Abstract: Abstract
This paper deals with dynamics particles moving on a Euclidean n-dimensional torus or in an n-dimensional parallelepiped box in a force field whose potential is proportional to the characteristic function of the region D with a regular boundary. After reaching this region, the trajectory of the particle is refracted according to the law which resembles the Snell -Descartes law from geometrical optics. When the energies are small, the particle does not reach the region D and elastically bounces off its boundary. In this case, we obtain a dynamical system of billiard type (which was intensely studied with respect to strictly convex regions). In addition, the paper discusses the problem of the existence of nontrivial first integrals that are polynomials in momenta with summable coefficients and are functionally independent with the energy integral. Conditions for the geometry of the boundary of the region D under which the problem does not admit nontrivial polynomial first integrals are found. Examples of nonconvex regions are given; for these regions the corresponding dynamical system is obviously nonergodic for fixed energy values (including small ones), however, it does not admit polynomial conservation laws independent of the energy integral. PubDate: 2014-04-01

Abstract: Abstract
Measure permutation formulas in Feynman’s operational calculi for noncommuting operators give relationships between the two operators
\(\mathcal{T}_{\mu 1,\mu 2} f\left( {\tilde A,\tilde B} \right)\)
and
\(\mathcal{T}_{\mu 2,\mu 1} f\left( {\tilde A,\tilde B} \right)\)
. We develop generalized and iterated measure permutation formulas in the Jefferies-Johnson theory of Feynman’s operational calculi. In particular, we apply our formulas to derive an identity for a function of the Pauli matrices. PubDate: 2014-04-01

Abstract: Abstract
The paper is devoted to the investigation of the relationship between the rate of decay of the coefficients of a trigonometric sum in two variables and the smoothness of the sum of the double series in the L
p
spaces. PubDate: 2014-04-01

Abstract: Abstract
We suggest a mathematical construction to define an individual state of a quantum particle which enables one to eliminate a contradiction between quantum mechanics of correlated particles and the Relativity Theory. PubDate: 2014-04-01

Abstract: Abstract
In this paper, we consider the q-analog of the Laplace transform and investigate some properties of the q-Laplace transform. In our investigation, we derive some interesting formulas related to the q-Laplace transform. PubDate: 2014-04-01

Abstract: Abstract
We consider problems of the linearized theory of hydrodynamic stability for the case in which the unperturbed plane-parallel-flow of a viscous incompressible fluid in a layer is substantially unsteady. We analyze the Orr-Sommerfeld equation, which is generalized for this case, with different combinations of the four boundary conditions specified on the straight parts of the boundaries of the layer. Using the apparatus of integral relations, including, in particular, the analysis of the minimization problem for quadratic functionals, we derive upper bounds for the growth or decay of kinematic perturbations with respect to the integral measure. A special attention is paid to the longitudinal oscillation mode of the layer, to the power-law acceleration or deceleration, and also to the process similar to the diffusion of the vortex layer. An investigation of the reducibility of the three-dimensional picture of perturbations imposed on a plane-parallel unsteady shift to a two-dimensional picture in the plane of this shift is carried out. Generalizations of the Squire theorem are established. PubDate: 2014-04-01

Abstract: Abstract
We obtain a representation, using a Feynman formula, for the operator semigroup generated by a second-order parabolic differential equation with respect to functions defined on the Cartesian product of the line ℝ and a graph consisting of n rays issuing from a common vertex. PubDate: 2014-04-01

Abstract: Abstract
We present a version of Maslov’s canonical operator to be used when constructing asymptotic solutions of hyperbolic equations degenerating in a special way on the boundary of the spatial domain. PubDate: 2014-04-01

Abstract: Abstract
The purpose of this paper is to investigate the connection between context-free grammars and normal ordered problem, and then to explore various extensions of the Stirling grammar. We present grammatical characterizations of several well known combinatorial sequences, including the generalized Stirling numbers of the second kind related to the normal ordered problem and the r-Dowling polynomials. Also, possible avenues for future research are described. PubDate: 2014-04-01

