Design problem of the profile or the cascade of profiles and construction of orthogonal networks using the Riemann surfaces for the multiform singularities

1989 ◽  
Vol 4 (4) ◽  
pp. 309-317
Author(s):  
T. S. Luu ◽  
G. Coulmy ◽  
B. Viney
Keyword(s):  
Design Problem ◽  
Riemann Surfaces ◽  
Orthogonal Networks

Design Problem Solving

2006 ◽  
Vol 214 (2) ◽  
pp. 73-86 ◽  
Author(s):  
Constance Winkelmann ◽  
Winfried Hacker
Keyword(s):  
Problem Solving ◽  
Design Problem

Zusammenfassung. In zwei experimentellen Studien werden mögliche alternative Bedingungen der lösungsgüteverbessernden Wirkung einer Frage-Antwort-Technik im Entwurfsdenken (Design Problem Solving) geprüft. In der ersten Studie (N = 84) wurden differential- und kognitionspsychologische Bedingungen geprüft. Eine Aussagenanalyse in einer zweiten Studie (N = 94) sollte die nachgewiesenen Ergebnisse vertiefen. Für die in beiden Studien ermittelten Lösungsgüteverbesserungen konnten keine differentialpsychologischen Abhängigkeiten von Vorgehensstilen nachgewiesen werden. In kognitionspsychologischer Hinsicht zeigte sich, dass sowohl Fragen, die die Forderungen der Aufgabe in Erinnerung bringen, als auch Fragen, deren Beantwortung die Auslösung von Denkprozessen fördern, zur Lösungsgüteverbesserung beitragen. Größere Effekte werden bei der Kombination beider Fragenarten nahe gelegt. Die Ergebnisse der Aussagenanalyse weisen auf eine sowohl beschreibende als auch begründende und bewertende Vorgehensweise der Untersuchungsteilnehmer hin, die Verbesserungen erzielten. Fragen für weiterführende Untersuchungserfordernisse werden abgeleitet.

Filling words in fundamental groups of Riemann surfaces

2013 ◽  
Vol 50 (1) ◽  
pp. 31-50
Author(s):  
C. Zhang
Keyword(s):  
Riemann Surface ◽  
Fundamental Group ◽  
Riemann Surfaces ◽  
Fundamental Groups ◽  
Closed Geodesics ◽  
New Words

The purpose of this article is to utilize some exiting words in the fundamental group of a Riemann surface to acquire new words that are represented by filling closed geodesics.

A Primer on Mapping Class Groups (PMS-49)

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Graduate Students ◽  
Moduli Space ◽  
Riemann Surfaces ◽  
Class Group ◽  
Mapping Class ◽  
Torelli Group ◽  
Surface Bundles ◽  
Surface Homeomorphisms ◽  
Low Dimensional

The study of the mapping class group Mod(S) is a classical topic that is experiencing a renaissance. It lies at the juncture of geometry, topology, and group theory. This book explains as many important theorems, examples, and techniques as possible, quickly and directly, while at the same time giving full details and keeping the text nearly self-contained. The book is suitable for graduate students. It begins by explaining the main group-theoretical properties of Mod(S), from finite generation by Dehn twists and low-dimensional homology to the Dehn–Nielsen–Baer–theorem. Along the way, central objects and tools are introduced, such as the Birman exact sequence, the complex of curves, the braid group, the symplectic representation, and the Torelli group. The book then introduces Teichmüller space and its geometry, and uses the action of Mod(S) on it to prove the Nielsen-Thurston classification of surface homeomorphisms. Topics include the topology of the moduli space of Riemann surfaces, the connection with surface bundles, pseudo-Anosov theory, and Thurston's approach to the classification.

On mappings with finite length distortion on Riemann surfaces

2019 ◽  
Vol 33 ◽  
Author(s):  
Serhii Volkov ◽  
Vladimir Ryazanov
Keyword(s):  
Finite Length ◽  
Riemann Surfaces ◽  
Boundary Behavior ◽  
Main Tool ◽  
Natural Generalization ◽  
Finite Distortion ◽  
Lipschitz Mappings ◽  
Geometric Function ◽  
Mappings With Finite Distortion ◽  
Length Distortion

The present paper is a natural continuation of our previous paper (2017) on the boundary behavior of mappings in the Sobolev classes on Riemann surfaces, where the reader will be able to find the corresponding historic comments and a discussion of many definitions and relevant results. The given paper was devoted to the theory of the boundary behavior of mappings with finite distortion by Iwaniec on Riemannian surfaces first introduced for the plane in the paper of Iwaniec T. and Sverak V. (1993) On mappings with integrable dilatation and then extended to the spatial case in the monograph of Iwaniec T. and Martin G. (2001) devoted to Geometric function theory and non-linear analysis. At the present paper, it is developed the theory of the boundary behavior of the so--called mappings with finite length distortion first introduced in the paper of Martio O., Ryazanov V., Srebro U. and Yakubov~E. (2004) in the spatial case, see also Chapter 8 in their monograph (2009) on Moduli in modern mapping theory. As it was shown in the paper of Kovtonyuk D., Petkov I. and Ryazanov V. (2017) On the boundary behavior of mappings with finite distortion in the plane, such mappings, generally speaking, are not mappings with finite distortion by Iwaniec because their first partial derivatives can be not locally integrable. At the same time, this class is a generalization of the known class of mappings with bounded distortion by Martio--Vaisala from their paper (1988). Moreover, this class contains as a subclass the so-called finitely bi-Lipschitz mappings introduced for the spatial case in the paper of Kovtonyuk D. and Ryazanov V. (2011) On the boundary behavior of generalized quasi-isometries, that in turn are a natural generalization of the well-known classes of bi-Lipschitz mappings as well as isometries and quasi-isometries. In the research of the local and boundary behavior of mappings with finite length distortion in the spatial case, the key fact was that they satisfy some modulus inequalities which was a motivation for the consideration more wide classes of mappings, in particular, the Q-homeomorphisms (2005) and the mappings with finite area distortion (2008). Hence it is natural that under the research of mappings with finite length distortion on Riemann surfaces we start from establishing the corresponding modulus inequalities that are the main tool for us. On this basis, we prove here a series of criteria in terms of dilatations for the continuous and homeomorphic extension to the boundary of the mappings with finite length distortion between domains on arbitrary Riemann surfaces.

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