Advances in Non-Archimedean Analysis and Applications: The p-adic Methodology in STEAM-H (STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health)

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. Foreword Preface Acknowledgments Contents Contributors Introduction: Advancing Non-Archimedean Mathematics References The p-adic Theory of Automata Functions 1 Introduction 2 Preliminaries 2.1 A Few Words About Words 2.2 p-adic Numbers 2.3 Automata: Basic Definitions and Properties 2.4 Automata Maps: p-adic View 3 Explicit Representations of General Automaton Function 3.1 Coordinate Representation 3.2 Representation via Mahler Series 3.3 Representation via van der Put Series 4 Special Classes of Automata Functions 4.1 Finite Automata Functions 4.2 (Locally) Analytic Automata Functions 4.2.1 C-Functions 4.2.2 B-Functions 4.2.3 Class A 5 The p-adic Ergodic Theory of Automata 5.1 Basics of (p-adic) Dynamics 5.1.1 Ergodic Theory 5.1.2 Topological Transitivity 5.2 Finite Dynamics 5.2.1 Heritable Dynamical Properties 5.3 1-Lipschitz Dynamics on Zpn 5.4 The p-adic Ergodic Theory of General Automata Functions 5.4.1 Ergodicity of Affine Mappings 5.4.2 Ergodicity and Measure-Preservation in Terms of Coordinate Functions 5.4.3 Ergodicity and Measure-Preservation in Terms of Mahler Expansion 5.4.4 Ergodicity and Measure-Preservation in Terms of van der Put Expansion 5.5 Measure-Preservation and Ergodicity of Uniformly Differentiable Automata Functions 5.5.1 Conditions for Measure-Preservation 5.5.2 Differentiable Ergodic Transformations on Zp 5.6 Automata Functions Which Are Ergodic on p-adic Balls and p-adic Spheres 5.7 Transitivity of Automata 6 Plots of Automata Functions in Rn 6.1 Automata 0-1 Law 6.1.1 Conditions for Complete Transitivity 6.1.2 Distribution of Orbits of Automata Functions in Rn 6.2 Plots of Finite Automata 7 Other Non-Archimedean Theories of Automata Functions 7.1 Automata Functions over Fp[[X]] 7.2 Automata over Continuous Time 7.2.1 General Considerations 7.2.2 Timed Automata 7.2.3 Finite Transducers over Continuous Time 7.2.4 Approximation of Automata over Continuous Time by Classical Ones 7.2.5 On the p-adic Time 8 Conclusion References Chaos in p-adic Statistical Lattice Models: Potts Model 1 Introduction 2 Preliminaries 2.1 p-adic Numbers 2.6 Dynamical Systems in Qp 2.8 Non-Archimedean Measure 2.13 Semi-Infinite Cayley Tree and Its Coordinate Structure 3 Construction of Generalized p-adic Gibbs Measure 4 Translation-Invariant Measures 5 Description the Set Off All Translation-Invariant p-adic Gibbs Measures 5.3 On Cardinality of the Set of All Translation-Invariant p-adic Gibbs Measures 6 Existence a Strong Phase Transition for q-State Potts Model 6.5 Behavior of the Dynamical System (6.1) 6.11 Boundedness of Generalized p-adic Gibbs Measures and Phase Transitions 7 Chaotic Behaviour of Potts-Bethe Mapping 7.1 p-adic Sub-Shift 7.4 Dynamics of p-adic Potts-Bethe Mapping 8 Conclusions References QFT, RG, and All That, for Mathematicians 1 Introduction 2 Scaling Limits 3 The Fundamental Problem 4 The RG Strategy 5 Hierarchical Models References Phase Operator on L2(Qp) and the Zeroes of Fisher and Riemann 1 Introduction 2 Quantum Spins in External Field 2.1 Phase Operator via Phase Eigenstates 3 The Case of Riemann Zeta Function 3.1 Aggregate Phase Operator for the Riemann Zeta Function 3.2 Total Phase Operator for the Riemann Zeta Function 4 Extension to the Dirichlet L-Functions References On Non-Archimedean Valued Fields: A Survey of Algebraic, Topological and Metric Structures, Analysis and Applications 1 Introduction 2 Preliminaries 2.1 Non-Archimedean Valued Fields 2.2 Ultrametric Spaces 2.3 Spherical Completeness 2.4 Completion of Valued Fields 3 Ordered Fields 3.1 Formally Real Fields 3.2 General Hahn Fields and the Embedding Theorem 3.3 Hahn Fields and Levi-Civita Fields 3.4 Real Closed Field Extensions of R 4 The Levi-Civita Fields R and C 5 Calculus on R and Rn 5.1 Locally Uniformly Differentiable and Weakly Locally Uniformly Differentiable Functions from R to R 5.2 WLUD Functions from Rn to Rm 6 Review of Power Series and Analytic Functions 6.1 Convergence of Sequences in Two Topologies 6.2 Power Series 6.3 Analytic Functions 7 Measure Theory and Integration 7.1 Measurable Sets 7.2 Measurable Functions and Integration on R 7.3 Integration on R2 and R3 7.4 Integrable Delta Functions 8 Optimization 8.1 One-Dimensional Optimization 8.2 Multidimensional Constrained Optimization 9 Computational Applications 10 Non-Archimedean Operator Theory References Non-Archimedean Models of Morphogenesis 1 Introduction 2 p-Adic Analysis: Essential Ideas 2.1 The Field of p-adic Numbers 2.2 Some Function Spaces 2.3 Fourier Transform 2.4 The Vladimirov Operator 2.4.1 The Spectrum of the Operator Dα 2.5 Two Spectral Problems 2.6 The p-adic Heat Equation 3 The Model 4 Turing Instability Criteria 5 Discrete Models of Morphogenesis 5.1 The Spaces DM-L 5.2 Discretization of the operator Dα 5.3 Computation of the Matrix AL,Mα 5.4 Discretization of the p-adic Turing System References p-Adic Wave Equations on Finite Graphs and T0-Spaces 1 Introduction 2 A Dictionary Between Graph Theory and p-Adic Analysis 3 Homogeneous Wave Equations on Undirected Graphs 3.1 Wave Equations Without Damping 3.2 Wave Equations with Damping 4 Application to Aggregation Maps Between Finite T0-Spaces 4.1 Brief Introduction to Finite Topological Spaces 4.2 Processes on Finite T0-Spaces and Their Aggregations References A Riemann-Roch Theorem on Infinite Graphs 1 Introduction 2 Riemann-Roch Theorem on a Weighted Finite Graph 3 Poincaré Inequality for Spectral Gap on an Infinite Graph 3.1 Basic Properties of Weighted Laplacians 3.2 Spectral Gaps and Poincaré Inequality 4 Proof of the Riemann-Roch Theorem on an Infinite Graph References Index