Title: Operator Theory And Analysis Of Infinite Networks: Theory and Applications: 7 (Contemporary Mathematics And Its Applications: Monographs, Expositions And Lecture Notes)
Publisher: World Scientific Publishing Company, Incorporated
Publication Date: 2023
Language: English
ISBN 10: 9811265518
ISBN 13: 9789811265518
Binding: Hardcover
Condition: New

About this title

Synopsis

This volume considers resistance networks: large graphs which are connected, undirected, and weighted. Such networks provide a discrete model for physical processes in inhomogeneous media, including heat flow through perforated or porous media. These graphs also arise in data science, e.g., considering geometrizations of datasets, statistical inference, or the propagation of memes through social networks. Indeed, network analysis plays a crucial role in many other areas of data science and engineering. In these models, the weights on the edges may be understood as conductances, or as a measure of similarity. Resistance networks also arise in probability, as they correspond to a broad class of Markov chains. The present volume takes the nonstandard approach of analyzing resistance networks from the point of view of Hilbert space theory, where the inner product is defined in terms of Dirichlet energy. The resulting viewpoint emphasizes orthogonality over convexity and provides new insights into the connections between harmonic functions, operators, and boundary theory. Novel applications to mathematical physics are given, especially in regard to the question of self-adjointness of unbounded operators. New topics are covered in a host of areas accessible to multiple audiences, at both beginning and more advanced levels. This is accomplished by directly linking diverse applied questions to such key areas of mathematics as functional analysis, operator theory, harmonic analysis, optimization, approximation theory, and probability theory.

About the Author

Palle E T Jorgensen is a Professor of Mathematics at the University of Iowa, USA. He has held faculty positions at Stanford University, and at the University of Pennsylvania. He is a member of the AMS and SIAM and an elected member of the Danish National Academy of Science, and of the New York Academy of Science. His recent distinction involved giving 10 lectures at CBMS that are published as Harmonic Analysis: Smooth and Non-smooth by the Conference Board of the Mathematical Sciences.

Erin P J Pearse is an Associate Professor of Mathematics at California Polytechnic State University, San Luis Obispo, USA. His dissertation was on connections between fractal geometry and convex geometry (Steiner's formula); he did his postdoctoral work on infinite networks and studied from the point of view of Hilbert spaces. His more recent work is on graph-based aspects of data science, including nonlinear dimensionality reduction, optimal transport, and machine learning. He also works on climate change and mathematical sustainability.

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