Effect of perturbation on the solitons of some nonlinear equations

The problem of the effect of perturbation on soliton solutions of some nonlinear equations is considered. It is shown that the relations for the shift (of the coordinate) and the velocity of the solution can be obtained in an elementary way without applying the inverse scattering problem.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 89, pp. 91–96, 1979.In conclusion, the author thanks the participants of the seminar of the laboratory for diffraction and propagation of waves of the Leningrad Branch of the Mathematics Institute (LOMI) for fruitful discussions and remarks and, in particular, V. M. Babich, V. S. Buldyrev, I. A. Molotkov, and V. Filippov. The author is grateful to Yu. Gorbachev and E. Silakov who called his attention to [6, 7, 8].     

The structure of orders all of whose representations are completely decomposable

The paper describes orders all of whose representations are completely decomposable, lying in a matrix algebra over the quotient field of a complete local Dedekind ring. For Morita-reduced orders an isomorphism criterion is provided in terms of the lattices of irreducible modules.Translated from Matematicheskie Zametki, Vol. 13, No. 2, pp. 325–335, February, 1973.The author wishes to thank V. V. Kirichenko, and also the participants in the Kiev seminar on representation theory, for their constant attention as well as for a host of valuable suggestions.     

Formal asymptotic integration for one class of weakly nonlinear infinite-dimensional Hamiltonian systems

An asymptotic integration technique is described for a class of infinite-dimensional Hamiltonian systems. The method will generate a solution with any desired accuracy. The solutions are constructed using the theory of generalized functions.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 148, pp. 42–51, 1985.In conclusion, I would like to acknowledge the helpful commments of V. B. Babich, I. A. Molotkov, V. F. Lazutkin, M. M. Popov and other participants of the LOMI wave diffraction seminar.     

Propagation of quasielastic Rayleigh waves in an inhomogeneous,anisotropic, elastic medium

The propagation of modulated oscillations along the smooth surface free from normal stresses of an inhomogeneous, anisotropic, elastic body with parameters which vary little over a characteristic wavelength is considered.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 89, pp. 234–245, 1979.The author thanks the participants of the seminar on the propagation and diffraction of waves of the Leningrad Branch of the Mathematics Institute (LOMI), in particular, V. S. Buldyrev and V. M. Babich, for fruitful discussions and their attention to the work.     

What Is the Philosophy of Mathematics,and What Should It Be?

This article was begun while the author was on sabbatical at the Université Catholique de Louvain and was partially supported by Lilly Faculty Open Fellowship grant number 910043. A preliminary version was presented in Belgium to the logic seminar, and the author thanks the participants in that seminar, especially Thierry Lucas, as well as Jan Denef for their suggestions.     

Justification of an asymptotic formula for solutions of the perturbed Klein-Fock-Gordon equation

A problem with periodic boundary conditions is considered for the Fock-Klein-Gordon equation perturbed by a small nonlinear operator R[t, u, u_x, u_xx]:. It is shown that under certain conditions the solution exists and is close to the known asymptotic solution on the interval 0 t l/.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 104, pp. 84–92, 1981.In conclusion, the author wishes to thank I. A. Molotkov, V. M. Babich, Ya. V. Kurylev, M. M. Popov, V. B. Filippov, and all the participants in the seminar on wave diffraction (LOMI) for valuable advice, pointers, and discussions.     

Local times for functions and Gaussian processes

Some necessary and sufficient conditions that a Gaussian process with continuous time have a local time are discussed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 85, pp. 104–112, 1979.The author is grateful to the participants of the seminar on local times and to Yu. A. Davydov in particular.     

Quadratic mappings of a special form

One gives sufficient conditions for the convexity of the image of the unit ball of the Euclidean space under a quadratic mapping.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 126, pp. 73–76, 1983.The author is grateful for numerous discussions to V. N. Sudakov, who, just as the author, would like to emphasize the beneficial influence of the scientific intercourse with V. P. Khavin, in particular, during the participation at the seminar on locally convex analysis.     

Equations of the Gel'fand-Levitan type in a multidimensional inverse problem for the wave equation

An approach to the solution of inverse problems using the ideas and results of the theory of boundary-value control (J. L. Lions and D. L. Russell) is developed in the paper.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 165, pp. 15–20, 1987.I am grateful to participants of A. S. Blagoveshchenskii's seminar for valuable advice and discussions. I wish to thank S. A. Avdonin for consultation on the TBVC.     

Subspaces of sequential spaces

It is proved that the space of continuous functions on the ordinary closed interval with the topology of pointwise convergence is not subsequential. In sequential spaces satisfying certain conditions, subspaces dense-in-themselves without convergent sequences are found; such subspaces are constructed in certain sequential compact spaces and semitopological groups. Translated fromMatematicheskie Zametki, Vol. 64, No. 3, pp. 407–413, September, 1998. The author thanks the participants of Arkhangel'skii's seminar held at Moscow State University for useful discussions of this work.     

A remark on equivariance of a moment mapping

It is known that the moment mapping of a strongly symplectic action of a Lie group on a symplectic manifold can be non-equivariant. It is proved in the paper that such non-equivariance can be eliminated in a canonical way; namely, a strongly symplectic action G × M → M of a connected Lie group has a Hamiltonian extension $ \tilde G It is known that the moment mapping of a strongly symplectic action of a Lie group on a symplectic manifold can be non-equivariant. It is proved in the paper that such non-equivariance can be eliminated in a canonical way; namely, a strongly symplectic action G × M → M of a connected Lie group has a Hamiltonian extension × M → M. Original Russian Text ? I.V. Mikityuk, A.M. Stepin, 2008, published in Vestnik Moskovskogo Universiteta, Matematika. Mekhanika, 2008, Vol. 63, No. 3, pp. 30–33.     

