Doklady Mathematics最新文献

The existence and uniqueness of a strong solution for an inhomogeneous incompressible Kelvin–Voigt fluid motion model is proved. It is not assumed that the initial value of the fluid density is separated from zero. To prove the existence of a solution, an approximation problem is considered, its solvability is proved, and strong a priori estimates independent of the approximation parameter are established for its solutions. After that, passing to the limit as the approximation parameter tends to zero, we show that the solutions of the approximation problem converge to a strong solution of the original problem as the approximation parameter tends to zero. The uniqueness of the solution is established using the Gronwall–Bellman inequality.

We investigate a nonlinear Schrödinger equation of general form in which the chromatic dispersion and the potential are given by two arbitrary functions. The equation is a natural generalization of a wide class of related nonlinear partial differential equations that are often encountered in various fields of theoretical physics and mechanics, including nonlinear optics and plasma physics. For several general nonlinear Schrödinger equations, new exact solutions in implicit form are found, which are expressed in terms of elementary and arbitrary functions. One-dimensional reductions are described, which reduce the considered partial differential equation to simpler ordinary differential equations or systems of such equations.

The paper describes a new method for constructing triangle-free graphs with an arbitrarily large chromatic number. The method is substantiated using properties of various types of ultrafilter extensions of functions and predicates.

The unique DipolarCalc software system is presented, which makes it possible to calculate the polarization of biomolecules through the total dipole moment of chemical bonds of “functional topological atoms” (FTA). The program implements a user-friendly interface that provides intuitive data entry and visualization of calculation results, as well as it uses analytical expressions that allow accurate calculations to be performed in real time. The presented software package is aimed at use in computational biology and provides a convenient tool for modeling intra- and intermolecular interactions of biological molecular structures.

Given a compact metric space with Carathéodory measure, we consider ergodic transformations of the space that are measure-preserving, but not necessarily invertible. The behavior of the Birkhoff sums for integrable and almost everywhere bounded functions with zero mean value in terms of the Carathéodory measure is studied. It is shown that for almost all points of the metric space there is an infinite sequence of “time instants” along which the Birkhoff sums tend to zero and the trajectory points at the these instants approach their initial position as close as possible (as in the Poincaré recurrence theorem). As an example, we consider the transformation (x mapsto 2x) (bmod 1) of the unit interval (0,leqslant ,x,leqslant ,1) closely related to Bernoulli trials.

We consider an even-order ordinary differential operator with a spectral parameter and integral conditions. An a priori estimate of solutions to the problem is obtained for sufficiently large values of the spectral parameter. The discreteness and the sectoral structure of the spectrum of the corresponding operators are proved.

The main methods and approaches of the theory of discontinuous systems are used to construct the theory of functional differential equations with a discontinuous right-hand side. In particular, methods for describing sets of discontinuity points and sliding modes of discontinuous systems with delay are considered using a special class of invariantly differentiable functionals.

We study the topological structure of the solution set of the Cauchy problem for semilinear differential inclusions of fractional order (alpha in (1,2)) in Banach spaces. It is assumed that the linear part of the inclusions is a linear closed operator generating a strongly continuous and uniformly bounded family of cosine operator functions. The nonlinear part is represented by an upper semicontinuous multivalued operator of Carathéodory type. It is established that the solution set of the problem is an ({{R}_{delta }})-set.

Holomorphic self-maps of the unit disk with boundary fixed points are investigated. In 1982, Cowen and Pommerenke established an interesting generalization of the classical Julia–Carathéodory theorem, which allowed them to derive a sharp estimate for the derivative at the Denjoy–Wolff point on the class of functions with an arbitrary finite set of boundary fixed points. In this paper, we obtain a new generalization of the Julia–Carathéodory theorem, which contains the Cowen–Pommerenke result as a special case; moreover, it is an effective tool for solving various problems on classes of functions with fixed points.

In classical works, the Hubble constant is defined via a metric. Here we define it, as it should be, via matter, following Milne and McCrea and extending their theory of the expanding Universe to the relativistic case. This allows us to explain the accelerated expansion as a simple relativistic effect without resorting to Einstein’s lambda, dark energy, or new particles, more specifically, as an exact consequence of the classical Einstein action. The well-verified fact of accelerated expansion allows us to determine the sign of the curvature in the Friedmann model: it turns out to be negative, and we live in Lobachevsky space.