Dirichlet series in function fields

We study certain functions from the p-adic integers to a locally compact field of characteristic p: finite linear combinations of exponentials and their uniform limits (which we call Dirichlet series). Our main result is that such a Dirichlet series is determined by its restriction to an arbitrarily small open subset of the p-adic integers.     

Recurrence and Transience Criteria for Symmetric Hunt Processes

In this paper, we will discuss recurrence, transience and other potential theoretic aspects based on symmetric regular Dirichlet space. We will first deal with Dirichlet space with the strong local property and give a recurrence criterion in terms of exhaustion function. This criterion shows that recurrence automatically provides us with an exhaustion function which is usable to verify a Liouville property on subharmonic functions. Secondly, a recurrence criterion and a transience criterion for a Nonlocal Dirichlet space will be presented. Those criteria can be applied to Albeverio–Karwowski"s random walks on p-adic number field. Lastly, we will prove the assertions which cover other potential theoretic aspect of p-adic number field such as Liouville property on harmonic functions.     

Multidimensional <Emphasis Type="Italic">p</Emphasis>-adic wavelets for the deformed metric

The approach to p-adic wavelet theory from the point of view of representation theory is discussed. p-Adic wavelet frames can be constructed as orbits of some p-adic groups of transformations. These groups are automorphisms of the tree of balls in the p-adic space. In the present paper we consider deformations of the standard p-adic metric in many dimensions and construct some corresponding groups of transformations. We build several examples of p-adic wavelet bases. We show that the constructed wavelets are eigenvectors of some pseudodifferential operators.     

Sobolev Space and Dirichlet Space Associated with Symmetric Markov Process on a Local Field

The space ℱ_r,p, which was designed so as to play similar roles to the ordinary Sobolev space W^r,p(ℝⁿ), introduced as a cornerstone for analyzing nonlinear potential theoretic features of the state space with a measure-symmetric transition probability semi-group. The aim of this article is revealing a sufficient condition for the counterpart of the Sobolev space to coincide with domain of some Dirichlet form on a local field and discussing some other features of those counterparts on the non-Archimedean metric space. For example, we will see a sufficient condition for the space ℱ_r,2 to be viewed as domain of some Dirichlet form and microscopic property such as polarity of singleton will be investigated. Mathematics Subject Classifications (2000) 31C45, 11D88 , 46E35, 60J25.     

Phase Transition of Mixed Type <Emphasis Type="Italic">p</Emphasis>-Adic λ-Ising Model on Cayley Tree

In the present paper, we consider an interaction of the nearest-neighbors and next nearest-neighbors for the mixed type p-adic λ-Ising model with spin values {?1, +1} on the Cayley tree of order two.We obtained the uniqueness and existence of the p-adic quasi Gibbs measures for the model. Thereafter, as a main result, we proved the occurrence of phase transition for the p-adic λ-Ising model on the Cayley tree of order two. To establish the results, we employed some properties of p-adic numbers. Therefore, our results are not valid in the real case.     

p-Adic elliptic quadratic forms,parabolic-type pseudodifferential equations with variable coefficients and Markov processes

In this article we study the Cauchy problem for a new class of parabolic-type pseudodifferential equations with variable coefficients for which the fundamental solutions are transition density functions of Markov processes in the four dimensional vector space over the field of p-adic numbers.     

Path continuity of fractional Dirichlet functionals

We prove that the quasi continuous version of a functional in E^p_r is continuous along the sample paths of the Dirichlet process provided that p>2, 0<r?1 and pr>2, without assuming the Meyer equivalence. Parallel results for multi-parameter processes are also obtained. Moreover, for 1<p<2, we prove that a n parameter Dirichlet process does not touch a set of (p,2n)-zero capacity. As an example, we also study the quasi-everywhere existence of the local times of martingales on path space.     

Logistic Dirichlet problems with discontinuous coefficients

This paper is devoted to the study of the existence of positive solutions of semilinear Dirichlet eigenvalue problems for diffusive logistic equations with discontinuous coefficients which model population dynamics in environments with spatial heterogeneity. The approach here is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the theory of singular integral operators. Moreover, we make use of an L^p variant of an estimate for the Green operator of the Dirichlet problem introduced in the study of Feller semigroups.     