Abstract: Abstract
We consider the main ideas that led the author to the construction of the new thermodynamics; this construction explains the Gibbs paradox and the new experimental effects of supercritical thermodynamics, as well as the law of preference of cluster formation before passage of gases to the liquid state. The law of invariance of the fractional number of degrees of freedom under constant temperature and the law of the maximal number K(T) of particles having the given energy level is established. PubDate: 2014-04-01

Abstract: Abstract
For quantum as well classical slow-fast systems, we develop a general method which allows one to compute the adiabatic invariant (approximate integral of motion), its symmetries, the adiabatic guiding center coordinates and the effective scalar Hamiltonian in all orders of a small parameter. The scheme does not exploit eigenvectors or diagonalization, but is based on the ideas of isospectral deformation and zero-curvature equations, where the role of “time” is played by the adiabatic (quantization) parameter. The algorithm includes the construction of the zero-curvature adiabatic connection and its splitting generated by averaging up to an arbitrary order in the small parameter. PubDate: 2014-04-01

Abstract: Abstract
Inspired by F. Wilczek’s QCD Lite, quantum Yang-Mills-Weyl Dynamics (YMWD) describes quantum interaction between gauge bosons (associated with a simple gauge group
\(\mathbb{G}\)
) and larks (massless chiral fields charged by an irreducible unitary representation of
\(\mathbb{G}\)
). Schrödinger representation of this quantum Yang-Mills-Weyl theory is based on a sesqui-holomorphic operator calculus of infinite-dimensional operators with variational derivatives.
The spectrum of quantum YMWD in a compact bag is a sequence of eigenvalues convergent to +∞. The eigenvalues have finite multiplicities with respect to a von Neumann algebra with a regular trace. The spectrum is inversely proportional to the square of the running coupling constant.
The rigorous mathematical theory is nonperturbative with a running coupling constant as the only ad hoc parameter. The application of the first mathematical principles is based on the properties of the compact simple Lie group
\(\mathbb{G}\)
. PubDate: 2014-04-01

Abstract: Abstract
We consider the Schrödinger operator with a periodic potential p plus a compactly supported potential q on the real line. We assume that both p and q have m ⩾ 0 derivatives. For generic p, the essential spectrum of the operator has an infinite sequence of open gaps. We determine the asymptotics of the S-matrix at high energy. PubDate: 2014-03-01

Abstract: Abstract
In the paper, we study uniqueness problems for solutions of a boundary value problem for a polyharmonic equation in the exterior of a compact set and in a half-space under the assumption that the generalized solution of the problem in question admits a finite Dirichlet integral with a weight of the form x
a
. In dependence on the values of the parameter a, we prove uniqueness theorems and also present precise formulas to evaluate the dimension of the space of solutions of this problem in the exterior of a compact set and in a half-space. PubDate: 2014-03-01

Abstract: Abstract
In this paper, we investigate some identities for Laguerre polynomials involving Bernoulli and Euler polynomials derived from umbral calculus. PubDate: 2014-03-01

Abstract: Abstract
We consider abelian groups formed by simply connected closed oriented smooth 6-manifolds with given 2-dimensional homology and given 2-dimensional Stiefel-Whitney class. In particular, an effective presentation of these groups is given. PubDate: 2014-03-01

Abstract: Abstract
The paper continues the first part (Russ. J. Math. Phys. 20 (3), 360–373). Let Ω be a John domain, let Γ ⊂ ∂Ω be an h-set, and let g and υ be weights on Ω that are distance functions to the set Γ of special form. In the paper, sufficient conditions are obtained under which the Sobolev weighted class W
p,g
r
(Ω) is continuously embedded in the space L
q,v
(Ω). Moreover, bounds for the approximation of functions in W
p,g
r
(Ω) by polynomials of degree not exceeding r − 1 in L
q,v
(
$\tilde \Omega $
) are found, where
$\tilde \Omega $
is a subdomain generated by a subtree of the tree T defining the structure of Ω. PubDate: 2014-03-01

Abstract: Abstract
We continue the study of automatic continuity conditions for finite-dimensional representations of connected Lie groups. In particular, we claim that every locally bounded finite-dimensional representation of a connected Lie group is continuous on the commutator subgroup in the intrinsic Lie topology of the subgroup and continuous on the intersection of the commutator subgroup with the radical of the group in the original topology of the Lie group, thus correcting one of our previous results. PubDate: 2014-03-01