Information Theory for Fields

Ramification Theory for Local Fields with Imperfect Residue Fields

Notation 1.1. Let l/k be a finite Galois extension of complete discretely valued fields. Let Ok, Ol, πk, πl, k̄, and l̄ be rings of integers, uniformizers, and residue fields, respectively. For an element a ∈ Ol, we use ā to denote its reduction in l̄. Let G = Gl/k be the Galois group. Use vl(·) to denote the valuation on l so that vl(πl) = 1. We call e = vl(πk) the näıve ramification degree; it i...

Yet Another Application of the Theory of ODE in the Theory of Vector Fields

In this paper we are supposed to define the θ−vector field on the n−surface S and then investigate about the existence and uniqueness of its integral curves by the Theory of Ordinary Differential Equations. Then thesubject is followed through some examples.

A variational theory for monotone vector fields

Monotone vector fields were introduced almost 40 years ago as nonlinear extensions of positive definite linear operators, but also as natural extensions of gradients of convex potentials. These vector fields are not always derived from potentials in the classical sense, and as such they are not always amenable to the standard methods of the calculus of variations. We describe here how the selfd...

Transition-rate theory for nongradient drift fields.

Ramification Theory for Higher Dimensional Local Fields

The paper contains a construction of ramification theory for higher dimensional local fields K provided with additional structure given by an increasing sequence of their “subfields of i-dimensional constants”, where 0 i n and n is the dimension of K. It is also announced that a local analogue of the Grothendieck Conjecture still holds: all automorphisms of the absolute Galois group of K, which...