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PHYSICS (590 journals)

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Journal Cover Russian Journal of Mathematical Physics
  [SJR: 0.858]   [H-I: 24]   [0 followers]  Follow
   Hybrid Journal Hybrid journal (It can contain Open Access articles)
   ISSN (Print) 1531-8621 - ISSN (Online) 1061-9208
   Published by Springer-Verlag Homepage  [2350 journals]
  • Waves on the Free Surface Described by Linearized Equations of
           Hydrodynamics with Localized Right-Hand Sides
    • Authors: S. Yu. Dobrokhotov; V. E. Nazaikinskii
      Pages: 1 - 16
      Abstract: A linearized system of equations of hydrodynamics with time-dependent spatially localized right-hand side placed both on the free surface (and on the bottom of the basin) and also in the layer of the liquid is considered in a layer of variable depth with a given basic plane-parallel flow. A method of constructing asymptotic solutions of this problem is suggested; it consists of two stages: (1) a reduction of the three-dimensional problem to a two-dimensional inhomogeneous pseudodifferential equation on the nonperturbed free surface of the liquid, (2) a representation of the localized right-hand side in the form of a Maslov canonical operator on a special Lagrangian manifold and the subsequent application of a generalization to evolution problems of an approach, which was recently suggested in the paper [A. Yu. Anikin, S. Yu. Dobrokhotov, V. E. Nazaikinskii, and M. Rouleux, Dokl. Ross. Akad. Nauk 475 (6), 624–628 (2017); Engl. transl.: Dokl. Math. 96 (1), 406–410 (2017)], to solving stationary problems with localized right-hand sides and its combination with “nonstandard” characteristics. A method of calculation (generalizing long-standing results of Dobrokhotov and Zhevandrov) of an analog of the Kelvin wedge and the wave fields inside the wedge and in its neighborhood is suggested, which uses the consideration that this method is the projection to the extended configuration space of a Lagrangian manifold formed by the trajectories of the Hamiltonian vector field issuing from the intersection of the set of zeros of the extended Hamiltonian of the problem with conormal bundle to the graph of the vector function defining the trajectory of motion of an equivalent source on the surface of the liquid.
      PubDate: 2018-01-01
      DOI: 10.1134/s1061920818010016
      Issue No: Vol. 25, No. 1 (2018)
  • Twisted Burnside–Frobenius Theory for Endomorphisms of Polycyclic
    • Authors: A. L. Fel’shtyn; E. V. Troitsky
      Pages: 17 - 26
      Abstract: Let R(ϕ) be the number of ϕ-conjugacy (or Reidemeister) classes of an endomorphism ϕ of a group G. We prove, for several classes of groups (including polycyclic ones), that the number R(ϕ) is equal to the number of fixed points of the induced mapping on an appropriate subspace of the unitary dual space Ĝ, when R(ϕ) < ∞. Applying the result to iterations of ϕ, we obtain Gauss congruences for Reidemeister numbers. In contrast to the case of automorphisms, studied previously, there are plenty examples having the above finiteness condition, even among groups with R∞ property.
      PubDate: 2018-01-01
      DOI: 10.1134/s1061920818010028
      Issue No: Vol. 25, No. 1 (2018)
  • New Trace Formulas in Terms of Resonances for Three-Dimensional
           Schrödinger Operators
    • Authors: H. Isozaki; E. L. Korotyaev
      Pages: 27 - 43
      Abstract: We consider the Schrödinger operator −Δ+V (x) in L2(R3) with a real shortrange (integrable) potential V. Using the associated Fredholm determinant, we present new trace formulas, in particular, on expressed in terms of resonances and eigenvalues only. We also derive expressions of the Dirichlet integral, and the scattering phase. The proof is based on a change of view the point for the above mentioned problems from that of operator theory to that of complex analytic (entire) function theory.
      PubDate: 2018-01-01
      DOI: 10.1134/s106192081801003x
      Issue No: Vol. 25, No. 1 (2018)
  • Degenerate r -Stirling Numbers and r -Bell Polynomials
    • Authors: T. Kim; Y. Yao; D. S. Kim; G. -W. Jang
      Pages: 44 - 58
      Abstract: The purpose of this paper is to exploit umbral calculus in order to derive some properties, recurrence relations, and identities related to the degenerate r-Stirling numbers of the second kind and the degenerate r-Bell polynomials. Especially, we will express the degenerate r-Bell polynomials as linear combinations of many well-known families of special polynomials.
