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  Subjects -> PHYSICS (Total: 801 journals)
    - ELECTRICITY AND MAGNETISM (9 journals)
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PHYSICS (578 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  [2345 journals]
  • On the distribution of energy of localized solutions of the Schrödinger
           equation that propagate along symmetric quantum graphs
    • Authors: A. I. Allilueva; A. I. Shafarevich
      Pages: 139 - 147
      Abstract: The theory of differential equations and differential operators on geometric graphs is actively developing in recent decades. One of the directions is devoted to the study of the socalled Gaussian packets, i.e., localized asymptotic solutions of the nonstationary Schrödinger equation. An interesting feature of such solutions is in their close connection with problems of analytic number theory and, in particular, with estimates for the number of integer points in polyhedra and the number of integer solutions of linear inequalities. At the same time, from the point of view of applications to quantum mechanics, it is natural to raise the question of the energy distribution of such solutions along the graph (in other words, the probabilities of finding a quantum particle in some area). Seemingly, this question is very complicated for general graphs, because the energy distribution is much more sensitive to the type of boundary conditions and to the initial state than the asymptotics of the number of localized functions. A similar problem is to describe the energy distribution of a solution of the wave equation on a geometric graph. For infinite regular trees, this question was studied in the paper [10, 11]; at the same time, the general case is practically unstudied. The main observation of the present paper is that the situation is considerably simplified if we consider strongly localized asymptotic solutions; in this case, a general unitary operator describing the scattering at a vertex is replaced by the operator of reflection from the subspace. In the simplest situations, this circumstance makes it possible to obtain comparatively simple formulas for the energy distribution along the edges.
      PubDate: 2017-04-01
      DOI: 10.1134/s1061920817020017
      Issue No: Vol. 24, No. 2 (2017)
       
  • Stabilizer of a function in the gage group
    • Authors: V. K. Beloshapka
      Pages: 148 - 152
      Abstract: It is proved that, for the dimension d of the stabilizer of an analytic function z(x, y) in the gage pseudogroup G = {z(x, y) → c(z(a(x), b(y))}, there are precisely four possibilities: (1) d = ∞ and the complexity of z is zero, (2) d = 3 and the complexity of z is equal to one, (3) d = 1 and z is equivalent the function r(x + y) − x of complexity two, (4) d = 0 in all remaining cases.
      PubDate: 2017-04-01
      DOI: 10.1134/s1061920817020029
      Issue No: Vol. 24, No. 2 (2017)
       
  • Topical problems of the theory of Transcendental numbers: Development of
           approaches to their solution in the works of Yu. V. Nesterenko
    • Authors: V. G. Chirskii
      Pages: 153 - 171
      Abstract: The present paper is a survey of a part of the theory devoted to certain problems concerning the algebraic independence of the values of analytic functions, to quantitative results on estimates for the measure of transcendence or the measure of algebraic independence of numbers, to functional analogs of these results on the algebraic independence of solutions of algebraic differential equations, and estimates for the multiplicities of zeros for polynomials in these solutions, which play an important role in the proof of numerical results. This choice is related to the fact that, in December 2016, the head of the Department of Number Theory of Moscow State University, Corresponding Member of the RAS Yu.V. Nesterenko, who did a lot to develop these directions of the theory Transcendental numbers and whose works are marked by many awards, became seventy. He is a laureate of the Markov RAS Prize, 2006, of the Ostrovsky international prize, 1997, of the Hardy–Ramanujan Society Prize, 1997, and the Alexander von Humboldt Prize, 2003. Since the article is dedicated to the 70th anniversary of the birth of Yurii Valentinovich, we preface the scientific part with a brief biography.
      PubDate: 2017-04-01
      DOI: 10.1134/s1061920817020030
      Issue No: Vol. 24, No. 2 (2017)
       