The p-adic quantum plane algebras and quantum Weyl algebra

Let q be a principal unit of the ring of valuation of a complete valued field K, extension of the field of p-adic numbers. Generalizing Mahler basis, K. Conrad has constructed orthonormal basis, depending on q, of the space of continuous functions on the ring of p-adic integers with values in K. Attached to q there are two models of the quantum plane and a model of the quantum Weyl algebra, as algebras of bounded linear operators on the space of p-adic continuous functions. For q not a root of unit, interesting orthonormal (orthogonal) families of these algebras are exhibited and providing p-adic completion of quantum plane and quantum Weyl algebras. The text was submitted by the authors in English.     

Simultaneous approximation problems of <Emphasis Type="Italic">p</Emphasis>-adic numbers and <Emphasis Type="Italic">p</Emphasis>-adic knapsack cryptosystems - Alice in <Emphasis Type="Italic">p</Emphasis>-adic numberland

In this paper we construct the multi-dimensional p-adic approximation lattices by using simultaneous approximation problems (SAP) of p-adic numbers and we estimate the l _∞ norm of the p-adic SAP solutions theoretically by applying Dirichlet’s principle and numerically by using the LLL algorithm. By using the SAP solutions as private keys, the security of which depends on NP-hardness of SAP or the shortest vector problems (SVP) of p-adic lattices, we propose a p-adic knapsack cryptosystem with commitment schemes, in which the sender Alice prepares ciphertexts and the verification keys in her p-adic numberland.     

Transcendence of some power series for Liouville number arguments

In this paper, we prove that some power series with rational coefficients take either values of rational numbers or transcendental numbers for the arguments from the set of Liouville numbers under certain conditions in the field of complex numbers. We then apply this result to an algebraic number field. In addition, we establish the p-adic analogues of these results and show that these results have analogues in the field of p-adic numbers.     

Topological geometrodynamics,Part I: General theory

The recent status of topological geometrodynamics (TGD) is reviewed. One can end up with TGD either by starting from the energy problem of general relativity or from the need to generalize hadronic or superstring models. The basic principle of the theory is `Do not quantize!' meaning that quantum physics is reduced to Kähler geometry and spinor structure of the infinite-dimensional space of 3-surfaces in 8-dimensional space H=M⁴₊×CP₂ with physical states represented by classical spinor fields. General coordinate invariance implies that classical theory becomes an exact part of the quantum theory and configuration space geometry and that space-time surfaces are generalized Bohr orbits. The uniqueness of the infinite-dimensional Kähler geometric existence fixes imbedding space and the dimension of the space-time highly uniquely and implies that superconformal and supercanonical symmetries acting on the lightcone boundary δM⁴₊×CP₂ are cosmologies symmetries.The work with the p-adic aspects of TGD, the realization of the possible role of quaternions and octonions in the formulation of quantum TGD, the discovery of infinite primes, and TGD inspired theory of consciousness encouraged the vision about TGD as a generalized number theory. The vision leads to a considerable generalization of TGD and to an extension of the symmetries of the theory to include superconformal and Super-Kac-Moody symmetries associated with the group P×SU(3)×U(2)_ew (P denotes the Poincaré group) acting as the local symmetries of the theory. Quantum criticality, which can be seen as a prediction of the theory, fixes the value spectrum for the coupling constants of the theory.The proper mathematical and physical interpretation of the p-adic numbers has remained a long-lasting challenge. Both TGD inspired theory of consciousness and the vision about physics as a generalized number theory suggest that p-adic space-time regions obeying p-adic counterparts of the field equations are geometric correlates of mind in the sense that they provide cognitive representations for the physics in the real space-time regions representing matter. Evolution identified as a gradual increase of the infinite p-adic prime characterizing the entire Universe is basic prediction of the theory.S-matrix elements can be identified as Glebsch–Gordan coefficients between interacting and free Super-Kac-Moody algebra representations and it is now possible to give Feynmann rules for the S-matrix in the approximation that elementary particles correspond to the so-called CP₂ type extremals.     

On recent results of ergodic property for p-adic dynamical systems

Theory of dynamical systems in fields of p-adic numbers is an important part of algebraic and arithmetic dynamics. The study of p-adic dynamical systems is motivated by their applications in various areas of mathematics, physics, genetics, biology, cognitive science, neurophysiology, computer science, cryptology, etc. In particular, p-adic dynamical systems found applications in cryptography, which stimulated the interest to nonsmooth dynamical maps. An important class of (in general) nonsmooth maps is given by 1-Lipschitz functions. In this paper we present a recent summary of results about the class of 1-Lipschitz functions and describe measure-preserving (for the Haar measure on the ring of p-adic integers) and ergodic functions. The main mathematical tool used in this work is the representation of the function by the van der Put series which is actively used in p-adic analysis. The van der Put basis differs fundamentally from previously used ones (for example, the monomial and Mahler basis) which are related to the algebraic structure of p-adic fields. The basic point in the construction of van der Put basis is the continuity of the characteristic function of a p-adic ball. Also we use an algebraic structure (permutations) induced by coordinate functions with partially frozen variables.     

