Diophantine representation of enumerable predicates

This paper is the author's abstract of dissertation in partial fulfillment of the requirements for the degree of Doctor of Physicomathematical Sciences. The dissertation was defended on February 24, 1972 at a session of the faculty council of the V. A. Steklov Mathematics Institute, Academy of Sciences of the SSSR. The official opponents were: Doctor of Physicomathematical Sciences S. I. Adyan; Corresponding Member of the Academy of Sciences of the USSR Yu. L. Ershov; and Corresponding Member of the Academy of Sciences of the USSR D. K. Faddeev.Translated from Matematicheskie Zametki, Vol. 12, No. 1, pp. 115–120, July, 1972.     

Studies in the matrix representability of abstract groups and in matrix groups

The author presents a summary of his dissertation in competition for the academic degree Doctor of Physicomathematical Sciences. The dissertation was defended on April 26, 1972, before the Academic Council of the Institute of Mathematics, Siberian Branch, Academy of Sciences of the USSR. The official opponents were Prof. D. M. Smirnov (Doctor of Physicomathematical Sciences), Prof. A. I. Starostin (Doctor of Physicomathematical Sciences of the Belorussian SSR and Doctor of Physicomathematical Sciences), and Prof. A. I. Shirshov (Corresponding Member of the Academy of Sciences of the USSR and Doctor of Physicomathematical Sciences).Translated from Matematicheskie Zametki, Vol. 14, No. 4, pp. 597–606, October, 1973.     

Certain problems of the theory of difference boundary value problems

This is an author's abstract of a dissertation for the degree of doctor of physicomathematical sciences. The dissertation was defended on December 2,1969,at a session of the Scientific Council of the Institute of Applied Mathematics, Academy of Sciences of the USSR. The official opponents were: Doctor of Physicomathematical Sciences Prof. K. I. Babenko, Doctor of Physicomathematical Sciences Prof. N. S. Bakhvalov, and Doctor of Physicomathematical Sciences E. A. Volkov.Translated from Matematicheskie Zametki, Vol. 7, No. 5, pp. 655–663, May, 1970.     

Prime extensions and categoricity in power

This paper deals with the notion of prime extension defined by Morley. A connection is established between the categoricity of a theory in a cardinal greater than the power of its language and the existence of prime extensions. For such theories we prove a minimality property of the prime extensions and we conclude by a property of the group of automorphisms of their models. Author working on Ph. D. thesis under supervision of Prof. M. O. Rabin. Author working at the CNRS under the supervision of Prof. D. Lacombe.     

Lyapunov first method for nonholonomic systems with circulatory forces

We consider the problem of instability of equilibrium states of scleronomic nonholonomic systems moving in a stationary field of conservative and circulatory forces. The applied methodology is based on the existence of solutions of differential equations of motion which asymptotically tend to the equilibrium state of the system. It is assumed that the forces in the neighbourhood of the equilibrium position can be presented in the form of the sum of two components, the first one being a homogeneous function of the position with the positive degree of homogeneity; the second one being infinitely small in comparison to the first one. The results obtained, which partially generalize results from [S.D. Taliaferro, Instability of an equilibrium in a potential field, Arch. Ration. Mech. Anal. 109 (2) (1990) 183–194; V.A. Vujičić, V.V. Kozlov, Lyapunov’s stability with respect to given state functions, J. Appl. Math. Mech. 55 (4) 9 (1991) 442–445; D.R. Merkin, Introduction to the Theory of the Stability of Motion, Nauka, Moscow, 1987 (in Russian); A.V. Karapetyan, On stability of equilibrium of nonholonomic systems, Prikl. Mat. Mekh. 39 (6) (1975) 1135–1140 (in Russian)], are illustrated by an example.     

Baryon string model. II. Special solutions of classical three-string equations of motion

Conclusions We have considered the simplest solutions of the three-string equations of motion; these are solutions with a finite number of excited degrees of freedom. It is of interest to construct the quantum theory of such motions of the relativistic three-string. Quantum theory of a meson string with finite number of degrees of freedom was constructed in [4]. Quantization of finite-mode solutions of the baryon string model will be considered in the third paper of the present work.Institute of High Energy Physics, Serpukhov. Translated from Teoreticheskaya i Matematicheskaya Fizika. Vol. 64, No. 2, pp. 245–258, August, 1985.     

