DOCSLIB.ORG

The Notion of Mixing and Rank One Examples

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Notion of Mixing and Rank One Examples THE NOTION OF MIXING AND RANK ONE EXAMPLES WENYU CHEN Abstract. This paper discusses the relationships among mixing, lightly mix- ing, weakly mixing and ergodicity. Besides proving the hierarchy of these notions, we construct examples of rank one transformations that exhibit each of these behaviors to demonstrate the differences. Contents 1. Preliminaries 1 1.1. Convergence 1 1.2. Sets of Density Zero 2 1.3. Dynamical Systems 2 2. Ergodicity 3 3. Mixing and Weakly Mixing 5 4. Mixing and Lightly Mixing 9 5. Rank One Maps 10 5.1. The Dyadic Odometer 11 5.2. Chac´on'sTransformation 15 5.3. Just Lightly Mixing Map 19 6. Discussion 24 Acknowledgments 25 References 26 1. Preliminaries We assume knowledge of measure theory (for example, chapters 1 and 2 of [5] would suffice). We use the same notation as in standard measure theory. 1.1. Convergence. Definition 1.1. A bounded sequence ai may converge to a number a in at least three ways. f g (1) Convergence: lim ai = a: i !1 (2) Strong Ces`aroconvergence: n 1 1 X− lim ai a = 0: n n j − j !1 i=0 Date: August 27 2015. 1 2 WENYU CHEN (3) Ces`aroconvergence: n 1 1 X− lim ai = a: n n !1 i=0 Lemma 1.2. Let ai be a bounded sequence. Then convergence implies Strong Ces`aro convergence,f andg Strong Ces`aro convergence implies Ces`aro convergence. Proof. The result follows easily by applying the - δ definition of limits. 1.2. Sets of Density Zero. Definition 1.3. A set D of nonnegative integers is said to be of zero density if n 1 1 X− lim ID(i) = 0; n n !1 i=0 where ID is the characteristic function defined by ID(x) = 1 if x D and ID(x) = 0 if x = D. 2 2 Definition 1.4. Let a R. We say a sequence ai of real numbers converges 2 f g in density to a if there exists a zero density set D N such that for any > 0, there is an integer N such that if i > N and i = D, then⊂ a a < . 2 j i − j Lemma 1.5. If bi is a bounded sequence of nonnegative real numbers, then bi converges Ces`arof tog0 if and only if b converges in density to 0. f g f ig Proof. The result follows easily by applying the definitions of convergences. 1.3. Dynamical Systems. Definition 1.6 (Dynamical System). A measurable dynamical system is a quadruple (X; ; µ, T ), where X is a space, a σ-algebra of subsets of X, µ a measure on Sand T : X X a measurableS transformation with respect to µ. Specifically, (SX; ; µ, T ) is a!Lebesgue measure-preserving dynamical system if (X; ; µ) is aS Lebesgue measurable space and T is a measure-preserving trans- formationS T : X X. We will refer to such system simply as measure-preserving dynamical system! in this paper. Though we are not going to study the \sameness" of dynamical systems, we define the notion of factor for technical reasons. A factor of a dynamical system carries some of its dynamical properties. Later we show if a dynamical system has an ergodic transformation, then so does its factor. Definition 1.7. Let (X; ; µ, T ) be a measurable dynamical system. A dynamical S system (X0; 0; µ0;T 0) is a factor of (X; ; µ, T ) if there exist measurable sets X S S 0 ⊂ X and X0 X0 of full measure with T (X ) X; T 0(X0 ) X0, and a factor map 0 ⊂ 0 ⊂ 0 ⊂ φ : X X0 such that: 0 ! 0 (1) φ is onto 1 (2) φ− (A) (X0) for all A X0 1 2 S 2 (3) µ(φ− (A)) = µ0(A) for all A 0(X0 ) 2 S 0 (4) φ(T (x)) = T 0(φ(x)). THE NOTION OF MIXING AND RANK ONE EXAMPLES 3 2. Ergodicity Definition 2.1 (Ergodicity). A measure-preserving transformation T is ergodic 1 if whenever A is a measurable set that is strictly T -invariant (i.e., T − (A) = A), then µ(A) is either 0 or 1. Theorem 2.2 (Birkhoff's ergodic theorem). Let (X; ; µ) be a probability space S and let T be a measure-preserving transformation on (X; ; µ). If f : X R is an integrable function, then S ! 