Estimates of the asphericity of sections of convex bodies

We define the function (n, k) to be the infimum of all such that any bounded centrally symmetric convex body inR ⁿ possesses an -asphericalk-dimensional central section. It is proved that (3, 2)=2–1 and (n, n-1)n-1-1. Several related functions are defined and their values on the pairs (n, n-1) are estimated.Translated from Ukrainskií Geometricheskií Sbornik, Issue 28, 1985, pp. 76–79.     

On Shikata's distance between differentiable structures

Shikata proved: there is a number (n) with the following property: If two compact homeomorphic n-dimensional manifolds have a distance less than (n), then they are diffeomorphic. We improve the known lower bound (n!)^–n for (n) to 1/3n ^–2.This work was done under the program Sonderforschungsbereich Theoretische Mathematik (SFB 40) at Bonn University while Shikata was SFB-guest at Bonn.     

Local Stochastic Channeling Theory: Kinetic Functions in the Case of Interaction Between Fast Particles and Lattice Atoms

We investigate the motion of high-energy particles in a crystal with regard to their interaction with the thermal vibrations of the lattice atoms using analytic methods in the theory of Markov processes including the local Fokker–Planck equation. We construct a local matrix of random actions, which is used to introduce the main kinetic functions in the traverse-energy space, namely, the function a() of energy losses due to the dynamic friction and the diffusion function b(). We show that the singularities of the functions a() and b() are related to the distinction between the contributions to the kinetics from particles moving in three different regimes, namely, in the channeling, quasichanneling, and chaotic motion modes.     

On the complexity of approximating a KKT point of quadratic programming

We present a potential reduction algorithm to approximate a Karush—Kuhn—Tucker (KKT) point of general quadratic programming (QP). We show that the algorithm is a fully polynomial-time approximation scheme, and its running-time dependency on accuracy (0, 1) is O((l/) log(l/) log(log(l/))), compared to the previously best-known result O((l/)²). Furthermore, the limit of the KKT point satisfies the second-order necessary optimality condition of being a local minimizer. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.Research support in part by NSF grants DDM-9207347 and DMI-9522507, and the Iowa Business School Summer Grant.     

How to approximate the solutions of certain free boundary problems for the Laplace equation by using the contraction principle

We show how the free boundary of an ideal fluid, subject to a generalized Bernoulli condition, can (under appropriate circumstances) be approximated. Our method is based on a class of free-boundary perturbation operatorsT , 0<<1, which are all contracting relative to a suitable norm and class of boundaries, and whose fixed points converge to the desired free boundary solution as 0+.

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Zusammenfassung Wir zeigen, wie der freie Rand einer idealen Flüssigkeit, welcher einer verallgemeinerten Bernoulli-Bedingung genügt, unter geeigneten Umständen approximiert werden kann. Unsere Methode stützt sich auf eine Klasse freier RandperturbationsoperatorenT , 0<<1, welche relativ zu einer geeigneten Norm und Ränderklasse kontrahierend sind und deren Fixpunkte gegen die gewünschte Lösung der freien Randaufgabe mit 0+ konvergieren.

     

The disappearance of criticality in the theory of thermal ignition

Summary In the study of thermal ignition, when reactant consumption is neglected, criticality is usually taken as the point at which there is a large jump in the maximum steady temperature for a small change in the external parameters. If the equations are taken in their exact form, without the Frank-Kamenetskii approximation RT _a E0, criticality vanishes for large. In this paper a model is introduced for which the value of at which criticality vanishes can be determined precisely for geometries of an infinite slab and infinite circular cylinder.

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Zusammenfassung Wird beim Studium thermischer Explosionen der Reaktionsverbrauch vernachlässigt, so spricht man üblicherweise von einem kritischen Punkt, wenn dort bei einer kleinen Änderung der äußeren Parameter ein großer Sprung der stationären Maximaltemperatur auftritt. Bei den exakten Gleichungen (ohne die Frank-Kamenetskii'sche Approximation RT _a E0) gibt es für große keinen kritischen Punkt mehr. In dieser Arbeit wird ein Modell eingeführt, bei dem der Wert von , für welchen der kritische Punkt verschwindet, im Fall der unendlichen Platte und des unendlichen Zylinders genau bestimmt werden kann.

