The cutting center theorem for trees

Ambarzumian’s Theorem for Trees

The classical Ambarzumian’s Theorem for Schrödinger operators −D2 + q on an interval, with Neumann conditions at the endpoints, says that if the spectrum of (−D2 + q) is the same as the spectrum of (−D2) then q = 0. This theorem is generalized to Schrödinger operators on metric trees with Neumann conditions at the boundary vertices.

Cutting Recursive Autoencoder Trees

Deep Learning models enjoy considerable success in Natural Language Processing. While deep architectures produce useful representations that lead to improvements in various tasks, they are often difficult to interpret. This makes the analysis of learned structures particularly difficult. In this paper, we rely on empirical tests to see whether a particular structure makes sense. We present an a...

A Periodicity Theorem for Trees

In this paper we extend the notion of periodicity from words to labelled ordered trees. This is achieved by deening an equivalence relation on the support of the tree (the set of the nodes) and by requiring that equivalent nodes have the same label. Another requirement is that corresponding sons of equivalent nodes are equivalent too. Since the support of a k-ary tree may be described by a pree...

A Combinatorial Theorem for Trees

On the defect theorem for trees

In this paper we extend the defect theorem from words to k-ary labeled trees. Indeed we prove that, if a set of n trees satisses a nontrivial identity, then these trees can be expressed as concatenation of at most n ? 1 trees. Some consequences in the theory of tree codes are derived.