1 Pn 1 i (1) limn − f(T (x)) exists for all x X N, for some null set N !1 n i=0 2 n depending on f. Denote this limit by f~. (2) f~(T x) = f~(x) a.e. (3) For any measurable set A that is T -invariant, Z Z fdµ = fdµ.~ A A In particular, if T is ergodic, then n 1 1 X− Z lim f(T i(x)) = fdµ a.e. n n !1 i=0 For the proof, see chapter 5 of [9]. Theorem 2.3. Let T be a finite measure-preserving transformation on a probability space (X; ; µ). T is ergodic if and only if for all measurable sets A and B, S n 1 1 X− i (2.4) lim µ(T − (A) B) = µ(A)µ(B): n n \ !1 i=0 Proof. First we prove the forward direction. Suppose T is ergodic and A, B mea- surable. Apply the Birkhoff ergodic theorem on T , A, B, and the characteristic map IA IB to get n 1 Z 1 X− i lim IA(T (x)) = IAdµ = µ(A) a.e. n n !1 i=0 Thus, n 1 1 X− i lim IA(T (x))IB(x) = µ(A)IB(x) a.e. n n !1 i=0 i Observe that IA(T (x))IB(x) 1 a:e:, which enables us to apply the dominated convergence theorem.j This givesj ≤ Z n 1 Z n 1 1 X− i 1 X− i lim IA(T (x))IB(x)dµ(x) = lim IA(T (x))IB(x)dµ(x) n n n n !1 i=0 !1 i=0 Z = µ(A)IB(x)dµ(x) = µ(A)µ(B): 4 WENYU CHEN Also, we know Z n 1 n 1 Z 1 X− i 1 X− i IA(T (x))IB(x)dµ(x) = IA(T (x))IB(x)dµ(x) n n i=0 i=0 n 1 1 X− i = µ(T − (A) B): n \ i=0 We take the limit of both sides and obtain what we want to show. Now we prove the reverse direction. Suppose (2.4) holds for all sets A and B in X. Then it also holds for A = B. Let A be T -invariant. By the defini- 1 Pn 1 i tion of T -invariant, limn − µ(T − (A) A) = µ(A): Let A = B. We get !1 n i=0 1 Pn 1 i 2 \ 2 limn − µ(T − (A) A) = µ(A) : This gives µ(A) = µ(A) , which implies !1 n i=0 \ µ(A) is 0 or 1. Thus T is ergodic. Proposition 2.5. Let T be defined on a probability space (X; (X); µ), T T defined on (X X; (X) (X); µ µ). If T T is an ergodicS transformation,× then T is also ergodic.× S × S × × 1 Proof. Let A be a strictly T -invariant measurable set. Then (T T )− (A A) = 1 1 × × T − (A) T − (A) = A A, so A A is (T T )-invariant. The property of product measure× shows × × × (µ µ)(A A) = µ(A)µ(A): × × Since T T is ergodic, (µ µ)(A A) is 1 or 0. Therefore µ(A) is either 1 or 0. × × × Hence T is ergodic. Theorem 2.6. Let (Y; 0(Y ); ν; S) be a factor of (X; (X); µ, T ). If T is ergodic, so is S. S S Proof. Let φ : X Y be a factor map. If A is a strictly S-invariant set, then ! 1 1 1 1 1 T − (φ− (A)) = φ− (S− (A)) = φ− (A): (The first equality holds by Definition 1.7 and the second holds because A is S- 1 1 invariant.) Thus φ− (A) is a T -invariant set. Also by Definition 1.7, µ(φ− (A)) = ν(A). Since T is ergodic, ν(A) is 0 or 1, i.e., S is ergodic. Definition 2.7 (Eigenvalue and Eigenfunction). Let (X; ; µ) be a probability space and let T : X X be a measure-preserving transformation.S We say a ! number µ C is an eigenvalue of T if there exists a function f L2(X; ; µ) which is nonzero2 a.e. such that 2 S f(T (x)) = λf(x) µ-a.e. The function f is called an eigenfunction. Lemma 2.8. If λ is an eigenvalue for a measure-preserving transformation T , then λ = 1. j j Proof. Let T be a measure-preserving transformation and f be its eigenfunction. Let λ be its eigenvalue. Then λ 2 R f 2dµ = R λ 2 f 2dµ = R f T 2dµ = R f 2 j j j j j j j j j ◦ j j j ◦ T dµ = R f 2dµ. j j Lemma 2.9. If T is ergodic and f is an eigenfunction, then f is constant a.e. j j THE NOTION OF MIXING AND RANK ONE EXAMPLES 5 Proof. Let f be an eigenfunction of T with eigenvalue λ, i.e., f T = λf a.e. By Lemma 2.8, we know λ = 1.
