Seminars in mathematics – serie

The present collection contains the results reported in 1970 at the Seminar on Approximate Com­ putations held by the Leningrad Section of the Mathematical Institute. Two trends are represented in the collection: automatic programming and numerical methods of analysis. V. N. Faddeeva CONTENTS On the Main Concepts of Parallel Sequencing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 T. A. Tushkina and K. V. Shakhbazyan The Solution of Certain Parallel Sequencing Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 T. A. Tushkina and K. V. Shakhbazyan Choice of Enumeration in Parallel Sequencing Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 13 K. V. Shakhbaz yan The PRORAB-Computer III (P, v) M-20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 T. N. Smirnova, A. A. Aleksandrova, Yu. V. Rybakova, and N. A. Solov'eva Application of the PRORAB-Computer III (P, v) M-20 to the Solving of Linear Programming Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 T. N. Smirnova On a Matrix Inversion Method. . • . . . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 V. D. Vulichevich The Solution of a Particular Eigenvalue Problem for Certain Matrices of Special Form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 V. D. Vulichevich and V. N. Kublanovskaya Solution of a Particular Eigenvalue Problem for a Polynomial Matrix. . . . . . . . . . . . . . . . . 65 M. I. Mavlyanova On a Method for Constructing the Matrix Solution for a Polynomial Matrix. . . . . . . . . . . . . . . 71 M. I. Mavlyanova On One Approach to the Solution of the Inverse Eigenvalue Problem. . . . . . . . . . . .. . . . . . . 80 V. N. Kublanovskaya Convergence of the Method of Lines when Solving Nonlinear Parabolic Boundary Value Problems with Discontinuous Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 A. P. Kubanskaya Some Applications of the Five-Point Scheme of the Method of Lines . . . . . . . . . . . . . . . . . . 93 A. P. Kubanskaya On Expansions into Nonminimal Sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 L. N.

This volume contains eight artieles by five authors. The common theme is indicated by the title, "Investigations in the Theory of Stochastic Processes. " The artiele by Vershik and Sudakov is a summary of severallectures delivered by the authors on the general aspects of measure theory in linear spaces. The main attention here is focused on the development of certain general concepts within whose framework are studied both the foundations of the theory of random processes as weIl as more specialized probIems. The published summary does not presume prior acquaintance with the subj eet. The group of artieles by Ibragimov and Solev is devoted to questions connected with the regular- ity of stationary random processes. Ibragimov, in "Conditions for the Complete Regularity of Station- ary Processes with Continuous Time," derives suffi"ient conditions for the complete regularity of a stationary random process with continuous time. The case of a rapid decrease of the regularity coef- ficient (in power or exponential form) is investigated.In his other artiele, "Complete Regularity of Generalized Stationary Random Processes," he shows that the problem of investigating the conditions for the complete regularity of generalized stationary random processes reduces to the analogous problem for general stationary random processes.

The papers comprising this collection are directly or indirectly related to an important branch of mathematical physics - the mathematical theory of wave propagation and diffraction. The paper by V. M. Babich is concerned with the application of the parabolic-equation method (of Academician V. A. Fok and M. A, Leontovich) to the problem of the asymptotic behavior of eigenfunc­ tions concentrated in a neighborhood of a closed geodesie in a Riemannian space. The techniques used in this paper have been föund useful in solving certain problems in the theory of open resonators. The topic of G. P. Astrakhantsev's paper is similar to that of the paper by V. M. Babich. Here also the parabolic-equation method is used to find the asymptotic solution of the elasticity equations which describes Love waves concentrated in a neighborhood of some surface ray. The paper of T. F. Pankratova is concerned with finding the asymptotic behavior of th~ eigenfunc­ tions of the Laplace operator from the exact solution for the surface of a triaxial ellipsoid and the re­ gion exterior to it. The first three papers of B. G. Nikolaev are somewhat apart from the central theme of the col­ lection; they treat the integral transforms with respect to associated Legendre functions of first kind and their applications. Examples of such applications are the use of this transform for the solution of integral equations with symmetrie kernels and for the solution of certain problems in the theory of electrical prospecting.