Showing papers on "Effective dimension published in 1996"
TL;DR: In this article, the Hausdorff and packing measures and dimensions of the limit sets of iterated function systems generated by countable families of conformal contractions are investigated.
Abstract: 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, proved 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 Perron-Frobenius operator, symbolic dynamics on a shift space with an infinite alphabet, 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 Hausdorff dimension are given. A large natural class of systems whose limit sets are 'dimensionless in the restricted sense' is described.
440 citations
TL;DR: In this article, the singular spectrum produced by rank-one perturbations was studied from a spectral point of view, and it was shown that the spectrum is always of dimension zero, albeit sometimes pure point and sometimes singular continuous.
Abstract: Although concrete operators with singular continuous spectrum have proliferated recently [7,11,13,17,34,35,37,39], we still don’t really understand much about singular continuous spectrum. In part, this is because it is normally defined by what it isn’t — neither pure point nor absolutely continuous. An important point of view, going back in part to Rodgers and Taylor [27,28], and studied recently within spectral theory by Last [22] (also see references therein), is the idea of using Hausdorff measures and dimensions to classify measures. Our main goal in this paper is to look at the singular spectrum produced by rank one perturbations (and discussed in [7,11,33]) from this point of view. A Borel measure μ is said to have exact dimension α ∈ [0, 1] if and only if μ(S) = 0 if S has dimension β < α and if μ is supported by a set of dimension α. If 0 < α < 1, such a measure is, of necessity, singular continuous. But, there are also singular continuous measures of exact dimension 0 and 1 which are “particularly close” to point and a.c. measures, respectively. Indeed, as we’ll explain, we know of “explicit” Schrödinger operators with exact dimension 0 and 1, but, while they presumably exist, we don’t know of any with dimension α ∈ (0, 1). While we’re interested in the abstract theory of rank one perturbations, we’re especially interested in those rank one perturbations obtained by taking a random Jacobi matrix and making a Baire generic perturbation of the potential at a single point. It is a disturbing fact that the strict localization (dense point spectrum with ‖xe−itHδ0‖2 = (e−itHδ0, x2e−itHδ0) bounded in t), that holds a.e. for the random case, can be destroyed by arbitrarily small local perturbations [7,11]. We’ll ameliorate this discovery in the present paper in three ways: First, we’ll see that, in this case, the spectrum is always of dimension zero, albeit sometimes pure point and sometimes singular continuous. Second, we’ll show that not
337 citations
TL;DR: In this paper, a quantum wave with probability density, confined by Dirichlet boundary conditions in a D-dimensional box of arbitrary shape and finite surface area, evolves from the uniform state.
Abstract: A quantum wave with probability density , confined by Dirichlet boundary conditions in a D-dimensional box of arbitrary shape and finite surface area, evolves from the uniform state . For almost all positions , the graph of the evolution of P is a fractal curve with dimension . For almost all times t, the graph of the spatial probability density P is a fractal hypersurface with dimension . When D = 1, there are, in addition to these generic time and space fractals, infinitely many special `quantum revival' times when P is piecewise constant, and infinitely many special spacetime slices for which the dimension of P is 5/4. If the surface of the box is a fractal with dimension , simple arguments suggest that the dimension of the time fractal is , and that of the space fractal is .
173 citations
TL;DR: Modified Monte Carlo methods are developed, using smoothing and dimension reduction, so that the convergence rate of nearly O (N^-^1) is regained and the effective dimension of the integration domain is drastically reduced.
173 citations
166 citations
Posted Content•
TL;DR: In this article, a linear probability model of binary choices over alternatives characterized by unobserved attributes is proposed to estimate preferences of congressmen as expressed in their votes on bills, and the model is applied to estimate the preferences of Congressmen.
Abstract: This paper formulates and estimates a rigorously-justified linear probability model of binary choices over alternatives characterized by unobserved attributes. The model is applied to estimate preferences of congressmen as expressed in their votes on bills. The effective dimension of the attribute space characterizing votes is larger than what has been estimated in recent influential studies of congressional voting by Poole and Rosenthal. Congressmen vote on more than ideology. Issue-specific attributes are an important determinant of congressional" voting patterns. The estimated dimension is too large for the median voter model to describe congressional voting
152 citations
TL;DR: The Hausdorff and Minkowski dimensions of self-affine Sierpinski sponges were determined in this paper, and it was shown that every compact set invariant under an expanding toral endomorphism which is a direct sum of conformal endomorphisms supports an invariant measure of full dimension.
Abstract: We determine the Hausdorff and Minkowski dimensions of some self-affine Sierpinski sponges, extending results of McMullen and Bedford. This result is used to show that every compact set invariant under an expanding toral endomorphism which is a direct sum of conformal endomorphisms supports an invariant measure of full dimension.
