Overview
- Features state-of-the-art developments, techniques, and applications of non- Archimedean analysis
- Gathers contributions by leading international experts in the field
- Introduces open problems and areas for future research
Part of the book series: STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health (STEAM)
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Table of contents (9 chapters)
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Front Matter
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Back Matter
Keywords
About this book
In the last forty years the connections between non-Archimedean mathematics and disciplines such as physics, biology, economics and engineering, have received considerable attention. Ultrametric spaces appear naturally in models where hierarchy plays a central role – a phenomenon known as ultrametricity. In the 80s, the idea of using ultrametric spaces to describe the states of complex systems, with a natural hierarchical structure, emerged in the works of Fraunfelder, Parisi, Stein and others. A central paradigm in the physics of certain complex systems – for instance,proteins – asserts that the dynamics of such a system can be modeled as a random walk on the energy landscape of the system. To construct mathematical models, the energy landscape is approximated by an ultrametric space (a finite rooted tree), and then the dynamics of the system is modeled as a random walk on the leaves of a finite tree. In the same decade, Volovich proposed using ultrametric spaces in physical models dealing with very short distances. This conjecture has led to a large body of research in quantum field theory and string theory. In economics, the non-Archimedean utility theory uses probability measures with values in ordered non-Archimedean fields. Ultrametric spaces are also vital in classification and clustering techniques. Currently, researchers are actively investigating the following areas: p-adic dynamical systems, p-adic techniques in cryptography, p-adic reaction-diffusion equations and biological models, p-adic models in geophysics, stochastic processes in ultrametric spaces, applications of ultrametric spaces in data processing, and more.
This contributed volume gathers the latest theoretical developments as well as state-of-the art applications of non-Archimedean analysis. It covers non-Archimedean and non-commutative geometry, renormalization, p-adic quantum field theory and p-adic quantum mechanics, as well as p-adic string theory and p-adic dynamics. Further topics include ultrametric bioinformation, cryptography and bioinformatics in p-adic settings, non-Archimedean spacetime, gravity and cosmology, p-adic methods in spin glasses, and non-Archimedean analysis of mental spaces. By doing so, it highlights new avenues of research in the mathematical sciences, biosciences and computational sciences.
Editors and Affiliations
About the editors
Bourama Toni is graduate of Universite de Montreal, and is presently a full Professor and Chair of the Department of Mathematics at Howard University, Washington DC, USA. He is also a founder and the Editor-in-Chief of the Springer's STEAM-H series. Dr. Toni's research interests are in differential and nonlinear analysis and related topics, including p-adic analysis, game theory and feedback loops analysis, and applications tonaval engineering and biosciences. He has published several books with Springer.
Bibliographic Information
Book Title: Advances in Non-Archimedean Analysis and Applications
Book Subtitle: The p-adic Methodology in STEAM-H
Editors: W. A. Zúñiga-Galindo, Bourama Toni
Series Title: STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health
DOI: https://doi.org/10.1007/978-3-030-81976-7
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
Hardcover ISBN: 978-3-030-81975-0Published: 02 December 2021
eBook ISBN: 978-3-030-81976-7Published: 02 December 2021
Series ISSN: 2520-193X
Series E-ISSN: 2520-1948
Edition Number: 1
Number of Pages: XVI, 318
Number of Illustrations: 23 b/w illustrations, 23 illustrations in colour
Topics: Number Theory, Dynamical Systems and Ergodic Theory, Field Theory and Polynomials, Real Functions, Partial Differential Equations