
Russian Journal of Mathematical Physics [SJR: 0.858] [HI: 24] [0 followers] Follow Hybrid journal (It can contain Open Access articles) ISSN (Print) 15318621  ISSN (Online) 10619208 Published by SpringerVerlag [2352 journals] 
 Localized asymptotic solutions of the wave equation with variable velocity
on the simplest graphs Authors: A. I. Allilueva; A. I. Shafarevich
Pages: 279  289
Abstract: Abstract The asymptotic behavior of the Cauchy problem for the wave equation with variable velocity and localized initial conditions on the line, semiaxis, and an infinite starlike graph is described. The solution consists of a shortwave and longwave parts; the shortwave part moves along the characteristics, while the longwave part satisfies the Goursat or Darboux problem. In the case of a starlike graph, the distribution of energy with respect to the edges is discussed; this distribution depends on the arrangement of the eigensubspaces of the unitary matrix that defines the boundary condition at the vertex of the star.
PubDate: 20170701
DOI: 10.1134/s1061920817030013
Issue No: Vol. 24, No. 3 (2017)
 Authors: A. I. Allilueva; A. I. Shafarevich
 Correction to the leading term of asymptotics in the problem of counting
the number of points moving on a metric tree Authors: V. L. Chernyshev; A. A. Tolchennikov
Pages: 290  298
Abstract: Abstract In the problem of determining the asymptotics for the number of points moving along a metric tree, a polynomial approximation that uses Barnes’ multiple Bernoulli polynomials is found. The connection between the second term of the asymptotic expansion and the graph structure is discussed.
PubDate: 20170701
DOI: 10.1134/s1061920817030025
Issue No: Vol. 24, No. 3 (2017)
 Authors: V. L. Chernyshev; A. A. Tolchennikov
 On some properties of multidimensional hyperbolic quasigasdynamic systems
of equations Authors: B. N. Chetverushkin; A. A. Zlotnik
Pages: 299  309
Abstract: Abstract We study a multidimensional hyperbolic quasigasdynamic (HQGD) system of equations containing terms with a regularizing parameter τ > 0 and 2nd order space and time derivatives; the body force is taken into account. We transform it to a form close to the compressible Navier–Stokes system of equations. Then we derive the entropy balance equation and show that the entropy production is similar to the latter system plus a term of the order of O(τ2). We analyze an equation for the total entropy as well. We also show that the corresponding residuals in the HQGD equations with respect to the compressible Navier–Stokes ones are of the order of O(τ2) too. Finally we treat the simplified barotropic HQGD system of equations with the general state equation and the stationary potential body force and obtain the corresponding results for it.
PubDate: 20170701
DOI: 10.1134/s1061920817030037
Issue No: Vol. 24, No. 3 (2017)
 Authors: B. N. Chetverushkin; A. A. Zlotnik
 Exact and asymptotic solutions of the Cauchy–Poisson problem with
localized initial conditions and a constant function of the bottom Authors: S. Yu. Dobrokhotov; S. Ya. SekerzhZen’kovich; A. A. Tolchennikov
Pages: 310  321
Abstract: Abstract In the paper, the asymptotic solutions for a problem of Cauchy–Poisson type with localized initial conditions are constructed. The bottom of the basin under consideration which was constant before the perturbation, is instantly perturbed at the initial time moment by a spatially localized function. Simplifications of the corresponding formulas are presented inside and outside the vicinity of the leading front, as well as in the case of a special choice of the initial condition. It is shown that, in the vicinity of the leading front, the asymptotic solution coincides with the asymptotic solution of the linear Boussinesq equation.
PubDate: 20170701
DOI: 10.1134/s1061920817030049
Issue No: Vol. 24, No. 3 (2017)
 Authors: S. Yu. Dobrokhotov; S. Ya. SekerzhZen’kovich; A. A. Tolchennikov
 Constitutive relations in multidimensional isotropic elasticity and their
restrictions to subspaces of lower dimensions Authors: D.V. Georgievskii
Pages: 322  325
Abstract: Abstract The mechanical meaning and the relationships among material constants in an ndimensional isotropic elastic medium are discussed. The restrictions of the constitutive relations (Hooke’s law) to subspaces of lower dimension caused by the conditions that an mdimensional strain state or an mdimensional stress state (1 ≤ m < n) is realized in the medium. Both the terminology and the general idea of the mathematical construction are chosen by analogy with the case n = 3 and m = 2, which is well known in the classical plane problem of elasticity theory. The quintuples of elastic constants of the same medium that enter both the ndimensional relations and the relations written out for any mdimensional restriction are expressed in terms of one another. These expressions in terms of the known constants, for example, of a threedimensional medium, i.e., the classical elastic constants, enable us to judge the material properties of this medium immersed in a space of larger dimension.
