（Co）Homology and Universal Central Extension of Hom-Leibniz Algebras

Hom-Leibniz algebra is a natural generalization of Leibniz algebras and Hom-Lie algebras. In this paper, we develop some structure theory (such as (co)homology groups, universal central extensions) of Hom-Leibniz algebras based on some works of Loday and Pirashvili.     

分裂的正则Hom-Leibniz-Rinehart代数

作为Hom-Leibniz代数胚的代数类比, 本文引入Hom-Leibniz-Rinehart代数的概念. 证明了分裂的正则Hom-Leibniz-Rinehart代数$L$写成$L=U+\sum_{\gamma}I_\gamma$, 其中$U$为极大交换子代数$H$的子空间和$I_\gamma$为$L$的理想, 若$[\gamma]\neq[d]$, 满足$[I_\gamma, I_d]=0$. 随后分别发展了分裂Hom-Leibniz-Rinehart代数的根和权的连通技术.最后研究了紧致的正则Hom-Leibniz-Rinehart代数的结构.     

Dynamically equilibrium shapes and directions of motion of ocean current rings

The dynamically equilibrium shapes of a uniform-density rotating mass of liquid (a ring) in the surface layer of a quiescent stratified ocean are determined. The examination is carried out in a plane tangential to the Earth, taking into account the vertical and horizontal projections of the angular velocity of its rotation. Exact solutions of the equations of motion of an ideal incompressibe fluid are obtained, making it possible, for a linearly stratified ocean, to determine the dynamic all equilibrium shape of the interfaces of water masses and the free boundaries of cyclonic and antocyclonic rings. These shapes comprise second-order surfaces inclined to the water level in the meridian plane, the type of surfaces depending on the governing parameters of the problem. Expressions are obtained for the angles of inclination of the principal axes. For small deviations from equilibrium, due to a difference in the gravitational forces and Archimedes forces, motion of the ring occurs, governed by the inclination of the principal axes and the nature of change (increase or reduction) in the average density of the ring, determined by the ratio of the rates of diffusion of heat and salt. The displacement along the parallel comprises geostrophic motion, for the velocity of which an analytical expression is obtained. The displacement along the meridian comprises motion over an inclined plane. An analytical expression is given that relates the change in the depth of the centre of mass of the ring to the velocity of motion along the meridian through the angle of inclination of the principal axes of the ring. This explains the motion of both types of Gulf Stream ring to the south-west and of the Oyasio ring to the north-east.     

The stability of the equilibrium of two-phase elastic solids

The distinctive features of the loss of stability of elastic solids which undergo phase transitions are investigated for the case of small deformations. The non-uniqueness of the solution of the boundary-value problem for the describing of the thermodynamic equilibrium of a two-phase body is caused by the non-linearity associated with the unknown interface. The solution can be chosen by comparing the potential energies of the body in the two-phase and single phase states and by analysing of the local stability of the two-phase states. A linearized boundary-value problem is formulated which describes infinitesimal small perturbations of an initial two-phase state which is in thermodynamic equilibrium. Analysis of the stability of the two-phase state reduces to an investigation of the bifurcation points and the behaviour of the small solutions of the system of integrodifferential equations in terms of functions describing the perturbations of the interface. The problem of the non-uniqueness and loss of stability of centrisymmetric equilibrium two-phase deformations is investigated as an example. A theorem concerning the number of centrisymmetric solutions is proved. The energy changes accompanying the formation and development of two-phase states and the stability of the solutions obtained are investigated. The concept of topological instability as a bifurcation is introduced, as a result of which the type of geometry of a solution of the boundary-value problem changes and surfaces of separation of the phases actually appear and disappear. Macrodiagrams of the deformational are constructed which demonstrate the effect of deformation softening in the path of a phase transition.     

Numerical inversion of Laplace transform by the method of asymptotic extension of the interval in dynamic viscoelasticity problems

Conclusions 1. A numerical investigation into the process of wave propagation in infinite and finite viscoelastic rods has been carried out by means of the method of asymptotic extension of the interval.2. It is established that use, in the calculations, of kernels of relaxation with weak singularity does not give rise to a stress jump at the wave front.3. The effect of parameters of the Rzhanitsyn kernel on the "erosion" of the wave front has been investigated. It is discovered that the instant of occurrence of a stress that is appreciably different from zero, at points far away from the beginning of the rod, is determined by the long-term modulus of elasticity of the material of the rod.4. The solution of the problem concerned with the propagation of a load impulse of finite duration shows a decrease of the maximum value of the stress in the rod, when the duration of the applied impulse decreases, and an intense change in the shape of the impulse while it propagates along the rod, as a consequence of wave dispersion.P. Stuchka Latvian State University, Riga. Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 5, pp. 864–870, September–October, 1976.     

