Zur Fortsetzbarkeit von Ordnungsfunktionen einer affinen Ebene

We show that every order function of a translation plane with kernel GF(2), GF(3) can he extended to an order function of the projective closure.

---

---

Herrn Professor Dr. Hanfried Lenz zum 70. Geburtstag     

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).

---

---

Helmut Karzel zum 60. Geburtstag gewidmet     

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.

---

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.

---

---

Dedicated to my colleague for many years, Professor Eduard Stiefel

---

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

Dualisierung der Hauptkurven in allgemeinen Cayley/Klein-Räumen

Dualizing the osculating hyperplanes of a curvek of an n-dimensional Cayley/Klein space (CK space) according to the duality principle of projective spaces yields a curve of the dual CK space, the dual curvek _d ofk. A point ofk with torsion 0 corresponds to a stationary point ofk _d . The dualization of the osculating hyperplanes ofk _d yieldsk. There are simple relations between the moving simplexes, the CK arc lengths and the CK curvatures of the two curves which are valid for all CK spaces, even for those which are not self-dual.

---

---

Herrn Prof. Dr.Johannes Böhm zum 70. Geburtstag gewidmet     

Über den Typenkonvergenzsatz auf zusammenhängenden Lie Gruppen

The Convergence of Types Theorem on ^d is wellknown as an important tool for investigations on the limit behaviour of normalized sums or r.v. It is natural to look for a generalization for group-valued r.v. While for simply connected nilpotent Lie groups the Theorem is valid in general the existence of non-trivial compact subgroups causes problems. For compact extensions of nilpotent groups we prove restricted versions of the Convergence of Types Theorem.

---

---

Herrn Prof. Dr. L. Schmetterer zum 70. Geburtstag gewidmet     

Winkelmessung und Orientierung

In an affine plane oriented distance d and oriented angle w (between non-oriented lines) are introduced as mapping into , fulfilling certain axioms. Examples show, that the use of w is of advantage in elementary geometry. Axiomatic consequences of generalizing d, w to mappings into ordered abelian groups are considered.

---

---

Herrn Helmut Salzmann zum 60. Geburtstag     

Über Distanzfunktionen mit Werten in angeordneten Halbgruppen

This paper presents a general investigation of the relations between structural properties of a totally ordered abelian semigroupS and the properties of various topological structures, such as topologies, bitopologies and (semi-)uniformities on a spaceX induced byS-valued distance functionsdX×XS satisfyingd(x,y)=0 iffx=y and the triangular inequalityd(x,z)d(x,y)+d(y,z), for allx,y,zX. Since a linearly ordered abelian semigroupS need not be a topological semigroup with respect to its order topology we have to consider two cases: the case where addition inS is continuous at 0S, and the case where it is not. For both cases, we state several metrization theorems, examples and applications. In this connection, we are also concerned with some special basis-properties of topological spaces. Closely connected is the program stated byAlexandroff-Bourbaki (amongst others) to investigate to what extent countability inherent in matrization theory can be replaced by order-theoretic properties.—Distinguishing between symmetric and not necessarily symmetric distancesd _S we obtain a theory containing the theory of _µ -metrics and _µ- quasimetrics.—As far as it concerns not necessarily symmetric distancesd onX, it seems adequate to study the bitopological structure ( _e , _r ) induced onX byd and the inverse distanced ^–1 respectively. This is done in § 4 where, in this respect, we also generalize a well-known theorem ofSion andZelmer.

---

---

Herrn Prof. Dr. E. Hlawka zum 60. Geburtstag gewidmet

---

Die ersten Anregungen zur grundsätzlichen Problemstellung erhielt der erste Autor in einem Seminar über Bewertungstheorie und Analysis in bewerteten Körpern, das unter der Leitung Herrn Prof. Dr.E. Hlawkas am Mathematischen Institut der Universität Wien abgehalten wurde.     

Temps locaux d'intersection de mouvements browniens perturbés et applications

Résumé On montre l'existence des temps locaux d'intersection pour les semimartingales continues à valeurs dans ^d dont la partie martingale est brownienne (d3). Pourd=2, on obtient une formule de Tanaka et un résultat de renormalisation du type de celui de Varadhan.

---

Summary Continuous semimartingales in ^d which martingale part is a brownian motion are shown to admit self-intersection local times ford3. Whend=2, we obtain a Tanaka formula and a Varadhan type renormalization.

