Bézier Curves Are Attractors of Iterated Function Systems
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Homeomorphisms Group of Normed Vector Space: Conjugacy Problems
HOMEOMORPHISMS GROUP OF NORMED VECTOR SPACE: CONJUGACY PROBLEMS AND THE KOOPMAN OPERATOR MICKAEL¨ D. CHEKROUN AND JEAN ROUX Abstract. This article is concerned with conjugacy problems arising in the homeomorphisms group, Hom(F ), of unbounded subsets F of normed vector spaces E. Given two homeomor- phisms f and g in Hom(F ), it is shown how the existence of a conjugacy may be related to the existence of a common generalized eigenfunction of the associated Koopman operators. This common eigenfunction serves to build a topology on Hom(F ), where the conjugacy is obtained as limit of a sequence generated by the conjugacy operator, when this limit exists. The main conjugacy theorem is presented in a class of generalized Lipeomorphisms. 1. Introduction In this article we consider the conjugacy problem in the homeomorphisms group of a finite dimensional normed vector space E. It is out of the scope of the present work to review the problem of conjugacy in general, and the reader may consult for instance [13, 16, 29, 33, 26, 42, 45, 51, 52] and references therein, to get a partial survey of the question from a dynamical point of view. The present work raises the problem of conjugacy in the group Hom(F ) consisting of homeomorphisms of an unbounded subset F of E and is intended to demonstrate how the conjugacy problem, in such a case, may be related to spectral properties of the associated Koopman operators. In this sense, this paper provides new insights on the relations between the spectral theory of dynamical systems [5, 17, 23, 36] and the topological conjugacy problem [51, 52]1. -
Learning from Benoit Mandelbrot
2/28/18, 7(59 AM Page 1 of 1 February 27, 2018 Learning from Benoit Mandelbrot If you've been reading some of my recent posts, you will have noted my, and the Investment Masters belief, that many of the investment theories taught in most business schools are flawed. And they can be dangerous too, the recent Financial Crisis is evidence of as much. One man who, prior to the Financial Crisis, issued a challenge to regulators including the Federal Reserve Chairman Alan Greenspan, to recognise these flaws and develop more realistic risk models was Benoit Mandelbrot. Over the years I've read a lot of investment books, and the name Benoit Mandelbrot comes up from time to time. I recently finished his book 'The (Mis)Behaviour of Markets - A Fractal View of Risk, Ruin and Reward'. So who was Benoit Mandelbrot, why is he famous, and what can we learn from him? Benoit Mandelbrot was a Polish-born mathematician and polymath, a Sterling Professor of Mathematical Sciences at Yale University and IBM Fellow Emeritus (Physics) who developed a new branch of mathematics known as 'Fractal' geometry. This geometry recognises the hidden order in the seemingly disordered, the plan in the unplanned, the regular pattern in the irregularity and roughness of nature. It has been successfully applied to the natural sciences helping model weather, study river flows, analyse brainwaves and seismic tremors. A 'fractal' is defined as a rough or fragmented shape that can be split into parts, each of which is at least a close approximation of its original-self. -
Iterated Function Systems, Ruelle Operators, and Invariant Projective Measures
MATHEMATICS OF COMPUTATION Volume 75, Number 256, October 2006, Pages 1931–1970 S 0025-5718(06)01861-8 Article electronically published on May 31, 2006 ITERATED FUNCTION SYSTEMS, RUELLE OPERATORS, AND INVARIANT PROJECTIVE MEASURES DORIN ERVIN DUTKAY AND PALLE E. T. JORGENSEN Abstract. We introduce a Fourier-based harmonic analysis for a class of dis- crete dynamical systems which arise from Iterated Function Systems. Our starting point is the following pair of special features of these systems. (1) We assume that a measurable space X comes with a finite-to-one endomorphism r : X → X which is onto but not one-to-one. (2) In the case of affine Iterated Function Systems (IFSs) in Rd, this harmonic analysis arises naturally as a spectral duality defined from a given pair of finite subsets B,L in Rd of the same cardinality which generate complex Hadamard matrices. Our harmonic analysis for these iterated function systems (IFS) (X, µ)is based on a Markov process on certain paths. The probabilities are determined by a weight function W on X. From W we define a transition operator RW acting on functions on X, and a corresponding class H of continuous RW - harmonic functions. The properties of the functions in H are analyzed, and they determine the spectral theory of L2(µ).ForaffineIFSsweestablish orthogonal bases in L2(µ). These bases are generated by paths with infinite repetition of finite words. We use this in the last section to analyze tiles in Rd. 1. Introduction One of the reasons wavelets have found so many uses and applications is that they are especially attractive from the computational point of view. -
Role of Nonlinear Dynamics and Chaos in Applied Sciences
