Numerical analysis of geoelectric fields in a medium with faults

An efficient method of electromagnetic field calculation is applied to the model of a multilayer nonhomogeneous medium with an arbitrary number of vertical faults. The model simulates a medium that includes the asthenosphere, the crustal layer, and the surface sedimentary layer, all joined by conducting faults. The numerical results are calculated and the effect of faults is assessed. This research was partially supported by the Russian Foundation for Basic Research (grant No. 96-05-64340) and by the Interuniversity Scientific Program “Russian Universities: Basic Research.” Translated from Chislennye Metody v Matematicheskoi Fizike, Moscow State University, pp. 85–93, 1998.     

Dirac γ-equation, classical gauge fields and Clifford algebra

An equation, we call Dirac γ-equation, is introduced with the help of the mathematical tools connected with the Clifford algebra. This equation can be considered as a generalization of the Dirac equation for the electron. Some features of Dirac γ-equation are investigated (plane waves, currents, canonical forms). Furthermore, on the basis of local gauge in variance regarding unitary group, a system of equations is introduced consisting of Dirac γ-equation and the Yang-Mills or Maxwell equations. This system of equations describes a Dirac’s field interacting with the Yang-Mills or Maxwell gauge field. Characteristics of this system of equations are studied for various gauge groups and the liaison between the new and the standard constructions of classical gauge fields is discussed. This paper is supported by the Russian Foundation for Basic Research, grant 95-10-00433a.     

The second initial boundary value problem for the gravitation-gyroscopic wave equation in exterior domains

We study the existence and uniqueness of the solution to the second initial boundary value problem for the gravitation-gyroscopic wave equation in an exterior multiply connected domain with various types of conditions at infinity. Translated fromMatematicheskie Zametki, Vol. 60, No. 1, pp. 40–57, July, 1996. This research was partially supported by the Russian Foundation for Basic Research under grant No. 96-01-01411.     

A singularly perturbed boundary value problem for a second-order equation in the case of variation of stability

A boundary value problem for a second-order nonlinear singularly perturbed differential equation is considered for the case in which there is variation of stability caused by the intersection of roots of the degenerate equation. By the method of differential inequalities, we prove the existence of a solution such that the limit solution is nonsmooth. Translated fromMatematicheskie Zametki, Vol. 63, No. 3, pp. 354–362, March, 1998. This research was partially supported by the Russian Foundation for Basic Research under grant No. 96-01-00694.     

An initial-boundary value problem on the half-line for the MKdV equation

The initial-boundary value problem on the half-line for the modified Korteweg-de Vries equation with zero boundary conditions and arbitrary rapidly decaying initial conditions is embedded in the scheme of the inverse scattering method. The corresponding inverse scattering problem is reduced to the Riemann problem on a system of rays in the complex plane. Supported by the Russian Foundation for Basic Research under grant 98-01-00576. Mathematical Institute, Russian Academy of Sciences, Ufa; e-mail: ihabib@imat.rb.ru. Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 34, No. 1, pp. 65–75, January–March, 2000. Translated by I. T. Habibullin     

Integrable boundary conditions for many-component burgers equations

Infinite series of boundary conditions that are consistent with even-order higher symmetries and ensure the integrability of a Burgers type equation are constructed. Translated fromMatematicheskie Zametki, Vol. 60, No. 6, pp. 888–901, December, 1996. This research was supported by the Russian Foundation for Basic Research under grant No. 93-011-16008 and by the International Science Foundation under grant Nos. MLY00 and RK2000.     

Generalized differentiable product measures

A class of measures on ℝ^∞ determined by sequences of functions of finitely many variables is considered. An existence theorem for such measures is proved, and their properties are examined. Examples are presented. Translated fromMatematicheskie Zametki, Vol. 63, No. 1, pp. 37–55, January, 1998. The author is greatly indebted to O. V. Zimina for many stimulating discussions. This research was supported by the Russian Foundation for Basic Research under grant No. 96-01 01701.     

An estimate of stability of the reconstruction of the normal multiplicative type

An estimate of the characterization stability of the normal distribution type is obtained for the case of the multiplicative type. Under some conditions, the estimate has the orderɛ ^1/3 L(ɛ), where L(∈) is a slowly varying function. Supported by the Russian Founation for Basic Research (grant Nos. 97-01-00273, 98-01-00621, and 98-01-00926) and by INTAS-RFBR (grant No. IR-97-0537). Proceedings of the Seminar on Stability Problems for Stochastic Models, Vologda, Russia, 1998, Part II.     

Singularity and absolute continuity of measures associated with the rotation of a circle

The problem of whether the infinite convolution of certain discrete distributions naturally associated with the rotation of a circle through an irrational angle of α is singular or absolutely continuous for different values of α is studied. Bibliography: 9 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 223, 1995, pp. 323–336. This paper is partially supported by the Russian Foundation for Basic Research, grant 94-01-00921. Translated by N. A. Sidorov.     

The asymptotic expansion of the solution of a boundary-value problem with rapidly alternating type of boundary conditions

We consider a boundary-value problem for the Poisson equation with rapidly alternating type of boundary conditions in a domain between two parallel hyperplanes. The solution of the problem is assumed to be 1-periodic with respect to all the coordinates but one. We construct the asymptotic expansion for the solution in powers of a small parameter characterizing the alteration frequency of the boundary conditon. We obtain an estimate for the remainder term. Bibliography: 16 titles. Dedicated to Olga Arsenievna Oleinik This research has been supported in part by the foundation “Cultural Initiative”, by the International Science Foundation (Grant MIEOOO), and by the Russian Foundation for Basic Research. Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 19, pp. 000-000, 0000.     

