The space-time caustic of the wave field in an anisotropic medium

We examine the smooth caustic of space-time rays in an anisotropic medium, assuming first-order tangency of the rays to the caustic. An analytical expression of the wave field near the caustic is derived.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 148, pp. 5–12, 1985.I would like to acknowledge the scientific supervision of Prof. V. M. Babich.     

Eikonal equations for an inhomogeneous anisotropic medium

This paper studies the wave propagation method in a many-dimensional medium in order to find the cases where the eiconal equation is integrated in an explicit compact form.     

A generalization of the Boussinesq-Papkovich functions to an anisotropic medium

Summary An extension of the Boussinesq-Papkovich displacement potentials to an anisotropic medium is presented. The representation is used to generate solutions to the problems of Boussinesq and Mindlin. The representation is shown to be complete.

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Zussamenfassung Eine Ausdehnung der Papkovich-Neuber-Verdrängungspotentiale auf eine querlaufende isotropische Substanz wird dargestellt. Es werden Potentiale erzeugt, die dem Problem von Boussinesq und dem ersten Problem von Mindlin entsprechen. Ausserdem wird die vollständige Lösung gezeigt.

     

A general central limit theorem under sublinear expectations

Under some weaker conditions, we give a central limit theorem under sublinear expectations, which extends Peng’s central limit theorem.     

Reconstruction of cracks in an inhomogeneous anisotropic elastic medium

In this paper we give in two and three dimensions a reconstruction formula for determining cracks buried in an inhomogeneous anisotropic elastic body by making elastic displacement and traction measurements at the boundary. The information is encoded in the local Neumann-to-Dirichlet map. With the help of the Runge property, the local Neumann-to-Dirichlet map is connected to the so-called indicator function. This function can be expressed as an energy integral involving some special solutions, called reflected solutions. The heart of our method lies in analyzing the blow-up behavior at the crack of the indicator function, which is by no means an easy task for the inhomogeneous anisotropic elasticity system. To overcome the difficulties, we construct suitable approximations of the reflected solutions that capture their singularities. The indicator function is then analyzed by the Plancherel formula.     

The central limit theorem without the condition of independence

Necessary and sufficient conditions are presented for sums of asymptotically independent random variables to converge to a normal random variable in the sense of total variation distance, uniform metric for characteristic functions. and mean metric of order q. Supported by the Russian Foundation for Fundamental Research (grant No. 96-01-01920). Proceedings of the Seminar on Stability Problems for Stochastic Models. Moscow. Russia. 1996. Part II.     

Elastic equilibrium of an anisotropic plate of arbitrary thickness with an elliptic cavity

We study the stressed state of an anisotropic plate of arbitrary thickness weakened by a cylindrical elliptic cavity and subject to forces that are independent of the thickness coordinate. The results of numerical computations are described. Four figures. Bibliography: 2 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 28, 1998, pp. 76–80.     

Thermohaline convection with cross-diffusion in an anisotropic porous medium

Using normal mode technique it has been shown that (i) values of the anisotropy parameter are important in deciding the mode of convection in a doubly diffusive fluid saturating a porous medium. (ii) Depending on the values of the Soret and Dufour parameters, an increase in anisotropy parameter either promotes or inhibits instability, (iii) cross-diffusion induces instability even in a potentially stable set-up and (iv) for certain values of the Dufour and Soret parameters there is a discontinuity in the critical thermal Rayleigh number, which disappears if the porous medium has horizontal isotropy.     

Existence results for quasi-equilibrium problems under a weaker equilibrium condition

By a quasi-equilibrium problem is understood an equilibrium problem with a constraint set depending on the current point. In this paper, we establish existence results for scalar and vector quasi-equilibrium problems, when the standard equilibrium condition is replaced by a weaker one.     

Behavior of Gaussian beams in an anisotropic medium with interface

A region that consists of two parts with anisotropic Riemannian metrics is considered. The metric has a jump on the interface. Asymptotic solutions of the wave equation, reflected and transmitted from the interface, i.e., Gaussian beams (“quasiphotons”), are constructed. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 324, 2005, pp. 77–109.     

An equilibrium penny-shaped crack in an inhomogeneous elastic medium

The problem of a penny-shaped tensile crack in a continuously-inhomogeneous space is considered. The problem reduces to a dual integral equation for which an approximate analytical solution is constructed. It is proved that the approximate solution of the integral equation is asymptotically exact for both small and large values of the dimensionless geometric parameter of the problem. The accuracy of the solution obtained is investigated. Expressions are presented for the stress intensity factor, the energy of the opening of the crack, the displacements of its sides and the normal components of the stress tensor in the neighbourhood of its contour. In the numerical analysis of the solution of the problem, special attention is paid to analysing of the problem when the first derivative of the change in the elastic properties of the material changes sign.     

Problem of determining the speed of sound and the memory of an anisotropic medium

Theoretical and Mathematical Physics - We consider the inverse problems of simultaneously determining two unknowns: the wave propagation velocity and the memory of a layered medium. To find them,...     

Gaussian beams of love waves in an anisotropic medium and their summation

Gaussian beams of surface waves are constructed by means of the Kravtsov ansatz. The Gaussian beams are summed which enables us to describe analytically the wave field of a surface Love wave in the vicinity of an arbitrary singularity. Bibliography: 13 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 186, pp. 6–13, 1990. Translated by V. D. Azhotkin.     

Perfect plasticity as a zero relaxation limit of plasticity with isotropic hardening

This work presents an approximation result for elasto‐perfect plasticity in the dynamical setting of the problem. We prove that the model of perfect plasticity with isotropic hardening converges to the Prandlt–Reuss model. Copyright © 2001 John Wiley & Sons, Ltd.