Advances in Non-Archimedean Analysis and Applications

1. Front Matter

   Pages i-xvi

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2. Introduction: Advancing Non-Archimedean Mathematics

   • Bourama Toni, W. A. Zúñiga-Galindo

   Pages 1-7

3. The p-adic Theory of Automata Functions

   • Vladimir Anashin

   Pages 9-113

4. Chaos in p-adic Statistical Lattice Models: Potts Model

   • Farrukh Mukhamedov, Otabek Khakimov

   Pages 115-165

5. QFT, RG, and All That, for Mathematicians

   • Abdelmalek Abdesselam

   Pages 167-183

6. Phase Operator on \(L^2(\mathbb {Q}_p)\) and the Zeroes of Fisher and Riemann

   • Parikshit Dutta, Debashis Ghoshal

   Pages 185-207

7. On Non-Archimedean Valued Fields: A Survey of Algebraic, Topological and Metric Structures, Analysis and Applications

   • Khodr Shamseddine, Angel Barría Comicheo

   Pages 209-254

8. Non-Archimedean Models of Morphogenesis

   • W. A. Zúñiga-Galindo

   Pages 255-274

9. p-Adic Wave Equations on Finite Graphs and T ₀-Spaces

   • Patrick Erik Bradley

   Pages 275-295

10. A Riemann-Roch Theorem on Infinite Graphs

    • Atsushi Atsuji, Hiroshi Kaneko

    Pages 297-312

11. Back Matter

    Pages 313-318

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This book provides a broad, interdisciplinary overview of non-Archimedean analysis and its applications. Featuring new techniques developed by leading experts in the field, it highlights the relevance and depth of this important area of mathematics, in particular its expanding reach into the physical, biological, social, and computational sciences as well as engineering and technology.


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.

• non-archimedean analysis
• p-adic pseudodifferential equations
• ultrametric spaces
• p-adic quantum field theory
• p-adic strings
• p-adic techniques in cryptography
• p-adic dynamical systems

• University of Texas Rio Grande Valley, Brownsville, USA

  W. A. Zúñiga-Galindo

• Department of Mathematics, Howard University, Washington, DC, USA

  Bourama Toni

Wilson Zúñiga-Galindo is a Research Professor in the Department of Mathematics of the CINVESTAV - Center for Research and Advanced Studies of the National Polytechnic Institute, Mexico. A member of the Colombian and Mexican Academy of Sciences, his research focuses on number theory, algebraic geometry, mathematical physics and non-Archimedean analysis. He has published more than fifty research articles and two books, including “Pseudodifferential Equations Over Non-Archimedean Spaces” (Springer, 2016, ISBN 978-3-319-46737-5).

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.

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