Stability of a laminated spherical shell

The stability "in the small" of a spherical shell consisting of alternating stiff and soft layers is investigated. The spectra of the bifurcation values of the load are found for a closed shell subject to hydrostatic pressure and their dependence on the buckling mode is studied. The effect of the relative stiffness of the layers on the buckling of the shell is investigated.Moscow Power Engineering Institute. Translated from Mekhanika Polimerov, No. 3, pp. 459–464, May–June, 1976.     

Mathematical programming models for some smallest-world problems

Given a weighted graph G, in the minimum-cost-edge-selection problem (MCES), a minimum weighted set of edges is chosen subject to an upper bound on the diameter of graph G. Similarly, in the minimum-diameter-edge-selection problem (MDES), a set of edges is chosen to minimize the diameter subject to an upper bound on their total weight. These problems are shown to be equivalent and proven to be NP-complete. MCES is then formulated as a 0–1 integer programming problem. The problems MCES and MDES provide models for determining smallest-world networks and for measuring the “small-worldness” of graphs.     

Analogies and the modeling of composites

On the basis of the solution of a series of problems of the distribution of static and dynamic fields of various physical kinds in the microstructure of composite materials reinforced with dispersed particles or fibers analogies are established between the stress, displacement, temperature, electromagnetic, and other fields subject to certain similarity criteria imposed on the properties of the components.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Mekhanika Polimerov, No. 6, pp. 1028–1037, November–December, 1969.     

Stability of cylindrical glass-reinforced plastic shells under external pressure

The stability of cylindrical glass-reinforced-plastic shells under external omnidirectional hydrostatic pressure has been the subject of an experimental investigation. The experimental data obtained are compared with calculations based on the theory of orthotropic shells.Mekhanika Polimerov, Vol. 3, No. 6, pp. 1089–1095, 1967     

Necessary conditions of optimality in constrained planar domain optimization

The article considers optimization of the shape of a planar domain with an integral functional subject to integral inequality constraints and constraints on domain location. The boundary-value problem is a Dirichlet problem for a quasilinear elliptical equation. A method is proposed for deriving necessary conditions of optimality for problems of this kind.Translated from Nelineinye Dinamicheskie Sistemy: Kachestvennyi Analiz i Upravlenie — Sbornik Trudov, No. 3, pp. 22–29, 1993.     

On one operator of convolution type in a quarter plane with an annihilatable symbol

We examine one class of Wiener-Hopf operators in a quarter plane, not subject to Noether's theory. We construct an unbounded inverse and describe the image of an operator from the class indicated.Translated from Matematicheskie Zametki, Vol. 20, No. 4, pp. 559–570, October, 1976.     

Formulation of scattering theory for the nonlinear Schrödinger equation with boundary conditions of the finite density type in a soliton-free sector

Asymptotic canonical variables and the classical S matrix for the nonlinear Schrödinger model with nonzero boundary conditions are calculated in terms of spectral data of an auxiliary linear operator, subject to the condition that a discrete spectrum does not exist.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 169, pp. 60–67, 1988.     

Functional equations for local normal zeta functions of nilpotent groups

We give explicit formulae for the local normal zeta functions of torsion-free, class-2-nilpotent groups, subject to conditions on the associated Pfaffian hypersurface which are generically satisfied by groups with small centre and sufficiently large abelianization. We show how the functional equations of two types of zeta functions – the Weil zeta function associated to an algebraic variety and zeta functions of algebraic groups introduced by Igusa – match up to give a functional equation for local normal zeta functions of groups. We also give explicit formulae and derive functional equations for an infinite family of class-2-nilpotent groups known as Grenham groups, confirming conjectures of du Sautoy.Received: September 2003 Revision: February 2004with Appendix by A. Beauville     

Estimates of Deviations for Generalized Newtonian Fluids

The paper is concerned with estimates of deviations from exact solutions for stationary models of viscous incompressible fluids. It is shown that if a function compared with exact solution is subject to the incompressibility condition, then the deviation majorant consists of terms that penalize the inaccuracy in the equilibrium equation and the rheological relation defined by a ceratin dissipative potential. If such a function does not satisfy the incompressibility condition, then the majorant includes an additional term. The factor of this term depends on the constant in the Ladyzhenskaya–Babuka–Brezzi condition. Bibliography: 27 titles.     

