Russian Journal of Mathematical Physics最新文献

A problem of a nonstationary incompressible viscous fluid flow along a plate with small fast-oscillating irregularities on the surface for a large Reynolds number is considered by using rigorous methods of mathematical physics. Depending on the scales of irregularities in the problem under study, there arises a solution that describes the double-deck or the triple-deck structure boundary layers on the plate. In the paper, we present a rigorous approach to the solution construction. It appears that, despite the long-term history, the triple-deck theory should be revised and the well-known Benjamin–Ono equation does not appear in this theory.

Broder introduced the (r)-Stirling numbers of the first kind and of the second kind which enumerate restricted permutations and respectively restricted partitions, the restriction being that the first (r) elements must be in distinct cycles and respectively in distinct subsets. Kim–Kim–Lee–Park constructed the degenerate (r)-Stirling numbers of both kinds as degenerate versions of them. The aim of this paper is to derive some identities and recurrence relations for the degenerate (r)-Stirling numbers of the first kind and of the second kind via boson operators. In particular, we obtain the normal ordering of a degenerate integral power of the number operator multiplied by an integral power of the creation boson operator in terms of boson operators where the degenerate (r)-Stirling numbers of the second kind appear as the coefficients.

The paper is devoted to the study of Nijenhuis operators of arbitrary dimension (n) in a neighborhood of a point at which the first (n-1) coefficients of the characteristic polynomial are functionally independent and the last coefficient (the determinant of the operator) is an arbitrary function. We prove a theorem on the general form of such Nijenhuis operators and also obtain their complete description for the case in which the determinant has a nondegenerate singularity.

We consider Schrödinger operators with complex decaying potentials (in general, not of trace class) on a lattice. We determine trace formulas in terms of the eigenvalues and the singular measure and some integrals of a Fredholm determinant. The proof is based on estimates of the free resolvent and analysis of functions in Hardy spaces. Moreover, we obtain an estimate for eigenvalues and singular measure in terms of potentials.

Solutions of the wave equation on a two-dimensional infinite cone are studied. The Laplacian is determined using the theory of extensions and is given by the boundary condition at a conical point. If the boundary condition corresponds to the Friedrichs extension (i.e., the domain of definition of the Laplacian consists of functions bounded in a neighborhood of a conical point), then the solution of the problem is known and is expressed in terms of Bessel functions. We obtain a solution to the Cauchy problem in the case of general boundary conditions using the spectral expansion of the Laplace operator.

Flows of incompressible continuous media with tensor-linear constitutive relations and an arbitrary scalar nonlinearity are investigated in the cases of absence of the yield stress (nonlinear-viscous liquids) and its presence (viscoplastic media with nonlinear viscosity). The occurrence of a finite yield stress of a medium is interpreted as a finite perturbation of the scalar dependence linking the stress intensity and the strain rate. A one-parameter family of such perturbed dependencies is proposed. As a test problem, a one-dimensional problem of stationary shear flow of a flat layer on an inclined plane in the field of gravity is given. The maximum velocities and consumptions are compared for different perturbation parameters. It is shown that, in the bounds thus arising, the sign of the convexity of the material function that specifies some nonlinear viscosity plays a great role.

A list of selected phenomena characterizing living systems on the main levels of the manifestation of life is considered. As a result of directed interpretation of these phenomena, life (in becomming of earthly reality) appears as an informational physico-mathematical process that solves the problem of optimal relations of its variability and heredity in a changing environment. Practical solutions found by living systems in the course of evolution can be used and are used in applied mathematics as approaches of working with information. Genetic and quantum algorithms, neural networks, annealing are important milestones of this use. On this way, mathematics discovers an unexpected efficiency in working with big data arrays, which represent the main content of modern biology.

Boutet de Monvel constructed an algebra of boundary value problems for pseudodifferential operators on a manifold with boundary. We define periodic cyclic cocycles on the algebra of symbols of Boutet de Monvel operators. Cocycles of this form enable one to interpret the index formula for elliptic pseudodifferential boundary value problems in the Boutet de Monvel calculus due to Fedosov as the Chern–Connes pairing with the classes in (K)-theory defined by elliptic symbols. We also consider the equivariant case. Namely, we construct periodic cyclic cocycles on the crossed product of the symbol algebra by a group acting on this algebra by automorphisms. Such crossed products arise in index theory of nonlocal boundary value problems with shift operators.

In the paper, a semiclassical asymptotics for the solution of the three-dimensional stationary Schrödinger equation with repulsive Coulomb potential and localized right-hand side is constructed. The asymptotics is based on the Lagrangian manifold woven from suitable Keplerian trajectories, and the asymptotic behavior itself is globally expressed through a combination of the Airy function and its derivative with composite arguments.

For a one-dimensional difference Schrödinger equation, we obtain uniform semiclassical asymptotic formulas for the transition matrix between basis solutions having simple asymptotic behavior in domains separated by two close turning points. We use no information on the behavior of the solutions near the turning points. Our method can be applied to a wide class of problems.