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.

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 ...

For many kinds of assets, the growth rate of the real asset stock is a nonlinear function of the economic owner’s decision whether to invest or extract the asset. Examples within the economy are primarily biological assets, both privately owned (such as those found in aquaculature and agriculture) and publicly owned or regulated (such as fish stocks, and in some case, timber stocks.) Optimal ex...

Valuations of dense near polygons were introduced in [9]. A valuation of a dense near polygon S = (P,L, I) is a map f from the point-set P of S to the set N of nonnegative integers satisfying very nice properties with respect to the set of convex subspaces of S. In the present paper, we give an alternative definition of the notion valuation and prove that both definitions are equivalent. In the...