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Якісні методи дослідження моделей математичної фізики

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Представлено Інститутом математики НАН України.

Автори: чл.к. Кочубей А.Н., д.ф.-м.н. Ребенко О.Л., д.ф.-м.н. Микитюк І.В., д.ф.-м.н. Прикарпатський А.К., д.ф.-м.н. Самойленко В.Г., чл.к. Фельдман Г.М., чл.к. Щербина М.В.

Роботу присвячено дослідженню актуальних задач теорії динамічних систем сучасної математичної і теоретичної фізики та побудові нових ефективних методів якісного аналізу широкого класу моделей математичної фізики та їх застосуванням. Авторами приділено значну увагу вивченню широкого кола математичних моделей, які використовуються у фізиці, механіці, теорії випадкових графів, квантовій інформатиці, теорії передачі інформації, інших галузях природознавства.

Розглянуто інтегровні системи, диференціальні рівняння з частинними похідними і дробово-диференціальні рівняння, що описують некласичну дифузію (аномально повільну дифузію на фракталах), методи p-адичного аналізу тощо. При цьому математичні об’єкти фізичного походження вивчено методами різноманітних розділів математики, від функціонального аналізу до алгебраїчної геометрії і теорії груп Лі.

Розвинено такі актуальні напрямки сучасної математичної фізики як методи неархімедового аналізу і неархімедової стохастики, якісно-аналітичний аналіз інтегровності нелінійних динамічних систем математичної фізики, якісно-аналітичні методи нескінченновимірного аналізу квантової теорії поля і статистичної механіки, імовірнісні методи в задачах спектральної теорії та на групах і дано їх застосування при дослідженні різноманітних моделей математичної і теоретичної фізики. Доведено гіпотезу Дайсона.

Кількість публікацій: 16 монографій (11 – закордонні), 122 статті (102 - у зарубіжних виданнях). Загальна кількість посилань на публікації авторів / h-індекс роботи згідно баз даних складає відповідно: Web of Science – 2165/24; Scopus – 1974/19; Google Scholar – 7435/37. За даною тематикою захищено 15 докторських та 37 кандидатських дисертацій.

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Коментарі

Alexander Balinsky

It is with great pleasure that I support the application for the State Prize of Ukraine
in Science and Technology by Prof. A.K. Prykarpatsky presented in the Collective Work.
I know Prof. A.K. Prykarpatsky for more than 30 years and read many of his seminal contributions
across many fields of mathematical physics, differential geometry and applied mathematics,
both in the theory and the applications. I would especially like to mention results obtained
by Prof. A.K. Prykarpatsky on the theory of integrability of nonlinear dynamical systems
on functional and operator varieties, the theory of transformation operators, differential-geometric
theory of invariant integral manifolds of completely integrable Hamiltonian systems
and generalisations of the Poincare-Melnikova-Samoilenko-Arnold theory.

Prof. A.K. Prykarpatsky is a highly respected international authority in his fields of research
with outstanding contributions to mathematical physics. His main strength is a unique combination
of mathematical rigour and deep understanding of application needs.

I strongly support the application by Prof. A.K. Prykarpatsky and I am confident that
he is very deserving of the State Prize of Ukraine in Science and Technology.

Professor Alexander Balinsky
WIMCS Chair in Mathematical Physics
Cardiff School of Mathematics
Cardiff University
Cardiff, CF24 4AG, United Kingdom

Yuri Kondratiev

The series of research works ªQualitative Methods of Studying Models of Mathematical Physicsªby the
well-known Ukrainian mathematicians G. Feldman, A. Kochubei, I. Mykytiuk, A. Prykarpatsky, O. Reben-
ko and M. Scherbyna represents several important branches of mathematics united by applications to ma-
thematical physics. Already the amount of publications (18 monographs and 122 research papers) shows
the scale of scienti®c activities by the authors. Of the work covered by the series, I know better the re-
search by Kochubei and Rebenko. Kochubei was among the ®rst mathematicians who began to work on
time-fractional evolution equations, in particular equations appearing in mathematical models of anoma-
lous diffusion. His work in this area had a key role in its development. His other research directions include
non-Archimedean analysis, pseudo-differential equations, extension of operators. In each of them, he has
made an essential contribution. Rebenko is an author of well-known publications on mathematical pro-
blems of quantum ®eld theory and statistical physics. I believe the group as a whole and their contributions
to mathematical physics clearly deserves the State Prize of Ukraine in Science and Technology.

Prof. Dr. Yuri Kondratiev
Bielefeld University, Germany,
Winner of the State Prize of Ukraine
in Science and Technology for 1998

Tony Dorlas

Prof. T. C. Dorlas
Dublin Institute for Advanced Studies
School of Theoretical Physics
Dublin 4, Ireland

The work of Prof. O. L. Rebenko is extensive and of the highest quality. I strongly believe that this project deserves the State Award in Science and Technology. Prof. Rebenko is an expert on mathematical physics, in particular in the area of statistical physics, both classical and quantum. I am particularly familiar with his work on quantum crystals and on classical interacting gases. He is particularly expert in cluster expansion techniques and has contributed much of importance to the development and application of these complicated but powerful methods.

Alexander D. Bendikov, IM UWr POLAND

I strongly support the application of professors G. Feldman, A. Kochubei and V. Samoilenko for State Award in Sciences and Technology.