Load more

Recommended publications
  • Algebraic Properties of the Richard Thompson's Group F
    ALGEBRAIC PROPERTIES OF THE RICHARD THOMPSON’S GROUP F AND ITS APPLICATIONS IN CRYPTOGRAPHY A THESIS SUBMITTED TO THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES OF MIDDLE EAST TECHNICAL UNIVERSITY BY HAKAN YETER IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN MATHEMATICS FEBRUARY 2021 Approval of the thesis: ALGEBRAIC PROPERTIES OF THE RICHARD THOMPSON’S GROUP F AND ITS APPLICATIONS IN CRYPTOGRAPHY submitted by HAKAN YETER in partial fulfillment of the requirements for the de- gree of Master of Science in Mathematics Department, Middle East Technical University by, Prof. Dr. Halil Kalıpçılar Dean, Graduate School of Natural and Applied Sciences Prof. Dr. Yıldıray Ozan Head of Department, Mathematics Assoc. Prof. Dr. Mustafa Gökhan Benli Supervisor, Mathematics Department, METU Examining Committee Members: Assoc. Prof. Dr. Fatih Sulak Mathematics Department, Atilim University Assoc. Prof. Dr. Mustafa Gökhan Benli Mathematics Department, METU Assist. Prof. Dr. Burak Kaya Mathematics Department, METU Date: I hereby declare that all information in this document has been obtained and presented in accordance with academic rules and ethical conduct. I also declare that, as required by these rules and conduct, I have fully cited and referenced all material and results that are not original to this work. Name, Surname: Hakan Yeter Signature : iv ABSTRACT ALGEBRAIC PROPERTIES OF THE RICHARD THOMPSON’S GROUP F AND ITS APPLICATIONS IN CRYPTOGRAPHY Yeter, Hakan M.S., Department of Mathematics Supervisor: Assoc. Prof. Dr. Mustafa Gökhan Benli February 2021, 69 pages Thompson’s groups F; T and V , especially F , are widely studied groups in group theory.
    [Show full text]
  • Mixing, Chaotic Advection, and Turbulence
    Annual Reviews www.annualreviews.org/aronline Annu. Rev. Fluid Mech. 1990.22:207-53 Copyright © 1990 hV Annual Reviews Inc. All r~hts reserved MIXING, CHAOTIC ADVECTION, AND TURBULENCE J. M. Ottino Department of Chemical Engineering, University of Massachusetts, Amherst, Massachusetts 01003 1. INTRODUCTION 1.1 Setting The establishment of a paradigm for mixing of fluids can substantially affect the developmentof various branches of physical sciences and tech- nology. Mixing is intimately connected with turbulence, Earth and natural sciences, and various branches of engineering. However, in spite of its universality, there have been surprisingly few attempts to construct a general frameworkfor analysis. An examination of any library index reveals that there are almost no works textbooks or monographs focusing on the fluid mechanics of mixing from a global perspective. In fact, there have been few articles published in these pages devoted exclusively to mixing since the first issue in 1969 [one possible exception is Hill’s "Homogeneous Turbulent Mixing With Chemical Reaction" (1976), which is largely based on statistical theory]. However,mixing has been an important component in various other articles; a few of them are "Mixing-Controlled Supersonic Combustion" (Ferri 1973), "Turbulence and Mixing in Stably Stratified Waters" (Sherman et al. 1978), "Eddies, Waves, Circulation, and Mixing: Statistical Geofluid Mechanics" (Holloway 1986), and "Ocean Tur- bulence" (Gargett 1989). It is apparent that mixing appears in both indus- try and nature and the problems span an enormous range of time and length scales; the Reynolds numberin problems where mixing is important varies by 40 orders of magnitude(see Figure 1).
