
Russian Journal of Mathematical Physics [SJR: 0.949] [HI: 23] [0 followers] Follow Hybrid journal (It can contain Open Access articles) ISSN (Print) 15318621  ISSN (Online) 10619208 Published by SpringerVerlag [2335 journals] 
 Contribution to the symplectic structure in the quantization rule due to
noncommutativity of adiabatic parameters Abstract: Abstract
A geometric construction of the `ala Planck action integral (quantization rule) determining adiabatic terms for fastslow systems is considered. We demonstrate that in the first (after zero) adiabatic approximation order, this geometric rule is represented by a deformed fast symplectic 2form. The deformation is controlled by the noncommutativity of the slow adiabatic parameters. In the case of one fast degree of freedom, the deformed symplectic form incorporates the contraction of the slow Poisson tensor with the adiabatic curvature.
The same deformed fast symplectic structure is used to represent the improved adiabatic invariant in a geometric form.
PubDate: 20160401
 Abstract: Abstract
A geometric construction of the `ala Planck action integral (quantization rule) determining adiabatic terms for fastslow systems is considered. We demonstrate that in the first (after zero) adiabatic approximation order, this geometric rule is represented by a deformed fast symplectic 2form. The deformation is controlled by the noncommutativity of the slow adiabatic parameters. In the case of one fast degree of freedom, the deformed symplectic form incorporates the contraction of the slow Poisson tensor with the adiabatic curvature.
The same deformed fast symplectic structure is used to represent the improved adiabatic invariant in a geometric form.
 Identities for Apostoltype Frobenius–Euler polynomials resulting
from the study of a nonlinear operator Abstract: Abstract
We introduce a special nonlinear differential operator and, using its properties, reduce higherorder Frobenius–Euler Apostoltype polynomials to a finite series of firstorder Apostoltype Frobenius–Euler polynomials and Stirling numbers. Interesting identities are established.
PubDate: 20160401
 Abstract: Abstract
We introduce a special nonlinear differential operator and, using its properties, reduce higherorder Frobenius–Euler Apostoltype polynomials to a finite series of firstorder Apostoltype Frobenius–Euler polynomials and Stirling numbers. Interesting identities are established.
 On rational functions of firstclass complexity
 Abstract: Abstract
It is proved that, for every rational function of two variables P(x, y) of analytic complexity one, there is either a representation of the form f(a(x) + b(y)) or a representation of the form f(a(x)b(y)), where f(x), a(x), b(x) are nonconstant rational functions of a single variable. Here, if P(x, y) is a polynomial, then f(x), a(x), and b(x) are nonconstant polynomials of a single variable.
PubDate: 20160401
 Abstract: Abstract
It is proved that, for every rational function of two variables P(x, y) of analytic complexity one, there is either a representation of the form f(a(x) + b(y)) or a representation of the form f(a(x)b(y)), where f(x), a(x), b(x) are nonconstant rational functions of a single variable. Here, if P(x, y) is a polynomial, then f(x), a(x), and b(x) are nonconstant polynomials of a single variable.
 Propagation and interaction of solitons for nonintegrable equations
 Abstract: Abstract
We describe an approach to the construction of multisoliton asymptotic solutions for nonintegrable equations. The general idea is realized in the case of N waves, N = 1, 2, 3, and for the KdVtype equation with nonlinearity u
4. A brief review of asymptotic methods as well as results of numerical simulation are included.
PubDate: 20160401
 Abstract: Abstract
We describe an approach to the construction of multisoliton asymptotic solutions for nonintegrable equations. The general idea is realized in the case of N waves, N = 1, 2, 3, and for the KdVtype equation with nonlinearity u
4. A brief review of asymptotic methods as well as results of numerical simulation are included.
 Nonsmooth nonoscillating WKB–Maslovtype asymptotics for linear
parabolic PDE Abstract: Abstract
We discuss the meaning of WKB–Maslovtype asymptotic solutions with nonsmooth phase function to parabolic PDE of Kolmogorov–Fellertype with a small parameter.
PubDate: 20160401
 Abstract: Abstract
We discuss the meaning of WKB–Maslovtype asymptotic solutions with nonsmooth phase function to parabolic PDE of Kolmogorov–Fellertype with a small parameter.
