
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 [2335 journals] 
 Weak and strong probabilistic solutions for a stochastic quasilinear
parabolic equation with nonstandard growth Authors: Z. I. Ali; M. Sango
Pages: 283  308
Abstract: Abstract In this paper, we investigate a class of stochastic quasilinear parabolic initial boundary value problems with nonstandard growth in the functional setting of generalized Sobolev spaces. The deterministic version of the equation was first introduced and studied by Samokhin in [45] as a generalized model for polytropic filtration. We establish an existence result of weak probabilistic solutions when the forcing terms do not satisfy Lipschitz conditions. Under the Lipschitz property of the forcing terms, we obtain the uniqueness of weak probabilistic solutions. Combining the uniqueness and the famous Yamada–Watanabe result, we prove the existence of a unique strong probabilistic solution of the problem.
PubDate: 20160701
DOI: 10.1134/s1061920816030018
Issue No: Vol. 23, No. 3 (2016)
 Authors: Z. I. Ali; M. Sango
 Unique solvability of a nonstationary problem of radiativeconductive heat
exchange in a system of semitransparent bodies Authors: A. A. Amosov
Pages: 309  334
Abstract: Abstract A nonstationary initial boundary value problem describing the radiativeconductive heat exchange in a system of semitransparent bodies is considered. The radiation transfer equation with boundary conditions of mirror reflection and refraction according to the Fresnel laws is used to describe the propagation of radiation. The dependence of the radiation intensity and the optical properties of bodies on the radiation frequency is taken into account. The existence and uniqueness of a weak solution are proved. A comparison theorem is proved. Some a priori estimates for the weak solution are derived and its regularity is proved.
PubDate: 20160701
DOI: 10.1134/s106192081603002x
Issue No: Vol. 23, No. 3 (2016)
 Authors: A. A. Amosov
 On the classes H p ω and Lip (α, p ) for trigonometric series
with monotone coefficients Authors: A. P. Antonov
Pages: 335  342
Abstract: Abstract The paper is devoted to the study of a relationship between the behavior of the coefficients of trigonometric series of many variables and the smoothness of the sums of these series in the spaces L p .
PubDate: 20160701
DOI: 10.1134/s1061920816030031
Issue No: Vol. 23, No. 3 (2016)
 Authors: A. P. Antonov
 Algebraic functions of complexity one, a Weierstrass theorem, and three
arithmetic operations Authors: V. K. Beloshapka
Pages: 343  347
Abstract: Abstract The Weierstrass theorem concerning functions admitting an algebraic addition theorem enables us to give an explicit description of algebraic functions of two variables of analytical complexity one. Their description is divided into three cases: the general case, which is elliptic, and two special ones, a multiplicative and an additive one. All cases have a unified description; they are the orbits of an action of the gauge pseudogroup. The first case is a 1parameter family of orbits of “elliptic addition,” the second is the orbit of multiplication, and the third of addition. Here the multiplication and addition can be derived from the “elliptic addition” by passages to a limit. On the other hand, the elliptic orbits correspond to complex structures on the torus, the multiplicative orbit corresponds to the complex structure on the cylinder, and the additive one to that on the complex plane. This work was financially supported by the Russian Foundation for Basic Research under grants nos. 1400709a and 130112417ofim2.
PubDate: 20160701
DOI: 10.1134/s1061920816030043
Issue No: Vol. 23, No. 3 (2016)
 Authors: V. K. Beloshapka
 Semiclassical spectral series of a quantum Schrödinger operator
corresponding to a singular circle composed of equilibria Authors: V. L. Chernyshev
Pages: 348  354
Abstract: Abstract On a twodimensional surface, a Schrödinger operator is considered with a potential whose critical points form a closed curve. We pose the problem of describing the semiclassical spectral series corresponding to this curve. The standard construction for describing the spectral series corresponding to isolated nondegenerate equilibria or to periodic trajectories of Hamiltonian systems is not applicable in this situation. In the paper, a description of semiclassical solutions of the spectral problem for the quantum Schrödinger operator that correspond to nonisolated equilibria are presented and, to calculate the splitting of the eigenvalues, it turns out to be necessary to find the spectral series with a higher order of accuracy than it is usually required.
PubDate: 20160701
DOI: 10.1134/s1061920816030055
Issue No: Vol. 23, No. 3 (2016)
 Authors: V. L. Chernyshev
 Negative energy, debts, and disinformation from the viewpoint of analytic
number theory Authors: V. P. Maslov
Pages: 355  368
Abstract: Abstract The number zero and negative numbers are added to analytical number theory which includes transcendents. New solutions of Diophantine equations are applied to thermodynamics, information theory and biology.