PubDate: 20170701
DOI: 10.1134/s1061920817030050
Issue No: Vol. 24, No. 3 (2017)
 Authors: D.V. Georgievskii
 Magnetodimensional resonance. Pseudospin phase and hidden quantum number
 Authors: M. V. Karasev
Pages: 326  335
Abstract: Abstract The Schrödinger operator for a spinless charge inside a layer with parabolic confinement profile and homogeneous magnetic field is considered. The Lorentz (cyclotron) and the confinement frequencies are assumed to be equal to each other. After inclination of the layer normal from the magnetic field direction there appears a pseudospin su(2)field removing the resonance degeneracy of Landau levels. Under deviations of the layer surface from the plane shape, a longitudinal geometric current is created. In circulations around surface warping, there is a nontrivial quantum phase transition generated by an element of the π1homotopy group and a hidden degree of freedom (spectral degeneracy) associated with a “charge” of geometric poles on the layer. The quantization rule contains an additional parity index related to the algebraic number of geometric poles and the Landau level number. The resonance pseudospin phaseshift represents an example of general Aharonov–Bohm type topologic phenomena in quantum (semiclassical or adiabatic) systems with deltafunction singularities in symplectic structure.
PubDate: 20170701
DOI: 10.1134/s1061920817030062
Issue No: Vol. 24, No. 3 (2017)
 Authors: M. V. Karasev
 Asymptotic completeness of scattering in the nonlinear Lamb system for
nonzero mass Authors: A. I. Komech; A. E. Merzon
Pages: 336  346
Abstract: Abstract We establish the asymptotic completeness in the nonlinear Lamb system with nonzero mass for hyperbolic stationary states. For the proof, we construct a trajectory of a reduced equation (which is a nonlinear nonautonomous ODE) converging to a hyperbolic stationary point, using the Banach space Inverse Function Theorem and a priori estimates. We give a counterexample showing that the hyperbolicity condition is essential.
PubDate: 20170701
DOI: 10.1134/s1061920817030074
Issue No: Vol. 24, No. 3 (2017)
 Authors: A. I. Komech; A. E. Merzon
 Topological complexity of certain classes of C *algebras
 Authors: A. I. Korchagin
Pages: 347  353
Abstract: Abstract We compute the topological complexity for some important classes of noncommutative C*algebras: AF algebras, AI algebras, and even Cuntz algebras.
PubDate: 20170701
DOI: 10.1134/s1061920817030086
Issue No: Vol. 24, No. 3 (2017)
 Authors: A. I. Korchagin
 A model of classical thermodynamics based on the partition theory of
integers, Earth gravitation, and semiclassical asymptotics. I Authors: V. P. Maslov
Pages: 354  372
Abstract: Abstract In the paper, a new construction of the theory of partitions of integers is proposed. The author defines entropy as the natural logarithm of the number of partitions of a number M into natural summands with repetitions allowed p(M) and repetitions forbidden q(M). The passage from ln p(M) to lnq(M) through the mesoscopic values M → 0 is studied. The topological transition from the mesoscopic lower levels of the Bohr–Kalckar construction to the macroscopic levels corresponding to the critical number of neutrons according to the consequence of Einstein’s inequality M ≤ c N c , where c is determined for the particles of the given atomic nucleus. The role of quantum mechanics in establishing the new world outlook in physics is analyzed. It is pointed out that the main equations of thermodynamics in the volume “Statistical Physics” of the Landau–Lifshits treatise are obtained without appealing to the socalled “three main principles of thermodynamics”. It is also pointed out that Niels Bohr’s liquid model of the nucleus does not involve any interaction of particles in the form of attraction and is based on the presence of a common potential trough for all elements of the nucleus. The author constructs a new approach to thermodynamics, using quantum mechanics and the Earth’s gravitational attraction as a common potential trough.
PubDate: 20170701
DOI: 10.1134/s1061920817030098
Issue No: Vol. 24, No. 3 (2017)
 Authors: V. P. Maslov
 Representations associated to quasirepresentations of amenable groups with
zero multiplication of defect operators Authors: A. I. Shtern
Pages: 373  375
Abstract: Abstract A direct construction of representations associated to dual quasirepresentations with zero multiplication of defect operators for amenable groups is given.
