>R</sup>***n*, *<sup>x</sup>* <sup>≤</sup> *<sup>y</sup>*.
*Proof.* (i) Let *α* ∈ **R**. Then we have
$$f(\lambda \otimes \mathbf{x} \oplus \mu \otimes \mathbf{y}) = f^T \otimes \lambda \otimes \mathbf{x} \oplus f^T \otimes \mu \otimes \mathbf{y}$$
$$= \lambda \otimes f^T \otimes \mathbf{x} \oplus \mu \otimes f^T \otimes \mathbf{y}$$
$$= \lambda \otimes f(\mathbf{x}) \oplus \mu \otimes f(\mathbf{y})$$
and the statement now follows.
(ii) Let *<sup>x</sup>*, *<sup>y</sup>* <sup>∈</sup> **<sup>R</sup>***<sup>n</sup>* such that *<sup>x</sup>* <sup>≤</sup> *<sup>y</sup>*. Since *<sup>x</sup>* <sup>≤</sup> *<sup>y</sup>*, we have
$$\max(\mathbf{x}) \le \max(y)$$
$$\iff f^T \otimes \mathbf{x} \le f^T \otimes y, \text{ for any } f \in \mathbb{R}^n$$
$$\iff \quad f(\mathbf{x}) \le f(y).$$
Note that it would be possible to convert equations to inequalities and conversely but this would result in an increase of the number of constraints or variables and thus increasing the computational complexity. The method we present here does not require any new constraint or variable.
We denote by
$$(A \otimes \mathfrak{x})\_{\mathfrak{i}} = \max\_{\mathfrak{j} \in N} (a\_{\mathfrak{i}\mathfrak{j}} + \mathfrak{x}\_{\mathfrak{j}})$$
A variable *xj* will be called *active* if *xj* = *f*(*x*), for some *j* ∈ *N*. Also, a variable will be called *active* on the constraint equation if the value (*A* ⊗ *x*)*<sup>i</sup>* is attained at the term *xj* respectively. It follows from Theorem 5 and Lemma 3 that *<sup>x</sup>*ˆ(*A*, *<sup>C</sup>*, *<sup>b</sup>*, *<sup>d</sup>*) <sup>∈</sup> *<sup>S</sup>*max(*A*, *<sup>C</sup>*, *<sup>b</sup>*, *<sup>d</sup>*). We now present a polynomial algorithm which finds *<sup>x</sup>* <sup>∈</sup> *<sup>S</sup>*min(*A*, *<sup>C</sup>*, *<sup>b</sup>*, *<sup>d</sup>*) or recognizes that *<sup>S</sup>*min(*A*, *<sup>B</sup>*, *<sup>c</sup>*, *<sup>d</sup>*) = <sup>∅</sup>. Due to Theorem 4 either *x*ˆ(*A*, *C*, *b*, *d*) ∈ *S*(*A*, *C*, *b*, *d*) or *S*(*A*, *C*, *b*, *d*) = ∅. Therefore, we assume in the following algorithm that *<sup>S</sup>*(*A*, *<sup>C</sup>*, *<sup>b</sup>*, *<sup>d</sup>*) �<sup>=</sup> <sup>∅</sup> and also *<sup>S</sup>*min(*A*, *<sup>C</sup>*, *<sup>b</sup>*, *<sup>d</sup>*) �<sup>=</sup> <sup>∅</sup>.
**Theorem 8.** The algorithm ONEMLP-EI is correct and its computational complexity is *O*((*k* + *r*)*n*2).
#### **Algorithm 1** ONEMLP-EI(Max-linear program with one-sided equations and inequalities)
**Input:** *<sup>f</sup>* = (*f*1, *<sup>f</sup>*2, ..., *fn*)*<sup>T</sup>* <sup>∈</sup> **<sup>R</sup>***n*, *<sup>b</sup>* = (*b*1, *<sup>b</sup>*2, ...*bk*)*<sup>T</sup>* <sup>∈</sup> **<sup>R</sup>***k*, *<sup>d</sup>* = (*d*1, *<sup>d</sup>*2, ...*dr*)*<sup>T</sup>* <sup>∈</sup> **<sup>R</sup>***r*, *<sup>A</sup>* = (*aij*) <sup>∈</sup> *<sup>R</sup>k*×*<sup>n</sup>* and *<sup>C</sup>* = (*cij*) <sup>∈</sup> *<sup>R</sup>r*×*n*. **Output:** *<sup>x</sup>* <sup>∈</sup> *<sup>S</sup>*min(*A*, *<sup>C</sup>*, *<sup>b</sup>*, *<sup>d</sup>*). 1. Find *x*¯(*A*, *b*), *x*¯(*C*, *d*), *x*ˆ(*A*, *C*, *b*, *d*) and *Kj*, *j* ∈ *J*;*J* = {*j* ∈ *N*; *x*¯*j*(*C*, *d*) ≥ *x*¯*j*(*A*, *b*)} 2. *x* := *x*ˆ(*A*, *C*, *b*, *d*) 3. *H*(*x*) := {*j* ∈ *N*; *fj* + *xj* = *f*(*x*)} 4. *J* := *J* \ *H*(*x*) 5. If � *j*∈*J Kj* �= *K*
then stop (*<sup>x</sup>* <sup>∈</sup> *<sup>S</sup>*min(*A*, *<sup>C</sup>*, *<sup>b</sup>*, *<sup>d</sup>*))
14 Will-be-set-by-IN-TECH
Suppose that the vector *<sup>f</sup>* = (*f*1, *<sup>f</sup>*2, ..., *fn*)*<sup>T</sup>* <sup>∈</sup> **<sup>R</sup>***<sup>n</sup>* is given. The task of minimizing [maximizing]the function *<sup>f</sup>*(*x*) = *<sup>f</sup> <sup>T</sup>* <sup>⊗</sup> *<sup>x</sup>* <sup>=</sup> max(*f*<sup>1</sup> <sup>+</sup> *<sup>x</sup>*1, *<sup>f</sup>*<sup>1</sup> <sup>+</sup> *<sup>x</sup>*2..., *fn* <sup>+</sup> *xn*) subject to (12) is called max-linear program with one-sided equations and inequalities and will be denoted
<sup>≤</sup> ]. We denote the sets of optimal solutions by *<sup>S</sup>*min(*A*, *<sup>C</sup>*, *<sup>b</sup>*, *<sup>d</sup>*) and
<sup>=</sup> *<sup>λ</sup>* <sup>⊗</sup> *<sup>f</sup> <sup>T</sup>* <sup>⊗</sup> *<sup>x</sup>* <sup>⊕</sup> *<sup>μ</sup>* <sup>⊗</sup> *<sup>f</sup> <sup>T</sup>* <sup>⊗</sup> *<sup>y</sup>*
= *λ* ⊗ *f*(*x*) ⊕ *μ* ⊗ *f*(*y*)
**5. Max-linear program with equation and inequality constraints**
**Lemma 3.** Suppose *<sup>f</sup>* <sup>∈</sup> **<sup>R</sup>***<sup>n</sup>* and let *<sup>f</sup>*(*x*) = *<sup>f</sup> <sup>T</sup>* <sup>⊗</sup> *<sup>x</sup>* be defined on **<sup>R</sup>***n*. Then,
(ii) *<sup>f</sup>*(*x*) is isotone, i.e. *<sup>f</sup>*(*x*) <sup>≤</sup> *<sup>f</sup>*(*y*) for every *<sup>x</sup>*, *<sup>y</sup>* <sup>∈</sup> **<sup>R</sup>***n*, *<sup>x</sup>* <sup>≤</sup> *<sup>y</sup>*.
(ii) Let *<sup>x</sup>*, *<sup>y</sup>* <sup>∈</sup> **<sup>R</sup>***<sup>n</sup>* such that *<sup>x</sup>* <sup>≤</sup> *<sup>y</sup>*. Since *<sup>x</sup>* <sup>≤</sup> *<sup>y</sup>*, we have
(i) *<sup>f</sup>*(*x*) is max-linear, i.e. *<sup>f</sup>*(*<sup>λ</sup>* <sup>⊗</sup> *<sup>x</sup>* <sup>⊕</sup> *<sup>μ</sup>* <sup>⊗</sup> *<sup>y</sup>*) = *<sup>λ</sup>* <sup>⊗</sup> *<sup>f</sup>*(*x*) <sup>⊕</sup> *<sup>μ</sup>* <sup>⊗</sup> *<sup>f</sup>*(*y*) for every *<sup>x</sup>*, *<sup>y</sup>* <sup>∈</sup> **<sup>R</sup>***n*.
max(*x*) ≤ max(*y*)
⇐⇒ *f*(*x*) ≤ *f*(*y*).
*<sup>f</sup>*(*<sup>λ</sup>* <sup>⊗</sup> *<sup>x</sup>* <sup>⊕</sup> *<sup>μ</sup>* <sup>⊗</sup> *<sup>y</sup>*) = *<sup>f</sup> <sup>T</sup>* <sup>⊗</sup> *<sup>λ</sup>* <sup>⊗</sup> *<sup>x</sup>* <sup>⊕</sup> *<sup>f</sup> <sup>T</sup>* <sup>⊗</sup> *<sup>μ</sup>* <sup>⊗</sup> *<sup>y</sup>*
⇐⇒ *<sup>f</sup> <sup>T</sup>* <sup>⊗</sup> *<sup>x</sup>* <sup>≤</sup> *<sup>f</sup> <sup>T</sup>* <sup>⊗</sup> *<sup>y</sup>*, for any, *<sup>f</sup>* <sup>∈</sup> **<sup>R</sup>***<sup>n</sup>*
Note that it would be possible to convert equations to inequalities and conversely but this would result in an increase of the number of constraints or variables and thus increasing the computational complexity. The method we present here does not require any new constraint
A variable *xj* will be called *active* if *xj* = *f*(*x*), for some *j* ∈ *N*. Also, a variable will be called *active* on the constraint equation if the value (*A* ⊗ *x*)*<sup>i</sup>* is attained at the term *xj* respectively. It follows from Theorem 5 and Lemma 3 that *<sup>x</sup>*ˆ(*A*, *<sup>C</sup>*, *<sup>b</sup>*, *<sup>d</sup>*) <sup>∈</sup> *<sup>S</sup>*max(*A*, *<sup>C</sup>*, *<sup>b</sup>*, *<sup>d</sup>*). We now present a polynomial algorithm which finds *<sup>x</sup>* <sup>∈</sup> *<sup>S</sup>*min(*A*, *<sup>C</sup>*, *<sup>b</sup>*, *<sup>d</sup>*) or recognizes that *<sup>S</sup>*min(*A*, *<sup>B</sup>*, *<sup>c</sup>*, *<sup>d</sup>*) = <sup>∅</sup>. Due to Theorem 4 either *x*ˆ(*A*, *C*, *b*, *d*) ∈ *S*(*A*, *C*, *b*, *d*) or *S*(*A*, *C*, *b*, *d*) = ∅. Therefore, we assume
**Theorem 8.** The algorithm ONEMLP-EI is correct and its computational complexity is *O*((*k* +
*<sup>j</sup>*∈*<sup>N</sup>* (*aij* <sup>+</sup> *xj*)
(*A* ⊗ *x*)*<sup>i</sup>* = max
in the following algorithm that *<sup>S</sup>*(*A*, *<sup>C</sup>*, *<sup>b</sup>*, *<sup>d</sup>*) �<sup>=</sup> <sup>∅</sup> and also *<sup>S</sup>*min(*A*, *<sup>C</sup>*, *<sup>b</sup>*, *<sup>d</sup>*) �<sup>=</sup> <sup>∅</sup>.
by MLPmin
*Proof.*
or variable. We denote by
*r*)*n*2).
<sup>≤</sup> and [MLPmax
*S*max(*A*, *C*, *b*, *d*), respectively.
(i) Let *α* ∈ **R**. Then we have
and the statement now follows.
*Proof.* The correctness follows from Theorem 5 and the computational complexity is computed as follows. In Step 1 *x*¯(*A*, *b*) is *O*(*kn*), while *x*¯(*C*, *d*), *x*ˆ(*A*, *C*, *b*, *d*) and *Kj* can be determined in *O*(*rn*), *O*(*k* + *r*)*n* and *O*(*kn*) respectively. The loop 3-7 can be repeated at most *n* − 1 times, since the number of elements in *J* is at most *n* and in Step 4 at least one element will be removed at a time. Step 3 is *O*(*n*), Step 6 is *O*(*kn*) and Step 7 is *O*(*n*). Hence loop 3-7 is *O*(*kn*2).
#### **5.1. An example**
Consider the following system max-linear program in which *<sup>f</sup>* = (5, 6, 1, 4, <sup>−</sup>1)*T*,
$$A = \begin{pmatrix} 3 \ 8 \ 4 \ 0 \ 1 \\ 0 \ 6 \ 2 \ 2 \ 1 \\ 0 \ 1 \ -2 \ 4 \ 8 \end{pmatrix}, b = \begin{pmatrix} 7 \\ 5 \\ 7 \end{pmatrix}.$$
$$\mathbf{C} = \begin{pmatrix} -1 \ 2 \ -3 \ 0 \ 6 \\ 3 \ 4 \ -2 \ 2 \ 1 \\ 1 \ 3 \ -2 \ 3 \ 4 \end{pmatrix} \text{ and } d = \begin{pmatrix} 5 \\ 5 \\ 6 \end{pmatrix}.$$
We now make a record run of Algorithm ONEMLP-EI. *<sup>x</sup>*¯(*A*, *<sup>b</sup>*)=(5, <sup>−</sup>1, 3, 3, <sup>−</sup>1)*T*, *<sup>x</sup>*¯(*C*, *<sup>d</sup>*) = (2, 1, 7, 3, <sup>−</sup>1)*T*, *<sup>x</sup>*ˆ(*A*, *<sup>C</sup>*, *<sup>b</sup>*, *<sup>d</sup>*)=(2, <sup>−</sup>1, 3, 3, <sup>−</sup>1)*T*, *<sup>J</sup>* <sup>=</sup> {2, 3, 4, 5} and *<sup>K</sup>*<sup>2</sup> <sup>=</sup> {1, 2}, *<sup>K</sup>*<sup>3</sup> <sup>=</sup> {1, 2}, *<sup>K</sup>*<sup>4</sup> <sup>=</sup> {2, 3} and *<sup>K</sup>*<sup>5</sup> <sup>=</sup> {3}. *<sup>x</sup>* :<sup>=</sup> *<sup>x</sup>*ˆ(*A*, *<sup>C</sup>*, *<sup>b</sup>*, *<sup>d</sup>*)=(2, <sup>−</sup>1, 3, 3, <sup>−</sup>1)*<sup>T</sup>* and *<sup>H</sup>*(*x*) = {1, 4} and *J* �⊆ *H*(*x*). We also have *J* := *J* \ *H*(*x*) = {2, 3, 5} and *K*<sup>2</sup> ∪ *K*<sup>3</sup> ∪ *K*<sup>5</sup> = *K*. Then set *x*<sup>1</sup> = *x*<sup>4</sup> = <sup>10</sup>−<sup>4</sup> (say) and *<sup>x</sup>* = (10−4, <sup>−</sup>1, 3, 10−4, <sup>−</sup>1)*T*. Now *<sup>H</sup>*(*x*) = {2} and *<sup>J</sup>* :<sup>=</sup> *<sup>J</sup>* \ *<sup>H</sup>*(*x*) = {3, 5}. Since *<sup>K</sup>*<sup>3</sup> <sup>∪</sup> *<sup>K</sup>*<sup>5</sup> <sup>=</sup> *<sup>K</sup>* set *<sup>x</sup>*<sup>2</sup> <sup>=</sup> <sup>10</sup>−4(say) and we have *<sup>x</sup>* = (10−4, 10−4, 3, 10−4, <sup>−</sup>1)*T*. Now *H*(*x*) = {3} and *J* := *J* \ *H*(*x*) = {5}. Since *K*<sup>5</sup> �= *K* then we stop and an optimal solution is *<sup>x</sup>* = (10−4, 10−4, 3, 10−4, <sup>−</sup>1)*<sup>T</sup>* and *<sup>f</sup>* min <sup>=</sup> 4.
#### **6. A special case of max-linear program with two-sided constraints**
Suppose *<sup>c</sup>* = (*c*1, *<sup>c</sup>*2, ..., *cm*)*T*, *<sup>d</sup>* = (*d*1, *<sup>d</sup>*2, ..., *dm*)*<sup>T</sup>* <sup>∈</sup> **<sup>R</sup>***m*, *<sup>A</sup>* = (*aij*) and *<sup>B</sup>* = (*bij*) <sup>∈</sup> **<sup>R</sup>***m*×*<sup>n</sup>* are given matrices and vectors. The system
$$A \otimes \mathfrak{x} \oplus \mathfrak{c} = \mathfrak{B} \otimes \mathfrak{x} \oplus \mathfrak{d} \tag{14}$$
is called non-homogeneous two-sided max-linear system and the set of solutions of this system will be denoted by *S*. Two-sided max-linear systems have been studied in [20], [21], [22] and [23].
Optimization problems whose objective function is max-linear and constraint (14) are called max-linear programs (MLP). Max-linear programs are studied in [24] and solution methods for both minimization and maximization problems were developed. The methods are proved to be pseudopolynomial if all entries are integer. Also non-linear programs with max-linear constraints were dealt with in [25], where heuristic methods were develeoped and tested for a number of instances.
Consider max-linear programs with two-sided constraints (minimization), MLPmin
$$\begin{aligned} f(\mathbf{x}) &= f^T \otimes \mathbf{x} \longrightarrow \min\\ \text{subject to} \\ A \otimes \mathbf{x} \oplus \mathbf{c} &= B \otimes \mathbf{x} \oplus d \end{aligned} \tag{15}$$
where *<sup>f</sup>* = (*f*1,..., *fn*)*<sup>T</sup>* <sup>∈</sup> **<sup>R</sup>***n*, *<sup>c</sup>* = (*c*1,..., *cm*)*T*, *<sup>d</sup>* = (*d*1,..., *dm*)*<sup>T</sup>* <sup>∈</sup> **<sup>R</sup>***m*, *<sup>A</sup>* = (*aij*) and *<sup>B</sup>* = (*bij*) <sup>∈</sup> **<sup>R</sup>***m*×*<sup>n</sup>* are given matrices and vectors. We introduce the following:
$$\begin{aligned} y &= (f\_1 \otimes \mathfrak{x}\_1, f\_2 \otimes \mathfrak{x}\_2, \dots, f\_n \otimes \mathfrak{x}\_n) \\ &= \text{diag}(f) \otimes \mathfrak{x} \end{aligned} \tag{16}$$
diag(*f*) means a diagonal matrix whose diagonal elements are *f*1, *f*2, ..., *fn* and off diagonal elements are −∞. It therefore follows from (16) that
$$\begin{aligned} f^T \otimes \mathbf{x} &= \mathbf{0}^T \otimes \mathbf{y} \\ \iff \mathbf{x} &= (f\_1^{-1} \otimes y\_1, f\_2^{-1} \otimes y\_2, \dots, f\_n^{-1} \otimes y\_n) \\ &= (\text{diag}(f))^{-1} \otimes \mathbf{y} \end{aligned} \tag{17}$$
Hence, by substituting (16) and (17) into (15) we have
$$\begin{aligned} \mathbf{0}^T \otimes \mathbf{y} &\longrightarrow \min\\ \text{subject to} \\ \mathbf{A}^\prime \otimes \mathbf{y} \oplus \mathbf{c} &= \mathbf{B}^\prime \otimes \mathbf{y} \oplus \mathbf{d}\_\prime \end{aligned} \tag{18}$$
where 0*<sup>T</sup>* is transpose of the zero vector, *A*� <sup>=</sup> *<sup>A</sup>* <sup>⊗</sup> (diag(*f*))−<sup>1</sup> and *<sup>B</sup>* � <sup>=</sup> *<sup>B</sup>* <sup>⊗</sup> (diag(*f*))−<sup>1</sup> Therefore we assume without loss of generality that *f* = 0 and hence (15) is equivalent to
$$f(\mathbf{x}) = \sum\_{j=1,\ldots,n} {}^{\oplus}\mathbf{x}\_{j} \longrightarrow \min$$
$\text{subject to}$
$$\mathcal{A}\otimes\mathfrak{x}\oplus\mathfrak{c}=\mathcal{B}\otimes\mathfrak{x}\oplus\mathfrak{d}$$
The set of feasible solutions for (19) will be denoted by *S* and the set of optimal solutions by *<sup>S</sup>*min. A vector is called *constant* if all its components are equal. That is a vector *<sup>x</sup>* <sup>∈</sup> **<sup>R</sup>***<sup>n</sup>* is constant if *x*<sup>1</sup> = *x*<sup>2</sup> = ··· = *xn*. For any *x* ∈ *S* we define the set *Q*(*x*) = {*i* ∈ *M*;(*A* ⊗ *x*)*<sup>i</sup>* > *ci*}. We introduce the following notation of matrices. Let *<sup>A</sup>* = (*aij*) <sup>∈</sup> *<sup>R</sup>m*×*n*, 1 <sup>≤</sup> *<sup>i</sup>*<sup>1</sup> <sup>&</sup>lt; *<sup>i</sup>*<sup>2</sup> <sup>&</sup>lt; ··· <sup>&</sup>lt; *iq* ≤ *m* and 1 ≤ *j*<sup>1</sup> < *j*<sup>2</sup> < ··· < *jr* ≤ *n*. Then,
$$A\begin{pmatrix} i\_{1\prime}i\_{2\prime}\dots\iota\_{\prime}i\_{\dot{q}}\\j\_{1\prime}j\_{2\prime}\dots\iota\_{\prime\dot{r}} \end{pmatrix} = \begin{pmatrix} a\_{i\_1j\_1}a\_{i\_1j\_2}\dots a\_{i\_1j\_r}\\a\_{i\_2j\_1}a\_{i\_2j\_2}\dots a\_{i\_2j\_r}\\\dots\\a\_{i\_qj\_1}a\_{i\_qj\_2}\dots a\_{i\_qj\_r} \end{pmatrix} = A(Q\_\prime R)$$
where, *Q* = {*i*1,..., *iq*}, *R* = {*j*1,..., *jr*}. Similar notation is used for vectors *c*(*i*1,..., *ir*) = (*ci*<sup>1</sup> ... *cir* )*<sup>T</sup>* <sup>=</sup> *<sup>c</sup>*(*R*). Given MLPmin with *<sup>c</sup>* <sup>≥</sup> *<sup>d</sup>*, we define the following sets
$$\begin{aligned} M^=&=\{i \in M; c\_i = d\_i\} \text{ and} \\ M^>&=\{i \in M; c\_i > d\_i\} \end{aligned}$$
We also define the following matrices:
$$\begin{aligned} A\_=&=A(M^\equiv,N), A\_> = A(M^>,N) \\ B\_=&=B(M^\equiv,N), B\_> = B(M^>,N) \\ c\_=&=c(M^\equiv), c\_> = c(M^>) \end{aligned} \tag{20}$$
An easily solvable case arises when there is a constant vector *x* ∈ *S* such that the set *Q*(*x*) = ∅. This constant vector *x* satisfies the following equations and inequalities
$$\begin{aligned} A &= \otimes \mathfrak{x} \le \mathfrak{c}\_{=} \\ A\_{>} \otimes \mathfrak{x} &\le \mathfrak{c}\_{>} \\ B\_{=} \otimes \mathfrak{x} &\le \mathfrak{c}\_{=} \\ B\_{>} \otimes \mathfrak{x} &= \mathfrak{c}\_{>} \end{aligned} \tag{21}$$
where *A*=, *A*>, *B*=, *B*>, *c*= and *c*> are defined in (20). The one-sided system of equation and inequalities (21) can be written as
$$\begin{cases} G \otimes \mathfrak{x} = p \\ H \otimes \mathfrak{x} \le q \end{cases} \tag{22}$$
where,
16 Will-be-set-by-IN-TECH
Suppose *<sup>c</sup>* = (*c*1, *<sup>c</sup>*2, ..., *cm*)*T*, *<sup>d</sup>* = (*d*1, *<sup>d</sup>*2, ..., *dm*)*<sup>T</sup>* <sup>∈</sup> **<sup>R</sup>***m*, *<sup>A</sup>* = (*aij*) and *<sup>B</sup>* = (*bij*) <sup>∈</sup> **<sup>R</sup>***m*×*<sup>n</sup>* are
is called non-homogeneous two-sided max-linear system and the set of solutions of this system will be denoted by *S*. Two-sided max-linear systems have been studied in [20], [21],
Optimization problems whose objective function is max-linear and constraint (14) are called max-linear programs (MLP). Max-linear programs are studied in [24] and solution methods for both minimization and maximization problems were developed. The methods are proved to be pseudopolynomial if all entries are integer. Also non-linear programs with max-linear constraints were dealt with in [25], where heuristic methods were develeoped and tested for
*<sup>f</sup>*(*x*) = *<sup>f</sup> <sup>T</sup>* <sup>⊗</sup> *<sup>x</sup>* −→ min
*A* ⊗ *x* ⊕ *c* = *B* ⊗ *x* ⊕ *d*
where *<sup>f</sup>* = (*f*1,..., *fn*)*<sup>T</sup>* <sup>∈</sup> **<sup>R</sup>***n*, *<sup>c</sup>* = (*c*1,..., *cm*)*T*, *<sup>d</sup>* = (*d*1,..., *dm*)*<sup>T</sup>* <sup>∈</sup> **<sup>R</sup>***m*, *<sup>A</sup>* = (*aij*) and
*y* = (*f*<sup>1</sup> ⊗ *x*1, *f*<sup>2</sup> ⊗ *x*2,..., *fn* ⊗ *xn*)
diag(*f*) means a diagonal matrix whose diagonal elements are *f*1, *f*2, ..., *fn* and off diagonal
<sup>−</sup><sup>1</sup> <sup>⊗</sup> *<sup>y</sup>*
�
⊗ *y* ⊕ *d*,
*xj* −→ min
<sup>=</sup> *<sup>A</sup>* <sup>⊗</sup> (diag(*f*))−<sup>1</sup> and *<sup>B</sup>*
�
<sup>=</sup> *<sup>B</sup>* <sup>⊗</sup> (diag(*f*))−<sup>1</sup>
<sup>2</sup> <sup>⊗</sup> *<sup>y</sup>*2,..., *<sup>f</sup>* <sup>−</sup><sup>1</sup> *<sup>n</sup>* <sup>⊗</sup> *yn*)
<sup>1</sup> <sup>⊗</sup> *<sup>y</sup>*1, *<sup>f</sup>* <sup>−</sup><sup>1</sup>
<sup>0</sup>*<sup>T</sup>* <sup>⊗</sup> *<sup>y</sup>* −→ min subject to
⊗ *y* ⊕ *c* = *B*
Therefore we assume without loss of generality that *f* = 0 and hence (15) is equivalent to
⊕
*j*=1,...,*n*
*A* ⊗ *x* ⊕ *c* = *B* ⊗ *x* ⊕ *d*
*f*(*x*) = ∑
subject to
Consider max-linear programs with two-sided constraints (minimization), MLPmin
subject to
*<sup>B</sup>* = (*bij*) <sup>∈</sup> **<sup>R</sup>***m*×*<sup>n</sup>* are given matrices and vectors. We introduce the following:
= diag(*f*) ⊗ *x*
= (diag(*f*))
*A*�
*<sup>f</sup> <sup>T</sup>* <sup>⊗</sup> *<sup>x</sup>* <sup>=</sup> <sup>0</sup>*<sup>T</sup>* <sup>⊗</sup> *<sup>y</sup>* ⇐⇒ *<sup>x</sup>* = (*<sup>f</sup>* <sup>−</sup><sup>1</sup>
elements are −∞. It therefore follows from (16) that
Hence, by substituting (16) and (17) into (15) we have
where 0*<sup>T</sup>* is transpose of the zero vector, *A*�
*A* ⊗ *x* ⊕ *c* = *B* ⊗ *x* ⊕ *d* (14)
(15)
(16)
(17)
(18)
(19)
**6. A special case of max-linear program with two-sided constraints**
given matrices and vectors. The system
[22] and [23].
a number of instances.
$$\begin{aligned} G = (B\_{>})\_{\prime} \; H &= \begin{pmatrix} A\_{=} \\ A\_{>} \\ B\_{=} \end{pmatrix} \\ p = c\_{>} \quad \text{and} \quad q = \begin{pmatrix} c\_{=} \\ c\_{>} \\ c\_{=} \end{pmatrix} \end{aligned} \tag{23}$$
Recall that *S*(*G*, *H*, *p*, *q*) is the set of solutions for (22).
**Theorem 9.** Let *<sup>Q</sup>*(*x*) = <sup>∅</sup> for some constant vector *<sup>x</sup>* = (*α*,..., *<sup>α</sup>*)*<sup>T</sup>* <sup>∈</sup> *<sup>S</sup>*. If *<sup>z</sup>* <sup>∈</sup> *<sup>S</sup>*min then *z* ∈ *S*(*G*, *H*, *p*, *q*).
*Proof.* Let *<sup>x</sup>* = (*α*,..., *<sup>α</sup>*)*<sup>T</sup>* <sup>∈</sup> *<sup>S</sup>*. Suppose *<sup>Q</sup>*(*z*) = <sup>∅</sup> and *<sup>z</sup>* <sup>∈</sup> *<sup>S</sup>*min. This implies that *<sup>f</sup>*(*z*) <sup>≤</sup> *f*(*x*) = *α*. Therefore we have, ∀*j* ∈ *N*, *z* ≤ *α*. Consequently, *z* ≤ *x* and (*A* ⊗ *z*)*<sup>i</sup>* ≤ (*A* ⊗ *x*)*<sup>i</sup>* for all *i* ∈ *M*. Since, *Q*(*z*) = ∅ and *z* ∈ *S*(*G*, *H*, *p*, *q*).
**Corollary 5.** If *<sup>Q</sup>*(*x*) = <sup>∅</sup> for some constant vector *<sup>x</sup>* <sup>∈</sup> *<sup>S</sup>* then *<sup>S</sup>*min <sup>⊆</sup> *<sup>S</sup>*min(*G*, *<sup>H</sup>*, *<sup>p</sup>*, *<sup>q</sup>*).
*Proof.* The statement follows from Theorem 9.
## **7. Some solvability concepts of a linear system containing of both equations and inequalities**
System of max-separable linear equations and inequalities arise frequently in several branches of Applied Mathematics: for instance in the description of discrete-event dynamic system [1, 4] and machine scheduling [10]. However, choosing unsuitable values for the matrix entries and right-handside vectors may lead to unsolvable systems. Therefore, methods for restoring solvability suggested in the literature could be employed. These methods include modifying the input data [11, 26] or dropping some equations [11]. Another possibility is to replace each entry by an interval of possible values. In doing so, our question will be shifted to asking about weak solvability, strong solvability and control solvability.
Interval mathematics was championed by Moore [27] as a tool for bounding errors in computer programs. The area has now been developed in to a general methodology for investigating numerical uncertainty in several problems. System of interval equations and inequalities in max-algebra have each been studied in the literature. In [26] weak and strong solvability of interval equations were discussed, control sovability, weak control solvability and universal solvability have been dealt with in [28]. In [29] a system of linear inequality with interval coefficients was discussed. In this section we consider a system consisting of interval linear equations and inequalities and present solvability concepts for such system.
An algebraic structure (*B*, ⊕, ⊗) with two binary operations ⊕ and ⊗ is called max-plus algebra if
$$B = \mathbb{R} \cup \{ -\infty \}, \ a \oplus b = \max\{a, b\}, \ a \otimes b = a + b$$
for any *a*, *b* ∈ **R**.
Let *m*, *n*,*r* be given positive integers and *a* ∈ **R**, we use throughout the paper the notation *<sup>M</sup>* <sup>=</sup> {1, 2, ..., *<sup>m</sup>*}, *<sup>N</sup>* <sup>=</sup> {1, 2, ..., *<sup>n</sup>*}, *<sup>R</sup>* <sup>=</sup> {1, 2, ...,*r*} and *<sup>a</sup>*−<sup>1</sup> <sup>=</sup> <sup>−</sup>*a*. The set of all *<sup>m</sup>* <sup>×</sup> *<sup>n</sup>*, *<sup>r</sup>* <sup>×</sup> *<sup>n</sup>* matrices over *B* is denoted by *B*(*m*, *n*) and *B*(*r*, *n*) respectively. The set of all *n*-dimensional vectors is denoted by *B*(*n*). Then for each matrix *A* ∈ *B*(*n*, *m*) and vector *x* ∈ *B*(*n*) the product *A* ⊗ *x* is define as
$$(A \otimes \mathfrak{x}) = \max\_{j \in N} \left( a\_{ij} + \mathfrak{x}\_j \right).$$
For a given matrix interval *A* = [*A*, *A*] with *A*, *A* ∈ *B*(*k*, *n*), *A* ≤ *A* and given vector interval *b* = [*b*, *b*] with *b*, *b* ∈ *B*(*n*), *b* ≤ *b* the notation
$$A \otimes \mathfrak{x} = \mathfrak{b} \tag{24}$$
represents an interval system of linear max-separable equations of the form
$$A \circledast \mathfrak{x} = b \tag{25}$$
Similarly, for a given matrix interval *C* = [*A*, *A*] with *C*, *C* ∈ *B*(*r*, *n*), *C* ≤ *C* and given vector interval *d* = [*d*, *d*] with *d*, *d* ∈ *B*(*n*), *b* ≤ *b* the notation
$$\mathcal{C} \otimes \mathfrak{x} \le d \tag{26}$$
represents an interval system of linear max-separable inequalities of the form
$$
\mathbb{C} \otimes \mathfrak{x} \le d \tag{27}
$$
Interval system of linear max-separable equations and inequalities have each been studied in the literature, for more information the reader is reffered to . The following notation
$$\begin{aligned} \mathbf{A} \otimes \mathbf{x} &= \mathbf{b} \\ \mathbf{C} \otimes \mathbf{x} &\le \mathbf{d} \end{aligned} \tag{28}$$
represents an interval system of linear max-separable equations and inequalities of the form
$$\begin{aligned} A \otimes \mathfrak{x} &= b \\ \mathfrak{C} \otimes \mathfrak{x} &\le d \end{aligned} \tag{29}$$
where *A* ∈ *A*, *C* ∈ *C*, *b* ∈ *b* and *d* ∈ *d*.
The aim of this section is to consider a system consisting of max-separable linear equations and inequalities and presents some solvability conditions of such system. Note that it is possible to convert equations to inequalities and conversely, but this would result in an increase in the number of equations and inequalities or an increase in the number of unknowns thus increasing the computational complexity when testing the solvability conditions. Each system of the form (29) is said to be a subsystem of (28). An interval system (29) has constant matrices if *A* = *A* and *C* = *C*. Similarly, an interval system has constant right hand side if *b* = *b* and *d* = *d*. In what follows we will consider *A* ∈ **R**(*k*, *n*) and *C* ∈ **R**(*r*, *n*).
#### **7.1. Weak solvability**
**Definition 8.** A vector *y* is a weak solution to an interval system (29) if there exists *A* ∈ *A*, *C* ∈ *C*, *b* ∈ *b* and *d* ∈ *d* such that
$$\begin{aligned} A \otimes y &= b \\ \mathbb{C} \otimes y &\le d \end{aligned} \tag{30}$$
**Theorem 10.** A vector *<sup>x</sup>* <sup>∈</sup> **<sup>R</sup>***<sup>n</sup>* is a weak solution of (29) if and only if
$$\mathfrak{x} = \mathfrak{x}\left(\frac{\underline{A}}{\underline{C}}\frac{\overline{b}}{\overline{d}}\right).$$
and
18 Will-be-set-by-IN-TECH
**Theorem 9.** Let *<sup>Q</sup>*(*x*) = <sup>∅</sup> for some constant vector *<sup>x</sup>* = (*α*,..., *<sup>α</sup>*)*<sup>T</sup>* <sup>∈</sup> *<sup>S</sup>*. If *<sup>z</sup>* <sup>∈</sup> *<sup>S</sup>*min then
*Proof.* Let *<sup>x</sup>* = (*α*,..., *<sup>α</sup>*)*<sup>T</sup>* <sup>∈</sup> *<sup>S</sup>*. Suppose *<sup>Q</sup>*(*z*) = <sup>∅</sup> and *<sup>z</sup>* <sup>∈</sup> *<sup>S</sup>*min. This implies that *<sup>f</sup>*(*z*) <sup>≤</sup> *f*(*x*) = *α*. Therefore we have, ∀*j* ∈ *N*, *z* ≤ *α*. Consequently, *z* ≤ *x* and (*A* ⊗ *z*)*<sup>i</sup>* ≤ (*A* ⊗ *x*)*<sup>i</sup>*
System of max-separable linear equations and inequalities arise frequently in several branches of Applied Mathematics: for instance in the description of discrete-event dynamic system [1, 4] and machine scheduling [10]. However, choosing unsuitable values for the matrix entries and right-handside vectors may lead to unsolvable systems. Therefore, methods for restoring solvability suggested in the literature could be employed. These methods include modifying the input data [11, 26] or dropping some equations [11]. Another possibility is to replace each entry by an interval of possible values. In doing so, our question will be shifted to asking
Interval mathematics was championed by Moore [27] as a tool for bounding errors in computer programs. The area has now been developed in to a general methodology for investigating numerical uncertainty in several problems. System of interval equations and inequalities in max-algebra have each been studied in the literature. In [26] weak and strong solvability of interval equations were discussed, control sovability, weak control solvability and universal solvability have been dealt with in [28]. In [29] a system of linear inequality with interval coefficients was discussed. In this section we consider a system consisting of interval linear equations and inequalities and present solvability concepts for such system.
An algebraic structure (*B*, ⊕, ⊗) with two binary operations ⊕ and ⊗ is called max-plus
*B* = **R** ∪ {−∞}, *a* ⊕ *b* = max{*a*, *b*}, *a* ⊗ *b* = *a* + *b*
Let *m*, *n*,*r* be given positive integers and *a* ∈ **R**, we use throughout the paper the notation *<sup>M</sup>* <sup>=</sup> {1, 2, ..., *<sup>m</sup>*}, *<sup>N</sup>* <sup>=</sup> {1, 2, ..., *<sup>n</sup>*}, *<sup>R</sup>* <sup>=</sup> {1, 2, ...,*r*} and *<sup>a</sup>*−<sup>1</sup> <sup>=</sup> <sup>−</sup>*a*. The set of all *<sup>m</sup>* <sup>×</sup> *<sup>n</sup>*, *<sup>r</sup>* <sup>×</sup> *<sup>n</sup>* matrices over *B* is denoted by *B*(*m*, *n*) and *B*(*r*, *n*) respectively. The set of all *n*-dimensional vectors is denoted by *B*(*n*). Then for each matrix *A* ∈ *B*(*n*, *m*) and vector *x* ∈ *B*(*n*) the product
> *j*∈*N aij* + *xj*
For a given matrix interval *A* = [*A*, *A*] with *A*, *A* ∈ *B*(*k*, *n*), *A* ≤ *A* and given vector interval
*A* ⊗ *x* = *b* (24)
(*A* ⊗ *x*) = max
**Corollary 5.** If *<sup>Q</sup>*(*x*) = <sup>∅</sup> for some constant vector *<sup>x</sup>* <sup>∈</sup> *<sup>S</sup>* then *<sup>S</sup>*min <sup>⊆</sup> *<sup>S</sup>*min(*G*, *<sup>H</sup>*, *<sup>p</sup>*, *<sup>q</sup>*).
**7. Some solvability concepts of a linear system containing of both**
about weak solvability, strong solvability and control solvability.
*z* ∈ *S*(*G*, *H*, *p*, *q*).
algebra if
for any *a*, *b* ∈ **R**.
*A* ⊗ *x* is define as
*b* = [*b*, *b*] with *b*, *b* ∈ *B*(*n*), *b* ≤ *b* the notation
for all *i* ∈ *M*. Since, *Q*(*z*) = ∅ and *z* ∈ *S*(*G*, *H*, *p*, *q*).
*Proof.* The statement follows from Theorem 9.
**equations and inequalities**
$$
\left(\overline{A} \otimes \bar{\underline{x}}\left(\frac{\underline{A}}{\underline{C}} \frac{\overline{b}}{\overline{d}}\right) \geq \underline{b}\right)
$$
*Proof.* Let *<sup>i</sup>* <sup>=</sup> {1, ..., *<sup>m</sup>*} be an arbitrary chosen index and *<sup>x</sup>* = (*x*1, *<sup>x</sup>*2, ..., *xn*)*<sup>T</sup>* <sup>∈</sup> **<sup>R</sup>***<sup>n</sup>* fixed. If *A* ∈ *A* then (*A* ⊗ *x*)*<sup>i</sup>* is isotone and we have
$$[(A \otimes \mathfrak{x})\_{\mathfrak{i}} \in [(\underline{A} \otimes \mathfrak{x})\_{\mathfrak{i}\mathfrak{i}} (\overline{A} \otimes \mathfrak{x})\_{\mathfrak{i}}] \subseteq \mathbb{R}$$
Hence, *x* is a weak solution if and only if
$$[(\underline{A}\otimes\mathfrak{x})\_{i\prime}(\overline{A}\otimes\mathfrak{x})\_{i}]\cap[\underline{b}\_{i\prime}\overline{b}\_{i}]\tag{31}$$
Similarly, if *C* ⊗ *x* ≤ *d* then *x* is obviously a weak solution to
$$\underline{A}\otimes\mathfrak{x}\leq b$$
$$\underline{C}\otimes\mathfrak{x}\leq\overline{d}$$
That is
$$\mathfrak{x} = \mathfrak{x}\left(\frac{\underline{A}}{\underline{C}}\frac{\overline{b}}{\underline{d}}\right).$$
Also from (31) *x* is a weak solution if and only if
$$[(\underline{A}\otimes\mathfrak{x})\_{i\prime}(\overline{A}\otimes\mathfrak{x})\_{\mathfrak{i}}]\cap[\underline{b}\_{i\prime}\overline{b}\_{\mathfrak{i}}]\neq\mathcal{O},\forall\mathfrak{i}=1,2,...,m$$
That is
$$
\overline{A} \otimes \overline{x} \left( \frac{\underline{A}}{\underline{C}} \frac{\overline{b}}{\overline{d}} \right) \geq \underline{b}
$$
**Definition 9.** An interval system (29) is weakly solvable if there exists *A* ∈ *A*, *C* ∈ *C*, *b* ∈ *b* and *d* ∈ *d* such that (29) is solvable.
**Theorem 11.** An interval system (29) with constant matrix *A* = *A* = *A*, *C* = *C* = *C* is weakly solvable if and only if
$$A \otimes \overline{\mathfrak{x}}\left(\begin{matrix} A \ \overline{b} \\ C \ \overline{d} \end{matrix}\right) \geq \underline{b}$$
*Proof.* The (if) part follows from the definition. Conversely, Let
$$A \otimes \mathfrak{F} \begin{pmatrix} A \ b \\ C \ d \end{pmatrix} = b$$
be solvable subsystem for *b* ∈ [*bi*, *bi*]. Then we have
$$A \otimes \mathfrak{x} \begin{pmatrix} A \ \overline{b} \\ \mathcal{C} \ \overline{d} \end{pmatrix} \ge A \otimes \mathfrak{x} \begin{pmatrix} A \ b \\ \mathcal{C} \ d \end{pmatrix} = b \ge \underline{b}$$
#### **7.2. Strong solvability**
**Definition 10.** A vector *x* is a strong solution to an interval system (29) if for each *A* ∈ *A*, *C* ∈ *C* and each *b* ∈ *b*, *d* ∈ *d* there is an *x* ∈ **R** such that (29) holds.
**Theorem 12.** a vector *x* is a strong solution to (29) if and only if it is a solution to
$$E \otimes \mathfrak{x} = f$$
$$\overline{\mathbb{C}} \otimes \mathfrak{x} \le \underline{d}$$
where
20 Will-be-set-by-IN-TECH
*Proof.* Let *<sup>i</sup>* <sup>=</sup> {1, ..., *<sup>m</sup>*} be an arbitrary chosen index and *<sup>x</sup>* = (*x*1, *<sup>x</sup>*2, ..., *xn*)*<sup>T</sup>* <sup>∈</sup> **<sup>R</sup>***<sup>n</sup>* fixed. If
(*A* ⊗ *x*)*<sup>i</sup>* ∈ [(*A* ⊗ *x*)*i*,(*A* ⊗ *x*)*i*] ⊆ **R**
*A* ⊗ *x* ≤ *b*
*C* ⊗ *x* ≤ *d*
*A b C d*
[(*A* ⊗ *x*)*i*,(*A* ⊗ *x*)*i*] ∩ [*bi*, *bi*] �= ∅, ∀*i* = 1, 2, ..., *m*
*A b C d* ≥ *b*
**Definition 9.** An interval system (29) is weakly solvable if there exists *A* ∈ *A*, *C* ∈ *C*, *b* ∈ *b*
**Theorem 11.** An interval system (29) with constant matrix *A* = *A* = *A*, *C* = *C* = *C* is weakly
*A b C d* ≥ *b*
*A b C d* = *b*
≥ *A* ⊗ *x*¯
*A b C d*
= *b* ≥ *b*
*x* = *x*¯
*A* ⊗ *x*¯
*A* ⊗ *x*¯
*A* ⊗ *x*¯
*Proof.* The (if) part follows from the definition. Conversely, Let
be solvable subsystem for *b* ∈ [*bi*, *bi*]. Then we have
*A* ⊗ *x*¯
*A b C d*
[(*A* ⊗ *x*)*i*,(*A* ⊗ *x*)*i*] ∩ [*bi*, *bi*] (31)
(32)
*A* ∈ *A* then (*A* ⊗ *x*)*<sup>i</sup>* is isotone and we have
Hence, *x* is a weak solution if and only if
Also from (31) *x* is a weak solution if and only if
and *d* ∈ *d* such that (29) is solvable.
solvable if and only if
That is
That is
Similarly, if *C* ⊗ *x* ≤ *d* then *x* is obviously a weak solution to
$$E = \begin{pmatrix} \overline{A} \\ \underline{A} \end{pmatrix}, f = \begin{pmatrix} \frac{b}{b} \\ \overline{b} \end{pmatrix} \tag{33}$$
*Proof.* If *x* is a strong solution of (29), it obviously satisfies (33). Conversely, suppose *x* satisfies (33) and let *<sup>A</sup>*˜ <sup>∈</sup> **<sup>A</sup>**, *<sup>C</sup>*˜ <sup>∈</sup> **<sup>C</sup>**, ˜ *<sup>b</sup>* <sup>∈</sup> **<sup>b</sup>**, ˜*<sup>d</sup>* <sup>∈</sup> **<sup>d</sup>** such that *<sup>A</sup>*˜ <sup>⊗</sup> *<sup>x</sup>* �<sup>=</sup> ˜ *<sup>b</sup>* and *<sup>C</sup>*˜ <sup>⊗</sup> *<sup>x</sup>* <sup>&</sup>gt; ˜*d*. Then <sup>∃</sup>*<sup>i</sup>* <sup>∈</sup> (1, 2, ..., *<sup>m</sup>*) such that either (*A*˜ <sup>⊗</sup> *<sup>x</sup>*)*<sup>i</sup>* <sup>&</sup>lt; ˜ *bi* or (*A*˜ <sup>⊗</sup> *<sup>x</sup>*)*<sup>i</sup>* <sup>&</sup>gt; ˜ *bi* and (*C*˜ <sup>⊗</sup> *<sup>x</sup>*)*<sup>i</sup>* <sup>&</sup>gt; ˜*di*. Therefore, (*<sup>A</sup>* <sup>⊗</sup> *<sup>x</sup>*)*<sup>i</sup>* <sup>&</sup>lt; (*A*˜ <sup>⊗</sup> *<sup>x</sup>*)*<sup>i</sup>* <sup>&</sup>lt; *bi*, (*<sup>A</sup>* <sup>⊗</sup> *<sup>x</sup>*)*<sup>i</sup>* <sup>≥</sup> (*A*˜ <sup>⊗</sup> *<sup>x</sup>*)*<sup>i</sup>* <sup>&</sup>gt; *bi* and (*<sup>C</sup>* <sup>⊗</sup> *<sup>x</sup>*)*<sup>i</sup>* <sup>&</sup>gt; (*C*˜ <sup>⊗</sup> *<sup>x</sup>*)*<sup>i</sup>* <sup>&</sup>gt; *di* and the theorem statement follows.
### **Acknowledgement**
The author is grateful to the Kano University of Science and Technology, Wudil for paying the publication fee.
#### **Author details**
Abdulhadi Aminu
*Department of Mathematics, Kano University of Science and Technology, Wudil, P.M.B 3244, Kano, Nigeria*
#### **8. References**
## **Efficient Model Transition in Adaptive Multi-Resolution Modeling of Biopolymers**
Mohammad Poursina, Imad M. Khan and Kurt S. Anderson
Additional information is available at the end of the chapter
http://dx.doi.org/10.5772/48196
## **1. Introduction**
22 Will-be-set-by-IN-TECH
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Multibody dynamics methods are being used extensively to model biomolecular systems to study important physical phenomena occurring at different spatial and temporal scales [10, 13]. These systems may contain thousands or even millions of degrees of freedom, whereas, the size of the time step involved during the simulation is on the order of femto seconds. Examples of such problems may include proteins, DNAs, and RNAs. These highly complex physical systems are often studied at resolutions ranging from a fully atomistic model to coarse-grained molecules, up to a continuum level system [4, 19, 20]. In studying these problems, it is often desirable to change the system definition during the course of the simulation in order to achieve an optimal combination of accuracy and speed. For example, in order to study the overall conformational motion of a bimolecular system, a model based on super-atoms (beads) [18, 22] or articulated multi-rigid and/or flexible body [21, 23] can be used. Whereas, localized behavior has to be studied using fully atomistic models. In such cases, the need for the transition from a fine-scale to a coarse model and vice versa arises. Illustrations of a fully atomistic model of a molecule, and its coarse-grained model are shown in Fig. (1-a) and Fig. (1-b).
Given the complexity and nonlinearity of challenging bimolecular systems, it is expected that different physical parameters such as dynamic boundary conditions and applied forces will have a significant affect on the behavior of the system. It is shown in [16] that time-invariant coarse models may provide inadequate or poor results and as such, an adaptive framework to model these systems should be considered [14]. Transitions between different system models can be achieved by intelligently removing or adding certain degrees of freedom (*do f*). This change occurs instantaneously and as such, may be viewed as model changes as a result of impulsively applied constraints. For multi-rigid and flexible body systems, the transition from a higher fidelity (fine-scale) model to a lower fidelity model (coarse-scale) using divide-and-conquer algorithm (DCA) has been studied previously in [8, 12]. DCA efficiently provides the unique states of the system after this transition. In this chapter, we focus on the transition from a coarse model to a fine-scale model. When the system is modeled in an articulated multi-flexible-body framework, such transitions may be achieved by two
©2012 Poursina et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ©2012 Poursina et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
(b) Mixed type multibody model
**Figure 1.** Illustration of a biomolecular system. a) Fully atomistic model. b) Coarse grain model with different rigid and flexible sub-domains connected to each other via kinematic joints.
different means. In the first, a fine-scale model is generated by adding flexible *do f* . This type of fine scaling may be necessary in order to capture higher frequency modes. For instance, when two molecules bind together, due to the impact, the higher frequency modes of the system are excited. The second type of fine scaling transition may be achieved through releasing the connecting joints in the multi-flexible-body system. In other words, certain constraints on joints are removed to introduce new *do f* in the model.
In contrast to those types of dynamic systems in which the model definition is persistent, and the total energy of the system is conserved, the class of problems discussed here experiences discontinuous changes in the model definition and hence, the energy of the system must also change (nominally increase) in a discontinuous fashion. During the coarse graining process, based on a predefined metric, one may conclude that naturally existing higher modes are less relevant and can be ignored. As such, the kinetic energy associated with those modes must be estimated and properly accounted for, when transitioning back to the fine-scale model. Moreover, any change in the system model definition is assumed to occur as a result of impulsively applied constraints, without the influence of external loads. As such, the generalized momentum of the system must also be conserved [6]. In other words, the momentum of each differential element projected onto the space of admissible motions permitted by the more restrictive model (whether pre- or post-transition) when integrated over the entire system must be conserved across the model transition. If the generalized momentum is not conserved during the transition, the results are non-physical, and the new initial conditions for the rest of the simulation of the system are invalid.
In the next section, a brief overview of the DCA and analytical preliminaries necessary to the algorithm development are presented. The optimization problem associated with the coarse to fine scale transitioning is discussed next. Then the impulse-momentum formulation for transitioning from coarse models to fine-scale models in the articulated flexible-body scheme is presented. Finally conclusions are made.
### **2. Theoretical background**
2 Will-be-set-by-IN-TECH
(a) Fully atomistic model
(b) Mixed type multibody model
different means. In the first, a fine-scale model is generated by adding flexible *do f* . This type of fine scaling may be necessary in order to capture higher frequency modes. For instance, when two molecules bind together, due to the impact, the higher frequency modes of the system are excited. The second type of fine scaling transition may be achieved through releasing the connecting joints in the multi-flexible-body system. In other words, certain constraints on
In contrast to those types of dynamic systems in which the model definition is persistent, and the total energy of the system is conserved, the class of problems discussed here experiences discontinuous changes in the model definition and hence, the energy of the system must also change (nominally increase) in a discontinuous fashion. During the coarse graining process, based on a predefined metric, one may conclude that naturally existing higher modes are less relevant and can be ignored. As such, the kinetic energy associated with those modes must be estimated and properly accounted for, when transitioning back to the fine-scale model. Moreover, any change in the system model definition is assumed to occur as a result of impulsively applied constraints, without the influence of external loads. As such, the generalized momentum of the system must also be conserved [6]. In other words, the momentum of each differential element projected onto the space of admissible motions permitted by the more restrictive model (whether pre- or post-transition) when integrated over the entire system must be conserved across the model transition. If the generalized
**Figure 1.** Illustration of a biomolecular system. a) Fully atomistic model. b) Coarse grain model with
different rigid and flexible sub-domains connected to each other via kinematic joints.
joints are removed to introduce new *do f* in the model.
In this section, a brief introduction of the basic divide-and-conquer algorithm is presented. The DCA scheme has been developed for the simulation of general multi-rigid and multi-flexible-body systems [5, 8, 9], and systems with discontinuous changes [11, 12]. The basic algorithm described here is independent of the type of problem and is presented so that the chapter might be more self contained. In other words, it can be used to study the behavior of any rigid- and flexible-body system, even if the system undergoes a discontinuous change. Some mathematical preliminaries are also presented in this section which are important to the development of the algorithm.
#### **2.1. Basic divide-and-conquer algorithm**
The basic DCA scheme presented in this chapter works in a similar manner described in detail in [5, 9]. Consider two representative flexible bodies *k* and *k* + 1 connected to each other by a joint *J<sup>k</sup>* as shown in Fig. (2-a). The two points of interest, *H<sup>k</sup>* <sup>1</sup> and *<sup>H</sup><sup>k</sup>* <sup>2</sup>, on body *k* are termed *handles*. A handle is any selected point through which a body interacts with the environment. In this chapter, we will limit our attention to each body having two handles, and each handle coincides with the joint location on the body, i.e. joint locations *Jk*−<sup>1</sup> and *J<sup>k</sup>* in case of body *k*. Similarly, for body *k* + 1, the points *Hk*+<sup>1</sup> <sup>1</sup> and *<sup>H</sup>k*+<sup>1</sup> <sup>2</sup> are located at the joint locations *<sup>J</sup><sup>k</sup>* and *Jk*<sup>+</sup>1, respectively. Furthermore, large rotations and translations in the flexible bodies are modeled as rigid body *do f* . Elastic deformations in the flexible bodies are modeled through the use of modal coordinates and admissible shape functions.
DCA is implemented using two main processes, hierarchic assembly and disassembly. The goal of the assembly process is to find the equations describing the dynamics of each body in the hierarchy at its two handles. This process begins at the level of individual bodies and adjacent bodies are assembled in a binary tree configuration. Using recursive formulations, this process couples the two-handle equations of successive bodies to find the two-handle equations of the resulting assembly. For example, body *k* and body *k* + 1 are coupled together to form the assembly shown in Fig. (2-b). At the end of the assembly process, the two-handle equations of the entire system are obtained.
The hierarchic disassembly process begins with the solution of the two-handle equations associated with the primary node of the binary tree. The process works from this node to the individual sub-domain nodes of the binary tree to solve for the two-handle equations of the constituent subassemblies. This process is repeated until all unknowns (e.g., spatial constraint forces, spatial constraint impulses, spatial accelerations, jumps in the spatial velocities) of the bodies at the individual sub-domain level of the binary tree are known. The assembly and disassembly processes are illustrated in Fig. (3).
(b) Assembly *k* : *k* + 1
**Figure 2.** Assembling of the two bodies to form a subassembly. a) Consecutive bodies *k* and *k* + 1. b) A fictitious subassembly formed by coupling bodies *k* and *k* + 1.
**Figure 3.** The hierarchic assembly-disassembly process in DCA.
#### **2.2. Analytical preliminaries**
For convenience, the superscript *c* shows that a quantity of interest is associated with the coarse model, while *f* denotes that it is associated with the fine model. For example, the column matrix � *v*1 *q*˙1 �*c* represents the velocity of handle-1 in the coarse model, and � *v*1 *q*˙1 �*f* represents the velocity of the same handle in the fine-scale model. In these matrices, *v*<sup>1</sup> and *q*˙1 are the spatial velocity vector of handle-1 and the associated generalized modal speeds, respectively.
As discussed previously, the change in the system model definition may occur by changing the number of flexible modes used to describe the behavior of flexible bodies, and/or the number of *do f* of the connecting joints. To implement these changes in the system model mathematically, the joint free-motion map is defined as follows.
The joint free-motion map *<sup>P</sup>J<sup>k</sup> <sup>R</sup>* can be interpreted as the 6 <sup>×</sup> *<sup>ν</sup><sup>k</sup>* matrix of the free-modes of motion permitted by the *<sup>ν</sup><sup>k</sup>* degree-of-freedom joint, *<sup>J</sup>k*. In other words, *<sup>P</sup>J<sup>k</sup> <sup>R</sup>* maps *<sup>ν</sup><sup>k</sup>* <sup>×</sup> <sup>1</sup> generalized speeds associated with relative free motion permitted by the joint into a 6 × 1 spatial relative velocity vector which may occur across the joint, *J<sup>k</sup>* [5]. For instance, consider a transition in which a spherical joint in the system is altered, where only one *do f* is locked about the first axis. The joint free-motion maps of the fine and coarse models in this case are shown in the following:
$$\begin{aligned} \;^{I}P\_{R}^{I^{k}f} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \;^{P\_{C}^{I^{k}c}} = \begin{bmatrix} 0 \; 0 \\ 1 \; 0 \\ 0 \; 1 \\ 0 \; 0 \\ 0 \; 0 \\ 0 \; 0 \end{bmatrix} \end{aligned} \tag{1}$$
We define the orthogonal complement of the joint free-motion map, *D<sup>k</sup> <sup>R</sup>*. As such, by definition one arises at the following
$$(D\_R^k)^T P\_R^{J^k} = 0 \tag{2}$$
#### **3. Optimization problem**
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(a) Consecutive bodies
(b) Assembly *k* : *k* + 1
**Figure 2.** Assembling of the two bodies to form a subassembly. a) Consecutive bodies *k* and *k* + 1. b) A
For convenience, the superscript *c* shows that a quantity of interest is associated with the coarse model, while *f* denotes that it is associated with the fine model. For example, the
fictitious subassembly formed by coupling bodies *k* and *k* + 1.
**Figure 3.** The hierarchic assembly-disassembly process in DCA.
**2.2. Analytical preliminaries**
Any violation in the conservation of the generalized momentum of the system in the transition between different models leads to non-physical results since the instantaneous switch in the system model definition is incurred without the influence of any external load. In other words, the momentum of each differential element projected onto the space of the admissible motions permitted by the more restrictive model (whether pre- or post-transition) when integrated over the entire system must be conserved across the model transition [6]. Jumps in the system partial velocities due to the sudden change in the model resolution result in the jumps in the generalized speeds corresponding to the new set of degrees of freedom. Since the model is instantaneously swapped, the position of the system does not change. Hence, the position dependent forces acting on the system do not change, and do not affect the generalized speeds. Any change in the applied loads (e.g., damping terms) which might occur due to the change in the model definition and the associated velocity jumps do not contribute to the impulse-momentum equations which describe the model transition. This is because these changes in the applied loads are bounded, and integrated over the infinitesimally short transition time.
Consider a fine-scale model with *n do f* . Let the *do f* of the model reduce to *n* − *m* after the imposition of certain instantaneous constraints. In this case, the conservation of the
#### 6 Will-be-set-by-IN-TECH 242 Linear Algebra – Theorems and Applications
generalized momentum of the system is expressed as
$$L^{c/c} = L^{f/c} \tag{3}$$
In the above equation, *Lc*/*<sup>c</sup>* and *L<sup>f</sup>* /*<sup>c</sup>* represent the momenta of the coarse and fine models, respectively, projected on to the space of partial velocity vectors of the coarse model. Equation (3) provides a set of *n* − *m* equations which are linear in the generalized speeds of the coarse model and solvable for the unique and physically meaningful states of the system after the transition to the coarser model.
Now consider the case in which, the coarse model is transitioned back to the fine-scale model. Equation (3) is still valid, and provides *n* − *m* equations with *n* unknown generalized speeds of the finer model. Furthermore, during the coarsening process, the level of the kinetic energy also drops because we chose to ignore certain modes of the system. However, in actual biomolecular systems such a decrease in energy does not happen. Consequently, it is important to realize the proper kinetic energy when transitioning back to the fine-scale model. Therefore, the following equation must be satisfied
$$KE^f = \frac{1}{2} (\boldsymbol{\mu}^f)^T \mathcal{M} \boldsymbol{u}^f \tag{4}$$
In the above equation *<sup>u</sup><sup>f</sup>* is the *<sup>n</sup>* <sup>×</sup> 1 column matrix containing the generalized speeds of the fine model, and M represents the generalized inertia matrix of the fine model. It is clear that Eqs. (3) and (4) provide *n* − *m* + 1 equations with *n* unknowns. This indicates that the problem is under-determined when multiple *do f* of the system are released. We may arrive at a unique or finite number of solutions, solving the following optimization problem
$$\text{Optimize} \qquad J(u^f, t) \tag{5}$$
$$\text{Subjected to }\quad \Theta\_i(\boldsymbol{\mu}^f, t) = 0, \; i = 1, \dots, k \tag{6}$$
In the above equation, *J* is the physics- or knowledge- or mathematics-based objective function to be optimized (nominally minimized) subjected to the constraint equations Θ*i*. In [1, 15], different objective functions are proposed for coarse to fine-scale transition problems. For instance, in order to prevent the generalized speeds of the new fine-scale model from deviating greatly from those of the coarse scale model, we may minimize the *L*<sup>2</sup> norm of the difference between the generalized speeds of the coarse and fine scale models as follows
$$J = (\mathfrak{u}^f - \mathfrak{u}^c)^T (\mathfrak{u}^f - \mathfrak{u}^c) \tag{7}$$
As indicated previously, (*n* − *m*) constraint equations governing the optimization problem are obtained from the conservation of the generalized momentum of the system within the transition. The rest of the constraint equations are obtained from other information about the system, such as the specific value of kinetic energy or the temperature of the system.
The generalized momenta balance equations from Eq. (3) are expressed as
$$A\mathfrak{u}^f = \mathfrak{b} \tag{8}$$
where *<sup>A</sup>* and *<sup>b</sup>* are (*<sup>n</sup>* <sup>−</sup> *<sup>m</sup>*) <sup>×</sup> *<sup>n</sup>* and *<sup>n</sup>* known matrices, respectively, and *<sup>u</sup><sup>f</sup>* is an *<sup>n</sup>* <sup>×</sup> 1 column matrix of the generalized speeds of the fine-scale system model. As a part of the optimization problem, one must solve this linear system for *n* − *m* dependent generalized speeds in terms of *m* independent generalized speeds. Therefore, the optimization is performed on a much smaller number (*m*) variables, with a cost of *O*(*m*3). For a complex molecular system, *n* could be very large, and *n* >> *m*, hence a significant reduction is achieved in the overall cost of optimization as compared to other traditional techniques, such as Lagrange multipliers [3]. However, the computations required to find the relations between dependent and independent generalized speeds can impose a significant burden on these simulations. It is shown in [2] that if traditional methods such as Gaussian elimination or LU factorization are used to find these relations, this cost tends to be *<sup>O</sup>*(*n*(*<sup>n</sup>* <sup>−</sup> *<sup>m</sup>*)2). The DCA scheme provided here finds these relations at the end of the hierarchic disassembly process with computational complexity of almost *O*(*n*) in serial implementation. In other words, in this strategy, DCA formulates the impulse-momentum equations of the system which is followed by providing the relations between dependent and independent generalized speeds of the system in a timely manner. As such, this significantly reduces the costs associated with forming and solving the optimization problem in the transitions to the finer models.
#### **4. DCA-based momenta balance for multi-flexible bodies**
6 Will-be-set-by-IN-TECH
In the above equation, *Lc*/*<sup>c</sup>* and *L<sup>f</sup>* /*<sup>c</sup>* represent the momenta of the coarse and fine models, respectively, projected on to the space of partial velocity vectors of the coarse model. Equation (3) provides a set of *n* − *m* equations which are linear in the generalized speeds of the coarse model and solvable for the unique and physically meaningful states of the system after the
Now consider the case in which, the coarse model is transitioned back to the fine-scale model. Equation (3) is still valid, and provides *n* − *m* equations with *n* unknown generalized speeds of the finer model. Furthermore, during the coarsening process, the level of the kinetic energy also drops because we chose to ignore certain modes of the system. However, in actual biomolecular systems such a decrease in energy does not happen. Consequently, it is important to realize the proper kinetic energy when transitioning back to the fine-scale model.
In the above equation *<sup>u</sup><sup>f</sup>* is the *<sup>n</sup>* <sup>×</sup> 1 column matrix containing the generalized speeds of the fine model, and M represents the generalized inertia matrix of the fine model. It is clear that Eqs. (3) and (4) provide *n* − *m* + 1 equations with *n* unknowns. This indicates that the problem is under-determined when multiple *do f* of the system are released. We may arrive at a unique
In the above equation, *J* is the physics- or knowledge- or mathematics-based objective function to be optimized (nominally minimized) subjected to the constraint equations Θ*i*. In [1, 15], different objective functions are proposed for coarse to fine-scale transition problems. For instance, in order to prevent the generalized speeds of the new fine-scale model from deviating greatly from those of the coarse scale model, we may minimize the *L*<sup>2</sup> norm of the difference between the generalized speeds of the coarse and fine scale models as follows
As indicated previously, (*n* − *m*) constraint equations governing the optimization problem are obtained from the conservation of the generalized momentum of the system within the transition. The rest of the constraint equations are obtained from other information about the
where *<sup>A</sup>* and *<sup>b</sup>* are (*<sup>n</sup>* <sup>−</sup> *<sup>m</sup>*) <sup>×</sup> *<sup>n</sup>* and *<sup>n</sup>* known matrices, respectively, and *<sup>u</sup><sup>f</sup>* is an *<sup>n</sup>* <sup>×</sup> 1 column matrix of the generalized speeds of the fine-scale system model. As a part of the optimization
)*T*(*u<sup>f</sup>* <sup>−</sup> *<sup>u</sup><sup>c</sup>*
*KE<sup>f</sup>* <sup>=</sup> <sup>1</sup> 2 (*u<sup>f</sup>*
or finite number of solutions, solving the following optimization problem
Optimize *J*(*u<sup>f</sup>*
Subjected to Θ*i*(*u<sup>f</sup>*
*<sup>J</sup>* = (*u<sup>f</sup>* <sup>−</sup> *<sup>u</sup><sup>c</sup>*
system, such as the specific value of kinetic energy or the temperature of the system.
The generalized momenta balance equations from Eq. (3) are expressed as
*Lc*/*<sup>c</sup>* = *L<sup>f</sup>* /*<sup>c</sup>* (3)
)*T*M*u<sup>f</sup>* (4)
, *t*) (5)
, *t*) = 0, *i* = 1, ··· , *k* (6)
*Au<sup>f</sup>* = *b* (8)
) (7)
generalized momentum of the system is expressed as
Therefore, the following equation must be satisfied
transition to the coarser model.
In this section, two-handle impulse-momentum equations of flexible bodies are derived. Mathematical modeling of the transition from a coarse model to a fine-scale model is discussed. For the fine-scale to coarse-scale model transition in multi-flexible-body system the reader is referred to [7, 17]. We will now derive the two-handle impulse-momentum equations when flexible degrees of freedom of a flexible body or the joints in the system are released. Then, the assembly of two consecutive bodies for which the connecting joint is unlocked is discussed. Finally, the hierarchic assembly-disassembly process for the multi-flexible-body system is presented.
#### **4.1. Two-handle impulse-momentum equations in coarse to fine transitions**
Now, we develop the two-handle impulse-momentum equations for consecutive flexible bodies in the transition from a coarse model to a fine-scale model. It is desired to develop the handle equations which express the spatial velocity vectors of the handles after the transition to the finer model as explicit functions of only newly introduced modal generalized speeds of the fine model. For this purpose, we start from the impulse-momentum equation of the flexible body as
$$
\Gamma^f \mathbf{v}\_1^f - \Gamma^c \mathbf{v}\_1^c = \begin{bmatrix} \gamma\_R \\ \gamma\_F \end{bmatrix}\_1^c \int\_{t\_\ell}^{t\_f} F\_{1c} dt + \begin{bmatrix} \gamma\_R \\ \gamma\_F \end{bmatrix}\_2^c \int\_{t\_\ell}^{t\_f} F\_{2c} dt \tag{9}
$$
where Γ*<sup>f</sup>* and Γ*<sup>c</sup>* are the inertia matrices associated with the fine-scale and coarse models, respectively. Also, *tc* and *tf* represent the time right before and right after the transition. The quantities *tf tc <sup>F</sup>*1*cdt* and *tf tc F*2*cdt* are the spatial impulsive constraint forces on handle-1 and handle-2 of the flexible body. The matrices *γ<sup>R</sup> γF c* 1 and *γ<sup>R</sup> γF c* 2 are the coefficients resulting from the generalized constraint force contribution at handle-1 and handle-2, respectively. Moreover, in Eq. (9), the impulses due to the applied loads are not considered since they represent a bounded loads integrated over an infinitesimal time interval. For detailed derivation of these quantities the reader is referred to [8]. It is desired to develop the handle equations which provide the spatial velocity vectors of the handles right after the transition to the fine-scale model in terms of newly added modal generalized speeds. Therefore, in Eq. (9), the inertia matrix of the flexible body is represented by its components corresponding to rigid and flexible modes, as well as the coupling terms
$$
\begin{aligned}
\begin{bmatrix}
\Gamma\_{RR}\,\Gamma\_{RF} \\
\Gamma\_{FR}\,\Gamma\_{FF}
\end{bmatrix}^{f}
\begin{bmatrix}
\upsilon\_{1} \\
\dot{\eta}
\end{bmatrix}^{f} &= \begin{bmatrix}
\gamma\_{R} \\
\gamma\_{F}
\end{bmatrix}^{c}\_{1} \int\_{t\_{c}}^{t\_{f}} F\_{1c} dt + \begin{bmatrix}
\gamma\_{R} \\
\gamma\_{F}
\end{bmatrix}^{c}\_{2} \int\_{t\_{c}}^{t\_{f}} F\_{2c} dt \\ &+ \begin{bmatrix}
\Gamma\_{RR}\,\Gamma\_{RF} \\
\Gamma\_{FR}\,\Gamma\_{FF}
\end{bmatrix}^{c} \begin{bmatrix}
\upsilon\_{1} \\
\dot{\eta}
\end{bmatrix}^{c}
\end{aligned}
\tag{10}
$$
which is decomposed to the following relations
$$
\Gamma^{\underline{f}}\_{\rm FF} \dot{\eta}^{\underline{f}} = \gamma^{\underline{c}}\_{\rm F1} \int\_{t\_{\rm c}}^{t\_{\rm f}} \mathcal{F}\_{\rm Ic} dt + \gamma^{\underline{c}}\_{\rm F2} \int\_{t\_{\rm c}}^{t\_{\rm f}} \mathcal{F}\_{\rm 2c} dt + \Gamma^{\underline{c}}\_{\rm FR} \upsilon^{\underline{c}}\_{\rm 1} + \Gamma^{\underline{c}}\_{\rm FF} \dot{\eta}^{\underline{c}} - \Gamma^{\underline{f}}\_{\rm FR} \upsilon^{\underline{f}}\_{\rm 1} \tag{11}
$$
$$
\Gamma^{f}\_{\rm RR} v^{f}\_{1} = \gamma^{\varepsilon}\_{\rm R1} \int\_{t\_{\varepsilon}}^{t\_{f}} \mathbf{F}\_{\rm lc} dt + \gamma^{\varepsilon}\_{\rm R2} \int\_{t\_{\varepsilon}}^{t\_{f}} \mathbf{F}\_{\rm 2c} dt + \Gamma^{c}\_{\rm RR} v^{c}\_{1} + \Gamma^{c}\_{\rm RF} \dot{\boldsymbol{\eta}}^{c} - \Gamma^{f}\_{\rm RF} \dot{\boldsymbol{\eta}}^{f} \tag{12}
$$
Since the generalized momentum equations are calculated based on the projection onto the space of the coarser model, the matrix Γ*<sup>f</sup>* is not a square matrix and thus Γ*<sup>f</sup> FF* is not invertible. However, we can partition Eq. (11) in terms of dependent (those associated with the coarser model) and independent (newly introduced) generalized speeds as
$$
\begin{bmatrix}
\Gamma\_{FF}^{f\_d} \vdots \Gamma\_{FF}^{f\_l} \end{bmatrix} \begin{bmatrix}
\dot{q}^{f\_d} \\
\vdots \\
\dot{q}^{f\_l}
\end{bmatrix} = \gamma\_{F1}^c \int\_{t\_c}^{t\_f} F\_{1c} dt + \gamma\_{F2}^c \int\_{t\_c}^{t\_f} F\_{2c} dt
$$
$$
+ \Gamma\_{FR}^c v\_1^c + \Gamma\_{FF}^c \dot{q}^c - \Gamma\_{FR}^f v\_1^f \tag{13}
$$
Using the above relation, the expression for the dependent generalized modal speeds is written as
$$\begin{split} \boldsymbol{\dot{q}^{f\_d}} &= (\boldsymbol{\Gamma}^{f\_d}\_{\mathrm{FF}})^{-1} [\boldsymbol{\gamma}^{\boldsymbol{c}}\_{\mathrm{F}1} \int\_{t\_{\boldsymbol{c}}}^{t\_{\boldsymbol{f}}} \boldsymbol{F}\_{\mathrm{L}} dt + \boldsymbol{\gamma}^{\boldsymbol{c}}\_{\mathrm{F}2} \int\_{t\_{\boldsymbol{c}}}^{t\_{\boldsymbol{f}}} \boldsymbol{F}\_{\mathrm{2c}} dt + \boldsymbol{\Gamma}^{\boldsymbol{c}}\_{\mathrm{FR}} \boldsymbol{v}^{\boldsymbol{c}}\_{\mathrm{1}} \\ &+ \boldsymbol{\Gamma}^{\boldsymbol{c}}\_{\mathrm{FF}} \boldsymbol{\dot{q}^{c}} - \boldsymbol{\Gamma}^{\boldsymbol{f}}\_{\mathrm{FR}} \boldsymbol{v}^{\boldsymbol{f}}\_{\mathrm{1}} - \boldsymbol{\Gamma}^{f\_{\mathrm{i}}}\_{\mathrm{FF}} \boldsymbol{\dot{q}^{f\_{\mathrm{i}}}}] \end{split} \tag{14}$$
Defining
$$
\Gamma\_{\rm RF}^f = \begin{bmatrix} \Gamma\_{\rm RF}^{f\_d} \vdots \Gamma\_{\rm RF}^{f\_l} \end{bmatrix}\_{\dots,\dots,\dots} \tag{15}
$$
$$\Lambda = \left[\Gamma\_{RR}^f - \Gamma\_{RF}^{f\_d} (\Gamma\_{FF}^{f\_d})^{-1} \Gamma\_{FR}^f\right]^{-1} \tag{16}$$
$$\mathcal{L}\_1 = \left[ \gamma\_{R1}^{\mathcal{L}} - \Gamma\_{RF}^{f\_d} (\Gamma\_{FF}^{f\_d})^{-1} \gamma\_{F1}^{\mathcal{L}} \right] \tag{17}$$
$$\mathcal{L}\_2 = \left[ \gamma\_{R2}^c - \Gamma\_{RF}^{f\_d} (\Gamma\_{FF}^{f\_d})^{-1} \gamma\_{F2}^c \right] \tag{18}$$
$$\mathcal{L}\_{\mathfrak{D}} = [\Gamma^{\mathfrak{c}}\_{RR} - \Gamma^{f\_{\mathfrak{d}}}\_{RF} (\Gamma^{f\_{\mathfrak{f}}}\_{FF})^{-1} \Gamma^{\mathfrak{c}}\_{FR}] v^{\mathfrak{c}}\_{1} + [\Gamma^{\mathfrak{c}}\_{RF} - \Gamma^{f\_{\mathfrak{d}}}\_{RF} (\Gamma^{f\_{\mathfrak{d}}}\_{FF})^{-1} \Gamma^{\mathfrak{c}}\_{FF}] \phi^{\mathfrak{c}} \tag{19}$$
$$\mathcal{L}\_4 = \left[ \Gamma\_{RF}^{f\_d} (\Gamma\_{FF}^{f\_d})^{-1} \Gamma\_{FF}^{f\_l} - \Gamma\_{RF}^{f\_l} \right] \tag{20}$$
$$
\lambda\_{1i} = \Lambda\_{\mathfrak{z}i}^{\tau}, \quad (i = 1, 2, 3, 4) \tag{21}
$$
and using Eqs. (12) and (14), the spatial velocity vector of handle-1 can be written in terms of the independent modal speeds
$$v\_1^f = \lambda\_{11} \int\_{t\_\ell}^{t\_f} F\_{1c} dt + \lambda\_{12} \int\_{t\_\ell}^{t\_f} F\_{2c} dt + \lambda\_{13} + \lambda\_{14} \dot{q}^{f\_l} \tag{22}$$
As such, the spatial velocity vector of handle-2 becomes
$$v\_2^f = (S^{k1k2})^T v\_1^f + \phi\_2^f \dot{q}^f \tag{23}$$
Employing the same partitioning technique, Eqs. (23) can be written as
$$\boldsymbol{v}\_{2}^{f} = (\boldsymbol{\mathcal{S}}^{\text{k}1\text{k}2})^{T} \boldsymbol{v}\_{1}^{f} + \left[ \boldsymbol{\phi}\_{2}^{f\_{d}} \vdots \boldsymbol{\phi}\_{2}^{f\_{l}} \right] \begin{bmatrix} \dot{\boldsymbol{q}}^{f\_{d}} \\ \cdots \\ \dot{\boldsymbol{q}}^{f\_{l}} \end{bmatrix} \tag{24}$$
$$\Rightarrow v\_2^f = (\mathbf{S}^{k1k2})^T v\_1^f + \phi\_2^{f\_d} \dot{\eta}^{f\_d} + \phi\_2^{f\_l} \dot{\eta}^{f\_l} \tag{25}$$
Using
8 Will-be-set-by-IN-TECH
equations which provide the spatial velocity vectors of the handles right after the transition to the fine-scale model in terms of newly added modal generalized speeds. Therefore, in Eq. (9), the inertia matrix of the flexible body is represented by its components corresponding to rigid
> Γ*RR* Γ*RF* Γ*FR* Γ*FF*
� *tf tc*
*F*1*cdt* +
�*<sup>c</sup>* � *v*1 *q*˙ �*c*
*F*2*cdt* + Γ*<sup>c</sup>*
*F*2*cdt* + Γ*<sup>c</sup>*
� *γ<sup>R</sup> γF* �*c* 2
*FRv<sup>c</sup>* <sup>1</sup> <sup>+</sup> <sup>Γ</sup>*<sup>c</sup> FFq*˙ *<sup>c</sup>* <sup>−</sup> <sup>Γ</sup>*<sup>f</sup> FRv<sup>f</sup>*
*RRv<sup>c</sup>* <sup>1</sup> <sup>+</sup> <sup>Γ</sup>*<sup>c</sup> RFq*˙ *<sup>c</sup>* <sup>−</sup> <sup>Γ</sup>*<sup>f</sup>*
*F*1*cdt* + *γ<sup>c</sup>*
*F*2 � *tf tc*
*F*2*cdt* + Γ*<sup>c</sup>*
*F*2*cdt*
*FRv<sup>c</sup>* 1
*FFq*˙ *fi* ] (14)
<sup>−</sup><sup>1</sup> (16)
*<sup>F</sup>*1] (17)
*<sup>F</sup>*2] (18)
*FF*)−1Γ*<sup>c</sup>*
*RF*] (20)
*FF*]*q*˙
*RF*(Γ*fd*
� *tf tc*
*F*2*cdt*
(10)
(15)
*<sup>c</sup>* (19)
<sup>1</sup> (11)
*RFq*˙ *<sup>f</sup>* (12)
*FF* is not invertible.
<sup>1</sup> (13)
and flexible modes, as well as the coupling terms
�*<sup>f</sup>* � *v*1 *q*˙ �*f* = � *γ<sup>R</sup> γF* �*c* 1
> + �
*F*1*cdt* + *γ<sup>c</sup>*
*F*1*cdt* + *γ<sup>c</sup>*
space of the coarser model, the matrix Γ*<sup>f</sup>* is not a square matrix and thus Γ*<sup>f</sup>*
⎤ <sup>⎦</sup> <sup>=</sup> *<sup>γ</sup><sup>c</sup> F*1 � *tf tc*
> + Γ*<sup>c</sup> FRv<sup>c</sup>* <sup>1</sup> <sup>+</sup> <sup>Γ</sup>*<sup>c</sup> FFq*˙ *<sup>c</sup>* <sup>−</sup> <sup>Γ</sup>*<sup>f</sup> FRv<sup>f</sup>*
Using the above relation, the expression for the dependent generalized modal speeds is
*F*1*cdt* + *γ<sup>c</sup>*
*F*2 � *tf tc*
model) and independent (newly introduced) generalized speeds as
�⎡ ⎣ *q*˙ *fd* ··· *q*˙ *fi*
*FF*)−1[*γ<sup>c</sup> F*1 � *tf tc*
*F*2 � *tf tc*
*R*2 � *tf tc*
Since the generalized momentum equations are calculated based on the projection onto the
However, we can partition Eq. (11) in terms of dependent (those associated with the coarser
Γ*RR* Γ*RF* Γ*FR* Γ*FF*
which is decomposed to the following relations
*F*1 � *tf tc*
*FFq*˙ *<sup>f</sup>* <sup>=</sup> *<sup>γ</sup><sup>c</sup>*
<sup>1</sup> <sup>=</sup> *<sup>γ</sup><sup>c</sup> R*1 � *tf tc*
> � Γ*fd FF* . . . Γ*fi FF*
*q*˙ *fd* = (Γ*fd*
+ Γ*<sup>c</sup> FFq*˙ *<sup>c</sup>* <sup>−</sup> <sup>Γ</sup>*<sup>f</sup> FRv<sup>f</sup>* <sup>1</sup> <sup>−</sup> <sup>Γ</sup>*fi*
*RR* <sup>−</sup> <sup>Γ</sup>*fd*
*<sup>R</sup>*<sup>1</sup> <sup>−</sup> <sup>Γ</sup>*fd*
*<sup>R</sup>*<sup>2</sup> <sup>−</sup> <sup>Γ</sup>*fd*
*RR* <sup>−</sup> <sup>Γ</sup>*fd*
*RF*(Γ*fd*
*RF*(Γ*fd*
*RF*(Γ*fd*
*RF*(Γ*fd*
*RF*(Γ*fd*
*FF*)−1Γ*fi*
*FF*)−1Γ*<sup>f</sup> FR*]
*FF*)−1*γ<sup>c</sup>*
*FF*)−1*γ<sup>c</sup>*
*FF*)−1Γ*<sup>c</sup>*
*FF* <sup>−</sup> <sup>Γ</sup>*fi*
*FR*]*v<sup>c</sup>*
<sup>1</sup> + [Γ*<sup>c</sup>*
*RF* <sup>−</sup> <sup>Γ</sup>*fd*
*λ*1*<sup>i</sup>* = Λ*ζi*, (*i* = 1, 2, 3, 4) (21)
Γ*f RF* = � Γ*fd RF* . . . Γ*fi RF* �
Λ = [Γ*<sup>f</sup>*
*ζ*<sup>1</sup> = [*γ<sup>c</sup>*
*ζ*<sup>2</sup> = [*γ<sup>c</sup>*
*ζ*<sup>3</sup> = [Γ*<sup>c</sup>*
*<sup>ζ</sup>*<sup>4</sup> = [Γ*fd*
�
Γ*f*
Γ*f RRv<sup>f</sup>*
written as
Defining
$$
\lambda\_{21} = \begin{bmatrix} (\mathbf{S}^{k1k2})^T \lambda\_{11} + \boldsymbol{\phi}\_2^{f\_d} (\boldsymbol{\Gamma}\_{FF}^{f\_d})^{-1} \boldsymbol{\gamma}\_{F1}^c - \boldsymbol{\phi}\_2^{f\_d} (\boldsymbol{\Gamma}\_{FF}^{f\_d})^{-1} \boldsymbol{\Gamma}\_{FR}^f \lambda\_{11} \end{bmatrix} \tag{26}
$$
$$\lambda\_{22} = \left[ (\mathbf{S}^{\rm k1k2})^T \lambda\_{12} + \phi\_2^{f\_d} (\Gamma\_{FF}^{f\_d})^{-1} \gamma\_{F2}^c - \phi\_2^{f\_d} (\Gamma\_{FF}^{f\_d})^{-1} \Gamma\_{FR}^f \lambda\_{12} \right] \tag{27}$$
$$\lambda\_{23} = \left[ \mathbf{S}^{\rm k1k2} \right)^T \lambda\_{13} + \phi\_2^{f\_d} (\Gamma\_{FF}^{f\_d})^{-1} \Gamma\_{FR}^c \upsilon\_1^c + \phi\_2^{f\_d} (\Gamma\_{FF}^{f\_d})^{-1} \Gamma\_{FF}^c \dot{\eta}^c$$
$$-\left[\phi\_2^{f\_d} (\Gamma\_{\rm FF}^{f\_d})^{-1} \Gamma\_{\rm FR}^f \lambda\_{13}\right] \tag{28}$$
$$
\lambda\_{24} = [\mathcal{S}^{\text{k}\text{l}\text{k}2}]^T \lambda\_{14} + \phi\_2^{f\_l} - \phi\_2^{f\_d} (\Gamma\_{\text{FF}}^{f\_d})^{-1} \Gamma\_{\text{FF}}^{f\_l} - \phi\_2^{f\_d} (\Gamma\_{\text{FF}}^{f\_d})^{-1} \Gamma\_{\text{FR}}^f \lambda\_{14} \tag{29}
$$
and Eq. (25), the spatial velocity vector of handle-2 can be written as
$$
v\_{2}^{f} = \lambda\_{21} \int\_{t\_{\mathcal{E}}}^{t\_{f}} F\_{1c} dt + \lambda\_{22} \int\_{t\_{\mathcal{E}}}^{t\_{f}} F\_{2c} dt + \lambda\_{23} + \lambda\_{24} \dot{q}^{f\_{1}} \tag{30}$$
Equations (22) and (30) are now in two-handle impulse-momentum form and along with Eq. (14), give the new velocities associated with each handle after the transition. These equations express the spatial velocity vectors of the handles of the body as well as the modal generalized speeds which have not changed within the transition in terms of the newly added modal generalized speeds. This important property will be used in the optimization problem to provide the states of the system after the transition to the finer models.
As such, the two-handle equations describing the impulse-momentum of two consecutive bodies, body *k* and body *k* + 1 are expressed as
$$v\_1^{(k)f} = \lambda\_{11}^{(k)} \int\_{t\_\ell}^{t\_f} F\_{1c}^{(k)} dt + \lambda\_{12}^{(k)} \int\_{t\_\ell}^{t\_f} F\_{2c}^{(k)} dt + \lambda\_{13}^{(k)} + \lambda\_{14}^{(k)} \dot{q}^{(k)f\_l} \tag{31}$$
$$v\_2^{(k)f} = \lambda\_{21}^{(k)} \int\_{t\_\ell}^{t\_f} F\_{1c}^{(k)} dt + \lambda\_{22}^{(k)} \int\_{t\_\ell}^{t\_f} F\_{2c}^{(k)} dt + \lambda\_{23}^{(k)} + \lambda\_{24}^{(k)} \dot{q}^{(k)f\_l} \tag{32}$$
$$w\_1^{(k+1)f} = \lambda\_{11}^{(k+1)} \int\_{t\_\varepsilon}^{t\_f} F\_{1\varepsilon}^{(k+1)} dt + \lambda\_{12}^{(k+1)} \int\_{t\_\varepsilon}^{t\_f} F\_{2\varepsilon}^{(k+1)} dt + \lambda\_{13}^{(k+1)} + \lambda\_{14}^{(k+1)} \dot{q}^{(k+1)f\_i} \tag{33}$$
$$v\_{2}^{(k+1)f} = \lambda\_{21}^{(k+1)} \int\_{t\_{\mathcal{c}}}^{t\_{f}} F\_{1\mathcal{c}}^{(k+1)} dt + \lambda\_{22}^{(k+1)} \int\_{t\_{\mathcal{c}}}^{t\_{f}} F\_{2\mathcal{c}}^{(k+1)} dt + \lambda\_{23}^{(k+1)} + \lambda\_{24}^{(k+1)} \dot{q}^{(k+1)f\_{i}} \tag{34}$$
#### **4.2. Assembly process and releasing the joint between two consecutive bodies**
In this section, a method to combine the two-handle equations of individual flexible bodies to form the equations of the resulting assembly is presented. Herein, the assembly process of the consecutive bodies is discussed only within the transition from a coarse model to a finer model. This transition is achieved by releasing the joint between two consecutive bodies. Clearly, this would mean a change in the joint free-motion map *<sup>P</sup>J<sup>k</sup> <sup>R</sup>* and its orthogonal complement *<sup>D</sup>J<sup>k</sup> <sup>R</sup>* . It will become evident that the assembly process of the consecutive bodies for the fine to coarse transition is similar, and the associated equations can be easily derived by following the given procedure.
From the definition of joint free-motion map, the relative spatial velocity vector at the joint between two consecutive bodies is expressed by the following kinematic constraint
$$
v\_1^{(k+1)f} - v\_2^{(k)f} = P\_R^{f^k f} \mathfrak{u}^{(k/k+1)f} \tag{35}$$
In the above equation, *u*(*k*/*k*+1)*<sup>f</sup>* is the relative generalized speed defined at the joint of the fine model. From Newton's third law of motion, the impulses at the intermediate joint are related by
$$\int\_{t\_{\mathcal{L}}}^{t\_f} F\_{2\mathcal{L}}^{(k)} dt = -\int\_{t\_{\mathcal{L}}}^{t\_f} F\_{1\mathcal{L}}^{(k+1)} dt \tag{36}$$
Substituting Eqs. (32), (33) and (36) into Eq. (35) results in
$$\begin{split} \left(\lambda\_{11}^{(k+1)} + \lambda\_{22}^{(k)}\right) \int\_{t\_{\mathcal{c}}}^{t\_{f}} F\_{1c}^{(k+1)} dt &= \lambda\_{21}^{(k)} \int\_{t\_{\mathcal{c}}}^{t\_{f}} F\_{1c}^{(k)} dt - \lambda\_{12}^{(k+1)} \int\_{t\_{\mathcal{c}}}^{t\_{f}} F\_{2c}^{(k+1)} dt \\ &+ \lambda\_{23}^{(k)} - \lambda\_{13}^{(k+1)} + \lambda\_{24}^{(k)} \dot{\eta}^{(k) f\_{\mathcal{i}}} - \lambda\_{14}^{(k+1)} \dot{\eta}^{(k+1) f\_{\mathcal{i}}} + \mathcal{P}\_{R}^{f^{k} f} u^{(k/k+1) f} \end{split} \tag{37}$$
Using the definition of the joint free-motion map, the spatial constraint impulses lie exactly in the space spanned by the orthogonal complement of joint free-motion map of the *coarser* model. These constraint impulses can be expressed as
$$\int\_{t\_{\mathcal{L}}}^{t\_f} \mathbf{F}\_{1c}^{(k+1)} dt = D\_{\mathcal{R}}^{J^k c} \int\_{t\_{\mathcal{L}}}^{t\_f} \mathbf{F}\_{1c}^{(k+1)} dt \tag{38}$$
In the above equation, *tf tc* **<sup>F</sup>**(*k*+1) <sup>1</sup>*<sup>c</sup> dt* is an ordered measure number of the impulsive constraint torques and forces. Pre-multiplying Eq. (37) by (*DJ<sup>k</sup> <sup>c</sup> <sup>R</sup>* )*T*, one arrives at the expression for *tf tc <sup>F</sup>*(*k*+1) <sup>1</sup>*<sup>c</sup> dt* as
$$\begin{split} \int\_{t\_{\mathcal{c}}}^{t\_{f}} F\_{1c}^{(k+1)} dt &= \mathcal{X} \lambda\_{21}^{(k)} \int\_{t\_{\mathcal{c}}}^{t\_{f}} F\_{1c}^{(k)} dt - \mathcal{X} \lambda\_{12}^{(k+1)} \int\_{t\_{\mathcal{c}}}^{t\_{f}} F\_{2c}^{(k+1)} dt \\ &+ \mathcal{X} \mathcal{Y} + \mathcal{X} \lambda\_{24}^{(k)} \dot{q}^{(k) f\_{i}} - \mathcal{X} \lambda\_{14}^{(k+1)} \dot{q}^{(k+1) f\_{i}} + \mathcal{X} P\_{\mathcal{R}}^{I^{k} f} u^{(k/k+1) f} \end{split} \tag{39}$$
where
$$X = D\_{\mathbb{R}}^{J^{\mathbb{k}}c} [(D\_{\mathbb{R}}^{J^{\mathbb{k}}c})^T (\lambda\_{11}^{(k+1)} + \lambda\_{22}^{(k)}) D\_{\mathbb{R}}^{J^{\mathbb{k}}c}]^{-1} (D\_{\mathbb{R}}^{J^{\mathbb{k}}c})^T \tag{40}$$
$$Y = \lambda\_{23}^{(k)} - \lambda\_{13}^{(k+1)} \tag{41}$$
Using Eqs. (31), (34), and (39), we write the two-handle equations for the assembly *k* : *k* + 1
$$\begin{split} v\_{1}^{(k:k+1)f} &= \Psi\_{11}^{(k:k+1)} \int\_{t\_{\mathcal{L}}}^{t\_{f}} F\_{1c}^{(k)} dt + \Psi\_{12}^{(k:k+1)} \int\_{t\_{\mathcal{L}}}^{t\_{f}} F\_{2c}^{(k+1)} dt \\ &+ \Psi\_{13}^{(k:k+1)} + \Psi\_{14}^{(k:k+1)} \dot{q}^{(k)f\_{i}} + \Psi\_{15}^{(k:k+1)} \dot{q}^{(k+1)f\_{i}} + \Psi\_{16}^{(k:k+1)} u^{(k/k+1)f} \end{split} \tag{42}$$
$$\begin{split} \boldsymbol{v}\_{2}^{(\boldsymbol{k}:\boldsymbol{k}+1)f} &= \mathbf{Y}\_{21}^{(\boldsymbol{k}:\boldsymbol{k}+1)} \int\_{t\_{\boldsymbol{\ell}}}^{t\_{f}} \boldsymbol{F}\_{1\boldsymbol{c}}^{(\boldsymbol{k})} dt + \mathbf{Y}\_{22}^{(\boldsymbol{k}:\boldsymbol{k}+1)} \int\_{t\_{\boldsymbol{\ell}}}^{t\_{f}} \boldsymbol{F}\_{2\boldsymbol{c}}^{(\boldsymbol{k}+1)} dt \\ &+ \mathbf{Y}\_{23}^{(\boldsymbol{k}:\boldsymbol{k}+1)} + \mathbf{Y}\_{24}^{(\boldsymbol{k}:\boldsymbol{k}+1)} \boldsymbol{\dot{q}}^{(\boldsymbol{k})f\_{i}} + \mathbf{Y}\_{25}^{(\boldsymbol{k}:\boldsymbol{k}+1)f\_{i}} + \mathbf{Y}\_{26}^{(\boldsymbol{k}:\boldsymbol{k}+1)} \boldsymbol{u}^{(\boldsymbol{k}/\boldsymbol{k}+1)f} \end{split} \tag{43}$$
where:
10 Will-be-set-by-IN-TECH
In this section, a method to combine the two-handle equations of individual flexible bodies to form the equations of the resulting assembly is presented. Herein, the assembly process of the consecutive bodies is discussed only within the transition from a coarse model to a finer model. This transition is achieved by releasing the joint between two consecutive
for the fine to coarse transition is similar, and the associated equations can be easily derived
From the definition of joint free-motion map, the relative spatial velocity vector at the joint
In the above equation, *u*(*k*/*k*+1)*<sup>f</sup>* is the relative generalized speed defined at the joint of the fine model. From Newton's third law of motion, the impulses at the intermediate joint are
> 21 *tf tc*
Using the definition of the joint free-motion map, the spatial constraint impulses lie exactly in the space spanned by the orthogonal complement of joint free-motion map of the *coarser*
*R*
<sup>1</sup>*<sup>c</sup> dt* <sup>−</sup> *<sup>X</sup>λ*(*k*+1) 12
(*k*)*fi* <sup>−</sup> *<sup>X</sup>λ*(*k*+1)
<sup>11</sup> <sup>+</sup> *<sup>λ</sup>*(*k*)
*tf tc*
**F**(*k*+1)
*tf tc*
<sup>14</sup> *q*˙
<sup>22</sup> )*DJ<sup>k</sup> <sup>c</sup> <sup>R</sup>* ]
<sup>1</sup>*<sup>c</sup> dt* is an ordered measure number of the impulsive constraint
*F*(*k*+1) <sup>2</sup>*<sup>c</sup> dt*
(*k*+1)*fi* <sup>+</sup> *XPJ<sup>k</sup> <sup>f</sup>*
<sup>−</sup>1(*DJ<sup>k</sup> <sup>c</sup>*
<sup>13</sup> (41)
*tf tc*
(*k*)*fi* <sup>−</sup> *<sup>λ</sup>*(*k*+1) <sup>14</sup> *q*˙
*F*(*k*+1)
*F*(*k*)
<sup>1</sup>*<sup>c</sup> dt* <sup>−</sup> *<sup>λ</sup>*(*k*+1) 12
(*k*+1)*fi* <sup>+</sup> *<sup>P</sup>J<sup>k</sup> <sup>f</sup>*
between two consecutive bodies is expressed by the following kinematic constraint
(*k*)*f* <sup>2</sup> <sup>=</sup> *<sup>P</sup>J<sup>k</sup> <sup>f</sup>*
*<sup>R</sup>* . It will become evident that the assembly process of the consecutive bodies
*<sup>R</sup>* and its orthogonal
*<sup>R</sup> <sup>u</sup>*(*k*/*k*+1)*<sup>f</sup>* (35)
<sup>1</sup>*<sup>c</sup> dt* (36)
*F*(*k*+1) <sup>2</sup>*<sup>c</sup> dt*
<sup>1</sup>*<sup>c</sup> dt* (38)
*<sup>R</sup>* )*T*, one arrives at the expression for
*<sup>R</sup> <sup>u</sup>*(*k*/*k*+1)*<sup>f</sup>* (37)
*<sup>R</sup> <sup>u</sup>*(*k*/*k*+1)*<sup>f</sup>* (39)
*<sup>R</sup>* )*<sup>T</sup>* (40)
*tf tc*
**4.2. Assembly process and releasing the joint between two consecutive bodies**
bodies. Clearly, this would mean a change in the joint free-motion map *<sup>P</sup>J<sup>k</sup>*
*v* (*k*+1)*f* <sup>1</sup> − *v*
> *tf tc*
*F*(*k*+1)
<sup>13</sup> <sup>+</sup> *<sup>λ</sup>*(*k*)
*F*(*k*+1)
Substituting Eqs. (32), (33) and (36) into Eq. (35) results in
<sup>23</sup> <sup>−</sup> *<sup>λ</sup>*(*k*+1)
*tf tc*
> 21 *tf tc*
+ *XY* + *<sup>X</sup>λ*(*k*)
*<sup>R</sup>* [(*DJ<sup>k</sup> <sup>c</sup>*
<sup>23</sup> <sup>−</sup> *<sup>λ</sup>*(*k*+1)
<sup>22</sup> ) *tf tc*
model. These constraint impulses can be expressed as
*tc* **<sup>F</sup>**(*k*+1)
torques and forces. Pre-multiplying Eq. (37) by (*DJ<sup>k</sup> <sup>c</sup>*
<sup>1</sup>*<sup>c</sup> dt* <sup>=</sup> *<sup>X</sup>λ*(*k*)
*<sup>X</sup>* = *<sup>D</sup>J<sup>k</sup> <sup>c</sup>*
*<sup>Y</sup>* = *<sup>λ</sup>*(*k*)
+ *<sup>λ</sup>*(*k*)
*F*(*k*) <sup>2</sup>*<sup>c</sup> dt* = −
<sup>1</sup>*<sup>c</sup> dt* <sup>=</sup> *<sup>λ</sup>*(*k*)
<sup>24</sup> *q*˙
<sup>1</sup>*<sup>c</sup> dt* <sup>=</sup> *<sup>D</sup>J<sup>k</sup> <sup>c</sup>*
*F*(*k*)
<sup>24</sup> *q*˙
*<sup>R</sup>* )*T*(*λ*(*k*+1)
complement *<sup>D</sup>J<sup>k</sup>*
related by
by following the given procedure.
