Construction of ɛ-nets

LetS be a set ofn points in the plane and let be a real number, 0<<1. We give a deterministic algorithm, which in timeO(n ^–2 log(1/)+ ^–8) (resp.O(n ^–2 log(1/)+ ^–10) constructs an-netNS of sizeO((1/) (log(1/))²) for intersections ofS with double wedges (resp. triangles); this means that any double wedge (resp. triangle) containing more thatn points ofS contains a point ofN. This givesO(n logn) deterministic preprocessing for the simplex range-counting algorithm of Haussler and Welzl [HW] (in the plane).We also prove that given a setL ofn lines in the plane, we can cut the plane intoO( ^–2) triangles in such a way that no triangle is intersected by more thann lines ofL. We give a deterministic algorithm for this with running timeO(n ^–2 log(1/)). This has numerous applications in various computational geometry problems.     

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.     

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.     

Die ɛ-Störung linearer und quadratischer Optimierungsaufgaben und ihre Anwendung auf das Hildreth-Verfahren

Zusammenfassung Durch eine -Störung in der Diagonalen der quadratischen Form kann man eine lineare oder quadratisch semidefinite Optimierungsaufgabe zu einer streng definiten quadratischen Aufgabe machen, so daß Lösungsverfahren, die die Formmatrix als nichtsingulär voraussetzen müssen, anwendbar werden. Bekanntlich konvergiert die Lösungx der -gestörten Aufgabe für 0 gegen den Lösungsvektorx ^m von minimalem Betrag der ursprünglichen Aufgabe. Wir zeigen darüber hinaus, daß im linearen Fall immer und im eigentlich quadratischen in gewissen Fällen schon für 0<<* die beiden Lösungenx undx ^m übereinstimmen. Im linearen Fall ist die obere Grenze * durch die Lösung eines linearen Ungleichungssystems gegeben.Im zweiten Abschnitt wenden wir dasHildreth-Verfahren mittels der -Störung auf lineare und quadratisch semidefinite Aufgaben an, diskutieren Konvergenz- und Genauigkeitsfragen und kommen zu dem Schluß, daß man in der Praxis sowohl bei Rechnung von Hand als auch bei maschineller Rechnung zu befriedigenden Ergebnissen kommt.

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Summary Linear and quadratic semidefinite programming problems may be transformed into strongly definite quadratic problems by means of an -perturbation of the quadratic form so that procedures which presuppose the matrix of the form to be nonsingular, may be applied. As is well known, the solutionx of the -perturbated problem converges to the solutionx ^m of minimal length of the original problem as 0. We show that always in the linear case and in the quadratic case under certain circumstances, both solutionsx andx ^m are equal if 0 <<*. In the linear case, the upper limit * is given by the solution of a system of linear inequalities.In the second part of this paper we apply the method ofHildreth to linear and quadratic semidefinite programming problems by the -perturbation. We discuss questions of convergence and exactness, and conclude that in practice calculation by hand as well as by computer leads to satisfying results.

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Der Verfasser ist Herrn Prof. Dr.W. Vogel, Bonn, für einen Hinweis zu Dank verpflichtet.

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Vorgel. v.:H. P. Künzi     

Functions with a finite dirichlet integral in a domain with cusp points on the boundary

Let be a domain in ⁿ, n >2, the boundary of which has a cusp point, pointing inside or outside the domain. The purpose of the paper is to characterize the traces on of the elements of the space H¹() of functions with a finite Dirichlet integral. As a consequence one establishes the existence of a linear continuous extension operator H¹ () H¹(ⁿ) under the presence of an interior cusp point on . Theorems on domains with cusps are proved with the aid of results on cylindrical domains. In the space of functions with a finite Dirichlet integral in the exterior or the interior of the cylinder one introduces the norm, depending on a small parameter and generating a norm of the trace on as an element of the quotient space. The latter is placed in correspondence with an explicitly described norm of functions on the boundary, uniformly equivalent relative to . One constructs an operator of extension of functions from the exterior of the cylinder to Rⁿ, preserving H¹, whose norm is uniformly bounded relative to . For the optimal operator of extension from the inside of the cylinder one finds the asymptotic behavior of the norm as 0. From these results there follow similar theorems on functions with a finite Dirichlet integral inside and outside a thin closed tube (of width ).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 126, pp. 117–137, 1983.     

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.     

On a Model Problem for the Orr–Sommerfeld Equation with Linear Profile

A model spectral problem of the form -i)y+xy= y on the finite interval [-1,1] with the Dirichlet boundary conditions is considered. Here is the spectral parameter and is positive. The behavior of the spectrum of this problem as 0 is completely investigated. The limit curves are found to which the eigenvalues concentrate and the counting eigenvalue functions along these curves are obtained.     

