On the approximation of singular integral equations by equations with smooth kernels

Singular integral equations with Cauchy kernel and piecewise-continuous matrix coefficients on open and closed smooth curves are replaced by integral equations with smooth kernels of the form(t–)[(t–) ²–n ² (t) ²]^–1,0, wheren(t), t , is a continuous field of unit vectors non-tangential to . we give necessary and sufficient conditions under which the approximating equations have unique solutions and these solutions converge to the solution of the original equation. For the scalar case and the spaceL ₂() these conditions coincide with the strong ellipticity of the given equation.This work was fulfilled during the first author's visit to the Weierstrass Institute for Applied Analysis and Stochastics, Berlin in October 1993.     

Multi-Faithful Spanning Trees of Infinite Graphs

For an end and a tree T of a graph G we denote respectively by m() and m _T () the maximum numbers of pairwise disjoint rays of G and T belonging to , and we define tm() := min{m _T(): T is a spanning tree of G}. In this paper we give partial answers — affirmative and negative ones — to the general problem of determining if, for a function f mapping every end of G to a cardinal f() such that tm() f() m(), there exists a spanning tree T of G such that m _T () = f() for every end of G.     

Parabolische Kollineationen,die eine ebene konvexe Menge invariant lassen

Collineations ₁, ₂ of PG(2, ) leaving invariant a compact convex setK ² are called parabolic if |K Fix _i|=1. Conditions are stated under which the existence of ₁, ₂ imply that K is an ellipse.

Herrn Helmut R. Salzmann zum 65. Geburtstag gewidmet     

Stability analysis of numerical methods for delay differential equations

Summary This paper deals with the stability analysis of step-by-step methods for the numerical solution of delay differential equations. We focus on the behaviour of such methods when they are applied to the linear testproblemU(t)=U(t)+U(t–) with >0 and , complex. A general theorem is presented which can be used to obtain complete characterizations of the stability regions of these methods.     

Birth times,death times and time substitutions in Markov chains

Summary Given a Markov chain (X _n ) _n0, random times are studied which are birth times or death times in the sense that the post- and pre- processes are independent given the present (X _–1, X ) at time and the conditional post- process (birth times) or the conditional pre- process (death times) is again Markovian. The main result for birth times characterizes all time substitutions through homogeneous random sets with the property that all points in the set are birth times. The main result for death times is the dual of this and appears as the birth time theorem with the direction of time reversed.Part of this work was done while the author was visiting the Department of Mathematics, University of California at San DiegoThe support of The Danish Natural Science Research Council is gratefully acknowledged     

On Idempotent Ranks of Semigroups of Partial Transformations

A subset U of a semigroup S is a generating set for S if every element of S may be written as a finite product of elements of U. The rank of S is the size of a minimal generating set of S, and the idempotent rank of S is the size of a minimal generating set of S consisting of idempotents in S. A partition of a q-element subset of the set X_n={1,2,..., n} is said to be of type if the sizes of its classes form the partition of q n. A non-trivial partition of a positive integer q consists of k < q elements. For a non-trivial partition of q n, the semigroup S(), generated by all the transformations with kernels of type , is idempotent-generated. It is known that if is a non-trivial partition of n, that is, S() consists of total many-to-one transformations, then the rank and the idempotent rank of S() are both equal to max{ⁿ_d, N()}, where N() is the number of partitions of X_n of type . We extend this result to semigroups of partial transformations, and prove that if is a non-trivial partition of q < n, then the rank and the idempotent rank of S() are both equal to N().     

Clean triangulations

A polyhedron on a surface is called a clean triangulation if each face is a triangle and each triangle is a face. LetS _p (resp.N _p ) be the closed orientable (resp. nonorlentable) surface of genusp. If (S) is the smallest possible number of triangles in a clean triangulation ofS, the results are: (N ₁)=20, (S ₁)=24, lim(S _p )p ^–1=4, lim(N _p )p ^–1=2 forp.     

