pADIC MATHEMATICAL PHYSICS: 2nd International Conference

Ergodic Transformations in the Space of p‐Adic Integers
View Description Hide DescriptionLet L _{1} be the set of all mappings f : Z _{ p } → Z _{ p } of the space of all p‐adic integers Z _{ p } into itself that satisfy Lipschitz condition with a constant 1. We prove that the mapping f ∈ L _{1} is ergodic with respect to the normalized Haar measure on Z _{ p } if and only if f induces a single cycle permutation on each residue ring Z/p^{k} Z modulo p^{k} , for all k = 1, 2, 3, …. The multivariate case, as well as measure‐preserving mappings, are considered also.
Results of the paper in a combination with earlier results of the author give explicit description of ergodic mappings from L _{1}. This characterization is complete for p = 2.
As an application we obtain a characterization of polynomials (and certain locally analytic functions) that induce ergodic transformations of p‐adic spheres. The latter result implies a solution of a problem due to A. Khrennikov about the ergodicity of a perturbed monomial mapping on a sphere.

p‐Adic and Adelic Cosmology: p‐Adic Origin of Dark Energy and Dark Matter
View Description Hide DescriptionA brief review of p‐adic and adelic cosmology is presented. In particular, p‐adic and adelic aspects of gravity, classical cosmology, quantum mechanics, quantum cosmology and the wave function of the universe are considered.
p‐Adic worlds made of p‐adic matters, which are different from real world of ordinary matter, are introduced. Real world and p‐adic worlds make the universe as a whole. p‐Adic origin of the dark energy and dark matter are proposed and discussed.

On the Chaotic Properties of Quadratic Maps Over Non‐Archimedean Fields
View Description Hide DescriptionWe study dynamic properties of the quadratic maps over arbitrary non‐archimedean fields. We find conditions under which these maps demonstrate the chaotic behavior. For the quadratic maps defined over a global field the chaos occurs only over a finite number of valuations.

On Ultrametricity and a Symmetry Between Bose‐Einstein and Fermi‐Dirac Systems
View Description Hide DescriptionWe present the results of computer simulations which give the evidence for the existence of an interesting symmetry in a multi‐agent model demonstrating, in special cases, both Bose‐Einstein and Fermi‐Dirac statistics. This symmetry is expressed in the coincidence of the degree of ultrametricity and the fraction of isosceles of the sets of agents memories coded by two different information loss coding schemes.

p‐Adic Strings and Their Applications
View Description Hide DescriptionThe theory of p‐adic strings is reviewed along with some of their applications, foremost among them to the tachyon condensation problem in string theory. Some open problems are discussed, in particular that of the superstring in 10 dimensions as the end‐stage of the 26‐dimensional closed bosonic string’s tachyon condensation.

Proof of the Kurlberg‐Rudnick Rate Conjecture
View Description Hide DescriptionIn this paper we present a proof of the Hecke quantum unique ergodicity conjecture for the Berry‐Hannay model, a model of quantum mechanics on a two dimensional torus. This conjecture was stated in Z. Rudnick’s lectures at MSRI, Berkeley, 1999 and ECM, Barcelona, 2000.

p‐Adic Description of Hierarchical Systems Dynamics
View Description Hide DescriptionWe show that p‐adic analysis provides a quite natural basis for the description of relaxation in hierarchical systems. For our purposes, we specify the Markov stochastic process considered by S. Albeverio and W. Karwowski. As a result we have obtained a random walk on the p‐adic integer numbers, which provides the generalization of Cayley tree proposed by Ogielski and Stein. The temperature‐dependent power‐law decay and the Kohlrausch law are derived.

Capacities and Function Spaces on the Local Field
View Description Hide DescriptionIn this article, we will compare non‐linear capacities with Hausdorff measure and present a trace theorem on Besov space over local field. These studies play important roles in existing fractal analysis on the Euclidean space.

p‐Adic Probability Theory and Its Generalizations
View Description Hide DescriptionThis is a review about p‐adic valued probabilities, applications to quantum physics and cognitive sciences. We also do the next step in development of non‐Kolmogorovian models and consider an analogue of probability theory for probabilities taking values in topological groups. The main attention is paid to statistical interpretation of p‐adic probabilities as well as more general probabilities with values in topological groups.

Ultrametric Analysis and Interbasin Kinetics
View Description Hide DescriptionWe discuss ultrametric pseudodifferential operators and wavelets and applications to models of interbasin kinetics. We show, that, using the language of ultrametric pseudodifferential operators, it is possible to describe interbasin kinetics for general complex landscape.

Critical Exponents in p‐Adic φ^{4}‐Model
View Description Hide DescriptionWe consider φ^{4}‐model with O(N)‐symmetry in d‐dimensional p‐adic space using the approach of renormalized projection Hamiltonians. Critical exponents ν and η are calculated up to three orders of perturbation theory using two types of expansions: (4 − d)‐expansion and (α − 3/2d)‐expansion, where α is a renormalization group parameter. Some resemblances and differences between the Euclidean and p‐adic models are discussed.

