On the first eigenfunction of the fixed membrane: Some extensions of results of Payne and Stakgold

Zusammenfassung Mit Hilfe der zwei Maximumprinzipien von E. Hopf [3], [4] werden Schranken für grad² w als Funktion des Quotientenw=u/h konstruiert. Dabei istu die erste Eigenfunktion einer am Rande eingespannten schwingenden MembranD, undh ist eine positive Lösung der Helmholz'schen Gleichung inD. Der Spezialfallh 1 führt auf die Resultate von L. E. Payne und I. Stakgold [7] zurück.

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Résumé A l'aide des deux principes de maximum de E. Hopf [3], [4], on obtient des bornes pour grad² w en fonction du quotientw=u/h, oùu est la première fonction propre d'une membrane vibranteD fixée sur son contour, et oùh est une solution positive de l'équation de Helmholz dansD. Dans le cas particulierh 1, on retrouve les résultats de L. E. Payne et I. Stakgold [7].

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This research was supported by the Swiss Nationalfonds.     

Gruppenuniversalität und Homogenisierbarkeit

Summary For a classC of structures there are two archetypical questions: 1) Is every group G the full automorphism group of some C C? and 2) May every C C embedded into some homogeneous H C, i.e. into a structure H enjoying some transitivity properties? Using model theoretic language and conditions on the existence of certain free constructions withinC, some rather general positive answers to these questions are obtained. These abstract results give some (methodical) unification to a variety of theorems for some very concrete classesC of combinatorial, geometrical or algebraic nature in the literature. To underline this point, the major part of the paper is devoted to a systematic survey of classesC of structures in which our general approach is applicable, included many classesC for which our type of questions has not yet been considered.

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Unserem lieben FreundeAdriano Barlotti zum 60. Geburtstag gewidmet.

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Die Autoren danken herzlich der Deutschen Forschungsgemeinschaft für die gewährte Unterstützung; der erstgenannte Autor bedankt sich darüber hinaus beim Consiglio Nazionale delle Ricerche für ein Stipendium an der Universität Bologna.     

Gruppi ortogonali di indice di Witt minimale generati da tre involuzioni

In this paper, we prove that the orthogonal groups ^–(2l+2, q) and P^–(2l+2, q), l3, q odd, are generated by three involutions.

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Dedicato a A. Wagner in occasione del suo 60° compleanno

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Lavoro eseguito nell'ambito del finanziamento del M.U.R.S.T. e dei Gruppi Nazionali del C.N.R. (G.N.S.A.G.A.).     

A team decision problem for a convex loss function over aBernoulli random variable

Summary Suppose each member of a team observes aBernoulli random variable,x, and the team outcome is the inner product, a,x, of the observation vectorx and the decision vectora. This paper considers the case when the loss function is convex. Two numerical examples are given.

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Zusammenfassung Angenommen, jedes Mitglied eines Teams beobachtet eineBernoulli-verteilte Zufallsvariablex und das Team-Ergebnis besteht aus dem skalaren Produkt a,x des Beobachtungsvektorsx und des Entscheidungsvektorsa. Die vorliegende Arbeit behandelt den Fall, bei dem die Nutzenfunktion konvex ist. Zwei numerische Beispiele werden vorgeführt.

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A Marketing Team

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Research supported in part by Deutsche Forschungsgemeinschaft while the first author was a guest professor at the Institut für Angewandte Mathematik, Universität Heidelberg. Additional support came from Studiengruppe für Systemforschung, Heidelberg.     

Eine Bemerkung über absolute Ebenen

By separating the compatibility conditions for incidence, order and congruence the axioms for absolute planes in [3] are put in a somewhat more lucid form.

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Die absoluten Ebenen aus [3] entsprechen den Ebenen, die die ebenen Axiome der Verknüpfung, die Axiome der Anordung und die Axiome der Kongruenz aus D.HilbertsGrundlagen der Geometrie erfüllen.

