arXiv subjects

Vladimir Yu. Protasov

Publications and source records attributed to Vladimir Yu. Protasov.

At least 19 recordsLinked to original sources

arXiv2025

Unique expansions in number systems via refinement equations

Using the subdivision schemes theory, we develop a criterion to check if any natural number has at most one representation in the $n$-ary number system with a set of non-negative integer digits $A=\{a_1, a_2,\ldots, a_n\}$ that contains zero. This uniqueness property is shown to be equivalent to a certain restriction on the roots of the trigonometric polynomial $\sum_{k=1}^n e^{-2πi a_k t}$. From this criterion, under a natural condition of irreducibility for $A$, we deduce that in case of prime $n$ the uniqueness holds if and only if the digits of $A$ are distinct modulo $n$, whereas for any composite $n$ we show that the latter condition is not necessary. We also establish the connection of this uniqueness to the semigroup freeness problem for affine integer functions of equal integer slope; this together with the two criteria allows to fill the gap in the work of D. Klarner on the question of P. Erdös about densities of affine integer orbits and establish a simple algorithm to check the freeness and the positivity of density when the slope is a prime number.

math.NT↗

arXiv2025

Chebyshev approximation by non-Chebyshev systems

We address the problem of the best uniform approximation by linear combinations of a finite system of functions. If the system is Chebyshev and the problem is unconstrained, then the classical Remez algorithm provides a fast and precise solution. For non-Chebyshev systems, this problem may offer a great resistance. The same happens to approximations under linear constraints. We propose a solution by modifying the concept of alternance and of the Remez iterative procedure. A criterion of the best approximation is proved and the full set of polynomials of best approximation (which may not be unique in the non-Chebyshev case) is characterized. The method of finding the best polynomial is applicable for arbitrary functional systems under arbitrary linear constraints. The efficiency is demonstrated in examples with systems of complex exponents, Gaussian functions, and lacunar polynomials. As an application, the Markov-Bernstein type inequalities are obtained for those systems. Applications to signal processing, linear ODEs, switching dynamical systems are considered.

math.NA↗

arXiv2025

Second-order linear switching systems with arbitrary control sets: stability and invariant norms

We show that the stability problem and the problem of constructing Barabanov norms can be resolved for planar linear switching systems in an explicit form. This can be done for every compact control set of $2 \times 2$ matrices. If the control set does not contain a dominant matrix with a real spectrum, then the invariant norm is always unique (up to a multiplier) and belongs to~$C^1$. Otherwise, there may be infinitely many such norms, including non-smooth ones. All of them can be found and classified. In particular, every symmetric convex body is a unit ball of the Barabanov norm of a suitable linear switching system. Several examples of control sets such as matrix Frobenius balls and matrix polyhedra are analysed.

math.FA↗

arXiv2025

Perron matrix semigroups

We consider multiplicative semigroups of real dxd matrices. A semigroup S is called Perron if each of its matrices has a Perron eigenvalue, i.e., an eigenvalue equal to the spectral radius. If all matrices of S leave a proper convex cone invariant, then S is Perron. Our main result asserts the converse: every irreducible Perron semigroup possesses a common invariant cone, provided that some mild assumptions are satisfied. This gives conditions for a set of matrices to share a common invariant cone, which is an important property widely studied in the literature. Then we address the problem to characterize the exceptions, when a Perron semigroup does not have an invariant cone. For d\le 4, all Perron semigroups are classified. For higher dimensions~$d$, several classes of such semigroups are found.

math.RA↗

arXiv2024

Autopolar conic bodies and polyhedra

An antinorm is a concave analogue of a norm. In contrast to norms, antinorms are not defined on the entire space $R^d$ but on a cone $K\subset R^d$. They are applied in the matrix analysis, optimal control, and dynamical systems. Their level sets are called conic bodies and (in case of piecewise-linear antinorms) conic polyhedra. The basic facts and notions of the "concave analysis" of antinorms such as separation theorems, duality, polars, Minkowski functionals, etc., are similar to those from the standard convex analysis. There are, however, some significant differences. One of them is the existence of many self-dual objects. We prove that there are infinitely many families of autopolar conic bodies and polyhedra in the cone $K=R^d_+$. For $d=2$, this gives a complete classification of self-dual antinorms, while for $d\ge 3$, there are counterexamples.

math.MG↗

arXiv2024

How do the lengths of switching intervals influence the stability of a dynamical system?

