On the algorithm of the solution of the signorini problem

An algorithm is proposed for solving the Signorini problem /1/ in the formulation of a unilateral variational problem for the boundary functional in the zone of possible contact /2/. The algorithm is based on a dual formulation of Lagrange maximin problems for whose solution a decomposition approach is used in the following sense: a Ritz process in the basis functions that satisfy the linear constraint of the problem, the differential equation in the domain, is used in solving the minimum problem (with fixed Lagrange multipliers); the maximum problem is solved by the method of descent (a generalization of the Frank-Wolf method) under convexity constraints on the Lagrange multipliers. The algorithm constructed can be conisidered as a modification of the well-known algorithm to find the Udzawa-Arrow-Hurwitz saddle points /3, 4/. The convergence of the algorithm is investigated. A numerical analysis of the algorithm is performed in the example of a classical contact problem about the insertion of a stamp in an elastic half-plane under approximation of the contact boundary by isoparametric boundary elements. The comparative efficiency of the algorithm is associated with the reduction in the dimensionality of the boundary value problem being solved and the possibility of utilizing the calculation apparatus of the method of boundary elements to realize the solution.     

Time-independence of the solution of the problem of determining the hydraulic conductivity field

We consider the determination of the hydraulic conductivity field of a nonhomogeneous layer with linear seepage of an incompressible fluid in a nonelastic layer. In mathematical terms, the problem is formulated as a Cauchy problem for a partial differential equation of a special form. A theorem is proved which establishes that the solutions obtained for different times are identical.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 57, pp. 67–70, 1985.     

Estimation of the measure of the image of the ball

Under study is the class of ring Q-homeomorphisms with respect to the p-module. We establish a criterion for a function to belong to the class and solve a problem that stems from M. A. Lavrentiev [1] on the estimation of the measure of the image of the ball under these mappings. We also address the asymptotic behavior of these mappings at a point.     

On the minimization of the product of the powers of several integrals

This paper considers the minimization of the product of the powers ofn integrals, each of which depends on a functiony(x) and its derivative . The necessary conditions for the extremum are derived within the frame of the Mayer-Bolza formulation of the calculus of variations, and it is shown that the extremal arc is governed by a second-order differential equation involvingn undetermined multipliers related to the unknown values of the integrals. After the general solution is combined with the definitions of the multipliers and the end conditions, a system ofn+2 algebraic equations is obtained; it involvesn+2 unknowns, that is, then undetermined multipliers and two integration constants.The procedure discussed here can be employed in the study of shapes which are aerodynamically optimum at supersonic, hypersonic, and free-molecular flow velocities, that is, wings and fuselages having the maximum lift-to-drag ratio or the minimum drag. The problem of a slender body of revolution having the minimum pressure drag in Newtonian hypersonic flow is developed as an example. First, a general solution is derived for any pair of conditions imposed on the length, the thickness, the wetted area, and the volume. Then, a particular case is treated, that in which the thickness and the wetted area are given, while the length and the volume are free; the shape minimizing the pressure drag is a cone.This research, supported by the Office of Scientific Research, Office of Aerospace Research, United States Air Force, Grant No. AF-AFOSR-828-67, is a condensed version of the investigation described in Ref. 1. The author is indebted to Messrs. H. Y. Huang, J. C. Heideman, and J. N. Damoulakis for analytical and numerical assistance.     

On the growth of the Betti sequence of the canonical module

We study the growth of the Betti sequence of the canonical module of a Cohen–Macaulay local ring. It is an open question whether this sequence grows exponentially whenever the ring is not Gorenstein. We answer the question of exponential growth affirmatively for a large class of rings, and prove that the growth is in general not extremal. As an application of growth, we give criteria for a Cohen–Macaulay ring possessing a canonical module to be Gorenstein. GJL was partly supported by a grant from the National Security Agency. An erratum to this article can be found at     

Effect of the properties of the components and the geometric characteristics of the structure on the thermal-conductivity coefficients of glass-reinforced plastics

A method is proposed for estimating the thermal-conductivity coefficients of orthotropic composite glass-reinforced plastics with a structure composed of continually repeating units, in which the contours of the reinforcing elements can be described by a sinusoidal law. The arguments in the formulas are the relative content and thermophysical characteristics of the components and the parameters of the microstructure of the materials considered.Mekhanika Polimerov, Vol. 4, No. 1, pp. 18–23, 1968     

On the Geometry of the Nodal Lines of Eigenfunctions of the Two-Dimensional Torus

The width of a convex curve in the plane is the minimal distance between a pair of parallel supporting lines of the curve. In this paper we study the width of nodal lines of eigenfunctions of the Laplacian on the standard flat torus. We prove a variety of results on the width, some having stronger versions assuming a conjecture of Cilleruelo and Granville asserting a uniform bound for the number of lattice points on the circle lying in short arcs.     

On the degree of the inverse of an automorphism of the affine space

We study the degree of the inverse of an automorphism f of the affine n-space over a -algebra k, in terms of the degree d of f and of other data. For n = 1, we obtain a sharp upper bound for deg (f^{− 1}) in terms of d and of the nilpotency index of the ideal generated by the coefficients of f^′'. For n = 2 and arbitrary d≥ 3, we construct a (triangular) automorphism f of Jacobian one such that deg(f^{− 1}) > d. This answers a question of A. van den Essen (see [3]) and enables us to deduce that some schemes introduced by authors to study the Jacobian conjecture are not reduced. Still for n = 2, we give an upper bound for deg (f^{− 1}) when f is triangular. Finally, in the case d = 2 and any n, we complete a result of G. Meisters and C. Olech and use it to give the sharp bound for the degree of the inverse of a quadratic automorphism, with Jacobian one, of the affine 3-space.     

