Hausdorff dimension assigns a dimension value to each subset of an arbitrary metric space. In Euclidean space, this concept coincides with our intuition that smooth curves have dimension 1 and smooth surfaces have dimension 2, but from its introduction in 1918 [23] Hausdorff noted that many sets have noninteger dimension, what he called “fractional dimension”. The development and applications o...

The level statistics of pseudointegrable fractal drums is studied numerically using periodic orbit theory. We find that the spectral rigidity ∆3(L), which is a measure for the correlations between the eigenvalues, decreases to quite small values (as compared to systems with only small boundary roughness), thereby approaching the behavior of chaotic systems. The periodic orbit results are in goo...

In this work, SiO2 –ZrO2 mixed oxides was prepared by the polymeric sol–gel route. The characterization of pore structure, which determines the permeation process of membrane, is of great importance. So far, most investigations have focused on such pore structure as specific surface area and pore size distribution, but the surface fractal, the important parameter reflecting the roughness of por...

In these two companion papers, we provide an overview and a brief history of the multiple roots, current developments and recent advances of integrative systems biology and identify multiscale integration as its grand challenge. Then we introduce the fundamental principles and the successive steps that have been followed in the construction of the scale relativity theory, which aims at describi...

The zeta-dimension of a set A of positive integers is Dimζ(A) = inf{s | ζA(s) < ∞}, where ζA(s) = ∑ n∈A n. Zeta-dimension serves as a fractal dimension on Z that extends naturally and usefully to discrete lattices such as Z, where d is a positive integer. This paper reviews the origins of zeta-dimension (which date to the eighteenth and nineteenth centuries) and develops its basic theory, with ...

The primary objective of this study was to evaluate a range of calculation points on water retention curves (WRC) instead of the singularity point at air-entry suction in the pore-solid fractal (PSF) model, which additionally considered the hysteresis effect based on the PSF theory. The modified pore-solid fractal (M-PSF) model was tested using 26 soil samples from Yangling on the Loess Plateau...

Modeling and forecasting dynamics of rainfall occurrences constitute one of the major topics, which have been largely treated by statisticians, hydrologists, climatologists and many other groups of scientists. In the same issue, we propose, in the present paper, a new hybrid method, which combines Extreme Values and fractal theories. We illustrate the use of our methodology for transformed Embe...

The fractal structure of the aggregates in food protein gels was analyzed. Three kinds of food protein gels were prepared: (1) β-lactoglobulin (β-LG) gel; (2) 11S soybean globulin gel; and (3) caseinate gel. From the concentration dependence of the gel elasticity, the fractal dimensions Df of the aggregates in the gels were evaluated, according to the theory of Shih et al. These gels showed the...

This thesis deals with fractal compression of still images based on the theory of iterated function systems IFS After an overview of the main lossy and lossless still image compression methods we introduce the IFS theory Then we detail the main fractal compression algorithms proposed in the literature A coder involves compari sons between two sets of blocks A block in the partition of the origi...