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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 з базами 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.**

**Громадське обговорення роботи відбудеться 28 вересня 2018 р. о 15.00 годині на засіданні Вченої ради Львівського національного університету імені Івана Франка за адресою: м. Львів, вул. Університетська, 1, аудиторія 377.**

## Коментарі

## Andrei Khrennikov

I am pleased to express my support the work “Qualitative Methods of Studying Models of Mathematical Physics" and its authors, G. Feldman, A. Kochubei, I. Mykytiuk, A. Prykarpatsky, O. Rebenko, V. Samoilenko and M. Scherbyna, nominated for the State Prize of Ukraine in Science and Technology. All its authors are distinguished scientists undoubtedly deserving the Prize. Most of them, I know A. Kochubei, with whom I have common interests in non-Archimedean analysis and its applications. Recently we developed a fruitful collaboration in the investigation of nonlinear equations coming from the p-adic model of a porous medium.

I believe his works, as well as those by the rest of the authors, should be supported.

Andrei Khrennikov, professor of Applied Mathematics,

Int. Center Math Modeling in Physics, Engineering, Economics, and Cognitive Science

Linnaeus University, Växjö, Sweden

## Vladimir Borisovich Matveev

The authors of Collective Research Work " Qualitative Methods of Studying Models of Mathematical Physics "

submitted for the competition for State Award in Sciences and Technology of Ukraine by the

Institute of Mathematics of the National Academy of Sciences of Ukraine are internationally recognised researchers, having a very wide scale of scientific interests and

making the decisive contributions in classical and quantum mechanics , algebro-geometric approach and inverse scattering

approach to the integrable systems, statistical physics and many other fields.

I personally know Professor A.K. Prykarpatsky with whom we maintain friendly scientific relations since more than 4O years. He has been one of the first to develop an algebro-geometric approach for solving the Landau-Lifshitz equation and made many others important

contributions in the area of integrable systems.

I am also familiar with remarkable achievements of some other authors of the project including

of M.V. Scherbina, I.V. Mykytiuk, and G.M. Feldman.

Besides their Scientific activities all the authors of the Collective Work made an important

contribution for developing a fruitful scientific collaborations between

various countries by organising the representative International conference and workshops both inside and outside of Ukraine.

Summarising, I strongly support the authors of Collective Work and I am convinced that it merits to be distinguished by attributing to the authors

the State Award in Sciences and Technology of Ukraine

Vladimir B. Matveev , Emeritus Professor at the Institut de Mathématiques de Bourgogne ,

Université de Bourgogne -Franche Comté, 21087, Dijon, France , e-mail: matveev@ubourgogne.

fr

## Yogesh Joshi

It is with immense pleasure that I would like to give my highest recommendation and my nomination to the Research Work

"Qualitative Methods of Studying Models of Mathematical Physics" by Authors: Kochubei A.N., Rebenko O.L., Mykytiuk I.V., Prykarpatsky A.K., Samoilenko V.H., Feldman G.M., Shcherbyna M.V. for the State Award In Science and Technology.

Dr. Yogesh Joshi

Assistant Professor

Kingsborough Community College (CUNY)

Brooklyn, New York, USA

## Michal Fečkan

I am pleased to write on support of the Project "Qualitative Methods of Studying Models of Mathematical Physics" and its investigators that are presented by the distinguished mathematicians: G. Feldman, A. Kochubei, I. Mykytiuk, A. Prykarpatsky, O. Rebenko, V. Samoilenko and M. Scherbyna which have received essential and challenging results in modern mathematics and its applications to problems of mathematical physics. In particular, I would like to mention the impressive and original results obtained by A. Prykarpatsky on the inverse spectral transform analysis and its important and breakthrough relationship to the differential geometric De Rham-Hodge theory and integrable nonlinear dynamical systems arising in modern mathematical physics with applications to multi-dimensional differential operators including three-dimensional Laplace operator, two-dimensional classical Dirac operator and its multidimensional affine extension, related with self-dual Yang-Mills equations.

I strongly support this project and a related group of prominent investigators for awarding them with a prestigious State Prize in Science and Technology.

