General theory of statistical decisions

Author's abstract of the dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Physical-Mathematical Sciences. The dissertation was defended on January 21, 1969 at the Scientific Council of the Institute of Applied Mathematics of the Academy of Sciences of the SSSR. The official opponents of the thesis were: Academician Yu. V. Linnik, Corresponding Member of the Academy of Sciences SSSR A. A. Lyapunov, Academician of the Academy of Science USSR B. V. Gnedenko, and Doctor of Physical-Mathematical Sciences Professor F. I. Karpelovich.Translated from Matematicheskie Zametki, Vol. 5, No. 5, pp. 635–648, May, 1969.     

Random flows generated by random processes (properties of random flows,level-crossing problems)

Author's summary of a dissertation submitted for the degree of Doctor of Physicomathematical Sciences. The dissertation was defended February 10, 1970, at a meeting of the Scientific Council of the Institute of Applied Mathematics, Academy of Sciences of the USSR. Official opponents: B. V. Gnedenko, Academician of the Academy of Sciences of the USSR, Yu. V. Prokhorov, Corresponding Member of the Academy of Sciences of the USSR, and N. N. Chentsov, Doctor of Physicomathematical Sciences.Translated from Matematicheskie Zametki, Vol. 8, No. 3, pp. 393–407, September, 1970.     

Branching processes

Author's abstract of a thesis for the attainment of the academic degree of Doctor of Physics and Mathematics. The thesis was defended on April 25, 1968 at a meeting of the Scientific Council of the V. A. Steklov Mathematical Institute, Academy of Sciences of the USSR. Official opponents: Academician A. N. Kolmogorov, Yu. V. Prokhorov, Corresponding Member of the Academy of Sciences of the USSR, and Prof. Yaglom, Doctor of Physics and Mathematics.Translated from Matematicheskie Zametki, Vol. 4, No. 2, pp. 239–251, August, 1968.     

Classes of functions and Fourier coefficients with respect to complete orthonormal systems

This is the author review of a dissertation presented in the competition for the academic degree of Doctor of Physicomathematical Sciences. The dissertation was defended on October 10, 1974 in a meeting of the academic council of the V. A. Steklov Mathematics Institute of the Academy of Sciences of the USSR. The official opponents were: A. A. Talalyan, Corresponding Member of the Academy of Sciences of the Armenian SSR; N. P. Kuptsov, Doctor of Physicomathematical Sciences; and E. M. Nikishin, Doctor of Physicomathematical Sciences.Translated from Matematicheskie Zametki, Vol. 20, No. 3, pp. 445–463, September, 1976.     

Numerical characteristics of the degree of anisotropy of the mechanical properties of materials

Conclusions 1. A unique procedure for determination of the numerical characteristics of the degree of anisotropy of the strength and deformation properties of materials is proposed.2. The concepts of the deformability surface and the anisotropy surface of the deformability permit representing graphically the deformation properties of materials.3. The system of integral characteristics of the mechanical properties of materials opens up new possibilities in connection with the analysis and optimization of different reinforcement schemes.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 4, pp. 601–609, July–August, 1978.     

Some extremal problems of theory of quadrature and approximation of functions

This is the author-review of the dissertation presented for the degree of the Doctor of Physicomathematical Sciences. The dissertation was defended on March 13, 1975 in a meeting of the scientists of the V. A. Steklov Mathematics Institute of the Academy of Sciences of the USSR. The official opponents were Doctor of Physicomathematical Sciences Prof. N. S. Bakhvalov, Doctor of Physicomathematical Sciences Prof. P. K.Suetin, and Doctor of Physicomathematical Sciences Prof. S. B. Stechkin.Translated from Matematicheskie Zametki, Vol. 19, No. 2, pp. 299–311, February, 1976.     

On spectral properties of multiparameter polynomial matrices

Spectral problems for multiparameter polynomial matrices are considered. The notions of the spectrum (including those of its finite, infinite, regular, and singular parts), of the analytic multiplicity of a point of the spectrum, of bases of null-spaces, of Jordan s-semilattices of vectors and of generating vectors, and of the geometric and complete geometric multiplicities of a point of the spectrum are introduced. The properties of the above characteristics are described. A method for linearizing a polynomial matrix (with respect to one or several parameters) by passing to the accompanying pencils is suggested. The interrelations between spectral characteristics of a polynomial matrix and those of the accompanying pencils are established. Bibliography: 12 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 229, 1995, pp. 284–321. Translated by V. B. Khazanov.     

Minimal surface as a diagram of sphere rotations

Based on I. N. Vekya's representation of the field of infinitely small (i. s.) bendings of a sphere in terms of analytic functions, we present a new proof of Liebmann's theorem to the effect that the diagram of rotations of i.s. bendings of a sphere is a minimal surface and, conversely, each minimal surface is the diagram of rotations of some i.s. bending of a sphere or of part of it. It is then established that all the minimal surfaces which are non-trivially locally isometric to a given minimal surface constitute an analytic single-parameter family, and explicit expressions for the surfaces of this family are given. The bibliography contains four titles.Translated from Matematicheskie Zametki, Vol. 2, No. 6, pp. 645–656, December, 1967.     

Cluster dynamics of a uniform chain of dissipatively coupled rotators

The existence and stability of stationary cluster structures in uniform chains of dissipatively coupled rotators is investigated. Cluster synchronization is interpreted as the classical synchronization of cluster rotators, which are elementary structure-forming objects. The complete set of types of cluster rotators and simple cells is defined. This definition is equivalent to the definition of the complete set of types of cluster structures. The completeness of this set is proved. The problem of the stability of cluster structures is solved. Physical examples of chains of rotators and a physical interpretation of the results of this research are given.     

Hausdorff and packing dimensions of non-normal tuples of numbers: Non-linearity and divergence points

We apply the results in [L. Olsen, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages, J. Math. Pures Appl. 82 (2003) 1591-1649; L. Olsen, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages. III, Aequationes Math. 71 (2006) 29-53; L. Olsen, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages. IV: Divergence points and packing dimension, Bull. Sci. Math. 132 (2008) 650-678; L. Olsen, S. Winter, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages. II: Non-linearity, divergence points and Banach space valued spectra, Bull. Sci. Math. 131 (2007) 518-558] to give a systematic and detailed account of the Hausdorff and packing dimensions of sets of d-tuples of numbers defined in terms of the asymptotic behaviour of the frequencies of strings of digits in their N-adic expansion.