  • On the asymptotical normality of statistical solutions for wave equations
           coupled to a particle
    • Authors: T. V. Dudnikova
      Pages: 172 - 194
      Abstract: We consider a linear Hamiltonian system consisting of a classical particle and a scalar field describing by the wave or Klein–Gordon equations with variable coefficients. The initial data of the system are supposed to be a random function which has some mixing properties. We study the distribution μ t of the random solution at time moments t ∈ R. The main result is the convergence of μ t to a Gaussian probability measure as t→∞. The application to the case of Gibbs initial measures is given.
      PubDate: 2017-04-01
      DOI: 10.1134/s1061920817020042
      Issue No: Vol. 24, No. 2 (2017)
       
  • On an inverse spectral problem
    • Authors: Yu. V. Egorov
      Pages: 195 - 206
      Abstract: We study an inverse spectral problem for the Sturm–Liouville problem. Such a problem arises in the study of micro- and nano-rods.
      PubDate: 2017-04-01
      DOI: 10.1134/s1061920817020054
      Issue No: Vol. 24, No. 2 (2017)
       
  • Generalized optical theorem to a multipole source excitation in the
           scattering theory
    • Authors: Yu. A. Eremin; A. G. Sveshnikov
      Pages: 207 - 215
      Abstract: In the present paper, the Optical Theorem is generalized to the case of a penetrable obstacle excited by a multipole of arbitrary order in the presence of a transparent substrate. This generalization allows one to test computer modules when wave scattering by lossless penetrable obstacle is considered. Besides, it enables one to evaluate the absorption cross-section by subtracting the scattering cross-section from the extinction cross-section. This seems to be important because, in this particular case, the far field does not involve a Sommerfeld integral.
      PubDate: 2017-04-01
      DOI: 10.1134/s1061920817020066
      Issue No: Vol. 24, No. 2 (2017)
       
  • Some features of bending of a rod under a strong longitudinal compression
    • Authors: A. A. Ershov; B. I. Suleimanov
      Pages: 216 - 233
      Abstract: Typical processes of rod bending under strong longitudinal compression are studied. The corresponding dynamic equation of bending is considered as a perturbation of the two-dimensional Laplace equation. It is established that, for these processes, the expantion of domains of rapid increase of bending begins in small neighborhoods of singularity points of solutions of the limiting Laplace equation. The initial stages of these increases are described using the Hardy integral.
      PubDate: 2017-04-01
      DOI: 10.1134/s1061920817020078
      Issue No: Vol. 24, No. 2 (2017)
       
  • Measures on the Hilbert space of a quantum system
    • Authors: A. Yu. Khrennikov; O. G. Smolyanov
      Pages: 234 - 240
      Abstract: The paper is the first in a series of papers on the use of measures and generalized measures in quantum theory. In particular, a survey of the proofs of equivalence of various definitions of the density operator is presented. The exposition is of algebraic nature, and analytic assumptions are usually omitted.
      PubDate: 2017-04-01
      DOI: 10.1134/s106192081702008x
      Issue No: Vol. 24, No. 2 (2017)
       
  • Degenerate Laplace transform and degenerate gamma function
    • Authors: T. Kim; D. S. Kim
      Pages: 241 - 248
      Abstract: In this paper, we introduce the degenerate Laplace transform and degenerate gamma function and investigate some of their properties. From our investigation, we derive some interesting formulas related to the degenerate Laplace transform and degenerate gamma function.
      PubDate: 2017-04-01
      DOI: 10.1134/s1061920817020091
      Issue No: Vol. 24, No. 2 (2017)
       
  • Topological phase transitions in the theory of partitions of integers
    • Authors: V. P. Maslov
      Pages: 249 - 260
      Abstract: In the paper, the problem of partitioning a natural number into summands is considered in connection with the BKT (Berezinskii–Kosterlitz–Thouless) phase transition and its two critical points. As examples, the passage from superfluid state to normal state and from a cell-like vortical state to turbulent state are considered.
      PubDate: 2017-04-01
      DOI: 10.1134/s1061920817020108
      Issue No: Vol. 24, No. 2 (2017)
       
  • Remark concerning Maslov’s theorem on homomorphisms of topological
           groups
    • Authors: A. I. Shtern
      Pages: 261 - 262
      Abstract: Maslov’s theorem on finally continuous sequences of homomorphisms of topological groups is presented for more general passages to the limit.
      PubDate: 2017-04-01
      DOI: 10.1134/s106192081702011x
      Issue No: Vol. 24, No. 2 (2017)
       