      PubDate: 2018-01-01
      DOI: 10.1134/s1061920818010041
      Issue No: Vol. 25, No. 1 (2018)
  • Passing from Mesoscopy to Macroscopy. The Mesoscopic Parameter $$\bar k$$
           k ¯
    • Authors: V. P. Maslov
      Pages: 59 - 66
      Abstract: In previous papers of the author it was shown that, depending on the hidden parameter, purely quantum problems behave like classical ones. In the present paper, it is shown that the Bose–Einstein and the Fermi–Dirac distributions, which until now were regarded as dealing with quantum particles, describe, for the appropriate values of the hidden parameter, the macroscopic thermodynamics of classical molecules.
      PubDate: 2018-01-01
      DOI: 10.1134/s1061920818010053
      Issue No: Vol. 25, No. 1 (2018)
  • On the Sharp Constant in the “Magnetic” 1D Embedding Theorem
    • Authors: A. I. Nazarov; A. P. Shcheglova
      Pages: 67 - 72
      Abstract: The problem of finding the sharp (exact) constant in the “magnetic” embedding theorem is considered.
      PubDate: 2018-01-01
      DOI: 10.1134/s1061920818010065
      Issue No: Vol. 25, No. 1 (2018)
  • Spectrum of a Problem of Elasticity Theory in the Union of Several
           Infinite Layers
    • Authors: S. A. Nazarov
      Pages: 73 - 87
      Abstract: The essential spectrum of the Dirichlet problem for the system of Lamé equations in a three-dimensional domain formed by three mutually perpendicular elastic layers occupies the ray [Λ†,+∞). The lower bound Λ† > 0 is the least eigenvalue (its existence is established) of the problem of elasticity theory in an infinite two-dimensional cross-shaped waveguide. It is proved that the discrete spectrum of the spatial problem is nonempty. Other configurations of layers and the scalar problem of the junction of quantum waveguides are also considered.
      PubDate: 2018-01-01
      DOI: 10.1134/s1061920818010077
      Issue No: Vol. 25, No. 1 (2018)
  • Existence and Asymptotic Stability of Periodic Solutions of the
           Reaction–Diffusion Equations in the Case of a Rapid Reaction
    • Authors: N. N. Nefedov; E. I. Nikulin
      Pages: 88 - 101
      Abstract: A singularly perturbed periodic in time problem for a parabolic reaction-diffusion equation in a two-dimensional domain is studied. The case of existence of an internal transition layer under the conditions of balanced and unbalanced rapid reaction is considered. An asymptotic expansion of a solution is constructed. To justify the asymptotic expansion thus constructed, the asymptotic method of differential inequalities is used. The Lyapunov asymptotic stability of a periodic solution is investigated.
      PubDate: 2018-01-01
      DOI: 10.1134/s1061920818010089
      Issue No: Vol. 25, No. 1 (2018)
  • Modeling of Pulse Signals in 3D Propagation Problems of Deep-Water
           Acoustics Based on the Modified Maslov’s Canonical Operator
    • Authors: P. S. Petrov; S. A. Sergeev; A. A. Tolchennikov
      Pages: 102 - 112
      Abstract: Formulae for an asymptotic solution of 3D problems of acoustical pulse signals propagation in the deep sea based on the modified Maslovs canonical operator are presented. The proposed formulae for acoustic pressure are applicable both in the regular case and in the situation when the receiver is located at a focal point of a family of rays. The asymptotics is compared with the solution computed using normal modes theory. It is shown that the derived formulae ensure very high precision of calculation of a pulse signal time series at the receiver point.
      PubDate: 2018-01-01
      DOI: 10.1134/s1061920818010090
      Issue No: Vol. 25, No. 1 (2018)
  • Countably Solvable Connected Pro-Lie Groups Are u -Amenable
    • Authors: A. I. Shtern
      Pages: 113 - 115
      Abstract: In contrast to some well-known discrete groups, countably solvable connected pro-Lie groups are u-amenable in the sense of de la Harpe’s 1973 paper.
