In his 1974 text, Commutative Ring Theory, Kaplansky states that among the examples of non-Dedekind Prüfer domains, the main ones are valuation domains, the ring of entire functions and the integral closure of a Prüfer domain in an algebraic extension of its quotient field [Kap74, p.72]. A similar list today would likely include Kronecker function rings, the ring of integervalued polynomials an...

The paper examines the existence of pseudo almost periodic solutions to some classes of partial hyperbolic evolution equations. Namely, sufficient conditions for the existence and uniqueness of pseudo almost periodic solutions to those classes of hyperbolic evolution equations are given. Applications include the existence of pseudo almost periodic solutions to the transport and heat equations w...

Let $V$ be a valuation domain with quotient field $K$. We show how to describe all extensions of $K(X)$ when the $V$-adic completion $\widehat{K}$ is algebraically closed, generalizing similar result obtained by Ostrowski in case one-dimensional domains. This accomplished realizing such means pseudo-monotone sequences, generalization pseudo-convergent sequences introduced Chabert. also that rin...

For the whole paper, K denotes an algebraically closed field endowed with a nontrivial non-archimedean complete absolute value | |. The corresponding valuation is v := − log | | with value group Γ := v(K). The valuation ring is denoted by K. Note that the residue field K̃ is algebraically closed. In Theorem 1.3, §8 and in the second part of §9, we start with a field K endowed with a discrete val...

We prove R = T theorems for certain reducible residual Galois representations. We answer in the positive a question of Gross and Lubin on whether certain Hecke algebras T are discrete valuation rings. In order to prove these results we determine (using the theory of Breuil modules) when two finite flat group schemes G and H of order p over an arbitrarily tamely ramified discrete valuation ring ...

We describe an alternate construction of some of the basic rings introduced by Fontaine in p-adic Hodge theory. In our construction, the central role is played by the ring of p-typical Witt vectors over a p-adic valuation ring, rather than theWitt vectors over a ring of positive characteristic. This suggests the possibility of forming a meaningful global analogue of p-adic Hodge theory.

A self stabilizing distributed system is a network of state machines which, regardless of its initial global state, once started will regain its consistency by itself in a nite number of steps. Two are the main issues in the design of a self-stabilizing system: stabilization time and memory requirements. In this paper we present an almost two state self stabilizing unidirectional system solving...