Russian Journal of Mathematical Physics最新文献

In this paper, we present a presumably new representation of the Bernoulli numbers. We also give an elementary proof of the Akiyama-Tanigawa algorithm for calculating the Bernoulli numbers.

One of the simplest and most important results following directly from the commutativity of the discontinuity group of a locally bounded homomorphism between connected Lie groups is the automatic continuity of every locally bounded homomorphism of a perfect Lie group which is proved here.

Convexity of (delta)-suns and (gamma)-suns is studied in asymmetric spaces with due consideration of geometric properties of the spaces. Known results for usual normed spaces are carried over to the case of general asymmetric normed spaces.

In the present discussion, we have studied the (mathbb{Z}_{2}-)(grading) of the quaternion algebra ((mathbb{H})). We have made an attempt to extend the quaternion Lie algebra to the graded Lie algebra by using the matrix representations of quaternion units. The generalized Jacobi identities of (mathbb{Z}_{2}-graded) algebra then result in symmetric graded partners ((N_{1},N_{2},N_{3})). The graded partner algebra ((mathcal{F})) of quaternions ((mathbb{H})) thus has been constructed from this complete set of graded partner units ((N_{1},N_{2},N_{3})), and (N_{0}=C). Keeping in view the algebraic properties of the graded partner algebra ((mathcal{F})), the (mathbb{Z}_{2}-graded) superspace ((S^{l,m})) of quaternion algebra ((mathbb{H})) has been constructed. It has been shown that the antiunitary quaternionic supergroup (UU_{a}(l;m;mathbb{H})) describes the isometries of (mathbb{Z}_{2}-graded) superspace ((S^{l,m})). The Superconformal algebra in (D=4) dimensions is then established, where the bosonic sector of the Superconformal algebra has been constructed from the quaternion algebra ((mathbb{H})) and the fermionic sector from the graded partner algebra ((mathcal{F})): asymmetric space, convex set, (delta)-sun, (gamma)-sun.

In the paper, formal asymptotic solutions of the initial-boundary value problem for the one-dimensional Klein–Gordon equation with variable coefficients on the semi-axis are constructed. Such a problem can be used, in particular, to simulate the propagation of plane acoustic waves in atmospheric gas initiated by a source at the lower boundary of the domain.

We define natural volume forms on (n)-dimensional oriented pseudo-Finslerian manifolds with nondegenerate (m)-th root metrics. Our definitions of the natural volume forms depend on the parity of the positive integer (m>1). The advantage of the stated definitions is their algebraic structure. The natural volume forms are expressed in terms of Cayley hyperdeterminants. In particular, the computation of the natural volume form does not require the difficult integration over the domain within the indicatrix in the tangent space (T_x M^n) of the pseudo-Finslerian manifold at a point (x).

Let (Y) be a random variable whose moment generating function exists in a neighborhood of the origin. The aim of this paper is to introduce and study the probabilistic extension of Bernoulli polynomials and Euler polynomials, namely the probabilistic Bernoulli polynomials associated (Y) and the probabilistic Euler polynomials associated with (Y). Also, we introduce the probabilistic (r)-Stirling numbers of the second associated (Y), the probabilistic two variable Fubini polynomials associated (Y), and the probabilistic poly-Bernoulli polynomials associated with (Y). We obtain some properties, explicit expressions, certain identities and recurrence relations for those polynomials. As special cases of (Y), we treat the gamma random variable with parameters (alpha,beta > 0), the Poisson random variable with parameter (alpha >0), and the Bernoulli random variable with probability of success (p).

DOI 10.1134/S106192084010072

Let ((M,g)) be a Riemannian manifold, (Omegasubset M) a domain with boundary (Gamma), and (phi) a smooth function such that (phi|_Omega > 0), ( varphi |_Gamma = 0), and (dphi|_Gammane 0). We study the geodesic flow of the metric (G=g/phi). The (G)-distance from any point of (Omega) to (Gamma) is finite, hence, the geodesic flow is incomplete. Regularization of the flow in a neighborhood of (Gamma) establishes a natural reflection law from (Gamma). This leads to a certain (not quite standard) billiard problem in (Omega).

DOI 10.1134/S106192084010047

In (L_2(mathbb{R}^d)), we consider a self-adjoint bounded operator ({mathbb A}_varepsilon), (varepsilon >0), of the form

It is assumed that (a(mathbf{x})) is a nonnegative function such that (a(-mathbf{x}) = a(mathbf{x})) and (int_{mathbb{R}^d} (1+| mathbf{x} |^4) a(mathbf{x}),dmathbf{x}<infty); (mu(mathbf{x},mathbf{y})) is (mathbb{Z}^d)-periodic in each variable, (mu(mathbf{x},mathbf{y}) = mu(mathbf{y},mathbf{x})) and (0< mu_- leqslant mu(mathbf{x},mathbf{y}) leqslant mu_+< infty). For small (varepsilon), we obtain an approximation of the resolvent (({mathbb A}_varepsilon + I)^{-1}) in the operator norm on (L_2(mathbb{R}^d)) with an error of order (O(varepsilon^2)).

DOI 10.1134/S106192084010114

We study the perturbation of the Schrödinger operator on the plane with a bounded potential of the form (V_1(x)+V_2(y),) where (V_1) is a real function and (V_2) is a compactly supported function. It is assumed that the one-dimensional Schrödinger operator ( mathcal{H} _1) with the potential (V_1) has two real isolated eigenvalues ( Lambda _0,) ( Lambda _1) in the lower part of its spectrum, and the one-dimensional Schrödinger operator ( mathcal{H} _2) with the potential (V_2) has a virtual level at the boundary of its essential spectrum, i.e., at (lambda=0), and a spectral singularity at the inner point of the essential spectrum (lambda=mu>0). In addition, the eigenvalues and the spectral singularity overlap in the sense of the equality ( lambda _0:= Lambda _0+mu= Lambda _1.) We show that a perturbation by an abstract localized operator leads to a bifurcation of the internal threshold ( lambda _0) into four spectral objects which are resonances and/or eigenvalues. These objects correspond to the poles of the local meromorphic continuations of the resolvent. The spectral singularity of the operator ( mathcal{H} _2) qualitatively changes the structure of these poles as compared to the previously studied case where no spectral singularity was present. This effect is examined in detail, and the asymptotic behavior of the emerging poles and corresponding spectral objects of the perturbed Schrödinger operator is described.

DOI 10.1134/S106192084010059