3,949 research outputs found
Square-free Words with Square-free Self-shuffles
We answer a question of Harju: For every n ≥ 3 there is a square-free ternary word of length n with a square-free self-shuffle.http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i1p
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.
Flows and stochastic Taylor series in Ito calculus
For stochastic systems driven by continuous semimartingales an explicit
formula for the logarithm of the Ito flow map is given. A similar formula is
also obtained for solutions of linear matrix-valued SDEs driven by arbitrary
semimartingales. The computation relies on the lift to quasi-shuffle algebras
of formulas involving products of Ito integrals of semimartingales. Whereas the
Chen-Strichartz formula computing the logarithm of the Stratonovich flow map is
classically expanded as a formal sum indexed by permutations, the analogous
formula in Ito calculus is naturally indexed by surjections. This reflects the
change of algebraic background involved in the transition between the two
integration theories
Automorphisms of graph products of groups from a geometric perspective
This article studies automorphism groups of graph products of arbitrary
groups. We completely characterise automorphisms that preserve the set of
conjugacy classes of vertex groups as those automorphisms that can be
decomposed as a product of certain elementary automorphisms (inner
automorphisms, partial conjugations, automorphisms associated to symmetries of
the underlying graph). This allows us to completely compute the automorphism
group of certain graph products, for instance in the case where the underlying
graph is finite, connected, leafless and of girth at least . If in addition
the underlying graph does not contain separating stars, we can understand the
geometry of the automorphism groups of such graph products of groups further:
we show that such automorphism groups do not satisfy Kazhdan's property (T) and
are acylindrically hyperbolic. Applications to automorphism groups of graph
products of finite groups are also included. The approach in this article is
geometric and relies on the action of graph products of groups on certain
complexes with a particularly rich combinatorial geometry. The first such
complex is a particular Cayley graph of the graph product that has a
quasi-median geometry, a combinatorial geometry reminiscent of (but more
general than) CAT(0) cube complexes. The second (strongly related) complex used
is the Davis complex of the graph product, a CAT(0) cube complex that also has
a structure of right-angled building.Comment: 36 pages. The article subsumes and vastly generalises our preprint
arXiv:1803.07536. To appear in Proc. Lond. Math. So
Shuffling and Unshuffling
We consider various shuffling and unshuffling operations on languages and
words, and examine their closure properties. Although the main goal is to
provide some good and novel exercises and examples for undergraduate formal
language theory classes, we also provide some new results and some open
problems
Homologies of Algebraic Structures via Braidings and Quantum Shuffles
In this paper we construct "structural" pre-braidings characterizing
different algebraic structures: a rack, an associative algebra, a Leibniz
algebra and their representations. Some of these pre-braidings seem original.
On the other hand, we propose a general homology theory for pre-braided vector
spaces and braided modules, based on the quantum co-shuffle comultiplication.
Applied to the structural pre-braidings above, it gives a generalization and a
unification of many known homology theories. All the constructions are
categorified, resulting in particular in their super- and co-versions. Loday's
hyper-boundaries, as well as certain homology operations are efficiently
treated using the "shuffle" tools
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