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20
SelfSimilar Measures And Intersections Of Cantor Sets
, 1997
"... . It is natural to expect that the arithmetic sum of two Cantor sets should have positive Lebesgue measure if the sum of their dimensions exceeds 1, but there are many known counterexamples, e.g. when both sets are the middleff Cantor set and ff 2 ( 1 3 ; 1 2 ). We show that for any compact set ..."
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Cited by 37 (12 self)
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. It is natural to expect that the arithmetic sum of two Cantor sets should have positive Lebesgue measure if the sum of their dimensions exceeds 1, but there are many known counterexamples, e.g. when both sets are the middleff Cantor set and ff 2 ( 1 3 ; 1 2 ). We show that for any compact set K and for a.e. ff 2 (0; 1), the arithmetic sum of K and the middleff Cantor set does indeed have positive Lebesgue measure when the sum of their Hausdorff dimensions exceeds 1. In this case we also determine the essential supremum, as the translation parameter t varies, of the dimension of the intersection of K + t with the middleff Cantor set. The same method yields an interesting property of infinite Bernoulli convolutions p (the distributions of random series P 1 n=0 \Sigma n ; where the signs are chosen independently with probabilities (p; 1 \Gamma p)). Let 1 q1 ! q2 2. For p 6= 1 2 near 1 2 and for a.e. in some nonempty interval, p is absolutely continuous and its den...
A generalization of Chaitin’s halting probability Ω and halting selfsimilar sets
 Hokkaido Math. J
, 2002
"... We generalize the concept of randomness in an infinite binary sequence in order to characterize the degree of randomness by a real number D> 0. Chaitin’s halting probability Ω is generalized to Ω D whose degree of randomness is precisely D. On the basis of this generalization, we consider the degree ..."
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Cited by 34 (12 self)
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We generalize the concept of randomness in an infinite binary sequence in order to characterize the degree of randomness by a real number D> 0. Chaitin’s halting probability Ω is generalized to Ω D whose degree of randomness is precisely D. On the basis of this generalization, we consider the degree of randomness of each point in Euclidean space through its basetwo expansion. It is then shown that the maximum value of such a degree of randomness provides the Hausdorff dimension of a selfsimilar set that is computable in a certain sense. The class of such selfsimilar sets includes familiar fractal sets such as the Cantor set, von Koch curve, and Sierpiński gasket. Knowledge of the property of Ω D allows us to show that the selfsimilar subset of [0,1] defined by the halting set of a universal algorithm has a Hausdorff dimension of one.
Multifractal Measures and a Weak Separation Condition
, 1999
"... We define a new separation property on the family of contractive similitudes that allows certain overlappings. This property is weaker than the open set condition of Hutchinson. It includes the wellknown class of infinite Bernoulli convolutions associated with the P.V. numbers and the solutions of ..."
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Cited by 28 (11 self)
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We define a new separation property on the family of contractive similitudes that allows certain overlappings. This property is weaker than the open set condition of Hutchinson. It includes the wellknown class of infinite Bernoulli convolutions associated with the P.V. numbers and the solutions of the twoscale dilation equations. Our main purpose in this paper is to prove the multifractal formalism under such condition.
Hierarchical Interpretation of Fractal Image Coding and Its Application to Fast Decoding
 in Proc. Digital Signal Processing Conference
, 1993
"... The basics of a block oriented fractal image coder, are described. The output of the coder is an IFS (Iterated Function System) code, which describes the image as a fixedpoint of a contractive transformation. A new hierarchical interpretation of the IFS code, which relates different scales of the f ..."
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Cited by 19 (4 self)
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The basics of a block oriented fractal image coder, are described. The output of the coder is an IFS (Iterated Function System) code, which describes the image as a fixedpoint of a contractive transformation. A new hierarchical interpretation of the IFS code, which relates different scales of the fixedpoint to the code, is presented and proved. The proof is based on finding a function of a continuous variable, from which different scales of the signal can be derived. Its application to a fast decoding algorithm is then described, leading typically to an order of magnitude reduction of computation time. I Introduction The use of fractal shapes to describe real world scenes has been shown to result in very realistic images [1]. This is due to the selfsimilarity property of fractal shapes, a property which is frequently encountered in real world scenes [2]. One way of creating a fractal shape is by considering it as a fixedpoint of a contractive Iterated Function System (IFS) [3]. Th...
Generalized Fractal Dimensions: Equivalences and Basic Properties
, 2000
"... Given a positive probability Borel measure , we establish some basic properties of the associated functions (q) and of the generalized fractal dimensions D for q 2 R. We rst give the connections between the generalized fractal dimensions, the Renyi dimensions and the meanq dimensions when q > ..."
