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In a previous paper (Funct. Anal. Appl., 3 (2015), 205–209), we defined quantum Markov states. Here we recall this definition and present a proof of the results from that paper (which are given there without proofs). We give a definition of a quantum hidden Markov state generated by a function of a quantum Markov process and show how it is related to other definitions of such states. Our definitions work for quantum Markov fields on ℤN and on graphs. We consider an example with the Cayley tree.

A map (S,G) is a closed Riemann surface S with embedded graph G such that S \G is the disjoint union of connected components, called faces, each of which is homeomorphic to an open disk. Tutte began a systematic study of maps in the 1960s and contemporary authors are actively developing it. In the present paper, after recalling the concept of a circular map introduced by the author and Mednykh, a relationship between bipartite maps and circular maps is demonstrated via the concept of the duality of maps. In this way an enumeration formula for the number of bipartite maps with a given number of edges is obtained. A hypermap is a map whose vertices are colored black and white in such a way that every edge connects vertices of different colors. The hypermaps are also known as dessins d’enfants (or Grothendieck’s dessins). A hypermap is self-equivalent with respect to reversing the colors of vertices if it is equivalent to the hypermap obtained by reversing the colors of its vertices. The main result of the present paper is an enumeration formula for the number of unrooted hypermaps, regardless of genus, which have n edges and are self-equivalent with respect to reversing the colors of vertices. Bibliography: 13 titles.

The problem of characterization of integrals as linear functionals is considered in the paper. It starts from the familiar results of F. Riesz (1909) and J. Radon (1913) on integral representation of bounded linear functionals by Riemann–Stieltjes integrals on a segment and by Lebesgue integrals on a compact in \( {\mathbb{R}^n} \), respectively. After works of J. Radon, M. Frechet, and F. Hausdorff the problem of characterization of integrals as linear functionals took the particular form of the problem of extension of Radon’s theorem from \( {\mathbb{R}^n} \) to more general topological spaces with Radon measures. This problem has turned out difficult and its solution has a long and rich history. Therefore, it may be naturally called the Riesz–Radon–Frechet problem of characterization of integrals. The important stages of its solution are connected with such mathematicians as S. Banach, S. Saks, S. Kakutani, P. Halmos, E. Hewitt, R. E. Edwards, N. Bourbaki, V. K. Zakharov, A. V. Mikhalev, et al. In this paper, the Riesz–Radon–Fr´echet problem is solved for the general case of arbitrary Radon measures on Hausdorff spaces. The solution is given in the form of a general parametric theorem in terms of a new notion of the boundedness index of a functional. The theorem implies as particular cases well-known results of the indicated authors characterizing Radon integrals for various classes of Radon measures and topological spaces.

Let μ be a Borel charge (i.e., a real Borel measure) on ℝ, and let $ {P_{(y)}}(t) = \frac{y}{\pi \left( {{y^2} + {t^2}} \right)},y > 0 $ , t ∈ ℝ, denote the Poisson kernel. Bourgain proved that for a nonnegative μ and for many points t ∈ ℝ, the variation of the function $ y \mapsto \left( \mu * {P_{ {(y)}}} \right)(x) $ on (0, 1] is finite. This is true, in particular, for so-called B-points x introduced in a previous author’s paper, In the present paper, we give new descriptions of B-points which are adjusted to some applications of this notion. Bibliography: 5 titles.

For every hyperbolic toral automorphism T, the present author has defined in his previous paper some unbounded T-invariant second-order difference operators related to the so-called homoclinic group of T. These operators were considered in the space L2 with respect to the Haar measure. It is shown in the present paper that such operators give rise to transition semigroups in the space of continuous functions on the torus and generate dynamically invariant Markov processes. This leads almost immediately to a family of invariant measures for the automorphism T.Along with a short discussion, some open questions about properties of these measures are posed. Bibliography: 9 titles.

