Visualization of Quaternions with Clifford Parallelism
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Spheres in Infinite-Dimensional Normed Spaces Are Lipschitz Contractible
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 88. Number 3, July 1983 SPHERES IN INFINITE-DIMENSIONAL NORMED SPACES ARE LIPSCHITZ CONTRACTIBLE Y. BENYAMINI1 AND Y. STERNFELD Abstract. Let X be an infinite-dimensional normed space. We prove the following: (i) The unit sphere {x G X: || x II = 1} is Lipschitz contractible. (ii) There is a Lipschitz retraction from the unit ball of JConto the unit sphere. (iii) There is a Lipschitz map T of the unit ball into itself without an approximate fixed point, i.e. inffjjc - Tx\\: \\x\\ « 1} > 0. Introduction. Let A be a normed space, and let Bx — {jc G X: \\x\\ < 1} and Sx = {jc G X: || jc || = 1} be its unit ball and unit sphere, respectively. Brouwer's fixed point theorem states that when X is finite dimensional, every continuous self-map of Bx admits a fixed point. Two equivalent formulations of this theorem are the following. 1. There is no continuous retraction from Bx onto Sx. 2. Sx is not contractible, i.e., the identity map on Sx is not homotopic to a constant map. It is well known that none of these three theorems hold in infinite-dimensional spaces (see e.g. [1]). The natural generalization to infinite-dimensional spaces, however, would seem to require the maps to be uniformly-continuous and not merely continuous. Indeed in the finite-dimensional case this condition is automatically satisfied. In this article we show that the above three theorems fail, in the infinite-dimen- sional case, even under the strongest uniform-continuity condition, namely, for maps satisfying a Lipschitz condition. -
Mostly Surfaces Richard Evan Schwartz
Mostly Surfaces Richard Evan Schwartz Department of Mathematics, Brown University Current address: 151 Thayer St. Providence, RI 02912 E-mail address: [email protected] 2010 Mathematics Subject Classification. Primary Key words and phrases. surfaces, geometry, topology, complex analysis supported by N.S.F. grant DMS-0604426. Contents Preface xiii Chapter 1. Book Overview 1 1.1. Behold, the Torus! 1 § 1.2. Gluing Polygons 3 § 1.3.DrawingonaSurface 5 § 1.4. Covering Spaces 8 § 1.5. HyperbolicGeometryandtheOctagon 9 § 1.6. Complex Analysis and Riemann Surfaces 11 § 1.7. ConeSurfacesandTranslationSurfaces 13 § 1.8. TheModularGroupandtheVeechGroup 14 § 1.9. Moduli Space 16 § 1.10. Dessert 17 § Part 1. Surfaces and Topology Chapter2. DefinitionofaSurface 21 2.1. A Word about Sets 21 § 2.2. Metric Spaces 22 § 2.3. OpenandClosedSets 23 § 2.4. Continuous Maps 24 § v vi Contents 2.5. Homeomorphisms 25 § 2.6. Compactness 26 § 2.7. Surfaces 26 § 2.8. Manifolds 27 § Chapter3. TheGluingConstruction 31 3.1. GluingSpacesTogether 31 § 3.2. TheGluingConstructioninAction 34 § 3.3. The Classification of Surfaces 36 § 3.4. TheEulerCharacteristic 38 § Chapter 4. The Fundamental Group 43 4.1. A Primer on Groups 43 § 4.2. Homotopy Equivalence 45 § 4.3. The Fundamental Group 46 § 4.4. Changing the Basepoint 48 § 4.5. Functoriality 49 § 4.6. Some First Steps 51 § Chapter 5. Examples of Fundamental Groups 53 5.1.TheWindingNumber 53 § 5.2. The Circle 56 § 5.3. The Fundamental Theorem of Algebra 57 § 5.4. The Torus 58 § 5.5. The 2-Sphere 58 § 5.6. TheProjectivePlane 59 § 5.7. A Lens Space 59 § 5.8. -
Examples of Manifolds
