USING THE RANDOM ITERATION ALGORITHM TO CREATE FRACTALS
|
|
- Avis Stewart
- 5 years ago
- Views:
Transcription
1 USING THE RANDOM ITERATION ALGORITHM TO CREATE FRACTALS UNIVERSITY OF MARYLAND DIRECTED READING PROGRAM FALL 205 BY ADAM ANDERSON
2
3 THE SIERPINSKI GASKET 2 Stage 0: A 0 = 2 22 A 0 = Stage : A = 2 = 4 A = Stage 2: 2 2 A 2 = = 3 3 = A 2 = Stage n: A n = 3 4 lim n 3 4 n n = 0 Sierpinski Gasket has zero area?
4 CAPACITY DIMENSION AND FRACTALS Let S R n where n =, 2, or 3 n-dimensional bo n = : Closed interval n = 2: Square n = 3: Cube Let N(ε) = smallest number of n-dimensional boes of side length ε necessary to cover S z y
5 CAPACITY DIMENSION AND FRACTALS n = ε Boes of length ε to cover line of length L: Length = L If L = 0cm and ε = cm, it takes 0 boes to cover L If ε = 0.5cm, it takes 20 boes to cover L N(ε) ε
6 CAPACITY DIMENSION AND FRACTALS n = 2 ε Boes of length ε to cover square S of side length L If L = 20cm, area of S = 400cm 2 ε = 2cm: each bo has area 4cm 2 It will take 00 boes to cover S ε = cm: each bo has area cm 2 It will take 400 boes to cover S N(ε) ε 2
7 CAPACITY DIMENSION AND FRACTALS N(ε) = C ε D ln (N(ε)) = D ln ε + ln(c) D = ln N ε ln ε ln C C just depends on scaling of S Capacity Dimension: dim c S = lim ε 0 + ln N ε ln ε
8 CAPACITY DIMENSION AND FRACTALS ε = 2 Stage : If ε = 2, N ε = 3 Stage 2: Stage n: If ε =, N ε = 9 If ε = n, N ε = 3n 4 2 ε = 2n ln N ε lim ε 0 + ln ε = lim ε 0 + ln 3 n n ln 3 ln 2n = = ln 3 n ln 2 ln
9 The Sierpinski Gasket is.585 dimensional A set with non-integer capacity dimension is called a fractal.
10 ITERATED FUNCTION SYSTEMS An Iterated Function System (IFS) F is the union of the contractions T, T 2 T n THEOREM: Let F be an iterated function system of contractions in R 2. Then there eists a unique compact subset A F in R 2 such that for any compact set B, the sequence of iterates F n B n= converges in the Hausdorff metric to A F A F is called the attractor of F. This means that if we iterate any compact set in R 2 under F, we will obtain a unique attractor (attractor depends on the contractions in F)
11 ITERATED FUNCTION SYSTEMS A function T : R 2 R 2 is affine if it is in the form f y = a b c d y + e f = a + by + e c + dy + f (linear function followed by translation) We will deal with iterated function systems of affine contractions.
12 RANDOM ITERATION ALGORITHM Drawing an attractor of IFS F:. Choose an arbitrary initial point v R 2 2. Randomly select one of the contractions T n in F 3. Plot the point T n ( v) 4. Let T n ( v) be the new v 5. Repeat steps 2-4 to obtain a representation of A F 000 Iterations 5000 Iterations 20,000 Iterations