(*λ*(*k*+1)
In the above equation, *tf*
*tf tc*
*F*(*k*+1)
*tf tc <sup>F</sup>*(*k*+1) <sup>1</sup>*<sup>c</sup> dt* as
where
<sup>11</sup> <sup>+</sup> *<sup>λ</sup>*(*k*)
$$\Psi\_{11}^{(k:k+1)} = \lambda\_{11}^{(k)} - \lambda\_{12}^{(k)} X \lambda\_{21}^{(k)} \tag{44}$$
$$\Psi\_{12}^{(k:k+1)} = \lambda\_{12}^{(k)} X \lambda\_{12}^{(k+1)} \tag{45}$$
$$\Psi\_{13}^{(k:k+1)} = \lambda\_{13}^{(k)} - \lambda\_{12}^{(k)} XY \tag{46}$$
$$\mathbf{Y}\_{14}^{(k:k+1)} = \lambda\_{14}^{(k)} - \lambda\_{12}^{(k)} X \lambda\_{24}^{(k)} \tag{47}$$
$$\Psi\_{15}^{(k:k+1)} = \lambda\_{12}^{(k)} X \lambda\_{14}^{(k+1)} \tag{48}$$
$$\Psi\_{16}^{(k:k+1)} = -\lambda\_{12}^{(k)} X P\_{\mathbb{R}}^{l^k f} \tag{49}$$
$$\Psi\_{21}^{(k:k+1)} = \lambda\_{21}^{(k+1)} X \lambda\_{21}^{(k)} \tag{50}$$
$$\Psi\_{22}^{(k:k+1)} = \lambda\_{22}^{(k+1)} - \lambda\_{21}^{(k+1)} X \lambda\_{12}^{(k+1)} \tag{51}$$
$$
\Psi\_{23}^{(k:k+1)} = \lambda\_{21}^{(k+1)} XY + \lambda\_{23}^{(k+1)} \tag{52}
$$
$$\Psi\_{24}^{(k:k+1)} = \lambda\_{21}^{(k+1)} X \lambda\_{24}^{(k)} \tag{53}$$
$$\mathbf{V}\_{25}^{(k:k+1)} = \lambda\_{24}^{(k+1)} - \lambda\_{21}^{(k+1)} X \lambda\_{14}^{(k+1)} \tag{54}$$
$$\Psi\_{26}^{(k:k+1)} = \lambda\_{21}^{(k+1)} X P\_R^{f^k f} \tag{55}$$
The two-handle equations of the resultant assembly express the spatial velocity vectors of the terminal handles of the assembly in terms of the spatial constraint impulses on the same handles, as well as the newly added modal generalized speeds of each constituent flexible body, and the newly introduced *do f* at the connecting joint. These are the equations which address the dynamics of the assembly when both types of transitions occur simultaneously. In other words, they are applicable when new flexible modes are added to the flexible constituent subassemblies and new degrees of freedom are released at the connecting joint. If there is no change in the joint free-motion map, the spatial partial velocity vector associated with *u*(*k*/*k*+1)*<sup>f</sup>* does not appear in the handle equations of the resulting assembly.
#### **5. Hierarchic assembly-disassembly**
The DCA is implemented in two main passes: assembly and disassembly [8, 9]. As mentioned previously, two consecutive bodies can be combined together to recursively form the handle equations of the resulting assembly. As such, the assembly process starts at the individual sub-domain level of the binary tree to combine the adjacent bodies and form the equations of motion of the resulting assembly. This process is recursively implemented as that of the binary tree to find the impulse-momentum equations of the new assemblies. In this process, the spatial velocity vector (after transition) and impulsive load of the handles at the common joint of the consecutive bodies are eliminated. The handle equations of the resulting assembly are expressed in terms of the constraint impulses and spatial velocities of the terminal handles, as well as the newly introduce modal generalized speeds and generalized speeds associated with the newly added degrees of freedom at the connecting joints. This process stops at the top level of the binary tree in which the impulse-momentum equations of the entire system are expressed by the following two-handle equations
$$\begin{split} \boldsymbol{w}\_{1}^{1f} &= \Psi\_{11}^{(1:n)} \int\_{t\_{\boldsymbol{c}}}^{t\_{f}} \boldsymbol{F}\_{1\boldsymbol{c}}^{1} dt + \Psi\_{12}^{(1:n)} \int\_{t\_{\boldsymbol{c}}}^{t\_{f}} \boldsymbol{F}\_{2\boldsymbol{c}}^{n} dt + \Psi\_{13}^{(1:n)} \\ &+ \Psi\_{14}^{(1:n)} \dot{\boldsymbol{q}}^{(1:n)f\_{\boldsymbol{t}}} + \Psi\_{15}^{(1:n)} \boldsymbol{u}^{(1:n)f} \\ \boldsymbol{w}\_{2}^{\boldsymbol{n}f} &= \Psi\_{21}^{(1:n)} \int\_{t\_{\boldsymbol{c}}}^{t\_{\boldsymbol{f}}} \boldsymbol{F}\_{1\boldsymbol{c}}^{1} dt + \Psi\_{22}^{(1:n)} \int\_{t\_{\boldsymbol{c}}}^{t\_{\boldsymbol{f}}} \boldsymbol{F}\_{2\boldsymbol{c}}^{\boldsymbol{n}} dt + \Psi\_{23}^{(1:n)} \end{split} \tag{56}$$
$$\begin{array}{cccc} & J \mathfrak{t}\_{\mathfrak{t}} & J \mathfrak{t}\_{\mathfrak{t}} \\ + & \Psi\_{24}^{(1:n)} \dot{\mathfrak{q}}^{(1:n)f\_{\mathfrak{t}}} + \Psi\_{25}^{(1:n)} u^{(1:n)f} \end{array} \tag{57}$$
Note that through the partial velocity vectors Ψ(1:*n*) *ij* ,(*i* = 1, 2 and *j* = 4, 5), these equations are linear in terms of the newly added generalized modal speeds as well as the generalized speeds associated with the released *do f* at the joints of the system.
The two-handle equations for the assembly at the primary node is solvable by imposing appropriate boundary conditions. Solving for the unknowns of the terminal handles initiates the disassembly process [1, 11]. In this process, the known quantities of the terminal handles of each assembly are used to solve for the spatial velocities and the impulsive loads at the common joint of the constituent subassemblies using the handle equations of each individual subassembly. This process is repeated in a hierarchic disassembly of the binary tree where the known boundary conditions are used to solve the impulse-momentum equations of the subassemblies, until the spatial velocities of the fine model and impulses on all bodies in the system are determined as a known linear function of the newly introduced generalized speeds of the fine model.
#### **6. Conclusion**
The method presented in this chapter is able to efficiently simulate discontinuous changes in the model definitions for articulated multi-flexible-body systems. The impulse-momentum equations govern the dynamics of the transitions when the number of deformable modes changes and the joints in the system are locked or released. The method is implemented in a divide-and-conquer scheme which provides linear and logarithmic complexity when implemented in serial and parallel, respectively. Moreover, the transition from a coarse-scale to a fine-scale model is treated as an optimization problem to arrive at a finite number of solutions or even a unique one. The divide-and-conquer algorithm is able to efficiently produce equations to express the generalized speeds of the system after the transition to the finer models in terms of the newly added generalized speeds. This allows the reduction in computational expenses associated with forming and solving the optimization problem.
### **Acknowledgment**
12 Will-be-set-by-IN-TECH
sub-domain level of the binary tree to combine the adjacent bodies and form the equations of motion of the resulting assembly. This process is recursively implemented as that of the binary tree to find the impulse-momentum equations of the new assemblies. In this process, the spatial velocity vector (after transition) and impulsive load of the handles at the common joint of the consecutive bodies are eliminated. The handle equations of the resulting assembly are expressed in terms of the constraint impulses and spatial velocities of the terminal handles, as well as the newly introduce modal generalized speeds and generalized speeds associated with the newly added degrees of freedom at the connecting joints. This process stops at the top level of the binary tree in which the impulse-momentum equations of the entire system
> <sup>1</sup>*cdt* <sup>+</sup> <sup>Ψ</sup>(1:*n*) 12
> <sup>1</sup>*cdt* <sup>+</sup> <sup>Ψ</sup>(1:*n*) 22
are linear in terms of the newly added generalized modal speeds as well as the generalized
The two-handle equations for the assembly at the primary node is solvable by imposing appropriate boundary conditions. Solving for the unknowns of the terminal handles initiates the disassembly process [1, 11]. In this process, the known quantities of the terminal handles of each assembly are used to solve for the spatial velocities and the impulsive loads at the common joint of the constituent subassemblies using the handle equations of each individual subassembly. This process is repeated in a hierarchic disassembly of the binary tree where the known boundary conditions are used to solve the impulse-momentum equations of the subassemblies, until the spatial velocities of the fine model and impulses on all bodies in the system are determined as a known linear function of the newly introduced generalized speeds
The method presented in this chapter is able to efficiently simulate discontinuous changes in the model definitions for articulated multi-flexible-body systems. The impulse-momentum equations govern the dynamics of the transitions when the number of deformable modes changes and the joints in the system are locked or released. The method is implemented in a divide-and-conquer scheme which provides linear and logarithmic complexity when implemented in serial and parallel, respectively. Moreover, the transition from a coarse-scale to a fine-scale model is treated as an optimization problem to arrive at a finite number of solutions or even a unique one. The divide-and-conquer algorithm is able to efficiently produce equations to express the generalized speeds of the system after the transition to the finer models in terms of the newly added generalized speeds. This allows the reduction in computational expenses associated with forming and solving the optimization problem.
(1:*n*)*fi* + Ψ(1:*n*)
(1:*n*)*fi* + Ψ(1:*n*)
*tf tc Fn*
*tf tc Fn*
<sup>2</sup>*cdt* <sup>+</sup> <sup>Ψ</sup>(1:*n*) 13
<sup>2</sup>*cdt* <sup>+</sup> <sup>Ψ</sup>(1:*n*) 23
<sup>15</sup> *<sup>u</sup>*(1:*n*)*<sup>f</sup>* (56)
<sup>25</sup> *<sup>u</sup>*(1:*n*)*<sup>f</sup>* (57)
*ij* ,(*i* = 1, 2 and *j* = 4, 5), these equations
are expressed by the following two-handle equations
<sup>1</sup> <sup>=</sup> <sup>Ψ</sup>(1:*n*) 11
<sup>2</sup> <sup>=</sup> <sup>Ψ</sup>(1:*n*) 21
Note that through the partial velocity vectors Ψ(1:*n*)
+ Ψ(1:*n*) <sup>14</sup> *q*˙
+ Ψ(1:*n*) <sup>24</sup> *q*˙
speeds associated with the released *do f* at the joints of the system.
*tf tc F*1
*tf tc F*1
*v* 1 *f*
*v n f*
of the fine model.
**6. Conclusion**
Support for this work received under National Science Foundation through award No. 0757936 is gratefully acknowledged.
### **Author details**
Mohammad Poursina, Imad M. Khan and Kurt S. Anderson *Department of Mechanical, Aeronautics, and Nuclear Engineering, Rensselaer Polytechnic Institute*
#### **7. References**
14 Will-be-set-by-IN-TECH
[14] Poursina, M. [2011]. *Robust Framework for the Adaptive Multiscale Modeling of Biopolymers*,
[15] Poursina, M., Bhalerao, K. D. & Anderson, K. S. [2009]. Energy concern in biomolecular simulations with discontinuous changes in system definition, *Proceedings of the ECCOMAS Thematic Conference - Multibody Systems Dynamics*, Warsaw, Poland. [16] Poursina, M., Bhalerao, K. D., Flores, S., Anderson, K. S. & Laederach, A. [2011]. Strategies for articulated multibody-based adaptive coarse grain simulation of RNA,
[17] Poursina, M., Khan, I. & Anderson, K. S. [2011]. Model transitions and optimization problem in multi-flexible-body modeling of biopolymers, *Proceedings of the Eighths International Conference on Multibody Systems, Nonlinear Dynam. and Control, ASME Design Engineering Technical Conference 2011, (IDETC11)*, number DETC2011-48383, Washington,
[18] Praprotnik, M., Site, L. & Kremer, K. [2005]. Adaptive resolution molecular-dynamics simulation: Changing the degrees of freedom on the fly, *J. Chem. Phys.*
[19] Scheraga, H. A., Khalili, M. & Liwo, A. [2007]. Protein-folding dynamics: Overview of
[20] Shahbazi, Z., Ilies, H. & Kazerounian, K. [2010]. Hydrogen bonds and kinematic mobility
[21] Turner, J. D., Weiner, P., Robson, B., Venugopal, R., III, H. S. & Singh, R. [1995]. Reduced variable molecular dynamics, *Journal of Computational chemistry* 16: 1271–1290. [22] Voltz, K., Trylska, J., Tozzini, V., Kurkal-Siebert, V., Langowski, J. & Smith, J. [2008]. Coarse-grained force field for the nucleosome from self-consistent multiscaling, *Journal*
[23] Wu, X. W. & Sung, S. S. [1998]. Constraint dynamics algorithm for simulation of semiflexible macromolecules, *Journal of Computational chemistry* 19(14): 1555–1566.
molecular simulation techniques, *Annu. Rev. Phys. Chem.* 58(1): 57–83.
of protein molecules, *Journal of Mechanisms and Robotics* 2(2): 021009–9.
PhD thesis, Rensselaer Polytechnic Institute, Troy.
*Methods in Enzymology* 487: 73–98.
*of Computational chemistry* 29(9): 1429–1439.
123(22): 224106–224114.
250 Linear Algebra – Theorems and Applications
DC.
## *Edited by Hassan Abid Yasser*
Linear algebra occupies a central place in modern mathematics. Also, it is a beautiful and mature field of mathematics, and mathematicians have developed highly effective methods for solving its problems. It is a subject well worth studying for its own sake. This book contains selected topics in linear algebra, which represent the recent contributions in the most famous and widely problems. It includes a wide range of theorems and applications in different branches of linear algebra, such as linear systems, matrices, operators, inequalities, etc. It continues to be a definitive resource for researchers, scientists and graduate students.
Photo by Selim Dönmez / iStock
Linear Algebra - Theorems and Applications
Linear Algebra
Theorems and Applications
*Edited by Hassan Abid Yasser*
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20-4-2021 17:35
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# **Climate Change and Air Pollution Effects on Forest Ecosystems**
Edited by Ovidiu Badea, Alessandra De Marco, Pierre Sicard and Mihai A. Tanase Printed Edition of the Special Issue Published in *Forests*
www.mdpi.com/journal/forests
## **Climate Change and Air Pollution Effects on Forest Ecosystems**
## **Climate Change and Air Pollution Effects on Forest Ecosystems**
Editors
**Ovidiu Badea Alessandra De Marco Pierre Sicard Mihai A. Tanase**
MDPI • Basel • Beijing • Wuhan • Barcelona • Belgrade • Manchester • Tokyo • Cluj • Tianjin
*Editorial Office* MDPI St. Alban-Anlage 66 4052 Basel, Switzerland
This is a reprint of articles from the Special Issue published online in the open access journal *Forests* (ISSN 1999-4907) (available at: https://www.mdpi.com/journal/forests/special issues/climate pollution ecosystem).
For citation purposes, cite each article independently as indicated on the article page online and as indicated below:
LastName, A.A.; LastName, B.B.; LastName, C.C. Article Title. *Journal Name* **Year**, *Volume Number*, Page Range.
**ISBN 978-3-0365-2666-9 (Hbk) ISBN 978-3-0365-2667-6 (PDF)**
Cover image courtesy of S, erban Mihai Chivulescu
© 2021 by the authors. Articles in this book are Open Access and distributed under the Creative Commons Attribution (CC BY) license, which allows users to download, copy and build upon published articles, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications.
The book as a whole is distributed by MDPI under the terms and conditions of the Creative Commons license CC BY-NC-ND.
## **Contents**
**Alexandru Claudiu Dobre, Ionut,-Silviu Pascu, Stefan , Leca, Juan Garcia-Duro, Carmen-Elena Dobrota, Gheorghe Marian Tudoran and Ovidiu Badea** Applications of TLS and ALS in Evaluating Forest Ecosystem Services: A Southern Carpathians Case Study Reprinted from: *Forests* **2021**, *12*, 1269, doi:10.3390/f12091269 .................... **169**
## **About the Editors**
**Ovidiu Badea** has a PhD ˆın Forestry and is a senior scientist and member of the Romanian Academy. He is a forest ecologist and specializes in dendrometrics, growth and forest ecosystems monitoring. He is an employee of the National Institute for Research and Development in Forestry (INCDS). Marin Dracea is a PhD advisor at Transylvania University of Brasov and has significant ˘ experience in climate change, air pollution and other stress factors effects on forests. Throughout his research career, he has led more than 45 competitively funded projects and published 58 ISI indexed journal articles, and the H-index according to WOS is 15. He has made significant contributions in the implementation of the Romanian National Strategies for Research, Forestry and Environment. Internationally, he developed an important collaboration with European institutions for use in monitoring the effects of climate change and air pollution on forest condition (ICP-Forests of UN/ECE) as head of the National Focal Center (NFC) since 1991. He is a representative for Romania in the IUFRO International Council, and since 2019, he has been the coordinator of the IUFRO Research WP 08.04.01—Detection and monitoring (between 2013 and 2019, he was the deputy of this IUFRO research group).
**Alessandra De Marco** is a biologist with a PhD in Bio-systematic and Plant Ecophysiology. She is a researcher at the Italian National Agency for New Technologies, Energy and Sustainable Economic Development (ENEA). She has been involved in IUFRO since 2014; from October 2019 she has been the coordinator of the RG 8.04.00 of Division 8 of the IUFRO. Since 2018, she has been the vice-chair of the Working Group on Effects under the LRTAP convention. She is a National Responsible for the implementation of ecosystem monitoring in the NEC Directive. She is involved in many national or European projects as Principal Investigator (FO3REST, MOTTLES, MITIMPACT, AIRFRESH, VEG-GAP, MODERN NEC). She is a member of the editorial board of *Environmental Pollution* and *Helyon*; she is an editor of Sustainability and Frontiers in *Forest* and *Global Change*. She is a member of the scientific committee of several meetings and is author of more than 100 scientific papers, with a Google h index of 37. Her main interest fields are: climate change and air pollution interactions and their synergistic impacts on natural and anthropogenic ecosystems; integrated assessment modeling to estimate beneficial effects of policies and measures to reduce air pollution; the modeling of air quality and its impacts on ecosystems.
**Pierre Sicard**, with a PhD in Atmospheric Chemistry, is working on air pollution and climate change impacts on forests ecosystems to reduce the risk for plant ecosystems. Currently, he is leading a European project AIRFRESH (2020-2025) to quantify and map the environmental and socio-economic benefits provided by urban trees at the city scale. He is involved in numerous national and EU-funded projects as coordinator (e.g., FO3REST, AIRFRESH) or Principal Investigator and steering committee (e.g., MOTTLES). He is very active in communication: he is Deputy Coordinator of the RG 8.04.00 "Air Pollution & Climate Change" under the International Union of Forest Research Organizations (IUFRO); he is involved in an UNECE Expert Panel on Clean Air in Cities and is active in the EU Clean Air Forum; he is a member of the Editorial Board of journals (e.g. *Environmental Research*, *Forests* and the *Atmosphere and Climate* journal); he is a member of the scientific committee of meetings and has published >80 papers, with an h-index of 32. He is also involved in a Regional Expert Group on Climate in the Provence-Alpes-Cote d'Azur region. ˆ
**Mihai A. Tanase** has a PhD in Spatial Planning and Environment, an MSc in Environmental Management, and a Diploma in Silviculture, which he attained at the University of Zaragoza (2010), CIHEAM (2006), and the University of Suceava (1999), respectively. He was a visiting scholar at CESBIO, Gamma Remote Sensing, University of Maryland and Boston University and was a postdoctoral fellow at the University of Melbourne. His research is focused on the use of remote sensing technologies for landscape monitoring, with an emphasis on forests. Dr. Tanase has been involved in 20+ national and international projects, has published over 65 scientific articles in JRC-indexed journals and conferences, has presented at 20+ national and international conferences, co-chaired sessions at two international conferences, was guest editor for three Special Issues and was part of the PLOS ONE journal editorial board. He led ten national and international projects and has supervised or co-supervised six master's and three PhD students.
## *Editorial* **Climate Change and Air Pollution Effect on Forest Ecosystems**
**Ovidiu Badea 1,2**
Climate change, air pollution, urbanization, globalization, demographic changes and changing consumption patterns affect forests and their social, cultural, ecological and economic functions, resulting in consequences for the social value of forests and for people's livelihoods, health and quality of life. These consequences are more acutely felt in regions where people are directly dependent on the environmental services provided by forests. Additionally, these consequences rapidly affect growing urban populations, as forests and trees make important contributions to urban resilience and human health and wellbeing.
Achieving a better understanding of the drivers of the changing relationship between forests and people is a major challenge of forest research and a prerequisite for the development of more sustainable relationships between forests and society. Additionally, scientific transdisciplinary and interdisciplinary research, at different regional levels, is needed in order to contribute to the successful adaptation of forests to climate change, and to the strengthening of tree health, resistance and resilience. At the same time, all scientific results are a precondition for maintaining and improving the potential to mitigate the effects of climate change and air pollution on the natural environment.
Climate change and air pollution have large negative impacts on physiological processes and functions at both an individual tree level and on the whole forest ecosystem. Our ability to take urgent measures to combat climate change and its impact on forest ecosystems, and conserve forest biodiversity, depends upon knowing the latest scientific results on the status of forest ecosystems.
At present, climate and air quality monitoring in forests around the globe is performed in different networks, by different organizations. Unfortunately, there are a lot of gaps in our knowledge concerning the detection and monitoring of the effects on forest ecosystems. There is a need to better understand the interactions and fluxes at an ecosystem level, and to understand how different pollutants and climate effects are reflected or transferred in quantifiable ecosystem variables, in both the short and long term. For the detection and monitoring of air pollution actions in the climate change context to be relevant, there is a need for better science–policy interactions. Using Earth Observation data for processing, validation and analysis, new technical developments may provide us with new results in air pollution investigations.
This Special Issue, "Effects of Climate Change and Air Pollution on Forest Ecosystems", includes 10 peer-reviewed contributions, dedicated to increasing the visibility of forest science in the European and global change research policy, and developing the link between forest science and practices in a changing environment and society. The topics addressed in these scientific articles refer to a range of themes, including: the promotion of adaptive management concepts; methods and techniques of restoring forest ecosystems; monitoring the forest's condition under climate change, atmospheric pollution and other biotic and abiotic stressors; the conservation of nature and forest-protected areas; forest genetic resource conservation; forest pests and diseases; and the value of the forest ecosystem services in the context of climate change, for the sustainable, adaptive, management of forests.
**Citation:** Badea, O. Climate Change and Air Pollution Effect on Forest Ecosystems. *Forests* **2021**, *12*, 1642. https://doi.org/10.3390/f12121642
Received: 23 November 2021 Accepted: 24 November 2021 Published: 26 November 2021
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**Copyright:** © 2021 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).
Nowadays, climate change and biodiversity losses are major challenges of global society. Forests, which cover one-third of the Earth's land surface, are an immense and renewable source of ecosystem services (ES). Understanding forest species interactions and their responses to past climates, as well as the different concepts of management, is critical in foreseeing forests' responses to future conditions, and in creating optimal strategies for climate change mitigation and adaptation. Thus, forest management, which is one of the main factors modifying forest structure and succession, can be used to promote resilient mixed forests, which are expected to accumulate a higher biomass quantity under intense climate change, and contribute to climate change mitigation and adaptation [1]. Additionally, research results suggest that climate change will alter the forests' composition and species abundance, with some forests being particularly vulnerable to climate change, e.g., F. Sylvatica forests in the Southern Carpathians. As far as productivity and forest composition changes are concerned, management practices should accommodate these new conditions, in order to mitigate the impacts of climate change [2]. In many cases, human activities change the condition of natural vegetation, leading to disturbances such as the degradation of vegetation, the erosion of soil, a decline in land productivity and even a reduction in ecosystem services. Gaining a better understanding of natural colonization with a pioneer woody species, for example by studying primary natural succession, can offer valuable knowledge about the species that are most adapted to these particular environmental conditions [3]. For the sustainable management of forests, knowledge on the increments of the main dendrometric characteristics of trees (diameter, height and volume) and the relationships among them, can contribute to adaptations in silvicultural work, with the purpose of reducing the risks generated by environmental factors at the stand level. Thus, the existence of stable stand structures is the main condition for an adaptive forest management [4].
Climate change and anthropic activities have given rise to serious environmental problems, and in an increasing number of ecosystems human influences are harming biodiversity and their functions. Through compiling the results of diagnostic assessments of damaged forest ecosystems by air pollution and reference information collected from intact natural forests, restoration plans have shown that ecological restoration is required urgently, as the extent of vegetation damage and soil acidification is very severe. However, tree growth recovery has been observed when the environmental condition has improved due to a significant reduction in air pollution [5,6]. Future climate change projections also underline the importance of hydrological assessments to investigate watershed behavior under climate-related risks and the endangered economic objectives, where it is necessary to intervene in protected forest areas. In this way, a hydrological model, the Soil and Water Assessment Tool, was built and tested in order to support decision makers in conceiving sustainable watershed management; thus, it has also contributed to guides that prioritize the most suitable measures to increase small river basin resilience, and ensure the water demand under climate change [7]. In order to ensure the benefits of forests in the future through the conservation and sustainable use of the forest tree species, silvicultural practices and forest adaptive management should increase, and maintain high genetic diversity and resilience within forest stands. One of the adaptive measures could be the selection, transfer and planting of highly productive and drought-resilient forest reproductive material in reforestation programs (assisted migration) [8].
To emphasize and maximize the ecological, social and economic benefits of forests, suitable assessment methods are required. Active remote-sensing technology, with proven advantages and characteristic limitations, can represent the foundation of multiple approaches, aiming to quantify the capacity of the forest ecosystem to provide services [9]. Representing an immense opportunity to mitigate climate change through carbon sequestration, soil stabilization and natural disaster mitigation, an integrative approach for valuing and assessing forest ES is needed, taking into account the many interdependent factors involving ES and their associated values, as well as the current challenges that people face [10].
All the scientific contributions to forest science in this Special Issue, "Effects of Climate Change and Air Pollution on Forest Ecosystems", will have an important role in promoting sustainable forest management based on mitigating the effects of climate change and air pollution on forests, and their adaptation to a changing environment and society in the global context.
This information will bring new knowledge concerning forests' conditions and their ecosystem service values in the context of climate change, air pollution and other biotic and abiotic factors. Additionally, they will promote adaptive management concepts, methods and techniques of restoring forest ecosystems based on nature and forest genetic resources conservation, for the sustainable and responsible adaptive management of forests.
**Acknowledgments:** We thank all the Guest Editors (representatives of IUFRO RG 08.04—Impacts of air pollution and climate change on forest ecosystems), reviewers and authors for their very fruitful work.
**Conflicts of Interest:** The author declares no conflict of interest.
#### **References**
## *Article* **Past and Future of Temperate Forests State under Climate Change Effects in the Romanian Southern Carpathians**
**Serban Chivulescu 1, Juan García-Duro 1,\*, Diana Pitar 1, S, tefan Leca <sup>1</sup> and Ovidiu Badea 1,2**
**\*** Correspondence: [email protected]
**Abstract:** Research Highlights: Carpathian forests hold high ecological and economic value while generating conservation concerns, with some of these forests being among the few remaining temperate virgin forests in Europe. Carpathian forests partially lost their original integrity due to their management. Climate change has also gradually contributed to forest changes due to its modification of the environmental conditions. Background and Objectives: Understanding trees' responses to past climates and forms of management is critical in foreseeing the responses of forests to future conditions. This study aims (1) to determine the sensitivity of Carpathian forests to past climates using dendrochronological records and (2) to describe the effects that climate change and management will have on the attributes of Carpathian forests, with a particular focus on the different response of pure and mixed forests. Materials and Methods: To this end, we first analysed the past climate-induced growth change in a dendrochronological reference series generated for virgin forests in the Romanian Curvature Carpathians and then used the obtained information to calibrate spatially explicit forest Landis-II models for the same region. The model was used to project forest change under four climate change scenarios, from mild to extreme. Results: The dendrochronological analysis revealed a climate-driven increase in forest growth over time. Landis-II model simulations also indicate that the amount of aboveground forest biomass will tend to increase with climate change. Conclusions: There are differences in the response of pure and mixed forests. Therefore, suitable forest management is required when forests change with the climate.
**Keywords:** temperate forests; climate change effects; Southern Carpathian forest management; forest growth; forest biomass; virgin forests
#### **1. Introduction**
Forests have a specific structure and function that determine their capacity to provide multiple ecosystem services, including C (carbon) sequestration [1,2], which has a main role in atmospheric CO2 (carbon dioxide) balance and, therefore, climate change mitigation strategies and agreements. Environmental conditions, particularly with regard to the soil and climate, play a significant role in forest structure development, forest succession, productivity, etc. Since the climate is continuously changing [3,4], forests also do [5,6]. Even when climate extremes do not have strong effects at the local level, forest changes are forced by direct [6] and indirect [7] climate change effects [8].
However, in terms of structure and composition, forests are often characterized by a high diversity and spatial heterogeneity [9,10]. Carpathian temperate forests, for instance, host both pure and mixed altitudinal forests [11,12], which have very different ecological preferences and properties [11,13,14]. Mixed stands proved to have high resistance and resilience and, often, a high level of productivity, compared to pure stands [15–19]. On the other hand, even-aged monospecific stands instead are less complex functionally and structurally complex [20]. Functional diversity, particularly in mixed forests, provides
**Citation:** Chivulescu, S.; García-Duro, J.; Pitar, D.; Leca, S, .; Badea, O. Past and Future of Temperate Forests State under Climate Change Effects in the Romanian Southern Carpathians. *Forests* **2021**, *12*, 885. https:// doi.org/10.3390/f12070885
Academic Editor: Steve Chhin
Received: 21 May 2021 Accepted: 4 July 2021 Published: 7 July 2021
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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).
redundancy [21,22], resistance [15,16,22] and resilience [16,17] to different disturbances. Having a relatively high forest integrity compared to other temperate regions in the world [10,18,19], the mixed forests in the Carpathians are usually constituted by unevenaged stands with high structural and functional diversity [9,23,24].
Characterized by their high stability and adaptability to climate change and stress factors [25] and by their high productivity [26], the mixed forest management model of the Carpathians might be a useful reference for other managed forests [23,27] in the region. Forest management, including its effect on forest structure and functioning, can be used to promote long-term forest resilience and resistance [22,28,29], contributing to C sequestration, as part of the atmospheric CO2 balance and climate change mitigation and adaptation strategies and specific programs [22,30]. However, a careful assessment of the forest's state is required.
Even though mixed stands benefit from functional diversity [31], they are also sensitive to management and changes in environmental conditions [31]. Proven drivers of mixed forest change include temperature rises [16] and drought [17]. Functional diversity guarantees greater stability at the stand and forest levels and can increase stress resilience and productivity under drought events [16], but climate change can trigger modifications in species abundance [16,32,33] and can, therefore, alter the structure and functioning of mixed forests, as well as the services they provide [1]. There is still high uncertainty regarding the local impacts of climate change and their potential changes in temperate forests [30], either pure or mixed, as well as their capacity for stocking biomass under regular management. There is also the need to reconstruct and to reanalyse past changes [8] in the context of climate change effects in order to explore future changes.
The complexity of the ecosystem processes and the need for reliable forecasts of climate change mitigation and adaptation actions make the use of a model approach that integrates plant–soil–climate relationships necessary. Landis-II models have been proven to be robust in simulating ecophysiological processes at the cohort level, including natural disturbances and management at the landscape spatial scale [34]. Landis-II models, coupled with the PnET model [35], cover local climate effects on vegetation development [35,36], and they are suitable tools for climate change forecasting [36]. However, their implementation is complex, because many parameters are intercorrelated, and many data inputs must be measured locally [36]. In addition, their calibration is complex and potentially affected by operator decisions [36]. Error propagation can affect the entire model, and the calibrated model can rarely be exported to other contexts, either geographical or temporal. Therefore, new alternatives for models' implementation are required, and this may be achieved by integrating ecological models with machine learning (ML) tools [37–39] and by utilizing the best data available, such as those from long-term ecological research databases [24], e.g., dendrochronological records.
It is well known that tree ring width is significantly influenced by climate variation [40], and therefore, tree ring cores provide a historical record of the climate variability (dendroclimatology) [41], forest growth and age structure, disturbances, site conditions (dendroecology) [42], etc. Climate reconstruction is commonly achieved through the study of the climate signals found in the tree rings [40–43] of open-grown trees [14,40,44], because they are more exposed to climate events and because isolated trees are less affected by competition and other ecological processes [14,45]. Non-isolated trees, mainly in virgin mixed forests, are the best option for tackling intra- and inter-specific interactions. Carpathian virgin forests, internationally recognized for their conservation value, for research and for the services they provide [22,25,46], together with other old-grown forests in the Curvature Carpathians, give an opportunity to deepen our understanding of the roles of mixed vs. pure forests role and their response to long-term processes [23,40–43], such as climate change [8,44,47], as well as their effects on biodiversity, productivity and C cycling [9,22,29]. Understanding species interactions and responses to past climates and forms of management is critical in foreseeing forest responses to future conditions and in creating optimal strategies for climate change mitigation and adaptation. Within
this context, this study aims (1) to determine the sensitivity of Carpathian forests to past climates using dendrochronological records and (2) to describe the future effects of forest management and climate change on Carpathian Forest biomass, with a particular focus on the different response of pure and mixed forests.
#### **2. Materials and Methods**
#### *2.1. Study Area*
The research area is located in the Penteleu Mountains within the Penteleu Forest District (Figure 1). It is a very representative area of Eastern Europe temperate forests specific to the Romanian Curvature Carpathians, at altitudes between 400 and 1800 m.a.s.l. The average annual temperature ranges between 4 ◦C and 6 ◦C. The minimum and maximum absolute temperatures recorded are −33.5 ◦C and +38 ◦C. The frost period ranges from 120 days at 800 m.a.s.l. to 220 days at the highest peaks. The average annual precipitation is 830 mm, mainly concentrated in summertime. The prevailing winds come from the NW, W and NE directions, with a 6–7 m·s−<sup>1</sup> annual average speed in high-altitude areas and 2.5–3.5 m·s−<sup>1</sup> in low-altitude areas.
**Figure 1.** Study area location in Southern Carpathians.
Most of the forest stands included in the research area are temperate montane mixed conifer–broadleaved forests. The main species in terms of timber volume are *Fagus sylvatica* L. (beech—34.6%), *Picea abies* (L.) H. Karst. (Norway spruce—38.1%) and *Abies alba* Mill. (silver fir—22.6%). Other deciduous and softwood species provide less than 4.7% of the overall volume [48]. There is no historical evidence that extreme events (such as windthrows, drought, insect damage, fires, etc.) had a relevant role in the development of the Penteleu forest over time.
Penteleu forest, similar to most of the Romanian Carpathian forests, suffered the effects of profound changes in its management. Being initially private property owned by local landowners and local communities, logging at the beginning of the 20th century was carried out at the pleasure of the owners and in accordance with market demand. However, forest integrity was not heavily affected, regardless of the massive harvests that occurred around a forest railway. Within this period, conifer stands were clear-cut and artificially regenerated afterwards. Mixed stands were subjected to selective cutting, extracting only conifers with a diameter greater than 30 cm, while beech was not extracted for commercial purposes, because it was considered, at that time, a species of low economic value. The Romanian forest nationalization of 1948 was an important step toward obtaining quality timber in maximum quantities, while, at the same time, satisfying the protection functions. Later, in 1975, the forest protection status generated by a change in the forest legislation (especially functional zoning of forests in particular) contributed to forest conservation. Nowadays, protected forests, in which no silvicultural prescriptions or only
special conservation prescriptions are allowed, occupy 39% of the total area [48]. The remaining areas are only eligible for selection cuttings (99.6%) and clear-cut prescriptions in a very small amount (0.4%). As a result, around 60% of the forest is older than 60 years old, with almost 20% of it being more than 100 years old [48], including over 200 hectares of virgin and quasi-virgin forests [49,50]. Within the context of temperate European forests, an important fraction of the Penteleu forest district area is occupied by relatively old forest.
The historical management of the different stands in Penteleu forest was reconstructed for the modelling approach following the ecological conditions and actual stand age structure and composition. In general, given the historical management changes and the information contained in the management plans [48], there is a high level of confidence in the historical management of the vast majority of the stands. An estimative historical management of Penteleu forest can be found in Supplementary Material S1.
The climate variables taken from van Oldenborgh et al.'s [51] multi-model mean climate change CMIP5 scenarios (https://climexp.knmi.nl/start.cgi; accessed on 1 July 2020) and additional climate series [43,52] for the region 25–26 E 45–46 N were statistically downscaled to 50 m resolution following Zorita et al. [53] using WorldClim 1.4 and 2.0 [54]. The downscaled monthly climate variables are the Tmin (monthly average of minimum temperature), Tmax (monthly average of maximum temperatures), PET (average monthly evapotranspiration), Prec (average monthly precipitation) and BAL (average monthly water balance). Spectral analysis of the climate variables is provided in Supplementary Material S2. Monthly radiation was also taken from WorldClim [54]. These climate products with past observations and future climate change forecasts until 2100 were used to analyse climate signal in dendrochronological series and in Landis-II simulations.
#### *2.2. Dendrochronological Series and Climate Signal*
In order to understand the principles of forests' structure and functioning, a deep analysis of the past and present structure and functionality of virgin forests was performed in a 1 ha control plot inside the Penteleu forest. This control plot (circular, with a 56.41 m radius) was installed and measured in 2014 in a representative temperate montane conifer– broadleaved mixed–virgin forest stand. The stand is unevenly aged, and the reference plot includes 439 trees from three species (beech—63%, Norway spruce—26% and silver fir—11%) [48]. Being unaffected by human activity, we assumed that this stand is optimal for detecting climate signals on tree growth and species interactions.
Tree spatial distribution, breast height diameter (DBH) and height (h) were measured for all living trees over 8 cm DBH. The aboveground tree volume (v) of each tree was calculated with the following regression equation: log v = b0 + b1·log d + b2·log2 d +b3·log h+b4·log2h, where b0, b1, b2, b3 and b4 are the species' national regression coefficients [55]. The overall stand volume measured was 803 m3. Tree aboveground biomass was computed using species wood density [55].
To build the reference dendrochronological series of silver fir, beech and Norway spruce, 406 radial core samples of living trees collected from the control plot were analysed and processed. The radial growth was calculated with CooRecorder 7.4 image analysis techniques [47,56]. Measurement checking and cross-dating was performed with COFECHA [42,57], and the growth series standardization was carried out with ASTRANwin [58]. The age structure was intensively analysed in an exploratory analysis to ascertain how stand development might have affected tree growth. DBH, measured through tree ring radial growth, was analysed by linear mixed models and analysis of variance in R software [59].
The statistical analysis of the dendrochronological series was performed with dplR [60], treeclim [61] and waveslim [62] packages in R software [59], and wavelet coherence among species dendrochronological series was explored with biwavelet [63]. The overall and moving correlations between climate variables (i.e., Tmin, Tmax, PET, Prec, BAL) and tree ring growth for the main species were calculated by treeclim dcc response and correlation function analysis [61]. In order to find interactions among species in mixed stands, the
moving correlation matrices were inspected. This was completed with a factorial analysis of the of the species' moving.
In addition, since there was a net change in the slope, the 1 ha plot was divided into 2 sectors during the fieldwork in order to explore and compare the relative species abundance in both sectors as a function of the local conditions. Being representative of 2 ecoregions, the reconstruction of the tree aboveground biomass in the 2 sectors was later used in the calibration of the Landis-II models.
#### *2.3. Climate Change Projections*
Landis-II [34,64], coupled with PnET models [35,65,66], was developed to simulate forest succession and the effect of climate [66] and management [67] under different climate change scenarios [66]. We built Landis-II models for the Penteleu forest district, aiming to assess management effects and the climate change impacts on Carpathian forest ecoregions, i.e., forests with homogeneous local conditions.
The model parameterization was performed following Gustafson [53] recommendations, using data extracted from scientific and technical works [12,68–71]. Since the full dataset had missing data issues, inconsistency, etc., the values were not used directly in the Landis-II model, but as input data for Landis-II calibration using the genetic algorithms (GA) [39,72] machine learning (ML) technique, with GA R package [38,39,59]. For calibration purposes, the GA fitness function for a given set of chromosome values used the difference in the biomass estimated for the 1 ha plot sectors (using the dendrochronological series) and the biomass predicted by the Landis-II models for those sectors (introduced in the model as fully independent ecoregions). In this, the supervised species calibration was bypassed, and species parameter values that are valid for the whole study area were set. Only the last period of the dendrochronological series was taken into account in order to minimize the effects of the stand dynamics effects on the calibration process.
After the calibration, the 50 m × 50 m resolution landscape was segmented in ecoregions combining the site type, forest types and relative productivity class in the management plans [48]. Soil properties for the ecoregions' parameterization were also extracted from the management plan [48]. Overall, 22 ecoregions, covering an altitudinal gradient and a wide range of site properties, were established. Every ecoregion was provided for monthly photosynthetically active radiation (PAR) from WorldClim 2.0 radiation [54], downscaled CMIP5 Tmax, Tmin and Prec data (https://climexp.knmi.nl/start.cgi; accessed on 1 July 2020), and atmospheric CO2 concentrations for the 1500–2100 period [3] under the 2.6, 4.5, 6.0 and 8.5 greenhouse gas (GHG) atmospheric concentration RCP scenarios [3].
The 1685 forest stands in the Penteleu forest district were aggregated into 568 management system types, from which 1369 treatment prescriptions (silvicultural regeneration interventions) were sequentially combined and implemented over time at the stand and management system levels, according to historical changes in forest management, species composition and abundance, silvicultural system and age structure. All of the spatial information was managed in R software [59] spatial libraries [73–76].
The silvicultural systems implemented in Landis-II are as follows: (i) strict protection (mostly mixed uneven-aged forests; all virgin forests, quasi-virgin forests and some pure stands are covered by this management type); (ii) old selection forests (predominantly mixed uneven-aged forests); (iii) selection forests (natural composition, often mixed stands, with an uneven-aged structure); (iv) quasi-selection systems (transformation to selection forests 60–70 years; natural composition with relative even to relative unevenaged structures); (v) group selection system (even aged with pure and mixed composition); (vi) clear-cut system (mostly even-age pure stands and some mixed stands with nonnative species).
The timeframe covered by the Landis-II models comprises four main management types and periods: (i) virgin forests and traditional selection forests (1900–1940); (ii) salvage logging (1941–1973); (iii) selection forests (1973–2020); (iv) increased protection of forests (2021–2100).
Disturbances (e.g., windthrows) were modelled through harvest extension, because (a) management plans show evidence of their low relevance and (b) the salvage logging, which is commonly practiced in disturbed stands, has similar effects to unplanned clearcuttings.
The Landis-II models, calibrated and validated for the entire research area, were computed for past conditions in the 1900–2010 period and for the 2010–2100 period under the RCP 2.6, 4.5, 6.0 and 8.5 climate change scenarios. Biomass and, implicitly, C stock, capitalizing on the energy and mass fluxes within the modelling approach, were the main target variables in this study. The ecoregions' biomass over time and the biomass in the natural distribution areas of pure and mixed forests were fitted and analysed using linear mixed models and an analysis of variance. Three key periods (1901–1910, 2001–2010 and 2091–2100) were targeted and compared. Comparisons between those periods were performed for every ecoregion and climate change scenario using Benjamini–Hochberg FDR correction [77]. The familywise and post hoc comparisons were carried out with nlme [78] and multcomp packages [79] in R software [59].
#### **3. Results**
#### *3.1. Forest Sensitivity to Past Climate*
Older trees in the 1 ha plot mixed–virgin forest stand are more than 350 years old and the maximum tree DBH is 110 cm. Exploratory analysis of the age structure and recruitment suggest that, despite the truncated diameter-based sampling, the age structure follows a geometric distribution, with a strong predominance of young age classes. This age structure has been quite stable over time, emphasized by the small peaks of recruitment detected over the last 300 years.
The analysis of the climate signal in a mixed–virgin forest stand indicates that tree radial growth depends on climate and interspecific relationships. Tree DBH, reconstructed from radial core samples collected in the Penteleu 1 ha plot, increased approximately linearly over time (Figure 2) at the stand level. The mean growth of the central 90% (0.05–0.95 quantiles) of trees in the plot ranged from 0.9 to 4.2 cm·year<sup>−</sup>1. At the tree level, radial growth has the general tendency to be constant over time, with a few exceptions. Radial growth changed strongly over time for a reduced number of trees: some trees with intense radial growth in the early stages of development gradually reduced their growth rate when they reached physiological maturity, while other trees with reduced radial growth in the early stages suddenly increased their growth rate in later stages. The dendrochronological dataset analysis showed a non-significant effect of the tree age on the tree ring width (*p*-value > 0.05).
**Figure 2.** DBH–age relationship in Penteleu 1 ha mixed–virgin forest plot.
All three biwavelets among pairs of species (Figure 3) contain regions with significant wavelet coherence among dendrochronological series of pairs of species. Anti-phase relationships dominate in the beech–silver fir biwavelet, while beech–Norway spruce and silver fir–Norway spruce biwavelets tend to be in phase when there is a lag in the phase.
**Figure 3.** Dendrochronological biwavelet between species, after 100 Monte Carlo randomizations. Arrows direction indicate that both dendrochronological series are in phase (pointing to the **right**) or anti-phase (pointing to the **left**), the black contour indicates ≤0.05 significance, and the transparency, the cone of influence.
Climate has a significant effect on tree ring growth. At the stand level, it was summarised the relationship between some of the main climate variables (i.e., Tmin, Tmax, PET, Prec and BAL) and the radial growth extracted from tree cores through moving correlations (details can be found in Supplementary Material S3), and then assessed the similarities between the responses of the three main species in the study area (silver fir, beech and Norway spruce) to the climate variables (Figure 4). The results showed synergic and antagonistic responses to the climate. Silver fir showed a similar trend to the Norway spruce, with the first not being as significant as the second. Silver fir tended to show negative correlations with climate variables in periods in which beech and Norway spruce growth were favoured. Beginning with the end of the 19th century and until the beginning of the 20th century, a consistently positive growth rate was recorded for beech, correlated with higher-than-average summer temperatures. Beech's correlation with September Prec and BAL varied over time, from a negative correlation to a positive one. Beech growth also had a positive correlation with late spring Prec. Norway spruce, in general, tended to show negative correlations with climate variables in the same periods in which beech was favoured by the climate. Beech moving correlations around 1750 showed significant negative correlations with September Prec and BAL, as in 1840. Additionally, around 1840, positive growth was associated with spring Prec and BAL. Later, around 1900, positive correlations were detected with spring Tmin, Tmax and PET. Around 1965, growth was positively correlated with September Prec and BAL and negatively correlated with July and September Tmin. Silver fir growth correlated negatively with spring Tmin and PET and positively with late-summer Prec and BAL around 1970. During the 19th century, silver fir growth was negatively correlated with May Tmin, Tmax and PET, although it was positively correlated with September and April Tmin and Tmax and also with May Prec and BAL. At the beginning of the 20th century, silver fir growth was negatively correlated with late-summer Tmin, Tmax and Prec and positively correlated with April Prec and BAL. Later, occasional positive significant positive correlations were detected with Tmin, Tmax and PET, while negative correlations were detected with July and August Prec and August BAL. Finally, no correlations were found for the 21st century. Norway spruce growth showed positive significant correlations with September Tmin, Tmax and PET around 1825, though negative correlations were found with April Prec and BAL. Later, until 1990, positive correlations tended to occur with spring and summer Tmin, Tmax and PET, particularly around 1875, 1925 and 1970. Significant positive correlations with April and June Prec were detected in the last decades of the 19th century and at the beginning of the 20th century. Finally, around 1945 and 1985, Norway spruce growth was eventually negatively correlated with late-summer Prec and BAL.
**Figure 4.** Saturation of the first and second components of the factorial analysis of beech, silver fir and Norway spruce moving correlations between tree ring growth and monthly climate variables (from April to September, respectively: Apr, May, Jun, Jul, Aug, Sep), here sorted by climate variable (i.e., Tmin, Tmax, BAL, PET, Prec).
Thus, the first and second components of the factorial analysis of the moving correlation matrices were built based on the antagonistic behaviour of beech and silver fir. Norway spruce displayed an intermediate behaviour, and consequently, less weight was placed on the construction of the first and second components.
Tree biomass reconstructed from tree cores showed that, at the stand level, around one third of the current biomass in the Penteleu 1 ha plot was produced more than 150 years ago and approximately half was formed in the last 50 years (Figure 5a). Regarding species biomass, before 1850, most of it belonged to beech. After 1900, only half of the biomass found was attributed to beech, with the remaining biomass being provided by conifers, particularly silver fir.
Due to the plot's heterogeneity and microtopography, we compared the biomass accumulation on steep and flat terrain. Before 1850, the accumulated tree biomass was higher in the flattest sector of the plot compared to the steepest one. After 1950, the biomass per hectare in the steepest sector (Figure 5b) was equal to the biomass in the flat sector. As extracted from the dendrochronological series, beech and silver fir are the dominant and codominant species in both sectors, with silver fir being dominant in the steep sector and beech in the flat. Norway spruce, having the lowest contribution in terms of biomass, accumulated more than 1.5 times the biomass per hectare in the flat sector, as compared to the amount of biomass measured in the steep sector.
**Figure 5.** Biomass (tonnes) reconstruction based on tree rings (**a**) at the stand level (continuous) and (**b**) the relative abundance of the main species for the steepest (dotted) and flattest (continuous) sectors in the Penteleu 1 ha plot (black), beech (orange), silver fir (green), Norway spruce (blue).
#### *3.2. Aboveground Living Biomass under Climate Change Scenarios*
Landis-II simulations show that the forest biomass changed from 1900 to 2010 in almost all of the ecoregions (Figure 6). Some ecoregions showed a moderate increase in biomass throughout the period. Other ecoregions hardly changed, while the remaining few displayed a reduced amount of biomass. Biomass reductions in the ecoregions were often associated with the harvesting process.
Harvesting caused very strong biomass reductions in some ecoregions, particularly through clearcutting during the 1941–1973 period. However, biomass reductions were also found in other periods, harvest types and management systems, such as selection cuttings in shelterwood systems. The rise in the level of forest protection, together with the aging of the forest, contributed to a reduction in biomass loss and to moderate biomass accumulation in recent decades. In ecoregions with null or very low harvest intensity, the amount of biomass tends to increase.
**Figure 6.** Average quantity of biomass (tonnes/ha) in the 22 ecoregions (different line styles) under the four climate change scenarios: RCP 2.6, RCP 4.5, RCP 6.0 and RCP 8.5.
Management, which, in the past, caused strong reductions in biomass through intense harvest, will contribute to biomass accumulation in the future through protection and low harvest pressure. The Landis-II projections show that, from 2010 to 2100, under a more protective management, the amount of biomass in most of the ecoregions will increase, particularly after 2025.
Apart from harvest effects, the different climate change scenarios (RCP 2.6 to RCP 8.5) had a significant role on forest biomass accumulation after 2010 (*p*-value ≤ 0.05). This increase in biomass was particularly noticeable in periods where harvesting was less intense. The intensity of the growth differed strongly among the ecoregions. After 2010, the accumulation of biomass is a result of a combination of both local ecoregion conditions, management and climate change scenarios. There is no common response to climate scenarios for all ecoregions, even though intense climate change scenarios, particularly RCP 8.5, tend to increase biomass accumulation. Indeed, some ecoregions have the lowest accumulated biomass under intermediate climate change scenarios (RCP 4.5 and RCP 6.0) and the highest under RCP 8.5. The amount of accumulated biomass barely changed over time in some ecoregions.
The ecoregions, aggregated according to their domination by pure and mixed forests (Figure 7), had different biomass accumulated at the beginning of the study period and over time. In 1900, when salvage logging had not begun, the biomass per hectare in pure stands was higher than in mixed stands. Around 2010, the amount of biomass was still higher in pure forest ecoregions. However, in the last period of the simulations (up to 2100), the biomass per hectare in mixed forests was similar to or even higher than that of pure forests, particularly in RCP 8.5, where the biomass per hectare in mixed forests broadly surpasses that of pure forests.
**Figure 7.** Average quantity of biomass (tonnes/ha—width lines) in pure and mixed forests from 1990 to 2100 and confidence intervals (shaded areas).
#### **4. Discussion**
This study shows that, even though virgin forests have a proven stability, both tree radial growth and biomass are affected by climate change and species interactions, thus providing long-term faithful records that are unaffected by management. In addition, it explores the reconstruction of the forest stand aboveground biomass in the past, projection in the future under different climate change scenarios and the C sequestration, as part of the atmospheric CO2 balance.
Tree DBH reconstructed from Penteleu dendrochronological series was approximately linear, as also found in other similar studies [8,31,80]. Thus, radial growth is not particularly affected by tree age. Radial growth and, thus, the DBH–age relationship, is conditioned by the environmentally limiting conditions [80] and by biological stress. Additionally, the change in the growth rates of individual trees often depends on their interactions with neighbouring trees (e.g., facilitation and competition) and other processes (e.g., mortality) [11,81], meaning that they are likely the cause of the changes found in the measured trees in Penteleu.
We found that small differences in the ecological preferences, local conditions and climate lead to several spatiotemporal patterns in the biomass accumulation over time. In mixed–virgin forest stands, intra- and inter-specific interactions have a strong impact on tree growth, while also having further consequences at the stand level. Such interactions differ in relation to local conditions and species dominance and abundance, which can lead to modifications even at short distances. The results showed that there are differences between growth rates of some trees in early vs. late stages, which (presumably) strongly depend on the physiological activity and dynamics of the in-stand competition and mortality [11,81] of the species. In the Penteleu virgin forest 1 ha plot, the steepest areas are dominated by silver fir, which is less susceptible to warmer and drier conditions than beech [13], while flat areas are occupied by beech, which is also accompanied with a higher abundance of Norway spruce. Having a higher abundance in the flattest area, beech, compared to silver fir and Norway spruce shows better resistance to different stress factors and also has a significant growth in the understory [23,81]. Due to withstand such conditions (including drought), beech growth usually has a low variability [11] compared to coniferous species. These heterogeneous responses to the climate are produced in mixed forests even when the climate conditions are favourable for all three species.
The dendrochronological series analysis proved that silver fir and beech tend to have opposite behaviours in response to climate in mixed stands, while Norway spruce, displaying intermediate behaviour, tends to negatively correlate with the climate when beech correlations are positive. Having different ecological preferences and life traits, the beech–silver fir association in mixed stands is advantageous in terms of their ability to endure drought [13,17,31,33,82] and also because of their efficient light use [83,84]. There
is also evidence that beech–Norway spruce mixed stands perform better than beech pure stands [13,33] due to the complementarity of the two species.
The overall biomass, capitalizing all ecological processes, is homogeneously distributed over the 1 ha plot, highlighting the stability and species complementarity in mixed forests, as other studies have suggested [13,18,19,33]. However, despite their high efficiency in terms of the use of resources, there are concerns on the response of mixed and pure forests to climate change (e.g., Piovesan et al. [32]). It was emphasized that pure beech forests have already been affected by climate-change-induced drought in recent decades [32], as Norway spruce shows high drought resistance when compared to beech [17,82,85]. Additionally, silver fir is expected to be less susceptible to warmer and drier conditions than beech [13].
Since tree ring measurements include the effects of past conditions, such types of long-term data can be used in a modelling approach to analyse past changes and to foresee, anticipate and manage future changes [86,87]. However, the complexity, variability and information gaps contained in biological records [87,88] often require solutions such as ML methods [37,86]. GA [38,39] provided a suitable mechanism to speed up and objectivize the modelling approach and was successfully used in this study for Landis-II model calibration in order to simulate landscape changes and for forest change forecasting under different climate change scenarios, which proved to be robust.
The Landis-II models demonstrate that management had a strong impact on forest structure and functioning over time, in agreement with official records and statistics [48]. Forest resilience to management in terms of relative biomass compared to the pre-harvest level is high at the ecoregion level. Biomass commonly recovers in less than 50 years, although impacts on other forest attributes, e.g., diversity, last much longer [48]. Forest resistance to management is very low, as denoted by both (a) the impact of punctual harvest events at the ecoregion level and (b) sustained biomass stock changes due to the change in the silvicultural system, following stakeholders and policymakers' decisions in the past and expectations after 2025, when even more protective management will be implemented.
Biomass change in the different ecological regions is not only related to management, but also to climate change effects. The most important impacts of climate change in the Carpathians will be caused by rising temperature and CO2 levels. Precipitation will be similar to the current conditions in some areas and may even slightly increase [4]; therefore, drought events are not expected to have a significant role, other than in a few ecological regions and marginal areas of the Carpathians [4]. Temperate forests will benefit from relative stability and better conditions for growth [2,4], contrary to Mediterranean forests, where droughts and fires will be the main drivers of change [89].
Some areas in the Carpathians will endure harsher conditions [4], and our results indicate that impacts will also differ at the local scale, with some ecoregions being more affected than others.
Uncertainty of climate change impacts and characteristics on forest ecosystems is relatively high [30,90], and our results show that temperate forest resistance to climate change is relatively low; however, forest biomass is predicted to be relatively similar under different climate change scenarios. Despite the findings of Nabuurs [2], projections from our Landis-II model also show that the Penteleu forest's C sequestration capacity is still far from saturation, and the aboveground biomass will continue to grow at least until the end of the current century, even when signs of deceleration appear at the end of the period in some ecoregions.
It is important to note that this C pool-growing trend is also related to a rise in the level of forest protection provided, forest aging and the decreasing harvest pressure throughout the period. Thus, the management of temperate forests has the potential to increase carbon sinks and mitigate climate change [1].
On the other hand, adaptation strategies can focus on sustainable management. The general interest is to promote C sequestration and management should be oriented toward forest adaptation to climate change, by reducing potential negative climate change impacts, e.g., drought events [2,89], etc.
In this regard, management can be used to promote mixed forests, which reach higher biomass accumulation under intense climate change scenarios, and thus, to maximize C sequestration. Mixed stands and their particularities contribute to forest resistance and resilience at the landscape level, in comparison to pure stands [15–17].
#### **5. Conclusions**
The dendrochronological series in mixed stands not only contain climate signal, but species interactions, and they are useful for past events and projections. Beech, Norway spruce and silver fir, whose radial growth was not dependent of tree age, have complementary responses in mixed stands.
PnET-Landis-II models, GA calibrated using dendrochronological records, applied to the Penteleu forest, which hosts virgin and managed stands, brings to light (i) the relevance of virgin forests in vegetation long-term monitoring and as a reference data source for all forests in the Carpathians, (ii) the similarities and differences in the structure and functioning of mixed and pure forest stands and their adaptative management and (iii) the effect of climate on tree growth and climate change impact on forests, in which mitigation and adaptation strategies, programs and actions must be implemented.
Landis-II models provide valuable insights into the medium- to long-term succession for forest management and forest planning. They provide a good approach to climate change impacts, which are becoming increasingly accentuated. Landis-II simulations revealed that aboveground forest biomass changed in almost all ecological regions in the Carpathian Curvature, often associated with harvesting. Forest projections demonstrate that climate change in the Carpathian Curvature will promote forest aboveground biomass and C sequestration. Climate change will modify species and cohort interactions, leading to changes in the forest structure and functioning. Climate change impacts will depend on climate change intensity and local conditions and management. Among forest ecosystems, mixed forests are capable of higher C sequestration and higher biomass accumulation under intense climate change.
In this study, management, which is one of the main factors modifying forest structure and succession, can be used to promote resilient mixed forests, which are expected to accumulate higher biomass quantity under intense climate change, and thus, contributes to climate change mitigation and adaptation.
**Supplementary Materials:** The following are available online at https://www.mdpi.com/article/10 .3390/f12070885/s1, Figure S1: Estimative historical management, Figure S2: Spectral analysis of climate variables (i.e., Tmin, Tmax, PET, Prec, BAL), Figure S3: Moving correlations between beech (*F. sylvatica* L.), Norway spruce (*P. abies* (L.) H. Karst.) and silver fir (*A. alba* Mill.) tree ring growth and climate variables (i.e., Tmin, Tmax, PET, Prec, BAL).
**Author Contributions:** Conceptualization, J.G.-D. and S.C.; methodology, J.G.-D., S.C. and S, .L.; software, J.G.-D. and S.C.; validation, J.G.-D., S.C., D.P. and O.B.; formal analysis, J.G.-D.; investigation, J.G.-D. and S.C.; resources, S.C. and J.G.-D.; data curation, J.G.-D.; writing—original draft preparation, J.G.-D. and S.C.; writing—review and editing, J.G.-D., S.C., D.P., S, .L. and O.B.; visualization, J.G.-D.; supervision, S.C. and O.B.; project administration, S.C. and O.B.; funding acquisition, S.C. and O.B. All authors have read and agreed to the published version of the manuscript.
**Funding:** This research was funded by the BIOSERV Programme, Project IDs PN19070103, PN19070102 and EO-ROFORMON project, ID P\_37\_651/SMIS 105058.
**Institutional Review Board Statement:** Not applicable.
**Informed Consent Statement:** Not applicable.
**Conflicts of Interest:** The authors declare no conflict of interest.
#### **References**
## *Article* **Shifts in Forest Species Composition and Abundance under Climate Change Scenarios in Southern Carpathian Romanian Temperate Forests**
**Juan García-Duro 1,\*, Albert Ciceu 1,2, Serban Chivulescu 1, Ovidiu Badea 1,2, Mihai A. Tanase 1,3 and Cristina Aponte 1,4**
**Abstract:** The structure and functioning of temperate forests are shifting due to changes in climate. Foreseeing the trajectory of such changes is critical to implementing adequate management practices and defining long-term strategies. This study investigated future shifts in temperate forest species composition and abundance expected to occur due to climate change. It also identified the ecological mechanisms underpinning such changes. Using an altitudinal gradient in the Romanian Carpathian temperate forests encompassing several vegetation types, we explored forest change using the Landis-II landscape model coupled with the PnET ecophysiological process model. We specifically assessed the change in biomass, forest production, species composition and natural disturbance impacts under three climate change scenarios, namely, RCP 2.6, 4.5 and 8.5. The results show that, over the short term (15 years), biomass across all forest types in the altitudinal gradient will increase, and species composition will remain unaltered. In contrast, over the medium and long terms (after 2040), changes in species composition will accelerate, with some species spreading (e.g., *Abies alba* Mill.) and others declining (e.g., *Fagus sylvatica* L.), particularly under the most extreme climate change scenario. Some forest types (e.g., *Picea abies* (L.) karst forests) in the Southern Carpathians will notably increase their standing biomass due to climate change, compared to other types, such as *Quercus* forests. Our findings suggest that climate change will alter the forest composition and species abundance, with some forests being particularly vulnerable to climate change, e.g., *F. sylvatica* forests. As far as productivity and forest composition changes are concerned, management practices should accommodate the new conditions in order to mitigate climate change impacts.
**Keywords:** LANDIS-II; PnET; climate change; Southern Carpathians; forest biomass; production; species composition; species abundance; Romanian temperate forests
#### **1. Introduction**
Human-induced climate change is one of the major processes affecting the global environment nowadays [1]. However, the impacts are so complex and diverse that the net effect of climate change on forest systems is still uncertain. For instance, while the increase in atmospheric CO2 concentrations often leads to greater productivity [2,3], the aridity caused by warming [4,5] and the intensification of disturbances regimes [6] usually have a negative impact on forests' structure and productivity. Recent studies have suggested that despite the positive climate change-associated effects [7,8], the negative ones often prevail [9,10].
Chivulescu, S.; Badea, O.; Tanase, M.A.; Aponte, C. Shifts in Forest Species Composition and Abundance under Climate Change Scenarios in Southern Carpathian Romanian Temperate Forests. *Forests* **2021**, *12*, 1434. https://doi.org/10.3390/ f12111434
**Citation:** García-Duro, J.; Ciceu, A.;
Academic Editor: Daniel J. Johnson
Received: 22 September 2021 Accepted: 18 October 2021 Published: 21 October 2021
**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.
**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).
Climate change effects, both positive and negative, will trigger quantitative and qualitative changes in forest composition, structure and functioning and will push for species adaptive responses [1,11]. Plant responses to climate change will be species specific [4], though species with similar vital attributes are expected to have close responses [5,12]. Thus, angiosperms and gymnosperms/conifers have different ecophysiological responses to raised CO2 concentrations, potentially affecting the regional response of each forest type [12].
Temperate forests occupy 1097 M ha worldwide, retain 118.6 Pg of terrestrial carbon [13] and provide 40% of the world's forest harvest [14], thus playing an important role in the overall carbon balance. Shifts in temperate forests' structure, functioning and distribution, such as those driven by climate change and mitigation measures, will affect their economic value [15] and will have notable consequences on their capacity to sequester carbon and to provide other ecosystem services [10,16–19]. Forest carbon stocks and sequestration primarily depend on forest productivity and disturbance regimes [10,14], both of which are expected to change with climate change [6,18]. A study by Nabuurs et al. [19] described the first signs of carbon sink saturation in European forests. Such weakening in C sink/sequestration capacity may be related to aging forests, decreased summer humidity, etc. A lower C sequestration capacity would also be related to the change in disturbance regimes. The main disturbances in temperate forests include drought, insects, windthrows, pathogens and fire [6]. However, disturbances are often interconnected and can generate significant feedback effects [6]. This is, for instance, the case of bark beetle attacks, which are secondary disturbances that tend to occur after drought, harvest or windthrows.
Other studies have already suggested that some temperate forests will eventually decline due to summer droughts [20] and their habitat distribution will be reduced to montane areas due to altitudinal shifts [21]. Some temperate forests will be more intensely affected than others [6,10,16], with vulnerability being related to species composition [16], health status, disturbances regimes [22] and the phytogeographical context [23]. In this context, the evolution of European temperate forests under climate change is largely uncertain, with studies suggesting that conifers will be severely damaged by climate change [16], while beech (*Fagus sylvatica* L.) forests, already affected by drought in recent decades [20], will be more sensitive to climate change. Given the gaps in the knowledge and the relevance of mixed (broadleaf and conifer) forests [24–26], such discrepancies require further research.
This study aimed to unveil the future effects of climate change and its interactions with natural disturbances and land management on the structure and composition of temperate forests by implementing a forest simulation modeling approach. The Southern Carpathian forests were used as a case study as they harbor a large diversity of forest types and are representative of European temperate forests. The forest simulation model LANDIS-II [27] coupled with the PnET model [28,29], which have been highlighted for their capacity to model complex forest dynamics under multiple interacting drivers (e.g., climate, management, pests, windthrows) [30–33], was used to increase our understanding of the potential impact of natural disturbances on temperate forest composition, structure and its productivity. In particular, this study addressed two main questions: (1) What future changes in forest species composition and abundance are expected to occur due to climate change? (2) What are the ecological mechanisms underpinning forest type and species (variable effects—geographical and species dependent) changes?
We hypothesized that species in the lower part of the altitudinal gradient (e.g., oaks) will grow in abundance and those in the high altitudes, particularly conifers, will be constrained, and that the main mechanisms driving the change are warmer climate and increments in droughts and disturbances, all of them modifying species interactions.
#### **2. Materials and Methods**
#### *2.1. Study Area*
The study area is located in the Fagara¸s Mountains, in the Romanian Southern Carpathians (Figure 1). It occupies around 6400 km2 and comprises a large geographical gradient that extends 110 km north to south and 73 km east to west, covering the main forested lands in the Arge¸s, Sibiu and Bra¸sov counties. The altitude ranges from 185 to 2544 m.a.s.l, with plain landscapes (c. 12% average slope, elevation < 500 m) on the southern sector and steep and rough topography (c. 32% average slope, elevation > 1000 m) on the high-altitude northern sector. The climate varies along the altitudinal gradient [23]: the annual mean temperature in the highlands is lower than 4 ◦C, with a winter mean temperature below −5 ◦C and a summer temperature around 14 ◦C. The precipitation approaches 1000 mm, with summer rainfall of around 315 mm. The climate in the plains is warmer and drier, with a 10.5 ◦C annual mean temperature, mean winter temperatures below 0 ◦C and summer temperatures over 20 ◦C. The annual precipitation here is about 700 mm, with summer rainfall of around 260 mm. In general, September and October are the driest months, while May and June are the wettest ones. Soil types in the area include protosols, spodozols, cambisols, argiluvisols and hydromorphic soils with pseudogleic properties [34,35].
**Figure 1.** (**a**) Location of the study area within Europe and (**b**) the Carpathians; (**c**) distribution of the five forest types within the study area.
Temperate forests, whose composition changes across the altitudinal belts, cover 52% of the total study area. There are five general forest types: (a) *Picea abies* (L.) karst forests, (b) mixed *Fagus sylvatica* L.–conifer forests, (c) *F. sylvatica* forests, (d) mixed broadleaved forests and (e) *Quercus* forests. *Picea* forests (Norway spruce) occupy the subalpine and montane superior belts. Mixed *Fagus*–conifer forests, mainly *Picea* and *Abies alba* Mill., dominate the intermediate montane belt, with the beech abundance increasing as the altitude decreases. Mixed *Fagus*–conifer forests are substituted by pure *Fagus* forests first and mixed broadleaved forest (*F. sylvatica* and *Carpinus betulus* L.) later, both of them in the inferior montane belt [24,36]. Oaks gradually appear in mixed broadleaved forests in the colline belt and, in low-altitude areas, reach the point where *Quercus* forests dominate the landscape. Within *Quercus* forests, there is an altitudinal transition from *Quercus petraea* (Matt.) Liebl. to *Quercus robur* L., *Quercus cerris* L. and *Quercus frainetto* Ten. in the most southern thermophilus areas [36].
Some of the forested lands are actively managed for timber production, with *F. sylvatica* providing 34% of the overall standing wood volume, *P. abies* 28%, *Quercus* spp. 17%, *A. alba* 5% and other species the remaining 16% [37–46]. Depending on forest species and conditions, different management systems are implemented including tree selection, shelterwood and clearcutting. Clearcutting is practiced only for small areas (lower than 3 ha) in Norway spruce (*P. abies*) and non-natural forest stands [47,48]. The rest of the forested land is either subjected to conservation management to protect forest health (e.g., phytosanitary felling, trees affected by small local windthrows) or is strictly protected with no management actions allowed. The most common natural disturbances in the area include windthrows and insect attacks [49]. Windthrow events mostly affect conifers in the north sector of the study area [50], particularly *Picea* forests, with insect outbreaks (*Hylobius abietis* L., *Ips duplicatus* Sahlberg and *I. typographus* L.) being a common secondary disturbance following windthrows [51] and drought [49,52] in conifer forests.
#### *2.2. Landis-II Model*
Landis-II is a collection of spatially explicit forest landscape models [53] that simulate forest change as a function of succession and disturbances [33]. The landscape in Landis-II is defined as a grid of cells (here 200 × 200 m), each of which belongs to an ecoregion (i.e., areas of homogeneous soil and climate) and can contain multiple species cohorts that can be independently killed by disturbances, competition or age-related mortality, as the succession progresses. Landis-II integrates a number of ecological process models through its modular design. Here, we used the PnET succession extension [54] to underpin tree species establishment, growth, mortality and decomposition. This extension embeds elements of the PnET ecophysiology model of [55] and accounts for competition for available light and water. Biomass growth is the result of a number of processes (e.g., photosynthesis, evapotranspiration) controlled by species ecophysiological parameters (e.g., foliar N concentration, photosynthetic rates) given a number of conditions that include precipitation, temperature and atmospheric CO2 concentration. Climate change and CO2 enrichment are interwoven as change in environmental parameters over time, making the PnET extension convenient for climate change modeling [30,56,57].
Landis-II interdependent disturbance extensions were used to model the impacts of harvest [31,58], wind [59] and insect outbreaks [60,61]. The harvest module [62] implements prescriptions in different management areas according to a temporal schedule, management systems and stand characteristics, including resource availability, age structure or stand composition. The biological disturbance agent module [63] implements insect impacts based on pest species preferences and resource availability. The wind module [64], which models windthrow events and operates independently of climate, was used to trigger insect outbreaks.
#### *2.3. Landscape Design: Ecoregions and Forest Communities*
The initial landscape for Landis-II simulations was built based on the local management plans that were available for the public forests (approximately 60% of the total forest land) [37–46]. Management plans contained the spatial delimitation of forest stands and information of their species composition and cohort ages, with stands ranging from 0.1 to 50 ha, the legal maximum allowed, with a median of 3.25 ha. Stands were classified into 14 ecoregions of homogeneous climate, soil type and tree species abundances (Supplementary Table S1; Supplementary Figure S1). Ecoregions were ascribed to one of the five forest types according to species composition (Figure 1). To constrain the number of communities, i.e., cohorts and species combinations that conform to a forest stand, a total of 223 initial communities, comprising from 1 to 4 species and a range of 1 to 17 age cohorts, were identified in the management plans and assigned, based on their frequency, to the corresponding stand and ecoregion in the simulated landscape. The area of private forest lands, for which information was not available, was delimited based on Corine Land Cover 2012 [65]. To ascribe ecoregions and initial communities in private forests, a random forest algorithm [66]
was trained using information from the management plans and environmental variables (DEM, [67], climate [68], soil properties and classification [34] and distance to infrastructure and populations [69,70]). The overall accuracy of the prediction reached 96.7% over the independent testing dataset.
#### *2.4. LANDIS-II Parameterization*
The modeled species included *Abies alba* Mill. (silver fir), *Alnus glutinosa* (L.) Gaertn. (European alder), *Alnus incana* (L.) Moench (grey alder), *C. betulus* (European hornbeam), *F. sylvatica* (European beech), *P. abies* (Norway spruce), *Q. cerris* (Turkey oak), *Q. frainetto* Ten. (Hungarian oak), *Q. petraea* (Matt.) Liebl. (sessile oak) and *Q. robur* L. (pedunculate oak). Species contributing less than 2% to the overall wood volume in the study area were not included in the simulation. Tree species parameters required by the model were measured in the study area [71], compiled from unpublished data or obtained from the local Romanian literature [36,72,73] and other international sources [74,75]. Model parameterization was performed following the recommendations of [29]. The three main background disturbances of the studied area were modeled to increase the accuracy of the simulated landscape and were considered a key element in this study.
The wind disturbance extension required the windthrow return interval and severity, which were modeled/introduced based on national historical records [76]. The average wind speed [68] and the area occupied by conifer forests were used to determine the windthrow local impact on each ecoregion. The mortality of age cohorts was based on Popa [76]. It was assumed that wind event occurrence will not be affected by climate change.
Bark beetle outbreaks were linked to windthrows and were followed by harvesting interventions as per standard practices in the study area. Three main bark beetle species were modeled: *I. typographus* and *I. duplicatus,* which target mature *Picea* stands [49,77], and *H. abietis*, which targets stands with a high density of *A. alba* saplings [78]. Pest species preferences (host species and age cohort) and impacts were defined based on expert knowledge, forest health status monitoring [79–81] and national Romanian research studies [49,51,52,82].
Harvesting prescriptions were defined following the national Romanian regulatory framework [47,48,83,84] which establishes that logging cycles for the main species in the study area are between 100 and 200 years for beech, Norway spruce, sessile oak and silver fir, 70 to 160 years for other oaks and up to 80 years for broadleaved softwood species, depending on the silvicultural system, forest function, site productivity and other specific situations. The simulated landscape was divided into management areas, composed of multiple stands, that included unmanaged, strictly protected forest (<1%), protected forest with special conservation prescription practices (less than 10 m3/ha, approximately 40%), selective logging (20%; with approximately half of the area dedicated to production of mixed beech–conifer, mixed beech–broadleaved and oak forests), shelterwood forests (28%; majority of Fagus, also the remaining mixed beech–conifer, mixed beech–broadleaved and oak forests) and clearcut forests (12%; mostly Norway spruce pure stands). Within each management area, management treatments (stand selection, harvesting periodicity, intensity, targeting species and cycle duration) were implemented based on stand composition and stand age.
#### *2.5. Climate Change Scenarios*
Three climate change scenarios were defined (Figure 2) based on the RCP 2.6, 4.5 and 8.5 emission scenarios [85]. For the model spin-up period (1901–2015), we used gridded climate data (precipitation, maximum and minimum temperature) from the Climate Research Unit Time Series (CRU TS) [86] high-resolution dataset. For the simulation period between 2015 and 2100, we used RCP projections from Climatic Data Generator (ClimGen) [87] based on the French National Centre for Meteorological Research CNRM-CN5 climate model [88]. For the simulation period between 2100 and 2140, for which there were no cli-
mate projections available, climate data were randomly resampled from the last 3 decades, the last climate normal period, of the corresponding CNRM-CN5 series. Climate series were extended to 2140 because of the species longevity and the long logging cycles used in Romanian forestry.
**Figure 2.** Comparison of climate change scenarios for the period 1970–2140 among the three RCP scenarios (RCP 2.6, RCP 4.5, RCP 8.5) in five main forest types representative of the altitudinal gradient (top to bottom). Precipitation (Prec), average temperature (Tavg) and standardized precipitation evapotranspiration index (SPEI index) (rows top to bottom represent RCP 2.6, RCP 4.5 and RCP 8.5, respectively).
Gridded 0.5◦ climate data were statistically downscaled following Moreno and Hasenauer [89] using a 30 resolution gridded climate dataset from WorldClim 1.4 and 2.0 [68]. Mean annual CO2 concentrations taken from van Vuuren et al. [85] differed among PCR scenarios but were assumed spatially constant. Similar to climate, CO2 concentrations after 2100 were kept constant at the level of the last climate normal to prevent inconsistencies with resampled climate series. Photosynthetically active radiation (PAR), calculated from WorldClim 2.0 solar radiation [68], changed seasonally and spatially but did not differ across climate change scenarios.
#### *2.6. Model Output and Validation*
We simulated changes in forest biomass and species composition over a period of 125 years (2015–2140) under three climate change scenarios (RCP 2.6, RCP 4.5 and RCP 8.5), using 22 replicates of each scenario at a spatial resolution of the 200 m cell size. Model outputs included annual live, dead and harvested biomass throughout the simulation period. Forest net productivity was calculated as the change in biomass accounting for the harvesting. Model outputs for the period 2010–2020, when the management plans were released, were validated using Romanian forest yield tables [72,73] and forest management plans of the study area [37–46]. The simulated initial biomass per hectare (Figure 3) and
the extracted timber volume were within the ranges recorded in the management plans for all five forest types. As expected, compared to management, background natural disturbances had minor effects on the current landscape. The simulated windthrow impact was in accordance with volumes estimated from Popa [76] and the area affected by intense windthrows during the period 1986–2016 [79–81]. Insect outbreak occurrence and impact were also in accordance with the area affected by intense insect attacks in the study area in the period 1986–2016 [79–81], and the harvested volumes were in agreement with the management plans [37–46].
**Figure 3.** Boxplot diagram comparing the modeled biomass in the five main forest types in the initial Landis-II simulated landscape (year 2015) and the field-measured biomass data recorded from the management plans.
#### *2.7. Statistical Analysis*
Biomass results were spatially aggregated for the five main forest types. Differences across scenarios and forest types were assessed by linear mixed models (LME) [90], with changes over time fit to polynomial functions (up to 5 terms) and simulation replicates as a random factor. Natural disturbance outputs (i.e., mean damaged area, number of killed cohorts and severity) and harvested biomass were analyzed for the entire study area. Shifts in forest species composition at the landscape level were analyzed through the change in species biomass, using constrained redundancy analysis [91] with year and climate change scenario as constraining variables.
#### **3. Results**
#### *3.1. Disturbances*
Windthrows affected a minor percentage of the forest area (1%), with all of the effect located in the Norway spruce forest type. The affected area significantly increased over time (*p* ≤ 0.05) such that by 2140, the area damaged was two-fold the area of 2015 (Figure 4a). A similar trend was observed in the severity and the number of cohorts killed by wind, with values increasing over time, and the number of cohorts killed being significantly lower for the RCP 8.5 scenario. These results are in accordance with the aging of Norway spruce forests, as the susceptibility to windthrows increases with tree age.
**Figure 4.** Variation in the average area damaged by (**a**) wind (**b**) and insects and (**c**) the harvested biomass across climate change scenarios. RCP 2.6 (green), RCP 4.5 (orange) and RCP 4.5 (red).
Insect outbreaks annually affected around 200 ha of the forest area (~1%), mostly in the spruce and mixed *Fagus* (beech)–conifer forests. The area and the number of cohorts killed by *Ips* insects, which target mature *P. abies* trees, increased over time (*p* < 0.05), also reflecting a change in the forest age structure. This trend was not observed for *H. abietis* (Figure 4b).
Harvesting occurred mostly in the *Fagus* and *Quercus* forest types, where annual means of 300,000 and 100,000 tons were extracted, respectively, with extraction rates of about 10 and 15% every 10 years (Figure 4c). Harvesting in the spruce and mixed beech–conifer forest types was mostly restricted to conservation works, with rates of extraction below 2%. Differences between climate change scenarios were noted from 2070, with biomass harvested in RCP 8.5 being greater, followed by RCP 4.5, and RCP 2.6, the lowest among all the scenarios. Given that harvesting prescriptions were constant over time, following the national Romanian regulatory framework, this indicates that differences in forest attributes developed over time among climate change scenarios.
#### *3.2. Changes in Biomass across Climate Change Scenarios*
Living aboveground biomass (Figure 5) tended to increase in all forest types and climate change scenarios throughout the simulation period, with fluctuations related to disturbance or harvesting events. Comparatively larger increases in biomass were observed for RPC 8.5, ranging from 21% in the *Quercus* forest type to 51% in the mixed beech–broadleaved forest type, than for RCP 4.5 (from 2% to 37%). The lowest change in biomass was observed in RCP 2.6, ranging from a decrease in total biomass (−15%) in the *Quercus* forest type to a 29% increase in the mixed beech–conifer forest type. The effect of climate varied across forest types: mixed beech–broadleaved forests showed the largest differences in the final biomass among scenarios, with RCP 8.5 showing a larger biomass than RCPs 4.5 and 2.6.
**Figure 5.** Mean (dashed line) and 95% confidence interval (shaded area) of the living biomass (−10<sup>3</sup> kg·ha<sup>−</sup>1) measured in the forest belts over time under the three climate change scenarios: RCP 2.6 (green), RCP 4.5 (orange) and RCP 4.5 (red).
The analysis of the biomass at the species level revealed changes in species biomass through the simulation timespan and among climate change scenarios (Figure 6). The five main species, *P. abies*, *A. alba*, *F. sylvatica, Q. petraea* and *Q. frainetto*, showed significant biomass increments during the succession (*F. sylvatica* up to 2050) (*p* < 0.05). Of those five, all but *F. sylvatica* showed significant differences in biomass among the scenarios, with the largest biomass under RCP 8.5 (*p* < 0.05). Three species, *C. betulus*, *Q. petraea* and particularly *F. sylvatica*, showed short periods with strong reductions in biomass after 2050, which co-occurred in time with low SPEI values, although they tended to recover later.
**Figure 6.** Mean (dashed line) and 95% confidence interval (shaded area) of the species living biomass (·10<sup>3</sup> kg·ha−1) measured over time under the three climate change scenarios: RCP 2.6 (green), RCP 4.5 (orange) and RCP 4.5 (red).
The percentage of dead wood biomass relative to the total biomass showed a slight increase in all the forest types over time (Figure 7). Whereas the living biomass of Norway spruce and *Quercus* spp. differed among scenarios, the RCP scenario did not affect the percentage of dead biomass in these forests, which also showed a trend with very small interdecadal variations. Forest types with participation of *F. sylvatica* were subjected to strong variations in dead biomass over time, usually within the range 2.5–5.0%, which is also related to fluctuations in living biomass (Figure 5) and a low 6-month SPEI (Figure 2). The mortality events after drought temporally altered the proportion of live and dead biomass and occurred with a higher intensity in the RCP 8.5 scenario.
**Figure 7.** Percentage dead biomass relative to total biomass in the different forest types. Shaded areas indicate 95% confidence interval for every decade and climate change scenario: RCP 2.6 (green), RCP 4.5 (orange) and RCP 4.5 (red).
#### *3.3. Productivity*
The main forest types showed different aboveground net productivities, with Norway spruce and mixed forest productivity being typically lower than 15 kg/ha, that of beech pure forests being around that value and that of *Quercus* being usually above it. The aboveground net productivity tended to decrease slightly over time for all forest types. Productivity also differed among climate change scenarios, with the magnitude of the differences depending on forest type (Figure 8). Three forest types, Norway spruce, mixed beech–broadleaved and *Quercus*, showed significantly (*p* < 0.05) higher productivity under RCP 8.5 than under the other two scenarios. Beech forest productivity, despite having high temporal variations, showed significant differences between RCP 8.5 and RCP 4.5 (*p* < 0.02), usually being larger in RCP 8.5. The mixed beech–conifer forest productivity only showed significant differences between the RCP 2.6 and RCP 4.5 scenarios (*p* = 0.035).
#### *3.4. Changes in Landscape Species Composition*
Forest species composition varied over time and across climate change scenarios (Figure 9). During the first half of the simulations, the landscape species composition was similar among scenarios; in the following decades, the position of samples started to drift with the displacement increasing over time. This resulted in a significant differentiation in the species composition between RCP 8.5 and the other two scenarios. As shown by the ordination, and in accordance with the biomass analysis, RCP 8.5 was characterized by a greater biomass of *Q. frainetto*, *P. abies* and *A. alba.*
**Figure 8.** Productivity of the five main forest types (*Picea*, *Fagus*–conifers, *Fagus*, *Fagus*–broadleaved, *Quercus*) for the forecasts for every decade and climate change scenario: RCP 2.6 (green), RCP 4.5 (orange) and RCP 4.5 (red).
**Figure 9.** Ordination-based redundancy analysis of species biomass for the whole landscape, every year in the period 2015–2140 and every Landis-II simulation obtained for climate change scenarios RCP 2.6 (green), RCP 4.5 (orange) and RCP 8.5 (red), with the constraining variables year and RCP scenario.
#### **4. Discussion**
The simulation model implemented for temperate forests in the Romanian Southern Carpathians showed that climate change produces changes in forest biomass, productivity and species abundance. Such changes were particularly noticeable under RCP 8.5, the most extreme climate change scenario, and rendered an overall increase in the carbon carrying capacity of the studied Carpathian forests.
#### *4.1. Forest Biomass and Productivity*
The results of our Landis-II model adapted to Southern Carpathian forests showed an overall increase in forest biomass over time, with significantly greater biomass accumulation in the RCP 8.5 scenario. This is in line with Hubau et al. [10], who found that contrary to some tropical forests, temperate forests maintain a certain capacity to keep stocking carbon. The increased forest biomass accumulation and productivity predicted for the study area with intense climate change point to the raised CO2 concentration [2,5] and warmer winters [3] as being the main drivers in the Carpathians, where precipitations do not change significantly for most of the area [23]. However, our simulations also indicate that the initial increase in biomass is limited, likely due to increasing drought and the stabilization of climate and atmospheric CO2 concentrations. This trend suggests that the equilibrium already observed in some tropical forests [10] might occur in temperate forests somewhere at the end of the current century. Indeed, other authors have reached similar conclusions and timing, pointing directly to the proliferation of bark beetles as one of the main natural disturbances limiting carbon sequestration associated with climate change [92]. In the Southern Carpathians, climate change conditions overcompensate the negative effects of the disturbances at the beginning of the period but may not be able to do that after 2040, for most of the forests.
The main forest types studied responded differently to climate change, according to the ecophysiology of the species. Norway spruce and *Quercus* spp. forests had low productivity but higher stability, while mixed beech–conifer forests, beech forests and mixed beech–broadleaved forests were more prone to biomass changes, particularly after 2050. Indeed, low- and medium-altitude forests in the Southern Carpathians are very likely to be more affected by drought, while the precipitation in mountain tops, often covered by Norway spruce forests, might increase slightly [23]. Bouriaud el al. [16] also predicted forest biomass increments for the Romanian Eastern Carpathians but found a decreasing trend for conifer forests. This discrepancy among Bouriaud et al.'s [16] results and ours is likely due to three causes: (a) climate change effects on forests vary in type and intensity at the global, regional and local levels [15,23,93], as do the local conditions and climate change projections of the study areas in the Eastern and Southern Carpathians; (b) the modeling approach, with PnET and Landis-II models also accounting for complex ecophysiological responses to photosynthetically active radiation, atmospheric CO2 concentration, etc.; and (c) the climate models used [16,94–97]. Indeed, the CNRM-CN5 model [87,88] used here is one of the least limiting climate models regarding water availability in the study area.
The predicted increase in the impact of the main natural disturbances in the Southern Carpathians: windthrows and bark beetle attacks, particularly *Ips* sp., is in agreement with other studies [6,98]. Even though some variations may be caused by harvest and aging forests, the periods with low productivity are likely associated with drought, since the SPEI value tends to decrease within the same periods. Drought events also occur simultaneously with raised mortality and deadwood biomass increments. This is the case of Norway spruce forests and beech forests after 2040. Natural disturbances induce organic carbon release, and hence it is common that disturbance regimes and ecosystem resilience determine forests' carbon sequestration capacity [10,99–102]. A net increase in temperate forest biomass caused by climate change has been forecasted for areas where disturbances and extreme climate events have a limited impact [54,103]. These disturbances usually damage conifer forests [6], and aging forests will likely contribute to it [52]. In our model, *H. abietis* had a minor impact, as a consequence of the low biomass of the cohorts they target, and because the selection harvesting management implemented in mixed beech and silver fir (*A. alba)* stands prevents *H. abietis* outbreaks [104]. The change in wood harvest observed here was also predicted by Bouriaud et al. [16], Ciceu et al. [105] and Chivulescu et al. [33] and concentrated in the intermediate- and low-altitude forests, having consequences for the forest structure. This increment in harvested timber is directly associated with both forest aging and the application of logging cycles [78]. As a result of all previous processes, around the year 2050, forests are expected to reach the carbon sequestration limit. Such
an idea is supported by Bouriaud et al.'s [16] and Hubau et al.'s [10] studies. Nabuurs et al. [19] also found evidence of carbon sink saturation in European forests.
#### *4.2. Forest Composition*
The Landis-II projections predicted shifts in forest species composition and abundance under the RCP scenarios in Romanian Southern Carpathian temperate forests, with the five main species showing significant biomass increments over time (*F. sylvatica* only up to 2040). Indeed, the reduction in the cohorts killed by windthrows under RCP 8.5 compared to RCP 4.5 and RCP 2.6 is likely due to the success of *A. alba* and the intense modification of the forest structure under RCP 8.5. However, only *F. sylvatica* showed differences among the RCP scenarios, with the highest biomass accumulation under RCP 8.5. Some species, particularly *F. sylvatica*, showed short periods with strong reductions in the biomass after 2050 that co-occurred, and thus are likely associated, with drought conditions. Climate change may alter local environmental conditions, making them no longer suitable for a given species and altering the interspecific competition [18,96], ultimately resulting in a change in species abundance [1]. For instance, Musselman and Fox [14] indicated that drought-tolerant species are expected to succeed at the global scale under the new conditions, and this is likely going to occur for *A. alba* in the Southern Carpathians, as predicted here.
