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Let rk(n) denote the number of lattice points on a k-dimensional sphere of radius $$ \sqrt{n} $$ . The generating function $$ {\zeta}_k(s)=\sum \limits_{n=1}^{\infty }{r}_k(n){n}^{-s},\kern0.5em k\ge 2, $$ is Epstein’s zeta function. The paper considers the Riesz mean of the type $$ {D}_{\rho}\left(x;{\zeta}_3\right)=\frac{1}{\Gamma \left(\rho +1\right)}\sum \limits_{n\le x}{\left(x-n\right)}^{\rho }{r}_3(n), $$ where ρ > 0; the error term Δρ(x; ζ3) is defined by $$ {D}_{\rho}\left(x;{\zeta}_3\right)=\frac{\uppi^{3/2}{x}^{\rho +3/2}}{\Gamma \left(\rho +5/2\right)}+\frac{x^{\rho }}{\Gamma \left(\rho +1\right)}{\zeta}_3(0)+{\Delta}_{\rho}\left(x;{\zeta}_3\right). $$ K. Chandrasekharan and R. Narasimhan (1962, MR25#3911) proved that $$ {\Delta}_{\rho}\left(x;{\zeta}_3\right)=\Big\{{\displaystyle \begin{array}{ll}O\Big({x}^{1/2+\rho /2\Big)}& \left(\rho >1\right),\\ {}{\Omega}_{\pm}\left({x}^{1/2+\rho /2}\right)& \left(\rho \ge 0\right).\end{array}} $$ In the present paper, it is proved that $$ {\Delta}_{\rho}\left(x;{\zeta}_3\right)=\Big\{{\displaystyle \begin{array}{ll}O\left(x\log x\right)& \left(\rho =1\right),\\ {}O\left({x}^{2/3+\rho /3+\varepsilon}\right)& \left(1/2

We study properties of sets having the minimum length (one-dimensional Hausdorff measure) in the class of closed connected sets Σ ⊂ ℝ2 satisfying the inequality max yϵM dist (y, Σ) ≤ r for a given compact set M ⊂ ℝ2 and given r > 0. Such sets play the role of the shortest possible pipelines arriving at a distance at most r to every point of M where M is the set of customers of the pipeline. In this paper, it is announced that every maximum distance minimizer is a union of finitely many curves having one-sided tangent lines at every point. This shows that a maximum distance minimizer is isotopic to a finite Steiner tree even for a “bad” compact set M, which distinguishes it from a solution of the Steiner problem (an example of a Steiner tree with infinitely many branching points can be found in a paper by Paolini, Stepanov, and Teplitskaya). Moreover, the angle between these lines at each point of a maximum distance minimizer is at least 2π/3. Also, we classify the behavior of a minimizer Σ in a neighborhood of any point of Σ. In fact, all the results are proved for a more general class of local minimizers.

Lately, the following problem attracted a lot of attention in various contexts: find the shortest factorization G = UU - UU - …U ± of a Chevalley group G = G(Φ, R) in terms of the unipotent radical U = U(Φ, R) of the standard Borel subgroup B = B(Φ, R) and the unipotent radical U - = U -(Φ, R) of the opposite Borel subgroup B - = B - (Φ, R). So far, the record over a finite field was established in a 2010 paper by Babai, Nikolov, and Pyber, where they prove that a group of Lie type admits the unitriangular factorization G = UU - UU - U of length 5. Their proof invokes deep analytic and combinatorial tools. In the present paper, we notice that from the work of Bass and Tavgen one immediately gets a much more general results, asserting that over any ring of stable rank 1 one has the unitriangular factorization G = UU - UU - of length 4. Moreover, we give a detailed survey of traingular factorizations, prove some related results, discuss prospects of generalization to other classes of rings, and state several unsolved problems. Another main result of the present paper asserts that, in the assumption of the Generalized Riemann’s Hypothesis, Chevalley groups over the ring $$ \mathbb{Z}\left[ {\frac{1}{p}} \right] $$ admit the unitriangular factorization G = UU - UU - UU - of length 6. Otherwise, the best length estimate for Hasse domains with infinte multiplicative groups that follows from the work of Cooke and Weinberger, gives 9 factors. Bibliography: 67 titles.

