Topology Without Tears1
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Groups and Geometry 2018 Auckland University Projective Planes
Groups and Geometry 2018 Auckland University Projective planes, Laguerre planes and generalized quadrangles that admit large groups of automorphisms G¨unterSteinke School of Mathematics and Statistics University of Canterbury 24 January 2018 G¨unterSteinke Projective planes, Laguerre planes, generalized quadrangles GaG2018 1 / 26 (School of Mathematics and Statistics University of Canterbury) Projective and affine planes Definition A projective plane P = (P; L) consists of a set P of points and a set L of lines (where lines are subsets of P) such that the following three axioms are satisfied: (J) Two distinct points can be joined by a unique line. (I) Two distinct lines intersect in precisely one point. (R) There are at least four points no three of which are on a line. Removing a line and all of its points from a projective plane yields an affine plane. Conversely, each affine plane extends to a unique projective plane by adjoining in each line with an `ideal' point and adding a new line of all ideal points. G¨unterSteinke Projective planes, Laguerre planes, generalized quadrangles GaG2018 2 / 26 (School of Mathematics and Statistics University of Canterbury) Models of projective planes Desarguesian projective planes are obtained from a 3-dimensional vector space V over a skewfield F. Points are the 1-dimensional vector subspaces of V and lines are the 2-dimensional vector subspaces of V . In case F is a field one obtains the Pappian projective plane over F. There are many non-Desarguesian projective planes. One of the earliest and very important class is obtain by the (generalized) Moulton planes. -
TOPOLOGY and ITS APPLICATIONS the Number of Complements in The
TOPOLOGY AND ITS APPLICATIONS ELSEVIER Topology and its Applications 55 (1994) 101-125 The number of complements in the lattice of topologies on a fixed set Stephen Watson Department of Mathematics, York Uniuersity, 4700 Keele Street, North York, Ont., Canada M3J IP3 (Received 3 May 1989) (Revised 14 November 1989 and 2 June 1992) Abstract In 1936, Birkhoff ordered the family of all topologies on a set by inclusion and obtained a lattice with 1 and 0. The study of this lattice ought to be a basic pursuit both in combinatorial set theory and in general topology. In this paper, we study the nature of complementation in this lattice. We say that topologies 7 and (T are complementary if and only if 7 A c = 0 and 7 V (T = 1. For simplicity, we call any topology other than the discrete and the indiscrete a proper topology. Hartmanis showed in 1958 that any proper topology on a finite set of size at least 3 has at least two complements. Gaifman showed in 1961 that any proper topology on a countable set has at least two complements. In 1965, Steiner showed that any topology has a complement. The question of the number of distinct complements a topology on a set must possess was first raised by Berri in 1964 who asked if every proper topology on an infinite set must have at least two complements. In 1969, Schnare showed that any proper topology on a set of infinite cardinality K has at least K distinct complements and at most 2” many distinct complements. -
Generalisations of the Fundamental Theorem of Projective
GENERALIZATIONS OF THE FUNDAMENTAL THEOREM OF PROJECTIVE GEOMETRY A thesis submitted for the degree of Doctor of Philosophy By Rupert McCallum Supervisor: Professor Michael Cowling School of Mathematics, The University of New South Wales. March 2009 Acknowledgements I would particularly like to thank my supervisor, Professor Michael Cowling, for conceiving of the research project and providing much valuable feedback and guidance, and providing help with writing the Extended Abstract. I would like to thank the University of New South Wales and the School of Mathematics and Statistics for their financial support. I would like to thank Dr Adam Harris for giving me helpful feedback on drafts of the thesis, and particularly my uncle Professor William McCallum for providing me with detailed comments on many preliminary drafts. I would like to thank Professor Michael Eastwood for making an important contribution to the research project and discussing some of the issues with me. I would like to thank Dr Jason Jeffers and Professor Alan Beardon for helping me with research about the history of the topic. I would like to thank Dr Henri Jimbo for reading over an early draft of the thesis and providing useful suggestions for taking the research further. I would like to thank Dr David Ullrich for drawing my attention to the theorem that if A is a subset of a locally compact Abelian group of positive Haar measure, then A+A has nonempty interior, which was crucial for the results of Chapter 8. I am very grateful to my parents for all the support and encouragement they gave me during the writing of this thesis. -
