
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 [2281 journals] 
 Thermodynamics and tropical mathematics. Definition of quasistatistical
processes Abstract: Abstract
We consider the relations between thermodynamics on the one hand and the (max,+)algebra and tropical mathematics on the other hand. The contribution of Grigorii Litvinov to tropical geometry is emphasized. Relations for a liquid in the negative pressure domain are given.
PubDate: 20160101
 Abstract: Abstract
We consider the relations between thermodynamics on the one hand and the (max,+)algebra and tropical mathematics on the other hand. The contribution of Grigorii Litvinov to tropical geometry is emphasized. Relations for a liquid in the negative pressure domain are given.
 The field theory of specific heat
 Abstract: Abstract
Finite temperature quantum field theory in the heat kernel method is used to study the heat capacity of condensed matter. The lattice heat is treated à la P. Debye as energy of the elastic (sound) waves. The dimensionless functional of free energy is rederived with a cutoff parameter and used to obtain the specific heat of crystal lattices. The new dimensionless thermodynamical variable is formed as Planck’s inverse temperature divided by the lattice constant. The dimensionless constant, universal for the class of crystal lattices, which determines the low temperature region of molar specific heat, is introduced and tested with the data for diamond lattice crystals. The low temperature asymptotics of specific heat is found to be the fourth power in temperature instead of the cubic power law of the Debye theory. Experimental data for the carbon group elements (silicon, germanium) and other materials decisively confirm the quartic law. The true low temperature regime of specific heat is defined by the surface heat, therefore, it depends on the geometrical characteristics of the body, while the absolute zero temperature limit is geometrically forbidden. The limit on the growth of specific heat at temperatures close to critical points, known as the Dulong–Petit law, appears from the lattice constant cutoff. Its value depends on the lattice type and it is the same for materials with the same crystal lattice. The Dulong–Petit values of compounds are equal to those of elements with the same crystal lattice type, if one mole of solid state matter were taken as the Avogadro number of the composing atoms. Thus, the Neumann–Kopp law is valid only in some special cases.
PubDate: 20160101
 Abstract: Abstract
Finite temperature quantum field theory in the heat kernel method is used to study the heat capacity of condensed matter. The lattice heat is treated à la P. Debye as energy of the elastic (sound) waves. The dimensionless functional of free energy is rederived with a cutoff parameter and used to obtain the specific heat of crystal lattices. The new dimensionless thermodynamical variable is formed as Planck’s inverse temperature divided by the lattice constant. The dimensionless constant, universal for the class of crystal lattices, which determines the low temperature region of molar specific heat, is introduced and tested with the data for diamond lattice crystals. The low temperature asymptotics of specific heat is found to be the fourth power in temperature instead of the cubic power law of the Debye theory. Experimental data for the carbon group elements (silicon, germanium) and other materials decisively confirm the quartic law. The true low temperature regime of specific heat is defined by the surface heat, therefore, it depends on the geometrical characteristics of the body, while the absolute zero temperature limit is geometrically forbidden. The limit on the growth of specific heat at temperatures close to critical points, known as the Dulong–Petit law, appears from the lattice constant cutoff. Its value depends on the lattice type and it is the same for materials with the same crystal lattice. The Dulong–Petit values of compounds are equal to those of elements with the same crystal lattice type, if one mole of solid state matter were taken as the Avogadro number of the composing atoms. Thus, the Neumann–Kopp law is valid only in some special cases.