Hydroelastic phenomena in the neighborhood of the bifurcation of the aorta

The mechanical behavior of the vessels and the blood near a bifurcation is analyzed. A single-layer homogeneous shell is taken as a model of the aorta on the assumption that the intima is much less stiff than the other layers. In analyzing the blood flow in large vessels, the blood is treated as a viscous Newtonian liquid, whose motion is described by the Navier-Stokes equations and the continuity equation.Zhukovskii Air Force Engineering Academy, Moscow; Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 1, pp. 164–166, January–February, 1971.     

Aspects of the simulation of the ferroelectric hysteresis

In this paper we present a thermodynamically consistent phenomenological model for an assumed transversely isotropic ferroelectric crystal. Such materials become spontaneously polarized within a certain range of temperature. On the macro scale a remanent polarization and remanent strains, due to the reorientation of the polarization vectors, are observed, if an electric field above the so-called coercive field is applied. The goal of this work is to construct a thermodynamically consistent formulation of the electro-mechanically coupled ceramic on a meso scale, that takes into account the orientation of the assumed transversely isotropic unit cell, see [4]. The anisotropic behavior is governed by isotropic tensor functions, which are formulated in terms of a finite set of invariants, see [3]. Considering a simple model problem we discuss the linearization of discrete weak forms, resulting from the electromechanical boundary value problem. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the regularity of the solution of the biharmonic variational inequality

It is shown that the solution of the biharmonic variational inequality has bounded second derivatives provided that the obstacle and the data are smooth.During the preparation of a portion of the paper, the author was a guest of the Scuola Normale Superiore in Pisa, supported by the German Research Association (DFG).     

On the explicit resolution of the mushy zone in the modelling of the continuous casting of alloys

In the continuous casting of alloys, it is well-known that the mushy zone is decisive for the final properties of the casting. Most numerical models for the process use enthalpy-based methods on fixed grids which determine the extent the mushy zone implicitly. Here, on the other hand, we develop a methodology for explicitly resolving the geometrical extent of the mushy zone; this involves a sharp-interface formulation to solve a dual moving boundary problem to locate the solidus and liquidus isotherms. The results compare favourably with those from enthalpy-based methods, and the advantages of our approach with respect to future multiphysics calculations are discussed.     

From the neutrino to the edge of the universe

Two recent findings necessitate a closer look at the existing standard models of particle physics and cosmology. These are the discovery of neutrino oscillation, and hence a non-zero mass on the one hand and, on the other, observations of distant supernovae which indicate that contrary to popular belief, the universe would continue to expand for ever, possibly accelerating in the process. In this paper it is pointed out that relatively recent studies which indicate a stochastic, quantum vacuum underpinning and a fractal structure for space time, reconcile both of the recent observations, harmoniously.     

On the case of the explicit integrability of the equation for SH-waves

The equation for SH-waves is considered for the following parameters μ and ρ: μ=a(x)·b(y), ρ=a(x)b(y)(c(x)+d(y)) (a,b,c,d are known functions). For such μ and ρ the variables in this equation can be separated. An explicit solution of the problem of the interaction of a whispering gallery wave with a vertical interface of two media is obtained. Bibliography: 3 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 203, 1992, pp. 12–16. Translated by N. S. Zabavnikova.     

On the determination of the control parameters of the optimal can-order policy

This paper considers the well-known class of can-order policies. This type of coordinated replenishment policies accounts for a joint set-up cost structure, where a major set-up cost is incurred for any order and an individual minor set-up cost is charged for each item in the replenishment. Recent comparative studies have pointed out that the performance of the optimal can-order policy is poor, compared to other coordinated replenishment strategies, when the major set-up cost is high. This paper shows that it is the approximate decomposition method to calculate the optimal canorder parameters which performs bad in such situations and not the policy itself. Attention is focused to a subclass of can-order policies, which is close to the optimal can-order policy for high major set-up costs. A solution procedure is developed to calculate the optimal control parameters of this policy. It is shown that a properly chosen combination of the solution procedures to calculate can-order parameters leads to a can-order strategy which performs as well as other coordinated replenishment policies.     

Asymptotics of the solution of the main problem of the integral fire theory

We obtain asymptotic representations at zero and infinity for the solution of the main problem of the integral theory of fire in a room with a single opening, compute the initial height of the equal pressure plane, and derive conditions under which the one-sided operation mode of the opening can be implemented.     

On the classification of the coadjoint orbits of the Sobolev Bott-Virasoro group

The purpose of this paper is to give the classification of the Bott-Virasoro coadjoint orbits, with nonzero central charge, in the functional analytic setting of smooth Hilbert manifolds. The central object of the paper is thus the completion of the Bott-Virasoro group with respect to a Sobolev topology, giving rise to a smooth Hilbert manifold and topological group, called the Sobolev Bott-Virasoro group. As a consequence of this approach, analytic and geometric properties of the coadjoint orbits are studied.     

Effect of inhomogeneity of the medium on the directivity of the simplest sources of elastic waves

The initial data for the first correction term of the ray expansion of a longitudinal wave excited by a point center of expansion are found by the boundary-layer method. The same is done for a transverse wave from a center of rotation.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 140, pp. 77–87, 1984.     

On the Volume of the Union of Balls

We prove that if some balls in the Euclidean space move continuously in such a way that the distances between their centers decrease, then the volume of their union cannot increase. The proof is based on a formula expressing the derivative of the volume of the union as a linear combination of the derivatives of the distances between the centers with nonnegative coefficients. Received September 6, 1996, and in revised form March 26, 1997.     

Localization of the identities of the number field

The image of the group of identities of the number field under a mapping into the reduced multiplicative group of a local field is investigated. The asymptotic behavior of this image under a cyclotomic p-extension of the ground field is studied.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 64, pp. 5–11, 1976.