Victor Kozyakin

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• Prof.
• Principal researcher at Moscow Institute of Physics and Technology

The Berger-Wang formula establishes equality between the joint and generalized spectral radii of a set of matrices. For matrix products whose multipliers are applied not arbitrarily but in accordance with some Markovian law, there are also known analogs of the joint and generalized spectral radii. However, the known proofs of the Berger-Wang formul...

In complex control systems containing sampled-data elements, it is possible that these elements operate asynchronously. In some cases asynchronous character of operation of sampled-data elements does not influence stability of system. In other cases any small desynchronization of the updating moments of sampled-data elements leads to dramatic chang...

In 1995 J.C. Lagarias and Y. Wang conjectured that the generalized spectral radius of a finite set of square matrices can be attained on a finite product of matrices. The first counterexample to this Finiteness Conjecture was given in 2002 by T. Bousch and J. Mairesse and their proof was based on measure-theoretical ideas. In 2003 V.D. Blondel, J....

It is shown that the assumption of D-stability of the interconnection matrix, together with the standard assumptions on the activation functions, guarantee the existence of a unique equilibrium under a synchronous mode of operation as well as a class of asynchronous modes. For the synchronous mode, these assumptions are also shown to imply local as...

The problem of construction of Barabanov norms for analysis of properties of the joint (generalized) spectral radius of matrix sets has been discussed in a number of publications. The method of Barabanov norms was the key instrument in disproving the Lagarias-Wang Finiteness Conjecture. The related constructions were essentially based on the study...

Recently, Bochi and Laskawiec constructed an example of a set of matrices {A,B} having two different (up to cyclic permutations of factors) spectrum maximizing products, AABABB and BBABAA. In this paper, we identify a class of matrix sets for which the existence of at least one spectrum maximizing product with an odd number of factors automatically...

Recently, Bochi and Laskawiec constructed an example of a set of matrices {A,B} having two different (up to cyclic permutations of factors) spectrum maximizing products, AABABB and BBABAA. In this paper, we identify a class of matrix sets for which the existence of at least one spectrum maximizing product with an odd number of factors automatically...

Описывается система преобразования библиографии из формата BibTEX в формат AMSBIB, рекомендованный общероссийским порталом Math-Net.Ru для исполь-зования при подготовке библиографии в публикациях в российских журналах математической направленности. Осуществление такого преобразования сводится к применению стандартной процедуры создания библиографии...

In the theory of linear switching systems with discrete time, as in other areas of mathematics, the problem of studying the growth rate of the norms of all possible matrix products with factors from a set of matrices A arises. So far, only for a relatively small number of classes of matrices A has it been possible to accurately describe the sequenc...

See preprint https://arxiv.org/abs/2112.00391 describing details of conducted numerical experiments and the website https://github.com/kozyakin/barnorm for the Python programs used in numerical testing, and some results of testing

In the theory of linear switching systems with discrete time, as in other areas of mathematics, the problem of studying the growth rate of the norms of all possible matrix products $A_{\sigma_{n}}\cdots A_{\sigma_{0}}$ with factors from a set of matrices $\mathscr{A}$ arises. So far, only for a relatively small number of classes of matrices $\maths...

We consider the question of the boundedness of matrix products $A_{n}B_{n}\cdots A_{1}B_{1}$ with factors from two sets of matrices, $A_{i}\in\mathscr{A}$ and $B_{i}\in\mathscr{B}$, due to an appropriate choice of matrices $\{B_{i}\}$. It is assumed that for any sequence of matrices $\{A_{i}\}$ there is a sequence of matrices $\{B_{i}\}$ for which...

An example of inconsistencies in information provided by popular bibliographic services is described and the reasons for these inconsistencies are discussed.

To estimate the growth rate of matrix products $A_{n}\cdots A_{1}$ with factors from some set of matrices $\mathcal{A}$, such numeric quantities as the joint spectral radius $\rho(\mathcal{A})$ and the lower spectral radius $\check{\rho}(\mathcal{A})$ are traditionally used. The first of these quantities characterizes the maximum growth rate of the...