Michal Fečkan, Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics of the Comenius University in Bratislava, Slovakia

## Robert Conte

Robert Conte,

E'cole normale supérieure de Cachan,

CNRS, Université Paris-Saclay.

Robert.Conte@cea.fr

Although I was aware of their great scientific achievements,

this is only last year that I had the great pleasure to meet some of the investigators

of this project, in particular Professor Anatolyi Prykarpatsky.

For a "Westerner" like me, it is always an immense pleasure to meet

former USSR scientists because, in their domain, they master both the physical background and the mathematical tools necessary to solve the problems. The applicants belong to one of the most famous schools of mathematical physics, and I strongly support the idea that they should be awarded the State prize of Ukraine in science and technology.

Robert Conte.

## Jean-Pierre Magnot

This is a great pleasure for me to support the project

Qualitative Methods of Studying Models of Mathematical Physics

and its distinguished investigators

G. Feldman, A. Kochubei, I. Mykytiuk, A. Prykarpatsky, O. Rebenko, V. Samoilenko and M. Scherbyna.

I have discovered this topic in 2003 when I what a post-doctoral fellow a the university of Bonn, Germany, just after finishing a PhD dissertation on infinite dimensional geometry. The great capacities of my supervisor, Prof. Dr. Sergio Albeverio, and of his collaborators, convinced me to study better current models in mathematical physics. Among great enlightening minds, I discovered the works of M. Scherbyna, G. Feldman and I. Mykytiuk whose works on probabilistic models in mathematical physics helped me, by their clarity, their deep insight and their well-suited perspectives, to carry a self-driven research in more deterministic models, keeping in mind the non-artificial (and yet actually too often heuristic) link between different approaches in mathematical physics. I can also highly recommand the work of O. Robenko with E. Lytvynov and G.V. Shchepan'ur, for its technical precision. The techniques developped in this medium length paper are central for analysis and probability on the Fock space, and to my opinion they may lead to future developments 20 years after their publication.

However, the researcher among all the investigators who I know better, through his great works and our frequent exchanges, is A. Prykarpatsky. We know each other since 2014, when we started to exchange around a differential geometric method that intends to prove well-posedness for Cauchy problems, in the field of integrable systems. What is striking for me is the great capacities of A. Prykarpatsky. He stands for me as a living example of a well-achieved researcher of the (ex-)USSR school of mathematics (and in particular of mathematical physics) which produced till nowaday leading international experts, with strong capacities in both fields of Mathematics and Physics. His knowledge in physics is far deeper from mine, while his more mathematical works are highly rigorous, and full of very original and very accurate ideas. His works on integrable systems and operator theory in relation with mathematical physics are for me highly stimulating, and his very open way of mind, in terms of specific fields of mathematics that have to be considered, often encourage me, as a younger researcher, to consider his advices as coming from a very smart senior researcher.

For all these reasons, I recommand as strongly as possible the works and the investigators under consideration.

Jean-Pierre Magnot

LAREMA

University of Angers

France

## Леонід Пастур

I support entirely the work “Qualitative Methods of Studying Models of Mathematical Physics " and its authors nominated for the State Prize of Ukraine in Science and Technology. The work includes the results that comprise a significant and widely recognized contribution into several leading and quite active branches of modern mathematical physics and adjacent fields. The result are describes in 18 monographs and 122 papers published in the leading international journals and presented in a number of prestigious international meetings.

My special support is for the contribution by G.M.Feldman and M.V.Shcherbina who presented an impressive collection of results on several probabilistic problems of mathematical physics which require new powerful analytic techniques. The totality of these results determines to large extend the modern state of the corresponding branches of mathematical physics.

In my opinion the spectrum and the significance of the results of the work leave no doubts that it possesses all the merits necessary for the State Prize to be awarded.

Leonid Pastur

Academician of National

Academy of Sciences of Ukraine

State Prize of Ukraine 1985

## Gabor J. Szekely

It is a great pleasure for me to recommend and support Dr. Gennady Fel'dman for the State of Ukraine Award in Sciences and Technology. Dr. Fel'dman is a leading expert of the important area "Probability measures on groups". His monograph on the arithmetic and characterizations of these distributions published by the American Mathematical Society is a classic by now. I had the opportunity to review many of Dr. Fel'dman's papers. They are examples of elegance and depth. He is a pillar of the world famous Kharkiv probability school.

