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 (1994) is different from the Hausdorff dimension. These results answer questions raised by Hu and Taylor. AMS Classification numbers: Primary 28A78, 28A80.

Let X = {X(t), t ∈ R+} be an operator stable Lévy process in R with exponent B, where B is an invertible linear operator on R. We determine the Hausdorff dimension and the packing dimension of the range X([0, 1]) in terms of the real parts of the eigenvalues of B. Running Title Dimension of Operator Stable Lévy Processes

We continue the investigation of the path-connected geometry on the Cantor space and the related notions of dilution and compressibility described in [1]. These ideas are closely related to the notions of effective Hausdorff and packing dimensions of reals, and we argue that this geometry provides the natural context in which to study them. In particular we show that every regular real can be m...

We have collected definitions and basic results for the (centered ball) density in metric space with respect to an arbitrary Hausdorff function. We have kept the definitions general: we do not assume the Hausdorff functions are continuous or blanketed, and we do not assume the metric space is a subset of Euclidean space. We discuss the covering measure (= centered Hausdorff measure) and packing...

Beef marbling level is the most important indicators in the evaluation of beef quality. The fractal dimension is closely related to marbling level. In this paper, the theory of fractal dimension is used to analyze beef marbling standards images in China and USA. After comparing several different dimension calculation method, the final method is improved box-counting dimension. Linear regression...

Recent empirical investigations suggest a universal scaling law for the spatial structure of social networks. It is found that the probability density distribution of an individual to have a friend at distance d scales as P(d) ∝ d−1. According to this spatial property, a scale invariant friendship (SIF) network model is introduced to deal with the navigation problem with non-uniform population ...

In this work we derive lower bounds for the Hausdorff and fractal dimensions of the global attractor of the Sabra shell model of turbulence in different regimes of parameters. We show that for a particular choice of the forcing term and for sufficiently small viscosity term ν, the Sabra shell model has a global attractor of large Hausdorff and fractal dimensions proportional to logν−1 for all v...

We introduce a fractal dimension for a metric space based on the persistent homology of subsets of that space. We exhibit hypotheses under which this dimension is comparable to the upper box dimension; in particular, the dimensions coincide for subsets of R whose upper box dimension exceeds 1.5.