Direct problem of calculating the magnetic dipole field in a quasilayered medium

A method is developed for calculating the electromagnetic field of a magnetic dipole in a quasilayered two-dimensional medium. The quasi-three-dimensional problem is reduced to a two-dimensional problem for the Fourier-transformed electromagnetic field. An equivalent system of integral equations on the layer boundaries is obtained. This research was partially supported by the State Scientific-Technical Program “Future Information Technologies” (grant No. 0201.06.010) and the Interuniversity Scientific Program “Russian Universities: Basic Research.” Translated from Chislennye Metody v Matematicheskoi Fizike, Moscow State University, pp. 94–110, 1998.     

Extension by zero of functions of several variables

We obtain sufficient conditions on a domainG ⊂ ℝn for functions defined onG to be extendable by zero to the entire space ℝn with smoothness preserved in an integral norm. Translated fromMatematicheskie Zametki, Vol. 64, No. 3, pp. 351–365, September, 1998. This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-00243 and by program “Leading Science Schools” under grant No. 96-15-96102.     

Asymptotics ast → ∞ of the solutions of nonlinear equations with nonsmall initial perturbations

A new method is proposed for finding asymptotics ast → ∞ of the solutions of the Cauchy problem for nonlinear evolution equations with nonsmall initial data. Translated fromMatematicheskie Zametki, Vol. 59, No. 6, pp. 855–864, June, 1996. This research was supported by the Russian Foundation for Basic Research under grant No. 93-011-134 and by the International Science Foundation under grant No. BX000.     

Estimates of trigonometric sums by the third derivative

The well-known Van der Corput estimate of a trigonometric sum by the third derivative is sharpened. Translated fromMatematicheskie Zametki, Vol. 60, No. 3, pp. 383–389, September, 1996. This research was supported by the Russian Foundation for Basic Research under grant No. 94-01-00002.     

Integral characteristics in the inverse problem of magnetotelluric sounding

This article examines some aspects of the one-dimensional inverse problem of magnetotelluric sounding. A uniqueness theorem is proved in the presence ofS-surfaces. A numerical algorithm based on transformation formulas is proposed. This research was partially supported by Russian Foundation for Basic Research (grant No. 96-01-00410) and by the State Scientific-Technical Program “Future Information Technologies” (grant No. 0201.06.010). Translated from Chislennye Metody v Matematicheskoi Fizike, Moscow State University, pp. 53–66, 1998.     

Differential equations with almost periodic or conditionally periodic coefficients: Recurrence and reducibility

Recurrence of an individual trajectory of a linear nonautonomous differential equation on a compact Lie group for the case of an almost periodic or conditionally periodic dependence of the right-hand side on time is proved. The relation between recurrence and reducibility is examined.Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 229–237, August, 1998.This research was supported by the Russian Foundation for Basic Research under grants No. 96-01-00378 and No. 96-15-96072.     

On continuous extension of locally homeomorphic simplicial maps of ℝ2 into itself byσ-processes

It is shown that the Klein bottle with two points removed can be embedded in the compactification of ℝ² by a finite tree. Translated fromMatematicheskie Zametki, Vol. 59, No. 6, pp. 821–831, June, 1996. I wish to express sincere gratitude to A. G. Vitushkin for posing the problem and guidance. This research was supported by the Russian Foundation for Basic Research under grant No. 93-01-00225 and by the Ministry of Scientific and Technological Research in the Mathematical Sciences of Italy.     

Three-page approach to knot theory. Universal semigroup

An explicit construction of a finitely presented semigroup whose central elements are in a one-to-one correspondence with the isotopy classes of unoriented links in the three-space is given, together with a finite presentation for the group of invertible elements of the semigroup. The group is presented by two generators and three relations. The commutator subgroup of the group is isomorphic to the braid group of infinite index. A similar construction is given for band-links. The kauffman theorems on the existence of polynomial band-link invariants satisfying some skein-relations are stated algebraically. This work is partially supported by Russian Foundation for Basic Research grant No. 99-01-00090. Moscow State University. Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 34, No. 1, pp. 29–40, January–March, 2000. Translated by I. a. Dynnikov     

Nondegeneracy of amplitude-frequency modulation for finite-gap solutions of integrable nonlinear equations

We study properties of complex finite-gap solutions of the nonlinear Schrödinger equation and the sine-Gordon model.Translated fromMatematicheskie Zametki, Vol. 59, No. 1, pp. 53–61, January, 1996.The work was partially supported by the Russian Foundation for Basic Research under grant No. 94-01-00193.     

Some remarks on homoclinic groups of hyperbolic toral automorphisms

The homoclinic group (an invariant with respect to topological conjugacy) for hyperbolic toral automorphisms is determined. Certain conditions are given for conjugacy of a homeomorphism of a compact space to hyperbolic toral automorphism. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 223, 1994, pp. 140–147. This paper is partially supported by Russian Foundation for Basic Research, grant 94-01-00921. Translated by V. V. Sadovskaya.