 Stokes–Leibenson problem for HeleShaw flow: a critical set in the
space of contours Abstract: Abstract
The Stokes–Leibenson problem for HeleShaw flow is reformulated as a Cauchy problem of a nonlinear integrodifferential equation with respect to functions a and b, linked by the Hilbert transform. The function a expresses the evolution of the coefficient longitudinal strain of the free boundary and b is the evolution of the tangent tilt of this contour. These functions directly reflect changes of geometric characteristics of the free boundary of higher order than the evolution of the contour point obtained by the classical Galin–Kochina equation. That is why we managed to uncover the reason of the absence of solutions in the sinkcase if the initial contour is not analytic at at least one point, to prove existence and uniqueness theorems, and also to reveal a certain critical set in the space of contours. This set contains one attractive point in the sourcecase corresponding to a circular contour centered at the sourcepoint. The main object of this work is the analysis of the discrete model of the problem. This model, called quasicontour, is formulated in terms of functions corresponding to a and b of our integrodifferential equation. This quasicontour model provides numerical experiments which confirm the theoretical properties mentioned above, especially the existence of a critical subset of codimension 1 in space of quasicontours. This subset contains one attractive point in the sourcecase corresponding to a regular quasicontour centered at the sourcepoint. The main contribution of our quasicontour model concerns the sinkcase: numerical experiments show that the above subset is attractive. Furthermore, this discrete model allows to extend previous results obtained by using complex analysis. We also provide numerical experiments linked to fingering effects.
PubDate: 20160101
 Abstract: Abstract
The Stokes–Leibenson problem for HeleShaw flow is reformulated as a Cauchy problem of a nonlinear integrodifferential equation with respect to functions a and b, linked by the Hilbert transform. The function a expresses the evolution of the coefficient longitudinal strain of the free boundary and b is the evolution of the tangent tilt of this contour. These functions directly reflect changes of geometric characteristics of the free boundary of higher order than the evolution of the contour point obtained by the classical Galin–Kochina equation. That is why we managed to uncover the reason of the absence of solutions in the sinkcase if the initial contour is not analytic at at least one point, to prove existence and uniqueness theorems, and also to reveal a certain critical set in the space of contours. This set contains one attractive point in the sourcecase corresponding to a circular contour centered at the sourcepoint. The main object of this work is the analysis of the discrete model of the problem. This model, called quasicontour, is formulated in terms of functions corresponding to a and b of our integrodifferential equation. This quasicontour model provides numerical experiments which confirm the theoretical properties mentioned above, especially the existence of a critical subset of codimension 1 in space of quasicontours. This subset contains one attractive point in the sourcecase corresponding to a regular quasicontour centered at the sourcepoint. The main contribution of our quasicontour model concerns the sinkcase: numerical experiments show that the above subset is attractive. Furthermore, this discrete model allows to extend previous results obtained by using complex analysis. We also provide numerical experiments linked to fingering effects.
 On local perturbations of waveguides
 Abstract: Abstract
The paper deals with an arbitrary sufficiently small localized perturbation of waveguides with different types of boundary conditions. We study both the qualitative structure of the spectrum of the perturbed operator and conditions for the occurrence of eigenvalues from the continuous spectrum. For the case in which eigenvalues occur, their asymptotic behavior is obtained.
PubDate: 20160101
 Abstract: Abstract
The paper deals with an arbitrary sufficiently small localized perturbation of waveguides with different types of boundary conditions. We study both the qualitative structure of the spectrum of the perturbed operator and conditions for the occurrence of eigenvalues from the continuous spectrum. For the case in which eigenvalues occur, their asymptotic behavior is obtained.
 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.
 New forms of the Cauchy operator and some of their applications
 Abstract: Abstract
In this paper, we first construct the Cauchy qshift operator T(a, b;D
xy
) and the Cauchy qdifference operator L(a, b; θ
xy
). We then apply these operators in order to represent and investigate some new families of qpolynomials which are defined in this paper. We derive some qidentities such as generating functions, symmetry properties and Rogerstype formulas for these qpolynomials. We also give an application for the qexponential operator R(bD
q
).
PubDate: 20160101
 Abstract: Abstract
In this paper, we first construct the Cauchy qshift operator T(a, b;D
xy
) and the Cauchy qdifference operator L(a, b; θ
xy
). We then apply these operators in order to represent and investigate some new families of qpolynomials which are defined in this paper. We derive some qidentities such as generating functions, symmetry properties and Rogerstype formulas for these qpolynomials. We also give an application for the qexponential operator R(bD
q
).
