Russian Journal of Mathematical Physics最新文献

We discuss homogenization of a strongly elliptic operator (mathcal A^varepsilon=-operatorname{div}A(x,x/varepsilon_#)nabla) on a bounded (C^{1,1}) domain in (mathbb R^d) with either Dirichlet or Neumann boundary condition. The function (A) is piecewise Lipschitz in the first variable and periodic in the second one, and the function (varepsilon_#) is identically equal to (varepsilon_i(varepsilon)) on each piece (Omega_i), with (varepsilon_i(varepsilon)to0) as (varepsilonto0). For (mu) in a resolvent set, we show that the resolvent ((mathcal A^varepsilon-mu)^{-1}) converges, as (varepsilonto0), in the operator norm on (L_2(Omega)^n) to the resolvent ((mathcal A^0-mu)^{-1}) of the effective operator at the rate (varepsilon_ {vee} ), where (varepsilon_ {vee} ) stands for the largest of (varepsilon_i(varepsilon)). We also obtain an approximation for the resolvent in the operator norm from (L_2(Omega)^n) to (H^1(Omega)^n) with error of order (varepsilon_ {vee} ^{1/2}).

A version of the Weyl complete reducibility theorem for finite-dimensional quasirepresentations of general connected Lie groups is proved.

We study some classes of noncommutative (C^*)-algebras and generalize some results which were originally obtained for commutative algebras in topological terms. In particular, we are interested in results obtained for topological spaces with properties close to separability and ( sigma )-compactness. To obtain the algebraic, noncommutative versions of corresponding properties, we define and use the notions of thick elements and states. In particular, an element is thick if the only element orthogonal to it is zero.

As is well known, on an infinite-dimensional Hilbert space, there is no countably additive sigma-finite locally finite nonzero translation-invariant nonnegative Borel measure (Andre Weil’s theorem, [1]). For this reason, to formalize the Feynman path integrals [2], one has to introduce a generalized translation-invariant measure (Lebesgue–Feynman in the sense of the definition in [2]) as a linear functional on some space of functions. The present paper proposes a natural extension of one of these functionals that were introduced in [3] and called there the generalized Lebesgue measure (henceforth, we call this (generalized) measure the Lebesgue–Feynman–Smolyanov). The extension makes it possible to give a precise mathematical meaning to the Schrödinger quantization of noncylindrical Hamiltonians for Hamiltonian systems with infinitely many degrees of freedom [3]: in particular, to give a correct mathematical solution to the problem of infinite vacuum energy at the bosonic quantization of the “free” electromagnetic field (N. N. Bogoliubov, D. V. Shirkov, Introduction to the Theory of Quantized Fields, John Wiley & Sons, New York–Chichester–Brisbane, 1980); invariant measures themselves have recently been used for the mathematical description of the phenomena of quantum anomalies [4, 5, 6].

An estimate for the domain of constant sign for a function harmonic in the unit disk is obtained under the condition that the function is represented on the boundary of the circle as a sine series with monotonic coefficients.

In the paper, an approach is discussed that makes it possible to obtain global formulas in terms of Airy functions ({rm Ai}) and ({rm Bi}) of compound argument for the asymptotics of the functions of parabolic cylinder (D_{nu}(z)) for real (z) and large (nu). The parabolic cylinder functions are determined from the Schrödinger equation, with potential in the form of a quadratic parabola, whose asymptotic solution can be constructed using the semiclassical approximation. In this case, the Bohr–Sommerfeld condition singles out the functions with an integer index whose asymptotics is determined only by the function ({rm Ai}). For noninteger indices, the function ({rm Bi}) also contributes into the asymptotics.

The paper is devoted to the analysis of relationships between principal objects of the spectral theory of dynamical systems (transfer and weighted shift operators) and basic characteristics of information theory and thermodynamic formalism (entropy and topological pressure). We present explicit formulas linking these objects with the (t)-entropy and spectral potential. Herewith we uncover the role of inverse rami-rate, the forward entropy along with an essential set, and the property of noncontractibility of a dynamical system.

In the paper, we study the singularities of smooth functions of two variables that are invariant under the action of a finite group (G) acting by rotations. A classification is obtained for critical points arising in typical parametric families of (G)-invariant smooth functions with at most two parameters, when (|G|ne4). A criterion is obtained for the reducibility of a smooth (G)-invariant function to a normal form (by means of a (G)-equivariant change of variables) when the Taylor polynomial of degree (|G|) of the function is not a polynomial in (x^2+y^2) and the Milnor (G)-multiplicity (the (G)-codimension, respectively) of the singularity is less than (|G|) (than (|G|/2), respectively). A criterion is obtained for the reducibility of a smooth parametric family of (G)-invariant functions to a normal form near the critical point of the type in question. The criteria are given in terms of partial derivatives of the function at the critical point.

In the paper, a systematic construction of the theory of “weighted” model surfaces using the Bloom–Graham–Stepanova concept of the type of a CR-manifold is given. The construction is based on the Poincaré construction. It is shown how the use of weighted model surfaces expands the abilities of the method. New questions are being posed.

The aim of this paper is to investigate some properties, recurrence relations and identities involving degenerate Stirling numbers of both kinds associated with degenerate hyperharmonic numbers and also with degenerate Bernoulli, degenerate Euler, degenerate Bell, and degenerate Fubini polynomials.