  • Singular sets of surfaces
    • Authors: I. G. Tsar’kov
      Pages: 263 - 271
      Abstract: Sets of values of the metric projection for an approximatively compact subset of Hilbert space are studied. The results obtained in this way are used to study the geometry of hypersurfaces in ℝ n and their singular sets.
      PubDate: 2017-04-01
      DOI: 10.1134/s1061920817020121
      Issue No: Vol. 24, No. 2 (2017)
       
  • Feynman formulas for semigroups generated by an iterated Laplace operator
    • Authors: M. S. Buzinov
      Pages: 272 - 277
      Abstract: In the present paper, we find representations of a one-parameter semigroup generated by a finite sum of iterated Laplace operators and an additive perturbation (the potential). Such semigroups and the evolution equations corresponding to them find applications in the field of physics, chemistry, biology, and pattern recognition. The representations mentioned above are obtained in the form of Feynman formulas, i.e., in the form of a limit of multiple integrals as the multiplicity tends to infinity. The term “Feynman formula” was proposed by Smolyanov. Smolyanov’s approach uses Chernoff’s theorems. A simple form of representations thus obtained enables one to use them for numerical modeling the dynamics of the evolution system as a method for the approximation of solutions of equations. The problems considered in this note can be treated using the approach suggested by Remizov (see also the monograph of Smolyanov and Shavgulidze on path integrals). The representations (of semigroups) obtained in this way are more complicated than those given by the Feynman formulas; however, it is possible to bypass some analytical difficulties.
      PubDate: 2017-04-01
      DOI: 10.1134/s1061920817020133
      Issue No: Vol. 24, No. 2 (2017)
       
  • Erratum to “A new class of Abelian theorems for the
           Mehler–Fock transforms”
    • Authors: H. M. Srivastava; B. J. González; E. R. Negrín
      Pages: 278 - 278
      PubDate: 2017-04-01
      DOI: 10.1134/s1061920817020145
      Issue No: Vol. 24, No. 2 (2017)
       
  • Double-deck structure of the boundary layer in the problem of flow in an
           axially symmetric pipe with small irregularities on the wall for large
           Reynolds numbers
    • Authors: V. G. Danilov; R. K. Gaydukov
      Pages: 1 - 18
      Abstract: The problem of flow of a viscous incompressible fluid in an axially symmetric pipe with small irregularities on the wall is considered. An asymptotic solution of the problem with the double-deck structure of the boundary layer and the unperturbed flow in the environment (the “core flow”) is obtained. The results of flow numerical simulation in the thin and “thick” boundary layers are given.
      PubDate: 2017-01-01
      DOI: 10.1134/s1061920817010010
      Issue No: Vol. 24, No. 1 (2017)
       
  • Tsallis p , q -deformed Touchard polynomials and Stirling numbers
    • Authors: O. Herscovici; T. Mansour
      Pages: 37 - 50
      Abstract: In this paper, we develop and investigate a new two-parametrized deformation of the Touchard polynomials, based on the definition of the NEXT q-exponential function of Tsallis. We obtain new generalizations of the Stirling numbers of the second kind and of the binomial coefficients and represent two new statistics for the set partitions.
      PubDate: 2017-01-01
      DOI: 10.1134/s1061920817010034
      Issue No: Vol. 24, No. 1 (2017)
       
  • On λ-Bell polynomials associated with umbral calculus
    • Authors: T. Kim; D. S. Kim
      Pages: 69 - 78
      Abstract: In this paper, we introduce some new λ-Bell polynomials and Bell polynomials of the second kind and investigate properties of these polynomials. Using our investigation, we derive some new identities for the two kinds of λ-Bell polynomials arising from umbral calculus.
      PubDate: 2017-01-01
      DOI: 10.1134/s1061920817010058
      Issue No: Vol. 24, No. 1 (2017)
       