Products of idempotent endomorphisms of free acts of infinite rank

In 1966, J. M. Howie characterized the self-maps of a set which can be written as a product (i.e., composite) of idempotent self-maps of that set. Using a wreath product construction introduced by V. Fleischer, the first-named author was recently able to describe products of idempotent endomorphisms of a freeS-act of finite rank whereS is any monoid. The purpose of the present paper is to extend this result to freeS-acts of infinite rank.Research supported by Natural Sciences and Engineering Research Council of Canada Research Grant A4494     

Middle dimension homologies of a codimension 2 submanifold. II

This paper is devoted to the estimates for the middle dimension homologies of codimension 2 submanifoldA of a smooth manifold X. Proofs are based on the information about homologies of the ramified covering of X branched alongA. This information is extracted from the Atiyah-Singer signature formulas and from some corollaries of the Smith theory.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 56, pp. 182–185, 1976.The author takes this oppotunity to express his gratitude to V. A. Rokhlin for the formulation of the problem and for his interest in this paper.     

Nonisothermal filtration and seismic acoustics in porous soil: Thermoviscoelastic equations and Lamé equations

We study applications of a new class of infinite-dimensional Lie algebras, called Lax operator algebras, which goes back to the works by I. Krichever and S. Novikov on finite-zone integration related to holomorphic vector bundles and on Lie algebras on Riemann surfaces. Lax operator algebras are almost graded Lie algebras of current type. They were introduced by I. Krichever and the author as a development of the theory of Lax operators on Riemann surfaces due to I. Krichever, and further investigated in a joint paper by M. Schlichenmaier and the author. In this article we construct integrable hierarchies of Lax equations of that type. Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2008, Vol. 263, pp. 216–226. Dedicated to S.P. Novikov on the occasion of his 70th birthday     

On the decidability of equational theories of varieties of rings

A result illustrating the complexity of describing the varieties of rings with undecidable equational theory is obtained.Translated fromMatematicheskie Zametki, Vol. 63, No. 6, pp. 873–881, June, 1998.The author wishes to thank Yu. M. Vazhenin for his supervision of this research.     

Entropy of stochastic processes homogeneous with respect to a commutative group of transformations

In order to define the entropy of a stochastic field homogeneous with respect to a countable commutative group of transformations G, one fixes a sequence {A_n} of finite subsets of the group G and considers the upper limit of the sequence of mean entropies of the iterates of the decomposition P. i.e., , where ¦A_n¦ is the number of elements in A_n. It is proved that for a fixed stochastic field and all sequences {A_n} satisfying the Folner condition, the limit of the means exists and is unique. If the sequence {A_n} is such that for all stochastic fields invariant under G, the entropy calculated in terms of it is the same as that calculated for a Folner-sequence, then {A_n} satisfies the Folner condition. In the case when G is a -dimensional lattice Z, the Folner condidition coincides with the Van Hove condition.Translated from Matematicheskie Zametki, Vol. 23, No. 3, pp. 447–462, March, 1978.The author thanks D. V. Anosov for valuable advice, and also B. M. Gurevich and A. M. Stepin for helpful discussions concerning this note.     

Stability of the characterization of the multivariate normal distribution in the Skitovich-Darmois theorem

A numerical estimate of stability is obtained in a multidimensional variant of the Skitovich-Darmois theorem when the original random vectors, as well as linear combinations of them, are assumed to be -independent.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 61, pp. 5–16, 1976.The author expresses his profound thanks to V. M. Zolotarev for his continuing attention to the work.     

Extreme problems on classes of polynomials and splines

By the methods of the theory of extremum problems, existence theorems for functions with given sequence of extreme values are proved.Translated fromMatematicheskie Zametki, Vol. 63, No. 4, pp. 494–502, April, 1998.The author wishes to express his sincere gratitude to Professor V. M. Tikhomirov for setting the problem and for his permanent attention and help and also to the referee for the remarks, which aided in the amelioration of the present paper.     

Stochastic homogenization of heat transfer in polycrystals with nonlinear contact conductivities

The heat transfer problem in a polycrystal with nonlinear jump conditions on the grain boundaries will be homogenized using the method of stochastic two-scale convergence developed by Zhikov and Pyatnitskii [V.V. Zhikov and A.L. Pyatnitskii, Homogenization of random singular structures and random measures, Izv. Math. 70(1) (2006), pp. 19–67] and recently extended by the author [M. Heida, An extension of stochastic two-scale convergence and application, Asympt. Anal. (2010) (in press)]. It will be shown that for monotone Lipschitz jump conditions differentiable in 0, the nonlinearity vanishes in the limit. Additionally, existing Poincaré inequalities will be extended to more general geometric settings with the only restriction of local C ¹-interfaces with finite intensity. In particular, the result can now be applied to the Poisson–Voronoi tessellation.     

On toric h-vectors of centrally symmetric polytopes

We prove tight lower bounds for the coefficients of the toric h-vector of an arbitrary centrally symmetric polytope generalizing previous results due to R. Stanley and the author using toric varieties. Our proof here is based on the theory of combinatorial intersection cohomology for normal fans of polytopes developed by G. Barthel, J.-P. Brasselet, K. Fieseler and L. Kaup, and independently by P. Bressler and V. Lunts. This theory is also valid for nonrational polytopes when there is no standard correspondence with toric varieties. In this way we can establish our bounds for centrally symmetric polytopes even without requiring them to be rational. Received: 24 March 2004