Realisability Problems in Algebraic Topology
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Mathematicians Fleeing from Nazi Germany
Mathematicians Fleeing from Nazi Germany Mathematicians Fleeing from Nazi Germany Individual Fates and Global Impact Reinhard Siegmund-Schultze princeton university press princeton and oxford Copyright 2009 © by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW All Rights Reserved Library of Congress Cataloging-in-Publication Data Siegmund-Schultze, R. (Reinhard) Mathematicians fleeing from Nazi Germany: individual fates and global impact / Reinhard Siegmund-Schultze. p. cm. Includes bibliographical references and index. ISBN 978-0-691-12593-0 (cloth) — ISBN 978-0-691-14041-4 (pbk.) 1. Mathematicians—Germany—History—20th century. 2. Mathematicians— United States—History—20th century. 3. Mathematicians—Germany—Biography. 4. Mathematicians—United States—Biography. 5. World War, 1939–1945— Refuges—Germany. 6. Germany—Emigration and immigration—History—1933–1945. 7. Germans—United States—History—20th century. 8. Immigrants—United States—History—20th century. 9. Mathematics—Germany—History—20th century. 10. Mathematics—United States—History—20th century. I. Title. QA27.G4S53 2008 510.09'04—dc22 2008048855 British Library Cataloging-in-Publication Data is available This book has been composed in Sabon Printed on acid-free paper. ∞ press.princeton.edu Printed in the United States of America 10 987654321 Contents List of Figures and Tables xiii Preface xvii Chapter 1 The Terms “German-Speaking Mathematician,” “Forced,” and“Voluntary Emigration” 1 Chapter 2 The Notion of “Mathematician” Plus Quantitative Figures on Persecution 13 Chapter 3 Early Emigration 30 3.1. The Push-Factor 32 3.2. The Pull-Factor 36 3.D. -
Arxiv:1907.00626V1 [Math.RT] 1 Jul 2019 Ru Srltdt Hto H Rp.Nml,W Prove: We Namely, 1.1
REPRESENTABILITY OF PERMUTATION REPRESENTATIONS ON COALGEBRAS AND THE ISOMORPHISM PROBLEM CRISTINA COSTOYA, DAVID MENDEZ,´ AND ANTONIO VIRUEL Abstract. Let G be a group and let ρ: G → Sym(V ) be a permutation representation of G on a set V . We prove that there is a faithful G-coalgebra C such that G arises as the image of the restriction of Aut(C) to G(C), the set of grouplike elements of C. Furthermore, we show that V can be regarded as a subset of G(C) invariant through the G-action, and that the composition of the inclusion G ֒→ Aut(C) with the restriction Aut(C) → Sym(V ) is precisely ρ. We use these results to prove that isomorphism classes of certain families of groups can be distinguished through the coalgebras on which they act faithfully. 1. Introduction Given X an object in a category C, the study of its automorphism group, Aut(X), is a difficult task. In fact, even deciding which groups arise as the automorphism groups of objects in C is far from trivial, see [13]. Nonetheless, it is also rewarding, as it can be expected that distinguished objects have distinguished automorphism groups, which in turn may give valuable information regarding the object X. Not only that, but if we know the automorphism groups of enough objects in C, we can also draw conclusions regarding the category itself. One clear example of this comes from representation theory, as the automorphism groups of the objects of a category tell us a lot about which groups may act on which objects. -
NOTICES Was Sent to Press, the Meeting Dates Which Fall Rather Far in the Future Are Subject Tr Change