Описаны некоторые классы линейных асинхронных мультиагентных систем с дискретным временем, для которых проблема устойчивости допускает конструктивное решение. Описан также общий аналитический подход к построению числовых характеристик, аналогичных обобщенному спектральному радиусу в теории устойчивости, которые предоставляли бы возможность анализир...

We describe certain classes of linear asynchronous multi-agent systems in discrete time for which the stability problem allows for a constructive solution. We also present a general analytic approach to constructing numerical characteristics similar to the generalized spectral radius in stability theory, which would provide an opportunity to analyz...

We describe mathematical methods for analyzing the stability, stabilizability and consensus of linear multiagent systems with discrete time. These methods are based on the idea of using the notion of joint/generalized spectral radius of matrix sets to analyze the rate of convergence of matrix products with factors from the sets of matrices with spe...

Описываются математические методы анализа устойчивости, стабилизируемости и консенсуса линейных мультиагентных систем с дискретным временем. В основе этих методов лежит идея привлечения понятия совместного/обобщенного спектрального радиуса наборов матриц для анализа скорости сходимости матричных произведений с сомножителями из множеств матриц со сп...

Представлен обзор результатов по моделям консенсуса в асинхронных мультиагентных системах с дискретным и непрерывным временем. Дается описание математических методов, разработанных в последние годы, которые используются при анализе проблем устойчивости, стабилизируемости и консенсуса для линейных мультиагентных систем с дискретным временем. В основ...

We present a survey of results on models of consensus in asynchronous multiagent systems with discrete and continuous time. We consider mathematical methods developed over recent years, which are used in the analysis of stability, stabilization, and consensus problems for linear multiagent systems with discrete time. These methods are based on the...

An example of inconsistencies in information provided by popular bibliographic services is described, and the reasons for such inconsistencies are discussed. This is the author's translation of article Козякин В. С. Осторожно, DOI! : “Библиографический детектив” эпохи цифровизации // Информационные процессы. 2019. Т. 19, № 1. С. 54–59.

Описывается пример несоответствия информации, предоставляемой популярными библиографическими сервисами, и обсуждаются причины такого несоответствия.

We consider the problem of convergence to zero of matrix products $A_{n}B_{n}\cdots A_{1}B_{1}$ with factors from two sets of matrices, $A_{i}\in\mathscr{A}$ and $B_{i}\in\mathscr{B}$, due to a suitable choice of matrices $\{B_{i}\}$. It is assumed that for any sequence of matrices $\{A_{i}\}$ there is a sequence of matrices $\{B_{i}\}$ such that t...

To estimate the growth rate of matrix products $A_{n}\cdots A_{1}$ with factors from some set of matrices $\mathcal{A}$, such numeric quantities as the joint spectral radius $\rho(\mathcal{A})$ and the lower spectral radius $\check{\rho}(\mathcal{A})$ are traditionally used. The first of these quantities characterizes the maximum growth rate of the...

We consider the problem of convergence to zero of matrix products $A_{n}B_{n}\cdots A_{1}B_{1}$ with factors from two sets of matrices, $A_{i}\in\mathscr{A}$ and $B_{i}\in\mathscr{B}$, due to a suitable choice of matrices $\{B_{i}\}$. It is assumed that for any sequence of matrices $\{A_{i}\}$ there is a sequence of matrices $\{B_{i}\}$ such that t...

We prove the minimax equality for the spectral radius $\rho(AB)$ of the product of matrices $A\in\mathcal{A}$ and $B\in\mathcal{B}$, where $\mathcal{A}$ and $\mathcal{B}$ are compact sets of non-negative matrices of dimensions $N\times M$ and $M\times N$, respectively, satisfying the so-called hourglass alternative. Full text: http://www.tandfonli...

