Vol 93, No 2 (2016)
- Year: 2016
- Articles: 29
- URL: https://journals.rcsi.science/1064-5624/issue/view/13729
Mathematics
Analytic continuation formulas and Jacobi-type relations for Lauricella function
Abstract
An approach for constructing a complete system of formulas for the analytic continuation of the Lauricella generalized hypergeometric function FD(N) with any N beyond the boundary of the unit polydisk is proposed. The approach is exposed in detail for the continuation of the function under consideration in neighborhoods of points whose all N components equal 1 or ∞. For the Lauricella function, differential relations being analogues of Jacobi’s formula for the Gaussian hypergeometric function are also presented. The results can be applied to solve the crowding problem for the Schwarz–Christoffel integral and to the theory of the Riemann–Hilbert problem.
Homogenization of a variational inequality for the p-Laplacian in perforated media with nonlinear restrictions for the flux on the boundary of isoperimetric perforations: p equal to the dimension of the space
Abstract
We address the homogenization of a variational inequality posed in perforated media issue from a unilateral problem for the p-Laplacian. We consider the n-Laplace operator in a perforated domain of ℝn, n ≥ 3, with restrictions for the solution and its flux (the flux associated with the n-Laplacian) on the boundary of the perforations which are assumed to be isoperimetric. The solution is assumed to be positive on the boundary of the holes and the flux is bounded from above by a negative, nonlinear monotone function multiplied by a large parameter. A certain non periodical distribution of the perforations is allowed while the assumption that their size is much smaller than the periodicity scale is performed. We make it clear that in the average constants of the problem, the perimeter of the perforations appears for any shape.
Large extremes of Gaussian chaos processes
Abstract
We study probabilities of large extremes of Gaussian chaos processes, that is, homogeneous functions of Gaussian vector processes. Important examples are products of Gaussian processes and quadratic forms of them. Exact asymptotic behaviors of the probabilities are found. To this aim, we use joint results of E. Hashorva, D. Korshunov and the author on Gaussian chaos, as well as a substantially modified asymptotical Double Sum Method.
Stochastic intermediate gradient method for convex optimization problems
Abstract
New first-order methods are introduced for solving convex optimization problems from a fairly broad class. For composite optimization problems with an inexact stochastic oracle, a stochastic intermediate gradient method is proposed that allows using an arbitrary norm in the space of variables and a prox-function. The mean rate of convergence of this method and the probability of large deviations from this rate are estimated. For problems with a strongly convex objective function, a modification of this method is proposed and its rate of convergence is estimated. The resulting estimates coincide, up to a multiplicative constant, with lower complexity bounds for the class of composite optimization problems with an inexact stochastic oracle and for all usually considered subclasses of this class.
On Geary’s theorem for the field of p-adic numbers
Abstract
Let ℚp, where p > 2, be a field of p-adic numbers. We consider two independent identically distributed random variables with values in ℚp and distribution μ with a continuous density. We prove that the sum and the squared difference of these random variables are independent if and only if μ is an idempotent distribution, i.e., a shift of the Haar distribution of a compact subgroup of the additive group of the field ℚp.
Nonasymptotic approach to Bayesian semiparametric inference
Abstract
The classical semiparametric Bernstein–von Mises (BvM) results is reconsidered in a non-classical setup allowing finite samples and model misspecication. We obtain an upper bound on the error of Gaussian approximation of the posterior distribution for the target parameter which is explicit in the dimension of the target parameter and in the dimension of sieve approximation of the nuisance parameter. This helps to identify the so called critical dimension pn of the sieve approximation of the full parameter for which the BvM result is applicable. If the bias induced by sieve approximation is small and dimension of sieve approximation is smaller then critical dimension than the BvM result is valid. In the important i.i.d. and regression cases, we show that the condition “pn2q/n is small”, where q is the dimension of the target parameter and n is the sample size, leads to the BvM result under general assumptions on the model.
