Structure of the acoustic field near the surface of a deep sea. I

A simple analytic expression is obtained for the acoustic field near the surface of a deep sea.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 117, pp. 98–111, 1981.The author is grateful to V. S. Buldyrev, D. M. Mil'rud, M. V. Perel', and especially M. I. Yavor for discussions of the results of the work.     

Short-wave scattering near the boundry inflection point

We investigate the problem of tangential incidence of short waves onto a surface with an inflection point. Formal solutions of the corresponding equation are constructed near the inflection point in the form of a quasihomogeneous function series. The formal solution is joined with the geometrical optics solution far from the inflection point of the boundary. The problem is restated as a scattering problem for the Schrodinger equation; existence, uniqueness, and smoothness theorems are proved. The formal asymptotic expansions are proved.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 148, pp. 152–166, 1985.In conclusion, I would like to thank M. M. Popov for suggesting the problem, and also V. M. Babich and M. M. Popov for useful comments.     

Multiplication operators and traces of commutators

Some new sufficient conditions are found on operators A, B in a Hilbert space, under which trace (AB–BA)=0.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskgo Instituta im. V. A. Steklova AN SSSR, Vol. 135, pp. 182–194, 1984.The author is profoundly grateful to V. I. Lomonosov, V. V. Peller, M. Z. Solomyak, and Yu. T. Turovskii for helpful discussions of the questions considered in the paper and for information on the results of [1, 8, 11, 14].     

Investigation of boundary value problems for elliptic systems of first order

This paper is the author's abstract of a dissertation submitted for the degree of Doctor of Physicomathematical Sciences. The dissertation was defended November 2, 1972 at a session of the Scientific Council of the V. A. Steklov Mathematical Institute of the Academy of Sciences, USSR. Official referees: Corresponding Member of the Academy of Sciences of the USSR Professor A. V. Bitsadze, Corresponding Member of the Academy of Sciences of the USSR Professor M. M. Lavrent'ev, and Doctor of Physicomathematical Sciences, Professor P. I. Lizorkin.Translated from Matematicheskie Zametki, Vol. 14, No. 2, pp. 291–304, August, 1973.     

Hausdorff metric structure of the space of probability measures

The Hausdorff distance between two bounded functions on the real line was introduced by B. Sendov and B. Penkov. On the class of distribution functions it agrees with the Levi distance. The goal of this article is to analyze the Levi-Prokhorov metric and a new metric in terms of the relation between their structure and the Hausdorff distance.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 87, pp. 87–103, 1979.In conclusion I would like to take this opportunity to thank V. M. Zolotarev for posing the problem and discussing the results. I express my gratitude to V. Senatov for helpful advice which helped improve the work.     

Undecidability of the elementary theory of groups of measure-preserving transformations

The undecidability of the elementary theory of the automorphism group for a Lebesgue space is proved. It is shown that arithmetic can be interpreted in this theory. The technique of proof can be carried over to certain other groups. Translated fromMatematicheskie Zametki, Vol. 63, No. 3, pp. 414–420, March, 1998. The author wishes to thank the participants of the seminar on dynamical systems headed by D. V. Anosov, R. I. Grigorchuk, and A. M. Stepin as well as the participants of the Kolmogorov seminar of the Mechanics and Mathematics Department of Moscow State University for discussion of this work. The author is also indebted to N. K. Vereshchagin and A. M. Stepin for support and valuable remarks as well as to V. V. Ryzhikov for setting the problem and for his assistance.     

Estimates of eigenfunctions of elliptic operators with constant coefficients

The Dirichlet problem for a symmetric elliptic operator with constant coefficients is studied. Estimates of the moduli of normalized eigenfunctions, uniform in a closed region, are obtained. These estimates generalize certain results of Kh. L. Smolitskii, O. A. Ladyzhenskaya, L. N. Slobodetskii, D. M. Éidus, V. A. Il'in, and I. A. Shishmarev.Translated from Trudy Seminara im. I. G. Petrovskogo, No. 12, pp. 229–237, 1987.     

Singularities and noncommutative Frobenius manifolds

We prove that the quaternionic miniversal deformations of an A _n singularity have the structure of a noncommutative Frobenius manifold in the sense of the extended cohomological field theory. Original Russian Text ? S.M. Natanzon, 2007, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2007, Vol. 259, pp. 143–155. Dedicated to V.I. Arnold     

Applications of fixed-point methods to discrete variational and quasi-variational inequalities

Summary In this paper, discrete analogues of variational inequalities (V.I.) and quasi-variational inequalities (Q.V.I.), encountered in stochastic control and mathematical physics, are discussed.It is shown that those discrete V.I.'s and Q.V.I.'s can be written in the fixed point formx=Tx such that eitherT or some power ofT is a contraction. This leads to globally convergent iterative methods for the solution of discrete V.I.'s and Q.V.I.'s, which are very suitable for implementation on parallel computers with single-instruction, multiple-data architecture, particularly on massively parallel processors (M.P.P.'s).This research is in part supported by the U.S. Department of Energy, Engineering Research Program, under Contract No. DE-AS05-84EH13145     

Computational and approximate methods of optimal control

Approximate methods of solving problems of optimal control are classified and analyzed, and their domain of applicability is indicated. Among the special problems the problem of the choice of optimal trajectories for aircraft is considered.Translated from Itogi Nauki i Tekhniki, Matematicheskii Analiz, Vol. 14, pp. 101–166, 1977.The authors of the survey are grateful to N. N. Bolotnik, M. Yu. Borodovskii, G. G. Egiyan, V. A. Korneev, V. M. Mamalyga, A. A. Mironov, Yu. R. Roshchin, and A. P. Seiranyan for their assistance in compiling the bibliography and to R. P. Soldatova and I. S. Kheiker for their help in the shaping of the paper.     