PubDate: 20170701
DOI: 10.1134/s1061920817030104
Issue No: Vol. 24, No. 3 (2017)
 Authors: A. I. Shtern
 Widths of weighted Sobolev classes with constraints f ( a ) = · · · = f
( k1 )( a ) = f ( k )( b ) = · · · = f ( r1 )( b ) = 0 and the
spectra of nonlinear differential equations Authors: A. A. Vasil’eva
Pages: 376  398
Abstract: Abstract In this paper we prove that the Kolmogorov widths of the weighted Sobolev class W p,g r,k [a, b] with restrictions f(a) = · · · = f( k1)(a) = f( k )(b) = · · · = f( r1)(b) = 0 in the weighted Lebesgue space L q,v [a, b] coincide with the spectral numbers of some nonlinear differential equation. We assume that 1 < q ≤ p < ∞, the weights are positive almost everywhere, and the Sobolev class is compactly embedded in the space L q,v [a, b].
PubDate: 20170701
DOI: 10.1134/s1061920817030116
Issue No: Vol. 24, No. 3 (2017)
 Authors: A. A. Vasil’eva
 Optimal control of the motion of a helical body in a liquid using rotors
 Authors: E. V. Vetchanin; I. S. Mamaev
Pages: 399  411
Abstract: Abstract The motion controlled by the rotation of three internal rotors of a body with helical symmetry in an ideal liquid is considered. The problem is to select controls that ensure the displacement of the body with minimum effort. The optimality of particular solutions found earlier is studied.
PubDate: 20170701
DOI: 10.1134/s1061920817030128
Issue No: Vol. 24, No. 3 (2017)
 Authors: E. V. Vetchanin; I. S. Mamaev
 Example of nonexistence of a positive generalized entropy solution of a
Cauchy problem with unbounded positive initial data Authors: L. V. Gargyants
Pages: 412  414
Abstract: Abstract A Cauchy problem for a firstorder quasilinear equation with polynomial flux function and exponential initial data is studied. We prove that this problem has no positive solution in the class of locally bounded functions.
PubDate: 20170701
DOI: 10.1134/s106192081703013x
Issue No: Vol. 24, No. 3 (2017)
 Authors: L. V. Gargyants
 Stabilizer of a function in the gage group
 Authors: V. K. Beloshapka
Pages: 148  152
Abstract: 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: 20170401
DOI: 10.1134/s1061920817020029
Issue No: Vol. 24, No. 2 (2017)
 Authors: V. K. Beloshapka
 Generalized optical theorem to a multipole source excitation in the
scattering theory Authors: Yu. A. Eremin; A. G. Sveshnikov
Pages: 207  215
Abstract: 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 crosssection by subtracting the scattering crosssection from the extinction crosssection. This seems to be important because, in this particular case, the far field does not involve a Sommerfeld integral.
PubDate: 20170401
DOI: 10.1134/s1061920817020066
Issue No: Vol. 24, No. 2 (2017)
 Authors: Yu. A. Eremin; A. G. Sveshnikov
 Measures on the Hilbert space of a quantum system
 Authors: A. Yu. Khrennikov; O. G. Smolyanov
Pages: 234  240
Abstract: 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: 20170401
DOI: 10.1134/s106192081702008x
Issue No: Vol. 24, No. 2 (2017)
 Authors: A. Yu. Khrennikov; O. G. Smolyanov
 Degenerate Laplace transform and degenerate gamma function
 Authors: T. Kim; D. S. Kim
Pages: 241  248
Abstract: 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: 20170401
DOI: 10.1134/s1061920817020091
Issue No: Vol. 24, No. 2 (2017)
 Authors: T. Kim; D. S. Kim
 Remark concerning Maslov’s theorem on homomorphisms of topological
groups Authors: A. I. Shtern
Pages: 261  262
Abstract: Abstract Maslov’s theorem on finally continuous sequences of homomorphisms of topological groups is presented for more general passages to the limit.
PubDate: 20170401
DOI: 10.1134/s106192081702011x
Issue No: Vol. 24, No. 2 (2017)
 Authors: A. I. Shtern
 Singular sets of surfaces
 Authors: I. G. Tsar’kov
Pages: 263  271
Abstract: 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: 20170401
DOI: 10.1134/s1061920817020121
Issue No: Vol. 24, No. 2 (2017)
 Authors: I. G. Tsar’kov
 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: 20170401
DOI: 10.1134/s1061920817020145
Issue No: Vol. 24, No. 2 (2017)
 Authors: H. M. Srivastava; B. J. González; E. R. Negrín