If a linear switching system with frequent switches is stable, will it be stable under arbitrary switches? In general, the answer is negative. Nevertheless, this question can be answered in an explicit form for any concrete system. This is done by finding the mode-dependent critical lengths of switching intervals after which any enlargement does not influence the stability. The solution is given in terms of the exponential polynomials of least deviation from zero on a segment (``Chebyshev-like'' polynomials). By proving several theoretical results on exponential polynomial approximation we derive an algorithm for finding such polynomials and for computing the critical switching time. The convergence of the algorithm is estimated and numerical results are provided.

math.OC↗

arXiv2024

Stability under dwell time constraints: Discretization revisited

We decide the stability and compute the Lyapunov exponent of continuous-time linear switching systems with a guaranteed dwell time. The main result asserts that the discretization method with step size~$h$ approximates the Lyapunov exponent with the precision~$C\,h^2$, where~$C$ is a constant. Let us stress that without the dwell time assumption, the approximation rate is known to be linear in~$h$. Moreover, for every system, the constant~$C$ can be explicitly evaluated. In turn, the discretized system can be treated by computing the Markovian joint spectral radius of a certain system on a graph. This gives the value of the Lyapunov exponent with a high accuracy. The method is efficient for dimensions up to, approximately, ten; for positive systems, the dimensions can be much higher, up to several hundreds.

math.DS↗

arXiv2023

Anisotropic refinable functions and the tile B-splines

The regularity of refinable functions has been analysed in an extensive literature and is well-understood in two cases: 1) univariate 2) multivariate with an isotropic dilation matrix. The general (non-isotropic) case offered a great resistance. It was done only recently by developing the matrix method. In this paper we make the next step and extend the Littlewood-Paley type method, which is very efficient in the aforementioned special cases, to general equations with arbitrary dilation matrices. This gives formulas for the higher order regularity in $W_2^k(\mathbb{R}^n)$ by means of the Perron eigenvalue of a finite-dimensional linear operator on a special cone. Applying those results to recently introduced tile B-splines, we prove that they can have a higher smoothness than the classical ones of the same order. Moreover, the two-digit tile B-splines have the minimal support of the mask among all refinable functions of the same order of approximation. This proves, in particular, the lowest algorithmic complexity of the corresponding subdivision schemes. Examples and numerical results are provided.

math.FA↗

arXiv2023

Closed simple geodesics on a polyhedron

It is well-known that every isosceles tetrahedron (disphenoid) admits infinitely many simple closed geodesics on its surface. They can be naturally enumerated by pairs of co-prime integers $n > m > 1$ with two additional cases $(1,0)$ and $(1,1)$. The (n,m)-geodesic is a broken line with $4(n+m)$ vertices, its length tends to infinity as $m\to \infty$. Are there other polyhedra possessing this property? The answer depends on convexity. We give an elementary proof that among convex polyhedra only disphenoids admit arbitrarily long closed simple geodesics. For non-convex polyhedra, this is not true. We present a counterexample with the corresponding polyhedron being a union of seven equal cubes. Several open problems are formulated

math.MG↗

arXiv2022

The length of switching intervals of a stable linear system

The linear switching system is a system of ODE with the time-dependent matrix taking values from a given control matrix set. The system is (asymptotically) stable if all its trajectories tend to zero for every control function. We consider possible mode-dependent restrictions on the lengths of switching intervals which keeps the stability of the system. When the stability of trajectories with short switching intervals implies the stability of all trajectories? To answer this question we introduce the concept of "cut tail points" of linear operators and study them by the convex analysis tools. We reduce the problem to the construction of Chebyshev-type exponential polynomials, for which we derive an algorithm and present the corresponding numerical results.

math.OC↗

arXiv2022

Generalized Markov-Bernstein inequalities and stability of dynamical systems

The Markov-Bernstein type inequalities between the norms of functions and of their derivatives are analysed for complex exponential polynomials. We establish a relation between the sharp constants in those inequalities and the stability problem for linear switching systems. In particular, the maximal discretization step is estimated. We prove the monotonicity of the sharp constants with respect to the exponents, provided those exponents are real. This gives asymptotically tight uniform bounds and the general form of the extremal polynomial. The case of complex exponent is left as an open problem.

math.FA↗

arXiv2022

Stability of linear systems with bounded switching intervals

We address the stability problem for linear switching systems with mode-dependent restrictions on the switching intervals. Their lengths can be bounded as from below (the guaranteed dwell-time) as from above. The upper bounds make this problem quite different from the classical case: a stable system can consist of unstable matrices, it may not possess Lyapunov functions, etc. We introduce the concept of Lyapunov multifunction with discrete monotonicity, which gives upper bounds for the Lyapunov exponent. Its existence as well as the existence of invariant norms are proved. Tight lower bounds are obtained in terms of a modified Berger-Wang formula over periodizable switching laws. Based on those results we develop a method of computation of the Lyapunov exponent with an arbitrary precision and analyse its efficiency in numerical results. The case when some of upper bounds can be cancelled is analysed.