PubDate: 20160701
DOI: 10.1134/s1061920816030067
Issue No: Vol. 23, No. 3 (2016)
 Authors: V. P. Maslov
 A series of irreducible unitary representations of a group of
diffeomorphisms of the halfline Authors: E. D. Romanov
Pages: 369  381
Abstract: Abstract We present a family of unitary representations of a group of diffeomorphisms of a finitedimensional real Euclidean space using a family of quasiinvariant measures. In the onedimensional case, for a special kind of group diffeomorphisms of the halfline, we prove the irreducibility of the representations thus obtained.
PubDate: 20160701
DOI: 10.1134/s1061920816030079
Issue No: Vol. 23, No. 3 (2016)
 Authors: E. D. Romanov
 Some families of generating functions and associated hypergeometric
transformation and reduction formulas Authors: H. M. Srivastava
Pages: 382  391
Abstract: Abstract Summation, transformation and reduction formulas for various families of hypergeometric functions in one, two and more variables are potentially useful in many diverse areas of applications. The main object of this paper is to derive several substantially more general results on this subject than those considered recently by Neethu et al. [7] in connection with Bailey’s transformation involving the Gauss hypergeometrc function 2F 1 (see [1]). The methodology used here is based essentially on some families of hypergeometric generating functions. Relevant connections of the results presented in this paper with those in the earlier works are also pointed out.
PubDate: 20160701
DOI: 10.1134/s1061920816030080
Issue No: Vol. 23, No. 3 (2016)
 Authors: H. M. Srivastava
 Embedding theorems for a weighted Sobolev class in the space L q,v with
weights having a singularity at a point: Case v ∉ L q 1 Authors: A. A. Vasil’eva
Pages: 392  424
Abstract: Abstract In the paper, embedding theorems for reduced weighted Sobolev classes in the space L q,v with weights having a singularity at a point are obtained. For a special class of weights, estimates for widths are also obtained. The case of v ∉ L q is considered.
PubDate: 20160701
DOI: 10.1134/s1061920816030092
Issue No: Vol. 23, No. 3 (2016)
 Authors: A. A. Vasil’eva
 Identities for Apostoltype Frobenius–Euler polynomials resulting from
the study of a nonlinear operator Authors: A. Bayad; T. Kim
Pages: 164  171
Abstract: Abstract We introduce a special nonlinear differential operator and, using its properties, reduce higherorder Frobenius–Euler Apostoltype polynomials to a finite series of firstorder Apostoltype Frobenius–Euler polynomials and Stirling numbers. Interesting identities are established.
PubDate: 20160401
DOI: 10.1134/s1061920816020023
Issue No: Vol. 23, No. 2 (2016)
 Authors: A. Bayad; T. Kim
 Belavkin filtering with squeezed light sources
 Authors: A. Dąbrowska; J. Gough
Pages: 172  184
Abstract: Abstract We derive the filtering equation for Markovian systems undergoing homodyne measurement in the situation where the output processes being monitored are squeezed. The filtering theory applies to case where the system is driven by Fock noise (that is, quantum input processes in a coherent state) and where the output is mixed with a squeezed signal. It also applies to the case of a system driven by squeezed noise, but here there is a physical restriction to emission/absorption coupling only. For the special case of a cavity mode where the dynamics is linear, we are able to derive explicitly the filtered estimate π t (a) for the mode annihilator a based on the homodyne quadrature observations up to time t.'
PubDate: 20160401
DOI: 10.1134/s1061920816020035
Issue No: Vol. 23, No. 2 (2016)
 Authors: A. Dąbrowska; J. Gough
 Asymptotic analysis of evolution of a neck in extended thin rigid plastic
solids Authors: D. V. Georgievskii; B. E. Pobedrya
Pages: 200  206
Abstract: Abstract We carry out an asymptotic analysis with natural small geometric parameter of the stressstrain state realized under the stretch of a rigidplastic rod of round section and of a flat sheet infinite in one direction, taking into account the presence of areas of thinning (a neck) and thickening with respect to the medium size. In examples, we stress qualitative effects of possible inertiafree development of a neck.
PubDate: 20160401
DOI: 10.1134/s1061920816020059
Issue No: Vol. 23, No. 2 (2016)
 Authors: D. V. Georgievskii; B. E. Pobedrya
 Contribution to the symplectic structure in the quantization rule due to
noncommutativity of adiabatic parameters Authors: M. V. Karasev
Pages: 207  218
Abstract: Abstract A geometric construction of the `ala Planck action integral (quantization rule) determining adiabatic terms for fastslow systems is considered. We demonstrate that in the first (after zero) adiabatic approximation order, this geometric rule is represented by a deformed fast symplectic 2form. The deformation is controlled by the noncommutativity of the slow adiabatic parameters. In the case of one fast degree of freedom, the deformed symplectic form incorporates the contraction of the slow Poisson tensor with the adiabatic curvature. The same deformed fast symplectic structure is used to represent the improved adiabatic invariant in a geometric form.
