2,899 research outputs found
New Representations of Matroids and Generalizations
We extend the notion of matroid representations by matrices over fields and
consider new representations of matroids by matrices over finite semirings,
more precisely over the boolean and the superboolean semirings. This idea of
representations is generalized naturally to include also hereditary
collections. We show that a matroid that can be directly decomposed as
matroids, each of which is representable over a field, has a boolean
representation, and more generally that any arbitrary hereditary collection is
superboolean-representable.Comment: 27 page
Unified theory for finite Markov chains
We provide a unified framework to compute the stationary distribution of any
finite irreducible Markov chain or equivalently of any irreducible random walk
on a finite semigroup . Our methods use geometric finite semigroup theory
via the Karnofsky-Rhodes and the McCammond expansions of finite semigroups with
specified generators; this does not involve any linear algebra. The original
Tsetlin library is obtained by applying the expansions to , the set of
all subsets of an element set. Our set-up generalizes previous
groundbreaking work involving left-regular bands (or -trivial
bands) by Brown and Diaconis, extensions to -trivial semigroups by
Ayyer, Steinberg, Thi\'ery and the second author, and important recent work by
Chung and Graham. The Karnofsky-Rhodes expansion of the right Cayley graph of
in terms of generators yields again a right Cayley graph. The McCammond
expansion provides normal forms for elements in the expanded . Using our
previous results with Silva based on work by Berstel, Perrin, Reutenauer, we
construct (infinite) semaphore codes on which we can define Markov chains.
These semaphore codes can be lumped using geometric semigroup theory. Using
normal forms and associated Kleene expressions, they yield formulas for the
stationary distribution of the finite Markov chain of the expanded and the
original . Analyzing the normal forms also provides an estimate on the
mixing time.Comment: 29 pages, 12 figures; v2: Section 3.2 added, references added,
revision of introduction, title change; v3: typos fixed and clarifications
adde
Identifiability of Large Phylogenetic Mixture Models
Phylogenetic mixture models are statistical models of character evolution
allowing for heterogeneity. Each of the classes in some unknown partition of
the characters may evolve by different processes, or even along different
trees. The fundamental question of whether parameters of such a model are
identifiable is difficult to address, due to the complexity of the
parameterization. We analyze mixture models on large trees, with many mixture
components, showing that both numerical and tree parameters are indeed
identifiable in these models when all trees are the same. We also explore the
extent to which our algebraic techniques can be employed to extend the result
to mixtures on different trees.Comment: 15 page
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