Bifurcation diagrams of Steiner tree topologies for four boundary points are constructed in this paper. Also some properties of such diagrams for an arbitrary number of points are considered.

The paper is devoted to the popularization of one of the topics at the border between analysis and number theory that is related to polynomial with integer coefficients.

The paper presents an explicit formula for the Hilbert pairing between the Milnor K-group of a higher-dimensional local field and a polynomial formal module. This formula generalizes similar results for the one-dimensional case and the higher-dimensional case of multiplicative group. The case where the field and its first residue field have different characteristics, is considered. Bibliography: 13 titles.

Let K be a multidimensional local field with characteristic different from the characteristic of its residue field, c be a unit of K, and Fc(X, Y) = X +Y +cXY be a polynomial formal group, which defines the formal module Fc( $$ \mathfrak{M} $$ ) over the maximal ideal of the ring of integers in K. Assume that K contains the group of roots of the isogeny [p m]c(X), which we denote by μ Fc, m . Let be the multiplicative group of Cartier curves and c be the formal analog of the module Fc( $$ \mathfrak{M} $$ ). In the present paper, the formal symbol { ·, · }c : K n ( )× c → μ Fc, m is constructed and its basic properties are checked. This is the first step in the construction of an explicit formula for the Hilbert symbol.

Let A and B be square nonsingular n-by-n matrices with entries being rational or rational Gaussian numbers. The paper describes a method for verifying whether these matrices are congruent. The method uses a finite number of arithmetic (and, in the complex case, also conjugation) operations.

Here and in what follows, we write "a b" instead of "a ≤ Cb for some uniform constant C" for brevity; we also write ab when a b and b a .T he symbol∂j, j =1 , 2, denotes the differentiation with respect to jth variable. To be more precise, we study estimates of the scalar product of two functions in some Hilbert space (in this paper, some Sobolev space of fractional order) in terms of the product of L1- norms of some differential polynomials applied to these functions. For the author, the interest in inequalities of such type originated from the work on nonisomorphism problems for Banach spaces of smooth functions and embedding theorems used there, see the short report (5) and preprint (6). We are going to use some formalism to make our statements shorter. Let k and l be natural numbers, let α and β be real nonnegative numbers, and let σ and τ be complex nonzero numbers. The symbol BE(k, l, α, β, σ, τ) means the statement that the inequality �

The paper deals with the Markov property for the occupation time process for a homogeneous Markov chain with continuous time and countable state space. The well-known result that the Markov property is valid for a random walk on a tree is generalized to the case of a random walk on an arbitrary graph. Bibliography: 7 titles.

In the paper, the scheme of independent particle allocation is considered. A connection is demonstrated between finite-dimensional distributions for the process of the number of particles assigned to the k-th cell and the conditional distribution of independent Poisson distributed random variables. Conditions are specified for the convergence of these distributions to finite-dimensional distributions of the Gaussian process. A certain scheme of dependent particle allocation on a countable set of cells is introduced and investigated.

Let Ω be a bounded open and oriented connected subset of ℝ n which has a compact topological boundary Γ, let C be the Dirac operator in ℝ n , and let ℝ0,n be the Clifford algebra constructed over the quadratic space ℝ n . An ℝ0,n -valued smooth function f : Ω → ℝ0,n in Ω is called monogenic in Ω if Df = 0 in Ω. The aim of this paper is to present the most general condition on Γ obtained so far for which a Holder continuous function f can be decomposed as F + − F − = f on Γ, where the components F ± are extendable to monogenic functions in Ω± with Ω+ := Ω, and Ω− := ℝ n \ (Ω ⋃ Γ), respectively.