Gabor J. Szekely

Program Director

National Science Foundation

Alexandria, VA

## Denis Blackmore

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, V. Samoilenko 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.

## Abram Kagan

I strongly support the candidatures of

G. Feldman, M. Shcherbyna, A. Kochubei, I.V. Mykytyuk, A. Prykarpatsky, O. Rebenko, V.H. Samoilenko for State of Ukraine Award in Sciences and Technology.

Feldman and Shcherbyna work in the area of probability closely related to mathematical physics. Their results are of the first class and often cited.

I have the first-hand knowledge of Feldman’s work, being a reviewer of some of his papers. Besides, he gave a colloquium talk at my university.

Feldman is one of the best experts in probability on algebraic structures and in arithmetic of probability distributions, continuing the tradition of the Kharkiv famous school. The methods he developed are an important tool for studying probability on algebraic structures.

It will be a real pleasure for me to congratulate Gennady Feldman and Mariya Shcherbyna (and her adviser, Leonid Pastur) with the well-deserved award.

Abram Kagan

Professor, Dept. of Mathematics

University of Maryland

College Park, MD 20742, USA

## Kamal N. Soltanov

Kamal Soltanov

Professor of Mathematical Sciences

Department of Mathematics

Hacettepe University

and

Department of Mathematics

Igdir University, Turkey

It is a pleasure for me to write in support of the project "Qualitative Methods of Studying Models of Mathematical Physics" and its investigators that are the group of distinguished scientists. This group of researcers: G. Feldman, A. Kochubei, I. Mykytiuk, A. Prykarpatsky, O. Rebenko and M. Scherbyna have received in this project results that are essential in mathematical physics. These results essentially advanced the area of analysis of finite and infinite-dimensional dynamical systems arising in mathematical physics.

I weell know Prof. Prykarpatsky, because I collaborated with his on the joint Project between TUBITA and NASU and on the joint works on investigations of some nonlinear dynamical systems. I think his creative ability and intellectual prowess, which identify he as among the elite thinkers and contributors in their field.

I think that this project and the group researchers that obtained very well results and continue to receive extraordinary results in their field has deserve the Ukrainian NAS State Prize in Science and Technology.

## Kamal Soltanov

It is a pleasure for me to write in support of the project "Qualitative Methods of Studying Models of Mathematical Physics" and its investigators that are the group of distinguished scientists. This group of researcers: G. Feldman, A. Kochubei, I. Mykytiuk, A. Prykarpatsky, V. H. Samoylenko, O. Rebenko and M. Scherbyna have received in this project results that are essential in mathematical physics. These results essentially advanced the area of analysis of finite and infinite-dimensional dynamical systems arising in mathematical physics.

I weell know Prof. Prykarpatsky, because I collaborated with his on the joint Project between TUBITA and NASU and on the joint works on investigations of some nonlinear dynamical systems. I think his creative ability and intellectual prowess, which identify he as among the elite thinkers and contributors in their field.

I think that this project and the group researchers that obtained very well results and continue to receive extraordinary results in their field has deserve the Ukrainian NAS State Prize in Science and Technology.

## Piotr Graczyk, Angers University France

I strongly support the candidature of the scientific team of professors

G. Feldman, A. Kochubei, I.V. Mykytyuk, A. Prykarpatsky, O. Rebenko, V.H. Samoilenko and M. Scherbyna, and their their Project “Qualitative Methods of Studying Models of Mathematical Physics”, for State Award in Sciences and Technology.

Our Maths Laboratory LAREMA at Angers University had a great pleasure to receive M. Scherbyna and G. Feldman at our seminar and conferences. We had several long-time Ukraine-France common projects and grants with this

world level team, which were fruitful and gave 5 common publications. We hope to continue this collaboration, towards mathematical physics of Bessel particles

and on sparse Wishart Ensembles, basic themes of modern mathematical physics.

The State Award in Sciences and Technology will surely profit to get new Ukraine-France grants in future.

