Doklady Mathematics

The global solvability and local uniqueness of a new boundary value problem for a stationary thermal diffusion model with variable coefficients, taking into account the Soret effect, are proven. A priori estimates of the norms of the main components of the solution are derived and analyzed, depending on the norms of the problem data and the leading coefficients of the model. A special dependence of the solution on the modulus of the Soret coefficient is established.

Modern optimization methods often encounter limited access to the values of the objective function, leading to the development of works dedicated to comparative oracles. This review provides an overview of contemporary algorithms for smooth, multivariate optimization that utilize only information about the order of the function values, rather than their numerical magnitudes. Both non-accelerated and accelerated methods are considered, including the latest advancements in optimization with comparative oracles. Special attention is paid to the diversity of approaches for designing algorithms that achieve convergence rates comparable to first-order methods (coordinate algorithms), as well as the first accelerated method within this oracle framework. Additionally, stochastic generalizations of the comparative oracle and their theoretical guarantees are discussed.

We proved that there are no polynomials f ∈ ℚ[x], deg f is odd, deg f ≥ 7, deg f ≠ 11,13, for which the corresponding hyperelliptic field ℚ(x)(√f) has a fundamental S-unit of degree 13 and for which the expansion of √f into a functional continued fraction is periodic. In the case deg f = 11,13, all polynomials f with the indicated properties are obtained. It is also proved that there exist at most finitely many pairwise nonequivalent polynomials f(x) ∈ ℚ[x] of degree 5 with such properties. Symbolic computations with Grobner bases play a significant role in proving the main results.

A parabolic initial-boundary value problem with homogeneous Zaremba boundary conditions in a cylinder whose base is a strictly Lipschitz bounded domain is considered, for which a theorem of the existence and uniqueness of the Green’s function is proved.

Digital technologies and the expansion of their application open up new opportunities for the development of the Russian economy, also form competitive advantages. However, the widespread adoption of digital technologies and the based digital business processes often entails the socio-economic complexities and challenges. The paper examines the information business as a new digital phenomenon. Also the paper examines the development of some regulate and support actions that provide relationship within the transition of digital transformations to a new technological level. Authors used the following scientific methods: system analysis, content analysis, sociological survey, comparative analysis, cluster analysis, mathematical modeling methods. The purpose of the paper is to substantiate and simulate the directions of state regulation of economic relations arising in the context of digital transformations. The result of the paper is a predictive economic and mathematical model of state regulation of the information business in Russia.