    [Show full text]
  • The Role of the Interval Domain in Modern Exact Real Airthmetic
    The Role of the Interval Domain in Modern Exact Real Airthmetic Andrej Bauer Iztok Kavkler Faculty of Mathematics and Physics University of Ljubljana, Slovenia Domains VIII & Computability over Continuous Data Types Novosibirsk, September 2007 Teaching theoreticians a lesson Recently I have been told by an anonymous referee that “Theoreticians do not like to be taught lessons.” and by a friend that “You should stop competing with programmers.” In defiance of this advice, I shall talk about the lessons I learned, as a theoretician, in programming exact real arithmetic. The spectrum of real number computation slow fast Formally verified, Cauchy sequences iRRAM extracted from streams of signed digits RealLib proofs floating point Moebius transformtions continued fractions Mathematica "theoretical" "practical" I Common features: I Reals are represented by successive approximations. I Approximations may be computed to any desired accuracy. I State of the art, as far as speed is concerned: I iRRAM by Norbert Muller,¨ I RealLib by Branimir Lambov. What makes iRRAM and ReaLib fast? I Reals are represented by sequences of dyadic intervals (endpoints are rationals of the form m/2k). I The approximating sequences need not be nested chains of intervals. I No guarantee on speed of converge, but arbitrarily fast convergence is possible. I Previous approximations are not stored and not reused when the next approximation is computed. I Each next approximation roughly doubles the amount of work done. The theory behind iRRAM and RealLib I Theoretical models used to design iRRAM and RealLib: I Type Two Effectivity I a version of Real RAM machines I Type I representations I The authors explicitly reject domain theory as a suitable computational model.
    [Show full text]
  • Stat 8112 Lecture Notes Stationary Stochastic Processes Charles J
    Stat 8112 Lecture Notes Stationary Stochastic Processes Charles J. Geyer April 29, 2012 1 Stationary Processes A sequence of random variables X1, X2, ::: is called a time series in the statistics literature and a (discrete time) stochastic process in the probability literature. A stochastic process is strictly stationary if for each fixed positive integer k the distribution of the random vector (Xn+1;:::;Xn+k) has the same distribution for all nonnegative integers n. A stochastic process having second moments is weakly stationary or sec- ond order stationary if the expectation of Xn is the same for all positive integers n and for each nonnegative integer k the covariance of Xn and Xn+k is the same for all positive integers n. 2 The Birkhoff Ergodic Theorem The Birkhoff ergodic theorem is to strictly stationary stochastic pro- cesses what the strong law of large numbers (SLLN) is to independent and identically distributed (IID) sequences. In effect, despite the different name, it is the SLLN for stationary stochastic processes. Suppose X1, X2, ::: is a strictly stationary stochastic process and X1 has expectation (so by stationary so do the rest of the variables). Write n 1 X X = X : n n i i=1 To introduce the Birkhoff ergodic theorem, it says a.s. Xn −! Y; (1) where Y is a random variable satisfying E(Y ) = E(X1). More can be said about Y , but we will have to develop some theory first. 1 The SSLN for IID sequences says the same thing as the Birkhoff ergodic theorem (1) except that in the SLLN for IID sequences the limit Y = E(X1) is constant.
