Differential Equations on Fractals (eBook)

A Tutorial

Robert S. Strichartz (Autor)

eBook Download: PDF

2018
192 Seiten
Princeton University Press (Verlag)
978-0-691-18683-2 (ISBN)

Infos

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 lots of 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.

Autor

Robert S. Strichartz is Professor of Mathematics at Cornell University. He is the author of The Way of Analysis and A Guide to Distribution Theory and Fourier Transforms.

Produkt-Details

• Erscheint lt. Verlag: 5.6.2018
• Verlagsort: Princeton
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Schlagworte: analytic continuation • analytic function • asymptotic analysis • Boundary (topology) • Boundary value problem • Calculation • Change of variables • coefficient • conformal map • continuous function • Continuous function (set theory) • Convergence of Fourier series • Decimation (signal processing) • Degeneracy (mathematics) • Derivative • Difference quotient • differentiable manifold • Differential Calculus • differential equation • Differential Geometry • Differential operator • Dimension (vector space) • Directional derivative • Dirichlet boundary condition • Dirichlet eigenvalue • Dirichlet kernel • Discrete Mathematics • Distribution (mathematics) • Eigenfunction • Eigenvalues and Eigenvectors • Equation • Euclidean space • existential quantification • Explicit formulae (L-function) • fractal analysis • function space • Green's Function • Harmonic Analysis • Harmonic function • Hausdorff measure • Heat kernel • hilbert space • Intersection (set theory) • Inverse function • Lebesgue constant (interpolation) • Lebesgue integration • limit of a function • linear algebra • linear equation • Linear Function • Lipschitz continuity • Mathematical Induction • Mathematical Optimization • Maxima and minima • nonlinear system • Numerical Integration • Numerology • ordinary differential equation • Orthonormal basis • Orthonormality • partial derivative • Peano existence theorem • Periodic function • Permutation • Piecewise linear function • Pointwise • Poisson kernel • polynomial • power series • probability measure • Probability Theory • Proportionality (mathematics) • pseudo-differential operator • quadratic function • Removable singularity • renormalization • Restriction (mathematics) • Riemannian manifold • Riemann sum • scientific notation • Self-similarity • Series (mathematics) • Simultaneous Equations • Sobolev Space • Special case • Spectral Gap • Spectral Theorem • Spline (mathematics) • Stone–Weierstrass theorem • symmetric function • system of linear equations • Tangent Space • Theorem • Three-dimensional space (mathematics) • Triangle Inequality • Trigonometric Functions • two-dimensional space • uniqueness theorem • Variable (mathematics) • Whitney extension theorem
• ISBN-10: 0-691-18683-9 / 0691186839
• ISBN-13: 978-0-691-18683-2 / 9780691186832