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Hausdorff Dimension (hausdorff + dimension)

Distribution by Scientific Domains
Mathematics and Statistics100%

Selected Abstracts

Growth of Self-Similar Graphs

JOURNAL OF GRAPH THEORY, Issue 3 2004
B. Krön
Abstract Locally finite self-similar graphs with bounded geometry and without bounded geometry as well as non-locally finite self-similar graphs are characterized by the structure of their cell graphs. Geometric properties concerning the volume growth and distances in cell graphs are discussed. The length scaling factor , and the volume scaling factor , can be defined similarly to the corresponding parameters of continuous self-similar sets. There are different notions of growth dimensions of graphs. For a rather general class of self-similar graphs, it is proved that all these dimensions coincide and that they can be calculated in the same way as the Hausdorff dimension of continuous self-similar fractals: . © 2004 Wiley Periodicals, Inc. J Graph Theory 45: 224,239, 2004 [source]

The long-time behaviour of the thermoconvective flow in a porous medium

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 8 2004
M. A. Efendiev
Abstract For the Boussinesq approximation of the equations of coupled heat and fluid flow in a porous medium we show that the corresponding system of partial differential equations possesses a global attractor. We give lower and upper bounds of the Hausdorff dimension of the attractor depending on a physical parameter of the system, namely the Rayleigh number of the flow. Numerical experiments confirm the theoretical findings and raise new questions on the structure of the solutions of the system. Copyright © 2004 John Wiley & Sons, Ltd. [source]

On the size of the algebraic difference of two random Cantor sets

RANDOM STRUCTURES AND ALGORITHMS, Issue 2 2008
Michel Dekking
Abstract In this paper we consider some families of random Cantor sets on the line and investigate the question whether the condition that the sum of Hausdorff dimension is larger than one implies the existence of interior points in the difference set of two independent copies. We prove that this is the case for the so called Mandelbrot percolation. On the other hand the same is not always true if we apply a slightly more general construction of random Cantor sets. We also present a complete solution for the deterministic case. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008 [source]

The attractor for a nonlinear reaction-diffusion system in an unbounded domain

COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 6 2001
Messoud A. Efendiev
In this paper the quasi-linear second-order parabolic systems of reaction-diffusion type in an unbounded domain are considered. Our aim is to study the long-time behavior of parabolic systems for which the nonlinearity depends explicitly on the gradient of the unknown functions. To this end we give a systematic study of given parabolic systems and their attractors in weighted Sobolev spaces. Dependence of the Hausdorff dimension of attractors on the weight of the Sobolev spaces is considered. © 2001 John Wiley & Sons, Inc. [source]