In this work, recursive expansions in Hilbert space H = L 2[0, 1] are considered. We discuss a related notion of frames in finite-dimensional spaces. We also suggest a constructive approach to extend an arbitrary basis to obtain a tight frame. The algorithm of extending is applied to bases of a special form, whose Gram matrix is circulant. A construction of a chain of nested subspaces $$ \left\{ {{V^n}} \right\}_{n = 1}^\infty $$ is given, and in its foundation lies an example of a function that can be expressed as a linear combination of its contractions and translations. The main result of the paper is the theorem that provides the uniform convergence of recursive Fourier series with respect to the chain $$ \left\{ {{V^n}} \right\}_{n = 1}^\infty $$ for continuous functions.

In this paper, we consider the functional differential equation with impulsive perturbations $$ \left\{ {\begin{array}{*{20}{c}} {{x^{\prime}}(t) = f\left( {t,{x_t}} \right),} \hfill & {t \geq {t_0},\quad t \ne {t_k},\quad x \in {\mathbb{R}^n},} \hfill \\ {\Delta x(t) = {I_k}\left( {t,x\left( {{t^{-} }} \right)} \right),} \hfill & {t = {t_k},\quad k \in {\mathbb{Z}^{+} }.} \hfill \\ \end{array} } \right. $$ Criteria on uniform asymptotic stability of sets are established for the above system using Lyapunov functions and the Razumikhin technique. Some examples are also discussed to illustrate the theorems.

The paper is dedicated to the memory of my teacher Nikolay Alexandrovich Shanin, who in his public speeches always emphasized the significance of research related to computational efficiency. During the last 30 years, many mathematicians working in the mathematical theory of computer science have begun to understand this efficiency as the computability in a polynomial number of steps by a Turing machine (the class FP). Bibliography: 4 titles.

Let L p , 1 ≤ p< ∞, be the space of 2π-periodic functions f with the norm $$ {\left\| f \right\|_p} = {\left( {\int\limits_{ - \pi }^\pi {{{\left| f \right|}^p}} } \right)^{{1 \mathord{\left/{\vphantom {1 p}} \right.} p}}} $$ , and let C = L ∞ be the space of continuous 2π-periodic functions with the norm $$ {\left\| f \right\|_\infty } = \left\| f \right\| = \mathop {\max }\limits_{e \in \mathbb{R}} \left| {f(x)} \right| $$ . Let CP be the subspace of C with a seminorm P invariant with respect to translation and such that $$ P(f) \leqslant M\left\| f \right\| $$ for every f ∈ C. By $$ \sum\limits_{k = 0}^\infty {{A_k}} (f) $$ denote the Fourier series of the function f, and let $$ \lambda = \left\{ {{\lambda_k}} \right\}_{k = 0}^\infty $$ be a sequence of real numbers for which $$ \sum\limits_{k = 0}^\infty {{\lambda_k}} {A_k}(f) $$ is the Fourier series of a certain function f λ ∈ L p . The paper considers questions related to approximating the function f λ by its Fourier sums S n (f λ) on a point set and in the spaces L p and CP. Estimates for $$ {\left\| {{f_\lambda } - {S_n}\left( {{f_\lambda }} \right)} \right\|_p} $$ and P(f λ − S n (f λ)) are obtained by using the structural characteristics (the best approximations and the moduli of continuity) of the functions f and f λ. As a rule, the essential part of deviation is estimated with the use of the structural characteristics of the function f. Bibliography: 11 titles.

We study in this paper Hardy-type spaces on a measure space ( $$ \mathbb{T} $$ , m) × (Ω, µ), where ( $$ \mathbb{T} $$ , m) is the unit circle with Lebesgue measure. There is a characterization of analytic stability for real interpolation of weighted Hardy spaces on $$ \mathbb{T} $$ × Ω, a complete proof of which was present in the literature only for the case where µ is a point mass. Here this gap is filled, and a proof of the general case is presented. In a previous work by Kislyakov, certain results concerning BMO-regular lattices on ( $$ \mathbb{T} $$ × Ω, m × µ) were proved under the assumption that the measure µ is discrete. Here this extraneous assumption is lifted. Bibliography: 9 titles.