 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.
 Electron velocity distribution moments for collisional inhomogeneous
plasma in crossed electric and magnetic fields Abstract: Abstract
We study a stationary kinetic equation describing the electron component of nonequilibrium plasma in crossed electric and magnetic fields. The collision integral is taken in the socalled relaxation (BGK) approximation. It is assumed that the plasma parameters vary only along the electric field. Using the Laplace method, asymptotic formulas for the moments of the distribution function including components of the stress tensor and heat flux vector are obtained with a qualified estimate of the remainder.
PubDate: 20151001
 Abstract: Abstract
We study a stationary kinetic equation describing the electron component of nonequilibrium plasma in crossed electric and magnetic fields. The collision integral is taken in the socalled relaxation (BGK) approximation. It is assumed that the plasma parameters vary only along the electric field. Using the Laplace method, asymptotic formulas for the moments of the distribution function including components of the stress tensor and heat flux vector are obtained with a qualified estimate of the remainder.
 Degenerate Euler zeta function
 Abstract: Abstract
Recently, T. Kim considered an Euler zeta function which interpolates Euler polynomials at negative integers (see [3]). In this paper, we study the degenerate Euler zeta function which is holomorphic on the complex splane and is associated with degenerate Euler polynomials at negative integers.
PubDate: 20151001
 Abstract: Abstract
Recently, T. Kim considered an Euler zeta function which interpolates Euler polynomials at negative integers (see [3]). In this paper, we study the degenerate Euler zeta function which is holomorphic on the complex splane and is associated with degenerate Euler polynomials at negative integers.
 On a problem in geometry of numbers arising in spectral theory
 Abstract: Abstract
We study the lattice point counting problem for a class of families of domains in a Euclidean space. This class consists of anisotropically expanding bounded domains that remain unchanged along some fixed linear subspace and expand in directions orthogonal to this subspace. We find the leading term in the asymptotics of the number of lattice points in such family of domains and prove remainder estimates in this asymptotics under various conditions on the lattice and the family of domains. As a consequence, we prove an asymptotic formula for the eigenvalue distribution function of the Laplace operator on a flat torus in adiabatic limit determined by a linear foliation with a nontrivial remainder estimate.
PubDate: 20151001
 Abstract: Abstract
We study the lattice point counting problem for a class of families of domains in a Euclidean space. This class consists of anisotropically expanding bounded domains that remain unchanged along some fixed linear subspace and expand in directions orthogonal to this subspace. We find the leading term in the asymptotics of the number of lattice points in such family of domains and prove remainder estimates in this asymptotics under various conditions on the lattice and the family of domains. As a consequence, we prove an asymptotic formula for the eigenvalue distribution function of the Laplace operator on a flat torus in adiabatic limit determined by a linear foliation with a nontrivial remainder estimate.
 On the behavior of the edge diffracted nonstationary wave in scattering by
wedges near the front Abstract: Abstract
We study the behavior of a wave diffracted by a wedge wave near its front in the twodimensional scattering of an incident plane harmonic wave with a Heaviside type profile. We find the asymptotics of the cylindrical wave diffracted by the edge of a wedge near the front in two cases: near the critical rays and far from the critical rays. The asymptotics turns out to be nonuniform and depends on the magnitude of the wedge. The cases of Dirichlet–Dirichlet, Dirichlet–Neumann and Neumann–Neumann boundary conditions are considered.
PubDate: 20151001
 Abstract: Abstract
We study the behavior of a wave diffracted by a wedge wave near its front in the twodimensional scattering of an incident plane harmonic wave with a Heaviside type profile. We find the asymptotics of the cylindrical wave diffracted by the edge of a wedge near the front in two cases: near the critical rays and far from the critical rays. The asymptotics turns out to be nonuniform and depends on the magnitude of the wedge. The cases of Dirichlet–Dirichlet, Dirichlet–Neumann and Neumann–Neumann boundary conditions are considered.
