Showing papers in "Russian Journal of Mathematical Physics in 2006"
TL;DR: In this article, the asymptotic behavior of the operator exponent related to the Cauchy problem for a parabolic equation with periodic coefficients is studied either under the reduction of the periodicity cell or for large times.
Abstract: The asymptotic behavior of the operator exponent related to the Cauchy problem for a parabolic equation with periodic coefficients is studied either under the reduction of the periodicity cell or for large times. Estimates for the closeness of the operator exponentials (the original and the limit) with respect to the L2-operator norm and the related H1-estimates are obtained under minimal assumptions concerning the smoothness of the heat matrix and of the initial data.
101 citations
TL;DR: In this paper, a twisted q-partial zeta function and some twisted two-variable q-L-functions that interpolate q-Bernoulli numbers, β fixme (q), and Bernoulli polynomials, β petertodd (x, q), respectively, at negative integers were constructed.
Abstract: In this paper, we construct a twisted q-partial zeta function and some twisted two-variable q-L-functions that interpolate q-Bernoulli numbers, β
(q), and Bernoulli polynomials, β
(x, q), respectively, at negative integers. Using these functions, we prove the existence of a p-adic interpolation function that interpolates the q-generalized polynomials β
(x, q) at negative integers. Consequently, we define a p-adic twisted q-L-function which is a solution of a question of Kim et al.
89 citations
TL;DR: In this paper, Kupershmidt constructed reflection symmetries of q-Bernoulli polynomials and q-Euler numbers using q-derivatives and qintegrals.
Abstract: Recently, B. A. Kupershmidt constructed reflection symmetries of q-Bernoulli polynomials (see [12]). In this paper, we study new q-extensions of Euler numbers and polynomials by using the method of Kupershmidt. We also investigate the properties of symmetries of these q-Euler polynomials by using q-derivatives and q-integrals.
77 citations
TL;DR: In this paper, an analytic function interpolating the multiple generalized Bernoulli numbers attached to a primitive Dirichlet character X at negative integers in the complex plane is studied, and the values of the partial derivative of this multiple p-adic L-function at s = 0 are given.
Abstract: An analytic function interpolating the multiple generalized Bernoulli numbers attached to a primitive Dirichlet character X at negative integers in the complex plane is studied. The multiple p-adic L-function is constructed as the p-adic analog of the above function. Finally, the values of the partial derivative of this multiple p-adic L-function at s = 0 are given.
52 citations
TL;DR: In this article, a mixed boundary value problem for a second-order strongly elliptic equation in a Lipschitz domain is considered, where the boundary condition on a part of the boundary is of the first order and contains a weight function and the spectral parameter, while on the remaining part the homogeneous Dirichlet condition is imposed.
Abstract: We consider a mixed boundary value problem for a second-order strongly elliptic equation in a Lipschitz domain. The boundary condition on a part of the boundary is of the first order and contains a weight function and the spectral parameter, while on the remaining part the homogeneous Dirichlet condition is imposed. The aim is to weaken the conditions sufficient for justifying the classical asymptotic formula for the eigenvalues. We show that it suffices to assume the boundary to be C 1 in a neighborhood of the support of the weight outside a closed subset of zero measure.
47 citations
TL;DR: In this article, the authors present, in a unified manner, a number of key results for fractional-calculus operators such as the Riemann-Liouville, the Weyl, and other fractional calculus operators based on the Cauchy-Goursat Integral Formula.
Abstract: In many recent works, several authors demonstrated the usefulness of fractional-calculus operators in many different directions. The main object of this paper is to present, in a unified manner, a number of key results for the general
$$\bar H$$
-function, which is associated with a certain class of Feynman integrals, involving the Riemann-Liouville, the Weyl, and other fractional-calculus operators such as those based on the Cauchy-Goursat Integral Formula. Various paricular cases and consequences of our main fractional-calculus results are also considered.