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Werner Leiner zum 60. Geburtstag     

Extremal Almost-Kähler Metrics and Seiberg–Witten Theory

We show that there exist compact non-Kähler almost-Kähler4-manifolds whose metrics minimize L ²-norm of(2/3) s + 2w among all metrics compatible with a fixeddecomposition H ²(M, = H ⁺ H ^–, where s is the scalar curvature and w is the lowest eigenvalue of self-dual Weyl curvature at each point. In particular, the moduli space of such metrics modulo diffeomorphisms is infinite dimensional. This example also shows that LeBrun's estimate of L ²-norm of (1 – )s + · 6won a compact oriented Riemannian4-manifold with a nontrivial Seiberg–Witten invariant cannot beextended over = 1/3.     

Euklidische d-Simplexe mit inhaltsgleichen k-Seiten

We give the answer of a question by H. Lenz, wether for 3k<d the k-faces of ad-simplex must be pairwise congruent if they all have the samek-volume.

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Herrn Professor Hanfried Lenz gewidmet     

Über die Quadratwurzel-Schranke für Quadratische-Rest-Codes

The minimal distanced of any QR-Code of lengthn 3mod4 over a prime fieldGF (p) with p3 mod4 satisfies the improved square root bound d(3d-2)4(n–1).

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Helmut Karzel zum 60. Geburtstag gewidmet     

Kac–Moody groups and cluster algebras

Let Q be a finite quiver without oriented cycles, let Λ be the associated preprojective algebra, let g be the associated Kac–Moody Lie algebra with Weyl group W, and let n be the positive part of g. For each Weyl group element w, a subcategory Cw of mod(Λ) was introduced by Buan, Iyama, Reiten and Scott. It is known that Cw is a Frobenius category and that its stable category is a Calabi–Yau category of dimension two. We show that Cw yields a cluster algebra structure on the coordinate ring C[N(w)] of the unipotent group N(w):=N∩(w−1N₋w). Here N is the pro-unipotent pro-group with Lie algebra the completion of n. One can identify C[N(w)] with a subalgebra of , the graded dual of the universal enveloping algebra U(n) of n. Let S^? be the dual of Lusztig?s semicanonical basis S of U(n). We show that all cluster monomials of C[N(w)] belong to S^?, and that S^?∩C[N(w)] is a C-basis of C[N(w)]. Moreover, we show that the cluster algebra obtained from C[N(w)] by formally inverting the generators of the coefficient ring is isomorphic to the algebra C[Nw] of regular functions on the unipotent cell Nw of the Kac–Moody group with Lie algebra g. We obtain a corresponding dual semicanonical basis of C[Nw]. As one application we obtain a basis for each acyclic cluster algebra, which contains all cluster monomials in a natural way.     

On the diameter of planar curves and Fourier coefficients

Summary LetF denote a complex valued function on , which is absolutely continuous and periodic with a period 2. Its complex Fourier coefficientsc _n satisfy the inequality 2|c _n |d,n0, whered denotes the diameter of the closed planar curve defined byF. In this paper those functionsF are considered, which give equality for a specificn.

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Zusammenfassung Es seiF eine komplexwertige Funktion auf , die absolut stetig und periodisch ist mit einer Periode 2. Sie definiert eine geschlossene ebene Kurve. Zwischen ihrem Durchmesserd und den komplexen Fourierkoeffizientenc _n besteht die Ungleichung 2|c _n |d fürn0. Es werden jene FunktionenF betrachtet, für die in dieser Ungleichung, für ein gegebenesn, Gleichheit besteht.

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Dedicated to my colleague for many years, Professor Eduard Stiefel

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This research was supported in part by the National Research Council of Canada A-7339     

Characterization of weakly efficient points

Weakly efficient points of a mappingF: S Y are characterized, where the feasible setS is given by infinitely many constraints, andY is equipped with an arbitrary convex ordering. In the linear and in the convex case a necessary and sufficient condition is given, which needs no constraint qualification.

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Zusammenfassung Es werden schwach effiziente Punkte einer AbbildungF: S Y charakterisiert, wobei der zulässige BereichS durch unendlich viele Restriktionen bestimmt wird undY mit einem beliebigen konvexen Ordnungskegel versehen ist. Im linearen und im konvexen Fall wird eine notwendige und hinreichende Bedingung angegeben, die keine Regularitätsvoraussetzung benötigt.