AMERICAN MATHEMATICAL SOCIETY Nottces Edited by GORDON L. WALKER Contents MEETINGS Calendar of Meetings • • • . • • • • • • • • . • • • • • . • • • • • • 4 Program of the February Meeting in Tucson • • • . • . • . • . 5 Abstracts for the Meeting, pp. 80-84 Program of the February Meeting in New York ••••••••• 10 Abstracts for the Meeting, pp. 85-94 PRELIMINARY ANNOUNCEMENTS OF MEETINGS ••••••.••• 15 ACTIVITIES OF OTHER ASSOCIATIONS ••••••••••••••••• 19 NEWS ITEMS AND ANNOUNCEMENTS •••••••••••••••••• 27 FOREIGN SCIENCE INFORMATION •••••••••.•••••••••• 38 PERSONAL ITEMS •••••••.•••••••••.••••••••••••• 47 NEW PUBLICATIONS ••••••••••••••••••••••••••••. 50 CATALOGUE OF LECTURE NOTES - Supplement No.1 . • • • • • 57 ABSTRACTS OF CONTRIBUTED PAPERS • • • • • • . • • • • • • • • 59 RESERVATION FORM . • . • • • • • • • • • • . • • • • • . • . • • • • • • 99 3 MEETINGS CALENDAR OF MEETINGS NOTE: This Calendar lists all of the meetings which have been approved by the Council up to the date at which this issue of the NOTICES was sent to press, The meeting dates which fall rather far in the future are subject tr change. This is particularly true of the meetings to which no numbers have yet been assigned, Meet Deadline ing Date Place for No, Abstracts* 567 April 14-16, 1960 New York, New York Mar. 1 568 April 22-23, 1960 Chicago, Illinois Mar, 1 569 April 22-23, 1960 Berkeley, California Mar, 570 June 18, 1960 Missoula, Montana May 5 571 August 29-September 3, 1960 East Lansing, Michigan July 15 (65th Summer Meeting) 572 October 22, 1960 Worcester, Massachusetts Sept. 8 January 24-27, 1961 Washington, D. C. (67th Annual Meeting) August, 1961 Stillwater, Oklahoma (66th Summer Meeting) January, 1962 Kansas City, Missouri (68th Annual Meeting) August, 1962 Vancouver, British Columbia (67th Summer Meeting) August, 1963 Boulder, Colorado (68th Summer Meeting) *The abstracts of papers to be presented at the meetings must be received i. -
Automorphism Groups of Simple Graphs
AUTOMORPHISM GROUPS OF SIMPLE GRAPHS LUKE RODRIGUEZ Abstract Group and graph theory both provide interesting and meaninful ways of examining relationships between elements of a given set. In this paper we investigate connections between the two. This investigation begins with automorphism groups of common graphs and an introduction of Frucht's Theorem, followed by an in-depth examination of the automorphism groups of generalized Petersen graphs and cubic Hamiltonian graphs in LCF notation. In conclusion, we examine how Frucht's Theorem applies to the specific case of cubic Hamiltonian graphs. 1. Group Theory Fundamentals In the field of Abstract Algebra, one of the fundamental concepts is that of combining particular sets with operations on their elements and studying the resulting behavior. This allows us to consider a set in a more complete way than if we were just considering their elements themselves outside of the context in which they interact. For example, we might be interested in how the set of all integers modulo m behaves under addition, or how the elements of a more abstract set S behave under some set of permutations defined on the elements of S. Both of these are examples of a group. Definition 1.1. A group is a set S together with an operation ◦ such that: (1) For all a; b 2 S, the element a ◦ b = c has the property that c 2 S. (2) For all a; b; c 2 S, the equality a ◦ (b ◦ c) = (a ◦ b) ◦ c holds. (3) There exists an element e 2 S such that a ◦ e = e ◦ a = a for all a 2 S. -
Automorphism Group of Graphs (Supplemental Material for Intro to Graph Theory)
Automorphism Group of Graphs (Supplemental Material for Intro to Graph Theory) Robert A. Beeler∗ January 15, 2018 1 Introduction and Preliminary Results In this supplement, we will assume that all graphs are undirected graphs with no loops or multiple edges. In graph theory, we talk about graph isomor- phisms. As a reminder, an isomorphism between graphs G and H is a bijec- tion φ : V (G) → V (H) such that uv ∈ E(G) if and only if φ(u)φ(v) ∈ E(H). A graph automorphism is simply an isomorphism from a graph to itself. In other words, an automorphism on a graph G is a bijection φ : V (G) → V (G) such that uv ∈ E(G) if and only if φ(u)φ(v) ∈ E(G). This definition gener- alizes to digraphs, multigraphs, and graph with loops. Let Aut(G) denote the set of all automorphisms on a graph G. Note that this forms a group under function composition. In other words, (i) Aut(G) is closed under function composition. (ii) Function composition is associative on Aut(G). This follows from the fact that function composition is associative in general. (iii) There is an identity element in Aut(G). This is mapping e(v) = v for all v ∈ V (G). ∗Department of Mathematics and Statistics, East Tennessee State University, Johnson City, TN 37614-1700 USA email: [email protected] 1 2 a c d b d b c a e α a c b d b d c a β αβ Figure 1: A graph and its automorphisms (iv) For every σ ∈ Aut(G), there is an inverse element σ−1 ∈ Aut(G). -