138 citations
TL;DR: In this article, the authors presented an algorithm which, given a finite set A of between 2 and N positive integers 2N, determines dim EA to within ±2−NusingO(N7) elementary bit operations.
92 citations
83 citations
TL;DR: In this paper, the Hausdorff and Minkowski dimensions of compact subsets of the 2-torus were determined under a linear endomorphism with integer eigenvalues and correspond to shifts of finite type or sofic shifts via some Markov partition.
Abstract: We determine the Hausdorff and Minkowski dimensions of compact subsets of the 2-torus which are invariant under a linear endomorphism with integer eigenvalues and correspond to shifts of finite type or sofic shifts via some Markov partition. This extends a result of McMullen (1984) and Bedford (1984), who considered full-shifts.
62 citations
TL;DR: In this article, it was shown that the Hausdorff dimension of the set of cut times of a subpath of a Brownian path is strictly greater than one in $R^3.
Abstract: Let $B$ be a Brownian motion in $R^d$, $d=2,3$. A time $t\in [0,1]$ is called a cut time for $B[0,1]$ if $B[0,t) \cap B(t,1] = \emptyset$. We show that the Hausdorff dimension of the set of cut times equals $1 - \zeta$, where $\zeta = \zeta_d$ is the intersection exponent. The theorem, combined with known estimates on $\zeta_3$, shows that the percolation dimension of Brownian motion (the minimal Hausdorff dimension of a subpath of a Brownian path) is strictly greater than one in $R^3$.
TL;DR: In this paper, it was shown that the Hausdorff dimension of the frontier of a two-dimensional Brownian motion is at most 2(1 - \alpha) where α is an exponent for Brownians called the two-sided disconnection exponent.
Abstract: Let $B$ be a two dimensional Brownian motion and let the frontier of $B[0,1]$ be defined as the set of all points in $B[0,1]$ that are in the closure of the unbounded connected component of its complement. We prove that the Hausdorff dimension of the frontier equals $2(1 - \alpha)$ where $\alpha$ is an exponent for Brownian motion called the two-sided disconnection exponent. In particular, using an estimate on $\alpha$ due to Werner, the Hausdorff dimension is greater than $1.015$.
TL;DR: For a map in which locally contracts k-dimensional volumes, the box dimension of any compact invariant set is less than k as mentioned in this paper, which was proved independently by Douady and Oesterl? and by Il'yashenko for Hausdorff dimension.
Abstract: We prove a conjecture of Il'yashenko, that for a map in which locally contracts k-dimensional volumes, the box dimension of any compact invariant set is less than k. This result was proved independently by Douady and Oesterl? and by Il'yashenko for Hausdorff dimension. An upper bound on the box dimension of an attractor is valuable because, unlike a bound on the Hausdorff dimension, it implies an upper bound on the dimension needed to embed the attractor. We also get the same bound for the fractional part of the box dimension as is obtained by Douady and Oesterl? for Hausdorff dimension. This upper bound can be characterized in terms of a local version of the Lyapunov dimension defined by Kaplan and Yorke.
01 May 1996
TL;DR: In this paper, it was shown that for arbitrary metric spaces X and Y, the following dimension inequalities hold: where X denotes Hausdorff dimension and Y denotes packing dimension.
Abstract: We show that for arbitrary metric spaces X and Y the following dimension inequalities hold:where ‘dim’ denotes Hausdorff dimension and ‘Dim’ denotes packing dimension. The main idea of the proof is to use modified constructions of the Hausdorff and packing measure to deduce appropriate inequalities for the measure of X × Y.
TL;DR: In this paper, the Hausdorff dimension of random sets of numbers with given averages of occurrences of digits in these expansions, as well as of general closed sets invariant with respect to these random transformations, generalizing the corresponding deterministic result of Furstenberg.
Abstract: I start with random base expansions of numbers from the interval [0, 1] and, more generally, vectors from [0, 1]d, which leads to random expanding transformations on the d-dimensional torus Td. As in the classical deterministic case of Besicovitch and Eggleston I find the Hausdorff dimension of random sets of numbers with given averages of occurrences of digits in these expansions, as well as of general closed sets “invariant” with respect to these random transformations, generalizing the corresponding deterministic result of Furstenberg. In place of the usual entropy which emerges (as explained in Billingsley’s book) in the Besicovitch-Eggleston and Furstenberg cases, the relativised entropy of random expanding transformations comes into play in my setup. I also extend to the case of random transformations the Bowen-Ruelle formula for the Hausdorff dimension of repellers.
01 Apr 1996
TL;DR: In this article, it was shown that the Hausdorff dimension of the Julia set of such a function is strictly greater than one, where the singularities of f−1 lie in a bounded set.
