Vol 54, No 2 (2019)
- Year: 2019
- Articles: 9
- URL: https://journals.rcsi.science/1068-3623/issue/view/14097
Algebra
Subsets in Linear Spaces over the Finite Field F3 Uniquely Determined by Their Pairwise Sums Collection
Abstract
Let F3n be an n-dimensional linear space over the finite field F3. Let A = {a1, a2,..., aN} be a set in F3n and A + A be the collection of sums of pairs of distinct elements in A. It is said, that A is uniquely determined by A + A if for any B ⊆ F3n such that |A| = |B| and A + A = B + B it follows that A = B. In this paper, we find those values of N for which any set A ⊂ F3n containing N elements is determined uniquely by A + A.
Probability Theory and Mathematical Statistics
Non-Asymptotic Guarantees for Sampling by Stochastic Gradient Descent
Abstract
Sampling from various kinds of distributions is an issue of paramount importance in statistics since it is often the key ingredient for constructing estimators, test procedures or confidence intervals. In many situations, the exact sampling from a given distribution is impossible or computationally expensive and, therefore, one needs to resort to approximate sampling strategies. However, it is only very recently that a mathematical theory providing non-asymptotic guarantees for approximate sampling problem in the high-dimensional settings started to be developed. In this paper we introduce a new mathematical framework that helps to analyze the Stochastic Gradient Descent as a method of sampling, closely related to Langevin Monte-Carlo.
Real and Complex Analysis
On a Riemann Boundary Value Problem in the Half-plane in the Class of Weighted Continuous Functions
Abstract
Let C(ρ) be the class of functions f such that f(x)ρ(x) is continuous on (−∞,+∞). In the upper half-plane of complex plane z we consider the Riemann boundary value problem in the weighted space C(ρ) with \(\rho \left( x \right) = \prod\limits_{k = 1}^m {{{\left| {\frac{{x - {x_k}}}{{x + i}}} \right|}^{{\alpha _k}}}} \), where αk and xk are real numbers, k = 1, 2,...,m. The problem is to determine an analytic in the upper and lower half-planes function Φ(z) to satisfy \(\mathop {\lim }\limits_{y \to + 0} {\left\| {{\Phi ^ + }\left( {x + iy} \right) - a\left( x \right){\Phi ^ - }\left( {x - iy} \right) - f\left( x \right)} \right\|_{C\left( \rho \right)}} = 0\), where f ∈ C(ρ), a(x) ∈ Cδ[−A;A] for any A > 0, a(x) ≠ 0, the limit \(\mathop {\lim }\limits_{\left| x \right| \to \infty } a\left( x \right) = a\left( \infty \right)\) exists and |a(x) − a(∞)| < C|x|-δ for |x| ≥ A > 0. The normal solvability of this problem is established.
On the Intersection Points of Two Plane Algebraic Curves
Abstract
We prove that a set X, #X = mn, m ≤ n, is the set of intersection points of some two plane algebraic curves of degrees m and n, respectively, if and only if the following conditions are satisfied: a) Any curve of degree m+ n − 3 containing all but one point of X, contains all of X, b) No curve of degree less than m contains all of X.
The conditions a) and b) in the“only if” direction of this result follow fromthe Ceyley-Bacharach and Noether theorems, respectively.
Limiting Embedding Theorems for Multianisotropic Functional Spaces
Abstract
The present paper is a continuation of the author’s previous papers devoted to the study of embedding theorems for functions belonging to Sobolev multianisotropic spaces. In the previous papers were considered the cases when the embedding index is less than one, while the present paper concerns the limiting case, that is, when the embedding index is equal to one.
The Gibbs Phenomenon for Stromberg’s Piecewise Linear Wavelet
Abstract
In this paper we study the Gibbs phenomenon for a Stromberg’s piecewise linear wavelet.We prove that the Gibbs phenomenon for partial sums of Fourier-Stromberg series occurs at all points of R and the Gibbs function is almost everywhere equal to \(\frac{{1 + 2\sqrt 3 }}{3}\) .
On the Quasi-greedy Constant of the Haar Subsystems in L1(0, 1)
Abstract
In this paper, we estimate the quasi-greedy constant for quasi-greedy subsystems of the Haar system in L1(0, 1). All quasi-greedy subsystems of the Haar system are characterized in [4]. The characterization is based on the length of the chains, which is introduced in [5]. In [4], the estimate G(H) ≤ C · 2H was obtainedwith H being the length of the longest chain of the subsystem. In this paper, we improve this estimate and show that H/16 ≤ G(H) ≤ 2H + 1.