36 citations
TL;DR: In this paper, quantum geometry, algebras with nonlinear commutation relations, representation theory, the coherent transform, and operator averaging are used to solve the perturbation problem for wave (quantum) systems near a resonance point or a resonance equilibrium ray in two and three-dimensional spaces.
Abstract: Quantum geometry, algebras with nonlinear commutation relations, representation theory, the coherent transform, and operator averaging are used to solve the perturbation problem for wave (quantum) systems near a resonance equilibrium point or a resonance equilibrium ray in two and three-dimensional spaces.
31 citations
TL;DR: In this paper, the Zipf law for frequency dictionaries is described and the principles from which these formulas follow are discussed, as well as their application in the context of dictionaries.
Abstract: Formulas specifying the Zipf law for frequency dictionaries are presented. Principles from which these formulas follow are discussed.
27 citations
TL;DR: Inverse spectral problems for non-selfadjoint matrix Sturm-Liouville differential operators on a finite interval and on the half-line are studied in this article, where the so-called Weyl matrix is introduced and it is shown that the specification of this matrix uniquely determines the matrix potential and the coefficients of the boundary conditions.
Abstract: Inverse spectral problems for nonselfadjoint matrix Sturm-Liouville differential operators on a finite interval and on the half-line are studied. As a main spectral characteristic, we introduce the so-called Weyl matrix and prove that the specification of the Weyl matrix uniquely determines the matrix potential and the coefficients of the boundary conditions. Moreover, for a finite interval, we also study the inverse problems of recovering matrix Sturm-Liouville operators from discrete spectral data (eigenvalues and “weight” numbers) and from a system of spectra. The results thus obtained are natural generalizations of the classical results in inverse problem theory for scalar Sturm-Liouville operators.
25 citations
TL;DR: In this article, the authors established soliton-like asymptotics for finite energy solutions to the Schrodinger equation coupled to a nonrelativistic classical particle, and showed that any solution with initial state close to the solitary manifold converges to a sum of a travelling wave and an outgoing free wave.
Abstract: We establish soliton-like asymptotics for finite energy solutions to the Schrodinger equation coupled to a nonrelativistic classical particle. Any solution with initial state close to the solitary manifold converges to a sum of a travelling wave and an outgoing free wave. The convergence holds in global energy norm. The proof uses spectral theory and the symplectic projection onto the solitary manifold in the Hilbert phase space.
23 citations
TL;DR: In this article, it is assumed that the principal part of L is a uniformly strongly elliptic operator and the coefficients cα,β with |α| + |β| < 2m are distributions.
Abstract: In this paper, we deal with operators of the form
$$L = \sum\limits_{\left| \alpha \right|,\left| \beta \right| \leqslant m} {D^\alpha c_{\alpha ,\beta } (x)D^\beta } $$
on the space ℝn. It is assumed that the principal part of L is a uniformly strongly elliptic operator and the coefficients c
α,β with |α| + |β| < 2m are distributions. We find sufficient conditions on these coefficients (in terms of generalized Sobolev spaces with negative smoothness indices to which these coefficients belong) for the operator in question to be well defined in the sense of quadratic forms.
TL;DR: For potential-type integral operators on a Lipschitz surface, an asymptotic formula for eigenvalues is proved in this article, based on the study of the rate of operator convergence as smooth surfaces approximate the Lipschnitz surface.
Abstract: For potential-type integral operators on a Lipschitz surface, an asymptotic formula for eigenvalues is proved. The reasoning is based upon the study of the rate of operator convergence as smooth surfaces approximate the Lipschitz surface.
TL;DR: The van der Waerden continuity theorem was proved in this paper, and the relation between the assertion of the theorem and some properties of the Bohr compactifications of topological groups was established.
Abstract: As was proved by van der Waerden in 1933, every finite-dimensional locally bounded representation of a semisimple compact Lie group is continuous. This is the famous “van der Waerden continuity theorem,” and it motivated a vast literature. In particular, relationships between the assertion of the theorem (and of the inverse, in a sense, to this theorem) and some properties of the Bohr compactifications of topological groups were established, which led to the introduction and the study of certain classes of the so-called van der Waerden groups and algebras. Until now, after more than 70 years have passed, the van der Waerden theorem appears in monographs and surveys in diverse forms; new proofs were found and then simplified in important special cases.