A perfect cuboid is a rectangular parallelepiped in which the lengths of all edges, the lengths of all face diagonals, and also the lengths of spatial diagonals are integers. No such cuboid has yet been found, but their nonexistence has also not been proved. The problem of a perfect cuboid is among unsolved mathematical problems. The problem has a natural S3-symmetry connected to permutations of edges of the cuboid and the corresponding permutations of face diagonals. In this paper, we give a survey of author’s results and results of J. R. Ramsden on using the S3 symmetry for the reduction and analysis of the Diophantine equations for a perfect cuboid.

In this paper, we present new properties of the space L1(M, τ) of integrable (with respect to the trace τ ) operators affiliated to a semifinite von Neumann algebra M. For self-adjoint τ-measurable operators A and B, we find sufficient conditions of the τ -integrability of the operator λI −AB and the real-valuedness of the trace τ (λI − AB), where λ ∈ ℝ. Under these conditions, [A,B] = AB − BA ∈ L1(M, τ) and τ ([A,B]) = 0. For τ -measurable operators A and B = B2, we find conditions that are sufficient for the validity of the relation τ ([A,B]) = 0. For an isometry U ∈ M and a nonnegative τ -measurable operator A, we prove that U − A ∈ L1(M, τ) if and only if I − A, I − U ∈ L1(M, τ). For a τ -measurable operator A, we present estimates of the trace of the autocommutator [A∗,A]. Let self-adjoint τ -measurable operators X ≥ 0 and Y be such that [X1/2, YX1/2] ∈ L1(M, τ). Then τ ([X1/2, YX1/2]) = it, where t ∈ ℝ and t = 0 for XY ∈ L1(M, τ).

Let SF be the set of continuous bounded selections from the set-valued mapping F : T → 2H with nonempty convex closed values; here T is a paracompact Hausdorff topological space, and H is a Hilbert space. In this paper, we obtain a criterion for the existence of a 1-Lipschitz selection from the metric projection onto the set SF in C(T, H).

Let r4(n) denote the number of representations of n as a sum of four squares. The generating function ζ4(s) is Epstein’s zeta function. The paper considers the Riesz mean $$ {D}_{\rho}\left(x;{\zeta}_4\right)=\frac{1}{\Gamma \left(\rho +1\right)}\sum \limits_{n\le x}{\left(x-n\right)}^{\rho }{r}_4(n) $$ for an arbitrary fixed ρ > 0. The error term Δρ(x; ζ4) is defined by $$ {D}_{\rho}\left(x;{\zeta}_4\right)=\frac{\uppi^2{x}^{2+\rho }}{\Gamma \left(\rho +3\right)}+\frac{x^{\rho }}{\Gamma \left(\rho +1\right)}{\zeta}_4(0)+{\Delta}_{\rho}\left(x;{\zeta}_4\right). $$ It is proved that $$ {\Delta}_4\left(x;{\zeta}_4\right)=\Big\{{\displaystyle \begin{array}{ll}O\left({x}^{1/2+\rho +\varepsilon}\right)& \left(1

Fix a certain class of perturbations of the coefficient matrix A(·) of a discrete linear homogeneous system of the form $$ x\left(m+1\right)=A(m)x(m),\kern1em m\in \kern0.5em \mathrm{N},\kern1em x\in {\mathrm{R}}^n, $$ where the matrix A(·) is completely bounded on ℕ. The spectral set of this system corresponding to a given class of perturbations is the collection of complete spectra of the Lyapunov exponents of perturbed systems when perturbations runs over the whole class considered. In this paper, we examine the class R of multiplicative perturbations of the form $$ y\left(m+1\right)=A(m)R(m)x(m),\kern1em m\in \mathrm{N},\kern1em y\in {\mathrm{R}}^n, $$ where the matrix R(·) is completely bounded on ℕ. We obtain conditions that guarantee the coincidence of the spectral set λ(R) corresponding to the class R with the set of all nondecreasing n-tuples of n numbers.

