An Erratum to this paper has been published: https://doi.org/10.1134/S1061920821010155

Abstract It is proved that if \(G\) is a divisible solvable group and \(\pi\) is a representation of \(G\) in a complex finite-dimensional vector space \(E\), then there is a basis in \(E\) in which the matrices of the representation operators have upper triangular form.

Abstract In the paper, a new class of integrable billiards, namely, billiards with slipping, is studied. At the reflection from the boundary, a billiard particle of such a system may not only change its velocity, but also move some distance along the border. Some laws of slipping preserve the integrability of flat confocal and circular billiards and billiard books glued from them, i.e., billiards on

Abstract The Dirichlet-Neumann operator for the linear water-wave problem describing oblique waves over a submerged horizontal cylinder of small (but otherwise, fairly arbitrary) cross-section in a layer of finite depth is constructed in the form of convergent series in powers of the small parameter characterizing the “thinness” of the cylinder. The terms of these series are expressed through the solution

Abstract Theoretical results of V.P. Maslov are applied in a new way to analyze empirical data in weakly formalized areas of human activity. The Maslov approach is used to solve the data clustering based on the behavior of rank distribution parameters. The updated Maslov law for rank distributions is used for the semiotic analysis of a computer language and for establishing the fact of its linguistic

Abstract Asymptotic solutions of the Cauchy problem with localized initial conditions are described for a hyperbolic equation describing waves in a channel of variable cross section. It is shown that, if the equation is reduced to a wave equation, then the solution remains localized, whereas, if the initial equation reduces to the Klein–Gordon equation, then nonlocalized corrections occur in the asymptotics

Abstract Congratulations to V. P. Maslov.

Abstract An initial boundary value problem with a free boundary for a third-order integro-differential equation for the unsteady flow of an aqueous polymer solution in a strip is studied. Questions concerning the solvability of the problem and the qualitative behavior of the solution in dependence on the initial data are investigated: time-local resolvability is established under natural conditions

Abstract The topic of the paper is the spectral analysis of a dynamical system representing a kinetic model for two-component rarefied plasma. In the present paper, we restrict ourselves to the case of two-dimensional one-particle phase space.

Abstract Recently, the degenerate Poisson random variable with parameter \(\alpha > 0\), whose probability mass function is given by \(P_{\lambda}(i) = e_{\lambda}^{-1} (\alpha) \frac{\alpha^{i}}{i!} (1)_{i,\lambda}\) \((i = 0,1,2,\dots)\), was studied. In probability theory, the zero-truncated Poisson distributions are certain discrete probability distributions whose supports are the set of positive

Abstract In the context of the averaging method for Poisson and symplectic structures and the theory of Hannay–Berry connections, we discuss some aspects of the semiclassical quantization for a class of slow-fast Hamiltonian systems with two degrees of freedom. For a pseudodifferential Weyl operator with two small parameters corresponding to the semiclassical and adiabatic limits, we show how to construct

Abstract We describe an approach that allows us to construct multi-soliton asymptotic solutions for essentially nonintegrable equations and analyze the type of wave collision. The general idea is demonstrated by the example of a generalization of Korteweg-de Vries equation with small dispersion in the cases of two and three waves. We also discuss the phenomenon of nonuniqueness that occurs in the framework

Abstract Linear evolution equations of mathematical physics admitting an invariant in the form of a positive quadratic form are considered. In particular, this includes the string vibration equation, the Liouville kinetic equation, the Maxwell system of equations and the Schrödinger equation. Conditions for the existence of an invariant Gaussian measure are indicated, which makes it possible to apply

Abstract A version of Maslov’s canonical operator on Lagrangian manifolds with singularities of special form (the so-called punctured manifolds) is described. These manifolds naturally arise in the construction of asymptotic solutions for a wide class differential and pseudodifferential equations; in particular, for Petrovsky hyperbolic systems or pseudo-differential equations of water waves.

Abstract It is proved that if \(G\) is a group and \(\pi\) is a pseudorepresentation of \(G\) in a Banach space \(E\) with a sufficiently small defect and if \(\pi\) is a sufficiently small perturbation of the identity representation of \(G\) in \(E\), then \(\pi(g)=1_E\) for all \(g\in G\).

