Maximal ordered fields of rank $n$

Rank Ordered Autoencoders

A new method for the unsupervised learning of sparse representations using autoencoders is proposed and implemented by ordering the output of the hidden units by their activation value and progressively reconstructing the input in this order. This can be done efficiently in parallel with the use of cumulative sums and sorting only slightly increasing the computational costs. Minimizing the diff...

The Maximal Rank Conjecture

Let C be a general curve of genus g, embedded in Pr via a general linear series of degree d. In this paper, we prove the Maximal Rank Conjecture, which determines the Hilbert function of C ⊂ Pr.

Dynamic Rank-Maximal Matchings

We consider the problem of matching applicants to posts where applicants have preferences over posts. Thus the input to our problem is a bipartite graph G = (A ∪ P , E), where A denotes a set of applicants, P is a set of posts, and there are ranks on edges which denote the preferences of applicants over posts. A matching M in G is called rank-maximal if it matches the maximum number of applican...

Rank-.n 1/ convexity and quasiconvexity for divergence free fields

We prove that rank-.n 1/ convexity does not imply quasiconvexity with respect to divergence free fields (so-called S-quasiconvexity) in M n for m > n, by adapting the well-known Šverák’s counterexample to the solenoidal setting. On the other hand, we also remark that rank-.n 1/ convexity and S-quasiconvexity turn out to be equivalent in the space of n n diagonal matrices.

On Partly Ordered Fields