AbstractWe study an \(h\)-pseudodifferential operator (in particular, differential) acting on the \(n\)-D physical space with a principal symbol determining an integrable Hamiltonian system and with nontrivial subprincipal symbol which destroys the integrability. We propose a pragmatic method for calculating asymptotic eigenvalues and eigenfunctions (quasimodes) of such an operator. Unlike the approach

AbstractThe main object of this paper is to establish some fixed point results for \(F(\psi,\varphi)\)-contractions in partially-ordered metric spaces. As an application of one of these fixed point theorems, we discuss the existence of a unique solution for a coupled system of higher-order fractional differential equations with multi-point boundary conditions. The results presented in this paper are

AbstractGiven a compact manifold with boundary, endowed with an isometric action of a discrete group of polynomial growth, we state an index theorem for elliptic elements in the algebra of nonlocal operators generated by the Boutet de Monvel algebra of pseudodifferential boundary value problems on the manifold and the shift operators associated with the group action.

AbstractRecently, the degenerate gamma functions were introduced as a degenerate version of the usual gamma function. In this paper, we investigate several properties of these functions. Namely, we obtain an analytic continuation as a meromorphic function on the whole complex plane, the difference formula, the values at positive integers, some expressions following from the Weierstrass and Euler formulas

AbstractWe give some sufficient conditions for the preservation of the second term in the spectral asymptotics of a compact operator under the perturbation of the metrics in the Hilbert space.

AbstractThe irreducible locally bounded finite-dimensional pure pseudorepresentations of connected locally compact groups are described.

AbstractA special asymptotic large parameter analysis is used to study the attractors of self-excited generators with different nonlinear delay elements. Asymptotic formulas for stable relaxation cycles and more complex attractors are derived.

AbstractIn this note, we compare the first integrals and exact solutions of equations of motion for scleronomic and rheonomic, and holonomic and nonholonomic oscillators.

AbstractA system of equations of two-dimensional shallow water over an uneven bottom is considered. An overdetermined system of equations for finding the corresponding symmetries is obtained. The compatibility of this overdetermined system of equations is investigated. A general form of the solution of the overdetermined system is found. The kernel of the symmetry operators is found. The cases of kernel

AbstractThis article deals with fundamental solutions of the Sturm–Liouville differential equations \(\left( P\left( x\right) y^{\prime }\right) ^{\prime }+\left( R\left( x\right) -\xi ^{2}Q\left( x\right) \right) y=0\) with a complex parameter \(\xi \), \(\left\vert \xi \right\vert \gg 1\), whose coefficients \(P\left( x\right) \) and \(Q\left( x\right) \) are positive piecewise continuous functions

AbstractAll solutions to the Burgers, Hopf, Helmholtz, Klein–Gordon, sine-Gordon, Schrödinger, and Monge–Ampere equations having analytical complexity one (simple solutions) are described. It turns out that all simple solutions of the Burgers and Hopf equation are represented by elementary functions. An example of a family of solutions of complexity two to the Burgers equation is presented. Simple

We study the properties of generalized solutions of the mixed Neumann–Robin problem for the linear system of elasticity theory in the exterior of a compact set and the asymptotic behavior of solutions of this problem at infinity under the assumption that the energy integral with weight |x|a is finite for such solutions. We use the variation principle and, depending on the value of the parameter a,

It is proved that an arbitrary continuous finite-dimensional pseudorepresentation of the group SL(2,ℚp) with sufficiently small defect is a multiple of the identity representation.

Studying degenerate versions of various special polynomials has became an active area of research and has yielded many interesting arithmetic and combinatorial results. Here we introduce a degenerate version of the polylogarithm function, the so-called degenerate polylogarithm function. Then we construct a new type of degenerate Bernoulli polynomial and number, the so-called degenerate poly-Bernoulli

The analogy between knots in mathematical knot theory and “knots” in medicine is studied for the example of a certain invariant of Legendrian knots. The connection between the “Chern class” and the “Maslov class” is analyzed. A conjecture on the connection between the “Connes-Chern character” and the “Maslov class” is proposed.