Examples of Manifolds Example 1 (Open Subset of IRn) Any open subset, O, of IRn is a manifold of dimension n. One possible atlas is A = (O, ϕid) , where ϕid is the identity map. That is, ϕid(x) = x. n Of course one possible choice of O is IR itself. Example 2 (The Circle) The circle S1 = (x,y) ∈ IR2 x2 + y2 = 1 is a manifold of dimension one. One possible atlas is A = {(U , ϕ ), (U , ϕ )} where 1 1 1 2 1 y U1 = S \{(−1, 0)} ϕ1(x,y) = arctan x with − π < ϕ1(x,y) <π ϕ1 1 y U2 = S \{(1, 0)} ϕ2(x,y) = arctan x with 0 < ϕ2(x,y) < 2π U1 n n n+1 2 2 Example 3 (S ) The n–sphere S = x =(x1, ··· ,xn+1) ∈ IR x1 +···+xn+1 =1 n A U , ϕ , V ,ψ i n is a manifold of dimension . One possible atlas is 1 = ( i i) ( i i) 1 ≤ ≤ +1 where, for each 1 ≤ i ≤ n + 1, n Ui = (x1, ··· ,xn+1) ∈ S xi > 0 ϕi(x1, ··· ,xn+1)=(x1, ··· ,xi−1,xi+1, ··· ,xn+1) n Vi = (x1, ··· ,xn+1) ∈ S xi < 0 ψi(x1, ··· ,xn+1)=(x1, ··· ,xi−1,xi+1, ··· ,xn+1) n So both ϕi and ψi project onto IR , viewed as the hyperplane xi = 0. Another possible atlas is n n A2 = S \{(0, ··· , 0, 1)}, ϕ , S \{(0, ··· , 0, −1)},ψ where 2x1 2xn ϕ(x , ··· ,xn ) = , ··· , 1 +1 1−xn+1 1−xn+1 2x1 2xn ψ(x , ··· ,xn ) = , ··· , 1 +1 1+xn+1 1+xn+1 are the stereographic projections from the north and south poles, respectively. -
Convolution on the N-Sphere with Application to PDF Modeling Ivan Dokmanic´, Student Member, IEEE, and Davor Petrinovic´, Member, IEEE
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 3, MARCH 2010 1157 Convolution on the n-Sphere With Application to PDF Modeling Ivan Dokmanic´, Student Member, IEEE, and Davor Petrinovic´, Member, IEEE Abstract—In this paper, we derive an explicit form of the convo- emphasis on wavelet transform in [8]–[12]. Computation of the lution theorem for functions on an -sphere. Our motivation comes Fourier transform and convolution on groups is studied within from the design of a probability density estimator for -dimen- the theory of noncommutative harmonic analysis. Examples sional random vectors. We propose a probability density function (pdf) estimation method that uses the derived convolution result of applications of noncommutative harmonic analysis in engi- on . Random samples are mapped onto the -sphere and esti- neering are analysis of the motion of a rigid body, workspace mation is performed in the new domain by convolving the samples generation in robotics, template matching in image processing, with the smoothing kernel density. The convolution is carried out tomography, etc. A comprehensive list with accompanying in the spectral domain. Samples are mapped between the -sphere theory and explanations is given in [13]. and the -dimensional Euclidean space by the generalized stereo- graphic projection. We apply the proposed model to several syn- Statistics of random vectors whose realizations are observed thetic and real-world data sets and discuss the results. along manifolds embedded in Euclidean spaces are commonly termed directional statistics. An excellent review may be found Index Terms—Convolution, density estimation, hypersphere, hy- perspherical harmonics, -sphere, rotations, spherical harmonics. in [14]. It is of interest to develop tools for the directional sta- tistics in analogy with the ordinary Euclidean. -
Minkowski Products of Unit Quaternion Sets 1 Introduction
Minkowski products of unit quaternion sets 1 Introduction The Minkowski sum A⊕B of two point sets A; B 2 Rn is the set of all points generated [16] by the vector sums of points chosen independently from those sets, i.e., A ⊕ B := f a + b : a 2 A and b 2 B g : (1) The Minkowski sum has applications in computer graphics, geometric design, image processing, and related fields [9, 11, 12, 13, 14, 15, 20]. The validity of the definition (1) in Rn for all n ≥ 1 stems from the straightforward extension of the vector sum a + b to higher{dimensional Euclidean spaces. However, to define a Minkowski product set A ⊗ B := f a b : a 2 A and b 2 B g ; (2) it is necessary to specify products of points in Rn. In the case n = 1, this is simply the real{number product | the resulting algebra of point sets in R1 is called interval arithmetic [17, 18] and is used to monitor the propagation of uncertainty through computations in which the initial operands (and possibly also the arithmetic operations) are not precisely determined. A natural realization of the Minkowski product (2) in R2 may be achieved [7] by interpreting the points a and b as complex numbers, with a b being the usual complex{number product. Algorithms to compute Minkowski products of complex{number sets have been formulated [6], and extended to determine Minkowski roots and powers [3, 8] of complex sets; to evaluate polynomials specified by complex{set coefficients and arguments [4]; and to solve simple equations expressed in terms of complex{set coefficients and unknowns [5]. -
