The Burnside ring of the infinite cyclic group and its relations to the necklace algebra, λ-rings, and the universal ring of Witt vectors

AbstractIt is shown that well-known product decompositions of formal power series arise from combinatorially defined canonical isomorphisms between the Burnside ring of the infinite cyclic group on the one hand and Grothendieck's ring of formal power series with constant term 1 as well as the universal ring of Witt vectors on the other hand

Noisy: Identification of problematic columns in multiple sequence alignments

Motivation Sequence-based methods for phylogenetic reconstruction from (nucleic acid) sequence data are notoriously plagued by two effects: homoplasies and alignment errors. Large evolutionary distances imply a large number of homoplastic sites. As most protein-coding genes show dramatic variations in substitution rates that are not uncorrelated across the sequence, this often leads to a patchwork pattern of (i) phylogenetically informative and (ii) effectively randomized regions. In highly variable regions, furthermore, alignment errors accumulate resulting in sometimes misleading signals in phylogenetic reconstruction. Results We present here a method that, based on assessing the distribution of character states along a cyclic ordering of the taxa, allows the identification of phylogenetically uninformative homoplastic sites in a multiple sequence alignment. Removal of these sites appears to improve the performance of phylogenetic reconstruction algorithms as measured by various indices of 'tree quality'. In particular, we obtain more stable trees due to the exclusion of phylogenetically incompatible sites that most likely represent strongly randomized characters. Software The computer program noisy implements this approach. It can be employed to improving phylogenetic reconstruction capability with quite a considerable success rate whenever (1) the average bootstrap support obtained from the original alignment is low, and (2) there are sufficiently many taxa in the data set – at least, say, 12 to 15 taxa. The software can be obtained under the GNU Public License from http://www.bioinf.uni-leipzig.de/Software/noisy/

Barriers in metric spaces

Defining a subset B of a connected topological space T to be a barrier (in T) if B is connected and its complement T-B is disconnected, we will investigate barriers B in the tight span View the MathML source Turn MathJax on of a metric D defined on a finite set X (endowed, as a subspace of RX, with the metric and the topology induced by the l8-norm) that are of the form B=Be(f)?{g?T(D):?f-g?8=e} Turn MathJax on for some f?T(D) and some e=0. In particular, we will present some conditions on f and e which ensure that such a subset of T(D) is a barrier in T(D). More specifically, we will show that Be(f) is a barrier in T(D) if there exists a bipartition (or split) of the e-support View the MathML source of f into two non-empty sets A and B such that f(a)+f(b)=ab+e holds for all elements a?A and b?B while, conversely, whenever Be(f) is a barrier in T(D), there exists a bipartition of View the MathML source into two non-empty sets A and B such that, at least, f(a)+f(b)=ab+2e holds for all elements a?A and b?B

Matroidizing set systems: a new approach to matroid theory

AbstractFor a finite nonempty set E we associate in a canonical way to every antichain B⊆P(E) a matroid M(B) such that M(B)=M0 if B is the set of bases of a matroid M0. We do this by first associating to B a closure operator <…>=<…>B:P(E)→P(E) and to a closure operator <…> the antichain B<…>, consisting of all minimal generating sets. For n≥0 we define new antichains Pn(B), where P0(B)≔B and Pn+1(B)≔Pn (B(<…>B)) for all such n. Then P1(B)=B if and only if B is the set of bases of some matroid. We show that there exists some m≥0, depending only on the cardinality #E, such that Pm+1(B)=Pm(B) for every antichain B⊆P(E) and, hence, may define M(B) to be the matroid with Pm(B) as its set of bases. This simple construction has many intriguing properties, which we believe deserve further study

Notes on Recovering Symbolically Dated, Rooted Trees from Symbolic Ultrametrics

A well known result from cluster theory states that there is a 1-to-1 correspondence between dated, compact, rooted trees and ultrametrics. In this paper, we generalize this result yielding a canonical 1-to-1 correspondence between symbolically dated trees and symbolic ultrametrics, using an arbitrary set as the set of (possible) dates or values. It turns out that a rather unexpected new condition is needed to properly define symbolic ultrametrics so that the above correspondence holds. In the second part of the paper, we use our main result to derive, as a corollary, a theorem by H. J. Bandelt and M. A. Steel regarding a canonical 1-to-1 correspondence between additive trees and metrics satisfying the 4-point condition, both taking their values in abelian monoids. All (di-)graphs G = (V; E ` V 2 ) studied in this paper will be finite (and -- by definition -- without multiple edges). For a vertex v, let d \Gamma (v) := #fw 2 V : (w; v) 2 Eg denote its in-degree, and d+ (v)..