P-adic Analysis Research Articles

Geometry, Topology and p-Adic Numbers in Geospatial Data Modelling, Management and Processing: Review and Future Approaches

Abstract. Geometry and topology are both central concepts in geospatial data modelling and management. While geometry is the central concept for computational geometry applications, e.g. to intersect surfaces and solids in 3D space, topology is helpful for many application classes starting from city modelling to subsurface modelling and indoor navigation. Furthermore, p-adic numbers are useful for describing hierarchical processes on topological models. In this paper we first shortly review geometry- and topology-based approaches used for geospatial applications. We then describe our approach on turning geometry “upside down” focusing on topology during the whole process of distributed geospatial computing, data modelling, data management, and simulation. Furthermore, the way to use p-adic numbers for the description of hierarchical processes on topological models is shown for the example of simulation and the idea of p-adic analysis in distributed simulations is presented in the context of topological city and building models. The approach opens new insights such as topological relationships, components, and efficiently studying of the approximate behaviour of processes in the built environment using distributed computational systems. Finally, exemplary applications are presented to underline the importance of this new approach.

Fonctions d'une variable p-adique et représentations de [formula omitted]

We extend the dictionary between Fontaine rings and p-adic functionnal analysis, and we give a refinement of the p-adic local Langlands correspondence for principal series representations of GL2(Qp).

STRATEGIC MANAGEMENT OF THE PORTFOLIO OF FINANCIAL ASSET

Until now, three approaches to asset portfolio management have been used. The first approach is the classic one, based on the "Efficient Market Hypothesis" (EMH). The second and more modern approach is related to the "Fractal Market Hypothesis" (FMH). Modern economic practice is characterized by the presence of structurally unstable markets, included as nodes in the network of the world economy, which functions in real time. The structure of the available financial instruments is heterogeneous and non-Markovian processes arise in them. The third approach is the formation of a dynamic strategy of investment management of the asset portfolio. Due to the complex structure of the modern global financial market, the heterogeneous structure of available financial instruments and traders using different approaches and time horizons, forecasts, as a rule, require a large number of observations, work poorly at the edges of bifurcations, and do not have a computer model that could build forecasts in real time. In these structures, slow diffusion-type processes with a memory phenomenon occur, i.e., non-Markovian processes. Therefore, the formation of a dynamic strategy using modern methods of mathematical and computer modeling is very promising.
 Real financial data is expressed in rational numbers. Computational models were developed on the basis of classical substance analysis. Therefore, p-adic analysis methods are increasingly used in financial mathematical modeling. These methods are used, in particular, in the construction of neural networks, cellular automata, and percolation models. The paper seems to have taken the first step towards building a "synthetic" model of dynamic asset portfolio management. The model has the form of a differential equation in fractional derivatives obtained using the so-called interbasin kinetics method. A general form of the energy market model is also offered, as one of the specific, especially nowadays, markets.

On the differential transcendentality of the Morita p-adic gamma function

In p-adic analysis one can find an analog of the classical gamma function defined on the ring of p-adic integers. In 1975, Morita defined the p-adic gamma function Gamma _p by a suitable modification of the function n mapsto n!. In this note we prove that for any given prime number p the Morita p-adic gamma function Gamma _p is differentially transcendental over {mathbb {C}}_p(X). The main result is an analog of the classical Hölder’s theorem, which states that Euler’s gamma function Gamma does not satisfy any algebraic differential equation whose coefficients are rational functions.

Formula omitted]-Adic mathematics and theoretical biology

The principal objective of this article is a brief overview of the main parts of p-adic mathematics, which have already had valuable applications and may have a significant impact in the near future on the further development of some fields of theoretical and mathematical biology. In particular, we present the basics of ultrametrics, p-adic numbers and p-adic analysis, as well as insight into their applications for modeling some cognitive processes, genetic code and protein dynamics. We also argue that ultrametric concepts and p-adic mathematics are natural tools for the viable description of biological systems and phenomena with a hierarchical structure.

Algebraic genericity and summability within the non-Archimedean setting

In this paper, we establish the analogue of some recent lineability and algebrability results on the sets of sequences and series within the context of p-adic analysis. More specifically, we prove (among several other results) that: (i) in the space of all p-adic sequences, the set of all convergent sequences for which Cesaro’s Theorem fails is lineable, (ii) the set of all non-absolutely convergent p-adic series considered with Cauchy product or pointwise product is algebrable in $$c_0$$ .