Abstract: Let f be a transcendental entire function such that the finite singularities of f−1 lie in a bounded set. We show that the Hausdorff dimension of the Julia set of such a function is strictly greater than one.
TL;DR: In this article, the authors obtained a critical function for which the Hausdorff measure of a branching set generated by a simple Galton-Watson process is positive and finite.
Abstract: We obtain a critical function for which the Hausdorff measure of a branching set generated by a simple Galton-Watson process is positive and finite.
Proceedings Article•
03 Sep 1996TL;DR: Using the the wellknown Hausdorff fractal dimension, some closed formulas are derived which allow us to predict the number of quadtree blocks, given some few parameters, in a real, spatial dataset.
Abstract: There is mounting evidence [Man77, SchSI] that real datasets are statistically self-similar, and thus, ‘fractal’. This is an important insight since it permits a compact statistical description of spatial datasets; subsequently, as we show, it also forms the basis for the theoretical analysis of spatial access methods, without using the typical, but unrealistic, uniformity assumption. In this paper, we focus on the estimation of the number of quadtree blocks that a real, spatial dataset will require. Using the the wellknown Hausdorff fractal dimension, we derive some closed formulas which allow us to predict the number of quadtree blocks, given some few parameters. Using our formulas, it is possible to predict the space overhead and the response time of linear quadtrees/z-ordering [OM88], which are widely used in practice. In order to verify our analytical model, we performed *This work was partially supported by the National Science Foundation under Grants No. CDR8803012, EEC-94-02384, IRI-8958546 and IRI-9205273), with matching funds from Empress Software Inc. and Thinking Machines Inc. Some of the work was performed while he was visiting AT&T Bell Labor+ tories, Murray Hill, NJ. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the VLDB copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Very Large Data Base Endowment. To copy otherwise, or to republish, requires a fee and/or special permission from the Endowment. Proceedings of the 22nd VLDB Conference Mumbai(Bombay), India, 1996 Volker Gaede Institut fiir Wirtschaftsinformatik Humboldt-Universitgt zu Berlin Spandauer Str. 1 10178 Berlin, Germany gaede@wiwi.hu-berlin.de an extensive experimental investigation using several real datasets coming from different domains. In these experiments, we found that our analytical model agrees well with our experiments as well as with older empirical observations on 2-d [Gae95b] and 3-d [ACF+94] data.
01 Jan 1996
TL;DR: For some compact sets K ⊂ R 2 of the Cantor type, the harmonic measure is supported by a set whose Hausdorff dimension is strictly smaller than the dimension of K as discussed by the authors.
Abstract: We show that for some compact sets K ⊂ R 2 of Cantor type the harmonic measure is supported by a set whose Hausdorff dimension is strictly smaller than the dimension of K.
TL;DR: In this article, the Hausdorff dimension of the complement of the frequency-locking set is less than 1 but not less than 3, and the rotation number is a Holder continuous function of the parameter.
Abstract: We estimate harmonic scalings in the parameter space of a one-parameter family of critical circle maps. These estimates lead to the conclusion that the Hausdorff dimension of the complement of the frequency-locking set is less than 1 but not less than 1/3. Moreover, the rotation number is a Holder continuous function of the parameter.
Journal Article•
TL;DR: In this paper, the authors calculate the box-counting dimension of the limit set of a general geometrically finite Kleinian group using the "global measure formula" for the Patterson measure and using an estimate on the horoball counting function.
Abstract: We calculate the box-counting dimension of the limit set of a general geometrically finite Kleinian group. Using the “global measure formula” for the Patterson measure and using an estimate on the horoball counting function we show that the Hausdorff dimension of the limit set is equal to both: the box-counting dimension and packing dimension of the limit set. Thus, by a result of Sullivan, we conclude that for a geometrically finite group these three different types of dimension coincide with the exponent of convergence of the group.
TL;DR: In this article, the authors generalize Ruelle's result for complex dimension two and show the real analytic dependence of the Hausdorff dimension of the corresponding Julia sets of hyperbolic Henon mappings.
Abstract: The Hausdorff dimension of closed invariant sets under diffeomorphisms is an interesting concept as it is a measure of their complexity. The theory of holomorphic dynamical systems provides us with many examples of fractal sets and, in particular, a theorem of Ruelle [Ru1] shows that the Hausdorff dimension of the Julia set depends real analytically on f if f is a rational function of ℂ and the Julia set J of f is hyperbolic. In this paper we generalize Ruelle's result for complex dimension two and show the real analytic dependence of the Hausdorff dimension of the corresponding Julia sets of hyperbolic Henon mappings.
TL;DR: In this paper, the Anderson transition in an effective dimension $d$ ($3 \leq d \LEq 11$) for one particle propagation in a model random and quasiperiodic potential was numerically investigated.
