Discrete Energy on Rectifiable Sets

Description

This book aims to provide an introduction to the broad and dynamic subject of discrete energy problems and point configurations. Written by leading authorities on the topic, this treatise is designed with the graduate student and further explorers in mind. The presentation includes a chapter of preliminaries and an extensive Appendix that augments a course in Real Analysis and makes the text self-contained. Along with numerous attractive full-color images, the exposition conveys the beauty of the subject and its connection to several branches of mathematics, computational methods, and physical/biological applications. This work is destined to be a valuable research resource for such topics as packing and covering problems, generalizations of the famous Thomson Problem, and classical potential theory in Rd. It features three chapters dealing with point distributions on the sphere, including an extensive treatment of Delsarte–Yudin–Levenshtein linear programming methods for lower bounding energy, a thorough treatment of Cohn–Kumar universality, and a comparison of 'popular methods' for uniformly distributing points on the two-dimensional sphere. Some unique features of the work are its treatment of Gauss-type kernels for periodic energy problems, its asymptotic analysis of minimizing point configurations for non-integrable Riesz potentials (the so-called Poppy-seed bagel theorems), its applications to the generation of non-structured grids of prescribed densities, and its closing chapter on optimal discrete measures for Chebyshev (polarization) problems. Front Matter ....Pages i-xviii An Overview: Discretizing Manifolds via Particle Interactions (Sergiy V. Borodachov, Douglas P. Hardin, Edward B. Saff)....Pages 1-11 Preliminaries (Sergiy V. Borodachov, Douglas P. Hardin, Edward B. Saff)....Pages 13-47 Basic Properties and Examples of Minimal Discrete Energy (Sergiy V. Borodachov, Douglas P. Hardin, Edward B. Saff)....Pages 49-75 Introduction to Best-Packing and Best-Covering (Sergiy V. Borodachov, Douglas P. Hardin, Edward B. Saff)....Pages 77-125 Continuous Energy and Its Relation to Discrete Energy (Sergiy V. Borodachov, Douglas P. Hardin, Edward B. Saff)....Pages 127-191 Linear Programming Bounds and Universal Optimality on the Sphere (Sergiy V. Borodachov, Douglas P. Hardin, Edward B. Saff)....Pages 193-260 Asymptotics for Energy Minimizing Configurations on \(S^d\) (Sergiy V. Borodachov, Douglas P. Hardin, Edward B. Saff)....Pages 261-327 Some Popular Algorithms for Distributing Points on \(S^2\) (Sergiy V. Borodachov, Douglas P. Hardin, Edward B. Saff)....Pages 329-353 Minimal Energy in the Hypersingular Case (Sergiy V. Borodachov, Douglas P. Hardin, Edward B. Saff)....Pages 355-410 Minimal Energy Asymptotics in the “Harmonic Series” Case (Sergiy V. Borodachov, Douglas P. Hardin, Edward B. Saff)....Pages 411-440 Periodic Riesz and Gauss-Type Potentials (Sergiy V. Borodachov, Douglas P. Hardin, Edward B. Saff)....Pages 441-478 Configurations with Nonuniform Distribution (Sergiy V. Borodachov, Douglas P. Hardin, Edward B. Saff)....Pages 479-496 Low-Complexity Energy Methods for Discretization (Sergiy V. Borodachov, Douglas P. Hardin, Edward B. Saff)....Pages 497-524 Best-Packing on Compact Sets (Sergiy V. Borodachov, Douglas P. Hardin, Edward B. Saff)....Pages 525-538 Optimal Discrete Measures for Potentials: Polarization (Chebyshev) Constants (Sergiy V. Borodachov, Douglas P. Hardin, Edward B. Saff)....Pages 539-602 Back Matter ....Pages 603-666