Errors in approximate solutions of Cauchy's problem for a first-order quasilinear equation

The proximity is investigated of the solution of Cauchy's problem for the equation u _t +((u))_x= u _xx ((u) > 0) to the solution of Cauchy's problem for the equation u_t+ ((u))_x= 0, when the solution of the latter problem has a finite number of lines of discontinuity in the strip 0 t T. It is proved that, everywhere outside a fixed neighborhood of the lines of discontinuity, we have |u–u| C, where the constant C is independent of. Similar inequalities are derived for the first derivatives of u–u.Translated from Matematicheskie Zametki, Vol. 8, No. 3, pp. 309–320, September, 1970.In conclusion we express our gratitude to L. A. Chudov for his valuable advice concerning this work.     

Bin packing games

We consider bin packing games introduced by Faigle and Kern (1993) and we restrict ourselves to the subclass of games for which all bins have unit capacity and all items are larger than 1/3. We adopt the taxation model of Faigle and Kern and we prove that for a tax-rate of = sk_7/1 the -core is always non empty. The bound is sharp, since for every < sk_7/1 there exist instances of the bin packing game within our sublass with an empty -core.     

Probabilities of moderate deviations

Let X₁, X₂, ... be a sequence of independent, identically distributed random variables, and let. The rate of convergence of probabilities of the form andis studied for any > ₀ and some r and ₀. Moreover, necessary and sufficient conditions are given that the relations be satisfied uniformly with respect to x in the region 0 x clog n, where and c are some positive constants, and. Local limit theorems are also presented.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituts im. V. A. Steklova AN SSSR, Vol. 85, pp. 6–16, 1979.     

The multiple-scale transport equation in one space dimension

This study examines the behavior of the one-dimensional non-homogeneous transport equation of the form u_t= u_x+f, «1. The solution consists of behavior which changes on two different time scales, one rapid and one slow. This time scale behavior is known. Additionally, however, we find here that because of the presence of the non-homogeneous forcing termf, and large wave speed 1/, there is a component of the solution which will vary only on a very large spatial scale. This large space-scale solution persists throughout all time, even after the source term of the solution has been shut off. The analysis of this large spacescale behavior is the focus of this paper. Numerical experiments highlight some of our results. These results have applications in fields such as meteorology, and other areas where multiple time scales are of interest.This work was supported in part by NSF grant NSF-DMS93-21728.     

Speed of convergence as a function of given accuracy for random search methods

The essence of this article lies in a demonstration of the fact that for some random search methods (r.s.m.) of global optimization, the number of the objective function evaluations required to reach a given accuracy may have very slow (logarithmic) growth to infinity as the accuracy tends to zero. Several inequalities of this kind are derived for some typical Markovian monotone r.s.m. in metric spaces including thed-dimensional Euclidean space ^d and its compact subsets. In the compact case, one of the main results may be briefly outlined as a constructive theorem of existence: if is a first moment of approaching a good subset of-neighbourhood ofx ₀=arg maxf by some random search sequence (r.s.s.), then we may choose parameters of this r.s.s. in such a way that E c(f) In² . Certainly, some restrictions on metric space and functionf are required.     

Sensitivity analysis of the greatest eigenvalue of a symmetric matrix via the ɛ-subdifferential of the associated convex quadratic form

LetA(·) be ann × n symmetric affine matrix-valued function of a parameteruR ^m , and let (u) be the greatest eigenvalue ofA(u). Recently, there has been interest in calculating (u), the subdifferential of atu, which is useful for both the construction of efficient algorithms for the minimization of (u) and the sensitivity analysis of (u), namely, the perturbation theory of (u). In this paper, more generally, we investigate the Legendre-Fenchel conjugate function of (·) and the -subdifferential (u) of atu. Then, we discuss relations between the set (u) and some perturbation bounds for (u).The author is deeply indebted to Professor J. B. Hiriart-Urruty who suggested this study and provided helpful advice and constant encouragement. The author also thanks the referees and the editors for their substantial help in the improvement of this paper.     

Perturbation of the poles of the scattering matrix under variation of the boundary condition

The behavior of the poles z_n(), n=1,2,... of the scattering matrix of the operatorl u=–u(x), x , (u/n)+(x)u|=0 as 0 is considered. It is proved that |z_n()–z_n|=0(^(1/2)qn), where q_n is the order of the pole of the scattering matrix for the operator ₀u=–u, u/=0.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 117, pp. 183–191, 1981.     