math.OC↗

arXiv2021

Surface dimension, tiles, and synchronising automata

We study the surface regularity of compact sets $G \subset R^n$ which is equal to the supremum of numbers $s\ge 0$ such that the measure of the set $G_{\varepsilon}\setminus G$ does not exceed $C\varepsilon^{s}, \varepsilon > 0$, where $G_{\varepsilon}$ denotes the $\varepsilon$-neighbourhood of~$G$. The surface dimension is by definition the difference between~$n$ and the surface regularity. Those values provide a natural characterisation of regularity for sets of positive measure. We show that for self-affine attractors and tiles those characteristics are explicitly computable and find them for some popular tiles. This, in particular, gives a refined regularity scale for the multivariate Haar wavelets. The classification of attractors of the highest possible regularity is addressed. The relation between the surface regularity and the Hölder regularity of multivariate refinable functions and wavelets is found. Finally, the surface regularity is applied to the theory of synchronising automata, where it corresponds to the concept of parameter of synchronisation.

math.CA↗

arXiv2021

Antinorms on cones: duality and applications

An antinorm is a concave nonnegative homogeneous functional on a convex cone. It is shown that if the cone is polyhedral, then every antinorm has a unique continuous extension from the interior of the cone. The main facts of the duality theory in convex analysis, in particular, the Fenchel-Moreau theorem, are generalized to antinorms. However, it is shown that the duality relation for antinorms is discontinuous. In every dimension there are infinitely many self-dual antinorms on the positive orthant and, in particular, infinitely many autopolar polyhedra. For the two-dimensional case, we characterise them all. The classification in higher dimensions is left as an open problem. Applications to linear dynamical systems, to the Lyapunov exponent of random matrix products, to the lower spectral radius of nonnegative matrices, and to convex trigonometry are considered.

math.MG↗

arXiv2021

The Barabanov norm is generically unique, simple, and easily computed

Every irreducible discrete-time linear switching system possesses an invariant convex Lyapunov function (Barabanov norm), which provides a very refined analysis of trajectories. Until recently that notion remained rather theoretical apart from special cases. In 2015 N.Guglielmi and M.Zennaro showed that many systems possess at least one simple Barabanov norm, which moreover, can be efficiently computed. In this paper we classify all possible Barabanov norms for discrete-time systems. We prove that, under mild assumptions, such norms are unique and are either piecewise-linear or piecewise quadratic. Those assumptions can be verified algorithmically and the numerical experiments show that a vast majority of systems satisfy them. For some narrow classes of systems, there are more complicated Barabanov norms but they can still be classified and constructed. Using those results we find all trajectories of the fastest growth. They turn out to be eventually periodic with special periods. Examples and numerical results are presented.

math.OC↗

arXiv2021

Tiling of polyhedral sets

A self-affine tiling of a compact set G of positive Lebesgue measure is its partition to parallel shifts of a compact set which is affinely similar to G. We find all polyhedral sets (unions of finitely many convex polyhedra) that admit self-affine tilings. It is shown that in R^d there exist an infinite family of such polyhedral sets, not affinely equivalent to each other. A special attention is paid to an important particular case when the matrix of affine similarity and the translation vectors are integer. Applications to the approximation theory and to the functional analysis are discussed.

math.MG↗

arXiv2021

Elliptic polytopes and invariant norms of linear operators

We address the problem of constructing elliptic polytopes in R^d, which are convex hulls of finitely many two-dimensional ellipses with a common center. Such sets arise in the study of spectral properties of matrices, asymptotics of long matrix products, in the Lyapunov stability, etc.. The main issue in the construction is to decide whether a given ellipse is in the convex hull of others. The computational complexity of this problem is analysed by considering an equivalent optimisation problem. We show that the number of local extrema of that problem may grow exponentially in d. For d=2,3, it admits an explicit solution for an arbitrary number of ellipses; for higher dimensions, several geometric methods for approximate solutions are derived. Those methods are analysed numerically and their efficiency is demonstrated in applications.

math.NA↗

arXiv2020

Self-affine 2-attractors and tiles

We study two-digit attractors (2-attractors) in $\mathbb{R}^d$ which are self-affine compact sets defined by two contraction affine mappings with the same linear part. They are widely studied in the literature under various names: twindragons, two-digit tiles, 2-reptiles, etc., due to many applications in approximation theory, in the construction of multivariate Haar systems and other wavelet bases, in the discrete geometry, and in the number theory. We obtain a complete classification of isotropic 2-attractors in $\mathbb{R}^d$ and show that they are all homeomorphic but not diffeomorphic. In the general, non-isotropic, case it is proved that a 2-attractor is uniquely defined, up to an affine similarity, by the spectrum of the dilation matrix. We estimate the number of different 2-attractors in $\mathbb{R}^d$ by analysing integer unitary expanding polynomials with the free coefficient $\pm 2$. The total number of such polynomials is estimated by the Mahler measure. We present several infinite series of such polynomials. For some of the 2-attractors, their Hölder exponents are found. Some of our results are extended to attractors with an arbitrary number of digits.

math.FA↗