Mathematical Notes最新文献

Abstract

We consider an (n)-dimensional system of first-order ordinary differential equations with a constant matrix having real, simple, and nonzero eigenvalues, with a discontinuous nonlinearity of two-position relay type with positive hysteresis and a continuous bounded perturbation function. We study continuous two-point oscillatory solutions with a certain period for the representative point to be returned to the switching hyperplane in the state space. When solving the Cauchy problem with initial condition at the switching point, we use the fitting method. We construct a system of transcendental equations for the switching instants and points. We prove a criterion for the existence and uniqueness of a solution with some fixed return period. For a system in the canonical form with diagonal matrix and with feedback vector of a special form, we obtain conditions for the solvability of a system of transcendental equations for the first switching instant for a given return period and formulas for the switching points. For a three-dimensional system, we give a numerical example to illustrate the theoretical results.

Abstract

On a uniform grid on the real axis, we study the Yanenko–Stechkin–Subbotin problem of extremal function interpolation in the mean in the space (L_p(mathbb R)), (1<p<infty), of two-way real sequences with the least value of the norm of a linear formally self-adjoint differential operator ({mathcal L}_n) of order (n) with constant real coefficients. In case of even (n), the value of the least norm in the space (L_p(mathbb R)), (1<p<infty), of the extremal interpolant is calculated exactly if the grid step (h) and the averaging step (h_1) are related by the inequality (h<h_1le 2h).

Abstract

In a finite-dimensional space, we consider a linear stochastic differential equation in Itô form with a singular constant matrix on the left-hand side. Taking into account various economic applications of such equations, they are classified as Leontief type equations, since under some additional assumptions, a deterministic analog of the equation in question describes the famous Leontief input–output balance model taking into account reserves. In the literature, these systems are more often called differential–algebraic or descriptor systems. In general, to study this type of equations, one needs higher-order derivatives of the right-hand side. This means that one must consider derivatives of the Wiener process, which exist in the generalized sense. In the previous papers, these equations were studied using the technique of Nelson mean derivatives of random processes, whose description does not require generalized functions. It is well known that mean derivatives depend on the (sigma)-algebra used to find them. In the present paper, the study of this equation is carried out using mean derivatives with respect to a new (sigma)-algebra that was not considered in the previous papers.

Abstract

We study a boundary value problem for a mathematical model describing the nonisothermal steady-state flow of a viscous fluid in a 3D (or 2D) bounded domain with locally Lipschitz boundary. The heat and mass transfer model considered here has the feature that a regularized Rayleigh dissipation function is used in the energy balance equation. This permits taking into account the energy dissipation due to the viscous friction effect. A theorem on the existence of a weak solution is proved under natural assumptions on the model data. Moreover, we establish extra conditions guaranteeing that the weak solution is unique and/or strong.

Abstract

Stechkin’s problem of the best approximation of differentiation operators by bounded linear operators on the half-line in the uniform norm is studied. The structure of the best approximation operator is investigated, and its relationship to the spline dual (in the sense of N. P. Kuptsov) to the extremal spline in the Landau–Kolmogorov inequality on the half-line is examined.

Abstract

We consider one thermal-electrical ((1+1))-dimensional model of heating a semiconductor in an electric field. For the corresponding Cauchy problem, we prove the existence of a classical solution nonextendable in time and obtain a global-in-time a priori estimate.

Abstract

In this paper, we study the ideal strong lacunary ward compactness of a subset of a 2-normed space (X) and the ideal strongly lacunary ward continuity of a function (f) on (X). Here a subset (E) of (X) is said to be ideal strong lacunary ward compact if any sequence in (E) has an ideal strong lacunary quasi-Cauchy subsequence. Additionally, a function on (X) is said to be ideal strong lacunary ward continuous if it preserves ideal strong lacunary quasi-Cauchy sequences; an ideal is defined to be a hereditary and additive family of subsets of (mathbb{N}). We find that a subset (E) of (X) with a countable Hamel basis is totally bounded if and only if it is ideal strong lacunary ward compact.

Abstract

We prove that if a quasi-Kähler manifold satisfies the (eta)-quasi-umbilical quasi-Sasakian hypersurfaces axiom, then it is a Kähler manifold. We also prove that the quasi-Sasakian structure on an (eta)-quasi-umbilical hypersurface in a quasi-Kähler manifold is either cosymplectic or homothetic to a Sasakian structure.

Abstract

The chief factor (H/K) of a group (G) is said to be (mathfrak{F})-central if

The (mathfrak{F})-hypercenter of a group (G) is defined to be a maximal normal subgroup of (G) such that all (G)-composition factors below it are (mathfrak{F})-central in (G). In 1995, at the Gomel algebraic seminar, L. A. Shemetkov formulated the problem of describing formations of finite groups (mathfrak{F}) for which, in any group, the intersection of (mathfrak{F})-maximal subgroups coincides with the (mathfrak{F})-hypercenter. In the present paper, new properties of such formations are obtained. In particular, a series of hereditary nonsaturated formations of soluble groups is constructed, which answer Shemetkov’s problem.

Abstract

Let (G = (V,E)) be a graph of order (n). For (S subseteq V(G)), the set (N_e(S)) is defined as the external neighborhood of (S) such that all vertices in (V(G)backslash S) have at least one neighbor in (S). The differential of (S) is defined to be (partial(S)=|N_e(S)|-|S|), and the 2-packing differential of a graph is defined as

A function (fcolon V(G) to {0,1,2}) with the sets (V_0,V_1,V_2), where

is a unique response Roman dominating function if (x in V_0 ) implies that (| N( x ) cap V_2 | = 1) and (x in V_1 cup V_2 ) implies that (N( x ) cap V_2 = emptyset). The unique response Roman domination number of (G), denoted by (mu_R(G)), is the minimum weight among all unique response Roman dominating functions on (G). Let (bar{G}) be the complement of a graph (G). The complementary prism (Gbar {G}) of (G) is the graph formed from the disjoint union of (G) and (bar {G}) by adding the edges of a perfect matching between the respective vertices of (G) and (bar {G}). The present paper deals with the computation of the 2-packing differential and the unique response Roman domination of the complementary prisms (Gbar {G}) by the use of a proven Gallai-type theorem. Particular attention is given to the complementary prims of special types of graphs. Furthermore, the graphs (G) such that (partial_{2p} ( Gbar G)) and (mu _R(Gbar G)) are small are characterized.