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Victor Kozyakin

Publications and source records attributed to Victor Kozyakin.

At least 19 recordsLinked to original sources
arXiv2025

Notes on Simplifying the Construction of Barabanov Norms

To answer the question about the growth rate of matrix products, the concepts of joint and generalized spectral radius were introduced in the 1960s. A common tool for finding the joint/generalized spectral radius is the so-called extremal norms and, in particular, the Barabanov norm. The goal of this paper is to try to combine the advantages of different approaches based on the concept of extremality in order to obtain results that are simpler for everyday use. It is shown how the Dranishnikov-Konyagin theorem on the existence of a special invariant body for a set of matrices can be used to construct a Barabanov norm. A modified max-relaxation algorithm for constructing Barabanov norms, which follows from this theorem, is described. Additional techniques are also described that simplify the construction of Barabanov norms under the assumption that

math.RA↗
arXiv2024

On pairs of spectrum maximizing products with distinct factor multiplicities

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 entails the existence of another spectrum maximizing product. Moreover, in addition to Bochi--Laskawiec's example, the number of factors of the same name (factors of the form $A$ or $B$) in these matrix products turns out to be different. The efficiency of the proposed approach is confirmed by constructing an example of a set of $2\times2$ matrices $\{A,B\}$ that has spectrum maximizing products of the form $BAA$ and $BBA$.

math.OC↗
arXiv2022

Non-Sturmian sequences of matrices providing the maximum growth rate of matrix products

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 $A_{σ_{n}}\cdots A_{σ_{0}}$ with factors from a set of matrices $\mathscr{A}$ arises. So far, only for a relatively small number of classes of matrices $\mathscr{A}$ has it been possible to accurately describe the sequences of matrices that guarantee the maximum rate of increase of the corresponding norms. Moreover, in almost all cases studied theoretically, the index sequences $\{σ_{n}\}$ of matrices maximizing the norms of the corresponding matrix products have been shown to be periodic or so-called Sturmian, which entails a whole set of "good" properties of the sequences $\{A_{σ_{n}}\}$, in particular the existence of a limiting frequency of occurrence of each matrix factor $A_{i}\in\mathscr{A}$ in them. In the paper it is shown that this is not always the case: a class of matrices is defined consisting of two $2\times 2$ matrices, similar to rotations in the plane, in which the sequence $\{A_{σ_{n}}\}$ maximizing the growth rate of the norms $\|A_{σ_{n}}\cdots A_{σ_{0}}\|$ is not Sturmian. All considerations are based on numerical modeling and cannot be considered mathematically rigorous in this part; rather, they should be interpreted as a set of questions for further comprehensive theoretical analysis.

math.RA↗
arXiv2021

On the boundedness of infinite matrix products with alternating factors from two sets of matrices

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 the sequence of matrix products $\{A_{n}B_{n}\cdots A_{1}B_{1}\}_{n=1}^{\infty}$ is norm bounded. Some situations are described in which in this case the norms of matrix products $A_{n}B_{n}\cdots A_{1}B_{1}$ are uniformly bounded, that is, $\|A_{n}B_{n}\cdots A_{1}B_{1}\|\le C$ for all natural numbers $n$, where $C>0$ is some constant independent of the sequence $\{A_{i}\}$ and the corresponding sequence $\{B_{i}\}$. In the general case, the question of the validity of the corresponding statement remains open.

math.RA↗
arXiv2017

Minimax joint spectral radius and stabilizability of discrete-time linear switching control systems

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 $ρ(\mathcal{A})$ and the lower spectral radius $\checkρ(\mathcal{A})$ are traditionally used. The first of these quantities characterizes the maximum growth rate of the norms of the corresponding products, while the second one characterizes the minimal growth rate. In the theory of discrete-time linear switching systems, the inequality $ρ(\mathcal{A})<1$ serves as a criterion for the stability of a system, and the inequality $\checkρ(\mathcal{A})<1 $ as a criterion for stabilizability. For matrix products $A_{n}B_{n}\cdots A_{1}B_{1}$ with factors $A_{i}\in\mathcal{A}$ and $B_{i}\in\mathcal{B}$, where $\mathcal{A}$ and $\mathcal{B}$ are some sets of matrices, we introduce the quantities $μ(\mathcal{A},\mathcal{B})$ and $η(\mathcal{A},\mathcal{B})$, called the lower and upper minimax joint spectral radius of the pair $\{\mathcal{A},\mathcal{B}\}$, respectively, which characterize the maximum growth rate of the matrix products $A_{n}B_{n}\cdots A_{1}B_{1}$ over all sets of matrices $A_{i}\in\mathcal{A}$ and the minimal growth rate over all sets of matrices $B_{i}\in\mathcal{B}$. In this sense, the minimax joint spectral radii can be considered as generalizations of both the joint and lower spectral radii. As an application of the minimax joint spectral radii, it is shown how these quantities can be used to analyze the stabilizability of discrete-time linear switching control systems in the presence of uncontrolled external disturbances of the plant.

math.OC↗
arXiv2017

On convergence of infinite matrix products with alternating factors from two sets of matrices

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 the corresponding matrix product $A_{n}B_{n}\cdots A_{1}B_{1}$ converges to zero. We show that in this case the convergence of the matrix products under consideration is uniformly exponential, that is, $\|A_{n}B_{n}\cdots A_{1}B_{1}\|\le C\lambda^{n}$, where the constants $C>0$ and $\lambda\in(0,1)$ do not depend on the sequence $\{A_{i}\}$ and the corresponding sequence $\{B_{i}\}$.