GEOMETRY Contents 1. Euclidean Geometry 2 1.1. Metric Spaces 2 1.2
GEOMETRY JOUNI PARKKONEN Contents 1. Euclidean geometry 2 1.1. Metric spaces 2 1.2. Euclidean space 2 1.3. Isometries 4 2. The sphere 7 2.1. More on cosine and sine laws 10 2.2. Isometries 11 2.3. Classification of isometries 12 3. Map projections 14 3.1. The latitude-longitude map 14 3.2. Stereographic projection 14 3.3. Inversion 14 3.4. Mercator’s projection 16 3.5. Some Riemannian geometry. 18 3.6. Cylindrical projection 18 4. Triangles in the sphere 19 5. Minkowski space 21 5.1. Bilinear forms and Minkowski space 21 5.2. The orthogonal group 22 6. Hyperbolic space 24 6.1. Isometries 25 7. Models of hyperbolic space 30 7.1. Klein’s model 30 7.2. Poincaré’s ball model 30 7.3. The upper halfspace model 31 8. Some geometry and techniques 32 8.1. Triangles 32 8.2. Geodesic lines and isometries 33 8.3. Balls 35 9. Riemannian metrics, area and volume 36 Last update: December 12, 2014. 1 1. Euclidean geometry 1.1. Metric spaces. A function d: X × X ! [0; +1[ is a metric in the nonempty set X if it satisfies the following properties (1) d(x; x) = 0 for all x 2 X and d(x; y) > 0 if x 6= y, (2) d(x; y) = d(y; x) for all x; y 2 X, and (3) d(x; y) ≤ d(x; z) + d(z; y) for all x; y; z 2 X (the triangle inequality). The pair (X; d) is a metric space. -
Lp Unit Spheres and the Α-Geometries: Questions and Perspectives
entropy Article Lp Unit Spheres and the a-Geometries: Questions and Perspectives Paolo Gibilisco Department of Economics and Finance, University of Rome “Tor Vergata”, Via Columbia 2, 00133 Rome, Italy; [email protected] Received: 12 November 2020; Accepted: 10 December 2020; Published: 14 December 2020 Abstract: In Information Geometry, the unit sphere of Lp spaces plays an important role. In this paper, the aim is list a number of open problems, in classical and quantum IG, which are related to Lp geometry. Keywords: Lp spheres; a-geometries; a-Proudman–Johnson equations Gentlemen: there’s lots of room left in Lp spaces. 1. Introduction Chentsov theorem is the fundamental theorem in Information Geometry. After Rao’s remark on the geometric nature of the Fisher Information (in what follows shortly FI), it is Chentsov who showed that on the simplex of the probability vectors, up to scalars, FI is the unique Riemannian geometry, which “contract under noise” (to have an idea of recent developments about this see [1]). So FI appears as the “natural” Riemannian geometry over the manifolds of density vectors, namely over 1 n Pn := fr 2 R j ∑ ri = 1, ri > 0g i Since FI is the pull-back of the map p r ! 2 r it is natural to study the geometries induced on the simplex of probability vectors by the embeddings 8 1 <p · r p p 2 [1, +¥) Ap(r) = :log(r) p = +¥ Setting 2 p = a 2 [−1, 1] 1 − a we call the corresponding geometries on the simplex of probability vectors a-geometries (first studied by Chentsov himself). -
Oriented Projective Geometry for Computer Vision