13 LET S DRAW SOME FRACTALS!
14 ITERATIONS AND FIXED POINTS Iterating = repeating the same procedure f = 2 + Let f() be a function. f f = f 2 is the second iterate of under f. f f(f( ) = f 3 is the third iterate of under f Eample: f = 2 + f 5 = 26 f f 5 = f 2 5 = f 26 = 677 f 2 = ( 2 +) 2 +
15 ITERATIONS AND FIXED POINTS A point p is a fied point for a function f if its iterate is itself f p = p Eample: f = 2 f 0 = 0 f f 0 = f 2 0 = f 0 =0 Therefore 0 is a fied point of f f = 2
16 METRIC SPACES Let S be a set. A metric is a distance function d, y that satisfies 4 aioms, y S. d, y 0 2. d, y = 0 if and only if = y 3. d, y = d y, 4. d, z d, y + d y, z y Eample: Absolute Value R d, y = y R, d is a metric space d, y = y
17 METRIC SPACES Let S, d be a metric space. A sequence n n= in S converges to S if lim d n, = 0 n This means that terms of the sequence approach a value s A sequence is Cauchy if for all ε > 0 there eists a positive integer N such that whenever n, m N, d n, m < ε This means that terms of the sequence get closer together S, d is a complete metric space if every Cauchy sequence in S converges to a member of S
18 CONTRACTION MAPPING THEOREM Let S, d be a metric space. A function T: S S is a contraction if q 0, such that d T(), T(y) q d(, y) T d, y d T(), T(y) = q d(, y)
19 CONTRACTION MAPPING THEOREM Contraction Mapping Theorem: If S, d is a complete metric space, and T is a contraction, then as n, T n unique fied point S T T
20 HAUSDORFF METRIC A set S is closed if whenever is the limit of a sequence of members of T, actually is in T. A set S is bounded if it there eists S and r > 0 such that s S, d, s < r Means S is contained by a ball of finite radius A set S R n is compact if it is closed and bounded Let K denote all compact subsets of R 2
21 HAUSDORFF METRIC If B is a nonempty member of K, and v is any point in R 2, the distance from v to B is d v, B = minimum value of v b b B (distance from point to a compact set) B v d v, B
22 HAUSDORFF METRIC If A and B are members of K, then the distance from A to B is d A, B = maimum value of d a, B for a A (distance between compact sets) Means we take the point in A that is most distant from any point in B and find the minimum distance between it an any point in B A B a d A, B
23 HAUSDORFF METRIC The Hausdorff metric on K is defined as: D A, B =maimum of d A, B and d B, A A a d B, A d A, B B b d A, B d B, A D A, B = d B, A
24 Iterated Function System: f y = 2 2 y 2 2 f 2 y = y + 0 Render Details: Point size: p # of iterations: 00,000
25 Iterated Function System: f y = y f 2 y = y f 3 y = 0 /2 /2 0 y + /2 Render Details: Point size: p # of iterations: 00,000
26 Iterated Function System: f f 2 f 3 f 4 f 5 y = y + 2/3 2/3 3 y = y + 2/3 2/3 3 y = y + 2/3 2/3 3 y = y + 2/3 2/ y = y Render Details: Point size: p # of iterations: 00,000
27 Iterated Function System: f y = y f 2 y = y f 3 y = y f 4 y = y Probabilities: f : % f 2 : 85% f 3 : 7% f 4 : 7%
28 Iterated Function System: f y = y f 2 f 3 f 4 f 5 f 6 f 7 y = y = y = y = y = y = y + / 3 /3 y + / 3 /3 y + / 3 /3 y + / 3 /3 y + 0 2/3 y + 0 2/3
29 THANKS Mentor Kasun Fernando Author and Book Provider Denny Gulick Some IFS Formulas from Agnes Scott College Graphs Created with Desmos Graphing Calculator
NOTES ON BARNSLEY FERN
NOTES ON BARNSLEY FERN ERIC MARTIN 1. Affine transformations An affine transformation on the plane is a mapping T that preserves collinearity and ratios of distances: given two points A and B, if C is
More informationMA2223 Tutorial solutions Part 1. Metric spaces
MA2223 Tutorial solutions Part 1. Metric spaces T1 1. Show that the function d(,y) = y defines a metric on R. The given function is symmetric and non-negative with d(,y) = 0 if and only if = y. It remains
More informationFractals and Dimension
Chapter 7 Fractals and Dimension Dimension We say that a smooth curve has dimension 1, a plane has dimension 2 and so on, but it is not so obvious at first what dimension we should ascribe to the Sierpinski
More informationTake a line segment of length one unit and divide it into N equal old length. Take a square (dimension 2) of area one square unit and divide
Fractal Geometr A Fractal is a geometric object whose dimension is fractional Most fractals are self similar, that is when an small part of a fractal is magnified the result resembles the original fractal
More informationFractals and iteration function systems II Complex systems simulation
Fractals and iteration function systems II Complex systems simulation Facultad de Informática (UPM) Sonia Sastre November 24, 2011 Sonia Sastre () Fractals and iteration function systems November 24, 2011
More informationThe Hausdorff Measure of the Attractor of an Iterated Function System with Parameter
ISSN 1479-3889 (print), 1479-3897 (online) International Journal of Nonlinear Science Vol. 3 (2007) No. 2, pp. 150-154 The Hausdorff Measure of the Attractor of an Iterated Function System with Parameter
More informationSlopes and Rates of Change
Slopes and Rates of Change If a particle is moving in a straight line at a constant velocity, then the graph of the function of distance versus time is as follows s s = f(t) t s s t t = average velocity
More informationFRACTAL GEOMETRY, DYNAMICAL SYSTEMS AND CHAOS