v.;.;.:.:.:.;.;.^ ROLE OF NONLINEAR DYNAMICS AND CHAOS IN APPLIED SCIENCES by Quissan V. Lawande and Nirupam Maiti Theoretical Physics Oivisipn 2000 Please be aware that all of the Missing Pages in this document were originally blank pages BARC/2OOO/E/OO3 GOVERNMENT OF INDIA ATOMIC ENERGY COMMISSION ROLE OF NONLINEAR DYNAMICS AND CHAOS IN APPLIED SCIENCES by Quissan V. Lawande and Nirupam Maiti Theoretical Physics Division BHABHA ATOMIC RESEARCH CENTRE MUMBAI, INDIA 2000 BARC/2000/E/003 BIBLIOGRAPHIC DESCRIPTION SHEET FOR TECHNICAL REPORT (as per IS : 9400 - 1980) 01 Security classification: Unclassified • 02 Distribution: External 03 Report status: New 04 Series: BARC External • 05 Report type: Technical Report 06 Report No. : BARC/2000/E/003 07 Part No. or Volume No. : 08 Contract No.: 10 Title and subtitle: Role of nonlinear dynamics and chaos in applied sciences 11 Collation: 111 p., figs., ills. 13 Project No. : 20 Personal authors): Quissan V. Lawande; Nirupam Maiti 21 Affiliation ofauthor(s): Theoretical Physics Division, Bhabha Atomic Research Centre, Mumbai 22 Corporate authoifs): Bhabha Atomic Research Centre, Mumbai - 400 085 23 Originating unit : Theoretical Physics Division, BARC, Mumbai 24 Sponsors) Name: Department of Atomic Energy Type: Government Contd...(ii) -l- 30 Date of submission: January 2000 31 Publication/Issue date: February 2000 40 Publisher/Distributor: Head, Library and Information Services Division, Bhabha Atomic Research Centre, Mumbai 42 Form of distribution: Hard copy 50 Language of text: English 51 Language of summary: English 52 No. of references: 40 refs. 53 Gives data on: Abstract: Nonlinear dynamics manifests itself in a number of phenomena in both laboratory and day to day dealings. -
Recursion, Writing, Iteration. a Proposal for a Graphics Foundation of Computational Reason Luca M
Recursion, Writing, Iteration. A Proposal for a Graphics Foundation of Computational Reason Luca M. Possati To cite this version: Luca M. Possati. Recursion, Writing, Iteration. A Proposal for a Graphics Foundation of Computa- tional Reason. 2015. hal-01321076 HAL Id: hal-01321076 https://hal.archives-ouvertes.fr/hal-01321076 Preprint submitted on 24 May 2016 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. 1/16 Recursion, Writing, Iteration A Proposal for a Graphics Foundation of Computational Reason Luca M. Possati In this paper we present a set of philosophical analyses to defend the thesis that computational reason is founded in writing; only what can be written is computable. We will focus on the relations among three main concepts: recursion, writing and iteration. The most important questions we will address are: • What does it mean to compute something? • What is a recursive structure? • Can we clarify the nature of recursion by investigating writing? • What kind of identity is presupposed by a recursive structure and by computation? Our theoretical path will lead us to a radical revision of the philosophical notion of identity. The act of iterating is rooted in an abstract space – we will try to outline a topological description of iteration. -
Mild Vs. Wild Randomness: Focusing on Those Risks That Matter
with empirical sense1. Granted, it has been Mild vs. Wild Randomness: tinkered with using such methods as Focusing on those Risks that complementary “jumps”, stress testing, regime Matter switching or the elaborate methods known as GARCH, but while they represent a good effort, Benoit Mandelbrot & Nassim Nicholas Taleb they fail to remediate the bell curve’s irremediable flaws. Forthcoming in, The Known, the Unknown and the The problem is that measures of uncertainty Unknowable in Financial Institutions Frank using the bell curve simply disregard the possibility Diebold, Neil Doherty, and Richard Herring, of sharp jumps or discontinuities. Therefore they editors, Princeton: Princeton University Press. have no meaning or consequence. Using them is like focusing on the grass and missing out on the (gigantic) trees. Conventional studies of uncertainty, whether in statistics, economics, finance or social science, In fact, while the occasional and unpredictable have largely stayed close to the so-called “bell large deviations are rare, they cannot be dismissed curve”, a symmetrical graph that represents a as “outliers” because, cumulatively, their impact in probability distribution. Used to great effect to the long term is so dramatic. describe errors in astronomical measurement by The good news, especially for practitioners, is the 19th-century mathematician Carl Friedrich that the fractal model is both intuitively and Gauss, the bell curve, or Gaussian model, has computationally simpler than the Gaussian. It too since pervaded our business and scientific culture, has been around since the sixties, which makes us and terms like sigma, variance, standard deviation, wonder why it was not implemented before. correlation, R-square and Sharpe ratio are all The traditional Gaussian way of looking at the directly linked to it. -
Summary of Unit 1: Iterated Functions 1
