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Metric Geometry in a Tame Setting
University of California Los Angeles Metric Geometry in a Tame Setting A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics by Erik Walsberg 2015 c Copyright by Erik Walsberg 2015 Abstract of the Dissertation Metric Geometry in a Tame Setting by Erik Walsberg Doctor of Philosophy in Mathematics University of California, Los Angeles, 2015 Professor Matthias J. Aschenbrenner, Chair We prove basic results about the topology and metric geometry of metric spaces which are definable in o-minimal expansions of ordered fields. ii The dissertation of Erik Walsberg is approved. Yiannis N. Moschovakis Chandrashekhar Khare David Kaplan Matthias J. Aschenbrenner, Committee Chair University of California, Los Angeles 2015 iii To Sam. iv Table of Contents 1 Introduction :::::::::::::::::::::::::::::::::::::: 1 2 Conventions :::::::::::::::::::::::::::::::::::::: 5 3 Metric Geometry ::::::::::::::::::::::::::::::::::: 7 3.1 Metric Spaces . 7 3.2 Maps Between Metric Spaces . 8 3.3 Covers and Packing Inequalities . 9 3.3.1 The 5r-covering Lemma . 9 3.3.2 Doubling Metrics . 10 3.4 Hausdorff Measures and Dimension . 11 3.4.1 Hausdorff Measures . 11 3.4.2 Hausdorff Dimension . 13 3.5 Topological Dimension . 15 3.6 Left-Invariant Metrics on Groups . 15 3.7 Reductions, Ultralimits and Limits of Metric Spaces . 16 3.7.1 Reductions of Λ-valued Metric Spaces . 16 3.7.2 Ultralimits . 17 3.7.3 GH-Convergence and GH-Ultralimits . 18 3.7.4 Asymptotic Cones . 19 3.7.5 Tangent Cones . 22 3.7.6 Conical Metric Spaces . 22 3.8 Normed Spaces . 23 4 T-Convexity :::::::::::::::::::::::::::::::::::::: 24 4.1 T-convex Structures . -
The Mathematical Structure of the Aesthetic
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 12 June 2017 doi:10.20944/preprints201706.0055.v1 Peer-reviewed version available at Philosophies 2017, 2, 14; doi:10.3390/philosophies2030014 Article A New Kind of Aesthetics – The Mathematical Structure of the Aesthetic Akihiro Kubota 1,*, Hirokazu Hori 2, Makoto Naruse 3 and Fuminori Akiba 4 1 Art and Media Course, Department of Information Design, Tama Art University, 2-1723 Yarimizu, Hachioji, Tokyo 192-0394, Japan; [email protected] 2 Interdisciplinary Graduate School, University of Yamanashi, 4-3-11 Takeda, Kofu,Yamanashi 400-8511, Japan; [email protected] 3 Network System Research Institute, National Institute of Information and Communications Technology, 4-2-1 Nukui-kita, Koganei, Tokyo 184-8795, Japan; [email protected] 4 Philosophy of Information Group, Department of Systems and Social Informatics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Aichi 464-8601, Japan; [email protected] * Correspondence: [email protected]; Tel.: +81-42-679-5634 Abstract: This paper proposes a new approach to investigation into the aesthetics. Specifically, it argues that it is possible to explain the aesthetic and its underlying dynamic relations with axiomatic structure (the octahedral axiom derived category) based on contemporary mathematics – namely, category theory – and through this argument suggests the possibility for discussion about the mathematical structure of the aesthetic. If there was a way to describe the structure of aesthetics with the language of mathematical structures and mathematical axioms – a language completely devoid of arbitrariness – then we would make possible a synthetical argument about the essential human activity of “the aesthetics”, and we would also gain a new method and viewpoint on the philosophy and meaning of the act of creating a work of art and artistic activities. -
A Guide to Topology