This review paper is devoted to the Jacobi bound for systems of partial differential polynomials. We prove the conjecture for the system of n partial differential equations in n differential variables which are independent over a prime differential ideal \(\mathfrak{p}\). On the one hand, this generalizes our result about the Jacobi bound for ordinary differential polynomials independent over a prime differential ideal \(\mathfrak{p}\) and, on the other hand, the result by Tomasovic, who proved the Jacobi bound for linear partial differential polynomials.

In this paper, we consider a new class of random dynamical systems that contains, in particular, neural networks and complicated circuits. For these systems, we consider the viability problem: we suppose that the system survives only the system state is in a prescribed domain Π of the phase space. The approach developed here is based on some fundamental ideas proposed by A. Kolmogorov, R. Thom, M. Gromov, L. Valiant, L. Van Valen, and others. Under some conditions it is shown that almost all systems from this class with fixed parameters are unstable in the following sense: the probability P t to leave Π within the time interval [0, t] tends to 1 as t → ∞. However, it is allowed to change these parameters sometimes (“evolutionary” case), then it may happen that P t

Let M be a fixed space of 2π-periodic functions, Lp or C, let ωr(f, h) be the continuity modulus of order r of the function f in the space M, and let ϕ(t) be a function such that ϕ(t) > 0 for t > 0. By Sn(f) we denote the Fourier sums and by Rn,r(f) we denote the Riesz sums (the Fejer sums for r = 1) of the function f. Set $$K_m (f) = K_{m,\varphi } (f) = \mathop {\sup }\limits_{0 < v < \infty } \frac{{\omega _m (f,v)}}{{\varphi (v)}}$$ . The paper studies the dependence of the behavior of the quantities $$K_m (f - S_n (f)) and K_m (f - R_{n,r} (f))$$ as n → ∞ on the structural properties of the function f expressed in terms of the continuity moduli. In this way, general results are established, which are applicable to other approximation methods as well. Bibliography: 10 titles.

In this paper, we introduce the class of graded Ω-groups, which includes: groups; associative, conformal and vertex algebras; Lie algebras and graded algebras. The graded prime radical of a graded Ω-group is defined, and its elementwise characterization is given. It is shown that the graded prime radical of a graded Ω-groups with a finiteness condition coincides with the lower weakly solvable (in the Parfyonov sense) radical.

In the paper, we prove that two definitions of E-functions present in mathematics are equivalent if E-functions satisfy second-order linear differential equations. For this set of functions, a weakened variant of the well-known Siegel conjecture about representability of any E-function by a polynomial of hypergeometric functions is also proved.

In this paper, we obtain atomic decompositions for the spaces H1,q, 1 < q < ∞ (this case has not been considered earlier) and show that a multiplier that satisfies the condition of the Marcinkiewicz theorem acts from the space H1 into H1,∞. Bibliography: 5 titles.

In this paper, we present a criterion for a Galois polynomial over a Galois ring to have full period represented in terms of coefficients of this polynomial. In contrast to [4–6, 12], we consider the case of a Galois ring of odd characteristic, which is neither a residue ring nor a finite field. Such a ring is said to be nontrivial. Computational experiments show that the criterion provided is useful and convenient for practical purposes.

In the present paper, we solve some problems on the maximum of the weighted sum $$\sum\limits_{\user1{k = }1}^\user1{n} {\alpha _\user1{k}^2 } M(D_\user1{k} ,\user1{a}_\user1{k} )$$ (M(Dk,ak) denotes the reduced module of the domain Dk with respect to the point ak∈ Dk in the family of all nonoverlapping simple connected domains Dk, ak ∈ Dk, k=1,... ,n, where the points a1,... ,an are free parameters satisfying certain geometric conditions. The proofs involve a version of the method of extremal metric, which reveals a certain symmetry of the extremal system of the points a1,... ,an. The problem on the maximum of the conformal invariant $$2\pi \sum\limits_{k = 1}^5 {M(D_\user1{k} ,\user1{b}_\user1{k} )} - \frac{1}{2}\sum\limits_{1 \leqslant \user1{b}_\user1{k} < \user1{b}_\user1{l} \leqslant 5} {\log \left| {\user1{b}_\user1{k} - \user1{b}_\user1{i} } \right|} $$ for all systems of points b1,... ,bs is also considered. In the case where the systems {b1,... ,b5} are symmetric with respect to a certain circle, the problem was solved earlier. A theorem formulated in the author's previous work asserts that the maximum of invariant (*) for all system of points {b1,... ,b5} is attained in a certain well-defined case. In the present work, it is shown that the proof of this theorem contains mistake. A possible proof of the theorem is outlined. Bibliography: 10 titles.