FRACTAL GEOMETRY, DYNAMICAL SYSTEMS AND CHAOS MIDTERM SOLUTIONS. Let f : R R be the map on the line generated by the function f(x) = x 3. Find all the fixed points of f and determine the type of their
More informationTopological properties of Z p and Q p and Euclidean models
Topological properties of Z p and Q p and Euclidean models Samuel Trautwein, Esther Röder, Giorgio Barozzi November 3, 20 Topology of Q p vs Topology of R Both R and Q p are normed fields and complete
More informationFractals. R. J. Renka 11/14/2016. Department of Computer Science & Engineering University of North Texas. R. J. Renka Fractals
Fractals R. J. Renka Department of Computer Science & Engineering University of North Texas 11/14/2016 Introduction In graphics, fractals are used to produce natural scenes with irregular shapes, such
More informationCMSC 425: Lecture 12 Procedural Generation: Fractals
: Lecture 12 Procedural Generation: Fractals Reading: This material comes from classic computer-graphics books. Procedural Generation: One of the important issues in game development is how to generate
More informationAnalytic Continuation of Analytic (Fractal) Functions
Analytic Continuation of Analytic (Fractal) Functions Michael F. Barnsley Andrew Vince (UFL) Australian National University 10 December 2012 Analytic continuations of fractals generalises analytic continuation
More informationITERATED FUNCTION SYSTEMS WITH CONTINUOUS PLACE DEPENDENT PROBABILITIES
UNIVERSITATIS IAGELLONICAE ACTA MATHEMATICA, FASCICULUS XL 2002 ITERATED FUNCTION SYSTEMS WITH CONTINUOUS PLACE DEPENDENT PROBABILITIES by Joanna Jaroszewska Abstract. We study the asymptotic behaviour
More informationTHE CONTRACTION MAPPING THEOREM, II
THE CONTRACTION MAPPING THEOREM, II KEITH CONRAD 1. Introduction In part I, we met the contraction mapping theorem and an application of it to solving nonlinear differential equations. Here we will discuss
More informationUNIVERSIDAD CARLOS III DE MADRID MATHEMATICS II EXERCISES (SOLUTIONS )
UNIVERSIDAD CARLOS III DE MADRID MATHEMATICS II EXERCISES (SOLUTIONS ) CHAPTER : Limits and continuit of functions in R n. -. Sketch the following subsets of R. Sketch their boundar and the interior. Stud
More informationEconomics 205 Exercises
Economics 05 Eercises Prof. Watson, Fall 006 (Includes eaminations through Fall 003) Part 1: Basic Analysis 1. Using ε and δ, write in formal terms the meaning of lim a f() = c, where f : R R.. Write the
More informationChapter 2 Metric Spaces
Chapter 2 Metric Spaces The purpose of this chapter is to present a summary of some basic properties of metric and topological spaces that play an important role in the main body of the book. 2.1 Metrics
More informationarxiv: v1 [math.ds] 31 Jul 2018
arxiv:1807.11801v1 [math.ds] 31 Jul 2018 On the interior of projections of planar self-similar sets YUKI TAKAHASHI Abstract. We consider projections of planar self-similar sets, and show that one can create
More informationHIDDEN VARIABLE BIVARIATE FRACTAL INTERPOLATION SURFACES
Fractals, Vol. 11, No. 3 (2003 277 288 c World Scientific Publishing Company HIDDEN VARIABLE BIVARIATE FRACTAL INTERPOLATION SURFACES A. K. B. CHAND and G. P. KAPOOR Department of Mathematics, Indian Institute
More informationVALUATION THEORY, GENERALIZED IFS ATTRACTORS AND FRACTALS
VALUATION THEORY, GENERALIZED IFS ATTRACTORS AND FRACTALS JAN DOBROWOLSKI AND FRANZ-VIKTOR KUHLMANN Abstract. Using valuation rings and valued fields as examples, we discuss in which ways the notions of
More informationON THE DIAMETER OF THE ATTRACTOR OF AN IFS Serge Dubuc Raouf Hamzaoui Abstract We investigate methods for the evaluation of the diameter of the attrac
ON THE DIAMETER OF THE ATTRACTOR OF AN IFS Serge Dubuc Raouf Hamzaoui Abstract We investigate methods for the evaluation of the diameter of the attractor of an IFS. We propose an upper bound for the diameter
More informationApproximation of the attractor of a countable iterated function system 1
General Mathematics Vol. 17, No. 3 (2009), 221 231 Approximation of the attractor of a countable iterated function system 1 Nicolae-Adrian Secelean Abstract In this paper we will describe a construction
More informationINTRODUCTION TO FRACTAL GEOMETRY
Every mathematical theory, however abstract, is inspired by some idea coming in our mind from the observation of nature, and has some application to our world, even if very unexpected ones and lying centuries
More informationAn Introduction to Self Similar Structures
An Introduction to Self Similar Structures Christopher Hayes University of Connecticut April 6th, 2018 Christopher Hayes (University of Connecticut) An Introduction to Self Similar Structures April 6th,
More informationDIFFERENTIAL EQUATIONS: EXISTENCE AND UNIQUENESS OF SOLUTIONS IN ECONOMIC MODELS
DIFFERENTIAL EQUATIONS: EXISTENCE AND UNIQUENESS OF SOLUTIONS IN ECONOMIC MODELS THOMAS ZHENG Abstract. The following paper aims to create an integration between differential equations and economics by
More information10 Typical compact sets
Tel Aviv University, 2013 Measure and category 83 10 Typical compact sets 10a Covering, packing, volume, and dimension... 83 10b A space of compact sets.............. 85 10c Dimensions of typical sets.............
