On two pursuit differential game problems with state and geometric constraints in a Hilbert space

This paper concerns with the study of two pursuit differential game problems of many pursuers and many evaders on a nonempty closed convex subset of R^n. Throughout the period of the games, players must stay within the given closed convex set. Players’ laws of motion are defined by certain first order differential equations. Control functions of the pursuers and evaders are subject to geometric constraints. Pursuit is said to be completed if the geometric position of each of the evader coincides with that of a pursuer. We proved two theorems each of which is solution to a problem. Sufficient conditions for the completion of pursuit are provided in each of the theorems. Moreover, we constructed strategies of the pursuers that ensure completion of pursuit.

A reaction-diffusion-advection competition model with a free boundary

In this paper, we study a competitive diffusion quasilinear system with a free boundary. First, we investigate the mathematical questions of the problem. A priori estimates of Schauder type are established, which are necessary for the solvability of the problem. One of two competing species is an invader, which initially exists on a certain sub-interval. The other is initially distributed throughout the area under consideration. Examining the influence of baseline data on the success or failure of the first invasion. We conclude that there is a dichotomy of spread and extinction and give precise criteria for spread and extinction in this model.

An optimal quadrature formula in the Sobolev space

This paper studies the problem of construction of optimal quadrature formulas for approximate calculation of integrals with trigonometric weight in the L(2m)(0, 1) space for any ω ൐= 0, ω ∈ R. Here explicit formulas for the optimal coefficients are obtained. We study the order of convergence of the optimal formulas for the case m = 1, 2. The obtained optimal quadrature formulas are exact for Pm−1(x), where Pm−1(x) is a polynomial of degree (m − 1).

An integral criterion for m − sh functions in terms of ellipsoids in Cn

In this paper we give a necessary and sufficient condition for the functions to be m - subharmonic in terms of integral mean by complex ellipsoids.

Initial-boundary value problem for a time-fractional subdiffusion equation on the torus

An initial-boundary value problem for a time-fractional subdiffusion equation with the Riemann-Liouville derivatives on N-dimensional torus is considered. Uniqueness and existence of the classical solution of the posed problem are proved by the classical Fourier method. Sufficient conditions for the initial function and for the right-hand side of the equation are indicated, under which the corresponding Fourier series converge absolutely and uniformly. It should be noted, that the condition on the initial function found in this paper is less restrictive than the analogous condition in the case of an equation with derivatives in the sense of Caputo.

Numerical solution of the system of Marchenko integral equations

In this paper, inverse scattering problems for a system of differential equations of the first order are considered. The Marchenko approach is used to solve the inverse scattering problem. The system of Marchenko integral equations is reduced to a linear system of algebraic equations such that the solution of the resulting system yields to the unknown coefficients of the system of first-order differential equations. Illustrative examples are provided to demonstrate the preciseness and effectiveness of the proposed technique. The results are compared with the exact solution by using computer simulations.

Structures of QTAG-modules determined by their submodules

A module M over an associative ring R with unity is a QTAG-module if every finitely generated submodule of any homomorphic image of M is a direct sum of uniserial modules. In this paper, we study the existence of several classes C of QTAG-modules which satisfy the property that M belongs to C uniquely when M/N belongs to C provided that N is a finitely generated submodule of the QTAG-module.

On the generalization of the central limit theorem for the least-squares estimator of the unknown parameter in the autoregressive process of order one (AR(1))

In this work it is proved central limit theorem for the least-squares estimator of the unknown parameter in the generalization autoregressive process of order one (AR(1)).

Inverse problems of determining an order of fractional Riemann-Liouville time-fractional derivative for the subdiffusion equation in RN

An initial-boundary value problem for the subdiffusion equation with an elliptic operator A(D) in RN is studied in the article. Existence and uniqueness theorems for the problem under study are proved by the Fourier method. Considering the order of the Riemann-Liouville time-fractional derivative as an unknown parameter, an inverse problem of determining this parameter is investigated. Likewise, the initial-boundary value problem was considered in the case of replacing the operator A(D) with its power Aσ.Then, existence and uniqueness theorems were proved for the solution of the inverse problem of determining the order of the fractional derivative and the power σ.