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Nomenclature A pre-exponential factor in Arrhenius law - d length scale (e.g. radius of cylinder and half-thickness of slab) - E Arrhenius activation energy - k thermal conductivity - Q exothermicity per unit mass - R gas constant - S surface of reacting material - T local absolute temperature - T _a surface temperature - V volume occupied by the reacting material - x dimensionless distance coordinate - =E(T-T _a )/RT _a ² dimensionless temperature - solution of Eqs. (6), (7) - maximum value of inV - =(1+)^–1 - maximum value of inV - density of material - =RT _a /E - ₀ smallest value of for which criticality vanishes - =QEAd ² exp (-E/RT _a )/kRT _a ² - _cr critical value of - ^* maximum value of for which positive solutions exist Supported in part by the Victoria Univeristy of Wellington Fellowship Committee and the European Research Office of the U.S. Army.     

An isolation theorem for decomposable forms of purely real algebraic fields of degree n⩾3

Let M be a complete module of a purely algebraic field of degree n3, let be the lattice of this module and let F(X) be its form. By we denote any lattice for which we have = , where is a nondiagonal matrix satisfying the condition ¦-I¦ , I being the identity matrix. The complete collection of such lattices will be denoted by {}. To each lattice we associate in a natural manner the decomposable form F(X). The complete collection of forms, corresponding to the set {}, will be denoted by {F} It is shown that for any given arbitrarily small interval (N–, N+), one can select an such that for each F(X) from {F} there exists an integral vector X₀ such that N– < F(X₀) < N+.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 112, pp. 167–171, 1981.     

Extensions and selections in subspaces ofC(K)

LetK be a compact Hausdorff space and letFK be a peak interpolation set for a function algebraAC(K). Let be a map fromK to the family of all convex subsets of such that the set {(z, x)zK, x(z)} is open inK×C and such thatg(z)(z) (zK) for somegA. We prove that everyfC(F) satisfyingf(s)(s) (sF) (f(s)closure (s) (sF)) admits an extensionfAA} satisfyingf(z)(z) (zK) (f(z))}closure (z) (zK), respectively). We prove a more general theorem of this kind and present various applications which generalize known dominated interpolation theorems for subspaces ofC(K).     

Singularly perturbed functional-differential inclusions

Differential inclusions of a retarded type with a small real parameter >0 in part of the derivatives are considered. We prove upper semicontinuity of the map set of solutions at =0⁺ inC[0, 1]×(L ²(0, 1)–weak) topology. In case of constant delay lower semicontinuity inC[0, 1]×(L ¹(0, 1)–strong) is shown.     

Approximating derivations on ideals ofC *-algebras

For each^*-derivation of a separableC ^*-algebraA and each >0 there is an essential idealI ofA and a self-adjoint multiplierx ofI such that (–ad(ix))|I< and x.     

On Some Isomorphism on the Category of b-Spaces

Given a nuclear b-space N, we show that if is a finite or -finite measure space and 1p, then the functors L _loc ^p (,N.) and NL ^p (,.) are isomorphic on the category of b-spaces of L. Waelbroeck.     

Shift invariant spaces inL 2

If is a complex, separable Hilbert space, letL ² () denote theL ²-space of functions defined on the unit circle and having values in . The bilateral shift onL ²() is the operator (U f)()=f(). A Hilbert spaceH iscontractively contained in the Hilbert spaceK ifHK and the inclusion mapH–K is a contraction. We describe the structure of those Hilbert spaces, contractively contained inL ²(), that are carried into themselves contractively byU . We also do this for the subcase of those spaces which are carried into themselves unitarily byU .     

On Logarithmic Smoothing of the Maximum Function

We consider the maximum function f resulting from a finite number of smooth functions. The logarithmic barrier function of the epigraph of f gives rise to a smooth approximation g of f itself, where >0 denotes the approximation parameter. The one-parametric family g converges – relative to a compact subset – uniformly to the function f as tends to zero. Under nondegeneracy assumptions we show that the stationary points of g and f correspond to each other, and that their respective Morse indices coincide. The latter correspondence is obtained by establishing smooth curves x() of stationary points for g , where each x() converges to the corresponding stationary point of f as tends to zero. In case of a strongly unique local minimizer, we show that the nondegeneracy assumption may be relaxed in order to obtain a smooth curve x().     

An asymptotic finite element method for improvement of solutions of boundary layer problems

Summary A scheme that uses singular perturbation theory to improve the performance of existing finite element methods is presented. The proposed scheme improves the error bounds of the standard Galerkin finite element scheme by a factor of O(ⁿ⁺¹) (where is the small parameter andn is the order of the asymptotic approximation). Numerical results for linear second order O.D.E.'s are given and are compared with several other schemes.     

