MULTIFRACTAL DECOMPOSITION OF FRACTALS WITH FINITE MEMORY

The authors study recursive structures with ＂finite memory＂ in Euclidean matric space and the multifractal decomposition of the corresponding fractals. For any two positive numbers q,β, and such a recursive structure, a linear operator V^q,β in finite dimensional space is defined. The multifractal spectrum is given by the spectral radius of Vq,^β.     

Multifractal analysis of the occupation measures of a kind of stochastic processes

Let {X(t), 0t1} be a stochastic process whose range is a random Cantor-like set depending on an -sequence (0<<1) and μ is the occupation measure of X(t). In this paper we examine the multifractal structure of μ and obtain the fractal dimensions of the sets of points of where the local dimension of μ is different from . It is interesting to notice that the final results of this paper are identical to those for the occupation measure of a stable subordinator with index , yet the stochastic process under consideration in this work is not even a Markov process.     

Multifractal analysis of a measure of multifractal exact dimension

We focus on the multifractal generalization of the centered Hausdorff measure and dimension. We analyze the correlation among different approaches to the definition of the multifractal exact dimension of locally finite and Borel regular measures on the basis of fractal analysis of essential supports of these measures. Using characteristic multifractal measures, we carry out the multifractal analysis of singular probability measures and prove theorems on the structural representation of these measures.     

Convergence of the Structure Function of a Multifractal Random Walk in a Mixed Asymptotic Setting

Some asymptotic properties of a Brownian motion in multifractal time, also called multifractal random walk, are established. We show the almost sure and L ¹ convergence of its structure function. This is an issue directly connected to the scale invariance and multifractal property of the sample paths. We place ourselves in a mixed asymptotic setting where both the observation length and the sampling frequency may go together to infinity at different rates. The results we obtain are similar to the ones that were given by Ossiander and Waymire [19 Ossiander , M. , and Waymire , E.C. 2000 . Statistical estimation for multiplicative cascades . Ann. Stat. 28 : 1533 – 1560 .[Crossref], [Web of Science ®] , [Google Scholar]] and Bacry et al. [1 Bacry , E. , Gloter , A. , Hoffmann , M. , and Muzy , J.F. Multifractal analysis in a mixed asymptotic framework . Ann. Appl. Prob. (to appear) . [Google Scholar]] in the simpler framework of Mandelbrot cascades.     

Multifractal Decomposition of Statistically Self-Similar Sets

Let K be a statistically self-similar set defined by Graf. In this paper, we construct a random measure ρ which is supported by K and study the multifractal decomposition for K with ρ. Under such a decomposition, we obtain the expression of the spectrum function ƒ(α). Received July 18, 1997, Accepted October 13, 2000     

统计自相似集的重分形分解

设K是由Graf定义的统计自相似集.本文构造了一支撑在K上的随机测 度p,并研究了结合P的K的重分形分解,得到了分形谱函数f(α)的表达式.     

Multifractal analysis of heartbeat time series in human races

We present a multifractal analysis of heartbeat and heart rate time series in human races. In order to improve the training of athletes, we compare heart rate multifractal behavior in free and constant speed 10,000 m runnings. We analyze also marathon races, free pace 42.195 km running, we compare the first and the second half heartbeat signals to measure the effect of fatigue. We find that freedom for choosing the own pace variation could be the racing condition for keeping good health conditions in an exhausting exercise.     

局部熵的高维重分形分析

对不变测度建立了高维形式的重分形分析,即考察与多维参数相关联的重分形分解.利用非紧集或非不变集的高维(q,μ)熵,给出了局部熵的高维重分形谱的一个关系式.     

The Validity of the Multifractal Formalism: Results and Examples

By obtaining a new sufficient condition for a valid multifractal formalism, we improve in this paper a result developed by L. Olsen (1995, Adv. Math.116, 82-196). In particular, we describe a large class of measures satisfying the multifractal formalism and for which the construction of Gibbs measures is not possible. Some of these measures are not unidimensional but have a nontrivial multifractal spectrum, giving a negative answer to a question asked by S. J. Taylor (1995, J. Fourier Anal. Appl., special issue). We also describe a necessary condition of validity for the formalism which is very close to the sufficient one. This necessary condition allows us to describe a measure μ for which the multifractal packing dimension function B_μ(q) is a nontrivial real analytic function but the multifractal formalism is nowhere satisfied. This example gives also a solution to a problem posed by Taylor (cited above).     

The multifractal structure of the product of two stable occupation measures

We study the multifractal structure of the product of two stable occupation measures and obtain the multifractal spectrum.     

