Solvable Leibniz algebras with triangular nilradical

A classification exists for Lie algebras whose nilradical is the triangular Lie algebra $T(n)$. We extend this result to a classification of all solvable Leibniz algebras with nilradical $T(n)$. As an example we show the complete classification of all Leibniz algebras whose nilradical is $T(4)$.Comment: arXiv admin note: text overlap with arXiv:1307.844

A Frattini Theory for Leibniz Algebras

A Frattini theory for non-associative algebras was developed by Towers and results for particular classes of algebras have appeared in various articles. Especially plentiful are results on Lie algebras. It is the purpose of this paper to extend some of the Lie algebra results to Leibniz algebras

Codes and shifted codes

The action of the Bernstein operators on Schur functions was given in terms of codes in [CG] and extended to the analog in Schur Q-functions in [HJS]. We define a new combinatorial model of extended codes and show that both of these results follow from a natural combinatorial relation induced on codes. The new algebraic structure provides a natural setting for Schur functions indexed by compositions

Inferring Species Trees Directly from Biallelic Genetic Markers: Bypassing Gene Trees in a Full Coalescent Analysis

The multi-species coalescent provides an elegant theoretical framework for estimating species trees and species demographics from genetic markers. Practical applications of the multi-species coalescent model are, however, limited by the need to integrate or sample over all gene trees possible for each genetic marker. Here we describe a polynomial-time algorithm that computes the likelihood of a species tree directly from the markers under a finite-sites model of mutation, effectively integrating over all possible gene trees. The method applies to independent (unlinked) biallelic markers such as well-spaced single nucleotide polymorphisms (SNPs), and we have implemented it in SNAPP, a Markov chain Monte-Carlo sampler for inferring species trees, divergence dates, and population sizes. We report results from simulation experiments and from an analysis of 1997 amplified fragment length polymorphism (AFLP) loci in 69 individuals sampled from six species of {\em Ourisia} (New Zealand native foxglove)