There are two approaches to the study of the cohomology of group algebras ℝ[G]: the Eilenberg-MacLane cohomology and the Hochschild cohomology. In the case of Eilenberg-MacLane cohomology, one has the classical cohomology of the classifying space BG. The Hochschild cohomology represents a more general construction, in which the so-called two-sided bimodules are considered. The Hochschild cohomology

We study unsteady shear flows realized in a half-plane with viscous incompressible fluid, where the law of motion of the boundary oscillating along itself is given. Either the longitudinal velocity of the boundary or the shear stress on it can be specified. The statement of the linearized problem with respect to small initial perturbations imposed on the kinematics in the entire half-plane is presented

We analyze the problem of estimating past quantum states of a monitored system from a mathematical perspective in order to ensure self-consistency with the principle of quantum nondemolition. Despite several claims of “measuring noncommuting observables” in the physics literature, we show that we are always measuring commuting processes. Our main interest is in the notion of quantum smoothing, or retrodiction

We consider the Cauchy problem with localized initial data for a linear pseudodifferential equation describing water waves with dispersion taken into account and obtain a uniform asymptotic solution in a neighborhood of regular points of the wave front, the size of the neighborhood being independent of the small parameters occurring in the problem.

This article deals with the solution of a fractionally-damped generalized Bagley–Torvik (BT) equation whose damping characteristics are well-defined by means of the fractional derivative (FD) of the Riemann–Liouville and the Liouville–Caputo types. The Ho-motopy Analysis Method (HAM) is implemented for computing the dynamic response (DR) analysis. Two external forces or loads (namely, the unit step

In the paper, a solution to the variational factors problem for systems of equations written in an extended Kovalevskaya form is given. The solution is presented locally in a coordinate form.

To describe the propagation of waves in media containing localized rapidly changing inhomogeneities (e.g., narrow underwater ridges or pycnoclines in the ocean, layers with sharply changing optical or acoustic density, etc.), it is natural to use the wave equation with a small parameter characterizing the ratio of the scales of the localized inhomogeneity and of the general change of velocity (e.g

In the paper, using a generalization of the Siegel-Shidlovskii method in the theory of transcendental numbers, we prove the infinite algebraic independence of elements, generated by generalized hypergeometric series, of direct products of the fields of \(\mathbb{K}_v\)-completions of an algebraic number field\(\mathbb{K}\) of finite degree over the field of rational numbers with respect to a valuation

We introduce new zeta functions related to an endomorphism ϕ of a discrete group Γ. They are of two types: counting numbers of fixed (ρ ~ ρ o ϕn) irreducible representations for iterations of ϕ from an appropriate dual space of Γ and counting Reidemeister numbers R(φn) of different compactifications. Many properties of these functions and their coefficients are obtained. In many cases, it is proved

In the paper, the model surface method is applied to arbitrary CR-manifolds of finite Bloom–Graham type. A family of basic assertions is proved. It is proved also that, for a model surface, the condition that the Bloom–Graham type is constant is a criterion for holomorphic homogeneity. Distinctions from the case of rigid models treated earlier are clarifies. A series of questions and conjectures is

Let DMAP be the class of locally compact groups that admit a (not necessarily continuous) embedding in a compact topological group, and let FIR be the class of locally compact groups all of whose continuous irreducible unitary representations are finite-dimensional. We prove that every almost connected DMAP group is an FIR group.

The main object of this paper is to present a family of q-series identities which involve some of the theta functions of Jacobi and Ramanujan. Each of these (presumably new) q-series identities reveals interesting relationships among three of the theta-type functions which stem from the celebrated Jacobi’s triple-product identity in a remarkably simple way. The q-results presented in this paper are

In the paper, we derive a formula expressing the reduced dynamics of the Wigner function using the dynamics of the Wigner function of a larger system. Here the latter system is assumed to be isolated and generating the dynamics of its subsystem.

The central factorial numbers of the second kind appear in the expansion of powers of x in terms of the central factorial sequence. In this paper, we introduce the central Bell numbers and polynomials associated with those central factorial numbers of the second kind and investigate some identities and properties of them.