Convergence of a penalty-finite element approximation for an obstacle problem

Summary This study establishes an error estimate for a penalty-finite element approximation of the variational inequality obtained by a class of obstacle problems. By special identification of the penalty term, we first show that the penalty solution converges to the solution of a mixed formulation of the variational inequality. The rate of convergence of the penalization is where is the penalty parameter. To obtain the error of finite element approximation, we apply the results obtained by Brezzi, Hager and Raviart for the mixed finite element method to the variational inequality.     

Integration of stiff mechanical systems by Runge-Kutta methods

The numerical integration of stiff mechanical systems is studied in which a strong potential forces the motion to remain close to a manifold. The equations of motion are written as a singular singular perturbation problem with a small stiffness parameter . Smooth solutions of such systems are characterized, in distinction to highly oscillatory general solutions. Implicit Runge-Kutta methods using step sizes larger than are shown to approximate smooth solutions, and precise error estimates are derived. As 0, Runge-Kutta solutions of the stiff system converge to Runge-Kutta solutions of the associated constrained system formulated as a differential-algebraic equation of index 3. Standard software for stiff initial-value problems does not work satisfactorily on the stiff systems considered here. The reasons for this failure are explained, and remedies are proposed.This work was supported in part by the Austrian Science Foundation, grant P8443-PHY.     

Errors of solutions of linear algebraic systems

Error estimates are derived for the following methods: the sweepout method for tridiagonal systems, the method of square roots, the bordering method, and the method of reflection matrices. The book of S. K. Godunov is devoted to the last method; he altered the method so that it takes any matrix into a bidiagonal matrix; a considerable part of that book is devoted to the error of this alteration. In the present paper the method of reflection matrices is studied in the form in which it is expounded in the familiar book of D. K. Faddeev and V. N. Faddeeva. Recurrent formulas are obtained for the sweepout method which make it possible to successively estimate errors of the components of the solution vector. In the method of square roots the error of the solution vector is estimated by the quantity Here _i and are small quantities; the first characterizes the accuracy of small arithmetics effects, and the second the round-off error in the reverse step. Further, A is the matrix of the system, m, is its order, f is the vector of free terms, and C and are constants with 0 <1. We shall not present here the rather involved estimates for the bordering method. The error of the solution vector obtained by the method of reflection matrices is estimated by the quantity (p_A is the conditioning number of the matrix A) All estimates are obtained up to terms of higher order of smallness than and ₁. The estimates themselves are related to the classification of errors of computing processes proposed by the author in recent years.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 139, pp. 94–110, 1984.     

Multiscale convergence and reiterated homogenization of parabolic problems

Reiterated homogenization is studied for divergence structure parabolic problems of the form u /t–div (a(x,x/,x/²,t,t/ ^k)u )=f. It is shown that under standard assumptions on the function a(x, y ₁,y ₂,t,) the sequence {u } of solutions converges weakly in L ² (0,T; H ₀ ¹ ()) to the solution u of the homogenized problem u/t– div(b(x,t)u)=f.This revised version was published online in April 2005 with a corrected missing date string.     

Rational Chebyshev approximation on the unit disk

Summary In a recent paper we showed that error curves in polynomial Chebyshev approximation of analytic functions on the unit disk tend to approximate perfect circles about the origin [23]. Making use of a theorem of Carathéodory and Fejér, we derived in the process a method for calculating near-best approximations rapidly by finding the principal singular value and corresponding singular vector of a complex Hankel matrix. This paper extends these developments to the problem of Chebyshev approximation by rational functions, where non-principal singular values and vectors of the same matrix turn out to be required. The theory is based on certain extensions of the Carathéodory-Fejér result which are also currently finding application in the fields of digital signal processing and linear systems theory.It is shown among other things that iff(z) is approximated by a rational function of type (m, n) for >0, then under weak assumptions the corresponding error curves deviate from perfect circles of winding numberm+n+1 by a relative magnitudeO( ^{m + n + 2} as 0. The CF approximation that our method computes approximates the true best approximation to the same high relative order. A numerical procedure for computing such approximations is described and shown to give results that confirm the asymptotic theory. Approximation ofe ^z on the unit disk is taken as a central computational example.     

