Analysis, Probability And Mathematical Physics On Fractals

In the 50 years since Mandelbrot identified the fractality of coastlines, mathematicians and physicists have developed a rich and beautiful theory describing the interplay between analytic, geometric and probabilistic aspects of the mathematics of fractals. Using classical and abstract analytic tools developed by Cantor, Hausdorff, and Sierpinski, they have sought to address fundamental questions: How can we measure the size of a fractal set? How do waves and heat travel on irregular structures? How are analysis, geometry and stochastic processes related in the absence of Euclidean smooth structure? What new physical phenomena arise in the fractal-like settings that are ubiquitous in nature?

This book introduces background and recent progress on these problems, from both established leaders in the field and early career researchers. The book gives a broad introduction to several foundational techniques in fractal mathematics, while also introducing some specific new and significant results of interest to experts, such as that waves have infinite propagation speed on fractals. It contains sufficient introductory material that it can be read by new researchers or researchers from other areas who want to learn about fractal methods and results.

Contents:

• A Capacity Approach to Box and Packing Dimensions of Projections and Other Images (K J Falconer)
• Minkowski Measurability Criteria for Compact Sets and Relative Fractal Drums in Euclidean Spaces (Michel L Lapidus, Goran Radunović and Darko Žubrinić)
• Orthonormal Coalescence Hidden-Variable Fractal Interpolation Functions (G P Kapoor and Srijanani Anurag Prasad)
• Fractal Transformed Doubly Reflected Brownian Motions (Tim Ehnes and Uta Freiberg)
• Harmonic Analysis of Fractal Measures: Basis and Frame Algorithms for Fractal L²-Spaces, and Boundary Representations as Closed Subspaces of the Hardy Space (John E Herr, Palle E T Jorgensen and Eric S Weber)
• Spectral Pairs and Positive-Definite-Tempered Distributions (Palle E T Jorgensen and Feng Tian)
• Asymptotic Behavior of Jacobi Matrices of IFS: A Long-Standing Conjecture (Giorgio Mantica)
• Random Walks and Induced Dirichlet Forms on Compact Spaces of Homogeneous Type (Shi-Lei Kong, Ka-Sing Lau and Ting-Kam Leonard Wong)
• Analysis on the Projective Octagasket (Yiran Mao, Robert S Strichartz, Levente Szabo and Wing Hong Wong)
• Measures Essentially of Finite Type and Spectral Asymptotics of Fractal Laplacians (Sze-Man Ngai)
• Eigenvalue Approximation for Krein–Feller-Operators (Uta Freiberg and Lenon Minorics)
• Spectral Asymptotics on the Hanoi Attractor (Elias Hauser)
• Sobolev Spaces and Calculus of Variations on Fractals (Michael Hinz, Dorina Koch and Melissa Meinert)
• Approximation of Magnetic Laplacians on the Sierpinski Gasket by Discrete Magnetic Laplacians (Olaf Post and Jan Simmer)
• Magnetostatic Problems in Fractal Domains (Simone Creo, Maria Rosaria Lancia, Paola Vernole, Michael Hinz and Alexander Teplyaev)
• Infinite Propagation Speed for Wave Solutions on Some Post-critically Finite Fractals (Yin Tat Lee)
• The Damped Stochastic Wave Equation on Post-critically Finite Fractals (Ben Hambly and Weiye Yang)
• Fractal AC Circuits and Propagating Waves on Fractals (Eric Akkermans, Joe P Chen, Gerald Dunne, Luke G Rogers and Alexander Teplyaev)

Readership: Academic mathematicians and physicists, both pure and applied, working on fractals or rough spaces or interested in fractal media; Scientists from the many areas in which fractal models are important, e.g., biology, chemistry, geology, hydrology, seismology.Analysis on Fractals;Fractal Measures;Post Critically Finite Fractal;P C F Fractal;Fractal Surfaces;Fractal Transformations;Complex Dimension;Fractal Zeta Function;Fractal String;Fractal Drum;Gauge Function;Sub-Gaussian Heat Kernel Bounds;Infinite Wave Propagation Speed;Dimension Profile;Projective Octagasket;Iterated Function System;AC Circuit;Complex Impedances;Self-Similar Measure;Spectral Measures;Harmonic Functions;Box Dimension;Packing Dimension;Dirichlet Form;Discrete Weighted Graphs;Discrete Laplacian;Eigenvalue Approximation;Eigenvalue Counting Function;Spectral Asymptotics;Schrodinger Operator;Magnetic Laplacian;Krein-Feller Operator;Martin Boundary;Na?m Kernel;Compact Space of Homogeneous Type;Heat Equation;Wave Equation;Spectral Pair;Hyperbolic Graph;Weighted Sobolev Spaces;Trace Theorems;Double Reflected Brownian Motion;P-Energies;Unitary One-Parameter Group;Harmonic Decomposition;Almost Periodic Jacobi Matrix;Isospectral Torus;Potential Theory;Capacity;Sobolev Spaces;Finite Element Method;Numerical Approximation;Fourier Series;Invariant Measure;Stochastic Partial Differential Equation;Spectral Projection;Reversible Random Walk;Stationary-Increment Stochastic Processes;Orthogonal Polynomials;Riesz Basis;Wavelets;Parseval Frame;Coalescence;Asymptotic Behavior;Multiresolution Analysis;Norm Resolvent Convergence;Interpolation;Reproducing Kernel;Tempered Distributions;Hilbert Space;Hardy Space0Key Features:

• The book presents the current frontiers in research on analysis, probability and mathematical physics on fractals
• The mathematics of fractals is a rapidly developing research area with a large and active community of scientists, many of whom contributed to this volume
• The authors include many leading researchers, and the book includes introductory material combined with more advanced topics that should make it accessible to newcomers to the field as well as interesting to experts