We describe a new class of positive linear discrete-time switching systems for which the problems of stability or stabilizability can be resolved constructively. This class generalizes the class of systems with independently switching state vector components. The distinctive feature of this class is that, for any system from this class, its compone...

Описывается новый класс линейных асинхронных положительных систем с дискретным временем, для которых задачи устойчивости и стабилизируемости допускают конструктивное решение. Системы, образующие этот класс, могут рассматриваться как естественное обобщение систем с так называемыми независимо переключающимися координатами векторов состояния. Отличите...

The investigation of the asymptotic behavior of trigonometric series near the origin is a prominent topic in mathematical analysis. For trigonometric series in one variable, this problem was exhaustively studied by various authors in a series of publications dating back to the work of G. H. Hardy, 1928. Trigonometric series in several variables hav...

Recent results on the theory of joint spectral radius are presented

The story: Despot and Tribune rule a country, inhabited by People. Despot aims to minimize People’s freedom, Tribune aims to maximize it. Despot issues a decree (which respects laws!), permitting/restricting activities and changing system state. People are then given some choice of activities (like go to circus, enrol). After that Tribune has contr...

We prove the minimax equality for the spectral radius $\rho(AB)$ of the product of matrices $A\in\mathcal{A}$ and $B\in\mathcal{B}$, where $\mathcal{A}$ and $\mathcal{B}$ are compact sets of non-negative matrices of dimensions $N\times M$ and $M\times N$, respectively, satisfying the so-called hourglass alternative.

Two intimately related new classes of games are introduced and studied: entropy games (EGs) and matrix multiplication games (MMGs). An EG is played on a finite arena by two-and-a-half players: Despot, Tribune and the non-deterministic People. Despot wants to make the set of possible People's behaviors as small as possible, while Tribune wants to ma...

Recently Blondel, Nesterov and Protasov proved that the finiteness conjecture holds for the generalized and the lower spectral radii of the sets of non-negative matrices with independent row/column uncertainty. We show that this result can be obtained as a simple consequence of the so-called hourglass alternative which can be used also to establish...

Recently Blondel, Nesterov and Protasov proved that the finiteness conjecture holds for the generalized and the lower spectral radii of the sets of non-negative matrices with independent row/column uncertainty. We show that this result can be obtained as a simple consequence of the so-called hourglass alternative used by the author and his companio...

The investigation of the asymptotic behavior of trigonometric series near the origin is a prominent topic in mathematical analysis. For trigonometric series in one variable, this problem was exhaustively studied by various authors in a series of publications dating back to the work of G. H. Hardy, 1928. Trigonometric series in several variables hav...

Исследование асимптотического поведения тригонометрических рядов в окрестности нулевой точки представляет важный раздел математического анализа. Для тригонометрических рядов от одной переменной эта проблема была исчерпывающе изучена различными авторами в серии публикаций, восходящих к работе Дж. Харди 1928-го года. Тригонометрические ряды нескольки...

The efficient markets hypothesis implies that arbitrage opportunities in markets such as those for foreign exchange (FX) would be, at most, short-lived. The present paper surveys the fragmented nature of FX markets, revealing that information in these markets is also likely to be fragmented. The "quant " workforce in the hedge fund featured in The...

One of fundamental results of the theory of joint/generalized spectral radius, the Berger-Wang theorem, establishes equality between the joint and generalized spectral radii of a set of matrices. Generalization of this theorem on products of matrices whose factors are applied not arbitrarily but are subjected to some constraints is connected with e...

This is a talk communicated during my visit to the Nanjing University, May 20-25, 2014.

For an up-to-the-minute version of this file see: https://kozyakin.github.io/jsrbib/JSRbib.html or https://kozyakin.github.io/jsrbib/JSRbib.pdf This is an attempt to systematize my personal knowledge about convergence of infinite matrix products and the rate of their growth/decrease when the number of multipliers tends to infinity.