Previous studies have indicated that the intensity of the change in climate is proportional to the magnitude of the impact on forests [16,33]. The intensity of such a change in the Southern Carpathians is expected to be relatively small, as the climate conditions do not strongly depart from the current conditions under the analyzed RCPs. However, some forest types are more vulnerable to changes than others [18,96]. Norway spruce forests at the upper limit will gain from the temperature rise [15], and, as suggested by Landis-II models, at the lower limit, they will also benefit from *F. sylvatica*'s vulnerability to drought. At lower altitudinal belts, conifers coexist with beech. Beech and conifer mixed forests are more successful than pure beech stands [25,26], but Norway spruce grows more and remains relatively unaffected compared to *F. sylvatica* during drought [106–108]. *A. alba* is also less susceptible to warmer and drier conditions than *F. sylvatica* [25]. Thus, at the stand level, the association between *F. sylvatica* and *A. alba* is not only advantageous for enduring drought [25,26,106,107] but also for sunlight use [104,109]. Our results for the Southern Carpathians suggest that climate change will contribute to *A. alba* encroachment in *F. sylvatica* pure forest stands. Indeed, there is strong evidence that beech forests have already been affected by climate change-induced drought in recent decades [20]. Moreover, Broadmeadow et al. [4] recently highlighted beech's vulnerability in temperate forests across Europe.
Conifer and beech forests aside, *Quercus* spp.-dominated forests will partially benefit from the temperature rise and *F. sylvatica* decline. *Quercus* species are relatively drought resistant. According to the Landis-II simulations, and as also supported by previous studies [110], the most thermophile and drought-resistant species in the Southern Carpathians such as *Q. frainetto* [36] will benefit from climate change and increase their biomass during succession, while *Q. petraea* will eventually suffer the impact of drought events.
#### **5. Conclusions**
Here, we simulated the changes in the composition and structure of the temperate forests of the Southern Carpathians under three climate change scenarios. Our results indicate that climate change may contribute, overall, to increasing temperate forest productivity and biomass, mainly due to the combination of increasing temperatures and thus extended growing periods in the uplands and only a mild reduction in rainfall across the landscape. Fluctuations in this trend associated with strong biomass reductions and temporary deadwood rise were related to drought periods. However, it is likely that increased drought events may eventually counteract such positive trends.
Climate change selectively affected areas and species, thus contributing to changes in species abundance. *P. abies* benefitted from warmer conditions, whereas forests dominated by *F. sylvatica* were vulnerable to climate change, with drought periods associated with large mortality events. The *A. alba* abundance increased under the new climate conditions at the expense of *Fagus*'s decline. Our results suggest that under the upcoming climate, mixed *F. sylvatica*–conifer (namely Norway spruce and silver fir) stands will have greater resistance and resilience than those of pure *F. sylvatica* stands, a finding that will help in guiding future forest management practices.
**Supplementary Materials:** The following are available online at https://www.mdpi.com/article/ 10.3390/f12111434/s1, Figure S1: Distribution of the ecoregions within the study area. Ecoregions represent areas of homogeneous climate, soil and forest types, Table S1: Characteristics of the defined ecoregions within each forest type: main species composition, elevation range, soil attributes and main natural disturbances affecting the ecoregion. Disturbances: It: *I. typografus*; Id: *I. duplicatus*; Ha: *H. abietis* [34,35,111,112].
**Author Contributions:** Conceptualization, C.A. and J.G.-D.; methodology, C.A. and J.G.-D.; software, J.G.-D.; validation, J.G.-D. and C.A.; formal analysis, J.G.-D.; investigation, J.G.-D. and C.A.; resources, J.G.-D., A.C. and S.C.; data curation, J.G.-D.; writing—original draft preparation, J.G.-D. and C.A.; writing—review and editing, C.A., J.G.-D., A.C., S.C., O.B. and M.A.T.; visualization, J.G.-D.; supervision, C.A.; project administration, M.A.T., C.A. and O.B.; funding acquisition, M.A.T., C.A. and O.B. All authors have read and agreed to the published version of the manuscript.
**Funding:** This work was supported financially by EO-ROFORMON project, ID P\_37\_651/SMIS 105058 and PN 19070101.
**Institutional Review Board Statement:** Not applicable.
**Informed Consent Statement:** Not applicable.
**Data Availability Statement:** The data presented in this study are available on request from the corresponding author.
**Acknowledgments:** The authors acknowledge the contribution of the panel of experts constituted by I. Seceleanu, M. Paraschiv, N. Olenici and D. Chira.
**Conflicts of Interest:** The authors declare no conflict of interest.
#### **References**
## *Article* **Photosynthesis Traits of Pioneer Broadleaves Species from Tailing Dumps in Călimani Mountains (Eastern Carpathians)**
**Andrei Popa 1,2 and Ionel Popa 1,2,\***
**Abstract:** The reforestation and stable ecological restoration of tailings dumps resulting from surface mining activities in the Călimani Mountains represent an ongoing environmental challenge. To assess the suitability of different tree species for restoration efforts, photosynthetic traits were monitored in four broadleaf pioneer species—green alder (*Alnus alnobetula* (Ehrh.) K. Koch), aspen (*Populus tremula* L.), silver birch (*Betula pendula* Roth.), and goat willow (*Salix caprea* L.)—that naturally colonized the tailings dumps. Green alder and birch had the highest photosynthetic rate, followed by aspen and goat willow. Water use efficiency parameters (WUE and iWUE) were the highest for green alder and the lowest for birch, with intermediary values for aspen and goat willow. Green alder also exhibited the highest carboxylation efficiency, followed by birch. During the growing season, net assimilation and carboxylation efficiency exhibited a maximum in late July and a minimum in late June. The key limitation parameters of the photosynthetic process derived from the FvCB model (*Vcmax* and *Jmax*) were the highest for green alder and exhibited a maximum in late July, regardless of the species. Based on photosynthetic traits, the green alder—a woody N2-fixing shrub—is the most well-adapted and photosynthetically efficient species that naturally colonized the tailings dumps in the Călimani Mountains.
**Keywords:** gas exchange; ecosystem restoration; mountain forests; photosynthesis
#### **1. Introduction**
Climate change and anthropic activities have given rise to the most serious environmental problems of the 21st century [1]. Today, in an increasing number of ecosystems, anthropic influences are harming biodiversity and ecosystem functioning. Human activities change the condition of natural vegetation, leading to disturbances such as degradation of vegetation, erosion of soil, decline in land productivity and even reduction of ecosystem services [2].
A large proportion of the Earth's geological resources (metals, minerals, fuel, etc.) are underground; mining activities to access these resources damage the above-ground landscape and have a significant impact on natural ecosystems such as forests, rivers and lakes [3]. Even after the mining activity is complete (especially in the case of surface mining), dump areas remain in the place of former natural ecosystems and have the potential to cause serious environmental problems. Due to social and political pressure to undertake more sustainable development, the restoration of mining areas has gradually become an important phase of mining activities [4]. Multiple restoration solutions are available; of these, soil amendment combined with phytoremediation (reforestation) is the most environmentally friendly method [5].
Due to high soil degradation (lack of nutrients and organic matter, small edaphic volume, high concentration of heavy metals), it is important to choose a mix of species that are well-adapted to the local climate and the unique conditions of the mining habitat [6].
**Citation:** Popa, A.; Popa, I. Photosynthesis Traits of Pioneer Broadleaves Species from Tailing Dumps in Călimani Mountains (Eastern Carpathians). *Forests* **2021**, *12*, 658. https://doi.org/10.3390/ f12060658
Academic Editor: Francois Girard
Received: 12 May 2021 Accepted: 21 May 2021 Published: 22 May 2021
**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.
**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).
Successful reforestation of these degraded landscapes depends on the adaptability of tree species to degraded soil and the capacity of plant associations to contribute to the restoration of soil proprieties and former environmental conditions [7].
The process of ecosystem reconstruction on degraded mining soils progresses gradually from dump consolidation and soil amendments to pioneering species installation and finally climax tree species establishment [8]. In many cases, human interventions through ecological reconstruction are informed by the natural colonization of pioneer species. The optimal species composition for reforestation is chosen based on criteria such as adaptative capacity (survival rate), growth and biomass production (photosynthesis), and the capacity to cover the land through vegetative or generative regeneration [7].
In mountainous areas (e.g., the Carpathians), pioneer species colonize open soils on degraded or abandoned lands as part of primary succession. These pioneer species include green alder (*Alnus alnobetula* (Ehrh.) K. Koch = *Alnus viridis* (Chaix) DC), aspen (*Populus tremula* L.), silver birch (*Betula pendula* Roth.), goat willow (*Salix caprea* L.) and rowan (*Sorbus aucuparia* L.) [9]. Green alder, an N2-fixing shrub, plays an important role in mountain ecosystems due to its capacity to stabilize slopes and prevent erosion [10,11]. On the other hand, silver birch is a fast-growing deciduous species that occurs naturally throughout most of Europe and is frequently observed on degraded lands [12,13]. Aspen and goat willow have high ecological amplitudes and are widely distributed throughout the temperate and boreal areas of Europe and Asia [14,15].
Carbon assimilation, as an indicator of species' adaptability to specific habitat conditions, is related to biomass production, CO2 storage, competitiveness and survival capacity [16]. Gas exchange measurements play a major role in understanding photosynthetic processes [17,18]. The performance of forest species in terms of photosynthetic traits is evaluated through gas exchange, which is measured under controlled experimental conditions or in situ conditions [19,20]. Particular growing conditions specific to mining-degraded soils modify the normal assimilation process [21–23].
To the best of our knowledge, no previous studies have measured gas exchange to analyse the photosynthetic traits of mountain pioneer species growing in former mining areas. However, analyses of photosynthetic parameters under different growing conditions (exposure to ozone, fertilization variants, enriched CO2, etc.) have been performed for *Populus* spp. [24–26], *Salix* spp. [27–29], *Betula* spp. [30–32], and *Alnus* spp. [33–35].
Our work aimed to quantify the eco-physiological performance of four broadleaf pioneer species that naturally colonized a mountain tailings dump area. The following scientific questions were examined: (i) Which of these four pioneer species is best adapted for use in reforestation processes from the point of view of photosynthetic traits? (ii) What are the seasonal patterns of the photosynthetic traits?
#### **2. Materials and Methods**
#### *2.1. Study Site*
This study took place in the Călimani Mountains (Eastern Carpathians, Romania; 47◦7 24 N and 25◦13 48 E; 1500 m a.s.l.) on degraded tailings dumps resulting from sulfur surface mining that occurred between 1969 and 1997. During this period, the former natural forest ecosystem (a mixture of Norway spruce and Swiss stone pine) and organic soil were destroyed, and dumps were formed by the successive storage of tailings. In 2007, an ecological reconstruction effort began with the stabilization of the slopes with high slants, alkaline amendment of the soil (debris from former buildings) and afforestation of some areas with Norway spruce (*Picea abies* L., H. Karst.) and green alder. Currently, the upper part of the mining dump is colonized by pioneer forest species (green alder, aspen, goat willow, birch) in combination with herbaceous species, and the area is characterized by a primary succession of vegetation [9,36].
The woody species present on the site comprise species specific to the upper mountain area: green alder, Norway spruce, dwarf mountain pine (*Pinus mugo* Turra), aspen, goat willow and birch. Shrubby vegetation is represented by blueberry (*Vaccinium myr-* *tillus* L.), cranberry (*Vaccinium vitis-idaea* L.), bog blueberry (*Vaccinium uliginosum* L.) and rhododendron (*Rhododendron myrtifolium* Schott & Kotschy).
Air temperature (2 m) and soil water content (10 cm depth) were measured using dedicated sensors (HOBO U23-001, Onset Computer Corporation, Bourne, MA, USA and CS650, Campbell Scientific, Logan, UT, USA). The average annual air temperature at the site in 2019 was 4.3 ◦C and, in the vegetation season, it was 10.7 ◦C (Figure 1). The average annual precipitation in the study area is 1000–1200 mm. The snow layer is present for an average of 180–200 days each year, the first snowfall can occur in early October, and the typical vegetation season starts in the second decade of May until middle October [37].
**Figure 1.** Air temperature (**a**) and soil water content (**b**) variation during the 2019 growing season (vertical lines represent measurements dates).
The tailings dump soils are characterized by high acidity (pH 3.2) and missing organic components [9,36].
#### *2.2. Gas Exchange Measurements*
Gas exchange during photosynthesis was measured using a portable photosynthesis system (LI-6800, LI-COR Inc., Lincoln, NE, USA) equipped with a standard infrared gasexchange analyser (IRGA) and a chamber for broadleaf species (area: 6 cm2). Instantaneous gas exchange was recorded at three time points in 2019 (29–30 June, 30–31 July and 30–31 August) between 9 am and 3 pm. Measurements, at each time point during the season, were performed on one leaf from five different exemplars of each broadleaf species: green alder, birch, aspen and goat willow. Due to difficult site accessibility (high mountain and degraded land) and the long time required for measurement (multiple dead times for stabilization and between measurements), we extended the day measurement period by
3 h, compared with the standard practice. We planned the experiment, in order to minimize the influence of diurnal variability of photosynthesis, by measuring the first exemplar of each species, followed by the second exemplar of each species and so on. We took into consideration that increasing the number of measurement days (now limited to two days for each time point during the season) could induce more variability because climatic parameters can change significantly (e.g., precipitation or temperature).
All studied species have C3 photosynthetic pathways. The selected trees were saplings, with a mean height of 1.5–2.0 m for silver birch, goat willow and aspen and 1.0 m for green alder. Measurements were performed on different leaves from one time point to another, but from the same exemplars. Selected leaves were completely developed, had no damage and were located on the upper part of the crown, in full exposure to light.
Instrument calibration was performed at the start of each session following the manufacturer's recommendations [38]. To assure measurement accuracy, the gas analyser was matched when the following conditions were met: elapsed time since the last match >10 min, CO2 measured by the reference analyser have changed by 100 <sup>μ</sup>mol·mol−<sup>1</sup> since the last match, difference between CO2 reference and CO2 sample <10 <sup>μ</sup>mol·mol−1, and difference in H2O <1 mmol·mol−1. While collecting the measurements, the mean temperature and relative humidity in the measurement chamber were 22.3 ± 1.8 ◦C and <sup>60</sup> ± 1.2%, respectively, the fan speed was 10,000 rpm and the flow rate was 500 <sup>μ</sup>mol·s<sup>−</sup>1. The irradiance photosynthetic photon flux density (PPFD) was set to 1000 <sup>μ</sup>mol·m−2·s−<sup>1</sup> and was kept constant for all measurements, with following light composition ratio of 0.9 red and 0.1 blue using the LI-COR 6800 light source.
To obtain response curves for net photosynthetic rate as a function of intercellular CO2 concentration, a chamber CO2 gradient consisting of 400, 200, 100, 50, 400, 600, 800, 1000, and 1200 <sup>μ</sup>mol·mol−<sup>1</sup> was used. Response curve measurements were performed on one leaf from five different exemplars for each species. Between each measurement time point during the season, the exemplars were kept the same, but the leaves differ. Steady-state values from each leaf at different CO2 concentrations were recorded after 2–3 min, an interval which allowed the leaf to adjust to the new environmental conditions (a stability point was reached when the standard deviation for CO2 and H2O differences was ≤0.1 for 20 s) [38,39].
Light-saturated net photosynthetic rate *<sup>A</sup>* (μmol CO2·m−2·s−1), transpiration rate *<sup>E</sup>* (mmol H2O·m−2·s<sup>−</sup>1), stomatal conductance to water vapour *gsw* (mol H2O·m−2·s<sup>−</sup>1), and intercellular CO2 concentration *Ci* (μmol·mol<sup>−</sup>1) and leaf temperature *Tleaf* ( ◦C) were measured. Using these measured parameters, three efficiency parameters were then calculated: water use efficiency *WUE* (as *<sup>A</sup>*/*E*, <sup>μ</sup>mol CO2·m−2·s<sup>−</sup>1/mmol H2O·m−2·s<sup>−</sup>1), intrinsic water use efficiency *iWUE* (as *<sup>A</sup>*/*gsw*, <sup>μ</sup>mol·mol<sup>−</sup>1) and instantaneous carboxylation efficiency *<sup>A</sup>*/*Ci* (μmol CO2·m−2·s−1/μmol·mol−1) [40]. During the gas exchange measurements, a uniform distribution of photosynthesis and transpiration over the leaf was assumed [41].
#### *2.3. The Farquhar–Von Caemmerer–Berry Model*
Data from the response curve of net photosynthetic rate to CO2 were analysed using the Farquhar–von Caemmerer–Berry model (the FvCB model) [42]. This model was developed to model leaf gas exchange and net photosynthetic rate (*A*) for C3 plants under any given environmental conditions. It has been used widely in recent decades to summarise the dependence of carbon assimilation rate on intercellular CO2 concentration (*Ci*) because of its simple form and the comparable metrics of photosynthetic capacity that are provided. The net photosynthetic rate predicted by the FvCB model is the minimum between the Rubisco limited rate (*Ac*), the ribulose 1,5-bisphosphate (RuBP)-regeneration or electron (e-) transport limited rate (*Aj*) and the triose phosphate utilization (TPU) limited rate (*Ap*) of CO2 assimilation [43,44]. Considering the three limitation phases, the net photosynthetic rate can be modelled as follows:
$$A = \min(Ac, Aj, Ap) \tag{1}$$
In the Rubisco limited phase, the response of net assimilation (*Ac*) to *Ci* is defined by:
$$Ac = \frac{V\_{\text{cmax}} \* (C\_i - \Gamma\_\*)}{C\_i + K\_C \* (1 + O/K\_O)} - R\_d \tag{2}$$
where *Rd* is day respiration (μmol·m−2·s−1), *Vcmax* is the maximum carboxylation rate of Rubisco (μmol CO2·m−2·s<sup>−</sup>1), <sup>Γ</sup>\* is the photosynthetic compensation point (μmol·mol<sup>−</sup>1), *KC* is the Michaelis–Menten constant of Rubisco for CO2 (μmol·mol<sup>−</sup>1), *KO* is the Michaelis– Menten constant of Rubisco for O2 (μmol·mol<sup>−</sup>1), and O is the intercellular partial pressure of O2 (mmol·mol<sup>−</sup>1) set to 21 KPa.
During the RuBP-regeneration or electron transport, the response of net assimilation (*Aj*) to *Ci* is defined by:
$$Aj = \frac{J \ast (C\_c - \Gamma\_\ast)}{4C\_c + 8\Gamma\_\ast} - R\_d \tag{3}$$
where *<sup>J</sup>* is the rate of electron transport (μmol·e−1·m−2·s<sup>−</sup>1).
In the TPU limited phase, the net assimilation rate (*Ap*) is defined as:
$$Ap = \mathfrak{F}T\_p - R\_d\tag{4}$$
where *Tp* is the rate of phosphate release in triose phosphate utilization (μmol·m−2·s<sup>−</sup>1).
The kinetic constants of Rubisco (Γ\*, *KC*, *KO*), which are temperature-dependent, were derived using Arrhenius-type equations using the leaf temperature measurements [45,46]. Traits of photosynthetic capacity (*Vcmax*, *Jmax*, TPU) were derived from the FvCB model and were corrected to a temperature of 25 ◦C [47]. More information and details about *A*/*Ci* data fitting can be found in the literature [48–50]. The FvCB model was applied for each leaf, and coefficients were analysed as mean.
#### *2.4. Data Analyses*
Differences between species were analysed using ANOVA followed by post hoc Tukey tests [51]. The dependence between photosynthetic parameters was quantified using the Pearson correlation coefficient. Statistical tests were considered significant at the *p* < 0.05 level. FvCB model parameters were estimated using the R package 'plantecophys' [47], and figures were constructed using the packages 'ggplot2' and 'cowplot'. All data processing was done using R 4.0.3 software [52].
#### **3. Results and Discussion**
#### *3.1. Photosynthetic Parameters for Deciduous Pioneer Species*
To compare the eco-physiological performance of the deciduous pioneer species that naturally colonized the mining dump areas in the Călimani Mountains, the mean values of photosynthetic traits during the 2019 vegetation period were analysed for CO2 concentration close to the actual environmental concentration (400 <sup>μ</sup>mol·mol<sup>−</sup>1).
Green alder and birch had the highest net assimilation rate for the entire season among the four species analysed (Table 1). The lowest net assimilation rate was recorded for goat willow. The net assimilation rate for green alder was significantly higher than those of aspen and goat willow, while the net assimilation rate of birch was significantly different only from that of goat willow. Similar values for net assimilation (ranging from 12.8 to 17.3 <sup>μ</sup>mol CO2·m−2·s−1) have been documented under similar measurement conditions (1400 <sup>μ</sup>mol·m−2·s−<sup>1</sup> PPFD and 320 <sup>μ</sup>mol·mol−<sup>1</sup> CO2 concentration) for the seedlings of different Alnus species [33]. However, our results showed slightly higher net assimilation values for silver birch compared with the range reported in the literature [31,32,53].
Birch had the highest rates of transpiration and stomatal conductance. The rates of transpiration for birch were significantly higher than those for green alder, but stomatal conductance was not significantly different between species. Similar transpiration rate (3.4 mmol·m−2·s−1) and stomatal conductance (0.4 mol·m−2·s−1) have been observed for mountain birch at the treeline on the Tibetan Plateau [54]. Birch proveniences and leaf types (early vs. late leaves) can induce differences in stomatal conductance [12].
Diffusion of CO2 into the intercellular spaces inside leaves occurs mainly through stomatal pores [55]. Stomatal density and distribution on leaves differ among species. As a result, we obtained different *Ci* values even though the CO2 concentration in the measurement chamber (400 <sup>μ</sup>mol·mol−1) was similar for all species. Green alder had the lowest intercellular CO2 concentration, which was significantly different from those of the other species. The variation in intercellular CO2 concentration allows us to understand whether the decline in net photosynthesis rate is due to stomatal limitations or to the reduction of photosynthetic activity in the leaves' cells [56].
Instantaneous carboxylation efficiency was the lowest for aspen and goat willow and the highest for green alder. This photosynthetic parameter can be considered as an estimate of Rubisco activity; generally, higher intercellular CO2 concentration is associated with lower stomatal conductance. Water use efficiency (WUE and iWUE) was also the highest for green alder and differed significantly from those of the other species.
Regardless of the species, the highest correlation was observed between transpiration rate and stomatal conductance (Table 2). For birch only, a negative correlation was found between net assimilation rate and intercellular CO2 concentration. The correlation coefficient between net assimilation rate and transpiration rate varied from 0.74 (birch) to 0.93 (goat willow) and was significant in all cases. Similar significant negative correlations between *Ci* and *A* have also been found for birch in a Lithuanian forest (boreal zone) [40].
#### *3.2. Variability of Photosynthetic Parameters during the Vegetation Season*
For broadleaf species, the chlorophyll content of leaves changes throughout the growing season, which induces variability in eco-physiological processes. Because of these biochemical changes in leaves, it is important to explore the variation in individual photosynthetic parameters during the vegetation season [12]. To avoid the systematic influence of diurnal variation, the gas exchange measurements were distributed during the day from 9 am to 3 pm. The net assimilation rate varied across the season for all four species, with the lowest values recorded at the end of June (Figure 2a). The lowest net assimilation rate was observed for goat willow at the end of June (7.55 <sup>μ</sup>mol CO2·m−2·s−1) and was significantly different from the values in other months. For green alder, the maximum value of net assimilation was measured at the end of July (18.64 <sup>μ</sup>mol CO2·m−2·s<sup>−</sup>1), and the minimum was recorded at the end of June (14.93 <sup>μ</sup>mol CO2·m−2·s<sup>−</sup>1). A similar trend in variation was found for goat willow. On the contrary, net assimilation rates for birch and aspen increased continuously during the season.
**Table 2.** Pearson correlation coefficients for relationships between gas exchange parameters (*A*–net photosynthetic rate, *E*–transpiration rate, *gsw*–stomatal conductance, *Ci*–intercellular CO2 concentration).
\*\* Significance at *p* < 0.01; \* Significance at *p* < 0.05.
Studies in boreal forests indicate that, for birch, the highest rates of leaf-mass net assimilation occur under light-saturated conditions in early May after the leaves unfold, and there is minimal variation during the vegetation season [12]. For other species, the maximum leaf-area net assimilation occurs in late June and early July [31]. Studies on different *Salix* spp. have highlighted a weak correlation between biomass yield and photosynthetic rate and a positive influence of total leaf area per plant [27].
Stomatal conductance increased continuously from the beginning of the season to autumn for all species, except goat willow (Figure 2b). A similar pattern was observed for transpiration rate, but with a higher rate of increase from the beginning to the end of the season. These two parameters had a similar trend in variation because transpiration is regulated mainly by stomatal conductance [56]. The stomatal conductance of aspen and goat willow in June was significantly different from that measured in July or August. The maximum values for transpiration rate were measured in August for green alder (2.96 mmol H2O·m−2·s<sup>−</sup>1), aspen (3.93 mmol H2O·m−2·s<sup>−</sup>1) and birch (4.22 mmol H2O·m−2·s<sup>−</sup>1) and in July for goat willow (3.35 mmol H2O·m−2·s<sup>−</sup>1) (Figure 2c).
iWUE and WUE decreased during the vegetation season for all studied species, possibly linked with the lower soil water content during the measurements in July and August (see Figure 1). Water use efficiency parameters were significantly different between all three measurement timepoints only in the case of aspen. Similarly, differences between monthly values of WUE have been reported for another poplar species (*Populus angustifolia*) from North America [57]. Goat willow exhibited a different trend in variation, with slightly higher iWUE values at the end of August compared to July. Green alder had the highest values of both WUE and iWUE compared to other species.
**Figure 2.** Variability in photosynthetic parameters during the 2019 growing season for deciduous pioneer species on a tailing dump in the Călimani Mountains. (**a**) Net assimilation rate, (**b**) stomatal conductance, (**c**) transpiration rate, (**d**) intrinsic water use efficiency, (**e**) instantaneous carboxylation efficiency, (**f**) water use efficiency (points are mean values, and whiskers represent standard errors).
Variation in iWUE and WUE during the season is a consequence of both changing environmental conditions and physiological changes in leaf structure due to ageing [58]. Both parameters reflect water use efficiency (how much carbon is fixed per unit of water loss), but WUE can also be used as a water stress indicator (drought indicator) [59]. Higher values of iWUE or WUE can be achieved through lower stomatal conductance or transpiration rate, higher assimilation capacity or a combination of both [60]. Water supply is one factor that can cause plants to have a lower stomatal conductance and higher WUE [12]. Our study was conducted in a mountain area where precipitation is not a limiting factor, but variations in WUE may still occur due to particular conditions induced by precipitation variation during vegetation season. In the second and third measurement time points, low soil water content was reported (see Figure 1), which may induce a decrease in water use efficiency parameters.
Green alder also demonstrated the highest instantaneous carboxylation efficiency compared with other deciduous pioneer species, with a maximum in late July (Figure 2e). It had a lower intercellular CO2 concentration but maximum values for assimilation rate. One possible explanation for this is that green alder is in its optimal distribution range, is acclimatized to cold environments and uses resources more efficiently. The maximum instantaneous carboxylation efficiency occurred in July for goat willow and at the end of August for aspen and birch.
The mining dump areas are exposed to high light intensity, large temperature variability and low nitrogen availability. Low availability of soil nitrogen can be an important limiting factor of photosynthesis [19,21]. N2-fixing plants that are capable of fixing atmospheric nitrogen, like green alder, can be more performant in terms of photosynthetic traits on degraded soils compared with other pioneer species [61]. Leaf size and structure, combined with crown size and branching patterns, play an important role in the assimilation performance of different species [62]. Field observations confirm that the leaf area and crown development of green alder are greater than for the other species.
#### *3.3. Photosynthesis Response Curve under Increasing CO2 Concentration*
Using a variable concentration of CO2 to understand assimilation performance is more efficient than using a constant CO2 concentration [63]. An *A*/*Ci* curve can be constructed to understand and interpret the biochemical processes of leaf photosynthesis under various environmental conditions. The measured values of net assimilation at different CO2 concentrations are used to estimate photosynthetic limitations and parameters of photosynthetic performance using two key parameters from the FvCB model: *Vcmax* and *Jmax*.
The response curve for the relationship between assimilation and CO2 concentration shows an initial increase followed by a relatively constant variation caused by light limitation (Figure 3). A clear difference between the three measurement time points was observed for all species, except for birch in June and July. The limitation of assimilation rate due to light availability occurred after reaching an intercellular CO2 concentration of <sup>500</sup> <sup>μ</sup>mol·mol<sup>−</sup>1. After passing this threshold, the highest assimilation rate was measured in August for all four species. For goat willow and aspen, variation in WUE in relation to CO2 concentration was higher in June.
Based on the FvCB model, the limitations of the photosynthetic process are characterized by two main parameters—*Vcmax* and *Jmax*—and occasionally also by TPU. Estimation of these parameters is regulated by the limitations of one of three curves of the FvCB model. *Vcmax* is the maximum rate of Rubisco activity and reflects a limitation in RuBPregeneration [48]. For green alder, the highest values of *Vcmax* were recorded in July, and the lowest in June, without significant differences during the season (Figure 4a). For birch and aspen, there were no significant differences in *Vcmax* during the season. The lowest *Vcmax* for goat willow was recorded in June, statistically different from the rest of the season, and was associated with lower assimilation levels. To derivate the seasonal mean of *Vcmax*, it is recommended to perform measurements in midsummer. Maximum values of *Vcmax* were obtained in the middle of the vegetation season (end July).
**Figure 3.** (**a**) Mean response curve of net assimilation relative to intercellular CO2 concentration as modelled by the FvCB model; (**b**) water use efficiency relative to intercellular CO2 concentration (shaded areas represent standard error).
**Figure 4.** Mean values of FvCB model parameters for deciduous pioneer species on tailings dumps in the Călimani Mountains: (**a**) *Vcmax*—maximum rate of Rubisco activity; (**b**) *Jmax*—maximum rate of electron transport for RuPB-regeneration (whiskers represent standard errors, and letters represent significant differences).
*Jmax* represents the maximum rate of electron transport for RuPB-regeneration at the light intensity used in the study (1000 <sup>μ</sup>mol·m−2·s−1) (Figure 4b) [64]. *Jmax* was the lowest in June and highest in August, regardless of the species. With exception of aspen, the *Jmax* differs significantly between time points during the season. *Vcmax* and *Jmax* are correlated with *Amax* (the maximum value of assimilation) because these parameters all reflect limitations in photosynthetic processes. Green alder exhibited the highest values of *Vcmax* and *Jmax* and also had the highest rate of photosynthesis at both ambient and saturating CO2, while goat willow presented the lowest values for all of these variables. The ratio of *Jmax* to *Vcmax* varied in the typical range (1.5 to 2.5) observed for other woody species [65].
TPU limitation occurred in less than half of the samples, as it generally requires a higher *Ci* concentration than that used in our study. This third state of photosynthesis limitation occurs when the chloroplast system's reaction is greater than the possibility of the leaf using the triose phosphate [43]. This is more likely to happen under experimental conditions than in natural situations, and this limitation state has not been taken into account in many studies [66].
Light dependence of *Jmax* varied by up to 40% across the different leaves of birch [30]. Meanwhile, *Vcmax* is dependent on the temperature and increases at higher temperatures, but with limitations at very high temperatures [30]. For *Populus* species, multiple studies have documented *Vcmax* and *Jmax* values lower than those reported in this study, even though the light radiance used was higher, at 200–500 <sup>μ</sup>mol·m−2·s−<sup>1</sup> [25]. A global study that analysed more than 350 species highlighted that, for tree species, the mean values of *Vcmax* and *Jmax* were 66.6 <sup>μ</sup>mol·m−2·s−<sup>1</sup> and 114.4 <sup>μ</sup>mol·m−2·s<sup>−</sup>1, respectively [67].
#### **4. Conclusions**
Reforestation and the stable ecological restoration of tailings dumps resulting from surface mining activities in the Călimani Mountains have been a high priority for regional and national administrations in the last two decades. Even though several solutions have been implemented (dumps stabilization, soil amendments, etc.), the complete recovery of the degraded habitats is still a challenge. Gaining a better understanding of natural colonization with pioneer woody species, such as through studying primary natural succession, can offer valuable knowledge about the species that are most adapted to these particular environmental conditions.
The most productive pioneer species in terms of photosynthetic traits was the green alder, a woody N2-fixing shrub. It showed the highest rates of net assimilation, carboxylation efficiency and water use efficiency and can be a suitable species for reforestation based on the study conditions. During the growing season, this species' maximum photosynthetic capacity is generally observed at the end of July, with a minimum in late June.
A detailed understanding of the variability and dynamics of the photosynthetic capacity of pioneer species that naturally occur on tailings dumps is essential, offering valuable data to the process of characterizing suitable species for ecological restoration systems to heal these open wounds in the landscape.
**Author Contributions:** Conceptualization, I.P. and A.P.; methodology, A.P.; writing—original draft preparation, A.P.; writing—review and editing, I.P. All authors have read and agreed to the published version of the manuscript.
**Funding:** This research was funded by the Romanian Ministry of Research and Innovation in Core Program for forestry–BIOSERV (2019)–Project PN 19070102.
**Data Availability Statement:** Data is available on request formed to the corresponding author.
**Acknowledgments:** We thank Margareta Grudnicki for valuably suggestions and the Forestry Faculty of Suceava for supporting this study.
**Conflicts of Interest:** The authors declare no conflict of interest.
#### **References**
## *Article* **Growth Relationships in Silver Fir Stands at Their Lower-Altitude Limit in Romania**
**Gheorghe-Marian Tudoran 1,\*, Avram Cics, a 1,2, Albert Ciceu 1,2 and Alexandru-Claudiu Dobre 1,2**
**Abstract:** This study presents the biometric relationships among various increments that is useful in both scientific and practical terms for the silvicultural of silver fir. The increments recorded in the biometric characteristics of trees are a faithful indicator of the effect of silvicultural work measures and of environmental conditions. Knowing these increments, and the relationships among them, can contribute to adaptations in silvicultural work on these stands with the purpose of reducing risks generated by environmental factors. We carried an inventory based on tree increment cores. The sample size was determined based on both radial increment and height increment variability of the trees. The sample trees were selected in proportion to their basal area on diameter categories. Current annual height increment (*CAIh*) was measured on felled trees from mean tree category. For *CAIh* we generated models based on the mean tree height. Percentages of the basal area increment and of form-height increment were used to compute the current annual volume increment percentage (*PCAIv*). For the mean tree, the *CAIh* estimated through the used models had a root-mean-square error (RMSE) of 0.8749 and for the current annual volume increment (*CAIv*) the RMSE value was 0.1295. In even-aged stands, the mean current volume increment tree is a hypothetical tree that may have the mean basal area of all the trees and the form-height of the stand. Conclusions: The diameter, height, and volume increments of trees are influenced by structural conditions and natural factors. The structures comprising several generations of fir mixed with beech and other deciduous trees, which have been obtained by the natural regeneration of local provenances, are stable and must become management targets. Stable structures are a condition for the sustainable management of stands.
**Keywords:** silver fir; current annual increment; percentage volume increment; basal area; tree diameter; tree form-height
#### **1. Introduction**
In Romania, the fir (*Abies alba* Mill.) is frequently found mixed with beech (*Fagus sylvatica* L.) at altitudes between 700 and 1200 m. Mixed of fir and beech forests are formations representative of the lower mountain zone in Romania. Of the coniferous trees, Norway spruce (*Picea abies* (L.) Karst.) descends to these altitudes sporadically, but heat and reduced precipitation become limiting factors to its lower range; the proportion of spruce increases with altitude. Fir and beech influence climatic factors differently—especially humidity, light, and heat—so that, under the shelter they provide, beech seedlings frequently become established under fir, and vice versa. Consequently, the structures of stands are greatly varied in terms of the relative proportions of the two species. The stands structures present a wide range of diameter categories and offer the most favorable conditions for promoting structure of the uneven-aged stand type, which is characteristic of the natural
**Citation:** Tudoran, G.-M.; Cics,a, A.; Ciceu, A.; Dobre, A.-C. Growth Relationships in Silver Fir Stands at Their Lower-Altitude Limit in Romania. *Forests* **2021**, *12*, 439. https://doi.org/10.3390/f12040439
Academic Editors: Alessandra De Marco and Timothy A. Martin
Received: 20 January 2021 Accepted: 3 April 2021 Published: 5 April 2021
**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.
**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).
selection system [1–6]. Long-term experiments [7–11] revealed the effects of long-term interventions on the structure, growth, and total production of stands [12]. In order to change even-aged stand structures into structures suitable for 65 forest selection systems, silvicultural conversion treatments are applied [13–16] For even-aged beech-coniferous mixed stands in the Postăvarul Massif, the interventions had a character of transformation to uneven-aged structure.
Knowledge of the growth and relationships among trees can explain the development of stands. Interventions carried out on stands result in regulation of the ratio between the number of trees and their growth area. Change in diameter is the most dynamic biometric characteristic due to its sensitivity to silvicultural practices. A reduction in density can result in an acceleration of the diameter increment.
Although a reduction in stocking degree can generate significant growth increases, these increases may be only temporary. If a certain stocking degree is exceeded, this can lead to losses in production and exploitability. The reactions of trees to a reduction in stocking degree, through the activation of diameter increments, are particularly notable in young, vigorous trees [17]. Stand structure also dictates the growing pattern, with radial stem increments having a linear variation in even-aged stands, whereas a curvilinear variation occurs in uneven-aged stands due to a decrease in the radial increment in trees with large diameters. Therefore, the largest stem increments occur in trees in central categories. The volume increment varies depending on the correlation between height and diameter increments. These aspects—for both even- and uneven-aged stands—have been discussed in detail in the forestry literature [18–25] and were expressed in growth functions used in forest modeling [26], developed at the level of the stand, size class, or single tree [27].
When no successive inventories are performed, the current annual volume increment (*CAIv*) can be determined based on the percentage current annual volume increment (*PCAIv*). *PCAIv* is based on the diameter and height increments. The models that have been developed express these increments in relation to variables such as diameter, height, and age of the trees. Following the experimentation of three variants of determining Δ*h* (measured on felled trees, estimated by dynamic height curve and by conventional height curve) the elaborated models estimate the current volume increment with reduced errors of only 4–9% [28].
Other studies relating to the physiological processes of trees highlighted that the size of aged trees [29,30] (but not necessarily their age [31]) becomes very important when considering a reduction in their height increments [32]. The correlation between radial and height increments is weak, such that the diameter increment cannot be used for estimating the height increment. The radial increment has maximum values at different points in a tree's lifespan, with fluctuations caused by silvicultural practices, site conditions, climatic factors, inter- and intraspecific competition, and niche development [33]. The correlation between severe periods of reduction in fir growth and climatic factors has also been highlighted by dendrochronological studies [34] through drought indices. This also has a significant influence on the radial increment in trees [35,36].
Diameter strongly influences the *CAIv* and, consequently, a tree's current annual basal area increment (*CAIg*) has an equally significant impact on the volume increment. One study on Romanian even-aged stands [19] highlighted a linear correlation relationship, with high values (*r* = 0.80–0.95) occurring in mid-diameter categories between the *CAIg* and *CAIv*.
It is important to note that all of these determinations, which are aimed at identifying the variability in stand growth and the relation between increments and tree characteristics, require the use of a large number of sample trees. To ensure a sampling error within 10%, for a coverage probability of 95%, it is necessary to take measurements from at least 40–60 trees. If measurements are taken from only 10 trees, the sampling error will be between 20 and 30% [19].
Understanding the growth behavior of the fir at the local level is essential to the adaptation of its management. The trends in annual increment are also useful for understanding tree vitality. There are differences between the current annual increments of trees and stands and the increment values determined as averages by periods (periodic annual increments). It is therefore useful to know the current increment determined for each year of increment. Furthermore, the size of the *CAIv* should result from directly measurable elements that intervene in its size. If *CAIg* and the *CAIh* or the current annual form height increment (*CAIfh*) participates in the *CAIv*, then, in the calculation relations, the annual current values of these characteristics can be entered. The *CAIg* participates with the largest share in the *CAIv*, but its contribution in terms of volume increment varies during tree development. Knowing each increment's contribution to the size of the *CAIv* during tree development provides information on opportunities for silvicultural work to stimulate the volume increment as well as each increment's effect on volume increment. The *CAIv* can be determined by direct measurements or increment cores. When height increment is not measured, it can be determined indirectly using the correlations between height increment and other biometric characteristics of trees. The extraction of increment cores from all the inventoried trees leads to the safest results, but the procedures based on sample trees are also accessible to practice. We wished to consider all these aspects at the structure level, so as to be able to develop biometric relationships that would be useful in silvicultural practices.
The main objectives of this study were (1) develop models to estimate the *PCAIh* based on the mean tree height of stand, (2) assess the accuracy of the relationship (based on the *PCAIg* and *PCAIfh*) for estimating the volume increment at the mean tree and stand level, and (3) assess the possibility of using the mean tree for determining the volume increment of the stand.
#### **2. Materials and Methods**
*2.1. Materials*
Table 1 shows the symbols of the variables used in this study.
**Table 1.** Abbreviations of variables used.
**Table 1.** *Cont*.
Study area: The research was carried out in stands of fir with beech situated in the Valea Cetăt,ii watershed in the Postăvarul Massif. The stands are situated at altitudes ranging between 700 and 950 m, on slopes with inclinations of between 20 and 35◦ and various slope aspects. The area is characterized by multiannual average temperatures of 7–8.3 ◦C and annual precipitation of around 780 mm. The soils are deep and intermediatedepth eutricambosoils, with a groundcover of *Galium odoratum*–*Cardamine bulbifera* type. Our study was limited to stands in which fir constituted at least 30% of stands composition (to ensure the sample representativeness). From the stands, we selected a key surface area comprising 10 stands covering 131.8 ha (please see the Supplementary material).
Field measurements: The stands had average ages of 100–130 years, with disseminated elements that may have reached 140–150 years old. The stands basal area are ranging between 35 and 45 m2ha−1. In each stand, observations were made and measurements performed on the site conditions, herbaceous flora and seedlings. To determine the structure of the stands, experimental areas of 0.25–1.0 ha were studied, in which all the trees with diameters greater than 3 cm were inventoried. For each inventoried tree, several biometric characteristics were measured (diameter, height, pruning height, diameter of the crown). In this study, we exemplify the main biometric characteristics for one stand (45◦37 05 N, 25◦35 39 E) representative of a sample area of 1 ha (100 × 100 m) (Table 2). Of 133 inventoried mature fir trees, we extracted increment cores from 64 trees, with two diametrically opposed cores taken from each, following a direction parallel to the contour line (from 45 trees), and four cores from each, in two perpendicular directions, from trees where the inclination of the ground also permitted the extraction of cores from downstream (from 19 trees, i.e., 166 cores). The trees from which the cores were extracted were selected in report to their basal area on diameter categories. Therefore, for large diameter categories (due to the smaller number of tree) samples were extracted also form outside the 1.0 ha plot. Core samples were extracted from 14 diameter categories (characterized by amplitude
that varied between 2 and 8 cm). Thus, the obtained radial growth and the curve of radial increments address the trees growth trend corresponding to all diameter categories. The size of the sample ensured an error of ±8% under the conditions of a probability of 95%, which was established following an examination of the variability of radial increments. In seven felled sample trees with dimensions close to the mean basal area tree, we determined height increments by measuring the stem internode distance from the top to the base. Towards the inferior part of trees (i.e., the first 3 m of stem) where the position of the verticil was not suitable, we used increment cores extracted to the pith (to establish the trees age at different heights).
**Table 2.** Biometric characteristics (100 × 100 m).
Seedlings: composition: fir (55%), beech (35%), spruce (8%), sycamore, and Norway maple (2%)—coverage: 40% of the stand area.
In what concern the stands included in the key area, we present general data in the Supplementary Table S1.
#### *2.2. Methods*
Calculating the growth of trees: The increment cores (extracted to the pith) of trees with different diameters were measured. The measurement of the diameter increment in trees of different ages enabled us to determine their development in relation to the diameter and basal area. Annual height increments, as determined in the sample trees, were expressed as a percentage in relation to the height at the end of the growing period. These data yielded models that expressed the variation in annual increments in relation to the heights the trees reached during their lifetimes. We used the ratio of the heights of mean trees to reconstitute the heights from the beginning of the growing period (0) to the end (*n*), *hg0/hgn*.
The diameter increment was determined on the basis of radial increments (*ir*), according to the relation *d*<sup>0</sup> = *dn* − 2*ir*, and the radial increments that we used in the calculations came from the equation for the radial increments curve. Radial increments were measured using a digital positiometer.
The *CAIg* (or *ig*) still resulted from the difference between the two moments (at the end and beginning of the chosen period), on the basis of a diameter increment, according to the relation:
$$\dot{\mathbf{i}}\_{\mathcal{S}} = \pi (d\_{\text{tr}} \dot{\mathbf{i}}\_{\mathcal{I}} \mathbf{k} - \dot{\mathbf{i}}\_{\mathcal{I}} \, ^2 \mathbf{k}^2) \tag{1}$$
We used a coefficient for the bark (*k*) of 1.054, which we determined experimentally as being specific to the stands studied. Bark thickness was measured in 142 fir trees.
*CAIh* were measured in the felled sample trees within the category of mean basal area tree.
The increment percentages of each biometric characteristic (*d*, *g*, *h*, *fh*, or *v*) were analyzed at the level of the sample trees from the category of the mean basal area tree (felled). To calculate the percentages of annual increments of the biometric characteristics at the level of average tree (diameter, basal area, height, volume), we applied Pressler's formula [22].
The *CAIv* was determined at the level of the mean tree of the stand and at the level categories of diameter as well as the entire stand. At the category-of-diameter and stand levels, the calculation was similar to that used for an individual tree. To obtain *CAIv* (or *iv*), we performed a single inventory based on increment cores.
Based on the diameter and height increments, the diameter and height of the trees could be determined at the beginning of the growth year. The *CAIv* was determined by subtracting the volume at the end (*n*) from that of the beginning (0) of the chosen period; in the case of the tree, this was
$$
\dot{v}\_v = \upsilon\_n - \upsilon\_0. \tag{2}
$$
The volume of the trees was determined based on diameter and height by regression equation used for this species in Romania [23]:
$$\log \upsilon = \mathbf{a}\_0 + \mathbf{a}\_1 \log d + \mathbf{a}\_2 \log^2 d + \mathbf{a}\_3 \log \mathfrak{h} + \mathbf{a}\_4 \log^2 \mathfrak{h} \tag{3}$$
In the Equation (3): a0 = −4.46414; a1 = 2.19479; a2 = −0.12498; a3 = 1.04645; a4 = −0.016848.
The relationships between the volume increments at the tree level (*iv*) and stand level (*IV*) were developed by introducing the basal areas (*g* and *G*, respectively) and form heights (*fh*) from the beginning and end of the chosen period. For the tree, we reached the known relation *iv* = *gnhnfn* − *g0h0f* 0. We expressed the form-height from the beginning of the chosen period (*fh*0), based on the form-height increment (*CAIfh* or *ifh*). We also introduced the *CAIg* (or *ig*) into the relation of the *CAIv*, respectively *ig = gn* − *g0* and we reached the known relationships: [18] *iv = igfh <sup>n</sup> + g0ifh* and [19,23]:
$$
\dot{a}\_{\upsilon} = \mathcal{g}\_n \dot{i}\_{\mathfrak{H}} + \dot{i}\_{\mathfrak{G}} \mathcal{f} \mathbf{i}\_n - \dot{i}\_{\mathfrak{G}} \mathbf{i}\_{\mathfrak{H}} \tag{4}
$$
The *CAIv* values we obtained by applying the Equations (2) and (4) were considered as reference values. With the *CAIv* determined by the two relations, we compared the *CAIv* obtained by applying the simplified relationship-based *PCAIv*.
The characteristics of the mean trees (of the basal area and of the volume) resulted, indirectly, from the following calculations: *dg* of the mean tree from the mean basal area (determined by the ratio *G/N*), the volume of the mean volume tree from *V/N*, and the form-height (FH) of the stand from *V/G*.
Finally, we present the value of the CAIs for fir mean tree and stand. They have been determined for a single year of growth, based on the values of the variables measured at the level of year 2017, which followed a period in which the health state of fir in the study area had continuously deteriorated.
#### **3. Results**
#### *3.1. Volume Increment of the Mean Tree*
The current increment percentages of the mean tree: The *CAIv* is the result of diameter increment, height, and changes in the stem shape of the trees. In the *PCAIv*, the *PCAIg* of the mean basal area tree can be used. The *PCAIg* has a relatively significant influence on the *PCAIv*.
At ages between 110 and 130 years, the *CAIh* models indicated a height increment (*CAIh*) ranging between 0.8 and 0.14 m. Increments of 0.14 m were recorded in structurally closed stands situated on the lower parts of slopes. The increments decreased toward the higher parts of slopes and toward stand densities less than 0.8. Models assist in predicting annual or periodic height increments in trees of various ages in relation to their heights. Such a model is represented by the equation:
$$PCAI\_h = 0.147097(h - 1.21892)^{-0.42450} \left(40.20507 - h\right)^{1.18566},\tag{5}$$
which characterizes the experimental distribution in Figure 1b, or
*PCAIh* = 0.020125(*h* − 1.181557) <sup>−</sup>0.359495(40.158443−*h*) 1.750728 (6)
**Figure 1.** Percentages of mean fir tree increment: (**a**) percentage of diameter annual increments, basal area and volume (determined by Pressler's formula); (**b**) percentage height increment of the mean tree by Equation (5) (in relation to the height of the mean tree at the end of the chosen period).
In Equations (5) and (6), the *CAIh* of the trees is expressed as a percentage in relation to their height at the end of the chosen period. The models become applicable if the height of the tree is known, based on a single inventory. In turn, the variation in height in relation to age can be explained through the development function. The height can also be determined by means of the height curve, which expresses the variation of the height in relation to the diameter of the trees. Equations (5) and (6) explained 84–86% of the variation units in the *CAIh*. The height and diameter significantly influence the percentage annual height increment (*PCAIhs*) and explain its tendency (*R*<sup>2</sup> = 0.91–0.93).
The percentages of the increment of each biometric characteristic (*d, h, g*, or *v*) diminish as a tree advances in age (Figure 1).
Relationships between the current increment percentages: Throughout a tree's development, the percentage increments *PCAIg* and *PCAIh* changes from one year to the next and from one development period to the next (Table 3). Together, *PCAIg* and *PCAIh* determine the tree's volume increment and ensure its tendency, as expressed by the functions of growth and development.
**Table 3.** Annual mean percentages of mean fir growth.
Results obtained from the analysis of the stem of the felled sample trees.
Fir trees presented the highest percentages of growth during the period between 15 and 55 years. The greatest contribution of the *PCAIh* in the *PCAIv* (42%) is made during the period between 56 and 85 years. After this interval, the contribution of *PCAIh* in *PCAIv* climbs as high as 20%. Between 85 and 115 years, *PCAIv* was 2.92% on average and this is largely (80%) attributable to the *PCAIg*.
The annual percentages of the increments established at the sample-tree level prove known relationships:
$$PCAI\_{\upsilon} = PCAI\_{\mathcal{J}} + PCAI\_{\mathcal{fl}} \tag{7}$$
and:
$$PCAI\_{\%} = 2PCAI\_d \tag{8}$$
The differences between the experimental values of the percentage of height increment and those estimated by the models diminished as the trees grew in height (Figure 2a). The average deviations of the experimental values of %ih, in comparison to the values estimated by the equation (BIAS), were +0.001483 and the average of the squares of the deviations RMSE was 0.87490. Using Equation (7) to calculate the *PCAIv* yielded an average deviation in the experimental values of *PCAIv*, compared to the values estimated by the relation, of −0.027425, with an average of the squares of deviations of 0.129521 (Figure 2b and Table 4). When *CAIfh* was related to the height of the trees at the end of period for the calculation of the *PCAIfh*, the BIAS was 0.019223 and the RMSE was 0.050536. Furthermore, the values of the deviations decreased for volume—just as they did for height—as the volume of the trees increased (Figure 2b). This can be explained by the reduced weight of the percentage *PCAIh* in relation to volume as the trees advanced in age. The accumulation in volume, then, is largely due to annual basal area increments.
**Figure 2.** Errors between the experimental values and those predicted by models: (**a**) between the experimental values (*PCAIh*) and those calculated based on percentages predicted by Equation (5); (**b**) between the experimental values (*PCAIv*) and those calculated using Equation (7). The differences calculated were expressed in relation to the height (**a**) and volume (**b**) of the mean tree.
**Table 4.** Proportion of the *PCAIfh* in the *PCAIv* equation (Equation (7)).
Results obtained from the analysis of the stem of the felled sample trees.
During tree development, the contribution of each biometric characteristic to the volume differs. Therefore, it is natural that, at different moments in tree development, the percentages of the biometric characteristics would have variable proportions relative to the *PCAIv*. For Equation (7) the participation of the respective percentages in the *PCAIv* was expressed by the coefficient *ki* (i = 0.1 to 1.0 h). For Equation (7) we noted the stability of the *ki*, which decreased throughout the development of the trees (Table 4). Thus, this relationship can be recommended for determining the current volume increment.
Current annual increment of the mean tree: The *CAIv* of the mean tree resulted from the application of Equations (2) and (4) adapted to the level of the individual trees, as well as Equation (7). Through the Equations (2) and (4), the same *CAIv* value resulted: 17,937 dm3.
*PCAIv*-based relationship (expressed by Equation (7)):
$$CAI\_{\mathcal{V}} = 0.01(PCAI\_{\mathcal{S}} + PCAI\_{fh})\upsilon\_{\mathcal{V}} \tag{9}$$
also led to a value close to the size of 17,937 dm3, the difference being +0.4%.
#### *3.2. Stand Growth*
The *CAIV* of the stand was produced by applying Equations (2), (4), and (7) to the category-of-diameter and entire-stand levels. The *CAIh* was determined by the regression Equation (5).
At the category-of-diameter and stand level, the calculation was similar to that used for an individual tree. At the level of the entire stand, we introduced into the calculations the values of the biometric characteristics from the level of the entire stand. By applying Equations (2) and (4) similar value for *CAIV* as those obtained (i.e., 2.38 m<sup>3</sup> yr−1). Of the simplified Equation (7), based on the percentage-of-form-height increment (*PCAIfh*), presented the greatest stability. This can be applied to determine the volume increment of individual trees (Figure 3a) and the stand (Figure 3b).
**Figure 3.** Current annual volume increment for individual trees (**a**) and stand (**b**).
By applying the simplified Equation (7), the volume difference compared to Equations (2) and (4) was +0.4%.
#### The Mean Tree of the Stand
Given the elements that impact volume increment, more attention should be paid to the basal area of the stand and the *CAIG* than to the form-height of the stand (FH).
When applying Equations (4) and (7), the form-height can be deduced from the volume of the mean volume tree or the stand volume. Using only the two trees (i.e., mean basal area tree or mean volume tree) in the calculation of volume increment leads to +8.6 and +8.9% errors (Table 5). The tree with the mean current annual volume increment (*iv*) is a hypothetical tree that may have the *g* and FH.
**Table 5.** Mean fir volume increment.
Furthermore, upon analyzing the results from Table 5, it results that, for the structural conditions of the studied stands (i.e., even-aged structures), the diameter of the real tree with represent about 95% of the *dg*. It results that the growth determination based on measurements of trees from the categories of the two trees (i.e., mean basal area tree and mean volume tree) can lead to values of current volume increment close to the values of the tree with *iv*. Thus, the field works can be simplified.
In Figure 3a, it is clear that the trees in the superior-diameter categories, due to their large exchange surfaces, also recorded the greatest volume increments. It was found that the *CAIv* per tree was influenced by weight of trees by category of diameter, and that the measure of the *CAIv* of a stand depended on the respective size of the growing stock on the distribution of tree volume by diameter category.
In Table 5, the *CAIvs* of the mean trees were determined using the Equation (4). For comparative purposes, we used the average value of the *CAIv* per tree: 17.937 dm3.
#### **4. Discussion**
#### *4.1. Relationships between Tree Growth and Biometric Characteristics*
Due to fluctuations in the radial increments, the maximum diameter increment occurred at different moments in the lifetimes of the trees. The trees with higher radial growths also have large volume growth. Remarkable volume growths were found in the diameter categories with the greatest volumes. However, these trees, in the conditions of the lower-altitude limit, do not have the largest radial increments. For older stands, the radial growth-diameter relationship can be expressed by a second-degree parabola. Expressing radial growth or diameter growth by a line or a logarithmic equation may overestimate the radial growths of older trees. Because stands are suited to uneven-aged structures, the study of increments is of interest, especially for establishing target diameters after which the stand structures may be modeled.
The maximum height increment was influenced by the position of the trees, their vitality and their stationary conditions. Just as with the diameter increment, the height increments of the studied stands, expressed in Equations (5) and (6), presented reduced values. The height increment values measured in the sample trees, expressed as a percentage in relation to the height, diameter or age, indicated the same decreasing tendency, with height and age explaining a variation in percentage of the height growth of 85%. Other studies [28,31] show that the age of trees does not statistically explain the reduced height gain, but the size of the trees significantly influences it.
Throughout the life of a tree, the rhythm of the diameter increment differs from that of the height increment. At the sample-tree level, the *PCAIv* is particularly influenced by *PCAIg*, with the correlation being very strong (*R*<sup>2</sup> = 0.99). The *PCAIh* also influences the volume increment (*R*<sup>2</sup> = 0.86). In the *PCAIv* of the trees, *PCAIh* has a low share, around 20% (Table 3). Therefore, the *CAIv* of trees is achieved mostly (approximately 80%) based on the *PCAIg*.
Trees distribute their maximum radial increments at different times in their lives, depending on the structural conditions. The measurements performed on the felled sample-
trees, from the category of the mean tree, show that the height increment contributed the greatest weight to the volume increment, among trees aged 55–65 years, when the trees grew 0.5–0.6 of the mean tree height at the end of the growing year, and 0.2 of their volume. At this point, the diameter increment had the lowest proportion of volume increment, but after this moment, the proportion of the diameter increment began to increase, while the proportion of height increment decreased. Thus, after 85 years, a progressively accentuated thickening of the stem occurs. In this period, trees grow 22% of their height increment and 16% of their form-height.
The form factor also decreased as the trees grew in diameter, from a value of 0.505 (for trees with *d* = 24 cm and *h* = 26 m) to 0.392 (for trees with *d* = 78 cm and *h* = 35.1 m). Fluctuations in the form factor depended on two variables: tree diameter and height. For short periods of time, the form factor gave reduced variations or remained constant [37], so that its value did not influence the measurement of the *CAIv*, which was determined by successive inventories. For the example presented, the value of the form factor of the stand remained almost unchanged (from 0.4486 in 2016 to 0.4482 in 2017).
The *PCAIh* determined on the basis of experimental data produced values of height increment close to those reported in the existing literature [38] for fir stands in Romania [39].
#### *4.2. Precision in Estimating the Current Volume Increment Using Simplified Relationship*
Efforts have been made to establish simplified relationships for determining the *CAIv*. In the literature, formulae based on the *PCAIv* have been proposed by Dvore¸tki, Tiurin, Anucin, Pressler, Breymann, Schneider, and Prodan (see Giurgiu, 1979). We have used only the equation for the *PCAIv*: *piv* = *pig* + *pih* + *pif* [22]. The precision of the different formulae varies. The formulae of Tretiacov and Dvore¸tki [19] lead to mean squared errors when determining the *PCAIV*, ranging between ±9 and ±13%, but the determination errors can reach up to 20%, especially for short observation periods of 3–5 years.
Simplified relationships are based on strong correlations between the percentages of mean-tree growth. It is known that, throughout a tree's lifetime, the percentages of biometric-characteristic increments vary from one year to the next, but, on the whole, they preserve a decreasing tendency. Elements such as *CAIg, CAIfh*, basal area, and form-height at the end of the growth period decisively influence the volume increment [18].
The precision of the *CAIv* determinations based on successive inventory (Equation (2)) was categorically influenced by errors in the respective volumes e*Vn*% and e*V*0%. These volumes were determined using Equation (3). Errors in the two volumes depended on the errors with which the variables introduced in their calculus were determined. For a probability of 95%, the interval of errors reaches of 8–10% [37]. Under our study conditions, the tree diameters were determined based on circumference, the heights were measured in all 133 samples, and the form factor was found using the volume of the trees, so we can consider that the error in volume increment determined by successive inventories reached a maximum of 10%.
According to Krenn's equation [18], in the case of successive inventories, the greater the volume of the stand and its determination error, the less the growth in the volume of the stand, and the greater the volume increment error. For periods of 10 years, the error in *CAIv*, obtained by applying the equation based on form-height and basal area (Equation (4)) from the beginning of the observation period, was ±8%, decreasing to ±2.4% when the period was extended to 40 years [18].
In the case of simplified relationship (Equation (7)), the error in *CAIv* (e*CAIv*) depends on the error involved in determining the volume (e*v*), the volume to which *PCAIv* is applied, and the error in the *PCAIv* (e*PCAIv*). Thus, the error in volume increment could be written:
$$\left| \mathbf{e}\_{\rm PCA} \right|^2 = \mathbf{e}\_{\upsilon}^2 + \mathbf{e}\_{\rm P\underline{C}Alv}^2 \tag{10}$$
The error involved in determining the volume of the trees and the stand (e*v*) can be considered to be ±10%. In determining the error in the *PCAIv*, the error in *PCAIg* (e*PCAIg*) and the error in the *PCAIfh* (e*PCAIfh*) intervene. By analyzing radial increments from 64 trees, the error e*PCAIg* was found to represent ±8%. Given that e*PCAIfh* represents around 25% of e*PCAIg*, we can consider that e*PCAIph* represents 2%. By introducing these values into each relationship (10), errors in volume increment fall within ±13% (in ~90% of the cases). These errors also include possible errors in determining the volume of the trees using Equation (3). Only the error in determining the *PCAIv* (e*PCAIv*), obtained by applying the relation (7), is ± 8%.
#### *4.3. Practical Utility of Volume Increment Relationship*
The relationships used to determine the *CAIv* can also be applied to cases where a single inventory is carried out at the end of the chosen period. In such cases, to determine *CAIg*, the extraction of increment cores is necessary. If the *CAIh* cannot be measured, it can be determined with the help of Equations (5) and (6). Because the form factor fluctuates throughout the lifetime of trees, it is best that the form-height be determined from the volume of the stand. However, the use of the volume, both at the end and beginning of the chosen period, leads to diminishing errors in determining volume increments.
At the level of the mean tree, using simplified Equation (7) to has led to values of the *CAIv* (17.937 dm3) that are close to those obtained using Equations (2) and (4) (Table 6). Annual increments in height of 0.08 m were obtained by other studies carried out on fir located at its southernmost distribution limit, but at tree heights of 42.5 m [32].
**Table 6.** Methods used to determine the current annual volume increment.
Biometric characteristics of stand: 115 years old, volume 299.1 m3yr−1, SD stand 0.74, SD fir 0.35, N 133 trees.
At the level of the stand, the *PCAIv* of the mean tree (0.80 from Table 3), obtained using Equation (7), applied to the volume of the fir stand, produced a measure of growth of 2.39 m3yr−1, which is 0.009 m3 greater than the (+0.4%) current growth of the tree (Table 6).
Limits to the simplified relationship: The simplified relationship, Equation (7), introduced the entire measure of *PCAIfh* into the calculation, together with the *PCAIg*. This made it applicable to any structural conditions, both at the level of individual trees, category of diameter and the entire stand.
The adaptation of silvicultural work in fir stands: This study showed that, at their lower-altitude limit, old, even-aged stands have lower growth. Future structures of fir and beech mixed stands will need to be adapted to ensure a better representation of younger generations in the stands [5,6]. These generations are capable of ensuring the continuity of the protection functions attributed to such stands as well as their stability. To create such structures, silvicultural interventions in the stands need not delay placing installed seedlings in sunlight or extracting trees in the central categories that hinder the stand's development. For stands in transformation, the necessity of installing and developing new generations dictates the timing and intensity of such interventions. By the age of around 80 years, silvicultural works should aim to stimulate the growth of trees in height—and after that age, growth in diameter. The deciduous species that naturally regenerate and are encountered in mixed stands, such as sycamore maple and mountain elm, need to be given the chance to contribute to the composition of these stands. The creation of diversified vertical structures must be a condition of sustainable silvicultural work on such stands. Thus, stands will be able to maintain their vitality and their capacity for regeneration at any moment in their existence. The natural regeneration of mixed stands also creates the premise of being able to capitalize on local provenances that are much better adapted to the prevailing environmental conditions [40]. Such characteristics should constitute targets for silvicultural work on the future structures of fir and beech mixed forests.
#### **5. Conclusions**
By applying silvicultural works, the stand structure and, implicitly, the relationships between the trees change, against a background of site conditions. These influences are recorded by the trees and are displayed in their growth. The *CAIv* is the result of diameter increment, height, and changes in the stem shape of the trees. The percentages of the increment of each biometric characteristic (*d, h, g, or v*) diminish as a tree advances in age. The height increment of the mean tree can be determined by the models expressed by Equations (5) and (6). The differences between the experimental values of the percentage of height increment and those estimated by the models diminished as the trees grew in height. The average deviations of the experimental values of *PCAIh*, in comparison to the values estimated by the equation, was 0.001483, and the RMSE was 0.87490. The *CAIv* of the tree was produced by applying relationship of the *PCAIv*. Using Equation (7) to calculate the *PCAIv* yielded an RMSE in the experimental values of *PCAIv*, compared to the values estimated by the relation, of 0.129521. When *CAIfh* was related to the height of the trees at the end of period for the calculation of the *PCAIfh* the RMSE was 0.050536. At the level of the mean tree and stand, the simplified equation (Equation (7)) has led to values of the *CAIv*, and respectively *CAIV*, that are close (0.4%) to those obtained using relationship based on basal area and form-height increments and relationship based on successive inventory.
The *CAIV* of a stand can be determined on the basis of the *CAIv* of the mean current volume increment tree. The tree with the average value of the *CAIv* of all trees is influenced by the structure of the stand, and its biometric characteristics are difficult to determine because they are specific to each structure. The characteristics of such a tree can be linked to the characteristics of the mean basal area tree and of the mean volume tree, which are easy to determine by means of a single inventory at the end of the chosen period. In even-aged stands, the mean current volume increment tree is a hypothetical tree that may have the mean basal area and the form-height of the stand. Such research should be continued on several felled sample trees in order to further clarify this statement in different structural conditions.
This study on the trees in the area shows that the fir trees maintain their ability to record active height increments until around the age of 80 (with a maximum between 50 and 60 years). After this age, the trees continue to accumulate significant increases in diameter. The reduction of stand density through silvicultural work must be correlated with these moments, which may prevent volume growth losses and increase final production.
**Supplementary Materials:** The following are available online at https://www.mdpi.com/article/10 .3390/f12040439/s1, Table S1 presents the general characteristics of the 10 research areas located in the stands included in the study. The sample plots of 0.5 ha have a circular shape and are allocated five to a survey (four in the direction of the cardinal points and one in the center). The sample plots of 1 ha have a regular shape (100 × 100 m).
**Author Contributions:** Conceptualization, G.-M.T.; Data curation, G.-M.T. and A.C. (Avram Cicsa); Formal analysis, G.-M.T.; Investigation, G.-M.T., A.C. (Avram Cicsa), A.C. (Albert Ciceu), and A.-C.D.; Methodology, G.-M.T.; Visualization, G.-M.T.; Writing—original draft, G.-M.T.; Writing—review & editing, G.-M.T., A.-C.D., and A.C. (Avram Cicsa). All authors have read and agreed to the published version of the manuscript.
**Funding:** This research received no external funding.
**Acknowledgments:** We are deeply indebted to N. Rucăreanu who, through his silvicultural management work, supported studies on transformation cuttings and initiated forestry biometry research
in the forests of this watershed, starting in 1960. We also owe thanks to the Romanian Ministry of Waters and Forests, which, since its endorsement of silvicultural management in these forests, has expressed its approval of maintaining experimental areas with a view to monitoring the structure, measurement and growth of the stands.
**Conflicts of Interest:** The authors declare no conflict of interest.
#### **References**
## *Article* **Impact of Industrial Pollution on Radial Growth of Conifers in a Former Mining Area in the Eastern Carpathians (Northern Romania)**
**Cristian Gheorghe Sidor 1, Radu Vlad 1, Ionel Popa 1,2, Anca Semeniuc 1, Ecaterina Apostol <sup>3</sup> and Ovidiu Badea 3,4,\***
**Abstract:** The research aims to evaluate the impact of local industrial pollution on radial growth in affected Norway spruce (*Picea abies* (L.) Karst.) and silver fir (*Abies alba* Mill.) stands in the Tarnit,a study area in Suceava. For northeastern Romania, the Tarnit,a mining operation constituted a hotspot of industrial pollution. The primary processing of non-ferrous ores containing heavy metals in the form of complex sulfides was the main cause of pollution in the Tarnit,a region from 1968 to 1990. Air pollution of Tarnit,a induced substantial tree growth reduction from 1978 to 1990, causing a decline in tree health and vitality. Growth decline in stands located over 6 km from the pollution source was weaker or absent. Spruce trees were much less affected by the phenomenon of local pollution than fir trees. We analyzed the dynamics of resilience indices and average radial growth indices and found that the period in which the trees suffered the most from local pollution was between 1978 and 1984. Growth recovery of the intensively polluted stand was observed after the 1990s when the environmental condition improved because of a significant reduction in air pollution.
**Keywords:** air pollution; increment cores; Norway spruce; radial growth series; silver fir
#### **1. Introduction**
Climate change and air pollution represent the main drivers of global change, significantly impacting forest health and sustainable development [1]. Accelerated industrial development after World War II increased pollutant inputs in many parts of the globe, and Central and Eastern Europe were significantly affected [2]. The changes in political regimes in Eastern Europe and national and world environmental policy after the 1900s allowed for a significant reduction in air pollution in most regions. Nonetheless, investigating the environmental impact of air pollution resulting from anthropogenic industrial activities remains critically important [3].
Air pollution negatively affects forest ecosystems and soil quality worldwide. Industrial emissions from the Ivano-Frankivsk and Chernivtsi regions (Ukraine) have led to high concentrations of heavy metals in forest soils, high levels of tree crown defoliation and ecosystem changes such as biodiversity decline or reduced productivity [4]. Soil acidification in the region has risen progressively due to the increased content of heavy metals [5], which indirectly influences forest ecosystem vegetation [6]. In Germany, the areas near recently halted mining operations were investigated to determine the uptake
**Citation:** Sidor, C.G.; Vlad, R.; Popa, I.; Semeniuc, A.; Apostol, E.; Badea, O. Impact of Industrial Pollution on Radial Growth of Conifers in a Former Mining Area in the Eastern Carpathians (Northern Romania). *Forests* **2021**, *12*, 640. https:// doi.org/10.3390/f12050640
Academic Editor: Riccardo Marzuoli
Received: 7 April 2021 Accepted: 13 May 2021 Published: 19 May 2021
**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.
**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).
of heavy metals by forest trees in heavily contaminated ecosystems. The researchers also described the levels of damage caused by heavy metal toxicity [7]. A similar study in the Czech Republic analyzed the health status of Norway spruce (*Picea abies*) forests according to the phenology and radial growth of trees in relation to air pollutants, especially NO2 and SO2 [8]. The distribution and accumulation of heavy metals in forest soils in the Rozocze National Park (SE Poland) was found to be related to anthropogenic pollution through local and background emission sources [9]. In the Carpathian Mountains, the results of long-term monitoring activities indicated that the combined effects of O3, SO2 and NO2 could negatively affect forest stands and highlighted the association between air pollution levels and tree growth [10]. Furthermore, elevated levels of N and S deposition at the levels found in the Carpathian Mountains may have negatively affected forest health status and biodiversity, including visible leaf injury, losses in stand growth and productivity and higher sensitivity to biotic and abiotic stressors [11,12]. The accumulation of heavy metals with accompanying S and associated soil and foliar nutrient imbalances and reduced soil water holding capacity can restrict the recolonization of plant communities in the forest ecosystem [13]. In the recent past, some industrial activities in Romania (Cops, a Mică, Zlatna, Baia Mare) were known to create regional hotspots of pollution, negatively affecting forest vegetation [14–17]. Toxicity thresholds for the forest environment in Romania were highlighted based on air quality analysis [18].
Trees are sensitive to environmental factors, and any changes in growing conditions are reflected in tree ring parameters. The reduction in tree growth is generally associated with unfavorable climate conditions and an increase in specific ecosystem competition. Furthermore, air pollution can be associated with narrow growth rings for several decades.
The evolution of forest ecosystems affected by past pollution in highly industrialized areas and damage dynamics assessments offer crucial knowledge needed to develop management strategies for the conservation and improvement of the environment. Thus, to accurately assess how severely trees or forests have been affected by pollutants, it is critical to study the issue in well-defined ecological areas. For the northeastern part of Romania, the Tarnit,a mining exploitation constituted a hotspot of industrial pollution. The primary processing activity of non-ferrous ores containing heavy metals in complex sulfide forms was the main cause of pollution in the Tarnit,a region. Tarnit,a mining exploitation began in 1968, and the amount of production increased until 1990 through the exploitation of new deposits. With the political regime change in Romania, concomitant with the economic recession after the fall of communism, there was a sharp decline in mining activity, followed by a cessation in 1998 [18].
Considering this specific pollution history, the aim of this research was to evaluate the impact of local past industrial pollution on radial growth in affected Norway spruce (*Picea abies*) and silver fir (*Abies alba*) stands in the Tarnit,a study area, Suceava (northern Romania).
#### **2. Materials and Methods**
A network of experimental plots was established in five representative yield management units in the Tarnit,a region, Suceava county, within the Stulpicani Forest District (FD) (Table 1). This network of experimental plots included 30 plots (15 for silver fir and 15 for Norway spruce) from which radial increment cores were collected (40 trees for each species per plot). In order to highlight the level of pollution intensity on the silver fir and Norway spruce stands in the Tarnit,a area, the 30 plots were located spatially at different distances from the main source of pollution. Taking into account previous research [19,20], in which several stands located at different distances from the source of pollution were analyzed and it was concluded that the 2 km length was the approximate distance to and from which the stands were affected by pollution to varying degrees of intensity, we tested this hypothesis and we considered in this study that intensively polluted stands are located at a maximum distance of 2 km from the main source of pollution, moderately polluted stands are located between 2 and 6 km and largely unpolluted stands are located at a distance greater than
6 km. Thus, based on the results obtained in these studies, in order to test our hypothesis, we considered these distances as limits between different pollution intensities.
**Forest Management Unit (u.a.) Yield Management Unit (UP) Distance from the Polluting Source (km) Area (ha) Age Composition \* Exposure Slope (Centesimal Degrees) Altitude (m) Canopy Cover Yield Class Standing Volume (m3/ha**−**1)** 73C V Tarnit,a 6.0 4.90 95 9MO1BR NE 28 1120 0.8 2 585 62A V Tarnit,a 3.1 6.34 105 4MO3BR2FA1PAM NE 25 985 0.7 2 416 39A V Tarnit,a 0.5 4.13 85 8MO2FA NE 16 860 0.4 2 304 111E V Tarnit,a 1.2 4.67 105 4MO4BR 2FA SE 26 980 0.6 3 396 118A V Tarnit,a 2.5 18.45 140 4MO4FA2BR S 26 1125 0.5 3 314 18F V Tarnit,a 3.2 19.54 140 5MO3FA2BR NW 36 1125 0.6 3 370 14C V Tarnit,a 1.6 6.87 95 5BR3FA2MO S 26 900 0.7 3 402 126I V Tarnit,a 1.2 5.2 100 6MO2BR 2FA SE 16 875 0.6 2 346 5A VI Botos,ana 4.5 29.84 95 7MO2BR1FA N 27 905 0.7 1 572 17B VI Botos,ana 3.1 10.66 75 8MO2BR N 18 1000 0.8 2 589 45A VI Botos,ana 6.9 8.57 115 6MO2BR2FA N 18 775 0.7 2 601 61A VI Botos,ana 6.0 16.22 110 4MO3FA2BR1PAM SE 22 1000 0.7 2 482 43B IV Porcăret, 6.2 24.2 95 6MO3BR1FA NE 24 990 0.7 2 513 22A II Negrileasa 9.7 28.78 95 5MO4BR1FA N 22 850 0.7 2 560 4B VIII Gemenea 12.0 18.10 110 8MO2BR NW 12 825 0.6 2 482 101C VIII Gemenea 12.1 13.03 125 8BR2MO SW 33 740 0.6 2 474
**Table 1.** The main characteristics of the experimental plots network from Tarnit,a region.
\* Degrees of participation (in tenths) of the species in the mix forest stand; Norway spruce (MO), silver fir (BR), European beech (FA), sycamore maple (PAM).
> Concerning the characteristics of the studied stands, the trees included in the forest stands of the network within the Tarnit,a study area were between 75 and 157 years old. The tree stand composition was generally a mixture of Norway spruce (the predominant species), silver fir and European beech (*Fagus sylvatica* L.). Plot altitude varied between 750 and 1150 m. The studied stands had a canopy cover between 0.6 and 0.8 and were mostly of higher productivity (classified in the 2nd relative yield class). The volume per hectare was between 346 and 601 m3.
> For the classification of the radial growth series in relation to the distance from the polluting source, both the distance from the source and the predominant direction of airmasses (NE–SW) were taken into account. Thus, there were six series of growth (three silver fir and three Norway spruce) located in the intensively polluted area, 14 series in the moderately polluted area (seven silver fir and seven Norway spruce) and 10 series in the largely unpolluted area (five silver fir and five Norway spruce). In the direction of the main valley, the stands were intensively polluted up to distances of 6–7 km (Figure 1).
> The following specific statistical parameters were calculated, both for radial growth series and tree ring index series [21,22]: the period covered by each series with a mini-mum replication of 10 individual series, sample depth, mean tree ring width, average sensitivity (average percentage change of annual ring width relative to the next annual ring [21]) and average Rbar (correlation coefficient between all individual series).
> The degree of reduction and recovery of growth due to the influence of local industrial pollution was determined through the resilience indices, presented as a 5-year moving average. Tree resilience is its post-disturbance ability to reach the level of radial growth it experienced before the disturbance, calculated as the ratio between the pre- and postdisturbance growth [23]. The tree resilience calculations were performed for each growth series analyzed, revealing the capacity of trees to grow after the disturbing events that caused reduced radial growth during certain periods.
> Radial growth indices of intensively and moderately polluted trees were compared to the radial growth indices of the unpolluted trees, considered reference values. The calculations and analyses were performed for the period common to all analyzed series from 1951 to 2018. The highlighting and quantification of the growth changes of the stands in the areas affected by the industrial pollution were performed using software such as CooRecorder 7.4 [24], CDendro 7.6 [24], TsapWin [25], COFECHA [26,27] and R studio [28].
**Figure 1.** Study location: (**a**) Silver fir from Tarnit,a region; (**b**) Norway spruce from Tarnit,a region (the red arrow indicates the source of pollution and the predominant direction of airmasses in the area is NE–SW, according to the valley orientation).
#### **3. Results**
#### *3.1. The Statistical Parameters of the Series of Average Radial Growth*
The main statistical parameters for all average radial growth series studied are shown in Table 2. The length of the analyzed silver fir series from the Tarnit,a area varied between 75 and 157 years, with an average ring width value between 1.704 and 3.346 mm. The mean values of sensitivity varied from 0.196 to 0.253, and the mean Rbar values of the residual series were between 0.219 and 0.411. The Norway spruce growth series lengths varied between 72 and 143 years, similar to silver fir. The average value of the tree ring width varied between 1.812 and 3.447 mm. The Norway spruce Rbar values were lower than the silver fir, varying between 0.194 and 0.364 (Table 2).
#### *3.2. Analysis of Growth Changes of Trees*
The average radial growth of the silver fir in the Tarnit,a area showed a similar dynamic regardless of the plot distance from the pollution source (Figure 2A). The exception was the period 1978 to 1990, during which the radial increments of intensively polluted silver fir were significantly lower than those from unpolluted areas. The average radial growth values of silver fir in moderately polluted areas during this period were intermediate between intensively polluted and unpolluted plots. According to average radial growth indices, the Norway spruce trees were the most affected by local pollution from 1978 to 1984 (Figure 2B). The silver fir trees in the Tarnit,a area demonstrated average radial growth dynamics (Figure 2A) that indicated they were more affected by the pollution than the Norway spruce.
**Table 2.** Statistical parameters of the average radial growth series for silver fir and Norway spruce in the Tarnit,a area.
**Figure 2.** The average series of radial growth indices developed for each of the 3 categories of stands studied (the dotted circle represents the time interval in which the trees were most affected by air pollution); (**A**) silver fir; (**B**) Norway spruce.
From 1978 to 1990 (Figure 3A), only those resilience indices corresponding to the analyzed trees in the intensively polluted area had negative values. The analysis of the resilience indices (Figure 3B) revealed that the Norway spruce trees were also affected by the local pollution from 1978 to 1990, but to a much lesser extent than the silver fir.
**Figure 3.** *Cont*.
**Figure 3.** Resilience indices of the average radial growth series of silver fir (**A**) and Norway spruce (**B**).
For intensely polluted silver fir trees, after the cessation of the polluting activity, the resilience index values were significantly higher than those of trees in unpolluted areas (Figure 4A). From 1978 to 1990, for the silver fir trees from the intensively polluted area, the resilience indices (Figure 3A) and average radial growth indices for Norway spruce (Figure 4B) were much lower than those of the silver fir trees in the unpolluted areas.
The quantification of growth losses for silver fir reflects reductions of up to almost 20% (in 1984 and 1989) in heavily polluted areas (Figure 5A). The growth losses of trees located in moderately polluted areas were not as significant (up to 5–7% relative to normal). The average losses throughout the highlighted period for heavily polluted silver fir were approximately 14%. Compared to the silver fir tree, the Norway spruce suffered much smaller diameter growth losses (Figure 5B). The average loss of diameter growth of the intensively polluted Norway spruce during the entire period of pollution exposure was 5%, and the loss was only 2% for the moderately polluted.
**Figure 4.** Average resilience indices for each of the 3 categories of silver fir (**A**) and Norway spruce (**B**) stands in the Tarnit,a area (the dotted circle represents the time interval in which the trees were most affected by the influence of pollution).
**Figure 5.** Radial growth losses recorded by the trees ((**A**) silver fir; (**B**) Norway spruce) in the Tarnit,a area affected by moderate and intensive air pollution.
At the stand level, the most affected trees by the local industrial pollution were the silver fir trees, while Norway spruce trees were less affected.
#### **4. Discussion**
As in the recent study developed in the same area [18], whose results confirmed that the frequency of growth events is determined by the distance from the sources of pollution, our results indicate that pollutant emissions near the local pollution zone significantly impacted the growth and development of coniferous trees in the Tarnit,a region, Suceava. In the studied area, the negative effect of pollution on the radial growth of coniferous trees (silver fir and Norway spruce) was greatest from 1978 to 1990. During this period, silver fir trees in the intensively polluted area experienced radial growth losses of up to almost 20% in 1984 and 1989. The growth losses of trees located in moderately polluted areas were not as significant, up to 5–7% compared to normal. The average loss throughout the highlighted period for heavily polluted silver fir was approximately 14%. In the case of Norway spruce, from 1978 to 1990, the trees were much less negatively affected by local pollution than the silver fir tree. The period in which the Norway spruce trees were most affected by local pollution was between 1978 and 1984, followed, except for 1987, by a period in which the trees did not experience as much growth reduction. Compared to the silver fir, the Norway spruce in this area showed much smaller radial growth losses. The average loss in radial growth of the intensively polluted and moderately polluted Norway spruce during the entire period of local pollution influence was 5 and 2%, respectively. After the 1990s, we observed a significant improvement in radial growth linked with the reduction in air pollution due to the closing of the mine. These results confirm those obtained in previous studies in this area [29]. Similarly, growth losses were registered as an effect of the long period of excessive drought [30]. The impact of air pollution on tree growth revealed by our results is slightly underestimated, because in the analysis were included only the trees that survived until the present. The growth decrease would likely be more evident in trees that did not survive the period of high industrial activity [18].
The effects of air pollution on forests are observed mainly as a direct impact on tree health, by crown damages and abnormal defoliation, favorable to losing tree vitality and in some cases, even death [31]. Coniferous trees are more sensitive to the effects of air pollution and acid rain than broad-leafed trees because of their greater capacity to intercept water from precipitation [32]. Similarly, in other regions of Europe with high air pollution, the silver fir is more pollution-sensitive than spruce [2,33].
Mihaljeviˇc et al. [34] analyzed the annual tree rings relative to the mining period and a potential source of contamination, a high concentration of cobalt (Co) that corresponded to maximum mining production. Hojdová et al. [35] assessed contaminated soils and vegetation surrounding mining areas. The authors found a strong correlation between Hg concentration in beech and mining metal production and no correlation with spruce trees located closer to the source of pollution. Numerous studies on this topic have proposed that emissions of heavy metals cause imbalances in the forest ecosystem and beyond. Shparyk et al. [4] showed that the highest levels of defoliation of trees were close to sources of industrial emissions. Biochemical investigations performed on leaves and phloem in tree trunks revealed decreased assimilative pigments in trees in those areas affected by pollution [17]. Changes in the processes of photosynthesis, respiration and transpiration occur differently in intensity both at the individual and species levels [36,37]. Barium mining activities can affect the quality of sediments and soil water through pollution with Fe, Hg and Pb, indicating an unacceptable risk to human health and the environment. After assessing the degree of contamination and the risks due to barite mining, the authors showed that the average concentrations of Fe, Hg and Pb were above allowable levels [38].
The radial growth of trees is certainly influenced by the variation and capacity of mining production. In our study, differences found in the dynamics of radial growth indices between the two species analyzed were determined by the sensitivity and reaction of each species to the imbalance of the ecophysiological process. The Norway spruce had much smaller diameter growth losses than the silver fir.
#### **5. Conclusions**
Air pollution from Tarnit,a mining operations induced strong growth reduction from 1978 to 1990, undoubtedly related to a decline in tree health and vitality due to airborne pollutants. The growth decline in stands further away (over 6 km) from the pollution source was weaker or absent, and the tree ring width variability was related to climate variation. Growth recovery of the intensively polluted stand was observed after the 1990s when the environmental condition improved because of a significant reduction in air pollution.
Of the two species analyzed (silver fir and Norway spruce), the silver fir demonstrated a higher sensitivity to local pollutants. Analyzing the dynamics of resilience indices and average radial growth indices in an integrated and comparative way allowed us to determine that the period in which the spruce trees suffered the most from the effect of local pollution was from 1978 to 1984, followed by, except for 1987, a period in which the trees experienced less growth reduction.
**Author Contributions:** Conceptualization, C.G.S.; methodology, C.G.S., R.V. and A.S.; software, C.G.S.; validation, C.G.S., O.B., I.P. and R.V.; formal analysis, C.G.S.; investigation, C.G.S.; resources, C.G.S.; data curation, C.G.S., R.V. and A.S.; writing—original draft preparation, C.G.S.; writing review and editing, C.G.S., E.A., O.B. and I.P.; visualization, C.G.S., E.A., O.B. and I.P.; supervision, C.G.S., E.A., O.B. and I.P.; project administration, C.G.S. and O.B.; funding acquisition, C.G.S. and O.B. All authors have read and agreed to the published version of the manuscript.
**Funding:** This study was funded by the Romanian Ministry of Research and Innovation, within the Nucleu National Programme, Project-PN-19070104/Contract no. 12N/2019.
**Acknowledgments:** We would like to thank the Stulpicani Forest District for permission to conduct field research and the research team within the project.
**Conflicts of Interest:** The authors declare no conflict of interest.
#### **References**
## *Article* **Diagnostic Assessment and Restoration Plan for Damaged Forest around the Seokpo Zinc Smelter, Central Eastern Korea**
**A Reum Kim 1, Bong Soon Lim 1, Jaewon Seol 1, Chi Hong Lim 2, Young Han You 3, Wan Sup Lee <sup>4</sup> and Chang Seok Lee 1,\***
- <sup>3</sup> Department of Biology, Kongju National University, Kongju 32588, Korea; [email protected]
- <sup>4</sup> Samseong Landscape Co., Ltd., Andong 36665, Korea; [email protected]
- **\*** Correspondence: [email protected]; Tel.: +82-2-970-5666
**Abstract:** *Research Highlights*: This study was carried out to diagnose the forest ecosystem damaged by air pollution and to then develop a restoration plan to be used in the future. The restoration plan was prepared by combining the diagnostic assessment for the damaged forest ecosystem and the reference information obtained from the conservation reserve with an intact forest ecosystem. The restoration plan includes the method for the amelioration of the acidified soil and the plant species to be introduced for restoration of the damaged vegetation depending on the degree of damage. *Background and Objectives*: The forest ecosystem around the Seokpo smelter was so severely damaged that denuded lands without any vegetation appear, and landslides continue. Therefore, restoration actions are urgently required to prevent more land degradation. This study aims to prepare the restoration plan. *Materials and Methods*: The diagnostic evaluation was carried out through satellite image analysis and field surveys for vegetation damage and soil acidification. The reference information was obtained from the intact natural forest ecosystem. *Results*: Vegetation damage was severe near the pollution source and showed a reducing trend as it moved away. The more severe the vegetation damage, the more acidic the soil was, and thereby the exchangeable cation content and vegetation damage were significantly correlated. The restoration plan was prepared by proposing a soil amelioration method and the plants to be introduced. The soil amelioration method focuses on ameliorating acidified soil and supplementing insufficient nutrients. The plants to be introduced for restoring the damaged forest ecosystem were prepared by compiling the reference information, the plants tolerant to the polluted environment, and the early successional species. The restoration plan proposed the *Pinus densiflora*, *Quercus mongolica*, and *Cornus controversa–Juglans mandshurica* communities as the reference conditions for the ridge, slope, and valley, respectively, by reflecting the topographic condition. *Conclusions*: The result of a diagnostic assessment showed that ecological restoration is required urgently as vegetation damage and soil acidification are very severe. The restoration plan was prepared by compiling the results of these diagnostic assessments and reference information collected from intact natural forests. The restoration plan was prepared in the two directions of soil amelioration and vegetation restoration.
**Keywords:** air pollution; diagnostic assessment; forest ecosystem; reference information; restoration plan
#### **1. Introduction**
Most developed countries have decreased anthropogenic air pollution emissions by implementing abatement polices [1,2]. Korea has also practiced such a policy, and thus environmental conditions around the industrial complexes of large scale have improved greatly [3,4].
**Citation:** Kim, R.A.; Lim, B.S.; Seol, J.; Lim, C.H.; You, Y.H.; Lee, W.S.; Lee, C.S. Diagnostic Assessment and Restoration Plan for Damaged Forest around the Seokpo Zinc Smelter, Central Eastern Korea. *Forests* **2021**, *12*, 663. https://doi.org/10.3390/ f12060663
Academic Editors: Ovidiu Badea, Alessandra De Marco, Pierre Sicard and Mihai A. Tanase
Received: 16 April 2021 Accepted: 21 May 2021 Published: 24 May 2021
**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.
**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).
However, factories of a small scale that are far away from the public's attention, such as the Seokpo smelter where this study was carried out, still emit air pollutants and thereby cause vegetation damage and acidify soil. The forest ecosystem around the Seokpo zinc smelter was severely destroyed and therefore landslides sometimes occur. Industrial activities have resulted in the enormous emissions of air pollutants for about 40 years since the 1970s when the smelter was constructed in Seokpo in central eastern Korea. The pollutants have continued to affect the surrounding forests and other ecosystems. Forest vegetation has become sparse and poor as trees have withered, undergrowth has disappeared, and bare ground has appeared throughout the wide area.
Pollutants discharged beyond the limits of the buffering capacity of an ecosystem prevent it from maintaining its normal structure and function. Excessive land and energy use and the ecological imbalance that it brings appear to be major factors that threaten environmental stability on local as well as global levels [4–8]. The vegetation decline and subsequent soil erosion and landslides observed in the vicinity of the Seokpo zinc smelter correspond to such an example. In fact, global environmental problems such as climate change are also due to this functional imbalance between the pollution source and the sink [9–11].
If the population grows and the land and energy use continue to intensify, such ecological imbalance is likely to increase even more in the future [7,12,13]. Indeed, industrialized and urbanized areas have been expanding steadily, and the real size of degraded vegetation, such as grassland and shrubland, has increased proportionally to such land transformation in industrial areas of Korea [4,7,8,14,15]. Moreover, vegetation decline induces the structural simplification and functional weakening of plant communities, consequently leading to negative effects on ecosystem service, which provides invaluable benefits to us [12,13]. In this respect, the restoration of degraded ecosystems is urgently required to prevent the spread of such additive pollution damage [4,7,13,16].
Ecological restoration is aimed at recovering the sound natural conditions before destruction. Ecological restoration is an ecological technology that heals the damaged nature by imitating the system and function of the integrated nature, thus providing habitats for various creatures and seeking to secure the future environment of humankind [17–20]. Ecological restoration has been considered as improving ecological productivity in degraded lands, conserving biological diversity, and mitigating lost or damaged ecosystems [19–28]. Human aids are often required to restore the damaged ecosystems and prevent further damage [16,19,20,29], and provisions of extra propagules and site amendment may initiate recovery processes [30].
In order to heal the damaged nature, we must first check what problems the target has, such as how much it is degraded or what is the cause of the damage. In other words, a diagnostic assessment of the restoration target should be made [19,20].
All restoration projects are with targets to reduce the negative ecological impacts of the past and to restore the natural potential of the restoration target as much as possible. Ecological restoration means copying nature by studying a system of the integrate nature. There are several planning steps, based on the results of diagnostic assessments, to find the deficits and the targets for planning to improve the degraded ecosystem. First of all, we have to prepare such measures by obtaining diverse ecological information on an area to be restored because specific restoration efforts have to be applied in the field [31–33].
The restoration of an ecosystem damaged by environmental pollution can be achieved either through improvement of the environment polluted or by establishing plants tolerant to the pollutants [7,8,13,16,29,34–36]. Species tolerant to environmental pollution can persist through growth and reproduction or even expand their distribution range in the polluted environment [5–8,37,38].
The Seokpo smelter is uniquely located on a small mountain village and is thus far away from the public's attention. Therefore, little is known about the situation and, moreover, no academic research has been carried out. However, due to the effects of air pollutants emitted over a long period of time and the soil pollution resulting from them,
the forest damage began to become visible, and it has led to landslides and damage to surrounding rivers in good condition, causing worry in recent years. There is, therefore, a growing demand for restoration. The major pollution source of the Seokpo smelters, which refines primarily processed ore rather than raw ore, is sulfur dioxide generated from the combustion of fossil fuels used as an energy source. The damage state was similar to that of Ulsan and Yeocheon industrial complexes, major industrial complexes in Korea [4,7,8,12,36].
Restoration is an ecological technology that ameliorates degraded nature by imitating integrated and healthy nature. Restoration is achieved through a series of procedures, such as a survey of the existing conditions, a statement of the goals and objectives, the designation and description of a reference, the preparation of a master plan, the establishment of a restoration plan, restoration practices, monitoring, adaptive management, and evaluation [39–42]. Such ecological restoration is common as a means to solve such problems in developed countries, which correctly recognize that the environmental problems at a global level, such as climate change, are due to the functional imbalance between the artificial environment as an environmental stress source and the natural environment as its sink. However, in most developing countries, including Korea, most restoration projects have neglected such procedures and thus have not met the restoration goals, in spite of great expense and labor [43–50]. A series of procedures are required to achieve successful ecological restoration. However, these procedures usually tend to be ignored in most restoration projects implemented in Korea. Diagnostic evaluation is generally omitted. Even if a diagnostic evaluation is made, there are very few cases in which the level and method of restorative treatment are determined based on the results, and most restoration projects progress only by active methods without any relation to the degree of damage [46,49]. Therefore, cost and energy are wasted, and the effect is very little [46,47,50]. In most restoration projects, the reference information is not used, and restoration is performed based on the subjective decisions of the project manager. Thus, restoration projects are conducted without any model or goals. Consequently, exotic species, which should be excluded thoroughly in a restoration project, are introduced frequently, and the spatial distribution range for plant species is barely considered [46,47,50]. Therefore, most restoration projects remain at the level of past afforestation or classical landscaping.
This study was attempted to implement ecological restoration of the advanced level, which is beyond this low level of restoration. This study conducted a diagnostic evaluation for the damaged forest from air and soil pollution, collected reference information from intact forest without any damage, and prepared a restoration plan by combining such information. In order to ecologically restore the area, the following situations need to be considered: first, since pollutants continue to be emitted, tolerant plants that can withstand such pollution should be selected and introduced. Second, soils contaminated by the effects of pollutants discharged for a long time should be improved. Third, many species have already disappeared due to environmental pollution, and thus we need to introduce species that have disappeared based on the ecological information obtained from the reference ecosystem. Finally, since the environment has been severely damaged to the point where landslides occur, measures should also be taken to prevent landslides when establishing the planting bed. This study aims to clarify the extent and the type of damaged forest and, furthermore, to recommend a restoration plan suitable for the ecological condition of the target site as well as the damage degree based on the principle of restoration ecology. In order to arrive at these goals, we assessed vegetation damage based on satellite image interpretation and field checks. Vegetation damage was also assessed based on species composition and species diversity. Furthermore, we also diagnosed the damaged state of soil based on its physic-chemical properties. We recommended the restoration plan by synthesizing the results of the diagnostic assessment and the reference information, including pollution tolerant species and early successional species considering the environmental condition of this area.