Summary of Unit 1: Iterated Functions 1 Summary of Unit 1: Iterated Functions David P. Feldman http://www.complexityexplorer.org/ Summary of Unit 1: Iterated Functions 2 Functions • A function is a rule that takes a number as input and outputs another number. • A function is an action. • Functions are deterministic. The output is determined only by the input. x f(x) f David P. Feldman http://www.complexityexplorer.org/ Summary of Unit 1: Iterated Functions 3 Iteration and Dynamical Systems • We iterate a function by turning it into a feedback loop. • The output of one step is used as the input for the next. • An iterated function is a dynamical system, a system that evolves in time according to a well-defined, unchanging rule. x f(x) f David P. Feldman http://www.complexityexplorer.org/ Summary of Unit 1: Iterated Functions 4 Itineraries and Seeds • We iterate a function by applying it again and again to a number. • The number we start with is called the seed or initial condition and is usually denoted x0. • The resulting sequence of numbers is called the itinerary or orbit. • It is also sometimes called a time series or a trajectory. • The iterates are denoted xt. Ex: x5 is the fifth iterate. David P. Feldman http://www.complexityexplorer.org/ Summary of Unit 1: Iterated Functions 5 Time Series Plots • A useful way to visualize an itinerary is with a time series plot. 0.8 0.7 0.6 0.5 t 0.4 x 0.3 0.2 0.1 0.0 0 1 2 3 4 5 6 7 8 9 10 time t • The time series plotted above is: 0.123, 0.189, 0.268, 0.343, 0.395, 0.418, 0.428, 0.426, 0.428, 0.429, 0.429. -
Towards a Theory of Social Fractal Discourse: Some Tentative Ideas
Page 1 of 16 ANZAM 2012 Towards a Theory of Social Fractal Discourse: Some Tentative Ideas Dr James Latham Faculty of Business and Enterprise, Swinburne University of Technology, Victoria, Australia Email: [email protected] Amanda Muhleisen PhD Candidate, Faculty of Business and Enterprise, Swinburne University of Technology, Victoria, Australia Email: [email protected] Prof Robert Jones Faculty of Business and Enterprise, Swinburne University of Technology, Victoria, Australia Email: [email protected] ANZAM 2012 Page 2 of 16 ABSTRACT The development of chaos theory and fractal geometry has made significant contribution to our understanding of the natural world. As students of organization, chaos theory provides an opportunity to broaden our thinking beyond the linear and predictable with its presumed stability, in order to embrace chaos rather than stability as the ‘default’ state for organization. This demands we think in new ways about organizational theory and practice. It also creates a schism in the fabric of our knowledge - a divide to be bridged. By taking the scientific concept of fractals and applying it to a social theory of discourse, this paper hopes to emerge some tentative ideas around developing a theory of social fractal discourse. Keywords: chaos theory, instability, organizing as process, critical discourse analysis, complexity The aim of this paper is to outline a number of tentative ideas around the development of a theory of social fractal discourse. It will focus on qualities and features of fractal and chaotic systems. Drawing on ideas from chaos and complexity theory, we will argue that discourse can be fractal. While there is some difference in theoretical constructs within the complexity literature, we will use the terms fractal and chaos interchangeably throughout this paper. -
Solving Iterated Functions Using Genetic Programming Michael D
Solving Iterated Functions Using Genetic Programming Michael D. Schmidt Hod Lipson Computational Synthesis Lab Computational Synthesis Lab Cornell University Cornell University Ithaca, NY 14853 Ithaca, NY 14853 [email protected] [email protected] ABSTRACT various communities. Renowned physicist Michael Fisher is An iterated function f(x) is a function that when composed with rumored to have solved the puzzle within five minutes [2]; itself, produces a given expression f(f(x))=g(x). Iterated functions however, few have matched this feat. are essential constructs in fractal theory and dynamical systems, The problem is enticing because of its apparent simplicity. Similar but few analysis techniques exist for solving them analytically. problems such as f(f(x)) = x2, or f(f(x)) = x4 + b are straightforward Here we propose using genetic programming to find analytical (see Table 1). The fact that the slight modification from these solutions to iterated functions of arbitrary form. We demonstrate easier functions makes the problem much more challenging this technique on the notoriously hard iterated function problem of highlights the difficulty in solving iterated function problems. finding f(x) such that f(f(x))=x2–2. While some analytical techniques have been developed to find a specific solution to Table 1. A few example iterated functions problems. problems of this form, we show that it can be readily solved using genetic programming without recourse to deep mathematical Iterated Function Solution insight. We find a previously unknown solution to this problem, suggesting that genetic programming may be an essential tool for f(f(x)) = x f(x) = x finding solutions to arbitrary iterated functions. -