More informationMany of you got these steps reversed or otherwise out of order.
Problem 1. Let (X, d X ) and (Y, d Y ) be metric spaces. Suppose that there is a bijection f : X Y such that for all x 1, x 2 X. 1 10 d X(x 1, x 2 ) d Y (f(x 1 ), f(x 2 )) 10d X (x 1, x 2 ) Show that if
More informationOn a Topological Problem of Strange Attractors. Ibrahim Kirat and Ayhan Yurdaer
On a Topological Problem of Strange Attractors Ibrahim Kirat and Ayhan Yurdaer Department of Mathematics, Istanbul Technical University, 34469,Maslak-Istanbul, Turkey E-mail: ibkst@yahoo.com and yurdaerayhan@itu.edu.tr
More informationHausdorff Measure. Jimmy Briggs and Tim Tyree. December 3, 2016
Hausdorff Measure Jimmy Briggs and Tim Tyree December 3, 2016 1 1 Introduction In this report, we explore the the measurement of arbitrary subsets of the metric space (X, ρ), a topological space X along
More informationChapter 2: Introduction to Fractals. Topics
ME597B/Math597G/Phy597C Spring 2015 Chapter 2: Introduction to Fractals Topics Fundamentals of Fractals Hausdorff Dimension Box Dimension Various Measures on Fractals Advanced Concepts of Fractals 1 C
More informationTube formulas and self-similar tilings
Tube formulas and self-similar tilings Erin P. J. Pearse erin-pearse@uiowa.edu Joint work with Michel L. Lapidus and Steffen Winter VIGRE Postdoctoral Fellow Department of Mathematics University of Iowa
More informationMaths 212: Homework Solutions
Maths 212: Homework Solutions 1. The definition of A ensures that x π for all x A, so π is an upper bound of A. To show it is the least upper bound, suppose x < π and consider two cases. If x < 1, then
More informationSelf-similar Fractals: Projections, Sections and Percolation
Self-similar Fractals: Projections, Sections and Percolation University of St Andrews, Scotland, UK Summary Self-similar sets Hausdorff dimension Projections Fractal percolation Sections or slices Projections
More informationCourse 212: Academic Year Section 1: Metric Spaces
Course 212: Academic Year 1991-2 Section 1: Metric Spaces D. R. Wilkins Contents 1 Metric Spaces 3 1.1 Distance Functions and Metric Spaces............. 3 1.2 Convergence and Continuity in Metric Spaces.........