Piotr Graczyk

professor

Maths Laboratory LAREMA

at Angers University

## Emin Özçağ

Emin Özçağ

Department of mathematics

Hacettepe Univ. Ankara-Turkey

It is great pleasure to write in support of the collective Project

" Qualitative Methods of Studing Models of Mathematical Physics "

application for the state prize of Ukraine in Science and Technology. The authors of the Project are well-known distinguised and leading scientists of Ukraine in the field of Mathematical Physics. I know A. K. Prykarpatsky on the occasion of the joint Project between Tubitak ( Scientific and Technological Research Council of Turkey) and NASU and the Project resulted important results in the theory of integrability of nonlinear dynamical systems. Prof. A. K. Prykarpatsky is highly respected authority in his field of the research.

I strongly believe that this Project deserves the State Prize of Ukraine in Science and Technology.

## Ufuk Taneri

Having worked with, published with, and knowing Prykarpatsky A.K.'s paramount contributions to research, it is my pleasure and honor to write this Comment in support of the nomination of the Collective Research Work on

"Qualitative Methods of Studying Models of Mathematical Physics"

by Kochubei A.N., Rebenko O.L., Mykytiuk I.V., Prykarpatsky A.K., Samoilenko V.H., Feldman G.M., Shcherbyna M.V.

This work is yet another outstanding research contribution by Anatolij Prykarpatsky and his co-workers as manifested by its nomination by the Kyiv Institute of Mathematics of NAS of Ukraine for

the State Award in Science and Technology

and submission by the Institute of Mathematics of the National Academy of Sciences of Ukraine.

It is no surprise that , ‘Comments’ before me also support the nomination and the submission, recognizing the significance of the work involved and contributions of all of the co-authors.

Please allow me to comment in full confidence about the work of Anatolij Prykarpatsky and his continued supreme contributions to science, as these values have always reflected in the work of all co-scientists with Prykarpatsky:

Talking about his extensive work is talking about oceans of research contributions to mathematics, physics and applied research be it in industry, engineering, space sciences, or health sciences. His work includes but is not limited to the Quantum Field Theory, Dynamical Systems, Bifurcation Analysis, Stability and Chaos Theories, Topology, the Electrodynamic Vacuum-Field Theory, the Theory of Mathematical Modelling, Group Theory, Harmonic Analysis, Fourier Analysis, Applied Analysis, Measure Theory, Statistical Mechanics, Computational Methods, Functional Analysis, Quantum Physics, Quantum Mechanics, the Theory of Reference Frames, and the Fixed-Point Theory. Each work has had its commemorative mark in research. His continued works will undoubtedly continue shed light on Science and Technology.

I strongly support the recognition of the work of Kochubei A.N., Rebenko O.L., Mykytiuk I.V., Prykarpatsky A.K., Samoilenko V.H., Feldman G.M., Shcherbyna M.V. on "Qualitative Methods of Studying Models of Mathematical

Physics". These co-researchers are well worthy of the acknowledgement of the prestigious Ukrainian State Award in Science and Technology.

Prof.Dr. Ufuk Taneri

Research Associate, UC Berkeley, Berkeley, CA, US

Professor of Mathematics,

Chancellor Emeritus, Eastern Mediterranean University, North Cyprus.

## Dr. Andrey Shoom, Max Plank Institute, Germany, and Universi

It is a great pleasure to support the collective project “Qualitative Methods of Studying Models of Mathematical Physics” application for the State Prize of Ukraine in Science and Technology. Undoubtedly, the authors of the project, Kochubei A. N., Rebenko O. L., Mykytiuk I. V., Prykarpatsky A. K., Samoilenko V. H., Feldman G. M., Shcherbyna M. V., are experienced and leading researchers in the field of mathematical physics, and, in particular, in the theory of dynamic systems of modern mathematical and theoretical physics. I am happy to know very well one of them, Prof. A. K. Prykarpatsky, who has been my research advisor and friend for more than twenty years. It was my pleasure to take a part in a few research projects led by him.

