In this article, we have studied the difference-difference Lotka-Volterra equations in p-adic number space and its p-adic valuation version. We pointed out that the structure of the space given by taking the ultra-discrete limit is the same as that of the p-adic valuation space.

We study the difference-difference Lotka-Volterra equations in p-adic number space and its p-adic valuation version. We point out that the structure of the space given by taking the ultra-discrete limit is the same as that of the p-adic valuation space. Since ultra-discrete limit can be regarded as a classical limit of a quantum object, it implies that a correspondence between classical and qua...

In this work we explore the p-adic valuation of Eulerian numbers. We construct a tree whose nodes contain information about the p-adic valuation of these numbers. Using this tree, and some classical results for Bernoulli numbers, we compute the exact p divisibility for the Eulerian numbers when the first variable lies in a congruence class and p satisfies some regularity properties.

Roughly speaking, the semialgebraic cell decomposition theorem for p-adic numbers describes piecewise the p-adic valuation of p-adic polyno-mials (and more generally of semialgebraic p-adic functions), the pieces being geometrically simple sets, called cells. In this paper we prove a similar cell decomposition theorem to describe piecewise the valuation of analytic functions (and more generally...

In this article, I have studied the ultra discrete limit, which is currently studied in soliton theory, from point of view of valuation theory. A quantity obtained after taking the ultra discrete limit should be regarded as non-archimedean valuation, which is related to the p-adic valuation in number theory. The ultra discrete difference-difference equations, whose domain and range are given by...

Kaplansky proved in 1942 that among ail fields with a valuation having a given divisible value group G, a given algebraically closed residue field R, and a given restriction to the minimal subfield (either the trivial valuation on Qor Fp , or the /?-adic valuation on Q), there is one that is maximal in the strong sensé that every other can be embedded in it. In this paper, we construct this fie...

A conjecture of G. McGarvey for the 2-adic valuation of the Schenker sums is established. These sums are n! times the sum of the first n+1 terms of the series for e. A certain analytic expression for the p-adic valuation of these sums is provided for a class of primes. Some combinatorial interpretations (using rooted trees) are furnished for identities that arose along the way.

Let R be a pseudo-valuation domain with associated valuation domain V and I a nonzero proper ideal of R. Let R̂ (resp., V̂ ) be the I-adic (resp., IV -adic) completion of R (resp., V ). We show that R̂ is a pseudo-valuation domain (which may be a field); and that if I 6= I2, then V̂ is the associated valuation domain of R̂. Let R be an SFT globalized pseudo-valuation domain with associated Prüfer do...

1. January 12th – Motivation (Bhargav Bhatt) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2. January 26th – Huber rings (Rankeya Datta). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.1. Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....

Let p be a fixed prime. Throughout this paper Zp, Qp, C and Cp will, respectively, denote the ring of p-adic rational integers, the field of p-adic rational numbers, the complex number field and the completion of algebraic closure of Qp, cf. [1], [3], [6], [10]. Let vp be the normalized exponential valuation of Cp with |p|p = p −vp(p) = p. Kubota and Leopoldt proved the existence of meromorphic...