If financial markets displayed the informational efficiency postulated in the efficient markets hypothesis (EMH), arbitrage operations would be self-extinguishing. The present paper considers arbitrage sequences in foreign exchange (FX) markets, in which trading platforms and information are fragmented. In Kozyakin et al. (2010) and Cross et al. (2...

Discrete-time discrete-state random Markov chains with a tridiagonal generator are shown to have a random attractor consisting of singleton subsets, essentially a random path, in the simplex of probability vectors. The proof uses the Hilbert projection metric and the fact that the linear cocycle generated by the Markov chain is a uniformly contract...

Continuous-time discrete-state random Markov chains generated by a random linear differential equation with a random tridiagonal matrix are shown to have a random attractor consisting of singleton subsets, essentially a random path, in the simplex of probability vectors. The proof uses comparison theorems for Carath\'eodory random differential equa...

In the author's article ``Algebraic unsolvability of problem of absolute stability of desynchronized systems'' (Automat. Remote Control 51 (1990), no. 6, pp. 754--759), it was shown that in general for linear desynchronized systems there are no algebraic criteria of absolute stability. In this paper, a few misprints occurred in the original version...

The efficient markets hypothesis implies that arbitrage opportunities in markets such as those for foreign exchange (FX) would be, at most, short-lived. The present paper surveys the fragmented nature of FX markets, revealing that information in these markets is also likely to be fragmented. The "quant" Â workforce in the hedge fund featured in The...

This paper investigates arbitrage chains involving four currencies and four foreign exchange trader-arbitrageurs. In contrast with the three-currency case, we find that arbitrage operations when four currencies are present may appear periodic in nature, and not involve smooth convergence to a "balanced" ensemble of exchange rates in which the law o...

Semi-hyperbolicity extends the classical concept of hyperbolicity for diffeomorphisms to noninvertible Lipschitz mappings on sets that need not be invariant. Moreover, the stable and unstable manifolds need to be mapped only approximately onto their counterparts. Semi-hyperbolic mappings are bi-shadowing, i.e., they also satisfy a converse form o...

In the paper, a new method allowing to `asynchronously stabilize' a broad class of control systems is developed. Viability of the proposed approach is demonstrated by examples among which an application to stabilizing unmanned aerial vehicle systems. Pros and cons of the proposed approach are discussed.

In the Leijonhufvud synthesis, economies are self-adjusting within some ‗corridor‘, but not outside. In a previous paper Cross et. al. (2010) used combinatorial analysis to see if arbitrage sequences involve a smooth convergence onto an equilibrium in which the law of one price holds. They found that arbitrage sequences tend to be periodic in natur...

In the paper, a simple condition guaranteing the finiteness property, for a bounded set S = {S k }k ∈ K of real or complex d × d matrices, is presented. It is shown that existence of a sequence of matrix products , guarantees the spectral finiteness property for S.

This is a talk during my visits to the Nanjing University and Sun Yat-Sen (Zhongshan) University, June 3-8, 2011.

The problem of computation of the joint (generalized) spectral radius of matrix sets has been discussed in a number of publications. In the paper an iteration procedure is considered that allows to build numerically Barabanov norms for the irreducible matrix sets and simultaneously to compute the joint spectral radius of these sets.

Steady state solutions are important in characterizing the asymptotic behaviour of epidemi- ological systems such as the ubiquitous SIR system, but they need not exist when the coefficients vary with time. Recent developments in the theory of nonautonomous dynamical systems provide the appropriate counterpart, time varying nonautonomous equilibria,...

В статье предлагаются условия, гарантирующие конечную достижимость обобщенного спектрального радиуса ограниченного множества вещественных или комплексных матриц. Доказано, что существование последовательности матричных произведений, обладающих тем свойством, что спектр каждого произведения равномерно субпериферийный.

Дается обзор результатов последнего времени, связанных с теоретическим анализом идеи ``асинхронности'' и ее применением в некоторых прикладных задачах. Первая часть работы посвящена описанию новых качественных и численных методов оценки совместного спектрального радиуса семейств матриц. Приводятся эффективные оценки скорости приближения спектраль...