Abstract: We numerically investigate the Anderson transition in an effective dimension $d$ ($3 \leq d \leq 11$) for one particle propagation in a model random and quasiperiodic potential. The found critical exponents are different from the standard scaling picture. We discuss possible reasons for this difference.
TL;DR: In this article, a new interpretation of the fractal dimension of a set is proposed, which concentrates on the visible complexity of the set, and uses counted boxes to define generators for adjacent self-similar sets.
Abstract: The fractal dimension of a set in the Euclidean n-space may depend on the applied concept of fractal dimension. Several concepts are considered here, and in a first part, properties are given for sets such that they have the same fractal dimension for all concepts. In particular, self-similar sets hold these properties. The second part deals with the measurement of fractal dimension. An often-used method to empirically compute the fractal dimension of a set E is the box-counting method where the slope of a regression line gives the estimate of the fractal dimension. A new interpretation, which concentrates on the visible complexity of the set, uses counted boxes to define generators for adjacent self-similar sets. Their maximal fractal dimension is assigned to the set as a measurement of fractal dimension or of visible complexity. The results from the first part guarantee that all measurements are independent of the considered concepts. The construction suggests a new method, which is called extended box-counting method, to estimate fractal dimension or to measure complexity of an image in a given range of magnification. The method works without linear regression and has the advantage to nearly preserve the union stability (maximum property).
TL;DR: A reaction–diffusion system capable of modeling ligand migration inside of proteins as well as conformational fluctuations of proteins is introduced, and a detailed analytical and numerical analysis of this system in general dimension is presented.
Abstract: We introduce a reaction–diffusion system capable of modeling ligand migration inside of proteins as well as conformational fluctuations of proteins, and present a detailed analytical and numerical analysis of this system in general dimension. The main observable, the probability of finding the system in the starting state, exhibits dimension‐dependent as well as dimension‐independent properties, allowing for sharp experimental tests of the effective dimension of the process in question. We discuss the application of this theory to ligand migration in myoglobin and to the description of gating fluctuations of ion channel proteins.
PARC1
TL;DR: This paper investigates how the Hausdorff distance changes as one set is transformed by some transformation group, and considers the graph of the directed Hausdorf distance under translation, rigid motion, translation and scaling, and affine transformation.
Abstract: The Hausdorff distance is a measure defined between two sets in some metric space. This paper investigates how the Hausdorff distance changes as one set is transformed by some transformation group. Algorithms to find the minimum distance as one set is transformed have been described, but few lower bounds are known. We consider the complexity of the graph of the Hausdorff distance as a function of transformation, and exhibit some constructions that give lower bounds for this complexity. We exhibit lower-bound constructions for both sets of points in the plane, and sets of points and line segments; we consider the graph of the directed Hausdorff distance under translation, rigid motion, translation and scaling, and affine transformation. Many of the results can also be extended to the undirected Hausdorff distance. These lower bounds are for the complexity of the graph of the Hausdorff distance, and thus do not necessarily bound algorithms that search this graph; however, they do give an indication of how complex the search may be.
TL;DR: In this paper, a real-space renormalization group transformation for dynamical triangulations is proposed, which preserves geometrical exponents such as string susceptibility and Hausdorff dimension.
TL;DR: In this article, it was shown that for any analytic set A in Rd, its packing dimension can be represented as supB{dimH(A×B) − dimH(B)}, where the supremum is over all compact sets B in Rd and dimH denotes Hausdorff dimension.
Abstract: We show that for any analytic set A in Rd, its packing dimension dimP (A) can be represented as supB{dimH(A×B) − dimH(B)} , where the supremum is over all compact sets B in Rd, and dimH denotes Hausdorff dimension. (The lower bound on packing dimension was proved by Tricot in 1982.) Moreover, the supremum above is attained, at least if dimP (A) < d. In contrast, we show that the dual quantity infB{dimP (A×B)− dimP(B)} , is at least the “lower packing dimension” of A, but can be strictly greater. (The lower packing dimension is greater than or equal to the Hausdorff dimension.)
01 Oct 1996
TL;DR: This article showed that the dimension adim introduced by R. Kaufman (1987) coincides with the packing dimension Dim, but the dimension aDim introduced by Hu and Taylor [7] is different from the Hausdorff dimension.
Abstract: We show that the dimension adim introduced by R. Kaufman (1987) coincides with the packing dimension Dim, but the dimension aDim introduced by Hu and Taylor [7] is different from the Hausdorff dimension. These results answer questions raised by Hu and Taylor.
TL;DR: In this paper, a general probabilistic approach was developed to get sharp estimates for the almost sure short-term behavior of hierarchical structured branching-diffusion processes, and the modulus-of-continuity-type results for hierarchical branching diffusion processes were used to derive upper estimates for Hausdorff dimension of support.