2,638 research outputs found
Polyhedral computational geometry for averaging metric phylogenetic trees
This paper investigates the computational geometry relevant to calculations
of the Frechet mean and variance for probability distributions on the
phylogenetic tree space of Billera, Holmes and Vogtmann, using the theory of
probability measures on spaces of nonpositive curvature developed by Sturm. We
show that the combinatorics of geodesics with a specified fixed endpoint in
tree space are determined by the location of the varying endpoint in a certain
polyhedral subdivision of tree space. The variance function associated to a
finite subset of tree space has a fixed algebraic formula within
each cell of the corresponding subdivision, and is continuously differentiable
in the interior of each orthant of tree space. We use this subdivision to
establish two iterative methods for producing sequences that converge to the
Frechet mean: one based on Sturm's Law of Large Numbers, and another based on
descent algorithms for finding optima of smooth functions on convex polyhedra.
We present properties and biological applications of Frechet means and extend
our main results to more general globally nonpositively curved spaces composed
of Euclidean orthants.Comment: 43 pages, 6 figures; v2: fixed typos, shortened Sections 1 and 5,
added counter example for polyhedrality of vistal subdivision in general
CAT(0) cubical complexes; v1: 43 pages, 5 figure
Fraisse Limits, Ramsey Theory, and Topological Dynamics of Automorphism Groups
We study in this paper some connections between the Fraisse theory of
amalgamation classes and ultrahomogeneous structures, Ramsey theory, and
topological dynamics of automorphism groups of countable structures.Comment: 73 pages, LaTeX 2e, to appear in Geom. Funct. Ana
Lehmer's Problem, McKay's Correspondence, and
This paper addresses a long standing open problem due to Lehmer in which the
triple 2,3,7 plays a notable role. Lehmer's problem asks whether there is a gap
between 1 and the next smallest algebraic integer with respect to Mahler
measure. The question has been studied in a wide range of contexts including
number theory, ergodic theory, hyperbolic geometry, and knot theory; and
relates to basic questions such as describing the distribution of heights of
algebraic integers, and of lengths of geodesics on arithmetic surfaces. This
paper focuses on the role of Coxeter systems in Lehmer's problem. The analysis
also leads to a topological version of McKay's correspondence
Freeness theorems for operads via Gr\"obner bases
We show how to use Groebner bases for operads to prove various freeness
theorems: freeness of certain operads as nonsymmetric operads, freeness of an
operad Q as a P-module for an inclusion P into Q, freeness of a suboperad. This
gives new proofs of many known results of this type and helps to prove some new
results.Comment: 15 pages, no figures, corrected typos and changed in parts the
structure of the pape
Notions of Relative Ubiquity for Invariant Sets of Relational Structures
Given a finite lexicon L of relational symbols and equality, one may view the collection of all L-structures on the set of natural numbers w as a space in several different ways. We consider it as: (i) the space of outcomes of certain infinite two-person games; (ii) a compact metric space; and (iii) a probability measure space. For each of these viewpoints, we can give a notion of relative ubiquity, or largeness, for invariant sets of structures on w. For example, in every sense of relative ubiquity considered here, the set of dense linear orderings on w is ubiquitous in the set of linear orderings on w
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