A functional integral with respect to the Wiener measure that represents a solution of the Schrödinger equation with signed Hamiltonian is obtained. In turn, this integral is derived, using the analytic continuation with respect to the argument, from a similar integral for the corresponding heat equation.

One shows that the Feynman’s Path Integral designed for quantum mechanics has an analogous in classical mechanics, the so-called (min, +) Path Integral. This former is build on (min, +)-algebra and (min, +)-analysis which permit to handle in a linear way non-linear problems occurring in mathematical physics. The Hamilton-Jacobi equations and their solutions within this mathematical framework, are introduced

We study a nonlinear problem of gas filtration in a porous and weakly conducting medium that is used to model the overheating process in reactors. We obtain explicit estimates for the critical values of the similarity parameter of the problem which determine a condition for the existence of a steady-state cooling regime for a system open to atmosphere and a condition for global overheating. The bifurcation

Massless Dirac equation for spinor multiplets is minimally coupled with a unitary representation of an arbitrary compact semisimple gauge group. The spectrum of the quantized coupled Dirac Hamiltonian has a positive mass gap running along the classical energy scale.

This paper considers the plane-parallel motion of an elliptic foil in a fluid with a nonzero constant circulation under the action of external periodic forces and torque. The existence of the first integral is shown for the case in which there is no external torque and an external force acts along one of the principal axes of the foil. It is shown that, in the general case, in the absence of friction

Using a solution concept defined in the setting of a product of distributions, we consider the nonlinear equation f(t)ut + (u2)x = 0, where f is a continuous function. This equation can be regarded a generalization of Burgers inviscid equation and allows to study several kinds of interaction of δ-waves. If f(t) ≠ 0 for all t, collisions of δ-waves cannot exist. If f(t) = 0 for certain values of t,

We consider 3-dimensonal Schrödinger operators with complex potential. We obtain new trace formulas with new terms, associated with singular measure. In order to prove these results, we study analytic properties of a modified Fredholm determinant as a function from Hardy spaces in the upper half-plane. In fact, we reformulate spectral theory problems as problems of analytic functions from Hardy spaces

In the paper, the problem of propagation of waves generated by localized source on a two-dimensional lattice is studied. The discrete problem is reduced to a continuous Cauchy problem, and the answer for this problem is constructed using the modified Maslov’s canonical operator, including a neighborhood of the leading wave front. The strong and weak dispersion effects depending on the relationship

In the one-dimensional case, an asymptotic expansion with respect to the parameter for the Feynman integral in the Lee—Pomeransky representation, is obtained. The previously stated conjecture on how to obtain asymptotic series is proved, and the form of the terms of the series and the relationship of the series with Newton’s polyhedron of the polynomial participating in the integrand are determined

In this paper, we consider \({\mathcal A}\)-Fredholm and semi-\({\mathcal A}\)-Fredholm operators on Hilbert C*-modules over a W*-algebra \({\mathcal A}\) defined in [3] and [9]. Using the assumption that \({\mathcal A}\) is a W*-algebra (rather than an arbitrary C*-algebra), we obtain a generalization of Schechter—Lebow characterization of semi-Fredholm operators and a generalization of the “punctured

A plane-parallel motion of a circular foil is considered in a fluid with a nonzero constant circulation under the action of external periodic force and torque. Various integrable cases are treated. Conditions for the existence of resonances of two types are found. In the case of resonances of the first type, the phase trajectory of the system and the trajectory of the foil are unbounded. In the case

In the paper, problems related to the theory of stochastic processes on nilpotent Lie groups are studied. In particular, a stochastic process on the Heisenberg group H3(ℝ) is considered such that the trajectories of this process, in the stochastic sense, satisfy the horizontality conditions with respect to the standard contact structure on H3(ℝ). The main result claims that the measure defined on the

For two discrete metric spaces X and Y, we consider metrics on X ⊔ Y compatible with the metrics on X and Y. As morphisms from X to Y, we consider Roe bimodules, i.e., the norm closures of bounded finite propagation operators from l2(X) to l2(Y). We study the corresponding category \(\mathcal{M}\), which is also a 2-category. We show that almost isometries determine morphisms in \(\mathcal{M}\). We

The stationary phase method is applied to investigate the asymptotic behavior at infinity of the Hankel transform of order zero.