#### **2. Materials and Methods**
#### *2.1. Study Site*
This study was carried out in the forest ecosystem around the Seokpo smelter, located on the central eastern Korea (Figure 1). The Seokpo smelter has been in operation since 1970. The smelter produces zinc, cadmium sulfate, copper sulfate, and manganese sulfate and has emitted many air pollutants around it [51]. This area is surrounded by mountainous areas with steep slopes and has topographical features that make it difficult for air pollutants to spread (Figure 2) [4,6,16]. In addition, the temperature inversion, which occurs as the cold air on the mountain descends along the surrounding mountain slopes and is trapped at the bottom of the basin after sunset, makes it more difficult to spread air pollutants, causing an increase in pollution damage in this narrow valley. During this temperature inversion time, dense smoke often settles in low-lying areas and becomes trapped due to temperature inversions—when a layer within the lower atmosphere acts as a lid and prevents vertical mixing of the air. Steep canyon walls act as a horizontal barrier, concentrating the smoke within the deepest parts of the canyon and increasing the strength of the inversion [52,53].
**Figure 1.** A map showing the study area, the Seokpo zinc smelter, which is located in central eastern Korea. Seokpo zinc smelter is composed of three factories. A colored map shows vegetation and land use types established around the smelter. Dots and the parts expressed with oblique lines around them indicate factories and factory sites.
As is shown in a vegetation map in Figure 1, the vegetation of this area is dominated by the *Quercus mongolica* community. However, the *Pinus densiflora* community is established
on the slopes of mountainous areas with steep slopes or mountain ridges and peaks with shallow soil depth due to edaphic characteristics. There is also a mixed forest in which two species forming a community together. On the other hand, there are plantations where *Larix kaempferi* and *P. koraiensis* are introduced artificially in some areas, and there are places where mixed forests are formed where natural vegetation is mixed with planted species. Meanwhile, agricultural and residential areas are established in the lowlands.
**Figure 2.** Maps showing the spatial distribution of elevation (m), slope (degree), and soil type in the study area.
The climate of Bonghwa is continental, with warm and moist summers and cold and dry winters. The mean annual temperature is 9.9 ◦C and the high and low mean temperatures are recorded as 28.6 ◦C and −10.3 ◦C in August and January, respectively. The mean annual precipitation is 1217.9 mm; about 60% falls in the rainy season from June to August and, including the typhoon season of September, about 70% is concentrated in both periods [54].
The elevation of the study area ranges from 400 to 900 m above sea level. The slope degree is as steep as more than 20◦ in most of the mountainous land except the valley. The parent rock of the study area consists mostly of granite, and in the flat land beside rivers and streams consists of alluvium. Soil in this area is composed of dry (B1), slightly dry (B2), and moderately moist brown forest soil (B3), which were developed on granite bedrock [55] (Figure 1).
The reference forest, used for comparison, was designated as the Korean red pine (*Pinus densiflora* Siebold & Zucc.) and oak (*Quercus mongolica* Fisch. Ex Ledeb., and *Q. variabilis* Blum communities), which are the representative late successional vegetation types in Korea, and *Cornus controversa* Hemsl. ex Prain, which represents the valley forest [56]. The reference forests were selected in the Uljin genetic resource reserve, which are about 15 km from this study area and therefore retain a healthy vegetation. The reference forests were selected as the forests that are from 50 to 100 years old, which is not an old growth forest but a stable forest. The number of plots chosen for the survey for the reference forests was 10, 10, and 10 for the *P. densiflora*, *Q. mongolica*, and *Cornus controversa* communities, respectively.
#### *2.2. Methods*
A vegetation map was made based on image interpretation and field checks. Aerial photo images (1:5000 scale) were used to identify the vegetation types and boundaries, which appear as a homogeneous patch. These vegetation types were confirmed by field checks. Vegetation types were overlapped onto topographical maps at a 1:5000 scale. Patches smaller than 1 mm on the map were excluded from this study because of the uncertainty of their sizes and shapes [57]. The final map was constructed with ArcGIS program (ver. 10.0, ESRI, Redlands, CA, USA) [58].
To determine the vitality of vegetation in the study area, Landsat images taken on 2 October 2018 were downloaded to analyze the normalized differential vegetation index (NDVI). Vegetation damage based on vitality was assessed through supervising analysis on the satellite image [59]. This study applied a supervised classification-maximum likelihood algorithm to classify the vegetation damage state around the Seokpo zinc smelter using Landsat images in the ArcGis10.1 program. The maximum likelihood algorithm is the most common method in remote sensing image data analysis [60], which is mainly controlled by selecting the pixels that are representative of the desired classes [61]. Using the signature file creation tool, vegetation damage was classified into five classes of very severe, severe, moderate, light, and none. The damage degrees classified were verified through field checks as follows.
Visible damage was investigated by recording the degree of necrosis that appeared on the leaf surface of plants appearing in the process of the vegetation survey. The damage degree was classified into five groups based on the percentage of injury shown on the leaf surface: very severe (V, more than 75% of total leaf area damaged), severe (IV 50–75% damaged), moderate (III, 25–50% damaged), light (II, less than 25% damaged), and none (I, 0%) [7].
The vegetation structure damage was assessed by the deformation of vegetation stratification based on [62]. More than 50% of the land, which is covered with barren ground was assessed as 'very severe'. Grassland or barren ground without any woody plants was assessed as 'severe'. Vegetation that had lost some stratification was assessed as 'moderate'. Vegetation with visible damage only to integrate structure, with all strata composed of canopy, understory, shrub, and herb layers shown without any loss of stratification was assessed as 'light'. A map expressing vegetation damage was prepared by applying the GIS program (ver. 10.0, ESRI, Redlands, CA, USA).
The vegetation survey was carried out from May to September in 2018 and 2019. The vegetation survey was carried out by recording the Braun–Blanquet's cover class of plant species appearing in quadrats of 2 m × 2 m, 5 m × 5 m, and 20 m × 20 m size in grassland, shrubland, and forest, respectively, installed randomly [63]. The vegetation survey was carried out in 99 plots (28, 28, 1, 5, 5, 19, and 13 plots for *Pinus densiflora* community, *Quercus mongolica* community, *Q. variabilis* community, valley forest, shrubland, grassland, and cut slope, respectively) from May to September in 2018 and 2019.
Soil samples were collected with a sampling spade in June–August 2019 from the top 10 cm after removing the litter at five random points in each plot, after which they were pooled, air dried at room temperature, and sieved through 2 mm mesh. A total of 18, 18, 1, 3, 3, 12, and 9 soil samples were collected from the *Pinus densiflora* community, the *Quercus mongolica* community, the *Q. variabilis* community, valley forest, shrubland, grassland, and cut slope, respectively. Soil properties were diagnosed for pH and Ca2+, Mg2+, and Al3+ content. Soil pH was measured with a bench top probe after mixing the soil with distilled water (1:5 ratio, w/v) and filtering the extract (Whatman No. 44 paper). Organic matter (OM) concentration was estimated by loss of dry mass on ignition at 400 ◦C. Total nitrogen was measured with the micro-Kjeldahl method [64]. Available P was extracted in 1-N ammonium fluoride (pH = 7.0) and exchangeable Ca2+, Mg2+, and Al3+ contents were extracted with 1N ammonium acetate (pH = 7.0 for Ca and Mg and pH = 4.0 for Al) and measured by ICP (inductively coupled plasma atomic emission spectrometry; Shimadzu ICPQ-1000) [65]. The results of the analysis on the physic-chemical properties of soil were
reinforced by the simple kriging model. Maps expressing the physic-chemical properties of the soil were prepared by applying the GIS program (Version 10.1). The soil properties (pH, OM, N, P, Ca, Mg and Al) of sites showing different degrees of damage to vegetation were compared with one-way analysis of variance (ANOVA) and Tukey's honestly significant difference (HSD) test at α = 0.05 [66].
The restoration plan was prepared by recommending a soil amelioration method and the plant species to be introduced depending on the degree of damage to the vegetation and soil. Dolomite and organic fertilizer were recommended for soil amelioration. The dolomite requirement was calculated by applying the following equation: dolomite requirement (t/ha) = (target pH–current pH) × soil texture factor. We decided to set the target pH as 5.5, based on the normal pH of the natural forest soil, and soil texture factor as 3, reflecting the soil texture of this area [67]. Dolomite raises the soil pH and increases available Ca2+ and Mg2+ due to the following chemical reactions in the soil solution [68]:
Initial chemical reaction: Ca• Mg(CO3)2 + 2H+ <sup>→</sup> 2HCO3 <sup>−</sup> + Ca2+ + Mg2+, (1)
$$\text{Second reaction:}\ 2\text{HCO}\_3^- + 2\text{H}^+ \rightarrow 2\text{CO}^2 + 2\text{H}\_2\text{O},\tag{2}$$
$$\text{Net reaction:}\,\text{Ca}\bullet\,\text{Mg(CO}^{3})\_2 + 4\text{H}^+ \rightarrow \text{Ca}^{2+} + \text{Mg}^{2+} + 2\text{CO}^2 + 2\text{H}\_2\text{O},\tag{3}$$
The amount of organic fertilizer applied was determined to be half the level of dolomite, referring to previous study results [8]. The chemical characteristics of the organic fertilizer are given in Appendix A, Table A1.
The introduction of vegetation for restoration took the form reinforcing the lost part in the vegetation stratification. Therefore, we planned to introduce the disappeared species compared to the species composition of the natural reference site. Furthermore, we added tolerant and early successional species in our restoration plan, considering the environmental condition of this area, where air and soil pollution continues and vegetation is so severely damaged that bare ground can appear and severe soil erosion occurs. The plant species to be introduced were selected by applying indicator species analysis. Indicator species analysis was carried out using the function 'vegan', 'indicspecies' of the R statistical package (version 4.0.2). In addition, we reinforced the plant species to be introduced by referring to national vegetation information [69] and existing research data conducted on the reference site of this study [70], considering that this study was conducted in a limited place.
#### **3. Results**
#### *3.1. Vegetation Damage*
The spatial distribution of NDVI showed that its value was lower near the factory and tended to increase as it moved away from it (Figure 3). The value was also related to the topographic condition; thus, it was low in the ridge and high in the valley (Figure 3).
**Figure 3.** Spatial distribution of NDVI in the study area.
Vegetation damage identified from the satellite image interpretation depended on the distance from the smelter and topography. The damage appeared was severer in sites closer to the smelter and decreased farther away (Figure 4). The degree of damage was also dominated by topographic conditions, and therefore damage was restricted within the first ridge from the pollution source, little damage thus appearing on the opposite slope or beyond the first ridge from the smelter (Figure 4).
**Figure 4.** Spatial distribution of vegetation damage based on satellite image interpretation in the study area.
Damage based on vegetation stratification showed a trend similar to the abovementioned results. Vegetation in the site where the damage was light showed the integrate structure with all strata composed of canopy, understory, shrub, and herb layers. However, vegetation structure became simplified with the increase of the damage, and thus grassland or barren ground appeared in sites where the damage was the severest (Figure 5).
**Figure 5.** Spatial distribution of vegetation damage based on vegetation stratification in the study area.
In a map where vegetation damage by damage class is shown (Figure 4), very severely damaged vegetation appeared in areas located in the northwestern direction of the first factory, the western and southern directions of the second and third factory, and surrounded by the first, second, and third factories. Severely damaged vegetation appeared in the areas farther than the very severely damaged vegetation from the three factories in all four directions of east, west, south, and north. Moderately damaged vegetation appeared in the areas located in the eastern and western directions, with the third factory at the center. Lightly damaged vegetation appeared in the areas far from them within the first ridge from the factories.
#### *3.2. Soil Degradation*
Spatial distribution of the physic-chemical properties of the soil reflected a trend of vegetation damage. Soil pH was usually low compared with the unpolluted area but was lower in sites close to pollution sources and became higher farther away (Figure 6).The
Ca2+ and Mg2+ content showed trends similar to that of the pH, whereas Al3+ content represented a reverse trend (Figure 6).
**Figure 6.** Spatial distribution of soil pH and Ca2+, Mg2+, and Al3+ contents of soil in the study area.
Soil pH tended to be relatively low in the northern and western directions of the three factories, the area surrounded by those factories, and the southern direction of the third factory, whereas it was relatively high in the southwestern and northeastern directions of the third factory (Figure 6).
The Ca2+ and Mg2+ content showed trends similar to that of the pH, while Al3+ content represented a reverse trend (Figure 6).
The physic-chemical properties of the soil were compared with those of the reference site and among the degrees of damage to the vegetation (Figure 7). The pH and Ca2+, Mg2+, and available phosphorus contents were lower than those in the reference site, whereas the total nitrogen content was vice versa. However, organic matter and Al3+ content did not show any significant difference between both sites. On the other hand, pH and Ca2+ and Mg2+ contents showed significant difference among the degrees of damage to the vegetation, but the other factors did not show any significant differences among the degrees of damage.
**Figure 7.** A comparison of the physic-chemical properties of the soil among damage degrees of vegetation and with that of the reference site. Very severe, severe, moderate, and light indicate damage degree and reference indicates the unpolluted site selected for comparison. OM: organic matter; TN: Total nitrogen; AP: available phosphorus. Each bar was expressed with mean and standard error of mean. Tukey's honestly significant difference (HSD) test was conducted on each of the parameters that show a statistically significant difference among the four types of damage degrees at *α* = 0.05; the means with the same alphabetical character (in superscript), for each parameter, were not different from each other.
#### *3.3. Species Composition*
As the result of stand ordination, arrangement of stands reflected vegetation damage (Figure 8). The reference stands were arranged on the left on the AXIS I and very severely or severally damaged stands on the right, and moderately and lightly damaged stands were arranged between both groups. Moderately and lightly damaged stands tended to be arranged depending on the topographical position as *P. densiflora* stands, *Q. Mongolica* stands, and stands established in the valley were arranged in the mentioned order as moves from bottom upward on the AXIS II. Meanwhile, cut slope stands were arranged in the left part on the AXIS I like the reference stands but separated from them on the AXIS II.
**Figure 8.** Ordination of vegetation stands established around the Seokpo smelter and on the natural reference forest, Uljin Forest Genetic Resources Conservation Reserve, central eastern Korea. Legends expressed as damage degree represent the damaged stands established around the Seokpo smelter. R\_Pd: *Pinus densiflora* stands established in the reference site; R\_Qm: *Quercus mongolica* stands established in the reference site; R\_Valley: stands established in the reference site; Cut slope: stands established on the incised slope along the forest road (*p* = 0.001, stress = 0.1702836).
#### *3.4. Species Diversity*
As a result of comparing the species diversity by the species rank–dominance curve, the species diversity of the damaged sites was lower than that in the reference sites (Figure 9). In the damaged sites, species diversity tended to decrease in proportion to the damage degree (Figure 8). The species diversity was also dominated by the topographic condition and thus high in the vegetation established in the valley and low in the pine forest on the ridge, and the species diversity of broad-leaved forests on the mountain slope was located between both vegetation types (Figure 9).
**Figure 9.** Species rank–dominance curves of vegetation stands established around the Seokpo smelter and on the natural reference forest, Uljin Forest Genetic Resources Conservation Reserve, southeastern Korea. Legends are the same as those in Figure 8.
#### *3.5. Selection of Plant Species for Vegetation Restoration*
We selected disappeared, tolerant, and early successional species by applying the indicator species analysis (Tables 1–4). First, we selected species to be introduced for vegetation restoration by comparing all vegetation data between polluted and natural reference sites and cut slope. We selected species which appear in the natural reference site but do not appear in the polluted site as the disappeared species. The disappeared species mean species that should be introduced for vegetation restoration. We selected species which showed the reverse trend to the disappeared species as the tolerant species. The species which appear characteristically on the cut slope were selected as the early successional species. Ecological restoration should copy the environment by studying a system of the integrate nature. However, considering the environmental condition of this area where air and soil pollution continues and vegetation is so severely damaged that bare ground can appear and severe soil erosion occurs, we added the tolerant and early successional species in our restoration plan. In addition, we added the plant species to be introduced by referring to the existing research data conducted around this study area [69,70] to enhance the stability of the restoration plan.
**Table 1.** The result of indicator species analysis for selecting the disappeared species, tolerant species to the polluted environment, and early successional species based on data collected in all study sites. Cut slope: early successional species; Severe, Moderate, Light: tolerant species in severely, moderately, and lightly damaged sites, respectively; Valley: tolerant species in valley; Reference(Pd), Reference(Qm), Reference(VA): disappeared species in the natural reference sites of *Pinus densiflora* forest, *Quercus mongolica* forest, and valley forest, respectively.
\*\*\* significant at 0.1% level, \*\* significant at 1% level, \* significant at 5% level.
**Table 2.** The result of indicator species analysis for selecting the disappeared species and tolerant species to the polluted environment based on data collected in both polluted and natural *Pinus densiflora* forests. Polluted: tolerant species in the polluted site of *Pinus densiflora* forest; Reference: disappeared species in the natural reference site of *Pinus densiflora* forest.
\*\*\* significant at 0.1% level, \*\* significant at 1% level, \* significant at 5% level.
**Table 3.** The result of indicator species analysis for selecting the disappeared species and tolerant species to the polluted environment based on data collected in both polluted and natural *Quercus mongolica* forests. Polluted: tolerant species in the polluted site of *Quercus mongolica* forest; Reference: disappeared species in the natural reference site of *Quercus mongolica* forest.
\*\*\* significant at 0.1% level, \*\* significant at 1% level, \* significant at 5% level.
**Table 4.** The result of indicator species analysis for selecting the disappeared species and tolerant species to the polluted environment based on data collected in both polluted and natural valley forests. Pol-luted: tolerant species in the polluted site of valley forest; Reference: disappeared species in the natural reference site of valley forest.
\*\*\* significant at 0.1% level, \* significant at 5% level.
#### *3.6. Zonning and Design for Restorative Treatment*
The spatial range for which the restoration was required was restricted within the first ridge from the pollution source, considering the impact extent of the air pollution. We divided the district for restoration into four zones of very severely, severely, moderately, and lightly damaged zones based on the damage degree (Table 5) and three zones of ridge, slope, and valley based on the topographic conditions (Table 6) [13].
**Table 5.** Level and method of restoration recommended based on a diagnostic evaluation of the forest ecosystem around Seokpo zinc smelter, central eastern Korea.
**Table 6.** Species to be introduced for restoration by layer of vegetation in each topographic condition. This species information was prepared by incorporating disappeared species compared with the natural reference site, tolerant species to the polluted environment, and early successional species. In very severely and severely damaged zones, plant species forming all layers of vegetation stratification including canopy tree, understory tree, shrub, and herb layers are recommended for restoration. Plant species forming shrub and herb layers are recommended in moderately damaged zone. Passive restoration is recommended in lightly damaged zone. Vegetation type, such as Korean red pine forest established as an edaphic climax type on the upper slope, usually lacks understory tree layer due to topographic condition, which is dry and infertile.
**Table 6.** *Cont.*
\*, \*\*, and \*\*\* indicate disappeared species, tolerant species, and pioneer species, respectively.
The restorative treatment was determined by reflecting the damaged level of vegetation and soil acidification. In very severely damaged zones, dolomite of 4.5 ton/ha and organic fertilizer of 2.25 ton/ha were recommended for soil amelioration. Plant species forming all layers of vegetation stratification, including canopy tree, understory tree, shrub, and herb layers, were recommended for vegetation restoration. In severely damaged zones, dolomite of 3.0 ton/ha and organic fertilizer of 1.5 ton/ha were recommended for soil amelioration. Plant species forming all layers of vegetation stratification were recommended for vegetation restoration, such as in the case of the very severely damaged zone. In moderately damaged zones, dolomite of 1.5 ton/ha and organic fertilizer of 0.75 ton/ha were recommended for soil amelioration. For restoring vegetation, plant species forming shrub and herb layers of vegetation stratification were recommended. Plant species forming shrub and herb layers were recommended in the moderately damaged zone. In the lightly damaged zone, dolomite of 1.0 ton/ha and organic fertilizer of 0.5 ton/ha were recommended for soil amelioration. Passive restoration was recommended for restoring vegetation.
In this restoration plan, we did not recommend for plant species forming the understory tree layer that vegetation to be introduced on the upper slope and ridge, as vegetation established there usually lacks the layer.
#### **4. Discussion**
#### *4.1. The Effects of Air Pollution on Forest Ecosystems*
Air pollution and atmospheric deposition emitted from industrial facilities have adverse effects on tree and forest health. Growing awareness of the air pollution effects on forests has, from the early 1980s on, led to intensive forest damage research and monitoring. This has fostered air pollution control, especially in developed countries, and also, to a smaller extent, in developing countries. At several forest sites, particularly in developed countries, there are the first indications of the recovery of forest soil and tree conditions that may be attributed to improved air quality [4,71]. This caused a decrease in the attention paid by the public to the air pollution effects on forests. However, air pollution continues to affect the structure and functioning of forest ecosystems just as when this study was conducted.
Air pollutants may impact trees as both wet and dry deposition. Wet deposition comprises rain, hail, and snow and is largely determined by atmospheric processes. Dry deposition consists of gases, aerosols, and dust and is largely influenced by the physical and chemical properties of the receptor surface. Forests receive higher deposition loads than open fields, depending on the tree species and canopy structure. A higher roughness of the canopy causes higher air turbulences and more intensive interactions between the air and the foliage. The interception of pollutants by the foliage in turn is determined by such factors as leaf area index, leaf shape, leaf surface roughness, and stomata size. Dry deposition accumulated on the foliage is washed off by precipitation and enhances the deposition under the canopy (throughfall) in comparison to deposition in an open field (bulk deposition). Moreover, throughfall is influenced by two components of canopy exchange: canopy leaching and canopy uptake of elements. The main air pollutants involved in forest damage are sulfur compounds, nitrogen compounds, ozone, and heavy metals [72–76].
Sulfur dioxide (SO2) was the first air pollutant found to cause damage to trees [77]. Its air concentrations increased rapidly when it was released into the atmosphere by the combustion of fossil fuels during the course of industrialization. While damaging trees directly via their foliage, SO2 also reacts with water in the atmosphere to form sulfurous acid (H2SO3) and sulfuric acid (H2SO4), thus contributing to the formation of acid precipitation and hence to the indirect damage of trees [72,73]. In Korea, forest decline has usually occurred around industrial areas, and SO2 has played a leading role [7,8,13]. However, Korea has shown declining trends in SO2 concentrations in recent years [4].
Nitrogen oxides (NOx) are released into the atmosphere in the course of various combustion processes in which nitrogen (N) in the air is oxidized mainly to nitrogen monoxide (NO), with a small admixture of nitrogen dioxide (NO2). In daylight, NO is easily converted to NO2 by photochemical reactions involving hydrocarbons present in the air. Both gases, especially NO, are also produced biologically by soil bacteria during nitrification, denitrification, and decomposition of nitrite (NO2 –) [78]. These substances are gaseous and act on trees as dry deposition directly via the foliage. Some of them are acidifying and lead—by means of chemical reactions with water in the atmosphere—to acid precipitation. Acidifying compounds such as SO2, NOx, and NH3, however, enhance the concentrations of protons and form sulfuric acid, nitric acid (HNO3), ammonium (NH4), and nitrate (NO3) [73,74,79].
Heavy metals result from most combustion processes and from many industrial production processes. They are released into the atmosphere by means of dust and, at high temperatures, also as gases. The main heavy metals considered to be detrimental to forest health are cadmium (Cd), lead (Pb), mercury (Hg), cobalt (Co), chromium (Cr), copper (Cu), nickel (Ni), and zinc (Zn). However, largely because of their impacts on human health, heavy metal emissions have been reduced greatly within the last 3 decades in many industrialized countries [80–82].
#### *4.2. Damage Status to Forest Ecosystem around Seokpo Smelter*
Vegetation in places close to smelters has been so severely damaged that only a few plants, such as *Athyrium yokoscense* (FR. Et SAV.) H.CHRIST, *Miscanthus sinensis* var. *purpurascens* RENDLE, and *Pteridium aquilinum* var. *latiusculum* (DESV.) UNDERW, sporadically exist; otherwise all of them have disappeared. As the distance from the smelter become farther, the damage decreases, resulting in shrubland and forest (Figures 4 and 5). The spatial distribution of vegetation in industrial areas usually reflects the degree of pollution damage [4,7,8,12]. If a forested ecosystem is being affected by air pollution, then the canopy stratum is generally impacted first and is stripped away. As canopy trees decline, shrubs and then the ground vegetation are affected. This syndrome of the sequential death of the horizontal strata of the terrestrial vegetation, described as a "peeling" of "layered vegetation effect" by [62], was observed clearly in this area (Figure 5).
Forests appeared as two types depending on the damage degree. Forests with moderate damage show poor development of vegetation stratum, while forests with light damage show visible damage in appearance and poor development of undergrowth (Figure 5). The damage was also reflected in species composition (Figure 8) and species diversity (Figure 9), showing a clear difference in species composition and species diversity compared to the reference area.
The soil is acidified and has a lower calcium and magnesium content compared to the reference area, while the aluminum content was higher in very severely damaged sites (Figure 6).
Synthesized results obtained from the diagnostic assessments on vegetation damage and soil acidification in the forest ecosystem around the Seokpo zinc smelter show that vegetation damage was so severe that denuded ground appeared throughout wide areas, and soil acidification was also relatively severe. In this respect, active restoration is required urgently to prevent follow-up damage, such as landslides [83–85]. However, the damage decreased with increasing distance from the pollution source and was restricted within the first ridge from the source. The results of this diagnostic assessment could help to determine the spatial range and level of restoration.
In the case of the Yeocheon industrial complex, passive restoration occurred in forests around industrial complexes [4], but the speed was slower than in the case of the Ulsan industrial complex where active restoration was applied [8]. In active restoration, dolomite and sludge treatment neutralized acidic soil and supplemented nutrients, thereby facilitating plant growth and contributing to the restorative effects [8]. In addition, the tolerant species, which was selected through field surveys and laboratory experiments, was well established and contributed to achieve successful restoration [8,36].
#### *4.3. Necessity and Recommendation of Ecological Restoration*
In Korea, most industrial complexes are located on the coastal areas [4,7,8]. However, the area where this study was conducted is uniquely located on a small mountain village. Most of this area is composed of a mountainous area with a steep slope, and it is therefore not easy to develop. Therefore, the nature is well conserved. The mountainous land of this area is very steep and shallow in soil depth. Thus, Korean red pine forest, which is adapted well to the dry and infertile environment, dominates the vegetation of this area. However, deciduous broad-leaved forests suitable for the climate condition of this region and the environmental characteristics of each site are well developed in lowlands below midslope, including mountainous valleys. The river that runs through this area corresponds to the upstream section of the Nakdong River, one of the longest rivers in Korea. As Korea is a mountainous nation where more than 65% of the total national territory is composed of mountainous areas, most riparian zones, including floodplains of rivers, are transformed into agricultural lands and urbanized areas, leaving few integrate rivers equipped with aquatic and riparian zones in the plains and lands with a gentle slope. However, as this area is not easy to develop due to the environmental characteristics composed of steep mountainous areas; the river also remains an almost completely intact status.
However, air pollutants emitted from the Seokpo smelter have destroyed forest vegetation established in the basin of this river, even leading to landslides. Therefore, its impact poses a great threat to the river ecosystem conserved so well. Considering these facts comprehensively, the restoration of the damaged forest ecosystem is absolutely necessary and urgently required. In particular, countermeasures such as restoring forests are important for ensuring future ecosystem services [7,8,13].
The continual growth of the human population and human land uses leads to declines in the quality of the environment. Further, the natural landscapes that provide many ecosystem services are being rapidly converted to agricultural, industrial, and urban areas, and even wastelands. The biodiversity and habitability of the planet are now more threatened than ever before. Therefore, it is imperative that degraded land be rehabilitated and that adjoining natural landscapes be protected. However, it is clear that degradation thresholds have been crossed in many habitats, and succession alone cannot restore viable and desirable ecosystems without intervention [86]. As was shown in our results (Figures 4 and 5), the forest has degraded through shrubland or grassland to denuded land, eventually producing continual landslides. In addition, the soil is acidified and becoming non-nutritive (Figure 6). Thus restoration actions are urgently required to prevent more land degradation.
To restore degraded ecosystems, in particular those degraded by pollution, we need to apply soil ameliorators, including dolomite [6,8,13,87,88]. Although these soil amendments contributed to improving the polluted environment and thereby achieved successful revegetation in the case of Sudbury [6], they may cause other problems, such as ground water pollution and eutrophication [73,89,90]. In this respect, we recommend planting tolerant plants or applying fertilizer plants rather than applying soil amendments as a restorative treatment in all cases [8].
#### *4.4. Soil Amelioration for Restoration*
The atmospheric environment in the industrial areas in Korea is improving due to a decrease in the emission of air pollutants [4]. However, the polluted soil has not improved so easily [7,8]. In fact, polluted soil usually provides the major challenge to most restoration programs [8,13,29,35]. Therefore, soil amendment was planned as a preparation for restoration. The soil amelioration focused on the neutralization of acidic soil and the enhancement of fertility. Previous research showed that dolomite was a superior ameliorator compared to lime [15], and it is also generally used [8,16]. The amount of dolomite applied was calculated with an equation being applied to determine the amount of dolomite required to improve varying soil pH to 5.5 [15]. In the present situation, the amounts of dolomite required were 4.5, 3.0, 1.5, and 1.0 ton/ha in very severely, severely, moderately, and lightly damaged zones, respectively (Table 1). These amounts were smaller than those used in the Sudbury region of Canada [16] and the Ulsan industrial area of Korea [13].
Soil neutralized by the addition of dolomite stimulates activities of soil microorganisms and enhances nutrient availability through the promoted decomposition of organic matter [8,91]. Organic fertilizer also may ameliorate acidified soil by raising pH and adding macronutrients, including phosphorus, which is often a limiting nutrient in acidic soil [13,88,92].
Aluminum toxicity results in rapid inhibition of root growth due to the impedance of both cell division and elongation [93–95], reduction of soil volume explored by the root system, and direct interference with the uptake of ions such as calcium and phosphate across the cell membrane of damaged roots [96,97]. The deficiency in soil nutrients, such as P, Ca2+, and Mg2+, exacerbate the problem of inefficient nutrient uptake due to restricted root growth and root damage [98,99].
Additions of undecomposed plant materials, such as pruning, to acid soils often increase soil pH, decrease Al3+ saturation, and improve conditions for plant growth generally [100–104]. Similarly, the addition of plant residue composts, urban waste compost, animal manure, and coal-derived organic products to acid soils increases soil pH, decreases Al3+ saturation, and improves conditions for plant growth [8,13,105–107]. The recycling of these waste products for soil amelioration has a double benefit for both the environment and the economy, provided that the waste materials are not contaminated with harmful impurities. These organic substances confer metal binding and pH buffering capacities, which are important determinants of the pH of the treated soil [104,108,109].
Treatment of sludge as a soil ameliorator contributed to the reduction of Al3+ content and resulted in increased plant growth [8]. This result suggests that the sludge is a chelating agent for Al3+ [104,109,110]. Although dolomite and sludge contribute to ameliorating the acidified soil, there are some serious concerns for the land application of dolomite and sewage sludge due to the potential for the contamination of ground water and eutrophication [73,88,89,110]. By stimulating the mineralization of soil organic matter, dolomitic liming causes ground water pollution by increasing nitrate release from the soil [8,89,90]. We, therefore, recommend restricting the use of those soil ameliorators.
Meanwhile, N-fixing plants have been used for enhancing soil fertility in revegetation projects elsewhere (e.g., [6,13,16]). Therefore, we recommend planting N-fixing plants as an initial step for revegetating in this area.
#### *4.5. Selection of Plant Species for Restoration*
This study corresponds to a diagnostic assessment which analyzed the damage status of the forest ecosystem as a preparatory step for realizing the ecological restoration of this area [19,20]. Furthermore, we carried out a vegetation survey to obtain the reference information in a conservation reserve with similar environmental conditions, which was designated as the forest genetic resource reserve from the Korea Forest Administration.
Studies for restoration have chosen the species for restoration on the basis of the following criteria: (1) species importance for restoring the ecosystem function, (2) species that are to be the main components of the final ecosystem, and (3) many plants that make up the final biodiversity of the ecosystem and should be able to recolonize by their own efforts [7,8,13,35].
In this study, we first selected the species lost in this study area by comparing and analyzing the vegetation data obtained from the damaged and natural reference areas.
Next, we selected tolerant species which can withstand the polluted environment. Tolerant plants were selected as species flourishing specifically in the damaged area and species showing higher frequency in the damaged area than in the reference area. Tolerant plants were selected by classifying the four vegetation layers of canopy tree, understory tree, shrub, and herb composing the vegetation profile.
Finally, we selected the pioneer species frequently invading the bare ground to restore the heavily damaged zone. Synthesized, the plant species to be introduced for restoration were selected by combining the tolerant species with the polluted environment, the pioneer species frequently invading with the bare ground, and the species disappeared from the damaged area compared with the species composition of the reference area (Table 6) [7,13,36].
#### **5. Conclusions**
The result of a diagnostic assessment for the damaged forests around the Seokpo zinc smelter showed that ecological restoration is required urgently, as vegetation damage and soil acidification are very severe within the first ridge and at a distance of about 5 km from the pollution source. Vegetation damage appeared variously, such as the level to which bare ground appears due to extreme damage and the landslides that follow, the degree to which some of the vegetation strata is lost, and the visible damage level. The degree of soil acidification tended to be proportional to vegetation damage. In relation to soil acidification, deficiencies in nutrients, such as Ca2+ and Mg2+, and an increase in toxic ion concentration, such as Al3+, were also identified. In particular, landslides continued around places where vegetation was severely destroyed, making ecological restoration urgent. The restoration plan was prepared by compiling the results of these diagnostic assessments and reference information collected from intact natural forests. The restoration plan was prepared in two directions: amelioration of acidified soil and vegetation restoration. In order to successfully complete this ecological restoration, the continuous monitoring and management of soil and air pollution need to be prepared, as well as the continuous monitoring of the establishment process of the vegetation that is to be introduced.
**Author Contributions:** Conceptualization, A.R.K., C.H.L. and C.S.L; methodology, A.R.K., C.H.L. and C.S.L; software, A.R.K. and C.H.L.; validation, B.S.L., J.S. and C.S.L.; formal analysis, A.R.K. and C.H.L.; investigation, A.R.K., B.S.L., J.S., C.H.L., W.S.L., Y.H.Y.. C.S.L.; resources, C.H.L. and C.S.L.; data curation, B.S.L and J.S.; writing—original draft preparation, A.R.K.; writing—review and editing, Y.H.Y. and C.S.L.; visualization, A.R.K., B.S.L.; supervision, C.S.L.; project administration, C.S.L.; funding acquisition, C.S.L. All authors have read and agreed to the published version of the manuscript.
**Funding:** This research received no external funding.
**Conflicts of Interest:** The authors declare no conflict of interest.
#### **Appendix A**
**Environmental Factors Content** Water content (%) 48.34 Organic matter (%) 33.76 Total Nitrogen (%) 1.24 Available Phosphorus (%) 1.04 Exchangeable Potassium (%) 0.26 C.E.C (cmol+/kg) 35.0 Sodium (%) 0.57
**Table A1.** Chemical Properties of Organic Fertilizer Planned as a Soil Ameliorator.
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## *Article* **SWAT Model Adaptability to a Small Mountainous Forested Watershed in Central Romania**
**Nicu Constantin Tudose 1,\*, Mirabela Marin 1, Sorin Cheval 2, Cezar Ungurean 1, Serban Octavian Davidescu 1, Oana Nicoleta Tudose 1, Alin Lucian Mihalache 1,3 and Adriana Agafia Davidescu <sup>1</sup>**
**Abstract:** This study aims to build and test the adaptability and reliability of the Soil and Water Assessment Tool hydrological model in a small mountain forested watershed. This ungauged watershed covers 184 km2 and supplies 90% of blue water for the Bras, ov metropolitan area, the second largest metropolitan area of Romania. After building a custom database at the forest management compartment level, the SWAT model was run. Further, using the SWAT-CUP software under the SUFI2 algorithm, we identified the most sensitive parameters required in the calibration and validation stage. Moreover, the sensitivity analysis revealed that the surface runoff is mainly influenced by soil, groundwater and vegetation condition parameters. The calibration was carried out for 2001–2010, while the 1996–1999 period was used for model validation. Both procedures have indicated satisfactory performance and a lower uncertainty of model results in replicating river discharge compared with observed discharge. This research demonstrates that the SWAT model can be applied in small ungauged watersheds after an appropriate parameterisation of its databases. Furthermore, this tool is appropriate to support decision-makers in conceiving sustainable watershed management. It also guides prioritising the most suitable measures to increase the river basin resilience and ensure the water demand under climate change.
**Keywords:** SWAT; hydrological model; sensitivity analysis; calibration; validation; small forested watershed
#### **1. Introduction**
Watershed behaviour is influenced by multiple factors such as its geomorphologic characteristics (e.g., slope, soil, land use) and climate conditions [1]. Evaluating the watershed response to these stressors is pivotal for achieving environmental sustainability [2], considering that worldwide, meaningful changes are projected by the Intergovernmental Panel on Climate Change (IPCC) in rainfall, temperatures and extreme events [3]. Additionally, for European regions, an increased risk of droughts and floods is projected [4]. Besides, flood events generated by faster snowmelt or compounded rain-snow events due to increased temperatures will be more frequent, particularly in the mountainous regions [5–7]. Those changes will jeopardise the future sustainability of natural resources and, accordingly, all activity sectors [8], particularly water resources, through changes in flow regime [9,10]. Alongside land use modifications due to urban development, water resources are more vulnerable to additional pressures [11,12], especially its quality and quantity [13]. It is noteworthy that there is an intensification of hydrological processes in urban watersheds simultaneous with increments in the degree of urbanisation [14]. Furthermore, as a climate change consequence, increments in water demand are forecasted [15].
**Citation:** Tudose, N.C.; Marin, M.; Cheval, S.; Ungurean, C.; Davidescu, S.O.; Tudose, O.N.; Mihalache, A.L.; Davidescu, A.A. SWAT Model Adaptability to a Small Mountainous Forested Watershed in Central Romania. *Forests* **2021**, *12*, 860. https://doi.org/10.3390/f12070860
Academic Editors: Ovidiu Badea, Alessandra De Marco, Pierre Sicard, Mihai A. Tanase and Timothy A. Martin
Received: 22 April 2021 Accepted: 24 June 2021 Published: 29 June 2021
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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).
Considering the resource interlinkages, the entire ecosystem and humanity's well-being are jeopardised by individual shifts with a spillover effect [16]. Hydrological modelling is a useful and valuable approach for understanding these interconnections at the watershed level and to assess the impact of multiple drivers (e.g., climate, land use, socio-economic) on ecosystems. Hydrological processes within different sizes and scales of watersheds can be understood, described and explored using hydrologic models [17]. Lately, many researchers have employed various models (like the Soil and Water Assessment Tool (SWAT), Distributed Hydrology Soil Vegetation Model (DHSVM), Hydrologic Engineering Center's Hydrologic Modelling System (HEC-HMS), Variable Infiltration Capacity (VIC), European Hydrological System Model (MIKE SHE) and so on) to investigate these cumulative impacts on hydrological processes within the watersheds aiming to anticipate and mitigate multiple challenges [18–21]. This action is crucial for the appropriate planning and management of natural resources in various environments [22]. Almost 40% of the worldwide population is located in mountainous watersheds [23]. The mountainous regions and urban areas are characterised by a high vulnerability to climate change [23]. Therefore, natural resources can be endangered by those changes [23]. Those environments assure up to 80% of freshwater resources [24] and are susceptible to numerous shortcomings related to water resources, particularly water supply reservoirs located in urban areas [25]. To capture the local or regional specificity of watersheds with high accuracy, the simulation must be performed under the regional climate model [26,27]. Moreover, small watersheds are more numerous than large ones and receive less attention globally [6,28]. Starting from a small scale is essential for accurate hydrological assessments [29,30], even if, unfortunately, limited hydrological data are available for local levels [17,31,32]. Therefore, considering the orographic influence and assessing the long-term impacts at an appropriate resolution is fundamental for mountainous watersheds [7,33]. For watersheds located in mountainous regions, as is the studied watershed, the topography and snowmelt significantly influence streamflow [33,34]. Consequently, not only heavy rainfall but also the snow melt process, amplified by increased temperatures [35], will generate higher downstream river flows [5,36], events that were already confirmed at the national level, particularly for the winter months and early spring [37,38].
Unfortunately, the National Strategy for Flood Risk Management is currently designed for large watersheds only; for small watersheds, no nationwide action plan exists [39]. Small watersheds, mostly without conventional gauges, have short response times and are therefore more vulnerable to flash flood events [40,41]. Hence, a new approach focused on small watersheds is mandatory for developing appropriate response strategies for these watersheds. In this respect, investigating small watersheds' behaviour under multiple challenges by assessing the negative impact on the local environment, and thus on the local society, is fundamental. Further, short-, medium- and long-term stream flow prediction is necessary to inform decision-makers and support them in achieving sustainable water management [42,43]. Amongst the wide range of hydrologic models developed to date, for this study, we chose the SWAT hydrological model due to its high adaptability and flexibility to investigate a wide range of water-related issues and supportive user groups that can be easily accessed. Constantly improved since the 1990s, SWAT is a physical open-sources model that, even if it was initially developed for large river basins, has also been proved to be suitable for watersheds up to 1000 km<sup>2</sup> [13]. Additionally, the model is recognised as suitable for investigating long-term impacts, particularly in watersheds without conventional gauges [1,44]. SWAT is considered a valuable tool that assists decisionmakers and enables them to project a series of impacts and, hence, identify and prioritise measures needed to alleviate future risks.
To our knowledge, the application and validation of the SWAT model for a small watershed represent a novelty for both the region being studied and the entire country. In this respect, the specific objectives of this research are: (1) to personalise the SWAT model databases and (2) to test its adaptability to the local specificity of a small mountain forested watershed. Given that a large local population depends on its reservoir, the calibrated and
validated SWAT model represents a valuable tool for local and national decision-makers, supporting them in designing new sustainable water resources management strategies, particularly because small watersheds are usually seen with reservoirs that ensure downstream water demand [15]. In this context, considering the multiple challenges that society faces nowadays, a new integrated approach for investigating the possible changes realistically and advocating for achieving sustainable management of those changes is required [15,25].
#### **2. Materials and Methods**
#### *2.1. Study Area*
The watershed is located in the central part of Romania (Figure 1) at 45◦30 56" N and 25◦48 13" E. Our research was performed in the Tărlung watershed upstream of the Săcele reservoir.
**Figure 1.** Study area location.
The watershed upstream of the Săcele reservoir covers 184 km<sup>2</sup> and represents the main source of water (90%) for the Bras,ov metropolitan area. The watershed elevation ranges between 724 and 1899 m. The study area is characterised by a continental climate that receives an annual precipitation of 700-800 mm and records an average temperature of 4–5 ◦C. The main land use within the watershed is forests (73%), followed by mountain meadows (12% of the area), pastures with scattered trees (8%), pastures (2%), meadows (4%) and water bodies (1%). Regarding the soil types, 84% of the watershed soils are included in the cambisoil class, followed by spodisols (11%), cernisols (2%) and protisols (1%).
#### *2.2. SWAT Hydrological Model*
SWAT is a basin-scale model that operates at a daily time step and is extensively used in gauged and ungauged watersheds to simulate long-term hydrological processes under different drivers [45]. The model divides watersheds into sub-basins, which sub-
sequently are delineated into multiple hydrological response units (HRUs) in agreement with homogeneous characteristics of soils, slopes and land use [46]. Thus, a more accurate physiographic description of the watershed will be ensured [1]. The model has a default database, but it also enables users to create a personalised database for the request inputs: soil, land use and weather database [47]. The flowchart to run the SWAT model is highlighted in Figure 2.
**Figure 2.** Diagram of SWAT model [48].
The model is an open-source software with low parameter requirements that enables users to customise their database and define elevation bands to adjust the orographic effect on precipitation and temperature, particularly for watersheds located in mountainous regions [46]. Moreover, for each elevation band, SWAT estimates accumulation, sublimation and snow melt [35] parameters with a large influence on hydrological processes within those river basins [36,49].
#### *2.3. Model Parameterisation*
To setup SWAT, four components are needed: The digital elevation model (DEM), weather, soil and a land use database. All model input data in vector and raster format (namely DEM, land use and soil) are in the EPSG 3844 projection (the projected coordinate system for Romania), datum Pulkovo 1942 (58)/Stereo70. DEM is the first and most important input considering that defining all the watershed characteristics relies on this component. We used a DEM with a 10-meter spatial resolution for our study, characterised by a 10-meter horizontal resolution and 5-meter vertical resolution. DEM has been supplied by the National Institute of Hydrology and Water Management (INHGA database). Using the ArcSWAT interface (an ArcGIS extension tool), the Tărlung watershed was delineated. Afterwards, we continued with HRU delineation by overlapping three spatial characteristics: land use, soil maps and slope. This procedure is based on similar characteristics of land use, soil and slopes that are lumped together after a threshold set
by the user. For the Tărlung watershed, we established a threshold level of 10% each for soil, slope and land use to minimise errors due to multiple HRUs covering minimal surfaces. The action allows the reallocation at the sub-basin level of those three basic characteristics, which cover areas lower than the threshold value [50]. In doing so, the studied watershed was delineated into 169 sub-basins and 2419 HRUs. Moreover, to encapsulate the orographic influence and obtain accurate results, we defined ten elevation bands. After stream delineation, the morphological parameters and flow direction were obtained at the sub-watershed level.
Weather data are the second input requested by the SWAT model. For our research, we utilised data retrieved from the ROCADA dataset V 1.0 [51,52] and covered the 1961–2013 period. ROCADA represents a state-of-the-art homogenised gridded climatic dataset encompassing Romania at a spatial resolution of 0.1◦. This database has been used and its accuracy has been confirmed in many studies [53,54]. We also two used other patchy datasets regarding precipitation (1988–2010) and river discharge (1974–2015) that were recorded inside of the watershed (Babarunca and Săcele Reservoir hydrometric stations). The river discharge measurements were used to calibrate and validate the model and minimise the model's uncertainty. These two hydrometric stations belong to the INHGA that provided us with the river discharge datasets. The INHGA is empowered to provide hydrological data for different types of research and development projects. The weather database comprises the precipitation, minimum and maximum temperature, average wind speed, solar radiation and relative humidity and was conceived in a particular format accepted by SWAT, and afterwards embedded in the model and used for performing simulations.
The soil database was updated based on the information retrieved from the forest and pastoral management plans (Forest Management Plan 2009 and 2013, Silvopastoral Management Plan 1989) compiled for the Tărlung watershed by the National Institute for Research and Development in Forestry (INCDS) database. The maps enclosed in the aforementioned studies were used to identify the spatial distribution of the soil types within the studied watershed (Figure 3). The database was developed at the forest management compartment level in vector format and subsequently converted into raster format.
Due to time and money constraints, we did not have information regarding some soil characteristics like bulk density (SOL\_BD), hydraulic conductivity (SOL\_K) and water content (SOL\_AWC) required when building the SWAT model. Instead, we used the soilplant-atmosphere-water model (SPAW), an open-source software [55]. The SPAW program automatically determines those parameters according to certain soil properties like organic matter, sand and clay percentage. The value of each soil characteristic was inserted into the SPAW application, which automatically delivered water content, bulk density and hydraulic conductivity for each soil layer. Other parameters like soil albedo (SOL\_ALB) and soil erodibility factor (K\_USLE) were computed considering the research performed by [56,57], respectively. After determining all the required parameters regarding soil characteristics, the user soil table was completed (Table S1). To connect the default database and the raster of soil types at the watershed level, we created a table (user soil .txt format) with codes for each soil type. The codes can be found both at the raster level and in the SWAT default database. Finally, the user soil table was fed into the model and soils were reclassified in agreement with the SWAT codes. Subsequently, soil types were classified by hydrological groups. This classification was made considering the research performed by [58] in accordance with sand and clay percentage and soil layer depth. The soils within the watershed were framed in two hydrological groups, namely Group B (90.57%) and Group C (9.43%), which are characterised by medium and low infiltration capacity, respectively [59].
**Figure 3.** Soil types within the Tărlung watershed.
The land use database was updated using the information collected from the management plans alluded to above and observations on satellite images regarding roads, buildings and water bodies. The dominant land use categories (Figure 4) identified in the watershed were forests (73%), followed by mountain meadows (12%) and pastures with scattered trees (8%). Small percentages within the watershed area were occupied by meadows (4%), pastures (2%) and water bodies (1%).
The land use look-up table was designed in the requested format *(.txt file)* and was uploaded in the model, and afterwards, the land use was reclassified accordingly with the codes defined in ArcSWAT. Soil and land databases were developed at the forest management unit level. After building the requested databases and feeding them into the model, we set SWAT to run at a monthly time step for the 1961–2013 period (i.e., 53 years). This procedure also implied setting a warm-up period, namely 1961–1965, a length of time following the recommendations regarding the warm-up period setting for hydrologic models [60]. Hence, we obtained the hydrological parameters at the sub-basin level for 48 years and were also able to identify potential errors.
**Figure 4.** Land use at the Tărlung watershed level.
#### *2.4. Model Performance Evaluation Criteria*
The performance of the SWAT model was automatically carried out using the SWAT-CUP software [61]. We selected the SUFI-2 (Sequential Uncertainty Fitting version 2) algorithm from the four distinct procedures provided by SWAT-CUP due to its ability to optimise the parameters with minimal repetitions [44]. Another advantage is that this procedure considers both the model uncertainty and the uncertainty between the SWAT parameters and those that are measured [61]. The following widely applied parameters in hydrological studies were used for evaluating the model performance [62]: The coefficient of determination (R2), percent bias (PBIAS), standard deviation rate (RSR) and Nash Sutcliffe Model Efficiency (NSE). Choosing a multiple statistics indicator has to "increase the likelihood of mixed interpretation of model performance" [63]. R2 reflects the degree of colinearity amongst simulated and observed values and is computed using Equation (1) [64]. This index ranges between 0 and 1, where 0 describes no correlation, while 1 shows a good agreement:
$$\mathbf{R}^2 = \frac{\left[\sum\_{i=1}^n \left(\mathbf{Q\_{obs}} - \mathbf{Q\_{obs,m}}\right) \left(\mathbf{Q\_{sim}} - \mathbf{Q\_{sim,m}}\right)\right]^2}{\left[\sum\_{i=1}^n \left(\mathbf{Q\_{obs}} - \mathbf{Q\_{obs,m}}\right)^2 \sum\_{i=1}^n \left(\mathbf{Q\_{sim}} - \mathbf{Q\_{sim,m}}\right)\right]^2} \tag{1}$$
where Qobs is the discharge measured, Qsim is the discharge simulated, Qobs, m is the mean of measured discharge, and Qsim, m is the mean of simulated discharge.
PBIAS calculates the model errors [65]. Expressed in percentage after using Equation (2), the good fit of the model is indicated through values close to 0 [66]. The underestimation
of the model results is highlighted by positive simulated PBIAS values, while negative simulated values suggest overestimation [63]:
$$\text{PBIAS} = \frac{\sum\_{i=1}^{n} \left(\chi\_{i}^{\text{obs}} - \chi\_{i}^{\text{sim}}\right) \* (100)}{\sum\_{i=1}^{n} \left(\chi\_{i}^{\text{obs}}\right)} \tag{2}$$
where Yobs is the measured value of considered variable, Ysim is the simulated value of considered variable.
RSR is computed as the ratio between root mean square error (RMSE) and standard deviation of observed values (STEDEVobs) using Equation (3) [63]. A value close to 0 of this parameter indicates a perfect model simulation [63]:
$$\text{RSR} = \frac{\text{RMSE}}{\text{STEDEV}\_{\text{obs}}} = \frac{\left[\sqrt{\sum\_{i=1}^{n} \left(\chi\_{\text{i}}^{\text{obs}} - \chi\_{\text{i}}^{\text{sim}}\right)^{2}}\right]}{\left[\sqrt{\sum\_{i=1}^{n} \left(\chi\_{\text{i}}^{\text{obs}} - \chi\_{\text{i}}^{\text{mean}}\right)^{2}}\right]} \tag{3}$$
where Yobs is the measured value of considered variable, Ysim is the simulated value of considered variable, Ymean is the mean of the measured and simulated value.
NSE highlights the 1:1 fit between observed and simulated values using Equation (4) [67]:
$$\text{NSE} = \frac{\left[\sum\_{i=1}^{n} (\mathbf{Q}\_{\text{sim}} - \mathbf{Q}\_{\text{obs}})\right]^2}{\left[\sum\_{i=1}^{n} (\mathbf{Q}\_{\text{obs}} - \mathbf{Q}\_{\text{obs},\text{m}})\right]^2} \tag{4}$$
where Qsim is the discharge simulated, Qobs is the discharge measured and Qobs, m is the mean of measured discharge.
Additionally, the model performance was evaluated using the *p-factor* and *r-factor*. The *p-factor* indicates the fraction of data bracketed by the 95PPu band, while the *r-factor* represents the ratio of the average width of the 95PPu band and the standard deviation of the measured variable [68–70]. For *p-factor*, better values are higher than 0.7, while for *r-factor* values between 0.7–1.5 are recommended [68–70].
#### **3. Results**
#### *3.1. Sensitivity Analysis*
The sensitivity analysis aims to identify the parameters with the largest influence on model outputs, thus influencing its successful application. Undertaken before calibration, this procedure has the role of identifying key parameters that subsequently will be used in model calibration [71]. The sensitivity analysis is a mathematical technique applied to enable users to examine how variations in the outputs of a numerical model can be attributed to variations of its inputs [45]. Alongside calibration and validation, this procedure is decisive for minimising the output uncertainty and efficiently perform the simulations [71]. The sensitivity analysis uses a *t*-test to assess the relative parameter significance, while the *p*-value indicates the sensitivity rank. After performing the global sensitivity analysis, the parameters with large *t*-test values and smallest *p*-values are the most sensitive [72]. We considered 12 parameters (defined in Table 1) with the largest influence on model outputs: CN2, REVAPMN, GW\_DELAY, SOL\_K, ESCO, GWQMN, CH\_N2, CH\_K2, GW\_REVAP, ALPHA\_BF, LAT\_TIME and SOL\_BD.
**Table 1.** The default range and adjusted values of parameters included in the calibration procedure.
#### *3.2. Model Calibration*
After sensitivity analysis, we performed the calibration procedure to minimise the discrepancies amongst simulated data and recorded values [73]. The automatic calibration was also conducted using the SWAT-CUP program under the SUFI-2 algorithm. The model performance was assessed in agreement with the model performance evaluation criteria alluded to above.
The calibration was done for 2001–2010. This period was chosen due to continuous measurements and the dry, average and wet years necessary to ensure a high model performance with a lower uncertainty in the predictions [74]. Previously, we set up a five-year warm-up period (1996–2000) requested for model initialisation [61]. In doing so, we obtained the monthly river discharge for 10 years (Figure 5).
To obtain the best estimates between simulated and observed flow (Figure 6), we used the parallel processing module and performed seven iterations of 2000 simulations each. The process stopped when the model achieved a good performance rating indicated by the values of the statistical parameters recommended by [63], which can be accepted and used for assessing future impacts.
**Figure 5.** Simulated river discharge (Qs), measured river discharge (Qm) and precipitations (PP) for the Tărlung watershed for the 2001-2010 period.
**Figure 6.** The 95PPU plot between observed and best simulated discharges after the calibration procedure.
The parameters that insert water into the system (e.g., snowmelt or canopy storage parameters) should be calibrated independently from the other parameters [73]. Therefore, we performed the first calibration, including only SFTMP, SMTMP, SMFMX, SMFMN, TIMP and CANMX parameters, and ran the model until the statistical indices reached the performance rating recommended [63]. After those parameters were adjusted and fixed, they were subsequently excluded for the following calibration simulations. The second calibration was done independently for the first one and for 17 parameters that concerned only the parameters related to soil, groundwater, watershed and management characteristics. The selected parameters, the default range and their adjusted values are given in Table 1. Similar to the first calibration, the procedure was repeated until the statistical indices met the performance level that proves model acceptance.
Overall, the calibration procedure revealed a satisfactory SWAT performance, indicated by the statistical parameter values, appraised after [63], namely: R<sup>2</sup> = 0.61 (satisfactory), NSE = 0.59 (satisfactory), RSR = 0.64 (satisfactory), PBIAS = −5.7, *p-factor* = 0.72, and *r-factor* = 1.22. Hence, the SWAT performance was satisfactory to very good, and the obtained values revealed the model acceptance for simulating hydrological processes within the Tărlung watershed.
#### *3.3. Model Validation*
The validation confirms the results obtained after calibration [73]. This stage is important for ensuring the accuracy of the outputs considering that these will be further used in the decision process [75]. In our study, the validation was carried out for the same parameters used in calibration and considering the 1996–1999 period after previously setting up a five-year period for model warm-up. The period adopted for validation followed the same characteristics as in the calibration, namely continuous river discharges measurements and the presence of wet, dry and average years. For obtaining the best estimates between simulated and observed river discharge during validation, we performed a single iteration of 2000 simulations (Figure 7).
Nonetheless, if the user performs more than one iteration, it will increase the uncertainty of the model output due to the iterative character of the SUFI-2 algorithm [69]. The model efficiency was assessed using the same statistical indices as in the calibration. For those indices we obtained the following values appraised in accordance with the recommended performance rating [63]: R<sup>2</sup> = 0.78 (very good), NSE = 0.62 (satisfactory), RSR = 0.62 (satisfactory), *p-factor* = 0.67 and *r-factor* = 1.22. Overall, the model performance was satisfactory to very good, and the validation procedure results indicate that SWAT is suitable for assessing future impacts within the Tărlung watershed.
#### **4. Discussion**
The SWAT model developed particularly for large watersheds was applied, for the first time, both to the case study area and nationwide for a small-sized watershed. In this respect, we first customised the SWAT database to the local specificity of the studied region. In the next step, we performed the sensitivity analysis procedure that reduces the time required for calibration. During this stage, the parameters with the largest influences on hydrological processes are identified. In doing so, it was revealed that snowmelt and canopy retention are parameters with large influences on water balance within the Tărlung watershed. Those parameters have triggered lately, in the mountainous area, perilous floods during the spring months [48], particularly when snowmelt is overlapped with rainfall [38]. Due to their meaningful influence on hydrological process parameters that directly insert water into the system, they should not be calibrated together with other parameters (e.g., groundwater delay time, the coefficient for groundwater revap, base flow alpha factor and so on) because, as [73] states, they can generate identifiability issues. Therefore, snowmelt and canopy retention parameters were calibrated separately from the rest of the parameters that describe the watershed characteristics. In this respect, the first calibration includes only the snowmelt and canopy retention parameters, and the second calibration was made only for parameters that illustrate the watershed characteristics. Comparing the maximum canopy storage (CANMX) for evergreen forests and deciduous forests, the lower value was obtained for evergreen forests (see Table 1). A similar situation was also reported by [76]. However, the maximum canopy storage of deciduous forests is quite similar to the value obtained for pasture (see Table 1). This result agrees with the findings reported by [77], who obtained for pastures a maximum canopy storage even higher than those obtained for forests. In the case study area, an extension of pasture will affect the water quality due to the turbidity increments. These increments are also favoured by the main soil types from the watershed (Eutric Cambisol and Dystric Cambisol), which have high percentages of clay and silt (see Table S1), particles that are retained longer in suspension and affect the quality of water [1]. Thus, the water treatment capacity of the water plant will be exceeded and the water demand will not be covered (as has previously happened in the case study area). To prevent turbidity increments, the decision-makers should consider promoting "close to nature" forest management. This management practice will help preserve biodiversity and achieve the objectives highlighted and promoted in the EU's Biodiversity Strategy for 2030 [78].
Afterwards, the model performance was appraised through calibration and validation procedures that provided a satisfactory rating. This result indicates that the hydrological processes within the Tărlung watershed are well captured. After performing both procedures we noticed that the values of R2 and NSE parameters increased in the validation compared with calibration. This is an unusual situation because the optimisation of parameters occurs during the first procedure, but this circumstance has been reported in other research [22,66,79–81]. This situation may be due to the symmetry regression of the SWAT model [80], the number of wet or dry years included in both procedures or most likely due to the iterative character of the model [79–81]. The model uncertainties were assessed through *p-factor* and *r-factor*. The values obtained for the *p-factor* showed that the 95PPU band envelops 72% of the measured river discharges in the calibration and 67% during validation. Those results indicate a minimum uncertainty for calibration compared with
validation. Although the *p-factor* value during validation was 0.67 (slightly less than the lower limit of the interval recommended in the literature), we preserved this result. We did not perform another iteration because this repetition would have increased the uncertainty of the model results [73]. The *r-factor* represents the thicknesses of the 95PPU envelope and was 1.22 both for calibration and validation. According to [61], the values obtained for these two indices during both procedures revealed lower uncertainties in model results. Overall, the SWAT performance evaluated using the R2, NSE, RSR, and PBIAS showed a satisfactory model performance.
After running the SWAT model, both overestimations and underestimations at the monthly level were revealed (see Figure 6). The most meaningful overestimations were observed during the spring season (e.g., March 2003, 2005, 2006, 2009) and can be attributed to the fast snowmelt process [17,38,80,82–85]. Overestimations were also noticed during the summer season after heavy rainfall events, with similar results being reported by other authors [11,86,87]. The most significant underestimations were noticed during May in 2003, 2005, 2006 and 2010. These deficiencies can be generated by rainfall spatial variability within the watershed [86,88] and underline the necessity of research infrastructure installation that is properly spatially distributed to capture the spatial variability of rainfalls inside the watershed with high accuracy. Another consequence can be an inaccurate simulation of some parameters included in the water balance equation like groundwater and evapotranspiration [89], highlighting the importance of using field measurements.
Nevertheless, the SWAT model proved its performance and reliability and is suitable scientific support for decision-makers in planning activities, particularly in watersheds located in mountainous regions. These environments are important sources of freshwater, food, energy and biodiversity, and therefore enhancing their resilience is imperative under climate and land use change [24]. This task is a priority mentioned in the SDG 15: "Protect, restore and promote sustainable use of terrestrial ecosystems, sustainably manage forests, combat desertification, and halt and reverse land degradation and halt biodiversity loss". SDG's target is in 15.4 "By 2030, ensure the conservation of mountain ecosystems, including their biodiversity, in order to enhance their capacity to provide benefits that are essential for sustainable development" [90]. Considering that mountain areas host 23% of the total forests [24], their protection will alleviate climate change effects [78]. Furthermore, the "Framework convention on the protection and sustainable development of the Carpathians," signed by our country, also highlights the importance of mountainous regions for all ecosystems and the role of the local community to achieve an integrated and balanced sustainability of these environments [91]. Therefore, the calibrated and validated SWAT model can be considered a valuable planning tool for designing action plans for small watersheds, which are currently neglected.
#### **5. Conclusions**
This research is an effort that can be considered a novel step for future studies investigating the hydrological behaviour of small watersheds. We presented the methodology used for customising the SWAT model to the local specificity for testing its ability to simulate the hydrological processes within a small forested ungauged watershed located in a mountainous region. The studied watershed has meaningful importance for Bras,ov city and its surrounding areas because it represents the main drinking and industrial water source. Future climate change projections published for the 21st century underline the importance of conducting such hydrological assessments to investigate watershed behaviour under climaterelated risks. Therefore, we focused on testing, for the first time (nationwide and for a small forested watershed), the applicability of the SWAT hydrological model in a small watershed located in a mountainous area. Given that we built a detailed and customised database, the calibration and validation procedures revealed that SWAT meets the requirements and is adequate to simulate the hydrological processes within the Tărlung watershed. The model was developed for large river basins and had certain deficiencies reported in the literature. Nevertheless, this study stresses the importance of several factors (e.g., the accuracy of input
data, choosing the proper interval for performing the model's calibration and validation procedures, and carefully selecting the parameters to perform those procedures) that contribute and ensure the SWAT's suitability for application in small ungauged watersheds. After running the SWAT model for 53 years, we noticed a good agreement in mirroring the hydrological process, which is accurately captured within the watershed. The contribution of this paper enables the local upgraded SWAT model to be further used as a guidance tool for management decisions that pursue sustainable and integrated watershed management under multiple challenges (climate, environmental and societal).
**Supplementary Materials:** The following are available online at https://www.mdpi.com/article/10 .3390/f12070860/s1, Table S1 presents the physicochemical characteristics across soils type under study case (Tărlung watershed).
**Author Contributions:** Conceptualisation, methodology, investigation, formal analysis, writing original draft, writing—review & editing, validation (N.C.T.); methodology, investigation, formal analysis, writing—review & editing (M.M.); review & editing, formal analysis (S.C., C.U., S.O.D., A.L.M., O.N.T. and A.A.D.). All authors have read and agreed to the published version of the manuscript.
**Funding:** The present research benefits from funding from the project Climate Services for Water-Energy-Land-Food Nexus (CLISWELN) funded by ERA4CS. ERA4CS is an ERA-NET initiated by JPI Climate, and CLISWELN is funded by BundesministeriumfürBildung und Forschung (BMBF-Germany), Executive Agency for Higher Education, Research, Development and Innovation Funding (UEFISCDI -Romania), BundesministeriumfürfürBildung, Wissenschaft und Forschung and ÖsterreichischeForschungsförderungsgesellschaft (BMBWF and FFG -Austria), and Ministerio de Economía y Competitividad (MINECO-Spain), with co-funding from the European Union's Horizon 2020 under Grant Agreement No 690462.This paper and the content included in it do not represent the opinion of the European Union, and the European Union is not responsible for any use that might be made of its content.
**Institutional Review Board Statement:** Not applicable.
**Informed Consent Statement:** Not applicable.
**Data Availability Statement:** Datasets generated and/or analysed during the current study are available from the corresponding authors on request.
**Acknowledgments:** The authors would like to express their gratitude to Roger Cremades, Hermine Mitter, Anabel Sanches, Annelies Broekman and Bernadette Kropf for comments and discussions that helped to improve the manuscript. We would also like to thank the editor and anonymous reviewers for their useful advice that helped to improve the manuscript.
**Conflicts of Interest:** The authors declare no conflict of interest.
#### **References**
## *Article* **Intraspecific Growth Response to Drought of** *Abies alba* **in the Southeastern Carpathians**
**Georgeta Mihai 1,\*, Alin Madalin Alexandru 1,2,\*, Emanuel Stoica <sup>1</sup> and Marius Victor Birsan <sup>3</sup>**
- 013686 Bucharest, Romania; [email protected]
**Abstract:** The intensity and frequency of drought have increased considerably during the last decades in southeastern Europe, and projected scenarios suggest that southern and central Europe will be affected by more drought events by the end of the 21st century. In this context, assessing the intraspecific genetic variation of forest tree species and identifying populations expected to be best adapted to future climate conditions is essential for increasing forest productivity and adaptability. Using a tree-ring database from 60 populations of 38-year-old silver fir (*Abies alba*) in five trial sites established across Romania, we studied the variation of growth and wood characteristics, provenancespecific response to drought, and climate-growth relationships during the period 1997–2018. The drought response of provenances was determined by four drought parameters: resistance, recovery, resilience, and relative resilience. Based on the standardized precipitation index, ten years with extreme and severe drought were identified for all trial sites. Considerable differences in radial growth, wood characteristics, and drought response parameters among silver fir provenances have been found. The provenances' ranking by resistance, recovery, and resilience revealed that a number of provenances from Bulgaria, Italy, Romania, and Czech Republic placed in the top ranks in almost all sites. Additionally, there are provenances that combine high productivity and drought tolerance. The correlations between drought parameters and wood characters are positive, the most significant correlations being obtained between radial growth and resilience. Correlations between drought parameters and wood density were non-significant, indicating that wood density cannot be used as indicator of drought sensitivity. The negative correlations between radial growth and temperature during the growing season and the positive correlations with precipitation suggest that warming and water deficit could have a negative impact on silver fir growth in climatic marginal sites. Silvicultural practices and adaptive management should rely on selection and planting of forest reproductive material with high drought resilience in current and future reforestation programs.
**Keywords:** silver fir; radial growth; wood characteristics; drought response; climate change
#### **1. Introduction**
Climate change is a major threat to forests in the 21st century. According to the reports of the Intergovernmental Panel on Climate Change [1,2], temperatures have increased globally, and the highest rates of warming have taken place in the last decades. Furthermore, recent evidence has shown a significant increase in the frequency of extreme weather events (prolonged droughts, heat waves, cold snaps, and floods) related to global climate change [3,4].
Among the extreme meteorological events, drought is considered to have the largest detrimental impact on forest ecosystems. Drought and heat stress associated with climate change could fundamentally alter the productivity, genetic diversity, and distribution of forest ecosystems [5–7]. In recent years, it was observed that drought frequency, severity,
**Citation:** Mihai, G.; Alexandru, A.M.; Stoica, E.; Birsan, M.V. Intraspecific Growth Response to Drought of *Abies alba* in the Southeastern Carpathians. *Forests* **2021**, *12*, 387. https://doi.org/10.3390/f12040387
Academic Editors: Alessandra De Marco, Ovidiu Badea, Pierre Sicard and Mihai A. Tanase
Received: 21 February 2021 Accepted: 22 March 2021 Published: 24 March 2021
**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.
**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).
and duration increased in many regions in Europe [8]. The most affected regions have been southern Europe with the Mediterranean region as a hotspot [9,10] and South-eastern Europe, particularly the Carpathian region [11–13].
The moderate scenario projections (RCP4.5) show that southern Europe, western Europe, and northern Scandinavia will be affected by a substantial increase in drought frequency by the end of the 21st century. However, the extreme emission scenario (RCP8.5) suggests that the entirety of Europe will be affected by more frequent and severe droughts compared to the last century. Under both scenarios, drought frequency is projected to increase in spring and summer everywhere in Europe, but especially in southern Europe [14,15].
The climate changes will also enhance the action of the new biotic (pest and disease) and abiotic disturbance factors (fire, windstorm) with major consequences for forest ecosystems. Increasing the extreme events, such as drought and disturbance factors, in the near future will pose serious threats to the growth and persistence of forest species than gradual climate changes [16,17]. There is a consensus that the ability of forest ecosystems to provide multiple goods and services will be impacted [18]. The mountain ecosystems and those located at the edges of forest species' distribution will be the most vulnerable.
Silver fir (*Abies alba* Mill.) is one of the main species of mountain ecosystems in Europe with multiple functions, including ecological, economic, and soil protection roles. European silver fir is a shade-tolerant species and can grow in an array of soil conditions, with various amounts of nutrients, but prefers humid and deep soils [19]. Results so far regarding the potential of silver fir to thrive under expected warmer and drier conditions are optimistic. The species distribution models (SDMs) suggest that the suitable distribution area of silver fir will decrease by the end of the century, particularly in the southern and eastern parts of its distribution, but the lowest decrease is projected for silver fir compared to other coniferous species [20,21]. On the other hand, paleoecological studies, as well as dynamic models accounting for biotic and abiotic disturbances, suggest that this species has a high potential to cope with the expected climate change [22] and can even expand in regions with summer water deficit from central and eastern Europe [23,24]. Additionally, other recent studies showed that European silver fir has high phenotypic plasticity [25] and is less vulnerable to drought stress than other conifers of temperate forests [26–31]. However, a possible decline may occur in the driest and warmest areas at the distribution edge [32–34]. Therefore, European silver fir could be one of the future species for consideration under changing climate conditions, particularly at lower altitudes in the mixed vegetation layer.
Considering that drought events will become more frequent and intense in the near future, the strategies to cope with climate change have to prepare forests by increasing the adaptive capacity of tree populations [35]. Recent research shows that selecting and transferring forest reproductive material adapted to the new environmental conditions of the planting site could increase genetic diversity in those areas and could facilitate the adaptation of forest species [36–38]. Therefore, assessment of intraspecific genetic variation and identifying populations expected to be best adapted to the future climate conditions is essential for increasing forest productivity and adaptability in the context of climate changes.
Many classical studies in the field of dendrochronology have investigated the potential impact of climate changes on tree growth [28,29,32]. Unfortunately, these studies do not take into account the existence of intraspecific genetic variation, considering genetically homogeneous species. Provenance trials, where tree populations throughout the entire distribution of a species are tested in different site conditions, can provide important data concerning intraspecific adaptive capacity and selection of suitable populations for reforestation programs. These genetic tests facilitate the identification of climatic variables that exert strong selective pressure on studied populations and the developing of the models that can be used to predict species' response to future climates [39,40].
Provenance trials have been used to analyze intraspecific variation in climate growth response in several tree species, such as *Pinus contorta* [39,41], *Pinus sylvestris*, *Fagus sylvatica* and *Quercus petraea* [42], *Quercus robur* [43], *Picea glauca* [44,45], *Pseudotsuga menziesii* var. *menziesii* [46,47], *Picea abies* [42,48–51], and *Abies alba* [34,52]. However, there are still gaps in our knowledge concerning the intraspecific genetic response of forest species to extreme climate events, such as severe or extreme droughts, because they require long-term experiments and growth and climate assessments over several decades. Recent studies have shown that there are significant genetic variations in drought response both within and among trees populations [53–56]. The provenance-specific drought response for silver fir has been investigated in even fewer studies [30,57,58].
The Romanian Carpathians represent the southeastern distribution limit of silver fir in Europe. Meteorological records show a warming trend and increasingly severe and extreme summer droughts in recent decades in this region. Considering that negative effects of predicted climate change will be more pronounced, especially at the xeric edge of species distribution range [14,15,34], knowing the adaptive capacity of silver fir becomes of major importance. In this context, the aim of this study was to investigate the genetic adaptive capacity and response of silver fir provenances originated from nine European countries to extreme drought events that have occurred in this region in the last 22 years. Understanding of the population's performance in relation to climate stress, selection of the best adapted seed sources to future climate conditions, and using them in reforestation programs (i.e., assisted migration) is essential for increasing forest productivity and adaptability.
Based on the assumption that drought will significantly impact silver fir ecosystems in southeastern Europe, in the near future, the objectives of this study were to (1) assess the genetic variation of radial growth and wood characteristics among silver fir provenances, (2) evaluate the provenances-specific drought response, (3) establish the climate–growth relationships, (4) determine correlations between radial growth and wood characteristics, and drought parameters, and (5) provide practical information for sustainable forest management in a changing climate context.
#### **2. Materials and Methods**
#### *2.1. Trial Site and Plant Material*
The study was conducted in a series of five provenances trials established in 1980 in Romania. The provenance trials were established in five geographic regions with different climatic conditions (Figure 1 and Table 1). Two trials are located outside of the natural range of silver fir in Romanian Carpathians, in the European beech zone, while three are within the natural range.
**Table 1.** Geographic and climatic variables for silver fir trial sites.
In these trials are tested 60 populations originating from the entire species distribution range in Europe (Figure 1). They were grouped as core, western, eastern, northern, southeastern, and southern according to their location within the natural distribution range (Table A1 in Appendix A). Forty-three provenances are common in all trials, and 17 provenances are tested additionally at the Sacele trial only. The silver fir tested provenances range from 38◦33 to 51◦07 N and from 4◦00 to 26◦40 E, and include both lower as well as higher mountain regions (altitudes between 130–1600 m above sea level). In four sites, the field layout was the randomized square lattice, type 7 × 7, with three repetitions and 25 trees per plot planted at 1.0 × 2.0 m, while at the Sacele trial, the field layout was the randomized square lattice, type 8 × 8, also with tree repetitions. The
five field trials were established with six-year-old bare root seedlings, which have been produced in the Sinaia nursery situated in the mountain beech zone, at 45◦29 N latitude, 25◦59 E longitude, and at 695 m a.s.l.
**Figure 1.** Location of silver fir provenances (triangles) and trials (circles). Gray area/dots—the natural distribution of silver fir (by EUFORGEN).
#### *2.2. Phenotypic and Climatic Data*
In each provenance trial, four dominant or (co)dominant trees per provenance and repetition (12 trees in total for each provenance) have been cored at 1.3 m breast height using 5 mm increment borers (Haglof, Sweden), from slope-parallel stem radii, to avoid tension and compression wood. In order to avoid tree damage, only one core per tree was taken. Cores were dried and progressively sanded [59]. Then, the core samples were scanned at 1200 dpi, using an Epson Expression 10,000 XL, and the ring width (RW), earlywood width (EW), and latewood width (LW) were measured using the Ligno Vision software package to the nearest 0.001 mm. Additionally, latewood proportion (LWP) was calculated as an indicator of wood quality.
For each trial, the tree-ring series has been cross-dated using COFECHA [60] to avoid dating errors due to missing or false rings, which could be present in an increment radial core. Only the tree-ring series that presented intercorrelation values > 0.328 (*p* < 0.01) were included in final tree-ring data. All tree-ring time series were standardized to a mean value of one to obtain a width index (RWI) [61,62]. The negative exponential regression in the R package dplR [63,64] was applied for each raw measurement series, because it is deterministic, meaning that it follows a model of tree growth. The final tree ring data set comprised 2699 tree-ring series, 669 from the Sacele trial and 500 from each of the other four trials. The analyzed period was 1997–2018, being the common interval for all tree-ring series.
Additionally, wood density (WD) in g/cm<sup>3</sup> was determined for each core sample and whole analyzed period using the [65] formula:
$$\rho\_{\rm c} = 1/\left[\left(\mathbf{M}\_{\rm max}/\mathbf{M}\_{\rm o}\right) - 1 + 1/\rho\_{\rm ml}\right] \tag{1}$$
where: ρ<sup>c</sup> = conventional density (g/cm3), Mmax = weight of saturated sample (g), Mo = weight of dried sample (g), ρml = wood density (1.53 g/cm3).
The climatic data have been calculated using a daily gridded climatic dataset covering the Romanian territory (ROCADA). The dataset used herein consists of a higher spatial resolution (1 × 1 km) for improved reproduction of the climatic spatial variability and has been made using state-of-the-art interpolation techniques [66]. The following climatic variables have been calculated for each trial site over the period 1997–2018: mean annual temperature (MAT); mean temperature during the growing season (April to September) (MTVEG); mean temperature for January (MTJAN) and July (MTJUL) (the coldest and the warmest months, respectively); mean temperature from October to December of the previous year (MTOCT-DEC); mean temperature from October of the previous year to March of the current year (MTOCT-MAR); mean temperature from January to March of the current year (MTJAN-MAR); mean annual precipitation amount (MAP); mean precipitation during the growing season (MPVEG); mean precipitation of the coldest (MPJAN) and the warmest (MPJUL) months; mean precipitation from October to December of the previous year (MPOCT-DEC); mean precipitation from January to March of the current year (MPJAN-MAR); mean precipitation from October of the previous year to March of the current year (MPOCT-MAR); annual precipitation amount (SAP); precipitation amount in the growing season (SPVEG); precipitation amount in the autumn-winter of the previous year (SPOCT-DEC); precipitation amount from January to March of the current year (SPJAN-MAR); precipitation amount from October of the previous year to March of the current year (SPOCT-MAR) (Table 2).
**Table 2.** Description of the wood and climatic characteristics.
#### *2.3. Determination of Drought Events and Drought Response Parameters*
As an indicator for the meteorological droughts, we calculated the standardized precipitation indices (SPI) [67], which account for anomalous low rainfall, over the period 1989–2018. Given that an extreme drought event obviously lasts two to three months, to identify drought years within the analyzed period, we calculated SPI for three consecutive drought months in each trial site. That allowed us to detect both seasonal and annual variation of drought events during the analyzed period. The drought years have been classified as follows: SPI ≤ −2—extreme drought year, SPI between −1.99 to −1.50—severe
drought year, SPI between −1.49 to −1.0—moderate drought year, SPI between −1.0 to +1.0—normal precipitation year [57].
The response of provenances to drought events was evaluated by four drought parameters [68]: resistance (Res), recovery (Rec), resilience (Rsl), and relative resilience (rRsl). Resistance was calculated as the ratio between ring width during (Dr) and before the drought event (preDr): Res = Dr/preDr and indicates how much the radial growth decreased during drought (Res ≥ 1 means high tolerance, Res < 1 means low tolerance). Recovery was calculated as the ratio between the ring width after drought event (postDr) and during drought (Rc = postDr/Dr) and indicates the revitalization capacity after a drought period. Resilience (Rsl) represents the ratio of the ring width after drought (postDr) and pre-drought (preDr): Rsl = postDr/preDr and describes the capacity of a provenance to reach pre-drought increment after a drought event (Rsl ≥ 1 means full restoration, Rs < 1 means long-term growth reductions). Relative resilience (rRsl) was calculated by rRsl = (postDr − Dr)/preDr. Pre-drought and post-drought ring widths were calculated as average values for three-year period before or after the drought year.
#### *2.4. Data Analysis*
Analyses of variance were performed at two levels, each trial site and among sites, using the GLM procedure (SPSS v20). The total amount of variation was divided into the following sources of variation: provenance, site, year, and the interaction between them. All effects were considered random, except for the trial location, which was considered fixed.
The following mixed model was applied:
$$Z\_{\rm ijkln} = \mu + \mathbf{P\_i} + \mathbf{S\_l} + \mathbf{B\_{\bar{l}}} + \mathbf{Y\_k} + \mathbf{PS\_{\bar{l}l}} + \mathbf{P}\mathbf{Y\_{ik}} + \mathbf{S}\mathbf{Y\_{lk}} + \mathbf{e\_{i\bar{j}kln}}\tag{2}$$
where: Zijkln = the trait (wood characters, drought parameters), μ = the overall mean, Pi, Sl, Bj,Yk, PSil, PYik, SYlk, and eijkln are the effect due to the ith provenance, lth site, jth repetition (block), kth year, interaction due to ith provenance and lth site, interaction due to ith provenance and kth year, interaction due to lth site and kth year, and random error associated with the ijklnth trees.
In order to investigate to what extent the local adaptation to climate conditions of origin location influence traits variation, Pearson correlations based on provenance means were computed between the wood characters, the drought parameters, and the geographical coordinates of the provenances' origin for each trial site.
Relationships between the wood characters and the climatic variable of trial sites were investigated by regression analysis. The growth response functions were developed to assess the impact of climate at trial sites on provenances radial growth. The quadratic models based on both temperature and precipitation were used to develop growth response functions, considering them more suitable [34,40,41,44]. We used seven temperature variables and 12 precipitation variables, and the best models were chosen based on the *R<sup>2</sup>* coefficient (SPSS program, stepwise selection method).
#### **3. Results**
#### *3.1. Identification of Drought Years*
Large variation in mean annual temperature and annual precipitation amount were recorded in each trial site (Figure 2). Based on SPI values we have identified the moderate, severe, and extreme drought years in each trial site in the period 1989–2018 (Figure 3). Ten years with extreme and severe drought during the analyzed period have been identified, in all trial sites. Most of the extreme and severe drought events occurred after the year 2000. Additionally, during this period, the most consecutive drought years were recorded, such as 2002–2003, 2011–2012, and 2013–2015.
The number of extreme drought years during the analyzed period (1997–2018) have varied among sites and ranged between three at Moinesti and Domnesti trials to five at the Strambu Baiut trial. However, 2000, 2002, and 2011 were the common extreme drought years in all testing sites.
The most significant drought event occurred in 2000 when the highest number of months with severe and extreme drought (nine at Moinesti, seven at Bucova, five at Sacele, four at Stambu Baiut, and three at Domnesti) has been recorded (Table 3). Among all extreme droughts, the 2000 drought had the longest duration in almost all sites. Furthermore, the drought overlapped with the growing season, in four of the five trial sites. The 2011 drought was characterized by the highest intensity and generally by two peaks, while the 2002 drought by lower duration and intensity (five to two months with extreme and severe drought).
**Figure 2.** Variation of the mean annual temperature (**a**) and annual precipitation amount (**b**) in trial sites.
#### *3.2. The Effect of Provenance, Site, and Year on Radial Growth and Wood Characteristics*
The analysis of variance for each trial site and analyzed period was presented in Table 4. Results have highlighted that both provenance and year effects were significant for studied traits in all trial sites. Provenance x year interaction was also significant in three testing sites. Multifactorial analysis of variance across sites highlighted significant provenance, year, and also site effects. Provenance x site and site x year interactions were very significant also (Table 5).
Results indicate that, during the analyzed period, the studied characters have varied significantly among sites (Figure 4). The highest values of average on experiment for RW were obtained at Strambu Baiut trial (3.8 mm) followed by Sacele (3.7 mm) and Domnesti (3.7 mm) trials. The lowest values were recorded at Moinesti trial (3.4 mm). Regarding the LWP and WD, the highest value of average on experiment was obtained at Domnesti trial (47% LWP and 0.36 g/cm<sup>3</sup> WD). The lowest values for LWP (41%) and WD (0.34 g/cm3) have been recorded at Sacele and Bucova, respectively.
**Figure 3.** Standardized precipitation index (SPI) for the trial sites. Trials are Bucova (**A**), Domnesti (**B**), Moinesti (**C**), Sacele (**D**), and Strambu Baiut (**E**).
**Table 3.** The months when occurred extreme and severe droughts during 2000, 2002, 2003, and 2011.
(1) Explanatory note: the year 2003 was extremely dry at Strambu Baiut only.
**Table 4.** Analysis of variance of wood traits for the period 1997–2018.
The level of significance is represented as follows: \* *p* < 0.05; \*\* *p* < 0.01; \*\*\* *p* < 0.001.
**Table 5.** Multifactorial analysis of variance of wood traits for the period 1997–2018.
The level of significance is represented as follows: \*\* *p* < 0.01; \*\*\* *p* < 0.001.
In all trials, it can be seen a strong relationship between the RW variation and severe and extreme drought years. The tree-ring pattern of provenances showed a strong increment drop in those years (Figure 4). The descriptive statistics of silver dendrochronology in each trial were presented in Table A2.
The average radial growth has varied between 4.83 mm (provenance 24-Devin at Sacele trial) to 2.91 mm (provenance 25-Kitilovo at Moinesti trial), latewood percentage between 53% (provenance 45-Le Joux at Bucova trial) to 33% (provenance 37-Liezen at Sacele trial), whereas wood density has varied from 0.45 g/cm3 (provenance 45-Le Joux at Strambu Baiut trial) to 0.31 g/cm3 (provenance 59-Banska Bystrica at Strambu Baiut trial) (Figure A1). However, despite this high variability among sites, there were some provenances that have obtained RW values over the average of the experiment in all trial sites: 63-Zarovice, 54-Strambu Baiut, 16-Toplita, 55-Valea Iadului, 21-Azuga, 26-St. Dimitrov, 51-Gura Putnei, 48-Pangarati, 47-Moinesti, and 14-Asau. The ranking by LWP has shown that the highest spatial stability had the provenances 43-Greseuss, 26-St. Dimitrov, 29-Vallombrosa, 45-Le Joux, 41-Enzklosterle, 42-Sulzburg, and 44-Lepilat.
#### *3.3. Genetic Variation in Drought Response*
The analysis of variance for all extreme drought years, taken together, revealed significant variation in drought response among silver fir provenances in all trial sites, except the Bucova trial (Table 6). Additionally, significant differences were obtained for the year's effect and provenance x year interaction. The highest variation among provenances was obtained for the resilience to drought, in four of the five trials. Significant differences for resistance capacity were obtained at Sacele and Stambu Baiut, while for recovery only at Domnesti.
**Figure 4.** *Cont*.
**Figure 4.** Variation of the radial growth and wood characteristics in trial sites during the analyzed period. Trials are Bucova (**a**), Domnesti (**b**), Moinesti (**c**), Sacele (**d**), and Strambu Baiut (**e**).
**Table 6.** Analysis of variance for drought parameters of silver fir provenances in all extreme drought years and each trial site.
The level of significance is represented as follows: \* *p* < 0.05; \*\* *p* < 0.01; \*\*\* *p* < 0.001.
Considering only the common extreme drought years in all testing sites 2000, 2002 (2003 at Strambu Baiut), and 2011, significant differences were found among drought parameters of silver fir provenances (Table 7). The provenance-specific drought response depended on the trial site and drought year. Thus, significant differences for all parameters and all extreme drought years were obtained at Domnesti and Moinesti trials. The highest differences in drought response were found in the year 2000, in all trial sites. Additionally, the 2011 drought caused a significant genetic variation in the drought response of silver fir provenances.
The ranking of silver fir provenances by drought parameters in the year 2000, as the most significant drought year, and in all sites, revealed a certain variation pattern (Figure 5). Thus, the provenances ranking by resistance, recovery, and resilience, taken together, have highlighted a best performing group placed at the top ranks in almost all sites. This group include the following silver fir provenances: 23-Rakitovo, 30-Paularo, 7-Vadul Dobri, 53- Botiza, 55-Valea Iadului, and 63-Zarovice. In terms of resistance and resilience, the most valuable provenances were 12-Naruja I, 25-Kitilovo, 33-Abeti Soprani, 43-Greseuss, and 45- Le Joux. Regarding the resistance and recovery, the most valuable provenances were 51-Gura Putnei, 54-Strambu Baiut, and 56-Ilisoara Mures, while regarding the recovery and resilience the provenance 6-Bucium obtained good results. Additionally, there are provenances that obtained a good response and high spatial stability for only one drought
parameter. For instance, 26- St. Dimitrov revealed high resistance capacity; 21-Azuga, 50-Malini, 52-Solca, and 59-Banska Bystrica revealed high recovery; while 22-Vallombrosa, 41-Enzklosterle, 44-Lepilat, 47-Moinesti, and 4-Avrig showed high resilience (Figure 5).
Making the ranking of drought parameters for all extreme drought years, the highest values of resistance were observed at Moinesti and Bucova trials (the drought-prone environments), while for recovery and resilience at the Strambu Baiut trial.
**Table 7.** ANOVA of drought parameters of silver fir provenances for the common extreme drought years during the analyzed period.
Explanatory note: (1) the year 2003 was extremely dry at Strambu Baiut only. The level of significance is represented as follows: \* *p* < 0.05; \*\* *p* < 0.01; \*\*\* *p* < 0.001.
#### *3.4. Phenotypic Correlations*
The correlations between wood characters and WD were negative in all trial sites, although statistically significant correlations were obtained in few trials only (Table A3).
Additionally, the correlations between wood characters and geographic coordinates of the provenances were few and indicate low pattern of local adaptation. The most significant correlations were found with LONG and, generally, they have been negative in almost all trials, except for RW and EW at Bucova trial, and for LW and LWP at Sacele trial, where they were positive. Statistically, correlations with LAT and ALT of seed origin were few and only at Moinesti trials and Strambu Baiut, respectively.
The correlations between drought parameters and wood characters were positive, and the most were obtained between RW and resilience (Table 8). Correlations between drought parameters and wood density were non-significant.
**Figure 5.** Variation of the drought parameters of silver fir provenances calculated for 2000 drought year in each trial site. Drought parameters are resistance (**a**), recovery (**b**), resilience (**c**), and relative resilience (**d**).
**Table 8.** Phenotypic correlations between wood characters and drought parameters of silver fir provenances.
The level of significance is represented as follows: \* *p* < 0.05; \*\* *p* < 0.01; \*\*\* *p* < 0.001.
#### *3.5. Growth Response Functions*
The influence of climate on the RW and LWP in each trial site has been investigated using quadratic regressions, and the best models obtained were presented in Tables 9 and 10. The main climatic drivers explaining the radial growth of silver fir were MTVEG, MPOCT-MAR, and MPJAN-MAR. The growth–climate relationship was moderate, *R<sup>2</sup>* ranging between 0.37 and 0.50, indicating that a substantial amount of the radial growth variation can be explained by these climatic factors. Partial *R2* indicates that silver fir is less sensitive to precipitation than to temperature. The influence of temperature during the growing season accounted for 29% and 48% of the total variation of RW. For latewood percentage, the response models founded were modest (*R2* varied between 0.09 and 0.15). MAT, MTVEG, MAP, and MTVEG are the main climatic factors that influence LWP of silver fir (Table 10). The temperature variables accounted for the greater part of LWP variation.
**Table 9.** Climatic response models for radial growth of silver provenances. MTVEG—the mean temperature of the growing season, MPOCT-MAR—the mean precipitation from October of the previous year to March of the current year, MPJAN-MAR—the mean precipitation from January to March of the current year.
The level of significance is represented as follows: \*\*\* *p* < 0.001.
**Table 10.** Climatic response models for late wood percentage of silver provenances. MAT—the mean annual temperature, MTVEG—the mean temperature of the growing season, MAP—the mean annual precipitation amount, MTVEG—the mean precipitation, during the growing season.
The level of significance is represented as follows: \* *p* < 0.05; \*\* *p* < 0.01; \*\*\* *p* < 0.001.
#### **4. Discussion**
In this study, we have analyzed the radial growth, wood characteristics, and drought response of 60 silver fir provenances tested in five long-term trials established in different geographic regions and climatic conditions across Romania. Considerable differences in radial growth and wood characteristics among silver fir provenances were found. The influence of the local site conditions of each experiment and interaction of provenance with site and year were also significant in three testing sites, suggesting that the stability over time of silver fir radial growth and wood characteristics depends on site conditions.
The analysis of climate data during the period 1997–2018 showed large variations in terms of temperature and precipitation at site and time scale too. Results revealed a warming trend and a decreasing in the sum of annual precipitation during the analyzed period. The De Martonne aridity index, calculated for each trial site at the entire year level and entire period, had a value between 50 to 67 indicating that climatic conditions of the trial sites fall into the wet category. However, the values of De Martonne aridity index calculated for the growing season ranged between 22 to 27 that classifies the sites climate into silvostepic.
Extreme drought events have increased their frequency during the last two decades and among all extreme droughts, the most significant in duration and intensity have been the 2000, 2002, and 2011 droughts, in all trial sites. Abrupt growth changes were detected in tree ring chronologies related to these drought events. The losses in RW caused by drought have varied depending on the site, drought year, and provenance. The highest losses in RW have occurred in 2011, characterized by the highest drought intensity and two peaks in all sites, ranging between 18% at Moinesti to 27% at the Sacele trial. The year 2011 exerted the highest water stress on vegetation over the half-century in many regions of Europe [1].
Results revealed significant genetic variation in drought response among tested provenances. The drought reaction of silver fir provenances varied significantly depending on the extreme drought year and site conditions. The highest response by almost all drought parameters was found in the year 2000 when a consistent pattern in provenances drought response across the sites was observed. Thus, the provenances ranking by resistance, recovery, and resilience revealed several provenances placed in the top ranks in almost all sites. This group include provenances from Bulgaria, Italy, Romania, and Czech Republic. Additionally, some provenances had a good tolerance and high spatial stability for two or only one drought parameter. The remarkable performance combining superior growth with high tolerance to drought events had the provenances 63-Zarovice, 53-Botiza, 54-Strambu Baiut, 55-Valea Iadului from core distribution range, 47-Moinesti, 50-Malini, 51-Gura Putnei from eastern edge, and 7-Vadul Dobrii and 26-St. Dimitrov from southeastern edge. The highest values of resistance to drought were observed at Moinesti and Bucova trials, in the drought-prone environments.
It is notable that our results highlight higher genetic variation in drought response among silver fir provenances compared to previous studies. Thus, George et al. [57], studying drought sensitivity of ten provenances of silver fir and four Mediterranean fir species in eastern Austria, found both intra- and inter-specific variation to drought. However, his results indicated that genetic variation in drought response among silver fir provenances is more reduced than among *Abies* species. Additionally, Sagnard et al. [69], analyzing growth traits and drought resistance of silver fir seedlings in France, found a low variation among provenances in drought response, while Sindelar and Beran [70] found little genetic differentiation among silver fir provenances for drought resilience. The high genetic differentiation of drought response revealed in our study can be explained by the broad geographic amplitude of the provenances tested in these trials. This geographic area comprises two putative glacial refugia in southern Europe where silver fir survived during the last glaciation: in the Appenines and in the Balkan Peninsula of southeastern Europe. The remarkable growth performances and drought resilience of some provenances from the eastern distribution range (48, 52, 53) and southeastern (7,23, 25, 26) and southern edge (33) indicate that these populations, most of them peripheral, possess high adaptive potential, most likely as a consequence of the selection pressure.
Forest species hold different adaptive capacity to withstand the impacts of drought according to their ecophysiological characteristics and evolutionary adaptation. For instance, Arend et al. [54] showed that *Quercus robur* needs a prolonged recovery phase after the drought, indicating a lower fitness for drought tolerance. Forner et al. [71] found that *Pinus nigra* was able to recover after the extreme event while *Quercus faginea* was not. Additionally, Gazol et al. [72] revealed that *Pinus ponderosa* and *Pseudotsuga menziesii* displayed greater plasticity in resistance to a drought that the two more frequently oaks (*Quercus alba* and *Quercus stellate*) in North America. Our study has demonstrated that the resilience and resistance to drought varied significantly among silver fir provenances.
