Russian Journal of Mathematical Physics最新文献

In the present paper, we review the (q)-analog of the Quantum Theory of Angular Momentum based on the (q)-algebra (su_q(2)) with a special emphasis on the representation of the Clebsch–Gordan coefficients in terms of (q)-hypergeometric series. This representation allows us to obtain several known properties of the Clebsch–Gordan coefficients in an unified and simple way.

DOI 10.1134/S106192084010023

We consider a linearization problem for Nijenhuis operators in dimension two around a point of scalar type in analytic category. The problem was almost completely solved in [8]. One case, however, namely the case of left-symmetric algebra (mathfrak b_{1, alpha}), proved to be difficult. Here we solve it and, thus, complete the solution of the linearization problem for Nijenhuis operators in dimension two. The problem turns out to be related to classical results on the linearization of vector fields and their monodromy mappings.

DOI 10.1134/S106192084010084

There are two ways to describe a geometric object (L): the object as an image of a mapping and the object as a preimage. Every method has its own advantages and shortcomings; together, they give a complete picture. In order to compare these descriptions by complexity, one can use Kolmogorov’s approach: i.e., after the clarification of the system of basic operations, the complexity of a description is the minimum length of the defining text. Accordingly, we obtain two Kolmogorov complexities: in the first case, (K^{+}(L)), and in the other, (K^{-}(L)). Let (Cl^n) be the class of functions of two variables that can be represented by analytic functions of one variable and by the addition of the depth not exceeding (n), and let (K^{+}(Cl^n)) and (K^{-}(Cl^n)) be their corresponding Kolmogorov complexities. There are arguments in favor of the fact that, for (n geq 2), the value of (K^{-}(Cl^n)) is very large, and the task of constructing a description of (Cl^n) in the form of a preimage (by defining relations) even for (n=2) is computationally unrealizable. Based on this observation, a signal encoding-decoding scheme is proposed, and arguments are given in favor of the fact that the decoding of a signal encoded using such a scheme is inaccessible to a quantum computer.

DOI 10.1134/S106192084010035

Given a graph (Gamma), one may consider the set (X) of its vertices as a metric space by assuming that all edges have length one. We consider two versions of homology theory of (Gamma) and their (K)-theory counterparts — the (K)-theory of the (uniform) Roe algebra of the metric space (X) of vertices of (Gamma). We construct here a natural mapping from homology of (Gamma) to the (K)-theory of the Roe algebra of (X), and its uniform version. We show that, when (Gamma) is the Cayley graph of (mathbb Z), the constructed mappings are isomorphisms.

DOI 10.1134/S106192084010102

Investigations concerning the extension of characters on normal subgroups to one-dimensional pure pseudorepresentations of the enveloping groups are continued. We prove necessary and sufficient conditions that an ordinary unitary character on the radical of a connected Lie group admits an extension to a one-dimensional pure pseudorepresentation of the group and prove the uniqueness of this pure pseudorepresentation if it exists.

DOI 10.1134/S106192084010126

We construct asymptotic solutions of a special type for the nonlinear system of shallow water equations in two-dimensional basins with gentle shores and depth function (D(x)), where (x=(x_1,x_2)). These solutions represent waves localized near the shorelines (coastal waves) and generalize the (linear) Stokes and Ursell waves. The waves we consider are periodic or close to periodic in time. The corresponding asymptotic solutions are represented in a parametric form based on the modification of the Carrier–Greenspan transformation and are generated by asymptotic eigenfunctions (quasimodes) of the operator (hat{H} = -nablacdot(gD(x)nabla)), where (g) is the gravity acceleration. These eigenfunctions are, in general, related to the trajectories of a Hamiltonian system with the Hamiltonian (H = gD(x)(p_1^2+p_2^2)), which forms billiards with “semi-rigid walls.” In the general case, the existence of such billiards assumes the integrability condition that is practically impossible to be satisfied in real situations. However, we consider a “degenerate” situation where the trajectories are localized in a very narrow vicinity of the boundary (Gamma_0={D(x)=0}), and the asymptotic eigenfunctions resemble the well-known “whispering gallery” wave functions in acoustics. In this case, the requirement of integrability is eliminated (the corresponding billiard is “almost integrable” for the considered set of trajectories). One important difference between the problem we study and the classical whispering gallery situation is that, due to the degeneracy of the depth function (D(x)) on the boundary (Gamma_0), the trajectories are always normal to the boundary, and the requirement of convexity of the domain of the considered problem is absent.

DOI 10.1134/S106192084010060

We study the problem of the decay of initial data in the form of a unit step for the Bogoyavlensky lattices. In contrast to the Gurevich–Pitaevskii problem of the decay of initial discontinuity for the KdV equation, it turns out to be exactly solvable, since the dynamics is linearizable due to termination on the half-line. The answer is written in terms of generalized hypergeometric functions, which serve as exponential generating functions for generalized Catalan numbers. This can be proved by the fact that the generalized Hankel determinants for these numbers are equal to 1, which is a well-known result in combinatorics. Another method is based on a nonautonomous symmetry reduction consistent with the dynamics. It reduces the lattice equation to a finite-dimensional system and makes it possible to solve the problem for a more general finite-parameter family of initial data.

DOI 10.1134/S106192084010011

Faddeev and Zakharov determined the trace formulas for the KdV equation with real initial conditions in 1971. We reprove these results for the KdV equation with complex initial conditions. The Lax operator is a Schrödinger operator with complex-valued potentials on the line. The operator has essential spectrum on the half-line plus eigenvalues (counted with algebraic multiplicity) in the complex plane without the positive half-line. We determine series of trace formulas. Here we have a new term: a singular measure, which is absent for real potentials. Moreover, we estimate of sum of the imaginary part of eigenvalues plus the singular measure in terms of the norm of potentials. The proof is based on classical results about the Hardy spaces.

DOI 10.1134/S106192084010096

The polyharmonic Navier problem is considered, the uniqueness (non-uniqueness) of its solution is studied in unbounded domains under the assumption that the generalized solution of this problem has a finite Dirichlet integral with weight (|x|^a). Depending on the values of the parameter (a), uniqueness theorems are proved and exact formulas are found for calculating the dimension of the space of solutions of the Navier problem for a polyharmonic equation in the exterior of a compact set and in a half-space.

DOI 10.1134/S1061920823040209

The paper considers the Cauchy problem for a multidimensional quasilinear hyperbolic system of differential equations with the data rapidly oscillating in time. This data do not explicitly depend on spatial variables. The method by N. M. Krylov–N. N. Bogolyubov is developed and justified for these systems. Also an algorithm is developed and justified, based on this method and the method of two-scale expansions, for constructing the complete asymptotics of solutions.

DOI 10.1134/S1061920823040118