TL;DR: In this article, the moduli space M(n, K) of the family of polynomial models was constructed, i.e., the space parametrizing the holomorphically nonequivalent models.
Abstract: In [6], to a completely nondegenerate germ of a real submanifold of a chosen C R-type (n, K) in a complex space, we assigned a tangent polynomial model of the submanifold. In the present paper, we construct the moduli space M(n, K) of the family of polynomial models, i.e., the space parametrizing the holomorphically nonequivalent polynomial models. The space thus obtained is used to construct C R-characteristic classes.
TL;DR: In this article, a p-adic approximation of E n,x for certain n was given, and the padic l-function of the Euler numbers and the k-adic measure for Euler number were treated.
Abstract: In this paper, we give some p-adic approximation of E n,x for certain n. Finally we will treat p-adic l-function of Kubota-Leopoldt’s type Euler numbers and p-adic measure for Euler numbers.
TL;DR: In this paper, the authors trace the main chronological stages in the mid nineteenth and early twentieth centuries concerning the understanding and proof of the Π-theorem treated both as a purely mathematical assertion and from the point of view of physics and the theory of experiment.
Abstract: The 100-year anniversary of the proof of one of the brightest and most universal theorems in mechanics and physics, the Π-theorem in dimension theory is approaching. In connection with this anniversary, it is of interest to trace the main chronological stages in the mid nineteenth and early twentieth centuries concerning the understanding and proof of the Π-theorem treated both as a purely mathematical assertion and from the point of view of physics and the theory of experiment. Below we reproduce and comment out these stages.
TL;DR: In this paper, a priori estimates for pseudohyperbolic PDOs with constant coefficients are obtained and the solvability of the Cauchy problem is developed.
Abstract: The first part of the paper focuses on a detailed description of strictly pseudohyperbolic polynomials. The classes of homogeneous and nonhomogeneous polynomials are considered separately. Based on this description, a priori estimates for pseudohyperbolic PDOs with constant coefficients are obtained and the solvability of the Cauchy problem is developed.
TL;DR: In this article, the authors studied the asymptotic behavior of solutions of semilinear abstract differential equations and showed that the solutions of these problems belong to the class of almost periodic, almost automorphic or almost periodic Banach space valued functions.
Abstract: In this paper, we study the asymptotic behavior of solutions of semilinear abstract differential equations (*) u′(t) = Au(t) + tnf(t, u(t)), where A is the generator of a C0-semigroup (or group) T(·), f(·, x) ∈ A for each x ∈ X, A is the class of almost periodic, almost automorphic or Levitan almost periodic Banach space valued functions ϕ: ℝ → X and n ∈ {0, 1, 2, ...}. We investigate the linear case when T(·)x is almost periodic for each x ∈ X; and the semilinear case when T(·) is an asymptotically stable C0-semigroup, n = 0 and f(·, x) satisfies a Lipschitz condition. Also, in the linear case, we investigate (*) when ϕ belongs to a Stepanov class Sp-A defined similarly to the case of Sp-almost periodic functions. Under certain conditions, we show that the solutions of (*) belong to Au:= A ∩ BUC(ℝ, X) if n = 0 and to tnAu ⊕ wnC0 (ℝ, X) if n ∈ ℕ, where wn(t) = (1 + |t|)n. The results are new for the case n ∈ ℕ and extend many recent ones in the case n = 0.
TL;DR: In this article, a real analytic isomorphism between periodic Jacobi operators and the spectral data formed by the gap lengths, the distances between the Dirichlet eigenvalues and the center of the corresponding gap, and some signs is constructed.