Associative rings

This survey mainly covers material abstracted in theReferativnyi Zhurnal Matematika during the period 1977–1983 and is a continuation of earlier surveys on the same subject, the theory of associative rings. An extensive bibliography of 1201 titles is included.Translated from Itogi Nauki i Tekhniki, Seriya Algebra, Topologiya, Geometriya, Vol. 22, pp. 3–115, 1984.     

Growth of arterial cracks in polymethyl methacrylate plates containing "silver" cracks

Conclusions 1. We have studied the influence of previously formed surface SC of various orientations on the force and kinetic characteristics of a slowly growing arterial FC. The subject of investigation was industrial glass based on polymethyl methacrylate. Steady growth of the FC was achieved in our experiments by applying torque in order to split lamellar samples.2. We have found that for no mutual disposition of the two kinds of cracks do surface SC ease the conditions of FC growth. On the contrary, in a number of cases the presence of SC having linear dimensions of only 3–5% of the front length interrupt the steady growth of the FC, reduce its average velocity, and sometimes stop it altogether. For the growth of the FC to be renewed the load has to be augmented, so that the limiting specific work of fracture W_lim increases by 10–40%. The growth of an FC is opposed most strongly by SC lying at right angles to it.3. The results of our experiments indicate comparatively easy fracture of the SC, and also show that FC prefer to develop along earlier-formed SC, subject to appropriate mutual orientation.4. By virtue of the presence of elastic interactions between different sections, any action applied to a small part of the leading edge (front) will alter the conditions of material fracture along the whole FC front.Institute of High-Molecular-Weight Compounds, Academy of Sciences of the USSR, Leningrad. Translated from Mekhanika Polimerov, No. 4, pp. 609–615, July–August, 1976.     

An Exact Procedure for the Resource-Constrained Weighted Earliness–Tardiness Project Scheduling Problem

In this paper we study the resource-constrained project scheduling problem with weighted earliness–tardinesss penalty costs. Project activities are assumed to have a known deterministic due date, a unit earliness as well as a unit tardiness penalty cost and constant renewable resource requirements. The objective is to schedule the activities in order to minimize the total weighted earliness–tardinesss penalty cost of the project subject to the finish–start precedence constraints and the constant renewable resource availability constraints. With these features the problem becomes highly attractive in just-in-time environments.We introduce a depth-first branch-and-bound algorithm which makes use of extra precedence relations to resolve resource conflicts and relies on a fast recursive search algorithm for the unconstrained weighted earliness–tardinesss problem to compute lower bounds. The procedure has been coded in Visual C++, version 4.0 under Windows NT. Both the recursive search algorithm and the branch-and-bound procedure have been validated on a randomly generated problem set.     

Anticipating stochastic differential systems with memory

This article establishes existence and uniqueness of solutions to two classes of stochastic systems with finite memory subject to anticipating initial conditions which are sufficiently smooth in the Malliavin sense. The two classes are semilinear stochastic functional differential equations (sfdes) and fully nonlinear sfdes with a sublinear drift term. For the semilinear case, we use Malliavin calculus techniques, existence of the stochastic semiflow and an infinite-dimensional substitution theorem. For the fully nonlinear case, we employ an anticipating version of the Itô–Ventzell formula due to Ocone and Pardoux [D. Ocone, E. Pardoux, A generalized Itô–Ventzell formula. Application to a class of anticipating stochastic differential equations, Annales de l’Institut Henri Poincaré. Probabilité s et Statistiques 25 (1) (1989) 39–71]. In both cases, the use of Malliavin calculus techniques is necessitated by the infinite dimensionality of the initial condition.     