Vladimir Miransky

Vladimir Miransky, Professor of the Department of Applied Mathematics of Western University, London, Ontario, Canada

I know quite well the high quality papers and monograph of Prof. Rebenko O.L.presented in this project. I strongly believe that this project deserves the State Award in Science and Technology.

Denis Blackmore

Denis Blackmore
Professor of Mathematical Sciences
New Jersey Institute of Technology
Newark, NJ 07102-1982, U.S.A.

It is indeed a pleasure for me to write in support of the project Qualitative Methods of Studying Models of Mathematical Physics and its distinguished investigators G. Feldman, A. Kochubei, I. Mykytiuk, A. Prykarpatsky, O. Rebenko and M. Scherbyna. Working on this project, the outstanding group of investigators has produced, and continue to produce, extraordinary results that have significantly advanced the state-of-the-art in mathematical physics – especially in the area of analysis of finite and infinite-dimensional dynamical systems arising in this discipline.
I collaborated on several projects with Profs. Mykytiuk, Prykarpatsky and Samoilenko involving such diverse subjects as Lax-type solutions of Hamilton—Jacobi equations, Delsarte—Lions transmutation operators and non-commutative and non-associative algebras related to the integrability of infinite-dimensional Hamiltonian dynamical systems. As a result, I have extensive first-hand knowledge of the extraordinary expertise, creative ability and intellectual prowess of these three researchers, which identifies them as among the elite thinkers and contributors in their field.
My most extensive collaboration is a long-standing and productive one with Anatoliy Prykarpatsky, which has produced over twenty published papers and a book on infinite-dimensional dynamical systems of mathematical physics (with V. Samoilenko as a coauthor). Most recently, Anatoliy, I and others have applied Lax—Sato theory to produce a novel solution of the classical problem of Buhl problem along with innovative methods for analyzing the integrability of heavenly equations. And we are currently working on several other things related to the integrability analysis of dispersionless dynamical systems, many of which look very promising – both theoretically and from an applications perspective. Moreover, Prof. Prykarpatsky, who developed the gradient holonomic method for analyzing complete integrability of the equations of mathematical physics, continues to make amazing breakthroughs. For example, I have just seen a submitted paper of his in which he obtains some remarkable new results on fractional differential-difference hierarchies of Hamiltonian dynamical systems.
I only know the other members of the research team by their publications and sterling international reputations as mathematical physics researchers. However, it is clear to me that the whole research team is of exceptional quality and prowess – that has already made significant contributions to mathematical physics and is apt to make many more. Consequently, this project and team of investigators has my strongest possible support for a Ukrainian NAS State Prize in Science and Technology.

Victor Enolski

Victor Enolski, Visiting Professor to Department of Physics of the University
of Oldenburg on leave of absence from the National University of Kyiv Mohyla Academy.

The group of authors of Collective Research Work
"Qualitative Methods of Studying Models of Mathematical Physics" submitted by the Institute of Mathematics of the National Academy of Sciences of Ukraine.) and nominated by the Kyiv Institute of Mathematics of NASU of Ukraine for the State Award In Science and Technology collected leading experts in the area of Mathematical physics. Collective Research Work resumes long-term work contributors and covers main internationally recognised fields of mathematical physics - completely integrable of non-linear dynamic system and associated partial-differential equations of physical interest, differential-geometric aspects of complete integrability, solvable models of quantum field theory and quantum statistic mechanics, quantum turbulence, evolution equations with fractal derivatives in time and others. The results are documented in 15 monographs
and more the hundreds of papers published in Ukraine and in various international journals
of high priority.

I better familiar with works of O.L.Rebenko, we both started at 70th years our researches
in the same group under supervision of academician NASU D.Ya. Petrina and were in scientific
contact since that time. Prof. O.L.Rebenko investigated model systems of quantum field theory
and infinite systems of statistical mechanics on the basis of Eucledian Scattering Matrix.
The international recognition was got by his works on equilibrium statistical mechanics of
infinite dimensional systems with Coulomb interaction, were existing of Debye screening was
proven by methods of cluster expansions for many-component ion-dipole systems. I also would
like to mention important monograph by O.L.Rebenko containing comprehensive introduction to
quantum field theory, I was using this monograph in my lectures on Non-Abelian Monopole in
the National University Kyiv Mohyla Academy and in my lecture courses given abroad, in
particular, in Oldenburg University and Tsuda University in Tokyo. That's extreme pleasure
for me to emphasise here that O.L.Rebenko, who was the same year student, now professor
have made the significant contribution to science on which we trained.

I also long time acquainted and communicated with two other applicants,
Member-corespondent NASU Prof. A.N.Kochubei and Prof. A.K.Prykarpatsky
and only mention here their high professional level and strong intetnational
reputation. The whole authors team, Kochubei A.N., Rebenko O.L., Mykytiuk I.V.,
Prykarpatsky A.K., Samoilenko V.H., Feldman G.M., Shcherbyna M.V. represents
they represent effectively working collective which has exerted a great influence
on development of science in Ukraine.

Summing up the reasoning above I consider that the achievements
of all applicants presented in the Collective Work deserve the
State Award In Science and Technology and strongly recommend to award applicants.

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