i i “topguide” — 2010/12/8 — 17:36 — page i — #1 i i A Guide to Topology i i i i i i “topguide” — 2011/2/15 — 16:42 — page ii — #2 i i c 2009 by The Mathematical Association of America (Incorporated) Library of Congress Catalog Card Number 2009929077 Print Edition ISBN 978-0-88385-346-7 Electronic Edition ISBN 978-0-88385-917-9 Printed in the United States of America Current Printing (last digit): 10987654321 i i i i i i “topguide” — 2010/12/8 — 17:36 — page iii — #3 i i The Dolciani Mathematical Expositions NUMBER FORTY MAA Guides # 4 A Guide to Topology Steven G. Krantz Washington University, St. Louis ® Published and Distributed by The Mathematical Association of America i i i i i i “topguide” — 2010/12/8 — 17:36 — page iv — #4 i i DOLCIANI MATHEMATICAL EXPOSITIONS Committee on Books Paul Zorn, Chair Dolciani Mathematical Expositions Editorial Board Underwood Dudley, Editor Jeremy S. Case Rosalie A. Dance Tevian Dray Patricia B. Humphrey Virginia E. Knight Mark A. Peterson Jonathan Rogness Thomas Q. Sibley Joe Alyn Stickles i i i i i i “topguide” — 2010/12/8 — 17:36 — page v — #5 i i The DOLCIANI MATHEMATICAL EXPOSITIONS series of the Mathematical Association of America was established through a generous gift to the Association from Mary P. Dolciani, Professor of Mathematics at Hunter College of the City Uni- versity of New York. In making the gift, Professor Dolciani, herself an exceptionally talented and successfulexpositor of mathematics, had the purpose of furthering the ideal of excellence in mathematical exposition. -
Analysis in Metric Spaces Mario Bonk, Luca Capogna, Piotr Hajłasz, Nageswari Shanmugalingam, and Jeremy Tyson
Analysis in Metric Spaces Mario Bonk, Luca Capogna, Piotr Hajłasz, Nageswari Shanmugalingam, and Jeremy Tyson study of quasiconformal maps on such boundaries moti- The authors of this piece are organizers of the AMS vated Heinonen and Koskela [HK98] to axiomatize several 2020 Mathematics Research Communities summer aspects of Euclidean quasiconformal geometry in the set- conference Analysis in Metric Spaces, one of five ting of metric measure spaces and thereby extend Mostow’s topical research conferences offered this year that are work beyond the sub-Riemannian setting. The ground- focused on collaborative research and professional breaking work [HK98] initiated the modern theory of anal- development for early-career mathematicians. ysis on metric spaces. Additional information can be found at https://www Analysis on metric spaces is nowadays an active and in- .ams.org/programs/research-communities dependent field, bringing together researchers from differ- /2020MRC-MetSpace. Applications are open until ent parts of the mathematical spectrum. It has far-reaching February 15, 2020. applications to areas as diverse as geometric group the- ory, nonlinear PDEs, and even theoretical computer sci- The subject of analysis, more specifically, first-order calcu- ence. As a further sign of recognition, analysis on met- lus, in metric measure spaces provides a unifying frame- ric spaces has been included in the 2010 MSC classifica- work for ideas and questions from many different fields tion as a category (30L: Analysis on metric spaces). In this of mathematics. One of the earliest motivations and ap- short survey, we can discuss only a small fraction of areas plications of this theory arose in Mostow’s work [Mos73], into which analysis on metric spaces has expanded. -
Structure” of Physics: a Case Study∗ (Journal of Philosophy 106 (2009): 57–88)
The “Structure” of Physics: A Case Study∗ (Journal of Philosophy 106 (2009): 57–88) Jill North We are used to talking about the “structure” posited by a given theory of physics. We say that relativity is a theory about spacetime structure. Special relativity posits one spacetime structure; different models of general relativity posit different spacetime structures. We also talk of the “existence” of these structures. Special relativity says that the world’s spacetime structure is Minkowskian: it posits that this spacetime structure exists. Understanding structure in this sense seems important for understand- ing what physics is telling us about the world. But it is not immediately obvious just what this structure is, or what we mean by the existence of one structure, rather than another. The idea of mathematical structure is relatively straightforward. There is geometric structure, topological structure, algebraic structure, and so forth. Mathematical structure tells us how abstract mathematical objects t together to form different types of mathematical spaces. Insofar as we understand mathematical objects, we can