PubDate: 20160401
DOI: 10.1134/s1061920816020060
Issue No: Vol. 23, No. 2 (2016)
 Authors: M. V. Karasev
 A finitedimensional version of Fredholm representations
 Authors: V. Manuilov
Pages: 219  224
Abstract: Abstract We consider pairs of mappings from a discrete group Γ to the unitary group. The deficiencies of these mappings from being homomorphisms may be great, but if they are close to each other, then we call such pairs balanced. We show that balanced pairs determine elements in the K 0 group of the classifying space of the group. We also show that a Fredholm representation of Γ determines balanced pairs.
PubDate: 20160401
DOI: 10.1134/s1061920816020072
Issue No: Vol. 23, No. 2 (2016)
 Authors: V. Manuilov
 Propagation and interaction of solitons for nonintegrable equations
 Authors: G. Omel’yanov
Pages: 225  243
Abstract: Abstract We describe an approach to the construction of multisoliton asymptotic solutions for nonintegrable equations. The general idea is realized in the case of N waves, N = 1, 2, 3, and for the KdVtype equation with nonlinearity u 4. A brief review of asymptotic methods as well as results of numerical simulation are included.
PubDate: 20160401
DOI: 10.1134/s1061920816020084
Issue No: Vol. 23, No. 2 (2016)
 Authors: G. Omel’yanov
 On the distribution of energy of localized solutions of the Schrödinger
equation that propagate along symmetric quantum graphs Authors: A. I. Shafarevich
Pages: 244  250
Abstract: Abstract From the point of view of applications to quantum mechanics, it is natural to pose a question concerning the distribution of energy of localized solutions of a nonstationary Schrödinger equation over the graph (in other words, the probability to find a quantum particle in a given area). This problem is apparently very complicated for general graphs, because the energy distribution is much more sensitive to the form of boundary conditions and to the initial state than the asymptotic behavior of the number of localized functions. Below, we present initial results concerning the distribution of energy in the case of symmetric quantum graphs (this means that the Schrödinger operators on different edges have the same structure). For general local selfadjoint boundary conditions, we describe the process of onestep scattering of the localized solutions and obtain a simple general result of the distribution of energy. Some special cases and specific examples are discussed.
PubDate: 20160401
DOI: 10.1134/s1061920816020096
Issue No: Vol. 23, No. 2 (2016)
 Authors: A. I. Shafarevich
 On rational functions of firstclass complexity
 Authors: M. Stepanova
Pages: 251  256
Abstract: Abstract It is proved that, for every rational function of two variables P(x, y) of analytic complexity one, there is either a representation of the form f(a(x) + b(y)) or a representation of the form f(a(x)b(y)), where f(x), a(x), b(x) are nonconstant rational functions of a single variable. Here, if P(x, y) is a polynomial, then f(x), a(x), and b(x) are nonconstant polynomials of a single variable.
PubDate: 20160401
DOI: 10.1134/s1061920816020102
Issue No: Vol. 23, No. 2 (2016)
 Authors: M. Stepanova
 Bloch principle for elliptic differential operators with periodic
coefficients Authors: V. V. Zhikov; S. E. Pastukhova
Pages: 257  277
Abstract: Abstract Differential operators corresponding to elliptic equations of divergent type with 1periodic coefficients are considered. The equations are put in Sobolev spaces with an arbitrary 1periodic Borel measure on the entire space R d . In the study of the spectrum of operators of this kind, the Bloch principle is of fundamental importance. According to this principle, all points of the desired spectrum are obtained when studying the equation on the unit cube with quasiperiodic boundary conditions. The proof of the Bloch principle for problems in the above formulation is proved, in several versions of the principle. Examples of the application of the principle to finding the spectrum of specific operators, for example, for the Laplacian in a weighted space or on a singular structure of lattice type.
PubDate: 20160401
DOI: 10.1134/s1061920816020114
Issue No: Vol. 23, No. 2 (2016)
 Authors: V. V. Zhikov; S. E. Pastukhova
 Thermodynamics, idempotent analysis, and tropical geometry as a return to
primitivism Authors: V. P. Maslov
Pages: 278  280
PubDate: 20160401
DOI: 10.1134/s1061920816020126
Issue No: Vol. 23, No. 2 (2016)
 Authors: V. P. Maslov
 Absence of two Paley–Wiener properties for semisimple Lie groups of
real rank one Authors: A. I. Shtern
Pages: 281  282
Abstract: Abstract The weak Paley–Wiener property and the topological Paley–Wiener property for connected semisimple Lie groups of real rank one with finite center are discussed.
PubDate: 20160401
DOI: 10.1134/s1061920816020138
Issue No: Vol. 23, No. 2 (2016)
 Authors: A. I. Shtern