Scale-invariant asymptotic (automodel) behavior in quantum field theory

We summarize the results of our 40-year investigations of scale-invariant (automodel) behavior of form factors in hadron-lepton deep-inelastic scattering processes at high energies and large transferred momenta within the Bogoliubov axiomatics of quantum field theory. This approach was conducive to the emergence of the notion of quark color in quantum chromodynamics.     

On the asymptotic behavior of nonlinear waves in the presence of a short-range potential

Consider the nonlinear wave equation with zero mass and a time-independent potential in three space dimensions. When it comes to the associated Cauchy problem, it is already known that short-range potentials do not affect the existence of small-amplitude solutions. In this paper, we focus on the associated scattering problem and we show that the situation is quite different there. In particular, we show that even arbitrarily small and rapidly decaying potentials may affect the asymptotic behavior of solutions.     

An asymptotic method of solving transient dynamic contact problems of the theory of elasticity for a strip

An asymptotic method is proposed for solving transient dynamic contact problems of the theory of elasticity for a thin strip. The solution of problems by means of the integral Laplace transformation (with respect to time) and the Fourier transformation (with respect to the longitudinal coordinate) reduces to an integral equation in the form of a convolution of the first kind in the unknown Laplace transform of contact stresses under the punch. The zeroth term of the asymptotic form of the solution of the integral equation for large values of the Laplace parameter is constructed in the form of the superposition of solutions of the corresponding Wiener-Hopf integral equations minus the solution of the corresponding integral equation on the entire axis. In solving the Wiener-Hopf integral equations, the symbols of the kernel of the integral equation in the complex plane is presented in special form — in the form of uniform expansion in terms of exponential functions. The latter enables integral equations of the second kind to be obtained for determining the Laplace-Fourier transform of the required contact stresses, which, in turn, is effectively solved by the method of successive approximations. After Laplace inversion of the zeroth term of the asymptotic form of the solution of the integral equations, the asymptotic solution of the transient dynamic contact problem is determined. By way of example, the asymptotic solution of the problem of the penetration of a plane punch into an elastic strip lying without friction on a rigid base is given. Formulae are derived for the active elastic resistance force on the punch of a medium preventing the penetration of the punch, and the law of penetration of the punch into the elastic strip is obtained, taking into account the elastic stress wave reflected from the strip face opposite the punch and passing underneath it.     

A problem in elasticity theory

The problem of determining the dimensions of the transverse cross-sections of a beam from the given frequencies of its natural vibrations is examined. Frequency spectra are indicated that determine the dimenions of the transverse cross-sections of the beam uniquely, an effective procedure is presented for solving the inverse problem, and a uniqueness theorem is proved. The method of standard models /1/ is used to solve the inverse problem.     

The common asymptotic nature of methods of solving problems of the theory of elasticity for slabs and plates

Four approaches to the analytical solution of the three-dimensional problem of the theory of elasticity for slabs and plates are given. All the proposed schemes have a common basis. It is proved that the asymptotic method is identical with the method of hypotheses and the method of successive approximations.     

Two-parameter asymptotic forms for space-time Gaussian beams in an elastic medium

Asymptotic equations are constructed which contain two small parameters, for the longitudinal and transverse space-time Gaussian beams.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 173, pp. 87–95, 1988.     

Methods of calculating further approximations in the asymptotic space-time Gaussian beam method

In the example of the equations of the inhomogeneous isotrophic theory of elasticity and of considerations of the longitudinal component of the displacement vector, a scheme is proposed for determining the successive approximations in the asymptotic expansion of the space-time Gaussian beam method.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 173, pp. 104–112, 1988.     

Asymptotic behavior of solutions to problems of elasticity theory at infinity in flat parabolic domains

Asymptotic representations of solutions to the boundary-value problems of elasticity theory are studied in domains with parabolic exit at infinity (or in bounded domains with singularities like polynomial zero sharpness). The procedure of derivating a formal asymptotic expansion looks like the algorithm of asymptotic analysis in domains. Under the Dirichlet conditions (displacements are prescribed on the boundary of a domain), it is not hard to justify the power asymptotic series. It follows from the theorem on the unique solvability of the problem in spaces of the type L₂ containing degrees of distance r=|x| as weight multipliers. For the Neumann conditions (stresses are prescribed on the boundary of a domain) an asymptotic expansion is justified by introducing the Eiry function Φ transforming the Lamé system to the biharmonic equation. Due to the appearance of the Dirichlet condition on Φ, the study of the asymptotic behavior of a solution to the last problem is simplified. The existence theorems and conditions for solvability of the “elastic” Neumann problem are presented. These results are based on the weighted Korn inequality. Bibliography: 29 titles. Translated fromProblemy Matematicheskogo Analiza. No. 15, 1995, pp. 162–200     

On the asymptotic linearization of acoustic waves

The initial value problem of a certain generalization of the nonlinear, dispersive wave equations with dissipation is rigorously studied. The solutions of the equations can be found exactly up to in certain norms. The essential use is made of the fact that this equation is asymptotically linearizable to i.e., the equations can be mapped to an equation which differs from a linearizable equation only in terms which are of An application of the equations to unidirectional small amplitude acoustic waves is discussed. The general methodology used here can also be applied to other asymptotically linearizable equations.

     

The asymptotic behavior of queueing systems: Large deviations theory and dominant pole approximation

This paper presents the exact asymptotics of the steady state behavior of a broad class of single-node queueing systems. First we show that the asymptotic probability functions derived using large deviations theory are consistent (in a certain sense) with the result using dominant pole approximations. Then we present an exact asymptotic formula for the cumulative probability function of the queue occupancy and relate it to the cell loss ratio, an important performance measure for service systems such as ATM networks. The analysis relies on a new generalization of the Taylor coefficients of a complex function which we call characteristic coefficients. Finally we apply our framework to obtain new results for the M/D/1 system and for a more intricate multiclass M/D/n system.     

On universal relations in elasticity theory

Zusammenfassung Es wird eine Methode zur Erhaltung allgemeingültiger Beziehungen für Verzerrungen kompressibler, isotroper, elastischer Körper dargestellt.     

A spectral theory of continuous functions and the Loomis-Arendt-Batty-Vu theory on the asymptotic behavior of solutions of evolution equations

In this paper we present a new approach to the spectral theory of non-uniformly continuous functions and a new framework for the Loomis-Arendt-Batty-Vu theory. Our approach is direct and free of C₀-semigroups, so the obtained results, that extend previous ones, can be applied to large classes of evolution equations and their solutions.