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430
Estimating fractal dimension
 Journal of the Optical Society of America A
, 1990
"... Fractals arise from a variety of sources and have been observed in nature and on computer screens. One of the exceptional characteristics of fractals is that they can be described by a noninteger dimension. The geometry of fractals and the mathematics of fractal dimension have provided useful tools ..."
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Cited by 120 (4 self)
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Fractals arise from a variety of sources and have been observed in nature and on computer screens. One of the exceptional characteristics of fractals is that they can be described by a noninteger dimension. The geometry of fractals and the mathematics of fractal dimension have provided useful tools for a variety of scientific disciplines, among which is chaos. Chaotic dynamical systems exhibit trajectories in their phase space that converge to a strange attractor. The fractal dimension of this attractor counts the effective number of degrees of freedom in the dynamical system and thus quantifies its complexity. In recent years, numerical methods have been developed for estimating the dimension directly from the observed behavior of the physical system. The purpose of this paper is to survey briefly the kinds of fractals that appear in scientific research, to discuss the application of fractals to nonlinear dynamical systems, and finally to review more comprehensively the state of the art in numerical methods for estimating the fractal dimension of a strange attractor. Confusion is a word we have invented for an order which is not understood.Henry Miller, "Interlude," Tropic of Capricorn Numerical coincidence is a common path to intellectual perdition in our quest for meaning. We delight in catalogs of disparate items united by the same number, and often feel in our gut that some unity must underlie it all.
Dimensions and measures in infinite iterated function systems
 PROC. LONDON MATH. SOC
, 1996
"... The Hausdorff and packing measures and dimensions of the limit sets of iterated function systems generated by countable families of conformal contractions are investigated. Conformal measures for such systems, reflecting geometric properties of the limit set, are introduced, proven to exist, and to ..."
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Cited by 114 (25 self)
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The Hausdorff and packing measures and dimensions of the limit sets of iterated function systems generated by countable families of conformal contractions are investigated. Conformal measures for such systems, reflecting geometric properties of the limit set, are introduced, proven to exist, and to be unique. The existence of a unique invariant probability equivalent to the conformal measure is derived. Our methods employ the concepts of the PerronFrobenius operator, symbolic dynamics on an infinite dimensional shift space, and the properties of the above mentioned ergodic invariant measure. A formula for the Hausdorff dimension of the limit set in terms of the pressure function is derived. Fractal phenomena not exhibited by finite systems are shown to appear in the infinite case. In particular a variety of conditions are provided for Hausdorff and packing measures to be positive or finite, and a number of examples are described showing the appearance of various possible combinations for these quantities. One example given special attention is the limit set associated to the complex continued fraction expansion  in particular lower and upper estimates for its Hausdor dimension are given. A large natural class of systems whose limit sets are "dimensionless in the restricted sense" is described.
Recent work connected with the Kakeya problem
 Prospects in mathematics (Princeton, NJ
, 1996
"... A Kakeya set in R n is a compact set E ⊂ R n containing a unit line segment in every direction, i.e. ∀e ∈ S n−1 ∃x ∈ R n: x + te ∈ E ∀t ∈ [ − 1 1 ..."
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Cited by 107 (2 self)
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A Kakeya set in R n is a compact set E ⊂ R n containing a unit line segment in every direction, i.e. ∀e ∈ S n−1 ∃x ∈ R n: x + te ∈ E ∀t ∈ [ − 1 1
Dimension in Complexity Classes
 SIAM Journal on Computing
, 2000
"... A theory of resourcebounded dimension is developed using gales, which are natural generalizations of martingales. When the resource bound (a parameter of the theory) is unrestricted, the resulting dimension is precisely the classical Hausdorff dimension (sometimes called "fractal dimension&qu ..."
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Cited by 104 (16 self)
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A theory of resourcebounded dimension is developed using gales, which are natural generalizations of martingales. When the resource bound (a parameter of the theory) is unrestricted, the resulting dimension is precisely the classical Hausdorff dimension (sometimes called "fractal dimension"). Other choices of the parameter yield internal dimension theories in E, E 2 , ESPACE, and other complexity classes, and in the class of all decidable problems. In general, if C is such a class, then every set X of languages has a dimension in C, which is a real number dim(X j C) 2 [0; 1]. Along with the elements of this theory, two preliminary applications are presented: 1. For every real number 0 1 2 , the set FREQ( ), consisting of all languages that asymptotically contain at most of all strings, has dimension H()  the binary entropy of  in E and in E 2 . 2. For every real number 0 1, the set SIZE( 2 n n ), consisting of all languages decidable by Boolean circuits of at most 2 n n gates, has dimension in ESPACE.
The Dimensions of Individual Strings and Sequences
 INFORMATION AND COMPUTATION
, 2003
"... A constructive version of Hausdorff dimension is developed using constructive supergales, which are betting strategies that generalize the constructive supermartingales used in the theory of individual random sequences. This constructive dimension is used to assign every individual (infinite, binary ..."