Abstract: We construct a real analytic isomorphism between periodic Jacobi operators and the spectral data formed by the gap lengths, the distances between the Dirichlet eigenvalues and the center of the corresponding gap, and some signs. This proves the uniqueness of the solution of the inverse problem and gives a characterization of the solution. Moreover, two-sided a priori estimates of periodic Jacobi operators in terms of gap lengths are obtained.
TL;DR: Asymptotic solutions of Navier-Stokes equations describing three-phase coherent structures of the Taylor scale are constructed in this paper, and the solutions turn out to be unstable with respect to small-scale perturbations, which finally leads to the appearance of perturbation of Kolmogorov scale.
Abstract: Asymptotic solutions of the Navier-Stokes equations describing three-phase coherent structures of the Taylor scale are constructed. These equations are related to topological invariants of the Liouville foliations on the three-dimensional torus into two-dimensional tori. The solutions turn out to be unstable with respect to small-scale perturbations, which finally leads to the appearance of perturbations of Kolmogorov scale. The flicker noise occurs at the corresponding frequencies.
TL;DR: In this article, the authors demonstrate a method that permits to obtain generalized solutions for some quasilinear equations and systems of hyperbolic type using the theory of equilibrium of a potential in an external field.
Abstract: We demonstrate a method that permits to obtain generalized solutions for some quasilinear equations and systems of hyperbolic type. The corresponding variational principle is constructed using the theory of equilibrium of a potential in an external field.
TL;DR: In this article, the problem of finding the number of independent compatibility equations in strains in n-dimensional Euclidean space is discussed and two counterexamples are presented that show the impossibility to transfer three "diagonal" or three "nondiagonal" Beltrami-Michell equations from the domain of an elastic solid to its boundary.
Abstract: The problem of finding the number of independent compatibility equations in strains in n-dimensional Euclidean space is discussed. The number in question coincides with the number of independent Beltrami-Michell compatibility equations in terms of stresses used when formulating the problem in elasticity theory and also with the number of components of the incompatibility tensor and of the Riemann-Christoffel tensor. For n = 3, two counterexamples are presented that show the impossibility to transfer three “diagonal” or three “nondiagonal” Beltrami-Michell equations from the domain of an elastic solid to its boundary. In this case, the formulation of the problem becomes nonequivalent to either classical or new formulating of the problem in terms of stresses.
TL;DR: In this paper, the spectral expansion for the differential operator generated in L 2 (−∞, ∞) by an ordinary differential expression of arbitrary order with periodic complex-valued coefficients Lebesgue integrable on bounded intervals was constructed.
Abstract: In the paper, we construct the spectral expansion for the differential operator generated in L2(−∞, ∞) by an ordinary differential expression of arbitrary order with periodic complex-valued coefficients Lebesgue integrable on bounded intervals
TL;DR: In this article, the technique of tensors with values in the Atiyah-Kahler algebra has been developed, which yields a new representation of the relativistic equations of field theory.
Abstract: A mathematical apparatus, the so-called technique of Λ-tensors, is developed in the paper (the technique of tensors with values in the Atiyah-Kahler algebra). This apparatus yields a new representation of the relativistic equations of field theory.
TL;DR: With the help of the apparatus of tensors with values in the Atiyah-Kahler algebra (the apparatus of lambda tensors), a new representation for the relativistic equation of field theory is proposed as discussed by the authors.
Abstract: With the help of the apparatus of tensors with values in the Atiyah-Kahler algebra (the apparatus of lambda tensors), a new representation for the relativistic equation of field theory is proposed. The circle of questions under consideration is standard for investigations concerning the theory of the Dirac equation (in this paper, we treat no problems dealing with second quantization).
TL;DR: In this paper, the trajectories of diffractive rays are regarded as geodesics in a Riemannian manifold, whose metric and topological properties are those induced by the refractive index.