math.OC↗
arXiv2016

Minimax theorem for the spectral radius of the product of non-negative matrices

We prove the minimax equality for the spectral radius $ρ(AB)$ 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.

math.RA↗
arXiv2016

Constructive stability and stabilizability of positive linear discrete-time switching systems

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 components or blocks can be arbitrarily connected in parallel or in series without loss of the `constructive resolvability' property. It is shown also that, for such systems, it is possible to build constructively the individual positive trajectories with the greatest or the lowest rate of convergence to the zero.

math.OC↗
arXiv2015

Entropy Games and Matrix Multiplication Games

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 make it as large as possible.An MMG is played by two players that alternately write matrices from some predefined finite sets. One wants to maximize the growth rate of the product, and the other to minimize it. We show that in general MMGs are undecidable in quite a strong sense.On the positive side, EGs correspond to a subclass of MMGs, and we prove that such MMGs and EGs are determined, and that the optimal strategies are simple. The complexity of solving such games is in NP\&coNP.

cs.GT↗
arXiv2015

Hourglass alternative and the finiteness conjecture for the spectral characteristics of sets of non-negative matrices

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 earlier used by the author and his companions to analyze the minimax relations between the spectral radii of matrix products. Axiomatization of the statements that constitute the hourglass alternative makes it possible to define a new class of sets of positive matrices having the finiteness property, which includes the sets of non-negative matrices with independent row uncertainty. This class of matrices, supplemented by the zero and identity matrices, forms a semiring with the Minkowski operations of addition and multiplication of matrix sets, which gives means to construct new sets of non-negative matrices possessing the finiteness property for the generalized and the lower spectral radii.

math.RA↗
arXiv2015

Hardy type asymptotics for cosine series in several variables with decreasing power-like coefficients

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 have got less attention. The aim of the work is to partially fill this gap by finding the asymptotics of trigonometric series in several variables with the terms, having a form of `one minus the cosine' up to a decreasing power-like factor: \[ \sum_{z\in\mathbb{Z}^{d}\setminus\{0\}}\frac{1}{\|z\|^{d+α}}\left(1-\cos\langle z,θ\rangle\right), \qquad θ\in\mathbb{R}^{d}, \] where $\langle\cdot,\cdot\rangle$ is the standard inner product and $\|\cdot\|$ is the max-norm on $\mathbb{R}^{d}$. The approach developed in the paper is quite elementary and essentially algebraic. It does not rely on the classic machinery of the asymptotic analysis such as slowly varying functions, Tauberian theorems or the Abel transform. However, in our case, it allows to obtain explicit expressions for the asymptotics and to extend to the general case $d\ge 1$ classical results of G. H. Hardy and other authors known for $d=1$.

math.CA↗
arXiv2014

Matrix products with constraints on the sliding block relative frequencies of different factors

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 essential difficulties since known proofs of the Berger-Wang theorem rely on the arbitrariness of appearance of different matrices in the related matrix products. Recently, X. Dai proved an analog of the Berger-Wang theorem for the case when factors in matrix products are formed by some Markov law. We introduce the concepts of the joint and generalized spectral radii for products of matrices subjected to constraints on the sliding block relative frequencies of occurrences of different matrices, and prove an analog of the Berger-Wang theorem for this case.

math.RA↗
arXiv2014

The Berger-Wang formula for the Markovian joint spectral radius

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 formula hardly can be directly applied in the case of Markovian products of matrices since they essentially rely on the arbitrariness of appearance of different matrices in the related matrix products. Nevertheless, as has been shown by X. Dai the Berger-Wang formula is valid for the case of Markovian analogs of the joint and the generalized spectral radii too, although the proof in this case heavily exploits the more involved techniques of multiplicative ergodic theory. In the paper we propose a matrix theory construction allowing to deduce the Markovian analog of the Berger-Wang formula from the classical Berger-Wang formula.

math.RA↗
arXiv2012

Double Exponential Instability of Triangular Arbitrage Systems

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. (2012) it was shown that sequences of triangular arbitrage operations in FX markets containing 4 currencies and trader-arbitrageurs tend to display periodicity or grow exponentially rather than being self-extinguishing. This paper extends the analysis to 5 or higher-order currency worlds. The key findings are that in a 5-currency world arbitrage sequences may also follow an exponential law as well as display periodicity, but that in higher-order currency worlds a double exponential law may additionally apply. There is an "inheritance of instability" in the higher-order currency worlds. Profitable arbitrage operations are thus endemic rather that displaying the self-extinguishing properties implied by the EMH.

q-fin.GN↗
arXiv2011

Periodic Sequences of Arbitrage: A Tale of Four Currencies

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 of one price holds. The goal of this article is to understand some interesting features of sequences of arbitrage operations, features which might well be relevant in other contexts in finance and economics.

q-fin.GN↗
arXiv2011

Finiteness Property of a Bounded Set of Matrices with Uniformly Sub-Peripheral Spectrum

In the paper, a simple condition guaranteing the finiteness property for a bounded set of matrices is presented. Given a bounded set S of real or complex matrices, it is shown that existence of a sequence of matrix products such that the spectrum of each matrix in this sequence is uniformly sub-peripheral and tends to the joint spectral radius of S, guarantees the spectral finiteness property for S.

math.FA↗
arXiv2011

Polynomial reformulation of the Kuo criteria for v-sufficiency of map-germs

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 to the Kuo conditions the conditions proved in the paper do not require to verify any inequalities in a so-called horn-neighborhood of the (a'priori unknown) set $f^{-1}(0)$. Instead, the proposed conditions reduce the problem on \textit{v-}sufficiency of jets to evaluating the local Łojasiewicz exponents for some constructively built polynomial functions.

math.AG↗