Which electric fields are realizable in conducting materials?

Which electric fields are realizable in conducting materials?

In this paper we study the realizability of a given smooth periodic gradient field ∇u defined in R, in the sense of finding when one can obtain a matrix conductivity σ such that σ∇u is a divergence free current field. The construction is shown to be always possible locally in R provided that ∇u is non-vanishing. This condition is also necessary in dimension two but not in dimension three. In fa...

Which electric fields are realizable in conducting materials ? January 8 , 2013

In this paper we study the realizability of a given smooth periodic gradient field ∇u defined in R, in the sense of finding when one can obtain a matrix conductivity σ such that σ∇u is a divergence free current field. The construction is shown to be always possible locally in R provided that ∇u is non-vanishing. This condition is also necessary in dimension two but not in dimension three. In fa...

Positively Oriented Matroids Are Realizable

We prove da Silva’s 1987 conjecture that any positively oriented matroid is a positroid; that is, it can be realized by a set of vectors in a real vector space. It follows from this result and a result of the third author that the positive matroid Grassmannian (or positive MacPhersonian) is homeomorphic to a closed ball.

A Realizable Learning Task which Exhibits Overfitting

In this paper we examine a perceptron learning task. The task is realizable since it is provided by another perceptron with identical architecture. Both perceptrons have nonlinear sigmoid output functions. The gain of the output function determines the level of nonlinearity of the learning task. It is observed that a high level of nonlinearity leads to overfitting. We give an explanation for th...

Stress Fields and Electric Potential in Piezoelectric Cylinders