On the two-gap elliptic potentials

The-parametrized family of two-gap elliptic potentials is constructed so that (i) 0<<1, (ii) for rational values of such potentials are elliptic (i.e., double-periodic), (iii) within the limit0 this family degenerates to the soliton potential, (iv) within the limit1 this family degenerates to the one-gap Lamé potential.Dedicated to the memory of J.-L. Verdier     

Weak convergence of probability measures relative to incompatible topology and σ-field with applications to renewal theory

Summary Necessary and sufficient conditions are given in order that a sequence of probability measures, weakly convergent relative to a given topology ₀ and associated -field ( ₀), are weakly convergent (and satisfy a continuity theorem) relative to the ( ₀)-measurable functions which are continuous in some finer topology ₁, even if does not extend to ( ₀). These conditions are shown to be applicable to a sequence of translated renewal measures. Alternate conditions (tightness, uniformity of weak convergence) are investigated and shown to be inappropriate.This research was partially supported by UMC Summer Faculty Research Fellowships     

Complementary permutations for abelian groups

Summary LetG be an additively written abelian group and leth: G G be a given function. M. Hall Jr. (1952) and L. Fuchs (1958) already answered the following question. For what functionsh: G G does the functional equation(x) + (x) = h(x) (x G) have as its solution a pair of permutations and ofG? In this paper, we give explicit constructions of such a pair, in a number of cases, in particular whenh(x) x andG is finite. We further determine the finite groupsG where the latter, can be chosen to be automorphisms.In the case whereG is an infinite topological group, we study in how far and can be chosen as Borel measurable permutations, given thath: G G itself is Borel measurable.     

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.     

Generalization of the additive problem of divisors

An asymptotic formula is obtained for a sum of the form _{m f (m) (n–m) where f(m) is a multiplicative function with additional constraints, and (m) is the number of divisors of m.Translated from Matematicheskie Zametki, Vol. 7, No. 4, pp. 477–482, April, 1970.}     

On the approximation of plane curves by parametric cubic splines

A plane curveC can be approximated by a parametric cubic spline as follows. Points (x _i ,y _i ) are chosen in order alongC and a monotonically increasing variable is assigned values _i at the points (x _i ,y _i ): _i = the cumulative chordal distance from (x ₁ ,y ₁ ). The points ( _i ,x _i ) and ( _i ,y _i ) are then fitted separately by cubic splinesx() andy(), to obtain : (x(),y()). This paper establishes estimates for the errors involved in approximatingC by . It is found that the error in position betweenC and decreases likeh ³, whereh is the maximum length of arc between consecutive knots onC. For first derivatives, the error behaves likeh ²; for second derivatives, likeh.     

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.     

Hypercubic 4 and 5-Designs from Double-Error-Correcting BCH Codes

An ordered orthogonal array OOA(, k, n) is a binary 2 ^k × n matrix with the property that for each complete -set of columns, each possible -tuple occurs in exactly 2 ^k– rows of those columns (for definition of a complete -set, see below). Constructions of OOA(, k, n) for = 4 and = 5 are given.     

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⁾.     

Automorphisms on a Lie group contracting modulo a compact subgroup and applications to semistable convolution semigroups

Let ( _t ) _t0 be a -semistable convolution semigroup of probability measures on a Lie groupG whose idempotent ₀ is the Haar measure on some compact subgroupK. Then all the measures ₁ are supported by theK-contraction groupC _K() of the topological automorphism ofG. We prove here the structure theoremC _K()=C()K, whereC() is the contraction group of . Then it turns out that it is sufficient to study semistable convolution semigroups on simply connected nilpotent Lie groups that have Lie algebras with a positive graduation.     

Stefan problem with a kinetic and the classical conditions at the free boundary

The Stefan problem is considered with the kinetic condition u⁺=u^–=k(y, )-v at the phase interface, where k(y, ) is the half-sum of the principal curvatures of the free boundary and v is the speed of its shifting in the direction of a normal. The solvability of a modified Stefan problem in spaces of smooth functions and the convergence of its solutions as 0 to a solution of the classical Stefan problem are proved.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 2, pp. 155–166, February, 1992.     

Über den algorithmischen Nachweis von Ungleichungen I

We shall develop a method to prove inequalities in a unified manner. The idea is as follows: It is quite often possible to find a continuous functional : ⁿ , such that the left- and the right-hand side of a given inequality can be written in the form (u)(v) for suitable points,v=v(u). If one now constructs a map ^{n n} , which is functional increasing (i.e. for each x ⁿ (which is not a fixed point of ) the inequality (x)<((x)) should hold) one specially gets the chain (u)( u))( ²(u))... ⁿ (u)). Under quite general conditions one finds that the sequence { ⁿ (u)} _n converges tov=v(u). As a consequence one obtains the inequality (u)(v).     

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 the asymptotic behaviour of first passage times for transient random walk

Summary Let _x denote the time at which a random walk with finite positive mean first passes into (x, ), wherex0. This paper establishes the asymptotic behaviour of Pr { _x >n} asn for fixedx in two cases. In the first case the left hand tail of the step-distribution is regularly varying, and in the second the step-distribution satisfies a one-sided Cramér type condition. As a corollary, it follows that in the first case Pr { _x >n}/Pr{ ₀ >n} coincides with the limit of the same quantity for recurrent random walk satisfying Spitzer's condition, but in the second case the limit is more complicated.     

The transposition of locally compact,connected translation planes

This note deals with the transposition of translation planes in the topological context. We show that a topological congruenceC of the real vector space ²ⁿ has the property that every hyperplane of ²ⁿ contains a component ofC. This makes it possible to define the transposeP of the topological translation planeP associated withC; it is proved that the translation planeP is topological also. The relationship between collineation groups and the relationship between coordinatizing quasifields ofP andP are also discussed.     

On the concentration of stresses due to a small elliptic inclusion on the neutral axis of a deep beam under constant bending moment

Zusammenfassung Es wird die Spannungsverteilung untersucht, die sich in einem breiten Balken mit konstanter Höhe unter einem konstanten Biegemoment ausbildet, wenn er eine kleine elliptische Einschliessung mit Zentrum auf der Neutralachse enthält. Insbesondere werden die Fälle eines sehr starren Einschlusses sowie eines elliptischen Loches im Detail diskutiert.

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Nomenclature x, y Cartesian coordinates - , elliptic coordinates - u, v (u ,u )=components of displacements - , unit elongations in -and -directions - shearing strain - , normal stress components in elliptical coordinates - shearing stress in elliptic coordinates - _x , _y normal stress in Cartesian coordinates - _xy shearing stress in Cartesian coordinates - E Young's modulus for the beam - v Poisson's ratio for the beam - 1/h ₁, 1/h ₂ stretch ratios - e _x + _y dilatation - 2 rotation - M bending moment     

Quadratic-Cycloidal Curves

Interesting curves can be represented in the space Q=span{1,t,t ²,cost,sint}, t[0,] (0<<2). In this paper we introduce quadratic-cycloidal B-splines associated to equally spaced knots and the properties of the generated curves for 0<<. It is proved that, when 0, the limit of a QC-B-spline curve approaches a B-spline curve of degree 4.     

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     

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.     

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