Infinitesimally flexible meshes and discrete minimal surfaces

We explore the geometry of isothermic meshes, conical meshes, and asymptotic meshes around the Christoffel dual construction of a discrete minimal surface. We present a discrete Legendre transform which realizes discrete minimal surfaces as conical meshes. Conical meshes turn out to be infinitesimally flexible if and only if their spherical image is isothermic, which implies that discrete minimal surfaces constructed in this way are infinitesimally flexible, and therefore possess reciprocal-parallel meshes. These are discrete minimal surfaces in their own right. In our study of relative kinematics of infinitesimally flexible meshes, we encounter characterizations of flexibility and isothermicity which are of incidence-geometric nature and are related to the classical Desargues configuration. The Lelieuvre formula for asymptotic meshes leads to another characterization of isothermic meshes in the sphere which is based on triangle areas.     

Konvergenz stetiger stochastischer Prozesse mit polnischem Zustands- und Parameterraum

Ohne Zusammenfassung Diese Arbeit ist Teil der Dissertation des Autors bei Prof. J.Heinhold am Institut für Angew. Math. der TU München.     

Large time behavior in Wasserstein spaces and relative entropy for bipolar drift-diffusion-Poisson models

We prove asymptotic stability results for nonlinear bipolar drift-diffusion-Poisson Systems arising in semiconductor device modeling and plasma physics in one space dimension. In particular, we prove that, under certain structural assumptions on the external potentials and on the doping profile, all solutions match for large times with respect to all q-Wasserstein distances. We also prove exponential convergence to stationary solutions in relative entropy via the so called entropy dissipation (or Bakry-émery) method. Authors’ addresses: Marco Di Francesco, Sezione di Matematica per L’Ingegneria, Dipartimento di Matematica Pura ed Applicata, Università di L'Aquila, Piazzale E. Pontieri, 2, Monteluco di Roio, 67040 L’Aquila, Italy; Marcus Wunsch, Fakult?t für Mathematik, Universit?t Wien, Nordbergstra?e 15, A-1090 Wien, Austria     

Projektive Punktskalen und Summationsformeln für Tschebyscheff-Polynome

Zusammenfassung. Mit Hilfe der projektiven Geometrie werden neuartige Summationsformeln hergeleitet, die die Tschebyscheff-Polynome und modifizierte Polynome enthalten; z. B. gilt für alle : Daraus ergeben sich wiederum interessante Summationsformeln für die Hyperbelfunktionen; es gilt z. B. für positive reelle Zahlen t: Eingegangen am 28.02.96, revidierte Fassung am 25.08.97 / Angenommen am 09.10.97     

The focal geometry of circular and conical meshes

Circular meshes are quadrilateral meshes all of whose faces possess a circumcircle, whereas conical meshes are planar quadrilateral meshes where the faces which meet in a vertex are tangent to a right circular cone. Both are amenable to geometric modeling – recently surface approximation and subdivision-like refinement processes have been studied. In this paper we extend the original defining property of conical meshes, namely the existence of face/face offset meshes at constant distance, to circular meshes. We study the close relation between circular and conical meshes, their vertex/vertex and face/face offsets, as well as their discrete normals and focal meshes. In particular we show how to construct a two-parameter family of circular (resp., conical) meshes from a given conical (resp., circular) mesh. We further discuss meshes which have both properties and their relation to discrete surfaces of negative Gaussian curvature. The offset properties of special quadrilateral meshes and the three-dimensional support structures derived from them are highly relevant for computational architectural design of freeform structures. Another aspect important for design is that both circular and conical meshes provide a discretization of the principal curvature lines of a smooth surface, so the mesh polylines represent principal features of the surface described by the mesh.      

The lattice discrepancy of bodies bounded by a rotating Lamé’s curv

This article provides an asymptotic result for the lattice point discrepancy of the special three-dimensional body for fixed k > 2 and large t. Authors’ addresses: Ekkehard Kr?tzel, Faculty of Mathematics, University of Vienna, Nordbergstra?e 15, 1090 Wien, ?sterreich; Werner Georg Nowak, Department of Integrative Biology, Institute of Mathematics, Universit?t für Bodenkultur Wien, Gregor Mendel-Stra?e 33, 1180 Wien, ?sterreich     

Conformal hexagonal meshes

We explore discrete conformal and discrete minimal surfaces whose faces are planar hexagons throughout. Discrete conformal meshes are built of conformal hexagons for which we establish a dual construction. We apply this dual construction to conformal hexagonal meshes covering the sphere and get discrete hexagonal minimal surfaces via a discrete analogue to the Christoffel dual construction. We compare the smooth and the discrete settings by means of limit considerations and also by a discussion of Möbius invariants.     

