Throughout this paper Z, Q, C, Zp, Qp and Cp will respectively denote the ring of rational integers, the field of rational numbers, the filed of complex numbers, the ring p-adic rational integers, the field of p-adic rational numbers, and the completion of the algebraic closure of Qp. Let vp be the normalized exponential valuation of Cp such that |p|p = p −vp(p) = p. If q ∈ Cp, we normally assu...

In this paper, we give an explicit description of the de Rham and p-adic polylogarithms for elliptic curves using the Kronecker theta function. We prove in particular that when the elliptic curve has complex multiplication and good reduction at p, then the specializations to torsion points of the p-adic elliptic polylogarithm are related to p-adic Eisenstein-Kronecker numbers, proving a p-adic ...

In this paper we investigate some relations between Bernstein polynomials and q-Euler numbers with weak weight α. From these relations, we derive some interesting identities on the q-Euler numbers with weak weight α. Let p be a fixed odd prime number. Throughout this paper, we always make use of the following notations: Z denotes the ring of rational integers, Zp denotes the ring of p-adic rati...

Many mathematicians have studied Euler numbers and Euler polynomials( see [1-11]). Euler polynomials posses many interesting properties and arising in many areas of mathematics and physics. In this paper we introduce the generalized q-Euler numbers and polynomials with weak weight α. Throughout this paper we use the following notations. By Zp we denote the ring of p-adic rational integers, Q de...

Rigged Hilbert space of the free coherent states is investigated. We prove that this rigged Hilbert space is isomorphous to the space of generalized functions over p–adic disk. We discuss the relation of the described isomorphism of rigged Hilbert spaces and noncommutative geometry and show, that the considered example realises the isomorphism of the noncommutative line and p–adic disk. In the ...

In this paper, we give an explicit description of the complex and p-adic polylogarithms for elliptic curves using the Kronecker theta function. We prove in particular that when the elliptic curve has complex multiplication and good reduction at p, then the specializations to torsion points of the p-adic elliptic polylogarithm are related to p-adic Eisenstein-Kronecker numbers, proving a p-adic ...

The p-adic formulation of replica symmetry breaking is presented. In this approach ultrametricity is a natural consequence of the basic properties of the p-adic numbers. Many properties can be simply derived in this approach and p-adic Fourier transform seems to be an promising tool. Unité Propre du Centre National de la Recherche Scientifique, associée à l’ Ecole Normale Supérieure et à l’Univ...

A conceptual framework for cluster analysis from the viewpoint of p-adic geometry is introduced by describing the space of all dendrograms for n datapoints and relating it to the moduli space of p-adic Riemannian spheres with punctures using a method recently applied by Murtagh (2004b). This method embeds a dendrogram as a subtree into the Bruhat-Tits tree associated to the p-adic numbers, and ...

A conceptual framework for cluster analysis from the viewpoint of p-adic geometry is introduced by describing the space of all dendrograms for n datapoints and relating it to the moduli space of p-adic Riemannian spheres with punctures using a method recently applied by Murtagh (2004b). This method embeds a dendrogram as a subtree into the Bruhat-Tits tree associated to the p-adic numbers, and ...