    [Show full text]
  • Pointwise and L1 Mixing Relative to a Sub-Sigma Algebra
    POINTWISE AND L1 MIXING RELATIVE TO A SUB-SIGMA ALGEBRA DANIEL J. RUDOLPH Abstract. We consider two natural definitions for the no- tion of a dynamical system being mixing relative to an in- variant sub σ-algebra H. Both concern the convergence of |E(f · g ◦ T n|H) − E(f|H)E(g ◦ T n|H)| → 0 as |n| → ∞ for appropriate f and g. The weaker condition asks for convergence in L1 and the stronger for convergence a.e. We will see that these are different conditions. Our goal is to show that both these notions are robust. As is quite standard we show that one need only consider g = f and E(f|H) = 0, and in this case |E(f · f ◦ T n|H)| → 0. We will see rather easily that for L1 convergence it is enough to check an L2-dense family. Our major result will be to show the same is true for pointwise convergence making this a verifiable condition. As an application we will see that if T is mixing then for any ergodic S, S × T is relatively mixing with respect to the first coordinate sub σ-algebra in the pointwise sense. 1. Introduction Mixing properties for ergodic measure preserving systems gener- ally have versions “relative” to an invariant sub σ-algebra (factor algebra). For most cases the fundamental theory for the abso- lute case lifts to the relative case. For example one can say T is relatively weakly mixing with respect to a factor algebra H if 1) L2(µ) has no finite dimensional invariant submodules over the subspace of H-measurable functions, or Date: August 31, 2005.
    [Show full text]
  • On Li-Yorke Measurable Sensitivity
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 143, Number 6, June 2015, Pages 2411–2426 S 0002-9939(2015)12430-6 Article electronically published on February 3, 2015 ON LI-YORKE MEASURABLE SENSITIVITY JARED HALLETT, LUCAS MANUELLI, AND CESAR E. SILVA (Communicated by Nimish Shah) Abstract. The notion of Li-Yorke sensitivity has been studied extensively in the case of topological dynamical systems. We introduce a measurable version of Li-Yorke sensitivity, for nonsingular (and measure-preserving) dynamical systems, and compare it with various mixing notions. It is known that in the case of nonsingular dynamical systems, a conservative ergodic Cartesian square implies double ergodicity, which in turn implies weak mixing, but the converses do not hold in general, though they are all equivalent in the finite measure- preserving case. We show that for nonsingular systems, an ergodic Cartesian square implies Li-Yorke measurable sensitivity, which in turn implies weak mixing. As a consequence we obtain that, in the finite measure-preserving case, Li-Yorke measurable sensitivity is equivalent to weak mixing. We also show that with respect to totally bounded metrics, double ergodicity implies Li-Yorke measurable sensitivity. 1. Introduction The notion of sensitive dependence for topological dynamical systems has been studied by many authors; see, for example, the works [3, 7, 10] and the references therein. Recently, various notions of measurable sensitivity have been explored in ergodic theory; see for example [2, 9, 11–14]. In this paper we are interested in formulating a measurable version of the topolog- ical notion of Li-Yorke sensitivity for the case of nonsingular and measure-preserving dynamical systems.
    [Show full text]
  • Chaos Theory
    By Nadha CHAOS THEORY What is Chaos Theory? . It is a field of study within applied mathematics . It studies the behavior of dynamical systems that are highly sensitive to initial conditions . It deals with nonlinear systems . It is commonly referred to as the Butterfly Effect What is a nonlinear system? . In mathematics, a nonlinear system is a system which is not linear . It is a system which does not satisfy the superposition principle, or whose output is not directly proportional to its input. Less technically, a nonlinear system is any problem where the variable(s) to be solved for cannot be written as a linear combination of independent components. the SUPERPOSITION PRINCIPLE states that, for all linear systems, the net response at a given place and time caused by two or more stimuli is the sum of the responses which would have been caused by each stimulus individually. So that if input A produces response X and input B produces response Y then input (A + B) produces response (X + Y). So a nonlinear system does not satisfy this principal. There are 3 criteria that a chaotic system satisfies 1) sensitive dependence on initial conditions 2) topological mixing 3) periodic orbits are dense 1) Sensitive dependence on initial conditions . This is popularly known as the "butterfly effect” . It got its name because of the title of a paper given by Edward Lorenz titled Predictability: Does the Flap of a Butterfly’s Wings in Brazil set off a Tornado in Texas? . The flapping wing represents a small change in the initial condition of the system, which causes a chain of events leading to large-scale phenomena.
