On the sums of a family of Kapteyn series

The series ₁ n ^r–1 J _n (n)J _n (n) (r 0, 0 < 1) arise in studying the emission and absorption of radiation by a charged particle on a Kepler orbit. For the first few even,r, the sums are obtained in closed form, and for oddr they are given in terms of a certain definite integral. The integral is expressed as a power series in for ||<1, and, near =1, an asymptotic expansion in powers of (1–²)^1/2 may be obtained.

---

Résumé La série ₁ n ^r–1 J _n (n)J _n (n) (r 0, 0 < 1) se trouve par l'émission et l'absorption du rayonnement d'une particule chargée sur l'orbite Keplerien. Pour les plus petites valuers paires der, les sommes s'obtienment en forme compacte, et pour les valuers impaires, elles se déterminent d'après une intégrale definie. Pour ||<1, cette intégrale se développe dans une série de puissances de , et dans le voisinage de =1, on obtient une série asymptotique et puissances de (1–²)^1/2.

     

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.     

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 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.     

Almost Fredholm operators in von Neumann algebras

LetA be a von Neumann algebra,J be the ideal of compact operators relative toA and letF ₊ be the left-Fredholm class ofA. We call almost left-Fredholm the class = {A A: if P A is a projection and AP J then P J}. Then and the inclusion is proper unlessA is semifinite and has a non-large center. satisfies all of the algebraic properties ofF ₊ but it is generally not open. IfA is semifinite then A iff there are central projectionsG with G = I such that AG F₊(AG). Let :A A/J. Then the left almost essential spectrum ofA A, , coincides with the set of eigenvalues of (A)     

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.

---

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.

---

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.     

Isolation theorem for forms corresponding to purely real algebraic fields

Let M be the complete module of a purely real algebraic field of degree n 3, let be a lattice in this module, and let F(X) be its form. We use to denote any lattice for which we have = , where is a nondiagonal matrix for which – I . With each lattice we can associate a factorizable formF (X) in a natural manner. We denote the complete set of forms corresponding to the set {} by {F (X)}. It is proved that for any > 0 there exists an > 0 such that for eachF (X) {F } we have |F (X₀₎| for some integer vector X₀ 0.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 185, pp. 5–12, 1990.In conclusion, the author would like to express his deep gratitude to B. F. Skubenko for stating the problem and for his constant attention.     

Compact operators whose singular numbers have powerlike asymptotics

For a compact operator in a Hilbert space, let s_n(A), n =1, 2,... be the singular numbers and let N(s; A) =card{n N:s_n(A)>, s>0. For 0

a _p and not on the individual elementAa, (H. Weyl's lemma); this allows us to write _p (a), _pp (a), a_p. One obtains certain results regarding the functionals _p, _p (and about the analogous functionals for the positive and negative eigenvalues in the casea=a ^*=A ^*:A a. In particular: I. Ifa ₁ a _2p, then. II.Let a ₁,a _{2 pP} ,.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matetmaticheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 126, pp. 21–30, 1983.     

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.     

Exchangeable random variables and the subsequence principle

Summary Call a sequence {X _n } of r.v.'s -exchangeable if on the same probability space there exists an exchangeable sequence {Y _n } such thatP(|X _n –Y _n |) for alln. We prove that any tight sequence {X _n } defined on a rich enough probability space contains -exchangeable subsequences for every >0. The distribution of the approximating exchangeable sequences is also described in terms of {X _n }. Our results give a convenient way to prove limit theorems for subsequences of general r.v. sequences. In particular, they provide a simplified way to prove the subsequence theorems of Aldous [1] and lead also to various extensions.     

The Chebyshev centers in a normed linear space

ＳＯＮＧＷＥＮＨＵＡ（宋文华）（ＩｎｓｔｉｔｕｔｅｏｆＭａｔｈｅｍａｔｉｃａｌＳｃｉｅｎｃｅ,ＤａｌｉａｎＵｎｉｖｅｒｓｉｔｙｏｆＴｅｃｈｎｏｌｏｇｙ,Ｄａｌｉａｎ１１６０２４,Ｃｈｉｎａ）Ａｂｓｔｒａｃｔ：ＡｃｈａｒａｃｔｅｒｉｚａｔｉｏｎｏｆＣｈ...     

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+.

---

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.

     

Asymptotic behavior of the solution of an elliptic boundary value problem in a thin domain

The asymptotic behavior as h0 of the solution of a mixed boundary value problem is investigated for an elliptic (in the sense of Petrovskii) system of second-order differential equations in the n-dimensional cylinder Q _h =×(–h/2, h/2) of small altitude h; is a domain in R ⁿ –1. The limit problem in contains a small parameter =h , (0, 1), for higher-order derivatives and degenerates regularly, as 0, into an elliptic problem of a lower order. It is shown that the limit problem and its corresponding degenerate problem (=0) are uniquely solvable. An estimate for the difference of solutions of the original and the limit problem in the energy norm is established. As an example, a problem on the deformation of a thin plate in the framework of the Cosserat continuum is considered.Translated from Problemy Matematicheskogo Analiza, No. 11, pp. 191–208, 1990.     

Higher order methods for a singularly perturbed problem

Summary A finite element method using piecewise polynomials of degree k is used to approximate the problem u+u=f, >0 a small parameter. A very irregular mesh is used. On this mesh error estimates of order0(h ^k+1) are obtained uniformly in ,h the maximum stepsize, fork2. The condition number of the system of linear equations one has to solve in order to get the approximation is estimated. Extension of the results to more complicated problems is briefly indicated. Finally, a numerical example is given.Work performed while visiting the IBM Thomas J. Watson Research Center, Yorktown Heights, N.Y.     

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.     

Singular perturbations in foliations

Summary We define a constraint system , [0,₀), which is a kind of family of vector fields on a manifold. This is a generalized version of the family of the equations , [0,₀),x ^m ,y ⁿ . Finally, we prove a singular perturbation theorem for the system , [0,₀).Dedicated to Professor Kenichi Shiraiwa on his 60th birthday     

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.     

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 .     

Exponential estimates for the Wiener sausage

Summary LetC (t) be the Wiener sausage of radius inR ^d up to timet. We obtain bounds on the asymptotics ofE exp (|C (t)|) ast, for all >0.     

Normal forms for vector fields with respect to an arbitrary dilation

Homogenous vector fields of degree one with respect to an arbitrary dilation ^r may be regarded as a natural extension of the linear vector fields. This paper is concerned with the problem of computing normal forms for vector fields that can be expanded in terms of homogeneous fields of degree greater than or equal to one with respect to an arbitrary dilation ^r . In particular, necessary and sufficient conditions are given for the existence of a local coordinate change that transforms an analytic field into a homogeneous field of degree one with respect to an arbitrary dilation ^r .This research was partially supported by National Science Foundation (Grant DMS-9301039), and by Istituto Nazionale di Alta Matematica F. Severi.