     

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.

---

---

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.

---

Werner Leiner zum 60. Geburtstag     

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.

---

---

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

---

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.).     

Generalized Cheeger sets related to landslides

We study the maximization problem, among all subsets X of a given domain , of the quotient of the integral in X of a given function f by the integral on the boundary of X of another function g. This is a generalization of the well-known Cheeger problem corresponding to constant functions f,g. The non-constant case is motivated by applications to landslides modeling where the the supremum given by a variational blocking problem appears as a safety coefficient. We prove that this coefficient is equal to the supremum of the shape optimization problem formerly mentioned. For constant data, this amounts to studying the first eigenvalue of the 1-laplacian operator.We prove existence of optimal sets, and give some differential characterization of their internal boundary. We study their symmetry properties using the Steiner symmetrization. In dimension two, we give explicit solutions for data depending only on one variable.

---

Résumé. Nous étudions le probléme de maximisation, parmi les ensembles X inclus dans un domaine fixé , du quotient de lintégrale dune fonction donnée f dans X par lintégrale dune autre fonction g sur le bord de X. Il sagit donc dune généralisation du célébre probléme de Cheeger (correspondant au cas f, g, constants). Le cas non-constant est motivé par des applications aux glissements de terrain, oú le supremum donné par un probléme variationnel de blocage, apparaít comme un coefficient de sreté. Nous démontrons que ce coefficient est égal á loptimum du probléme doptimisation de formes mentionné précédemment. Dans le cas de données constantes, cela revient á étudier la premiére valeur propre de lopérateur 1-laplacien.Nous démontrons lexistence densembles optimaux, et donnons une caractérisation différentielle de leur bord intérieur. Nous étudions leur symétrie á laide de la symétrisation de Steiner. En dimension deux, nous exhibons des solutions explicites dans le cas oú les données ne dépendent que dune variable.

---

---

Received: 18 June 2004, Accepted: 12 July 2004, Published online: 10 December 2004Mathematics Subject Classification (2000): 49J40, 49Q10     

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.

---

---

Mit 3 Abbildungen

---

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

Sur le comportement asymptotique des potentiels d'une chaîne de Markov sur ℝ d

Résumé Dans cet article j'étudie le comportement à l'infini des potentiels des chaînes de Markov sur ^d (d3) proches du mouvement brownien, tout spécialement le cas des marches aléatoires, ainsi que des critères de transience et de récurrence inspirés de la méthode utilisée.

---

We study the asymptotic behaviour of potentials of Markov chains on ^d (d3), closed to Brownian motion, and particularly the case of random walks. Following a similar approach, we give transience and recurrence criteria.

     

Un phénomène de concentration de genre

We show that the main part of the genus of the preimages of a projective line by a generic holomorphic endomorphism of goes to the support of the Green measure.

---

Résumé Nous montrons que le genre des préimages dune droite projective par un endomorphisme holomorphe générique de se concentre sur le support de la mesure de Green.A phenomenon of concentration of genus

---

---

Mathematics Subject Classification (2000): 32H50, 32U40, 32Q45     

Zur Geometrie höherer Planetenumschwungbewegungen

The product ofn rotations with constant velocities around parallel axes in the Euclidean 3-space is called a planetary motion of degreen. The paper discusses spatial motions (P-motions) defined as the result of a planetary motion and a special translation in the direction of the axes; this translation is the sum of a finite number of harmonic translations with different frequencies.

---

---

Mit 2 Abbildungen     

Global bifurcation for a special Hill equation with nonlinear boundary conditions

Summary We introduce a new bifurcation model which consists of a linear second order ordinary differential equation together with nonlocal nonlinear boundary conditions. These are periodic boundary conditions in the case of the trivial solutiony=0 and semi-periodic boundary conditions for y tending to infinity. We use an elementary method to prove the existence of global solution branches connecting the periodic eigenvalues and the trivial solutiony=0 with the semi-periodic eigenvalues and the solutiony=.

---

Zusammenfassung Wir stellen ein neues Verzweigungsmodell, bestehend aus einer linearen gewöhnlichen Differentialgleichung zweiter Ordnung und nichtlokalen nichtlinearen Randbedingungen, vor. Periodische Randbedingungen ergeben sich für den Fall der trivialen Lösungy=0 und semi-periodische Randbedingungen falls y gegen unendlich strebt. Mit Hilfe elementarer Methoden weisen wir die Existenz globaler Lösungszweige nach, die die periodischen Eigenwerte und die triviale Lösungy=0 mit den semi-periodischen Eigenwerten und der Lösungy= verbinden.