Littlewood-paley operators on the generalized Lipschitz spaces

Littlewood-Paley operators defined on a new kind of generalized Lipschitz spaces ₀ ^,p are studied. It is proved that the image of a function under the action of these operators is either equal to infinity almost everywhere or is in ₀ ^,p , where –n<<1 and 1<p<.     

The Parametric Buffer Phenomenon for a Singularly Perturbed Telegraph Equation with a Pendulum Nonlinearity

We consider the boundary-value problem u _tt + u _t + (1 + cos2)sin u =² u _xx, u _x|_x=0=u_x|_x==0, where 0<1, =(1+)t, ,> 0, and the sign of is arbitrary. It is proved that for an appropriate choice of the external parameters and and for sufficiently small the number of exponentially stable solutions 2-periodic in can be made equal to an arbitrary predefined number.     

Finite activation energy reactions Part II: A reactive ridge

Summary The effect is determined of a large,O( ^–2),1, activation energy on the thermal stability of a reactant in the form of a two-dimensional ridge, undergoing a steady zero-order exothermic reaction. The reactive ridge decreases in depth at a rate ofO( ²) from a maximum ofO(1). The Biot number of the uniform lower surface of the reactant is taken to be zero and of the upper surface to beO( ^–2). The critical Frank-Kamenetskii parameter is determined toO( ²).     

Simple modules over the multiparameter quantum function algebra at roots of 1

We construct essentially all the irreducible modules for the multiparameter quantum function algebraF [G], whereG is a simple simply connected complex algebraic group, and is a root of unity.     

Blow-up solutions of nonlinear elliptic equations in ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> with critical exponent <Emphasis Type="Italic">Dedicated to Philippe Dufour,whose exquisite horology artwork ‘‘Simplicity’’ inspires so much.</Emphasis>

For an integer n3 and any positive number , we establish the existence of smooth functions K on ⁿ {0} with |K–1|, such that the equation has a smooth positive solution which blows up at the origin (i.e., u does not have slow decay near the origin). Furthermore, we show that in some situations K can be extended as a Lipschitz function on ⁿ . These provide counter-examples to a conjecture of C.-S. Lin when n>4, and a question of Taliaferro. Mathematics Subject Classification (2000)Primary 35J60; Secondary 53C21     

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     

On a variational problem with lack of compactness: the topological effect of the critical points at infinity

We study the subcritical problemsP :–u=u ^p–,u>0 on;u=0 on , being a smooth and bounded domain in ^N,N–3,p+1=2N/N–2 the critical Sobolev exponent and >0 going to zero — in order to compute the difference of topology that the critical points at infinity induce between the level sets of the functional corresponding to the limit case (P₀).

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Résumé Nous étudions les problèmes sous-critiquesP :–u=u ^p–,u > 0 sur;u=0 sur –où est un domaine borné et régulier de ^N,N–3,p + 1=2N/N –2 est l'exposant critique de Sobolev, et >0 tend vers zéro, afin de calculer la différence de toplogie induite par les points critiques à l'infini entre les ensembles de niveau de la fonctionnelle correspondant au cas limite (P₀).

     

On the shape of the convex hull of random points

Summary Denote by E _n the convex hull of n points chosen uniformly and independently from the d-dimensional ball. Let Prob(d, n) denote the probability that E _n has exactly n vertices. It is proved here that Prob(d, 2 ^d/2 d ^-)1 and Prob(d, 2 ^d/2 d ^(3/4)+)0 for every fixed >0 when d. The question whether E _n is a k-neighbourly polytope is also investigated.     

Representation of distributions with compact support

Distributions having compact support are represented as the boundary value of Cauchy and Poisson integrals corresponding to tubular radial domains T^C in ⁿ where C is an open convex cone. The Cauchy integral of U is shown to be an analytic function in T^C which satifies a certain boundedness condition. All analytic functions in T^C having this boundedness condition have a distributional boundary value which can be used to determine an distribution. The results are extended to vector valued distributions.     

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 remark on the strong law of large numbers

The strong law of large numbers for independent and identically distributed random variablesX _i ,i=1, 2, 3,... with finite expectationE|X ₁| can be stated as, for any >0, the number of integersn such that \varepsilon $$ " align="middle" border="0"> ,N is finite a. s. It is known thatEN < iffEX ₁ ² < and that ² EN var X₁ as 0, ifE X ₁ ² <. Here we consider the asymptotic behaviour ofEN (n) asn, whereN (n) is the number of integerskn such that \varepsilon $$ " align="middle" border="0"> andE N ₁ ² =.     

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.     

Invariance principles for Gaussian sequences

An almost sure invariance principle is proved for stationary Gaussian sequences whose covariances r(n) satisfy r(n) = O (n ^–1–)for some >0.