Oriented Projective Geometry for Computer Vision St~phane Laveau Olivier Faugeras e-mall: Stephane. Laveau@sophia. inria, fr, Olivier.Faugeras@sophia. inria, fr INRIA. 2004, route des Lucioles. B.R 93. 06902 Sophia-Antipolis. FRANCE. Abstract. We present an extension of the usual projective geometric framework for computer vision which can nicely take into account an information that was previously not used, i.e. the fact that the pixels in an image correspond to points which lie in front of the camera. This framework, called the oriented projective geometry, retains all the advantages of the unoriented projective geometry, namely its simplicity for expressing the viewing geometry of a system of cameras, while extending its adequation to model realistic situations. We discuss the mathematical and practical issues raised by this new framework for a number of computer vision algorithms. We present different experiments where this new tool clearly helps. 1 Introduction Projective geometry is now established as the correct and most convenient way to de- scribe the geometry of systems of cameras and the geometry of the scene they record. The reason for this is that a pinhole camera, a very reasonable model for most cameras, is really a projective (in the sense of projective geometry) engine projecting (in the usual sense) the real world onto the retinal plane. Therefore we gain a lot in simplicity if we represent the real world as a part of a projective 3-D space and the retina as a part of a projective 2-D space. But in using such a representation, we apparently loose information: we are used to think of the applications of computer vision as requiring a Euclidean space and this notion is lacking in the projective space. -
17 Measure Concentration for the Sphere
17 Measure Concentration for the Sphere In today’s lecture, we will prove the measure concentration theorem for the sphere. Recall that this was one of the vital steps in the analysis of the Arora-Rao-Vazirani approximation algorithm for sparsest cut. Most of the material in today’s lecture is adapted from Matousek’s book [Mat02, chapter 14] and Keith Ball’s lecture notes on convex geometry [Bal97]. n n−1 Notation: We will use the notation Bn to denote the ball of unit radius in R and S n to denote the sphere of unit radius in R . Let µ denote the normalized measure on the unit sphere (i.e., for any measurable set S ⊆ Sn−1, µ(A) denotes the ratio of the surface area of µ to the entire surface area of the sphere Sn−1). Recall that the n-dimensional volume of a ball n n n of radius r in R is given by the formula Vol(Bn) · r = vn · r where πn/2 vn = n Γ 2 + 1 Z ∞ where Γ(x) = tx−1e−tdt 0 n−1 The surface area of the unit sphere S is nvn. Theorem 17.1 (Measure Concentration for the Sphere Sn−1) Let A ⊆ Sn−1 be a mea- n−1 surable subset of the unit sphere S such that µ(A) = 1/2. Let Aδ denote the δ-neighborhood n−1 n−1 of A in S . i.e., Aδ = {x ∈ S |∃z ∈ A, ||x − z||2 ≤ δ}. Then, −nδ2/2 µ(Aδ) ≥ 1 − 2e . Thus, the above theorem states that if A is any set of measure 0.5, taking a step of even √ O (1/ n) around A covers almost 99% of the entire sphere. -
On Expansive Mappings
mathematics Article On Expansive Mappings Marat V. Markin and Edward S. Sichel * Department of Mathematics, California State University, Fresno 5245 N. Backer Avenue, M/S PB 108, Fresno, CA 93740-8001, USA; [email protected] * Correspondence: [email protected] Received: 23 August 2019; Accepted: 17 October 2019; Published: 23 October 2019 Abstract: When finding an original proof to a known result describing expansive mappings on compact metric spaces as surjective isometries, we reveal that relaxing the condition of compactness to total boundedness preserves the isometry property and nearly that of surjectivity. While a counterexample is found showing that the converse to the above descriptions do not hold, we are able to characterize boundedness in terms of specific expansions we call anticontractions. Keywords: Metric space; expansion; compactness; total boundedness MSC: Primary 54E40; 54E45; Secondary 46T99 O God, I could be bounded in a nutshell, and count myself a king of infinite space - were it not that I have bad dreams. William Shakespeare (Hamlet, Act 2, Scene 2) 1. Introduction We take a close look at the