Wesentliche Beiträge Österreichs zur Strömungsmechanik bis 1958

Das erste Privileg (Patent) für eine Schraube zum Vortrieb eines Schiffes erhielt der als Förster in der Österreichisch-Ungarischen Monarchie tätige J. Ressel im Jahre 1827. Die Fahrt mit einem Versuchsschiff im Hafen von Triest wurde wegen eines Schadens der Dampfmaschine abgebrochen und konnte nicht wiederholt werden. Der weltweit als Erfinder der nach ihm benannten Wasserturbine bekannte V. Kaplan wirkte an der Technischen Hochschule Brünn 1903-1934. R. Knoller wurde 1909 an die neu gegründete Lehrkanzel für „Luftschiffahrt und Automobilwesen“ berufen. Seine wesentlichen Beiträge sind der Bau eines aeromechanischen Laboratoriums an der Technischen Hochschule Wien, eines Luftschraubenprüfstandes in Fischamend und zahlreiche Flugzeugkonstruktionen. R. Katzmayr war Leiter dieses Laboratoriums und nach dem Tode Knollers leitete er auch die Lehrkanzel. Er veröffentlichte die „Berichte der Aeromechanischen Versuchsanstalt in Wien“. Nach 1945 kam das aeromechanische Laboratorium zur Lehrkanzel für Strömungslehre, die mit F. Magyar besetzt wurde. Seine wesentlichen Beiträge sind Untersuchungen von Fliehkraftreinigern und Dralldüsen. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Group invariant Colombeau generalized functions

Colombeau generalized functions invariant under smooth (additive) one-parameter groups are characterized. This characterization is applied to generalized functions invariant under orthogonal groups of arbitrary signature, such as groups of rotations or the Lorentz group. Further, a one-dimensional Colombeau generalized function with two (real) periods is shown to be a generalized constant, when the ratio of the periods is an algebraic nonrational number. Finally, a nonstandard Colombeau generalized function invariant under standard translations is shown to be constant. Supported by research grants M949 and Y237 of the Austrian Science Foundation (FWF). Author’s address: Institut für Grundlagen der Bauingenieurwissenschaften, Technikerstra?e 13, 6020 Innsbruck, Austria     

A curvature theory for discrete surfaces based on mesh parallelity

We consider a general theory of curvatures of discrete surfaces equipped with edgewise parallel Gauss images, and where mean and Gaussian curvatures of faces are derived from the faces’ areas and mixed areas. Remarkably these notions are capable of unifying notable previously defined classes of surfaces, such as discrete isothermic minimal surfaces and surfaces of constant mean curvature. We discuss various types of natural Gauss images, the existence of principal curvatures, constant curvature surfaces, Christoffel duality, Koenigs nets, contact element nets, s-isothermic nets, and interesting special cases such as discrete Delaunay surfaces derived from elliptic billiards.     

The isodiametric problem with lattice-point constraints

In this paper, the isodiametric problem for centrally symmetric convex bodies in the Euclidean d-space containing no interior non-zero point of a lattice L is studied. It is shown that the intersection of a suitable ball with the Dirichlet-Voronoi cell of 2L is extremal, i.e., it has minimum diameter among all bodies with the same volume. It is conjectured that these sets are the only extremal bodies, which is proved for all three dimensional and several prominent lattices. Authors’ addresses: M. A. Hernández Cifre, Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100-Murcia, Spain; A. Schürmann, Institut für Algebra und Geometrie, Otto-von-Guericke Universit?t Magdeburg, 39106 Magdeburg, Germany; F. Vallentin, Centrum voor Wiskunde en Informatica (CWI), Kruislaan 413, 1098 SJ Amsterdam, The Netherlands     

Herausarbeiten funktionaler und dynamischer Aspekte von Parameterdarstellungen durch die Erstellung von Computeranimationen