Silver fir is a species that highlights low genetic variability among populations, but high genetic diversity within populations, even in marginal populations [33,73], which could be a benefit for adapting to climate warming. Heer et al. [74] analyzed dendroecological and genetic data of surviving silver fir trees to the drought episodes of the 1970s and 1980s that caused forest dieback in Central Europe and found fifteen genes associated with the dendrophenotypes, including genes linked to photosynthesis and drought stress. Therefore, besides the so-called "avoidance strategy" of silver fir through bud cessation at the end of July and deep root system [75], there is a genetic basis of adaptation to drought.
The correlations between drought parameters and wood characters of silver fir provenances are positive. The most significant correlations have been obtained between radial growth and resilience. Our results are in accordance with findings from Eilmann et al. [56], while other studies have shown that drought-tolerant provenances were less productive [76]. Correlations between drought parameters and wood density were non-significant, indicating that wood density cannot be used as an indicator of drought sensitivity. Results can be explained by lower genetic variation of WD compared to RW among silver fir provenances at this age. Similar results for silver fir have been obtained by George at al. [57], while for other species like *Picea abies* and *Pseudotsuga menziesii*, correlations between wood density and trees sensitivity to drought have been found to be moderate to strong negative [55,77].
The wood characteristics varied, especially along the longitude, which represents an important gradient of increasing aridity eastward within Romania. In the Bucova trial, located in Banat Mountains with a warmer climate and an increasing deficit in rainfall, the best-performing provenances come from Eastern Carpathians.
The growth response functions revealed that the climatic variables of the trial sites were the significant drivers of the growth performance of the silver fir provenances. The main climatic variables explaining the radial growth of silver fir were MTVEG, MPOCT-MAR, and MPJAN-MAR, while for latewood percentage were MAT, MTVEG, MAP, and MTVEG. The negative correlations between RW and temperature during the growing season and positive correlations with precipitation suggest that warming and water deficit could have a negative impact on silver fir growth in climatic marginal sites, the more so because precipitation patterns are projected to change more than temperature in near future.
#### **5. Conclusions**
Even though silver fir experienced the most stressful droughts over the last two decades, it has revealed a plastic response to drought. Results revealed significant genetic variation among silver fir provenances by resistance, recovery, and resilience to drought. The provenance-specific response depended on the climatic conditions of the planting site and drought year. However, there are some local and foreign provenances that combine high radial growths and high drought tolerance.
Silvicultural practices and forest adaptive management should increase and maintain a high genetic diversity and resilience within forest stands. One of the adaptive measures could be selection, transfer, and planting of high-productive and drought resilient forest reproductive material in reforestation programs (assisted migration). Assisted migration may support adaptation process and help to conserve and increase genetic diversity, especially at the species distribution edges.
Finally, we argue that silver fir holds a great potential to thrive under warmer and drier conditions at the eastern limit of its distribution, in the southeastern Carpathians.
**Author Contributions:** Conceptualization: G.M.; methodology: G.M., A.M.A. and M.V.B.; software: G.M., A.M.A., E.S. and M.V.B.; validation: G.M., A.M.A. and M.V.B.; statistical analysis: G.M. and A.M.A.; resources: G.M. and M.V.B.; writing-original draft preparation: G.M.; writing-review and editing: G.M., A.M.A., E.S. and M.V.B.; project administration: G.M. and A.M.A.; funding acquisition: G.M. All authors have read and agreed to the published version of the manuscript.
**Funding:** This research was funded by Ministry of Research, Innovation and Digitization, grant number PN 19070303, Nucleu Program.
**Institutional Review Board Statement:** Not applicable.
**Informed Consent Statement:** Not applicable.
**Data Availability Statement:** Datasets generated and/or analyzed during the current study are available from the corresponding author on request.
**Acknowledgments:** We would like to thank the editor and anonymous reviewers for their useful advice that helped to improve the manuscript.
**Conflicts of Interest:** The authors declare no conflict of interest.
#### **Appendix A**
**Table A1.** List of silver fir provenances tested in comparative trials.
**Table A1.** *Cont.*
**Figure A1.** Variation of mean radial growth for period 1997–2018 of silver fir provenances in trial sites. 145
**Table A2.** Descriptive statistics of silver fir dendrochronology.
**Table A3.** Phenotypic correlations between the wood characters and geographic coordinates of the origin place of silver fir provenances.
The level of significance is represented as follows: \* *p* < 0.05; \*\* *p* < 0.01; \*\*\* *p* < 0.001.
#### **References**
## *Review* **Valuing Forest Ecosystem Services. Why Is an Integrative Approach Needed?**
**Gabriela Elena Baciu 1,2, Carmen Elena Dobrotă 3,4,\* and Ecaterina Nicoleta Apostol 2,\***
**Abstract:** Among the many types of terrestrial ecosystems, forests have some of the highest levels of biodiversity; they also have many interdependent economic, ecological and social functions and provide ecosystem services. They supply a range of tangible, marketable goods, as well as a variety of nonmarketable and intangible services derived from various forest functions. These translate into social, cultural, health and scientific benefits for people's quality of life. However, because they cannot be traded on a market, nonmarketable and intangible services are often perceived as free, inexhaustible and, as a result, underestimated. The human–nature interaction has affected both nature (via resource consumption) and society (via development of human welfare and well-being). Decision-makers, both public and private, often manage natural capital for multiple aims. In recent years it has been found that the single, individual approach estimating the value for these goods and services is not able to provide information that generates and supports decisions and policies in complex areas of current relevance such as the constant loss of biodiversity, climate change and global warming in close connection with the need for social development and ensuring an acceptable level of well-being for the greatest part of humanity. An integrated assessment with advanced techniques and methods using a pluralist framework of a heterogeneous set of values is considered a better approach to the valuation of such complex nature of the ecosystem goods and services. This assessment should take into account both costs and benefits trade-off issues among the multiple uses of ecosystem goods and/or services, especially the relationships between them and how they influence or determine the economic, social and cultural development of society. It should also consider the estimation of the complex inverse effect, from society to nature, whose goods and services can be diminished to exhaustion by the extensive and intensive anthropization of natural ecosystems with major impact on the number and quality of goods and services provided by ecosystems. Research has shown that applying an integrative assessment approach that utilizes tools developed by sustainability sciences could be an important component of future environmental policy making.
**Keywords:** biodiversity; forests; valuing ecosystem services; climate change; policy making
#### **1. Introduction**
The Earth's population relies on the benefits provided by ecosystems, including ecosystem provisioning, regulation and cultural and support services [1]. Over time, humans have transformed ecosystems to meet their needs and desires. Nowadays, climate change and biodiversity loss are major challenges for both developed and developing countries. According to the Intergovernmental Panel on Climate Change (IPCC, 2018), if global warming
**Citation:** Baciu, G.E.; Dobrot ˘a, C.E.; Apostol, E.N. Valuing Forest Ecosystem Services. Why Is an Integrative Approach Needed? *Forests* **2021**, *12*, 677. https://doi.org/ 10.3390/f12060677
Academic Editors: Alessandra De Marco, Pierre Sicard and Mihai A. Tanase
Received: 6 April 2021 Accepted: 19 May 2021 Published: 25 May 2021
**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.
**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).
exceeds 1.5 ◦C, climate change will severely affect humanity and ecosystems. An analysis of how people's use and management of natural resources affects ecosystem resilience is necessary because people's daily choices will result in continued biodiversity loss and new social costs [2]. Forests, which cover one-third of Earth's land surface, are an immense and renewable source of ecosystem services (ESs) [3,4]. They represent an extraordinary opportunity to mitigate climate change through carbon sequestration [5,6], soil stabilization and natural disaster mitigation [5]; forest conservation efforts (e.g., establishing protected areas) do not contradict territorial and regional development objectives [3] since changes in land cover and land use are among the major drivers of forest area reduction, biodiversity loss and land and ecosystem degradation at the global, regional and local levels [7–12]. In this respect, all these aspects should be kept together, to establish correlation among services and their impact on communities' development. In addition, the emergence of states of necessity (e.g., economic crises and social, political and military conflicts) could potentially intensify the use of resources and ESs offered by forests [13]. There are a number of less visible services provided by forests that support local development through cultural services [14] or sustainable tourism services [3]. Depending on the goals of the valuation of the ESs, some services should be seen and evaluated in a strict correlation and an integrative manner. Many studies have addressed how cultural services can be integrated into spatial planning methods; they showed that using spatial mapping and integrating information on habitat types, landscape features and land-use methods with information on existing infrastructure, number of visitors to the area and proximity to local communities during stakeholder consultations often led to increased stakeholder involvement in the planning process [15].
Recent research has analyzed ESs in relation to bioeconomic strategy objectives. This trend reflects how ESs and bioeconomy strategy, two key concepts in sustainability science, must be addressed together, especially given the effects of bioeconomy strategies on ESs [16,17]. Recent sustainable development initiatives have embraced the concept of a circular economy; this paradigm challenges the current linear behavioral model of take–do–consume–throw, which produces excessive waste and inefficiently uses natural resources [18]. The new EU Forest Strategy (2021–2027) emphasizes the need to ensure that the multifunctional potential of EU forests and their vital ESs are managed sustainably.
However, when discussing natural capital (NC), ecosystems and ESs, it is important to integrate concepts and methods that give a perceptible expression of their value. Depending on the final purpose of the analysis and evaluation, at least one of the following types of value can be assigned to NC and then calculated or estimated: philosophical value, economic value, social value, aesthetic value, inheritance value (for future generations), altruistic value [19,20], egoistic value [19], biospheric value [19,21] or intangible and cultural value [22]. Previous research has shown that everything is valuable but in different ways. Art objects often have sentimental value, historical value or financial value [23]. Landscapes, mountains and forests can have economic value and recreational value. In addition, great works of art, as well as natural landscapes, possess a distinct noninstrumental and nonutilitarian value, which is a central concern when works of art or landscapes are evaluated. Though some may think the value of art and landscapes comes from their beauty, others may not consider them beautiful. As such, beauty is a particular case of aesthetic value [23,24]. Aesthetic value is defined as the value possessed by an object, event or state of affairs by virtue of its ability to cause pleasure (positive value) or dissatisfaction (negative value) [23]. It is often seen as more subjective than other types of value and is usually of low priority in policy debates [24]. An example of this is the complex relationship between human aesthetic experience and the development of ethical attitudes towards the environment [25,26]. For ESs, their value often reflects contributions to human welfare and well-being, and a distinction can be made between use value derived from direct or indirect use of ESs and nonuse value derived from the intrinsic value of ecosystems and their biodiversity [27]. Currently, macro-indicators such as GDP report the values of goods and services exchanged in the market, but they do not reflect the values of nonmarket ESs, the deterioration of ESs or the loss of biodiversity. The inclusion of ES indicators in national accounts would allow for not only an economic assessment, but also an environmental and social assessment of a country's development [27,28]. Additionally, mapping ESs and establishing assessment indicators [29] is an important and current issue, with the EU Biodiversity Strategy explicitly calling for this action under Action 5 [26,30–32].
A holistic approach, using sustainability science methods and techniques developed for ES valuation, seems to be now the challenge for value pluralism of forest ecosystems, including well-known services and other indirect benefits such as health, education, equality and governance [17,32–34]. There are many complexities that have to be taken into account in order to value ESs. Their resources provide multifaceted benefits, and for some of them, it is difficult to quantify their value. Cost–benefit analysis allows the aggregation of the values of ESs on a single monetary scale of measurement [35]. However, public sector entities are deeply involved in such efforts. A plethora of multinational organizations are involved, including TEEB, WAVES (Wealth Accounting for the Value of Ecosystem Services, a World Bank program) and IGPBES. National governments are more involved in assessing ESs. The United Kingdom conducts an evaluation of national ecosystems that includes the assessment of several ESs. In the United States, all departments and agencies in the executive branch are now directed to "develop and institutionalize policies to promote the consideration of ecosystem services... and, where appropriate, monetary or non-monetary values for those services" [36] (p. 8/32). In Romania, the ES valuation process is at the beginning; up to now, several exploratory studies have been conducted related to the value of ESs in natural protected areas, and a case study on Piatra Craiului National Park has been conducted [37,38]. The studies revealed that even though the Piatra Craiului protected area generates significant ESs, very low economic values are mirrored in the earnings of the park administration. Thus, in-depth studies combining biodiversity aspects with economic evaluations of ESs will be a strong base for decision-makers for promoting sustainable development public policies in this area.
This paper aims to explore why an integrative approach for valuing and assessing forest ESs is needed, taking into account the many interdependent factors involving ESs and their associated values, as well as current challenges people face.
#### **2. Evaluation of Ecosystem Services—Why Is It Necessary?**
Natural resources associated with production (such as wood, food and energy resources), as well as services associated with protection (such as air quality), are assets that help increase the efficiency of services provided to people by NC [39]. The exploitation of NC produces social costs and benefits, referred to as externalities [39]. From an economic viewpoint, externalities occur when a variable (not the price) generated by an economic unit influences the production processes of other economic units or of the population. For example, the construction of a slaughterhouse could produce water, land or air pollution, all of which are negative externalities that affect other economic units and the local population. Due to the difficulty in measuring total benefits or already proven multiple benefits, decision-makers are often required to depend on cost-effectiveness analyses of different management options. More importantly, trade-offs of benefits and burden distribution happen between space, time and social groups, and in general, the perceived value of ecosystems has not been accounted for all of the services the ecosystems provide. One study assessed the monetary and nonmonetary values of forest ecosystems in eight Mediterranean countries and found that wood and wood fuel represent less than one-third of the total economic value (TEV) of forests in the countries under study. The other, nontimber services offered by the assessed ecosystems—recreational activities, fishing, protection provided by the river network and carbon sequestration—made up between 25 and 96% of the ecosystems' TEV.
Scientists have long reported the implications of biodiversity loss. In 1872, Yellowstone National Park became the first geographic area defined as a protected area due to the initiative of several scientists [40]. The economic view that people's survival depends on natural resources, which are limited, has been held since the 18th century (Malthus 1888); the concept of ESs, or services offered by nature to people, was developed in the 1960s and 1970s [39–42]. Many natural processes improve human well-being [43] and welfare, but human activity negatively affects ecosystems through ecosystem conversion, habitat fragmentation, landscape alteration and the anthropization of the natural environment over time [26] and biodiversity loss, which ultimately harms human well-being [31,32,43].
Globally, the importance of protecting and sustainably managing forest ESs has been recognized through a series of UN-adopted documents. These include the 'Rio Forest Principles' from the 1992 United Nations Conference on Environment and Development [44]; the United Nations Framework Convention on Climate Change (UNFCCC) [45], which emphasizes the importance of forests in terms of the global greenhouse gas (GHG) balance; the Convention on Biological Diversity [46,47], which addresses forest biodiversity; the United Nations Forum on Forests (UNFF); the UN Convention to Combat Desertification (UNCCD) [36]; and the Paris Agreement [48], which calls for major reforms in order to fight global warming.
However, in recent years, ESs emerged as an important issue on the public agenda through discussions on topics such as biodiversity loss [2,40,41], land-use and spatial planning [7,9,10], climate change [42], circular economies [49,50] and bioeconomies [16,51] and public policies [36,52–54] and strategies [16]. To address all of these challenges requires sound decision-making [55]; the development of a tool for measuring TEV is necessary to support the political decision-making process and to inform both citizens and businesses about the benefits and costs inherent in projects, programs and policies [56]. There is a growing consensus that in spatial planning, land management and other decision-making contexts, the economic valuation of ESs is essential for the development of efficient public policies and strategies [57]. The value of ESs and biodiversity is assigned based on what societies are ready to offer in exchange for nature conservation [25,58] because the valuation of ESs can vary with time and spaces [59], ranging from simply raising awareness to analyzing various policy choices and scenarios in detail [60]. The estimated loss of ESs from 1997 to 2011 due to land-use change is \$4.3–20.2 trillion per year [19].
#### **3. Ecosystem Services and Natural, Socioeconomic and Public Policy Challenges**
In recent decades, the concept of nature and ESs as capital has gained visibility [61], as society can receive important goods and services, such as clean air and water, flood control and crop pollination, by conserving and restoring natural habitats [56]. These goods and services, if properly considered, may be valuable enough to justify the protection of forest ESs [62]. Public debates on ESs have hit a sensitive chord. For some, the concept of ESs presents an opportunity to include all of the environmental benefits that the market failed to account for in public and private decision-making. For others, the possibility of structuring payments for ESs that assign and respect property rights and bring the power of the market to a bearable level may seem just as attractive [36].
Addressing climate change requires mitigation and action to adapt to new conditions. Forests and the forestry sector play a significant role in mitigating climate change by capturing CO2 and producing timber products, as well as by substituting materials whose processing requires high energy consumption [63–65]. They also provide services that can help people adapt to both current and future climate risks [42]. While ESs are part of the solution to climate change, they are also affected by climate change. Climate change will impact forests and may impair their ability to provide essential ecosystem services in the decades to come. Addressing this challenge requires adjustments to forest management strategies as of now, but it is still unclear to what extent this is already in progress [66]. An EFI study found that forests and the role of the forestry sector could be significantly enhanced through Climate-Smart Forestry [63]. This approach aims to increase the climate benefits of forests and the forestry sector in a way that creates synergies with other forestrelated needs. It is based on three pillars: (1) reducing or eliminating GHG emissions to mitigate climate change, (2) adapting forest management to build resilient forests and
(3) actively managing forests in order to sustainably increase productivity and provide all of the benefits that forests can offer [62–64,67]. The European Environmental Bureau, an international nonprofit association that has assembled over 160 civil society organizations from more than 35 European countries, stated in 2021 that "the global material footprint is already beyond ecological limits, being over 100 billion tonnes per year and, if we continue 'business as usual', is expected to double in the next 40 years. The impact of excessive consumption is significant. In the European Green Deal, the European Commission states that 'resource extraction and processing account for more than 90% of the global impact on biodiversity loss and water quality and about half of global climate change emissions'" [50].
In this context, sustainable development has become a global concept that transcends different sciences with environmental, social, cultural and economic dimensions. A bioeconomy is currently being promoted both for policymakers and businesses as a sustainable action plan for reconciling environmental, social and economic goals [16,68,69]. Human activity has led to the degradation of the natural environment, which has had a far-reaching impact on society and the economy and has created new conceptual frameworks for how people interact with and depend on the environment. A bioeconomy generally involves replacing fossil fuels with bio-based ones, so three main goals—involving resources, biotechnology and agroecology—are becoming more prevalent in the scientific literature [16]. In 2020, a review of 45 documents and articles showed that, although the publications were diverse and the approaches used were still quite new, eight topics were predominant: (a) the technical and economic feasibility of biomass extraction and use; (b) the potential and challenges of a bioeconomy; (c) frames and tools; (d) the sustainability of biologybased processes, products and services; (e) the ecological sustainability of a bioeconomy; (f) the governance of a bioeconomy; (g) biosecurity; and (h) bioremediation [16]. Though both the bioeconomy and NC combine economics and natural sciences and propose new interdisciplinary frameworks for environmental sustainability, the two concepts are rarely applied together [51]. A circular economy would positively impact ecological systems by not exhausting or overburdening them with technological and productive tasks. This is reflected in the environmental benefits of the circular economy. For example, a circular economy would emit less GHGs; the soil, air and water would remain unchanged; and natural reservations would be preserved [18,53]. Forest ecosystems provide services and products such as wood, pollination and clean drinking water. In a linear economy, these services will eventually be depleted by the constant extraction of products from ecosystems or will be affected by the release of toxins from technological processes [53,69]. If the products extracted from an ecosystem are used in a rational and intelligent technological and economic cycle, and the technological processes do not discharge toxic substances into the environment, then the soil, air and water will remain resistant and productive [52,69,70]. Understanding ESs and their economic applications offers a number of environmental and economic advantages because assessing NC and ES flows provides a powerful economic engine for nature conservation and nature-based solutions to current economic challenges, processes and industrial systems [49].
#### **4. Ecosystem Valuation: Utilitarian vs. Nonutilitarian Approaches**
The importance of ESs for human society has multiple dimensions: ecological, sociocultural and economic [71]. Over time, concerns related to ES valuation have led to the development of various methods for conducting these assessments, from mapping and modeling supply and demand for ESs to determine their market value (utilitarian approach) to social and environmental assessment techniques to assess their nonmarket value (nonutilitarian approach).
#### *Utilitarian Approach*
The utilitarian approach is intrinsically linked with cost–benefit analysis and welfare economics since they approach human well-being in terms of individual satisfaction based on the individual utility of goods and services. At the same time, environmental psychology research confirms that the relevance of ESs for human well-being is more than the satisfaction of individual needs and consists of physical and psychological health, social integration and cultural identity (ACB). While market valuation is relatively simple to perform, challenges arise when estimating the nonmarket value of an ecosystem. From the seminal classification of Krutilla (1967), the utilitarian approach divides the TEV of ESs into two types of value: the use value, which relates to ESs associated with production and protection functions for which market prices usually exist, and the nonuse value, which reflects the satisfaction of knowing that biodiversity and ESs are preserved and that future generations will also benefit from them [58]. Both of these categories have subsequently been disaggregated into multiple components. Use value was broken up into direct use, indirect use, optional, quasi-optional and bequest values; nonuse value was split into existence or intrinsic, aesthetic, altruist, bequest, moral and religious values [21,40,58,59,72,73]. Direct use value is associated with the benefits of using ESs, such as raw materials. Indirect use value is associated with regulating services like water quality regulation. The optional and quasi-optional values are the values of ESs based on the option to use the services at a certain time in the future. Of the nonuse values, existence or intrinsic value is usually presented as the value attributed by an individual to the continued existence of a service or good, regardless of its current or possible uses [58]. Both use and nonuse values are associated with the utilitarian approach, which primarily aims to express the associated values of ESs in monetary terms and takes into account the utility of NC for humans and for the socioeconomic system [71]. This includes ecosystem resources that can be used or are used by the population and by economic units in their daily activities.
In a neoclassical economy, on which environmental economics and assessment methods are based, the nonuse values are defined and measured in monetary units based on a willingness to pay (WTP) or a willingness to accept (WTA) [19,39,58]. Nonuse values such as WTP are estimated by methods of preference declared in questionnaires or interviews, including both the contingent assessment method (CVM) and direct choice experiments (DCEs) [39]. Two assessment approaches are commonly used to estimate nonuse values. The first approach asks how many respondents would be willing to pay for ESs (or their attributes in the case of DCE) if they were absolutely certain they would never use them. In this case, the interviews would be based on nonusers. The second approach asks respondents, including users, to divide the total WTP for ESs into different categories, such as inheritance, existence and own use. Such statement decomposition approaches have been applied in many CVM-related ES applications and have been useful in understanding the relative quotas of value categories in WTP estimates [39,74] or in identifying warm glow effect in willingness to pay (WTP) responses [75]. In most cases, the proportions of nonuse values in WTP are considered to be quite substantial, representing between 40 and 90% of the total WTP [39,74]. Despite its popularity, the approach to decomposition stated in interviews has substantial shortcomings and is highly controversial, mainly due to the cognitive difficulty of addressing the components of an unfamiliar and inseparable value. An individual's total WTP for an ES is usually a consequence of different overlapping and correlated motivations that may be inseparable and, as such, inaccessible to the researcher [76]. In most cases, the ES assessment is completed when a choice must be made among different services.
Over time, the desire to conduct a comprehensive economic assessment of ESs has led to the identification and refinement of various measurement methods. The first significant economic assessment of ESs, including from a nonmonetary perspective, was made by Costanza in 1997 based on the fact that ecosystems provide benefits to populations through ecosystem functions and components (i.e., services). Ecosystems are unique and irreplaceable, which makes them invaluable. Based on this, the author grouped ESs into categories and calculated their unit values, using assessment techniques based mainly on people's WTP. The resulting values were then multiplied by the area occupied by all US ecosystems and totalled \$33 trillion per year, more than double the annual GDP, which was estimated at \$16 trillion [20]. Fourteen years later, the value of ESs globally was estimated
at \$18 billion per year, of which 19% came from ES climate regulation and 4% came from raw materials related to productive functions. ES contributions to recreation, protection against extreme phenomena, the water supply, erosion control, nutrient cycling, habitat, genetic resources and nonwood products represent the rest of the value [20].
Costanza's work can be considered pioneering. From other perspectives, however, the proposed methodology was both technically and ethically challenged because ecosystems, as a support for life, are constantly evolving and cannot be measured monetarily. There is skepticism about the association of ecology with the economy; many specialists consider a strong involvement in the economic sector for the conservation of ecosystems dangerous, which could lead to an increase in nature depreciation. For example, developing countries could request and receive financial compensation in accordance with the estimated value of the ESs they provide, as long as they preserve them. Costanza's approach produced much debate and criticism, but it is better to have debate and criticism among scientists, policy-makers and stakeholders than to have nothing. However, despite the interest in making monetary assessments of ESs, these are not the only possible value assessments. In 2010, TEEB, published by The Ecological and Economic Foundations, developed the concept of TEV and presented a classification of TEV components and assessment tools that can be used to assess various components of ESs. The authors hypothesized that the value of ESs and biodiversity is determined by what a society is willing to offer in exchange for nature conservation. Society and policy-makers need to understand that ecosystems are unique and limited resources and that depreciation or degradation involves costs to society. From an economic point of view, when a resource is limited, an opportunity cost exists, representing the value of the best of the sacrificed chances (i.e., the one that is given up when a choice is made). However, the difficulty of conducting a monetary assessment of ESs is due to the fact that the changes to ecosystems are irreversible or are reversible for a prohibitive cost. The estimated economic value is a cumulation of choices of the buyer, which includes a multitude of preferences for ecology, society, health, technology and expectations regarding the future [58]. The modification of any of the factors listed influences the estimated economic value [58,77] and could lead to different scenarios being planned [77].
The evaluation methods identified in the VET methodology fall into three categories: (a) direct market valuation approaches, such as the price-based method, cost-based method and production function-based method; (b) revealed preference approaches, including the travel cost method and hedonic pricing method; and (c) simulated valuation, such as the contingent valuation method, choice modeling and group valuation.
Price-based methods are most often used to calculate the value of provided goods and services. Because they are traded on the market, their value is relatively easy to calculate. Examples include the value of wood, honey or tourist services [58]. Cost-based methods [39] are based on several identified techniques, such as the avoided costs method, which assesses the costs that would have occurred in the absence of the ES. The replacement cost method estimates the costs of replacing ESs with artificial technologies, the restoration cost method assesses the costs of counteracting the effects of ecosystem loss or restoration and the production function-based method estimates how much of the nonmarket ESs contribute to other services or goods traded on the market, noting how much the services contribute to increasing the productivity or price of those goods or services.
The travel cost method is relevant mainly for determining the value of recreational services associated with biodiversity and ESs. The method is based on the principle that recreational experiences can be associated with a cost that consists of direct costs and opportunity costs. In the case of tourism, changing ecosystem biodiversity can influence the demand to visit that location. The hedonic pricing method is based on the added value that a landscape, or location near an ecosystem, can bring to a market, such as the real estate market. Changing the biodiversity of an ecosystem can change the market value of a property. The revealed preference approaches require a large amount of complex data and statistics and so are expensive and time-consuming. In addition, since these methods are based on direct observation of clients, they can provide an image at a certain moment in time [58].
The contingent valuation method uses questionnaires through which respondents provide information on how they would be willing to pay to protect ESs and how much they would be willing to pay to accept ecosystem loss or degradation. The choice modeling method focuses on modeling human behavior in particular contexts; this method starts with the supposition that people must choose from two or more alternatives when making a decision, one of which is the price in money. The group valuation method combines the use of questionnaires with elements of the deliberative process from political science and is becoming a widespread method for collecting values such as the uniqueness of ecosystems and social justice, as well as altruism towards other people and towards future generations compared to the species that live in the ecosystem. These methods should be applied carefully, and their limitations should be considered, especially when evaluating the nonuse value of a service that does not have a corresponding price on the market [36,54].
Extensive research conducted in Europe through the study Operationalisation of Natural Capital and Ecosystem Services Integrated (OpenNESS) [78] classified the methods used for evaluating ESs into the following categories: (i) biophysical methods, which are used for mapping ESs and include matrix approaches, ecosystem modelling with InVEST (Integrated Valuation of Ecosystem Services and Tradeoffs [79], E-Tree [80] or ESTIMAP [81,82]; (ii) integrated mapping-modelling approaches; (iii) land-use scoring [83]; (iv) participatory mapping; (v) sociocultural methods for understanding social preferences or values for ESs, such as deliberative assessment methods, preference prioritization methods, multicriteria analysis methods and photo-elicitation surveys; (vi) monetary methods for estimating the economic value of services, such as preference methods, revealed preference methods and travel cost methods [58] or hedonic pricing methods [58,84]; and (vii) integrative approaches [85]. The selection of a particular method for a specific case can depend on many factors, including the decision-making context; the strengths and limitations of each method; and pragmatic reasons such as available data, resources and expertise. Each method has specific features that inform its relevance or appropriateness to certain decisions or problems in the context of the study. The ability of a method to address a specific purpose may be the primary factor influencing method selection. Most methods are able to characterize the current state of ecosystem service demand or supply, but only a few are able to explore potential future service provision through modeling approaches and participatory scenario development (which was specifically designed to address this purpose). Some methods focus on specific ESs, such as biophysical models of soil erosion, or specific groups of services, such as photo-series analyses of cultural ESs. Other methods attempt to provide a more holistic or strategic overview of multiple ESs, which may be used to assess trade-offs [86] between the supply of different services (e.g., matrix-based approaches) or the demand for services by different stakeholders (e.g., PGIS, preference assessment methods, photo-elicitation or MCDA). The integration of ES assessment with life cycle assessment (LCA) is important for developing decision support tools for environmental sustainability. LCA methods have traditionally been employed as environmental management tools to assess the environmental impacts of production processes from 'cradle to grave' [87]. The method was developed in the 1960s in reaction to the 'Limits to Growth' discourse, which raised concerns about natural resource finiteness. The assessments were initially limited to energy efficiency and emissions and were information for internal use by companies.
After the 1980s, academia and governments began using LCA as well; methodological development progressed and was supported by formal attempts at international standardization [88]. LCA has since become a reference tool for the assessment of sustainability issues in the context of production–consumption systems, obviously bearing both strengths and weaknesses [89,90]. Despite emerging interest in the topic, additional work is needed for tackling the integration of ES issues in LCA approaches [91].
The nonutilitarian approach identifies four types of value: ecological value, sociocultural value, value with direct economic significance and intrinsic value [40,92].
The ecological value is determined by the integrity of the regulation and habitat functions of the ecosystem and by various ecosystem parameters such as complexity, diversity and scarcity (de Groot). The most appropriate methods to evaluate the ecological value are the biophysical methods mentioned above as well as integrated mapping–modeling approaches and land-use scoring [92]. Sociocultural value is mainly related to aspects such as physical and mental health, education, cultural diversity and identity (heritage value), freedom and spiritual values. The most used methods to evaluate it are participatory mapping and sociocultural methods described above [92].
As regards the economic value, the monetary methods, such as direct methods of valuation based on market prices or indirect valuation methods (e.g WTP, WTA, Replacement cost, travel cost, Hedonic pricing), are the most commonly identified. [92].
For determining intrinsic value, the most adequate methods could be preference prioritization methods, multicriteria analysis methods and photo-elicitation surveys, combined with biophysical methods such as ecological models.
In conclusion, the utilitarian approach is in line with the philosophy of environmental economists who are in favor of extension of monetary valuation methods to nonmarket ESs, while the nonutilitarian approach is aligned with the concepts of ecological economists who consider the substitutability and valuation of NC controversial. Boundaries between utilitarian and nonutilitarian approaches (Figure 1) are blurred, and they benefit from an abundant and expanding body of literature [2].
**Figure 1.** Utilitarian and nonutilitarian frameworks for valuing ESs. (adapted after TEEB).
The nonutilitarian approach is recognized as an important component of the ES valuation and an important motivation for increasing conservation efforts, but using monetary units to raise awareness of policymakers about their importance is a powerful tool [71].
#### **5. Cost–Benefit Analysis of ES**
Data on each ecosystem and each service highlighted the need to preserve ecosystems to ensure sustainable development. Even if ecosystems are subject to intensive and extensive exploitation, people must take care of them to ensure continuity. Therefore, in response to the exploitation of resources, plans must be made to conserve ecosystems. An environmental cost–benefit analysis (CBA) is best suited for this purpose [62]. First, because a CBA presents the territorial distribution of benefits and costs and compares this distribution with the distribution of biodiversity, it allows for the identification of important areas for both people and biodiversity (win–win areas), as well as the identification of areas of potential conflict and areas in need of compromises (negotiations). In these areas, the net economic benefits of ecosystem conservation are low, but biodiversity values are high, or vice versa. Second, a CBA highlights which areas have the highest unit cost benefits, thus indicating the most effective places for conservation efforts. Third, maps with ESs could help identify providers and consumers of ESs, enabling the identification of efficient and equitable payment mechanisms for financing conservation projects [62,93]. The core activity in an environmental CBA is estimating monetary values of the environment, especially the economic value of nonmarketable goods and services; the objective of the analysis is to estimate the TEV that arises from a policy proposal [94]. In 1970, CBAs were introduced for use on publicly financed projects with an environmental impact in the US. Since then, CBAs have been continuously adapted and applied to different methods and techniques, such as stated preference methods (which include the contingent valuation method, WTP, WTA, choice experiments, deliberative group valuation and health risk valuation) and revealed preference methods (which include the travel cost and hedonic price methods) [35]. At the same time, an important aspect that has to be taken into consideration when performing CBA is spatiotemporal frames, meaning that ESs are generated at different scales from short-term site level to long-term global level, and any slight change in the spatial or temporal frame approached in CBA can generate different consequences and stakeholders included in the CBA.
#### **6. Ecosystem Service Valuation—What Is Next?**
ES approaches and assessment efforts have changed the discourse on issues such as nature conservation, natural resource management and other areas of public policy. It is now accepted that in order to create a win–win situation rather than a compromise between environment and development, strategies for natural resource management and conservation through investment in the conservation, restoration [68] and sustainable use of ecosystems should be based on a combination of all values that occur when estimating the TEV [3,56,95,96] (Figure 2). Nonmarket assessments and methods used for cultural and environmental services have been criticized for their inability to provide values that represent or substantiate the total value of an ecosystem, but economists' efforts to involve interdisciplinary teams and incorporate a variety of methods and information into their research have demonstrated their flexibility, which reinforces the idea that they are effective in the process of diluting public policy decisions [13,57,76,97]. At the same time, actions have to be based on evidence, data and analysis to assess how public policies are beneficial for both people and nature [98], and the valuation methods have to be adapted to the local conditions and stakeholders involved [99]. The moving from conceptual frameworks and theory to practical integration of ESs into credible, replicable, scalable and sustainable public policies will require radical transformations [100] towards systematical integration of the ESs in decision-making at the individual, corporate or governmental level [101].
The ways in which ESs can be included in national accounts have generated a great deal of debate because it is, to some extent, a matter of choice [102,103]. In 2002, the UN's System of Environmental–Economic Accounting—Experimental Ecosystem Accounting (SEEA EEA) showed that the concept of valuation has made a significant difference in attempts to incorporate the generated ES values into national accounts [104,105]. Accounting ESs supposedly quantifies the amount of ESs provided by an ecosystem to socioeconomic
systems [25,106]. This can highlight ES contributions to the economy, social well-being, jobs and livelihoods.
**Figure 2.** Integrative approach of valuing ESs.
SEEA methodology gave rise to the concept of the information pyramid (Figure 3), which combines basic economic, ecological and sociodemographic data. These data can be collected, centralized, processed and used for the development of analyses and studies that provide evidence for public policies and lead to the development of aggregate key indicators at the macro level.
**Figure 3.** Information pyramid for ES key indicators (from SEEA).
However, creating such key indicators is a challenge. Using exchange value methods based on market techniques to quantify ESs [107] is easier because these are already compatible with the Systems on National Accounts; well-being value-based methods are difficult to translate into exchange value terms [106,108]. This shows that more effort should be put into the development of a pluralistic value-based approach able to capture both monetary and nonmonetary values [105].
In addition, the development of experimental ES accounts revealed the need to develop different indicators for separate ESs since each service has different characteristics. For forest ecosystems, the main indicators are related to timber production, biomass harvesting for energy, wild food provision, climate regulation, fire management, air quality regulation, noise reduction, water purification and recreational and aesthetic values. The accounts developed at the EU level [109] face many challenges, such as a lack of data and a lack of availability at the required spatial resolution [106], because natural, historical and cultural resources do not have an explicit monetary value. A different conclusion is reached if the cost of living with regard to the maintenance of nature in acceptable conditions is compared to conditions in which nature is allowed to degrade [20], showing that the single-value approaches are not an option anymore [110].
#### **7. Conclusions**
NC produces multiple ecosystem services with differences in values in human life and measurement requirements. The values vary between time and space. Valuation of an individual service or by a single method may result in the overestimation of values of some of them. At the same time, the exploitation of NC generates costs that translate into negative externalities or trade-offs for the environment and for society.
In real life, people do compromise between them. Policy and management decisionmaking requires information of different dimensions. Information from integrated valuation methods would provide information from different aspects and help policymakers to make informed and pragmatic decisions.
ES valuation does not aim to establish prices in order to capitalize on ESs through the market. Instead, it highlights how ESs contribute to human well-being and welfare and how they are an essential tool for developing efficient public policies and strategies based on scientific evidence. Utilitarian and nonutilitarian approaches to NC have developed multiple methods and techniques for assessing different types of value for ecosystems. However, there is still a significant lack of reliable evidence on nonuse values of ESs. Many approaches to ES assessment remain controversial because they raise concerns related to the availability and accuracy of data. Establishing accurate methods for calculating VET of ESs, as well as indicators and methods for their modeling and calculation, is a topic that requires further research. Using a pluralist framework composed of a set of decision-making instruments adapted to specific spatial and temporal scales involved, in which CBA is an important component, will allow identifying win–win areas and areas of potential conflicts, both for people and for the environment. Such techniques may be the best solution for supporting the public policy measures needed to mitigate current challenges. In recent years, there has been an increased focus on how climate change affects ecosystems, as well as on how ESs connect to sustainability science topics like environmental economies, bioeconomies and circular economies. Further research that utilizes an integrative approach to connect ES valuation to sustainability science is needed in order to support the decision-making process and public policies.
**Author Contributions:** Conceptualization, G.E.B., C.E.D. and E.N.A.; methodology, G.E.B., software, G.E.B.; validation, G.E.B.; formal analysis, G.E.B.; investigation, G.E.B.; resources, G.E.B.; data curation, G.E.B. and E.N.A.; writing—original draft preparation, G.E.B.; writing—review and editing, G.E.B., C.E.D. and E.N.A.; visualization, C.E.D.; supervision, E.N.A.; project administration, G.E.B.; funding acquisition, G.E.B. and E.N.A. All authors have read and agreed to the published version of the manuscript.
**Funding:** This study was funded by the Romanian Ministry of Research and Innovation, within the Nucleu National Programme, the Project PN-19070109, Contract No. 12N/2019.
**Acknowledgments:** We would like to thank to Bogdan Apostol, the project coordinator, for his support in preparing this paper.
**Conflicts of Interest:** The authors declare no conflict of interest.
#### **References**
## *Article* **Applications of TLS and ALS in Evaluating Forest Ecosystem Services: A Southern Carpathians Case Study**
**Alexandru Claudiu Dobre 1,2, Ionut,-Silviu Pascu 1,2,\*, S, tefan Leca 2, Juan Garcia-Duro 2, Carmen-Elena Dobrota 3,4, Gheorghe Marian Tudoran <sup>1</sup> and Ovidiu Badea 1,2**
**Abstract:** Forests play an important role in biodiversity conservation, being one of the main providers of ecosystem services, according to the Economics of Ecosystems and Biodiversity. The functions and ecosystem services provided by forests are various concerning the natural capital and the socioeconomic systems. Past decades of remote-sensing advances make it possible to address a large set of variables, including both biophysical parameters and ecological indicators, that characterize forest ecosystems and their capacity to supply services. This research aims to identify and implement existing methods that can be used for evaluating ecosystem services by employing airborne and terrestrial stationary laser scanning on plots from the Southern Carpathian mountains. Moreover, this paper discusses the adaptation of field-based approaches for evaluating ecological indicators to automated processing techniques based on airborne and terrestrial stationary laser scanning (ALS and TLS). Forest ecosystem functions, such as provisioning, regulation, and support, and the overall forest condition were assessed through the measurement and analysis of stand-based biomass characteristics (e.g., trees' heights, wood volume), horizontal structure indices (e.g., canopy cover), and recruitment-mortality processes as well as overall health status assessment (e.g., dead trees identification, deadwood volume). The paper, through the implementation of the above-mentioned analyses, facilitates the development of a complex multi-source monitoring approach as a potential solution for assessing ecosystem services provided by the forest, as well as a basis for further monetization approaches.
**Keywords:** ecosystem services; natural capital; socio-economic system; ecological indicators; terrestrial laser scanning; aerial laser scanning
#### **1. Introduction**
Forest is playing a crucial role in biological diversity, local welfare, the balance of carbon emissions, and the global economy [1–3]. In the context of climate change, the understanding of forest ecosystem processes' importance is essential in assuring sustainable management and economic development [4]. Toward this purpose, forest monitoring was established as the main tool for studying the dynamics of forest structure and functioning and its response to anthropogenic influences [3,5]. The necessity of this tool is highlighted by decisional factors' requirements and forest governance [6]. Due to the high
**Citation:** Dobre, A.C.; Pascu, I.-S.; Leca, S, .; Garcia-Duro, J.; Dobrota, C.-E.; Tudoran, G.M.; Badea, O. Applications of TLS and ALS in Evaluating Forest Ecosystem Services: A Southern Carpathians Case Study. *Forests* **2021**, *12*, 1269. https://doi.org/10.3390/f12091269
Academic Editor: Jarosław Socha
Received: 20 August 2021 Accepted: 13 September 2021 Published: 17 September 2021
**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.
**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).
complexity of the forest dynamics, a high amount of warranted information is needed in the characterization process.
The primary mechanism of forest monitoring in assuring the data integration is developing forest inventories focused on parameters related to the main dendrometric characteristics of trees (e.g., diameter at breast height (DBH), height-DBH ratio, crown width). Besides these variables, the monitoring also has to take into consideration information regarding the climate (temperature and precipitations) and pollution (atmospheric depositions). However, it is a well-known fact that the traditional forest inventory can be expensive, time-consuming, and requires a large amount of qualified personnel [3]. Moreover, forest inventory is limited to statistically established sample plots, resulting in a weaker representativity at larger scales [7–9].
To overcome the mentioned limitations, alternative solutions and measuring methodologies were sought in the remote-sensing field. In the past decades, remote-sensing systems have evolved, ensuring a large variety of applications [10]. As expected, the remote-sensing portfolio already contains several techniques addressing forest ecology and management [11]. From their beginning, remote-sensing systems were mostly equivalated to satellite imagery. New instruments of interest here, airborne laser scanning, unmanned aerial vehicles, digital photography systems, and terrestrial laser scanners, have more recently captured the researchers' attention, gradually gaining visibility through a large number of scientific studies.
Land cover analysis [12,13], biomass estimation [14–17], hazard identification [18–20], structure assessment [21–26], and ecological indicators are just some of the most frequent applications of remote sensing in forestry. The major advantages of remote sensing are related to its capability of capturing a large amount of data and the possibility of revisiting in relatively short periods, as well as the plurality of the associated analyses [24].
A keen interest in remote sensing was shown toward biophysical parameters, such as DBH, tree height, volume, and implicitly biomass. The majority of these parameters were initially computed employing regression models, with input data derived from crown projections and height measurements from passive sensors [27–29], calibrated with ground samples. New technologies, as is the case of terrestrial laser scanning, propose different approaches for estimating tree characteristics. These provide a more direct method that involves point cloud classification, tree segmentation, and stem reconstruction [30–32]. Besides the biophysical parameters, active remote-sensing systems are used to describe stands' structure through indirect analyses of the number of trees, canopy stratification, and trees distribution. As described in the work of [24,33,34], airborne and terrestrial laser scanning represent optimal solutions in describing forest stands through structural indicators based on point cloud processing.
Regarding this matter, the literature offers a rich variety of active remote-sensing-based forest variables, from foliage indices [24,35,36] (leaf area index—LAI, gap probability pgap) to trees spatial distribution [37,38] (mostly distance and angles between trees, but also the position itself for marginal trees detection, sampling plot edge effect mitigation, etc.). Satellite imagery also proposes indicators related to the status of forest stand health [39–42], an aspect that will not be detailed here since passive remote sensing does not make the subject of our study.
Disregarding the plethora of variables and its promising evolution, passive remotesensing technology still demands innovative approaches to address the requirements of ecological relevant indicators [11]. The constant need for ground measurement calibration represents the main disadvantage of most passive remote-sensing systems. Furthermore, the applications based on regression models can lead to important errors due to potentially incorrect assumptions regarding the relationship between forest characteristics [43,44].
In the ecological research field, active remote-sensing data are increasingly being used. Quantifying forest ecosystems information from indices based on active remote-sensing highlights the need for further analysis and adaptation. The processing and uptake of these data are necessary for linking the indicators to the capacity of forest ecosystems to provide
benefits. These benefits materialize in what we call ecosystem services and represent the ecosystems' benefits, processes, and assets for providing human well-being [45].
In the field of research, the relationship between ecology and economy has been attributed with great importance, a fact that is corroborated by the very nature of ecosystem services. This has made it possible to develop the concept of natural capital on an environmental basis [46] and led to the idea of value, from a monetary point of view, of the ecosystem services and goods [47]. The need to exploit the benefits of ecosystems derives from their contribution to the human economy [48,49] and their expression in services and commercial goods [50,51].
Nowadays, there is a multitude of methods for evaluating and monetizing services, most of them being subjective. The methods are based on human preferences or physical costs upon which ecosystem services can be integrated [46]. The established methods are based on damaged cost avoided, replacement cost, market price, productivity cost, hedonic pricing, benefits transfer, and contingent evaluation method [51–53].
Despite the difficulties encountered in the process of applying ecosystem evaluation methods, they have an essential role in communicating the value of nature to the decisional factors and policymakers [54]. In this regard, there is an absolute need for objective ecological indicators that can provide information about ecosystem health status and structure.
This paper intends to identify and test several methods and variables applicable to airborne and terrestrial stationary laser scanning to quantify the capacity of the forest ecosystem in providing benefits. The identification of suitable ecosystem services will be performed according to The Economics of Ecosystem and Biodiversity (TEEB) classification [51,55]. Alongside, Millennium Ecosystem Assessment classification (MEA) and Common International Classification of Ecosystem Services (CICES), TEEB represents one of the widely known ecosystem services classification networks. The latter is a global campaign aiming at raising awareness regarding biodiversity's economic benefits and the rising costs of ecosystem degradation. The final purpose of this initiative is to analyze and explain in a mainstream approach the importance of taking action [56]. This classification was adopted because it corresponds faithfully to the functions attributed to the studied stands according to the Romanian forest legislation. The majority of ecosystem functions will be analyzed in relation to the existing indicators, as well as other variables adapted to active remote-sensing sampling. The paper does not intend to calibrate or to validate existing methodologies but to showcase a minimal set of indicators computed through active remote-sensing methods that can offer sufficient information about the ecosystems' capacity to provide services. Furthermore, as mentioned above, the paper aims only at information obtained through the use of ALS and stationary TLS measurements, excluding any other potential data based on satellite imagery or other passive remote-sensing technologies.
#### **2. Materials and Methods**
#### *2.1. Study Site*
To analyze the identified methods and variables, ten stands were considered in the current study, each of them being designed as a one-hectare rectangular plot with three 15 m-radius circular subplots within them.
The ten one-hectare plots are located in two different areas of the Southern Carpathian mountains, thus covering three of the most representative tree species of Romania. These are sessile oak (*Quercus petraea*) and beech (*Fagus sylvatica*) in the hill region and Norway spruce (*Picea abies*) in the mountainous region (Figure 1).
**Figure 1.** (**a**) Location of the sampled forest stands, (**b**) detailed position (coordinates in WGS 84 projection system).
Both deciduous and coniferous forest plots were considered in the process of assessing the applicability of the studied methods as well as in the evaluation of the different structural characteristics of the plots. Therefore, the plots were chosen in relation to species, age, and applied silvicultural interventions (Table 1).
**Table 1.** Sample plots characteristics.
#### *2.2. Conventional Field Data Collection*
In order to ensure control over the LiDAR data sets, a classical inventory was also carried out in the plots. Field measurements included DBH, tree height, crown height, crown width, and position of each tree (XYZ coordinates) and targeted all the trees with a DBH equal to or greater than 6 cm. To acquire these variables, an integrated GIS field software and electronic mapping and dendrometrics sensors [57] for recording tree positions and canopy characteristics were used.
#### *2.3. Terrestrial Laser Scanner Data*
In each 15 m circular subplot, five terrestrial scans were performed accordingly to a cardinal point sampling scheme to compensate for the shadowing (Figure 2a). The scanning process was achieved with a phase shift terrestrial laser scanner [58]. The resulting point clouds were characterized by 8 μs per scan point and over 44 million points per 360◦ sweep.
(**a**)
**Figure 2.** *Cont*.
**Figure 2.** (**a**) TLS ground-based data collection (**b**) terrestrial laser scan of the subplots.
Regarding the TLS pre-processing methods for classification and segmentation of the point cloud prior to obtaining the stems and the foliage, an approach proposed by Pascu et al. was followed [24,30,32,59] (Figure 2b).
#### *2.4. Airborne Laser Scanner Data*
The airborne LiDAR data for the one-hectare plots were collected through the use of a full-wave airborne laser scanner [60]. The discrete points extraction was conducted by the provider of the data sets, according to the standard processing procedure. Following processing, an average point density of 6 points/m2 was reached (Figure 3).
Further analyses, such as the ground-non-ground classification, were performed using filtering algorithms by means of dedicated software [61], as shown in the work of [62]. The digital terrain model (DTM) was generated through an inverse distance-weighting interpolation, which ensured a 1 × 1 m spatial resolution. The DTM was further used as support in the computation of several parameters (e.g., tree height, canopy height).
**Figure 3.** Airborne laser scan of the studied area.
#### *2.5. Ecosystem Services Identification and Evaluation*
The literature proposes an entire series of ecosystem functions and services assessment methods (monetary, non-monetary, and integrated methods). In this research, the interest was to gather reliable information needed in applying those evaluation methods. The ecosystem services identification is presented according to the ecosystem functions stated by TEEB, and the paper intends to cover the majority of the functions.
#### **3. Results**
#### *3.1. Provisioning Services*
Wood products are one of the most prominent resources provided by forest ecosystems [63], being a direct economic benefit that can be easily assessed from a monetary point of view. In literature, wood products, equated to above-ground biomass, are an important variable that can be estimated through remote-sensing techniques. Between the implementation of biophysical parameters relationships [64,65] to allometric models and direct measurements [30,66–68], the above-ground biomass estimation gained impressive interest in research due to the associated accuracy.
In our study, the applied methodology was the one proposed by Pascu et al. in the work of [30]. Therefore, the above-ground estimation implied the use of stand volume derived from number of trees, DBH, and tree height (Figure 4). Even though other studies [69,70] show volume underestimation when based on terrestrial laser scanning data, this was due to low stand heterogeneity. The accuracy presented by Pascu et al. in what concerns the number of trees and DBH is more than satisfactory (errors under 5%). Moreover, the use of terrestrial laser scanning proved to be an adequate approach for the above-ground volume computation [71].
Height values computed through this active remote-sensing technology show biases and errors, also highlighted by several research papers [30,69,70,72,73]. To overcome this limitation, compensations were applied based on airborne laser scanning.
**Figure 4.** Canopy height model from airborne laser scanning (**a**) SMTM (**b**) SGTM.
For computing above-ground tree volume, the logarithmic regression equation (Equation (1)) described in the work of [74] was used:
$$
\log \upsilon = a\_0 + a\_1 \log d + a\_2 \log^2 d + a\_3 \log h + a\_4 \log^2 h \tag{1}
$$
where:
*d—*tree diameter at breast height;
*v*—tree volume;
*h*—tree height;
*a*0, *a*1, *a*2, *a*3, *a*4*—*species-specific regression coefficients.
The above-ground volumes for each plot are presented in Table 2. When compared to the field measurements based on the same methodology (Equation (1)), the errors are between 4.6% and 13.3%.
V—stand volume; dm—stand mean diameter; hm—stand mean height; vm—tree mean volume.
Considering the biophysical parameters, differences in mean stand volume can be observed between the plots where silvicultural interventions were applied and those without interventions. The reduced volume, specific to the young forest stands and to those targeted by interventions, confirms the viability of the methods and results and makes it possible to compare them in terms of wood product provisioning.
#### *3.2. Regulating Services*
At the moment, the ecosystem services specific to regulating functions represent a great challenge in the evaluating processes [54]. This function includes services for air quality regulation, moderation of extreme events, erosion prevention, and carbon
sequestration [51,55,75]. In the context of evaluating the related services, specific indicators were developed in the field of ecological research.
The assessment methods tend to use indirect measurements and quantify the relationship between different variables. Tree canopy cover, canopy structure indices (e.g., leaf area index), and trees distribution are the most used parameters in the majority of the evaluating approaches [76–78].
#### 3.2.1. Structural Indices
As previously mentioned, forest structure characteristics and biodiversity are the main sources of information for the assessment of ecosystem services. To establish the capacity of ecosystems in supplying regulating services, indices such as Clark-Evans nearest neighbor index (CE), uniform angle index (*UAI*), and relative dominance diameter index were computed at the subplot level.
Clark-Evans nearest neighbor index (CE) describes the horizontal trees distribution by using the mean distance between a reference tree and the nearest neighbors and the mean distance defined by a Poison distribution [79]. CE can range from 0, when the stand is characterized by tree clustering, to 2.1491 [79] in the case of regular distribution.
Uniform angle index (*UAI*) describes the uniform distribution of the nearest neighboring trees in relation to the reference tree [38]. The method is based on the angles between trees, compared to a uniform dispersion angle of 72◦ (Equation (2)). The interpretation of these values is made according to the confidence interval of 0.475–0.517 [38], describing a random distribution.
$$
\natural IIAI = \frac{1}{n} \sum\_{i=1}^{n} \natural IIAI\_i = \frac{1}{4n} \sum\_{i=1}^{n} \sum\_{j=1}^{4} z\_{ij} \tag{2}
$$
where:
*n*—number of reference trees
*zij*—angle coefficients in relation to the reference (72◦), 1 if <72◦, 0 if > 72◦
*UAIi*—uniform angle index
The relative dominance diameter index (IDR) is defined as the ratio between the number of trees with a diameter greater than the reference tree. The value of this indicator reaches values in the range (0–1) and is interpreted in relation to five default thresholds. Thus, in relation to the number of trees with a diameter larger than the reference, the indicator falls into the following categories: shade tolerant, dominated, co-dominant, dominant, predominant. These categories correspond to the Kraft classes, a method used for validating the obtained values. The variable considered in the evaluation of the dominancy indicator may be substituted by other tree characteristics such as height or species.
In the structural indices computation process, the edge effect was removed in order to ensure accurate results. This was performed by selecting only the trees within an inner buffer, defining an area smaller than that of the circular subplots (Figure 5).
The interpretation of these indices made it possible to identify the supplied services and the level to which they could be quantified. CE values greater than 1 suggested that the studied subplots were characterized by a more uniform horizontal structure. An exception was identified in the SFTM-3 subplot, which was characterized by a mean value of 0.4. This could be explained by the smaller number of trees clustered together and by the fact that this circle is crossed by a forest harvesting road. Based on the calculated t-values for the CE, according to the work of [80], the subplots that overpass 1.96 can be described as having a regular distribution (Table 3).
**Figure 5.** Nearest neighbors identification and reference trees selection; **green**—reference tree; **red** marginal tree.
**Table 3.** Horizontal structure indices in the 15 m-radius subplot.
Nref—number of reference trees, \* *t*—CE value.
When uniform angle index values were analyzed, differences between plots could be observed, thus detecting structural differences between the corresponding stands. The uniform angle index values ranged between 0.393 and 0.725, values covering the entire interpretation interval. Within the old sessile oak stand, without interventions (SGTM), the corresponding subplots reached values equivalent to a rather random distribution. This was the case with the 2.3 (0.510) subplot, reaching values quite different than its counterpart, subplots 2.1 and 2.2, characterized by a clustered structure (Table 2). In the case of the Norway spruce (SMTM), the reached values defined a uniform structure, while the young beech stand with interventions (SFR) was characterized by a clustered structure in all subplots.
Also, from this analysis, the difference between the plots with interventions and those without could be observed. The plots covered with silvicultural treatments tend to describe more clustered structures, an effect caused by the increased distance between trees after harvesting.
Figure 6 facilitates the interpretation of the structure and conditions similarity within a plot. As stated before, discrepancies appeared in SFTM for the CE index and in SMRM for the uniform angle index. The latter was a consequence of a windthrow event that had affected the SMRM-3 subplot.
**Figure 6.** Clark-Evans nearest neighbors index and uniform angle index.
Analyses of relative dominance diameter index described the vertical stand structure at the subplot level. A similarity could be observed between old Norway spruce (SMTM) subplots (Figure 7), indicating a uniform structure within the stand and a uniform tree distribution between classes. The sessile oak stand is characterized by a lower degree of heterogeneity and more unevenness between classes.
**Figure 7.** Vertical stand structure (Kraft classes) based on IDR.
#### 3.2.2. Carbon Storage
Carbon is stocked in forest stands in the following five pools: above and below-ground living biomass, soil, litter, and deadwood [81,82]. Apart from the variables used for the above-ground volume, carbon stock evaluation (Table 4) required another set of parameters, namely the theoretical number of trees per hectare, wood density, root-to-shoots ratio, and biomass expansion factor (Equation (3)). These were retrieved from specific yield tables and international guides [83,84].
$$\mathcal{C}\_{\text{stock}} = \sum V \ast D \ast (1 + R) \ast BEF \ast \mathcal{C}F \tag{3}$$
where:
*Cstock*—carbon stock [tC] *V*—tree volume [m3] *D*—wood density [t/m3] *R*—root-to-shoot ratio *BEF*—biomass expansion factor *CF*—carbon fraction
**Table 4.** Carbon stock required variables.
<sup>1</sup> [83] <sup>2</sup> [84] <sup>3</sup> [85].
Due to the methodology for the above-ground volume, for the sessile oak and beech species, the biomass expansion factor (BEF) was omitted, as the regression equation for volume already took into consideration the branches' volume. Including BEF would have led to biased results.
The obtained carbon stock values ranged between 74.68 tC·ha−<sup>1</sup> (273.82 tCO2·ha−1) in the case of the young Norway spruce plot covered with silvicultural intervention and 221.06 tC·ha−<sup>1</sup> (810.55 tCO2·ha<sup>−</sup>1) in the case of the old sessile oak plot. The upper values
of the storage capacity interval of the studied plots are in accordance with those stated in the work of [86]. The lower values are a consequence of age and species characteristics (wood density, root-to-shoot ratio, and carbon fraction).
#### 3.2.3. Foliage Indices
Active remote-sensing technology advances allowed for the development of multiple applications addressing the canopy structure, crown dynamics, and phenology [21,24,29,87–91]. These applications based on active remote-sensing data are a powerful tool in the decisional process associated with forestry and ecology sectors. From the variety of indices computed through remote sensing, in the research field, the leaf area index (LAI) is the most commonly used. Furthermore, along with the LAI, an important role in improving the canopy description is held by leaf area density (*LAD*), which offers detailed information regarding the stand vertical structure. Leaf area index estimation as the ratio between leaves (single-faced) area and area of the studied plot, was measured over time through various indirect methods (orbital sensors, hemispherical photography, and light intensity attenuation) [92–94], and still require improvement in what concerns the stability and robustness of their results. Alternatively, airborne laser scanning, despite its limitations related to penetration capability, has promising results in forestry indices and parameters computation, including those above-mentioned [70,95].
In this study, LAI and *LAD* were estimated through the MacArthur and Horn equation [96] developed on the principle of the Beer–Lambert law [97,98] and following methodologies proposed in other related research papers [95,99–102]. Thus, to each voxel from the processed point cloud (voxel—5 × 5 × 1 m), the following proposed equation was applied [102]:
$$LAD\_{i-1,i} = \ln\left(\frac{S\_\varepsilon}{S\_t}\right) \frac{1}{k\Delta z} \tag{4}$$
where:
*Se*—number of pulses entering the voxel;
*St*—number of pulses exiting the voxel;
*k—*Beer–Lambert law extinction coefficient;
*z—*voxel height (1 m).
From the variety of estimated indices resulting when applying derivatives of the above-mentioned methodology, of most interest to our study were the total LAI values, the height of the mean *LAD*, and standard deviation corresponding to each voxel (cell of a three-dimensional grid) column taken into consideration.
As shown in the case of the IDR, sessile oak (SGTM) is characterized by an uneven structure, a fact also illustrated in the LAI and *LAD* values. In the northwest part of the plot, the higher density of smaller trees impacted the LAI and height of the mean *LAD*, reaching values in the range 1–3, respectively, 5–10 m.
As expected, the Norway spruce plot is characterized by smaller standard deviation values, suggesting a constant horizontal structure throughout the plot (Figure 8). In the case of the sessile oak plot, the standard deviation trend highlights a generation individualization through higher variation within the upper levels of the canopy.
**Figure 8.** Mapped values of (**a**) LAI; (**b**) height of the mean *LAD*; (**c**) standard deviation corresponding to each voxel column.
#### *3.3. Supporting Services*
In the majority of the research papers, this function is not a self-contained one. Millennium Ecosystem Assessment classification (MEA) [103] presents the support function as integration between provisioning, regulating, and cultural functions, quantifying benefits that ensure the rest of the services. In the Common International Classification of Ecosystem Services (CICES), the support function is not promoted as one and is considered an underlying structure that provides indirect outputs [104,105].
Understory biomass has an important ecological significance in forest ecosystem stability and in assessing the relationships between wildlife and their habitat. Despite the low proportion in above-ground volume, the understory biomass represents a tool for the researchers in evaluating the food provisioning and the quality of the environment [106–114].
The understory biomass computation implies a complex and expensive forest inventory due to multiple variables that should be taken into consideration. Active remotesensing applications that aim at assessing understory biomass were proposed. Terrestrial and airborne laser scanning data were analyzed in order to estimate the understory, following [106,112,115].
This study addressed the methodology proposed by the authors of [116] that aims to predict the presence of shrub layers from aerial-based point clouds. In the mentioned thesis, two indices were computed: (a) undergrowth return fraction and (b) undergrowth cover density. For our case, of most interest was the undergrowth return fraction, expressed as the ratio between the number of points in the 0.5–5 m range and the total number of points (Figure 9).
The old Norway spruce plot, in comparison with the sessile oak, is characterized by a sparse distribution of the shrub layer of lower intensities, with no understory clusters identified. In the case of the sessile oak plot, a central area with a high density of understory vegetation could be observed, mirrored in the northern part, by the lower values of the canopy height model. Overall, the sessile oak plot recorded a value of 0.20, which according to the work of [116], is indicative of a medium-to-high shrub cover intensity (Figure 10).
**Figure 9.** Mapping of the shrub layer (5 × 5 m pixel)—undergrowth return fraction.
The majority of the rest of the plots have a low-to-minim shrub cover, covering 6% to 10%. An issue identified through field observations was in the case of SMRM. The plot is characterized by a high shrub coverage, but due to the lower age and high tree density, the laser beams could not penetrate the canopy layer. This resulted in a small number of points near the ground and an underestimation of the shrub layer. This shortcoming can be compensated by using TLS data to complete the ALS points cloud.
#### *3.4. Structure Analysis for Cultural Services Assessment*
The cultural services are the most problematic in what concerns the evaluation processes. The services provided by the one-hectare plots are not traded on the market, and therefore the methods of valuation applied tend to be more subjective. In addition, the evaluation of the forest ecosystem's capacity in providing these benefits is a challenging one due to public preferences and the number of variables involved.
The forest structure indices computed under the regulating function section and part of the health status information can be used in quantifying the human preferences regarding the ideal distribution and biodiversity. Tree clusters, number of trees, sparse distribution, higher canopy density, light penetration, visibility, understory volume, and snags volume can all be indirectly assessed through tree distribution characteristics and mortality analysis.
The snags identification and mortality characteristics were analyzed based on airborne laser scanning data according to the work of [117] methodology (Figure 11).
After processing, the point clouds were classified into four classes, namely live trees, small snags, live crown edge snags, and higher canopy snags. Due to the small proportion of dead trees in the studied plots, not all classes were well represented. Moreover, following the analysis, none of the snag classes were identified in the Norway spruce plots, apart from sparse, unrepresentative small snags in the understory. By way of comparison, the sessile oak plot presents a higher proportion corresponding to the live crown edge snags class.
A crucial role is attributed to the higher canopy snags class, which makes possible the identification of dead treetops. A higher proportion of snags would have allowed for the evaluation of a ratio between deadwood and the above-ground biomass. This information could have then been used in the carbon sequestration estimation or the mortality rate of the forest stand.
**Figure 11.** Snags identification and classification (SGTM).
#### **4. Discussion**
Forest ecosystems are characterized by various structures and complex processes defined by a plethora of intra- and inter-plot relationships. The assessment of all services provided by these ecosystems' characteristics is still a challenging subject for the research field [118]. Therefore, this study aimed to highlight some of the most important and quantifiable services employing the latest applications of active remote-sensing technology [119].
Taking advantage of the terrestrial stationary laser scanning, the obtained values for above-ground volume, at tree and stand level, was within the characteristic tolerances [30]. Moreover, we compensated the height-specific bias caused by the terrestrial laser scanner's inability to penetrate dense canopies, a well know feature relevant to Romanian forests [24], by deriving a canopy height model from aerial laser scanning.
Compared to the rest of the functions, provisioning could be evaluated most straightforward [76,120]. By only knowing the above-ground volume and market price, this service could be monetized. To better understand this service, a technical approach can further be used by classifying the wood in relation to the type of final product or the quality of the timber, information that can be extracted from management plans.
The assessment of carbon sequestration is a more complex process, and it partially uses wood volume calculations. The results are influenced by multiple biophysical parameters (wood density, the ratio between above and below-ground biomass, or the biomass expansion factor) [84]. All these parameters depend on the species composition, a feature that cannot be presently assessed on a large scale by means of close-range active remote sensing [121,122]. Furthermore, as described in the IPCC Guideline [84,123], forest ecosystems stock a large amount of carbon not only in living biomass, so other pools (soil and dead organic matter) should also be taken into consideration for a real carbon emission/removals analysis. On the other hand, assumptions are made even within the country-level estimation. Pools as soil, deadwood, or litter are considered to be neutral in the carbon emission and removals balance. Therefore, only considering the living biomass can represent a viable solution for carbon sequestration and stock assessment.
The utility of structural indices was highlighted in a long list of studies [24,119,124–126], and the indirect quantification made by means of these indices could be considered a proper method for evaluating forest ecosystem capacity to provide services. Forest structure
represents a valuable source of information, and relating structure characteristics to specific services is the approach used in this study.
Soil and water regulation services assured by the forest ecosystem are quantifiable through trees distribution, LAI, *LAD*, and canopy projection [127,128]. The distances, the angles, and the relationship between different individuals define the rate of success of the forest to ensure the regulation function. Indices such as uniform angle index, Clark-Evans nearest neighbor index, and relative dominance diameter index can describe the ideal structure, which prevents gaps or corridors from occurring. The resulting values, corresponding to the studied plots, describe a uniform tree distribution. According to the uniform angle index values, the Norway spruce plot has a better capacity to assure the regulation function if we consider the ideal structure being a random one.
When analyzing the CE, the obtained values tend to reach the upper half of the range, close to a perfectly regular hexagonal distribution (2.1491). There are some dissimilarities between CE and *UAI* regarding, in particular, the sessile oak plot due to the high number of trees in the sampled area. However, of great importance in soil and water regulation services is the fact that none of the plots is characterized by clustered trees that would facilitate soil erosion and low water retention.
Air regulation function was evaluated in this study through the use of LAI and *LAD* [129]. The capacity of the forest ecosystem to provide these services is directly proportional to LAI values. For the studied plots, LAI values are within the 0–6 range, with a considerable proportion in the upper classes throughout the entire old Norway spruce plot. Due to lower values in the canopy height model and implicit smaller crown volumes, the sessile oak recorded lower intensities in the northern part of the plot.
The support function assessment was evaluated based on the shrub cover, indicating the capacity of the studied forests to provide various species' habitat requirements. Important differences were observed between the studied areas. The sessile oak plot (SGTM) recorded a larger area covered by understory vegetation. This analysis can also provide additional information on the forest structure, information that can be used in further biodiversity assessments.
The results regarding mortality have not allowed any further analyses regarding the dead matter stratification. However, it offered enough information to assess the overall health status of the forest ecosystem [117,123,130]. By following the presented structural indices, as well as the ones addressing foliage, while simultaneously considering the human preferences toward the ideal forest structure, a suitable evaluation of the cultural services could be deployed.
Given the results corresponding to our studied plots, the lack of clustered trees, large gaps, and overall canopy structure, an appropriate scale for referencing forest ecosystems' capability to provide cultural benefits could have also been developed.
#### **5. Conclusions**
In order to emphasize and maximize the ecological, social, and economic benefits of forests, suitable assessment methods are required. Active remote-sensing technology, with the proven advantages and characteristic limitations, can represent the foundation for multiple approaches aiming to quantify the capacity of the forest ecosystem to provide services. This study highlighted the possibility of using two different active remote-sensing data sets and several techniques to assess the main ecosystem functions according to TEEB classification.
To estimate the key biophysical parameters of a tree, terrestrial laser scanning point clouds proved to be a viable solution. The processing of this data source led to errors associated with DBH of below 1 cm [30] at the subplot level when analyzing the mean tree. Precisions associated with tree coordinates are comparable to those obtained through the electronic field mapping system. Using the TLS-based variables as well as the airborne laser scanning data, the provisioning function, particularly wood products, was evaluated. This was performed by means of above-ground volume, characterized by errors smaller than 6%.
Combining the terrestrial and aerial laser scans, the evaluation of regulating function was also possible. The indices computed by processing the above-mentioned data sources proved to be a suitable basis for acquiring the forest's horizontal structure and the distribution of trees. CE, *UAI*, and IDR implementations through active remote-sensing approaches can represent the link between ecosystem services and human preferences, but also the qualitative parameters for assessing the degree of ensuring certain services, namely soil stability, air quality, and water regulation.
Challenges still exist in applying active remote-sensing techniques due to the complex ecosystems' intra- and inter-plot relationships. However, the development of tools to address the environmental assessment requirements is encouraged by the stakeholders and decisional factors. Thus, it can be stated that active remote-sensing applications have a significant role in forestry, a role that translates to an overall improvement of human well-being.
Therefore, implementing the described methodologies highlighted the necessity of developing custom reference scales relevant in the assessment processes of the relative capacity of forest ecosystems to supply benefits. To achieve this, the study should be extended to address further stands of different structures, species compositions, and microclimates.
**Author Contributions:** Conceptualization, A.C.D., I.-S.P., G.M.T. and O.B.; Formal analysis, A.C.D. and I.-S.P.; Investigation, A.C.D., I.-S.P., S, .L. and J.G.-D.; Methodology, A.C.D. and I.-S.P.; Software, I.-S.P.; Validation, S, .L., C.-E.D., G.M.T. and O.B.; Visualization, I.-S.P.; Writing—original draft, A.C.D. and I.-S.P.; Writing—review and editing, A.C.D., J.G.-D., G.M.T., I.-S.P. and O.B. All authors have read and agreed to the published version of the manuscript.
**Funding:** This study was conducted under the project CRESFORLIFE (SMIS 105506), subsidiary contract no. 18/2020, co-financed by the European Regional Development Fund through the 2014– 2020 Competitiveness Operational Program.
**Conflicts of Interest:** The authors declare no conflict of interest.
#### **References**
MDPI St. Alban-Anlage 66 4052 Basel Switzerland Tel. +41 61 683 77 34 Fax +41 61 302 89 18 www.mdpi.com
*Forests* Editorial Office E-mail: [email protected] www.mdpi.com/journal/forests
MDPI St. Alban-Anlage 66 4052 Basel Switzerland
Tel: +41 61 683 77 34 Fax: +41 61 302 89 18
www.mdpi.com
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# FRONTIERS IN BRAIN BASED THERAPEUTIC INTERVENTIONS AND BIOMARKER RESEARCH IN CHILD AND ADOLESCENT PSYCHIATRY
EDITED BY: Paul E. Croarkin and Stephanie H. Ameis PUBLISHED IN: Frontiers in Psychiatry
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ISSN 1664-8714 ISBN 978-2-88919-954-9 DOI 10.3389/978-2-88919-954-9
### About Frontiers
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# **FRONTIERS IN BRAIN BASED THERAPEUTIC INTERVENTIONS AND BIOMARKER RESEARCH IN CHILD AND ADOLESCENT PSYCHIATRY**
Topic Editors: **Paul E. Croarkin,** Mayo Clinic, USA **Stephanie H. Ameis,** Centre for Addiction and Mental Health & The Hospital for Sick Children & University of Toronto, Canada
Developmental neuroscience research is on the cusp of unprecedented advances in the understanding of how variations in brain structure and function within neural circuits confer risk for symptoms of childhood psychiatric disorders. Novel dimensional approaches to illness classification, the availability of non-invasive, diverse and increasingly sophisticated methods to measure brain structure and function in humans in vivo, and advances in genetics, animal model and multimodal research now place brain-based biomarkers within reach in the field of psychiatry. These advances hold great promise for moving neuroscience research into the clinical realm. One exciting new area of translational research in child and adolescent psychiatry, is in the use of a variety of neuroscience research tools to track brain response to clinical intervention. Examples of this include: using longitudinal neuroimaging techniques to track changes in white matter microstructure following a training intervention for children with poor reading skills, or using functional imaging to compare brain activity before and after children with bipolar disorder begin taking psychotropic medication treatment. Brain stimulation is another cutting-edge research area where brain response to therapeutic intervention can be closely tracked with electroencephalography or other brain imaging modalities. Research using neuroscience tools to track brain response to clinical interventions is beginning to yield novel insights into the etiopathogenesis of psychiatric illness, and is providing preliminary feedback around how therapeutic interventions work in the brain to bring about symptom improvement. Using these novel approaches, neuroscience research may soon move into the clinical realm to target early pathophysiology, and tailor treatments to both individuals and specific neurodevelopmental trajectories, in an effort to alter the course of development and mitigate risk for a lifetime of morbidity and ineffective treatments. Excitement and progress in these areas must be tempered with safety and ethical considerations for these vulnerable populations. This research topic focuses on efforts to use neuroscience research tools to identify brain-based biomarkers of therapeutic response in child and adolescent psychiatry.
**Citation:** Croarkin, P. E., Ameis, S. H., eds. (2016). Frontiers in Brain Based Therapeutic Interventions and Biomarker Research in Child and Adolescent Psychiatry. Lausanne: Frontiers Media. doi: 10.3389/978-2-88919-954-9
# Table of Contents
### **Chapter 1 Editorial**
### *05 Editorial: Frontiers in Brain-Based Therapeutic Interventions and Biomarker Research in Child and Adolescent Psychiatry*
Paul E. Croarkin and Stephanie H. Ameis
### **Chapter 2 Neurodevelopmental Disorders**
### *08 Biomarkers in autism*
Andre A. S. Goldani, Susan R. Downs, Felicia Widjaja, Brittany Lawton and Robert L. Hendren
*21 A comparison of neuroimaging findings in childhood onset schizophrenia and autism spectrum disorder: a review of the literature*
Danielle A. Baribeau and Evdokia Anagnostou
*36 Upregulated GABA inhibitory function in ADHD children with child behavior checklist–dysregulation profile: 123I-iomazenil SPECT study*
Shinichiro Nagamitsu, Yushiro Yamashita, Hitoshi Tanigawa, Hiromi Chiba, Hayato Kaida, Masatoshi Ishibashi, Tatsuyuki Kakuma, Paul E. Croarkin and Toyojiro Matsuishi
### **Chapter 3 Affective Disorders**
*43 A preliminary study of white matter in adolescent depression: relationships with illness severity, anhedonia, and irritability*
Sarah E. Henderson, Amy R. Johnson, Ana I. Vallejo, Lev Katz, Edmund Wong and Vilma Gabbay
*55 Meta-analyses of developing brain function in high-risk and emerged bipolar disorder*
Moon-Soo Lee, Purnima Anumagalla, Prasanth Talluri and Mani N. Pavuluri
*65 Stress, inflammation, and cellular vulnerability during early stages of affective disorders: biomarker strategies and opportunities for prevention and intervention*
Adam J. Walker, Yesul Kim, J. Blair Price, Rajas P. Kale, Jane A. McGillivray, Michael Berk and Susannah J. Tye
### **Chapter 4 Eating Disorders**
### *73 Neural responses during social and self-knowledge tasks in bulimia nervosa* Carrie J. McAdams and Daniel C. Krawczyk
### *85 Altered SPECT 123I-iomazenil Binding in the Cingulate Cortex of Children with Anorexia Nervosa*
Shinichiro Nagamitsu, Rieko Sakurai, Michiko Matsuoka, Hiromi Chiba, Shuichi Ozono, Hitoshi Tanigawa, Yushiro Yamashita, Hayato Kaida, Masatoshi Ishibashi, Tatsuki Kakuma, Paul E. Croarkin and Toyojiro Matsuishi
### **Chapter 5 Treatment Innovation in Children and Youth**
*94 Assessing and stabilizing aberrant neuroplasticity in autism spectrum disorder: the potential role of transcranial magnetic stimulation*
Pushpal Desarkar, Tarek K. Rajji, Stephanie H. Ameis and Zafiris Jeff Daskalakis
*100 Neurocognitive effects of repetitive transcranial magnetic stimulation in adolescents with major depressive disorder*
Christopher A. Wall, Paul E. Croarkin, Shawn M. McClintock, Lauren L. Murphy, Lorelei A. Bandel, Leslie A. Sim and Shirlene M. Sampson
# Editorial: Frontiers in Brain-Based Therapeutic Interventions and Biomarker Research in Child and Adolescent Psychiatry
*Paul E. Croarkin1 \* and Stephanie H. Ameis2,3,4*
*1Noninvasive Brain Stimulation Program, Department of Psychiatry and Psychology, Mayo Clinic Depression Center, Mayo Clinic, Rochester, MN, USA, 2 The Margaret and Wallace McCain Centre for Child, Youth and Family Mental Health, Temerty Centre for Therapeutic Brain Intervention, Slaight Centre for Youth in Transition, Campbell Family Mental Health Research Institute, Centre for Addiction and Mental Health, Toronto, ON, Canada, 3Centre for Brain & Mental Health, The Hospital for Sick Children, Toronto, ON, Canada, 4Child & Youth Mental Health Division, Department of Psychiatry, Faculty of Medicine, University of Toronto, Toronto, ON, Canada*
Keywords: adolescent, biomarker, child, child and adolescent psychiatry, neuroimaging, non-invasive brain stimulation, pediatric
**The Editorial on the Research Topic**
**Frontiers in Brain-Based Therapeutic Interventions and Biomarker Research in Child and Adolescent Psychiatry**
*Edited and Reviewed by: Stefan Borgwardt, University of Basel, Switzerland*
> *\*Correspondence: Paul E. Croarkin [email protected]*
#### *Specialty section:*
*This article was submitted to Neuroimaging and Stimulation, a section of the journal Frontiers in Psychiatry*
> *Received: 19 June 2016 Accepted: 27 June 2016 Published: 05 July 2016*
#### *Citation:*
*Croarkin PE and Ameis SH (2016) Editorial: Frontiers in Brain-Based Therapeutic Interventions and Biomarker Research in Child and Adolescent Psychiatry. Front. Psychiatry 7:123. doi: 10.3389/fpsyt.2016.00123*
Childhood psychiatric disorders present challenges given the heterogeneity of presentations, instability of phenotypes, and nascent understanding of neurodevelopment. Recent efforts, such as the National Institute of Mental Health Research Domain Criteria, aim to hone precision medicine approaches for psychiatric disorders (1). Elucidating the ontogeny of psychiatric illnesses and underlying neurobiology is a mandate for advancing modern clinical practice. Recent advances in neuroimaging, preclinical studies, genomics, and non-invasive brain stimulation may soon provide improved monitoring of development in health and disease. These tools also hold great promise for developing biological markers of illness that may be targeted through treatment innovation. This research topic surveys recent developmentally informed clinical neuroscience efforts focused on conditions that affect children and adolescents. Broadly, this includes studies focusing on neurodevelopmental disorders, eating disorders, mood disorders, and treatment innovations.
### NEURODEVELOPMENTAL DISORDERS
Recent changes in descriptive diagnostic criteria, such as DSM-5 (2), aim to bridge basic science findings with clinical practice (3). Goldani et al. examine existing literature focused on putative biomarkers of autism spectrum disorder (ASD). Markers of mitochondrial function, oxidative stress, genetic clustering, and inflammation are promising approaches. However, at present, there is insufficient evidence to embed these markers into clinical practice (Goldani et al.). In other efforts to better understand neurobiology in ASD and related neurodevelopmental disorders, Baribeau and Anagnostou review neuroimaging correlates of ASD and schizophrenia. Volumetric changes, cortical thickness differences, and white matter changes in childhood onset schizophrenia (COS) appear to attenuate with age. Impaired local connectivity may also be coupled with amplified long-range connectivity in this condition. Neuroimaging findings in ASD collectively suggest an initial period of brain verdancy followed by dysmorphogenesis in adolescence. Furthermore in ASD, patterns of local hyper-connectivity are coupled with impaired long-range neural communication (Baribeau and Anagnostou). Nagamitsu et al. present recent dimensional work with brain single-photon emission computed tomography (SPECT) in participants with attention deficit hyperactivity disorder (ADHD). Children with ADHD, and higher scores on the child behavior checklist-dysregulation profile had significantly increased 123I-iomazenil biding in the posterior cingulate cortex. These data suggest that GABAergic inhibitory neurons are involved in the pathophysiology of ADHD (Nagamitsu et al.).
## AFFECTIVE DISORDERS
Recent controversies surrounding the phenotyping of childhood mood disorders underscore the necessity of ongoing work focused on the neurobiological characterization of affective disorders during neurodevelopment (4). Henderson et al. examine white matter microorganization as assessed with diffusion tensor imaging (DTI) in adolescents with depression and healthy controls. Anhedonia and irritability were associated with unique neuroanatomical signatures. This promising early work suggests that prefrontal and limbic tracts are disrupted in depression and unique symptom presentations may have DTI signatures (Henderson et al.). Lee et al. examine brain function of healthy control participants, high-risk offspring, and youth with bipolar disorder by means of a meta-analytic approach. The authors postulated that greater activity in high-risk participants signifies potential compensatory mechanisms, whereas more widespread findings in bipolar patients signified chronic disease processes (Lee et al.). Finally, Walker et al. propose a compelling model for the mood disorder prodrome. These authors posit that early life stress, inflammation, and allosteric load are key contributors to disease burden, disease progression, and neuropathology (Walker et al.).
### EATING DISORDERS
Eating disorders are severely impairing psychiatric illnesses, with high mortality rates, and profound neurobiologic underpinnings (5). Herein, McAdams and Krawczyk describe findings from a study of patients with anorexia, patients with bulimia, and healthy controls. Participants were exposed to social attribution, social identity, and physical identity fMRI tasks. Consistently throughout each region of interest, average activation levels for bulimic participants were greater than the group of patients with anorexia, but less than healthy participants. The authors concluded that patients with eating disorders could have a similar biological substrate in terms of social functioning, yet a distinctive functional characterization is a plausible pursuit for future work (McAdams and Krawczyk). Nagamitsu et al. also present intriguing work focused on developing SPECT biomarkers to guide the treatment of children with anorexia nervosa. In children with anorexia nervosa, decreased 123I-iomazenil binding in the anterior cingulate and left parietal cortices was associated with a suboptimal response to treatment. Successful weight restoration was associated with increased relative binding of 123I-iomazenil in the posterior cingulate and occipital cortices (Nagamitsu et al.).
## TREATMENT INNOVATION IN CHILDREN AND YOUTH
Transcranial magnetic stimulation (TMS) is a powerful therapeutic and neurophysiological probe. Neurocognitive outcomes are key both in terms of safety and for intervention development, as they may serve as optimal clinical outcome measures in youth (6, 7). Wall et al. report on a study in which eighteen depressed adolescents received 30 sessions of 10-Hz rTMS, applied to the left dorsolateral prefrontal cortex. Participants demonstrated improvements in delayed verbal recall and memory. Furthermore, there were no decrements in other neurocognitive dimensions (Wall et al.). Desarkar et al. postulate that imbalances in excitatory and inhibitory neurotransmission could underlie aberrant neuroplasticity in ASD. At the receptor level, this may involve excessive NMDA and deficient GABAmediated neurotransmission. Interventions with high frequency rTMS may have a role in stabilizing dysregulated neuroplasticity in ASD (Desarkar et al.).
In conclusion, the synthesis of neuroscience with child and adolescent psychiatry is yielding important discoveries and new directions for treatment innovation. However, we have yet to make the discoveries necessary to bring neuroscience research into the clinical realm through specific biomarker discovery that could pave the way for precision medicine where biomarkers are profiled in the clinic and individualized treatments are selected to optimize neurodevelopmental trajectories, mitigate the long-term effects of psychiatric illness, and maximize functioning for individuals. Novel research tools, innovative study designs that go beyond the case–control model, longitudinal research that identifies developmental trajectories within heterogeneous conditions, and largescale studies with the power to detect small effects are likely the next frontier in research focused on advancing our understanding of neurobiological underpinnings and developing biologically informed treatments for children and youth with mental illness.
## AUTHOR CONTRIBUTIONS
All authors listed have made substantial, direct, and intellectual contribution to the work and approved it for publication.
## ACKNOWLEDGMENTS
Dr. PC has received financial support from the Brain and Behavior Research Foundation, the Mayo Foundation, and NIMH (grant K23 MH100266). The content of this editorial and research topic is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health. Dr. SA receives financial support from the CAMH Foundation *via* the O'Brien Scholarship Fund, the University of Toronto, Faculty of Medicine, Dean's Fund New Staff Grant, and the Ontario Mental Health Foundation.
## REFERENCES
7. Croarkin PE, Daskalakis ZJ. Could repetitive transcranial magnetic stimulation improve neurocognition in early-onset schizophrenia spectrum disorders? *J Am Acad Child Adolesc Psychiatry* (2012) 51(9):949–51. doi:10.1016/j. jaac.2012.05.012
**Conflict of Interest Statement:** Dr. PC has received in-kind support for research (supplies and genotyping) from Assurex Health. He has received in-kind support for equipment from Neuronetics. He has received travel support and research support (investigator-initiated trial and serves as a site primary investigator for a multicenter study) from Neuronetics. He has received research support from Pfizer (investigator-initiated grant). Dr. SA has no financial conflicts of interest to declare.
*Copyright © 2016 Croarkin and Ameis. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.*
# Biomarkers in autism
#### **Andre A. S. Goldani <sup>1</sup> , Susan R. Downs <sup>2</sup> , FeliciaWidjaja<sup>2</sup> , Brittany Lawton<sup>2</sup> and Robert L. Hendren<sup>2</sup>\***
<sup>1</sup> Universidade Federal do Rio Grande do Sul, Porto Alegre, Brazil
<sup>2</sup> Department of Psychiatry, University of California San Francisco, San Francisco, CA, USA
#### **Edited by:**
Paul Croarkin, Mayo Clinic, USA
**Reviewed by:**
Randi Hagerman, UC Davis Medical Center, USA Richard Eugene Frye, Harvard University, USA
#### **\*Correspondence:**
Robert L. Hendren, University of California San Francisco, 401 Parnassus Avenue, San Francisco, CA 94143-0984, USA e-mail: [email protected]
Autism spectrum disorders (ASDs) are complex, heterogeneous disorders caused by an interaction between genetic vulnerability and environmental factors. In an effort to better target the underlying roots of ASD for diagnosis and treatment, efforts to identify reliable biomarkers in genetics, neuroimaging, gene expression, and measures of the body's metabolism are growing. For this article, we review the published studies of potential biomarkers in autism and conclude that while there is increasing promise of finding biomarkers that can help us target treatment, there are none with enough evidence to support routine clinical use unless medical illness is suspected. Promising biomarkers include those for mitochondrial function, oxidative stress, and immune function. Genetic clusters are also suggesting the potential for useful biomarkers.
**Keywords: biomarker, autism spectrum disorders, epigenetics, treatment targets, neuroimaging, genetics**
### **INTRODUCTION**
Several neurodevelopmental disorders have complex genetic and epigenetic features that lead to their phenotype and for some there is no single genetic marker for the diagnosis; therefore, the diagnosis is made phenotypically as in schizophrenia, ADHD, and autism spectrum disorder (ASD). While phenotypic characterization of neurodevelopmental disorders is an integral part of advances in clinical practice and research, a given phenotype may arise from a diverse set of biochemical processes (especially when the disorder is caused by numerous genetic and epigenetic factors). Therefore, the treatment of a "phenotypic diagnosis" with a specific drug or intervention might be extremely effective for one "phenotypically characterized" individual with a given set of genetic and/or epigenetic biomarkers, but completely ineffective for another with a different pattern of biomarkers. An important goal of ongoing research in ASD, therefore, is to more precisely identify the many different abnormal genetic and epigenetic processes that underlie the phenotype of the disorder. This might allow individuals with ASD to be characterized into subsets with certain biomarker profiles that would respond more favorably to specific treatments. It also has the potential to elucidate the abnormal physiology that leads to autism, which could improve the understanding of the disorder and lead to earlier diagnosis and more targeted treatments.
A significant challenge in identifying biomarkers in ASD is that biomarkers may reflect genetic and neurobiological changes or epigenetic (broadly defined, see below) processes that may be active only during particular periods of time and do not define the disorder, only the process that led to it. In addition, treatment research should ideally include biomarkers that are believed to predict improvements in clinical symptoms from clinical interventions (1) to know if an intervention is altering or targeting an active biomedical process that relates to response in the subject at that time. Indeed, the National Institute of Mental Health (NIMH) has changed how they fund clinical trials so that "trial proposals will need to identify a target or mediator; a positive result will require not only that an intervention ameliorated a symptom but also that it had a demonstrable effect on a target, such as a neural pathway implicated in the disorder or a key cognitive operation"(2).
Traditionally, research in psychiatry has been guided by DSM symptom based diagnoses and selection criteria for clinical trials were based on these symptom clusters. Biomarkers have not been reliable or valid markers of response to treatment in past trials, and this may be due to the wide variety of genetic and epigenetic processes that underlie the DSM-based diagnosis. Recently, progress in biomarker research has led to the commitment to the Research Domain Criteria project (RDoC) as a basis for future NIMH funding for biomarker based research (3, 4). The RDoC goal is to define basic dimensions of functioning to be studied across multiple units of analysis, from genes to neural circuits to behaviors, cutting across disorders as traditionally defined. The intent is to translate rapid progress in basic neurobiological and behavioral research to an improved integrative understanding of psychopathology and the development of new and/or optimally matched treatments for mental disorders (5).
In this article, we review the literature on biomarkers for ASD including genetic, epigenetic, brain based, and body metabolism biomarkers. This is a huge area and this review is not intended to be comprehensive. New potential biomarkers for ASD are being identified every day so the list needs to be updated frequently. We do extensively review the literature at the time of this writing, report on methodologically sounds studies, offer summary tables, and summarize what we know.
#### **GENETIC BIOMARKERS**
The literature supports a hereditary component in the susceptibility to ASDs, there are much higher concordance rates of ASDs in monozygotic twins (92%) than dizygotic twins (10%), and a recent estimate of the sibling recurrence risk ratio (λs) is 22 for autism. Despite being highly heritable, ASDs show heterogeneous clinical symptoms and genetic architecture, which have hindered the identification of common genetic susceptibility factors. Although previous linkage studies, candidate gene association studies, and cytogenetic studies have implicated several chromosomal regions for the presence of autism susceptibility loci, they have not consistently identified and replicated common genetic variants that increase the risk of ASDs other than some clearly genetic disorders such as fragile X, tuberous sclerosis, and RASopathies whose phenotypes meet the ASD category description (5). As autism is not a single clinical entity, it can be viewed as a behavioral manifestation of tens or perhaps hundreds of genetic and genomic disorders (6). It has been estimated that there are over 500 distinct genetic loci that may be related to ASD (7) (**Figure 1**).
In addition, recent research has shown that there are many epigenetic mechanisms that could account for hereditary influences. A study by Hallmayer et al. (9) reports that the environment may actually account for more of the etiology of autism than genetics. Their study, the largest population-based twin study of autism that used contemporary standards for the autism diagnosis, found that heritability estimated at 38%, while shared environmental component was 58% (9). Heritability of ASD and autistic disorder is estimated to be approximately 50% (10).
Being one of the most familial psychiatric disorders, autism has garnered inquiries about possible genetic biomarkers (11); however, progress has been slow until recently with the introduction of genome-wide association studies (GWAS) and microarrays (12). Research into the microbiological underpinnings of ASDs suggests that it is not a monogenic disorder following Mendelian tendencies, with a few studied individuals and families as notable exceptions (11). In fact, the literature suggests that the risk of developing autism is derived by variations across many genes, none of which have been conclusively, definitively responsible for ASDs although some individuals with single gene disorders such as fragile X also meet the criteria for ASD.
Genome-wide association studies have identified, with replication, *de novo* variations that are strongly associated (with sufficient
power) with ASDs (**Table 1**): deletions at the Neurexin 1 (NRXN1) locus, duplications at 7q11.23, duplications at 15q11-13, and deletions and duplications at 16p11.2. Earlier studies found rare, functional mutations in genes encoding for NRXN1, SHANK3, and SHANK2, all of which are proteins that affect the functioning of synapses and have been linked to other, known genetic disorders (12). In addition, whole exome sequencing verified by four reports have found genetic mutations associated with autism including SNC2A, CHD8, DYRKIA, POG2, GRIN2B, and KATNAL2 (13).
Studying particular genes in certain, recognized disorders with social deficits, such as fragile X syndrome and tuberous sclerosis, may shed light on the genetic underpinnings of ASDs. This strategy gives credence to the idea that ASD is the result of many variations among genes that converge to a similar phenotype. A prime example of implementation of such a strategy is with contactin 4 (CNTN4), and its association with social and intellectual disability in a recurrent deletion syndrome. Mutations in the respective genes are identified in idiopathic ASDs. Similarly, mutations in CNTNAP2 are linked to a variety of results, such as language delay, functional connectivity abnormalities, selective mutism, and anxiety. More importantly in the scope of ASDs, alterations in CNTNAP2 are noted in consanguineous pedigrees (12). Research shows an increased prevalence of ASDs in families that are consanguineous (11).
In a study published by Nature in 2009, Wang and colleagues completed a genetic analysis in a large number of ASD individuals and families, with a combined sample set of more than 10,000 subjects of European ancestry. They identified common genetic variants on 5p14.1 that are associated with susceptibility to ASDs and replicated these findings in separate analyses. The contribution of chromosome 5p14 to cell adhesion and its connection to autism susceptibility supports the conclusion that specific genes in this class help create the connectivity and structure of the brain that ultimately leads to ASD (14). Besides the potential role of the nearby CDH10 and CDH9 genes, pathway-based association analysis lend further support to neuronal cell-adhesion molecules in conferring susceptibility to ASDs, suggesting that specific genetic variants in this gene class may be involved in shaping the physical structure and functional connectivity of the brain that leads to the clinical manifestations of ASDs (14).
Among the common polymorphisms found to be associated with autism risk, the methylenetetrahydrofolate reductase (MTHFR) polymorphism is one of the most widely studied genetic correlations with autism. The MTHFR 677C > T polymorphism causes a reduction in enzyme activity, which results in higher production of 5-formyltetrahydrofolate (5-FTHF) necessary for DNA synthesis and repair along with lower 5-MTHF production. The MTHFR 677C > T polymorphism causes decline of normal enzyme activity to 35% (15). The MTHFR 677T-variant allele is correlated with a 2.79-fold increased risk for autism. However, this study also found that MTRR 66A and SHMT 1420T alleles demonstrated protective roles against autism risk (16). MTHFR also has a strong interaction with maternal folic acid intake before and during pregnancy, which is associated with autism risk. Children with high autism risk whose mothers carried MTHFR 677 TT allele and were reported taking prenatal vitamins had fewer diagnoses of autism than the children whose mothers with the same allele and did not take prenatal vitamins (17).
In several GWAS (14, 18–20), four genes have been associated with ASDs. These genes, cadherin (CDH9), cadherin 10 (CDH10), semaphorin 5A (SEMA5A), and taste receptor, type 2, member 1 (TAS2R1), are found on chromosome 5p14, which regulates axon growth and cell adhesion. While gene networks could not be established from the small number of genes, these findings do suggest that these genes and the dysregulation of synaptic connection may be a key feature in ASDs (21).