In various problems of control theory, non-autonomous and multivalued dynamical systems, wavelet theory and other fields of mathematics information about the rate of growth of matrix products with factors taken from some matrix set plays a key role. One of the most prominent quantities characterizing the exponential rate of growth of matrix product...

The goal of this article is to understand some interesting features of sequences of arbitrage operations, which look relevant to various processes in Economics and Finances. In the second part of the paper, analysis of sequences of arbitrages is reformulated in the linear algebra terms. This admits an elegant geometric interpretation of the problem...

The problem of construction of Barabanov norms for analysis of properties of the joint (generalized) spectral radius of matrix sets has been discussed in a number of publications. In previous papers of the author the method of Barabanov norms was the key instrument in disproving the Lagarias-Wang Finiteness Conjecture. The related constructions wer...

In theory of dynamical systems and nonlinear analysis quite a number of problems depending on parameters require analyzing the structure of the set of solutions of nonlinear equations, the number of variables in which exceeds the number of equations. As a rule, arising equations are rather complicated for investigation and need to be simplified in...

In the paper a set of necessary and sufficient conditions for \textit{v-}sufficiency (equiv. \textit{sv-}sufficiency) of jets of map-germs $f:(\mathbb{R}^{n},0)\to (\mathbb{R}^{m},0)$ is proved which generalize both the Kuiper-Kuo and the Thom conditions in the function case ($m=1$) so as the Kuo conditions in the general map case ($m>1$). Contrary...

In 2002, Wirth has proved that the joint spectral radius of irreducible compact sets of matrices is locally Lipschitz continuous as a function of the matrix set. In the paper, an explicit formula for the related Lipschitz constant is obtained.

The famous Gelfand formula $\rho(A)= \limsup_{n\to\infty}\|A^{n}\|^{1/n}$ for the spectral radius of a matrix is of great importance in various mathematical constructions. Unfortunately, the range of applicability of this formula is substantially restricted by a lack of estimates for the rate of convergence of the quantities $\|A^{n}\|^{1/n}$ to $\...

In 2002 F. Wirth has proved that the joint spectral radius of irreducible compact sets of matrices is locally Lipschitz continuous as a function of the matrix set. In the paper, an explicit formula for the related Lipschitz constant is obtained.

One of the most prominent tool to compute the the joint spectral radius of a matrix set is the so-called generalized Gelfand formula which represents the joint spectral radius as a limit of the weighted norms of matrix products with factors taken from the given matrix set. Unfortunately, the range of applicability of this formula is substantially r...

Различные задачи теории неавтономных и многозначных линейных динамических систем, теории вейвлетов и многих других разделов математики приводят к необхо-димости анализа скорости роста матричных произведений с сомножителями из некоторого набора матриц. Одной из характеристик, описывающих экспоненциальную скорость роста произведений матриц, является...

It is shown how known results for autonomous difference equations can be adapted to definitions of semi-hyperbolicity and bi-shadowing generalized to nonautonomous difference equations with Lipschitz continuous mappings. In particular, invertibility and smoothness of the mappings are not required and, for greater applicability, the mappings are all...

We consider discrete time systems $x_{k+1}=U(x_{k};\lambda)$, $x\in R^{N}$, with a complex parameter $\lambda$, and study their trajectories of large amplitudes. The expansion of the map at infinity contains a principal linear term, a bounded positively homogeneous nonlinearity, and a smaller vanishing part. We study Arnold tongues: the sets of par...

We consider discrete time systems x k+1 = U (x k ; λ), x ∈ R N , with a complex parameter λ. The map U (·; λ) at infinity contains a principal linear term, a bounded positively homogeneous nonlinearity, and a smaller part. We describe the sets of parameter values for which the large-amplitude n-periodic trajectories exist for a fixed n. In the rela...