Zusammenfassung Die Beschr?nkung des Unterrichts in analytischer Geometrie auf die Behandlung von Geraden und Ebenen führt zu einer Formenarmut des Unterrichts. Vielfach gewinnen Schüler zudem nur „statische“ Vorstellungen von Parameterdarstellungen und erfassen insbesondere die damit verbundenen funktionalen Beziehungen zwischen Parameterwerten und Punkten nicht. Die Einbeziehung von Computervisualisierungen und einfachen Animationen kann dazu beitragen, bei der Behandlung von Parameterdarstellungen oft vernachl?ssigte Gesichtspunkte „mit Leben zu erfüllen“. Zudem lassen sich dadurch Modellbildungen anregen, die zu Parametrisierungen interessanter Kurven führen. Es werden hierfür anhand von Geraden sowie als Bahnkurven aufgefassten Kreisen, Spiralen, Schraubenlinien und Wurfparabeln Vorschl?ge unterbreitet und entsprechende Vorgehensweisen skizziert.     

Semi-Discrete Isothermic Surfaces

We study mappings of the form ${x : \mathbb{Z}\times\mathbb{R}\to\mathbb{R}^3}$ which can be seen as a limit case of purely discrete surfaces, or as a semi-discretization of smooth surfaces. In particular we discuss circular surfaces, isothermic surfaces, conformal mappings, and dualizability in the sense of Christoffel. We arrive at a semi-discrete version of Koenigs nets and show that in the setting of circular surfaces, isothermicity is the same as dualizability. We show that minimal surfaces constructed as a dual of a sphere have vanishing mean curvature in a certain well-defined sense, and we also give an incidence-geometric characterization of isothermic surfaces.     

Zur Rolle des Erlanger Programms im Geometrieunterricht

Zusammenfassung. Es wird untersucht, inwieweit das Erlanger Programm Felix Kleins, die Systematisierung der Geometrie mittels Abbildungsgruppen, auch für den Geometrieunterricht der Schule bedeutsam geworden ist. Dabei werden sowohl historische Entwicklungen der Inhalte des Geometrieunterrichts als auch inhaltliche Aspekte berücksichtigt. Weiterhin werden Auswirkungen auf die Schulgeometrie angesprochen, die auf das Erlanger Programm „im weiteren Sinne” zurückgehen. Eingegangen am 3. 11. 1995 / Angenommen am 3. 1. 1996     

Average growth-behavior and distribution properties of generalized weighted digit-block-counting functions

We introduce a generalized weighted digit-block-counting function on the nonnegative integers, which is a generalization of many digit-depending functions as, for example, the well known sum-of-digits function. A formula for the first moment of the sum-of-digits function has been given by Delange in 1972. In the first part of this paper we provide a compact formula for the first moment of the generalized weighted digit-block-counting function and show that a (weak) Delange type formula holds if the sequence of weights converges. The question, whether the converse is true as well, can only be answered partially at the moment. In the second part of this paper we study distribution properties of generalized weighted digit-block-counting sequences and their d-dimensional analogues. We give an if and only if condition under which such sequences are uniformly distributed modulo one. Roswitha Hofer, Recipient of a DOC-FFORTE-fellowship of the Austrian Academy of Sciences at the Institute of Financial Mathematics at the University of Linz (Austria). Friedrich Pillichshammer, Supported by the Austrian Science Foundation (FWF), Project S9609, that is part of the Austrian National Research Network “Analytic Combinatorics and Probabilistic Number Theory”. Dedicated to Prof. Robert F. Tichy on the occasion of his 50th birthday Authors’ address: Roswitha Hofer, Gerhard Larcher and Friedrich Pillichshammer, Institut für Finanzmathematik, Universit?t Linz, Altenbergerstra?e 69, A-4040 Linz, Austria     

Congruence of minimal surfaces and higher fundamental forms

 We show that minimal surfaces in space forms are determined, up to ambient isometries, by the induced metric and certain invariants which are defined in terms of the higher fundamental forms and the complex structure. Received: 6 May 2002 / Revised version: 5 July 2002 Current address: Institut für Mathematik, Universit?t Augsburg, Universit?tsstrasse 12, D-86135 Augsburg, Germany. e-mail: Theodoros.Vlachos@Math.Uni-Augsburg.De The author was supported by the Alexander von Humboldt Foundation Mathematics Subject Classification (2000): 53A10, 53C42 Acknowledgements. This work was written during the author's stay at the University of Augsburg as a research fellow of the Alexander von Humboldt Foundation. I wish to express my gratitude to the Alexander von Humboldt Foundation for its generous support. Moreover, I wish to thank Professor J.H. Eschenburg for many fruitful and stimulating conversations.     