Griswold and coworkers found a significantly higher burden in the number and size of deletions carried by ASD individuals when compared with controls (22). Among the copy-number variations (CNVs) identified were several that overlapped with well-established autism-associated regions and candidate genes. They isolated four large, novel deletions on 2q22.1, 3p26.3, 4q12, and 14q23 that include new genes and regions linked to ASDs. Scattered findings related to NLGN4 and autism susceptibility occur across cultures. In the Chinese ASD cases, there were no significant findings regarding SNPs along NLGN4 gene and autism risk (23), yet in Greek ASD cases, nine nucleotide changes in NLGN4X are found to be associated with autism (24).
Copy-number variations has unveiled the overexpression of rare, *de novo* structural variations in the genome of simplex families (families which have one affected offspring) when compared to families with multiple affected offspring, and especially control families. Furthermore, these results have been replicated in later studies, bolstering the confidence in which discoveries can be made about genetic ties with common diseases and autism (12); however, *de novo* CNVs have been found in only 5–10% of researched subjects, and thus, do not make up the majority of affected, researched individuals. Despite this finding, it seems as though large (>100 kb), multigenic *de novo* CNVs are the most indicative of ASD risk at this time.
The genetic component of a disorder can be transmitted or acquired through *de novo* ("new") mutations. A study based on a 343 family subset of the Simons Simplex collection did not find significantly greater numbers of *de novo* missense mutations in affected versus unaffected children, but gene-disrupting mutations (nonsense, splice site, and frame shifts) were twice as frequent (59 versus 28) (25). They found that the father is more frequently the parent of origin for *de novo* mutations than the mother (50/17) for single nucleotide variants (SNVs). Parental age also appears to play a role in mutation rate. A study published in *Nature* found that the rate of *de novo* SNVs increases with paternal age (*p* = 0.008) and that paternal and maternal ages are highly correlated (*p* < 0.0001) (26). Overall these data demonstrate that non-synonymous *de novo* SNVs, and particularly highly disruptive nonsense and splice-site *de novo* mutations, are associated with ASD.
Several companies are marketing genetic testing for autism based on clusters of genes with a strong clustering for ASD risk (27, 28). In the future, there may be biomarkers that can pinpoint for high risk for ASD diagnosis. For example, a mother who may be high risk for immune dysfunction leading to ASD in a second child once the first child has ASD (29) or the increase in the AktmTOR pathway, which can be seen in fragile X syndrome and in other ASD subtypes (30).
#### **EPIGENETICS**
Considerable symptom severity differences within ASDconcordant monozygotic twins, strongly implicates a role for non-genetic epigenetic factors (31). Epigenetics refers to the study of heritable changes in gene activity that are not caused by changes in the DNA sequence; it also can be used to describe the study of stable, long-term alterations in the transcriptional potential of a cell that are not necessarily heritable. Epigenetic changes in ASD occur through methylation, histone modification (31), chromatin remodeling, transcriptional feedback loops, and RNA silencing (32). Processes in the gene × environment interaction that influence gene expression include metabolic processes such as oxidative stress, mitochondrial function, methylation, immune function, and inflammation that are byproducts of influences such as the mothers and fathers immune systems, environmental toxicants, and diet to name a few. This section will review these epigenetic influences associated with ASD.
Studies show that DNA methylation differences can occur in many loci including AFF2, AUTS2, GABRB3, NLGN3, NRXN1, SLC6A4, UBE3A (31), the oxytocin receptor (33), MeCP2 (a cause for most cases of Rett syndrome) in the frontal cortex (34), and changed chromatin structure in prefrontal cortex neurons at hundreds of loci (35). The severity of the autistic phenotype is related to DNA methylation at specific sites across the genome (31). Environmental and physiological influences are important factors accounting for interindividual DNA methylation differences, and these influences differ across the
genome (36). The following sections describe markers for metabolic pathways and environmental influences that can effect epigenetic changes.
#### **METABOLIC BIOMARKERS**
There are no autism-defining, metabolic biomarkers, but examining the biomarkers of pathways associated with ASD can point to potentially treatable metabolic abnormalities and provide a baseline that can be tracked over time. Each child may have different metabolic pathologies related to SNPs, nutrient deficiencies, and toxic exposures. Examples of metabolic disorders that can lead to an autistic-like presentation include phenylketonuria (PKU) (37), disorders of purine metabolism (38), biotinidase deficiency (39), cerebral folate deficiency (40), creatine deficiency (41), and excess propionic acid (which is produced by *Clostridium*) (42, 43).
A recent review assessed the research on physiological abnormalities associated with ASD (44). The authors identified four main mechanisms that have been increasingly studied during the past decade: immunologic/inflammation, oxidative stress, environmental toxicants, and mitochondrial abnormalities. In addition, there is accumulating research on the lipid, GI systems, microglial activation, and the microbiome, and how these can also contribute to generating biomarkers associated with ASD (45, 46).
Pathways are interconnected with a defect in one likely leading to dysfunction in others. Many metabolic disorders can lead to endpoints such as impaired methylation, sulfuration, and detoxification pathways and nutritional deficiencies. Mitochondrial dysfunction, environmental risk factors, metabolic imbalances, and genetic susceptibility can all lead to oxidative stress (47), which in turn leads to inflammation, damaged cell membranes, autoimmunity (48), impaired methylation (49), cell death (48), and neurological deficits (50). The brain is highly vulnerable to oxidative stress (51), particularly in children (52) during the early part of development (47). As environmental events and metabolic imbalances affect oxidative stress and methylation, they also can affect the expression of genes.
Several studies have detected altered levels of a large collection of substances in body-based fluids from ASD subjects compared to controls (e.g., serum, whole-blood, and CSF) (53). These findings encompass either of two main disease-provoking mechanisms: a CNS disorder that is being detected peripherally [e.g., serotonin and its metabolites, sulfate (54), low platelet levels of gammaaminobutyric acid (GABA) (55), low oxytocin (which affects social affiliation) (56), and low vitamin D levels (57, 58)] or a systemic abnormality that has repercussions in the brain (59).
Serotonin in the brain promotes prosocial behavior and correct assessment of emotional, social cues (60) and can contribute to immune abnormalities (61). Oxytocin can affect social affiliation and social communication deficits (62). Vitamin D has many effects including regulating serotonin synthesis, reducing maternal antibodies that attack the fetal brain, modulating oxytocin synthesis, lowering GI inflammation by lowering gut serotonin (58), DNA repair, anti-inflammatory actions, anti-autoimmune activities, antiseizure activity, increase in regulatory T cells, mitochondrial protection,stimulation of antioxidant pathway (63),and increasing glutathione (64).
#### **OXIDATIVE STRESS MARKERS**
Oxidative stress can be detected by studying antioxidant status, antioxidant enzymes, lipid peroxidation, and protein/DNA oxidation, all of which have been found to be elevated in children with autism (**Table 2**). Different subgroups of children with ASD have different redox abnormalities, which may arise from various sources (65). A recent meta-analysis from 29 studies of blood samples from subjects with ASD shows that reduced levels of glutathione, glutathione peroxidase, methionine, and cysteine along with increased levels of oxidized glutathione are statistically different in ASD (66). The level of antioxidants excreted in urine was found to be significantly lower than normal in autistic children. These findings correlated with the severity of the ASD (67).
Measurements of antioxidant status include measurement of *glutathione*, the primary antioxidant in the protection against oxidative stress, neuroinflammation, and mitochondrial damage (68, 69). Glutathione is instrumental in regulating detoxification pathways and modulates the production of precursors to advanced glycation end products (AGEs) (70). Measuring reduced glutathione, oxidized glutathione, or the ratio of reduced glutathione to oxidized glutathione helps determine the patient's oxidation status. In many patients with ASD, the ratio of reduced glutathione to oxidized glutathione is decreased, indicating a poor oxidation status (71).
The enzyme glutathione peroxidase has been used as a marker and is typically reduced. There are mixed results concerning the enzyme levels of *superoxide dismutase (SOD)* (72). Other markers for glutathione inadequacy include alpha hydroxybutyrate, pyroglutamate, and sulfate, which can be assessed in an organic acid test. Lipid peroxidation refers to the oxidative degradation of cell membranes. There is a significant correlation between the severity autism and urinary lipid peroxidation products (67), which are increased in patients with ASD.
*Plasma F2t-Isoprostanes (F2-IsoPs)* are the most sensitive indicator of redox dysfunction and are considered by some to be the gold standard measure of oxidative stress (73). They are increased in patients with ASD and are even higher when accompanied by gastrointestinal dysfunction (73). F2t-isoprostanes (F2-IsoPs) can be measured in the urine as well.
*Urine 8-OHdG* is biomarker for oxidative damage to DNA. It is commonly used although there are confounding factors and intra individual variations (74) and some researchers have reported that the increases in urine 8-OHdG in patients with ASD is not significant. The increases in urine 8-OHdG did not reach statistical significance (75).
Decreased levels of major antioxidant serum proteins *transferrin* (iron-binding protein) and *ceruloplasmin* (copper binding protein) have been observed in patients with ASD. The levels of reduction in these proteins correlate with loss of previously acquired language (47) although there are mixed reviews of the significance of this (66).
Plasma *3-chlortyrosine (3CT)*, a measure of reactive nitrogen species and myeloperoxidase activity, is an established biomarker of chronic inflammatory response. Plasma 3CT levels reportedly increased with age for those with ASD and mitochondrial dysfunction but not for those with ASD without mitochondrial dysfunction (65).
#### **Table 2 | Oxidative stress biomarkers in ASD (see text for references)**.
**Table 3 | Mitochondrial function biomarkers in ASD (see text for references)**.
3-Nitrotyrosine (3NT) is a plasma measure of chronic immune activation and is a biomarker of oxidative protein damage and neuron death. This measure correlates with several measures of cognitive function, development, and behavior for subjects with ASD and mitochondrial dysfunction but not for subjects with ASD without a mitochondrial dysfunction (65).
#### **MITOCHONDRIAL DYSFUNCTION MARKERS**
Mitochondrial dysfunction is marked by impaired energy production. Some children with ASD are reported to have a spectrum of mitochondrial dysfunction of differing severity (44) (**Table 3**). Mitochondrial dysfunction, most likely an early event in neurodegeneration (76), is one of the more common dysfunctions found in autism (77) and is more common than in typical controls (78). There is no reliable biomarker to identify all cases of mitochondrial dysfunction (79). It is possible that up to 80% of the mitochondrial dysfunction in patients with both ASD and a mitochondrial disorder are acquired rather than inherited (44).
Mitochondrial dysfunction can be a downstream consequence of many proposed factors including dysreactive immunity and altered calcium (Ca2+) signaling (80), increased nitric oxide and peroxynitrite (68), propionyl CoA (81), malnutrition (82), vitamin B6 or iron deficiencies (83), toxic metals (83), elevated nitric acid (84, 85), oxidative stress (86), exposure to environmental toxicants, such as heavy metals (87–89), chemicals (90), polychlorinated
biphenyls (PCBs) (91), pesticides (92, 93), persistent organic pollutants (POPs) (94), and radiofrequency radiation (95). Other sources of mitochondrial distress include medications such as valproic acid (VPA), which inhibits oxidative phosphorylation (96) and neuroleptics (97, 98).
Markers of mitochondrial dysfunction include lactate, pyruvate and lactate-to-pyruvate ratio, carnitine (free and total), quantitative plasma amino acids, ubiquinone, ammonia, CD, AST, ALT, CO<sup>2</sup> glucose, and creatine kinase (CK) (44). Many studies of ASD report elevations in lactate and pyruvate, others report a decrease in carnitine, while others report abnormal alanine in ASD patients (44) or elevations in aspartate aminotransferase and serum CK (99). Increases in lactate are not specific and may only occur during illness, after exercise or struggling during a blood draw (100).
Rossignol and Frye (44) recommend a mitochondrial function screening algorithm. This includes fasting morning labs of lactate, pyruvate, carnitine (free and total), acyl carnitine panel, quantitative plasma amino acids, ubiquinone, ammonia, CK, AST/ALT, CO2, and glucose (44). The interpretation of such a panel and the indications for specific treatments has not yet been established.
#### **METHYLATION**
The methylation pathway provides methyl groups for many functions, including the methylation of genes, which can result in the epigenetic changes of turning genes on and off (**Table 4**). This transfer occurs when *S*-adenosylmethionine (SAM) donates a methyl group and is transformed to *S*-adenosylhomocysteine (SAH). SAH can be transferred to homocysteine, which can either be re-methylated to methionine or be transferred by the sulfuration pathway to cysteine to create glutathione. With increased oxidative stress, SAH might be diverted away from the methylation pathway to the sulfuration pathway in order to make more glutathione. This will result in less methionine and less methylation ability.
Impaired methylation may reflect the effects of toxic exposure on sulfur metabolism. Oxidative stress initiated by environmental factors in genetically vulnerable individuals, can lead to impaired methylation and neurological deficits (49) both of which may contribute to the manifestation of autism (71).
A marker of methylation dysfunction is decreased SAM/SAH ratio in patients with ASD. Fasting plasma methionine decreases since through SAM it is the main methyl donor. Fasting plasma cysteine, a sulfur containing amino acid is the rate-limiting step in the production of glutathione and is significantly decreased. Plasma sulfate is decreased, which may impair detoxification pathways. Homocysteine is generally increased, but the studies are mixed (66). Vitamin B12 and folate are required for the methylation pathway. The MTHFR genetic SNP is reported to heavily influence the methylation pathway (66).
#### **IMMUNE DYSREGULATION**
#### **Cytokine evaluation**
Chronic inflammation and microglia cell activation is present in autopsied brains of people with ASD (101, 102) (**Table 5**). Factors that increase the risk of activating brain microglia include traumatic brain injury (TBI) (103) reactive oxygen species (104) and a dysfunctional blood brain barrier (105).
#### **Table 4 | Methylation biomarkers in ASD (see text for references)**.
S-adenosylmethionine (SAM)/S-adenosylhomocysteine (SAH) Homocysteine MTHFR
#### **Table 5 | Immune biomarkers in ASD (see text for references)**.
The blood brain barrier can be compromised by oxidative stress (106), acutely stressful situations (107), elevated homocysteine (108), diabetes (109), and hyperglycemia (110). Cytokines can pass through a permeable blood brain barrier and start this process (111). Hence, cytokines can serve as a marker of the immune dysregulation, which can further complicate ASD.
Irregular cytokines profiles are found in ASD (112, 113) and elevations in plasma cytokines are reportedly correlated with regressive onset and severity of autistic and behavioral symptoms (113). Altered pro-inflammatory cytokines, complement proteins, chemokines, adhesion molecules, and growth factors are correlated with ASD. More specifically, altered TGF-beta, CCL2, and CCL5, IgM and IgG classes of immunoglobulin circulating levels are linked with a worsening of behavioral scores (114). An imbalance in Th1/Th2 has are found as well, which may play a role in the pathogenesis of autism (115).
*Neopertin* as a urine marker of immune dysfunction and activation. Neopterin is associated with increased production of reactive oxygen systems and can be considered as a measurement of the oxidative stress elicited by the immune system. Neopterin levels are found to be significantly higher in children with autism than in the comparison subjects (116).
Increased *S100B protein*, a calcium binding protein produced primarily by astrocytes, is a biomarker reflecting neurological/brain damage found elevated in ASD and correlated to autistic severity (117).
#### **AUTOIMMUNITY AND MATERNAL ANTIBODIES**
Autoimmune autistic disorder is proposed as a major subset of autism (118), and autoimmunity may play a role in the pathogenesis of language and social developmental abnormalities in a subset of children with these disorders (119). There are many autoantibodies found in the nervous system of children with ASD who have a high level of brain antibodies (120, 121). These can be measured as biomarkers in this subset of ASD patients. The anti ganglioside M1 antibodies (122), antineuronal antibodies (123), and serum anti-nuclear antibodies (123, 124) correlate with the severity of autism. Other autoantibodies postulated to play a pathological role in autism include: anti neuron-axon filament protein (anti-NAFP) and glial fibrillary acidic protein (anti-GFAP) (125), antibodies to brain endothelial cells and nuclei (119), antibodies against myelin basic protein (126, 127), and anti myelin associated glycoprotein, an index for autoimmunity in the brain (128). BDNF antibodies were found higher in ASD (129), and low BDNF levels may be involved in the pathophysiology of ASD (130).
Antibodies in patients with autism are found to cells in the caudate nucleus (131), cerebellum (132, 133), hypothalamus and thalamus (121), the cingulate gyrus (134), and to cerebral folate receptors (135). Children with cerebellar autoantibodies had lower adaptive and cognitive function as well as increased aberrant behaviors compared to children without these antibodies (132).
#### **MOTHER'S IMMUNE STATUS**
Research studies indicate an association between viral or bacterial infections in expectant mothers and their ASD offspring (136, 137). Maternal antibodies cross the underdeveloped blood brain barrier of the fetus (138) leading to impaired fetal neurodevelopment and long-term neurodegeneration, neurobehavioral, and cognitive difficulties (139).
A maternal infection or immune response includes cytokines, which affect aspects of fetal neurogenesis, neuronal migration (140), synaptic plasticity, and stem cell fate (141). Elevated serum IFN-γ, IL-4, and IL-5 were more common in women who gave birth to a child subsequently diagnosed with ASD (142). Fetal IL-6 exposure, especially in late pregnancy, leads abnormalities of hippocampal structural and morphology, and decreased learning during adulthood (139).
Some of the antibodies that cross the fetal developing blood brain barrier recognize and attack the brain (138). The presence of fetal brain protein antibodies in ASD can result in an inappropriate approach to unfamiliar peers (143).
Braunschweig et al. developed a panel of clinically significant maternal autoantibody-related autoantibody biomarkers with over 99% specificity for autism risk (144). This panel is suggested to lead to an early diagnosis of maternal autoantibodyrelated autism, allow for interventions that limit fetal exposure to these antibodies and allow for early behavioral intervention.
#### **DYSBIOSIS**
When the gut becomes inflamed, it breaks down and becomes permeable, sometimes referred to as dysbiosis. Dysbiosis is reported to be an upstream contributing factor to autoimmune conditions and inflammation. Markers under consideration include circulating antibodies against tight junction proteins, LPS, actomyosin (145)
#### **Table 6 | Other potential biomarkers in ASD**.
calprotectin (146), and lactoferrin (147). Dysbiosis was found in 25.6% of patients with ASD (148). It is proposed to have a direct effect on the brain as it is a hypothesized source of inflammation (149–151) and autoimmunity (152, 153), possibly through molecular mimicry (154). Diet is one source of dysbiosis (155).
#### **AMINO ACIDS AND NEUROPEPTIDES**
Platelet hyperserotonemia is considered one of the most consistent neuromodulator findings in patients with ASD (**Table 6**). As for other neuropeptides, a recent review reported approximately 15 components that are altered in ASD compared to controls (53). Among them, interesting research has been done on glutamate, GABA, BDNF, and dopamine and noradrenaline systems. A recent study reported a positive correlation between severity of clinical symptoms and plasma GABA levels in patients with ASD, supporting the idea of a disrupted GABAergic system (156). Additionally, a similar grouping of substances measured in the urine is suggested as a more convenient and less invasive way to draw information on these patients (41).
#### **FATTY ACID ANALYSIS**
Abnormal fatty acid metabolism may play a role in the pathogenesis of ASD and may suggest some metabolic or dietary abnormalities in the regressive form of autism (42,157). There is evidence of a relationship between changes in brain lipid profiles and the occurrence of ASD-like behaviors using a rodent model of autism (42). Hyperactivity in patients was inversely related to the fluidity of the erythrocyte membrane and membrane polyunsaturated fatty acid (PUFA) levels (158). Imbalances of membrane fatty acid composition and PUFA loss can affect ion channels and opiate, adrenergic, insulin receptors (159) and the modulation of (Na + K)-ATPase activity (160). Analysis of red blood cell membrane fatty acids is a very sensitive indicator of tissue status and may reflect the brain fatty acid composition (161).
Seventeen percent of children with ASD manifest biomarkers of abnormal mitochondrial fatty acid metabolism, the majority of which are not accounted for by genetic mechanisms (162). Patients with ASD had reduced percentages of highly unsaturated fatty acids (163) and an increase in ω6/ω3 ratio (158).
#### **ENVIRONMENTAL TOXICANTS**
For environmental toxicant biomarkers, it is difficult to interpret abnormal levels in ASD. For instance, a high burden of aluminum, cadmium, lead, mercury, and arsenic was found in a subgroup of a sample of over 500 patients with ASD (164). Other studies have described decreased levels of some of these heavy metals in urine and in hair samples, which may imply that the body is not excreting the heavy metals adequately (41).
A systematic review of toxicant-related studies in ASD found that pesticides, phthalates, PCBs, solvents, toxic waste sites, air pollutants, and heavy metals were implicated in ASD, with the strongest evidence found for air pollutants and pesticides (165).
#### **BRAIN FOCUSED BIOMARKERS**
#### **MAGNETIC RESONANCE IMAGING**
Like other areas in psychiatry, new approaches are being devised to tackle ASD in a "bottom-up paradigm" – that is, identifying genetic or biological alterations, which are associated with the clinical manifestations of symptoms. In neuroimaging, much progress has been made toward understanding the condition, but only very few observed biomarkers have sufficient evidence to suggest that they might hold diagnostic or treatment significance.
One of the best-replicated brain findings from subjects with ASD is an early-accelerated brain volume growth. The increase is usually around 10%, peaking between 2 and 4 years of age followed by a plateau (166). Head circumference (HC), an adequate proxy for brain size, is being investigated for diagnostic relevance for ASD (167). However, recent findings on HC in ASD show that there might be an unrelated growth in HC in both patients and controls. Thus, the abnormal overgrowth observed in older studies might be because of a biased Center for Disease Control (CDC) HC norm, which is commonly used as the control group (168).
Gray matter thickness and surface areas and white matter integrity are also being studied. A general trend demonstrating increased gray matter thickness in subjects with ASD compared to controls is observed with an age-dependent effect (166). Even though there are studies correlating symptom severity with altered thickness there are several limitations such as using a cross-sectional approach and a small number of subjects that hinder clinical application (169). Likewise, diffusion tensor imaging (DTI) studies on white matter connectivity are not yet conclusive across studies.
Early studies using functional magnetic resonance imaging (fMRI) focus on task specific cognitive networks (e.g., face recognition, theory of mind, imitation, language processing, and proxies for receptive behavior) (166). In these cognitive network studies, individuals with ASD and controls perform a task while the fMRI is monitored. More recently, researchers are investigating the connectivity between these network and resting-state methods where fMRI is obtained while a subject is at rest and not performing a task. These more recent studies reveal a pattern that suggests less activity in the brain areas that typically perform executive function tasks (such as organization or planning). This combination of activity patterns in ASD is often called a "high noise-information ratio," supporting an excitatory/inhibitory imbalance theory of ASD (170). Conversely, even though all these fMRI findings shed light on the pathophysiology of ASD, they also are not mature enough to translate into a reliable biomarker that can be used in clinical practice.
#### **ELECTROENCEPHALOGRAPHY**
Aligned with the notion that ASD is an abnormal connectivity disorder, studies using electroencephalography (EEG) have explored local changes in signal complexities in patients (171). Some studies were able to detect abnormalities as early as 6 months of age, suggesting an important tool for early detection and risk group assessment (172). However, despite findings like multi-scale entropy differences being proposed as an early diagnostic biomarker, EEG has not yet been established as a reliable tool for diagnosis or to document clinical changes (173).
#### **NEUROCHEMISTRY**
Neuroimaging techniques also are used to monitor*in vivo* concentration of substances in the brain, and include positron emission tomography (PET), single photon emission tomography (SPECT), and magnetic resonance spectroscopy (MRS). So far, the majority of studies report abnormalities in several of neurotransmitter networks and their respective metabolites (e.g., dopamine, GABA, serotonin, glutamate, and *N*-acetyl-aspartate), varying from synthesis, transport, and receptor activity in different regions of the brain in the glutamate–glutamine system, in particular, there appears to be either hyper (174) or hypoglutamatergic (175) states depending on the brain region, which could be interpreted as an excitatory increase relative to inhibition in key neural circuits (176). In addition, studies pointing toward GABA alterations also are accumulating, with findings of reduced levels of GABA in the frontal lobes of subjects with ASD. Using MRS (177), corroborated the histopathologic research on altered density and distribution of the GABA receptors (178).
### **BIOMEDICAL INTERVENTIONS**
There are no published studies of interventions for ASD that use neuroimaging or genetic biomarkers in a prospective manner to guide treatment. Biomedical interventions based on body fluid/product biomarkers have been used in a small but growing numbers of well designed, published studies. Several recent reviews summarize these (179–181).
#### **FUTURE RESEARCH DIRECTIONS**
A common feature of all prior studies of these putative biomarkers is that most consist of small samples of patients, and therefore, do not grasp the heterogeneity that characterizes ASD. Also, since they mainly compare subjects with ASD to typically developing controls, it is uncertain whether these biomarker profiles are unique to ASD – they may be present in other neurodevelopmental disorders. A promising new method that is designed to increase specificity of biomarkers in ASD is the multiplex immunoassay, a method that analyzes sets of biomarkers to create a diagnostic profile (182, 183). Furthermore, advances in chromatographic and proteomic techniques are also contributing to the progress of the field, allowing easier assessment of several substances (184, 185).
Thus far, numerous studies examining a diverse set of potential biomarkers have found a large number of genetic, imaging, and metabolic tests that are abnormal in children with ASD compared to control subjects. For most of these measures, it is not yet clear if the abnormal biomarker is a contributing factor to the development of ASD or a result of another underlying abnormality (i.e., causal or merely associated). Not surprisingly, the conclusion is that more studies are needed to further explore these possible mechanisms individually. However, the future in the ASD research might involve a broader view of these biomarkers, which might hold more value in combination than in isolation. As a result of new technological advances, it is possible to use a machine learning technique that is trained to identify complex patterns of data that can be applied to new individuals to make predictions (186). A recent study pooled regional white and gray matter volumes of whole-brain MRI scans in ASD subjects using this computer algorithm program, known as super vector machine. As a result, they could classify a new patient as having an ASD diagnosis or not with a high true positive rate (187). Although exemplified with neuroimaging, this approach could be generalized to other biomarkers (53, 188). In other words, individually insignificant biomarkers when analyzed together might generate a pattern of clinical relevance like diagnosis, severity staging, or response to treatment. These techniques might also be able to identify the most relevant or most predictive biomarkers among the many candidate biomarkers described above.
Although the maxim that "further studies are needed" still holds, ASDs may be witnessing the emergence of clinically relevant biomarkers in the near future.
#### **REFERENCES**
a nested whole-brain analysis. *Brain* (2005) **128**:213–26. doi:10.1093/brain/ awh330
**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
*Received: 18 December 2013; accepted: 22 July 2014; published online: 12 August 2014. Citation: Goldani AAS, Downs SR, Widjaja F, Lawton B and Hendren RL (2014) Biomarkers in autism. Front. Psychiatry 5:100. doi: 10.3389/fpsyt.2014.00100*
*This article was submitted to Neuropsychiatric Imaging and Stimulation, a section of the journal Frontiers in Psychiatry.*
*Copyright © 2014 Goldani, Downs, Widjaja, Lawton and Hendren. This is an openaccess article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.*
REVIEW ARTICLE published: 20 December 2013 doi: 10.3389/fpsyt.2013.00175
# A comparison of neuroimaging findings in childhood onset schizophrenia and autism spectrum disorder: a review of the literature
### **Danielle A. Baribeau<sup>1</sup> and Evdokia Anagnostou<sup>2</sup>\***
<sup>1</sup> Department of Psychiatry, University of Toronto, Toronto, ON, Canada
<sup>2</sup> Autism Research Centre, Bloorview Research Institute, University of Toronto, Toronto, ON, Canada
#### **Edited by:**
Stephanie Ameis, University of Toronto, Canada
**Reviewed by:**
Meng-Chuan Lai, University of Cambridge, UK Ossama Yassin Mansour, Alexandria University Hospital, Egypt Peter G. Enticott, Deakin University, Australia
#### **\*Correspondence:**
Evdokia Anagnostou, Holland Bloorview Kids Rehabilitation Hospital, 150 Kilgour Road, Toronto, ON M4G 1R8, Canada e-mail: eanagnostou@ hollandbloorview.ca
**Background**: Autism spectrum disorder (ASD) and childhood onset schizophrenia (COS) are pediatric neurodevelopmental disorders associated with significant morbidity. Both conditions are thought to share an underlying genetic architecture. A comparison of neuroimaging findings across ASD and COS with a focus on altered neurodevelopmental trajectories can shed light on potential clinical biomarkers and may highlight an underlying etiopathogenesis.
**Methods**: A comprehensive review of the medical literature was conducted to summarize neuroimaging data with respect to both conditions in terms of structural imaging (including volumetric analysis, cortical thickness and morphology, and region of interest studies), white matter analysis (include volumetric analysis and diffusion tensor imaging) and functional connectivity.
**Results**: In ASD, a pattern of early brain overgrowth in the first few years of life is followed by dysmaturation in adolescence. Functional analyses have suggested impaired long-range connectivity as well as increased local and/or subcortical connectivity in this condition. In COS, deficits in cerebral volume, cortical thickness, and white matter maturation seem most pronounced in childhood and adolescence, and may level off in adulthood. Deficits in local connectivity, with increased long-range connectivity have been proposed, in keeping with exaggerated cortical thinning.
**Conclusion**: The neuroimaging literature supports a neurodevelopmental origin of both ASD and COS and provides evidence for dynamic changes in both conditions that vary across space and time in the developing brain. Looking forward, imaging studies which capture the early post natal period, which are longitudinal and prospective, and which maximize the signal to noise ratio across heterogeneous conditions will be required to translate research findings into a clinical environment.
**Keywords: autism spectrum disorder, childhood onset schizophrenia, neuroimaging, magnetic resonance imaging, child development, review**
### **INTRODUCTION**
Autism spectrum disorder (ASD) is a neurodevelopmental disorder of increasing prevalence in the modern era. Presently, this condition is reported to affect 1 in 88 individuals (1). Manifested by social communication deficits and restricted or repetitive interests and behaviors, children with ASD present along a wide spectrum of clinical severity, from mild social difficulties to severe functional impairment. This condition typically presents in the first 3 years of life, manifested by a failure to gain, or a loss of, social communication milestones.
Childhood onset schizophrenia (COS), on the other hand, is a relatively rare disorder, affecting 1 in 10,000–30,000 children (2). The diagnostic criteria are the same as in adult onset schizophrenia, including the presence of positive and/or negative symptoms (3), but with onset occurring prior to the 13th birthday (4). Despite clinical heterogeneity, COS typically presents with psychotic symptoms after age seven, and is associated with a more severe course and poorer outcomes as compared to adult onset schizophrenia (2).
Although presently considered to separate clinical entities, prior to the twentieth century, catatonia, social withdrawal, bizarre behavior, and/or psychosis in children were considered undifferentiated conditions, labeled as "hereditary insanity," "dementia praecox," or "developmental idiocy" (5). With the onset of contemporary nosology, "autistic behavior and social withdrawal" were initially specified as features of "childhood schizophrenia" in the first and second editions of the Diagnostic and Statistical Manual of Mental Disorder (DSM-I and -II). Although formally defined as separate entities in DSM-III (6), at present the DSM-5 permits concurrent diagnosis of both conditions, should an
individual with ASD subsequently develop prominent delusions or hallucinations (3).
In the current review, a comparison between ASD and COS was chosen for several reasons. Firstly, children with co-occurring and overlapping symptoms complicate a diagnosis (2, 4). At times, a period of medication washout and inpatient observation is required to achieve a diagnostic consensus (7), further supporting a need for brain based biomarkers of disease state and treatment response. Indeed, over one quarter of patients diagnosed with COS display prodromal neurodevelopmental disturbances, meeting criteria for pervasive developmental disorder, or ASD (8, 9). Children diagnosed with ASD are more likely to report psychotic symptoms in adolescence and adulthood (10, 11), although the exact incidence of a subsequent diagnosis of schizophrenia varies by study, ranging from 0 to 7% (12–14). From a neuroimaging perspective, analysis of atypical brain "growth curves" may afford an opportunity for early identification and risk stratification; consistent with the present goal of moving toward biologically based diagnostic categories in neuropsychiatric disease.
Secondly, a growing body of literature supports a neurodevelopmental origin of both schizophrenia and autism, with a shared genetic architecture contributing to, or precipitating, the development of both conditions (15, 16). Some have hypothesized that ASD and schizophrenia are diametrically opposed with respect to underlying pathology (17). While adult onset schizophrenia and ASD have been compared in previous reviews [see Ref. (18)], a focus on COS specifically permits a more indepth analysis of aberrant neurodevelopmental trajectories across comparable age ranges, which may provide insight into disease pathogenesis.
This review intends to translate several decades of neuroimaging research for a clinical audience, to highlight our current understanding of similarities and differences in the clinicopathogenesis of ASD and COS from a neuroimaging perspective. To our knowledge, this is the first focused review of neuroimaging findings in ASD and COS.
### **STRUCTURAL MRI STUDIES (VOLUMETRIC ANALYSIS, CORTICAL THICKNESS AND MORPHOLOGY, AND REGION OF INTEREST STUDIES)**
#### **VOLUMETRIC ANALYSIS**
Structural magnetic resonance imaging (MRI) analysis for neuropsychiatric diseases began to emerge in the 1990s. Early trials employed manual delineation of gray and white matter to investigate specific regions of interest. With advancement in high resolution MRI technology and automated analysis, voxel-based morphometry (VBM) made it possible to quantify the specific gray matter content of each voxel (a volumetric pixel) in an image, allowing large data sets to be processed more efficiently (19). For statistical comparisons between case and control populations, images are "warped" onto a common template, and the degree of transposition of each voxel can be quantified. Inferences must be heeded with the consideration that the relative volumetric differences by region can vary by age, gender, whole brain volume, and by IQ, thus the degree to which these factors have been controlled for must be kept in mind.
#### **Volumetric analysis in COS**
Initial trials conducted by the National Institute of Mental Health (NIMH) on a cohort of children with COS, identified a pattern of reduced cerebral volumes and larger ventricles, consistent with findings in the adult onset schizophrenia population (20). With expansion and longitudinal analysis of this patient sample, investigators were able to localize and describe patterns of change in brain structure and volume over time. While typically developing children were found to have a small decrease in cortical gray matter (~2%) in the frontal and parietal regions throughout adolescence, children with COS displayed exaggerated gray matter losses (~8%), involving the frontal, parietal, and temporal lobes. Of note, baseline IQ varied significantly between case and control groups in this data set (70 vs. 124) (21).
Subsequent analysis on the same NIMH sample (*n* = 60 patients), suggested that this pattern took on a "back to front" trajectory, with losses originating in the parietal lobes and spreading anteriorly over time (22). This pattern persisted after controlling for IQ and medication administration (23). Despite significant differences at an early age, the rate of gray matter loss was shown to level off in early adulthood, implicating adolescent neurodevelopment as a key window in disease pathogenesis (22, 24). This data is consistent with hypotheses pertaining to exaggerated synaptic pruning as a feature of schizophrenia (25).
Later work by the same group demonstrated that the abovedescribed pattern was specific for COS. Using VBM, 23 COS patients were compared to 38 age and gender matched healthy control subjects and 19 patients with other psychotic symptoms but not meeting criteria for COS, defined as "multidimensionally impaired" (MDI). MRI scans were conducted at study intake, and at 2.5 years follow up. The MDI group had equal exposure to neuroleptics at study intake, and had a similar degree of cognitive impairment. Total gray matter loss between the two time points demonstrated 5.1% loss for COS patients, 0.5% loss for MDI patients, and 1.5% loss for healthy control subjects. Thus, exaggerated gray matter loss during adolescence was considered to be a potential biomarker of COS (26).
There is very little literature looking at infants or toddlers who subsequently develop schizophrenia, given the methodological complexities of such a study. That being said, offspring of mothers with schizophrenia were found on average to have *larger* intracranial volumes, greater volumes of CSF, and greater gray matter volume on structural MRI in male neonates, compared to controls, although controlling for total intracranial volume resulted in all differences being non-significant (27).
#### **Volumetric analysis in ASD**
In ASD, earlier studies suggested a pattern of increased total brain volume, as well as increased ventricle size (28–30). Analyses across age ranges helped to further elucidate the chronology of this brain overgrowth picture. Indeed, exaggerated gray and white matter volumes seemed most pronounced in younger children, while older children with ASD had more typically appearing brains, when compared to their peers (31, 32) (see **Figure 1**). The hypothesis of brain overgrowth correlated with the measureable increase in rate of growth of head circumference during the first few years of life as well in this population (33, 34).
In 2005, a meta-analysis of published data on brain volume, head circumference, and post-mortem brain weight in ASD, further described the effect of age, with most marked differences occurring in the first few years of life. In adulthood, however, brain sizes did not vary from controls (35). Subsequent longitudinal and cross-sectional data from hundreds of children and adults with ASD documented volume enlargement during preschool years, most prominently in the anterior regions, followed by possible growth arrest or exaggerated losses later in childhood (36–38). Using cross-sectional age-adjusted data, Schumann et al. (36), for example, showed that children with ASD had 10% greater white matter volume, 6% greater frontal gray matter volume, and 9% greater temporal gray matter volume at 2 years of age. Longitudinal data showed altered growth trajectories at follow up scans (36).
Volumetric differences did not hold true in all ASD studies however, for example, when structural MRI from children with ASD were compared to children with other developmental delays (39, 40). Similarly, a recent systematic review of published data on head circumference overgrowth in children with ASD suggests differences may be much more subtle than previously thought. The authors attribute exaggerated differences to biased normative data in the CDC head circumference growth curves, to the selection of control groups from non-local communities, as well as to a failure to control for head circumference confounders such as weight and ethnicity (41).
Recently, a small study looked at whether volumetric MRI might be predictive of a subsequent diagnosis of ASD, prior to the development of clinical symptoms. A group of 55 infants (33 of which were considered high risk given that they had a sibling with ASD) were scanned prospectively at three time points prior to 24 months of age. At 24 and 36 months, they underwent detailed developmental assessments, at which point 10 infants were identified as having a diagnosis of ASD, and 11 were noted to have other developmental delays. The authors found increased extraaxial fluid volume in infants who developed ASD, and quantified the difference through manual delineation of CSF compartments. They were able to show that a ratio of fluid:brain volume of
>0.14 yielded 79% specificity and 78% sensitivity in 12–15 month old infants regarding a subsequent diagnosis of ASD (42) (see **Figure 2**). The finding remains to be replicated.
*Summary and comparison.* In summary, volumetric analyses in ASD describe early brain overgrowth in the first few years of life, a finding that is difficult to contrast to COS, given the methodological complexity of acquiring neuroimaging data in very young children or neonates who subsequently develop this condition. During childhood and adolescence, volumetric data suggests that individuals with ASD may have attenuated brain growth or exaggerated volume loss, since adults with ASD have comparable brain volumes to their typically developing peers. Some similarities emerge with the COS population, given findings of exaggerated gray matter loss during adolescent years.
#### **CORTICAL THICKNESS AND MORPHOLOGY**
With advancements in computational statistics, it became possible extract a more detailed analysis of the cortical gray matter with respect to surface morphology. Specifically, the transposition of cortical imaging data onto a common surface template allowed cortical gray matter volume to be further quantified in terms of cortical thickness, surface area, and gyrification. More recently, complex statistical approaches employing mathematical algorithms and machine-learning models have manipulated neuroimaging data collected from both volumetric and cortical thickness measurements, in efforts to generate diagnostic classifiers of ASD/COS.
Cortical measurements are of interest for neurodevelopmental disorders as they are thought to represent distinct embryological processes under tight regulatory control (43). Cortical surface area, for example, reflects to the process of neural stem cell proliferation and migration early in embryologic development (44). Cortical thickness, on the other hand, reflects axon and dendrite remodeling, myelination, and synaptic pruning, in a dynamic process lasting from birth into adulthood (45).
#### **Cortical thickness and morphology in COS**
In the NIMH-COS sample (46), a combination of cross-sectional and longitudinal data from 70 patients compared to controls revealed diffuse decreases in mean cortical thickness in childhood (~7.5% smaller), which became localized specifically to the frontal and temporal lobes with increasing age. Statistical significance survived correction for covariates such as sex, socioeconomic status, and IQ. Accordingly, while individuals with COS displayed global gray matter and cortical thickness losses in childhood, with age these losses became similar to those observed in adult onset schizophrenia, with deficits localizing more anteriorly (see **Figure 3**).
Interestingly, in two separate samples, non-affected siblings of COS probands also demonstrated a pattern of decreased cortical thickness in the frontal, temporal and parietal lobes during childhood and adolescence, which then normalized in early adulthood, implicating some sort of compensatory mechanism despite underlying genetic risk (47, 48).
With hospitalization and medication management, symptom remission correlated with localized increases in cortical thickness measurable in specific subregions of the cortex (49), irrespective of choice of antipsychotic (50). Children who had other psychiatric conditions with comorbid psychotic symptoms but not meeting full criteria for COS demonstrated cortical deficits in prefrontal/temporal pattern as well, but deficits were smaller and less striking than in COS patients (51).
As mentioned in the introduction to this section, complex algorithms and mathematical protocols have been designed to identify and combine measurements that may be predictive of disease state. A multivariate machine-learning algorithm applied to cortical thickness data from the NIMH cohort was able to correctly classify 73.7% of patients with COS and controls. Through this method, 74 "important" regions were identified. Areas with the most predictive power clustered in frontal regions (primarily the superior and middle frontal gyris), and the left temporoparietal region (52). Given the rarity of COS in the general population, and the case-control study design, these results were not validated in a separate study population, precluding any calculation of positive or negative predictive value, and thus limiting any inferences regarding clinical utility.
#### **Cortical thickness and morphology in ASD**
There is significant heterogeneity in the literature with respect to cortical thickness and morphology in ASD, with at times seemingly contradictory results depending on the age, IQ, and clinical severity of the study population.
In a very young group of patients with ASD, cortical volume, and surface area (but not thickness) were found to be increased compared to controls at the age of 2 years. The rate of cortical growth between ages 2 and 5 years did not differ between groups, further implicating the prenatal and early postnatal periods as central to disease pathogenesis (53).
In slightly older age groups, many authors have observed evidence of exaggerated cortical thinning in ASD. For example, Hardan et al. (54) demonstrated that children with ASD ages 8–13 years had increased cortical thickness, particularly in the temporal lobe, as compared to aged matched controls. The small sample size (*n* = 17 cases), however, precluded co-variation for IQ, or analysis of age-related interactions (54). Longitudinal imaging 2-years later on seemingly the same cohort, showed that those with a diagnosis of ASD underwent exaggerated cortical thinning compared to controls, and that the degree of thinning correlated with the severity of symptoms. Differences, however, were mostly non-significant after controlling for multiple comparisons
and variation in IQ (55). In a comparable age group (6–15 years). Mak-Fan et al. (56) showed a similar pattern of increased cortical thickness, surface area, and gray matter volume in children with ASD at earlier ages (6–10 years), that then underwent exaggerated losses compared to controls, such that by 12–13 years of age, controls surpassed patients on all three measures (56). Wallace et al. (57), on the other hand, found baseline *deficits* in cortical thickness for adolescents with ASD, but also observed exaggerated rates of cortical thinning during adolescence and early adulthood (57). In the same study population, no differences in overall surface area were noted, but more overall gyrification in the ASD group, particularly in the occipital and parietal regions was observed. Both groups showed a decline in gyrification overtime (58).
On the other hand, several authors have noted deficits in cortical thinning in ASD. Looking over a wide age range, Raznahan et al. (59) used cross-sectional MRI data from 76 patients with ASD (primarily Asperger's syndrome) and 51 controls from ages 10 to 60 years to study the effects of age on cortical thickness and surface area. While surface area was relatively stable and comparable between both groups, they found significant differences with respect to cortical thickness. Typically developing individuals had greater cortical thickness in adolescence, which thinned steadily overtime. Individuals with ASD had reduced cortical thickness early in life, which underwent relatively little cortical thinning overtime, such that by middle age, they had surpassed their typically developing peers (59). ASD associated deficits in expected age-related cortical thinning during adolescence and adulthood has been shown in several other studies as well, both diffusely and in specific subregions (60, 61).
Recently, Ecker et al. (62) sought to tease apart the relative contributions of cortical thickness and cortical surface area to overall differences in cortical volume in a group of adult males (mean age of 26 years) with ASD compared to controls. While total brain volume and mean cortical thickness measurements were not significantly different between the two groups, several regional clusters emerged with both increased and decreased cortical volumes. The authors found that these relative differences were accounted for by variability primarily in cortical surface area, and less so from cortical thickness. As well, differences in cortical thickness/surface area were largely non-overlapping, and were deemed to be spatially independent from each other (62).
As in COS, several groups have aimed to combine the predictive power of multiple measurements by applying mathematical algorithms to neuroimaging data. Ecker et al. (63), for example, included five parameters (cortical convexity, curvature, folding, thickness and surface area) in their support vector machine analytic approach. These combined measurements were able to correctly classify patients with ASD (*n* = 20) and controls (*n* = 20) with 80–90% specificity and sensitivity, with cortical thickness being the most predictive measurement. This approach also demonstrated proof of principle in separating patients with ASD from patients with ADHD, despite the small sample size, and lack of reproduction in a separate group of patients with ASD from which the algorithm was generated (63). Similarly, Jiao et al. (64) incorporated cortical thickness and volume data from children with ASD and controls (ages 7–13) into a machine-learning model with the aims of predicting presence or absence of ASD. One
algorithm was able to predict diagnostic stratification with 87% accuracy based on cortical thickness measurements. The most predictive regions included both areas of decreased cortical thickness (in the left pars triangularis, orbital frontal gyrus, parahippocampalgyrus,and left frontal pole) and increased cortical thickness (left anterior cingulate and left precuneus) (64). Again, the case control design was not representative of true population prevalence, precluding calculation of positive predictive values.
*Summary and comparison.* In ASD, a small number of studies support a pattern of very early overgrowth in cortical surface area and volume (<2 years of age), which is immediately followed by cortical dysmaturation throughout childhood and adolescence, with evidence suggesting both exaggerated and impaired cortical thinning, depending on the study. Changes in cortical thickness and surface area seem to occur in non-overlapping regions. In COS on the other hand, cortical thickness is reduced diffusely in childhood, although data from very young patients (<8 years) are lacking. During adolescence, reductions in cortical thickness become more localized to frontal regions, although less has been written about the specific rates of cortical thinning in this patient group.
#### **REGIONS OF INTEREST**
Studies seeking out and investigating specific regions of interest in both COS and ASD have employed several different approaches. On the one hand, a general approach simultaneously comparing dozens of regions of interest or thousands of specific points in the absence of an *a priori* defined hypothesis has been used to survey for areas associated with the greatest differences between patient and control samples, and can help guide future areas of research. On the other hand, a predefined hypothesis regarding volumetric differences in a particular region allows optimization of statistical power, to more precisely elucidate candidate regions.
#### **Regions of interest in COS**
A meta-analysis of studies conducted in adult onset schizophrenia patients describes global deficits in volume, most consistently in the left superior temporal gyrus and the left medial temporal lobe (65). Looking specifically at COS, in the NIMH cohort, an automated and longitudinal analysis of over 40,000 points across the cortical surface found that the superior and middle frontal gyris showed the greatest overall reduction in cortical thickness compared to controls (46). In a different sample COS population from UCLA, specific analysis of the right posterior superior temporal gyrus (Wernicke's area, involved in verbal comprehension), found volume to be *increased* in this region (66). Investigations conducted by the same group on the anterior cingulate gyrus, a central and highly connected structure in the prefrontal cortex involved in many functions including error monitoring, yielded volume reductions (67).
Hypothesis driven approaches in the NIMH-COS cohort have been able to identify specific regional volume deficits as well. The insular cortex, for example, has been implicated in schizophrenia, given its role in distinguishing self from non-self, in visceral somatosensory interpretation, in processing of emotional experiences, and in salience. Patients with COS were found to have smaller insular volumes, whereas COS-siblings and controls were not statistically different, suggesting reduced insular size as an indicator of disease state. Additionally, level of functioning and severity of symptoms correlated with insular volume (68).
The cerebellum, classically understood to be involved in motor coordination and planning, has been implicated in schizophrenia given its association with learning and cognition. In longitudinal data from the NIMH cohort, smaller overall and regional cerebellar volumes were detected in affected individuals, with siblings falling between patients and controls on various measures (69).
Regarding subcortical structures, enlargement of the caudate (70) has been shown. In the limbic system, increased amygdala volume (71), but volume loss in the hippocampus and fornix (72, 73) has also been found in COS.
#### **Regions of interest in ASD**
Brain regions proposed to play a role in social cognition, communication, and "theory of mind" have been a focus of investigation in ASD. The region of the temporoparietal junction in particular, is thought to be central to the integration of social information and empathy, as well as selective attention to salient stimuli (74). Thinning of several areas in the temporoparietal region, particularly on the left side, has been shown in children, adolescents, and adults with ASD (38, 57, 59, 61, 75).
The orbital frontal cortex, in the ventromedial prefrontal region, is thought to play a role in sensory processing, goal directed behavior, adaptive learning, and attachment formation (76). Patients with autism, despite increased overall cortical thickness in the frontal region, have been shown to have specific deficits in cortical thickness (38), volume, and surface area (62) in the orbital frontal cortex, which correlated with symptoms severity (62). Other frontal lobe structures showing reduced cortical thickness in ASD include the inferior and middle frontal gyri, and the prefrontal cortex, depending on the study (38, 64, 77).
The anterior cingulate is a highly connected part of the social brain network situated along the medial aspect of the frontal cortex. Its role in self-perception, social processing, error monitoring, and reward based learning has been described (78). Relative increases (60, 64) and decreases (62, 75, 77) in volume and thickness of the anterior cingulate have been shown in ASD. Given that different regions may grow at different rates in individuals with ASD vs. controls (60, 61), variation in the age and distribution of study populations may account for some inconsistencies.
Volume deficits in the insular cortex have been demonstrated in young adults with pervasive developmental disorders (79). In adults with ASD, those who had a history of psychotic symptoms also demonstrated reduced insular volumes, particularly on the right side, as well as reduced cerebellar volumes (80).
Looking at subcortical structures, the caudate has been shown to be enlarged in ASD, across whole brain volumetric metaanalyses (81–83), and in targeted ROI analysis, even after controlling for confounding medication administration (84). Volume loss in the putamen has been shown across whole brain meta-analyses in adults with ASD (81, 83, 85), but enlargement of the putamen has also been observed in younger populations (86). In the amygdala, volume losses emerge across whole brain meta-analytic approaches (83, 85, 87), but volume gains are noted in younger patient groups as well (88). From a functional perspective, enlargement of the caudate may be associated with repetitive or self-injurious behavior (89–92), while volume loss in the amygdala may pertain to impaired emotional perception and regulation (93).
*Summary and comparison.* Volume losses have been noted in some overlapping prefrontal regions in both ASD and COS, particularly along the middle frontal gyrus. The anterior cingulate is also implicated in both conditions, although bidirectional changes in volume have been noted in ASD, depending on age of study participants. The area of the temporal-parietal junction shows volume loss in ASD, and was an area strongly predictive of diagnosis in group of individuals with COS (discussed in see Cortical Thickness and Morphology in COS). The insula is implicated in patients with COS, and in those with ASD who have comorbid psychotic symptoms. Looking at deep structures, both conditions are associated with volume gains in the caudate, which may pertain to repetitive behaviors, or concomitant neuroleptic treatment.
#### **STRUCTURAL WHITE MATTER ANALYSIS (VOLUMETRIC ANALYSIS AND DIFFUSION TENSOR IMAGING)**
Magnetic resonance imaging analyses that incorporate diffusion measurements allow for further sub-characterization of white matter microstructure, above volumetric differences. The diffusion of water molecules is measurable with MRI technology, and the magnitude and direction of diffusion within each individual voxel can be modeled mathematically with vector algebra. Axial diffusivity (AD) is the measurement of diffusion occurring *parallel* to white matter fibers; increased AD occurs in diseases involving axonal degeneration, and is thought to reflect both the integrity and density of axon structures. Radial diffusivity (RD) on the other hand, is a measurement of diffusion occurring *perpendicular* to the white matter fibers; it is used as a measure of myelination, and is increased in demyelinating diseases. Mean diffusivity (MD) (also known as the apparent diffusion coefficient, ADC) is a measure of average diffusion in absence of a directional gradient (94).
A summary ellipsoid vector incorporating the overall spherical nature of the combined vectors is termed "fractional anisotropy" (FA). A perfectly "isotropic" solution (FA = 0), such as free water, contains molecules that diffuse freely in all directions, whereas an anisotropic solution (i.e., a white matter fiber bundle) would restrict diffusion in one direction resulting in an elongated ellipsoid and FA values closer to 1. In white matter tract analysis, increased FA is thought to be a sensitive but not specific measure of fiber myelination, the integrity of cell membranes as well as the diameter of the fibers (95). Typically developing individuals show age related increases in FA and decreases in MD throughout development, in keeping with increasing white matter maturation (96). As in gray matter analyses, DTI can be applied to the whole brain in a voxel-based approach, or alternatively, specific regions of interest can be investigated with this method. Along these lines, specific anatomic white matter tracts can be reconstructed and analyzed from DTI data, in a method known as tractography. DTI data can also be transposed onto a common FA template, in tract-based spatial statistics (TBSS) (97).
Magnetic resonance imaging data collected in the absence of diffusion measurements can still be utilized in studying white matter integrity and growth. Similar to gray matter analysis, simple volumetric studies on white matter structures have been employed. Alternatively, 3D mapping of volumetric changes in white matter tracts via tensor-based morphometry (TBM) has been validated as a method of studying white matter development over time. In brief, TBM applies initial and follow up scans to a standardized brain template to ensure precise anatomical alignment. Next, an elastic-deformation algorithm is used to calculate the specific degree of volume expansion in a set area, represented by an expansion factor called the "Jacobian determinant." Growth rates are calculated by comparing the Jacobian determinant measures across patient and control samples.
#### **WHITE MATTER ANALYSIS IN COS**
The corpus callosum is the largest white matter structure in the human brain, and is central for connectivity and relay of information between hemispheres. Deficits in the corpus callosum have been inconsistently demonstrated in adult onset schizophrenia populations (95). In a longitudinal analysis of children and young adults with COS, differences in the midsagittal area of the splenium of the corpus callosum emerged around age 22, with patients having significantly smaller structures (98). Later analysis looking at volumetric differences in subsections of the corpus callosum revealed no differences between NIMH-COS patients, their siblings and controls with respect to overall volume, and/or volume change over time (99).
Comparison of whole brain TBM data between 12 patients with COS and 12 age matched controls followed over a 5-year interval revealed aberrant white matter development between ages 13 and 19 years. Specifically, at baseline MRI, patients had a 15% deficit in white matter volume in the frontal regions. At follow up, control patients showed an average of 2.6% growth in white matter per year, while COS patient had only 0.4% white matter growth
**Table 1 | Summary of white matter findings in ASD and COS**.
**COS vs. controls ASD vs. controls White matter volume in COS DTI in COS Meta-analysis on white matter volume in ASD Meta-analysis on DTI in ASD** Study (160); (99); (98) (102) (109) (110) Mean age of patient group (160) 14.1–18.7; (99) 17.3; (98) 14.8 14.7 21.4 15.2 Whole brain white matter ↓ (160) ND FA ND – Corpus callosum ↓ (98); ND (99) ND FA; ↑ RD/AD in LI ↓ ↓ FA; ↑ MD Superior longitudinal fasciculus – ND FA; ↑ RD/AD in LI (L) – ↓ FA (L); ↑ MD Arcuate fasciculus – ND FA ↑ – Inferior longitudinal fasciulus – ND FA; ↑ RD/AD in LI (L) – ND FA Inferior fronto-occipital fasciculus – ND FA; ↑ RD/AD in LI (L) ↑ ND FA Cingulum ↓ (160) ND FA ↓ ND FA Uncinate fasciulus – ND FA ↑ ↓ FA (L); ND MD
Note that for ASD, significant findings are reported from meta-analyses only. COS, childhood onset schizophrenia; ASD, autism spectrum disorder; DTI, diffusion tensor imaging; ND, no difference; FA, fractional anisotropy; RD, radial diffusivity; MD, mean diffusivity; AD, axial diffusivity; L, left side; R, right side; LI, COS patients with language impairment.
per year. The white matter deficits in the COS sample seemed to progress in a front to back pattern, opposite to previous findings regarding gray-matter deficits, but consistent with expected growth patterns in healthy adolescent brains (100). Unaffected siblings of children with COS showed delayed white matter growth at younger ages (<14 years) but not at older ages (14–18 years) as measured by TBM. Delayed white matter growth was most significant in the parietal regions for siblings, but normalized by age 18 (101).
There are relatively few DTI studies in specific COS populations. Clark et al. (102) found no significant differences in FA diffusely between 18 children and adolescents with COS, and 25 controls. Of note, five COS patients had a comorbid diagnosis of ASD, of which four were tested as having a linguistic impairment. Increased RD and AD was noted for patient vs. control groups in several white matter tracts (see **Table 1**). Increases in RD and AD in these regions were explained primarily by the presence of a linguistic impairment, and not the diagnosis COS, however (102).
There is a growing body of literature, however, on diffusion tensor imaging in adult onset schizophrenia and early-onset schizophrenia (EOS: defined as symptom onset prior to age 18 years). Findings investigating these patient groups are summarized in several reviews (103, 104). Given the paucity of literature applying DTI in COS, some conclusions may be extrapolated from the earlyonset schizophrenia literature; therefore they will be discussed briefly.
In general, while results have varied, the corpus callosum, superior and inferior longitudinal fasciculus, cingulum, and the uncinate fasciculus have been suggested as areas most affected with respect to white matter integrity as measured by decreases in FA (103, 104). Some studies have attempted to correlate DTI findings with symptomatology. Ashtari et al. (105), for example, found decreased FA in the left inferior longitudinal fasciculus was more pronounced for EOS patients with a history of visual hallucinations (105). As in volumetric imaging, studies that incorporate
analyses for age effects provide evidence of dynamic white matter abnormalities as well, in EOS. For example, FA in the anterior cingulate region increased with age in the healthy control population, but decreased with age in the early onset psychosis population (106). Similarly, patients with EOS showed decreased FA in parietal regions, while patients with adult onset schizophrenia had findings localizing to the frontal, temporal, and cerebellar regions (107).
#### **WHITE MATTER ANALYSIS IN ASD**
Earlier volumetric analyses suggested a pattern of accelerated of white matter volume and growth in younger children, particularly in the frontal regions, but that adolescents with ASD had similar or reduced white matter volume compared to controls (108). Meta-analysis of 13 VBM studies on white matter volume found no differences globally in white matter volume, and no differences between child/adolescent groups and adults groups, although no studies included very young children (<6 years). Some regional differences emerged, however (109) (see **Table 1**).
With respect to diffusion tensor imaging, a recent systematic review and meta-analysis, combining DTI data from 14 studies, including both children and adults with ASD, summarized some areas of consensus and heterogeneity in the literature. In summary, decreased FA was most consistently demonstrated in the corpus callosum, left uncinate fasciculus, and left superior longitudinal fasciculus of individuals with ASD. Mean diffusivity was increased in the corpus callosum, and bilaterally in the superior longitudinal fasciculus (110). This meta-analysis included data from ROI and tractography studies only, however, excluding whole brain TBSS and voxel-based analyses. A recent literature review on DTI in ASD by Travers et al. (97), identified decreased FA, increased MD, and RD as the most common finding across methods, with the corpus callosum, cingulum, arcuate fasciculus, superior longitudinal, and uncinate fasciculus showing the greatest differences (97).
Most imaging studies in autism to date, as well as those included in the above-described meta-analyses, have been conducted in older children,adolescents,or adults. In these age groups, decreased FA and increased MD have been repeatedly documented in many white matter regions. The specific rate of change in white matter markers, as well as the effect of age on white matter maturation seems to vary by study, however. For example, Mak-Fan et al. (56) showed RD and MD measurements stayed stable between the ages 6 and 14 years in subjects with ASD, while control subjects showed expected decreases with age (111). Ameis et al. (112) found the between group differences in RD, AD, and MD, but not FA, which were more pronounced in childhood than in adolescence (112).
Few studies have been conducted in very young children, however, and less consistency emerges in the data from this age range. Contrary to literature in older populations, Weinstein et al. (113), reported that FA was *greater* for children ages 1.5–6 years with ASD compared to controls in the areas of the corpus callosum, superior longitudinal fasciculus, and cingulum. Differences in FA were attributable to decreased RD, while AD was the same between cases and controls (113). Similarly, Ben Bashat et al. (114), found evidence of accelerated white matter maturation marked by increased FA and reduced displacement values in a small sample of children with ASD ages 1.8–3.3 years, most prominently in frontal regions
(114). Abdel Razek and colleagues (115), found ADC scores to be greater for preschool children with ASD in several regions, which correlated with severity of autistic symptoms as measured by the childhood autism rating scale (115). Walker et al. (116) on the other hand, found that 39 children between ages 2 and 8 years with ASD had decreased MD and FA compared to controls, accompanied by an attenuated rate of increase in FA, as well an accelerated rate of decreased MD compared to controls (116). Longitudinal data looking at high risk infants found evidence of higher FA at 6 months in children who were subsequently diagnosed with ASD, but that they had then had a slower rate of change such that by 24 months typically developing children had surpassed them in this measure (117).
For most studies, although differences have been statistically significant for certain regions, the magnitude of these differences has been quite small, on the range of 1–2%, thus limiting the predictive ability of any individual measurement. Lange et al. (118) generated a discriminant function that was able to distinguish between individuals with and without ASD with 94% sensitivity, 90% specificity, and 92% accuracy, by combining the predictive ability of DTI data points centered primarily around the superior temporal gyrus and the temporal stem. The sensitivity and specificity was reproduced in a replicate sample as well, however the case-control design was not reflective of true population prevalence, again precluding inferences regarding predictive ability in a real life clinical setting (118).
Emerging efforts have tried to correlate neuroimaing findings to functional and behavioral outcomes. For example, increased MD in the superior longitudinal fasciculus correlated with degree of language impairment in children and adolescents (119). Increase FA and decreased RD in the arcuate fasciculus correlated with greater language abilities in another group of children with ASD (120). Similarly, lower FA in the dorsal lateral prefrontal region was associated with increased social impairment in a group of children with ASD in Japan (121). Attempts to identify structural deficits in areas involved socio-emotional processing have yielded mixed results as well. Further focus on understanding the functional connectivity between distant regions is described in the next section.
*Summary and comparison.* White matter development in COS patients compared to controls appears marked by global deficits in white matter volume and decreased rates of white matter growth/integrity in adolescence, although the specific chronology, most affected regions and the relation to symptoms continues to be explored. In ASD, meta-analyses suggest no differences overall in white matter volume in adults, although early white matter volumetric overgrowth may occur in younger patient samples. Looking at specific white matter regions, volume losses have been noted in both ASD and COS in the corpus callosum and cingulum. In both conditions, decreased white matter integrity as measured though DTI has been observed in the superior longitudinal fasciculus, which may pertain to comorbid language impairments.
### **FUNCTIONAL CONNECTIVITY**
While imaging of white matter tracts through techniques like DTI permits the quantification of structural connectivity between regions, *functional* connectivity requires *in vivo* analysis of brain activation. Functional magnetic resonance imaging (fMRI) measures regional changes in blood oxygen level dependent (BOLD) signaling, given the subtle differences in magnetic field strength between oxygenated and deoxygenated blood. Brain activation patterns may be analyzed in subjects at rest (termed resting state) or during a specific cognitive or behavioral task performed in an MRI scanner. Data can be analyzed with respect to a specific region of interest (seed technique), where connections to and from an *a priori* defined region are studied. Alternatively, independent component analysis (ICA), or similar techniques, look at overall activation patterns across all regions, and can comment on patterns in functional networks (i.e., default mode network, salience network). Data from functional neuroimaging studies are often analyzed using graph theory. In this approach, the relationship between certain areas of central activation (termed "nodes") and the vectors of connectivity between nodes (termed "edges") are described using discrete mathematics (122). Short-range connectivity (i.e., within a specific lobe, or to a neighboring lobe) and long-range connectivity between remote regions can be quantified in this manner.
#### **FUNCTIONAL CONNECTIVITY IN COS**
Two separate analyses in the NIMH cohort of COS have suggested exaggerated long-range connectivity, and impaired short-range connectivity, in keeping with a hypothesis of exaggerated synaptic pruning. Resting state fMRI data was used to graph the connectivity between 100 regional nodes for 13 patients and 19 controls. Data showed that patients with COS had signals that were less clustered with more disrupted modularity marked by fewer edges between nodes of the same module. On the other hand, they showed greater global connectedness and greater global efficiency (123). Subsequent analyses with a slightly larger sample again found reduced connectivity at short distances and increased connectivity at long distances for patients with COS compared to controls on resting state fMRI. Relative to healthy controls, patients with COS had several regions in the frontal and parietal lobes that were "nodes" of over-connectedness with respect to long-range associations (124). White et al. (125) on the other hand, interpreted an opposite pattern from a study using a visual stimulus to analyze connectivity in the occipital lobe of children and adolescents with early onset schizophrenia (125). Similarly, structural connectivity analysis in neonates at high risk for schizophrenia found decreased global efficiency, increased local efficiency, and fewer nodes and edges overall compared to control infants (126).
#### **FUNCTIONAL CONNECTIVITY IN ASD**
In ASD on the other hand, there is an abundance of recent literature on functional connectivity. An emerging hypothesis suggests that frontoparietal under connectivity in ASD results in reduced "bandwidth" in long-range circuits [reviewed by Just et al. (127)]. Some propose that this coincides with local increases in connectivity within a specific lobe, resulting in a failure to integrate and regulate multiple sources of information (128). This hypothesis is consistent with structural white matter deficits in long-range association fibers, as well as structural patterns in gray matter showing increased local, but deficits in global modularity (129).
With respect to functional analyses, impaired synchronization, and under connectivity between large-scale networks has been shown in fMRI studies incorporating various task-based assessments, including those pertaining to language comprehension and auditory stimuli (130–132), executive functioning (133), visual spatial processing (134), and response to emotional cues (135, 136). Under connectivity has not been the only finding however, with many functional MRI studies showing evidence of increased connectivity or altered developmental trajectories with respect to integrated neural networks (137–139). For example, a recent meta-analysis of fMRI studies found greater activation in children with ASD in response to a social task in certain specific regions (i.e., in the left-precentral gyrus) but relative under activation compared to controls in other areas (superior temporal gyrus, parahippocampal gyrus, amygdala, and fusiform gyrus). In adults with ASD, activation was greater in the superior temporal gyrus, but less in the anterior cingulate during social processing (140).
The literature is also divided with respect to functional neuroimaging in resting state MRI, in the absence of any particular stimulus or task. Some have proposed that methodological issues may be contributing to observed inconsistencies (141). While hypoconnectivity seems most prevalent in the literature, [Ref. (142, 143); reviewed by Uddin et al. (144)], Uddin et al. (144) observed long-range hyperconnectivity via ICA across remote regions in 20 children ages 7–12 years with autism compared to controls. Hyperconnectivity was noted to involve the default mode network, frontotemporal, motor, visual, and salience networks. Hyperconnectivity of the salience network (which involves the anterior cingulate and insula) was most predictive of the diagnosis of ASD and was able to discriminate between cases and controls with 83% accuracy, a finding that was reproduced in a separate image dataset (145). Other resting state fMRI studies have also observed mixed patterns, which vary by region, network, and by age of the sample (146, 147).
The literature in very young patients with ASD is relatively sparse but seems to suggest altered developmental trajectories for affected children beginning at very young ages. A recent publication observed increased functional connectivity at 3 months, which disappeared by 12 months in high risk infants (148). Alternatively, Redcay and Courchesne (139) found increased connectivity between hemispheres in 2–3 year old children with ASD compared to chronological age matched controls, however the opposite pattern emerged when they were compared to mental age matched controls (139). Dinstein et al. (132) observed hypoconnectivity between hemispheres and in language regions in toddlers with ASD in response to auditory stimuli (132).
A recent review article by Uddin et al. (144) summarizes the literature to date with respect to resting state functional connectivity analyses. While intrinsic connectivity and seed-based analyses across 17 published studies suggest both hyper- and hypo-connectivity, Uddin and colleagues propose that the developmental age of the sample may be one explanatory factor with respect to variability in results. They describe a hypothesis in which increased functional connectivity in prepubescent children with ASD as compared to their peers is then met with altered maturational trajectories such that adults with ASD seem to have reduced connectivity compared to controls (144).
A recent publication put forth by a data sharing initiative entitled "autism brain imaging data exchange" (ABIDE) proposes to remedy disagreement in the literature through a large-scale international collaboration combining 1112 resting state fMRI scans. Analysis of 360 male subjects with ASD compared to controls found hypo connectivity in cortical networks but hyper connectivity in subcortical networks. They also identified localized differences in connectivity in certain regions, including the insula, cingulate, and thalamus. They did not perform specific analyses looking for age-associated differences, however, given that the majority of included participants were adolescents or adults (146).
*Summary and comparison.* There are only a handful of studies looking at functional connectivity in COS, but data from fMRI suggest a pattern of increased long-range connectivity, with disrupted short-range connectivity, in keeping with pathology of exaggerated synaptic pruning. In comparison, data from fMRI in ASD suggest to some extent an opposite pattern, with increased local but decreased global connectivity. fMRI data sharing between research centers reveal hyperconnectivity in subcortical networks, and hypoconnectivity in cortical networks in adult males with ASD. Smaller studies in younger age groups suggest important age effects regarding the connectivity hypothesis as well, with younger children with ASD seemingly showing more "over-connectedness" than adults.
### **DISCUSSION**
This review compares and contrasts neuroimaging findings in ASD and COS. Overall, across volumetric, structural, and functional neuroimaging data, there arises evidence for a dynamic changes in both conditions. In ASD, a pattern of early brain overgrowth is seemingly met with dysmaturation in adolescence, although the literature in this regard is far from certain. Functional analyses have suggested impaired long-range connectivity as well as increased local and/or subcortical connectivity, which may also progress with age. In COS, global deficits in cerebral volume, cortical thickness, and white matter maturation seem most pronounced in childhood and adolescence, and may level off in early adulthood. Deficits in local connectivity, with increased long-range connectivity have been proposed, in keeping with exaggerated cortical pruning; however the opposite has also been shown. Symptom and neuroimaging overlap across conditions was illustrated via a meta-analysis of fMRI data in both schizophrenia and ASD, which identified shared deficits in regions involved in social cognition (149).
The significance of these findings is tempered, however, by heterogeneity in results across other pediatric onset neurodevelopmental disorders. In ADHD for example, longitudinal MRI analyses in children suggest overall reduced cortical thickness prior to the onset of puberty (158) with peak cortical thickness and onset of cortical thinning occurring at later ages (159). In the future, clinical neuroimaging must be able to identify not only the presence of aberrant neurodevelopment, but also be able to discern across overlapping conditions.
While there is heterogeneity in the literature in both conditions, findings regarding COS at times appear more consistent. It is important to note that, given the rarity of this condition, these findings emerge from relatively few research samples, and are derived primarily from data collected from the same population of individuals. In ASD on the other hand, there has been an international explosion of investigation at numerous institutions, across ages, IQ ranges, and diagnostic severity, which has resulted in at times seemingly contradictory results. A call for collaboration (150) has been met with a first international compilation of neuroimaging datasets, which has helped to clarify some discrepancies in the literature with respect to fMRI (146). Going forward, ongoing collaboration to facilitate large scale, prospective, longitudinal neuroimaging studies, will be necessary to separate signals from noise in these complex and heterogeneous diseases. A focus on genetic subtypes may help to unite synapse pathology with neuroimaging findings and network dysfunction, to permit some degree of hypothesis generation with respect to molecular pathogenesis.
In ASD, for example, a loss of inhibitory control leading to exaggerated growth, premature cortical thinning, and then early stabilization of cortical structures has led some to suggest that overall the developmental curve has been "shifted to the left" along the time axis in this condition, with respect to brain maturation (75, 151). Current genetic investigations suggest alterations in structural scaffolding at the excitatory synapse could be contributory in ASD (152). Single gene disorders associated with autism may shed light on underlying final common pathways (153). Fragile X syndrome (FXS), for example, is a genetic condition comorbid with ASD in 20–30% of cases (154). Individuals afflicted with this condition have dysfunction or absence of the fragile X mental retardation protein (FMRP). FMRP is now understood to play a critical role in regulation of protein synthesis at the excitatory synapse, and without it, exaggerated receptor cycling and dysfunctional neuroplasticity can results (153). A similar mechanism in idiopathic ASD would hypothetically results in a loss of inhibitory control on expected maturational changes, uncoupling the structural and temporal timeline of synaptic neurodevelopment.
In schizophrenia, exaggerated synaptic pruning has been a long held hypothesis with respect to an etiology (25), which is consistent with aspects of the neuroimaging literature in COS. On the other hand, a small study in high risk infants suggests enlarged cerebral volumes may exist early in life, implying that some type of early dysregulated growth may be at play in this condition as well, similar to the process occurring in ASD (27). Investigations in 22q11.2 deletion syndrome (DS), a genetic disorder associated with schizophrenia in 20–25% of cases (155), permits longitudinal and prospective analysis of children at high risk for schizophrenia. Interestingly, MRI data collected in children as young as 6 years old with 22q11.2 DS found early increases in cortical thickness and deficits in cortical thinning in preadolescence, which are then met with exaggerated cortical thinning during adolescent years. Patients who subsequently developed schizophrenia indeed had more exaggerated deficits in cortical thickness (156).
In studies recruiting adolescents, it is difficult to tease out the possible influence of confounders such as substance abuse on both clinical and radiologic findings. While comorbid substance abuse is common in adult onset schizophrenia populations (occurring in 50–80% cases),the rate of substance abuse in COS,while presumed lower, has not been described (157). Ongoing study of clinical, environmental, and cultural confounding factors in both ASD and COS is needed.
Many investigators have sought to use neuroimaging protocols as predictors of diagnosis in case-control studies. The accuracy, sensitivity, and specificity of these analyses have on average ranged between 60 and 90%, and some groups have been able to reproduce high levels of diagnostic accuracy in separate patient samples. The clinical utility of these algorithms, however, remains uncertain in the absence of their application to populations reflecting realistic disease prevalence (i.e., positive predictive values are low or not reported). The development of clinically useful, cost-effective wide scale diagnostic tests for neurodevelopment conditions remains a common goal, and several groups have initiated prospective trials on high risk patient populations which may perhaps yield some hopeful results in the next decade.
#### **AUTHORS CONTRIBUTION**
Danielle A. Baribeau authored the manuscript. Evdokia Anagnostou developed the research topic, provided guidance, editing, and supervision.
#### **ACKNOWLEDGMENTS**
Funding for this research was provided by the Province of Ontario Neurodevelopmental Disorders Network (POND), supported by the Ontario Brain Institute.
#### **REFERENCES**
spectrum disorder. *J Am Acad Child Adolesc Psychiatry* (2010) **49**(6):e1–4. doi:10.1016/j.jaac.2010.02.012
(22q11DS): a cross-sectional and longitudinal study. *Schizophr Res* (2009) **115**(2–3):182–90. doi:10.1016/j.schres.2009.09.016
**Conflict of Interest Statement:** Danielle A. Baribeau has no financial conflicts to disclose. Evdokia Anagnostou has consulted to Seaside Therapeutics and Novartis. She has received grant funding from Sanofi Canada. The Guest Associate Editor – Stephanie Ameis – declares that, in spite of her adjunct affiliation with the same institution as authors Danielle A. Baribeau and Evdokia Anagnostou, the review process was handled objectively and no conflict of interest exists.
*Received: 29 September 2013; accepted: 09 December 2013; published online: 20 December 2013.*
*Citation: Baribeau DA and Anagnostou E (2013) A comparison of neuroimaging findings in childhood onset schizophrenia and autism spectrum disorder: a review of the literature. Front. Psychiatry 4:175. doi: 10.3389/fpsyt.2013.00175*
*This article was submitted to Neuropsychiatric Imaging and Stimulation, a section of the journal Frontiers in Psychiatry.*
*Copyright © 2013 Baribeau and Anagnostou. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.*
# **Upregulated GABA inhibitory function in ADHD children with child behavior checklist–dysregulation profile: 123I-iomazenil SPECT study**
*Shinichiro Nagamitsu<sup>1</sup> \*, Yushiro Yamashita<sup>1</sup> , Hitoshi Tanigawa<sup>2</sup> , Hiromi Chiba<sup>3</sup> , Hayato Kaida<sup>2</sup> , Masatoshi Ishibashi <sup>2</sup> , Tatsuyuki Kakuma<sup>4</sup> , Paul E. Croarkin<sup>5</sup> and Toyojiro Matsuishi <sup>1</sup>*
*<sup>1</sup> Department of Pediatrics and Child Health, Kurume University School of Medicine, Fukuoka, Japan, <sup>2</sup> Department of Radiology, Kurume University School of Medicine, Fukuoka, Japan, <sup>3</sup> Department of Psychiatry, Kurume University School of Medicine, Fukuoka, Japan, <sup>4</sup> Biostatistics Center, Kurume University School of Medicine, Fukuoka, Japan, <sup>5</sup> Department of Psychiatry and Psychology, Mayo Clinic, Rochester, MN, USA*
#### *Edited by:*
*Stephanie Ameis, The Hospital for Sick Children and University of Toronto, Canada*
#### *Reviewed by:*
*Rosana Lima Pagano, Hospital Sírio-Libanês, Brazil Mehereen Bhaijiwala, The Hospital for Sick Children, Canada*
#### *\*Correspondence:*
*Shinichiro Nagamitsu, Department of Pediatrics and Child Health, Kurume University School of Medicine, 67 Asahi-machi Kurume City, Fukuoka 830-0011, Japan [email protected]*
#### *Specialty section:*
*This article was submitted to Neuropsychiatric Imaging and Stimulation, a section of the journal Frontiers in Psychiatry*
> *Received: 16 February 2015 Accepted: 18 May 2015 Published: 02 June 2015*
#### *Citation:*
*Nagamitsu S, Yamashita Y, Tanigawa H, Chiba H, Kaida H, Ishibashi M, Kakuma T, Croarkin PE and Matsuishi T (2015) Upregulated GABA inhibitory function in ADHD children with child behavior checklist–dysregulation profile: 123I-iomazenil SPECT study. Front. Psychiatry 6:84. doi: 10.3389/fpsyt.2015.00084* The child behavior checklist–dysregulation profile (CBCL–DP) refers to a pattern of elevated scores on the attention problems, aggression, and anxiety/depression subscales of the child behavior checklist. The aim of the present study was to investigate the potential role of GABA inhibitory neurons in children with attention deficit/hyperactivity disorder (ADHD) and dysregulation assessed with a dimensional measure. Brain single photon emission computed tomography (SPECT) was performed in 35 children with ADHD using 123I-iomazenil, which binds with high affinity to benzodiazepine receptors. Iomazenil binding activities were assessed with respect to the presence or absence of a threshold CBCL–DP (a score *≥*210 for the sum of the three subscales: Attention Problems, Aggression, and Anxiety/Depression). We then attempted to identify which CBCL–DP subscale explained the most variance with respect to SPECT data, using "age," "sex," and "history of maltreatment" as covariates. Significantly higher iomazenil binding activity was seen in the posterior cingulate cortex (PCC) of ADHD children with a significant CBCL–DP. The Anxiety/Depression subscale on the CBCL had significant effects on higher iomazenil binding activity in the left superior frontal, middle frontal, and temporal regions, as well as in the PCC. The present brain SPECT findings suggest that GABAergic inhibitory neurons may play an important role in the neurobiology of the CBCL–DP, in children with ADHD.
**Keywords: CBCL-dysregulation profile, iomazenil, GABA, ADHD**
## **Introduction**
Severe behavioral and affective dysregulation with symptoms, such as hyperactivity, aggression, irritability, mood instability, and anxiety, contribute to significant academic and psychosocial impairment in children. Some of these symptoms are consistent with attention deficit hyperactivity disorder (ADHD). ADHD is the most frequent neuropsychiatric disorder in children and often presents with co-occurring disruptive behavior disorders, anxiety disorders, and bipolar disorder. Hyperactivity, irritability, and impulsivity place children at risk of maltreatment as a result of strained parent–child interactions (1, 2). The insecure parent–child relationship further exacerbates the behavioral and affective dysregulation observed in children.
The child behavior checklist-dysregulation profile (CBCL–DP) refers to a pattern of elevated scores on the Attention Problems, Aggression, and Anxiety/Depression subscales of the child behavior checklist (CBCL) (3). The CBCL–DP was originally proposed as a means of identifying youth with bipolar disorder (4). However, recent studies suggest that the results of the CBCL–DP are not simply an early manifestation of a single disease process, but rather that the CBCL–DP can be used as a developmental risk marker for a persisting deficit in self-regulation of affect and behavior (5, 6). The CBCL–DP may be best interpreted as an indicator of symptom severity and functional impairment (7, 8). Children with ADHD who had a threshold level CBCL–DP score (*≥*210) showed higher rates of comorbidity disorders, including oppositional defiant disorder (ODD), conduct disorder (CD), anxiety disorder, bipolar disorder, and depression (9). The CBCL–DP is also associated with mood, anxiety, disruptive behavior disorders, and substance use in adulthood (3).
The underlying neurobiological defects or aberrant neuronal activity leading to the dysregulation profile in children with ADHD are elusive. Reducing serotoninergic function in children with ADHD and a significant CBCL–DP resulted in slower cognitive performance compared to children with ADHD who did not have the CBCL–DP, indicating that serotoninergic function could play a decisive role in the etiology of the CBCL–DP (10). In addition, the CBCL subscale of "Aggression" was found to be the main discriminator of ADHD children with CBCL–DP versus those without CBCL–DP with respect to serotoninergic dysfunction. Conversely, prior translational work with magnetic resonance spectroscopy and transcanial magnetic stimulation paradigms suggest that GABAergic neurochemistry and neurotransmission are dysregulated in children with ADHD (11). Ongoing work also suggests that defects in the GABAergic system in adults increase an individual's vulnerability to severe psychiatric illnesses due to aberrant regulation of serotoninergic and/or dopaminergic neurons (12, 13). Previous biochemical and pharmacological studies indicate that deficits in GABA receptor function, induced by intravenous infusion of iomazenil followed by a serotoninergic agonist, predispose healthy volunteers to increased anxiety and dissociative disturbances, suggesting that deficits in the GABAergic system may contribute to the pathophysiology of serotonininduced psychosis (12).
123I-iomazenil is a radioactive ligand for central-type benzodiazepine receptors that forms a complex with GABA(A) receptors. Thus, 123I-iomazenil single photon emission computed tomography (SPECT) can indirectly index GABA receptor function. 123Iiomazenil is a frequently used radionuclide tracer for presurgical evaluation of patients with refractory partial epilepsy (14, 15). Moreover, recent neuroimaging studies have explored the role of GABAergic inhibitory function in psychiatric disorders such as schizophrenia, anxiety disorders, and developmental disorders (16–20). To our knowledge, there is no previous work which characterizes GABA receptor functioning with 123I-iomazenil SPECT among children with ADHD. The working hypothesis of the present study was that behavioral and affective symptoms in children with ADHD, reflected in CBCL–DP scores, would correlate with changes in cortical GABAergic neuronal activity. To confirm this hypothesis, brain SPECT was performed using 123Iiomazenil in ADHD children with or without CBCL–DP. Further, we tried to identify which of the three significant scales in the CBCL–DP explains the most variance with respect to SPECT data using "age," "sex," and "history of maltreatment" as covariates.
### **Materials and Methods**
### **Ethics Statement**
The design of the study and procedures for obtaining informed consent were approved by the Medical Ethics Committee of Kurume University School of Medicine (#10081). Informed consent was obtained from each child and his/her parents prior to their participation in the study.
### **Participants**
Thirty-five children with ADHD (23 boys, 12 girls) enrolled in the study. Participants were recruited after visits from the Department of Pediatrics, Kurume University, for the management of externalizing symptoms (e.g., difficulty maintaining attention, restlessness, hyperactivity, and aggressive behavior) or internalizing symptoms (e.g., anxiety, dissociation, and depressive symptoms). A diagnosis of ADHD was made using the *Diagnostic and Statistical Manual of Mental Disorders, 4th Edition, Text Revision (Dsm-Iv-Tr)* (21). Children who had anxious or depressive symptoms, but did not have ADHD symptoms were excluded in this study. Of the 35 child participants with ADHD, 15 had the combined type, 11 had hyperactive-impulsive type, and 9 had inattentive type. Seventeen children (7 male, 10 females) had experienced an obvious maltreatment, such as physical (*n* = 9), psychological (*n* = 6), or sexual abuse (*n* = 1), or sexual assault (*n* = 1) during preschool. In 11 of these instances, a child-welfare consultation center had previously supported the families in hopes of preventing maltreatment. Two children stayed in a child residential care institution. However, none of the participants met the diagnostic criteria for posttraumatic stress disorder (PTSD) on assessment. Seven of the 35 subjects (20%) had comorbid disorders, such as ODD (*n* = 2), CD (*n* = 1), anxiety disorder (*n* = 2), and depression (*n* = 2). The mean age of the children at the time of their hospital visit was 10.4 years. All participants were medication naïve prior to enrollment.
### **Child Behavior Checklist**
Behavioral and psychiatric assessments of the children included the CBCL, ADHD rating scale (hyperactivity/impulsive and inattention scores, as well as total score) (22, 23), the Child Depression Inventory (CDI), the Child Dissociative Checklist (CDC), and the Wechsler Intelligence Scale for Children (WISC-III). The CBCL was used to evaluate children's emotional and behavioral functioning, competencies, and social problems, with specific items evaluating internalizing and externalizing symptoms, as well as attention and thought problems. Items evaluating internalizing symptoms focus on withdrawal, somatic complaints, and anxiety/depression. Items evaluating externalizing symptoms focus on delinquent or aggressive behavior. The CBCL–DP refers to Nagamitsu et al. GABA in children with ADHD
a pattern of elevated scores on the Attention Problems, Aggression, and Anxiety/Depression subscales of the CBCL. A threshold CBCL–DP was defined as a score *≥*210 for the sum of three subscales. (4) Physicians rated the participants using the ADHD rating scale, CDI, and CDC, and the parents rated their children using the CBCL.
### **Iomazenil Single Photon Emission Computed Tomography and Analysis of Regions of Interest**
All 35 children underwent iomazenil SPECT imaging of the brain. Briefly, children were injected intravenously with a bolus of 95–117 MBq 123I-iomazenil (Nihon Medi-Physics, Tokyo, Japan), which binds with high affinity to benzodiazepine receptors. The SPECT scan was performed 3 h after injection of the tracer, without any sedation, using a large field-of-view dual-detector camera and a computer system equipped with a low-energy, high-resolution, parallel-hole collimator. The dual detector camera rotated over 180° in a circular orbit and in 32 steps of 40 s each to cover 360° in about 22 min. Brain magnetic resonance imaging (MRI) was performed using a superconducting magnet operating at 1.5 T. For coregistered SPECT and MRI analysis, a method of image integration was applied using Fusion Viewer software (Nihon Medi-Physics) with a registration algorithm based on maximum mutual information (**Figure 1**). Subsequently, the cortical and subcortical regions of interest (ROIs) in the acquired SPECT data were defined. Using elliptical templates, the ROIs were placed over the following regions: the superior frontal, middle frontal, parietal, temporal, and occipital regions in each hemisphere; the midbrain; and the anterior and posterior cingulate cortex (ACC and PCC, respectively; **Figure 1**). Each relative iomazenil binding activity in ROIs was expressed as a ratio of that in the occipital cortex. As 123I-iomazenil affinity in the occipital region was maximum and stable in brain cortex, the occipital region was used as a reference (24).
### **Data Analysis**
The differences of each CBCL subscale, ADHD-RS, CDI score, CDC score, and Intelligence scale between ADHD children with/without CBCL–DP were compared by student's *t*-test. We first analyzed correlations between the relative iomazenil binding activity expressed as a ratio in each ROI and psychometric profiles after controlling for the effects of age, sex, and history of maltreatment. Further, we compared iomazenil binding activity with respect to the presence or absence of a threshold CBCL–DP score in these children and tried to identify which of the three CBCL–DP subscales explained the most variance with respect to the SPECT data. Associations between each of the CBCL–DP subscales and iomazenil binding activity in each brain area were evaluated using liner regression models, with "age," "sex," and "history of maltreatment" as covariates.
## **Results**
Behavioral and psychiatric assessments of the participants were shown in **Table 1**. Of the 35 participants, 15 had a threshold CBCL–DP score (i.e., a score *≥*210) and 20 had CBCL–DP scores *<*210. The group with threshold CBCL–DP scores had a lower
**FIGURE 1 | Designated regions of interest (ROIs) in fusion images of 123I-iomazenil SPECT and MRI**. The top panel shows brain MRI (transverse and sagittal T<sup>1</sup> sequences), the middle panel shows corresponding results of 123I-iomazenil SPECT, and the bottom panel shows fusion imaging. Outlined regions in the bottom panel indicate designated ROIs, namely (a) the superior frontal, (b) parietal, (c) middle frontal, (d) temporal, (e) occipital regions, (f) anterior, and (g) posterior cingulate gyrus.
ratio of male to female participants and more instances of maltreatment. Each ADHD rating scale, all subscales of CBCL with the exception of somatic problems, and CDC score were significantly higher in ADHD children with threshold CBCL–DP scores. Four participants with threshold level CBCL–DP scores had comorbidity disorders, including depression (*n* = 2), ODD (*n* = 1), and CD (*n* = 1). Three participants without threshold CBCL–DP scores had comorbidity disorders, including anxiety disorder (*n* = 2) and ODD (*n* = 1). There was a difference in the CBCL–DP scores between participants with/without comorbidity disorders; however, this difference did not reach statistical significance (*n* = 28, 199 *±* 20, and *n* = 7, 209 *±* 12, respectively, *p* = 0.10).
Analyses of all participants (*n* = 35) revealed correlations between iomazenil binding activity in several brain regions and some part of the CBCL profile, after controlling for the effects of age, sex, and a history of maltreatment (**Table 2**). In both ACC and PCC, iomazenil binding activity had a statistically significant positive correlation with scores on the Anxiety/Depressed (**Table 2**; **Figure 2**), Internalizing, and Withdrawal Problems subscales of the CBCL (**Table 2**). In addition, significant positive correlations were noted for iomazenil binding activity in the ACC and Thought Problems on the CBCL, as well as for iomazenil binding activity in the PCC and Attention problems and Social Problems on the CBCL (**Table 2**). These significant correlations were not seen for other combinations in other brain regions, except for iomazenil binding activity in the midbrain and Thought Problems on the CBCL, and iomazenil binding activity in the right temporal region and Social Problems on the CBCL. There were no significant correlations between iomazenil binding activities in any brain region and any of the ADHD rating scales, CDI score, and CDC score
*CBCL–DP, child behavior checklist–dysregulation profile; ADHD-RS, attention deficit hyperactivity disorder rating scale; CDI, child depression inventor; CDC, children dissociative checklist; WISC, Wechsler intelligence scale for children.*
*Significant difference from children without significant CBCL–DP (\*indicates p < 0.05, \*\*indicates p < 0.001).*
(data not shown). Iomazenil binding activity in the PCC was significantly higher in ADHD children with a threshold CBCL–DP score than in ADHD children with scores *<*210 after controlling for the effects of age, sex, and a history of maltreatment (**Table 3**, *F*-value = 4.36, *p <* 0.05). Of the three CBCL–DP subscales, the Anxiety/Depression subscale had significant effects on higher iomazenil binding activity in the left superior frontal, middle frontal, and temporal regions, as well as in the PCC (**Table 4**).
## **Discussion**
This is the first neuroimaging study showing that behavioral and affective symptoms in children with ADHD, reflected in CBCL–DP scores, are correlated with changes in cortical GABAergic neuronal activity. Overall, increased iomazenil activity in the ACC and PCC was associated with higher scores on many of the CBCL subscales. In ADHD children with a significant CBCL–DP, iomazenil activity was upregulated in the PCC. Of the three CBCL–DP subscales, the Anxiety/Depression subscale had a significant effect on iomazenil binding activity in many brain regions. These results suggest that behavioral and affective dysregulation in ADHD children may be characterized by changes of GABAergic neural activity. In this section, we discuss the role of the cingulate cortex in GABA function, the association between CBCL–DP scores and GABA function, and age-dependent differences in GABA function.
The cingulate cortex is one of the largest parts of the limbic lobe and the prefronto-limbic circuitry. The ACC plays key roles in emotion, motivation, and motor functions, whereas the PCC is involved in emotion, facial recognition, and memory functions (25–27). In the present study, we found higher iomazenil binding activity in ACC and PCC that was associated with higher scores on many of the CBCL subscales in ADHD children with and without CBCL–DP. Similar findings have been reported in healthy adults. For example, Kim et al. (28) found a positive correlation in healthy subjects between high GABA concentrations in the ACC and a high harm avoidance temperament, characterized by worrying about potential problems, fearful of uncertainties, and being shy in unfamiliar environments. Moreover, increased activity in the PCC has been observed in emotional disorders, including obsessive–compulsive disorder, major depression, and social phobia (29, 30). Because the cingulate cortex has been suggested to have an important role in modulating human fear and anxiety by modulating the activity of other limbic structures, including the amygdala (31), the increased GABAergic function in the cingulate cortex of ADHD children in the present study may have inhibited excessive excitation of the limbic system, which contributes to the development of behavioral and affective dysregulation.
We found that the Anxiety/Depression subscale of the CBCL–DP explains the most variance with respect to SPECT data in various brain regions using "age," "sex," and "history of maltreatment" as covariates. The Aggression and Attention Problem subscales of the CBCL–DP had no significant effects on SPECT data in various brain regions. These findings strongly support previous converging lines of evidence regarding the association between GABAergic activation and increased anxiety (32). Conversely, several biochemical and genetic studies have provided evidence of a significant role of serotoninergic function in aggressive behavior. For example, an inverse correlation has been reported between downregulated platelet or CSF 5-hydroxyindolecetic acid (5-HIAA), a major metabolite of serotonin, and levels of aggression and impulsivity (33, 34). Furthermore, Haberstick et al. (35) reported an association between certain promoter polymorphisms in the serotonin transporter (5HTTLPR) and greater aggressive behavior in middle childhood, suggesting that differences in serotonergic functioning may be a contributing factor to different levels of aggressive behavior. In terms of the biological mechanism underlying attention function, an important role for dopaminergic neurons has been proposed. Several neuroimaging studies have shown aberrant dopamine transporter (DAT) levels in the nucleus accumbens, caudate, and midbrain, as well as a positive relationship between DAT levels in the putamen and inattention scores in ADHD patients (36–38). Together, these findings suggest that changes in several neurotransmitter systems, including serotoninergic, dopaminergic, and GABAergic neurons, are likely to be involved in constructing the clinical manifestations of the CBCL–DP.
Significant positive correlations between GABAergic inhibitory function and the Anxiety/Depression subscale were also seen in our study. Although previous neuroimaging studies have reported those correlations in adulthood with psychiatric disorders (17, 28), the present study is the first report of the correlation in childhood with psychiatric disorders. Despite the positive correlation in childhood, previous neuroimaging studies using iomazenil SPECT have revealed negative correlations between
#### **TABLE 2 | Partial correlation coefficients between CBCL profiles and the iomazenil binding activites in each brain region**.
*CBCL, child behavior checklist; ACC, anterior cingulate cortex; PCC, posterior cingulate cortex; R, right; L, left.*
*\*Indicates p < 0.05; \*\*indicates p < 0.01.*
*a Indicates subscale which comprises CBCL–DP (dysregulation profile).*
**TABLE 3 | Effect of the significant CBCL–DP (score** *≥***210) on iomazenil binding activities in each brain region**.
*CBCL–DP, child behavior checklist–dysregulation profile.*
GABA–benzodiazepine receptor binding activity and the severity of anxiety symptoms in adults with panic or traumatic disorders (17–19). It is well known that there are considerable changes in the number of GABA receptors and in subunit expression during brain development (39). Specifically, the greatest number of GABA receptors is found in the youngest children, with numbers decreasing exponentially with age, and there are age-related increases in α1-subunit-containing GABA receptors (40). These age-related changes in GABA receptors may affect outcomes when assessing increases and/or decreases in overall iomazenil binding activity in children.