ADAPTED BOX-COUNTING METHOD FOR ASSESSMENT OF THE FRACTAL DIMENSION OF FINANCIAL TIME SERIES

This paper continues research within the framework of the scientific direction in econophysics at the Department of Information Systems and Mathematical Methods in Economics of the faculty of Economics of PSU. Modeling and prediction of financial time series is quite a perspective area of research, because it allows participants of financial processes to reduce risks and make effective decisions. For example, we could research financial processes with the help of fractal analysis. In the article there is studied and worked out in detail one of the methods of fractal analysis of financial time series – the box-counting method for assessment of the fractal dimension. This method is often used in studies conducted by domestic authors, but the authors do not delve into the characteristics and problems of using the box-counting method for analysis of time series, that means that the answers to the interested questions have not yet been given. The main problem is that, as a rule, the analyzed object in the tasks of applying the box-counting method to time series is a computer image of the plot of series. In the article there is proposed the procedure of adaptation of the box-counting method for assessment of the fractal dimension of time series, the procedure does not require the formation of a computer image of the plot. In the article there is considered following difficulties developed from this adaptation: 1) high sensitivity of the resulting estimation of the dimension to the input parameters of the method (the ratio of the sides of the covered by cells plane with the plot; the used range of lengths of the side of the cell; the number of partitions of the plane into cells); 2) the non-obviousness of choosing the optimal values ​​of these parameters. In the article there are analyzed approaches to the selection of these parameters that were proposed by other authors, and there are determined the most suitable approaches for the adapted box-counting method. Also there are developed unique methods for determining the ratio of the sides of the plane with the plot. In the paper there is written the computer program that implements the developed method, and this program is tested on the generated data. The study obtained the following results. The fact of sensitivity of the adapted box-counting method to input parameters is confirmed, that indicates the high importance of the correct choice of these parameters. According to the study, there is found out inability of the proposed methods of automatic determination the ratio of the sides of the plane in relation to artificial time series. There are obtained the most precise (in a statistical sense) estimates of fractal dimension, those found by means of the adapted box-counting method, with the fixed ratio of the sides 1:1. According to comparing the adapted box-counting method and R/S analysis, there are obtained the most precise estimates by the second method (R/S analysis). Finally in the paper there are formulated the possible directions for further research: 1) comparison of the accuracy of various methods for assessment of the fractal dimension on series of different lengths; 2) comparison of the methods of fractal analysis and p-adic analysis for modeling and prediction of financial time series; 3) determination of the conditions of applicability of various methods; 4) approbation of the developed methods for determining of the ratio of the sides of the plane with the plot on real economic data.

The Integral Points On A Class Of Elliptic Curve

It is very important to study the integer points of various elliptic curves in elementary number theory. On the one hand, as an important research object in elementary number theory, elliptic curve plays an indispensable role in the development of mathematics. On the other hand, as an important research object in elementary number theory, elliptic curve has practical application in many aspects. So far, there are some conclusions about elliptic curve y 2 = x 3 + a x + b, a, b ∈ Z. In 1987, D. Zagier proposed the integer points problem on y 2 = x 3 - 27x + 62, which is important for studying the arithmetic and geometric properties of elliptic curves. In 2009, Zhu H L and Chen J H used the methods of P-adic analysis and algebraic number theory to solve the problem of the integer points on y 2 = x 3 - 27x + 62. In 2010, Wu H M used the elementary methods to find all the integral points of elliptic curves y 2 = x 3 - 27x - 62. In 2015, Li Y Z and Cui B J used the elementary methods to solve the problem of the integer points on y 2 = x 3 - 21x - 90. In 2016, Guo J used the elementary methods to solve the problem of the integer points on y 2 = x 3 + 27x + 62. In 2017, Guo J used the elementary method proves that y 2 = x 3 - 21x + 90 has no integer points. Up to now, there is no relevant conclusions on the integral points of elliptic curves y 2 = x 3 + 19x - 46, which is the subject of this paper. By using congruence and Legendre Symbol, it can be proved that elliptic curve y 2 = x 3 + 19x - 46 has only one integer point: (x, y) = (2,0).

