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.     

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.     

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).     

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     

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⁾.     

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.     

On exact D-optimal designs for regression models with correlated observations

Let ^* be an exact D-optimal design for a given regression model Y = X + Z . In this paper sufficient conditions are given for sesigning how the covariance matrix of Z may be changed so that not only ^* remains D-optimal but also that the best linear unbiased estimator (BLUE) of stays fixed for the design ^*, although the covariance matrix of Z * is changed. Hence under these conditions a best, according to D-optimality, BLUE of is known for the model with the changed covariance matrix. The results may also be considered as determination of exact D-optimal designs for regression models with special correlated observations where the covariance matrices are not fully known. Various examples are given, especially for regression with intercept term, polynomial regression, and straight-line regression. A real example in electrocardiography is treated shortly.     

Tangential Limits of Potentials on Homogeneous Trees

Let T be a homogeneous tree of homogeneity q+1. Let denote the boundary of T, consisting of all infinite geodesics b=[b ₀,b ₁,b ₂,] beginning at the root, 0. For each b, 1, and a0 we define the approach region _,a (b) to be the set of all vertices t such that, for some j, t is a descendant of b _j and the geodesic distance of t to b _j is at most (–1)j+a. If >1, we view these as tangential approach regions to b with degree of tangency . We consider potentials Gf on T for which the Riesz mass f satisfies the growth condition _T f ^p (t)q ^–|t|<, where p>1 and 0<<1, or p=1 and 0<1. For 11/, we show that Gf(s) has limit zero as s approaches a boundary point b within _,a (b) except for a subset E of of -dimensional Hausdorff measure 0, where H (E)=sup_>0inf _i q ^{–|t i|}:E a subset of the boundary points passing through t _i for some i,|t _i |>log _q (1/).     

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.     

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.     

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     

Noetherian Equations for Matrices

Let L|K be a finite Galois extension. Using central simple algebras we deal with the crossed representations of G = Gal(L|K) over L which are defined as mappings X of G into GL_n(L) satisfying X = X X. The last equation is the Noetherian equation in case n=1. Furtheron, more general crossed projective representations are considered which obey an equation X X = Xf_, where f_, L.     

Opérateurs de mise en mémoire et traduction de Gödel

Résumé En-calcul, la stratégie de réduction à gauche (appel par nom) a, comme on sait, de bonnes propriétés mathématiques; en particulier, elle termine toujours si on l'applique à un terme normalisable. Mais, avec cette stratégie, l'argument d'une fonction est recalculé à chaque utilisation.Pour éviter ce défaut, on définit la notion «d'opérateur de mise en mémoire» (pour un type de données). SiT est un opérateur de mise en mémoire, pour les entiers par exemple, on remplace l'évaluation, par réduction gauche, de (où est un entier et un -terme quelconque) par celle deT; et celle-ci revient à ramener d'abord à une forme réduite ₀, puis à appliquer à ₀. On a donc ainsi simulé «l'appel par valeur» dans la stratégie de réduction à gauche.Le théorème principal (Corollaire du Théorème 4.1) montre que, dans un 1-calcul typé du second ordre, en utilisant la traduction de Gödel de la logique classique en logique intuitionniste, on peut trouver un type (spécification) très simple pour les opérateurs de mise en mémoire. Il donne donc aussi un moyen d'obtenir ces opérateurs, à savoir de démontrer ce type dans le calcul des prédicats intuitionniste du second ordre.

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In-calculus, the strategy of leftmost reduction (call-by-name) is known to have good mathematical properties; in particular, it always terminates when applied to a normalizable term. On the other hand, with this strategy, the argument of a function is re-evaluated at each time it is used.To avoid this drawback, we define the notion of storage operator, for each data type. IfT is a storage operator for integers, for example, let us replace the evaluation, by leftmost reduction, of (where is an integer, and any-term) by the evaluation oft. Then, this computation is the same as the following: first compute up to some reduced form ₀, and then apply to ₀. So, we have simulated call-by-value evaluation within the strategy of leftmost reduction.The main theorem of the paper (Corollary of Theorem 4.1) shows that, in a second order-calculus, using Gödel's translation of classical intuitionistic logic, we can find a very simple type (or specification) for storage operators. Thus, it gives a way to get such operators, which is to prove this type in second order intuitionistic predicate calculus.

     

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.     

Complemented Topologies on Abelian Groups

A topology on a group G is complemented if there exists an indiscrete topology ' on G such that UV={0} for suitable neighborhoods of zero U and V in the topologies and . The authors give a complementation test for an arbitrary topology. Locally compact groups with complemented topologies have been described. A group all of whose continuous homomorphic images are complete is proved to be compact. A family of 2 topologies that are pairwise complementary to one another is defined for an arbitrary group.     

Finite Extensions of Topogenities

Let X = Y Z, Y Z = Ø, < be a topogenity on Y, a topology on X. A (<, )-extension is a topogenity < on X such that < ¦Y = <, (<) = . We establish some properties of (<, )-extensions and construct all of them in the case of a finite Z.     

Folgenräume als universelle Generatoren für Klassen lokalkonvexer Räume

Motivated by the known characterizations of equicontinuity in the dual of a Schwartz space, a nuclear space, or a strongly nuclear space,we introduce the concepts of a -sequence and of a ()-sequence in the dual of an arbitrary lcs [E,], and we investigate the corresponding topologies and ₍₎ on E of uniform convergence on these sequences. Here is a normal sequence space such that . Under favorable enough conditions on , including the nuclearity of its normal topology , [,] acts as a universal generator for those lcs [E,] which satisfy =. Under somewhat weaker assumptions on , [,₍₎] is a universal generator for the lcs [E,] with =₍₎. These results cover e.g. the cases of -nuclear spaces and of nuclear spaces known from the recent literature. As an application we show that every non-trivial ultrabornological lcs is representable as an inductive limit of isomorphic copies of [, ( , )], where is any nuclear power series space of infinite type with stable exponent sequence.     

Analogues of hadwiger's theorem in space ℝ n —Sufficient conditions for a convex domain to enclose another

We estimate the kinematic measure of one convex domain moving to another under the groupG of rigid motions in ⁿ . We first estimate the kinematic formula for the total scalar curvature D ₀gD ₁ Rdv of then–2 dimensional intersection submanifold D ₀gD ₁. Then we use Chern and Yen's kinematic fundamental formula and our integral inequality to obtain a sufficient condition for one convex domain to contain another in ⁿ (4). Forn=4, we directly obtain another sufficient condition in ⁴.     

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.