Boundedness of the solution of the third problem for the Laplace equation

A necessary and sufficient condition for the boundedness of a solution of the third problem for the Laplace equation is given. As an application a similar result is given for the third problem for the Poisson equation on domains with Lipschitz boundary.Supported by GACR, Grant No. 201/00/1515.     

Effect of the incompressibility condition on the stability of the solutions of ill-posed problems of the theory of elasticity

One considers the energy functional with density depending only on the first invariant of the strain tensor. Under natural constraints on the density and in the absence of exterior actions, the zero displacements yield a minimum to the energy functional both for the compressible and the incompressible media. In the paper one proves the isolation of this minimum for incompressible media and its nonisolation for compressible media.Translated from Problemy Matemticheskogo Analiza, No. 10, pp. 62–71, 1986.     

On the regularity of the axisymmetric solutions of the Navier-Stokes equations

We obtain improved regularity criteria for the axisymmetric weak solutions of the three dimensional Navier-Stokes equations with nonzero swirl. In particular we prove that the integrability of single component of vorticity or velocity fields, in terms of norms with zero scaling dimension give sufficient conditions for the regularity of weak solutions. To obtain these criteria we derive new a priori estimates for the axisymmetric smooth solutions of the Navier-Stokes equations. Received: 11 April 2000; in final firm: 26 November 2000 / Published online: 28 February 2002     

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.     

Singular cases of the problem of the continuation of the boundary layer

One obtains conditions for the existence and uniqueness of the solution of the continuation of the boundary layer near a solid wall in the case when the pressure gradient P_x is positive and satisfies one of the inequalities p_xp_x(0).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Maternaticheskogo Instituts im. V. A. Steklova AN SSSR, Vol. 138, pp. 86–89, 1984.The author is grateful to V. V. Pukhnachev for his assistance and also to N. V. Khusnutdinova for useful remarks.     

Numerical analysis of the inverse problem of estimation of the parameters of the ionosphere

The article simulates the process of signal propagation in the ionosphere using the authors' method for solving the inverse problem of estimation of the ionospheric parameters and the algorithm for computing the signal trajectory when the state of the ionosphere changes.Translated from Matematicheskoe Modelirovanie i Reshenie Obratnykh Zadach. Matematicheskoi Fiziki, pp. 30–36, 1993.     

On the bound of proximity of the binomial distribution to the normal one

Bounds for the error of the Gaussian approximation for the binomial distribution are stated, depending from the probability of success and the number of observations. As a consequence, the upper bound for a constant in the Berry-Esseen inequality is deduced which differs from asymptotical one slightly more than 0.01.     

On the distribution of the ratio of the largest eigenvalue to the trace of a Wishart matrix

The ratio of the largest eigenvalue divided by the trace of a p×p random Wishart matrix with n degrees of freedom and an identity covariance matrix plays an important role in various hypothesis testing problems, both in statistics and in signal processing. In this paper we derive an approximate explicit expression for the distribution of this ratio, by considering the joint limit as both p,n→∞ with p/n→c. Our analysis reveals that even though asymptotically in this limit the ratio follows a Tracy-Widom (TW) distribution, one of the leading error terms depends on the second derivative of the TW distribution, and is non-negligible for practical values of p, in particular for determining tail probabilities. We thus propose to explicitly include this term in the approximate distribution for the ratio. We illustrate empirically using simulations that adding this term to the TW distribution yields a quite accurate expression to the empirical distribution of the ratio, even for small values of p,n.     

A functional approach to the theory of the structure of the surface of metals

A self-consistent field equation obtained by the method of functional integration is applied to investigate the structure of the lattice near the surface of a simple metal. Formulas are obtained for the displacements of ions relative to the nodes of an ideal lattice near the surface of the metal. Formulas are presented for the surface potential and the work of exit for one-dimensional and three-dimensional models.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 120, pp. 3–11, 1982.     

On the Passage to the Limit in the Construction of Geometric Solutions of the Riemann Problem

Mathematical Notes - A method for constructing geometric solutions of the Riemann problem for an impulsively perturbed conservation law is described. A complete classification of the possible...     

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.     

On the second term of the spectral asymptotics of the transmission problem

This article is devoted to the investigation of spectral asymptotics for the second-order differential operator with transmission boundary conditions. A two-term asymptotic formula is obtained for the distribution function of eigenvalues under some assumptions not excluding the reach set of periodic billiard trajectories.     

On the stability of the solution of the thermistor problem

The stability of the stationary solution of the thermistor problem 1s proved using a Liapunov functional for a class of physically relevant electrical conductivity.     

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.     

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.     

On the complexity of the Leibniz hierarchy

We prove that the problem of determining whether a finite logical matrix determines an algebraizable logic is complete for EXPTIME. The same result holds for the classes of order algebraizable, weakly algebraizable, equivalential and protoalgebraic logics. Finally, the same problem for the class of truth-equational logic is shown to be hard for EXPTIME.     

Operator methods in the problem of estimating the asymptotics of the time of the first hit for the birth and death process

Estimates are established for the exponential asymptotics of the time of the first hit of a certain leveln for the birth and death process.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 8, pp. 1104–1108, August, 1993.     

On the application of the residual method for the correction of inconsistent problems of convex programming

For the correction of a convex programming problem with potentially inconsistent constraint system (an improper problem), we apply the residual method, which is a standard regularization procedure for ill-posed optimization models. A problem statement typical for the residual method is reduced to a minimization problem for an appropriate penalty function. We apply two classical penalty functions: the quadratic penalty function and the exact Eremin-Zangwill penalty function. For each of the approaches, we establish convergence conditions and bounds for the approximation error.     

Aftermath of the Chernobyl Catastrophe from the Point of View of the Security Concept

The paper deals with uncertainty relations for time and energy operators, and the aftermath of the Chernobyl catastrophe is considered as an example. The mathematical approach developed by Holevo is analyzed, which allows us to assign the corresponding observables to non-self-adjoint operators and to establish uncertainty relations for nonstandard canonical conjugate pairs.

Relations for calculating the minimal time interval in which the energy jump can be discovered are given. Based on the intensity parameter introduced by the author, which is related to a special statistics called Gentile statistics and to the polylogarithm function, properties of stable chemical elements, such as time fluctuations and the jump of specific energy in the transition from the Bose—Einstein distribution to the Fermi—Dirac distribution, are mathematically described with regard to experimental data. The obtained data are arranged in a table for 255 stable chemical elements.

The mathematical approach developed by the author of the present paper allows one to describe the “antipode” (in a certain sense) of the standard thermodynamics, i.e., the thermodynamics of nuclear matter. This field of nuclear physics is very important for the study of properties of radioactive elements and, accordingly, from the standpoint of ensuring nuclear safety.

     

Investigation of the nonunifcrmity of the acoustic properties of the human tibia

Mechanics of Composite Materials - A method is proposed for determining the propagation velocity and coefficient of attenuation of ultrasonic vibrations in both the whole diaphysis of the human...