
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 [2351 journals] 
 Further extended Caputo fractional derivative operator and its
applications Authors: P. Agarwal; S. Jain; T. Mansour
Pages: 415  425
Abstract: In this paper, our principle aim is to establish a new extension of the Caputo fractional derivative operator involving the generalized hypergeometric type function F p (a, b; c; z; k), introduced by Lee et al. Some extensions of the generalized hypergeometric functions and their integral representations are also presented. Furthermore, linear and bilinear generating relations for the extended hypergeometric functions are obtained. We also present some properties of the extended fractional derivative operator.
PubDate: 20171001
DOI: 10.1134/s106192081704001x
Issue No: Vol. 24, No. 4 (2017)
 Authors: P. Agarwal; S. Jain; T. Mansour
 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
 A new approach to Catalan numbers using differential equations
 Authors: D. S. Kim; T. Kim
Pages: 465  475
Abstract: In this paper, we introduce two differential equations arising from the generating function of the Catalan numbers which are ‘inverses’ to each other in a certain sense. From these differential equations, we obtain some new and explicit identities for Catalan and higherorder Catalan numbers. In addition, by other means than differential equations, we also derive some interesting identities involving Catalan numbers which are of arithmetic and combinatorial nature.
PubDate: 20171001
DOI: 10.1134/s1061920817040057
Issue No: Vol. 24, No. 4 (2017)
 Authors: D. S. Kim; T. Kim
 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
 On the kernel of the Laplace operator on twodimensional polyhedra
 Authors: E. N. Lukzen; A. I. Shafarevich
Pages: 488  493
Abstract: The structure of spaces of harmonic functions on polyhedra is studied.
PubDate: 20171001
DOI: 10.1134/s1061920817040070
Issue No: Vol. 24, No. 4 (2017)
 Authors: E. N. Lukzen; A. I. Shafarevich
 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
 Transport through a network of capillaries from ultrametric diffusion
equation with quadratic nonlinearity Authors: K. Oleschko; A. Khrennikov
Pages: 505  516
Abstract: This paper is about a novel mathematical framework to model transport (of, e.g., fluid or gas) through networks of capillaries. This framework takes into account the tree structure of the networks of capillaries. (Roughly speaking, we use the treelike system of coordinates.) As is well known, treegeometry can be topologically described as the geometry of an ultrametric space, i.e., a metric space in which the metric satisfies the strong triangle inequality: in each triangle, the third side is less than or equal to the maximum of two other sides. Thus transport (e.g., of oil or emulsion of oil and water in porous media, or blood and air in biological organisms) through networks of capillaries can be mathematically modelled as ultrametric diffusion. Such modelling was performed in a series of recently published papers of the authors. However, the process of transport through capillaries can be only approximately described by the linear diffusion, because the concentration of, e.g., oil droplets, in a capillary can essentially modify the dynamics. Therefore nonlinear dynamical equations provide a more adequate model of transport in a network of capillaries. We consider a nonlinear ultrametric diffusion equation with quadratic nonlinearity  to model transport in such a network. Here, as in the linear case, we apply the theory of ultrametric wavelets. The paper also contains a simple introduction to theory of ultrametric spaces and analysis on them.
PubDate: 20171001
DOI: 10.1134/s1061920817040094
Issue No: Vol. 24, No. 4 (2017)
 Authors: K. Oleschko; A. Khrennikov
 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
 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: 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
 Asymptotic completeness of scattering in the nonlinear Lamb system for
nonzero mass Authors: A. I. Komech; A. E. Merzon
Pages: 336  346
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: 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: 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: 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
 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: 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: 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