A family of extension operators

The definition of the degree of a family of extension operators is given. The connection of lower estimates with general properties is established. The problem of approximation of the segment [0, 1] by finite subsets is studied. In the class of homogeneous operators the independence of norms of operators from the degree of the family of extension operators is proved.Translated from Matematicheskie Zametki, Vol. 22, No. 2, pp. 215–219, August, 1977.     

A solution of the problem of oscillations of a die on a two-phase base under a seismic impact

A method of studying nonstationary seismic oscillations of a cylindrical die on a two-phase base is developed, which is modeled with an application of the theory of M. Bio-Ya. I. Frenkel'—V.N. Nikolaevskii. In the solution of the problem integral transforms are used in combination with the method of orthogonal polynomials for a representation of required contact stresses. Numerical results are given, which characterize a change of coefficients of expansion of the series of contact stresses with respect to time, including their resultant, under different masses of the die. Transformations of the projection diagram of normal contact stresses are shown as time passes.Translated from Dinamicheskie Sistemy, No. 7, pp. 32–41, 1988.     

The multidimensional plateau problem in Riemannian varieties

This paper is the author's abstract of his dissertation for the degree of Doctor of Physico-Mathematical Sciences. The dissertation was defended on September 29, 1972 at a session of the Council of the Mechanico-Mathematical Faculty of M. V. Lomonosov Moscow University. The official opponents were Prof. V. M. Alekseev, Doctor of Phys.-Mat. Sci.; Prof. D. V. Anosov, Doctor of Phys.-Mat. Sci.; and Prof. M. M. Postnikov, Doctor of Phys.-Mat. Sci.Translated from Matematicheskie Zametki, Vol. 13, No. 1, pp. 159–167, January, 1973.     

Approximations relative to Hausdorff distance

The present paper is an abstract of a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Physical and Mathematical Sciences. The dissertation was defended on November 30, 1967 before the faculty of the V. A. Steklov Mathematical Institute of the Academy of Sciences of the USSR. The official opponents were Professors N. P. Korneichuk, P. P. Korovkin, and S. B. Stechkin, Doctors of Physical and Mathematical Sciences.     

Concerning the investigation of the supermolecular structure of polymers with a three-dimensional network

The results of an investigation of the structure and properties of a series of crosslinked polymers are presented. It is shown that the structure is determined by the chemical structure of the starting components and the method of synthesis.Institute of Chemistry of High-Molecular Compounds, Academy of Sciences of the Ukrainian SSR, Kiev. Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Mekhanika Polimerov, No. 5, pp. 934–936, September–October, 1970.     

Load-bearing capacity of compressible plates of glass-fiber-reinforced plastic with different placement of the reinforcement

The effect of placement of reinforcement on the load-bearing capacity of circular and rectangular plates compressible in the reinforcement plane was evaluated experimentally. Four placement schemes for circular plates and nine for rectangular plates were investigated. Different schemes of supporting the edges were studied. Two mechanisms of exhaustion of the bearing capacity were revealed. A numerical evaluation and comparison of different placement schemes with respect to the value of the critical and breaking load are given. The role of the actual realization of clamping and nonlinearity of the material is shown. A comparison with analytic relations taking into account the characteristics of the material is given.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 4, pp. 646–653, July–August, 1972.     

Some laws of the elastic properties of oriented fibrous polymers and fibers

The author considers the results of calculations of the limiting values of the elastic properties (modulus of elasticity and load-extension diagram) of the main types of chemical fibers, using a model with "ideal orientation" of the molecules and the derived laws of deformation of polymer chains. A method is proposed for calculating the elastic properties of "ideally oriented" polymers from the velocity of propagation of an elastic deformation pulse and the effective density of the "skeletons" of the polymer chains. Values of the moduli of elasticity of the amorphous regions of the structure of oriented polymers are calculated. The calculated results are compared with experimental data on the elastic properties of fibers.Mekhanika Polimerov, Vol. 2, No. 1, pp. 34–42, 1966Paper read at the XIV All-Union Conference on High-Molecular Compounds, Oriented State.     

