Dissertation Discrete-Time Topological Dynamics
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WHAT IS a CHAOTIC ATTRACTOR? 1. Introduction J. Yorke Coined the Word 'Chaos' As Applied to Deterministic Systems. R. Devane
WHAT IS A CHAOTIC ATTRACTOR? CLARK ROBINSON Abstract. Devaney gave a mathematical definition of the term chaos, which had earlier been introduced by Yorke. We discuss issues involved in choosing the properties that characterize chaos. We also discuss how this term can be combined with the definition of an attractor. 1. Introduction J. Yorke coined the word `chaos' as applied to deterministic systems. R. Devaney gave the first mathematical definition for a map to be chaotic on the whole space where a map is defined. Since that time, there have been several different definitions of chaos which emphasize different aspects of the map. Some of these are more computable and others are more mathematical. See [9] a comparison of many of these definitions. There is probably no one best or correct definition of chaos. In this paper, we discuss what we feel is one of better mathematical definition. (It may not be as computable as some of the other definitions, e.g., the one by Alligood, Sauer, and Yorke.) Our definition is very similar to the one given by Martelli in [8] and [9]. We also combine the concepts of chaos and attractors and discuss chaotic attractors. 2. Basic definitions We start by giving the basic definitions needed to define a chaotic attractor. We give the definitions for a diffeomorphism (or map), but those for a system of differential equations are similar. The orbit of a point x∗ by F is the set O(x∗; F) = f Fi(x∗) : i 2 Z g. An invariant set for a diffeomorphism F is an set A in the domain such that F(A) = A. -
MA427 Ergodic Theory
MA427 Ergodic Theory Course Notes (2012-13) 1 Introduction 1.1 Orbits Let X be a mathematical space. For example, X could be the unit interval [0; 1], a circle, a torus, or something far more complicated like a Cantor set. Let T : X ! X be a function that maps X into itself. Let x 2 X be a point. We can repeatedly apply the map T to the point x to obtain the sequence: fx; T (x);T (T (x));T (T (T (x))); : : : ; :::g: We will often write T n(x) = T (··· (T (T (x)))) (n times). The sequence of points x; T (x);T 2(x);::: is called the orbit of x. We think of applying the map T as the passage of time. Thus we think of T (x) as where the point x has moved to after time 1, T 2(x) is where the point x has moved to after time 2, etc. Some points x 2 X return to where they started. That is, T n(x) = x for some n > 1. We say that such a point x is periodic with period n. By way of contrast, points may move move densely around the space X. (A sequence is said to be dense if (loosely speaking) it comes arbitrarily close to every point of X.) If we take two points x; y of X that start very close then their orbits will initially be close. However, it often happens that in the long term their orbits move apart and indeed become dramatically different. -
A Finiteness Result for Circulant Core Complex Hadamard Matrices
A FINITENESS RESULT FOR CIRCULANT CORE COMPLEX HADAMARD MATRICES REMUS NICOARA AND CHASE WORLEY UNIVERSITY OF TENNESSEE, KNOXVILLE Abstract. We show that for every prime number p there exist finitely many circulant core complex Hadamard matrices of size p ` 1. The proof uses a 'derivative at infinity' argument to reduce the problem to Tao's uncertainty principle for cyclic groups of prime order (see [Tao]). 