 Electromagnetic field generated by a modulated moving point source in a
planarly layered waveguide Abstract: Abstract
In the present work, we consider a modulated point source in an arbitrary motion in an isotropic planarly layered waveguide. The radiation field generated by this source is represented in the form of double oscillatory integrals in terms of the time and the frequency, depending on the large parameter λ. By means of the stationary phase method, we analyze, in the waveguide, the Doppler effect, the retarded time, and the Vavilov–Cherenkov radiation. Numerically, the problem of the moving source is approached by the method of spectral parameter power series.
PubDate: 20160401
 Abstract: Abstract
In the present work, we consider a modulated point source in an arbitrary motion in an isotropic planarly layered waveguide. The radiation field generated by this source is represented in the form of double oscillatory integrals in terms of the time and the frequency, depending on the large parameter λ. By means of the stationary phase method, we analyze, in the waveguide, the Doppler effect, the retarded time, and the Vavilov–Cherenkov radiation. Numerically, the problem of the moving source is approached by the method of spectral parameter power series.
 Bloch principle for elliptic differential operators with periodic
coefficients Abstract: Abstract
Differential operators corresponding to elliptic equations of divergent type with 1periodic coefficients are considered. The equations are put in Sobolev spaces with an arbitrary 1periodic Borel measure on the entire space R
d
. In the study of the spectrum of operators of this kind, the Bloch principle is of fundamental importance. According to this principle, all points of the desired spectrum are obtained when studying the equation on the unit cube with quasiperiodic boundary conditions. The proof of the Bloch principle for problems in the above formulation is proved, in several versions of the principle. Examples of the application of the principle to finding the spectrum of specific operators, for example, for the Laplacian in a weighted space or on a singular structure of lattice type.
PubDate: 20160401
 Abstract: Abstract
Differential operators corresponding to elliptic equations of divergent type with 1periodic coefficients are considered. The equations are put in Sobolev spaces with an arbitrary 1periodic Borel measure on the entire space R
d
. In the study of the spectrum of operators of this kind, the Bloch principle is of fundamental importance. According to this principle, all points of the desired spectrum are obtained when studying the equation on the unit cube with quasiperiodic boundary conditions. The proof of the Bloch principle for problems in the above formulation is proved, in several versions of the principle. Examples of the application of the principle to finding the spectrum of specific operators, for example, for the Laplacian in a weighted space or on a singular structure of lattice type.
 Absence of two Paley–Wiener properties for semisimple Lie groups of
real rank one Abstract: Abstract
The weak Paley–Wiener property and the topological Paley–Wiener property for connected semisimple Lie groups of real rank one with finite center are discussed.
PubDate: 20160401
 Abstract: Abstract
The weak Paley–Wiener property and the topological Paley–Wiener property for connected semisimple Lie groups of real rank one with finite center are discussed.
 Thermodynamics, idempotent analysis, and tropical geometry as a return to
primitivism PubDate: 20160401
 PubDate: 20160401
 On the distribution of energy of localized solutions of the
Schrödinger equation that propagate along symmetric quantum graphs Abstract: Abstract
From the point of view of applications to quantum mechanics, it is natural to pose a question concerning the distribution of energy of localized solutions of a nonstationary Schrödinger equation over the graph (in other words, the probability to find a quantum particle in a given area). This problem is apparently very complicated for general graphs, because the energy distribution is much more sensitive to the form of boundary conditions and to the initial state than the asymptotic behavior of the number of localized functions. Below, we present initial results concerning the distribution of energy in the case of symmetric quantum graphs (this means that the Schrödinger operators on different edges have the same structure). For general local selfadjoint boundary conditions, we describe the process of onestep scattering of the localized solutions and obtain a simple general result of the distribution of energy. Some special cases and specific examples are discussed.
PubDate: 20160401
 Abstract: Abstract
From the point of view of applications to quantum mechanics, it is natural to pose a question concerning the distribution of energy of localized solutions of a nonstationary Schrödinger equation over the graph (in other words, the probability to find a quantum particle in a given area). This problem is apparently very complicated for general graphs, because the energy distribution is much more sensitive to the form of boundary conditions and to the initial state than the asymptotic behavior of the number of localized functions. Below, we present initial results concerning the distribution of energy in the case of symmetric quantum graphs (this means that the Schrödinger operators on different edges have the same structure). For general local selfadjoint boundary conditions, we describe the process of onestep scattering of the localized solutions and obtain a simple general result of the distribution of energy. Some special cases and specific examples are discussed.