The congruent centralizer of a complex n × n matrix A is the set of n × n matrices Z such that Z*AZ = A. This set is an analog of the classical centralizer in the case where the similarity relation on the space of n × n matrices is replaced by the congruence relation. The study of the classical centralizer C A reduces to describing the set of solutions of the linear matrix equation AZ = ZA. The structure of this set is well known and is explained in many monographs on matrix theory. As to the congruent centralizer, its analysis amounts to a description of the solution set of a system of n 2 quadratic equations for n 2 unknowns. The complexity of this problem is the reason why there is still no description of the congruent centralizer $$ {C}_J^{\ast } $$ even in the simplest case of the Jordan block J = J n (0) with zeros on the principal diagonal. This paper presents certain facts concerning the structure of matrices in $$ {C}_J^{\ast } $$ for an arbitrary n and then gives complete descriptions of the groups $$ {C}_J^{\ast } $$ for n = 2, 3, 4, 5.

It is well known that for a standard Brownian motion (BM) {B(t), t ≥ 0} with values in R d, its convex hull V (t) = conv{B(s), s ≤ t} with probability 1 for each t > 0 contains 0 as an interior point. We also know that the winding number of a typical path of a two-dimensional BM is equal to +∞. The aim of this paper is to show that these properties are not specifically “Brownian,” but hold for a much larger class of d-dimensional self-similar processes. This class contains, in particular, d-dimensional fractional Brownian motions and (concerning convex hulls) strictly stable Levy processes. Bibliography: 10 titles.

In this paper, we calculate the maximal dimension of the Lie group of automorphisms of an n-dimensional Riemann–Cartan manifold.

The paper is devoted to some problems associated with a probabilistic representation and probabilistic approximation of the Cauchy problem solution for the family of equations $$ \frac{\partial u}{\partial t}=\frac{\sigma^2}{2}\varDelta u $$ with a complex parameter σ such that Re σ2 ≥ 0. The above family includes as a particular case both the heat equation (Im σ = 0) and the Schrodinger equation (Re σ2 = 0).

In the present paper, we start with a criterion of elementary equivalence of linear groups over rings with just a finite number of central idempotents. Then we study elementary equivalence of linear groups over Boolean algebras. We prove that two linear groups over Boolean algebras are elementarily equivalent if and only if their dimensions coincide and these Boolean algebras are elementarily equivalent.

In this paper, we consider a group of transformations of the space of trajectories of symmetric α-stable Levy laws with stability constant α ∈ [0; 2). For α = 0, the true analog of a stable Levy process (the so-called 0-stable process) is the gamma process, whose law is quasi-invariant under the action of the group of multiplicators $ \mathcal{M}\equiv \left\{ {{M_a}:a\geq 0,\,\log a\in {L^1}} \right\} $ ; the action of Ma on a trajectory ω(∙) is given by (Maω)(t) = a(t)ω(t). For every α < 2, an appropriate conjugacy transformation sends the group $ \mathcal{M} $ to the group $ {{\mathcal{M}}_a} $ of nonlinear transformations of trajectories, and the law of the corresponding stable process is quasi-invariant under this group. We prove that for α = 2, an appropriate change of coordinates reduces the group of symmetries to the Cameron-Martin group of nonsingular translations of trajectories of the Wiener process.

In this paper, the main properties of the partial indices of the Riemann boundary-value problem are considered. This important invariant point of view gives a modern approach to two central problems of complex analysis.

In the paper, upper estimates for sums of the form $$ \sum\limits_{{\left( {{n_1},{n_2}} \right)}} {\sum\limits_{{\in \varOmega }} {\psi \left( {f\left( {{n_1},{n_2}} \right)} \right),} } $$ where $ \psi (x)=x-\left[ x \right]-\frac{1}{2},\,\,f\left( {x,y} \right) $ is a polynomial, (n 1, n 2) ∈ ℤ2, and Ω is a region in ℝ2, are obtained. One of the upper estimates is of interest, particularly, in connection with the lattice point problem considered in Theorem 2. Bibliography: 10 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.