1. Introduction A complex Hadamard matrix is a matrix H P MnpCq having all entries of absolute value 1 and all rows ?1 mutually orthogonal. Equivalently, n H is a unitary matrix with all entries of the same absolute value. For ij 2πi{n example, the Fourier matrix Fn “ p! q1¤i;j¤n, ! “ e , is a Hadamard matrix. In the recent years, complex Hadamard matrices have found applications in various topics of mathematics and physics, such as quantum information theory, error correcting codes, cyclic n-roots, spectral sets and Fuglede's conjecture. A general classification of real or complex Hadamard matrices is not available. A catalogue of most known complex Hadamard matrices can be found in [TaZy]. The complete classification is known for n ¤ 5 ([Ha1]) and for self-adjoint matrices of order 6 ([BeN]). Hadamard matrices arise in operator algebras as construction data for hyperfinite subfactors. A unitary ?1 matrix U is of the form n H, H Hadamard matrix, if and only if the algebra of n ˆ n diagonal matrices Dn ˚ is orthogonal onto UDnU , with respect to the inner product given by the trace on MnpCq. Equivalently, the square of inclusions: Dn Ă MnpCq CpHq “ ¨ YY ; τ˛ ˚ ˚ ‹ ˚ C Ă UDnU ‹ ˚ ‹ ˝ ‚ is a commuting square, in the sense of [Po1],[Po2], [JS]. -
Robert De Montessus De Ballore's 1902 Theorem on Algebraic
Robert de Montessus de Ballore’s 1902 theorem on algebraic continued fractions : genesis and circulation Hervé Le Ferrand ∗ October 31, 2018 Abstract Robert de Montessus de Ballore proved in 1902 his famous theorem on the convergence of Padé approximants of meromorphic functions. In this paper, we will first describe the genesis of the theorem, then investigate its circulation. A number of letters addressed to Robert de Montessus by different mathematicians will be quoted to help determining the scientific context and the steps that led to the result. In particular, excerpts of the correspondence with Henri Padé in the years 1901-1902 played a leading role. The large number of authors who mentioned the theorem soon after its derivation, for instance Nörlund and Perron among others, indicates a fast circulation due to factors that will be thoroughly explained. Key words Robert de Montessus, circulation of a theorem, algebraic continued fractions, Padé’s approximants. MSC : 01A55 ; 01A60 1 Introduction This paper aims to study the genesis and circulation of the theorem on convergence of algebraic continued fractions proven by the French mathematician Robert de Montessus de Ballore (1870-1937) in 1902. The main issue is the following : which factors played a role in the elaboration then the use of this new result ? Inspired by the study of Sturm’s theorem by Hourya Sinaceur [52], the scientific context of Robert de Montessus’ research will be described. Additionally, the correlation with the other topics on which he worked will be highlighted, -
Hadamard Equiangular Tight Frames
Air Force Institute of Technology AFIT Scholar Faculty Publications 8-8-2019 Hadamard Equiangular Tight Frames Matthew C. Fickus Air Force Institute of Technology John Jasper University of Cincinnati Dustin G. Mixon Air Force Institute of Technology Jesse D. Peterson Air Force Institute of Technology Follow this and additional works at: https://scholar.afit.edu/facpub Part of the Discrete Mathematics and Combinatorics Commons Recommended Citation Fickus, M., Jasper, J., Mixon, D. G., & Peterson, J. D. (2021). Hadamard Equiangular Tight Frames. Applied and Computational Harmonic Analysis, 50, 281–302. https://doi.org/10.1016/j.acha.2019.08.003 This Article is brought to you for free and open access by AFIT Scholar. It has been accepted for inclusion in Faculty Publications by an authorized administrator of AFIT Scholar. For more information, please contact [email protected]. Hadamard Equiangular Tight Frames Matthew Fickusa, John Jasperb, Dustin G. Mixona, Jesse D. Petersona aDepartment of Mathematics and Statistics, Air Force Institute of Technology, Wright-Patterson AFB, OH 45433 bDepartment of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221 Abstract An equiangular tight frame (ETF) is a type of optimal packing of lines in Euclidean space. They are often represented as the columns of a short, fat matrix. In certain applications we want this matrix to be flat, that is, have the property that all of its entries have modulus one. In particular, real flat ETFs are equivalent to self-complementary binary codes that achieve the Grey-Rankin bound. Some flat ETFs are (complex) Hadamard ETFs, meaning they arise by extracting rows from a (complex) Hadamard matrix. -