The present study has several limitations that require consideration in future studies. For example, in the present study, SPECT exhibited poor resolution around some limbic regions, such as the amygdala and hippocampus, which are important for emotion processing. In these small regions, the obtained radioactivity might differ from the true activity because of partial volume effect (PVE). The PVE can be defined as the underestimation of binding per unit brain volume in small objects or regions because of the blurring of the radioactivity (spillout and spill-in) between regions. These regions need to be resolved using MR imaging-based correction for PVE (41, 42). Brain imaging data from normal healthy children are not available
*CBCL–DP, child behavior checklist–dysregulation profile.*
*\*Indicates significant effects on higher iomazenil binding activity.*
because of ethical concerns with SPECT studies of this population. Therefore, we focused our research questions on the correlation between GABAergic inhibitory function in specific brain regions and psychometric profiles. It is possible that, in addition to the
### **References**
population of people with ADHD, our result is generalizable to the normal population. Furthermore, we selected iomazenil activity in the occipital regions as a reference, meaning that we could not evaluate inhibitory function in occipital regions. We did not clarify how putative dopaminergic or serotoninergic changes are involved in other subscales, such as the Aggression and Attention Problem subscales, of the CBCL–DP. It is possible that investigations incorporating the simultaneous assessment of benzodiazepine receptor binding activity and homovanillic acid (HVA) and 5HIAA in the urine (principal metabolites of dopamine and serotonin, respectively) could provide new insights into the underlying neurobiological defects or aberrant neuronal activity leading to the dysregulation profile in children.
In conclusion, the present 123I-iomazenil brain SPECT study provides evidence that changes in GABAergic inhibitory neuronal activity correlate with some elements of function measured by the CBCL–DP. Brain SPECT may be useful for the evaluation of the possible pathogenesis of neuropsychiatric symptoms observed in children.
### **Author Contributions**
SN participated in the design of this study and compiled the manuscript. SN, YY, and HC saw the patients and obtained informed consent and their agreement to participate in the study. Diagnosis of comorbidity disorders was made by HC. SN and HC summarized participant behavioral and psychiatric assessments, including CBCL–DP data. Three radiologists (HT, HK, and MI) were in charge of radioactive measurements and calculations of iomazenil activity using ROIs. TK, a statistician, conducted the statistical analyses. PC and TM supervised the preparation of the manuscript.
### **Acknowledgments**
This work was supported by grants from the Ministry of Education, Culture, Sports, Science, and Technology (#22591143, #25460643).
suicidality, and functional impairment: a longitudinal analysis. *J Child Psychol Psychiatry* (2011) **52**:139–47. doi:10.1111/j.1469-7610.2010.02309.x
**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
*Copyright © 2015 Nagamitsu, Yamashita, Tanigawa, Chiba, Kaida, Ishibashi, Kakuma, Croarkin and Matsuishi. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.*
# A preliminary study of white matter in adolescent depression: relationships with illness severity, anhedonia, and irritability
#### **Sarah E. Henderson<sup>1</sup> , Amy R. Johnson<sup>1</sup> , Ana I. Vallejo<sup>1</sup> , Lev Katz <sup>1</sup> , EdmundWong<sup>1</sup> and Vilma Gabbay 1,2\***
<sup>1</sup> Department of Psychiatry, Icahn School of Medicine at Mount Sinai, New York, NY, USA <sup>2</sup> Nathan S. Kline Institute for Psychiatric Research, Orangeburg, NY, USA
#### **Edited by:**
Paul Croarkin, Mayo Clinic, USA
#### **Reviewed by:**
Faisal Al-Otaibi, Alfaisal University, Saudi Arabia Niels Bergsland, Buffalo Neuroimaging Analysis Center, USA Kirti Saxena, Baylor College of Medicine, USA Susannah J. Tye, Mayo Clinic, USA
#### **\*Correspondence:**
Vilma Gabbay, Department of Psychiatry, Icahn School of Medicine at Mount Sinai, One Gustave L. Levy Place, New York, NY 10029, USA e-mail: [email protected]
Major depressive disorder (MDD) during adolescence is a common and disabling psychiatric condition; yet, little is known about its neurobiological underpinning. Evidence indicates that MDD in adults involves alterations in white and gray matter; however, sparse research has focused on adolescent MDD. Similarly, little research has accounted for the wide variability of symptom severity among depressed teens. Here, we aimed to investigate white matter (WM) microstructure between 17 adolescents with MDD and 16 matched healthy controls (HC) using diffusion tensor imaging. We further assessed within the MDD group relationships between WM integrity and depression severity, as well as anhedonia and irritability – two core symptoms of adolescent MDD. As expected, adolescents with MDD manifested decreased WM integrity compared to HC in the anterior cingulum and anterior corona radiata. Within the MDD group, greater depression severity was correlated with reduced WM integrity in the genu of corpus callosum, anterior thalamic radiation, anterior cingulum, and sagittal stratum. However, anhedonia and irritability were associated with alterations in distinctWM tracts. Specifically, anhedonia was associated with disturbances in tracts related to reward processing, including the anterior limb of the internal capsule and projection fibers to the orbitofrontal cortex. Irritability was associated with decreased integrity in the sagittal stratum, anterior corona radiata, and tracts leading to prefrontal and temporal cortices. Overall, these preliminary findings provide further support for the hypotheses that there is a disconnect between prefrontal and limbic emotional regions in depression, and that specific clinical symptoms involve distinct alterations in WM tracts.
**Keywords: depression, adolescent, white matter, diffusion tensor imaging, anhedonia, irritability**
### **INTRODUCTION**
Major depressive disorder (MDD) in adolescence is a prevalent and disabling psychiatric illness associated with serious consequences including academic failure, social withdrawal, substance abuse, and most critically, suicide (1–5). Converging evidence derived from neuroimaging studies suggests that adolescent MDD entails morphological, functional, and neurochemical alterations (6–10). Importantly, since adolescence represents a critical period of rapid neuroplasticity [e.g., increased myelination, synaptic pruning; (11–13)], white matter (WM) alterations can contribute to the neurobiology of adolescent MDD. Indeed, in our prior investigation of gamma-Aminobutyric acid in the anterior cingulate cortex (ACC), we incidentally found reduced WM volume in adolescents withMDD compared to healthy controls [HC; (9)]. However, there has been sparse research focusing on WM alterations in this age group.
Diffusion tensor imaging (DTI) enables the non-invasive examination of *in vivo* structural connectivity by providing measures of WM microstructure and integrity based on the extent of water diffusion (14). Several DTI measures are typically quantified, with fractional anisotropy (FA) being the most commonly used
to reflect WM integrity. Higher FA values suggest greater diffusion in the direction of the axon, and thus greater WM integrity. Other measures, including mean diffusivity (MD), radial diffusivity (RD), and axial diffusivity (AD), can also be determined to investigate different aspects of WM microstructure.
To date, most DTI research in MDD has investigated adults and consistently reported decreased FA in tracts connected to the prefrontal cortex (PFC) or tracts connecting the two hemispheres within the PFC (15, 16). Only one DTI study was carried out in adolescents with depression, demonstrating a similar pattern to adult MDD of reduced FA in tracts connected to the subgenual ACC and the PFC [i.e., uncinate fasciculus, inferiorfronto-occipital fasciculus, anterior cingulum, superior longitudinal fasciculus; (17)]. However, results may have been impacted by the concurrent use of psychotropic medication and past substance abuse in some subjects. Relatedly, medication-naïve adolescents with a familial risk for unipolar depression also demonstrated reduced FA compared to HC in similar tracts (18).
In this study, we aimed to expand on prior work by examining WM integrity in psychotropic medication-free adolescents with MDD compared to HC. Given the inherent heterogeneity of adolescent MDD, we further sought to identify specific WM alterations in relation to MDD severity as well as anhedonia and irritability – two core symptoms of adolescent MDD. Due to our desire to explore a range in depression severity, we included patients with mild to moderate severity. Based on others' and our prior findings (8, 15–17), we hypothesized that adolescents with MDD would manifest less restricted diffusion (i.e., decreased WM integrity) compared to HC in tracts connecting frontal, striatal, and limbic regions. We also predicted that similar tracts would be associated with depression severity. For anhedonia, we expected reduced WM integrity in tracts that have been implicated in reward-related processing in the ventral striatum [i.e., subgenual cingulate, forceps minor, inferior-fronto-occipital fasciculus, anterior thalamic radiation (ATR), anterior limb of the internal capsule; (19)], and that these would be distinct from those associated with irritability. Finally, only one study has examined irritability in a clinical population [i.e., Huntington's disease; (20)] and found involvement of the amygdala. As such, we predicted the impairment of tracts connecting to the amygdala (e.g., uncinate fasciculus, inferior-fronto-occipital fasciculus, inferior longitudinal fasciculus).
### **MATERIALS AND METHODS**
#### **PARTICIPANTS**
The sample population partially overlapped with that used for our previously published resting state functional magnetic resonance imaging research (8), and was comprised of 17 adolescents with MDD (ages 13–20 years, *M* = 16.8, *SD* = 2.2, 8 female) and 16 HC (ages 13–19 years, *M* = 16.4, *SD* = 1.4, 10 female) group matched for age, and all were right-handed. Depressed adolescents were recruited through the New York University (NYU) Child Study Center, the Bellevue Hospital Center Department of Psychiatry, and local advertisements in the NY metropolitan area. HC were recruited from the greater NY metropolitan area through local advertisements and from the families of NYU staff. The study was approved by the NYU School of Medicine and the Icahn School of Medicine at Mount Sinai institutional review boards. Prior to enrollment, study procedures were explained to the subjects and parents. Written informed consent was provided by participants age 18 and older; those under age 18 provided signed assent and a parent/guardian provided signed informed consent.
#### **Inclusion and exclusion criteria**
All subjects were≤20 years old and did not present with any significant medical or neurological disorders. Other exclusionary criteria consisted of an IQ < 80, MRI contraindications as assessed by a standard screening form, a positive urine toxicology test or a positive urine pregnancy test in females.
All MDD subjects met the DSM-IV-TR diagnosis of MDD with a current episode ≥8 weeks duration, and raw severity score ≥40 (i.e., T score ≥ 63) during the initial evaluation on the Children's Depression Rating Scale-Revised (CDRS-R), and were psychotropic medication-free for at least 7 half-lives of the medication. To explore a wider range of depression severity we included patients presenting with mild to severe depression on the date of the scan. Exclusionary criteria for the MDD group included current or past DSM-IV-TR diagnoses of bipolar disorder, schizophrenia, pervasive developmental disorder, panic disorder, obsessive-compulsive disorder, conduct disorder, Tourette's disorder, or a substance-related disorder in the past 12 months. Current diagnoses of post-traumatic stress disorder or an eating disorder were also exclusionary. In addition, acute suicidality requiring immediate inpatient admission was exclusionary. HC subjects did not meet the criteria for any major current or past DSM-IV-TR diagnoses and had never received psychotropic medication.
#### **CLINICAL ASSESSMENTS**
All subjects were assessed by a board-certified child and adolescent psychiatrist or a clinical psychologist at the NYU Child Study Center. Clinical diagnoses were established using the Schedule for Affective Disorders and Schizophrenia for School-Age Children-Present and LifetimeVersion [KSADS-PL; (21)], a semi-structured interview performed with both the subjects and their parents. Depression severity was assessed by the CDRS-R and the Beck Depression Inventory, second edition [BDI-II; (22)]. Additionally, suicidality and anxiety were assessed using the Beck Scale for Suicidal Ideation [BSSI; (23)] and the Multidimensional Anxiety Scale for Children [MASC; (24)], respectively. The Kaufman Brief Intelligence Test (25) or the Wechsler Abbreviated Scale of Intelligence (26) were used to estimate IQ. Urine toxicology and pregnancy tests were administered on the day of the scan.
#### **Anhedonia**
Our approach to quantifying anhedonia allows for clinician- and self-rated assessments to contribute equally to the anhedonia score (range 1–13). As in our previous studies (8, 9, 27), the score for each subject was computed by summing the responses to three items associated with anhedonia from the clinician-rated CDRS-R (item 2: "difficulty having fun;" scale of 1–7) and the self-rated BDI-II (items 4: "loss of pleasure" and 12: "loss of interest;" scales of 0–3).
#### **Irritability**
Our approach again combined both clinician- and self-rated assessments to contribute to the irritability score (range 1–10). The score for each subject was computed by summing the responses to the items associated with irritability from the CDRS-R (item 8: "irritability;" scale of 1–7) and the BDI-II (item 17: "irritability;" scale of 0–3).
#### **DATA ACQUISITION AND ANALYSIS**
Diffusion data were acquired on a Siemens Allegra 3.0 T scanner at the NYU Center for Brain Imaging using a single-channel head coil. Diffusion-weighted echo-planar images (EPI) were acquired along 12 diffusion gradient directions for acquisition of 35 slices through the whole brain (TR = 6000 ms, TE = 82 ms, flip angle = 90°,*b* value = 1000 s/mm<sup>2</sup> , FOV = 192 mm,128 × 128 matrix, slice thickness = 2.5 mm, with four averages). Highresolution T1-weighted anatomical images were acquired using a magnetization-prepared gradient echo sequence (TR = 2530 ms; TE = 3.25 ms; TI = 1100 ms; flip angle = 7°; 128 slices; FOV = 256 mm; acquisition voxel size = 1.3 mm × 1 mm × 1.3 mm).
All preprocessing was performed using FMRIB's Software Library (FSL; Oxford, UK): FMRIB's Diffusion Toolbox (FDT). Preprocessing began with eddy current correction to correct for gradient-coil distortions and small head motions, using affine registration to a b0 reference volume. A diffusion tensor model was fitted to each voxel along the principal λ1, λ2, λ<sup>3</sup> directions to generate FA, MD, RD, and AD. We then implemented FSL's tract-based spatial statistics pipeline [TBSS; (28)], in which a nonlinear registration aligned each subject to the FMRIB58\_FA template in 1 mm × 1 mm × 1 mm standard space, and then warped each subject into standard Montreal Neurological Institute space (MNI152). A WM "skeleton" was then generated representing a single line running down the centers of all of the common WM fibers by using an FA cut-off of 0.2, and relevant diffusivity measures (i.e., FA, MD, RD, AD) were projected onto the skeleton. Group statistical analysis was then conducted only on voxels within the WM skeleton mask, therefore restricting the voxel-wise analysis only to voxels with high confidence of lying within equivalent major WM pathways in each individual.
In order to assess differences in FA, MD, RD, and AD between the MDD and HC groups, we used FSL's Randomise with 5000 permutations to perform voxel-wise independent samples *t*-tests using voxel-based thresholding while controlling for age and sex. Group comparisons did not withstand stringent correction for multiple comparisons using family-wise error correction (FWE; implemented by Randomise) or FDR correction (implemented by FSL's FDR program). As such, we performed a second, more exploratory analysis in which we accepted clusters of at least 10 contiguous voxels at *p* < 0.001.
To investigate relationships with depression severity, anhedonia, and irritability, Randomise was again used to perform a series of one-sample *t*-tests using the scorefor each category, respectively, while controlling for age and gender. Due to the low variability in scores in the HC group, as well as the nature of the study topic, the analysis was limited to the MDD group whose scores were normally distributed. Once again, tests did not withstand correction for multiple comparisons and we used the more exploratory approach of accepting clusters of at least 10 contiguous voxels at *p* < 0.001. We used FSL's cluster program to extract all clusters across the brain, and anatomical localization of each cluster was determined using the FSLView atlas toolbox and the relevant gray matter (Harvard-Oxford Cortical/Subcortical) and WM (Johns Hopkins UniversityWM tractography) atlases. Brain imaging results were prepared for display using FSL's tbss\_fill script, which displays results superimposed upon the WM skeleton from the group TBSS analysis.
#### **RESULTS**
#### **PARTICIPANTS**
Demographic and clinical characteristics are summarized in **Table 1**. One subject with MDD had been treated with Lexapro and Ambien for 7 months, but was medication-free for approximately 14 months prior to scanning. A second subject with MDD had a brief trial with Prozac which was self-discontinued prior to participation in this study. All other subjects were psychotropic medication-naïve. Fifteen subjects with MDD had experienced **Table 1 | Demographic and clinical characteristics of adolescents with major depressive disorder (MDD) and healthy controls.**
<sup>a</sup>Respective percentages (may not add up to 100% due to rounding).
<sup>b</sup>Children's depression rating scale – revised.
<sup>c</sup>Beck depression inventory, 2nd ed.
<sup>d</sup>Beck scale for suicidal ideation.
<sup>e</sup>Multidimensional anxiety scale for children.
<sup>f</sup>Attention deficit hyperactivity disorder.
<sup>g</sup>Generalized anxiety disorder.
only one episode of depression, with length of episode ranging from 4 to 48 months, and two patients reported having two distinct episodes. Two Shapiro–Wilk tests revealed that anhedonia and irritability were both normally distributed within the MDD group, *p*s = 0.81, 0.48, respectively. Depression severity (excluding the anhedonia- or irritability-related items) was significantly correlated with both severity of anhedonia (*r* = 0.66, *p* < 0.005) and irritability (*r* = 0.66, *p* < 0.005) within our MDD sample. Anhedonia and irritability were not correlated (*r* = 0.37, *p* = 0.15).
#### **WHOLE-BRAIN GROUP COMPARISON**
No voxel-wise group comparisons for FA, MD, RD, or AD withstood correction for multiple comparisons. An exploratory analysis using a threshold of *p* < 0.001, uncorrected with clusters exceeding 10 contiguous voxels, revealed 4 significant clusters (**Table 2**, **Figure 1**). Compared with HC, the MDD group had lower FA in the anterior cingulum, and lower AD in the anterior corona radiata (ACR). However, the MDD group also had greater FA and lower RD in the posterior cingulum compared to HC.
#### **Table 2 | Voxel-wise group comparison results.**
Units for AD and RD = mm<sup>2</sup> /s. Coordinates in MNI space. Threshold p < 0.001, uncorrected, k > 10. COG, center of gravity; FA, fractional anisotropy; MD, mean diffusivity; RD, radial diffusivity; AD, axial diffusivity; WM, white matter; L, left; R, right; ACR, anterior corona radiata.
**FIGURE 1 | (A)** Increased WM integrity in the MDD group vs. HC in the posterior cingulum near the hippocampus; **(B)** decreased WM integrity in the MDD group vs. HC in the anterior cingulum near the precuneus.
#### **WHOLE-BRAIN CORRELATIONS WITH DEPRESSION SEVERITY IN MDD**
Exploratory analyses revealed a total of 16 uncorrected clusters, with 5 overlapping clusters between the 4 diffusivity measures (**Table 3**, **Figure 2**). As depression severity increased, FA decreased in the genu of the corpus callosum, the sagittal stratum, the ATR, and the anterior cingulum. Additionally, a positive correlation between MD and depression severity was found in the same sagittal stratum cluster as the FA analysis, as well as in clusters in the ATR and corticospinal tract. Similarly, illness severity was positively correlated with RD in the same sagittal stratum cluster as the FA and MD analyses, the same genu of the corpus callosum and
ATR clusters as the FA analysis, the same ATR cluster as the MD analysis, and a cluster in the superior longitudinal fasciculus (SLF). Finally, increased illness severity was associated with increased AD in the same corticospinal cluster as the MD analysis as well as in clusters in the inferior-fronto-occipital fasciculus (IFOF), the SLF, and fibers projecting to the orbitofrontal cortex (OFC).
#### **WHOLE-BRAIN CORRELATIONS WITH ANHEDONIA IN MDD**
Analyses revealed a total of 14 uncorrected clusters, with 2 overlapping clusters between FA and RD (**Table 4**,**Figure 3**). As anhedonia increased, FA increased in a cluster in the posterior cingulum near
#### **Table 3 | Voxel-wise correlations with depression severity (CDRS-R).**
Coordinates in MNI space.Threshold p < 0.001, uncorrected, k > 10. COG, center of gravity; L, left; R, right; FA, fractional anisotropy; MD, mean diffusivity; RD, radial diffusivity; AD, axial diffusivity; WM, white matter; PHG, parahippocampal gyrus; ACC, anterior cingulate cortex; MFG, medial frontal gyrus; OFC, orbitofrontal gyrus; ILF; inferior longitudinal fasciculus; IFOF, inferior-fronto-occipital fasciculus; ATR, anterior thalamic radiation; SLF, superior longitudinal fasciculus.
the hippocampus – similar to a cluster from the group comparison analysis – and decreased in the anterior limb of the internal capsule, OFC projection fibers, and the posterior cingulum near the precuneus. Furthermore, anhedonia was positively correlated with MD in OFC projection fibers, the external capsule, and the sagittal stratum. Additionally, increased anhedonia severity was associated with greater RD in the same posterior limb of the internal capsule and posterior cingulum clusters as the FA analysis, the same OFC projection fibers as the MD analysis, and clusters in the ATR and corticospinal tract. Finally, there were positive correlations between anhedonia and AD in the corticospinal tract and projection fibers into the occipital cortex.
#### **WHOLE-BRAIN CORRELATIONS WITH IRRITABILITY IN MDD**
Analyses revealed a total of 14 uncorrected clusters, with 2 overlapping clusters (**Table 5**, **Figure 4**). As irritability increased, FA decreased in clusters in the sagittal stratum and IFOF, while MD increased in the same sagittal stratum cluster as well as in clusters in the ACR, SLF, and IFOF. For RD, positive correlations with irritability were evident in the same sagittal stratum cluster, the anterior limb of the internal capsule, and the SLF. Positive correlations were also found between AD and irritability in the same ACR cluster as the MD analysis as well as in the IFOF, the corticospinal tract, and the SLF.
### **DISCUSSION**
Consistent with our hypotheses, analyses revealed reduced WM integrity (i.e., decreased FA, and increased MD, RD, AD) in the MDD group compared to HC as well as in the more severely depressed, anhedonic, and irritable patients. Furthermore, despite significant correlations between the two dimensional measures and depression severity, we found distinct WM alterations for both anhedonia and irritability that differed from those for depression severity. Reduced integrity was found in fronto-striatal and thalamic tracts, the corpus callosum, and tracts connected to the inferior temporal (IT) cortex. Additionally, reduced integrity in the sagittal stratum was consistently found in our analyses to be correlated with increasing depression severity, anhedonia, and
irritability. Unexpectedly, analyses also revealed a cluster in the posterior cingulum near the hippocampus which demonstrated more anisotropic diffusion, both as anhedonia increased and in the MDD group vs. HC. Finally, it is interesting to note that in many of our analyses there was overlap in the clusters demonstrating a relationship with FA and RD, potentially suggesting that structural issues related to RD are driving the observed relationships with FA in this and other studies.
#### **GROUP DIFFERENCES IN WM INTEGRITY**
Group differences were observed in both the posterior and anterior cingulum. Specifically, depressed adolescents demonstrated decreased WM integrity in the anterior cingulum near the precuneus, and increased integrity in the posterior cingulum near the hippocampus, compared to HC. The cingulum connects the cingulate and entorhinal cortices and is broadly involved in attention, memory, and emotions (29, 30). The anterior portion of the cingulate has been implicated in emotional processing and depression (31). Altered functioning, connectivity, and diffusion around the precuneus are frequently reported in MDD (32, 33). Given the role of the precuneus in self-related processes, and that self-processing is typically altered in depression (34), this potentially suggests that reduced WM integrity contributes to altered functioning in this region early in the course of the disease.
The MDD group also demonstrated more coherent diffusion in the posterior cingulum near the hippocampus. The posterior cingulate is involved in cognitive functions including attention and memory (35). Functional hyperactivity in the hippocampus (36–38), as well as decreased hippocampal volume (39), are consistent findings in adult MDD. Given the role of the hippocampus in learning and memory (40), but also in the regulation of motivation and emotion (41, 42), this region is critical to carrying out normal behaviors that may be altered in depression. Furthermore, greater WM integrity in tracts leading to the hippocampus would be consistent with the literature demonstrating hyperactivity of this region in non-medicated MDD patients. Overall, the categorical comparison between depressed adolescents and HC revealed differences in an important tract connecting prefrontal and limbic regions.
### **DEPRESSION SEVERITY AND WM INTEGRITY**
Our use of an approach that accounts for a range of depression severity in our sample revealed a pattern of reduced WM integrity as depression severity increased. Specifically, we found reduced integrity in the genu of the corpus callosum, a region that connects prefrontal and orbitofrontal cortices. Many studies have documented altered diffusivity in the genu (16) as well as reduced volume (6, 43–45). Given that the prefrontal and orbitofrontal cortices are involved in critical processes, including decision-making, attention, reward processing, and the evaluation and regulation of emotion (46–48), an interruption in communication between these areas has implications for depression and mood disorders.
Additionally, we found decreased integrity in the sagittal stratum with increasing severity, not only in this analysis, but also in the dimensional analyses within the MDD population. The sagittal stratum is a complex fiber bundle connecting the occipital cortex to the rest of the brain, and includes fibers from many major tracts including the ILF and IFOF (49). The ILF and IFOF both connect the occipital cortex to temporal limbic structures (i.e., amygdala, hippocampus) and the PFC, although the IFOF connects directly
#### **Table 4 | Voxel-wise correlations with anhedonia.**
Coordinates in MNI space. Threshold p < 0.001, uncorrected, k > 10. COG, center of gravity; L, left; R, right; FA, fractional anisotropy; MD, mean diffusivity; RD, radial diffusivity; AD, axial diffusivity; WM, white matter; OFC, orbitofrontal gyrus; IC, internal capsule; ATR, anterior thalamic radiation; IFOF, inferior-fronto-occipital fasciculus; ILF, inferior longitudinal fasciculus.
to the OFC and the ILF does so indirectly through the uncinate fasciculus (50). Therefore, both tracts are involved in connecting visual information with areas involved in emotional memories, judgments, and behaviors. A meta-analysis of diffusion studies of patients with MDD found WM alterations in both the ILF and IFOF (16). Additionally, alterations in WM have been found in the IFOF for depressed adolescents (17), adolescents with a familial risk for depression (18), and adults with MDD (51).
We also observed reduced integrity with increasing severity in bilateral clusters in the ATR near the pallidum. The ATR connects thalamic nuclei with the PFC through the anterior limb of the internal capsule. Reduced WM integrity has been reported in the ATR in several studies of depressed adults (16). Furthermore, given the role of the thalamus in motivation and goal pursuit (52, 53), altered connectivity within this circuit could contribute to the motivational deficits associated with depression.
Additionally, increased illness severity was associated with reduced integrity in the corticospinal tract near the postcentral gyrus. The corticospinal tract transmits motor impulses from the motor and premotor cortices to the spinal cord. Although this was an unexpected finding, motor disturbances and retardation are a relevant clinical symptom of depression (54). In this way, altered diffusivity may be related to the observed slowing and impairment of motor functions. Finally, we again found decreased integrity with increasing severity in the previously described anterior cingulum near the precuneus, which is consistent with the findings from our group analysis. Overall, our analysis with varied levels of depression severity was more robust than the categorical comparison and revealed a more extensive network of reduced WM integrity.
#### **ANHEDONIA AND WM INTEGRITY**
The dimensional analysis with anhedonia revealed an association between increased anhedonia and reduced integrity in the anterior limb of the internal capsule near the thalamus, a tract implicated in reward processing (19). The anterior limb of the internal capsule connects the thalamus with cingulate and prefrontal cortices, which are heavily involved in motivation, decision-making, and evaluating the saliency of emotional and rewarding stimuli. Additionally,increased anhedonia was correlated with reduced integrity in tracts (i.e., IFOF, projection fibers) connected to the posterior lateral OFC (BA 47), an area involved in many functions including emotional and reward processing, complex learning, and the inhibition of responses (46, 47, 55). Depressed patients
have demonstrated reduced gray matter volume in the posterior lateral OFC as well as altered functional responses to emotional stimuli, reward processing, and reversal learning (56).
We also found reduced integrity in the external capsule as anhedonia increased. The external capsule contains cholinergic fibers projecting from the basal forebrain to the cerebral cortex. Reduced integrity in the external capsule has been found previously in adult MDD (57, 58). Furthermore, we once again found reduced integrity in the previously discussed sagittal stratum and posterior cingulum near the precuneus with greater symptom severity. Finally, the analysis revealed increased integrity with increased anhedonia in the posterior cingulum near the hippocampus, in an area fairly close to the cluster that showed increased integrity in the MDD group in our categorical comparison. In this way, it is possible that anhedonic symptoms are related to the group differences we observed. Given the previously discussed role of the hippocampus and limbic system in the regulation of motivation and emotion, the relationship between hippocampal functioning and anhedonia represents an important area for future research.
#### **IRRITABILITY AND WM INTEGRITY**
As predicted, we found decreased integrity as irritability increased in a tract near the amygdala (i.e., sagittal stratum including the ILF and IFOF). However, increased irritability was correlated with decreased integrity in tracts primarily connecting to prefrontal and occipital cortices. We also found clusters in the previously discussed IFOF, although one was in the lingual gyrus while the other was in the middle frontal gyrus. The lingual gyrus has been implicated in processing emotional faces (59), which is typically altered in MDD (60). Altered cerebral blood flow and resting state connectivity have been demonstrated in the lingual gyrus in adults with MDD (61, 62). Additionally, decreased integrity in WM has previously been found around the middle frontal gyrus (63), an area broadly involved in a variety of higher-level cognitive processes (64) which are often compromised in MDD.
Reduced integrity related to elevated irritability was also found in the ACR, which connects the striatum to the ACC. Reduced integrity has previously been demonstrated in the ACR in pediatric bipolar patients (65), and dysfunctional activity in the ACC is typically considered a hallmark of depression (37, 42, 66–70). Furthermore, altered intrinsic functional connectivity (i.e., resting state) between the striatum and ACC has been documented for depressed adolescents (8). Finally, a cluster in the SLF in the IT gyrus was found. The SLF is a major bidirectional association tract connecting large parts of the frontal cortex with the parietal, temporal, and occipital lobes. Less restricted diffusion in the SLF has been previously demonstrated for depressed adolescents (17), adolescents with a genetic risk for depression (18), and adults with MDD (71).
#### **MEASURES OF WM INTEGRITY**
Although a complete discussion of RD and AD goes beyond the scope of this paper, it is interesting to note that for both the categorical and dimensional analyses we found overlap in
#### **Table 5 | Voxel-wise correlations with irritability.**
Coordinates in MNI space.Threshold p < 0.001, uncorrected, k > 10. COG, center of gravity; L, left; R, right; FA, fractional anisotropy; MD, mean diffusivity; RD, radial diffusivity; AD, axial diffusivity; WM, white matter; IT, inferior temporal gyrus; MFG, medial frontal gyrus; MT, middle temporal gyrus; ILF, inferior longitudinal fasciculus; IFOF, inferior-fronto-occipital fasciculus; ACR, anterior corona radiata; ATR, anterior thalamic radiation; SLF, superior longitudinal fasciculus; IC, internal capsule.
**FIGURE 4 | DecreasedWM integrity as irritability increased in the (A) sagittal stratum in the IT; (B) IFOF near the MFG**. Key: IT, inferior temporal cortex; IFOF, inferior-fronto-occipital fasciculus; MFG, medial frontal gyrus.
clusters with reduced FA and increased RD, but no overlapping relationships with AD. Increased RD may be caused by disturbances in myelin, whereas decreased AD has been suggested to reflect disrupted axonal integrity (72–74). As such, our findings and those from previous research may reflect that alterations in FA for MDD are being driven more by issues of myelination than axonal integrity. However, further research is needed to replicate and expand upon a possible mechanism.
#### **LIMITATIONS AND FUTURE DIRECTIONS**
Although our findings are consistent with other clinical studies investigating altered WM in depressed adolescents, it should be noted that very liberal thresholds were used for the analyses and the inability to correct for multiple comparisons is an issue of concern. Although our statistical methodology and sample size were comparable to those of other studies of clinical populations using DTI (17, 63, 75), it is possible that the sample sizes used in many clinical studies are not large enough to produce adequate statistical power. In this way, it is difficult to adequately balance the concerns of committing a Type I error by not correcting while also avoiding a Type II error due to small sample sizes and reduced statistical power. Therefore, our findings are considered preliminary. Furthermore, the inclusion of patients with milder symptomatology may have weakened our ability to detect group differences.
Although small sample sizes may be a possible explanation for the relatively weak results in our and other clinical studies of adolescent depression, another possibility is that the adolescent brain is still malleable and the alterations in WM structure may not fully take hold until adulthood (11). Therefore, it is even more pressing to understand a neuroimmunological model of depression and the factors that may contribute to changes in WM before chronicity begins to take effect. For example, given past findings that depressed adolescents exhibit higher levels of circulating inflammatory cytokines (76), one possible explanation for the observed reduction in FA in adult MDD is that it may reflect effects of chronic low grade inflammation. Additionally, given our previous research on fronto-striatal functional connectivity in MDD,future studies should investigate altered WM microstructure using a targeted tractography approach. Finally, further research is needed to investigate this hypothesis and other models of the systemic consequences of depression. To this end, a better understanding of what FA, MD, AD, and RD illustrate in an adolescent population, as well as the factors that contribute to these diffusivity measures, is needed in the field.
#### **CONCLUSION**
Our investigation of altered WM microstructure in medicationfree adolescents with MDD revealed a general pattern of impaired WM integrity in the depressed adolescents, and as depression severity, anhedonia, and irritability increased. Our findings are consistent with an overall hypothesis that depression, even in adolescence, involves a disconnection of prefrontal, striatal, and limbic emotional areas (16). Although this represents a good step toward understanding depression during this critical period, more research is needed to understand the factors that ultimately contribute to alteredWM microstructure in order to develop potential interventions.
#### **ACKNOWLEDGMENTS**
We would like to thank Chao-Gan Yan of the Child Mind Institute for his help with this project. This study was supported by grants from the NIH (AT004576, MH095807), the Brain & Behavior Research Foundation (formerly NARSAD), the Hope for Depression Research Foundation, and generous gifts from the Leon Levy Foundation.
### **REFERENCES**
**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
*Received: 04 October 2013; accepted: 08 November 2013; published online: 25 November 2013.*
*Citation: Henderson SE, Johnson AR, Vallejo AI, Katz L, Wong E and Gabbay V (2013) A preliminary study of white matter in adolescent depression: relationships with illness severity, anhedonia, and irritability. Front. Psychiatry 4:152. doi: 10.3389/fpsyt.2013.00152*
*This article was submitted to Neuropsychiatric Imaging and Stimulation, a section of the journal Frontiers in Psychiatry.*
*Copyright © 2013 Henderson, Johnson, Vallejo, Katz, Wong and Gabbay. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.*
# Meta-analyses of developing brain function in high-risk and emerged bipolar disorder
#### **Moon-Soo Lee1,2, Purnima Anumagalla<sup>1</sup> , Prasanth Talluri <sup>1</sup> and Mani N. Pavuluri <sup>1</sup>\***
<sup>1</sup> Pediatric Brain Research and Intervention Center, University of Illinois at Chicago, Chicago, IL, USA
<sup>2</sup> College of Medicine, Korea University, Seoul, South Korea
#### **Edited by:**
Stephanie Ameis, The Hospital for Sick Children and University of Toronto, Canada
#### **Reviewed by:**
Meng-Chuan Lai, University of Cambridge, UK Nicholas Neufeld, University of Toronto, Canada Annette Beatrix Bruehl, University of Cambridge, UK
#### **\*Correspondence:**
Mani N. Pavuluri, Pediatric Brain Research and Intervention Center, University of Illinois at Chicago, M/C 747, 1747 West Roosevelt Road, Chicago, IL 60608, USA e-mail: [email protected]
**Objectives:** Identifying early markers of brain function among those at high risk (HR) for pediatric bipolar disorder (PBD) could serve as a screening measure when children and adolescents present with subsyndromal clinical symptoms prior to the conversion to bipolar disorder. Studies on the offspring of patients with bipolar disorder who are genetically at HR have each been limited in establishing a biomarker, while an analytic review in summarizing the findings offers an improvised opportunity toward that goal.
**Methods:** An activation likelihood estimation (ALE) meta-analysis of mixed cognitive and emotional activities using the GingerALE software from the BrainMap Project was completed. The meta-analysis of all fMRI studies contained a total of 29 reports and included PBD, HR, and typically developing (TD) groups.
**Results:**The HR group showed significantly greater activation relative to theTD group in the right DLPFC–insular–parietal–cerebellar regions. Similarly, the HR group exhibited greater activity in the right DLPFC and insula as well as the left cerebellum compared to patients with PBD. Patients with PBD, relative to TD, showed greater activation in regions of the right amygdala, parahippocampal gyrus, medial PFC, left ventral striatum, and cerebellum and lower activation in the right VLPFC and the DLPFC.
**Conclusion:** The HR population showed increased activity, presumably indicating greater compensatory deployment, in relation to both the TD and the PBD, in the key cognition and emotion-processing regions, such as the DLPFC, insula, and parietal cortex. In contrast, patients with PBD, relative to HR and TD, showed decreased activity, which could indicate a decreased effort in multiple PFC regions in addition to widespread subcortical abnormalities, which are suggestive of a more entrenched disease process.
**Keywords: pediatric bipolar disorder, high risk, meta-analysis, GingerALE, dorsolateral prefrontal cortex, amygdala**
### **INTRODUCTION**
The relationship between pediatric and adult bipolar disorder has been the subject of controversy. It is not clear whether pediatric bipolar disorder (PBD) is the pediatric form of the typical adult bipolar disorder or an entity of its own, as bipolar disorder usually manifests differently in childhood than in adulthood. Some studies in adults have reported that a portion of adults with bipolar I disorder experienced childhood or adolescent onset, and some of them began showing symptoms even before 12 years of age (1, 2). Identifying early markers of brain function among those at high risk (HR) for PBD could serve as a screening measure when children and adolescents present with subsyndromal clinical symptoms prior to the conversion to bipolar disorder (pediatric or adult form). These biomarkers can also aid as a stand-alone bio-signature for the identification of risk even prior to the emergence of any clinical symptoms and could allow an opportunity to prevent the onset of full-blown illness (3). One way to begin identifying the biomarkers is to examine the brain function in the genetically HR offspring of patients with bipolar disorder. While some studies of HR have been published (4–11), due to
their small sample sizes and corrections for multiple comparisons, the findings remain inconclusive.
To offer robust and reliable findings, we used a recently developed activation likelihood estimation (ALE) technique. This method assumes that the peak co-ordinates reported by each study represent the activation maps from which they are derived and uses the reported co-ordinates in voxel-wise analysis to assess the consistency of activation in any given set of studies (12–14). By performing the quantitative voxel-wise meta-analysis of already published results from the HR population and comparing them with those from the converted PBD and typically developing (TD) youth, we can provide objective, unbiased, and statistically based quantified evidence.
Ideally, a separate meta-analysis would be conducted for each individual domain, such as emotion processing or attention, as they relate to bipolar disorder diathesis. However, given the infancy of the current literature regarding HR patients, this is not practical, as no individual construct has included a sufficient number of studies to date. Instead, it is more feasible to study the commonalities probed across multiple domains in a systematic and
statistically driven fashion. There is a certain advantage to combining all the studies that include multi-domain probes. First, the brain does not work in isolation across individual domains; therefore, it is necessary to examine the brain's function as a whole while it is engaged in affective, cognitive, and motor control tasks (15). Furthermore, pooling several pilot studies produces an exploratory power of how the brain functions in a larger sample, eventually offering the possibility of correlating the results with the clinical manifestations of domains and disorders presenting with combined affective, cognitive, and motoric symptoms (16). This approach is a segue into future studies that can explore the interface of multiple domain functions in individual studies.
We consider emotional systems and circuits, in illness or wellness, to be closely linked to cognitive and motor control circuits of attention, working memory, and response inhibition (17). These systems interface at three tiers as shown in animal (18) and human studies of PBD (19): (1) at the prefrontal level between the ventrolateral prefrontal cortex [VLPFC; inferior frontal gyrus; Brodmann areas (BAs) 45, 47] and the dorsolateral prefrontal cortex (DLPFC; middle frontal gyrus; BAs 9, 9, 46), (2) at the intermediary cortex in the anterior cingulate cortex (ACC), such as between the dorsal (BA 32) and pregenual ACC (BA 24), and (3) at the subcortical level between the amygdala and striatum (19). While we could not determine which probe or domain dysfunction would contribute to activity in any given co-ordinate in this meta-analysis, we developed our hypotheses based on knowledge derived from the emerging literature. Emotion-processing tasks probing the affective systems entered into our meta-analysis would contribute to the increased prefrontal activity at the interface of VLPFC and DLPFC in HR and the decreased activity in PBD relative to TD (19). Increased subcortical amygdala activity would be a specific marker of PBD (20) relative to HR and TD. Based on our knowledge of attention and working memory task response, the DLPFC will manifest with increased activity in HR (6) and decreased activity in PBD (21),relative to TD. Impaired subcortical striatal activity would be a more entrenched specific marker of PBD's cognitive and motor dysfunction (20, 22, 23) relative to the HR and TD groups.
### **MATERIALS AND METHODS SEARCH STRATEGY**
We identified primary studies through a comprehensive literature search of the MEDLINE (using both free-text and MeSH search) and PsychINFO databases using the following keywords: pediatric or child or adolescent, plus bipolar disorder or highrisk or at risk, and plus functional magnetic resonance imaging or fMRI. In addition, manual searches were conducted via reference sections of review articles and individual studies to check for any missing studies that were not identified using computerized searches. There were no language restrictions; in fact, all the included manuscripts were written in English. Only fMRI studies were chosen for review. An initial list of studies was produced that included any report of fMRI studies of PBD and HR offspring published in print or online by December 31, 2013. The selection process for the final list of primary studies for the planned meta-analyses in this study was very specific. The first-level literature search yielded 235 unique published articles with 49 studies meeting the initial inclusion criteria. A further manual search leads to eight other studies. After a second-level review of these 57 studies, only 29 contained the co-ordinates essential for inclusion in our meta-analysis (**Figure 1**). Any ambiguity in inclusion was resolved through a consensus decision by the authors of this manuscript. Study data (e.g., co-ordinates, participant numbers, and imaging spaces) were entered and crosschecked by participating authors.
### **SELECTION CRITERIA**
"High risk" in this project refers to adolescents who have a biological parent diagnosed with BD. We selected studies with participants whose mean age was less than 19 years. Every study that we included had participants between the ages of 7 and 18 except for the study performed by Thermenos et al. (11), where the ages ranged up to 24. All reports included in the meta-analysis satisfied the following criteria: (1) a healthy comparison group is included, (2) the studies conducted whole-brain analyses, (3) all studies provided standard Talairach or Montreal Neurological Institute (MNI) spatial co-ordinates for the key findings, (4) patient participants had been diagnosed with bipolar disorder, and (5) there were at least five members in each of the participant groups. We included only those studies that reported activation foci as 3D co-ordinates in stereotactic space, examined active task constructs, and presented results for groups of participants.
Excluded manuscripts consisted of the following: (1) reviews or meta-analyses, (2) those with subject overlap, and (3) other MRI modalities (e.g., structural imaging, spectroscopy, diffusion tensor imaging, and functional connectivity studies).
#### **ACTIVATION LIKELIHOOD ESTIMATION METHODS AND PAIRWISE ALE META-ANALYSIS**
GingerALE software version (version 2.3.1) from the BrainMap project was used to conduct ALE meta-analysis of eligible studies (13, 14, 43). Meta-analyses were performed using the revised ALE software (i.e., GingerALE 2.3). The key modification in the revised ALE software included the change from fixed-effects (convergence between foci) to random-effects inference (convergence between studies but not individual foci reported for the same study), as well as greater meta-analytic weighting for primary studies that involved more participants. In line with our goal of gaining insight on the whole brain's function through tasks that probe combined domains, we performed exploratory analyses using all eligible data in the HR offspring, BD patient, and TD groups in the pediatric age group. Conversely, we did not separate the analyses by the type of the task or the brain domain probed. This method also helped to harness sample size and power. Activation co-ordinates reported in the MNI space were converted to Talairach co-ordinates using the Lancaster transform (icbm2tal) in GingerALE. Our meta-analysis was conducted in Talairach space. Co-ordinates originally presented as MNI space were transformed into Talairach space using Lancaster transformation. For uniformity, Talairach co-ordinates expressed by the previous Brett transformation (44) were converted into MNI space and re-transformed into Talairach space. The meta-analysis was performed using pairwise ALE meta-analysis.
Pairwise ALE meta-analyses included the following comparisons at first: greater activation in PBD versus HR, in HR versus PBD, in PBD versus TD, in TD versus PBD, in HR versus TD, and in TD versus HR. However, two pairwise ALE meta-analyses (greater activation in PBD versus HR and greater activation in TD versus HR) were not performed due to the lack of available data. The input co-ordinates were weighted to form estimates of activation likelihood for each intracerebral voxel. The activation likelihood of each voxel in standard space was then combined to form a statistic map of the ALE score at each voxel. Statistical significance of the ALE scores was determined by a permutation test controlling the false discovery rate (FDR) at *p* < 0.05 (45). The statistic maps were thresholded by default at this critical value, and a recommended minimum cluster size was suggested from the cluster statistics. By using this minimum cluster size for the supra-threshold voxels, we can obtain the thresholded ALE image. Pairwise ALE analyses results were reported at *p* = 0.05 and were whole-brain corrected. A Talairach Daemon was used for anatomical locations for significant clusters.
### **RESULTS**
The meta-analysis of all fMRI reports included 29 studies (PBD, HR, and TD). There was no overlap in patients who completed the same task across the selected studies. The primary studies included in the meta-analysis are listed in **Table 1**. Findings are summarized in **Table 2** and **Figure 2**.
#### **HR AND TD: RECOGNIZING HIGH-RISK PARTICIPANTS**
Participants in the HR group showed significantly greater activation in the right DLPFC, insula, inferior parietal lobule, and left cerebellum relative to TD. No other group differences were found. In case of greater activation in the TD group relative to HR, the analysis was not performed due to the lack of a large enough sample size and of experiments showing significant results.
### **PBD AND HR: RECOGNIZING THE EMERGENCE OF THE DISORDER**
The HR group showed significant greater activation of the right DLPFC, insula, and left cerebellum than PBD. No other group differences were identified. In case of greater activation in the PBD group relative to HR, the analysis was also not performed due to a small sample size and few experiments showing significant results.
#### **PBD AND TD: RECOGNIZING THE ILLNESS FROM WELLNESS**
Patients with PBD demonstrated greater activation in the subcortical regions of the right amygdala, the parahippocampal gyrus, the subgenual ACC, and the medial PFC, and in the left ventral striatum, VLPFC, and cerebellum relative to TD. The TD group showed greater activation in the right VLPFC, DLPFC, superior frontal gyrus, dorsal ACC, and striatum than patients with PBD.
### **DISCUSSION**
We found the recently published developmental meta-analysis of bipolar disorder performed by Wegbreit et al. The researchers compared different age groups with bipolar disorder (youths and adults). PBD youths showed increased activation in the amygdala, the inferior frontal gyrus, and precuneus compared to bipolar disorder adults during tasks using emotional stimuli. These findings revealed that these structures are underdeveloped and work less efficiently when compared with those of adults (46). However, our meta-analysis was conducted using the comparison between participants of the same age (participants' mean age is less than 19 years). The central findings of the meta-analyses of brain function among the PBD, HR, and TD groups, during the performance of mixed cognitive and emotional activities, illustrated a coherent pattern of group differences in line with our *a priori* hypothesis. The HR group showed a significantly greater activation in the *right DLPFC–insular–parietal–cerebellar regions* relative to TD, and this may be a bio-signature – an earlier sign of potential PBD development. At the junction of the DLPFC and VLPFC regions, where prefrontal systems interface in voluntary modulation of cognition, emotion, and motor control, brain function was amplified in the HR group (6, 7). Large future studies of symptomatic HR population (47) and genetic HR population must be compared both at a symptomatic and brain functional level to look at the definitive predictability of symptoms and the correlation of brain activity patterns.
#### **Table 1 | Primary fMRI studies of participants with pediatric bipolar disorder (PBD), those at high risk (HR) for PBD, and typically developing (TD): children included in meta-analysis**.
(Continued)
#### **Table 1 | Continued**
<sup>a</sup>Studies including HR groups.
Some studies were missing age range information and showed only the mean age. Accordingly, that information could not be included within the table. Specific medications were heterogeneous when reported and at times went unreported. Hence, we were only able to comment on participants' medicated/unmedicated status. Similarly, the mood and affect of participants were also largely unreported and, therefore, could not be included in the table.
A repeated and important observation of hemodynamics of the fMRI studies is the increased activity in the brain that reflects increased effort (48). If one construes TD as the reference point of normative activity, then the HR group showed increased effort to get the same work done by deploying the right DLPFC–insular– parietal regions relative to TD, while in PBD, these same regions went offline relative to TD. This finding is akin to the analogy of "stretching an elastic band"with increased DLPFC activity (requiring a greater effort than TD) in the HR group, whereas those with PBD who had a more severe illness had reached a breaking point with decreased right VLPFC and DLPFC activity (with no effort to spare relative to TD). We could not explain the increased left VLPFC activity in PBD relative to TD. While such a finding is not unexpected in a meta-analytic study, it was largely based upon the
**Table 2 | Activation likelihood estimation (ALE) meta-analysis findings for fMRI studies comparing pediatric bipolar disorder (PBD) patients, participants with a high risk (HR) for PBD, and typically developing (TD) children**.
R: right, L: left.
participants of only one study (21). However, it can be explained by bilateral disturbances in the VLPFC in PBD, albeit with the common and prominent right-sided abnormality than the left (32, 37). In the end, while one can postulate with explanations consistent with repeatedly published findings, definitive interpretations are not possible in understanding the nature of abnormal hemodynamic activity. For example, decreased (5) or increased (6) activation of the striatum with failed trials cannot easily differentiate HR from PBD based on any individual study. It could be mediated by the severity of illness in case of PBD, subsyndromal symptoms in HR, type of task, or hemodynamic relationship between the striatum and the PFC control regions.
With regard to recognizing the fully formed illness, typically noted underactivity of the higher cortical regions of emotion modulation (i.e., the interfacing dyad of the right VLPFC and DLPFC in the prefrontal regions) and overactivity of the subcortical amygdala consistently reported in BD Type I participants relative to TD adolescents (19) has also emerged as a significant finding in the current meta-analyses. The VLPFC is believed to serve the dual function of emotion (49) and motor (50) control via top–down regulation of the amygdala (51) and striatum (52), respectively. The DLPFC also serves a dual function, but it is predominantly through diverse cognitive functions involving executive control, response selection, problem solving, and emotion (53), and by being closely connected to the medial PFC, VLPFC, and the subcortical regions directly (54) as well as indirectly (52). The cognitive and emotion control regions in the PFC are not able to moderate the overactive subcortical regions, a consistent finding that was further underscored in our meta-analysis. In addition to the top–down *affect modulation circuitry* problems, increased activity is lateralized to the left side in the evaluative medial PFC, pregenual ACC, and the striatal loop (55); furthermore, all these regions are known to be closely connected to the amygdala (56). This subcortical and medial PFC loop is the *affective evaluation circuit* that is overactive in PBD. These findings could explain the excessive reactivity to negative emotions reported in PBD (21, 57) and are also in line with the concept suggested for bipolar disorder in general, including adult patients. Phillips and Swartz conceptualized bipolar disorder as multiple dysfunctions in prefrontal hippocampal–amygdala, emotion processing,
and emotion-regulation circuits, together with an "overactive," left-sided ventral striatal-ventrolateral, and orbitofrontal cortical reward-processing circuit (58). These results attest to the fact that, in relative terms of group comparison from fMRI studies, cognitive
DLPFC and the corresponding dorsal circuitry hub that includes the parietal region and the insula are more involved in the HR population, while the wider multiple cortico (VLPFC, DLPFC, and
**(A)** DLPFC: dorsolateral prefrontal cortex, x = 46, y = 8, z = 22, cluster
medial PFC) and subcortical (limbic and basal ganglia) regions are
ventrolateral prefrontal cortex.
implicated in PBD. Published structural and fMRI studies of HR have not been conclusive and are limited to a comparison with the TD at times (7, 11). Singh et al. (59) reported that 8- to 12-year-old children with a familial risk for mania did not exhibit any statistically significant volumetric differences in the PFC, thalamus, striatum, or amygdala
compared with the TD group. However, they concluded that longitudinal studies will be needed to examine whether structural changes over time may be associated with a HR for BD (59). Bechdolf et al. (60) reported volume reduction in emotion-processing regions (i.e., the insula and amygdala) in HR, relative to TD, that corresponded to the functional abnormality involving increased amygdala activity in HR (9). While we found abnormal function in the insula in HR in this meta-analysis, three-way comparison did not reveal increased amygdala activity in HR. Existing studies consistently reported smaller amygdala and hippocampus (61), larger basal ganglia (62), and reduced PFC gray matter (63) in PBD. Hemodynamic (64) and resting state connectivity (65) findings in PBD relative to TD also point to frontolimbic and frontostriatal functional disturbance in PBD. Such uniformity in multi-modal imaging findings attests to the high reliability in establishing a significant pattern of brain dysfunction specific to PBD.
Limitations of this study include fewer and unequal numbers of participants in the HR group and the inclusion of studies that employed variable tasks used to probe multiple domains. However, due to the broad array of daily functions that draws from the active involvement of multiple and highly integrated networks, and the dual engagement of VLPFC, DLPFC, ACC, and the striatum in both cognitive and emotional tasks, this study was a reasonable first attempt to examine the entire brain's level of functionality from the existing data.
#### **ACKNOWLEDGMENTS**
We thank all the authors of the included published papers for their important contributions to the field. Support for this work was provided by endowed funds by the Colbeth Foundation and Berger-Colbeth Chair funds. All authors had full access to the data in the study. Dr. Pavuluri takes responsibility for directing the analysis and interpretation of the data. Dr. Moon-Soo Lee ensured the accuracy of the data analysis. Drs. Anumagalla, Talluri, and Lee entered the data and checked them for accuracy. Inclusion and exclusion criteria were applied following the full consensus of all authors.
#### **REFERENCES**
64. Wegbreit E, Passarotti AM, Ellis JA, Wu M, Witowski N, Fitzgerald JM, et al. Where, when, how high, and how long? The hemodynamics of emotional response in psychotropic-naive patients with adolescent bipolar disorder.*J Affect Disord* (2013) **147**(1–3):304–11. doi:10.1016/j.jad.2012.11.025
65. Wu M, Lu LH, Passarotti AM, Wegbreit E, Fitzgerald J, Pavuluri MN. Altered affective, executive and sensorimotor resting state networks in patients with pediatric mania. *J Psychiatry Neurosci* (2013) **38**(4):232–40. doi:10.1503/jpn. 120073
**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
*Received: 10 June 2014; accepted: 24 September 2014; published online: 03 November 2014.*
*Citation: Lee M-S, Anumagalla P, Talluri P and Pavuluri MN (2014) Meta-analyses of developing brain function in high-risk and emerged bipolar disorder. Front. Psychiatry 5:141. doi: 10.3389/fpsyt.2014.00141*
*This article was submitted to Neuropsychiatric Imaging and Stimulation, a section of the journal Frontiers in Psychiatry.*
*Copyright © 2014 Lee, Anumagalla, Talluri and Pavuluri. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.*
# Stress, inflammation, and cellular vulnerability during early stages of affective disorders: biomarker strategies and opportunities for prevention and intervention
#### **Adam J.Walker 1,2,Yesul Kim1,2, J. Blair Price<sup>1</sup> , Rajas P. Kale1,3, Jane A. McGillivray <sup>2</sup> , Michael Berk 4,5,6,7 and Susannah J. Tye1,2,8\***
<sup>1</sup> Department of Psychiatry and Psychology, Mayo Clinic, Rochester, MN, USA
<sup>2</sup> School of Psychology, Deakin University, Melbourne, VIC, Australia
<sup>3</sup> School of Engineering, Deakin University, Geelong, VIC, Australia
<sup>4</sup> School of Medicine, Deakin University, Geelong, VIC, Australia
<sup>5</sup> Department of Psychiatry, University of Melbourne, Melbourne, VIC, Australia
<sup>6</sup> Orygen Youth Health Research Centre, Melbourne, VIC, Australia
<sup>7</sup> The Florey Institute of Neuroscience and Mental Health, Melbourne, VIC, Australia
<sup>8</sup> Department of Psychiatry, University of Minnesota, Minneapolis, MN, USA
#### **Edited by:**
Stephanie Ameis, University of Toronto, Canada
#### **Reviewed by:**
Vilma Gabbay, Mount Sinai School of Medicine, USA Ellen Grishman, University of Texas Southwestern Medical Center, USA
#### **\*Correspondence:**
Susannah J. Tye, Department of Psychiatry and Psychology, Mayo Clinic, 200 First Street SW, Rochester, MN 55905, USA e-mail: [email protected]
The mood disorder prodrome is conceptualized as a symptomatic, but not yet clinically diagnosable stage of an affective disorder. Although a growing area, more focused research is needed in the pediatric population to better characterize psychopathological symptoms and biological markers that can reliably identify this very early stage in the evolution of mood disorder pathology. Such information will facilitate early prevention and intervention, which has the potential to affect a person's disease course.This review focuses on the prodromal characteristics, risk factors, and neurobiological mechanisms of mood disorders. In particular, we consider the influence of early-life stress, inflammation, and allostatic load in mediating neural mechanisms of neuroprogression. These inherently modifiable factors have known neuroadaptive and neurodegenerative implications, and consequently may provide useful biomarker targets. Identification of these factors early in the course of the disease will accordingly allow for the introduction of early interventions which augment an individual's capacity for psychological resilience through maintenance of synaptic integrity and cellular resilience. A targeted and complementary approach to boosting both psychological and physiological resilience simultaneously during the prodromal stage of mood disorder pathology has the greatest promise for optimizing the neurodevelopmental potential of those individuals at risk of disabling mood disorders.
**Keywords: prodrome, depression, bipolar, biomarker, stress, inflammation, cellular resilience, plasticity**
#### **INTRODUCTION**
There is increasing appreciation for the need to both identify and treat mood disorders during their earliest stages (1). Although some dispute remains, maladaptive changes in mood and behavior first become evident during the prodromal period (2). However, the low specificity of these changes makes the prodromal stage difficult to definitively characterize prior to disease onset (3). Observable changes in mood and general physiologic functioning can include increases in sadness, anhedonia, irritability, anger, and anxiety, together with alterations in sleep and energy (4). Correlating these symptoms with prodromal biomarkers offers an exciting juncture whereby targeted interventions could be opportunistically employed to prevent neurodegenerative changes from accruing as the disease progresses (5). The potential to intervene during the prodromal stage of psychiatric illness through the detection and remediation of novel biomarkers has perhaps been best studied in schizophrenia, wherein most individuals experience a lengthy prodromal period prior to the full emergence of diagnosable psychotic symptoms (6). As an exemplar, low levels of nervonic acid appear to be a risk factor for conversion from
high-risk to frank psychosis (7), and this risk of conversion may be reduced by targeted omega-3 fatty acid supplementation (8). Encouraging results from this work have renewed interest in the early detection of affective disorders, particularly bipolar disorder, with the hope that earlier and more targeted interventions might slow disease progression (3, 9–12). This can significantly impact neuroprogression and subsequent disease course for the individual (13). This concept of "neuroprogression" refers to the cumulative restructuring of the central nervous system which in turn mediates the development and persistence of psychiatric illness (14, 15). This process results from disturbances in inflammatory mediators, neurotrophins, oxidative stress, and energy regulation (14, 15).
#### **BIOMARKER STRATEGIES FOR PRODROMAL MOOD DISORDERS**
#### **STRESS AND ALLOSTATIC LOAD Stress sensitization and early detection**
Stress is one of the best-studied mediators by which genetic vulnerabilities are translated into mood disorder pathology through the process of neuroprogression (16–18). Numerous studies have demonstrated that both depression and bipolar disorder are more prevalent in individuals who have experienced adverse early-life events. This is partly because such experiences prime future physiologic and neural responses to stress, elicit a state of chronic inflammation (19), alter cellular mediators of plasticity and energy metabolism, and increase cellular "wear and tear" (20–22). Earlylife stress (2) can be particularly deleterious because of its potential to influence the programing of the hypothalamic–pituitary– adrenal (HPA) axis (23) to induce persistent sensitization of neuroendocrine, autonomic, oxidative, and immune responses to stress. Over time these sensitized systems cumulatively contribute to the cellular and synaptic alterations underlying neuroprogression (21, 24–26). Specific examples include changes in reactivity of inflammatory cytokines [e.g., interleukin 6 (IL-6)] (25), alterations in markers for lipid peroxidation [e.g., 8-iso-prostaglandin F (2α)],oxidative damage to DNA (8-hydroxy-2<sup>0</sup> -deoxyguanosine) and RNA (8-hydroxyguanosine) (24), as well as altered cortisol, adrenocorticotropic hormone, and corticotrophin releasing factor responses (26). Identification of the state of physiologic and cellular resilience or sensitivity to stress may provide an important indicator of the level of neuroprogression and stress-mediated disease pathology for affective disorders,potentially prior to the initial manifestation of the mood episode (22).
One mechanism whereby HPA axis sensitization is likely to occur is through epigenetic regulation of stress response processes (21, 27). Evidence shows that exposure to various forms of stress result in multiple epigenetic changes in limbic regions as well as the HPA axis (21, 27). Interestingly, a recent study by Klendel and colleagues (18) found that only individuals who exhibited allele-specific DNA demethylation in functional glucocorticoid response elements of FK506 binding protein 5 (*FKBP5*), were prone to developing persistent cortisol dysregulation (18, 21). Further, this association was found to be dependent on an interaction effect with trauma in early life, suggesting that key developmental stages are directly related to stability of the observed effects across time (18). In another study, significant interactions between peripheral *FKBP5* mRNA expression and disease progression were reported, suggesting that polymorphisms in the gene directly impact the extent of neuroendocrine dysregulation, and corresponding neuroprogression (28). The *FKBP5* risk allele and corresponding levels of mRNA expression may represent useful biomarkers. These markers could be employed to identify individuals in the prodromal stages of stress-sensitive psychiatric disorders, such as major depression or bipolar disorder. Such detection would facilitate early intervention and could improve resilience and alleviate allostatic load in the prodromal individual.
#### **Early-life stress and accumulation of allostatic load**
Accumulation of allostatic load is a key mechanism through which early-life stress is thought to result in psychopathology (29). This is mediated via a series of enduring adaptive changes across a range of systems primed both to respond rapidly to challenge, as well as to restore homeostatic equilibrium (30). Adaptive allostatic mechanisms may fail when chronically challenged or when regulatory systems falter. This leads to a state of allostatic overload, which is thought to considerably impact the clinical course of mood disorders (31–33). Without sufficient opportunity for recovery, the brain and body are repeatedly exposed to molecular mediators of stress that can increase the level of cellular "wear and tear" (33). These mediators, which include metabolic factors, inflammatory cytokines, neurotrophins, and oxidative species, collectively impact an individual's mental and physical resilience as outlined below [for more detailed reviews see Ref. (6, 34, 35)]. Both physiological (i.e., immune and/or metabolic) and psychological (i.e., bullying) stressors contribute significantly to allostatic load, and thus need to be considered together when assessing both risk and relative staging of mood disorder pathology (6, 34).
Enhancing an individual's capacity to buffer the physiologic toll that accumulates through allostatic overload should be considered an important early intervention strategy. As allostatic load accumulates and attempts to maintain cellular homeostasis fail, cell danger signals are propagated and pro-apoptotic cell signaling pathways become increasingly engaged (36–39). This may play a role in medical comorbidities such as heart disease (40), as well as interfere with the therapeutic mechanisms of antidepressants and mood stabilizers to impair treatment efficacy (41–43). Internal stressors that activate the HPA axis and associated allostatic systems can limit an individual's capacity for allostasis even prior to the onset of external stressors (36). For example, an endogenous load can build through the expression of homocysteine or inflammatory cytokines,limiting the capacity of adaptive responses in the face of subsequent stressors. Interventions that counter this load and reduce levels of proinflammatory mediators or interfere with their neuromodulatory actions could limit neuroprogression in both bipolar and unipolar depression, as well as enhance capacity for antidepressant efficacy (44–46).
#### **INFLAMMATORY PROFILE**
Stress during earlier life is not only associated with disruption of the HPA axis, but may also serve to sensitize proinflammatory responses to future insults (47–49). Inflammatory mechanisms are increasingly appreciated for their critical role in mood disorder pathophysiology, in particular via their regulation of neuronal excitability, synaptic transmission, synaptic plasticity and neuronal survival (41, 50, 51). Of specific interest are proinflammatory mediators, such as cytokines [i.e., interleukin 1, IL-6, and tumor necrosis factor alpha (TNF-α)] and C-reactive protein (CRP). CRP is often used as a biomarker for inflammation in studies due to its relationship with proinflammatory cytokines and role in the immune response. As demonstrated by Slopen and colleagues (49), individuals at ages 10 and 15 who reported adverse life events at critical stages between the ages of 1.5 and 8 years were found to have significantly increased levels of CRP and IL-6. These heightened concentrations were correlated with immune activation and depressive-like symptoms. Notably, increased CRP levels have been used previously to predict depression severity and recurrence rates in males (48, 52).
There is a growing literature supporting the use of inflammatory biomarkers as predictors of ensuing mood disorder pathology (22). Research to date has been focused on investigating the relationship between inflammatory cytokines and affective disorders in adults; however, their specific role in early onset/adolescent psychopathology is less well explored (53). Cytokines are thought to influence neurodevelopment during key stages, such as adolescence, interacting with biological systems including those of stress hormones and gonadal hormones (53). As such, perturbation of inflammatory balance in adolescents may significantly contribute to neuroprogression and development of psychiatric illness (19, 53, 54). For example, elevated serum levels of TNF-α, IL-6, and interleukin-10 (IL-10) have been reported during the early stages of bipolar disorder (55), and CRP appears to be a biomarker of *de novo* depression risk (56).
As the mood disorder pathology progresses, an increasing number of proinflammatory cytokines are observed, including elevated levels of interferon gamma (IFN-γ) (22, 54, 55). Notably, increases in IFN-γ are associated with dysregulation of the tryptophan metabolite pathway via direct role in indoleamine 2,3-dioxygenase (IDO) activation. Activation of IDO is commonly found in later stages of mood disorders, and is a biomarker of depression-like behavior mediated by neural inflammation in animal models (48). Proinflammatory cytokines activate IDO, resulting in depletion of serotonin and augmentation of quinolinic acid (QUIN) metabolism over kynurenic acid (KYNA). Tryptophan metabolites (kynurenine, KYNA, 3-hydroxykynurenine, and QUIN) act as neuromodulators to influence behavioral, neuroendocrine, and neurochemical aspects of depression (57–60). Consequently, this accumulation of QUIN facilitates neurodegeneration over neuroprotection, impacting mood disorder neuroprogression and resultant disability (61).
It is noteworthy to mention several other findings regarding altered inflammation in youth with psychiatric pathology. Increased mRNA and protein expression levels of IL-1β, IL-6, and TNF-α were reported in the anterior prefrontal cortex of adolescent suicide victims compared with normal control subjects (62). Elevated levels of inflammatory cytokines (among others: TNF-α, IL-1β, IL-6, and IFN-γ) were also observed in the serum of pediatric patients who experienced first-episode psychosis, in addition to increased leukocyte counts and evidence of blood–brain barrier damage (63). Quantification of inflammatory biomarkers (e.g., TNF-α, IL-6, IL-10, or CRP) may thus prove useful for detecting individuals at risk for developing a mood disorder. A recent study by Byrne and colleagues (64) suggests that levels of peripheral cytokines (e.g., IFN-γ) and CRP in salivary samples may correlate with serum samples in young people. Salivary assay may prove to be a simpler, less invasive method of estimating peripheral levels of inflammatory markers in adolescents (64). This provides one avenue whereby prodromal individuals could potentially be identified and their disease onset delayed.
#### **DIMINISHED SYNAPTIC INTEGRITY**
Homeostatic control of synaptic connections within key moodrelated circuits plays a critical role in the etiology of mood disorders (65). Stress and inflammation as discussed in previous sections are implicated in disruption of synaptic signaling and integrity during the early stages of mood disorder pathogenesis. This is mediated in part through the inhibition of neurotrophin function, of which brain derived neurotrophic factor (BDNF) is the most thoroughly characterized. BDNF plays an important role in neuronal development, survival, and function, including activity-dependent synaptic plasticity (66). Synaptic plasticity is characterized by various processes, including synaptic remodeling, synaptogenesis, long-term potentiation, and long-term depression, all of which critically mediate the flow of electrochemical information throughout the central nervous system (67, 68). Stress, allostatic load, inflammation, antidepressants, and mood stabilizers exert major effects on signaling pathways that regulate cellular plasticity, suggesting these are critical neurobiological mediators of mood dysfunction and therapeutic intervention (69–72).
Glycogen synthase kinase-3 (GSK-3), part of the signaling cascade regulated by BDNF, plays an important role in synaptic homeostasis through regulation of synaptic deconsolidation (pruning) and glutamate receptor cycling (73). Increased GSK-3-mediated synaptic deconsolidation has been suggested to be an important factor contributing to reduced spine density in mood disorders (74). Additionally, levels of activated GSK-3 are increased in postmortem brain tissue from individuals with unipolar and bipolar depression (74). In addition to BDNF, GSK-3 is deactivated by signals originating from numerous signaling pathways demonstrated to be dysregulated in mood disorders (e.g., Wnt and PI3K pathways), and is either the direct or downstream target of many mood stabilizer and antidepressant medications (75). GSK-3 activity is modulated by serotonin and dopamine, and is a critical node at the intersection of multiple neurotransmitter and cell signaling cascades (68). As a result, GSK-3 modulates not only synaptic plasticity but also apoptotic mechanisms and, in turn, plays a critical role in mediating cellular resilience (75). For this reason, GSK-3 has received much attention for its potential to be targeted as an early intervention strategy during the prodrome period.
#### **IDENTIFYING IMPAIRED CELLULAR RESILIENCE**
Stress, allostatic overload, and neuroinflammation function together to impair synaptic plasticity and cellular resilience. Disrupted plasticity along with increased cellular vulnerability contributes significantly to the pathophysiology of mood disorders and directly to the neuroprogressive nature of the disease course (3, 76). Some of the key mechanisms of disease progression affecting cellular resilience include: oxidative stress, decreased neurotrophic factor expression, reduced neurogenesis, impaired regulation of calcium, altered endoplasmic reticulum and mitochondrial function, together with dysregulated energy metabolism and insulin signaling. Each of these mechanisms are mediated by allostatic overload and neuroinflammation [for detailed reviews see Ref. (3, 36, 76–78)]. Together, these processes demonstrate that in addition to synaptic integrity, maintenance of cellular homeostasis is critical for facilitating cellular resilience and attenuating mood disorder pathogenesis (79), which is also likely to enhance the capacity for treatment response during later stages of the disorder (80).
Cellular vulnerability and resilience are mediated by apoptotic and anti-apoptotic intracellular signaling cascades, respectively. Apoptosis is important for the regulation of developmental processes and prevention of cancerous growths. Excessive apoptosis in neuronal systems, however, leads to neurodegeneration and certain cell populations are at increased risk of stress-mediated apoptotic cell death (80). Apoptosis is a tightly regulated and energy-dependent process, which coordinates programed cell death in response to different stimuli (81). This can occur through stimulation of death receptor proteins [i.e., tumor necrosis factor (TNF) receptor] by cytokines of the TNF superfamily or in response to mitochondrial degradation. These stimuli result in activation of executioner caspases that function to coordinate cellular process necessary for apoptosis, including cessation of cell repair processes and cell cycle progression, cytoskeletal and nuclear disassembly, and flagging the cell for phagocytosis (82). Distinct classes of antidepressants and mood stabilizers have been demonstrated to facilitate cellular resilience to prevent progression of pro-apoptotic processes, and novel treatments are currently being developed to target these specific mechanisms (83). Biomarkers that characterize the level of neuronal vulnerability relative to resilience may prove useful as biomarkers of prodromal mood disorder pathology. This has been demonstrated for later stages of bipolar disorder (84), however more studies are needed to determine the utility of such cell danger biomarkers during the mood disorder prodrome (22).
## **OPPORTUNITIES FOR PREVENTION AND INTERVENTION**
#### **IDENTIFYING VULNERABILITIES AND BUILDING RESILIENCE AT THE CELLULAR LEVEL**
Identification of individuals at risk of developing a mood disorder, or those in the prodromal stage, provides a potential opportunity to target these mechanisms for neuroprotective interventions that enhance cellular resilience, maintain synaptic plasticity and boost psychological resilience (**Figure 1**) (85). One of the longest held notions of brain plasticity is that certain critical periods or windows exist in development, during which circuitry is consolidated for lifetime functionality. Recently, there is a rising consensus that developmentally induced plasticity can, to an extent, be reversed by "re-opening" those windows of plasticity (86). Hyman and Nestler (87) have underscored the importance of shifting the brain into an "adaptive state" to necessitate the antidepressant response. Their theory of "initiation and adaption" is exemplified by psychotropic drugs wherein primary molecular targets that initiate alterations in brain function activate homeostatic mechanisms that return the system to an adaptive and treatment responsive state (87). Plasticity and cellular resilience are thus necessary for the efficacy of antidepressants and mood stabilizing treatments. McGorry and colleagues (6, 88) and others (89) have demonstrated this concept with pre-psychotic interventions, and repeatedly emphasized the need to take advantage of the "windows of opportunity" present within the prodromal stages of psychiatric disease (6, 88, 89). During this stage, the course of the disease remains theoretically plastic and amenable to intervention (90). Previous literature indicates that once risk or prodromal symptoms of mood disorders are identified, there is some (91), but not unequivocal (92) evidence that early intervention in adolescents can significantly reduce mood-related symptoms and incidence of fully diagnosable psychiatric disorders such as depression (93– 95). Neuroprotective pharmacotherapies together with appropriate psychotherapy may reduce the risk of neuropsychiatric disease progression in young people which, together with allostatic load reducing behavioral interventions, may significantly slow the trajectory of the disease course into adulthood (6, 36, 96). Such interventions may include reducing lifestyle mediators of allostatic load (19, 97).
**FIGURE 1 | A representation of the conceptual balance between vulnerability and resilience in prodromal individuals**. The scale's balance beam teeters between vulnerability and resilience as scale pans are loaded with different positive and negative biological, psychological and social factors. The presence or absence of these factors influence the ability of the individual to cope with stressors, and maintain allostasis. **(A)** Prodromal individuals are somewhat predisposed to vulnerability; but with intervention **(B)** an individual may adopt more adaptive environmental coping strategies, support mechanisms, general healthy lifestyle choices, and/or receive pharmacological interventions that collectively enhance physiological and psychological resilience.
#### **COGNITIVE AND BEHAVIORAL INTERVENTIONS TO BUFFER STRESS AND BUILD RESILIENCE**
Individuals provided with effective social and emotional support to help cope with stressors that are adverse and potentially taxing will be much better placed to limit associated biological costs and maintain allostasis (98). The absence of emotional or social support and the implementation of maladaptive coping strategies can enhance the toxic effects of stress and contribute to allostatic overload (98). Exposure to regular and controllable stressors over the course of childhood and adolescence is essential for the development of effective coping strategies. Through such exposure, an individual can develop a repertoire of these coping strategies. Mathew and Nanoo (99) found that adaptive coping strategies (e.g., employing self-control, accepting responsibilities, problem solving, seeking social support, or positive reappraisal) are protective for suicide risk in adolescents. Conversely, maladaptive coping strategies, such as confrontation, distancing, and escape-avoidance were reported to be significant risk factors associated with adolescent suicide attempts (99). These findings provide evidence to support the notion that coping strategies can act as protective factors against both the development and progression of mood disorders. Importantly, educating children and adolescents in protective coping skills may be a promising intervention that could be implemented as early as elementary school. In recent years, patterns of threat perception such as optimism have attracted much attention in relation to later mood, coping, and immune change in response to stress (100, 101). Moreover, it has been found to be protective against the development of depressive symptoms in later life (102). Its potential role in buffering against the negative emotional consequence of adverse events has led to a view of optimism as an index of resilience (103). Optimists may also choose lifestyles that promote physical as well as mental health, thereby reducing other aspects of allostatic load.
Healthy lifestyle, similar to optimism, provides a solid foundation for adaptation, and increases available resources for buffering the neurodegenerative effects of stress. Specifically, previous literature highlights the importance of healthy diet, adequate sleep, avoidance of smoking, and sufficient exercise (104). A populationbased study reported higher emotional well-being among physically active youths, independent of social class and health status (105). Across a 2-year period, Motl and colleagues (106) found changes in physical activity were inversely related to a change in depressive symptoms. Levels of physical activity in childhood can modulate the risk of adult depression (107). Exercise modulates many of the core biomarkers of neuroprogression, including inflammation, oxidative stress, and neurotrophins (108). Poor eating habits and sleep have been linked to the manifestation of toxic stress and unhealthy growth in pediatrics by disrupting the architecture of the plastic, adaptive brain (109). There is now extensive evidence that poor diet quality is a risk for adolescent depression (110), and new data suggests that maternal diet influences the mental health of offspring (111). Similarly, smoking increases the risk of mood and anxiety disorders, and appears to influence similar biological pathways (112, 113). Parents and care givers of younger children need to be informed of the potential impact that a healthy lifestyle can have in mitigating mood-related symptoms and problematic behaviors. Low-risk interventions such as those aforementioned are critical for enhancing both psychological and biological resilience to stress. When such perspectives and lifestyle health behaviors are consolidated early in childhood and adolescence, the cumulative effect may be meaningful (103).
#### **CONCLUSION**
Early intervention offers the possibility of altering the trajectory of mood disorder pathology. In so doing, we may curtail the progressive nature of the illness, both through neuroprotection and maintenance of peripheral health. Prevention and intervention treatments should go beyond stabilizing mood to include various and complementary strategies for reducing allostatic load, perhaps through psychoeducation and lifestyle-related interventions, including effective stress management. The combination of these techniques with specific pharmacotherapies may significantly improve functional outcomes by both reducing cellular insults and enhancing resilience. In so doing, this optimizes the capacity for maintenance of synaptic integrity and cellular resilience, which must be aggressively targeted as a therapeutic strategy during the prodromal stage of mood disorder pathology (90). This neuroprotective approach not only slows neuroprogression associated with the disease, but lays a foundation for more treatment-responsive outcomes during later stages.
#### **AUTHOR CONTRIBUTIONS**
Adam J. Walker, Yesul Kim, J. Blair Price, Rajas P. Kale, Jane A. McGillivray, Michael Berk, and Susannah J. Tye each made contributions to the writing of this manuscript.
#### **ACKNOWLEDGMENTS**
This work was supported by a Mayo Minnesota Partnership grant to Susannah J. Tye. Adam J. Walker is supported by an Australian Postgraduate Award. Yesul Kim is supported by a Deakin University Award. Michael Berk is supported by a NHMRC Senior Principal Research Fellowship 1059660.
#### **REFERENCES**
of depressive symptoms in adolescents. *J Am Acad Child Adolesc Psychiatry* (2004) **43**(3):298–306. doi:10.1097/00004583-200403000-00011
5 years: a prospective cohort study. *J Am Acad Child Adolesc Psychiatry* (2013) **52**(10):1038–47. doi:10.1016/j.jaac.2013.07.002
**Conflict of Interest Statement:** Michael Berk has received grant/research support from the NIH, Cooperative Research Centre, Simons Autism Foundation, Cancer Council of Victoria, Stanley Medical Research Foundation, MBF, NHMRC, Beyond Blue, Rotary Health, Geelong Medical Research Foundation, Bristol Myers Squibb, Eli Lilly, GlaxoSmithKline, Meat and Livestock Board, Organon, Novartis, Mayne Pharma, Servier and Woolworths, has been a speaker for Astra Zeneca, Bristol Myers Squibb, Eli Lilly, GlaxoSmithKline, Janssen Cilag, Lundbeck, Merck, Pfizer, Sanofi Synthelabo, Servier, Solvay, and Wyeth, and served as a consultant to Astra Zeneca, Bristol Myers Squibb, Eli Lilly, GlaxoSmithKline, Janssen Cilag, Lundbeck, Merck, and Servier. The other authors have no conflicts to report.
*Received: 29 October 2013; accepted: 17 March 2014; published online: 09 April 2014. Citation: Walker AJ, Kim Y, Price JB, Kale RP, McGillivray JA, Berk M and Tye SJ (2014) Stress, inflammation, and cellular vulnerability during early stages of affective disorders: biomarker strategies and opportunities for prevention and intervention. Front. Psychiatry 5:34. doi: 10.3389/fpsyt.2014.00034*
*This article was submitted to Neuropsychiatric Imaging and Stimulation, a section of the journal Frontiers in Psychiatry.*
*Copyright © 2014 Walker, Kim, Price, Kale, McGillivray, Berk and Tye. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.*
# Neural responses during social and self-knowledge tasks in bulimia nervosa
### **Carrie J. McAdams <sup>1</sup>\* and Daniel C. Krawczyk 1,2**
<sup>1</sup> Department of Psychiatry, The University of Texas Southwestern Medical Center, Dallas, TX, USA
<sup>2</sup> School of Behavioral and Brain Sciences, Center for Brain Health, The University of Texas at Dallas, Dallas, TX, USA
#### **Edited by:**
Paul Croarkin, Mayo Clinic, USA
#### **Reviewed by:**
Peter G. Enticott, Monash University, Australia Jamie Morris, University of Virginia, USA
#### **\*Correspondence:**
Carrie J. McAdams, Department of Psychiatry, The University of Texas Southwestern Medical Center, 6363 Forest Park Road BL6.110E, Dallas, TX 75390-8828, USA e-mail: carrie.mcadams@ utsouthwestern.edu
Self-evaluation closely dependent upon body shape and weight is one of the defining criteria for bulimia nervosa (BN). We studied 53 adult women, 17 with BN, 18 with a recent history of anorexia nervosa (AN), and 18 healthy comparison women, using three different fMRI tasks that required thinking about self-knowledge and social interactions: the Social Identity task, the Physical Identity task, and the Social Attribution task. Previously, we identified regions of interest (ROI) in the same tasks using whole-brain voxel-wise comparisons of the healthy comparison women and women with a recent history of AN. Here, we report on the neural activations in those ROIs in subjects with BN. In the Social Attribution task, we examined activity in the right temporoparietal junction (RTPJ), an area frequently associated with mentalization. In the Social Identity task, we examined activity in the precuneus (PreC) and dorsal anterior cingulate (dACC). In the Physical Identity task, we examined activity in a ventral region of the dACC. Interestingly, in all tested regions, the average activation in subjects with bulimia was more than the average activation levels seen in the subjects with a history of anorexia but less than that seen in healthy subjects. In three regions, the RTPJ, the PreC, and the dACC, group responses in the subjects with bulimia were significantly different from healthy subjects but not subjects with anorexia. The neural activations of people with BN performing fMRI tasks engaging social processing are more similar to people with AN than healthy people.This suggests biological measures of social processes may be helpful in characterizing individuals with eating disorders.
**Keywords: mentalization, identity, theory of mind, eating disorders, anorexia, bulimia, neuroimaging, social behavior**
#### **INTRODUCTION**
Bulimia nervosa (BN) is an eating disorder characterized by frequent binge-eating followed by purging behaviors in concert with a self-esteem that is overly associated with body shape and weight (1). The symptoms of many eating disorder patients change during their lives (2, 3). For example, a patient may develop restricting behaviors with weight loss in high school, begin binging and purging behaviors at a low weight, continue binge-purge behaviors at a healthy weight throughout college, and then cease the purging behaviors but have occasional binge-eating problems. Such a patient would have met criteria for anorexia nervosa (AN), restricting subtype initially, then AN, binge-purge subtype, then BN, and finally binge-eating disorder. This diagnostic instability makes clinical treatment as well as research into eating disorders challenging (4, 5). A better understanding of biological and cognitive similarities and differences that contribute to eating disorders may improve clinical treatment. Currently, treatment of BN leads to sustained recovery in only about half of the patients (6, 7). Through the use of fMRI, we examined neural activations related to social processes in BN.
A specific set of neural regions is modulated in response to tasks that require thinking about people in healthy subjects (8, 9); this provides a framework to assess differences related to psychiatric illnesses. Severe impairments in social interaction are one of the diagnostic criteria for autistic spectrum disorders (1), but problems in social cognition have been reported in many psychiatric illnesses (10–14). Decreased social cognition has been reported in a variety of behavioral tasks in adults with AN (15–17). In BN, recent studies have concluded that there was little evidence of social cognition differences in psychological tasks (18, 19), although far fewer studies of social cognition have been completed in BN than in AN.
Because both BN and AN include in their diagnostic criteria an association between appearance and self-esteem (1), these experiments focused on neural pathways related to thinking about oneself. Self-esteem is a term used to describe one's overall sense of one's own value as a person, and is generally considered a fairly stable psychological characteristic (20). Although the diagnostic criteria in eating disorders connect self-esteem specifically to physical appearance, similarly unrealistic social expectations are reported and observed in eating disorder patients (21, 22). Selfknowledge, as used in MRI tasks, relates to the ability to evaluate oneself, and is expected to be a process that involves self-esteem as well as other criteria. For example, an individual whose self-esteem is highly related to appearance might be very good at her work, and correctly describe herself as a competent employe, but maintain an overall low self-esteem because of perceived inadequacies of appearance. Furthermore, low self-esteem has been related to
prognosis in AN (23,24) as well as onset of bulimic symptoms (25). Negative beliefs about one's self, unrelated to physical appearance, have been observed in eating disorders (26, 27), and neural differences in the processing of these negative self-beliefs have been seen in BN (28). These data show not only that psychological similarities in self-esteem are present in AN and BN, but also that self-esteem is an important factor in assessing the prognosis and severity of eating disorders.
In healthy people, midline cortical structures, including the cingulate (Cing), dorsal anterior cingulate (dACC), and precuneus (PreC), have been specifically associated with thinking about oneself, using a variety of self-knowledge, appraisal, and viewing tasks (29). Most commonly, these areas show activation during neuroimaging tasks that ask healthy subjects to reflect upon whether specific characteristics describe oneself (30). Performance of this type of task is likely to acutely stimulate similar cognitive processes as those that generate one's longer-term sense of self-esteem. We recently reported differences in brain activations in AN and CN based on differences in self-knowledge using two neuroimaging tasks that required self-evaluations, one using social adjectives and the other physical descriptors (31). In that study, we identified regions in the dACC, PreC, and Cing with different activations in subjects with AN compared to the healthy controls. Here, we consider the responses of subjects with BN in the same self-evaluative tasks.
In addition to self-evaluative tasks, we included a more general social processing neuroimaging task that robustly engages additional regions in the social processing network associated with considering other people (32). This task, the Social Attribution Task, strongly activates the right temporoparietal junction (RTPJ), a region that has been closely associated with theory of mind (TOM), and mentalization [for reviews, see (9, 33)], as well as the fusiform gyrus, a region closely associated with facial processing (34, 35). Furthermore, differences in fMRI activations in both adult participants with AN (36) as well as adolescent participants with AN (37) have been examined using this task. In this manuscript, we describe the neural activations of subjects with BN during the Social Attribution Task.
#### **METHODS AND MATERIALS ETHICS STATEMENT**
This study was approved by the institutional review boards at both the University of Texas Southwestern Medical Center and The University of Texas at Dallas. Additionally, the study adhered to the guidelines as set out in the Declaration of Helsinki. Written informed consent was required from all participants, and subjects were reimbursed for time spent participating.
#### **PARTICIPANTS**
A total of 53 female participants, between 18 and 42 years of age, were recruited for this study from the general public, from treatment providers, and support groups in the Dallas–Fort Worth area. Subjects volunteered to spend 2 h in clinical assessments and completing questionnaires and 1.5 h completing behavioral tasks in the MRI scanner, and were compensated for their time financially. The participant groups consisted of 18 healthy controls (CN), 18 individuals with a recent history of anorexia but were currently in
the process of recovering from AN, and 17 individuals recovering from BN. All AN and BN participants had met full DSM-IV criteria for either AN or BN within the previous 2 years. The AN subjects were required to be maintaining a minimum BMI of 17.5 with no weight loss for the 3 months preceding the MRI scans. This was based primarily on a detailed eating disorder symptom and weight history obtained at the initial screening interview, only after which was it divulged that low or unstable weight was an exclusion factor for MRI scans. One of the AN and one of the CN participants did not complete the Social Attribution task and one of the BN participants did not complete the Social and Physical Identity neuroimaging task; these subjects were excluded in the analyses involving those tasks. Another BN participant was excluded from the neuroimaging analyses of the Social and Physical Identity tasks due to excessive movement. Eleven of the AN subjects had the restricting subtype and seven had the binge-purge subtype of AN. All subjects were recruited and scanned between 2009 and 2012. Because of difficulty recruiting BN subjects, data collection of AN and CN subjects finished nearly 1 year before the last four BN subjects were obtained. Therefore, the AN and CN data were analyzed and published in two earlier papers, one describing results obtained from the Social Attribution task and the other results from the two Identity tasks (31, 36).
Subjects provided written informed consent to participate in this study at an initial appointment. All subjects were then interviewed using the Structured Clinical Interview for DSM-IV disorders (SCID-RV). Participants were also screened for MRI compatibility. Some of the subjects had a history of recurrent MDD (1, CN; 7, AN; 9 BN) but none had met symptom criteria for an MDE for at least 3 months prior to the neuroimaging studies. No participants had a current or past diagnosis of any psychotic disorders or bipolar disorder based on the SCID-RV; no participants were currently taking mood-stabilizers, antipsychotics, or benzodiazepines. Participants on antidepressants whose dosage had not changed for at least 3 months prior to their MRI scans were included (1 CN; 8 AN; 7 BN).
Participants also completed the Quick Inventory of Depression, Self-Report (QIDS-SR), a self-report questionnaire consisting of 16 items to assess current symptoms of depression (38), and the Eating Attitudes Test-26 (EAT-26), a self-report questionnaire consisting of 26 items that relate to current eating behaviors (39). Subjects also completed the Self-Liking and Self-Competence Self-Esteem Questionnaire (SLCS), a 16 item self-report questionnaire that provides two measures of self-esteem (40), and the Social Problem-Solving Inventory (SPSI-R), a 26 item self-report questionnaire (41).
#### **NEUROIMAGING TASKS**
Three fMRI tasks were employed, the Social Attribution Task, the Social Identity task, and the Physical Identity task. Most subjects (35 of 53) preferred to complete the tasks in two scanner sessions, the first session consisting of the Social Attribution Task, and the second both the Identity Tasks. If all tasks were completed in 1 day, the Identity tasks were run before the Social Attribution task, and the total scan time was 80 min. When completed on separate days, the first day lasted about 30 min and the second session was about 50 min.
The Social Attribution Task presented short videos of moving shapes (32, 36). Briefly, subjects were asked to view the shapes in two conditions: the visuospatial or Bumper condition, preceded by the question "Bumper cars: Same weight?" and the social attribution or People condition, preceded by the question "People: All friends?". Each animation consisted of a moving display of three white shapes (circle, triangle, and square) and a white box with one side that opened as if hinged on a black background. Although the same shapes were presented in both conditions, the movements of the shapes in the two tasks differed. During the visuospatial task, the shapes moved around the box for the duration of the animation periodically bumping into one another. During the social task, the shapes moved in ways that suggested social behavior was occurring among the shapes (e.g., playing, fighting, avoiding etc.). We recorded responses to the weight and friendship questions about the animations to determine accuracy and maintenance of concentration.
The Identity tasks consisted of the presentation of written appraisal statements projected onto a screen within the MRI scanner (31). For both the Social Identity and the Physical Identity task, three different types of appraisals were shown: self (evaluation of an attribute about one's own identity based on one's own opinion), Friend (evaluation of an attribute about a close female friend), and Reflected (evaluation of an attribute about one's self from one's friend's perspective). Each statement was presented above a scale reading 1 "Strongly Disagree," 2 "Slightly Disagree," 3, "Slightly Agree," and 4 "Strongly Agree." Subjects were asked to read each statement and select a rating via a hand-held button. The Friend and Reflected statements were personalized to contain the name of a specific female friend of each subject. Each task was conducted separately, with all runs of the Social task preceding any runs of the Physical task. In the Social task, the statements were presented in a format ending with a socially descriptive adjective (ex. Self Statement"I believe I am nice,"Friend statement"I believe my friend is mean," Reflected statement, "My friend believes I am responsible"). For the Physical task, the statements were presented in the format ending with a physical body part and a descriptor (ex. Self statement "I believe my arms are toned," Friend statement "I believe my friend's eyes are bloodshot," Reflected statement "My friend believes my stomach is flabby"). In all cases my friend was replaced with the name of a close female friend of the subject.
#### **MRI ACQUISITION AND ANALYSIS**
All images were acquired with a 3T Philips MRI scanner. High resolution MP-RAGE 3D T1-weighted images were acquired for anatomical localization with the following imaging parameters: repetition time (TR) = 2100 ms, echo time (TE) = 3.7 ms; slice thickness of 1 mm with no gap, a 12°flip angle, and 1 mm<sup>3</sup> voxels. For both fMRI tasks, each slice was acquired with a 22.0 cm<sup>2</sup> field of view, a matrix size of 64 × 64, and a voxel size of 3.4 mm × 3.4 mm × 3 mm using a one-shot gradient T2\* weighted echoplanar (EPI) image sequence sensitive to blood oxygen level-dependent (BOLD) contrast. Head motion was limited using foam head-padding.
For the Social Attribution task, images were acquired during four runs, each lasting 128 s and presenting four 17-s videos, two in each condition (People or Bumper). These sequences were
acquired using a TR of 1.5 s, an TE of 25 ms, and a flip angle of 60°, and volumes were composed of 33 tilted axial slices (3 mm thick, 1 mm slice gap) designed to maximize whole-brain coverage while minimizing signal dropout in the ventral anterior brain regions. For the Identity tasks, images were acquired during eight runs (four for Social and four for Physical), each lasting 360 s, and presenting 12 statements of each condition (Self, Friend, and Reflected). These sequences were acquired using a TR of 2 s, an TE of 35 ms, and a flip angle of 0°, and volumes were composed of 36 axial slices (4 mm thick, no gap).
Prior to statistical analyses, preprocessing for all tasks consisted of spatial realignment to the first volume of acquisition, normalization to the MNI standard template, and spatial smoothing with a 6 mm 3D Gaussian kernel. fMRI task data were analyzed using Statistical Parametric Mapping software (SPM5, Wellcome Department of Imaging Neuroscience London)<sup>1</sup> run in MATLAB 7.4<sup>2</sup> , and viewed with xjview<sup>3</sup> .
The fMRI data were analyzed separately for each of the three tasks. For the Social Attribution task, the data were analyzed using a general linear model to create contrast images with a block design (blocks: People and Bumper); the Identity tasks were analyzed separately using an event-related design, in which each type of event (events: Self, Friend, and Reflected) corresponded to the BOLD signal during the 4 s presentation of each statement. With both techniques, the general linear model was used to create contrast images with activation of each condition assessed using a multiple regression analysis set as boxcar functions. Each regressor was convolved with a canonical hemodynamic response function (HRF) provided in SPM5 and entered into the modified general linear model of SPM5. Parameter estimates (e.g., beta values) were extracted from this GLM analysis for the regressors. Resulting single-subject one-sample *t*-test contrast images were created for each participant for each of the three tasks. These contrast images were combined for group map analyses.
#### **REGIONS OF INTEREST**
Previously we identified four regions showing group differences with whole-brain voxel-wide comparisons of the AN and CN group maps using the contrasts of conditions in the three tasks (31, 36). These were the *a priori* regions of interest (ROI) for this study focusing on BN. In the Social Attribution Task, the wholebrain voxel-wide comparisons of the AN and CN groups led to identification of a 94 voxel region in the RTPJ (MNI 52, −64, 20) that showed more modulation in the People condition than the Bumper condition in the CN subjects compared to the AN subjects. In the Social Identity Task, the whole-brain voxel-wide comparisons of the Reflected–Self contrast for the AN and CN groups led to identification of a 379 voxel region in the dACC (MNI 6, 26, 36) with the opposite modulation in the CN subjects compared to the AN subjects. In the Social Identity Task, the whole-brain voxel-wide comparisons of the Self–Friend contrast for the AN and CN groups led to identification of a 43 voxel region in the PreC (MNI −8, −48, 46) with more modulation in the Self
<sup>1</sup>www.fil.ion.ucl.ac.uk/spm
<sup>2</sup>http://www.mathworks.com <sup>3</sup>http://www.alivelearn.net/xjview8/
condition than the Friend condition in the CN subjects compared to the AN subjects. In the Physical Identity Task, the whole-brain voxel-wide comparisons of the Self–Friend contrast for the AN and CN groups led to identification of a 61 voxel region in a ventral region of the dACC adjacent to the corpus callosum (cc-dACC, MNI −6, 20, 24) showing more modulation in the CN subjects than the AN subjects. In addition to the ROIs defined by group differences in these tasks, we also examined activations in medial prefrontal cortex (MPFC; vmPFC) and dorsolateral prefrontal cortex (DLPFC) based on prior reports of differences in these areas with similar tasks in eating disorder subjects. We created 5 mm spherical ROIs centered on the published coordinates for MPFC [10, 64, 18, (37)], vmPFC [−12, 44, −12, (42)], and DLPFC [−48, 6, 38 (28)]. For all ROI analyses, we extracted the percent signal change occurring within each of these regions for each subject using the MarsBar toolbox<sup>4</sup> and transferred this data to (SPSS, Inc., Chicago). In SPSS, we first conducted a three-group ANOVA to identify whether differences were present across the three subject groups for each ROI, and conducted follow-up analyses of significant results using between-group *t*-tests.