Programmed iteration method and sets of positional absorption
Abstract
A pursuit–evasion differential game is considered, and the programmed iteration method is used to construct a set of positional absorption corresponding to the Krasovskii–Subbotin alternative theorem. The case is considered where the set of positions determining the state constraints may not be closed (in the position space), but has closed sections corresponding to fixed times. Properties are established that are interpreted as the (one-sided) continuity of the positional absorption set from above, and the relation to the solution of the game in the class of set-valued quasi-strategies is shown.
Extremal polynomials related to Zolotarev polynomials
Abstract
Algebraic polynomials bounded in absolute value by M > 0 in the interval [–1, 1] and taking a fixed value A at a > 1 are considered. The extremal problem of finding such a polynomial taking a maximum possible value at a given point b < −1 is solved. The existence and uniqueness of an extremal polynomial and its independence of the point b < −1 are proved. A characteristic property of the extremal polynomial is determined, which is the presence of an n-point alternance formed by means of active constraints. The dependence of the alternance pattern, the objective function, and the leading coefficient on the parameter A is investigated. A correspondence between the extremal polynomials in the problem under consideration and the Zolotarev polynomials is established.
New approach to optimality conditions for degenerate nonlinear programming problems
Abstract
In the work, a new approach to constructing optimality conditions for degenerate smooth optimization problems with inequality constraints is proposed. The approach is based on the theory of p-regularity. A special case of degeneracy, when the first derivatives of some function-constraints are equal to zero up to some order, is considered. Optimality conditions for the general case of degeneracy with p = 2 are presented. Proposed constructions and optimality conditions are illustrated by an example. A general case of degeneracy is considered and optimality conditions for the case of p ≥ 2 are proposed.
Deficiency numbers of operators generated by infinite Jacobi matrices
Abstract
New conditions for minimality, maximality, and nonmaximality of deficiency numbers of the minimal operator generated by the infinite Jacobi matrix with m × m matrix entries in the Hilbert space of mdimensional vectors are presented. Special attention is given to the case m = 1, i.e., to conditions on the elements of a tridiagonal numerical Jacobi matrix under which the determinate case of the classical power moment problem is realized.
Canonical and existential groups in universal classes of Abelian groups
Abstract
Universal classes of Abelian groups are classified in terms of sets of finitely generated groups closed with respect to the discrimination operator. The notions of a principal universal class and a canonical group for such a class are introduced. For any universal class K, the class Kec of existentially closed groups generated by the universal theory of K is described. It is proved that Kec is axiomatizable and, therefore, the universal theory of K has a model companion.
Standard finite difference scheme for a singularly perturbed elliptic convection–diffusion equation on a rectangle under computer perturbations
Abstract
A singularly perturbed elliptic convection–diffusion equation with a perturbation parameter ε (ε ∈ (0, 1]) is considered on a rectangle. As applied to this equation, a standard finite difference scheme on a uniform grid is studied under computer perturbations. This scheme is not ε-uniformly stable with respect to perturbations. The conditions imposed on a “computing system” are established under which a converging standard scheme (referred to as a computer difference scheme) remains stable.
On bases for direct decomposition
Abstract
A method for constructing a basis for a Banach space from bases for its subspaces is proposed. The case of isomorphic subspace bases and the case when no corresponding isomorphisms are required are considered separately. The completeness, minimality, uniform minimality, and basis property with parentheses of the corresponding systems are studied. This approach has wide applications in the spectral theory of discontinuous differential operators.
Invariants of four- and three-dimensional singularities of integrable systems
Abstract
A relationship between invariants of four-dimensional singularities of integrable Hamiltonian systems (with two degrees of freedom) and invariants of two-dimensional foliations on three-dimensional manifolds being the “boundaries” of these four-dimensional singularities is discovered. Nonequivalent singularities which, nevertheless, have equal three-dimensional invariants are found.
Behavior of the formal Fourier solution of the wave equation with a summable potential
Abstract
The convergence of the formal Fourier solution to a mixed problem for the wave equation with a summable potential is analyzed under weaker assumptions imposed on the initial position u(x, 0) = φ(x) than those required for a classical solution up to the case φ(x)∈ Lp[0,1] for p > 1. It is shown that the formal solution series always converges and represents a weak solution of the mixed problem.