      PubDate: 2018-01-01
      DOI: 10.1134/s1061920818010107
      Issue No: Vol. 25, No. 1 (2018)
  • Some Families of the Incomplete H -Functions and the Incomplete
           $$\overline H $$ H ¯ -Functions and Associated Integral Transforms and
           Operators of Fractional Calculus with Applications
    • Authors: H. M. Srivastava; R. K. Saxena; R. K. Parmar
      Pages: 116 - 138
      Abstract: Our present investigation is inspired by the recent interesting extensions (by Srivastava et al. [35]) of a pair of the Mellin–Barnes type contour integral representations of their incomplete generalized hypergeometric functions p γ q and p Γ q by means of the incomplete gamma functions γ(s, x) and Γ(s, x). Here, in this sequel, we introduce a family of the relatively more general incomplete H-functions γ p,q m,n (z) and Γ p,q m,n (z) as well as their such special cases as the incomplete Fox-Wright generalized hypergeometric functions p Ψ q (γ) [z] and p Ψ q (Γ) [z]. The main object of this paper is to study and investigate several interesting properties of these incomplete H-functions, including (for example) decomposition and reduction formulas, derivative formulas, various integral transforms, computational representations, and so on. We apply some substantially general Riemann–Liouville and Weyl type fractional integral operators to each of these incomplete H-functions. We indicate the easilyderivable extensions of the results presented here that hold for the corresponding incomplete \(\overline H \) -functions as well. Potential applications of many of these incomplete special functions involving (for example) probability theory are also indicated.
      PubDate: 2018-01-01
      DOI: 10.1134/s1061920818010119
      Issue No: Vol. 25, No. 1 (2018)
  • Nonrelativistic limit for 2 p × 2 p –Dirac operators with point
           interactions on a discrete set
    • Authors: V. Budyika; M. Malamud; A. Posilicano
      Pages: 426 - 435
      Abstract: We consider two families of realizations of the 2p×2p–Dirac differential expression with point interactions on a discrete set X = {x n } n=1 ∞ ⊂ ℝ on a half–line (line) and generalize certain results from [10] to the matrix case. We show that these realizations are always self-adjoint. We investigate the nonrelativistic limit as the velocity of light tends to infinity.
      PubDate: 2017-10-01
      DOI: 10.1134/s1061920817040021
      Issue No: Vol. 24, No. 4 (2017)
  • On the optimal form of nano-rods
    • Authors: Yu. V. Egorov
      Pages: 436 - 453
      Abstract: The Euler–Bernoulli model of elastic nano-rods is considered in the framework of nonlocal elasticity. Existence and uniqueness theorems are proved, the structure of the optimal form under an axial charge is studied. A method of numerical solution is indicated.
      PubDate: 2017-10-01
      DOI: 10.1134/s1061920817040033
      Issue No: Vol. 24, No. 4 (2017)
  • Bi-states and 2-level systems in rectangular Penning traps
    • Authors: M. Karasev; E. Novikova; E. Vybornyi
      Pages: 454 - 464
      Abstract: We introduce a notion of semiclassical bi-states. They arise from pairs of eigenstates corresponding to tunnel-split eigenlevels and generate 2-level subsystems in a given quantum system. As an example, we consider the planar Penning trap with rectangular electrodes assuming the 3: (−1) resonance regime of the charge dynamics. We demonstrate that, under a small deviation of the rectangular shape of electrodes from the square shape (symmetry breaking), there appear instanton pseudoparticles, semiclassical bi-states, and 2-level subsystems in such a quantum trap.
      PubDate: 2017-10-01
      DOI: 10.1134/s1061920817040045
      Issue No: Vol. 24, No. 4 (2017)
  • Asymptotic expansions for some integrals of quotients with degenerated
    • Authors: S. Kuksin
      Pages: 476 - 487
      Abstract: We study the asymptotic expansion as ν → 0 for integrals over R2d = {(x, y)} of the quotients F(x, y)/((x · y)2 + (νΓ(x, y))2), where Γ is strictly positive and F decays at infinity sufficiently fast. Integrals of this kind appear in the description of the four-waves interactions.