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Rudolf Henn zum 65. Geburtstag gewidmet     

Fault-tolerant routing in circulant networks and cycle prefix networks

Reliability and efficiency are important criteria in the design of interconnection networks. Connectivity is a widely used measurement for network fault-tolerance capacities, while diameter determines routing efficiency along individual paths. In practice, we are interested in high-connectivity, small-diameter networks. Recently, Hsu introduced the notion ofw-wide diameter, which unifies diameter and connectivity. This paper investigates thew-wide diameterd _w (G) and two related parameters:w-fault diameterD _w (G) andw-Rabin numberr _w (G). In particular, we determined _w (G) andD _w (G) for 2wK(G) andG is a circulant digraphG(d ⁿ ; 1,d,...,d ^n–1) or a cycle prefix digraph.Supported in part by the National Science Council under grant NSC86-2115-M009-002.     

Über eine Klasse von Brezelknoten

The paper gives a classification of pretzel knots (p,qq,rr) with one even cross number. The Alexander polynomial is computed in general and it is proved that its degree is always twice the genus of the knot. The classification does not include amphicheirality.

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Mit 3 Abbildungen

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Diese Resultate sind in meiner Dissertation Die Längskreisinvariante und Brezelknoten, Frankfurt a. M. 1978, enthalten.     

Über lineare Unabhängigkeit

It is the aim of the present work to prove, under appropriate conditions, lower estimates for the dimension of w ₁ + ... + w _m over , wherew ₁,...,w _m are given real numbers. In particular, if this dimension ism, i.e. ifw ₁,...,w _m are linearly independent over , we are also interested in a quantitative version of this fact. Our qualitative theorems generalize a result of Nesterenko. Its formulation is quite similar to the axiomatization of methods for algebraic independence, as it became usual during the last decade.

     

The Measure of Noncompactness of Linear Operators between Spaces of Strongly C 1 Summable and Bounded Sequences

We give necessary and sufficient conditions for infinite matrices to map a sequence space X into a sequence space Y where X = l ₁ and Y = w ^p , w ^p , w ₀ ^p (1 p < ), or X = w ₀, w, w and Y = l _p (1 p ), or X = w ₀, w, w and Y = w ₀ ^p , w ^p , w ^p (1 p < ). Furthermore the Hausdorff measure of noncompactness is applied to give necessary and sufficient conditions for a linear operator between these spaces to be compact.     

Comparison of duality models in fractional linear programming

Summary This paper deals with the duality models in fractional linear programming presented in the last years bySwarup, Kaka, Sharma andSwarup and other authors.

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Zusammenfassung Der Aufsatz befaßt sich mit Dualitätsmodellen für Linear Fractional Programming, die in den letzten Jahren vonSwarup, Kaka, Sharma undSwarup sowie von anderen Autoren angegeben wurden.

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This work was sponsored by the Grant No. A 7329 from the National Research Council of Canada.     

Omogeneizzazione stocastica di funzionali non coercivi

Summary We give a -convergence theorem, for sequences of non coercive integral functionals associated to a stochastic homogenization process. We make use of recent results in Ergodic Theory.

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Lavoro eseguito nell'ambito del G.N.A.F.A. del C.N.R.     

Alcune caratterizzazioni della γ-convergenza multipla

Summary The main features considered in this paper are characterisations of multiple -convergence in the saddle function case and in the general case. The main results for saddle case are theorems II.1.1 II.1.2 II.2.1 III.3.3 III.3.4and for general case theorem IV.1.2. In the fifth section we introduce vF-convergence and give equivalence results between vF.

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Lavoro eseguito nell'ambito del G.N.A.F.A. del C.N.R.     

Eine charakterisierung der Permutationsgruppe PSL(2, K) über einem quadratisch abgeschlossenen Körper K der Charakteristik ≠2

This paper gives a characterization of the permutation group PSL(2, K) over a quadratic closed field of characteristic 2.

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Herrn Professor Dr. H. Salzmann zum 60. Geburtstag gewidmet.     

Berechnung von Beulwerten mit Hilfe von Mehrstellenoperatoren

Summary Buckling of a plate leads to an eigenvalue problem for which a very accurate finite difference approximation is given. The resulting algebraic eigenvalue problemAx= Bx is solved for specific problems by an improved iteration method. Numerical results are discussed.

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Die vorliegende Arbeit wurde ermöglicht durch die Schweizerische Volkswirtschaftsstiftung. Die Verfasser sind dieser Institution zu bestem Dank verpflichtet.