p-Adic mathematical physics is a branch of modern mathematical physics based on the application of p-adic mathematical methods in modeling physical and related phenomena. It emerged in 1987 as a result of efforts to find a non-Archimedean approach to the spacetime and string dynamics at the Planck scale, but then was extended to many other areas including biology. This paper contains a brief review of main achievements in some selected topics of p-adic mathematical physics and its applications, especially in the last decade. Attention is mainly paid to developments with promising future prospects.

p-Adic mathematical physics : the first 30 years

This paper is devoted to new methods of p-adic and ultrametric analysis and their applications in physics and molecular biology. Although the basic definitions are given in the text, it is recommended that the reader be familiar with the material of the first two chapters in Vladimirov, Volovich and Zelenov’s book p-Adic Analysis andMathematical Physics [1]. Our purpose in this paper is to present some new methods and applications of p-adic and ultrametric analysis developed from 19941 till 2007. The overlap of material between this paper and [1] is minimized. In particular, we do not discuss p-adic quantum mechanics and string theory. This paper covers several topics of p-adic and ultrametric analysis and applications. Most of the results given here are new and have been presented earlier only in journal articles. The first topic is purely mathematical; it is related to the development of the theory of ultrametric wavelets and the study of it relationship to the spectral analysis of ultrametric pseudodifferential operators. The role of ultrametric pseudodifferential operators (such as the Vladimirov operator of p-adic fractional differentiation) in ultrametric analysis is similar to that played by the differentiation operator in real analysis. The use of wavelet bases makes it possible to calculate spectra of ultrametric pseudodifferential operators. In general, in the ultrametric case, wavelet theory is substantially simpler and more powerful and convenient than in the real case. We consider here both the p-adic case and the case of general locally compact ultrametric spaces. An important place in the paper is occupied by a discussion of applications of ultrametric analysis. We discuss applications to the theory of spin glasses, dynamics of macromolecules, and genetics. A common feature of the applications of ultrametric analysis considered here is that they describe collective effects in complex systems. The main destination of physics is describing phenomena based on first principles (physical laws). In many cases, such a description involves difficulties. Thus, it is important to develop various approximations relying on patterns of the effective behavior of systems under consideration in various regimes. A well-known example is the passage from a microscopic description of a many-particle system to a kinetic and, then, a hydrodynamic description [2]. Another example is the stochastic limit of quantum theory, in which the dynamics of a quantum system interacting with the environment is approximated by some quantum random process [3]. In this paper, we consider three examples of application to complex systems, where an important role in the description of a system is played by the ultrametric specifics. In general, complexity, which is often discussed in applications to biological systems (as well as in different contexts), does not lead to farreaching conclusions by itself. The examples considered in what follows show that, as supposed in [1], ultrametric analysis makes it possible to give a simple description of some complex systems, for which methods of real analysis are essentially insufficient. Certainly, we do not claim that ultrametric methods

Methods and Applications of Ultrametric and p-Adic Analysis: From Wavelet Theory to Biophysics

The fault diagnosis of the gearbox of wind turbines is a crucial task for wind turbine operation and maintenance. Although a convolutional neural network can extract the related information of adjacent sampling points using kernels, traditional deep learning methods have not leveraged related information from points with a large span of vibration signal data. In this article, a novel fast deep graph convolutional network is proposed to diagnose faults in the gearbox of wind turbines. First, the original vibration signals of the wind turbine gearbox are decomposed by wavelet packet, which presents time–frequency features as graphs. Then, graph convolutional networks are introduced to extract the features of points with a large span of the defined graph samples. Finally, the fast graph convolutional kernel and the particular pooling improvement are used to reduce the number of nodes and achieve fast classification. Experiments on two data sets are performed to verify the efficacy of the proposed method.

Fault Diagnosis of Wind Turbine Gearbox Using a Novel Method of Fast Deep Graph Convolutional Networks

Starting from physical motivations and leading to practical applications, this book provides an interdisciplinary perspective on the cutting edge of ultrametric pseudodifferential equations. It shows the ways in which these equations link different fields including mathematics, engineering, and geophysics. In particular, the authors provide a detailed explanation of the geophysical applications of p-adic diffusion equations, useful when modeling the flows of liquids through porous rock. p-adic wavelets theory and p-adic pseudodifferential equations are also presented, along with their connections to mathematical physics, representation theory, the physics of disordered systems, probability, number theory, and p-adic dynamical systems. Material that was previously spread across many articles in journals of many different fields is brought together here, including recent work on the van der Put series technique. This book provides an excellent snapshot of the fascinating field of ultrametric pseudodifferential equations, including their emerging applications and currently unsolved problems.

Ultrametric Pseudodifferential Equations and Applications

http://www.math.tau.ac.il/~levin/ACHA2014.pdf

Regularity of generalized Daubechies wavelets reproducing exponential polynomials with real-valued parameters

https://iopscience.iop.org/article/10.1070/RM1998v053n06ABEH000089/pdf

Basic wavelet theory

This paper is devoted to the study of the asymptotics of roots of a sequence of Bernstein polynomials approximating a piecewise linear function. This sequence arises in the construction of modified compactly supported wavelets that, in contrast to classical Daubechies wavelets, preserve localization with the growth of smoothness. It is proved that the limiting curve for roots is the boundary of the domain of convergence of the Bernstein polynomials on the complex plane.

Asymptotics of the Roots of Bernstein Polynomials Used in the Construction of Modified Daubechies Wavelets

In this paper, one-parameter families of integer translates of the Gaussian and Lorentz functions are studied. For a Lorentz function, we obtain formulas for the coefficients of the series defining node functions and show that the limit value of node functions is given by a sample function. For systems of translates generated by the Gaussian and Lorentz functions as well as by the node functions related to them, we obtain explicit expressions for the Riesz constants and study the parameter-dependent behavior of these constants. While proving some of the results of this paper, we establish the monotonicity of a special ratio of two Jacobi theta functions, a fact which is of interest in itself.

On the Riesz Constants for Systems of Integer Translates

On infinitely smooth compactly supported almost-wavelets

In this paper we study single-parametric systems of integer shifts of Gauss and Lorenz functions. In case of Cauchy--Lorenz system we explicitly calculate nod functions and prove that it tends to sinc function in limit. For both Gauss and Cauchy--Lorenz systems and corresponding nod functions we explicitly calculate Riesz constants via trigonometric, hyperbolic and Jacobi theta-functions, also limit behavior of this values is found depending on parameters. 
A special result is a sharp monotonicity property proved for a special ratio of Jacobi theta-functions which is important in many areas. Brief discussion of numerical methods is included with additional references. Different applications are outlined to coherent states and atomic spectra. 
Some unsolved problems and comments are added.

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I. Ya. Novikov

Papers

Basic wavelet theory

Asymptotics of the Roots of Bernstein Polynomials Used in the Construction of Modified Daubechies Wavelets

On the Riesz Constants for Systems of Integer Translates

On infinitely smooth compactly supported almost-wavelets

On Evaluation of Riesz Constants for Systems of Shifted Gaussians