Asymptotic stability analysis in uncertain multi-delayed state neural networks via Lyapunov–Krasovskii theory

This paper presents a new approach to the analysis of asymptotic stability of artificial neural networks (ANN) with multiple time-varying delays subject to polytope-bounded uncertainties. This approach is based on the Lyapunov–Krasovskii stability theory for functional differential equations and the linear matrix inequality (LMI) technique with the use of a recent Leibniz–Newton model based transformation without including any additional dynamics.Three examples with numerical simulations are used to illustrate the effectiveness of the proposed method. The first example considers the neural network with multiple time-varying delays, which may be seen as a particular case of the second example where it is subject to uncertainties and multiple time-varying delays. Finally, the third example analyzes the stability of the neural network with higher numbers of neurons subject to a single time-delay. The Hopf bifurcation theory is used to verify the stability of the system when the origin falls into instability in the bifurcation point.     

Equilibrium glass-plastic pressure vessels of minimum mass with a nongeodesic winding

A lens-shaped vessel having various central apertures made by a continuous winding process is considered. Using the methods of variational calculus, the optimum winding trajectory (yielding a structure of minimum mass) is established, subject to certain limitations regarding the angle of geodesic deviation and the stress level in the filaments. The shape of the base is determined with due allowance for the trajectory found in this way, which comprises sections of geodesic lines and lines of limiting deviation.Translated from Mekhanika Polimerov, No. 6, pp. 983–987, November–December, 1975.     

Generalized Mehler Semigroups and Catalytic Branching Processes with Immigration

Skew convolution semigroups play an important role in the study of generalized Mehler semigroups and Ornstein–Uhlenbeck processes. We give a characterization for a general skew convolution semigroup on a real separable Hilbert space whose characteristic functional is not necessarily differentiable at the initial time. A connection between this subject and catalytic branching superprocesses is established through fluctuation limits, providing a rich class of non-differentiable skew convolution semigroups. Path regularity of the corresponding generalized Ornstein–Uhlenbeck processes in different topologies is also discussed.     

Conditions for unique solvability of inverse problems for controllable dynamical systems

Problems arising in the applications of control theory of dynamical systems, when part of the variables of the mathematical model of the system is unknown and is subject to determination from information on the output of the system, are considered. Fundamental among them are the observation and identification problems, when unknown are the state of the system and its parameters, respectively, and also the problem of inverting the system, in which one seeks the control. Based on an analysis of mappings that are generated by an extended measurement vector, conditions for the unique solvability of the afore-mentioned problems with respect to a single trajectory or a set of trajectories are obtained.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 10, pp. 1359–1366, October, 1992.     

Locally semisimple algebras. Combinatorial theory and the K0-functor

Survey is devoted to theory of locally finite algebras and approximately finite-dimensional ¦AF-¦C^*-algebras which has been developed intensively in recent years. It can serve as an introduction to the subject. Both known and new results are contained in it.Translated from Itogi Nauki i Tekhniki, Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 26, pp. 3–56, 1985.     

Energy capacity of rotating composite designs (review)

Conclusions Theoretical investigations that have been performed and experimental confirmation on models have conclusively confirmed the high efficiency of composite flywheels. The most promising design at the present time is the chorded flywheel, especially for pulse energy sources. It can be assumed that the flexibility of the chords makes it possible to achieve a balanced design more simply. Problems of balancing, as, in general, the dynamics of energy-storing rotating composite components, however, will be the subject of further independent research.Translated from Mekhanika Kompozitnykh Materialov, No. 1, pp. 65–78, January–February, 1986.     

On the Index of Differential Operators on Manifolds with Conical Singularities

The paper contains the proof of the index formula for manifolds with conical points. For operators subject to an additional condition of spectral symmetry, the index is expressed as the sum of multiplicities of spectral points of the conormal symbol (indicial family) and the integral from the Atiyah–Singer form over the smooth part of the manifold. The obtained formula is illustrated by the example of the Euler operator on a two-dimensional manifold with conical singular point.