A theorem on the distribution of quadratic residues having applications in the ergodic method of Yu. V. Linnik

A theorem is proved which is used in the next paper of the collection. In the paper the following theorem is proved.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 50, pp. 169–178, 1975.I wish to thank A. V. Malyshev for posing the problem and for attention to the work and D. N. Andreev for help in reviewing the paper of A. I. Vinogradov and Yu. V. Linnik.     

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].     

On existence of global solutions to the Navier-Stokes equations for compressible and viscous flows on the surface of a sphere

Following tecniques proposed by A. V. Khazikhov and V. A. Weigant in 1995, we prove the global, with respect to time, existence and uniqueness of the solution to the Navier-Stokes equations for a compressible, viscous and barotropic fluid which moves on the surface of a sphere. In obtaining the main estimates we make use of the Hodge decomposition and the generalized potential theory due to K. Kodaira.

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Sunto Seguendo le tecniche proposte da A. V. Khazikhov and V. A. Weigant nel 1995, si prova l'esistenza ed unicità della soluzione per le equazioni di Navier-Stokes per un fluido comprimibile, viscoso e barotropico che si muova sulla superficie di una sfera. Nell'ottenere le principali stime si utilizzano la decomposizione di Hodge e la teoria del potenziale generalizzato, dovuta a K. Kodaira.

     

Application of the discrete ergodic method to the analytical arithmetic of indefinite ternary quadratic forms

One obtains generalizations and refinements of ergodic and mixing theorems [Yu. V. Linnik, Ergodic Properties of Algebraic Fields, Springer-Verlag, New York (1968)] for the distribution of integral points on a two-sheeted hyperboloid. One gives a survey of the applications of the discrete ergodic method to indefinite ternary quadratic forms; in particular, one describes the results of the papers of A. M. Istamov, E. V. Podsypanin, A. V. Malyshev, and U. M. Pachev, published in this issue.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im V. A. Steklova AN SSSR, Vol. 93, pp. 5–24, 1980.     

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.     

Solutions of nonlinear equations concentrated near curves in the plane

A method of successive approximations is considered which makes it possible to construct formal solutions of some nonlinear equations concentrated near curves in the plane.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 89, pp. 84–90, 1979.In conclusion, the author thanks V. M. Babich, A. Kachalov, and V. Nomofilov for fruitful discussions and valuable suggestions.     

Diffraction of short waves on a screen of variable transparency

A model problem of a point source in the field of which there is a screen of variable transparency is considered. The character of the change of transparency of the screen makes it possible to construct an exact solution of the problem by the method of separation of variables. A shortwave asymptotic expression for the wave field is obtained.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 104, pp. 24–32, 1981.In conclusion, the author thanks V. A. Borovikov for posing the problem and V. M. Babich for assistance with the work.     

Scattering at a strongly singular potential

Hubert's equation for the resolvent and the maximum principle for the equation u + V(x)u=0 is used to construct a self-adjoint extension of the operator + V(x) with a positive singular potential V(x). The properties of this operator are investigated.Translated from Matematicheskie Zametki, Vol. 8, No. 2, pp. 217–228, August, 1970.In conclusion I wish to thank sincerely A. N. Tikhonov for his interest in the work, and V. A. Il'in, V. P. Maclov, A. A. Samarskii, A. G. Sveshinkov, I. A. Shishmarev, and especially A. A. Arsen'ev for the interesting and informative discussions we had concerning this work.     

Isomonodromic deformations and self-similar solutions of the Einstein-Maxwell equations

It is shown that the self-similar solutions of the Einstein-Maxwell equations in the cylindrical case describe the isomonodromic deformations of ordinary linear differential equations with rational coefficients. New types of such solutions, expressed in terms of the fifth Painlevé transcendent, are found.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 181, pp. 65–92, 1990.The author is grateful to A. R. Its and V. V. Matveev for the discussion of the results and for support.     

Application of the ray method to the computation of complex eigenvalues

The ray method is used to obtain asymptotic equations for the complex eigenvalues of the Helmholtz equation. The problem of the transparent disk and oscillations of transparent whispering gallery type are considered in detail.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 42, pp. 228–235, 1974.The author is grateful to V. M. Babich for a detailed discussion of the work and to A. L. Alimov, V. S. Buldyrev, and V. N. Krasil'nikov for useful discussions.     

The theory of thermal potentials and its applications

A review by the author of his dissertation presented for the academic degree of doctor of physico-mathematical sciences. A defense of the dissertation was given on February 29, 1968 before the science committee of the V. A. Steklov Institute of Mathematics of the Academy of Sciences of the USSR. Official challengers were A. V. Bitsadze and A. A. Samarskii, Corresponding Members of the Academy of Sciences of the USSR, and V. P. Mikhailov, Doctor of Physico-Mathematical Sciences.Translated from Matematicheskie Zametki, Vol. 4, No.1, pp. 113–123, July, 1968.     