One-dimensional branching extremals of Lagrangian-type functionals are considered. Such extremals appear as solutions to the classical Steiner problem on a shortest network, i.e., a connected system of paths that has the smallest total length among all the networks spanning a given finite set of terminal points in the plane. In the present paper, the Manhattan-length functional is investigated, with Lagrangian equal to the sum of the absolute values of projections of the velocity vector onto the coordinate axes. Such functionals are useful in problems arising in electronics, robotics, chip design, etc. In this case, in contrast to the case of the Steiner problem, local minimality does not imply extremality (however, each extreme network is locally minimal). A criterion of extremality is presented, which shows that the extremality with respect to the Manhattan-length functional is a global topological property of networks. Bibliography: 95 titles.

In [5], the author investigated the asymptotic unbiasedness and consistency of a kernel estimator of the distribution density of interarrival times in a renewal process. The present paper continues this research. Here the problem of estimation of the distribution density of the sojourn time in a xed state for semi-Markov processes from the observation of the process on the interval [0;T ]w ithT large enough is considered. Let (t) be a regular semi-Markov process with nite set of states and (0) = i be the initial state. The time t =0 coincides with the time of falling of the process (t )i nto the statei. Introduce the following random variables related to the process (t): i is the time of a single stay in the state i; ik is the sojourn time of the semi-Markov process in the state i during the kth visit of this state; i and ik are identically distributed; tik is the time between the kth and (k + 1)th returns to the state i; ik =tik ik. Denote F (x )= Pfi1 xg, G(x )= Pfi1 xg ,a ndQ(x )=( FG)(x )= Pfti1 xg. The sequence of random variablesftikg forms a simple renewal process. Construct a kernel estimator for the distribution density of the random variable i in the form f i(T )(u )= 1 i(T )h(T ) i(T ) X k=1 K u ik h(T )

The boundedness conditions for the differentiation operator in Hilbert spaces of entire functions (Branges spaces) and conditions under which the embedding Kи⊂L2(μ) holds in spaces Kи associated with the Branges spacesH(E) are studied. Measure μ such that the above embedding is isometric are of special interest. It turns out that the condition E'/E∈H∞(C+) is sufficient for the boundedness of the differentiation operator inH(E). Under certain restrictions on E, this condition is also necessary. However, this fact fails in the general case, which is demonstrated by the counterexamples constructed in this paper. The convex structure of the set of measures μ such that the embedding KE*/E⊂L2(μ) is isometric (the set of such measures was described by de Brages) is considered. Some classes of measures that are extreme points in the set of Branges measures are distinguished. Examples of measures that are not extreme points are also given. Bibliography: 7 titles.

It is known that in the case of the unit disk the invertibility of the orthogonal projection of one subspace of H2 which is co-invariant with respect to the shift operator onto another such subspace is connected with the Helson-Szego theorem and the Muckenhoupt condition. In the present paper, we consider the same problem in character-automorphic Hardy spaces on a finitely connected planar domain. The problem is reduced to estimating the angles between certain subspaces of the weighted L2-space on the boundary of the domain. The answer is given in terms of the Muckenhoupt condition for certain weights. Bibliography: 29 titles.