nature of expansive mappings on certain metric spaces (compact, totally bounded, and bounded), provide a finer classification for such mappings, and use them to characterize boundedness. When finding an original proof to a known result describing all expansive mappings on compact metric spaces as surjective isometries [1] (Problem X.5.13∗), we reveal that relaxing the condition of compactness to total boundedness still preserves the isometry property and nearly that of surjectivity. We provide a counterexample of a not totally bounded metric space, on which the only expansion is the identity mapping, demonstrating that the converse to the above descriptions do not hold. -
Correlators of Wilson Loops on Hopf Fibrations
UNIVERSITA` DEGLI STUDI DI PARMA Dottorato di Ricerca in Fisica Ciclo XXV Correlators of Wilson loops on Hopf fibrations in the AdS5=CFT4 correspondence Coordinatore: Chiar.mo Prof. PIERPAOLO LOTTICI Supervisore: Dott. LUCA GRIGUOLO Dottorando: STEFANO MORI Anno Accademico 2012/2013 ויאמר אלהיM יהי אור ויהי אור בראשׂיה 1:3 Io stimo pi`uil trovar un vero, bench´edi cosa leggiera, che 'l disputar lungamente delle massime questioni senza conseguir verit`anissuna. (Galileo Galilei, Scritti letterari) Contents 1 The correspondence and its observables 1 1.1 AdS5=CFT4 correspondence . .1 1.1.1 N = 4 SYM . .1 1.1.2 AdS space . .2 1.2 Brane construction . .4 1.3 Symmetries matching . .6 1.4 AdS/CFT dictionary . .7 1.5 Integrability . .9 1.6 Wilson loops and Minimal surfaces . 10 1.7 Mixed correlation functions and local operator insertion . 13 1.8 Main results from the correspondence . 14 2 Wilson Loops and Supersymmetry 17 2.1 BPS configurations . 17 2.2 Zarembo supersymmetric loops . 18 2.3 DGRT loops . 20 2.3.1 Hopf fibers . 23 2.4 Matrix model . 25 2.5 Calibrated surfaces . 27 3 Strong coupling results 31 3.1 Basic examples . 31 3.1.1 Straight line . 31 3.1.2 Cirle . 32 3.1.3 Antiparallel lines . 33 3.1.4 1/4 BPS circular loop . 33 3.2 Systematic regularization . 35 3.3 Ansatz and excited charges . 36 3.4 Hints of 1-loop computation . 37 v 4 Hopf fibers correlators 41 4.1 Strong coupling solution . 41 4.1.1 S5 motion . -
Metric Spaces and Continuity This Publication Forms Part of an Open University Module
M303 Further pure mathematics Metric spaces and continuity This publication forms part of an Open University module. Details of this and other Open University modules can be obtained from the Student Registration and Enquiry Service, The Open University, PO Box 197, Milton Keynes MK7 6BJ, United Kingdom (tel. +44 (0)845 300 6090; email [email protected]). Alternatively, you may visit the Open University website at www.open.ac.uk where you can learn more about the wide range of modules and packs offered at all levels by The Open University. Note to reader Mathematical/statistical content at the Open University is usually provided to students in printed books, with PDFs of the same online. This format ensures that mathematical notation is presented accurately and clearly. The PDF of this extract thus shows the content exactly as it would be seen by an Open University student. Please note that the PDF may contain references to other parts of the module and/or to software or audio-visual components of the module. Regrettably mathematical and statistical content in PDF files is unlikely to be accessible using a screenreader, and some OpenLearn units may have PDF files that are not searchable. You may need additional help to read these documents. The Open University, Walton Hall, Milton Keynes, MK7 6AA. First published 2014. Second edition 2016. Copyright c 2014, 2016 The Open University All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, transmitted or utilised in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without written permission from the publisher or a licence from the Copyright Licensing Agency Ltd.