      PubDate: 2017-10-01
      DOI: 10.1134/s1061920817040069
      Issue No: Vol. 24, No. 4 (2017)
  • A model of classical thermodynamics and mesoscopic physics based on the
           notion of hidden parameter, Earth gravitation, and quasiclassical
           asymptotics. II
    • Authors: V. P. Maslov
      Pages: 494 - 504
      Abstract: This paper presents a new approach to thermodynamics based on two “first principles”: the theory of partitions of integers and Earth gravitation. The self-correlated equation obtained by the author from Gentile statistics is used to describe the effect of accumulation of energy at the moment of passage from the boson branch of the partition to its fermion branch. The branch point in the passage from bosons to fermions is interpreted as an analog of a jump of the spin. A hidden parameter–the measurement time as time of the G¨odel numbering–is introduced.
      PubDate: 2017-10-01
      DOI: 10.1134/s1061920817040082
      Issue No: Vol. 24, No. 4 (2017)
  • Explicit determination of certain periodic motions of a generalized
           two-field gyrostat
    • Authors: A. A. Oshemkov; P. E. Ryabov; S. V. Sokolov
      Pages: 517 - 525
      Abstract: The case of motion of a generalized two-field gyrostat found by V. V. Sokolov and A.V. Tsiganov is known as a Liouville integrable Hamiltonian system with three degrees of freedom. For this system, we find some special periodic motions at which the momentum mapping has rank 1. For such motions, all phase variables can be expressed in terms of algebraic functions of a single auxiliary variable and a set of constants. This auxiliary variable satisfies a differential equation which can be integrated in elliptic functions of time. As an application, the explicit formulas of characteristic exponents for determining the Williamson type of the special periodic motions are obtained.
      PubDate: 2017-10-01
      DOI: 10.1134/s1061920817040100
      Issue No: Vol. 24, No. 4 (2017)
  • Estimation of accuracy of an asymptotic solution of the generalized Cauchy
           problem for the Boussinesq equation as applied to the potential model of
           tsunami with a “simple” source
    • Authors: S. Ya. Sekerzh-Zen’kovich
      Pages: 526 - 533
      Abstract: Statement of the hydrodynamic problem in the framework of the potential tsunami model with “simple” source whose solution is chosen as the reference one. Generalized Cauchy problem for the Boussinesq equation and its reduction to the classical one. Analytical solution of the Cauchy problem for the Boussinesq equation. An explanation of the incorrectness of the formulation of the problem. Derivation of an approximate equation for the correct setting of the Cauchy problem. The known reference solution of the problem. An analytical solution of the correct problem and the derivation of its asymptotic representation in the “far zone.” Comparison of the graphs of the temporal history of wave height calculated by the formulas of the asymptotic and reference solutions. Estimation of the accuracy of the asymptotic solution by a three-level scale. Discussion. A remark concerning the referees.
      PubDate: 2017-10-01
      DOI: 10.1134/s1061920817040112
      Issue No: Vol. 24, No. 4 (2017)
  • A family of pseudo-differential operators on the Schwartz space associated
           with the fractional Fourier transform
    • Authors: H. M. Srivastava; S. K. Upadhyay; K. Khatterwani
      Pages: 534 - 543
      Abstract: Motivated essentially by their potential for applications in the mathematical, physical, and engineering sciences, the authors introduce and investigate various useful properties of a family of pseudo-differential operators on the Schwartz space S(ℝ) (i.e., the space of rapidly decreasing infinitely differentiable functions on ℝ), which are associated with the fractional Fourier transform of order α (0 < α ≦ 1). Relevant connections with several related recent developments on this subject are also pointed out.
      PubDate: 2017-10-01
      DOI: 10.1134/s1061920817040124
      Issue No: Vol. 24, No. 4 (2017)
  • Asymptotics of linear water waves generated by a localized source near the
           focal points on the leading edge
    • Authors: S. Yu. Dobrokhotov; V. E. Nazaikinskii; A. A. Tolchennikov
      Pages: 544 - 552
      Abstract: Consider a liquid in a uniform gravitational field in a horizontally infinite basin of variable finite depth. Assume that the liquid performs a potential motion and its state at the initial time is characterized by a given free surface elevation decaying at infinity and by the zero vertical velocities of the free surface (which, for example, corresponds to the so-called piston model of tsunami generation [1, 2]). The problem is to determine the free surface elevation at the subsequent times.
      PubDate: 2017-10-01
      DOI: 10.1134/s1061920817040136
      Issue No: Vol. 24, No. 4 (2017)
School of Mathematical and Computer Sciences
Heriot-Watt University
Edinburgh, EH14 4AS, UK
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