The present paper is devoted to further development and refinement of previous results due to A. V. Malyshev and the author concerning the so-called discrete ergodic method of Yu. V. Linnik. An “ergodic theorem” and a “mixing theorem” for flows of positive binary quadratic forms are proved; these theorems describe the asymptotic distribution of the coefficients of these forms over the residue classes and over the corresponding surface. Bibliography: 12titles.

We study a wide class of minimax problems of signal detection under nonparametric alternatives and a modification of these problems for a special class of loss functions. Under rather general assumptions, we obtain the exact asymptotics (of Gaussian type) for the minimax error probabilities and the structure of asymptotically minimax tests. The methods are based on a reduction of the problems under consideration to extremal problems of minimization of a certain Hilbert norm on a convex set of sequences of probability measures on the real line. These extremal problems are investigated in a paper by I. A. Suslina for alternatives having the type of lq-ellipsoids with lp-balls removed. Bibliography: 16 titles.

Let B be a Blaschke product with simple zeros in the unit disk, let Λ be the set of its zeros, and let ϕ∈H∞. It is known that ϕ+BH∞ is a weak* generator of the algebra H∞/BH∞ if (for B that satisfy the Carleson condition (C)) and only if the sequence ϕ(Λ) is a weak* generator of the algebra l∞. In this paper, we show that for any Blaschke product B with simple zeros that does not satisfy condition (C), there exists B=B1·…·BN, where N ∈ℕ, and B1, …, BN are Blaschke products satisfying condition (C), there exists a function ϕ∈H∞ such that ϕ(Λ) is a weak* generator of the algebra l∞, and ϕ+BH∞ is not a weak* generator of the algebra H∞/BH∞. Bibliography: 12 titles.

In this paper, we characterize the Hermite polynomials via stochastic tools: the conditional mean value and the Wiener process.

Let u1=1, u2=2, u3,... be the sequence of Fibonacci numbers. A Fibonacci partition of a natural number n is a partition of n into different Fibonacci numbers. In this paper it is proved that the set of Fibonacci partitions of a natural number, partially ordered with respect to refinement is the lattice of ideals of a multizigzag. On the basis of this theorem we obtain some results concerning the coefficients of the Taylor series of infinite products $$\prod\limits_{i = 1}^{ + \infty } {\left( {1 + zq^{ui} } \right) = 1 + \sum\limits_{k = 1}^{ + \infty } {a_k \left( z \right)q^k } } $$ where $$z = \pm 1, - \tfrac{1}{2} \pm i\tfrac{{\sqrt 3 }}{2}$$ , ±i. Bibliography: 6 titles.

In the present paper we consider the relation between the distributions of probability of attaining the boundary point of a polynomial plan and that of a Poisson plan. This relation is used to compare the moments of the functions of the boundary point coordinates, the closedness and completeness conditions of plans, the classes of the unbiased estimates and the unbiased estimated functions of the unknown parameters.

Consider the following stochastic model. Let the resource consumption in discrete time be represented in the form {sup {eta}}{sub k} = Y(k) + n{sub k} where y(k) are deterministic components of consumption and w{sub k} are independent Gaussian random variables with zero mean. Correspondingly, the resource supply is given (in discrete time) by the process {xi}{sub k} = z(k) - {tau}{sub k}, where z(k) are its deterministic components and {tau}{sub k} are quantities characterizing the undersupply of the resource associated with breakdowns, repairs, and so on; their structure will be described below in detail. This model describes a quite realistic system of transporting and storage of gas, oil-products, and so on [1]. The supply to the users is conducted over pipelines (main and local), the main deficits and losses being associated with the seasonal variations in consumption and accidents; to decrease the losses along the main pipelines, storage facilities are constructed which are used as compensators. The model considered in this paper describes the case of a terminal storage, i.e., of a district or an industrial region.

This paper is the continuation of [1], [2]. In it questions of approximation of functions are discussed, when a realization of the homogeneous Gaussian field belongs to the class Hpr(G)0.