The Milnor–Totaro Theorem for Space Polygons

If denotes the curvature and the torsion of a closed, generic, and oriented polygonal space curve X in , then we show that _X (² + ²) ds = _X ds + _X | | ds > 4 if is positive. We also show that _X (² + ²) ds 2n if no four consecutive vertices lie in a plane and X has linking number n with a straight line. These extend theorems of Milnor and Totaro.     

Asymptotic Formulae and Divisor Problems

We investigate the asymptotic behaviour of the summatory functions of _z(n, ), _k(n, ) z ⁽ⁿ⁾ and _k(n, ) z ⁽ⁿ⁾.     

C r -finite elements of Powell-Sabin type on the three direction mesh

Let be the triangulation generated by a uniform three direction mesh of the plane. Let ₆ be the Powell-Sabin subtriangulation obtained by subdividing each triangleT by connecting each vertex to the midpoint of the opposite side.Given a smooth functionu, we construct a piecewise polynomial function C ^r (²) of degreen=2r (resp. 2r+1) forr odd (resp. even) in each triangle of ₆, interpolating derivatives ofu up to orderr at the vertices of.     

Das Zeitverhalten von einfachen offenen exponentiellen Warteschlangensystemen mit unendlich vielen Warteplätzen

Zusammenfassung Die zeitabhängige (instationäre) Lösung für die Zustandswahrscheinlichkeiten und für einige Kenngrößen von Warteschlangensystemen mit einer Bedienungsstation, unendlich vielen Warteplätzen, exponentiellem Zu- und Abgang und beliebigem Anfangszustand wird bestimmt. Die ZustandswahrscheinlichkeitenP _v (), d. h. die Wahrscheinlichkeiten für Einheiten im System zur Zeit, ergeben sich als Integrale, in denen modifizierteSessel-Funktionen 1. Art auftreten. Der ErwartungswertL () und die VarianzV() der Zahl von Einheiten im System lassen sich als Integrale darstellen, in denen nur die ZustandswahrscheinlichkeitP ₀() auftritt.Für<1 und erreichen die Systeme einen stationären Zustand (für den die Lösung bekannt ist); für1 und giltP _v ()0 für alle, L(),V().Ist>1, dann wachsenL() undV() für große linear mit; ihre Asymptoten werden berechnet. Ist=1, dann wachsenL() und die Standardabweichung() für große mit ; einfache Näherungsformeln werden gefunden.

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Summary The time dependent solution is determined for the state probabilities and for some characteristic values of queuing systems with a single server, an infinite number of waiting places, exponentially distributed inter-arrival and service times, and any initial state. The state probabilitiesP _v (), i.e. the probabilities for units in the system at time, are given in the form of integrals in which modifiedBessel functions of the first kind occur. Integrating the state probalityP ₀() over leads to the meanL() and the varianceV() of the number of units in the system.For<1 and the systems tend to a steady state (for which the solution is known); for1 and we haveP _v ()0 for all, L(),V().If>1 asymptotic expansions for large are found givingL() andV() proportional to. If=1 simple approximate formulas for large are obtained givingL() and the standard deviation() proportional to .

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Vorgel. v.:J. Nitsche.     

On a theorem of lovász on covers inr-partite hypergraphs

A theorem of Lovász asserts that (H)/*(H)r/2 for everyr-partite hypergraphH (where and * denote the covering number and fractional covering number respectively). Here it is shown that the same upper bound is valid for a more general class of hypergraphs: those which admit a partition (V ₁, ...,V _k ) of the vertex set and a partitionp ₁+...+p _k ofr such that |eV _i |p _i r/2 for every edgee and every 1ik. Moreover, strict inequality holds whenr>2, and in this form the bound is tight. The investigation of the ratio /* is extended to some other classes of hypergraphs, defined by conditions of similar flavour. Upper bounds on this ratio are obtained fork-colourable, stronglyk-colourable and (what we call)k-partitionable hypergraphs.Supported by grant HL28438 at MIPG, University of Pennsylvania, and by the fund for the promotion of research at the Technion.This author's research was supported by the fund for the promotion of research at the Technion.     

