719 research outputs found
Proto-exact categories of matroids, Hall algebras, and K-theory
This paper examines the category of pointed matroids
and strong maps from the point of view of Hall algebras. We show that
has the structure of a finitary proto-exact category -
a non-additive generalization of exact category due to Dyckerhoff-Kapranov. We
define the algebraic K-theory of
via the Waldhausen construction, and show that it is
non-trivial, by exhibiting injections from the stable homotopy groups of spheres for
all . Finally, we show that the Hall algebra of is
a Hopf algebra dual to Schmitt's matroid-minor Hopf algebra.Comment: 29 page
Renormalization : A number theoretical model
We analyse the Dirichlet convolution ring of arithmetic number theoretic
functions. It turns out to fail to be a Hopf algebra on the diagonal, due to
the lack of complete multiplicativity of the product and coproduct. A related
Hopf algebra can be established, which however overcounts the diagonal. We
argue that the mechanism of renormalization in quantum field theory is modelled
after the same principle. Singularities hence arise as a (now continuously
indexed) overcounting on the diagonals. Renormalization is given by the map
from the auxiliary Hopf algebra to the weaker multiplicative structure, called
Hopf gebra, rescaling the diagonals.Comment: 15 pages, extended version of talks delivered at SLC55 Bertinoro,Sep
2005, and the Bob Delbourgo QFT Fest in Hobart, Dec 200
Recipe theorem for the Tutte polynomial for matroids, renormalization group-like approach
Using a quantum field theory renormalization group-like differential
equation, we give a new proof of the recipe theorem for the Tutte polynomial
for matroids. The solution of such an equation is in fact given by some
appropriate characters of the Hopf algebra of isomorphic classes of matroids,
characters which are then related to the Tutte polynomial for matroids. This
Hopf algebraic approach also allows to prove, in a new way, a matroid Tutte
polynomial convolution formula appearing in W. Kook {\it et. al., J. Comb.
Series} {\bf B 76} (1999).Comment: 14 pages, 3 figure
Cumulants, Spreadability and the Campbell-Baker-Hausdorff Series
We define spreadability systems as a generalization of exchangeability
systems in order to unify various notions of independence and cumulants known
in noncommutative probability. In particular, our theory covers monotone
independence and monotone cumulants which do not satisfy exchangeability. To
this end we study generalized zeta and M\"obius functions in the context of the
incidence algebra of the semilattice of ordered set partitions and prove an
appropriate variant of Faa di Bruno's theorem. With the aid of this machinery
we show that our cumulants cover most of the previously known cumulants. Due to
noncommutativity of independence the behaviour of these cumulants with respect
to independent random variables is more complicated than in the exchangeable
case and the appearance of Goldberg coefficients exhibits the role of the
Campbell-Baker-Hausdorff series in this context. In a final section we exhibit
an interpretation of the Campbell-Baker-Hausdorff series as a sum of cumulants
in a particular spreadability system, thus providing a new derivation of the
Goldberg coefficients.Comment: some minor corrections, 48 page
A New Family of Diagonal Ade-Related Scattering Theories
We propose the factorizable S-matrices of the massive excitations of the
non-unitary minimal model perturbed by the operator .
The massive excitations and the whole set of two particle S-matrices of the
theory is simply related to the unitary minimal scattering theory. The
counting argument and the Thermodynamic Bethe Ansatz (TBA) are applied to this
scattering theory in order to support this interpretation. Generalizing this
result, we describe a new family of NON UNITARY and DIAGONAL -related
scattering theories. A further generalization suggests the magnonic TBA for a
large class of non-unitary \G\otimes\G/\G coset models
(\G=A_{odd},D_n,E_{6,7,8}) perturbed by , described by
non-diagonal S-matrices.Comment: 13 pages, Latex (no macros), DFUB-92-12, DFTT/30-9
Hypertree posets and hooked partitions
We adapt here the computation of characters on incidence Hopf algebras
introduced by W. Schmitt in the 1990s to a family mixing bounded and unbounded
posets. We then apply our results to the family of hypertree posets and
partition posets. As a consequence, we obtain some enumerative formulas and a
new proof for the computation of the Moebius numbers of the hypertree posets.
Moreover, we compute the coproduct of the incidence Hopf algebra and recover a
known formula for the number of hypertrees with fixed valency set and edge
sizes set.Comment: 18 page
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