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Russian Journal of Mathematical Physics

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ISSN (Print) 1531-8621 - ISSN (Online) 1061-9208
Published by Springer-Verlag

[2216 journals]

Ideal gas/liquid transition as a generalization of the problem of “partitio numerorum”
- Abstract: Abstract It is shown that the Boltzmann-Maxwell distribution is not applicable to the generalized notion of ideal gas and corresponds to this ideal gas at low density only. The behavior of an ideal gas and its transition to liquid is studied as a result of a kind of generalization of the problem of “partitio numerorum”. The reflection of particles of the ideal gas from the walls of the vessel walls is taken into account (as a kind of “trap” for the Bose gas). Two constants are additionally introduced, namely, the constant Λ inversely proportional to the mass and the heat g of the phase transition as T → 0 (the constant of the Clausius-Clapeyron relation), and also the experimental value of the critical compressibility factor for the given gas and the experimental value of the pressure at the triple point, from which one can find the value of the constant Λ.
  PubDate: 2012-10-01
Cubic algebra and averaged Hamiltonian for the resonance 3: (−1) Penning-ioffe trap
- Abstract: Abstract For the 3: (−1) resonance Penning trap, we describe the algebra of symmetries which turns out to be a non-Lie algebra with cubic commutation relations. The irreducible representations and coherent states of this algebra are constructed explicitly. The perturbing inhomogeneous magnetic field of Ioffe type, after double quantum averaging, generates an effective Hamiltonian of the trap. In the irreducible representation, this Hamiltonian becomes a second-order ordinary differential operator of the Heun type.
  PubDate: 2012-10-01
Continuity conditions for finite-dimensional representations of connected locally compact groups
- Abstract: Abstract We obtain simple necessary and sufficient conditions for the continuity of a locally bounded finite-dimensional representation of a connected locally compact group. We also prove that the discontinuity group of a locally bounded finite-dimensional representation of a connected locally compact group is a central subgroup in the closure of the image.
  PubDate: 2012-10-01
Remarks on strongly elliptic systems in Lipschitz domains
- Abstract: Abstract We present some remarks to the general theory of strongly elliptic second-order systems in bounded Lipschitz domains. The most important remarks are related to the use of the “Weyl decomposition” of the solution space. In particular, we suggest a simplified approach to the unique choice of the right-hand side of the system and the conormal derivative in the Neumann problem and obtain two-sided a priori estimates for the solutions. We consider the transmission problem for two systems in domains with a common Lipschitz boundary without the assumption that the coefficients do not have jumps on that boundary. We construct examples of strongly elliptic second-order systems for which the Neumann problem is not Fredholm.
  PubDate: 2012-10-01
A new mathematical problem related to quantization of fields
- Abstract: Abstract This paper is a survey of author’s mathematical and logical studies concerning the quantization of fields.
  PubDate: 2012-10-01
Transformation of supermeasures and their logarithmic derivatives under the action of a flow of diffeomorphisms of the superspace
- Abstract: Abstract The paper presents formulas for the logarithmic derivatives mentioned in the title.
  PubDate: 2012-10-01
Analytical complexity: Development of the topic
- Abstract: Abstract The complexity of analytic functions of two variables is studied in terms of the order of complexity suggested in [1]. This paper continues [1]. An estimate for the complexity of a polynomial using its degree is given. Examples of homogeneous and harmonic functions are treated. An estimate for the complexity of a power series in terms of the geometry of the support of the series is given. Differential equations defining classes of complexity are considered. For polynomial mappings of complexity one, the Jacobian conjecture is proved. In this connection, the complexity of mappings of the plane is discussed. The research was supported by the Russian Foundation for Basic Research under grants no. 11 - 01 - 00495 a and no. 11 - 01 - 12033_ofi_m_2011.
  PubDate: 2012-10-01
On the critical behavior in nonlinear evolutionary PDEs with small viscosity
- Abstract: Abstract The problem of general dissipative regularization of the quasilinear transport equation is studied. We argue that the local behavior of solutions to the regularized equation near the point of gradient catastrophe for the transport equation is described by the logarithmic derivative of the Pearcey function; this statement generalizes a result of Il’in [12]. We provide some analytic arguments supporting the conjecture and test it numerically.
  PubDate: 2012-10-01
On the existence of solutions of the first boundary value problem for elliptic equations on unbounded domains
- Abstract: Abstract The problem mentioned in the title is studied.
  PubDate: 2012-10-01
Eigenvalue problems modelling the stability of a plane-parallel shear in a two-layer viscous composite
- Abstract: Abstract We present the derivation of a precise statement of a linearized eigenvalue problem that models the stability of a plane-parallel shear flow in a composite consisting of two different layers of heavy Newtonian media. The peculiarities of the influence on the stability of five dimensionless criteria participating in the equations and boundary conditions are discussed. The specific feature of the power conditions of contact is that a spectral parameter α enters these conditions nonlinearly. As applied to the composite structure, we develop the apparatus of integral relations, which makes it possible to derive the lower integral bounds for the stability parameter (the real part of α), and hence also upper bounds of combinations of critical numbers.
