Discrete Energy on Rectifiable Sets

Sergiy V. Borodachov is a Professor of Mathematics at Towson University, which he joined in 2008. Prof. Borodachov's primary research interests include approximation theory, numerical analysis, and minimal energy problems. He authored or co-authored more than 30 research articles and gave more than 90 talks at research conferences and seminars.Douglas P. Hardin is a Professor of Mathematics and a Professor of Biomedical Informatics at Vanderbilt University.  His research interests include discrete minimum energy problems, fractals, harmonic analysis (wavelets), inverse problems, and machine learning.  Hardin has authored or co-authored over 115 research publications, 2 monographs, and co-edited 3 research journal special issues.Edward B. Saff is a Professor of Mathematics at Vanderbilt University and Director of the Center for Constructive Approximation. His research areas include approximation theory, numerical analysis, and potential theory. Saff is a Fellow of the American Mathematics Society, a Foreign Member of the Bulgarian Academy of Science, and was a recipient of both a Guggenheim and a Fulbright Fellowships. He has authored or co-authored over 270 research articles, 4 research monographs and 4 textbooks, and is co-Editor-in-Chief and Managing Editor of the research journal Constructive Approximation. Prof. Saff also serves on the boards of 3 other research journals.

“The authors have done an excellent job at completeness in showcasing the range of techniques one can bring to such a seemingly simple subject. This is a book for every practicing mathematician who is interested in the subject. I am happy it has a place on my bookshelf.” (Jeff Ibbotson, MAA Reviews, July 4, 2021)“The book is mostly self-contained and has been written with graduate students in mind. … The reviewer certainly thinks that this book has the potential to become a standard text for students new to the rich and vibrant area of discrete energy problems and point configurations and a valuable resource for researchers.” (Johann S. Brauchart, Mathematical Reviews, April, 2021)“The authors have done an excellent work by taking the reader, who is primarily supposed to be a graduate student, from the basics of Real Analysis to the frontiers of research on several mathematical topics, what turns the text of interest for both students and research professionals. The vast content of the book will certainly provide the reader with an extremely valuable source on this fascinating subject.” (Antonio Roberto da Silva, zbMATH 1437.41002, 2020)

• 0. An Overview: Discretizing Manifolds via Particle Interactions.-1. Preliminaries.- 2. Basics of Minimal Energy.- 3.-Introduction to Packing and Covering.- 4. Continuous and Discrete Energy.- 5. LP Bounds on the Sphere.- 6. Asymptotics for Energy Minimizing Congurations on Sd.- 7. Some Popular Algorithms for Distributing Points on S2.- 8. Minimal Energy in the Hypersingular Case.-  9. Minimal Energy Asymptotics in the "Harmonic Series" Case.-  10. Periodic Riesz Energy.- 11. Congurations with non-Uniform Distribution.-  12. Low Complexity Energy Methods for Discretization.- 13. Best-Packing on Compact Sets.- 14. Optimal Discrete Measures for Potentials: Polarization (Chebyshev) Constants.- Appendix.- References.- List of Symbols.- Index.