Group Actions on Dendrites and Curves 3
GROUP ACTIONS ON DENDRITES AND CURVES BRUNO DUCHESNE AND NICOLAS MONOD ABSTRACT. We establish obstructions for groups to act by homeomorphisms on dendrites. For instance, lattices in higher rank simple Lie groups will always fix a point or a pair. The same holds for irreducible lattices in products of connected groups. Further results include a Tits alternative and a description of the topological dynamics. We briefly discuss to what extent our results hold for more general topological curves. 1. INTRODUCTION 1.A — Dendrites. Recall that a dendrite is a locally connected continuum without simple closed curves. Other equivalent definitions and some basic facts are recalled in Section 2 below. A simple example of a dendrite is obtained by compactifying a countable simplicial tree (Section 12), but the typical dendrite is much more complicated: certain Julia sets are den- drites [BT07] and Wazewski’s˙ universal dendrite [Waz23a]˙ can be identified with the Berkovich projective line over Cp [HLP14]. The homeomorphisms group of some dendrites is enor- mous (Section 12), whilst for others it is trivial [dGW58, p. 443]. In fact, it was recognized early on [Why28, Why30] that any continuum has a canoni- cal dendrite quotient reflecting its cut-point structure, see [Bow98], [PS06] for the general statement. Bowditch made remarkable use of this quotient [Bow99b], leading to the solu- tion of the cut-point conjecture for hyperbolic groups [Bow99a, Swa96]. In this application, Bowditch’s dendrites retained a decidedly geometric aspect inherited from the hyperbolic group, namely the dynamical convergence property, allowing him to reconstruct a metric tree with an isometric action. -
Connected Sum at Infinity and Cantrell-Stallings Hyperplane Unknotting
CONNECTED SUM AT INFINITY AND CANTRELL-STALLINGS HYPERPLANE UNKNOTTING JACK S. CALCUT, HENRY C. KING, AND LAURENT C. SIEBENMANN Dedicated to Ljudmila V. Keldysh and the members of her topology seminar on the occasion of the centenary of her birth.1 1. Introduction We give a general treatment of the somewhat unfamiliar operation on manifolds called connected sum at infinity or CSI for short. A driving ambition has been to make the geometry behind the well defi- nition and basic properties of CSI as clear and elementary as possible. CSI then yields a very natural and elementary proof of a remark- able theorem of J.C. Cantrell and J.R. Stallings [Can63, Sta65]. It as- serts unknotting of cat embeddings of Rm−1 in Rm with m 6= 3, for all three classical manifold categories: topological (= top), piecewise linear (= pl), and differentiable (= diff) — as defined for example in [KS77]. It is one of the few major theorems whose statement and proof can be the same for all three categories. We give it the acronym HLT, which is short for “Hyperplane Linearization Theorem” (see The- orem 6.1 plus 7.3). We pause to set out some common conventions that are explained in [KS77] and in many textbooks. By default, spaces will be assumed metrizable, and separable (i.e. having a countable basis of open sets). arXiv:1010.2707v1 [math.GT] 13 Oct 2010 Simplicial complexes will be unordered. A pl space (often called a polyhedron) has a maximal family of pl compatible triangulations by Date: September 17, 2008 (revised October 12, 2010). -
Degree (Graph Theory) from Wikipedia, the Free Encyclopedia Contents