Hermitian K-theory on a Theorem of Giffen

Let be an exact category with duality. In [1] a category () was introduced and the authors asserted that the loop space of the topological realization of () is homotopy equivalent to Karoubis U-theory space of when = (R), the category of finitely generated projective modules over a ring R with an involution if 2 is invertible in R. Unfortunately, their proof contains a mistake. We present a different proof which avoids their argument.Mathematics Subject Classifications (1991): 19DO6, 19G38, 11E70.     

A certain periodic control problem for differential equations with impulses in the space of bounded numerical sequences

One considers the differential equation dx/dt=f(t, x) with the impulse action ¦_t=t_i=H_i(t_i,x) in the space of bounded numerical sequences, where f(t, x), H_i(t, x) are T-periodic, countable-dimensional vector-valued functions, is a positive parameter. One gives conditions for the existence of a control (₁,₂) such that the solution of the equation dx/dt=f(t, x)–₁ with impulse action x¦_t=t_i=H_i(t_i,x)–₂ assuming for t= the value x=x₀, be T-periodic.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 2, pp. 271–275, February, 1990.     

Zur krümmungsverwandtschaft von zwangläufen in affinen ck — Ebenen II

In part I we have studied a map of osculating elements of an affine Cayley-Klein (CK-) plane into the Lie algebra A₄(²) of the aequiform transformations A₄(²) of the given plane A₂(, ²). If we use the real projective space P₃() over A₄(²) each osculating element defines a straight line in P₃(). We now give a one parameter motion in A₄(²) and study second order properties and their analogon in the Lie algebra and P₃(), respectively. We show that the wellknown relationship between the points of the moving frame and the osculating circles of the point paths in the fixed frame may be interpreted as part of a quadratic map of certain straight Lines of P₃(). An analogous result holds for the curvature of pairs of envelopes; the mapV induced in P₃() than is contained in a cubic relationship of straight lines.

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Herrn Professor Oswal Giering zum 60. Geburtstag gewidmet     

ɛ-duality theorem of nondifferentiable nonconvex multiobjective programming

Necessary Kuhn-Tucker conditions up to precision without constraint qualification for -Pareto optimality of multiobjective programming are derived. This article suggests the establishment of a Wolfe-type -duality theorem for nondifferentiable, nonconvex, multiobjective minimization problems. The -vector Lagrangian and the generalized -saddle point for Pareto optimality are studied.     

On some approximately balanced combinatorial cooperative games

A model of taxation for cooperativen-person games is introduced where proper coalitions Are taxed proportionally to their value. Games with non-empty core under taxation at rate-balanced. Sharp bounds on in matching games (not necessarily bipartite) graphs are estabLished. Upper and lower bounds on the smallest in bin packing games are derived and euclidean random TSP games are seen to be, with high probability,-balanced for0.06.     

On the completeness of a system of functions on a segment

The question is considered of the completeness of the systems of functions {Xⁿ[1÷_n]}, where _n(x) are small, in the spaces C and L_p on the segment [0, a].Translated from Matematicheskie Zametki, Vol. 4, No. 5, pp. 557–568, November, 1968.     

Zur Krümmungsverwandtschaft Von Zwangläufen in Affinen CK — Ebenen I

We study a map of osculating elements of an affine Cayley- Klein (CK-) plane into the Lie algebraA ₄(²) of the aequiform transformationsA ₄(²) of the given plane. If we use the real projective spaceP ₃() overA ₄(²) each osculating element defines a straight line inP ₃(). In the first part of this paper this map is studied in detail. In the second part we study second order properties of one- parameter motions and their corresponding properties in the Lie algebraA ₄(²). This is done by considering the analogen to the formula of EULERSAVARY in the image spaceP ³() overA ₄(²).     

A Five-term Exact Sequence for Opext of Crossed Modules in Lie Algebras

Extensions of crossed modules in Lie algebras with abelian kernel are studied, particularly backward and forward induced extensions and related properties. The set Opext((U, Q, ), (R, K, )) of congruence classes of extensions of (R, K, ) by (U, Q, ) is endowed with a K-vector space structure. This K-vector space appears in a five-term natural and exact sequence associated with an extension of crossed modules.2000 Mathematics Subject Classification: 17B56, 17B99, 18G99