 Asymptotic analysis of evolution of a neck in extended thin rigid plastic
solids Abstract: Abstract
We carry out an asymptotic analysis with natural small geometric parameter of the stressstrain state realized under the stretch of a rigidplastic rod of round section and of a flat sheet infinite in one direction, taking into account the presence of areas of thinning (a neck) and thickening with respect to the medium size. In examples, we stress qualitative effects of possible inertiafree development of a neck.
PubDate: 20160401
 Abstract: Abstract
We carry out an asymptotic analysis with natural small geometric parameter of the stressstrain state realized under the stretch of a rigidplastic rod of round section and of a flat sheet infinite in one direction, taking into account the presence of areas of thinning (a neck) and thickening with respect to the medium size. In examples, we stress qualitative effects of possible inertiafree development of a neck.
 A finitedimensional version of Fredholm representations
 Abstract: Abstract
We consider pairs of mappings from a discrete group Γ to the unitary group. The deficiencies of these mappings from being homomorphisms may be great, but if they are close to each other, then we call such pairs balanced. We show that balanced pairs determine elements in the K
0 group of the classifying space of the group. We also show that a Fredholm representation of Γ determines balanced pairs.
PubDate: 20160401
 Abstract: Abstract
We consider pairs of mappings from a discrete group Γ to the unitary group. The deficiencies of these mappings from being homomorphisms may be great, but if they are close to each other, then we call such pairs balanced. We show that balanced pairs determine elements in the K
0 group of the classifying space of the group. We also show that a Fredholm representation of Γ determines balanced pairs.
 Belavkin filtering with squeezed light sources
 Abstract: Abstract
We derive the filtering equation for Markovian systems undergoing homodyne measurement in the situation where the output processes being monitored are squeezed. The filtering theory applies to case where the system is driven by Fock noise (that is, quantum input processes in a coherent state) and where the output is mixed with a squeezed signal. It also applies to the case of a system driven by squeezed noise, but here there is a physical restriction to emission/absorption coupling only. For the special case of a cavity mode where the dynamics is linear, we are able to derive explicitly the filtered estimate π
t
(a) for the mode annihilator a based on the homodyne quadrature observations up to time t.'
PubDate: 20160401
 Abstract: Abstract
We derive the filtering equation for Markovian systems undergoing homodyne measurement in the situation where the output processes being monitored are squeezed. The filtering theory applies to case where the system is driven by Fock noise (that is, quantum input processes in a coherent state) and where the output is mixed with a squeezed signal. It also applies to the case of a system driven by squeezed noise, but here there is a physical restriction to emission/absorption coupling only. For the special case of a cavity mode where the dynamics is linear, we are able to derive explicitly the filtered estimate π
t
(a) for the mode annihilator a based on the homodyne quadrature observations up to time t.'
 Asymptotic stability of stationary states in the wave equation coupled to
a nonrelativistic particle Abstract: Abstract
We consider the Hamiltonian system consisting of a scalar wave field and a single particle coupled in a translation invariant manner. The point particle is subjected to an external potential. The stationary solutions of the system are a Coulomb type wave field centered at those particle positions for which the external force vanishes. It is assumed that the charge density satisfies the Wiener condition, which is a version of the “Fermi Golden Rule.” We prove that in the large time approximation, any finite energy solution, with the initial state close to the some stable stationary solution, is a sum of this stationary solution and a dispersive wave which is a solution of the free wave equation.
PubDate: 20160101
 Abstract: Abstract
We consider the Hamiltonian system consisting of a scalar wave field and a single particle coupled in a translation invariant manner. The point particle is subjected to an external potential. The stationary solutions of the system are a Coulomb type wave field centered at those particle positions for which the external force vanishes. It is assumed that the charge density satisfies the Wiener condition, which is a version of the “Fermi Golden Rule.” We prove that in the large time approximation, any finite energy solution, with the initial state close to the some stable stationary solution, is a sum of this stationary solution and a dispersive wave which is a solution of the free wave equation.