Math 501. Homework 2 September 15, 2009 ¶ 1. Prove Or
Math 501. Homework 2 September 15, 2009 ¶ 1. Prove or provide a counterexample: (a) If A ⊂ B, then A ⊂ B. (b) A ∪ B = A ∪ B (c) A ∩ B = A ∩ B S S (d) i∈I Ai = i∈I Ai T T (e) i∈I Ai = i∈I Ai Solution. (a) True. (b) True. Let x be in A ∪ B. Then x is in A or in B, or in both. If x is in A, then B(x, r) ∩ A , ∅ for all r > 0, and so B(x, r) ∩ (A ∪ B) , ∅ for all r > 0; that is, x is in A ∪ B. For the reverse containment, note that A ∪ B is a closed set that contains A ∪ B, there fore, it must contain the closure of A ∪ B. (c) False. Take A = (0, 1), B = (1, 2) in R. (d) False. Take An = [1/n, 1 − 1/n], n = 2, 3, ··· , in R. (e) False. ¶ 2. Let Y be a subset of a metric space X. Prove that for any subset S of Y, the closure of S in Y coincides with S ∩ Y, where S is the closure of S in X. ¶ 3. (a) If x1, x2, x3, ··· and y1, y2, y3, ··· both converge to x, then the sequence x1, y1, x2, y2, x3, y3, ··· also converges to x. (b) If x1, x2, x3, ··· converges to x and σ : N → N is a bijection, then xσ(1), xσ(2), xσ(3), ··· also converges to x. (c) If {xn} is a sequence that does not converge to y, then there is an open ball B(y, r) and a subsequence of {xn} outside B(y, r). -
How to Study Mathematics – the Manual for Warsaw University 1St Year Students in the Interwar Period
TECHNICAL TRANSACTIONS CZASOPISMO TECHNICZNE FUNDAMENTAL SCIENCES NAUKI PODSTAWOWE 2-NP/2015 KALINA BARTNICKA* HOW TO STUDY MATHEMATICS – THE MANUAL FOR WARSAW UNIVERSITY 1ST YEAR STUDENTS IN THE INTERWAR PERIOD JAK STUDIOWAĆ MATEMATYKĘ – PORADNIK DLA STUDENTÓW PIERWSZEGO ROKU Z OKRESU MIĘDZYWOJENNEGO Abstract In 1926 and in 1930, members of Mathematics and Physics Students’ Club of the Warsaw University published the guidance for the first year students. These texts would help the freshers in constraction of the plans and course of theirs studies in the situation of so called “free study”. Keywords: Warsaw University, Interwar period, “Free study”, Study of Mathematics, Freshers, Students’ clubs, Guidance for students Streszczenie W 1926 r. i w 1930 r. Koło Naukowe Matematyków i Fizyków Studentów Uniwersytetu War- szawskiego opublikowało poradnik dla studentów pierwszego roku matematyki. Są to teksty, które pomagały pierwszoroczniakom w racjonalnym skonstruowaniu planu i toku ich studiów w warunkach tzw. „wolnego stadium”. Słowa kluczowe: Uniwersytet Warszawski, okres międzywojenny, „wolne stadium”, studio wanie matematyki, pierwszoroczniacy, studenckie koła naukowe, poradnik dla studentów DOI: 10.4467/2353737XCT.15.203.4408 * L. & A. Birkenmajetr Institute of History of Science, Polish Academy of Sciences, Warsaw, Poland; [email protected] 14 This paper is focused primarily on the departure from the “free study” in university learning in Poland after it regained its independence in 1918. The idea of the “free study” had been strongly cherished by professors and staff of the Philosophy Department of Warsaw University even though the majority of students (including the students of mathematics and physics) were not interested in pursuing an academic career. The concept of free study left to the students the decision about the choice of subjects they wished to study and about the plan of their work. -
Isolated Points, Duality and Residues
JOURNAL OF PURE AND APPLIED ALGEBRA Journal of Pure and Applied Algebra 117 % 118 (1997) 469-493 Isolated points, duality and residues B . Mourrain* INDIA, Projet SAFIR, 2004 routes des Lucioles, BP 93, 06902 Sophia-Antipolir, France This work is dedicated to the memory of J. Morgenstem. Abstract In this paper, we are interested in the use of duality in effective computations on polynomials. We represent the elements of the dual of the algebra R of polynomials over the field K as formal series E K[[a]] in differential operators. We use the correspondence between ideals of R and vector spaces of K[[a]], stable by derivation and closed for the (a)-adic topology, in order to construct the local inverse system of an isolated point. We propose an algorithm, which computes the orthogonal D of the primary component of this isolated point, by integration of polynomials in the dual space K[a], with good complexity bounds. Then we apply this algorithm to the computation of local residues, the analysis of real branches of a locally complete intersection curve, the computation of resultants of homogeneous polynomials. 0 1997 Elsevier Science B.V. 1991 Math. Subj. Class.: 14B10, 14Q20, 15A99 1. Introduction Considering polynomials as algorithms (that compute a value at a given point) in- stead of lists of terms, has lead to recent interesting developments, both from a practical and a theoretical point of view. In these approaches, evaluations at points play an im- portant role. We would like to pursue this idea further and to study evaluations by themselves. -