Localization of microfailure in fibrous composites

Conclusion When a fibrous composite is loaded, the process of microfailures becomes localized in consequence of the nonuiformity of internal stresses. The degree of localization can be quantitatively characterized by the magnitude of the parameter of localization whose determination was provided in the present work. The dependence of the parameter of localization on the stress applied to the specimen can be measured experimentally from the data on the location of the coordinates of the sources of AE, and it can be calculated theoretically on the basis of the model of failure of the composite. A comparison of the theoretical model with the experimental data makes it possible to determine the magnitude of the overstresses in the fibers of the composite material and the form of the distribution function of these overstresses.Translated from Mekhanika Kompozitnykh Materialov, No. 3, pp. 437–443, May–June, 1989.     

Bending of a semi-infinite cantilevered plate weakened by a cut with contacting edges

We solve the problem of the bending of a semi-infinite cantilevered plate containing a cut perpendicular to a clamped edge. Contact of the edges of the cut is taken into account in the two-dimensional formulation on the basis of the model of contacting edges on the face of the plate. We study the effect of the boundary on the distribution of the contact reaction and compute the coefficients of force and moment intensity and determine the breakinge load. We compare the results obtained with the solution of the problem not taking account of the contact of the edges of the cut. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 2, 1997, pp. 83–86.     

Problem of an elastic half-plane with a crack emerging orthogonally at the boundary

The problem of the half-plane, in which a finite crack emerges orthogonally at the boundary, is studied. On the edges of the crack a self-balancing load is applied. A detailed investigation is carried out for an integral equation with respect to the unknown derivative of the displacement jump, to which the problem can be reduced. The exact solution of the integral equation is constructed with the aid of the Mellin transform and the Riemann boundary value problem for the halfplane. The asymptotic behavior of the solution at both ends of the crack is elucidated. First the asymptotic behavior of the solution at the point of emergence of the crack is obtained and the dependence of this asymptotic behavior on the type of the load is established. For a special form of the load one obtains a simple expression of the stress intensity coefficient. In the case of a general load, the asymptotic behavior is used for the construction of an effective approximate solution on the basis of the method of orthogonal polynomials. As a result, the problem reduces to an infinite algebraic system, solvable by the reduction method.Translated from Dinamicheskie Sistemy, No. 4, pp. 45–51, 1985.     

天然心瓣关闭机理研究

本文在二维模型假定的基础上,从一个新的角度,通过理论分析和实验研究,解决了心脏瓣膜在射血加速相的部分关闭机理以及旋涡在瓣关闭中的作用这两个心瓣关闭机理研究中长期以来一直悬而未决的问题,并指出了窦(或瓣叶后空腔)对有效瓣膜关闭的决定性作用.此外,本文还研究了瓣长、频率等对瓣运动的影响.估计本文的研究对人工心瓣的研制有一定参考作用.     

A study of the causes of the reduced strength of glass-reinforced phenolformaldehyde plastics

A study of the curing kinetics of phenolformaldehyde resin in the presence of glass and quartz has shown that one of the chief causes of the reduced strength of glass-reinforced plastics based on phenol-formal-dehyde resin is the difference in the rate and degree of cure in layers close to the fibers and in the bulk of the resin. This is caused by the presence on the surface of the fibers of a hydrate sheath with increased concentration of hydroxyl ions and by the presence of hydrogen bonds between the oxyphenyl groups of the resin and the silanol groups on the surface of the fibers. Chemical treatment of the glass fibers has the effect of diminishing those factors responsible for the reduced rate and degree of cure, and in spite of the lower surface energy of the fibers, the strength of the glass-reinforced plastic increases.Mekhanika Polimerov, Vol. 1, No. 3. pp. 8–14, 1965     

Function integrals corresponding to a solution of the Cauchy-Dirichlet problem for the heat equation in a domain of a Riemannian manifold

A solution of the Cauchy-Dirichlet problem is represented as the limit of a sequence of integrals over finite Cartesian powers of the domain of the manifold considered. It is shown that these limits coincide with the integrals with respect to surface measures of Gauss type on the set of trajectories in the manifold. Moreover, the integrands are a combination of elementary functions of the coefficients of the equation considered and geometric characteristics of the manifold. Also, a solution of the Cauchy-Dirichlet problem in the domain of the manifold is represented as the limit of a solution of the Cauchy problem for the heat equation on the whole manifold under an infinite growth of the absolute value of the potential outside the domain. The proof uses some asymptotic estimates for Gaussian integrals over Riemannian manifolds and the Chernoff theorem. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 6, pp. 3–15, 2006.     

