
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 [2350 journals] 
 Waves on the Free Surface Described by Linearized Equations of
Hydrodynamics with Localized RightHand Sides Authors: S. Yu. Dobrokhotov; V. E. Nazaikinskii
Pages: 1  16
Abstract: A linearized system of equations of hydrodynamics with timedependent spatially localized righthand 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 planeparallel flow. A method of constructing asymptotic solutions of this problem is suggested; it consists of two stages: (1) a reduction of the threedimensional problem to a twodimensional inhomogeneous pseudodifferential equation on the nonperturbed free surface of the liquid, (2) a representation of the localized righthand 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 righthand sides and its combination with “nonstandard” characteristics. A method of calculation (generalizing longstanding 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: 20180101
DOI: 10.1134/s1061920818010016
Issue No: Vol. 25, No. 1 (2018)
 Authors: S. Yu. Dobrokhotov; V. E. Nazaikinskii
 Twisted Burnside–Frobenius Theory for Endomorphisms of Polycyclic
Groups Authors: A. L. Fel’shtyn; E. V. Troitsky
Pages: 17  26
Abstract: Let R(ϕ) be the number of ϕconjugacy (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: 20180101
DOI: 10.1134/s1061920818010028
Issue No: Vol. 25, No. 1 (2018)
 Authors: A. L. Fel’shtyn; E. V. Troitsky
 New Trace Formulas in Terms of Resonances for ThreeDimensional
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: 20180101
DOI: 10.1134/s106192081801003x
Issue No: Vol. 25, No. 1 (2018)
 Authors: H. Isozaki; E. L. Korotyaev
 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 rStirling numbers of the second kind and the degenerate rBell polynomials. Especially, we will express the degenerate rBell polynomials as linear combinations of many wellknown families of special polynomials.
PubDate: 20180101
DOI: 10.1134/s1061920818010041
Issue No: Vol. 25, No. 1 (2018)
 Authors: T. Kim; Y. Yao; D. S. Kim; G. W. Jang
 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: 20180101
DOI: 10.1134/s1061920818010053
Issue No: Vol. 25, No. 1 (2018)
 Authors: V. P. Maslov
 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: 20180101
DOI: 10.1134/s1061920818010065
Issue No: Vol. 25, No. 1 (2018)
 Authors: A. I. Nazarov; A. P. Shcheglova
 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 threedimensional 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 twodimensional crossshaped 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: 20180101
DOI: 10.1134/s1061920818010077
Issue No: Vol. 25, No. 1 (2018)
 Authors: S. A. Nazarov
 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 reactiondiffusion equation in a twodimensional 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: 20180101
DOI: 10.1134/s1061920818010089
Issue No: Vol. 25, No. 1 (2018)
 Authors: N. N. Nefedov; E. I. Nikulin
 Modeling of Pulse Signals in 3D Propagation Problems of DeepWater
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: 20180101
DOI: 10.1134/s1061920818010090
Issue No: Vol. 25, No. 1 (2018)
 Authors: P. S. Petrov; S. A. Sergeev; A. A. Tolchennikov
 Countably Solvable Connected ProLie Groups Are u Amenable
 Authors: A. I. Shtern
Pages: 113  115
Abstract: In contrast to some wellknown discrete groups, countably solvable connected proLie groups are uamenable in the sense of de la Harpe’s 1973 paper.
PubDate: 20180101
DOI: 10.1134/s1061920818010107
Issue No: Vol. 25, No. 1 (2018)
 Authors: A. I. Shtern
 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 Hfunctions γ p,q m,n (z) and Γ p,q m,n (z) as well as their such special cases as the incomplete FoxWright 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 Hfunctions, 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 Hfunctions. 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: 20180101
DOI: 10.1134/s1061920818010119
Issue No: Vol. 25, No. 1 (2018)
 Authors: H. M. Srivastava; R. K. Saxena; R. K. Parmar
 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 selfadjoint. We investigate the nonrelativistic limit as the velocity of light tends to infinity.
PubDate: 20171001
DOI: 10.1134/s1061920817040021
Issue No: Vol. 24, No. 4 (2017)
 Authors: V. Budyika; M. Malamud; A. Posilicano
 On the optimal form of nanorods
 Authors: Yu. V. Egorov
Pages: 436  453
Abstract: The Euler–Bernoulli model of elastic nanorods 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: 20171001
DOI: 10.1134/s1061920817040033
Issue No: Vol. 24, No. 4 (2017)
 Authors: Yu. V. Egorov
 Bistates and 2level systems in rectangular Penning traps
 Authors: M. Karasev; E. Novikova; E. Vybornyi
Pages: 454  464
Abstract: We introduce a notion of semiclassical bistates. They arise from pairs of eigenstates corresponding to tunnelsplit eigenlevels and generate 2level 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 bistates, and 2level subsystems in such a quantum trap.
PubDate: 20171001
DOI: 10.1134/s1061920817040045
Issue No: Vol. 24, No. 4 (2017)
 Authors: M. Karasev; E. Novikova; E. Vybornyi
 Asymptotic expansions for some integrals of quotients with degenerated
divisors 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 fourwaves interactions.
PubDate: 20171001
DOI: 10.1134/s1061920817040069
Issue No: Vol. 24, No. 4 (2017)
 Authors: S. Kuksin
 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 selfcorrelated 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: 20171001
DOI: 10.1134/s1061920817040082
Issue No: Vol. 24, No. 4 (2017)
 Authors: V. P. Maslov
 Explicit determination of certain periodic motions of a generalized
twofield gyrostat Authors: A. A. Oshemkov; P. E. Ryabov; S. V. Sokolov
Pages: 517  525
Abstract: The case of motion of a generalized twofield 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: 20171001
DOI: 10.1134/s1061920817040100
Issue No: Vol. 24, No. 4 (2017)
 Authors: A. A. Oshemkov; P. E. Ryabov; S. V. Sokolov
 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. SekerzhZen’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 threelevel scale. Discussion. A remark concerning the referees.
PubDate: 20171001
DOI: 10.1134/s1061920817040112
Issue No: Vol. 24, No. 4 (2017)
 Authors: S. Ya. SekerzhZen’kovich
 A family of pseudodifferential 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 pseudodifferential 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: 20171001
DOI: 10.1134/s1061920817040124
Issue No: Vol. 24, No. 4 (2017)
 Authors: H. M. Srivastava; S. K. Upadhyay; K. Khatterwani
 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 socalled piston model of tsunami generation [1, 2]). The problem is to determine the free surface elevation at the subsequent times.
PubDate: 20171001
DOI: 10.1134/s1061920817040136
Issue No: Vol. 24, No. 4 (2017)
 Authors: S. Yu. Dobrokhotov; V. E. Nazaikinskii; A. A. Tolchennikov