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, it is proved that in a Coxeter group of extra large type every torsion-free, finitely generated subgroup is free.

The paper considers the L-function associated with the Kubota-Patterson cubic theta function. A method for computing values of the L-function is developed, and some conjectures on the distribution of its zeros are advanced and supported by numerical results. Bibliography: 12 titles.

We give necessary and sufficient conditions for cofactor expansibility of determinants along a row or column for Boolean square matrices over an arbitrary Boolean algebra. First of all we define a natural decomposition of an arbitrary Boolean matrix by interior, exterior, and determinate parts. The introduced notions allow us to establish the main result of this paper. It is shown that the formulas of the cofactor expansion along a row (column) of determinants of an arbitrary square Boolean matrix hold if and only if the formulas of the cofactor expansion along the corresponding row (column) hold for determinants of its interior part.

The paper considers the function defined by the Dirichlet series $$L(\tau ;s) = \sum\limits_\nu {\frac{{\tau (\nu )}}{{\left\| \nu \right\|^s }}} , s \in \mathbb{C}$$ , where τ(ν) is the νth Fourier coefficient of the cubic Kubota-Patterson theta function. The zeros of the function L(τ ·) in the domain |Im s| < 444 are computed. Bibliography: 4 titles.

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.

The paper studies the function L(τ;·) defined by the Dirichlet series $$L(\tau ;s) = \sum\limits_\nu {\frac{{\tau (\nu )}}{{\left\| \nu \right\|^s }}, s \in \mathbb{C}} $$ , where τ(υ) is the υth Fourier coefficient of the Kubota-Patterson cubic theta function. For this function, an exact and an approximate functional equations are derived. It is established that the function does not vanish in the halfplane Re s ≥ 1.3533 and has no singularities except for a simple pole at the point 5/6. Issues related to computing the coefficients τ(υ) and values of the special functions arising in the approximate functional equation are considered. Bibliography: 11 titles.

We study the limiting behavior of the probability with which the path of a centered Poisson process of high intensity gets into a small ball with a receding center. The results of this paper are restricted to the simplest case where the variation of the shift function (center of the ball) is finite. The estimates are obtained under the optimal conditions for the intensity of the process. Bibliography: 19 titles.

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.

Let \(P_k^{(\alpha ,\beta )} \) be the Jacobi polynomials and let C[a,b] be the space of continuous functions on [a,b] with the uniform norm. In this paper, we study sequences of Lebesgue constants, i.e., of the norms of linear operators \({\mathcal{U}}_n^\Lambda :C\left[ { - 1,1} \right] \to C\left[ { - 1,1} \right]\)generated by a multiplier matrix \(\Lambda = \left\{ {\lambda _k^{\left( n \right)} } \right\}\) defined by the following relations: $$f \sim \sum\limits_{k = 0}^\infty {a_k P_k^{\left( {\alpha ,\beta } \right)} } ,{\mathcal{U}}_n^\Lambda f \sim \sum\limits_{k = 0}^\infty {\lambda _k^{\left( n \right)} } a_k P_k^{\left( {\alpha ,\beta } \right)} ,$$ and $${\mathfrak{L}}_n^{\left( {\alpha ,\beta } \right)} \left( \Lambda \right) = \mathop {sup}\limits_{y \in \left[ { - 1,1} \right]} \mathop {sup} \limits_{\left\| f \right\| \leqslant 1} \left| {\mathcal{U}_n^\Lambda f\left( y \right)} \right|.$$ In the case |α| = |β| = 1/2, we prove the following statements for the Jacobi polynomials (these statements are similar to known results for the trigonometrical system). Consider the cases $$(1)\;\;\;\alpha = \beta = - {1}/{2}\;\;and\;\;\lambda _k^{\left( n \right)} = \varphi \left( {k/n} \right);$$ $$\left({2} \right)\;\;\;\alpha = \beta = {1}/{2}\;\;and\;\;\lambda _k^{\left( n \right)} = \varphi \left( {\left( {k + 1} \right)/n} \right);$$ and $$\left( {3} \right)\;\;\;\alpha = - \beta = \pm {1}/{2}\;\;and\;\;\lambda _k^{\left( n \right)} = \varphi \left( {\left( {k + {1}/{2}} \right)/n} \right).$$ Under some conditions on a function ϕ, the values \(\mathop {\sup }\limits_{n \in {\mathbb{N}}} {\mathfrak{L}}_n^{\left( {\alpha ,\beta } \right)} \left( \Lambda \right)\) and \(\mathop {\lim }\limits_{n \to \infty } {\mathfrak{L}}_n^{\left( {\alpha ,\beta } \right)} \left( \Lambda \right)\) equal $$\frac{2}{\pi }\int_{0}^\infty {\kern 1pt} {\kern 1pt} \left| {\int_{0}^\infty {\varphi \left( t \right)\cos zt\;dt} } \right|dz\;\;\;\left( {case\left({1} \right)} \right)$$ and $$\frac{2}{\pi }\int_0^\infty z \left| {\int_0^\infty {t\varphi \left( t \right)\sin zt\;dt} } \right|dz\;\;\left( {cases\;\left({2} \right)\;and\;\left( \right)} \right).$$ In addition, we show that for the Fourier–Legendre summation methods (α = β = 0) generated by the multiplier function \(\lambda _k^{\left( n \right)} = \varphi \left( {k/n} \right)\), the limit and supremum of the sequence of Lebesgue constants may differ. Bibliography: 11 titles.