    [Show full text]
  • Turbulence, Fractals, and Mixing
    Turbulence, fractals, and mixing Paul E. Dimotakis and Haris J. Catrakis GALCIT Report FM97-1 17 January 1997 Firestone Flight Sciences Laboratory Guggenheim Aeronautical Laboratory Karman Laboratory of Fluid Mechanics and Jet Propulsion Pasadena Turbulence, fractals, and mixing* Paul E. Dimotakis and Haris J. Catrakis Graduate Aeronautical Laboratories California Institute of Technology Pasadena, California 91125 Abstract Proposals and experimenta1 evidence, from both numerical simulations and laboratory experiments, regarding the behavior of level sets in turbulent flows are reviewed. Isoscalar surfaces in turbulent flows, at least in liquid-phase turbulent jets, where extensive experiments have been undertaken, appear to have a geom- etry that is more complex than (constant-D) fractal. Their description requires an extension of the original, scale-invariant, fractal framework that can be cast in terms of a variable (scale-dependent) coverage dimension, Dd(X). The extension to a scale-dependent framework allows level-set coverage statistics to be related to other quantities of interest. In addition to the pdf of point-spacings (in 1-D), it can be related to the scale-dependent surface-to-volume (perimeter-to-area in 2-D) ratio, as well as the distribution of distances to the level set. The application of this framework to the study of turbulent -jet mixing indicates that isoscalar geometric measures are both threshold and Reynolds-number dependent. As regards mixing, the analysis facilitated by the new tools, as well as by other criteria, indicates en- hanced mixing with increasing Reynolds number, at least for the range of Reynolds numbers investigated. This results in a progressively less-complex level-set geom- etry, at least in liquid-phase turbulent jets, with increasing Reynolds number.
    [Show full text]
  • Blocking Conductance and Mixing in Random Walks
    Blocking conductance and mixing in random walks Ravindran Kannan Department of Computer Science Yale University L´aszl´oLov´asz Microsoft Research and Ravi Montenegro Georgia Institute of Technology Contents 1 Introduction 3 2 Continuous state Markov chains 7 2.1 Conductance and other expansion measures . 8 3 General mixing results 11 3.1 Mixing time estimates in terms of a parameter function . 11 3.2 Faster convergence to stationarity . 13 3.3 Constructing mixweight functions . 13 3.4 Explicit conductance bounds . 16 3.5 Finite chains . 17 3.6 Further examples . 18 4 Random walks in convex bodies 19 4.1 A logarithmic Cheeger inequality . 19 4.2 Mixing of the ball walk . 21 4.3 Sampling from a convex body . 21 5 Proofs I: Markov Chains 22 5.1 Proof of Theorem 3.1 . 22 5.2 Proof of Theorem 3.2 . 24 5.3 Proof of Theorem 3.3 . 26 1 6 Proofs II: geometry 27 6.1 Lemmas . 27 6.2 Proof of theorem 4.5 . 29 6.3 Proof of theorem 4.6 . 30 6.4 Proof of Theorem 4.3 . 32 2 Abstract The notion of conductance introduced by Jerrum and Sinclair [JS88] has been widely used to prove rapid mixing of Markov Chains. Here we introduce a bound that extends this in two directions. First, instead of measuring the conductance of the worst subset of states, we bound the mixing time by a formula that can be thought of as a weighted average of the Jerrum-Sinclair bound (where the average is taken over subsets of states with different sizes).