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Cited by 14 (7 self)
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Given a positive probability Borel measure , we establish some basic properties of the associated functions (q) and of the generalized fractal dimensions D for q 2 R. We rst give the connections between the generalized fractal dimensions, the Renyi dimensions and the meanq dimensions when q > 0. We then use these relations to prove some regularity properties for (q); we also provide some estimates for these functions (in particular estimates on their behaviour at 1), as well as for the dimensions corresponding to convolution of two measures. We nally present some calculations for speci c examples. 1
Heat kernels on metricmeasure spaces and an application to semilinear elliptic equations
 Trans. Amer. Math. Soc
, 2003
"... Abstract. We consider a metric measure space (M, d,µ) andaheat kernel pt(x, y) on M satisfying certain upper and lower estimates, which depend on two parameters α and β. We show that under additional mild assumptions, these parameters are determined by the intrinsic properties of the space (M, d, µ) ..."
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Cited by 13 (4 self)
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Abstract. We consider a metric measure space (M, d,µ) andaheat kernel pt(x, y) on M satisfying certain upper and lower estimates, which depend on two parameters α and β. We show that under additional mild assumptions, these parameters are determined by the intrinsic properties of the space (M, d, µ). Namely, α is the Hausdorff dimension of this space, whereas β, called the walk dimension, is determined via the properties of the family of Besov spaces W σ,2 on M. Moreover, the parameters α and β are related by the inequalities 2 ≤ β ≤ α +1. We prove also the embedding theorems for the space W β/2,2, and use them to obtain the existence results for weak solutions to semilinear elliptic equations on M of the form −Lu + f(x, u) =g(x), where L is the generator of the semigroup associated with pt. The framework in this paper is applicable for a large class of fractal domains, including the generalized Sierpiński carpet in Rn. 1.
The Multiscale Nature of Network Traffic: Discovery, Analysis, and Modelling
, 2002
"... The complexity and richness of telecommunications traffic is such that one may despair to find any regularity or explanatory principles. Nonetheless, the discovery of scaling behavior in teletraffic has provided hope that parsimonious models can be found. The statistics of scaling behavior present ..."
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Cited by 11 (3 self)
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The complexity and richness of telecommunications traffic is such that one may despair to find any regularity or explanatory principles. Nonetheless, the discovery of scaling behavior in teletraffic has provided hope that parsimonious models can be found. The statistics of scaling behavior present many challenges, especially in nonstationary environments. In this paper, we overview the state of the art in this area, focusing on the capabilities of the wavelet transform as a key tool for unravelling the mysteries of traffic statistics and dynamics.
L^qSpectrum Of The Bernoulli Convolution Associated With The Golden Ratio
, 1995
"... . Based on the higher order selfsimilarity of the Bernoulli convolution measure for ( p 5 \Gamma 1)=2 proposed by Strichartz et al, we derive a formula for the L q spectrum, q ? 0 of the measure. This formula is the first one obtained in the case where the open set condition does not hold. x1. ..."
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Cited by 5 (4 self)
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. Based on the higher order selfsimilarity of the Bernoulli convolution measure for ( p 5 \Gamma 1)=2 proposed by Strichartz et al, we derive a formula for the L q spectrum, q ? 0 of the measure. This formula is the first one obtained in the case where the open set condition does not hold. x1. Introduction. Let ¯ be a positive bounded regular Borel measure on R d with compact support. For h ? 0 and q ? 0; we define the L q (moment) spectrum of ¯ by ø(q) = lim h!0 + ln P i ¯(Q i (h)) q ln h (1.1) where fQ i (h)g i is a family of hmesh cubes. We also define the (lower) L q dimension of ¯ by dim q (¯) = ø(q)=(q \Gamma 1); q ? 1: These notions were first used by Renyi [R'e] to extend the entropy dimension (corresponding to q = 1). For some variants of the definition one can refer to [St], [LN1]. We Key words and phrases: Bernoulli convolution, golden ratio, L q spectrum, L q dimension, multifractal measure, renewal equation, selfsimilarity. . Typeset by A...
L^pSpectrum Of The Bernoulli Convolutions Associated With The P.V. Numbers
, 1998
"... The class of Bernoulli convolutions associated with the P.V. numbers are singular measures. The associated iterated function systems do not satisfy the open set condition and the wellknown formula to calculate the L^qspectrum of singular measures does not apply. In this paper we use a Markov mat ..."
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The class of Bernoulli convolutions associated with the P.V. numbers are singular measures. The associated iterated function systems do not satisfy the open set condition and the wellknown formula to calculate the L^qspectrum of singular measures does not apply. In this paper we use a Markov matrix technique to obtain an algorithm to calculate the exact L^qspectrum of such measures for positive integral q.