Within the presented collective work for the Award, I ought to mention that Prof. A.K. Prykarpatsky obtained many important results in the theory of integrability of nonlinear dynamical systems, in functional and operator varieties, in the theory of transformation operators, in differential-geometric theory of invariant integral manifolds of Liouville completely integrable Hamiltonian systems, and in generalizations of the Poincare-Melnikova-Samoilenko-Arnold theory. His methods devised for describing transversal splitting of separatrix manifolds of weakly and adiabatically perturbed nonlinear dynamical systems proved both new and effective for many applications in modern mathematical physics.

In particular, extremely important results were obtained by the Prof. A.K. Prykarpatsky on the study

of mathematical properties of operator dynamical systems and their Delsarte-Lions transmutation properties, arising in modern mathematical physics. Here I need to mention deeply studied by A.K. Prykarpatsky differential-geometric aspects of generalized de Rham-Hodge complexes naturally related with integrable multi-dimensional differential systems of M. Gromov type, as well as the geometric structure of Chern characteristic classes. In particular, special differential invariants of the Chern type constructed in his works are of great importance for the modern integrability of multi-dimensional nonlinear differential systems on Riemannian manifolds. As an example, he analyzed in detail the three-dimensional Davey-Stewartson type nonlinear integrable differential system and studied its Cartan type connection mapping, as well as the related Chern type differential invariants.

In a wide cycle of works of Prof. A. K. Prykarpatsky on the theory of nonlinear dynamical systems he developed an effective gradient-holonomic approach to studying the complete integrability of a wide class of nonlinear Hamiltonian systems of mathematical physics, which are presented by means of nonlinear differential equations in partial derivatives, and, in particular, constructed their exact solutions by means of modern methods of algebraic geometry, devised before in classical work by S. Novikov, B. Dubrovin, B. Matveev. The devised approach to studying the hydrodynamic Hamiltonian structures, based on the suggested before by A. Balinsky and S. Novikov non-commutative and non-associative algebras, Prof. A.K. Prykarpatsky, jointly with his coworkers D. Blackmore and O. Artemovych, described structural properties of wide classes of these algebras, as well as constructed their new types, including Leibniz and Riemann algebras.

As the main mathematical achievements in the analysis of complete integrability of nonlinear dynamical systems were previously obtained by specialists for the case of spatially-one-dimensional functional manifolds, in the case of so called multi-dimnensional dispersionless nonlinear dynamical systems, Prof. A.K. Prykarpatsky devised jointly with his coworkers D. Blackmore and O. Hentosh new powerful methods for analyzing their integrability, by creatively developing the modern results of such mathematicians as P. Lax and M. Sato, as well as ones obtained by such well-known specialists in the field as E. Ferapontov, O. Mokhov, L. Bogdanov, B. Konopelchenko, M. Dunajski and M. Pavlov. He extended the outstanding classical results of French and Ukrainian mathematicians of the last century, among which are those by Prof. M.A. Buhl from France and academician VUAN, Prof. G. Pfeiffer from Kiev.

Based on the theory of the Japanese mathematician M. Sato and his students, Prof. A. K. Prykarpatsky has shown that the Buhl problem can be effectively described by means of the so-called specially constructed Casimir type invariants, generating systems of special vector fields, which automatically satisfy the Lax-Sato compatibility conditions. Moreover, he has shown that in these cases the compatibility conditions are equivalent to an infinite hierarchy of the so-called 'heavenly' dispersiveless nonlinear partial differential equations.

During the last years Prof. A. K. Prykarpatsky with his coworkers systematically studied many classes of compatible systems of the Lax-Sato vector fields and associated with them Lie-algebraic structures, that are important for the complete integrability of a wide class of nonlinear many-dimensional dynamical systems, important for applications in modern mathematical physics and first introduced by the Polish mathematical physicist Ye. Plebanski. Based on these results Prof. A.K. Prykarpatsky developed a new and effective Lie-algebraic method for studying completely integrable multi-dimensional nonlinear dynamical systems. This method is widely applied for solving many important nonlinear models of modern mathematical physics.

With no doubts, I consider that the achievements of Prof. A. K. Prykarpatsky presented in the Collective Work deserve with no reservation the State Award In Science and Technology.