Reaction-diffusion equations on complex networks and Turing patterns, via p-adic analysis

Nakao and Mikhailov proposed using continuous models (mean-field models) to study reaction-diffusion systems on networks and the corresponding Turing patterns. This work aims to show that p-adic analysis is the natural tool to carry out this program. This is possible due to the fact that the discrete Laplacian attached to a network turns out to be a particular case of a p-adic Laplacian. By embedding the graph attached to the system into the field of p-adic numbers, we construct a continuous p-adic version of a reaction-diffusion system on a network. The existence and uniqueness of the Cauchy problems for these systems are established. We show that Turing criteria for the p-adic continuous reaction-diffusion system remain essentially the same as in the classical case. However, the properties of the emergent patterns are very different. The classical Turing patterns consisting of alternating domains do not occur in the p-adic continuous case. Instead of this, several domains (clusters) occur. Multistability, that is, coexistence of a number of different patterns with the same parameter values occurs. Clustering and multistability have been observed in the computer simulations of large reaction-diffusion systems on networks. Such behavior can be naturally explained in the p-adic continuous model, in a rigorous mathematical way, but not in the original discrete model.

A Common q-Analogue of Two Supercongruences

We give a q-congruence whose specializations q=-1 and q=1 correspond to supercongruences (B.2) and (H.2) on Van Hamme’s list (in: p-Adic Functional Analysis (Nijmegen, 1996), Lecture Notes in Pure and Applied Mathematics, vol 192. Dekker, New York, pp 223–236, 1997): ∑k=0(p-1)/2(-1)k(4k+1)Ak≡p(-1)(p-1)/2(modp3)and∑k=0(p-1)/2Ak≡a(p)(modp2),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned}&\\sum _{k=0}^{(p-1)/2}(-1)^k(4k+1)A_k\\equiv p(-1)^{(p-1)/2}\\;({\\text {mod}}p^3) \\quad \\text {and}\\quad \\\\&\\sum _{k=0}^{(p-1)/2}A_k\\equiv a(p)\\;({\\text {mod}}p^2), \\end{aligned}$$\\end{document}where p>2 is prime, Ak=∏j=0k-1(1/2+j1+j)3=126k2kk3fork=0,1,2,…,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} A_k=\\prod _{j=0}^{k-1}\\biggl (\\frac{1/2+j}{1+j}\\biggr )^3=\\frac{1}{2^{6k}}{\\left( {\\begin{array}{c}2k\\\\ k\\end{array}}\\right) }^3 \\quad \\text {for}\\; k=0,1,2,\\ldots , \\end{aligned}$$\\end{document}and a(p) is the pth coefficient of the modular form qprod _{j=1}^infty (1-q^{4j})^6 (of weight 3). We complement our result with a general common q-congruence for related hypergeometric sums.

Survey on the Kakutani problem in p-adic analysis I

Survey on the Kakutani problem in p-adic analysis I

Some Results on Arithmetic Properties of p-Adic Liouville Numbers

In 1906, Clark defined and studied the set of p-adic Liouville numbers and, in 1985, Schikhof also studied this set in his book Ultrametric Calculus. In this paper, we introduce the set of weak p-adic Liouville numbers, which is a set of transcendental p-adic numbers that contains the p-adic Liouville numbers, and we show some properties about these numbers. In particular, we shall prove an analogous result to a classic theorem of Maillet about Liouville numbers.

Asymptotic Semicircular Law Induced by p-Adic Number Fields $$\mathbb {Q}_{p}$$ Q p Over Primes

In this paper, we study asymptotic semicircular law, and corresponding free-distributional data, induced by p-adic analysis on the p-adic number fields $$\mathbb {Q}_{p}$$ , as $$p \rightarrow \infty $$ in the set $$ \mathcal {P}$$ of all primes.

Some Identities on Degenerate Bernstein and Degenerate Euler Polynomials

In recent years, intensive studies on degenerate versions of various special numbers and polynomials have been done by means of generating functions, combinatorial methods, umbral calculus, p-adic analysis and differential equations. The degenerate Bernstein polynomials and operators were recently introduced as degenerate versions of the classical Bernstein polynomials and operators. Herein, we firstly derive some of their basic properties. Secondly, we explore some properties of the degenerate Euler numbers and polynomials and also their relations with the degenerate Bernstein polynomials.

Non-Archimedean replicator dynamics and Eigen’s paradox

We present a new non-Archimedean model of evolutionary dynamics, in which the genomes are represented by p-adic numbers. In this model the genomes have a variable length, not necessarily bounded, in contrast with the classical models where the length is fixed. The time evolution of the concentration of a given genome is controlled by a p-adic evolution equation. This equation depends on a fitness function f and on mutation measure Q. By choosing a mutation measure of Gibbs type, and by using a p-adic version of the Maynard Smith ansatz, we show the existence of threshold function Mc(f,Q), such that the long term survival of a genome requires that its length grows faster than Mc(f,Q). This implies that Eigen’s paradox does not occur if the complexity of genomes grows at the right pace. About twenty years ago, Scheuring and Poole, Jeffares and Penny proposed a hypothesis to explain Eigen’s paradox. Our mathematical model shows that this biological hypothesis is feasible, but it requires p-adic analysis instead of real analysis. More exactly, the Darwin–Eigen cycle proposed by Poole et al takes place if the length of the genomes exceeds Mc(f,Q).