Degree (graph theory) From Wikipedia, the free encyclopedia Contents 1 Degree (graph theory) 1 1.1 Handshaking lemma .......................................... 1 1.2 Degree sequence ............................................ 2 1.3 Special values ............................................. 3 1.4 Global properties ........................................... 3 1.5 See also ................................................ 4 1.6 Notes ................................................. 4 1.7 References ............................................... 4 2 Graph operations 5 2.1 Unary operations ........................................... 5 2.1.1 Elementary operations .................................... 5 2.1.2 Advanced operations ..................................... 5 2.2 Binary operations ........................................... 5 2.3 Notes ................................................. 6 3 Regular graph 7 3.1 Existence ............................................... 7 3.2 Algebraic properties .......................................... 7 3.3 Generation .............................................. 8 3.4 See also ................................................ 8 3.5 References .............................................. 8 3.6 External links ............................................. 8 3.7 Text and image sources, contributors, and licenses .......................... 9 3.7.1 Text .............................................. 9 3.7.2 Images ............................................ 9 3.7.3 Content license ....................................... -
Fixing Numbers of Graphs and Groups Courtney Gibbons Hamilton College, [email protected]
Hamilton College Hamilton Digital Commons Articles Works by Type 2009 Fixing Numbers of Graphs and Groups Courtney Gibbons Hamilton College, [email protected] Joshua D. Laison Follow this and additional works at: https://digitalcommons.hamilton.edu/articles Part of the Algebra Commons, and the Discrete Mathematics and Combinatorics Commons This document is the publisher's version of an article published in: The Electronic Journal of Combinatorics., vol. 16, no. 1, (2009): #R39 1-13. http://www.combinatorics.org/ojs/index.php/eljc/article/ view/v16i1r39 Citation Information Gibbons, Courtney and Laison, Joshua D., "Fixing Numbers of Graphs and Groups" (2009). Hamilton Digital Commons. https://digitalcommons.hamilton.edu/articles/59 This work is made available by Hamilton College for educational and research purposes under a Creative Commons BY-NC-ND 4.0 license. For more information, visit http://digitalcommons.hamilton.edu/about.html or contact [email protected]. Fixing Numbers of Graphs and Groups Courtney R. Gibbons Joshua D. Laison University of Nebraska – Lincoln Mathematics Department Department of Mathematics Willamette University 228 Avery Hall 900 State St. PO Box 880130 Salem, OR 97301 Lincoln, NE 68588-0130 [email protected] [email protected] Submitted: Sep 11, 2006; Accepted: Mar 12, 2009; Published: Mar 20, 2009 Mathematics Subject Classification: 05C25 Abstract The fixing number of a graph G is the smallest cardinality of a set of vertices S such that only the trivial automorphism of G fixes every vertex in S. The fixing set of a group Γ is the set of all fixing numbers of finite graphs with automorphism group Γ. -
A Counterexample in the Theories of Compactness and of Metrization 1)
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector MATHEMATICS A COUNTEREXAMPLE IN THE THEORIES OF COMPACTNESS AND OF METRIZATION 1) BY FRANKLIN D. TALL (Communicated by Prof. H. FEEUDENTHAL at the meeting of May 26, 1973) The fact that the Baire category theorem held for both compact Haus- doti spaces and complete metric spaces led to a search for a natural class of spaces encompassing both of these classes and satisfying the theorem. The first successful attempt was due to TECH [5]-the so-called de& complete spaces. There have been numerous additional attempts since then. In addition to Tech completeness, we shall only deal with two concepts closely related to each other, basis-compactness [lo] and co- compactness [l]. A comprehensive survey [2] of these and many more completeness properties by J. M. AARTS and D. J. LUTZER is in preparation. I am indebted to them for many stimulating