ROTATION THEORY These Notes Present a Brief Survey of Some
ROTATION THEORY MICHALMISIUREWICZ Abstract. Rotation Theory has its roots in the theory of rotation numbers for circle homeomorphisms, developed by Poincar´e. It is particularly useful for the study and classification of periodic orbits of dynamical systems. It deals with ergodic averages and their limits, not only for almost all points, like in Ergodic Theory, but for all points. We present the general ideas of Rotation Theory and its applications to some classes of dynamical systems, like continuous circle maps homotopic to the identity, torus homeomorphisms homotopic to the identity, subshifts of finite type and continuous interval maps. These notes present a brief survey of some aspects of Rotation Theory. Deeper treatment of the subject would require writing a book. Thus, in particular: • Not all aspects of Rotation Theory are described here. A large part of it, very important and deserving a separate book, deals with homeomorphisms of an annulus, homotopic to the identity. Including this subject in the present notes would make them at least twice as long, so it is ignored, except for the bibliography. • What is called “proofs” are usually only short ideas of the proofs. The reader is advised either to try to fill in the details himself/herself or to find the full proofs in the literature. More complicated proofs are omitted entirely. • The stress is on the theory, not on the history. Therefore, references are not cited in the text. Instead, at the end of these notes there are lists of references dealing with the problems treated in various sections (or not treated at all). -
The Early View of “Global Journal of Computer Science And
The Early View of “Global Journal of Computer Science and Technology” In case of any minor updation/modification/correction, kindly inform within 3 working days after you have received this. Kindly note, the Research papers may be removed, added, or altered according to the final status. Global Journal of Computer Science and Technology View Early Global Journal of Computer Science and Technology Volume 10 Issue 12 (Ver. 1.0) View Early Global Academy of Research and Development © Global Journal of Computer Global Journals Inc. Science and Technology. (A Delaware USA Incorporation with “Good Standing”; Reg. Number: 0423089) Sponsors: Global Association of Research 2010. Open Scientific Standards All rights reserved. Publisher’s Headquarters office This is a special issue published in version 1.0 of “Global Journal of Medical Research.” By Global Journals Inc. Global Journals Inc., Headquarters Corporate Office, All articles are open access articles distributed Cambridge Office Center, II Canal Park, Floor No. under “Global Journal of Medical Research” 5th, Cambridge (Massachusetts), Pin: MA 02141 Reading License, which permits restricted use. Entire contents are copyright by of “Global United States Journal of Medical Research” unless USA Toll Free: +001-888-839-7392 otherwise noted on specific articles. USA Toll Free Fax: +001-888-839-7392 No part of this publication may be reproduced Offset Typesetting or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information Global Journals Inc., City Center Office, 25200 storage and retrieval system, without written permission. Carlos Bee Blvd. #495, Hayward Pin: CA 94542 The opinions and statements made in this United States book are those of the authors concerned. -
Asymptotic Existence of Hadamard Matrices
