24,473 research outputs found
Polishness of some topologies related to word or tree automata
We prove that the B\"uchi topology and the automatic topology are Polish. We
also show that this cannot be fully extended to the case of a space of infinite
labelled binary trees; in particular the B\"uchi and the Muller topologies are
not Polish in this case.Comment: This paper is an extended version of a paper which appeared in the
proceedings of the 26th EACSL Annual Conference on Computer Science and
Logic, CSL 2017. The main addition with regard to the conference paper
consists in the study of the B\"uchi topology and of the Muller topology in
the case of a space of trees, which now forms Section
Labeled binary planar trees and quasi-Lie algebras
We study the natural map eta between a group of binary planar trees whose
leaves are labeled by elements of a free abelian group H and a certain group
D(H) derived from the free Lie algebra over H. Both of these groups arise in
several different topological contexts. The map eta is known to be an
isomorphism over Q, but not over Z. We determine its cokernel and attack the
conjecture that it is injective.Comment: This is the version published by Algebraic & Geometric Topology on 9
August 200
Tropical Geometry of Phylogenetic Tree Space: A Statistical Perspective
Phylogenetic trees are the fundamental mathematical representation of
evolutionary processes in biology. As data objects, they are characterized by
the challenges associated with "big data," as well as the complication that
their discrete geometric structure results in a non-Euclidean phylogenetic tree
space, which poses computational and statistical limitations. We propose and
study a novel framework to study sets of phylogenetic trees based on tropical
geometry. In particular, we focus on characterizing our framework for
statistical analyses of evolutionary biological processes represented by
phylogenetic trees. Our setting exhibits analytic, geometric, and topological
properties that are desirable for theoretical studies in probability and
statistics, as well as increased computational efficiency over the current
state-of-the-art. We demonstrate our approach on seasonal influenza data.Comment: 28 pages, 5 figures, 1 tabl
- …