  PubDate: 2012-10-01
Linear and nonlinear liftings of states of quantum systems
- Abstract: Abstract In this paper, we study the representability of an arbitrary quantum state ρ ∈ Σ(H) as the reduction of a vector state r ∈ Σ(H) of the extended system. We extend the operation of lifting from the set of states Σ n (H) to the set of generalized states Σ(H). A method of constructing the Hilbert space H and the affine linear lifting Σ(H) → Σ(H) is studied. The construction of individual expansion H ρ of the space H for which the state ρ is a reduction of a vector state H ρ is of special interest.
  PubDate: 2012-10-01
Probability theory for random variables with unboundedly growing values and its applications
- Abstract: Abstract We develop the unbounded probability theory on the basis of Kolmogorov complexity and show its connections to thermodynamics, economics, and its role in the study of the Web.
  PubDate: 2012-07-01
On the genus two free energies for semisimple Frobenius manifolds
- Abstract: Abstract We represent the genus two free energy of an arbitrary semisimple Frobenius manifold as the sum of contributions associated with dual graphs of certain stable algebraic curves of genus two plus the so-called “genus two G-function.” Conjecturally, the genus two G-function vanishes for a series of important examples of Frobenius manifolds associated with simple singularities, as well as for ℙ1-orbifolds with positive Euler characteristics. We explain the reasons for the conjecture and prove it in particular cases.
  PubDate: 2012-07-01
Analytical study of a potential model of tsunami with a simple source of piston type. 1. Exact solution. Creation of tsunami
- Abstract: Abstract A linear hydrodynamic problem concerning the generation of gravitational waves on the free surface of a liquid by a source (defined as an initial instant vertical displacement of the bottom of the basin) is studied, where the displacement is defined by a rather simple axially symmetric function of the horizontal coordinates. A solution to the problem is obtained in the form of single integrals and is regarded as a distribution (a “generalized function”) with respect to time. These integrals are evaluated numerically and asymptotically. In this part of the paper, using the results of numerical evaluation carried out for each source (having a given characteristic radius in a wide range of values), we find the initial instantaneous displacement of the fluid, determine the parameters of the leading crest of the created surface wave, and estimate the minimal radius which a source must have to be referred to tsunami generators.
  PubDate: 2012-07-01
Feynman formula for Schrödinger-type equations with time- and space-dependent coefficients
- Abstract: Abstract Feynman formulas for a Schrödinger-type equation whose right-hand side is a second-order differential operator are proved.
  PubDate: 2012-07-01
Normal forms of twisted wire knots
- Abstract: Abstract This article is a continuation of the study of new types of knot energy undertaken in [1, 2] (but is formally independent of those articles); it describes some experiments with mechanical models of knots (that we call twisted wire knots), contains rigorous definitions of their mathematical counterparts, formulations of a series of problems and conjectures. Different energy functionals for various classes of knot types and the corresponding normal forms are discussed and compared.
  PubDate: 2012-07-01
On the convergence of the amplitude of the diffracted nonstationary wave in scattering by wedges
- Abstract: Abstract We make more precise the Limiting Amplitude Principle in the two-dimensional scattering of an incident plane harmonic wave by a wedge. We find the long-time asymptotic regime of convergence of the amplitude of the cylindrical wave diffracted by the vertex of a wedge to the limiting amplitude of the solution to the corresponding stationary problem. The asymptotics turns out to be uniform on compacta and depends on the magnitude of the wedge and the profile of the incident wave. The cases of Dirichlet-Dirichlet and Dirichlet-Neumann boundary conditions are considered.
  PubDate: 2012-07-01
Bose condensate in the D-Dimensional case, in particular, for D = 2
- Abstract: Abstract In the paper, the problem of Bose condensation into the zero energy of particles is investigated using methods of number theory.
  PubDate: 2012-07-01
Approximate formulas for eigenvalues of the Laplace operator on a torus arising in linear problems with oscillating coefficients
- Abstract: In the paper, using relatively simple formulas derived in the abstract perturbation theory of selfadjoint operators, we obtain explicit asymptotic formulas for a family of elliptic operators of Laplace type that arise in linear problems with rapidly oscillating coefficients.
  PubDate: 2012-07-01
Hall quantum Hamiltonians and electric 2D-curvature
- Abstract: Abstract For the Dirac 2D-operator in a constant magnetic field with perturbing electric potential, we derive Hamiltonians describing the quantum quasiparticles (Larmor vortices) at Landau levels. The discrete spectrum of this Hall-effect quantum Hamiltonian can be computed to all orders of the semiclassical approximation by a deformed Planck-type quantization condition on the 2D-plane; the standard magnetic (symplectic) form on the plane is corrected by an “electric curvature” determined via derivatives of the electric field. The electric curvature does not depend on the magnitude of the electric field and vanishes for homogeneous fields (i.e., for the canonical Hall effect). This curvature can be treated as an effective magnetic charge of the inhomogeneous Hall 2D-nanosystem.
  PubDate: 2012-07-01