Abstract We consider a generalization of the notion of Hausdorff operator on Lebesgue spaces and, under natural conditions, prove that such an operator is not a Riesz operator provided it is non zero. In particular, it cannot be represented as a sum of a quasinilpotent and a compact operators.

Abstract Using symbolic calculations, a method of local asymptotics is developed, which describes the diffraction focusing of electromagnetic and acoustic fields for edge and corner catastrophes. Explicit expressions are found for the unfolding coefficients, the functional module, and the phase of the traveling wave, when the family of primary (geometrical-optical) and secondary (edge) rays form focuses

Abstract We consider the operators of the following boundary problems $$\mathbb{D}_{\boldsymbol{A,}\Phi,\mathfrak{B}}\boldsymbol{u}=\left\{ \begin{array} [c]{c} \mathfrak{D}_{\boldsymbol{A},\Phi}\boldsymbol{u}\text{ on }\Omega\\ \mathfrak{B}\boldsymbol{u}_{\partial\Omega}\text{ on }\partial\Omega \end{array} \right. $$(1) in unbounded domains \(\Omega\subset\mathbb{R}^{3}\), where \(\mathfrak{D} _{\boldsymbol{A}

Abstract We extend the inequalities originally obtained by D. Hundertmark and B. Simon for \(L_p-\)bounds, \(1\leq p\leq \infty\), for the Krein spectral shift function to the setting of general semifinite von Neumann algebras. We also complete these results by showing that in the quasi-normed setting, for example, for \(L^p\)-spaces with \(0

Abstract Let \(00\). For any \(T>0\) and \(k\in\overline{\mathbb{N}}=\mathbb{N}\cup\{\infty\}\), let \(N(k,T)\) be the number of solutions \(\{n_j\}_{j=1}^k\), \(n_j\in \mathbb{N}_0=\mathbb{N}\cup\{0\}\), of the inequality \(\sum_{j=1}^{k}n_jt_j\le T\). We are interested in the asymptotics of \(\log N(k,T)\) as \(T\to\infty\).

Abstract We construct formal asymptotic solutions describing shock waves and tangential and weak discontinuities for the nonlinear system of gas dynamics of a fluid with small viscosity.

Abstract In the present paper, we discuss the notion of “tropical mathematics” in view of the fact that, up to the present time, there is no generally accepted correct understanding of this notion. The paper presents, rather than mathematical proofs, humanitarian and social justification of the correct understanding of the term. The author proposes to describe the term “tropical mathematics” by means

Abstract In this paper, order estimates for the linear widths of some function classes are obtained; these classes are defined by restrictions on the weighted \(L_{p_1}\)-norm of the \(r\)th derivative and the weighted \(L_{p_0}\)-norm of zero derivatives.

Abstract We discuss a specific model, which we refer to as RandLOE, of a large multi-agent network whose dynamic is prescribed via a combination of deterministic local laws and random exogenous factors. The RandLOE approach lies outside the framework of Stochastic Differential Equations, but lends itself to analytic examination as well as to stable simulation even for relatively large networks. RandLOE

Abstract The action on the space \(\mathbb C^4\) of the 7-dimensional Lie group of infinitesimal holomorphic automorphisms of a completely nondegenerate cubic model surface \(Q\) of CR-type \((1,3)\) is considered. All orbits of the given action are found and their biholomorphic classification is given. One of the orbits coincides with the surface \(Q\) (the 5-dimensional orbit), two orbits are 6-dimensional

Abstract In the paper, the dynamics is investigated of an uncontrolled multi-link wheeled vehicle consisting of trolleys with one wheel pair that are attached to the previous link with metal frames and are capable of rotating around their anchor points. The mechanical system under consideration is nonholonomic, since it has additional nonholonmic kinamic connections. A system of differential equations

Abstract In this paper, we describe multigraded generalizations of certain constructions useful for the mathematical understanding of gauge theories: we perform a near-at-hand generalization of the Aleksandrov–Kontsevich–Schwarz–Zaboronsky procedure; we also extend the formalism of \(Q\)-bundles first introduced by A. Kotov and T. Strobl. We compare these approaches by studying certain supersymmetric

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.