12,026 research outputs found
Phylogenetic mixtures and linear invariants for equal input models
The reconstruction of phylogenetic trees from molecular sequence data relies on modelling site substitutions by a Markov process, or a mixture of such processes. In general, allowing mixed processes can result in different tree topologies becoming indistinguishable from the data, even for infinitely long sequences. However, when the underlying Markov process supports linear phylogenetic invariants, then provided these are sufficiently informative, the identifiability of the tree topology can be restored. In this paper, we investigate a class of processes that support linear invariants once the stationary distribution is fixed, the âequal input modelâ. This model generalizes the âFelsenstein 1981â model (and thereby the JukesâCantor model) from four states to an arbitrary number of states (finite or infinite), and it can also be described by a ârandom clusterâ process. We describe the structure and dimension of the vector spaces of phylogenetic mixtures and of linear invariants for any fixed phylogenetic tree (and for all treesâthe so called âmodel invariantsâ), on any number n of leaves. We also provide a precise description of the space of mixtures and linear invariants for the special case of n=4 leaves. By combining techniques from discrete random processes and (multi-) linear algebra, our results build on a classic result that was first established by James Lake (Mol Biol Evol 4:167â191, 1987).Peer ReviewedPostprint (author's final draft
Spectral Properties of Quantum Walks on Rooted Binary Trees
We define coined Quantum Walks on the infinite rooted binary tree given by
unitary operators on an associated infinite dimensional Hilbert space,
depending on a unitary coin matrix , and study their spectral
properties. For circulant unitary coin matrices , we derive an equation for
the Carath\'eodory function associated to the spectral measure of a cyclic
vector for . This allows us to show that for all circulant unitary coin
matrices, the spectrum of the Quantum Walk has no singular continuous
component. Furthermore, for coin matrices which are orthogonal circulant
matrices, we show that the spectrum of the Quantum Walk is absolutely
continuous, except for four coin matrices for which the spectrum of is
pure point
Parallel Construction of Wavelet Trees on Multicore Architectures
The wavelet tree has become a very useful data structure to efficiently
represent and query large volumes of data in many different domains, from
bioinformatics to geographic information systems. One problem with wavelet
trees is their construction time. In this paper, we introduce two algorithms
that reduce the time complexity of a wavelet tree's construction by taking
advantage of nowadays ubiquitous multicore machines.
Our first algorithm constructs all the levels of the wavelet in parallel in
time and bits of working space, where
is the size of the input sequence and is the size of the alphabet. Our
second algorithm constructs the wavelet tree in a domain-decomposition fashion,
using our first algorithm in each segment, reaching time and
bits of extra space, where is the
number of available cores. Both algorithms are practical and report good
speedup for large real datasets.Comment: This research has received funding from the European Union's Horizon
2020 research and innovation programme under the Marie Sk{\l}odowska-Curie
Actions H2020-MSCA-RISE-2015 BIRDS GA No. 69094
Commutative combinatorial Hopf algebras
We propose several constructions of commutative or cocommutative Hopf
algebras based on various combinatorial structures, and investigate the
relations between them. A commutative Hopf algebra of permutations is obtained
by a general construction based on graphs, and its non-commutative dual is
realized in three different ways, in particular as the Grossman-Larson algebra
of heap ordered trees.
Extensions to endofunctions, parking functions, set compositions, set
partitions, planar binary trees and rooted forests are discussed. Finally, we
introduce one-parameter families interpolating between different structures
constructed on the same combinatorial objects.Comment: 29 pages, LaTEX; expanded and updated version of math.CO/050245
The Combinatorics of Iterated Loop Spaces
It is well known since Stasheff's work that 1-fold loop spaces can be
described in terms of the existence of higher homotopies for associativity
(coherence conditions) or equivalently as algebras of contractible
non-symmetric operads. The combinatorics of these higher homotopies is well
understood and is extremely useful.
For the theory of symmetric operads encapsulated the corresponding
higher homotopies, yet hid the combinatorics and it has remain a mystery for
almost 40 years. However, the recent developments in many fields ranging from
algebraic topology and algebraic geometry to mathematical physics and category
theory show that this combinatorics in higher dimensions will be even more
important than the one dimensional case.
In this paper we are going to show that there exists a conceptual way to make
these combinatorics explicit using the so called higher nonsymmetric
-operads.Comment: 23 page
Hopf Algebras of m-permutations, (m+1)-ary trees, and m-parking functions
The m-Tamari lattice of F. Bergeron is an analogue of the clasical Tamari
order defined on objects counted by Fuss-Catalan numbers, such as m-Dyck paths
or (m+1)-ary trees. On another hand, the Tamari order is related to the product
in the Loday-Ronco Hopf algebra of planar binary trees. We introduce new
combinatorial Hopf algebras based on (m+1)-ary trees, whose structure is
described by the m-Tamari lattices.
In the same way as planar binary trees can be interpreted as sylvester
classes of permutations, we obtain (m+1)-ary trees as sylvester classes of what
we call m-permutations. These objects are no longer in bijection with
decreasing (m+1)-ary trees, and a finer congruence, called metasylvester,
allows us to build Hopf algebras based on these decreasing trees. At the
opposite, a coarser congruence, called hyposylvester, leads to Hopf algebras of
graded dimensions (m+1)^{n-1}, generalizing noncommutative symmetric functions
and quasi-symmetric functions in a natural way. Finally, the algebras of packed
words and parking functions also admit such m-analogues, and we present their
subalgebras and quotients induced by the various congruences.Comment: 51 page
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