Reversible linear systems and their versal deformations

Let R be a fixed linear involution (R ²=id) of the spaceR ⁿ . A linear operator M is said to bereversible with respect to R if RM R=M^–1 and infinitesimally reversible with respect to R if M R=–RM. A linear differential equation dx/dt=B(t)x is said to be reversible with respect to R if V(t)R –RV(–t). We construct normal forms and versal deformations for reversible and infinitesimally reversible operators. The results are applied to describe the homotopy classes of strongly stable reversible linear differential equations with periodic coefficients. The analogous theory for linear Hamiltonian systems was developed by J. Williamson, M. G. Krein, I.M. Gel'fand, V. B. Lidskii, D. M. Galin, and H. Koçak.Translated fromTrudy Seminara imeni I. G. Petrovskogo, No. 15, pp. 33–54, 1991. Original article submitted April 27, 1988.     

To the reconstruction of two unknown coefficients in a multidimensional isotropic inverse problem

A procedure is suggested for recovering two unknown coefficients in an isotropic elliptic operator given in a bounded domain. The BC-method developed by M. I. Belishev is used for solving this problem. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 210, 1994, pp. 158–163. Translated by Ya. V. Kurylev.     

Partitions and centralizers in the theory of finite groups

This paper is the author's abstract of a dissertation for the degree of Doctor of Physicomathematical Sciences. The dissertation was defended on February 10, 1969 at a united council for adjudicating academic degrees in mathematical sciences at the A. M. Gor'kii Urals State University. Official opponents: Corresponding Member of the Academy of Sciences of the USSR M. I. Kargapolov, Corresponding Member of the Academy of Sciences of the Ukrainian SSR S. N. Chernikov, Doctor of Physicomathematical Sciences Professor B. I. Plotkin, and Doctor of Physicomathematical Sciences V. S. Charin.Translated from Matematicheskie Zametki, Vol. 6, No. 4, pp. 499–511, October, 1969.     

Book review

Book Review

Book reviewV. S. Mikhalevich, A. M. Gupal, and V. I. Norkin (in Russian), Nauka, Moscow, 1987, 280 pp. Price: 3 rubls 10 kop     

三角代数上Jordan高阶导子的刻画

设u=Tri(A,M,B)是含单位元I的三角代数,()={()_n}_(n∈N)是u上一簇线性映射.本文证明了:如果对任意U,V∈u且UV=VU=I,有()_n(UV+VU)=∑_(i+j=n)(()_i(U)_(()_j)(V)+()_i(V)()_j(U)),则()={()_n}_(n∈N)是u上高阶导子.作为应用,得到了套代数上Jordan高阶导子的一个刻画.     

Approximation of smooth functions by S-splines

The paper contains a method for recovering a function from a table of its values. The method ensures smoothing of measured data, can be easily transformed into an algorithm and used in automatic information processing. The reconstruction method described in the paper is related to the approximation of smooth functions by means of some special splines, called S-splines, introduced by one of the authors. Conditions for stability and convergence of S-splines are obtained.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 10, pp. 197–206, 1984.The authors would like to thank Prof. V. M. Tikhomirov for advice and help with this work; thanks are also due to V. A. Gruzdev and E. V. Logacheva who provided their programs and computations.     

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.     

A three-chamber model of the cochlea

We construct a three-chamber hydrodynamic model of the cochlea, based on the two-chamber model with Reissner's membrane added. The anisotropic stiffness values of both membranes are obtained from Bekesy's hair probe experiments. The membrane response in the two- and three-chamber models is examined as a function of some of the cochlea parameters.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 148, pp. 133–143, 1985.I would like to acknowledge the help of Prof. V. M. Babich in the course of this research.     

Steenrod squares on conjugation spaces

We prove that the coefficients of the so-called conjugation equation for conjugation spaces in the sense of Hausmann–Holm–Puppe are completely determined by Steenrod squares. This generalises a result of V.A. Krasnov for certain complex algebraic varieties. It also leads to a generalisation of a formula given by Borel and Haefliger, thereby largely answering an old question of theirs in the affirmative. To cite this article: M. Franz, V. Puppe, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Matroids describing Bruhat cells

The aim of this paper is to generalize results of A. V. Borovik and I. M. Gelfand to the case of two-sided small cells, which turns out to be surprisingly much simpler. Bibliography: 16 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 319, 2004, pp. 244–260.