understand mathematical structure. Of course, what to say about the nature of mathematical objects is not easy. But there seems to be no further problem for understanding mathematical structure. ∗For comments and discussion, I am extremely grateful to David Albert, Frank Arntzenius, Gordon Belot, Josh Brown, Adam Elga, Branden Fitelson, Peter Forrest, Hans Halvorson, Oliver Davis Johns, James Ladyman, David Malament, Oliver Pooley, Brad Skow, TedSider, Rich Thomason, Jason Turner, Dmitri Tymoczko, the philosophy faculty at Yale, audience members at The University of Michigan in fall 2006, and in 2007 at the Paci c APA, the Joint Session of the Aristotelian Society and Mind Association, and the Bellingham Summer Philosophy Conference. -
Group Theory
Appendix A Group Theory This appendix is a survey of only those topics in group theory that are needed to understand the composition of symmetry transformations and its consequences for fundamental physics. It is intended to be self-contained and covers those topics that are needed to follow the main text. Although in the end this appendix became quite long, a thorough understanding of group theory is possible only by consulting the appropriate literature in addition to this appendix. In order that this book not become too lengthy, proofs of theorems were largely omitted; again I refer to other monographs. From its very title, the book by H. Georgi [211] is the most appropriate if particle physics is the primary focus of interest. The book by G. Costa and G. Fogli [102] is written in the same spirit. Both books also cover the necessary group theory for grand unification ideas. A very comprehensive but also rather dense treatment is given by [428]. Still a classic is [254]; it contains more about the treatment of dynamical symmetries in quantum mechanics. A.1 Basics A.1.1 Definitions: Algebraic Structures From the structureless notion of a set, one can successively generate more and more algebraic structures. Those that play a prominent role in physics are defined in the following. Group A group G is a set with elements gi and an operation ◦ (called group multiplication) with the properties that (i) the operation is closed: gi ◦ g j ∈ G, (ii) a neutral element g0 ∈ G exists such that gi ◦ g0 = g0 ◦ gi = gi , (iii) for every gi exists an −1 ∈ ◦ −1 = = −1 ◦ inverse element gi G such that gi gi g0 gi gi , (iv) the operation is associative: gi ◦ (g j ◦ gk) = (gi ◦ g j ) ◦ gk. -
General Topology
General Topology Tom Leinster 2014{15 Contents A Topological spaces2 A1 Review of metric spaces.......................2 A2 The definition of topological space.................8 A3 Metrics versus topologies....................... 13 A4 Continuous maps........................... 17 A5 When are two spaces homeomorphic?................ 22 A6 Topological properties........................ 26 A7 Bases................................. 28 A8 Closure and interior......................... 31 A9 Subspaces (new spaces from old, 1)................. 35 A10 Products (new spaces from old, 2)................. 39 A11 Quotients (new spaces from old, 3)................. 43 A12 Review of ChapterA......................... 48 B Compactness 51 B1 The definition of compactness.................... 51 B2 Closed bounded intervals are compact............... 55 B3 Compactness and subspaces..................... 56 B4 Compactness and products..................... 58 B5 The compact subsets of Rn ..................... 59 B6 Compactness and quotients (and images)............. 61 B7 Compact metric spaces........................ 64 C Connectedness 68 C1 The definition of connectedness................... 68 C2 Connected subsets of the real line.................. 72 C3 Path-connectedness.......................... 76 C4 Connected-components and path-components........... 80 1 Chapter A Topological spaces A1 Review of metric spaces For the lecture of Thursday, 18 September 2014 Almost everything in this section should have been covered in Honours Analysis, with the possible exception of some of the examples. For that reason, this lecture is longer than usual. Definition A1.1 Let X be a set. A metric on X is a function d: X × X ! [0; 1) with the following three properties: • d(x; y) = 0 () x = y, for x; y 2 X; • d(x; y) + d(y; z) ≥ d(x; z) for all x; y; z 2 X (triangle inequality); • d(x; y) = d(y; x) for all x; y 2 X (symmetry). -
