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.     

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     

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).     

Blow-up rates for parabolic systems

Let ⁿ be a bounded domain andB _R be a ball in ⁿ of radiusR. We consider two parabolic systems: ut=u +f(), i= +g(u) in × (0,T) withu=v=0 on × (0,T) andu _t =u, v _t =v inB _r × (0,T) withe/v=f (v), e/v=g(u) onB _R × (0,T). Whenf(v) andg(u) are power law or exponential functions, we establish estimates on the blow-up rates for nonnegative solutions of the systems.     

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⁾.     

Links between the Slater Index and the Ryser Index of Tournaments

Given a tournament T, the Slater index i(T) of T is the minimum number of arcs that must be reversed to make T transitive. The Ryser index (T) of T, defined from the out-degrees of T, measures a remoteness between T and the transitive tournaments of same order. In this paper, we study some links between i(T) and (T). More precisely, calling I(n, ) the maximum value of i(T) over the set of tournaments on n vertices and such that (T)=, we compute an upper bound of I(n,) for every value of . Then we use this upper bound to study a conjecture stated by J.-C. Bermond on the regularity (i.e., the fact that all the out-degrees are equal or almost equal) of the tournaments with a maximum Slater index by showing that the out-degrees of such tournaments cannot be too far from the ones of the regular tournaments.     

Reconstruction of a wiener field from its realizations on a curve

Let w(x, y), x 0 and y 0 be a Wiener field on the plane; be a curve given parametrically x=x() and y=y(), [0, 1], where x() is a positive, continuous, nondecreasing function; y() is a positive, continuous, nonincreasing function. A best estimate in the mean-square sense is constructed for w(u, v)(u, v) , based on the values w(x, y), (x, y) and its error is found.Translated from Teoriya Sluchainykh Protsessov, No. 16, pp. 87–93, 1988.     

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     

Thermorelaxation resonance in the quantum theory of Brownian motion

The Brownian motion of a quantum particle in a thermal reservoir possessing a finite correlation time _c is considered. Non-Markov Langevin equations for a stationary nonequilibrium state are obtained. At low temperaturesT of the thermal reservoir, the correlation time _c =/2T is fairly long. It is shown that allowance for the damping of the particle momentum over the correlation times _c : _c 1, leads to an oscillating temperature dependence of the relaxation coefficient (1/T) in the region of low temperatures of the thermal reservoir.N. I. Lobachevski Nizhegorod State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 92, No. 1, pp. 119–126, July, 1992.     

τ-Pseudocompact Mappings

We consider the problem of extending the notion of -pseudocompactness from spaces to continuous mappings, obtain conditions under which the product of -pseudocompact mappings is -pseudocompact. Since any space X can be considered as a continuous mapping from X into a singleton, we obtain consequences of the theorems on multiplicativity of -pseudocompactness for spaces. Thus, we study the notion of -pseudocompact mapping and some its properties similar to those of a pseudocompact space as well as consequences of the obtained assertions for spaces.     

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.)     

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.     

Structures of topological type

In this paper we continue the study of structures of various types initiated by the author in the earlier paper Structures of extensions (Ref. Zh. Mat., 1974, 4A361). The present paper is devoted to the so-called structure of topological type. By a structure of topological type on the set X is meant a topological structure, defined on some set obtained from X, and possibly additional sets, by a totally ordered sequence of operations of unions of sets, products of sets, and passage to the set of subsets. We study certain structures of topological type: bitopological (Sec. 2) and settopological (Sec. 3). A bitopological structure on the set X is any topological structure on the set X×X, and a bitopological space is a pair (X,). This concept is a natural extension of the concept of a bitopological space as a set X on which there are given two topological structures ₁ and ₂-these structures define a structure =₁×₂ on the set X×X. A settopological structure on the set X is any topological structure on the set={A¦A. There are given representations of piecewise-linear structures (Sec. 4) and smooth structures (Sec. 5) as settopological structures.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 83, pp. 5–62, 1979.     

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.     

Torsion tensor and critical metrics on contact (2n+1)-manifolds

Contact Riemannian manifolds (M, ,g) satisfying the condition (1) =0, where is the torsion introduced byChern andHamilton [6] and is the characteristic vector field, have interesting geometric properties (see [6], [9], [11]). In this paper we give a variational characterization of compact contact Riemannian manifolds which satisfy (1). Moreover we study the tangent sphere bundles (T ₁ M, , g), where (,g) is the standard contact Riemannian structure, which satisfy the condition (1); in particular in the 3-dimensional case we find a surprising result (see Corollary 5.3).Supported by funds of the M.U.R.S.T.     