ASYMPTOTIC EXISTENCE OF HADAMARD MATRICES by Ivan Livinskyi A Thesis submitted to the Faculty of Graduate Studies of The University of Manitoba in partial fulfilment of the requirements of the degree of MASTER OF SCIENCE Department of Mathematics University of Manitoba Winnipeg Copyright ⃝c 2012 by Ivan Livinskyi Abstract We make use of a structure known as signed groups, and known sequences with zero autocorrelation to derive new results on the asymptotic existence of Hadamard matrices. For any positive odd integer p it is obtained that a Hadamard matrix of order 2tp exists for all ( ) 1 p − 1 t ≥ log + 13: 5 2 2 i Contents 1 Introduction. Hadamard matrices 2 2 Signed groups and remreps 10 3 Construction of signed group Hadamard matrices from sequences 18 4 Some classes of sequences with zero autocorrelation 28 4.1 Golay sequences .............................. 30 4.2 Base sequences .............................. 31 4.3 Complex Golay sequences ........................ 33 4.4 Normal sequences ............................. 35 4.5 Turyn sequences .............................. 36 4.6 Other sequences .............................. 37 5 Asymptotic Existence results 40 5.1 Asymptotic formulas based on Golay sequences . 40 5.2 Hadamard matrices from complex Golay sequences . 43 5.3 Hadamard matrices from all complementary sequences considered . 44 6 Thoughts about further development 48 Bibliography 51 1 Chapter 1 Introduction. Hadamard matrices A Hadamard matrix H is a square (±1)-matrix such that any two of its rows are orthogonal. In other words, a square (±1)-matrix H is Hadamard if and only if HH> = nI; where n is the order of H and I is the identity matrix of order n. -
The Mathematical Heritage of Henri Poincaré
http://dx.doi.org/10.1090/pspum/039.1 THE MATHEMATICAL HERITAGE of HENRI POINCARE PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS Volume 39, Part 1 THE MATHEMATICAL HERITAGE Of HENRI POINCARE AMERICAN MATHEMATICAL SOCIETY PROVIDENCE, RHODE ISLAND PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 39 PROCEEDINGS OF THE SYMPOSIUM ON THE MATHEMATICAL HERITAGE OF HENRI POINCARfe HELD AT INDIANA UNIVERSITY BLOOMINGTON, INDIANA APRIL 7-10, 1980 EDITED BY FELIX E. BROWDER Prepared by the American Mathematical Society with partial support from National Science Foundation grant MCS 79-22916 1980 Mathematics Subject Classification. Primary 01-XX, 14-XX, 22-XX, 30-XX, 32-XX, 34-XX, 35-XX, 47-XX, 53-XX, 55-XX, 57-XX, 58-XX, 70-XX, 76-XX, 83-XX. Library of Congress Cataloging in Publication Data Main entry under title: The Mathematical Heritage of Henri Poincare\ (Proceedings of symposia in pure mathematics; v. 39, pt. 1— ) Bibliography: p. 1. Mathematics—Congresses. 2. Poincare', Henri, 1854—1912— Congresses. I. Browder, Felix E. II. Series: Proceedings of symposia in pure mathematics; v. 39, pt. 1, etc. QA1.M4266 1983 510 83-2774 ISBN 0-8218-1442-7 (set) ISBN 0-8218-1449-4 (part 2) ISBN 0-8218-1448-6 (part 1) ISSN 0082-0717 COPYING AND REPRINTING. Individual readers of this publication, and nonprofit librar• ies acting for them are permitted to make fair use of the material, such as to copy an article for use in teaching or research. Permission is granted to quote brief passages from this publication in re• views provided the customary acknowledgement of the source is given. -
One Method for Construction of Inverse Orthogonal Matrices
ONE METHOD FOR CONSTRUCTION OF INVERSE ORTHOGONAL MATRICES, P. DIÞÃ National Institute of Physics and Nuclear Engineering, P.O. Box MG6, Bucharest, Romania E-mail: [email protected] Received September 16, 2008 An analytical procedure to obtain parametrisation of complex inverse orthogonal matrices starting from complex inverse orthogonal conference matrices is provided. These matrices, which depend on nonzero complex parameters, generalize the complex Hadamard matrices and have applications in spin models and digital signal processing. When the free complex parameters take values on the unit circle they transform into complex Hadamard matrices. 