卷 54, 编号 2 (2019)

In this paper we generalize Belousov’s theorem on linearity of invertible algebras with ∀∃(∀) identities. The obtained results can be used for classification of medial hyperidentities.

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.

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 α_k and x_k 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) ∈ C^δ[−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.

In this paper we describe the meromorphic solutions of the equations fⁿ(z) + (f')ⁿ (z) = e^αz+β and fⁿ(z) + fⁿ(z + c) = e^αz+β (c ≠ 0) over the complex plane C for integers n ≥ 1.

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 \(\frac{{1 + 2\sqrt 3 }}{3}\) .