Различные задачи теории управления, неавтономных и многозначных линейных динамических систем, теории вейвлетов и многих других разделов математики приводят к необходимости анализа скорости роста матричных произведений с сомножителями, взятыми из некоторого набора матриц. Одной из характеристик, описывающих экспоненциальную скорость роста произведен...

Исследовано значение теоремы В. А. Котельникова для решения практических задач восстановления непрерывных сигналов по дискретным отсчетам. Рассмотрены вопросы потери информации при дискретизации непрерывных сигналов по времени и в пространстве.

In the paper some problems related to computer studies of continuous objects are considered. Examples are presented which demonstrate that even in trivial situations results of computer modelling can differ drastically from properties of original continuous objects. This observation is aggravated by the fact that often the situation cannot be impro...

В предлагаемой обзорной статье обсуждаются вопросы, связанные с проблемами компьютерного моделирования непрерывных объектов. Приводятся примеры, показывающие, что даже в самых, казалось бы, тривиальных ситуациях результаты компьютерного моделирования могут кардинально отличаться от свойств исходных непрерывных объектов. Данный факт усугубляется тем...

Consider a finite-dimensional discrete time dynamical system $x_{k+1} =U_{\lambda}x_k$, $x\in\mathbb{R}^{n}$ depending on a complex parameter $\lambda$. In a neighbourhood of an equilibrium (say, at $x_* =0$) its properties are determined by its linear part $A_{\lambda}x$. If the matrix $A_{\lambda}$ has eigenvalues near the unit circle, then in a...

In 1995 J. C. Lagarias and Y. Wang conjectured that the generalized spectral radius of a finite set of matrices can be attained on a finite product of matrices. The first counterexample to this Finiteness Conjecture was given in 2002 by T. Bousch and J. Mairesse. In 2003 V. D. Blondel, J. Theys and A. A. Vladimirov proposed another proof of a count...

В 1995г. Дж. Лагариас и Янг Ванг высказали предположение о том, что обобщенный спектральный радиус конечного набора матриц всегда достигается на некотором конечном произведении матриц. Первый контрпример к этой ``гипотезе о конечности'' был построен в 2002г. Т. Бушем и Ж. Мерессом, а соответствующее доказательство существенно опиралось на конструкц...

The theory of circle homeomorphisms has a great number of deep results. However, sometimes continuity or single-valuedness of a circle map may be restrictive in theoretical constructions or applications. In this paper it is shown that some principal properties of circle homeomorphisms are inherited by the class of orientation-preserving circle maps...

In 1995 J.C. Lagarias and Y. Wang conjectured that the generalized spectral radius of a finite set of square matrices can be attained on a finite product of matrices. The first counterexample to this Finiteness Conjecture was given in 2002 by T. Bousch and J. Mairesse and their proof was based on measure-theoretical ideas. In 2003 V.D. Blondel, J....

В 1995г. Дж. Лагариас и Янг Ванг высказали предположение о том, что обобщенный спектральный радиус конечного набора матриц всегда достигается на некотором конечном произведении матриц. Первый контрпример к этой ``гипотезе о конечности'' был построен в 2002г. Т. Бушем и Ж. Мерессом, а соответствующее доказательство существенно опиралось на конструкц...

В 1995г. Дж. Лагариас и Янг Ванг высказали предположение о том, что обобщенный спектральный радиус конечного набора матриц всегда достигается на некотором конечном произведении матриц. Первый контрпример к этой ``гипотезе о конечности'' был построен в 2002г. Т. Бушем и Ж. Мерессом, а соответствующее доказательство существоенно опиралось на конструк...

Известно, что гомеоморфизмы окружности обладают множеством нетривиальных и сильных свойств. В то же время требование непрерывности отображения окружности в ряде приложений может оказаться ограничительным. В связи с этим актуальность приобретает вопрос о выделении такого класса отображений окружности, который с одной стороны был бы достаточно широки...

