Effective dimension

About: Effective dimension is a research topic. Over the lifetime, 2408 publications have been published within this topic receiving 63811 citations.

Fractal Geometry: Mathematical Foundations and Applications

Abstract: Part I Foundations: mathematical background Hausdorff measure and dimension alternative definitions of dimension techniques for calculating dimensions local structure of fractals projections of fractals products of fractals intersections of fractals. Part II Applications and examples: fractals defined by transformations examples from number theory graphs of functions examples from pure mathematics dynamical systems iteration of complex functions-Julia sets random fractals Brownian motion and Brownian surfaces multifractal measures physical applications.

Fractals: Form, Chance and Dimension

Abstract: This is the most extraordinarily beautiful book in thought and in form that I have read for many years, and that is all the more peculiar for its being a somewhat technically mathematical treatise. Fractals is a whole new field of mathematics that models the most interdisciplinary grab-bag of naturally occurring forms, such as coastlines and clouds, crystals, snowflakes and cosmological structures. This new English edition with its triking format and illustrations doe justice to the idio yncratic geniu of the author in a way that the parsimonious French ver ion did not. Mandelbrot, who ha had chairs in economics, engineering, physiology, as well as everal of the choice t plums of the world of mathematics, is a jack-of-all-mathematical trades to IBM. Both the term and the field of fractals are his invention and pet, though he is also reverently anecdotal about the' prior history of the concepts of wiggliness involved in Brownian motion and the work of Edmund Fournier D'Albe, Lewis Fry Richard on and the many cia ical mathematicians who hav contributed to this theory. The idea of fractals extend to both random and non-random sets, and is defined as a set for which the Hausdorff-Be icovitch dimension trictly exceeds the topologica] dimen ion. Both of the e dimensionalities lie between zero and the dimension of the Euclidean space in which one works. The topological dimension i always an integer for Brownian motion, for example, it is unity, whereas the Hausdorff-Besicovitch i of value two, and for other fractals its value is not necessarily integral. Mandelbrot gives an intere ting and simple illu tration of the way in which dimensionalities of thi art depend on an interaction between the object observed and the resolving power of the observer. Consider, he suggests, a ball 10 cm in diameter, wound of a thick thread 1mm in diameter. To an observer at a di tance of 10 m it appears as a zerodimensional point. At 10 cm it i perceived a a three-dimensional ball. At 10 mm it seems to be a one-dimensional mess of thread. At 0.1 mm each thread would be seen as a column which is again a three-dimensional figure. At 0.01 mm each column

The geometry of fractal sets

Abstract: This book contains a rigorous mathematical treatment of the geometrical aspects of sets of both integral and fractional Hausdorff dimension. Questions of local density and the existence of tangents of such sets are studied, as well as the dimensional properties of their projections in various directions. In the case of sets of integral dimension the dramatic differences between regular 'curve-like' sets and irregular 'dust like' sets are exhibited. The theory is related by duality to Kayeka sets (sets of zero area containing lines in every direction). The final chapter includes diverse examples of sets to which the general theory is applicable: discussions of curves of fractional dimension, self-similar sets, strange attractors, and examples from number theory, convexity and so on. There is an emphasis on the basic tools of the subject such as the Vitali covering lemma, net measures and Fourier transform methods.

Scaling and percolation in the small-world network model

Abstract: In this paper we study the small-world network model of Watts and Strogatz, which mimics some aspects of the structure of networks of social interactions. We argue that there is one nontrivial length-scale in the model, analogous to the correlation length in other systems, which is well-defined in the limit of infinite system size and which diverges continuously as the randomness in the network tends to zero, giving a normal critical point in this limit. This length-scale governs the crossover from large- to small-world behavior in the model, as well as the number of vertices in a neighborhood of given radius on the network. We derive the value of the single critical exponent controlling behavior in the critical region and the finite size scaling form for the average vertex-vertex distance on the network, and, using series expansion and Pade approximants, find an approximate analytic form for the scaling function. We calculate the effective dimension of small-world graphs and show that this dimension varies as a function of the length-scale on which it is measured, in a manner reminiscent of multifractals. We also study the problem of site percolation on small-world networks as a simple model of disease propagation, and derive an approximate expression for the percolation probability at which a giant component of connected vertices first forms (in epidemiological terms, the point at which an epidemic occurs). The typical cluster radius satisfies the expected finite size scaling form with a cluster size exponent close to that for a random graph. All our analytic results are confirmed by extensive numerical simulations of the model.

Fractals: Form, Chance, and Dimension.

Abstract: Some people may be laughing when looking at you reading in your spare time. Some may be admired of you. And some may want be like you who have reading hobby. What about your own feel? Have you felt right? Reading is a need and a hobby at once. This condition is the on that will make you feel that you must read. If you know are looking for the book enPDFd fractals form chance and dimension as the choice of reading, you can find here.