Metric Spaces We Have Talked About the Notion of Convergence in R
Mathematics Department Stanford University Math 61CM – Metric spaces We have talked about the notion of convergence in R: Definition 1 A sequence an 1 of reals converges to ` R if for all " > 0 there exists N N { }n=1 2 2 such that n N, n N implies an ` < ". One writes lim an = `. 2 ≥ | − | With . the standard norm in Rn, one makes the analogous definition: k k n n Definition 2 A sequence xn 1 of points in R converges to x R if for all " > 0 there exists { }n=1 2 N N such that n N, n N implies xn x < ". One writes lim xn = x. 2 2 ≥ k − k One important consequence of the definition in either case is that limits are unique: Lemma 1 Suppose lim xn = x and lim xn = y. Then x = y. Proof: Suppose x = y.Then x y > 0; let " = 1 x y .ThusthereexistsN such that n N 6 k − k 2 k − k 1 ≥ 1 implies x x < ", and N such that n N implies x y < ". Let n = max(N ,N ). Then k n − k 2 ≥ 2 k n − k 1 2 x y x x + x y < 2" = x y , k − kk − nk k n − k k − k which is a contradiction. Thus, x = y. ⇤ Note that the properties of . were not fully used. What we needed is that the function d(x, y)= k k x y was non-negative, equal to 0 only if x = y,symmetric(d(x, y)=d(y, x)) and satisfied the k − k triangle inequality. -
MATH 3210 Metric Spaces
MATH 3210 Metric spaces University of Leeds, School of Mathematics November 29, 2017 Syllabus: 1. Definition and fundamental properties of a metric space. Open sets, closed sets, closure and interior. Convergence of sequences. Continuity of mappings. (6) 2. Real inner-product spaces, orthonormal sequences, perpendicular distance to a subspace, applications in approximation theory. (7) 3. Cauchy sequences, completeness of R with the standard metric; uniform convergence and completeness of C[a; b] with the uniform metric. (3) 4. The contraction mapping theorem, with applications in the solution of equations and differential equations. (5) 5. Connectedness and path-connectedness. Introduction to compactness and sequential compactness, including subsets of Rn. (6) LECTURE 1 Books: Victor Bryant, Metric spaces: iteration and application, Cambridge, 1985. M. O.´ Searc´oid,Metric Spaces, Springer Undergraduate Mathematics Series, 2006. D. Kreider, An introduction to linear analysis, Addison-Wesley, 1966. 1 Metrics, open and closed sets We want to generalise the idea of distance between two points in the real line, given by d(x; y) = jx − yj; and the distance between two points in the plane, given by p 2 2 d(x; y) = d((x1; x2); (y1; y2)) = (x1 − y1) + (x2 − y2) : to other settings. [DIAGRAM] This will include the ideas of distances between functions, for example. 1 1.1 Definition Let X be a non-empty set. A metric on X, or distance function, associates to each pair of elements x, y 2 X a real number d(x; y) such that (i) d(x; y) ≥ 0; and d(x; y) = 0 () x = y (positive definite); (ii) d(x; y) = d(y; x) (symmetric); (iii) d(x; z) ≤ d(x; y) + d(y; z) (triangle inequality). -
Euclidean Space - Wikipedia, the Free Encyclopedia Page 1 of 5
Euclidean space - Wikipedia, the free encyclopedia Page 1 of 5 Euclidean space From Wikipedia, the free encyclopedia In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions. The term “Euclidean” distinguishes these spaces from the curved spaces of non-Euclidean geometry and Einstein's general theory of relativity, and is named for the Greek mathematician Euclid of Alexandria. Classical Greek geometry defined the Euclidean plane and Euclidean three-dimensional space using certain postulates, while the other properties of these spaces were deduced as theorems. In modern mathematics, it is more common to define Euclidean space using Cartesian coordinates and the ideas of analytic geometry. This approach brings the tools of algebra and calculus to bear on questions of geometry, and Every point in three-dimensional has the advantage that it generalizes easily