Dimensional Properties of Fractional Brownian Motion

Let B^α = {B^α（t）,t E R^N} be an （N,d）-fractional Brownian motion with Hurst index α∈ （0, 1）. By applying the strong local nondeterminism of B^α, we prove certain forms of uniform Hausdorff dimension results for the images of B^α when N 〉 αd. Our results extend those of Kaufman for one-dimensional Brownian motion.     

Logarithmic corrections for dilute uniaxial ferromagnets at the critical dimension

The leading logarithmic corrections to the critical behavior of a dilute uniaxial (Ising) ferromagnet in the disordered phase are derived using renormalization group methods. The values of the exponents in the logarithmic terms differ from those given by previous authors.     

An infinite dimensional affine nil algebra with finite Gelfand-Kirillov dimension

The famous 1960's construction of Golod and Shafarevich yields infinite dimensional nil, but not nilpotent, algebras. However, these algebras have exponential growth. Here, we construct an infinite dimensional nil, but not locally nilpotent, algebra which has polynomially bounded growth.

     

基于小波变换及布朗分形模型的纹理分割

在研究小波变换及布朗分形模型的基础上，定义了水平分维数，垂直分维数，对角分维数，并将它们作为纹理特征量来实现纹理的分割．实验结果表明，这些特征量能够很好地反映纹理特征，从而分割时不需有关图象中纹理类别的先验信息，较以往方法在准确度和计算复杂程度上都有很大提高．     

Some path properties of generalized Lévy sheet

Let X(t) be an N parameter generalized Lévy sheet taking values in ℝ^d with a lower index α, ℜ = {(s, t] = ∏ _i=1 ^N (s _i, t _i], s _i < t _i}, E(x, Q) = {t ∈ Q: X(t) = x}, Q ∈ ℜ be the level set of X at x and X(Q) = {x: ∃t ∈ Q such that X(t) = x} be the image of X on Q. In this paper, the problems of the existence and increment size of the local times for X(t) are studied. In addition, the Hausdorff dimension of E(x, Q) and the upper bound of a uniform dimension for X(Q) are also established.     

Dimension Theory and Nonstable K 1

This article provides an introduction to A. Bak's theory of group-valued functors on categories with structure and dimension and applies this theory to the algebraic K-theory functors nonstable K ₁ and K ₂.     

Computing the dimension ofN-free ordered sets is NP-complete

We show that the problem of deciding whether anN-free ordered set has dimension at most 3 is NP-complete.Both authors supported by Office of Naval Research contract N00014-85K-0494.     

On classical analogues of free entropy dimension

We define a classical probability analogue of Voiculescu's free entropy dimension that we shall call the classical probability entropy dimension of a probability measure on Rn. We show that the classical probability entropy dimension of a measure is related with diverse other notions of dimension. First, it can be viewed as a kind of fractal dimension. Second, if one extends Bochner's inequalities to a measure by requiring that microstates around this measure asymptotically satisfy the classical Bochner's inequalities, then we show that the classical probability entropy dimension controls the rate of increase of optimal constants in Bochner's inequality for a measure regularized by convolution with the Gaussian law as the regularization is removed. We introduce a free analogue of the Bochner inequality and study the related free entropy dimension quantity. We show that it is greater or equal to the non-microstates free entropy dimension.     

Size distribution of fractured areas in one-dimensional systems

We study a one-dimensional model for fracture, identifying fractured areas with intervals on which a stress field exceeds a threshold value. When is a diffusion process, the cumulative numberN(l) of fractured areas whose length is greater thanl obeys a power lawCl ^–p asl0 with probability one. The exponentp and the constantC are determined. The exponentp agrees with the Hausdorff dimension of the end points of fractured areas, i.e., ^–1(). Even if is self-similar with parameterH>0, i.e.,(cx)– is equivalent toc ^H {(x)–} for anyc>0, the exponentp does not depend solely onH;p=H, where(0, 1/H) is another parameter characterizing. Non-diffusion processes are given whereN(l) does not follow a power law.     

Resistivity exponent of two-dimensional lattice animals

We calculate the average resistanceR(L) of lattice animals spanningL×L cells on the square lattice using exact and Monte Carlo methods. The dynamical resistivity exponent, defined asR(L) L , is found to be =1.36±0.07. This contradicts the Alexander-Orbach conjecture, which predicts 0.8. Our value for differs from earlier measurements of this quantity by other methods yielding =1.17±0.05 and 1.22±0.08 by Havlin et al.On leave from the Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing, China.