We simplify and strengthen Abrahamse's result on the Nevanlinna–Pick interpolation problem in a finitely connected planar domain, according to which the problem has a solution if and only if the Pick matrices associated with character-automorphic Hardy spaces are positive semidefinite for all characters in ℝn−1/ℤn−1, where n is the connectivity of the domain. The main aim of the paper is to reduce the indicated procedure (verification of the positive semidefiniteness) for the entire real (n−1)-torus ℝn−1/ℤn−1 to a part of it, whose dimension is, possibly, less than n−1. Bibliography: 14 titles.

The random vectorD= (D1, …, Dn) is called lβ-isotropic (lβ-norm symmetric) if and only if $$D\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{d} } U_\beta \theta $$ for Uβ with the Dirichlet distribution with parameters (β, 1) on the positive rectangle of S β n ={x∈R n: ‖x‖β-1}, Θ being a nonnegative random variable independent of Uβ. Equivalently we can say that D is lβ-isotropic if and only if ‖D‖β and D/‖D‖β are independent with D/‖D‖β, having Dirichlet distributions on Sβ. The lβ-isotropic random vectors have density level curves of the form rSβ r≥0. In this paper, we show that if the continuous time process {Xt: t≥0} has the property that for every T>0 and every n∈N the sequence of powers of increments {D n,k α =(X[k/n]T−X[(k−1)/n]T)α: k=1, …, n} is lβ-isotropic for some α,β>0, then {Xt} must be a scale mixture of gamma processes. The gamma process is especially useful for modelling deterioration processes that proceed in one direction and in random jumps. On the basis of a gamma process, many tailor-made maintenance decision models can be developed.

The present paper contains an exact topological classification of all nondegenerate Hamiltonian systems on smooth closed two-dimensional surfaces. Bibliography: 8 titles.

In the paper, the sets of limit points for trajectories of continuous stationary Gaussian random fields are investigated. Subsets of the parametric set such that they generate the full set of limit points are described. Bibliography: 4 titles.

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.

The paper presents upper bounds for the largest eigenvalue of a block Jacobi scaled symmetric positive-definite matrix which depend only on such parameters as the block semibandwidth of a matrix and its block size. From these bounds we also derive upper bounds for the smallest eigenvalue of a symmetric matrix with identity diagonal blocks. Bibliography: 4 titles.

In this paper we present a generalization of the Kolmogorov goodness-of-fit test for the case of many samples. This generalization is based on the theory of multivariate Poisson plans of random walks (PPRW) [1]. The method of calculation of the precise distribution of the test statistic is given. Moreover, it is suggested to use multisample homogeneous tests of Kolmogotrov-Smirnov type as goodness-of-fit ones.