The p-adic analysis of Stirling numbers via higher order Bernoulli numbers

In this paper, we use our previous study of the higher order Bernoulli numbers [Formula: see text] to investigate [Formula: see text]-adic properties of Stirling numbers of the second kind [Formula: see text]. For example we give a new greatly simplified proof of the formula [Formula: see text] if [Formula: see text], and generalize this result to arbitrary primes [Formula: see text]. We also extend these results to a translation invariance property. Finally, we consider the Stirling numbers of the first kind [Formula: see text], with new results analogous to those for the Stirling numbers of the second kind. New mod [Formula: see text] congruences for Stirling numbers of both kinds are also given.

Towards Three-Dimensional Conformal Probability

In this outline of a program, based on rigorous renormalization group theory, we introduce new definitions which allow one to formulate precise mathematical conjectures related to conformal invariance as studied by physicists in the area known as higher-dimensional conformal bootstrap which has developed at a breathtaking pace over the last few years. We also explore a second theme, intimately tied to conformal invariance for random distributions, which can be construed as a search for very general first and second-quantized Kolmogorov-Chentsov Theorems. First-quantized refers to regularity of sample paths. Second-quantized refers to regularity of generalized functionals or Hida distributions and relates to the operator product expansion.We formulate this program in both the Archimedean and p-adic situations. Indeed, the study of conformal field theory and its connections with probability provides a golden opportunity where p-adic analysis can lead the way towards a better understanding of open problems in the Archimedean setting. Finally, we present a summary of progress made on a p-adic hierarchical model and point out possible connections to number theory. Parts of this article were presented in author’s talk at the 6th International Conference on p-adicMathematical Physics and its Applications,Mexico 2017.

Elliptic Curve Integral Points on y2 = x3 + 3x − 14

The positive integer points and integral points of elliptic curves are very important in the theory of number and arithmetic algebra, it has a wide range of applications in cryptography and other fields. There are some results of positive integer points of elliptic curve y2 = x3 + ax + b, a, b ∈ Z In 1987, D. Zagier submit the question of the integer points on y2 = x3 − 27x + 62, it count a great deal to the study of the arithmetic properties of elliptic curves.In 2009, Zhu H L and Chen J H solved the problem of the integer points on y2 = x3 − 27x + 62 by using algebraic number theory and P-adic analysis method.In 2010, By using the elementary method, Wu H M obtain all the integral points of elliptic curves y2 = x3 − 27x − 62.In 2015, Li Y Z and Cui B J solved the problem of the integer points on y2 = x3 − 21x − 90 By using the elementary method.In 2016, Guo J solved the problem of the integer points on y2 = x3 + 27x + 62 by using the elementary method.In 2017, Guo J proved that y2 = x3 − 21x + 90 has no integer points by using the elementary method.Up to now, there is no relevant conclusions on the integral points of elliptic curves y2 = x3 + 3x − 14, which is the subject of this paper.By using congruence and Legendre Symbol, it can be proved that elliptic curve y2 = x3 + 3x − 14 has only one integer point: (x, y) = (2, 0).

A heat equation on some adic completions of ℚ and ultrametric analysis

For each finite set S of prime numbers there exists a unique completion ℚ S of ℚ, which is a second countable, locally compact and totally disconnected topological ring. This topological ring has a natural ultrametric that allows to define a pseudodifferential operator D α and to study an abstract heat equation on the Hilbert space L 2(ℚ S ). The fundamental solution of this equation is a normal transition function of a Markov process on ℚ S . The techniques developed provides a general framework for these kind of problems on different ultrametric groups.

Constraction of p-Adic Gibbs Measure for p-Adic λ-Ising Model

In this work we establish the existence of the p-adic Gibbs measure for the p-adic λ-Ising model on the Cayley tree of order two. In the previous studies, p-adic Ising model and p-adic λ-model were distinctly studied in many papers with the various properties. In this work, it is the first we combined both p-adic Ising model and p-adic λ-model on the Cayley tree of order two. In this research we only establish the model and we proved the existence of the p-adic Gibbs measures in the p-adic case. Here, we use the methods of p-adic analysis, and then, our results do not work in the real case.