Enlargement of Monotone Operators with Applications to Variational Inequalities

Given a point-to-set operator T, we introduce the operator T defined as T(x)= {u: u – v, x – y – for all y Rn, v T(y)}. When T is maximal monotone T inherits most properties of the -subdifferential, e.g. it is bounded on bounded sets, T(x) contains the image through T of a sufficiently small ball around x, etc. We prove these and other relevant properties of T, and apply it to generate an inexact proximal point method with generalized distances for variational inequalities, whose subproblems consist of solving problems of the form 0 H(x), while the subproblems of the exact method are of the form 0 H(x). If k is the coefficient used in the kth iteration and the k's are summable, then the sequence generated by the inexact algorithm is still convergent to a solution of the original problem. If the original operator is well behaved enough, then the solution set of each subproblem contains a ball around the exact solution, and so each subproblem can be finitely solved.     

Remarks on the perturbation of analytic matrix functions III

As in [N], [LN] the Newton diagram is used in order to get information about the first terms of the Puiseux expansions of the eigenvalues () of the perturbed matrix pencilT(, )=A()+B(, ) in the neighbourhood of an unperturbed eigenvalue () ofA(). In fact sufficient conditions are given which assure that the orders of these first terms correspond to the partial multiplicities of the eigenvalue ₀ ofA().     

The regularity problem for generalized harmonic maps into homogeneous spaces

Let be a Riemannian surface and be a standard sphere, or more generally a Riemannian manifold on which a Lie group,, acts transitively by isometries. We define generalized harmonic maps by extending the notion of weakly harmonic maps in a natural way (motivated by Noether's Theorem), to mapsu W _loc ^1,1 (, ). We prove that, under some slight technical restrictions, for 1 <-p < 2, there are generalized harmonic mapsu W ^1,p(, ) that are everywhere discontinuous (in particular, this solves an open problem proposed by F. Bethuel, H. Brezis and F. Hélein, in [BBH]). We also show that the natural -regularity condition for such maps is to require <u to belong to the Lorentz space L(2, ). To prove this -regularity result we extend a compensated compactness result of R. Coifman, P.-L. Lions, Y. Meyer and S. Semmes, proved in [CLMS], to the case of Lorentz spaces in duality.     

S-Continued Fraction Method for the Investigation of a Complex Dielectric Constant of Two-Components Composite

The aim of this contribution is to examine the S-continued fraction method of obtaining bounds on the effective dielectric constant _e of a two-phase composite for the case where the dielectric coefficients ₁and ₂ of both components are either complex or real. The starting point for our study is a power expansion of _e (z) at(z)=0 (z)=₂/₁-1. The obtained S-continued fraction bounds have an interesting mathematical structure convenient for theoretical and numerical investigations of _e. They also agree with the earlier estimations reported by Bergman and Milton. Specific examples of calculation of bounds on _e by theS-continued fraction method are also provided.     

On the distribution of square-full and cube-full integers

LetN _r (x) be the number ofr-full integers x and let _r (x) be the error term in the asymptotic formula forN _r (x). Under Riemann's hypothesis, we prove the estimates ₂(x)x^1/7+, ₃(x)x^97/804+(>0), which improve those of Cao and Nowak. We also investigate the distribution ofr-full andl-free numbers in short intervals (r=2,3). Our results sharpen Krätzel's estimates.     

Über die Anzahl der Lösungen der diskreten Theodorsen-Gleichung

Summary Discretization of the Theodorsen integral equation (T) yields the discrete Theodorsen-equation (T _d ), a system of 2N nonlinear equations. A so-called -condition may be fulfilled. It is known that (T) has exactly one continuous solution. This solution gives the boundary correspondence of the normalized conformal map of the unit disc onto a given domainG. It is also known that (T _d ) has one and only one solution if <1 and at least one solution if 1. We show here that for every 1 and N\ {1} there is a domainG satisfying an -condition such that (T _d ) has an infinite number of solutions. Moreover, givenK>0 and any domainG that fulfills an -condition, we will construct a domainG ₁ in the neighbourhood ofG that fulfills a max (1, +K)-condition such that (T _d ) forG ₁ has an infinite number of solutions. The underlying idea of the construction of those domains allows also to give important new facts about iterative methods for the solution of (T _d ), even in the case <1.