Investigations in the theory of Stochastic equations

This is the author-review of a dissertation presented in the competition of the degree of the Doctor of Physicomathematical Sciences. The dissertation was defended on February 27, 1975 at a meeting of the Scientific Council of V. A. Steklov Mathematics Institute of the Academy of Sciences of the USSR. The official opponents were: A. A. Borovkov, Corresponding Member of the Academy of Sciences of the USSR; A. V. Skorokhod, Corresponding Member of the Academy of Science of the Ukrain. SSR, and V. N. Sudakov, Doctor of Physicomathematical Sciences.Translated from Matematicheskie Zametki, Vol. 20, No. 1, pp. 147–159, July, 1976.     

On the conditions that the net method for the Laplace equation converges with speed h2

The paper gives a uniform estimate of order h² of the error in the net method of solving the Dirichlet problem for the Laplace equation under the assumptions that the modulus of continuity of the second derivatives of the boundary values and the modulus of continuity of the curvature of the region's boundary do not exceed a function satisfying the Dini condition. It is shown that with the removal of the Collatz boundary values the constraints on the boundary values cannot be significantly relaxed in terms of the moduli of continuity of the second derivatives, while the constraints on the moduli of continuity of the curvature of the boundary cannot be completely lifted.Translated from Matematicheskie Zametki, Vol. 6, No. 6, pp. 669–679, December, 1969.The author wishes to thank S. B. Stechkin under whose influence the present work was performed.     

On procedures for constructing solutions in differential games on a finite interval of time

Many problems of conflict-control theory can be reduced to approach-avoidance games with a certain terminal set. One of the main approaches to solution of such problems is the approach suggested by N. N. Krasovskii, which is based on positional constructions. The basis of these constructions consists of the extremal aiming principle at stable bridges. In this connection, the problem of constructing the maximal stable bridge, the set of all positions from which the problem of approaching the terminal set (which is the main task of one of the problem) is solvable, is important. The paper considers the approach-avoidance game on a finite interval of time in which the first player must ensure the attainment by the state vector of the control system of the terminal set on this interval, and the second player must ensure the avoidance of the terminal set. The main subject of the study is the maximal stable bridge, which is the set of positional consumption in this game. The problem of exactly constructing the set of positional consumption is solvable only in simple cases, and it is more realistic to consider and solve the problem of approximate construction of this set. The paper proposes approaches to approximate construction of the set of positional consumption based on sampling the time interval of the game and the technique of backward constructions, which has been developed at the scientific school of N. N. Krasovskii since the 1980s. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 23, Optimal Control, 2005.     

Micromechanics of polymer fracture

Small-angle x-ray scattering has been used to investigate the formation of embryonic submicroscopic cracks in polymers under a load. The main characteristics of crack formation in various loading regimes are analyzed. It is shown that there is a relation between the submicrocrack concentration and the deformation of the loaded polymer. The principal parameters of crack formation determining the strength properties of the polymer are found to be the transverse dimension of the initial submicrocracks relative to the loading axis, which is determined by the structural heterogeneity of the material, and the submicrocrack concentration in the prefracture state. The principles of the micromechanics of polymer fracture are formulated on the basis of the results of an analysis of the quantitative relationship between these parameters. The dominant role of the surface in the fracture process is demonstrated by comparing the parameters of crack formation in the interior and at the surface of the loaded polymer.A. F. Ioffe Physicotechnical Institute, Academy of Sciences of the USSR, Leningrad. Translated from Mekhanika Polimerov, No. 5, pp. 792–801, September–October, 1974.     