Nonzero-sum differential games (the noncooperative variant)

One presents the fundamental results of the theory of noncooperative differential games: necessary and sufficient equilibrium conditions, existence theorems, properties of equilibrium solutions, numerical methods. One indicates applications to concrete problems in economics, the mechanics of controlled motions, and to strategic games.Translated from Itogi Nauki i Tekhniki, Matematicheskii Analiz, Vol. 15, pp. 199–266, 1977.The authors are grateful to É. M. Vaisbord, A. F. Koaonenko, V. S. Molostvov, V. P. Patsyukov, V. A. Plotnikov, V. V. Podinovskii, R. A. Polyak, and R. T. Yanushevskii for the problems in Sec. 11 and for the remarks which have been taken into account in the final editing of the survey.     

Regions of stability of queuing systems

A method of stability analysis is considered for the solution of the inverse queuing problem, and quantitative estimates are obtained.Translated from Problemy Ustoichivosti Stokhasticheskikh Modelei — Trudy Seminara, pp. 107–115, 1980.The author expresses his gratitude to V. V. Kalashnikov for a useful discussion of this paper.     

Investigations of multidimensional,infinite-dimensional and limit theorems of the theory of probabilities

Author's abstract of a dissertation in competition for the scientific degree of Doctor of Physical-Mathematical Sciences. The dissertation was defended on May 30, 1968 at a meeting of the Scientific Council of the V. A. Steklov Mathematical Institute, Academy of Sciences of the USSR. The official opponents were: V. S. Korolyuk, Corresponding Member, Academy of Sciences of the Ukrainian SSR, A. V. Skorokhod, Corresponding Member, Academy of Sciences of the Ukrainian SSR, and V. A. Statulyavichus, Doctor of Physical-Mathematical Sciences.Translated from Matematicheskie Zametki, Vol. 4, No. 5, pp. 611–619, November, 1968.     

Singular cases of the problem of the continuation of the boundary layer

One obtains conditions for the existence and uniqueness of the solution of the continuation of the boundary layer near a solid wall in the case when the pressure gradient P_x is positive and satisfies one of the inequalities p_xp_x(0).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Maternaticheskogo Instituts im. V. A. Steklova AN SSSR, Vol. 138, pp. 86–89, 1984.The author is grateful to V. V. Pukhnachev for his assistance and also to N. V. Khusnutdinova for useful remarks.     

Representation of square matrices by the product of diagonal,orthogonal, triangular,and integral unimodular matrices (in connection with the inhomogeneous Minkowski conjecture)

It is shown that for all sufficiently large numbers n nonsingular square matrices of order n are found for which the representation indicated in the title is impossible.Translated by L. Guy and J. P. Jones.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 67, pp. 86–94, 1977.In closing, I express deep gratitude to B. F. Skubenko for posing the problem and his attention to the work, and also to L. V. Kuz'min and A. V. Malysev for remarks and assistance in the final formulation of this paper.     

An analytic form of differential equations

A review by the author of his dissertation presented for the academic degree of Doctor of Physico-Mathematical Sciences. This dissertation was defended on May 13, 1969 before the Academic Committee of the Institute of Applied Mathematics of the Academy of Sciences of the USSR. The official challengers were the following Doctors of Physico-Mathematical Sciences: Professor V. I. Arnol'd, Professor V. A. Pliss, and Professor V. V. Rumyantsev.Translated from Matematicheskie Zametki, Vol. 6, No. 6, pp. 771–778, December, 1969.     

Sharpening stability estimates for Cramér's theorem

Estimates of stability in Cramér's theorem on the decomposition of the normal law due to N. A. Sapogov and V. M. Zolotarev are improved somewhat.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematichesko Instituta Im. V. A. Steklova AN SSSR, Vol. 61, pp. 125–134, 1976.In conclusion, let us take this opportunity to thank V. M. Zolotarev for his constant attention to the work.     

Asymptotic behavior of the fundamental solution of an elliptic equation with respect to a complex parameter

Maslov's canonical operator method is used for constructing the asymptotic behavior with respect to a complex parameter of the fundamental solution of a secondorder elliptic equation with smooth finite coefficients. The asymptotic form is constructed on the assumption that all trajectories of the corresponding Hamiltonian system depart to infinity. The asymptotic form is used for investigating the analytic properties of the fundamental solution.Translated from Matematicheskie Zametki, Vol. 21, No. 3, pp. 377–390, March, 1977.The author thanks V. V. Kucherenko for suggesting the problem and continued interest, V. P. Maslov, V. P. Mikhailov, and L. A. Muravei for valuable advice and comments, and M. V. Karasev for his interest.     

Scalar mesons in the Skyrme model and solitons with baryon number

On the basis of the effective meson Lagrangian for chiral and scalar fields composed of quarks, we show the possibility of describing a stable chiral solitonbaryon having quantized baryon number and two scales: a scale R_c of chiral symmetry restoration and a hadronizaton scale R_h.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 169, pp. 3–11, 1988.The authors express thanks to A. A. Andrianov and Yu. V. Novozhilov for discussions and their attention to the present work, as well as to V. N. Popov for allowing it to be published in the present collection.