  • Differential-geometric structures associated with Lagrangians
           corresponding to scalar physical fields
    • Authors: A. K. Rybnikov
      Pages: 111 - 121
      Abstract: The paper is devoted to the investigation, using the method of Cartan–Laptev, of the differential-geometric structure associated with a Lagrangian L, depending on a function z of the variables t, x 1,...,x n and its partial derivatives. Lagrangians of this kind are considered in theoretical physics (in field theory). Here t is interpreted as time, and x 1,...,x n as spatial variables. The state of the field is characterized by a function z(t, x 1,..., x n ) (a field function) satisfying the Euler equation, which corresponds to the variational problem for the action integral. In the present paper, the variables z(t, x 1,..., x n are regarded as adapted local coordinates of a bundle of general type M with n-dimensional fibers and 1-dimensional base (here the variable t is simultaneously a local coordinate on the base). If we agree to call t time, and a typical fiber an n-dimensional space, then M can be called the spatiotemporal bundle manifold. We consider the variables t, x 1,...,x n , z (i.e., the variables t, x 1,...,x n with the added variable z) as adapted local coordinates in the bundle H over the fibered base M. The Lagrangian L, which is a coefficient in the differential form of the variational action integral in the integrand, is a relative invariant given on the manifold J 1 H (the manifold of 1-jets of the bundle H). In the present paper, we construct a tensor with components Λ00, Λ0i , Λ ij (Λ ij = Λ ji ) which is generated by the fundamental object of the structure associated with the Lagrangian. This tensor is an invariant (with respect to admissible transformations the variables t, x 1,...,x n , z) analog of the energy-momentum tensor of the classical theory of physical fields. We construct an invariant I, a vector G i , and a bivalent tensor G jk generated by the Lagrangian. We also construct a relative invariant of E (in the paper, we call it the Euler relative invariant) such that the equation E = 0 is an invariant form of the Euler equation for the variational action integral. For this reason, a nonvariational interpretation of the Euler equation becomes possible. Moreover, we construct a connection in the principal bundle with base J 2 H (the variety of 2-jets of the bundle H) and with the structure group GL(n) generated by the structure associated with the Lagrangian.
      PubDate: 2017-01-01
      DOI: 10.1134/s1061920817010083
      Issue No: Vol. 24, No. 1 (2017)
       
  • A new class of Abelian theorems for the Mehler–Fock transforms
    • Authors: H. M. Srivastava; B. J. González; E. R. Negrín
      Pages: 124 - 126
      Abstract: The main object of this paper is to derive several new Abelian theorems for the Mehler–Fock transforms. The results presented here are compared with those given earlier by R. S. Pathak and R. N. Pandey [Math. Soc. 3 (1987), 91–95]. Some applications and particular cases are also considered.
      PubDate: 2017-01-01
      DOI: 10.1134/s1061920817010101
      Issue No: Vol. 24, No. 1 (2017)
       
  • Propagation of a linear wave created by a spatially localized perturbation
           in a regular lattice and punctured Lagrangian manifolds
    • Authors: S. Yu. Dobrokhotov; V. E. Nazaikinskii
      Pages: 127 - 133
      Abstract: The following results are obtained for the Cauchy problem with localized initial data for the crystal lattice vibration equations with continuous and discrete time: (i) the asymptotics of the solution is determined by Lagrangian manifolds with singularities (“punctured” Lagrangian manifolds); (ii) Maslov’s canonical operator is defined on such manifolds as a modification of a new representation recently obtained for the canonical operator by the present authors together with A. I. Shafarevich (Dokl. Ross. Akad. Nauk 46 (6), 641–644 (2016)); (iii) the projection of the Lagrangian manifold onto the configuration plane specifies a bounded oscillation region, whose boundary (which is naturally referred to as the leading edge front) is determined by the Hamiltonians corresponding to the limit wave equations; (iv) the leading edge front is a special caustic, which possibly contains stronger focal points. These observations, together with earlier results, lead to efficient formulas for the wave field in a neighborhood of the leading edge front.
      PubDate: 2017-01-01
      DOI: 10.1134/s1061920817010113
      Issue No: Vol. 24, No. 1 (2017)
       
 
 
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