Math 3283W: Solutions to Skills Problems Due 11/6 (13.7(Abc)) All
Math 3283W: solutions to skills problems due 11/6 (13.7(abc)) All three statements can be proven false using counterexamples that you studied on last week's skills homework. 1 (a) Let S = f n jn 2 Ng. Every point of S is an isolated point, since given 1 1 1 n 2 S, there exists a neighborhood N( n ; ") of n that contains no point of S 1 1 1 different from n itself. (We could, for example, take " = n − n+1 .) Thus the set P of isolated points of S is just S. But P = S is not closed, since 0 is a boundary point of S which nevertheless does not belong to S (every neighbor- 1 hood of 0 contains not only 0 but, by the Archimedean property, some n , thus intersects both S and its complement); if S were closed, it would contain all of its boundary points. (b) Let S = N. Then (as you saw on the last skills homework) int(S) = ;. But ; is closed, so its closure is itself; that is, cl(int(S)) = cl(;) = ;, which is certainly not the same thing as S. (c) Let S = R n N. The closure of S is all of R, since every natural number n is a boundary point of R n N and thus belongs to its closure. But R is open, so its interior is itself; that is, int(cl(S)) = int(R) = R, which is certainly not the same thing as S. (13.10) Suppose that x is an isolated point of S, and let N = (x − "; x + ") (for some " > 0) be an arbitrary neighborhood of x. -
Perfect Mappings and Their Applications in the Theory of Compact Convex Sets Contents
Perfect mappings and their applications in the theory of compact convex sets Ji°í Spurný Charles University in Prague, Faculty of Mathematics and Physics Sokolovská 83, 186 75 Prague, Czech Republic Contents 1. Introduction 6 2. Summary of Chapter 2 9 2.1. Perfect images of absolute Souslin and absolute Borel Tychono spaces . 9 2.2. Extending Baire one functions on topological spaces . 17 2.3. Borel sets and functions in topological spaces . 23 3. Summary of Chapter 3 9 3.1. A solution of the abstract Dirichlet problem for Baire one functions . .9 3.2. Descriptive properties of elements of biduals of Banach spaces . 17 4. List of articles (sections) 18 Bibliography 20 Summary We study an interplay between descriptive set theory and theory of compact convex sets. Theory of compact convex sets serves as a general framework for an investigation of Banach spaces as well as more general objects as subsets of Banach spaces, sets of measures etc. Descriptive set theory provides a way how to measure a complexity of given ob- jects. Connections between these two dierent mathematical disciplines provide a useful insight in both of them. We recall that theory of Banach spaces is in a way subsumed by theory of compact convex sets via the following procedure. If E is a Banach space, its dual unit ball BE∗ endowed with the weak* topology is a compact convex set and E can be viewed as an isometric subspace of the space of all ane continuous functions on BE∗ . Further, BE∗ is a natural example of a compact topological space. -
Adolf Lindenbaum: Notes on His Life, with Bibliography and Selected References
CORE Metadata, citation and similar papers at core.ac.uk Provided by Springer - Publisher Connector Log. Univers. 8 (2014), 285–320 c 2014 The Author(s). This article is published with open access at Springerlink.com 1661-8297/14/030285-36, published online December 3, 2014 DOI 10.1007/s11787-014-0108-2 Logica Universalis Adolf Lindenbaum: Notes on his Life, with Bibliography and Selected References Jan Zygmunt and Robert Purdy Abstract. Notes on the life of Adolf Lindenbaum, a complete bibliography of his published works, and selected references to his unpublished results. Mathematics Subject Classification. 01A60, 01A70, 01A73, 03-03. Keywords. Adolf Lindenbaum, biography, bibliography, unpublished works. This paper is dedicated to Adolf Lindenbaum (1904–1941)—Polish- Jewish mathematician and logician; a member of the Warsaw school of mathe- matics under Waclaw Sierpi´nski and Stefan Mazurkiewicz and school of math- ematical logic under JanLukasiewicz and Stanislaw Le´sniewski;1 and Alfred Tarski’s closest collaborator of the inter-war period. Our paper is divided into three main parts. The first part is biograph- ical and narrative in character. It gathers together what little is known of Lindenbaum’s short life. The second part is a bibliography of Lindenbaum’s published output, including his public lectures. Our aim there is to be complete and definitive. The third part is a list of selected references in the literature attesting to his unpublished results2 and delineating their extent. Just to confuse things, we name the second and third parts of our paper, respectively, “Bibliography Part One” and “Bibliography Part Two”. Why, we no longer remember. -