Pontryagin maximum principle, relaxation, and controllability
Abstract
The relations between the necessary minimum conditions in an optimal control problem (Pontryagin maximum principle), the minimum conditions in the corresponding relaxation (weakened) problem, and sufficient conditions for the local controllability of the controlled system specifying the constraints in the original formulation are studied. An abstract optimization problem that models the basic properties of the optimal control problem is considered.
Arak’s inequalities for concentration functions and the Littlewood–Offord problem
Abstract
In this paper we study the behavior of concentration functions of weighted sums of independent random variables with respect to the arithmetic structure of coefficients. Recently, Tao and Vu formulated a so-called Inverse Principle in the Littlewood–Offord problem. We discuss the relations between this Inverse Principle and a similar principle formulated for sums of arbitrarily distributed independent random variables formulated by T. Arak in the 1980’s.
On the absence of the basis property for the root function system of the Sturm–Liouville operator with degenerate boundary conditions
Abstract
The eigenvalue problem generated by the Sturm–Liouville equation on the interval (0, π) with degenerate boundary conditions is considered. Under certain conditions imposed on the spectrum, it is shown that the system of eigen- and associated functions is not a basis in L2(0, π).
Lμ → Lν equiconvergence of spectral decompositions for a Dirac system with Lκ potential
Abstract
It is proved that if P ∈ Lκ[0, π], κ ∈ (1, ∞], then the expansions of any function f ∈ Lμ[0, π], μ ∈ [1, ∞], in the generalized eigenfunctions of the perturbed and unperturbed operators are equiconvergent in the norm of the space Lν[0, π], provided that ν ∈ [1, ∞] satisfies the inequality \(\frac{1}{\kappa } + \frac{1}{\mu } - \frac{1}{\nu } \leqslant 1\), except in the case where κ = ν = ∞ and μ = 1.
Mathematical Physics
Composite electromagnetic waves in magnetodielectric systems
Abstract
The concept of a composite electromagnetic field in multilayered magnetodielectric (MD) systems is introduced. The concept is based on an “algebraic” description of plane electromagnetic fields relying on the phase approach [1], which is useful in optimizing film covers and solving inverse problems for MD systems [4–6].
Computer Science
Mathematical knowledge ontologies and recommender systems for collections of documents in physics and mathematics
Abstract
An approach to the implementation of a recommender system based on ontologies of mathematical knowledge is presented. On the basis of a document browsed by a user, the system forms on line a list of recommendations, which include similar documents, key words, and definitions of these words from ontology and other terminological sources. The method of recommendations yields a vector representation of documents, taking into account the position of terms in the logical structure of the document and their ontological connections. On the basis of the cosine measure, a measure of proximity between documents is calculated. The order of documents in the list of recommendations is determined by values of the proximity measure. Various adaptations of the system to user scenarios aimed at the preparation of personalized recommendations are discussed.
Threshold model of a neural ensemble
Abstract
Networks of threshold elements whose inputs are assigned positive and negative (inhibitory) weights and outputs take the values 0 and 1 are considered. A stationary ensemble is defined as a connected subnetwork of a threshold network for which the unit state (1, 1, …, 1) = 1 is stable. The transfer of an ensemble into the state 1 is called switching on. Necessary and sufficient conditions for a network to be an ensemble are given. It is shown that, in the proposed model, the switching on of one of two ensembles having common elements does not necessarily lead to the switching on of the other.
Control Theory
Anisotropy-based bounded real lemma for discrete-time systems with multiplicative noise
Abstract
A model of a discrete-time system with multiplicative noise is considered. For this model, a condition is derived under which the anisotropic norm of the system is bounded by the anisotropic norm of an auxiliary linear discrete-time stationary system with parametric uncertainty. Conditions for the anisotropic norm of the system with multiplicative noise to be bounded by a given positive number are obtained in terms of solutions of linear matrix inequalities and a single equation.
Localization of invariant compact sets of families of discrete-time systems
Abstract
The functional method of localization of invariant compact sets developed for continuous- and discrete-time dynamical systems is extended to families of discrete-time systems. Positively invariant compact sets are considered. As an example, the method is applied to the Hénon system with uncertain parameters.