 Freudenthal–Weil theorem for proLie groups
 Abstract: Abstract
An analog of the Freudenthal–Weil theorem holds for the discontinuous homomorphisms of a connected proLie group into a compact group if and only if the radical of the proLie group is amenable.
PubDate: 20160101
 Abstract: Abstract
An analog of the Freudenthal–Weil theorem holds for the discontinuous homomorphisms of a connected proLie group into a compact group if and only if the radical of the proLie group is amenable.
 A simple probabilistic model of ideal gases
 Abstract: Abstract
We describe a discrete 3D model of ideal gas based on the idea that, on the microscopic level, the particles move randomly (as in ASEP models), instead of obeying Newton’s laws as prescribed by Boltzmann.
PubDate: 20160101
 Abstract: Abstract
We describe a discrete 3D model of ideal gas based on the idea that, on the microscopic level, the particles move randomly (as in ASEP models), instead of obeying Newton’s laws as prescribed by Boltzmann.
 On solutions of the mixed Dirichlet–Navier problem for the
polyharmonic equation in exterior domains Abstract: Abstract
We study the unique solvability of the mixed Dirichlet–Navier problem for the polyharmonic equation in exterior domains under the assumption that a generalized solution of this problem has a bounded Dirichlet integral with weight x
a
. Depending on the value of the parameter a, we prove a uniqueness theorem or present exact formulas for the dimension of the solution space of the mixed Dirichlet–Navier problem in the exterior of a compact set.
PubDate: 20160101
 Abstract: Abstract
We study the unique solvability of the mixed Dirichlet–Navier problem for the polyharmonic equation in exterior domains under the assumption that a generalized solution of this problem has a bounded Dirichlet integral with weight x
a
. Depending on the value of the parameter a, we prove a uniqueness theorem or present exact formulas for the dimension of the solution space of the mixed Dirichlet–Navier problem in the exterior of a compact set.
 Stability of the autoresonance in a dissipative system
 Abstract: Abstract
The autoresonance problem is to distinguish solutions with unboundedly increasing amplitude for model equations of principal resonance. At the level of formal constructions, the problem is solved by constructing an asymptotic solution in the form of a power series with constant coefficients. As is known, such a series represents the asymptotic behavior of the exact solution. However, for this solution to be related to the description of a physical phenomenon, the stability of the solution is required both with respect to perturbations of the initial data and with respect to the relatively constantly acting perturbations. These two properties are established using the Lyapunov function.
PubDate: 20160101
 Abstract: Abstract
The autoresonance problem is to distinguish solutions with unboundedly increasing amplitude for model equations of principal resonance. At the level of formal constructions, the problem is solved by constructing an asymptotic solution in the form of a power series with constant coefficients. As is known, such a series represents the asymptotic behavior of the exact solution. However, for this solution to be related to the description of a physical phenomenon, the stability of the solution is required both with respect to perturbations of the initial data and with respect to the relatively constantly acting perturbations. These two properties are established using the Lyapunov function.
 A note on nonlinear Changhee differential equations
 Abstract: Abstract
In this paper, we study nonlinear Changhee differential equations and derive some new and explicit identities of Changhee and Euler numbers from those nonlinear differential equations.
PubDate: 20160101
 Abstract: Abstract
In this paper, we study nonlinear Changhee differential equations and derive some new and explicit identities of Changhee and Euler numbers from those nonlinear differential equations.
 Creation of spectral bands for a periodic domain with small windows
 Abstract: Abstract
We consider a Schrödinger operator in a periodic system of striplike domains coupled by small windows. As the windows close, the domain decouples into an infinite series of identical domains. The operator similar to the original one, and defined on one copy of these identical domains, has an essential spectrum. We show that once there is a virtual level at the threshold of this essential spectrum, the windows turn this virtual level into the spectral bands for the original operator. We study the structure and the asymptotic behavior of these bands.
PubDate: 20160101
 Abstract: Abstract
We consider a Schrödinger operator in a periodic system of striplike domains coupled by small windows. As the windows close, the domain decouples into an infinite series of identical domains. The operator similar to the original one, and defined on one copy of these identical domains, has an essential spectrum. We show that once there is a virtual level at the threshold of this essential spectrum, the windows turn this virtual level into the spectral bands for the original operator. We study the structure and the asymptotic behavior of these bands.