A fast ‘Monte-Carlo cross-validation’ procedure for large least squares problems with noisy data

Summary We propose a fast Monte-Carlo algorithm for calculating reliable estimates of the trace of the influence matrixA involved in regularization of linear equations or data smoothing problems, where is the regularization or smoothing parameter. This general algorithm is simply as follows: i) generaten pseudo-random valuesw ₁, ...,w _n , from the standard normal distribution (wheren is the number of data points) and letw=(w ₁, ...,w _n ) ^T , ii) compute the residual vectorw–A w, iii) take the normalized inner-product (w ^T (w–A w))/(w ^T w) as an approximation to (1/n)tr(I–A ). We show, both by theoretical bounds and by numerical simulations on some typical problems, that the expected relative precision of these estimates is very good whenn is large enough, and that they can be used in practice for the minimization with respect to of the well known Generalized Cross-Validation (GCV) function. This permits the use of the GCV method for choosing in any particular large-scale application, with only a similar amount of work as the standard residual method. Numerical applications of this procedure to optimal spline smoothing in one or two dimensions show its efficiency.     

Upper bound for transversals of tripartite hypergraphs

We prove 2 7/9v for 3-partite hypergraphs. (This is an improvement of the trivial bound 3v.)     

Über einen Homomorphiesatz von G.Köthe

The main object of this paper is to generalize a homomorphism theorem of Köthe [5] for a wide class of not necessarily locally convex topological vector spaces. A sequence =(B_n) of balanced sets B_n in a vector space E(), such that the union of the B_n's equals E and such that B_n+B_nB_n+1 for all nN, is called an absorbent sequence. E() is called -locally topological, if it possesses an absorbent sequence =(B_n) of bounded sets and if a linear mapping A from E() into any other vector space is continuous if all restrictions are continuous at 0. It is shown, that a continuous linear mapping A from a vector space E() with an absorbant sequence of compact sets into a boundedly summing -locally topological space F() is a homomorphism if A(E) is closed in F().     

On the Pseudospectra of Multidimensional Integral Operators with Homogeneous Kernels of Degree -n

We study the limit behavior of the spectral characteristics of truncated multidimensional integral operators whose kernels are homogeneous of degree –n and invariant under the rotation group SO(n). We prove that the limit of the -pseudospectra of the truncated operators K as 0 is equal to the union of the -pseudospectra of the original operator K and the transposed operator .     

Asymptotic Distribution of Extremes of Randomly Indexed Random Variables

Let {X _n} be a sequence of i.i.d. random variables and let {_k} be a sequence of random indexes. We study the problem of the existence of non-degenerated asymptotic distribution for min{X ₁,..., X _n}.     

A condition for the absence of the singular continuous spectrum in the Friedrichs model. II

One considers a self-adjoint operator H for which one has a unitary group U such that the operator H UHU ^–1 is analytic with respect to . Under certain additional restrictions on H, one proves the absence of the singular continuous spectrum of H. In this connection one admits such a behavior of the essential spectrum of H for Im 0 which excludes the application of the method of analytic dilatations. In our analysis, analogies with the method of the inverse scattering problem play an important role.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 127, pp. 3–6, 1983.     

Divisor Functions in the Ring of Gaussian Integers Weighted by the Generalized Klosterman Sum

Let _a,b,c () be the number of factorizations of a Gaussian number in the form = ₁ ^a ₂ ^b ₃ ^c , where a, b, and c are natural numbers. In the ring of Gaussian numbers, we construct an asymptotic formula for a summatory function of _a,b,c () weighted by the generalized Klosterman function.     

The asymptotic stability of one-parameter methods for neutral differential equations

This paper deals with the asymptotic stability of theoretical solutions and numerical methods for systems of neutral differential equationsx=Ax(t–)+Bx(t)+Cx(t–), whereA, B, andC are constant complexN ×N matrices, and >0. A necessary and sufficient condition such that the differential equations are asymptotically stable is derived. We also focus on the numerical stability properties of adaptations of one-parameter methods. Further, we investigate carefully the characterization of the stability region.