5,223 research outputs found
Analytical study of the effect of recombination on evolution via DNA shuffling
We investigate a multi-locus evolutionary model which is based on the DNA
shuffling protocol widely applied in \textit{in vitro} directed evolution. This
model incorporates selection, recombination and point mutations. The simplicity
of the model allows us to obtain a full analytical treatment of both its
dynamical and equilibrium properties, for the case of an infinite population.
We also briefly discuss finite population size corrections
Combinatorial Hopf algebra structure on packed square matrices
We construct a new bigraded Hopf algebra whose bases are indexed by square
matrices with entries in the alphabet , , without
null rows or columns. This Hopf algebra generalizes the one of permutations of
Malvenuto and Reutenauer, the one of -colored permutations of Novelli and
Thibon, and the one of uniform block permutations of Aguiar and Orellana. We
study the algebraic structure of our Hopf algebra and show, by exhibiting
multiplicative bases, that it is free. We moreover show that it is self-dual
and admits a bidendriform bialgebra structure. Besides, as a Hopf subalgebra,
we obtain a new one indexed by alternating sign matrices. We study some of its
properties and algebraic quotients defined through alternating sign matrices
statistics.Comment: 35 page
Hopf algebras and Markov chains: Two examples and a theory
The operation of squaring (coproduct followed by product) in a combinatorial
Hopf algebra is shown to induce a Markov chain in natural bases. Chains
constructed in this way include widely studied methods of card shuffling, a
natural "rock-breaking" process, and Markov chains on simplicial complexes.
Many of these chains can be explictly diagonalized using the primitive elements
of the algebra and the combinatorics of the free Lie algebra. For card
shuffling, this gives an explicit description of the eigenvectors. For
rock-breaking, an explicit description of the quasi-stationary distribution and
sharp rates to absorption follow.Comment: 51 pages, 17 figures. (Typographical errors corrected. Further fixes
will only appear on the version on Amy Pang's website, the arXiv version will
not be updated.
Avoiding and Enforcing Repetitive Structures in Words
The focus of this thesis is on the study of repetitive structures in words, a central topic in the area of combinatorics on words. The results presented in the thesis at hand are meant to extend and enrich the existing theory concerning the appearance and absence of such structures. In the first part we examine whether these structures necessarily appear in infinite words over a finite alphabet. The repetitive structures we are concerned with involve functional dependencies between the parts that are repeated. In particular, we study avoidability questions of patterns whose repetitive structure is disguised by the application of a permutation. This novel setting exhibits the surprising behaviour that avoidable patterns may become unavoidable in larger alphabets. The second and major part of this thesis deals with equations on words that enforce a certain repetitive structure involving involutions in their solution set. Czeizler et al. (2009) introduced a generalised version of the classical equations u` Æ vmwn that were studied by Lyndon and Schützenberger. We solve the last two remaining and most challenging cases and thereby complete the classification of these equations in terms of the repetitive structures appearing in the admitted solutions. In the final part we investigate the influence of the shuffle operation on words avoiding ordinary repetitions. We construct finite and infinite square-free words that can be shuffled with themselves in a way that preserves squarefreeness. We also show that the repetitive structure obtained by shuffling a word with itself is avoidable in infinite words
Gravitational Dynamics of an Infinite Shuffled Lattice: Particle Coarse-grainings, Non-linear Clustering and the Continuum Limit
We study the evolution under their self-gravity of infinite ``shuffled
lattice'' particle distributions, focussing specifically on the comparison of
this evolution with that of ``daughter'' particle distributions, defined by a
simple coarse-graining procedure. We consider both the case that such
coarse-grainings are performed (i) on the initial conditions, and (ii) at a
finite time with a specific additional prescription. In numerical simulations
we observe that, to a first approximation, these coarse-grainings represent
well the evolution of the two-point correlation properties over a significant
range of scales. We note, in particular, that the form of the two-point
correlation function in the original system, when it is evolving in the
asymptotic ``self-similar'' regime, may be reproduced well in a daughter
coarse-grained system in which the dynamics are still dominated by two-body
(nearest neighbor) interactions. Using analytical results on the early time
evolution of these systems, however, we show that small observed differences
between the evolved system and its coarse-grainings at the initial time will in
fact diverge as the ratio of the coarse-graining scale to the original
inter-particle distance increases. The second coarse-graining studied,
performed at a finite time in a specified manner, circumvents this problem. It
also makes more physically transparent why gravitational dynamics from these
initial conditions tends toward a ``self-similar'' evolution. We finally
discuss the precise definition of a limit in which a continuum (specifically
Vlasov-like) description of the observed linear and non-linear evolution should
be applicable.Comment: 21 pages, 8 eps figures, 2 jpeg figures (available in high resolution
at http://pil.phys.uniroma1.it/~sylos/PRD_dec_2006/
From Aztec diamonds to pyramids: steep tilings
We introduce a family of domino tilings that includes tilings of the Aztec
diamond and pyramid partitions as special cases. These tilings live in a strip
of of the form for some integer , and are parametrized by a binary word that
encodes some periodicity conditions at infinity. Aztec diamond and pyramid
partitions correspond respectively to and to the limit case
. For each word and for different types of boundary
conditions, we obtain a nice product formula for the generating function of the
associated tilings with respect to the number of flips, that admits a natural
multivariate generalization. The main tools are a bijective correspondence with
sequences of interlaced partitions and the vertex operator formalism (which we
slightly extend in order to handle Littlewood-type identities). In
probabilistic terms our tilings map to Schur processes of different types
(standard, Pfaffian and periodic). We also introduce a more general model that
interpolates between domino tilings and plane partitions.Comment: 36 pages, 22 figures (v3: final accepted version with new Figure 6,
new improved proof of Proposition 11
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