In this paper, we develop a new multiscale modeling framework for characterizing positive-valued data with longrange-dependent correlations (1=f noise). Using the Haar wavelet transform and a special multiplicative structure on the wavelet and scaling coefficients to ensure positive results, the model provides a rapid O(N) cascade algorithm for synthesizing N-point data sets. We study both the second-order and multifractal properties of the model, the latter after a tutorial overview of multifractal analysis. We derive a scheme for matching the model to real data observations and, to demonstrate its effectiveness, apply the model to network traffic synthesis. The flexibility and accuracy of the model and fitting procedure result in a close fit to the real data statistics (variance-time plots and moment scaling) and queuing behavior. Although for illustrative purposes we focus on applications in network traffic modeling, the multifractal wavelet model could be useful in a number of other areas involving positive data, including image processing, finance, and geophysics.

This paper investigates the dynamics in a simple present discounted value asset pricing model with heterogeneous beliefs. Agents choose from a finite set of predictors of future prices of a risky asset and revise their `beliefs' in each period in a boundedly rational way, according to a `fitness measure' such as past realized profits. Price fluctuations are thus driven by an evolutionary dynamics between different expectation schemes (`rational animal spirits'). Using a mixture of local bifurcation theory and numerical methods, we investigate possible bifurcation routes to complicated asset price dynamics. In particular, we present numerical evidence of strange, chaotic attractors when the intensity of choice to switch prediction strategies is high.

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 two most important notions of fractal dimension are Hausdorff dimension, developed by Hausdorff (1919), and packing dimension, developed independently by Tricot (1982) and Sullivan (1984). Both dimensions have the mathematical advantage of being defined from measures, and both have yielded extensive applications in fractal geometry and dynamical systems. Lutz (2000) has recently proven a simple characterization of Hausdorff dimension in terms of gales, which are betting strategies that generalize martingales. Imposing various computability and complexity constraints on these gales produces a spectrum of effective versions of Hausdorff dimension, including constructive, computable, polynomial-space, polynomial-time, and finite-state dimensions. Work by several investigators has already used these effective dimensions to shed significant new light on a variety of topics in theoretical computer science. In this paper we show that packing dimension can also be characterized in terms of gales. Moreover, even though the usual definition of packing dimension is considerably more complex than that of Hausdorff dimension, our gale characterization of packing dimension is an exact dual

Given a set S of n sites (points), and a distance measure d, the nearest neighbor searching problem is to build a data structure so that given a query point q, the site nearest to q can be found quickly. This paper gives a data structure for this problem; the data structure is built using the distance function as a “black box”. The structure is able to speed up nearest neighbor searching in a variety of settings, for example: points in low-dimensional or structured Euclidean space, strings under Hamming and edit distance, and bit vector data from an OCR application. The data structures are observed to need linear space, with a modest constant factor. The preprocessing time needed per site is observed to match the query time. The data structure can be viewed as an application of a “kd-tree ” approach in the metric space setting, using Voronoi regions of a subset in place of axis-aligned boxes. 1

By presenting simple theorems for the absence of positive eigenvalues for certain one-dimensional Schrödinger operators, we are able to construct explicit potentials which yield purely singular continuous spectrum.

In this paper we investigate effective versions of Hausdorff dimension which have been recently introduced by Lutz. We focus on dimension in the class E of sets computable in linear exponential time. We determine the dimension of various classes related to fundamental structural properties including different types of autoreducibility and immunity. By a new general invariance theorem for resource-bounded dimension we show that the class of p-m-complete sets for E has dimension 1 in E. Moreover, we show that there are p-m-lower spans in E of dimension H(β) for any rational β between 0 and 1, where H(β) is the binary entropy function. This leads to a new general completeness notion for E that properly extends Lutz’s concept of weak completeness. Finally we characterize resourcebounded dimension in terms of martingales with restricted betting ratios and in terms of prediction functions. 1.

Let B be a Brownian motion in R d , d = 2; 3. A time t 2 [0; 1] is called a cut time for B[0; 1] if B[0; t)"B(t; 1] = ;: We show that the Hausdorff dimension of the set of cut times equals 1 \Gamma i, where i = i d is the intersection exponent. The theorem, combined with known estimates on i 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 R³.

We show that the Hausdorff dimension equals the logarithmic loss unpredictability for any set of infinite sequences over a finite alphabet. Using computable, feasible, and finite-state predictors, this equivalence also holds for the recently introduced computable, feasible, and finite-state dimensions [Lutz (2000) and Dai, Lathrop, Lutz, and Mayordomo (2001)]. Combining this with recent results of Fortnow and Lutz (2002), we have a tight relationship between prediction with respect to logarithmic loss and absolute loss.

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 \Gamma ff) where ff is an exponent for Brownian motion called the two-sided disconnection exponent. In particular, using an estimate on ff due to Werner, the Hausdorff dimension is greater than 1:015. 1 Introduction Let B(t) be a Brownian motion taking values in IR 2 which we also consider as I C. Let B[0; 1] be the image of [0; 1]. For any compact A ae I C we define the frontier of A, fr(A) to be the set of points in A connected to infinity. More precisely, fr(A) is the set of x 2 A such that x is in the closure of the unbounded connected component of I C n A. Take a typical point x 2 fr(B[0; 1]). Then locally at x the Brownian motion looks like two independent Brownian motions starting at x with the condition that x is...