Wavelet bases in the Lebesgue spaces on the field of p-adic numbers

In this paper we prove that p-adic wavelets form an unconditional basis in the space L ^r (? _p ⁿ ) and give the characterization of the space L ^r (? _p ⁿ ) in terms of Fourier coefficients of p-adic wavelets.Moreover, the Greedy bases in the Lebesgue spaces on the field of p-adic numbers are also established.     

Some aspects of <Emphasis Type="Italic">m</Emphasis>-adic analysis and its applications to <Emphasis Type="Italic">m</Emphasis>-adic stochastic processes

In this paper we consider a generalization of analysis on p-adic numbers field to the m case of m-adic numbers ring. The basic statements, theorems and formulas of p-adic analysis can be used for the case of m-adic analysis without changing. We discuss basic properties of m-adic numbers and consider some properties of m-adic integration and m-adic Fourier analysis. The class of infinitely divisible m-adic distributions and the class of m-adic stochastic Levi processes were introduced. The special class of m-adic CTRW process and fractional-time m-adic random walk as the diffusive limit of it is considered. We found the asymptotic behavior of the probability measure of initial distribution support for fractional-time m-adic random walk.     

The successive minima in the geometry of numbers and the distinction between algebraic and transcendental numbers

This paper discusses an application of Minkowski's theory of the successive minima in the geometry of numbers to the problem of the approximation of an algebraic or transcendental number a by algebraic numbers. I consider for simplicity only real numbers a. However, it is obvious that an analogous theory can be established for complex numbers, and also for p-adic numbers, as well as for the field of formal ascending or descending Laurent series with coefficients in an arbitrary field.     

Dirichlet Forms on Totally Disconnected Spaces and Bipartite Markov Chains

We use Dirichlet form methods to construct and analyze a general class of reversible Markov precesses with totally disconnected state space. We study in detail the special case of bipartite Markov chains. The latter processes have a state space consisting of an interior with a countable number of isolated points and a, typically uncountable, boundary. The equilibrium measure assigns all of its mass to the interior. When the chain is started at a state in the interior, it holds for an exponentially distributed amount of time and then jumps to the boundary. It then instantaneously re-enters the interior. There is a local time on the boundary. That is, the set of times the process is on the boundary is uncountable and coincides with the points of increase of a continuous additive functional. Certain processes with values in the space of trees and the space of vertices of a fixed tree provide natural examples of bipartite chains. Moreover, time-changing a bipartite chain by its local time on the boundary leads to interesting processes, including particular Lévy processes on local fields (for example, the p-adic numbers) that have been considered elsewhere in the literature.     

p-Adic valued quantization

This review covers an important domain of p-adic mathematical physics — quantum mechanics with p-adic valued wave functions. We start with basic mathematical constructions of this quantum model: Hilbert spaces over quadratic extensions of the field of p-adic numbers ? _p , operators — symmetric, unitary, isometric, one-parameter groups of unitary isometric operators, the p-adic version of Schrödinger’s quantization, representation of canonical commutation relations in Heisenberg andWeyl forms, spectral properties of the operator of p-adic coordinate.We also present postulates of p-adic valued quantization. Here observables as well as probabilities take values in ? _p . A physical interpretation of p-adic quantities is provided through approximation by rational numbers.     

Gauss sums and multinomial coefficients

Consider a Gauss sum for a finite field of characteristic p, where p is an odd prime. When such a sum (or a product of such sums) is a p-adic integer we show how it can be realized as a p-adic limit of a sequence of multinomial coefficients. As an application we generalize some congruences of Hahn and Lee to exhibit p-adic limit formulae, in terms of multinomial coefficients, for certain algebraic integers in imaginary quadratic fields related to the splitting of rational primes. We also give an example illustrating how such congruences arise from a p-integral formal group law attached to the p-adic unit part of a product of Gauss sums.     

On Carlitz's q-Bernoulli numbers

Properties of q-extensions of Bernoulli numbers and polynomials which generalize those satisfied by B_k and B_k(x) are used to construct q-extensions of p-adic measures and define a q-extension of p-adic Dirichlet L-series.