thoughts on the subject. We shall present an example to show that neither basis-compactness nor cocompactness is implied by Tech completeness. The example has a number of other interesting properties- it is a metacompact locally metrizable Moore space which is not metrizable, but may, consistently with the axioms of set theory, be assumed normal. The example’s sig- nificance in metrization theory is dealt with in [9]. Here we will confine ourselves to completeness questions. DEFINITION. A space (we assume all spaces completely regular) is tech complete if it is a Ga in its Stone-Tech compactification. -
Arxiv:1807.04372V1 [Math.CO] 11 Jul 2018 Fixing Numbers of Graphs and Groups
Fixing Numbers of Graphs and Groups Courtney R. Gibbons Joshua D. Laison University of Nebraska – Lincoln Mathematics Department Department of Mathematics Willamette University 228 Avery Hall 900 State St. PO Box 880130 Salem, OR 97301 Lincoln, NE 68588-0130 [email protected] [email protected] Submitted: September 2006; Accepted: March 2009 Mathematics Subject Classification: 05C25 Abstract The fixing number of a graph G is the smallest cardinality of a set of vertices S such that only the trivial automorphism of G fixes every vertex in S. The fixing set of a group Γ is the set of all fixing numbers of finite graphs with automorphism group Γ. Several authors have studied the distinguishing number of a graph, the smallest number of labels needed to label G so that the automorphism group of the labeled graph is trivial. The fixing number can be thought of as a variation of the distinguishing number in which every label may be used only once, and not every vertex need be labeled. We characterize the fixing sets of finite abelian groups, and investigate the fixing sets of symmetric groups. 1 Introduction In this paper we investigate breaking the symmetries of a finite graph G by labeling its vertices. There are two standard techniques to do this. The first is to label all arXiv:1807.04372v1 [math.CO] 11 Jul 2018 of the vertices of G with k distinct labels. A labeling is distinguishing if no non- trivial automorphism of G preserves the vertex labels. The distinguishing number of G is the minimum number of labels used in any distinguishing labeling [1, 13]. -
Moving Average Network Examples for Asymptotically Stable Periodic Orbits of Monotone Maps
Electronic Journal of Qualitative Theory of Differential Equations 2018, No. 52, 1–18; https://doi.org/10.14232/ejqtde.2018.1.52 www.math.u-szeged.hu/ejqtde/ Moving average network examples for asymptotically stable periodic orbits of monotone maps To Professor László Hatvani, with respect and affection Barnabás M. Garay B 1, 2 and Judit Várdai1 1Faculty of Information Technology and Bionics, Pázmány Péter Catholic University, Práter utca 50/A, Budapest, H–1083, Hungary 2SZTAKI Computer and Automation Research Institute, Lágymányosi utca 11, Budapest, H-1111, Hungary Received 27 February 2018, appeared 26 June 2018 Communicated by Tibor Krisztin Abstract. For a certain type of discrete-time nonlinear consensus dynamics, asymptoti- cally stable periodic orbits are constructed. Based on a simple ordinal pattern assump- tion, the Frucht graph, two Petersen septets, hypercubes, a technical class of circulant graphs (containing Paley graphs of prime order), and complete graphs are considered – they are all carrying moving average monotone dynamics admitting asymptotically stable periodic orbits with period 2. Carried by a directed graph with 594 (multiple and multiple loop) edges on 3 vertices, also the existence of asymptotically stable r-periodic orbits, r = 3, 4, . is shown. Keywords: consensus dynamics, periodic orbits, monotone maps, graph eigenvectors, ordinal patterns 2010 Mathematics Subject Classification: 05C50, 37C65. 1 Introduction and the main result Let G be a (simple, undirected) graph with vertices V(G) = f1, 2, . , Ng and edges E(G). As usual, AG denotes the adjacency matrix of G (defined by letting aij = 1 if (i, j) 2 E(G) and 0 if (i, j) 62 E(G)).