to Euclidean Euclidean space is determined by three spaces of more than three dimensions. coordinates. From the modern viewpoint, there is essentially only one Euclidean space of each dimension. In dimension one this is the real line; in dimension two it is the Cartesian plane; and in higher dimensions it is the real coordinate space with three or more real number coordinates. Thus a point in Euclidean space is a tuple of real numbers, and distances are defined using the Euclidean distance formula. Mathematicians often denote the n-dimensional Euclidean space by , or sometimes if they wish to emphasize its Euclidean nature. Euclidean spaces have finite dimension. Contents 1 Intuitive overview 2 Real coordinate space 3 Euclidean structure 4 Topology of Euclidean space 5 Generalizations 6 See also 7 References Intuitive overview One way to think of the Euclidean plane is as a set of points satisfying certain relationships, expressible in terms of distance and angle. -
Distance Metric Learning, with Application to Clustering with Side-Information
Distance metric learning, with application to clustering with side-information Eric P. Xing, Andrew Y. Ng, Michael I. Jordan and Stuart Russell University of California, Berkeley Berkeley, CA 94720 epxing,ang,jordan,russell ¡ @cs.berkeley.edu Abstract Many algorithms rely critically on being given a good metric over their inputs. For instance, data can often be clustered in many “plausible” ways, and if a clustering algorithm such as K-means initially fails to find one that is meaningful to a user, the only recourse may be for the user to manually tweak the metric until sufficiently good clusters are found. For these and other applications requiring good metrics, it is desirable that we provide a more systematic way for users to indicate what they con- sider “similar.” For instance, we may ask them to provide examples. In this paper, we present an algorithm that, given examples of similar (and, if desired, dissimilar) pairs of points in ¢¤£ , learns a distance metric over ¢¥£ that respects these relationships. Our method is based on posing met- ric learning as a convex optimization problem, which allows us to give efficient, local-optima-free algorithms. We also demonstrate empirically that the learned metrics can be used to significantly improve clustering performance. 1 Introduction The performance of many learning and datamining algorithms depend critically on their being given a good metric over the input space. For instance, K-means, nearest-neighbors classifiers and kernel algorithms such as SVMs all need to be given good metrics that -
The (Homo)Morphism Concept: Didactic Transposition, Meta-Discourse and Thematisation Thomas Hausberger
The (homo)morphism concept: didactic transposition, meta-discourse and thematisation Thomas Hausberger To cite this version: Thomas Hausberger. The (homo)morphism concept: didactic transposition, meta-discourse and the- matisation. International Journal of Research in Undergraduate Mathematics Education, Springer, 2017, 3 (3), pp.417-443. 10.1007/s40753-017-0052-7. hal-01408419 HAL Id: hal-01408419 https://hal.archives-ouvertes.fr/hal-01408419 Submitted on 20 Feb 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Noname manuscript No. (will be inserted by the editor) The (homo)morphism concept: didactic transposition, meta-discourse and thematisation Thomas Hausberger Received: date / Accepted: date Abstract This article focuses on the didactic transposition of the homomor- phism concept and on the elaboration and evaluation of an activity dedicated to the teaching of this fundamental concept in Abstract Algebra. It does not restrict to Group Theory but on the contrary raises the issue of the teaching and learning of algebraic structuralism, thus bridging Group and Ring Theo- ries and highlighting the phenomenon of thematisation. Emphasis is made on epistemological analysis and its interaction with didactics. The rationale of the isomorphism and homomorphism concepts is discussed, in particular through a textbook analysis focusing on the meta-discourse that mathematicians offer to illuminate the concepts.