1. INTRODUCTION In a seminal paper [1] Sylvester defined a general class of orthogonal matrices named inverse orthogonal matrices and provided the first examples of what are nowadays called (complex) Hadamard matrices. A particular class of inverse orthogonal matrices Aa()ij are those matrices whose inverse is given 1 t by Aa(1ij ) (1 a ji ) , where t means transpose, and their entries aij satisfy the relation 1 AA nIn (1) where In is the n-dimensional unit matrix. When the entries aij take values on the unit circle A–1 coincides with the Hermitian conjugate A* of A, and in this case (1) is the definition of complex Hadamard matrices. Complex Hadamard matrices have applications in quantum information theory, several branches of combinatorics, digital signal processing, etc. Complex orthogonal matrices unexpectedly appeared in the description of topological invariants of knots and links, see e.g. Ref. [2]. They were called two- weight spin models being related to symmetric statistical Potts models. These matrices have been generalized to two-weight spin models, also called , Paper presented at the National Conference of Physics, September 10–13, 2008, Bucharest-Mãgurele, Romania. -
Introduction to Linear Algebra
Introduction to linear algebra Teo Banica Real matrices, The determinant, Complex matrices, Calculus and applications, Infinite matrices, Special matrices 08/20 Foreword These are slides written in the Fall 2020, on linear algebra. Presentations available at my Youtube channel. 1. Real matrices and their properties ... 3 2. The determinant of real matrices ... 19 3. Complex matrices and diagonalization ... 35 4. Linear algebra and calculus questions ... 51 5. Infinite matrices and spectral theory ... 67 6. Special matrices and matrix tricks ... 83 Real matrices and their properties Teo Banica "Introduction to linear algebra", 1/6 08/20 Rotations 1/3 Problem: what’s the formula of the rotation of angle t? Rotations 2/3 2 x The points in the plane R can be represented as vectors y . The 2 × 2 matrices “act” on such vectors, as follows: a b x ax + by = c d y cx + dy Many simple transformations (symmetries, projections..) can be written in this form. What about the rotation of angle t? Rotations 3/3 A quick picture shows that we must have: ∗ ∗ 1 cos t = ∗ ∗ 0 sin t Also, by paying attention to positives and negatives: ∗ ∗ 0 − sin t = ∗ ∗ 1 cos t Thus, the matrix of our rotation can only be: cos t − sin t R = t sin t cos t By "linear algebra”, this is the correct answer. Linear maps 1/4 2 2 Theorem. The maps f : R ! R which are linear, in the sense that they map lines through 0 to lines through 0, are: x ax + by f = y cx + dy Remark. If we make the multiplication convention a b x ax + by = c d y cx + dy the theorem says f (v) = Av, with A being a 2 × 2 matrix. -
January 3, 2019 Dynamical Systems B. Solomyak LECTURE 11
January 3, 2019 Dynamical Systems B. Solomyak LECTURE 11 SUMMARY 1. Hyperbolic toral automorphisms (cont.) d d d Let A be d × d integer matrix, with det A = ±1. Consider T = R =Z (the d- dimensional torus). d Definition 1.1. The map TA(x) = Ax (mod Z ) is called the toral automorphism asso- ciated with A. d Proposition 1.2. TA is a group automorphism; in fact, every automorphism of T has such a form. Definition 1.3. The automorphism is called hyperbolic if A is hyperbolic, that is, A has no eigenvalues of absolute value one. " # 2 1 Example 1.4 (Arnold's \cat map"). This is TA for A = . 1 1 Theorem 1.5 (see [1, x2.4]). Every hyperbolic toral automorphism is chaotic. We will see the proof for d = 2, for simplicity, following [1, 2]. The proof will be broken into lemmas. The first one was already covered on December 27. m n 2 Lemma 1.6. The points with rational coordinates f( k ; k ); m; n; k 2 Ng (mod Z ) are 2 dense in T , and they are all periodic for TA. Lemma 1.7. Let A be a hyperbolic 2×2 matrix. Thus it has two real eigenvalues: jλ1j > 1 and jλ2j < 1. Then the eigenvalues are irrational and the eigenvectors have irrational slopes. Let Es, Eu be the stable and unstable subspaces for A, respectively; they are one- dimensional and are spanned by the eigenvectors. s u Consider the point 0 = (0; 0) on the torus; it is fixed by TA.