---

---

Dedicated to the memory of Maria Adelaide Sneider and Johann Kreyenberg

---

Supported in part by CONICYT (Grant 89-576) in Concepción/Chile.     

Fonctions holomorphes à croissance modérée et vecteurs distributions

The space of distribution vectors of a holomorphic discrete series representation of scalar type can be identified to a space of holomorphic functions with moderate growth. This is the main result of the paper. We give an application of it to the Hankel transform and to multivariate Laguerre series expansions.

---

Resume. Lespace des vecteurs distributions dune représentation de la série discrète holomorphe scalaire dun domaine borné symétrique sidentifie à lespace des fonctions holomorphes à croissance modérée. Cest le principal résultat de cet article. Nous en donnons une application à la transformation de Hankel et aux développements en série de fonctions de Laguerre de plusieurs variables.

---

---

Mathematics Subject Classification (2000): 32M15, 22E46in final form: 19 January 2004     

Die Zykliden 3. Ordnung des Flaggenraumes

According to H. Brauner a flag space is a three-dimensional affine space with an absolute {, f, F}, where f is a line in the plane at infinity and F a point on f. A cyclide of order 3 is an algebraic surface of order 3 that intersects only at f. In this paper we determine all cyclides of order 3, we give normal forms and CAD-pictures of a great part of these surfaces. We show that exactly 28 types exist — among them two ruled surfaces.

---

---

In memoriam Herrn o.Univ.-Prof. Dr Dr H. Brauner     

Zur globalen Raumkinematik

In this paper we continue the considerations of [8], i.e. the study of one-parametric closed equiform motions in Euclidean 3-spaceE ³. Thereby the Steiner-formula on the aperture-distances of ruled surfaces, traced by lines of the moving space, is discussed and applied in different ways. This leads to new Holditch-theorems and to results in the field of global geometry of ruled surfaces and curves inE ³. On the other hand these theorems can be specialized to important formulas of global spherical kinematics.

---

---

Herrn Prof. Dr. F. Hohenberg zum 80. Geburtstag gewidmet     

Diffraction of solitary waves

Whitham's extension of geometrical optics to nonlinear diffraction is applied to solitary waves of reference amplitudea ₀ in water of uniform depthd ₀ on the hypotheses thata ₀d ₀ and that the angle through which a diffracted wave is turned is of the order of (a ₀/d ₀)^1/2. The equations governing the amplitude and direction of the waves are reduced to a quasi-linear, hyperbolic systemof two first-order partial differential equations. Explicit results are obtained for diffraction by a convex bend and by a concave corner, and it is found that a solitary wave of initial amplitudea ₀ cannot be turned through a convex angle greater than (3a ₀/d ₀)^1/2 without separating or otherwise losing its identity. An empirical generalization for larger amplitudes and turning angles is proposed. General solutions are obtained (in an appendix) through a hodograph transformation.

---

Zusammenfassung Whitham's Erweiterung der geometrischen Optik auf nichtlineare Diffraktion wird auf Einzelwellen (solitary waves) angewendet. Dabei wird angenommen, dass die Referenzamplitudea ₀ viel kleiner als die konstante, ungestörte Wassertiefed ₀ sei und dass der Ablenkungswinkel die Grössenordnung (a ₀/d ₀)^1/2 habe. Die Gleichungen für Amplitude und Richtung der Wellen werden auf ein quasi-lineares, hyperbolisches System von zwei partiellen Differentialgleichungen erster Ordnung reduziert. Explizite Resultate für die Diffraktion an einer konvex gekrümmten Wand und an einer konkaven Ecke werden angegeben. Dabei wird gefunden, dass eine Welle der ursprünglichen Amplitudea ₀ durch eine konvexe Biegung nicht mehr als (3a ₀/d ₀)^1/2 abgelenkt werden kann, ohne dass sie ablöst oder ihre Identität verliert. Eine empirische Verallgemeinerung für grössere Amplituden und Ablenkwinkel wird vorgeschlagen. Im Anhang werden mit einer Hodographentransformation allgemeine Lösungen gegeben.

---

---

Dedicated to my good friend Nikolaus Rott on the happy occasion of his sixtieth birthday.