#### **RESULTS**
#### **PSYCHOLOGICAL SCALES AND DEMOGRAPHIC DATA**
The three groups were not significantly different in age or years of education. The AN group had a significantly lower body mass index than either the CN and BN groups (**Table 1**). The BN and AN groups both scored higher than the CN group on measures of depression and eating behaviors but were not significantly different from each other. The AN and BN subjects also reported lower levels of both self-liking and self-competence compared to the CN subjects. On the SPSI-R, the AN and BN groups had lower overall scores on social problem solving as well as lower levels of positive
**Table 1 | Sociodemographic and symptom scale values for the participants.**
4 sourceforge.net/projects/marsbar problem orientation and higher levels of negative problem orientation than the CN groups. The AN subjects also showed higher levels of avoidance than the CN subjects, whereas the BN subjects had lower levels of rational-problem solving than the CN subjects. However, there were no significant differences in the AN and BN groups in comparisons for any of the SPSI-R subscales.
#### **SOCIAL ATTRIBUTION TASK**
The Social Attribution Task required subjects to respond to a question about each video. There were no differences in the accuracy of the subjects in response to either the Bumper visuospatialweight question [mean percent correct, CN 59%, AN 64%, BN 69%,*F*(50) = 2.17, *p* = 0.13], or the People social-friendship question [mean percent correct, CN 77%, AN 81%, and BN 84%, *F*(50) = 1.67, *p* = 0.20]. There were also no differences in reaction times for subjects in either task [Bumper, mean reaction time in seconds, CN 1.22, AN 1.30, BN 1.35, *F*(50) = 0.28, *p* = 0.76; People, mean reaction time in seconds, CN 1.30, AN 1.35, BN 1.40, *F*(50) = 0.15, *p* = 0.86].
The People–Bumper contrast of the Social Attribution Task resulted in significant clusters of activation in the middle temporal gyri, and temporoparietal junctions (TPJ) in all three groups (**Table 2**). Additionally, the CN subjects had bilateral activations in inferior frontal gyri, the fusiform gyri, the medial frontal gyrus, and the PreC. The AN and BN subjects also had activations in the right inferior frontal gyrus, but not the left inferior frontal gyrus. The AN subjects also showed modulation of the medial frontal gyrus like the CN subjects, and also activated a region in the ventral anterior Cing. The BN subjects did not modulate MPFC, like the CN and AN groups, but did modulate the PreC and the fusiform gyri, like the CN subjects but differing from the AN subjects.
In **Figure 1**, we show the percent signal change occurring in the ROI for this contrast, the RTPJ, in the CN,AN, and BN groups during the Social Attribution task [means People–Bumper, CN 0.35,
<sup>a</sup>All entries under the subject groups contain the mean (range).
<sup>b</sup>Mean and range for AN subjects exclude one higher weight outlier.
<sup>c</sup>Statistical values obtained using a three group ANOVA for each metric; p values provided for significant differences (<0.05).
<sup>d</sup>SLSC: self-Liking and self-competence scale.
AN 0.09, BN 0.17, *F*(50) = 9.7, *p* < 0.001]. The BN group showed significantly less modulation of this region than the CN group [*t*(33) = 2.7,*p* = 0.01; Cohen's *d* = −0.85; effect size = −0.39] and no difference compared to the AN group [*t*(33) = −1.2, *p* = 0.23]. Similar to the AN group, the differences in activation are primarily the result of less activation of this region during the People condition. We also examined percent signal change by subject group in a MPFC ROI previously described in a similar task as related to outcomes in adolescent AN (37). Although the BN subjects had less activation in this ROIs than the other groups, it was not statistically different from either of the other groups [means, CN 0.30, AN 0.22, BN 0.17, *F*(50) = 2.1, *p* = 0.13].
#### **SOCIAL IDENTITY TASK**
The Social Identity Task required subjects to read and respond to social adjectives presented in three different conditions (Self, Friend, and Reflected) in the scanner. For each statement, we obtained a response on a four point scale and a reaction time. There were no significant differences across the three groups in any condition for either average response [mean response, Self: CN 2.53, AN 2.44, BN, 2.49, *F*(51) = 0.178, *p* = 0.84; Friend: CN 2.49, AN 2.46, BN 2.46, *F*(51) = 1.54, *p* = 0.22; Reflected: CN 2.53, AN 2.38, BN 2.41, *F*(51) = 0.75, *p* = 0.48] or the reaction times [mean reaction times in seconds, Self: CN 2.07, AN 2.18, BN 2.26, *F*(51) = 0.35, *p* = 0.70; Friend: CN 2.03, AN 1.97, BN 2.10, *F*(51) = 1.97, *p* = 0.15; Reflected: CN 2.17, AN 2.15, BN 2.16, *F*(51) = 0.42, *p* = 0.66].
The Social Identity Task activates regions associated with selfknowledge and personal mentalization. In the personal mentalization contrast (Social Reflected–Self), subjects were asked to imagine what a close friend thinks about their social characteristics in contrast to their own belief about themselves. This contrast differs somewhat from the mentalization processes activated in the Social Attribution task because the mentalization is now attributed to a known individual. The largest clusters of activation occurred in the PreC in all subject groups (**Table 3**). For both the CN and BN groups, this cluster also included a portion of the posterior Cing, but the AN group had a smaller PreC cluster and an additional cluster in the posterior Cing. The BN and CN groups also had other activation clusters including some consistent with activations seen in the impersonal mentalization task (CN subjects, cluster in left medial temporal gyrus; BN subjects, bilateral clusters in the TPJs). In **Figure 2**, the BN group showed a lower degree of modulation of the ROI from this task contrast, the dACC, than the CN group [means, Reflected–Self, CN 0.076, BN −0.011, *t*(31) = 2.2, *p* = 0.03, Cohen's *d* = −0.79, effect size = −0.37], and no difference from the AN group [means, Reflected–Self, AN −0.091, BN −0.01, *t*(31) = −2.0, *p* = 0.06].
The self-knowledge comparison (Social Self–Friend) led to very different activation patterns in the AN and BN subjects compared to the CN subjects (**Table 4**). Notably, the CN subjects only activated clusters in the occipital lobes, whereas the AN and BN subjects had many clusters with the largest in occipital, parietal, and frontal cortex. In **Figure 2**, the BN group also showed significantly less modulation of the ROI from this task contrast, the PreC, than the CN group [means, Self–Friend, CN 0.083, BN −0.001, *t*(31) = 2.7, *p* = 0.01, Cohen's *d* = −0.94, effect size = −0.43], and no difference from the AN group [means, Social Self–Friend, AN −0.039, BN −0.001, *t*(31) = −0.9, *p* = 0.36]. We also examined percent signal change in the vmPFC and DLPFC but found no differences in either region across the three groups [vmPFC, means CN 0.02, AN 0.02, BN 0.18, *F*(50) = 1.75, *p* = 0.18; DLPFC, means CN 0.08, AN 0.07, BN 0.14, *F*(50) = 1.18, *p* = 0.31].
#### **PHYSICAL IDENTITY TASK**
The Physical Identity Task required subjects to read and respond to physical descriptive phrases presented in three different conditions (Self, Friend, and Reflected) in the scanner. For each statement, we obtained a response related to agreeing or disagreeing with the description using a four point scale and a reaction time. There were no significant differences across the three groups in any condition for either average response [mean response, Self: CN 2.39, AN 2.40, BN, 2.36, *F*(51) = 0.178, *p* = 0.84; Friend: CN 2.36, AN 2.40, BN 2.31, *F*(51) = 1.54, *p* = 0.22; Reflected: CN 2.31, AN 2.31, BN 2.24, *F*(51) = 0.75, *p* = 0.48) or the reaction times [mean reaction times in milliseconds, Self: CN 2339, AN 2294, BN 2367, *F*(51) = 0.35, *p* = 0.70; Friend: CN 2392, AN 2267, BN 2440, *F*(51) = 1.97, *p* = 0.15; Reflected: CN 2493, AN 2414, BN 2492, *F*(51) = 0.42, *p* = 0.66].
In the Physical Identity self-knowledge contrast (Physical Self– Friend), very different activation patterns were present in the three groups (**Table 5**). The CN subjects had several clusters in the anterior and middle Cing; the AN subjects had clusters in the inferior frontal gyri; and the BN subjects showed no activation clusters at all. In **Figure 2**, the BN group showed no differences in the modulation of the ROI for this task contrast, the cc-dACC, with either the CN group [means Physical Self–Friend, CN 0.12, BN 0.04,*t*(31) = 1.8, *p* = 0.09] or the AN group [means, Physical Self– Friend, AN −0.014, BN 0.04, *t*(31) = −1.3]. We also examined percent signal change in vmPFC and DLPFC but found no differences in either region across the three groups [vmPFC, means CN 0.06, AN 0.04, BN −0.04, *F*(50) = 0.44, *p* = 0.65; DLPFC, means CN 0.06, AN 0.00, BN 0.05, *F*(50) = 1.35, *p* = 0.27].
**FIGURE 2 | ROIs and the percent signal change in each group for the Social and Physical Identity contrasts**. **(A)** The upper row shows the extent of each ROI in red; from left to right, the dACC (MNI x = 6; Social Identity Reflected–Self); the precuneus (x = −8; Social Identity Self–Friend); and the cc-dACC (x = −6; Physical Identity Self–Friend). **(B)** The bar graphs shows the average percent signal modulation within each ROI for the CN (green), AN (blue), and BN (red) groups. Percent signal change was computed for the contrast and task that defined each ROI; the Social Identity task for both the dACC and PreC; the Physical Identity task for the cc-dACC.
### **DISCUSSION**
Neuroimaging work in the last decade has shown that neural regions involved in self-knowledge are often also activated in social cognitive processing, so the same brain regions that enable understanding one's own self may also be involved in understanding others (8, 9, 30). One diagnostic criterion for both AN and BN is related to self-knowledge: body shape or weight having undue influence on self-esteem (1). Additionally, problems related to understanding self and others have long been observed in AN (14, 43, 44). Recently, neural evidence of differences in social processing has been reported in AN subjects (36, 37). Here, we assessed whether BN subjects showed more similarities to AN or CN subjects in their neural activations in response to fMRI tasks requiring social processing.
First,it is worth observing that the psychiatric and demographic data for the subjects with AN and BN were only significantly different from each other in that the AN subjects had a lower body mass index. On all other scales, including measures of self-esteem and social behavior, the two subject groups did not differ from one another. There were two differences in comparisons with the CN group on subscales of the SPSI-R: AN subjects had a higher avoidance style and BN subjects showed less rational-problem solving than the CN subjects, but there were no significant differences on these measures in the direct comparisons of the AN and BN subjects. These results are consistent with studies of clinical and personality characteristics in the literature that have examined both AN and BN subjects: few differences are identified, supporting a theory that similar psychological processes underlie
both disorders (45–49). Many self-report and clinical measures of psychiatric symptoms depend both upon a subject's willingness to admit to their symptoms and concerns as well as their ability to recognize and report on their actual symptoms (50). In eating disorders, minimization and denial of symptoms are frequently observed, making psychological and cognitive assessments challenging (51, 52). Neuroimaging data is less likely to be affected by these problems. This study suggests that neural data may provide increased sensitivity for the detection of altered brain function in eating disorders.
Few studies have examined social cognition in BN (19). Interestingly, nearly all of these studies have examined social cognition using facial stimuli, either in a recognition of feelings portrayed by faces (16, 18), an identification of emotions in faces (53, 54), or through an emotional facial Stroop task (48). Amongst these tasks, only the emotional Stroop task, showed strong differences in direct comparisons of BN and CN subjects. Akin to these studies, the neural data from the Social Attribution task showed more similar activation clusters in the BN and CN group maps than in the AN and CN group maps. Notably, both the CN and BN subjects showed significant activations bilaterally in the fusiform face areas in the People–Bumper contrast but the AN group did not have activation in this region. In concert with the numerous behavioral observations of differences related to facial emotion processing in AN (15–17, 55), our neuroimaging data suggest that the neural regions that subserve the processing of facial expressions may be intact in BN but not in the AN (**Figure 1**; **Table 1**).
However, the BN subjects did show less modulation than the CN subjects within the RTPJ, the ROI previously identified as showing differences in the task activations using the whole-brain comparisons of the AN and CN groups. This area has been most consistently associated with TOM across a wide variety of imaging tasks that include imagining human movement, interpreting stories, and viewing complex videos (32, 56–59). Our demonstration of reduced modulation of this region in the BN group suggests that there are similarities in the neural processing of TOM in both types of eating disorders. This finding further highlights the fact that neuroimaging markers for cognitive processes may be more sensitive to measuring certain aspects of processing, as the behavioral studies have not detected mentalization differences in BN subjects.
Bydlowski (60) reported reduced TOM in BN subjects using the Levels of Emotional Awareness Scale (LEAS). This is a TOM task that involves answering questions about one's own emotions and another person's emotional state based on responses to short vignettes, rather than viewing faces or videos. In PET imaging studies, LEAS scores has been positively correlated with emotional arousal in the dACC in healthy people (61–63) but negatively correlated in post-traumatic stress disorder patients (63). Interestingly, the Social Identity mentalization contrast showed an opposite pattern of modulation in the dACC in the AN and BN groups compared to the CN group. In concert with our data, these results suggest that neural differences in the dACC related to social and emotional processing are present in both AN and BN. Interestingly, activations of the dACC are more commonly observed in tasks with personal relevance (30), a condition present in our Identity task mentalization condition but not the Social Attribution task.
#### **Table 4 | Clusters in the CN, AN, and BN group maps during the Self–Friend contrast of the social identity task.**
**Table 5 | Clusters in the CN, AN, and BN group maps during the Self–Friend contrast of the physical identity task.**
One of the most intriguing findings relates to the differences seen in both eating disorder patients and healthy people with a mentalization process that is personal (*my friend thinks*. . .) compared to the impersonal task (*People: All friends?*). Very different neural regions are engaged in these two tasks, demonstrating that tasks that separate personal and impersonal mentalization may be important for examination of psychopathology related to social processing. Our neural data implies that the consideration of one's own self may fundamentally alter social cognitive processing. Recognition of the specific neurocognitive demands of both imaging and behavioral tasks may be essential in detecting psychological and biological differences in eating disorders. Further research with more complicated behavioral and neuroimaging tasks that assess personal and impersonal mentalization are warranted in BN.
Stein and Corte (64) described identity as the stable yet evolving set of memory structures relating to one's own experiences, and dissociable into different dimensions, which they referred to as self-schemas. They proposed that in eating disorders, the selfschemas related to emotional and physical understanding of one's own self-state are impaired (64–66). Limitations in assessment of their own emotions are seen in the elevated levels of alexithymia reported in both AN and BN (46, 60, 67–69). Problems in selfesteem are also present and often precede the development of both AN and BN (20–22, 70, 71). The presence of negative selfbeliefs unrelated to shape and weight has been proposed as a core component of eating disorders (26, 27).
In Social Identity self-knowledge contrast (Social Self–Friend), the group maps of the AN and BN subjects were very different from the CN subjects. The CN subjects only activated areas in
the occipital and lingual cortex, whereas the AN and BN subjects showed significant activations not only in those areas, but also in frontal, parietal, and temporal regions (**Table 4**). Additionally, we observed reduced activation of the dorsal PreC in AN and BN subjects, the area previously identified with greater modulation in the CN subjects than the AN. Consistent with our findings, reduced modulation of the PreC has been reported in two other imaging tasks in BN. Ashworth and colleagues (72) examined cognitive processing related to social emotional appraisals by asking subjects to remember and match negative facial expressions, and Pringle and colleagues (28) asked subjects to consider whether negative eating and depression words were self-relevant or not. Together with our data, these studies support an idea that PreC activity in response to self-evaluation may be altered in eating disorders. The PreC has connections with temporal, limbic, and parietal regions, suggesting that it serves to integrate current physical and emotional status with prior experiences (73). The reduced modulation of the PreC in the eating disorder subjects observed here implies a reduced connection between physiological state and personal experience, supporting an idea that the psychological processes that mediate identity formation are disrupted in eating disorders.
In the physical self-knowledge contrast, subjects were asked to think about their own physical appearance. This task strongly activated a ventral region of the dACC, immediately adjacent to the corpus callosum in the CN subjects, with little modulation in the AN subjects, and no clusters identified in the BN subjects. In this comparison, BN subjects were not significantly different from either the AN or the CN subjects. The variance of the BN group was nearly twice that of either the AN or CN group for this ROI, supporting an idea that some subjects with BN may have problems activating this cortical area and others may not. For the other ROIs, the variance of the BN group was similar to that of the AN and CN groups. Interestingly, Marsh and colleagues have focused on the neural circuits involved in self-regulation in BN, and also reported differences in the activation of this area of the dACC in BN (74, 75). From a cognitive perspective, the differences in the cc-dACC suggests that some subjects with BN, but not all, think about their current physical or physiological state differently than healthy people. This neural difference may correspond to less information about physiological needs being available in the minds of eating disorder patients, making it easier for these individuals to develop feeding behaviors that are removed from nutritional needs. Future studies may want to focus on whether the activation of the cc-dACC can be related to psychological measures of body shape perception and interoceptive awareness.
Recently, Schulte-Ruther and colleagues (37) used an fMRI task similar to the Social Attribution task to examine whether longitudinal changes in social cognitive regions were associated with weight recovery in adolescents being treated for AN. They observed reduced activation of temporal and medial frontal regions both before and after weight recovery in AN subjects compared to CN subjects. They also reported that stronger modulation of one social cognition region, the MPFC at the start of treatment, was predictive of outcome. We also examined responses in this MPFC ROI, but found no differences in the three subject groups. One major difference in the studies is that all our subjects were at a stable weight when scanned, whereas the earlier study had observed changes in this region related to outcomes following treatment. Nevertheless, the observation that MPFC may be relevant to recovery is particularly exciting when considered in the context of a study by Somerville and colleagues (76) in which healthy people with low self-esteem showed modulation of responses in their vACC and MPFC in response to social feedback whereas people with high self-esteem showed no changes in this region in response to feedback. Low self-esteem has previously been shown to be predictive of the development of eating disorders (20, 22, 25) as well as an indicator of outcome and severity (77, 78). Neural responses within the ACC and MPFC to social feedback may provide a biological mechanism that connects social cognitive responses and self-esteem with eating pathology; understanding how biological factors impact specific patients may lead to improved outcomes by providing more individualized treatments in the future.
Additionally, two earlier studies have identified frontal cortical regions associated with eating disorders and processing verbal stimuli. Pringle and colleagues (28) found differences in dorsolateral PFC in response to negative emotional words considered in the context of oneself, and Miyake et al. (42) has reported differences in vmPFC associated with responses to selecting a negative body image words compared to selecting the most neutral of a random word sets. Although we did not observe differences across our subject groups in these same regions in either the Social Identity task or the Physical Identity task, this is likely related to differences in task design. Our subjects performed Self-appraisals, Friend-appraisals, and Reflected-appraisals, using the same sets of adjectives. As such, neither of the studies showing effects in frontal regions had a comparison situation involving the same stimuli words referenced to a different person. This suggests that the areas we have identified in the dACC, cc-dACC, and PreC may be specific for altered cognitions related to one's own self in eating disorder subjects, whereas frontal activation differences may relate to the cognitions evoked by physical and emotional descriptive terms.
There are a number of limitations to these studies. First, the sample groups were small and as such the study may be underpowered to identify both differences and similarities that are present. Furthermore, the BN group showed more variability in their neural activations than the AN and CN groups, and that variability may warrant collecting a larger group to identify specific differences. Additionally, a larger study could explore the relationships between neural activity and clinical symptoms for both AN and BN. Potentially, neural data may provide an additional tool to assess the severity and symptoms present in a specific patient with an ED.
Another limitation of any single-time point psychiatric study is that the presence of neural differences does not determine whether these differences are a cause or an effect of the disorder. In eating disorders, medical issues are likely to alter brain function. Purging behaviors alter electrolytes, a critical factor for neuronal signaling (79). Restriction leads to nutritional deficiencies and hormonal changes, additional factors that alter brain function (80). Neurodevelopment is critically dependent on myelination (81, 82), and that process may be impacted with the presence of an eating disorder in adolescence and young adulthood. The size of the anterior Cing, a brain region fundamental in self-processing, decreases with the severity of starvation in AN (83). Our studies show differences in neural activations in response to social tasks in patients currently or recently with eating disorders, including both AN and BN.
Differences in neural activations in psychiatric populations may be a result of a variety of processes. They can emerge because of pre-existing biological differences that lead to the disorder, because of the effects of having the disorder such as electrolyte changes, or may merely be a reflection of current psychological differences related to processing stimuli relevant to the disorder. Most commonly, differences are viewed as a biological predisposition to the illness but it is impossible to determine if these differences existed before the eating disorder and if they will still be present following recovery from the eating disorder. Our data show that there are differences in the neural activity that underlies social processing in people with BN. Clinically, this implies that social processing pathways, including TOM and self-knowledge, are engaged differently during an eating disorder. This reinforces choosing treatment models with a focus on issues related to social interaction and function in addition to disordered eating behaviors (84–86). Neural evidence of social processing differences in eating disorders may be important in helping patients accept treatments that appear indirectly related to alteration of eating patterns.
In summary, these experiments examined the neural modulations in response to fMRI tasks focusing on self-knowledge and social cognition in BN. We observed modulation in the BN group that was consistently intermediate between the AN and CN groups. In three ROIs, all of which were activated in CN during MRI tasks involving social evaluation, the BN subjects were significantly different from the CN subjects but not from the AN subjects. This suggests that neural processes that mediate social thinking are similar in AN and BN. Recently, Lavender and colleagues (87)
#### **REFERENCES**
examined outcomes for BN using a group therapy focusing on emotional and social mind training, and found recovery rates similar to more established cognitive behavior therapy. That study, in concert with our neural data, suggest that further exploration of social processing interventions may lead to improved outcomes in BN. One interesting observation in that pilot treatment study was that subjects in the emotional and social training group were more likely to attend sessions, suggesting that this type of treatment may help to engage patients in treatment. The fourth region, cc-dACC, was identified in a contrast of the Physical Identity task. In this area, the BN subjects were not significantly different from either the AN or the CN subjects. This suggests that cognitions surrounding physical appearance may be altered less in BN than in AN, or less consistently altered amongst patients with BN. Overall, our studies demonstrate similarities in neural processing in BN and AN, and suggest that there may be shared biological mechanisms related to processing social concepts that differ systematically from neural modulations seen in healthy CN subjects.
#### **ACKNOWLEDGMENTS**
We thank M. Michelle McClelland, Sunbola Ashimi, Lauren Monier, and Joanne Thambuswamy for assistance in the execution of this project. We thank The Elisa Project and Texas Health Presbyterian Hospital of Dallas for assistance in recruitment of subjects. This work was supported by unspecified funds from the School of Behavioral and Brain Science at the University of Texas at Dallas, the UT Southwestern Medical Center Physician Scientist Training Program, and National Institutes of Mental Health grant K23 MH093684-01A1.
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**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
*Received: 14 June 2013; accepted: 28 August 2013; published online: 12 September 2013.*
*Citation: McAdams CJ and Krawczyk DC (2013) Neural responses during social and self-knowledge tasks in bulimia nervosa. Front. Psychiatry 4:103. doi: 10.3389/fpsyt.2013.00103*
*This article was submitted to Neuropsychiatric Imaging and Stimulation, a section of the journal Frontiers in Psychiatry.*
*Copyright © 2013 McAdams and Krawczyk. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.*
# Altered SPECT 123I-iomazenil Binding in the Cingulate Cortex of Children with Anorexia Nervosa
*Shinichiro Nagamitsu1 \*, Rieko Sakurai2 , Michiko Matsuoka3 , Hiromi Chiba3 , Shuichi Ozono1 , Hitoshi Tanigawa4 , Yushiro Yamashita1 , Hayato Kaida5 , Masatoshi Ishibashi6 , Tatsuki Kakuma7 , Paul E. Croarkin8 and Toyojiro Matsuishi1*
*1Department of Pediatrics and Child Health, Kurume University School of Medicine, Fukuoka, Japan, 2Graduate School of Medicine, Kurume University, Fukuoka, Japan, 3Department of Psychiatry, Kurume University School of Medicine, Fukuoka, Japan, 4Center of Diaginostic Imaging, Kurume University Hospital, Fukuoka, Japan, 5Department of Radiology, Kinki University Faculty of Medicine, Osakasayama, Japan, 6Department of Radiology, Kurume University School of Medicine, Fukuoka, Japan, 7Biostatistics Center, Kurume University School of Medicine, Fukuoka, Japan, 8Department of Psychiatry and Psychology, Mayo Clinic, Rochester, MN, USA*
#### *Edited by:*
*Gregor Hasler, University of Bern, Switzerland*
#### *Reviewed by:*
*Annette Beatrix Bruehl, University of Zürich, Switzerland Jochen Kindler, University of Bern, Switzerland*
> *\*Correspondence: Shinichiro Nagamitsu [email protected]*
#### *Specialty section:*
*This article was submitted to Neuroimaging and Stimulation, a section of the journal Frontiers in Psychiatry*
*Received: 25 August 2015 Accepted: 01 February 2016 Published: 16 February 2016*
#### *Citation:*
*Nagamitsu S, Sakurai R, Matsuoka M, Chiba H, Ozono S, Tanigawa H, Yamashita Y, Kaida H, Ishibashi M, Kakuma T, Croarkin PE and Matsuishi T (2016) Altered SPECT 123I-iomazenil Binding in the Cingulate Cortex of Children with Anorexia Nervosa. Front. Psychiatry 7:16. doi: 10.3389/fpsyt.2016.00016*
Several lines of evidence suggest that anxiety plays a key role in the development and maintenance of anorexia nervosa (AN) in children. The purpose of this study was to examine cortical GABA(A)-benzodiazepine receptor binding before and after treatment in children beginning intensive AN treatment. Brain single-photon emission computed tomography (SPECT) measurements using 123I-iomazenil, which binds to GABA(A) benzodiazepine receptors, was performed in 26 participants with AN who were enrolled in a multimodal treatment program. Sixteen of the 26 participants underwent a repeat SPECT scan immediately before discharge at conclusion of the intensive treatment program. Eating behavior and mood disturbances were assessed using Eating Attitudes Test with 26 items (EAT-26) and the short form of the Profile of Mood States (POMS). Clinical outcome scores were evaluated after a 1-year period. We examined association between relative iomazenil-binding activity in cortical regions of interest and psychometric profiles and determined which psychometric profiles show interaction effects with brain regions. Further, we determined if binding activity could predict clinical outcome and treatment changes. Higher EAT-26 scores were significantly associated with lower iomazenil-binding activity in the anterior and posterior cingulate cortex. Higher POMS subscale scores were significantly associated with lower iomazenil-binding activity in the left frontal, parietal cortex, and posterior cingulate cortex (PCC). "Depression–Dejection" and "Confusion" POMS subscale scores, and total POMS score showed interaction effects with brain regions in iomazenil-binding activity. Decreased binding in the anterior cingulate cortex and left parietal cortex was associated with poor clinical outcomes. Relative binding increases throughout the PCC and occipital gyrus were observed after weight gain in children with AN. These findings suggest that cortical GABAergic receptor binding is altered in children with AN. This may be a state-related change, which could be used to monitor and guide the treatment of eating disorders.
Keywords: anorexia nervosa, cingulate cortex, GABA, children, iomazenil SPECT
## INTRODUCTION
Anorexia nervosa (AN) typically presents in females during adolescence. It is a serious psychiatric illness conferring substantial morbidity and mortality, which manifests as disturbances in eating habits, excessive preoccupation with weight, restricted caloric intake, and body image distortion (1). Although some research regarding the outcome of childhood AN is encouraging in terms of mortality and recovery from AN (2), long-term comorbid psychiatric disorders, such as anxiety disorders and affective disorders, represent unfavorable prognostic factors (3). Anxiety is present in the majority of children with AN prior to abnormal eating or body image distortions (4). Anxiety in children with AN is also associated with decreased body mass index (BMI) (5, 6). Moreover, trait anxiety scales in children show significant positive correlations with eating disorder psychopathology such as "drive for thinness," "body dissatisfaction," and "perfectionism" (5).
Several lines of evidence implicate gamma-aminobutyric acid (GABA)ergic neurotransmission in the pathophysiology of anxiety (7). Recently, a large-scale candidate gene study found that allele frequency differences in the GABA receptor SNP, *GABRG1*, are related to levels of trait anxiety in AN and bulimia nervosa (8). Furthermore, elevated GABA(A) receptor levels in the amygdala were reported in activity-based anorexia (ABA), an animal model of the behavioral phenotype of AN (9). Upregulated GABA(A) receptor function may be associated with anxiety in ABA animals. Several neuroimaging studies have shown negative correlations between GABA-benzodiazepine receptor binding activity and severity of anxiety symptoms in adults with panic or traumatic disorders (10, 11). However, to date, no study has examined GABA(A) receptor binding or function in AN. It is possible that GABAergic neurons may play an important role in both premorbid anxiety of AN and the pathogenesis of childhood AN.
Single-photon emission computed tomography (SPECT) is a nuclear medicine tomographic modality employing gamma rays, and in which, injected radionuclides are attached to ligands selective for specific receptors of interest. 123I-iomazenil is a radioactive ligand for central-type benzodiazepine receptors, which form a complex with GABA(A) receptors. Thus, 123I-iomazenil SPECT measures GABA(A) receptor binding and indirectly assays GABA(A) receptor function. 123I-iomazenil is a frequently used radionuclide tracer for presurgical evaluation of patients with refractory partial epilepsy (12). Moreover, recent neuroimaging studies have explored the role of GABAergic inhibitory function in psychiatric disorders, such as schizophrenia, Alzheimer's disease, and developmental disorders, as well as anxiety disorders including panic and traumatic stress disorders (10, 11, 13–17). In these reports, significant correlations between GABAergic function and dimensional scales measuring anxiety, panic, negative cognitions, and psychiatric status were found.
The aims of this study were to (1) determine if GABA(A) receptor binding is associated with AN symptoms and anxiety in children initiating clinical treatment for AN; (2) determine which brain regions are involved; (3) determine if measures of GABA(A) receptor binding can predict a participant's clinical outcome; and (4) determine if these measures change with successful treatment. We hypothesized that lower cortical iomazenil-binding activity is associated with greater baseline symptom severity and poor clinical outcome in children with AN.
### MATERIALS AND METHODS
### Participants
The study complies with the Declaration of Helsinki and informed consent was obtained from participants and parents or legal guardians prior to enrollment in the imaging study. The procedures for assent, informed consent, and study design were approved by the Medical Ethical Committee of Kurume University School of Medicine. Twenty-six female participants were recruited who fulfilled the Diagnostic and Statistical Manual of Mental Disorders, 4th Edition (DSM-IV) criteria for AN, and had been admitted to the Department of Pediatrics, Kurume University School of Medicine between 2007 and 2012 for clinical treatment in an eating disorders program. All were restricted-type AN. The flow diagram for the study is shown in **Figure 1**. The eating disorder treatment program has a multimodal approach, which includes parenteral nutrition, psychotherapy, and behavioral intervention. On initiation of treatment, participants and families have extensive psychoeducation focused on major physical risk factors associated with restricted body weight and therapeutic goals for hospitalization. Individual behavior therapy with reward reinforcement is used to facilitate recovery. Although oral feeding was sufficient for the majority of participants, parenteral nutrition was implemented for select participants with severe AN. Behavioral therapy was combined with nutritional counseling and individual psychotherapy to target difficult emotions and family relational
stress. As part of the treatment, participants completed the Eating Attitude Test (EAT-26), a standardized, self-report measure of eating disorder symptoms, which is widely used for screening and measurement of symptoms and characteristics of eating disorders (18). Participants rated their mood using the short form of the Profile of Mood States (POMS), a validated measure that consists of 30 items describing six moods: "Tension–Anxiety," "Depression–Dejection," "Anger–Hostility," "Vigor," "Fatigue', and "Confusion" (19). High Vigor scores reflect a good mood or emotion, and low scores in the other subscales reflect a good mood or emotion. Total mood disturbance (TMD) was obtained by subtraction of the Vigor score from the sum of Tension–Anxiety, Depression–Dejection, Anger–Hostility, Fatigue, and Confusion scores. Each original POMS score was converted to a *T*-score (20). We selected the POMS for measurement of anxiety, as we have neither a Japanese version of State-Trait Anxiety Inventory for Children (STAIC) nor other validated Japanese psychometric scales for anxiety. Upon enrollment in the study, a diagnosis of AN and comorbidities was confirmed in all participants by semi-structured interviews using the Mini-International Neuropsychiatric Interview (MINI) (21), which were performed by two psychiatrists (Michiko Matsuoka and Hiromi Chiba). All participants underwent cognitive assessment using the Wechsler Intelligence Scale for Children (WISC-III). Psychometric profiles were performed before treatment. Participants were medication naïve and did not receive pharmacological treatment during the course of the study. Twenty-three participants had secondary amenorrhea and three had not yet reached menarche. In all participants, brain magnetic resonance imaging (MRI) examination was performed on admission to identify any structural abnormalities, e.g., regional brain atrophy. Participants with severe, co-occurring medical illnesses (such as superior mesenteric artery syndrome) were excluded. Ethical concern regarding the use of ionizing radiation in healthy children precluded the enrollment of a control group for this study.
### Clinical Outcome Measures
Follow-up clinical assessments were performed 1 year after hospital discharge by one pediatrician (Shinichiro Nagamitsu) and two psychiatrists (Michiko Matsuoka and Hiromi Chiba). A structured approach was used to define clinical outcome *a priori*. Clinical outcome score was based on eight items, as defined in prior work (22). This included weight change, menstrual status, abnormal eating behavior, body image, binge eating or purging behaviors, insight, school attendance, and quality of family relationship. Improved or impaired answers were scored "0" and "1," respectively. For items of weight change and school attendance, improved and unimproved ratings were scored "0" and "2," respectively. The middle score "1" indicates "unchanged condition." The outcome was considered good with total scores less than "4" and poor with total scores of "4" or over. The clinical outcome raters (Shinichiro Nagamitsu, Michiko Matsuoka, and Hiromi Chiba) were blinded regarding SPECT data.
### Iomazenil SPECT
All 26 children underwent brain imaging using SPECT. The first 123I-iomazenil SPECT examination was performed before treatment and the second one immediately before discharge (16 of 26 participants). Mean duration between the first and second SPECT examinations was approximately 4 months. Briefly, participants were injected intravenously with a bolus of 95–117 MBq 123I-iomazenil (Nihon Medi-Physics Co., Tokyo, Japan), which binds with high affinity to the GABA(A)-benzodiazepine receptor. The SPECT scan was performed 3 h after injection of the tracer without any sedation, using a large field-of-view dual-detector camera and a computer system equipped with a low-energy, highresolution, and parallel-hole collimator. The dual detector camera rotated over 180° in a circular orbit and in 32 steps of 40 s each to cover 360° in approximately 22 min.
### Image and Statistical Analyses
Images of 123I-iomazenil scintigraphs were analyzed by threedimensional stereotactic surface projections (3D-SSP) using iSSP3 software (Nihon Medi-Physics Co.). Stereotactic anatomical standardization was performed as described previously (23). Briefly, rotational correction of the SPECT data set and three-dimensional centering were performed, followed by realignment to the anterior commissure–posterior commissure line. Differences in individual brain size were accounted for by linear scaling and regional anatomical differences minimized using a non-linear warping technique (24). Consequently, each brain was anatomically standardized to match a standard atlas brain. Brain MRI was performed using a superconducting magnet operating at 1.5 T. For coregistered SPECT and MRI analysis, a method of image integration was applied using Fusion Viewer software (Nihon Medi-Physics Co.) with a registration algorithm based on maximum mutual information (**Figure 2**). Subsequently, cortical and subcortical regions of interest (ROIs) in the acquired SPECT
FIGURE 2 | Designated regions of interest (ROIs) in fusion images of 123I-iomazenil SPECT and MRI. The top panel shows brain MRI (transverse and sagittal T1 sequences), the middle panel the corresponding results of 123I-iomazenil SPECT, and the bottom panel fusion imaging. Outlined regions in the bottom panel indicate designated ROIs, namely, (a) the superior frontal, (b) parietal, (c) frontal, (d) middle temporal, and (e) occipital regions; (f) anterior and (g) posterior cingulate gyrus.
data were defined. Using elliptical templates, ROIs were drawn manually for the major cortical and subcortical brain regions in a representative subject. To eliminate the disadvantage of lower reliability with manual operations, the same elliptical templates were used to define ROIs in other subjects. ROIs were placed over the following regions: superior frontal, middle frontal, parietal, middle temporal, and occipital regions; the cerebellum in each hemisphere; and the anterior and posterior cingulate cortex (ACC and PCC, respectively; **Figure 2**). Two neuroradiologists (Hitoshi Tanigawa and Masatoshi Ishibashi), blinded to clinical symptoms, independently drew ROIs. Each relative iomazenilbinding activity in ROIs was expressed as a ratio of that in the cerebellum, as patients with AN have no cerebellar symptoms. Spearman's correlation was used to determine correlations between relative iomazenil binding in each region on baseline SPECT scan and age, BMI-standard deviation score (BMI-SDS), EAT-26, and POMS subscale score. To test for possible differential relationships between POMS and iomazenil-binding activity in brain regions, the brain regions were classified into three groups: center region, left hemisphere region, and right hemisphere region. ROIs were grouped accordingly: ACC and PCC as the center region; left of superior frontal, parietal, frontal, temporal, and occipital as the left hemisphere region; and right of superior frontal, parietal, frontal, temporal, and occipital as the right hemisphere region. POMS subscales were separately analyzed using the mixed-effect model (SAS 9.3 PROC MIXED). Regions, POMS subscale, and their interactions were treated as fixed effects in the model, while the intercept was treated as a random effect, therefore accounting for correlations among iomazenil-binding activities. When the interaction between all three regions and POMS subscale was significant, ROIs were analyzed within the region and POMS using mixed model regression to determine significances between ROIs. The Mann–Whitney *U* test was used to compare between participants with good and poor outcomes.
Neurological Statistical Image Analysis software (NEUROSTAT, Stat\_1tZ), which can perform a paired *t* test between two corresponding groups using cross-sectional images, was adopted to examine changes in iomazenil binding between the first and second SPECT. Statistical significance was set at *Z*-score >2, a level commonly used to discriminate abnormalities. The regions identified were transformed into three-dimensional anatomical data and Talairach coordinates to show brain landmarks.
### RESULTS
### Participants' Characteristics
The mean and SD of age before treatment was 14.1 (1.3) years of age (range, 10.5–15.6 years of age) (**Table 1**). Mean (SD) BMI before and after treatment were 13.7 (2.0) and 15.7 (0.9), respectively. Mean (SD) BMI-SDS before and after treatment were −3.7 (1.9) and −2.3 (1.1), respectively. Mean (SD) EAT-26 score before therapy was 22.4 (12.0), which was higher compared with the reference control value (25). Five participants had co-occurring psychiatric disorders including two with autism spectrum disorders, one with learning disability, and two with selective mutism. Mean (SD) scores for each of the POMS subscales were TABLE 1 | Clinical characteristics and POMS scores for subjects with relative iomazenil-binding activity in each brain region.
*SPECT, single-photon emission computed tomography; BMI, body mass index; EAT-26, Eating Attitude Test with 26 items; POMS, Profile of Mood States; WISC-III, Wechsler Intelligence Scale for Children; R, right; L, left.*
*\*Significant difference compared with all subjects (P* < *0.05). † Significant difference compared with subjects at second SPECT (P* < *0.05).*
*‡ Trend toward significance in all subjects (P* < *0.1).*
46.1 (10.3) for Tension–Anxiety, 53.1 (11.8) for Depression– Dejection, 50.3 (11.9) for Anger–Hostility, 43.4 (11.8) for Vigor, 47.4 (11.4) for Fatigue, and 52.1 (15.5) for Confusion. Mean (SD) score for TMD of POMS was 205.6 (59.4). The subscale scores of Tension–Anxiety and Depression–Dejection were higher than normal ranges (26), but the differences did not reach significance. Mean (SD) IQ was 105 (13). None of the participants showed regional brain atrophy on brain MRI examination. Mean duration of hospitalization was approximately 4 months. Mean (SD) age on the second SPECT examination was 14.8 (1.1). The range of duration between the first and second SPECT was from 86 to 250 days (mean, 128 days).
### Participants' Outcome
Clinical outcome scores 1 year after treatment were examined in 20 out of 26 participants. Three participants did not complete the treatment. It was not possible to examine three other participants, as two were transferred to locked psychiatric units and one was transferred to a local hospital. After the evaluations, 13 participants were classified as having a good outcome and 7 with a poor outcome.
## Baseline Correlations between 123I-iomazenil Binding and Clinical
### Measures
Relative iomazenil-binding activities in each brain region are summarized in **Table 1**. There were significant associations between some clinical measures and relative iomazenil-binding activity in several brain regions. Higher EAT-26 scores were significantly associated with lower iomazenil-binding activity in the ACC (**Table 2**). Higher "Tension–Anxiety" score at the beginning of therapy was significantly associated with lower iomazenilbinding activity in the left superior frontal, parietal, middle frontal cortex, and PCC (**Table 2**). Higher "Anger–Hostility," "Confusion," and "Total" scores at the beginning of therapy were significantly associated with lower iomazenil-binding activity in the same regions and left occipital cortex (**Table 2**). Furthermore, "Depression–Dejection" and "Fatigue" scores were also significantly associated with lower iomazenil-binding activity in the PCC (**Table 2**). There were no associations between BMI-SDS and iomazenil-binding activity in any brain region. However, there were significant positive correlations between age and iomazenil-binding activity in the left and right middle frontal, left parietal, and PCC (*r* = 0.415, 0.392, 0.430, and 0.454, respectively, *P* < 0.05) (data not shown).
### Interactions between POMS Subscales and Brain Regions in Iomazenil-Binding Activity
The mixed-effect model detected significant interaction effects between three main brain regions and POMS total scale and subscales of "Depression–Dejection" and "Confusion" (**Table 3**). In the three main brain regions, significant differences were identified between the left and right hemisphere regions on POMS total score and subscales of "Depression–Dejection" and "Confusion" (**Table 4**). Furthermore, significant differences between left and right hemisphere regions were identified in the superior frontal region on POMS total score (*t* = −2.63, *P* = 0.0094), superior frontal and occipital regions on the subscale of "Confusion" (*t* = −3.16, *P* = 0.0018; *t* = −3.49, *P* = 0.0062, respectively), and superior frontal region on the subscale of "Depression–Dejection" (*t* = −2.75, *P* = 0.0065). This finding remained significant after Bonferroni correction for multiple comparisons (*P* = 0.01).
### Comparison of 123I-iomazenil-Binding Activity Before and After Treatment
Relative iomazenil-binding activity after treatment was significantly increased in the ACC, right occipital, and bilateral middle temporal gyrus (**Table 1**). Comparisons of adjusted iomazenilbinding activity before and after weight gain were examined in the same 16 participants using NEUROSTAT. There were significant increases in iomazenil-binding activity after treatment in the ACC, PCC, frontal gyrus, occipital gyrus, and hippocampus (**Figure 3**). By contrast, there was a significant decrease in iomazenil-binding activity after treatment in the bilateral inferior temporal cortex (data not shown). The Talairach coordinates of sites with *Z*-scores >3.0 are listed with the associated brain regions in **Table 5**.
### Association between 123I-iomazenil-Binding Activity in the ACC and Clinical Outcome
There was a significant baseline difference in iomazenil-binding activity between participants with good and poor clinical outcome scores in the ACC (1.48 ± 0.09 vs. 1.32 ± 0.12, respectively, *P* < 0.05) (**Figure 4**) and left parietal gyrus (1.57 ± 0.12 vs. 1.40 ± 0.15, respectively, *P* < 0.05). Relative baseline iomazenilbinding activity in the ACC and left parietal gyrus in participants with a poor clinical outcome were significantly lower than those with a good clinical outcome.
### DISCUSSION
To our knowledge, this is the first investigation on SPECT 123I-iomazenil brain imaging in children with AN. Using SPECT
*POMS, Profile of Mood States; TA, Tension–Anxiety; D, Depression–Dejection; AH, Anger–Hostility; V, Vigor; F, Fatigue; C, Confusion; EAT-26, Eating Attitude Test with 26 items. \*P* < *0.05 and \*\*P* < *0.01 indicate significant correlations.*
123I-iomazenil brain imaging, we found association between cortical GABAergic receptor binding and clinical manifestations of childhood AN. Higher EAT-26 and mood disturbance scores were significantly associated with lower GABAergic inhibitory binding in various brain regions. Poor clinical outcome was also associated with lower GABAergic receptor binding in the ACC and left parietal region. GABAergic receptor binding was mainly activated in the ACC, PCC, frontal gyrus, and occipital gyrus after treatment.
TABLE 3 | Interaction effects between POMS subscales and three main regions (center, left, and right hemispheres).
*POMS, Profile of Mood States.*
*\*Indicates significance.*
TABLE 4 | Interaction between three regions and POMS subscales.
*\*indicates significance.*
We found significant correlation between reduced GABAergic receptor binding in various brain regions and mood disturbances, as assessed using POMS subscales. Mixed model regression showed significant effects for the interactions between brain regions and POMS total scale and subscales of "Depression–Dejection" and "Confusion." Furthermore, the effect of these POMS profiles showed significantly different binding activities between the left and right hemispheres, especially in the superior frontal region. These results indicate that GABAergic neuronal activity correlates to mood disturbances in children with AN. The potential involvement of GABAergic neurotransmission in the pathophysiology of AN was recently investigated by genetic allele frequency analysis of *GABRG1* in AN patients. This study showed significant correlation between specific allele frequency in this GABA receptor SNP and levels of trait anxiety in the patients (8). Further, in an animal model of AN, elevated GABA(A) receptor expression in the amygdala was associated with increased anxiety (9). It remains unclear whether GABA(A) receptor function is associated with the underlying pathophysiology of childhood AN or a result of long-term starvation. However, a relative strength of our present findings is that at repeat SPECT scan, the participants were not completely weight-restored [mean BMI 15.7 (0.9 SD)], suggesting that changes in brain 123I-iomazenil binding may be related to clinical improvement rather than mere weight restoration. Nonetheless, this does suggest that GABAergic receptor-mediated inhibitory function may be associated with mood disturbances in children with AN.
We also found a significant negative correlation between 123I-iomazenil-binding activity in the ACC and abnormal eating attitude described by EAT-26 score, with a greater decrease in activity in the ACC of AN children with poor clinical outcomes,
FIGURE 3 | Image analysis (1tZ) of increased iomazenil-binding changes before and after weight gain in the brain of children with AN. Significant increases in iomazenil-binding activity before and after weight gain are shown in the anterior and posterior cingulate cortex, occipital cortex, frontal cortex, and hippocampus, as indicated by the bright orange color.
TABLE 5 | Brain regions and Talairach coordinates showing significantly increased and decreased iomazenil binding in children with anorexia nervosa before and after weight gain.
compared with those with good clinical outcomes. Furthermore, binding activity in the ACC was significantly increased after treatment. As the present clinical outcome score was composed of current BMI and presence of menstruation, as well as changes of eating behavior and social interaction, GABAergic functional activity in the ACC may be related to biological vulnerability of recovery from AN symptoms. Converging lines of evidence suggest correlations between morphological or functional neural changes and differential clinical outcomes in AN. For example, McCormick et al. (27) reported that although the dorsal ACC gray matter volume is significantly reduced in patients with AN compared with normal controls, greater normalization of the right dorsal ACC volume following weight restoration prospectively predicted sustained remission at 1 year post-hospitalization. Functional MRI studies have shown that increased activation in the dorsal ACC and prefrontal cortex in response to food stimuli differentiates recovered AN patients from chronically ill AN patients (28). Further, subcallosal cingulate deep brain stimulation has recently been applied as a treatment strategy for treatment-refractory AN and associated with improvement in mood, anxiety, affective regulation, and increased BMI (29). Taken together, our findings contribute to emerging evidence that variations in functional activities of the ACC may be predictors of outcomes of AN.
Similar to the ACC, neural activities in the PCC may play important roles in the pathophysiology of AN. The PCC is functionally coupled with other brain regions as a default mode network and involved in self-related aspects of cognitive processing such as self-reference and self-reflection (30). Functional brain imaging suggests that dysfunction in resting-state functional connectivity in regions involved in self-referential processing might be associated with development of AN (31). Further, several lines of evidence show that less activation in the PCC is associated with altered inhibitory processing, which might represent a behavioral characteristic and impairment of emotional processing in AN (32, 33). In the present study, several higher mood disturbance scores were significantly associated with lower GABAergic inhibitory binding, mainly in the PCC. Similarly, the PCC was one of the brain regions in which iomazenil-binding activity increased after treatment. In a previous neuroimaging study in children with AN, increased cerebral blood flow (CBF) was observed in the parietal cortex and PCC after inpatient treatment (34). Taken together, increased GABAergic inhibitory function in the PCC after weight gain in our study might indicate improved self-referential processing and cognitive control, which were missing during their starvation period.
We found evidence of increased 123I-iomazenil-binding activity in the occipital cortex after treatment in children with AN. In general, iomazenil-binding activity is strongest in the occipital cortex, indicating that GABA receptors are densely distributed in this area. GABA is involved in interocular suppression in the visual cortex and plays a central role in determining visual cortex selectivity (35). As the brain has a limited capacity, attention allows relevant incoming information to be selectively enhanced while suppressing irrelevant information, the processing for which may be modulated by GABAergic inhibitory function (36, 37). A recent MR spectroscopy study revealed negative correlation between the amount of occipital GABA and cognitive failure in healthy patients, indicating that the inhibitory capacity in sensory areas affects their ability to ignore information that is irrelevant to current behavioral goals (37). In AN patients, cognitive deficits in impaired visuospatial ability, impaired complex visual memory, and impaired selective attention have been reported (38, 39). As GABA plays an important part in stimulus processing and suppression in sensory areas, increased GABAergic inhibitory activation observed in the occipital cortex in AN participants in our study may reflect enhanced suppression of visual information processing, possibly resulting in the improvement of cognitive deficits.
The present study has several limitations that require consideration in future studies. First, brain imaging data from normal healthy children are not available because ethics approval was not feasible for SPECT studies in healthy control children. Therefore, we focused our research on the correlation between GABAergic inhibitory function in brain regions and psychometric profiles, and changes in these functions before and after treatment in AN participants. Thus, we could not determine if the basic GABAergic inhibitory function is upregulated or downregulated, compared to healthy controls. Second, it is possible that changes in iomazenil binding after treatment might be associated with confounding effects, secondary to starvation and restored body weight. However, as the average period for repeat SPECT was short (4 months), weight restoration was not complete. Furthermore, our previous neuroimaging report regarding CBF changes after treatment in AN patients showed no global increase in CBF changes except specific brain regions (bilateral parietal lobe and PCC) (34). Consequently, confounding effects are unlikely to be a cause of our SPECT findings. Third, SPECT exhibits poor resolution around some limbic regions, such as the amygdala and hippocampus, which are important for emotion processing. In these small regions, the obtained radioactivity might differ from the true activity because of a partial volume effect (PVE). The PVE can be defined as underestimation of binding per unit brain volume in small objects or regions because of blurring of radioactivity (spill-out and spill-in) between regions (40, 41). These regions need to be resolved using MR imaging-based correction for PVE. Fourth, we did not examine the participants' psychometric profiles at the end of hospitalization. Although the majority of participants obtained proper eating habits, attended school, and improved parental relationships by the discharge period, these socio-emotional behaviors were not evaluated using the psychometric profile. To confirm our understanding of altered 123I-iomazenil-binding activity due to therapeutic intervention, improvement of socio-emotional difficulties should be consecutively measured.
### REFERENCES
In conclusion, GABAergic inhibitory receptor function in the brain may play an important role in manifesting clinical symptoms of childhood AN. Lower GABAergic 123I-iomazenil binding in specific brain regions at initiation of treatment is associated with clinical severity of mood disturbances and abnormal eating attitude in this sample of participants. Decreased binding in the ACC and left parietal cortex were associated with poor clinical outcomes. Conversely, increased changes in GABAergic receptor binding in the ACC, PCC, and occipital gyrus might be important for the recovery process of childhood AN. Although GABAergic function in the cingulate cortex might be evaluated as a potential predictor of clinical outcome, it will be important to determine the association between function and long-term prognosis. Further research focused on GABAergic receptor-mediated function among participants with eating disorders is warranted.
### AUTHOR CONTRIBUTIONS
SN participated in the design of this study and compiled the manuscript. SN, MM, HC, SO, and YY saw the patients and obtained informed consent and their agreement to participate in the study. Three radiologists (HT, HK, and MI) were in charge of radioactive measurements and calculations of iomazenil activity using ROIs. RS and TK, statistician, conducted the statistical analyses. PC and TM supervised the preparation of the manuscript.
### ACKNOWLEDGMENTS
This work was supported by grants from the Ministry of Education, Culture, Sports, Science, and Technology (#22591143 and #25460643) and Ministry of Health, Labour and Welfare, (#14428000).
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**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
The Reviewer, JK, and handling Editor, GH, declared their shared affiliation, and the handling Editor states that the process nevertheless met the standards of a fair and objective review.
*Copyright © 2016 Nagamitsu, Sakurai, Matsuoka, Chiba, Ozono, Tanigawa, Yamashita, Kaida, Ishibashi, Kakuma, Croarkin and Matsuishi. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.*
# Assessing and stabilizing aberrant neuroplasticity in autism spectrum disorder: the potential role of transcranial magnetic stimulation
*Pushpal Desarkar1,2\*, Tarek K. Rajji1,2, Stephanie H. Ameis1,2,3,4 and Zafiris Jeff Daskalakis1,2*
*1Department of Psychiatry, Centre for Addiction and Mental Health, University of Toronto, Toronto, ON, Canada, 2 Temerty Centre for Therapeutic Brain Intervention, Centre for Addiction and Mental Health, Toronto, ON, Canada, 3Department of Psychiatry, The Hospital for Sick Children, University of Toronto, Toronto, ON, Canada, 4Research Imaging Centre, Campbell Family Mental Health Research Institute, The Centre for Addiction and Mental Health (CAMH), Toronto, ON, Canada*
#### *Edited by:*
*Lena Palaniyappan, University of Nottingham, UK*
#### *Reviewed by:*
*Kwang-Hyuk Lee, University of Sheffield, UK Lindsay M. Oberman, Harvard Medical School, USA Melissa Kirkovski, Deakin University, Australia*
#### *\*Correspondence:*
*Pushpal Desarkar, Department of Psychiatry, Centre for Addiction and Mental Health, University of Toronto, 1001 Queen Street West, Unit 4-4, Toronto, ON M6J 1H4, Canada [email protected]*
#### *Specialty section:*
*This article was submitted to Neuropsychiatric Imaging and Stimulation, a section of the journal Frontiers in Psychiatry*
*Received: 09 July 2015 Accepted: 25 August 2015 Published: 09 September 2015*
#### *Citation:*
*Desarkar P, Rajji TK, Ameis SH and Daskalakis ZJ (2015) Assessing and stabilizing aberrant neuroplasticity in autism spectrum disorder: the potential role of transcranial magnetic stimulation. Front. Psychiatry 6:124. doi: 10.3389/fpsyt.2015.00124*
Exciting developments have taken place in the neuroscience research in autism spectrum disorder (ASD), and results from these studies indicate that brain in ASD is associated with aberrant neuroplasticity. Transcranial magnetic stimulation (TMS) has rapidly evolved to become a widely used, safe, and non-invasive neuroscientific tool to investigate a variety of neurophysiological processes, including neuroplasticity. The diagnostic and therapeutic potential of TMS in ASD is beginning to be realized. In this article, we briefly reviewed evidence of aberrant neuroplasticity in ASD, suggested future directions in assessing neuroplasticity using repetitive TMS (rTMS), and discussed the potential of rTMS in rectifying aberrant neuroplasticity in ASD.
Keywords: autism spectrum disorder, transcranial magnetic stimulation, neuroplasticity, EEG, treatment
### Introduction
Autism spectrum disorder (ASD) is a complex neurodevelopmental disorder characterized by persistent deficits in social communication and interaction and stereotyped behaviors, interests, and activities [*Diagnostic and Statistical Manual of Mental Disorders*, 5th Edition (DSM-5)] (1). The most recent US Centers for Disease Control and Prevention data estimate that ASD now affects 1 in 68 children (2). These data establish ASD as the most common neurodevelopmental disorder. Thus, the social, clinical, and economic burden of ASD is tremendous.
Since the turn of the century, significant advancements have been made in ASD research, and a range of macro- and micro-structural, neurochemical, functional, anatomic, and genetic abnormalities have been proposed [see reviews by Rubenstein and Merzenich (3), Parellada et al. (4), Chen et al. (5), Ameis and Catani (6)]; however, despite gaining important leads, the exact etiology of ASD is still unknown and successful treatment remains elusive. Thus, there is an urgent need to explore novel and effective investigational and mechanism-driven treatment paradigms for ASD.
One mechanism that has recently received a large amount of support suggesting its role in the pathophysiology of ASD is aberrant neuroplasticity (7, 8) In fact, several lines of evidence from genetic (9–13) to animal model (7, 14), neuroimaging (15, 16), and brain stimulation (17, 18) research have all begun to implicate aberrant neuroplasticity in ASD. One neuroscientific tool that has become a widely used, safe, and non-invasive way to probe aberrant neuroplasticity is transcranial magnetic stimulation (TMS) and repetitive TMS (rTMS). Perhaps a fair example of this is the use of
TMS/rTMS in the study of Parkinson's disease [see review by Shukla and Vaillancourt (19)], depression (20), and schizophrenia (21). The diagnostic and therapeutic potential of rTMS in ASD is beginning to be realized. In this article, we will briefly review evidence of aberrant neuroplasticity in ASD, suggest future directions in assessing neuroplasticity using rTMS, and discuss the potential of rTMS in rectifying aberrant neuroplasticity in ASD.
### Aberrant Neuroplasticity in ASD
Before describing the evidence in favor of aberrant neuroplasticity in ASD, it may be worthwhile briefly revisiting neuroplasticity first. Neuroplasticity refers to neuron's ability to reorganize and alter their anatomical and functional connectivity in response to the environmental input. Long-term potentiation (LTP), which involves a net increase in synaptic efficacy, and long-term depression (LTD), which indicates a net decrease in synaptic efficacy, are the two prototypes of neuroplasticity (22).
In a simplistic model, LTP is mediated by glutamate via *N*-methyl-d-aspartate (NMDA) receptors (23). The basic process of LTP generation involves the removal of the Mg2<sup>+</sup> block of the post-synaptic NMDA receptors by a strong wave of depolarization in the dendritic spine, leading to a rapid inflow of Ca2<sup>+</sup> that activates several kinases, eventually leading to the generation of LTP. Similarly, LTD too perhaps is dependent on NMDA receptors. The mechanism of LTD generation, however, requires milder activation of post-synaptic NMDA receptors, which leads to an intermediate intracellular Ca2<sup>+</sup> elevation (23). One key regulator of LTP and LTD is gamma-aminobutyric acid (GABA) released by the inhibitory interneurons (24). At the synaptic level, the fine balance between excitation (mediated by glutamate) and inhibition (mediated by GABA) could be crucial for optimal level of neuroplasticity (25).
### Evidence from the Structural Neuroimaging Studies in ASD
Most of the symptoms of ASD develop in the first few years of life when synaptic development and maturation are occurring at a rapid rate, and one of the most consistent morphological findings that emerged from the structural neuroimaging studies in ASD is early brain overgrowth (15) [also see review by Courchesne et al. (16)]. Such atypical brain enlargement appears to be most pronounced between 2 and 5 years of age (16), and it preferentially affects the frontal and temporal cortices (5). Furthermore, recent evidence indicates that atypical cortical development in ASD subjects persists beyond toddlerhood. In particular, evidence of cortical thinning has been observed among adolescents and young adults (26). These observations led to the hypothesis that ASD is associated with a significant disruption of the typical synaptic maturation and plasticity (5).
### Evidence from the Genetic Studies in ASD
Of all the proposed neurobiological theories of ASD, the potential contribution of genetic factors is backed by a large body of evidence [see review by Chen et al. (5)]. It is important to note that many ASD-associated genes reported by genome-wide association studies encode proteins related to synaptic formation, transmission, and neuroplasticity, and results from recent genetic studies involving ASD clients have consistently linked mutations involving several genes supporting synaptic maturation and neuroplasticity. The examples of such mutations involve genes critically involved in (a) synaptic maturation, e.g., neuroligin 3 and 4 (10), c3orf58, NHE9, and PCDH10 (13); (b) neuronal migration, e.g., CNTNAP2 (12); and (c) dendritic development, e.g., SHANK3 (12).
### Evidence from Animal Models of ASD
Further evidence of aberrant neuroplasticity in ASD is shown by animal models. Perhaps one of the best known among these models is the valproic acid (VPA) rat model of autism. This model predicts that brain in ASD is likely to be hyperplastic. It has been found that, following a Hebbian Pairing Stimulation protocol, the amount of post-synaptic LTP measured in the neocortex and the amygdala doubled in VPA-treated rats compared with controls (14). However, other animal models utilizing genetically modified mice showed that ASD brain could be characterized by both impairment and enhancement of neuroplasticity. For example, Shank3(G/G) mice (27) and mice with MECP2 mutations (model of Rett's syndrome) (28) were shown to have cellular hypoplasticity, but mice with neuroligin-3 mutation were associated with hyperplasticity (29). Such divergent outcomes with regard to the direction of neuroplasticity in these animal experiments could be due to the nature of the genetic modifications used and their impact on the brain substrates of neuroplasticity. Nevertheless, a key insight emerging from these animal models is that if the brain becomes too much or too less plastic (i.e., hyper or hypo), cognition and behavior will be affected. It has been suggested that an optimum level of plasticity is necessary for optimal performance (30), and this process essentially involves keeping excitability within a normal physiological range (31).
### Excitation/Inhibition Imbalance in ASD
Perhaps one of the widely cited neurobiological models in ASD over the past decade is the increased excitation/inhibition ratio in ASD brain (3). It has been suggested that the excitation–inhibition imbalance could be the key determinant of neuroplasticity abnormalities in neurodevelopmental disorders such as ASD (32), and a deficit in the inhibitory neurotransmission has been implicated in the etiopathogenesis of ASD [see review by Baroncelli et al. (25)]. It is believed that such deficits could develop during neuronal maturation (25). At the synaptic level, abnormally increased NMDA-mediated state of excitation, and/or abnormally reduced GABA-mediated inhibition, may lead to abnormally increased neuronal excitability and neuroplasticity. In fact, studies involving subjects with ASD have shown that excitatory glutamate receptors (NMDA and metabotropic glutamate receptor 5) are overexpressed, whereas inhibitory gamma aminobutyric acid A (GABAA) and B (GABAB) receptors are underexpressed in the ASD brain (25, 33). Additionally, post-mortem studies of minicolumnar morphometry in subjects with ASD also demonstrate a significant reduction of the peripheral neuropil space, which is the site of GABA-ergic lateral inhibition in the brain (34).
Transcranial magnetic stimulation has also been used to investigate excitation–inhibition imbalance in ASD. Specifically, paired-pulse TMS paradigms, involving the "pairing" of a "conditioning stimulus" with a "test stimulus" at different interstimulus intervals, have been used to assess cortical inhibition (CI) and facilitation. CI is the neurophysiological process in which inhibitory GABA-ergic interneurons selectively attenuate the activity of pyramidal neurons in the cortex. It has been suggested that CI is key to the regulation of neuroplasticity, and the therapeutic effects of rTMS could be mediated by the induction of local changes in CI (35). Emerging evidence indicates that post-synaptic GABAB receptor-mediated CI is crucial for the regulation of neuroplasticity. GABAB regulates neuroplasticity in two ways: (a) they contribute to the regulation of inhibition by mediating long-lasting inhibitory post-synaptic potentials (IPSPs) and (b) they reduce GABAA receptor-mediated inhibition through presynaptic autoinhibition of inhibitory interneurons (36). Using paired-pulse TMS paradigms, studies have found evidence for excitation– inhibition imbalance in a subgroup of individuals with ASD (37, 38). Other studies have shown no abnormality in CI (18, 39) or a heterogeneous response to this paradigm (40). The heterogeneity in these findings reflects the known heterogeneity of ASD at both the behavioral and the physiological level.
### rTMS in the Assessment of Neuroplasticity in ASD
Repetitive TMS, which involves repetitive delivery of pulses (>1 Hz), is used to modulate cortical activity for investigative and therapeutic purposes [see review by Kobayashi and Pascual-Leone (41)]. rTMS has been increasingly used to study neuroplasticity in humans. The basic premise is that rTMS can modulate activity in the targeted brain region for a duration that can outlast the effects of stimulation itself (30). It is believed that rTMS induces such lasting changes in the brain through altering neuroplasticity mechanisms (42). So far, two rTMS paradigms – theta-burst stimulation (TBS) (17) and paired associative stimulation (PAS) (18) – have been used to assess neuroplasticity in ASD.
### Theta-Burst Stimulation
Theta-burst stimulation involves the delivery of a burst of three pulses at 50 Hz (i.e., 20 ms between stimulus) repeated at intervals of 200 ms (i.e., 5 Hz, hence called theta-burst) (43). TBS comprises two well-established patterned stimulation protocols – continuous TBS (also known as cTBS) and intermittent TBS or iTBS. cTBS paradigm involves the delivery of continuous uninterrupted TBS for 40 s. In the iTBS paradigm, a 2-s train of TBS is repeated every 10 s for a total of 190 s. However, the total number of pulses delivered may vary from one study to another. In the original study, Huang et al. (43) used 600 pulses. iTBS produces sustained enhancement, whereas cTBS is associated with lasting suppression of cortical activity, indexed by potentiation and suppression of motor-evoked potential (MEP) following single-pulse TMS in the contralateral thumb muscle, respectively (43). It is believed that such lasting changes induced by iTBS and cTBS reflect LTP- and LTD-like mechanisms in the brain (43), and in previous experiments, they have been found to be mediated by NMDA receptor (44) and GABA receptor pathways (45), respectively.
### Paired Associative Stimulation
Paired associative stimulation is another well-established rTMS paradigm that has been associated with the induction of LTP-like neuroplasticity (PAS-LTP). It has been shown that PAS-LTP is mediated by NMDA receptors (46). The PAS protocol involves the repetitive delivery of two paired (180 pairs at 0.1 Hz for 30 min) stimulations: (1) an electrical peripheral nerve stimulation of the right median nerve, and 25 ms later, a (2) TMS pulse delivered to the contralateral motor cortex (M1) (hence PAS-25). PAS-25 results in LTP-like neuroplasticity that manifests as the potentiation of MEP in the thumb muscle following single-pulse TMS (46).
### Safety of rTMS in ASD
Available limited data indicate that rTMS, when applied within established safety guidelines, is well tolerated and safe in both adult and pediatric ASD populations (47, 48). There is no current evidence of increased risk of seizure (48).
### rTMS Studies Assessing Neuroplasticity in ASD
Asperger's disorder (AD), which was a subtype of the DSM-IV Pervasive Developmental Disorder, has now been subsumed under ASD in DSM-5 (1). A more direct evidence of aberrant neuroplasticity in AD subjects has been shown by recent rTMS studies using TBS and PAS paradigms. All these studies, however, have assessed neuroplasticity in the motor cortex (M1). One group found greater and long-lasting modulation of neuroplasticity (reflective of aberrant hyperplasticity) following both forms of TBS (cTBS and iTBS) in a small cohort (40) and, subsequently, in a relatively bigger sample of adults with AD (17). Another group, examining LTP-like neuroplasticity in a mixed cohort of adolescents and adults with AD using PAS, obtained similar results, i.e., aberrant neuroplasticity (18); however, the direction of aberrant neuroplasticity was different. In this study, it was found that, compared to typically developing subjects, PASinduced LTP-like plasticity was significantly deficient (reflective of aberrant hypoplasticity) in the AD group.
### Assessing Neuroplasticity in ASD Subjects Using rTMS: Future Considerations
At present, research assessing neuroplasticity using rTMS in ASD population is at an early stage. Studies so far have only tested highfunctioning ASD subjects at the motor cortex (M1). Furthermore, findings obtained in the adult population may not be generalized to the pediatric population. For example, Oberman et al. (47) found a "paradoxical facilitatory effect" to cTBS in more than one-third of their sample consisting of children and adolescents. Therefore, to what extent current findings can be generalized is certainly not very clear at present. The potential factors that need to be considered by future research are heterogeneity in the ASD population, potential impact of the presence/absence of comorbidities including intellectual disabilities, medication use, developmental age, site of stimulation, stimulation parameters (e.g., TBS versus PAS), etc.
The other important point for consideration is that all existing studies utilizing rTMS have assessed neuroplasticity at the motor cortex (M1) of ASD brain. In the future, studies need to look at neuroplasticity in other potential areas of interest in the ASD brain. Information regarding which sites to choose for assessing neuroplasticity in ASD brain may come from existing rTMS intervention studies. So far, studies that used rTMS for therapeutic purposes to improve either symptoms or physiological and cognitive indices have focused on four areas of ASD brain – the dorsolateral prefrontal cortex (DLPFC), medial prefrontal cortex (mPFC), supplementary motor area, and right pars triangularis and pars opercularis [for a review see Oberman et al. (49)]. The DLPFC was chosen due to its extensive network connection with other specialized distributed and local networks in brain (34). Dorsomedial PFC (dmPFC) is another key area for stimulation since it is believed to be uniquely linked with the mentalizing ability (50). A recent trial of deep rTMS delivered bilaterally to the dmPFC significantly improved social relatedness in ASD subjects (51). Therefore, both DLPFC and mPFC could be potential sites of interest for studying neuroplasticity in ASD. Other brain areas related to mentalizing, such as the temporoparietal junction (TPJ) (52), and facial processing, such as superior temporal sulcus (53), could be potential sites for stimulation as well.
Establishing a stimulation paradigm to reliably assess neuroplasticity from these key areas of brain is challenging; however, the combination of TMS with electroencephalography (TMS–EEG) offers researchers an exciting opportunity to gather a more direct measure of neuroplasticity from these areas of brain. Previously, our group established that TMS–EEG can be a reliable method to measure neuroplasticity from M1 and also DLPFC (54). More recently, using a pioneering technique that combines PAS with EEG – "PAS–EEG," our group assessed and successfully demonstrated PAS-induced potentiation of cortical evoked activity, which is reflective of LTP-like neuroplasticity, in DLPFC (55). A similar TMS–EEG approach may be useful for studying neuroplasticity in other key areas of brain. For example, TBS can be combined with EEG to investigate neuroplasticity measures.
In the future, TMS–EEG can also be combined with various social–cognitive tasks and functional neuroimaging to better elucidate the brain–behavior relationship in ASD. Ultimately, TMS– EEG will be combined with genetic research to better elucidate the link between underlying genetic factors (i.e., polymorphisms) and aberration in neuroplasticity captured more directly by TMS–EEG cortical readout. Results from a few early exploratory studies assessing the impact of single-nucleotide polymorphisms, e.g., brain-derived neurotrophic factor valine-to-methionine substitution at codon 66 (Val66Met) genotype (56), on TMSinduced plasticity measures have so far been encouraging.
### Can rTMS be Used as a Therapeutic Tool to Rectify Aberrant Neuroplasticity in ASD?
Repetitive TMS affords researchers to design specific stimulation protocols that can modulate neuroplasticity, and such neuroplasticity-based brain stimulation interventions look promising. Recently, in a randomized double-blind sham-controlled study, our group demonstrated that application of 1,500 pulses/session of high-frequency (20 Hz) rTMS to DLPFC can "normalize" working memory deficits in schizophrenia (57). One possible mechanism of such improvement is enhancement of neuroplasticity in the DLPFC. There is a need to explore similar approach to treat aberrant neuroplasticity in ASD.
### What rTMS Stimulation Protocol to Choose for Stabilizing Aberrant Neuroplasticity in ASD?
Since aberrant neuroplasticity has been linked with the pathogenesis of ASD (7, 8), there is an urgent need to explore treatment paradigms that can stabilize aberrant neuroplasticity and thus potentially facilitate optimal social and cognitive performance and improve restricted and repetitive behaviors in ASD. In this regard, we would like to propose the potential role of extended dosing (i.e., 6,000 pulses) of high-frequency (i.e., 20 Hz) rTMS (58).
In healthy adults, rTMS applied on M1 has been shown to enhance GABA-mediated inhibitory neurotransmission indexed by lengthening of the cortical silent period (CSP), a CI measure reflective of GABAB-mediated inhibitory neurotransmission, with increased stimulation frequency. Our group found that the enhancement was maximal at 20 Hz (31). This finding breaks with convention that high-frequency stimulation results in excitation, whereas low-frequency stimulation results in inhibition, as 20-Hz rTMS, but not 1-Hz rTMS, resulted in a CSP prolongation (31, 58). One explanation is that 20-Hz rTMS may exert its inhibitory effect by selectively affecting networks involving fastspiking inhibitory interneurons that mainly oscillate at higher (i.e., 30–70 Hz) frequencies (58). A recent study by our group investigating differing durations or doses of rTMS on CI in M1 in healthy subjects found that even a single session of extended dosing (6,000 pulses) with high-frequency (20 Hz) pulses led to significant lengthening of the GABAB-mediated CSP compared with other paradigms (58). This effect was not seen with active or sham 1- or 20-Hz rTMS at 1,200 pulses or 3,600 pulses.
It has been suggested that, depending on the direction and magnitude of inhibition, GABAB receptor-mediated neurotransmission may attenuate neuroplasticity. In fact, baclofen, a GABAB agonist, significantly attenuated LTP-like neuroplasticity in M1 induced by PAS (59). Since extended dosing (i.e., 6,000 pulses) of such specific high-frequency (20 Hz) rTMS protocol (58) appears to maximally enhance GABAB-mediated inhibitory neurotransmission, one approach would be to assess if such protocols are able to stabilize aberrant hyperplasticity seen in ASD. This line of approach is also consistent with the excitation–inhibition imbalance in ASD, i.e., a general deficit in GABA-ergic inhibition, an increased excitation/inhibition ratio (3), and an evidence of reduced expression of GABAB receptors (33). In the future, proofof-principle studies are needed to test this assumption. Because of its simplicity and reliability, such experiments may begin at M1 to see if the delivery of 6,000 pulses at 20 Hz can stabilize aberrant neuroplasticity in ASD subjects. If successful, further pilot studies will be required to assess whether rectifying aberrant neuroplasticity translates into actual clinical improvement or not. These pilot studies may potentially stimulate key areas of ASD brain discussed above, i.e., DLPFC, TPJ, and dmPFC, and determine key stimulation parameters, duration of sessions, etc.
In summary, existing genetic and animal studies of ASD and evidence emerging from human rTMS studies have consistently indicated aberrant neuroplasticity in ASD brain. However, at this point, there are many unanswered questions regarding the exact etiopathological connection between aberrant neuroplasticity in the brain and development of autistic symptoms. Nevertheless, existing evidence still indicates that aberrant neuroplasticity could play a critical role in the pathogenesis of ASD. Therefore, it can be postulated that it may be possible to attain optimal social and cognitive performance in ASD by stabilizing aberrant neuroplasticity. In this context, we discussed a novel mechanism-driven approach toward achieving such goal using rTMS. If successful, this information will not only help us better understand the brain mechanisms involved in ASD but also stimulate trials testing mechanism-driven novel brain stimulation treatment paradigms for ASD.
### Acknowledgments
PD is supported by the Innovation Fund from the Alternate Funding Plan of the Academic Health Sciences Centres of
### References
Ontario and Dean's Fund, Faculty of Medicine, University of Toronto. TR receives research support from Brain Canada, Brain and Behavior Research Foundation (previously known as NARSAD), Canadian Foundation for Innovation, Canadian Institutes of Health Research (CIHR), Ontario Ministry of Health and Long-Term Care, Ontario Ministry of Research and Innovation, the US National Institute of Health, and the Weston Brain Institute. SA receives financial support from the O'Brien Scholar's Program, an Ontario Mental Health Foundation New Investigator Fellowship, and The University of Toronto Dean's Fund New Staff Grant. In the last 5 years, ZD received research and equipment in-kind support for an investigator-initiated study through Brainsway Inc. ZD has also served on the advisory board for Hoffmann-La Roche Limited and Merck and received speaker support from Sepracor and Eli Lilly. This work was supported by the Ontario Mental Health Foundation (OMHF), the Canadian Institutes of Health Research (CIHR), the Brain and Behaviour Research Foundation, the Temerty Family and Grant Family, the Centre for Addiction and Mental Health (CAMH) Foundation, and the Campbell Institute. We would like to thank Dr. Amanda Sawyer for her contribution toward preparing this manuscript.
**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
*Copyright © 2015 Desarkar, Rajji, Ameis and Daskalakis. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.*
# Neurocognitive effects of repetitive transcranial magnetic stimulation in adolescents with major depressive disorder
#### **Christopher A.Wall 1,2\*, Paul E. Croarkin1,2, Shawn M. McClintock 3,4, Lauren L. Murphy <sup>1</sup> , Lorelei A. Bandel <sup>1</sup> , Leslie A. Sim1,2 and Shirlene M. Sampson<sup>1</sup>**
<sup>1</sup> Department of Psychiatry and Psychology, Mayo Clinic, Rochester, MN, USA
<sup>2</sup> Division of Child and Adolescent Psychiatry, Mayo Clinic, Rochester, MN, USA
<sup>3</sup> Neurocognitive Research Laboratory, Division of Brain Stimulation and Neurophysiology, Department of Psychiatry and Behavioral Sciences, Duke University School of Medicine, Durham, NC, USA
<sup>4</sup> Department of Psychiatry, University of Texas Southwestern Medical Center, Dallas, TX, USA
#### **Edited by:**
Stephanie Ameis, University of Toronto, Canada
#### **Reviewed by:**
Peter G. Enticott, Deakin University, Australia Daniel Blumberger, Centre for Addiction and Mental Health, Canada
#### **\*Correspondence:**
Christopher A. Wall, Department of Psychiatry and Psychology, Mayo Clinic, 200 First Street South West, Rochester, MN 55905, USA e-mail: [email protected]
**Objectives:** It is estimated that 30–40% of adolescents with major depressive disorder (MDD) do not receive full benefit from current antidepressant therapies. Repetitive transcranial magnetic stimulation (rTMS) is a novel therapy approved by the US Food and Drug Administration to treat adults with MDD. Research suggests rTMS is not associated with adverse neurocognitive effects in adult populations; however, there is no documentation of its neurocognitive effects in adolescents. This is a secondary post hoc analysis of neurocognitive outcome in adolescents who were treated with open-label rTMS in two separate studies.
**Methods:** Eighteen patients (mean age, 16.2 ± 1.1 years; 11 females, 7 males) with MDD who failed to adequately respond to at least one antidepressant agent were enrolled in the study. Fourteen patients completed all 30 rTMS treatments (5 days/week, 120% of motor threshold, 10 Hz, 3,000 stimulations per session) applied to the left dorsolateral prefrontal cortex. Depression was rated using the Children's Depression Rating Scale-Revised. Neurocognitive evaluation was performed at baseline and after completion of 30 rTMS treatments with the Children's Auditory Verbal Learning Test (CAVLT) and Delis–Kaplan Executive Function System Trail Making Test.
**Results:** Over the course of 30 rTMS treatments, adolescents showed a substantial decrease in depression severity. Commensurate with improvement in depressive symptoms was a statistically significant improvement in memory and delayed verbal recall. Other learning and memory indices and executive function remained intact. Neither participants nor their family members reported clinically meaningful changes in neurocognitive function.
**Conclusion:** These preliminary findings suggest rTMS does not adversely impact neurocognitive functioning in adolescents and may provide subtle enhancement of verbal memory as measured by the CAVLT. Further controlled investigations with larger sample sizes and rigorous trial designs are warranted to confirm and extend these findings.
**Keywords: adolescents, depression, neurocognition, memory, learning,TMS**
#### **INTRODUCTION**
Repetitive transcranial magnetic stimulation (rTMS) is a novel treatment approach for medication-resistant patients with major depressive disorder (MDD). Repetitive TMS has been approved by the US Food and Drug Administration for the treatment of adults with MDD who fail to achieve satisfactory improvement from one prior adequate antidepressant treatment trial. Although several sham-controlled studies have indicated that rTMS is efficacious in adults with MDD (1–4), there have been few studies in adolescents. We recently reported results of an open-label pilot study that found rTMS to be a potentially effective adjunctive therapy for adolescents with treatment-resistant MDD (5). Adolescents showed statistically significant improvement in the Children's Depression Rating Scale-Revised (CDRS-R) from baseline through the rTMS treatment series (30 sessions) and at 6-month follow-up. A second, recently completed replication trial in 10 adolescents revealed similar findings for clinical improvement in treatment completers (submitted). In both studies, assessments of cognitive functioning were performed at baseline and treatment completion.
A number of studies have indicated that rTMS treatment for MDD is not associated with adverse effects on neuropsychological functions such as attention, learning, and memory (2, 3, 6–9), but these investigations have only included adults. Not only have studies not shown any deterioration in neuropsychological functioning from rTMS, but several investigations have shown an improvement in neurocognitive function in adult patients
with MDD. For example, in a sham-controlled study by Avery et al. (1), adults with MDD showed considerably improved performance on measures of attention, learning and memory, and cognitive flexibility following 10 sessions of 10 Hz rTMS applied to the left dorsolateral prefrontal cortex (L-DLPFC). Furthermore, two sham-controlled studies reported improvement in verbal memory performance as a result of multiple sessions of 10 Hz rTMS treatment to the L-DLPFC in adults with MDD (10, 11). Also, following multiple sessions of 10 Hz rTMS to the L-DLPFC in depressed adults, Fitzgerald et al. (12) found that there was significant improvement in neuropsychological function, including autobiographical memory; and Martis et al. (13) reported improvement in working memory and executive function. Recently, Luber and Lisanby described a review of over 60 TMS studies that reported "significant improvements in speed and accuracy in a variety of tasks involving perceptual, motor, and executive processing" (14).
In the present study we evaluated the neurocognitive effects of rTMS when used as an adjunctive treatment for adolescents with treatment-resistant MDD. We hypothesized that adolescents would demonstrate no difference in measures of memory, executive functioning, or auditory and visual learning tasks following a robust course of left-sided, high-frequency rTMS.
### **MATERIALS AND METHODS**
#### **PARTICIPANTS**
Participants were diagnostically assessed by a board-certified child and adolescent psychiatrists (Paul E. Croarkin and Christopher A. Wall). This included a comprehensive clinical evaluation and standardized diagnostic interview that utilized the Kiddie Schedule for Affective Disorders and Schizophrenia for School-Age Children – Present and Lifetime Version (K-SADS-PL) (15). At the time of enrollment, all participants were receiving active antidepressant treatment for an MDD episode according to the *Diagnostic and Statistical Manual of Mental Disorders*, Fourth Edition, Text Revision (*DSM-IV-TR*) (16). Clinically significant depressive symptoms were defined by CDRS-R (17) total score of at least 40 (*t* score >63). Participants included those with treatment failure/non-response to at least one adequate antidepressant trial [i.e., treated with stable selective serotonin reuptake inhibitor (SSRI) dose regimen for at least 6 weeks as defined by a score of ≥3 on the Antidepressant Treatment History Form] (18). All participants continued treatment with a stable dose of their prestudy antidepressant during the rTMS course. Participants in psychotherapy were ineligible if they had changed therapists, type of psychotherapy, or providers in the 4 weeks prior to rTMS initiation. Participants were allowed to continue previous sleep aids such as melatonin, trazodone, or diphenhydramine during treatment. Stimulants, antipsychotics, mood stabilizers, and tricyclic antidepressants were not permitted during the active treatment phase.
Patients with comorbid secondary diagnoses of dysthymia, attention-deficit/hyperactivity disorder, or anxiety disorders were eligible for enrollment. However, patients with schizophrenia, schizoaffective disorder, bipolar spectrum disorders, substance abuse or dependence, somatoform disorders, dissociative
disorders, post-traumatic stress disorder, obsessive-compulsive disorder, eating disorders, mental retardation, or pervasive developmental disorder/autism spectrum disorders were excluded from participation. Medical exclusions included preexisting seizure disorders or active neurologic conditions (e.g., brain tumor, dyskinesias, or paralysis). The screening process included a urine toxicology screen for drugs of abuse and a urine pregnancy test. All participants and treaters wore earplugs during the sessions to minimize the risk of auditory threshold changes.
#### **STUDY OVERVIEW**
Both trials were prospective, open, multicenter pilot trial of active rTMS in adolescents with MDD confirmed by the K-SADS-PL. Both studies received institutional review board approval and were performed under United States Food and Drug Administration Investigational Device Exemptions: trial #1 – G060269 and trial #2 – G110091. All patients provided written informed assent, and parents provided written informed consent per institutional review board-approved guidelines. Recruitment, outcomes, and potential adverse effects were monitored by a Data and Safety Monitoring Board comprised of clinicians with no direct involvement in the study.
#### **rTMS PROCEDURES**
Identification of the treatment site and stimulus dosing were based on previously defined techniques and guidelines noted in adult rTMS trials (4, 19). In both trials, the motor cortex was identified, using the rTMS machinery via a single pulse administered every 3–5 s, at the location that produced a localized contraction of the contralateral abductor pollicis brevis muscle. Once this site was defined, the resting motor threshold (MT) was determined using a computer-assisted maximum likelihood threshold-hunting algorithm (MT Assist, Neuronetics Inc., Malvern, PA, USA). Repeat MT determinations occurred at least once every 10 treatments to assess for possible changes that could produce safety issues due to changes in cortical excitability.
In trial #1, the L-DLPFC treatment location was determined by moving the treatment coil 5 cm anterior to the MT location along a left superior oblique plane (20). In trial #2, the L-DLPFC treatment location was determined via an MRI-based neurolocalization technique. The identified treatment site was then marked and spatial coordinates were recorded with a mechanical coil positioning system to ensure reproducibility of the coil placement.
In both trials, a total of 30 treatments were administered across a range of 6–8 weeks. This range was chosen for potential variation in patient schedules related to school and family events. Thus, each patient was offered a total of 40 treatment opportunities in which to complete 30 treatments. Each treatment was titrated to 120% of calculated MT, at a frequency of 10 Hz, with stimulus train duration of 4 s and an inter-train interval of 26 s, for a total of 3,000 stimulations per treatment session. In trial #1, rTMS was delivered using the Neuronetics Model 2100 Therapy System; in trial #2 treatments were delivered using the NeuroStar System (Neuronetics, Inc., Malvern, PA, USA).
#### **NEUROCOGNITIVE ASSESSMENTS**
Neurocognitive testing was administered by trained psychometrists at baseline and upon completion of the active rTMS treatments. Testing was typically performed in the afternoon hours, although not universally due to scheduling accommodations related to subject school obligations, family work hours, and clinician/psychometrist availability. A doctorate-level child psychologist (Leslie A. Sim) analyzed the results. The neurocognitive battery was tailored to assess a variety of neurocognitive domains including psychomotor speed, simple attention, learning, memory, and executive function. Specifically, the battery included the Children's Auditory Verbal Learning Test-2 (CAVLT-2) (21) and the Delis–Kaplan Executive Function System (D-KEFS) (22) Trail Making Test.
The CAVLT-2 (21) is a measure designed to quantify a child's (ages 6.6–17.11) verbal learning and memory abilities. The measure is comprised of a 16-item word list that is administered across five trials, and the participant is asked to recall the words after each trial. A different set of words is then presented and the participant is asked to immediately recall the items from the new list (Interference Trial). Following the interference list, the participant is instructed to recall as many items as possible from the original list (immediate recall). Following a 15 min delay, the participant is asked to recall the original list for a final time (delayed recall). Finally, the participant is presented with a 32-item word list and asked to recognize the 16 words from the original list (Recognition Trial). The CAVLT-2 yields multiple indices of learning and memory, including immediate memory span, level of learning, immediate recall, delayed recall, recognition accuracy, and total intrusions.
The D-KEFS Trail Making Test is used to assess components of cognitive flexibility on a visual-motor sequencing task (23). It consists of five conditions: (a) Visual Scanning, (b) Number Sequencing, (c) Letter Sequencing, (d) Motor Speed, and (e) Number-Letter Switching. The D-KEFS Trail Making Test was selected to assess aspects of executive function, particularly cognitive flexibility, as it is influenced by mood and anxiety states. Importantly, the D-KEFS has been normed for the age group of children in this trial and provides standardized scores (24). Based on adequate reliability and validity of these tests along with appropriate developmental normative data, these neuropsychological measures (CAVLT-2 and D-KEFS TMT) were thought to be developmentally appropriate tools to assess subtle changes in cognitive function of adolescents receiving rTMS treatments.
Safety and participant comfort were assessed and recorded before and after each study visit with prompted opportunities to report adverse events.
#### **STATISTICAL ANALYSIS**
The primary aim of this study was to assess whether adjunctive rTMS is a safe and feasible treatment approach in adolescents. This question was evaluated by neurocognitive assessments that occurred at baseline and immediately following treatment number 30. Within patient changes from baseline to treatment completion were examined with comparative statistics.
Neurocognitive measurements obtained at baseline and immediately following treatment were summarized using mean and standard deviation (±SD). The paired *t*-test was used to assess whether scores changed significantly from baseline to end of treatment. For these analyses, two-tailed *P*-values of ≤0.05 were considered statistically significant. In addition to the primary analyses which included all enrolled subjects, a subset analysis was performed that was restricted to subjects who completed treatment. All analyses were performed using SAS software, version 9.2 (SAS Institute Inc., Cary, NC, USA).
### **RESULTS**
Of the 18 adolescents enrolled in both trials, 14 adolescents (5 males and 9 females; ages 13.9–17.8 years; mean age, 16.3 ± 1.1 years) completed the entire rTMS treatment course (clinicaltrials.gov Identifier: NCT00587639). Fourteen out of 14 adolescents who completed the entire treatment course also completed neurocognitive testing at baseline and treatment completion. Subjectively, no reportable changes in memory, cognitive functioning, or attention were noted by any of the participating adolescents or their families. Objectively, subtle but statistically significant improvement was observed in immediate memory and delayed recall as measured by the CAVLT when measured in all study participants (**Table 1**; **Figure 1**) and participants who completed all 30 rTMS sessions of the treatment protocol (**Table 2**; **Figure 2**). All other CAVLT indices of learning and memory (interference, immediate recall, or level of learning) remained stable over the course of treatment (**Table 1**).
No significant changes were noted on the D-KEFS Trail Making Test indices from baseline to treatment completion for either the all participant group (**Table 3**; **Figure 3**) or the treatment completers only group (**Table 4**; **Figure 4**).
### **DISCUSSION**
To our knowledge, this is the first report on neurocognitive outcomes within a clinical trial of rTMS in depressed adolescents. The neurocognitive safety findings of this case series of high-frequency rTMS in treatment-resistant depressed adolescents are consistent with previously reported findings in clinical trials of rTMS in adults with psychiatric illness. Previous adult trials have frequently included cognitive assessments that demonstrated rTMS to have no adverse effects on cognitive functions (2, 3, 6–9, 25). Interestingly, a number of these clinical trials in adults have shown time-limited improvements in various aspects of cognitive function, mainly in attention, concentration, working memory, and processing speed (10– 13). Similarly, modest improvements in attention, and verbal and learning and memory were observed in this cohort of adolescents.
#### **LIMITATIONS**
Clearly, these findings must be interpreted with caution due to the small total number of participants and the lack of a control group. However, if there was a distinct pattern of clinically and psychometrically meaningful adverse cognitive effects – as could
**Table 1 | Children's Auditory Verbal Learning Task (CAVLT) results (all participants)**.
BL, baseline; PT, post-treatment.
be found in a robust course of electroconvulsive therapy – we would expect to see these findings even in this small group of participants. It is reassuring to note that none of the participants or their family members described any impairments (or marked improvements) in learning, memory, or other untoward cognitive effects.
### **CONCLUSION**
Collectively, the findings of this study combined with our prior clinical findings suggest that rTMS may be a safe, feasible, and potentially efficacious adjunctive therapy for adolescents with MDD given the lack of negative changes in cognitive functioning and reduced overall side-effect burden (4, 5). Future studies of rTMS in adolescents will need to monitor for cognitive changes, which would benefit from the use of a comprehensive, standardized, and validated neurocognitive battery that will be sensitive to cognitive changes, particularly in those cognitive domains essential for continued academic maturation and instrumental activities of daily living. Such a neurocognitive battery should assess domains of intellectual ability, processing speed, attention,
BL, baseline; PT, post-treatment.
**Table 3 | Delis–Kaplan Executive Function System (D-KEFS) Trail Making Test results (all participants)**.
learning and memory, working memory, and executive function. Indeed, Semkovska and McLoughlin recommended the development of standardized and validated tools to measure the ability to recall autobiographical events in patients with depression (26). Such a measure would be of particular value in the adolescent population.
#### **ACKNOWLEDGMENTS**
Dr. Paul E. Croarkin has received grant/research support from the National Alliance for Research on Schizophrenia and Depression (NARSAD), Pfizer Inc., and Stanley Medical Research Foundation. Drs. Christopher A. Wall and Shirlene M. Sampson have received equipment support (disposable e-shields and Senstar shields) from Neuronetics, Malvern, PA, USA. Dr. Shawn M. McClintock has received grant/research support from NARSAD, the National Institute of Mental Health, and was a consultant for Shire. Dr. Lauren L. Murphy, Dr. Leslie A. Sim, Dr. Shirlene M. Sampson and Ms. Lorelei A. Bandel have no potential conflicts to declare. The authors would like to thank Data and Safety Monitoring Board (DSMB) members from both trials including Louis Kraus, MD (DSMB Chair, Rush University Medical Center), John E. Huxsahl, MD (Division of Child and Adolescent Psychiatry and Psychology, Mayo Clinic), Kirti Saxena, MD (University of Texas Southwestern), Simon Kung, MD (Mayo Clinic), and Tanya Brown Ph.D. (Mayo Clinic) for providing oversight during the conduct of these studies. We would also like to thank the O'Shaughnessy (Woolls') Foundation for providing generous infrastructure support of the neurostimulation program.
#### **REFERENCES**
**Conflict of Interest Statement:** Study 1 was funded by an American Academy of Child and Adolescent Psychiatry/Eli Lilly Pilot Research Award (Dr. Christopher A. Wall); a Small Grant Award from the Department of Psychiatry and Psychology, Mayo Clinic (Dr. Christopher A. Wall); and a Stanley Medical Research Institute Center Grant (Dr. Paul E. Croarkin). For Study 2 Dr. Christopher A. Wall received funding via funded by the Klingenstein Third Generation Foundation Fellowship in Child and Adolescent Depression; the Mayo Clinic CReFF award and Mayo Clinic Kids' Cup scholarship. Funding was also provided by the National Institute of Mental Health (K23 MH087739, Dr. Shawn M. McClintock). Neuronetics, Malvern, PA, USA provided equipment support, but had no involvement in the protocol design or analysis. The other co-authors declare that the research was conducted in the
absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
*Received: 01 October 2013; paper pending published: 18 October 2013; accepted: 25 November 2013; published online: 12 December 2013.*
*Citation: Wall CA, Croarkin PE, McClintock SM, Murphy LL, Bandel LA, Sim LA and Sampson SM (2013) Neurocognitive effects of repetitive transcranial magnetic stimulation in adolescents with major depressive disorder. Front. Psychiatry 4:165. doi: 10.3389/fpsyt.2013.00165*
*This article was submitted to Neuropsychiatric Imaging and Stimulation, a section of the journal Frontiers in Psychiatry.*
*Copyright © 2013 Wall, Croarkin, McClintock, Murphy, Bandel, Sim and Sampson. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.*
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2025-04-07T04:13:04.061639
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11-2-2021 15:09
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ffd71d14-d514-40a1-a7c5-3184c8dde886.0
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**Iron and Cobalt Catalysts**
• Wilson D. Shafer and Gary Jacobs
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2025-04-07T04:13:04.094791
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1-5-2021 17:26
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"book_id": "ffd71d14-d514-40a1-a7c5-3184c8dde886",
"url": "https://mdpi.com/books/pdfview/book/2436",
"author": "",
"title": "Iron and Cobalt Catalysts",
"publisher": "MDPI - Multidisciplinary Digital Publishing Institute",
"isbn": "9783039283880",
"section_idx": 0
}
|
ffd71d14-d514-40a1-a7c5-3184c8dde886.1
|
**Iron and Cobalt Catalysts**
Printed Edition of the Special Issue Published in *Catalysts* Wilson D. Shafer and Gary Jacobs Edited by
www.mdpi.com/journal/catalysts
**Iron and Cobalt Catalysts**
|
doab
|
2025-04-07T04:13:04.094871
|
1-5-2021 17:26
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"book_id": "ffd71d14-d514-40a1-a7c5-3184c8dde886",
"url": "https://mdpi.com/books/pdfview/book/2436",
"author": "",
"title": "Iron and Cobalt Catalysts",
"publisher": "MDPI - Multidisciplinary Digital Publishing Institute",
"isbn": "9783039283880",
"section_idx": 1
}
|
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