On Phase Transitions for p‐Adic Potts Model with Competing Interactions on a Cayley Tree
View Description Hide DescriptionIn the paper we consider three state p‐adic Potts model with competing interactions on a Cayley tree of order two. We reduce a problem of describing of the p‐adic Gibbs measures to the solution of certain recursive equation, and using it we will prove that a phase transition occurs if and only if p = 3 for any value (non zero) of interactions. As well, we completely solve the uniqueness problem for the considered model in a p‐adic context. Namely, if p ≠ 3 then there is only a unique Gibbs measure the model.

From Data to the Physics Using Ultrametrics: New Results in High Dimensional Data Analysis
View Description Hide DescriptionWe begin with pervasive ultrametricity due to high dimensionality and/or spatial sparsity. How extent or degree of ultrametricity can be quantified leads us to the discussion of varied practical cases when ultrametricity can be partially or locally present in data. We show how the ultrametricity can be assessed in text or document collections, in time series signals, and in other areas. We conclude with a discussion of ultrametricity in astrophysics, relating to observational cosmology.

The Arithmetic of Discretized Rotations
View Description Hide DescriptionWe consider the problem of planar rotation by an irrational angle, where the space is discretized to a lattice by means of a round‐off procedure which preserves invertibility. For a dense set of values of the rotational angle, this mapping admits an embedding into a dynamical system which is expanding with respect to a non‐archimedean metric, and which has a complete symbolic dynamics. We consider the arithmetical phenomena that arise in such systems, and their relation to the question of pseudo‐randomness in discrete dynamics. The exposition is organized around the concept of minimal modules, the lattices of minimal complexity which support periodic orbits.

p‐Adic Models of Turbulence
View Description Hide DescriptionIt is shown that p‐adic numbers and ultrametric topology are very useful for describing turbulence. p‐Adic numbers are proved to give us a natural and systematic approach to cascade models. p‐Adic scalar model of turbulence is suggested.

Pseudo‐Differential Operators in the p‐Adic Lizorkin Space
View Description Hide DescriptionThe p‐adic Lizorkin type spaces of test functions and distributions are introduced and a class of pseudo‐differential operators on this spaces are constructed. The p‐adic Lizorkin spaces are invariant under the above‐mentioned pseudo‐differential operators. This class of pseudo‐differential operators contains the Taibleson fractional operators. Solutions of pseudo‐differential equations are also constructed.

Sequence‐Spaces and Applications
View Description Hide DescriptionWe give a survey of the general theory of p‐adic sequence spaces (special attention is paid to Köthe spaces) and show how this theory can be applied to the solution of various problems in p‐adic functional analysis.

Ultrametric Gelfand Transforms
View Description Hide DescriptionLet K be an algebraically closed field, complete for a non‐trivial ultrametric absolute value, and let A be a commutative normed K‐algebra with identity. We call multiplicative spectrum of A the set Mult(A, ‖ ⋅ ‖) of continuous multiplicative semi‐norms on A. We denote by Mult(K[x]) the set of multiplicative semi‐norms on the polynomial algebra K[x]. Both sets of semi‐norms are endowed with the topology of pointwise convergence. We also denote by X(A, K) the set of K‐algebra homomorphism from A onto K. Unlike in complex analysis, there might exist some maximal ideals in A which are not the kernel of elements of X(A, K). In A, we can define two kinds of Gelfand transform. G_{A} and GM_{A} . The first one, denoted by G_{A} is similar to that in complex analysis, consisting of associating to each element f of A the mapping f̂ from X(A, K) to K defined as f̂(χ) = χ(f), (χ ∈ X(A, K)). The second, denoted by GM_{A} consists of associating to each element f of A the mapping f ^{*} from Mult(A, ‖ _{⋅} ‖) to Mult(K[x]) defined as f ^{*}(φ)(P) = φ(P ∘ f). This transform allows us to interpret any element of A as a continuous function defined on a compact space (Mult(A, ‖ ⋅ ‖)) and with value in a locally compact space (Mult(K[x])). We study these transforms and particularly the injectivity of the second. We deduce some spectral properties of this injectivity. Given φ ∈ Mult(A, ‖ _{⋅} ‖), we will denote by Z _{φ} the mapping from A into Mult(K[x]) defined as Z _{φ}(f) = f ^{*}(φ). We show that each function Z _{φ} is continuous. Moreover we put a metric topology δ on (Mult(K[x])) such that the family of functions Z _{φ}, φ ∈ Mult(A, ‖ _{⋅} ‖) is equicontinuous.

p‐Adic Multiple Zeta Values: a Précis
View Description Hide DescriptionThis is an expose of the theory of p‐adic multiple zeta value developed by the author. This theory is almost parallel to the story of the complex case. p‐adic multiple polylogarithm, which is a p‐adic function, is constructed by Coleman’s p‐adic iterated integration theory. p‐adic multiple zeta value is defined to be its special value at 1. Many nice properties of p‐adic multiple zeta value is shown. p‐adic KZ equation is also introduced and the p‐adic Drinfel’d associator is constructed from two fundamental solutions of the p‐adic KZ equation. A relation of p‐adic multiple zeta value with the p‐adic Drinfel’d associator as well as its relation with the fundamental group of the projective line minus three points is established.