Iterated Function System
IARJSET ISSN (Online) 2393-8021 ISSN (Print) 2394-1588 International Advanced Research Journal in Science, Engineering and Technology ISO 3297:2007 Certified Vol. 3, Issue 8, August 2016 Iterated Function System S. C. Shrivastava Department of Applied Mathematics, Rungta College of Engineering and Technology, Bhilai C.G., India Abstract: Fractal image compression through IFS is very important for the efficient transmission and storage of digital data. Fractal is made up of the union of several copies of itself and IFS is defined by a finite number of affine transformation which characterized by Translation, scaling, shearing and rotat ion. In this paper we describe the necessary conditions to form an Iterated Function System and how fractals are generated through affine transformations. Keywords: Iterated Function System; Contraction Mapping. 1. INTRODUCTION The exploration of fractal geometry is usually traced back Metric Spaces definition: A space X with a real-valued to the publication of the book “The Fractal Geometry of function d: X × X → ℜ is called a metric space (X, d) if d Nature” [1] by the IBM mathematician Benoit B. possess the following properties: Mandelbrot. Iterated Function System is a method of constructing fractals, which consists of a set of maps that 1. d(x, y) ≥ 0 for ∀ x, y ∈ X explicitly list the similarities of the shape. Though the 2. d(x, y) = d(y, x) ∀ x, y ∈ X formal name Iterated Function Systems or IFS was coined 3. d x, y ≤ d x, z + d z, y ∀ x, y, z ∈ X . (triangle by Barnsley and Demko [2] in 1985, the basic concept is inequality). -
Aspects of Functional Iteration Milton Eugene Winger Iowa State University
Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1972 Aspects of functional iteration Milton Eugene Winger Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Statistics and Probability Commons Recommended Citation Winger, Milton Eugene, "Aspects of functional iteration " (1972). Retrospective Theses and Dissertations. 5876. https://lib.dr.iastate.edu/rtd/5876 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. INFORMATION TO USERS This dissertation was produced from a microfilm copy of the original document. While the most advanced technological means to photograph and reproduce this document have been used, the quality is heavily dependent upon the quality of the original submitted. The following explanation of techniques is provided to help you understand markings or patterns which may appear on this reproduction. 1. The sign or "target" for pages apparently lacking from the document photographed is "Missing Page(s)". If it was possible to obtain the missing page(s) or section, they are spliced into the film along with adjacent pages. This may have necessitated cutting thru an image and duplicating adjacent pages to insure you complete continuity. 2. When an image on the film is obliterated with a large round black mark, it is an indication that the photographer suspected that the copy may have moved during exposure and thus cause a blurred image. -
Study of Behavior of Human Capital from a Fractal Perspective
http://ijba.sciedupress.com International Journal of Business Administration Vol. 8, No. 1; 2017 Study of Behavior of Human Capital from a Fractal Perspective Leydi Z. Guzmán-Aguilar1, Ángel Machorro-Rodríguez2, Tomás Morales-Acoltzi3, Miguel Montaño-Alvarez1, Marcos Salazar-Medina2 & Edna A. Romero-Flores2 1 Maestría en Ingeniería Administrativa, Tecnológico Nacional de México, Instituto Tecnológico de Orizaba, Colonia Emiliano Zapata, México 2 División de Estudios de Posgrado e Investigación, Tecnológico Nacional de México, Instituto Tecnológico de Orizaba, Colonia Emiliano Zapata, México 3 Centro de Ciencias de la Atmósfera, UNAM, Circuito exterior, CU, CDMX, México Correspondence: Leydi Z. Guzmán-Aguilar, Maestría en Ingeniería Administrativa, Tecnológico Nacional de México, Instituto Tecnológico de Orizaba, Ver. Oriente 9 No. 852 Colonia Emiliano Zapata, C.P. 94320, México. Tel: 1-272-725-7518. Received: November 11, 2016 Accepted: November 27, 2016 Online Published: January 4, 2017 doi:10.5430/ijba.v8n1p50 URL: http://dx.doi.org/10.5430/ijba.v8n1p50 Abstract In the last decade, the application of fractal geometry has surged in all disciplines. In the area of human capital management, obtaining a relatively long time series (TS) is difficult, and likewise nonlinear methods require at least 512 observations. In this research, we therefore suggest the application of a fractal interpolation scheme to generate ad hoc TS. When we inhibit the vertical scale factor, the proposed interpolation scheme has the effect of simulating the original TS. Keywords: human capital, wage, fractal interpolation, vertical scale factor 1. Introduction In nature, irregular processes exist that by using Euclidean models as their basis for analysis, do not capture the variety and complexity of the dynamics of their environment.