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Of the various theoretical frameworks [2, 5, 8, lo] that have been proposed for the purpose of explaining in a unified way the combinatorial significance of algebraic operations on formal power series, regarded as generating functions, the most general and flexible, if not always the most transparent, has been and remains the notion of incidence algebra as… (More)
This paper is an initial inquiry into the structure of the Hopf algebra of matroids with restriction-contraction coproduct. Using a family of matroids introduced by Crapo in 1965, we show that the subalgebra generated by a single point and a single loop in the dual of this Hopf algebra is free.
We introduce a noncommutative binary operation on matroids, called free product. We show that this operation respects matroid duality, and has the property that, given only the cardinalities, an ordered pair of matroids may be recovered, up to isomorphism, from its free product. We use these results to give a short proof of Welsh's 1969 conjecture, which… (More)
We study the combinatorial, algebraic and geometric properties of the free product operation on matroids. After giving cryptomorphic definitions of free product in terms of independent sets, bases, circuits, closure, flats and rank function, we show that free product, which is a noncommutative operation, is associative and respects matroid duality. The free… (More)
The Hopf algebra of renormalisation in quantum field theory is described at a general level. The products of fields at a point are assumed to form a bialgebra B and renormalisation endows T (T (B) +), the double tensor algebra of B, with the structure of a noncommutative bialgebra. When the bialgebra B is commutative, renormalisation turns S(S(B) +), the… (More)
A theorem about closure operators on partially ordered sets is given, and applications to the counting of colorings of graphs according to partition type are derived.
The Hopf algebra of renormalization in quantum field theory is described at a general level. The products of fields at a point are assumed to form a bialgebra B and renormalization endows T (T (B) +), the double tensor algebra of B, with the structure of a noncommutative bialgebra. When the bialgebra B is commutative, renormalization turns S(S(B) +), the… (More)
We introduce the matroid-minor coalgebra C, which has labeled matroids as distinguished basis, and coproduct given by splitting a matroid into a submatroid and complementary contraction all possible ways. We introduce two new bases for C; the first of these is is related to the distinguished basis by Möbius inversion over the rank-preserving weak order on… (More)
The critical processing parameters affecting average particle size, particle size distribution, yield, and level of residual carrier solvent using the supercritical anti-solvent method (SAS) were identified. Carbon dioxide was used as the supercritical fluid. Methylprednisolone acetate was used as the model solute in tetrahydrofuran. Parameters examined… (More)