More informationMetric Spaces Math 413 Honors Project
Metric Spaces Math 413 Honors Project 1 Metric Spaces Definition 1.1 Let X be a set. A metric on X is a function d : X X R such that for all x, y, z X: i) d(x, y) = d(y, x); ii) d(x, y) = 0 if and only
More informationANALYSIS WORKSHEET II: METRIC SPACES
ANALYSIS WORKSHEET II: METRIC SPACES Definition 1. A metric space (X, d) is a space X of objects (called points), together with a distance function or metric d : X X [0, ), which associates to each pair
More informationNonarchimedean Cantor set and string
J fixed point theory appl Online First c 2008 Birkhäuser Verlag Basel/Switzerland DOI 101007/s11784-008-0062-9 Journal of Fixed Point Theory and Applications Nonarchimedean Cantor set and string Michel
More informationCharacterisation of Accumulation Points. Convergence in Metric Spaces. Characterisation of Closed Sets. Characterisation of Closed Sets
Convergence in Metric Spaces Functional Analysis Lecture 3: Convergence and Continuity in Metric Spaces Bengt Ove Turesson September 4, 2016 Suppose that (X, d) is a metric space. A sequence (x n ) X is
More informationFractal Geometry Time Escape Algorithms and Fractal Dimension
NAVY Research Group Department of Computer Science Faculty of Electrical Engineering and Computer Science VŠB- TUO 17. listopadu 15 708 33 Ostrava- Poruba Czech Republic Basics of Modern Computer Science
More informationIntroduction to Real Analysis Alternative Chapter 1
Christopher Heil Introduction to Real Analysis Alternative Chapter 1 A Primer on Norms and Banach Spaces Last Updated: March 10, 2018 c 2018 by Christopher Heil Chapter 1 A Primer on Norms and Banach Spaces
More informationChanges to the third printing May 18, 2013 Measure, Topology, and Fractal Geometry
Changes to the third printing May 18, 2013 Measure, Topology, and Fractal Geometry by Gerald A. Edgar Page 34 Line 9. After summer program. add Exercise 1.6.3 and 1.6.4 are stated as either/or. It is possible
More information9.3 Path independence
66 CHAPTER 9. ONE DIMENSIONAL INTEGRALS IN SEVERAL VARIABLES 9.3 Path independence Note:??? lectures 9.3.1 Path independent integrals Let U R n be a set and ω a one-form defined on U, The integral of ω
More informationAlgorithmic interpretations of fractal dimension. Anastasios Sidiropoulos (The Ohio State University) Vijay Sridhar (The Ohio State University)
Algorithmic interpretations of fractal dimension Anastasios Sidiropoulos (The Ohio State University) Vijay Sridhar (The Ohio State University) The curse of dimensionality Geometric problems become harder
More informationConstruction of recurrent fractal interpolation surfaces(rfiss) on rectangular grids
Construction of recurrent fractal interpolation surfacesrfiss on rectangular grids Wolfgang Metzler a,, Chol Hui Yun b a Faculty of Mathematics University of Kassel, Kassel, F. R. Germany b Faculty of
More informationTHE INVERSE FUNCTION THEOREM
THE INVERSE FUNCTION THEOREM W. PATRICK HOOPER The implicit function theorem is the following result: Theorem 1. Let f be a C 1 function from a neighborhood of a point a R n into R n. Suppose A = Df(a)
More informationFractals list of fractals - Hausdorff dimension
3//00 from Wiipedia: Fractals list of fractals - Hausdorff dimension Sierpinsi Triangle -.585 3D Cantor Dust -.898 Lorenz attractor -.06 Coastline of Great Britain -.5 Mandelbrot Set Boundary - - Regular
More informationHAUSDORFF DIMENSION AND ITS APPLICATIONS
HAUSDORFF DIMENSION AND ITS APPLICATIONS JAY SHAH Abstract. The theory of Hausdorff dimension provides a general notion of the size of a set in a metric space. We define Hausdorff measure and dimension,
More informationThe Order in Chaos. Jeremy Batterson. May 13, 2013
2 The Order in Chaos Jeremy Batterson May 13, 2013 Fractal geometry will make you see everything differently. There is a danger in reading further. You risk the loss of your childhood vision of clouds,
More informationCMSC 425: Lecture 11 Procedural Generation: Fractals and L-Systems
CMSC 425: Lecture 11 Procedural Generation: ractals and L-Systems Reading: The material on fractals comes from classic computer-graphics books. The material on L-Systems comes from Chapter 1 of The Algorithmic
More informationMetric Spaces and Topology
Chapter 2 Metric Spaces and Topology From an engineering perspective, the most important way to construct a topology on a set is to define the topology in terms of a metric on the set. This approach underlies
More informationLanguage-Restricted Iterated Function Systems, Koch Constructions, and L-systems
Language-Restricted Iterated Function Systems, Koch Constructions, and L-systems Przemyslaw Prusinkiewicz and Mark Hammel Department of Computer Science University of Calgary Calgary, Alberta, Canada T2N
More informationMath Subject GRE Questions
Math Subject GRE Questions Calculus and Differential Equations 1. If f() = e e, then [f ()] 2 [f()] 2 equals (a) 4 (b) 4e 2 (c) 2e (d) 2 (e) 2e 2. An integrating factor for the ordinary differential equation
More informationReal Analysis. Joe Patten August 12, 2018
Real Analysis Joe Patten August 12, 2018 1 Relations and Functions 1.1 Relations A (binary) relation, R, from set A to set B is a subset of A B. Since R is a subset of A B, it is a set of ordered pairs.