## Гість

В циклі робіт “Якісні методи дослідження моделей математичної фізики” колективу відомих українських математиків Кочубея А.Н, Ребенка О.Л., Микитюка І.В., Прикарпатського А.К., Самойленка В.Г., Фельдмана Г.М. і Щербини М.В., який включає 16 монографій і 122 наукових статей, досліджені актуальні задачі сучасної математичної і теоретичної фізики.

Я найкраще знайомий з роботами О.Л. Ребенка, який в 70-ті роки працював в групі академіка Д.Я. Петрини в Інституті теоретичної фізики в Києві. Ним були отримані вкрай важливі результати для квантової теорії поля і статистичної фізики, зокрема, розвинуто квантову теорію евклідової матриці розсіювання і запропоновано операторний підхід до розв’язання проблеми усунення ультрафіолетових розбіжностей, який еквівалентний відомій операції Боголюбова-Парасюка. Дуже суттєвим є вклад О.Л. Ребенко у розвиток статистичної механіки, де ним вперше в рамках евклідової квантової теорії поля описано рівноважну статистичну механіку нескінченних класичних систем з кулонівською взаємодією і побудовано термодинамічний граничний перехід її фізичних характеристик. Результати з квантової теорії поля увійшли до його монографії “Основи сучасної теорії взаємодіючих квантових полів”, яку я мав нагоду рецензувати і яку використовую при читанні лекцій в Київському Національному університеті імені Тараса Шевченка.

Публікації авторів колективу в престижних міжнародних виданнях засвідчують високий рівень даного циклу робіт. Я безумовно підтримую цей цикл робіт для присудження Державної премії в галузі науки і техніки України за 2018 рік.

Завідувач відділу астрофізики та елементарних частинок

Інституту теоретичної фізики ім. М.М. Боголюбова НАН України,

д-р фіз.-мат. наук, професор, член-кореспондент НАНУ

В.П. Гусинін

## Pedro M Gadea

Ukrainian researchers G. Feldman, A. Kochubei, I.V. Mykytyuk, A. Prykarpatsky, O. Rebenko and M. Scherbyna, have made, working in their Project “Qualitative Methods of Studying Models of Mathematical Physics,” substantial advances in mathematical physics, among them in the related field of analysis of finite and infinite-dimensional dynamical systems.

I have collaborated in several research projects with Prof. I.V. Mykytyuk, jointly either with Professor Marco Castrillon Lopez (University Complutense, Madrid), in two published papers on structures linked to the spinor Lie group Spin(9); or Professor Jaime Munoz Masque (CSIC, Madrid), in a book on topics of differential geometry, published in 2013 by Springer-Verlag, London, UK; or Professor Carmelo Gonzalez-Davila (University of La Laguna, Tenerife, Canary Islands, Spain), with who we are currently writing, in an advanced stage, a paper on “Invariant Ricci-flat Kaehler structures on semisimple Riemannian symmetric spaces,” with both motivation and applications to geometric quantization.

I would underline here the wonderful mixture of wide knowledge, attractive and useful ideas, expert calculations and clear exposition, characteristics of Prof. Mykytyuk. I would like to add that Professor Mykytyuk gave a short course to professors and researchers in the University of Valencia, Spain, in the last week of September 2016, and seeing the high level and clear exposition of the course, both the Department of Prof. Castrillon and Prof. Gonzalez-Davila have planned to invite him to the Univ. of Madrid and the Univ. of Tenerife to impart there two more short courses for professors and researchers.

Fruitfully collaborating during past years with Prof. I.V. Mykytyuk, I have become familiar with the mathematical investigations with strong international reputation of his Ukrainian coworkers entering in the team for the Award. Compiled in two high level mathematical monographs, authored by Prof. I.V. Mykytyuk and Prof. A. Prykarpatsky, these investigations demonstrate an impressive impact on the research in the modern theory of nonlinear dynamical systems on manifolds.

In conclusion, I strongly support this project and team of investigators for the State Award in Science and Technology.