Alice in the Wonderful Land of Logical Notions)
The Mystery of the Fifth Logical Notion (Alice in the Wonderful Land of Logical Notions) Jean-Yves Beziau University of Brazil, Rio de Janeiro, Brazilian Research Council and Brazilian Academy of Philosophy [email protected] Abstract We discuss a theory presented in a posthumous paper by Alfred Tarski entitled “What are logical notions?”. Although the theory of these logical notions is something outside of the main stream of logic, not presented in logic textbooks, it is a very interesting theory and can easily be understood by anybody, especially studying the simplest case of the four basic logical notions. This is what we are doing here, as well as introducing a challenging fifth logical notion. We first recall the context and origin of what are here called Tarski- Lindenbaum logical notions. In the second part, we present these notions in the simple case of a binary relation. In the third part, we examine in which sense these are considered as logical notions contrasting them with an example of a non-logical relation. In the fourth part, we discuss the formulations of the four logical notions in natural language and in first- order logic without equality, emphasizing the fact that two of the four logical notions cannot be expressed in this formal language. In the fifth part, we discuss the relations between these notions using the theory of the square of opposition. In the sixth part, we introduce the notion of variety corresponding to all non-logical notions and we argue that it can be considered as a logical notion because it is invariant, always referring to the same class of structures. -
Geometry and Teaching
Andrew McFarland Joanna McFarland James T. Smith Editors Alfred Tarski Early Work in Poland – Geometry and Teaching This book is dedicated to Helen Marie Smith, in gratitude for her advice and support, and to Maria Anna McFarland, as she enters a world of new experiences. Andrew McFarland • Joanna McFarland James T. Smith Editors Alfred Tarski Early Work in Poland—Geometry and Teaching with a Bibliographic Supplement Foreword by Ivor Grattan-Guinness Editors Andrew McFarland Joanna McFarland Páock, Poland Páock, Poland James T. Smith Department of Mathematics San Francisco State University San Francisco, CA, USA ISBN 978-1-4939-1473-9 ISBN 978-1-4939-1474-6 (eB ook) DOI 10.1007/978-1-4939-1474-6 Springer New York Heidelberg Dordrecht London Library of Congress Control Number: 2014945118 Mathematics Subject Classification (2010): 01A60, 01A70, 01A75, 03A10, 03B05, 03E75, 06A99, 28-03, 28A75, 43A07, 51M04, 51M25, 97B50, 97D40, 97G99, 97M30 © Springer Science+Business Media New York 2014 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. -
Arxiv:1009.4249V12 [Math.MG] 17 Oct 2012 H Dniyof Identity the a K Ulig Fsmsml Leri Ruswihetbihsti.W O P Now We This
A LOCAL-TO-GLOBAL RESULT FOR TOPOLOGICAL SPHERICAL BUILDINGS RUPERT McCALLUM Abstract. Suppose that ∆, ∆′ are two buildings each arising from a semisimple algebraic group over a field, a topological field in the former case, and that for both the buildings the Coxeter diagram has no isolated nodes. We give conditions under which a partially defined injective chamber map, whose domain is the subcomplex of ∆ generated by a nonempty open set of chambers, and whose codomain is ∆′, is guaranteed to extend to a unique injective chamber map. Related to this result is a local version of the Borel-Tits theorem on abstract homomorphisms of simple algebraic groups. Keywords: topological geometry, local homomorphism, topological building, Borel-Tits the- orem Mathematicsl Subject Classification 2000: 51H10 1. Introduction Throughout the history of Lie theory there has been a notion of “local isomorphism”. Indeed, the original notion of Lie groups considered by Sophus Lie in [9] was an essentially local one. Suppose that G is an algebraic group defined over a Hausdorff topological field k. One may consider the notion of a local k-isogeny from the group G to a group G′ defined over k of the same dimension. This is a mapping defined on a nonempty open neighbourhood of the identity of G(k) in the strong k-topology, (see Definition 1.10), which “locally” acts as a k-isogeny, in the sense that it is a local k-homomorphism and its range is Zariski dense. As far as I know this notion has not been investigated systematically. In this paper we shall produce a local version of the Borel-Tits theorem which hints at the possibility that it may be fruitful to investigate this notion.