More informationSelf-Referentiality, Fractels, and Applications
Self-Referentiality, Fractels, and Applications Peter Massopust Center of Mathematics Research Unit M15 Technical University of Munich Germany 2nd IM-Workshop on Applied Approximation, Signals, and Images,
More informationAperiodic Substitution Tilings
Aperiodic Substitution Tilings Charles Starling January 4, 2011 Charles Starling () Aperiodic Substitution Tilings January 4, 2011 1 / 27 Overview We study tilings of R 2 which are aperiodic, but not completely
More informationFINAL PROJECT TOPICS MATH 399, SPRING αx 0 x < α(1 x)
FINAL PROJECT TOPICS MATH 399, SPRING 2011 MARIUS IONESCU If you pick any of the following topics feel free to discuss with me if you need any further background that we did not discuss in class. 1. Iterations
More informationExpectations over Fractal Sets
Expectations over Fractal Sets Michael Rose Jon Borwein, David Bailey, Richard Crandall, Nathan Clisby 21st June 2015 Synapse spatial distributions R.E. Crandall, On the fractal distribution of brain synapses.
More informationExamensarbete. Fractals and Computer Graphics. Meritxell Joanpere Salvadó
Examensarbete Fractals and Computer Graphics Meritxell Joanpere Salvadó LiTH - MAT - INT - A - - 20 / 0 - - SE Fractals and Computer Graphics Applied Mathematics, Linköpings Universitet Meritxell Joanpere
More informationScalar functions of several variables (Sect. 14.1)
Scalar functions of several variables (Sect. 14.1) Functions of several variables. On open, closed sets. Functions of two variables: Graph of the function. Level curves, contour curves. Functions of three
More informationSpectral Properties of an Operator-Fractal
Spectral Properties of an Operator-Fractal Keri Kornelson University of Oklahoma - Norman NSA Texas A&M University July 18, 2012 K. Kornelson (U. Oklahoma) Operator-Fractal TAMU 07/18/2012 1 / 25 Acknowledgements
More informationMASS TRANSFERENCE PRINCIPLE: FROM BALLS TO ARBITRARY SHAPES. 1. Introduction. limsupa i = A i.
MASS TRANSFERENCE PRINCIPLE: FROM BALLS TO ARBITRARY SHAPES HENNA KOIVUSALO AND MICHA L RAMS arxiv:1812.08557v1 [math.ca] 20 Dec 2018 Abstract. The mass transference principle, proved by Beresnevich and
More informationIn last semester, we have seen some examples about it (See Tutorial Note #13). Try to have a look on that. Here we try to show more technique.
MATH202 Introduction to Analysis (2007 Fall and 2008 Spring) Tutorial Note #4 Part I: Cauchy Sequence Definition (Cauchy Sequence): A sequence of real number { n } is Cauchy if and only if for any ε >
More informationLecture 3. Econ August 12
Lecture 3 Econ 2001 2015 August 12 Lecture 3 Outline 1 Metric and Metric Spaces 2 Norm and Normed Spaces 3 Sequences and Subsequences 4 Convergence 5 Monotone and Bounded Sequences Announcements: - Friday
More informationFractal Geometry Mathematical Foundations and Applications
Fractal Geometry Mathematical Foundations and Applications Third Edition by Kenneth Falconer Solutions to Exercises Acknowledgement: Grateful thanks are due to Gwyneth Stallard for providing solutions
More informationMAS331: Metric Spaces Problems on Chapter 1
MAS331: Metric Spaces Problems on Chapter 1 1. In R 3, find d 1 ((3, 1, 4), (2, 7, 1)), d 2 ((3, 1, 4), (2, 7, 1)) and d ((3, 1, 4), (2, 7, 1)). 2. In R 4, show that d 1 ((4, 4, 4, 6), (0, 0, 0, 0)) =
More informationFractals list of fractals by Hausdoff dimension
from Wiipedia: Fractals list of fractals by Hausdoff dimension Sierpinsi Triangle D Cantor Dust Lorenz attractor Coastline of Great Britain Mandelbrot Set What maes a fractal? I m using references: Fractal
More informationMath 118B Solutions. Charles Martin. March 6, d i (x i, y i ) + d i (y i, z i ) = d(x, y) + d(y, z). i=1
Math 8B Solutions Charles Martin March 6, Homework Problems. Let (X i, d i ), i n, be finitely many metric spaces. Construct a metric on the product space X = X X n. Proof. Denote points in X as x = (x,
