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'Differential Equations on Fractals' opens the door to understanding the recently developed area of analysis on fractals, focusing on the construction of a Laplacian on the Sierpinski gasket and related fractals. Written in a lively and informal style, with numerous intriguing exercises on all levels of difficulty, the book is accessible to advanced undergraduates, graduate students, and mathematicians who seek an understanding of analysis on fractals. Robert Strichartz takes the reader to the frontiers of research, starting with carefully motivated examples and constructions. One of the great accomplishments of geometric analysis in the nineteenth and twentieth centuries was the development of the theory of Laplacians on smooth manifolds. But what happens when the underlying space is rough? Fractals provide models of rough spaces that nevertheless have a strong structure, specifically self-similarity. Exploiting this structure, researchers in probability theory in the 1980s were able to prove the existence of Brownian motion, and therefore of a Laplacian, on certain fractals. An explicit analytic construction was provided in 1989 by Jun Kigami. 'Differential Equations on Fractals' explains Kigami's construction, shows why it is natural and important, and unfolds many of the interesting consequences that have recently been discovered. This book can be used as a self-study guide for students interested in fractal analysis, or as a textbook for a special topics course.

Fractals --- Differential equations --- Fractales --- Equations différentielles --- Differential equations. --- Fractals.

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Fractals. --- Natural disasters. --- Geology --- Research.

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This book is devoted to the structure of the Mandelbrot set - a remarkable and important feature of modern theoretical physics, related to chaos and fractals and simultaneously to analytical functions, Riemann surfaces, phase transitions and string theory. The Mandelbrot set is one of the bridges connecting the world of chaos and order. The authors restrict consideration to discrete dynamics of a single variable. This restriction preserves the most essential properties of the subject, but drastically simplifies computer simulations and the mathematical formalism. The coverage

Mandelbrot sets. --- Molecules, Mu --- Mu molecules --- Sets, Mandelbrot --- Fractals

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The dynamics of complex systems can clarify the creation of structures in Nature. This creation is driven by the collective interaction of constitutive elements of the system. Such interactions are frequently nonlinear and are directly responsible for the lack of prediction in the evolution process. The self-organization accompanying these processes occurs all around us and is constantly being rediscovered, under the guise of a new jargon, in apparently unrelated disciplines. This volume offers unique perspectives on aspects of fractals and complexity and, through the examination of compleme

Fractals. --- Multifractals. --- Fractals --- Fractal geometry --- Fractal sets --- Geometry, Fractal --- Sets, Fractal --- Sets of fractional dimension --- Dimension theory (Topology) --- Multifractals --- lemac

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This book is devoted to an important branch of the dynamical systems theory : the study of the fine (fractal) structure of Poincare recurrences -instants of time when the system almost repeats its initial state. The authors were able to write an entirely self-contained text including many insights and examples, as well as providing complete details of proofs. The only prerequisites are a basic knowledge of analysis and topology. Thus this book can serve as a graduate text or self-study guide for courses in applied mathematics or nonlinear dynamics (in the natural sciences). Moreover, the bo

Fractals. --- Poincaré series. --- Series, Poincaré --- Fractal geometry --- Fractal sets --- Geometry, Fractal --- Sets, Fractal --- Sets of fractional dimension --- Dimension theory (Topology) --- Poincare series.