Professor Pedro M. Gadea

Emeritus Researcher

Institute of Fundamental Physics

Spanish National Research Council

28006 Madrid, Spain

## Pedro M Gadea

Ukrainian researchers G. Feldman, A. Kochubei, I.V. Mykytyuk, A. Prykarpatsky, O. Rebenko, V.H. Samoilenko and M. Scherbyna, have made, working in their Project “Qualitative Methods of Studying Models of Mathematical Physics,” substantial advances in mathematical physics, among them in the related field of analysis of finite and infinite-dimensional dynamical systems.

I have collaborated in several research projects with Prof. I.V. Mykytyuk, jointly either with Professor Marco Castrillon Lopez (University Complutense, Madrid), in two published papers on structures linked to the spinor Lie group Spin(9); or Professor Jaime Munoz Masque (CSIC, Madrid), in a book on topics of differential geometry, published in 2013 by Springer-Verlag, London, UK; or Professor Carmelo Gonzalez-Davila (University of La Laguna, Tenerife, Canary Islands, Spain), with who we are currently writing, in an advanced stage, a paper on “Invariant Ricci-flat Kaehler structures on semisimple Riemannian symmetric spaces,” with both motivation and applications to geometric quantization.

I would underline here the wonderful mixture of wide knowledge, attractive and useful ideas, expert calculations and clear exposition, characteristics of Prof. Mykytyuk. I would like to add that Professor Mykytyuk gave a short course to professors and researchers in the University of Valencia, Spain, in the last week of September 2016, and seeing the high level and clear exposition of the course, both the Department of Prof. Castrillon and Prof. Gonzalez-Davila have planned to invite him to the Univ. of Madrid and the Univ. of Tenerife to impart there two more short courses for professors and researchers.

Fruitfully collaborating during past years with Prof. I.V. Mykytyuk, I have become familiar with the mathematical investigations with strong international reputation of his Ukrainian coworkers entering in the team for the Award. Compiled in two high level mathematical monographs, authored by Prof. I.V. Mykytyuk and Prof. A. Prykarpatsky, these investigations demonstrate an impressive impact on the research in the modern theory of nonlinear dynamical systems on manifolds.

In conclusion, I strongly support this project and team of investigators for the State Award in Science and Technology.

Professor Pedro M. Gadea

Emeritus Researcher

Institute of Fundamental Physics

Spanish National Research Council

28006 Madrid, Spain

## 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.

## Vladimir Borisovich Matveev

Vladimir B. Matveev , Emeritus Professor at the Institut de Mathématiques de Bourgogne ,

Université de Bourgogne -Franche Comté, 21087, Dijon, France , e-mail: matveev@u-bourgogne.fr

The authors of Collective Research Work " Qualitative Methods of Studying Models of Mathematical Physics "

submitted for the competition for State Award in Sciences and Technology of Ukraine by the Institute of Mathematics of the National Academy of Sciences of Ukraine

are internationally recognised researchers, having a very wide scale of scientific interests and making the decisive contributions

in classical and quantum mechanics , algebro-geometric approach and inverse scattering approach to the integrable systems,

statistical physics and many other fields.

I personally know Professor A.K. Prykarpatsky with whom we maintain friendly scientific relations

since more than 4O years. He has been one of the first to develop an algebro-geometric approach for solving the Landau-Lifshitz equation and made many others important contributions in the area of integrable systems.

I am also familiar with remarkable achievements of some other authors of the project including of M.V. Scherbina, I.V. Mykytiuk, and G.M. Feldman.

Besides their Scientific activities all the authors of the Collective Work made an important contribution for developing a fruitful scientific collaborations between

various countries by organising the representative International conference and workshops both inside and outside of Ukraine.

Summarising, I strongly support the authors of Collective Work and I am convinced that it merits to be distinguished by attributing to the authors

theState Award in Sciences and Technology of Ukraine

## Vladimir Borisovich Matveev

Dear members of the State Award in Science and Technology of Ukraine ,

I was rather surprised being informed by Professor Prykarpatsky that my letter of support above disappeared from your site.

Nevertheless I was able to see it perfectly well after looking again on your site a few minutes ago. It was preceded by the the letter of support by Professor

VZ Enolsk'i and I see no reason to write the same letter again - I believe that it is just a technical problem with your sited I am not able to solve it on my own.

Please comment.

Sincerely yours ,

Vladimir Matveev ,

Professor of Mathematics

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