More informationThe spectral decimation of the Laplacian on the Sierpinski gasket
The spectral decimation of the Laplacian on the Sierpinski gasket Nishu Lal University of California, Riverside Fullerton College February 22, 2011 1 Construction of the Laplacian on the Sierpinski gasket
More informationIterated Function Systems
1 Iterated Function Systems Section 1. Iterated Function Systems Definition. A transformation f:r m! R m is a contraction (or is a contraction map or is contractive) if there is a constant s with 0"s
More information3.5 Continuity of a Function One Sided Continuity Intermediate Value Theorem... 23
Chapter 3 Limit and Continuity Contents 3. Definition of Limit 3 3.2 Basic Limit Theorems 8 3.3 One sided Limit 4 3.4 Infinite Limit, Limit at infinity and Asymptotes 5 3.4. Infinite Limit and Vertical
More informationReal Analysis Math 125A, Fall 2012 Sample Final Questions. x 1+x y. x y x. 2 (1+x 2 )(1+y 2 ) x y
. Define f : R R by Real Analysis Math 25A, Fall 202 Sample Final Questions f() = 3 + 2. Show that f is continuous on R. Is f uniformly continuous on R? To simplify the inequalities a bit, we write 3 +
More informationIntroduction to Hausdorff Measure and Dimension
Introduction to Hausdorff Measure and Dimension Dynamics Learning Seminar, Liverpool) Poj Lertchoosakul 28 September 2012 1 Definition of Hausdorff Measure and Dimension Let X, d) be a metric space, let
More informationFractals. Krzysztof Gdawiec.
Fractals Krzysztof Gdawiec kgdawiec@ux2.math.us.edu.pl 1 Introduction In the 1970 s Benoit Mandelbrot introduced to the world new field of mathematics. He named this field fractal geometry (fractus from
More informationMA677 Assignment #3 Morgan Schreffler Due 09/19/12 Exercise 1 Using Hölder s inequality, prove Minkowski s inequality for f, g L p (R d ), p 1:
Exercise 1 Using Hölder s inequality, prove Minkowski s inequality for f, g L p (R d ), p 1: f + g p f p + g p. Proof. If f, g L p (R d ), then since f(x) + g(x) max {f(x), g(x)}, we have f(x) + g(x) p
More informationconverges as well if x < 1. 1 x n x n 1 1 = 2 a nx n
Solve the following 6 problems. 1. Prove that if series n=1 a nx n converges for all x such that x < 1, then the series n=1 a n xn 1 x converges as well if x < 1. n For x < 1, x n 0 as n, so there exists
More informationPart II. Geometry and Groups. Year
Part II Year 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2014 Paper 4, Section I 3F 49 Define the limit set Λ(G) of a Kleinian group G. Assuming that G has no finite orbit in H 3 S 2, and that Λ(G),
More informationMATH 51H Section 4. October 16, Recall what it means for a function between metric spaces to be continuous:
MATH 51H Section 4 October 16, 2015 1 Continuity Recall what it means for a function between metric spaces to be continuous: Definition. Let (X, d X ), (Y, d Y ) be metric spaces. A function f : X Y is
More informationMetric Spaces Math 413 Honors Project
Metric Spaces Math 413 Honors Project 1 Metric Spaces Definition 1.1 Let X be a set. A metric on X is a function d : X X R such that for all x, y, z X: i) d(x, y) = d(y, x); ii) d(x, y) = 0 if and only
More informationMAS3706 Topology. Revision Lectures, May I do not answer enquiries as to what material will be in the exam.
MAS3706 Topology Revision Lectures, May 208 Z.A.Lykova It is essential that you read and try to understand the lecture notes from the beginning to the end. Many questions from the exam paper will be similar
More informations P = f(ξ n )(x i x i 1 ). i=1
Compactness and total boundedness via nets The aim of this chapter is to define the notion of a net (generalized sequence) and to characterize compactness and total boundedness by this important topological
More informationSequence of maximal distance codes in graphs or other metric spaces
Electronic Journal of Graph Theory and Applications 1 (2) (2013), 118 124 Sequence of maximal distance codes in graphs or other metric spaces C. Delorme University Paris Sud 91405 Orsay CEDEX - France.