Isothermic Surfaces in Spaces of Arbitrary Dimension: Integrability, Discretization, and Bäcklund Transformations—A Discrete Calapso Equation

A vector analog of the classical Calapso equation governing isothermic surfaces in R ^{n +2} is introduced. It is shown that this vector Calapso system admits a nonlocal) scalar Lax pair based on the classical Moutard equation. The analog of Darboux's Bäcklund transformation for isothermic surfaces in R³ is derived in a systematic manner and shown that it may be formulated in terms of the classical Moutard transformation acting on the scalar Lax pair. A permutability theorem for isothermic surfaces is set down that manifests itself in an explicit superposition principle for the vector Calapso system. This superposition principle in vectorial form is shown to constitute an integrable discretization of the vector Calapso system and, therefore, defines discrete isothermic surfaces in R ^{n +2}. The discrete Calapso equation is related to the discrete Korteweg–de Vries equation and discrete holomorphic functions. A matrix Lax pair based on Clifford algebras and a scalar Lax pair are derived for the discrete Calapso equation. A discrete Moutard-type transformation for the discrete Calapso equation is obtained, and it is shown that the discrete Calapso equation may be specialized to an integrable discrete version of the O( n +2) nonlinear σ-model.     

Die Geometrischen Arbeiten von Leopold Vietoris. Ein Kommentar

Da es zur Zeit fraglich ist, ob Gesammelte Werke von Leopold Vietoris (4.6.1891–9.4.2002) im Rahmen der Österreichischen Akademie der Wissenschaften erscheinen können, folgt der Autor dem ihm aus dem Herausgebergremium nahegelegten Rat, den vorliegenden Beitrag zunächst einmal an anderer Stelle zugänglich zu machen. – Auswahl der Arbeiten und hierfür verwendete Nummern waren dem Autor vorgegeben. Es handelt sich um die folgenden Veröffentlichungen von L. Vietoris:

1. Eine besondere Erzeugungsweise der Raumkurven vierter Ordnung zweiter Art. Wien. Ber. 125 (1916) 259–283.
2. Zur Geometrie ebener Massenanziehungsprobleme. Math. Zeitschr. 19 (1923) 130–135.
3. Zur Geometrie der ebenen analytischen Kurven. Anz. Akad. Wiss. Wien. Math.–Nat. Kl. 83 (1946) 17–20.
4. Ein einfacher Beweis des Vierscheitelsatzes der ebenen Kurven. Arch. Math. 3 (1952) 304–306.
5. Zur konformen Geometrie der ebenen Kurven. Rev. Math. Pures Appl. 1 (1956), no. 3, 73–77.

     

Systems of quadratic Diophantine inequalities and the value distribution of quadratic forms

Let Q ₁,…,Q _r be quadratic forms with real coefficients. We prove that the set is dense in , provided that the system Q ₁(x) = 0,…,Q _r (x) = 0 has a nonsingular real solution and all forms in the real pencil generated by Q ₁,…,Q _r are irrational and have rank larger than 8r. Moreover, we give a quantitative version of the above assertion. As an application we study higher correlation functions of the value distribution of a positive definite irrational quadratic form. Author’s address: Institut für Statistik, Technische Universit?t Graz, A-8010 Graz, Austria     

在不可定向的流形网格曲面上计算测地距离的一般方法

不可定向的流形曲面不仅在拓扑学中占据重要的地位,在可视化和极小曲面等问题中也有很多的应用.从拓扑学的观点来看,二流形曲面的每个局部与圆盘同胚,该性质与曲面的全局可定向性无关.但在离散化的网格表示上,可定向的二流形曲面常用半边结构来表达,而不可定向的二流形曲面大多表达成若干多边形的集合,这给以可定向网格曲面为主要研究对象的数字几何处理带来很多不便.本文提出了把不可定向的二流形网格曲面上的测地距离问题转化到可定向曲面上进行处理的一般算法框架.该框架有望在不可定向的二流形网格曲面与传统数字几何处理方法之间搭起一座桥梁.为了展示该算法框架的普适性,本文将其应用于不可定向曲面上的三个重要场合,包括测地距离的求解、离散指数映射和最远点采样.