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.     

Computability by nondeterministic program and the Moschovakis search computability

This work has been completed under a financial support from the Russian Fund for Basic Research (Grant No. 93-011-16004).     

Weakly symmetric spaces and spherical varieties

Weakly symmetric homogeneous spaces were introduced by A. Selberg in 1956. We prove that, for a real reductive algebraic group, they can be characterized as the spaces of real points of affine spherical homogeneous varieties of the complexified group. As an application, under the same assumption on the transitive group, we show that weakly symmetric spaces are precisely the homogeneous Riemannian manifolds with commutative algebra of invariant differential operators.Supported by the Alexander von Humboldt Foundation and Russian Foundation for Basic Research, Grant No. 95-01-01263.Supported by the U. S. Civilian Research and Development Foundation, Award No. 206, Russian Foundation for Basic Research, Grant No. 98-01-00598, and the Alexander von Humboldt Foundation.     

Metric spaces of semicontinuous functions

This work was supported by the Russian Foundation of Fundamental Research under Grant No. 93-011-236.     

On the stability of characterization by the distribution of any order statistic

Some upper and lower estimates of stability of characterization of distribution by the distribution of any order statistic are obtained. Supported by the Russian Foundation for Fundamental Research (grant Nos. 96-01-01919 and 97-01-00273) and INTAS-RFBR (grant No. IR-98-0537). Proceedings of the Seminar on Stability Problems for Stochastic Models, Vologda, Russia, 1998, Part I.     

A three-time-level explicit difference scheme of the second order of accuracy for parabolic equations

The difference schemes of Richardson [1] and of Crank-Nicolson [2] are schemes providing second-order approximation. Richardson's three-time-level difference scheme is explicit but unstable and the Crank-Nicolson two-time-level difference scheme is stable but implicit. Explicit numerical methods are preferable for parallel computations. In this paper, an explicit three-time-level difference scheme of the second order of accuracy is constructed for parabolic equations by combining Richardson's scheme with that of Crank-Nicolson. Restrictions on the time step required for the stability of the proposed difference scheme are similar to those that are necessary for the stability of the two-time-level explicit difference scheme, but the former are slightly less onerous.Translated fromMatematicheskie Zametki, Vol. 60, No. 5, pp. 751–759, November, 1996.This research was supported by the Russian Foundation for Basic Research under grant No. 95-01-00489 and by the International Science Foundation under grants No. N8Q300 and No. JBR100.     

Weight of faces in plane maps

Precise upper bounds are obtained for the minimum weight of minor faces in normal plane maps and 3-polytopes with specified maximum vertex degree. Translated fromMatematicheskie Zametki, Vol. 64, No. 5, pp. 648–657, November, 1998. The research of the first named author was supported in part by the Visiting Fellowship Research Grant GR/K00561 from the Engineering and Physical Sciences Research Council and by the Russian Foundation for Basic Research under grant No. 96-01-01614 and No. 97-01-01075.     

Solutions of the heat equation in domains with singularities

We study the asymptotics of solutions to the Dirichlet problem for the heat equation in time-dependent domains with singular points.Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 163–179, August, 1998.This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-00504 and by INTAS under grant No. 93-351.     

Asymptotics of a multiple Cauchy integral with rapidly oscillating exponential

An asymptotic expansion uniform with respect to a parameter is obtained for a multiple Cauchy integral with a rapidly oscillating exponential in the integrand under the assumption that the singularity lines are in a general position.Translated fromMatematicheskie Zametki, Vol. 58, No. 2, pp. 231–242, August, 1995.The work was financially supported by the Russian Foundation for Basic Research under grant No. 94-01-00193-a.     

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.     

Use of the Lagrange method in nonlinear analysis

The problem of the spectrum of a nonlinear system and its relation to ellipsoid transform widths are studied.Translated fromMatematicheskie Zametki, Vol. 59, No. 3, pp. 334–342, March, 1996.This research was partially supported by the Russian Foundation for Basic Research under grant No. 93-01-00237 and by the International Science Foundation under grant No. MP1000.     

Justification of Dipole Approximation in the Problem of Nonlinear Wave Generation by a Submerged Sphere

Some nonlinear dipole approximation is constructed for the nonstationary problem of a solid sphere motion under a free surface. The approximation is justified in the class of analytic functions decaying at infinity.Original Russian Text Copyright © 2005 Pyatkina E. V.The author was supported by the State Maintenance Program for the Leading Scientific Schools (Grant NSh-440.2003.1) and the Russian Foundation for Basic Research (Grant 05-01-00250).__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 907–927, July–August, 2005.     

Optimal guaranteed control of linear systems under disturbances

This paper is devoted to the problem of the minimax control of a dynamical system with quadratic performance functional under external disturbances and geometric control constraints. The optimal guaranteed control strategy is obtained in explicit form.Translated fromMatematicheskie Zametki, Vol. 60, No. 2, pp. 198–205, August, 1996.This research was supported by the Russian Foundation for Basic Research under grant No. 95-01-000771a.     

Simulating the effect of external helical magnetic fields on MHD stability of tokamak plasma

The effect of external helical magnetic fields on the stability of a rotating plasma in an axially symmetric tokamak machine is simulated numerically. Threshold amplitude effects are investigated using the NFTC code for the complete nonlinear dynamical 3-D system of MHD equations. The research was partially funded by Russian Foundation of Basic Research (grant 96-07-89110) and Ministry of Science grants 201.03.002 and 0201.03.003. Translated from Chislennye Metody i Vychislitel'nyi Eksperiment, Moscow State University, pp. 4–14, 1998.     

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.     

Regular sets in residual classes

Sets regular modulo a fixed odd prime power are explicitly constructed under the condition that their cardinalities do not exceed an arbitrarily small positive power of the modulus.Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 224–228, August, 1998.This research was supported by the Russian Foundation for Basic Research under grant No. 97-01-00721 and by the Professor B. Novak grant (Karlov University, Prague).     

Modules over quantum polynomials

We describe the structure of finitely generated modules over general quantum Laurent polynomials and prove that Artin modules over general quantum Laurent polynomials are cyclic. Translated fromMatematicheskie Zametki, Vol. 59, No. 4, pp. 497–503, April, 1996. This research was partially supported by the Russian Foundation for the Basic Research under grant No. 93-011-1544 and by the INTAS program under grant No. 93-2618.     

Some identities for the integro-differential equations describing quasiparticles on an isoenergetic surface

The equations describing quasiparticles that correspond to classical self-consistent fields are reduced to a form symmetric with respect to an indefinite metric.Translated fromMatematicheskie Zametki, Vol. 60, No. 5, pp. 692–707, November, 1996.This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-01544.     

Some conic bundless

A representation of the anticanonical K3 surface of a singular pencil of conics is described. This generalizes the well-known Shokurov theorem.Translated fromMatematicheskie Zametki, Vol. 63, No. 6, pp. 903–910, June, 1998.The author is greatly indebted to V. A. Iskovskikh, Yu. G. Prokhorov, and I. A. Chel'tsov for fruitful discussions.This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-00820 and by INTAS under grant No. 93-2805.00-00-00.     

A weakened asymptotic monogeny condition

Functions of one complex variable that satisfy a weakened asymptotic monogeny condition are proved to be holomorphic. Translated fromMatematicheskie Zametki, Vol. 60, No. 6, pp. 902–911, December, 1996. The author thanks Proffessor Dolzhenko for discussing the results of the paper. The research was supported by the Russian Foundation for Basic Research under grant No. 93-01-00236.