The main result is that a Lie group admitting a (not necessarily continuous) embedding in an amenable Lie group is amenable.

The notions of “hole” and “vacuum” in various branches of science are considered. A philosophical generalization of these notions on the basis of examples from physics, history, and linguistics is presented. Some aspects of the further development of these notions in mathematical logic, thermodynamics of nuclear matter, and other branches of science are sketched.

In the present note, we show that, for some special classes of external flows, the short-wave asymptotic behavior of the solution of linearized equations of relativistic gas dynamics can be described rather explicitly.

We consider Sobolev problems (problems for an elliptic operator on a closed manifold with conditions on a closed submanifold) for the case in which these conditions are of nonlocal nature and include weighted spherical means of the unknown function over spheres of a given radius. For such problems, we establish a criterion for the Fredholm property and, in some special cases, obtain index formulas

We consider a d-dimensional harmonic crystal, d ⩾ 1, and study the Cauchy problem with random initial data. The distribution μt of the solution at time t ∈ ℝ is studied. We prove the convergence of correlation functions of the measures μt to a limit for large times. The explicit formulas for the limiting correlation functions and for the energy current density (in the mean) are obtained in terms of

A method for computing the solution to the boundary value problem for the symmetric Thomas-Fermi model of a heavy atom or ion at zero absolute temperature.

New order estimates for the Kolmogorov, linear, and Gel’fand diameters of the Sobolev weight classes on a domain satisfying John’s condition in the Lebesgue weighted space are obtained.

Short-wave asymptotics of solutions of linearized equations of relativistic hydrodynamics are described.

In this paper, we consider the spectral problem for the magnetic Schrödinger operator on the 2-D plane (x1, x2) with the constant magnetic field normal to this plane and with the potential V having the form of a harmonic oscillator in the direction x1 and periodic with respect to variable x2. Such a potential can be used for modeling a long molecule. We assume that the magnetic field is quite large

In the context of normal forms, we study a class of perturbed Hamiltonian systems on phase spaces with symmetry which arise from deformation of Poisson brackets. By combining the averaging method with some facts on the Poisson invariant cohomology, we derive various normalization criteria. In particular, we compute the invariant normal forms of first order for systems of adiabatic type on Poisson fibrations

In the paper, the role of the product formula for nonzero elements of an algebraic number field of finite degree over the field of rationals in several problems of number theory is discussed.

In the present paper, an analytic description of nonlinear delta-shock waves interaction is described by using the weak asymptotics method.

In the paper, eight classes of integrable billiards are studied; in particular, classes introduced by the authors: elementary, topological, billiard books, billiards on the Minkowski plane, geodesic billiards on quadrics in three-dimensional Euclidean space, billiards in a magnetic field, and also a class containing all of the ones above. It turns out that, in the class of billiard books, topological

The problem of viscous compressible fluid flow in an axially symmetric pipe with small periodic irregularities on the wall is considered for large Reynolds numbers. An asymptotic solution with double-deck structure of the boundary layer and unperturbed core flow is obtained. Numerical investigations of the influence of the density of the core flow on the flow behavior in the near-wall region are presented

Formulas for the asymptotics of some class of integrals of rapidly oscillating functions that generalize the well-known stationary phase method, which were obtained in the previous paper of the author, are applied to integrals arising in the well-known tsunami hydrodynamic piston model in the case of a constant pool bottom. As a result, asymptotic formulas are obtained for the head part of the wave

In the framework of the Poisson geometry of twistor space we consider a family of perturbed 3-dimensional Kepler systems. We show that Hamilton equations of these systems can be integrated in quadratures. Their solutions for some subcases are given explicitly in terms of Jacobi elliptic functions.

The problem of the Zeemann-Stark effect for the hydrogen atom in electromagnetic fields is considered using the irreducible representations of the Karasev-Novikova algebra with quadratic commutation relations. An asymptotics of the series of eigenvalues and the asymptotic eigenfunctions are obtained near the upper boundaries of resonance spectral clusters which are formed near the energy levels of