 Planar Penning trap with combined resonance and top dynamics on quadratic
algebra Abstract: Abstract
We study the dynamics of a charge in the planar Penning trap in which the direction of the magnetic field does not coincide with the trap axis. Under a certain combined resonance condition on the deviation angle and magnitudes of magnetic and electric fields, the trajectories of a charge are nearperiodic.We describe the reduction to a toplike system with one degree of freedom on the space with quadratic Poisson brackets and study the stability of the equilibrium points of this system.
PubDate: 20151001
 Abstract: Abstract
We study the dynamics of a charge in the planar Penning trap in which the direction of the magnetic field does not coincide with the trap axis. Under a certain combined resonance condition on the deviation angle and magnitudes of magnetic and electric fields, the trajectories of a charge are nearperiodic.We describe the reduction to a toplike system with one degree of freedom on the space with quadratic Poisson brackets and study the stability of the equilibrium points of this system.
 Asymptotic solutions of linearized Navier–Stokes equations localized
in small neighborhoods of curves and surfaces Abstract: Abstract
In the paper, the asymptotic behavior of solutions of the Cauchy problem is described for the linearized Navier–Stokes equation with the initial condition localized in a neighborhood of a curve or a twodimensional surface in threedimensional space. In particular, conditions for the growth of the perturbation in planeparallel, twodimensional, and helical external flows are obtained.
PubDate: 20151001
 Abstract: Abstract
In the paper, the asymptotic behavior of solutions of the Cauchy problem is described for the linearized Navier–Stokes equation with the initial condition localized in a neighborhood of a curve or a twodimensional surface in threedimensional space. In particular, conditions for the growth of the perturbation in planeparallel, twodimensional, and helical external flows are obtained.
 On the smoothness of generalized solutions to boundary value problems for
strongly elliptic differentialdifference equations on a boundary of
neighboring subdomains Abstract: Abstract
In this paper, we study the smoothness of generalized solutions to boundary value problems for strongly elliptic differentialdifference equations on a boundary of neighboring subdomains of smoothness.
PubDate: 20151001
 Abstract: Abstract
In this paper, we study the smoothness of generalized solutions to boundary value problems for strongly elliptic differentialdifference equations on a boundary of neighboring subdomains of smoothness.
 Testing of solutions for the Boussinesq wave equations on a solution of a
potential tsunami model with “simple” source Abstract: Abstract
The Cauchy problem for the wave equations of Boussinesq type is treated by considering the initial conditions taken from the solution of generalized Cauchy problem for the potential model of tsunami with some “simple” impulsive source under the assumption that the depth of the liquid is constant. The solutions of the problem under consideration are derived in the form of a single integral giving the wave height at every point of observation at any time moment after the pulsed action of the source. The results of comparing the time history of the the height of tsunami waves at different distances from the source for different values of its characteristic radius (these histories are calculated using two solutions, namely, the solution derived here and the solution known for the potential tsunami model) are described. Conclusions concerning the accuracy of the tested solutions are made.
PubDate: 20151001
 Abstract: Abstract
The Cauchy problem for the wave equations of Boussinesq type is treated by considering the initial conditions taken from the solution of generalized Cauchy problem for the potential model of tsunami with some “simple” impulsive source under the assumption that the depth of the liquid is constant. The solutions of the problem under consideration are derived in the form of a single integral giving the wave height at every point of observation at any time moment after the pulsed action of the source. The results of comparing the time history of the the height of tsunami waves at different distances from the source for different values of its characteristic radius (these histories are calculated using two solutions, namely, the solution derived here and the solution known for the potential tsunami model) are described. Conclusions concerning the accuracy of the tested solutions are made.