Random time change and an integral representation for marked stopping times

Summary Consider the set of all possible distributions of triples (, , ), such that is a finite stopping time with associated mark in some fixed Polish space, while is the compensator random measure of (, ). We prove that is convex, and that the extreme points of are the distributions obtained when the underlying filtration is the one induced by (, ). Moreover, every element of has a corresponding unique integral representation. The proof is based on the peculiar fact that EV _, =0 for every predictable processV which satisfies a certain moment condition. From this it also follows thatT _, isU(0, 1) wheneverT is a predictable mapping into [0, 1] such that the image of , a suitably discounted version of , is a.s. bounded by Lebesgue measure. Iterating this, one gets a time change reduction of any simple point process to Poisson, without the usual condition of quasileftcontinuity. The paper also contains a very general version of the Knight-Meyer multivariate time change theorem.Research supported by NSF grant DMS-8703804     

Über eine Verallgemeinerung des Plateauschen Problems

Let T be the domain in ^N defined by the inequalities O < ₁ < ... < _N < +. Put _N+k = /2(1+k) (k=1,2,3), _N+4=₁+2, and denote byF() the set of functions x=x(u,v)=(x¹(u,v),...,x^p(u,v)), (p2) of class , where B is the unit disk u²+v²<1, which maps the circular arcs _k={w=eⁱ:_k<<_K+1} (k=1,..., N+3) into the straight lines containing the edges a_k, a_k+1 (a_N+4=a₁) of a polygon IR^p. Then we show that the function ()= _inf ^xF() D(x) is analytic in T. This generalizes and sharpens an unproved result of I. Marx and M. Shiffman (see [4]).

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Hans Lewy und Charles B. Morrey gewidmet     

On the torsion theories of Morita equivalent rings

We generalize the well-known fact that for a pair of Morita equivalent ringsR andS their maximal rings of quotients are again Morita equivalent: If _n (M) denotes the torsion theory cogenerated by the direct sum of the firstn+1 injective modules forming part of the minimal injective resolution ofM then _n (R)= _n (S) where is the category equivalenceR-ModS-Mod. Consequently the localized ringsR _{n (R)} andS n _(S) are Morita equivalent.     

Parameter-dependent renewal theorems with applications

Let ₁, ₂, ... be a sequence of i.i.d. random variables with positive mean and finite variance and letr(b), b0, be real numbers tending to 0 asb . Definings _n=₁+...+_n andS _n=S_n(b)=s_n+r(b)n, the stopping time =(b)=inf {n>/1:S_n >b} whereb=b(b) , will be considered with special regard to the excess over the boundaryR _b=s+r(b)–b. It turns out that the limiting distribution ofR _b is the same as in the caser(b)0 for allb. Proving this, Blackwell's renewal theorem and its integral version have to be established first in the above stated situation. Finally, an expansion ofE to vanishing terms asb will be provided and applied to some examples arising in economics.

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Zusammenfassung Seien ₁, ₂, ... unabhängige identisch verteilte Zufallsgrößen mit positivem Erwartungswert und endlicher Varianz sowier(b), b0, reelle Zahlen mitr(b)0 für b. Sei ferners ₁, s₂, ... der zugehörige Summenprozeß,S _n= S_n(b)=s_n+r(b)n fürn1 und =(b)=inf {n1: S_n>b, wobeib=b(b) fürb . Es wird gezeigt, daß die asymptotische Verteilung des ExzessesR _b=s +r(b) –b mit der im Fallr(·)0 übereinstimmt. Dazu werden sowohl das Blackwellsche Erneuerungstheorem als auch seine Integralversion in der vorher beschriebenen parameterabhängigen Situation geeignet formuliert und bewiesen. Als Folgerung ergibt sich dann eine asymptotische Entwicklung vonE(b) fürb bis zu Termen o(1). Anh- and einiger Beispiele aus dem ökonomischen Bereich wird schließlich noch aufgezeigt, wo Approximationen fürE(b) von Interesse sein können.

     

Modular Transformations of Ramanujan's Fifth and Seventh Order Mock Theta Functions

In his last letter to Hardy, Ramanujan defined 17 functions F(q), where |q| < 1. He called them mock theta functions, because as q radially approaches any point e ^2ir (r rational), there is a theta function F _r(q) with F(q) – F _r(q) = O(1). In this paper we obtain the transformations of Ramanujan's fifth and seventh order mock theta functions under the modular group generators + 1 and –1/, where q = e ⁱ. The transformation formulas are more complex than those of ordinary theta functions. A definition of the order of a mock theta function is also given.