    [Show full text]
  • Dyadic Rationals and Surreal Number Theory C
    IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 16, Issue 5 Ser. IV (Sep. – Oct. 2020), PP 35-43 www.iosrjournals.org Dyadic Rationals and Surreal Number Theory C. Avalos-Ramos C.U.C.E.I. Universidad de Guadalajara, Guadalajara, Jalisco, México J. A. Félix-Algandar Facultad de Ciencias Físico-Matemáticas de la Universidad Autónoma de Sinaloa, 80010, Culiacán, Sinaloa, México. J. A. Nieto Facultad de Ciencias Físico-Matemáticas de la Universidad Autónoma de Sinaloa, 80010, Culiacán, Sinaloa, México. Abstract We establish a number of properties of the dyadic rational numbers associated with surreal number theory. In particular, we show that a two parameter function of dyadic rationals can give all the trees of n-days in surreal number formalism. --------------------------------------------------------------------------------------------------------------------------------------- Date of Submission: 07-10-2020 Date of Acceptance: 22-10-2020 --------------------------------------------------------------------------------------------------------------------------------------- I. Introduction In mathematics, from number theory history [1], one learns that historically, roughly speaking, the starting point was the natural numbers N and after a centuries of though evolution one ends up with the real numbers from which one constructs the differential and integral calculus. Surprisingly in 1973 Conway [2] (see also Ref. [3]) developed the surreal numbers structure which contains no only the real numbers , but also the hypereals and other numerical structures. Consider the set (1) and call and the left and right sets of , respectively. Surreal numbers are defined in terms of two axioms: Axiom 1. Every surreal number corresponds to two sets and of previously created numbers, such that no member of the left set is greater or equal to any member of the right set .
    [Show full text]
  • Chaos and Chaotic Fluid Mixing
    Snapshots of modern mathematics № 4/2014 from Oberwolfach Chaos and chaotic fluid mixing Tom Solomon Very simple mathematical equations can give rise to surprisingly complicated, chaotic dynamics, with be- havior that is sensitive to small deviations in the ini- tial conditions. We illustrate this with a single re- currence equation that can be easily simulated, and with mixing in simple fluid flows. 1 Introduction to chaos People used to believe that simple mathematical equations had simple solutions and complicated equations had complicated solutions. However, Henri Poincaré (1854–1912) and Edward Lorenz [4] showed that some remarkably simple math- ematical systems can display surprisingly complicated behavior. An example is the well-known logistic map equation: xn+1 = Axn(1 − xn), (1) where xn is a quantity (between 0 and 1) at some time n, xn+1 is the same quantity one time-step later and A is a given real number between 0 and 4. One of the most common uses of the logistic map is to model variations in the size of a population. For example, xn can denote the population of adult mayflies in a pond at some point in time, measured as a fraction of the maximum possible number the adult mayfly population can reach in this pond. Similarly, xn+1 denotes the population of the next generation (again, as a fraction of the maximum) and the constant A encodes the influence of various factors on the 1 size of the population (ability of pond to provide food, fraction of adults that typically reproduce, fraction of hatched mayflies not reaching adulthood, overall mortality rate, etc.) As seen in Figure 1, the behavior of the logistic system changes dramatically with changes in the parameter A.
    [Show full text]
  • Generalized Bernoulli Process with Long-Range Dependence And
    Depend. Model. 2021; 9:1–12 Research Article Open Access Jeonghwa Lee* Generalized Bernoulli process with long-range dependence and fractional binomial distribution https://doi.org/10.1515/demo-2021-0100 Received October 23, 2020; accepted January 22, 2021 Abstract: Bernoulli process is a nite or innite sequence of independent binary variables, Xi , i = 1, 2, ··· , whose outcome is either 1 or 0 with probability P(Xi = 1) = p, P(Xi = 0) = 1 − p, for a xed constant p 2 (0, 1). We will relax the independence condition of Bernoulli variables, and develop a generalized Bernoulli process that is stationary and has auto-covariance function that obeys power law with exponent 2H − 2, H 2 (0, 1). Generalized Bernoulli process encompasses various forms of binary sequence from an independent binary sequence to a binary sequence that has long-range dependence. Fractional binomial random variable is dened as the sum of n consecutive variables in a generalized Bernoulli process, of particular interest is when its variance is proportional to n2H , if H 2 (1/2, 1). Keywords: Bernoulli process, Long-range dependence, Hurst exponent, over-dispersed binomial model MSC: 60G10, 60G22 1 Introduction Fractional process has been of interest due to its usefulness in capturing long lasting dependency in a stochas- tic process called long-range dependence, and has been rapidly developed for the last few decades. It has been applied to internet trac, queueing networks, hydrology data, etc (see [3, 7, 10]). Among the most well known models are fractional Gaussian noise, fractional Brownian motion, and fractional Poisson process. Fractional Brownian motion(fBm) BH(t) developed by Mandelbrot, B.
    [Show full text]