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Cited by 99 (11 self)
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A constructive version of Hausdorff dimension is developed using constructive supergales, which are betting strategies that generalize the constructive supermartingales used in the theory of individual random sequences. This constructive dimension is used to assign every individual (infinite, binary) sequence S a dimension, which is a real number dim(S) in the interval [0, 1]. Sequences that
The Lipschitz continuity of the distance function to the cut locus
 Trans. Amer. Math. Soc
"... Abstract. Let N be a closed submanifold of a complete smooth Riemannian manifold M and Uν the total space of the unit normal bundle of N. Foreach v ∈ Uν,letρ(v) denote the distance from N to the cut point of N on the geodesic γv with the velocity vector ˙γv(0) = v. The continuity of the function ρ ..."
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Cited by 52 (2 self)
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Abstract. Let N be a closed submanifold of a complete smooth Riemannian manifold M and Uν the total space of the unit normal bundle of N. Foreach v ∈ Uν,letρ(v) denote the distance from N to the cut point of N on the geodesic γv with the velocity vector ˙γv(0) = v. The continuity of the function ρ on Uν is well known. In this paper we prove that ρ is locally Lipschitz on which ρ is bounded; in particular, if M and N are compact, then ρ is globally Lipschitz on Uν. Therefore, the canonical interior metric δ may be introduced on each connected component of the cut locus of N, and this metric space becomes a locally compact and complete length space. Let N be an immersed submanifold of a complete C ∞ Riemannian manifold M and π: Uν → N the unit normal bundle of N. For each positive integer k and vector v ∈ Uν, letanumberλk(v) denote the parameter value of γv, whereγv denotes the geodesic for which the velocity vector is v at t =0, such that γv(λk(v)) is the kth focal point (conjugate point for the case in which N is a point) of N along γv, counted with focal (or conjugate) multiplicities. A unit speed geodesic segment
Menger curvature and rectifiability
 Ann. of Math
, 1999
"... Let us first introduce some basic definitions needed to better understand this introduction and the rest of the paper. For a Borel set E ⊂ Rn, we call “total Menger curvature of E ” the nonnegative number c(E) defined by c 2 ∫ ∫ ∫ ..."
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Cited by 52 (0 self)
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Let us first introduce some basic definitions needed to better understand this introduction and the rest of the paper. For a Borel set E ⊂ Rn, we call “total Menger curvature of E ” the nonnegative number c(E) defined by c 2 ∫ ∫ ∫
Scale Invariance in Biology: Coincidence Or Footprint of a Universal Mechanism?
, 2001
"... In this article, we present a selfcontained review of recent work on complex biological systems which exhibit no characteristic scale. This property can manifest itself with fractals (spatial scale invariance), flicker noise or 1}fnoise where f denotes the frequency of a signal (temporal scale i ..."
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Cited by 51 (1 self)
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In this article, we present a selfcontained review of recent work on complex biological systems which exhibit no characteristic scale. This property can manifest itself with fractals (spatial scale invariance), flicker noise or 1}fnoise where f denotes the frequency of a signal (temporal scale invariance) and power laws (scale invariance in the size and duration of events in the dynamics of the system). A hypothesis recently put forward to explain these scalefree phenomomena is criticality, a notion introduced by physicists while studying phase transitions in materials, where systems spontaneously arrange themselves in an unstable manner similar, for instance, to a row of dominoes. Here, we review in a critical manner work which investigates to what extent this idea can be generalized to biology. More precisely, we start with a brief introduction to the concepts of absence of characteristic scale (powerlaw distributions, fractals and 1}fnoise) and of critical phenomena. We then review typical mathematical models exhibiting such properties : edge of chaos, cellular automata and selforganized critical models. These notions are then brought together to see to what extent they can account for the scale invariance observed in ecology, evolution of species, type III epidemics and some aspects of the central nervous system. This article also discusses how the notion of scale invariance can give important insights into the workings of biological systems.
Mellin Transforms And Asymptotics: Digital Sums
, 1993
"... Arithmetic functions related to number representation systems exhibit various periodicity phenomena. For instance, a well known theorem of Delange expresses the total number of ones in the binary representations of the first n integers in terms of a periodic fractal function. We show that such perio ..."
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Cited by 47 (12 self)
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Arithmetic functions related to number representation systems exhibit various periodicity phenomena. For instance, a well known theorem of Delange expresses the total number of ones in the binary representations of the first n integers in terms of a periodic fractal function. We show that such periodicity phenomena can be analyzed rather systematically using classical tools from analytic number theory, namely the MellinPerron formulae. This approach yields naturally the Fourier series involved in the expansions of a variety of digital sums related to number representation systems.