The rolling of a wheel along a corrugated rail

The rolling without detachment of a rigid massive wheel, carrying a static load, along a rail with undulations on the running surface, which arises as a result of non-uniform wear, is investigated. The rail is supported by an elastoviscous base. Because of the inertia of the wheel and the carriage the horizontal component of the velocity of the wheel centre differs only slightly from a constant quantity, and hence the motion of the wheel along the rail is assumed to be uniform. Steady vertical vibration of the wheel is considered. The vertical coordinate of the wheel centre, and also the difference between the longitudinal coordinates of the wheel centre and the point of contact of the wheel and the rail, are periodic and, correspondingly, even and odd functions of the longitudinal coordinate of the wheel centre, and their period is equal to the wave length on the rail surface. The periodic force of interaction of the wheel and the rail is given in the form of a Fourier series. Short waves, the amplitude of which is much less than their length, are often observed on the rail surface, and this length is much less than the wheel radius. In this case the coefficients of the Fourier series are expressed in terms of Bessel functions of the first kind of integer order. Observations show that the depth of the short wave on the rail surface increases until the radius of curvature in the rail trough approximates to the wheel radius, and hence it is assumed that these radii are close to or equal to one another. In this case the trajectory of the wheel centre differs considerably from the wave on the rail surface.     

Electron diffraction investigation of the fracture surfaces of filled mixtures of SKS-85 butadiene-styrene copolymer

As the result of an electron microscope study of replicas from the fracture surface during the disintegration of samples of mixtures of SKS-85 butadiene-styrene copolymer with 20 vol.% DG-100 carbon black and chalk, in the temperature range from +60 to –60°C, it is shown that, in the region of vitrification of the copolymer and of the loss of the reinforcing effect of the carbon black, there is a sharp change in the character of the fracture micro-surface, and the particles of carbon black become the sites of additional fracture. In the region of the temperature transition of the copolymer, connected with the mobility of the phenyl groups in the same way as in the vitrification region, there is a sharp decrease in the number of particles of carbon black at the fracture surface, while the fracture surface of a mixture with chalk passes selectively along the contact boundary of a polymer with larger particles. The microstructure of the Wallner lines on the surface of a mixture with carbon black, at Tst, is made up of traces of secondary fracture, arising around the particles of carbon black.Moscow Institute of Technology for the Meat and Dairy Industry. M. V. Lomonosov Institute for the Technology of Fine Chemicals, Moscow. É. L. Ter-Gazaryan State Scientific-Research and Design Institute, Kirovakan. Translated from Mekhanika Polimerov, No. 3, pp. 437–444, May–June, 1970.     

Structural theory of elastomeric composites based on fiber systems — Invariant description

A three-dimensional theory of elastomeric composites with elastomeric matrices reinforced by systems of fibers is presented. The theory is based on a structural approach in which the matrix and the reinforcement of the composite are considered separately without reduction to a medium having continuously changing characteristics. The approach is based on the idea of a vector field of macroscopic displacements given by the positions of the axial lines of the fibers in the curret (deformed) configuration of the composite. The vector field determines the current macroscopic configuration, the tensor fields of the measures of macroscopic strain, and the field of the macroscopic stress tensor in the composite. The displacement, strain, and stress fields in the elastomeric matrix and the fibers of the reinforcing systems are regarded as derivatives of the field of macroscopic displacements of the medium. Relations are presented to describe the kinematics of the fibers in the current configuration of the composite, including the evolution of their orientation and the frequency of their planar and spatial distribution. Equations are obtained for the macroscopic motion of the fiber-reinforced matrix, and the dynamic variational principle that governs this motion is established. The elastic macroscopic potential of the matrix is found and related to the components of the macroscopic stress tensor. The procedure to be followed in constructing the constitutive equations of the composite is described. The proposed system of equations, relations, and algorithms is closed and can be used to solve problems involving the deformation of products made of fiber-reinforced elastomers and the creation of elastomeric composite products, based on fiber systems, that possess the requisite properties.     

Kohonen neural networks and genetic classification

We discuss the property of a.e. and in mean convergence of the Kohonen algorithm considered as a stochastic process. The various conditions ensuring a.e. convergence are described and the connection with the rate decay of the learning parameter is analyzed. The rate of convergence is discussed for different choices of learning parameters. We prove rigorously that the rate of decay of the learning parameter which is most used in the applications is a sufficient condition for a.e. convergence and we check it numerically. The aim of the paper is also to clarify the state of the art on the convergence property of the algorithm in view of the growing number of applications of the Kohonen neural networks. We apply our theorem and considerations to the case of genetic classification which is a rapidly developing field.