More informationSupplement. The Extended Complex Plane
The Extended Complex Plane 1 Supplement. The Extended Complex Plane Note. In section I.6, The Extended Plane and Its Spherical Representation, we introduced the extended complex plane, C = C { }. We defined
More informationExistence and Uniqueness
Chapter 3 Existence and Uniqueness An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect
More informationRandom measures, intersections, and applications
Random measures, intersections, and applications Ville Suomala joint work with Pablo Shmerkin University of Oulu, Finland Workshop on fractals, The Hebrew University of Jerusalem June 12th 2014 Motivation
More informationSOLUTIONS TO SOME PROBLEMS
23 FUNCTIONAL ANALYSIS Spring 23 SOLUTIONS TO SOME PROBLEMS Warning:These solutions may contain errors!! PREPARED BY SULEYMAN ULUSOY PROBLEM 1. Prove that a necessary and sufficient condition that the
More informationThe Kakeya problem. The University of Manchester. Jonathan Fraser
Jonathan M. Fraser The University of Manchester Kakeya needle sets A subset of the plane is called a Kakeya needle set if a unit line segment can be smoothly rotated within it by 360 degrees. Kakeya needle
More informationIntroduction to Proofs
Real Analysis Preview May 2014 Properties of R n Recall Oftentimes in multivariable calculus, we looked at properties of vectors in R n. If we were given vectors x =< x 1, x 2,, x n > and y =< y1, y 2,,
More informationCONTRACTION MAPPING & THE COLLAGE THEOREM Porter 1. Abstract
CONTRACTION MAPPING & THE COLLAGE THEOREM Porter 1 Abstract The Collage Theorem was given by Michael F. Barnsley who wanted to show that it was possible to reconstruct images using a set of functions.
More informationTHE HAUSDORFF DIMENSION OF THE BOUNDARY OF A SELF-SIMILAR TILE
THE HAUSDORFF DIMENSION OF THE BOUNDARY OF A SELF-SIMILAR TILE P DUVALL, J KEESLING AND A VINCE ABSTRACT An effective method is given for computing the Hausdorff dimension of the boundary of a self-similar
More informationMaximum and Minimum Values
Maimum and Minimum Values y Maimum Minimum MATH 80 Lecture 4 of 6 Definitions: A function f has an absolute maimum at c if f ( c) f ( ) for all in D, where D is the domain of f. The number f (c) is called
More informationFixed Point Theorem and Sequences in One or Two Dimensions
Fied Point Theorem and Sequences in One or Two Dimensions Dr. Wei-Chi Yang Let us consider a recursive sequence of n+ = n + sin n and the initial value can be an real number. Then we would like to ask
More informationSet Inversion Fractals
Set Inversion Fractals by Bryson Boreland A Thesis Presented to The University of Guelph In partial fulfilment of requirements for the degree of Master of Science in Applied Mathematics Guelph, Ontario,
More informationThe problems I left behind
The problems I left behind Kazimierz Goebel Maria Curie-Sk lodowska University, Lublin, Poland email: goebel@hektor.umcs.lublin.pl During over forty years of studying and working on problems of metric
More informationOn some non-conformal fractals
On some non-conformal fractals arxiv:1004.3510v1 [math.ds] 20 Apr 2010 Micha l Rams Institute of Mathematics, Polish Academy of Sciences ul. Śniadeckich 8, 00-950 Warszawa, Poland e-mail: rams@impan.gov.pl
More information+ ε /2N) be the k th interval. k=1. k=1. k=1. k=1
Trevor, Angel, and Michael Measure Zero, the Cantor Set, and the Cantor Function Measure Zero : Definition : Let X be a subset of R, the real number line, X has measure zero if and only if ε > 0 a set
More informationl(y j ) = 0 for all y j (1)
Problem 1. The closed linear span of a subset {y j } of a normed vector space is defined as the intersection of all closed subspaces containing all y j and thus the smallest such subspace. 1 Show that
More informationTake-Home Final Examination of Math 55a (January 17 to January 23, 2004)
Take-Home Final Eamination of Math 55a January 17 to January 3, 004) N.B. For problems which are similar to those on the homework assignments, complete self-contained solutions are required and homework
More informationMath 421, Homework #9 Solutions
Math 41, Homework #9 Solutions (1) (a) A set E R n is said to be path connected if for any pair of points x E and y E there exists a continuous function γ : [0, 1] R n satisfying γ(0) = x, γ(1) = y, and
More informationLimits and the derivative function. Limits and the derivative function
The Velocity Problem A particle is moving in a straight line. t is the time that has passed from the start of motion (which corresponds to t = 0) s(t) is the distance from the particle to the initial position
More informationLecture 2: A crash course in Real Analysis
EE5110: Probability Foundations for Electrical Engineers July-November 2015 Lecture 2: A crash course in Real Analysis Lecturer: Dr. Krishna Jagannathan Scribe: Sudharsan Parthasarathy This lecture is
More information