Nonautonomous difference equations are formulated as difference cocycles driven by an autonomous dynamical system. Pullback attractors are the appropriate generalization of autonomous attractors to cocycles and their existence follows when the difference cocycle has a pullback absorbing set. The effects of perturbing the driving autonomous system o...

A nonautonomous or cocycle dynamical system that is driven by an autonomous dynamical system acting on a compact metric space is assumed to have a uniform pullback attractor. It is shown that discretization by a one-step numerical scheme gives rise to a discrete time cocycle dynamical system with a uniform pullback attractor, the component subsets...

Looking at the dynamics of the system described by the equation\[ x(n+1)=f(x(n)) \] one may say that coordinates of the vector $x=\{x_{1},x_{2},\ldots,x_{N}\}$ are updated \emph{synchronously}. What happens with the system if coordinates of the vector $x$ are updated \emph{asynchronously}, i.e., if at a given moment $n$ only coordinates with indice...

Looking at the dynamics of the system described by the equation x(n + 1) = f (x(n)) one may say that coordinates of the vector x = {x 1 , x 2 , . . . , x N } are updated synchronously. What happens with the system if co-ordinates of the vector x are updated asynchronously, i.e., if at a given moment n only coordinates with indices i from some set ω...

The theory of circle homeomorphisms has a great number of deep results. However, sometimes continuity of a circle map may be restrictive in theoretical constructions or applications. In this paper it is shown that some principal properties of circle homeomorphisms are inherited by the class of order preserving circle maps. The latter class is rathe...

Рассматривается задача о сходимости, ограниченности или неограниченности множества всех возможных произведений матриц с сомножителями из некоторой конечной совокупности, к которой сводятся многие вопросы теории управления и математики. Установлена неопределимость данной задачи в o-минимальных структурах, содержащих полуалгебраические множества, что...

There are plenty of known non-trivial properties of circle home-omorphisms. Sometimes, continuity of a circle map may be re-strictive in applications. Therefore, the problem of distinguishing a class of circle maps retaining as much properties of homeo-morphisms as possible while remaining rather broad is urgent. Clearly, by discarding continuity o...

This paper is concerned with the convergence and boundedness or unboundedness of the set of all possible matrix products with coefficients belonging to some finite set, i.e., the problem to which many problems of control theory and mathematics are reduced. The indefinability of this problem in o-minimal structures containing semialgebraic sets, whi...

The problem is considered about convergence, boundedness or unboundedness of a set of all possible matrices product wish multipliers from some finite set to which many problems of the control theory and mathematics are reduced. The problem indeterminability is found in o-minimal structures containing semi-algebraic sets. It is shown that the proble...

The influence of the driving system on a skew-product flow generated by a triangular system of differential equations can be perturbed in two ways, directly by perturbing the vector field of the driving system component itself or indirectly by perturbing its input variable in the vector field of the coupled component. The effect of such perturbatio...

The problem on asymptotic of the value π(m,n)=m!σm(p(1,n),p(2,n),…,p(n,n)) is considered, where σm(x1,x2,…,xn) is the mth elementary symmetric function of n variables. The result is interpreted in the context of nonequiprobable random mappings theory.

Discrete time nonautonomous dynamical systems generated by nonautonomous difference equations are formulated as discrete time skew—product systems consisting of cocycle state mappings that are driven by discrete time autonomous dynamical systems. Forwards and pullback attractors are two possible generalizations of autonomous attractors to such syst...

Many original ideas and techniques that were proposed, developed and applied by Mark Alexandrovich Krasnosel'skii during his long career as a mathematician are being used repeatedly today throughout mathematics, often without knowledge or acknowledgement of their source. In this paper we will see how useful several of his ideas and techniques are i...

In this paper we introduce the concept of inflation of an attractor, which will give us a larger positively invariant set for the original system that contains the maximal attractors of all possible autonomous perturbations and discretizations of a certain size, as well as even the attracting objects of corresponding nonautonomous perturbations and...

Description of goals and achievments of the laboratory