多重分形的统计物理方法在证券市场中的应用

本文研究了多重分形的统计物理方法,以上证综合指数长达四年的一分钟高频数据为研究对象,计算了实际交易数据的多重分形谱及其特征参数,并确定了权重因子的取值范围。结果表明奇异指数和相应的谱函数作为多重分形谱的重要参数,一定程度上反映了股指本身的变化范围和高低价位出现频率的变化,然而谱函数可以预测股市趋势的断言在沪市并不成立。     

基于符号序列方法的股价指数预测研究

用两种符号序列方法分别对上证指数进行实证统计分析。一种是以日收盘价计算的指数价差的符号序列,另一种是以每日5分钟高频数据计算的多重分形谱参数符号序列。统计结果表明,指数的涨落不是完全随机的,两种方法都能以一定的条件概率来预测指数的涨落;进一步地,在引入股价指数大涨落的阈值和条件平均增益后,发现大幅涨落时,条件与指数变化的关联性比小的涨落要强得多,而且将两种符号序列方法结合可以更好地预测指数的大涨落。     

Thermodynamic Formalism and Multifractal Analysis of Conformal Infinite Iterated Function Systems

We develop the thermodynamic formalism for equilibrium states of strongly Hölder families of functions. These equilibrium states are supported on the limit set generated by iterating a system of infinitely many contractions. The theory of these systems was laid out in an earlier paper of the last two authors. The first five sections of this paper except Section 3 are devoted to developing the thermodynamic formalism for equilibrium states of Hölder families of functions. The first three sections provide us with the tools needed to carry out the multifractal analysis for the equilibrium states mentioned above assuming that the limit set is generated by conformal contractions. The theory of infinite systems of conformal contractions is laid out in [13]. The multifractal analysis is then given in Section 7. In Section 8 we apply this theory to some examples from continued fraction systems and Apollonian packing.     

Multifractal Formalism for Self-Similar Functions Under the Action of Nonlinear Dynamical Systems

We study functions which are self-similar under the action of some nonlinear dynamical systems. We compute the exact pointwise H{?}lder regularity, then we determine the spectrum of singularities and the Besov ``smoothness' index, and finally we prove the multifractal formalism. The main tool in our computation is the wavelet analysis. October 1, 1996. Date revised: May 13, 1997. Date re-revised: January 10, 1998. Date accepted: February 27, 1998.     

A simple version of Karzanov's blocking flow algorithm

Dinic has shown that the classic maximum flow problem on a graph of n vertices and m edges can be reduced to a sequence of at most n ? 1 so-called ‘blocking flow’ problems on acyclic graphs. For dense graphs, the best time bound known for the blocking flow problems is O(n²). Karzanov devised the first O(n²)-time blocking flow algorithm, which unfortunately is rather complicated. Later Malhotra, Kumar and Maheshwari devise another O(n²)-time algorithm, which is conceptually very simple but has some other drawbacks. In this paper we propose a simplification of Karzanov's algorithm that is easier to implement than Malhotra, Kumar and Maheshwari's method.     

A branch and bound algorithm for the maximum diversity problem

This article begins with a review of previously proposed integer formulations for the maximum diversity problem (MDP). This problem consists of selecting a subset of elements from a larger set in such a way that the sum of the distances between the chosen elements is maximized. We propose a branch and bound algorithm and develop several upper bounds on the objective function values of partial solutions to the MDP. Empirical results with a collection of previously reported instances indicate that the proposed algorithm is able to solve all the medium-sized instances (with 50 elements) as well as some large-sized instances (with 100 elements). We compare our method with the best previous linear integer formulation solved with the well-known software Cplex. The comparison favors the proposed procedure.     

On Recognition of the Finite Simple Orthogonal Groups of Dimension 2m, 2m+1, and 2m+2 over a Field of Characteristic 2

The spectrum (G) of a finite group G is the set of element orders of G. A finite group G is said to be recognizable by spectrum (briefly, recognizable) if HG for every finite group H such that (H)=(G). We give two series, infinite by dimension, of finite simple classical groups recognizable by spectrum.     

Multifractal structure and product of matrices

There is a well established multifractal theory for self-similar measures generated by non-overtapping contractive similutudes. Our report here concerns those with overlaps. In particular we restrict our attentionto the important classes of self-similar measures that have matrix representations. The dimension spectra andthe L-spectra are analyzed through the product of matrices. There are abnormal behaviors on the multifrac-tal structure and they will be discussed in detail.     

A greedy algorithm for multicut and integral multiflow in rooted trees

We present an O(min(Kn,n²)) algorithm to solve the maximum integral multiflow and minimum multicut problems in rooted trees, where K is the number of commodities and n is the number of vertices. These problems are NP-hard in undirected trees but polynomial in directed trees. In the algorithm we propose, we first use a greedy procedure to build the multiflow then we use duality properties to obtain the multicut and prove the optimality.     

Recognition of the Group O+ 10(2) from Its Spectrum

We prove that if the set of orders of elements of a finite group G coincides with the set of orders of elements of the group D=O₁₀ ⁺(2), then G is isomorphic to D. In other words, O ₁₀ ⁺(2) is recognizable from its spectrum.