Abstract: In this article, the ray tracing method is studied beyond the classical geometrical theory. The trajectories are here regarded as geodesics in a Riemannian manifold, whose metric and topological properties are those induced by the refractive index (or, equivalently, by the potential). First, we derive the geometrical quantization rule, which is relevant to describe the orbiting bound-states observed in molecular physics. Next, we derive properties of the diffractive rays, regarded here as geodesics in a Riemannian manifold with boundary. A particular attention is devoted to the following problems: (i) modification of the classical stationary phase method suited to a neighborhood of a caustic; (ii) derivation of the connection formulas which enable one to obtain the uniformization of the classical eikonal approximation by patching up geodesic segments crossing the axial caustic; (iii) extension of the eikonal equation to mixed hyperbolic-elliptic systems, and generation of complex-valued rays in the shadow of the caustic. By these methods, we can study creeping waves in diffractive scattering, describe the orbiting resonances present in molecular scattering beside the orbiting bound-states, and, finally, describe the generation of evanescent waves, which are relevant in the nuclear rainbow.
TL;DR: In this article, the equilibrium set of the potential of an extremal measure for some energy minimization problem is studied for the energy functional with the logarithmic kernel provided that the external field exists, and the minimum of the functional is sought on a set of measures bounded by a given measure.
Abstract: The equilibrium set, in the form of an interval on the real line, of the potential of an extremal measure for some energy minimization problem is studied for the energy functional with the logarithmic kernel provided that the external field exists. The minimum of the functional is sought on a set of measures bounded by a given measure. In particular, we prove that, under a special dependence of the external field on time, the ends of the interval of equilibrium satisfy a system of partial differential equations, the so-called continuum limit of the Toda lattice. Another result of the paper is a system of integral equations for the ends of the interval of equilibrium.
TL;DR: Asymptotic formulas for the eigenvalues (energy levels) of the 2D and 3D stationary Schrodinger operators describing the states of a quantum particle in a waveguide formed by soft walls characterized by a periodic parabolic confinement potential slowly varying along the waveguide axis are presented in this paper.
Abstract: Asymptotic formulas for the eigenvalues (energy levels) of the 2D and 3D stationary Schrodinger operators describing the states of a quantum particle in the waveguide formed by soft walls characterized by a periodic parabolic confinement potential slowly varying along the waveguide axis are presented The formulas are derived by a unified procedure based on adiabatic approximation and are illustrated in the first part of this paper by an example of a 2D straight waveguide This waveguide can be used for simulating some effects in nanostructures and can be viewed as a simple linear model describing electronic transport in a long molecule consisting of so-called “sites” The accuracy of the obtained asymptotic eigenvalues and the possibility of using the adiabatic approximation are discussed Examples of quantum states with large energies to which the adiabatic approximation does not apply are given
TL;DR: In this article, the authors consider the dynamics of a harmonic crystal in n dimensions with d com- ponents, where d and n are arbitrary, d;n > 1 The initial data are given by a random function with finite mean energy density which also satisfies a Rosenblatt- or Ibragimov-type mixing condition.
Abstract: We consider the dynamics of a harmonic crystal in n dimensions with d com- ponents, where d and n are arbitrary, d;n > 1 The initial data are given by a random function with finite mean energy density which also satisfies a Rosenblatt- or Ibragimov-type mixing condition The random function is close to diverse space-homogeneous processes as xn ! §1, with the distributions "§ We prove that the phase flow is mixing with respect to the limit measure of statistical solutions equations and harmonic crystals Here we study the properties of the equilibrium measures for harmonic crystals Consider a discrete subgroup i of R n which is isomorphic to Z n We may assume that i =Z n after a suitable change of coordinates A lattice inR n is a set of the points of the form r‚(x) = x+»‚, where x 2 Z n , »‚ 2 R n , ‚ = 1;:::;⁄ The points of the lattice represent equilibrium positions of the atoms (molecules, ions, ::: ) of the crystal Denote by r‚(x;t) the positions of the atoms in dynamics Then the dynamics of the displacements r‚(x;t)ir‚(x) is governed by the equations of the form 8 > > ¨(x;t) = i X y2Zn V (x i y)u(y;t); x 2Z n ; t 2R;