Some negative characteristics of fiber-reinforced materials

The effect of low resistance to shear and tension-compression at right angles to the layers of reinforcement on the state of stress and strain of elements composed of fiber-reinforced materials is analyzed. It is shown that to take these characteristics into account the calculations must be refined. In certain cases this leads to the appearance of qualitatively new effects. Typical problems in which the negative characteristics of the material must be allowed for are examined.Review of research conducted in the Polymer Structures Laboratory of the Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, Vol. 5, No. 1, pp. 140–149, January–February, 1969.     

Structural damage accumulation and stable postcritical deformation of composite materials

The macroscopic failure of composite materials is preceded by complex multilevel processes accompanied by accumulation and localization of damaged centers and formation of a failure cluster. Therefore, the study of these mechanisms is one of the basic problems for the mechanics of modern composite materials used in aerospace engineering. The formation of a theory of the stable postcritical deformation of the work-softening media is considered. The pseudo-plastic deformation affected by structural damage of granular composites is investigated within the framework of the considered two-level structurally phenomenological model of heterogeneous media. The stable evolution of the interconnected processes is accompanied by stress redistributions, partial or complete unloading, and strain or damage localization that are one of the main causes of implementation of the postcritical deformation stage. The numerical calculation results of inelastic deformation and failure of the periodic unidirectional fiber-reinforced composites are presented under conditions of the displacement-controlled transverse proportional loading mode. The main mechanisms of the work-softening behavior for the indicated type of materials are described in the macro-homogeneous stress-strain states. Macroscopically, the failure of heterogeneous media as a result of postcritical deformation and the loss of stability of damage accumulation depends on the stiffness of the loading system. When a deformable body is fixed on the closed surface with sufficiently but not infinitely large coefficients of stiffness, it is possible to observe the equilibrium development of the localized volumes of work-softening and damage. The constitutive equations for the work-softening isotropic, transverse isotropic, and orthotropic media are presented. The effect of the loading system on the stability of deformation, damage accumulation, and failure under monotone and nonmonotone triaxial loading was studied. The growth of failure strains with increase in stiffness of the loading system and unequal resistance of heterogeneous body are registered and investigated. A preventive unloading method is offered for the mathematical modeling of the damage accumulation during the testing of the materials on the servo-controlled systems. The displacement-controlled mode is simulated by a series of soft loading and unloading cycles. The detected phenomenon of failure where the unloading leads to stress-strain diagrams with a negative slope of the descending branch was not found either in the displacement or stress-controlled monotone loading mode.Submitted to the 10th International Conference on Mechanics of Composite Materials, April 20–23, 1998, Riga, Latvia.Perm' State Technical University, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 2, pp. 234–250, March–April, 1998.     

Iterative algorithms for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of families and semigroups of nonexpansive mappings

We introduce iterative algorithms for finding a common element of the set of solutions of a system of equilibrium problems and of the set of fixed points of a finite family and a left amenable semigroup of nonexpansive mappings in a Hilbert space. We prove the strong convergence of the proposed iterative algorithm to the unique solution of a variational inequality, which is the optimality condition for a minimization problem. Our results extend, for example, the recent result of [V. Colao, G. Marino, H.K. Xu, An Iterative Method for finding common solutions of equilibrium and fixed point problems, J. Math. Anal. Appl. 344 (2008) 340–352] to systems of equilibrium problems.     

A mixed dynamic problem of the theory of elasticity for a piecewise homogeneous plane

We study the process of splitting of a compressed piecewise homogeneous medium under high-speed motion of a wedge with the formation of a crack of unknown extent ahead of the wedge. The motion of the wedge occurs along the interface between physico-mechanical properties of the piecewise homogeneous plane. We carry out numerical studies. We obtain asymptotic representations of the elastic potentials at the limiting velocities of motion of the wedge. Two figures. One table. Bibliography: 6 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 26, 1996, pp. 48–58.