A free subalgebra of the algebra of matroids

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.Comment: 19 pages, 3 figures. Accepted for publication in the European Journal of Combinatorics. This version incorporates a few minor corrections suggested by the publisher

The Free product of Matroids

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 provides a progressive lower bound for the number of isomorphism classes of matroids on an n-element set.Comment: 5 pages, 1 figure. Accepted for publication in the European Journal of Combinatorics. See also arXiv:math.CO/040902

A unique factorization theorem for matroids

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 product of matroids $M$ and $N$ is maximal with respect to the weak order among matroids having $M$ as a submatroid, with complementary contraction equal to $N$. Any minor of the free product of $M$ and $N$ is a free product of a repeated truncation of the corresponding minor of $M$ with a repeated Higgs lift of the corresponding minor of $N$. We characterize, in terms of their cyclic flats, matroids that are irreducible with respect to free product, and prove that the factorization of a matroid into a free product of irreducibles is unique up to isomorphism. We use these results to determine, for K a field of characteristic zero, the structure of the minor coalgebra $\cal C$ of a family of matroids $\cal M$ that is closed under formation of minors and free products: namely, $\cal C$ is cofree, cogenerated by the set of irreducible matroids belonging to $\cal M$.Comment: Dedicated to Denis Higgs. 25 pages, 3 figures. Submitted for publication in the Journal of Combinatorial Theory (A). See arXiv:math.CO/0409028 arXiv:math.CO/0409080 for preparatory work on this subjec

Permanents by Möbius inversion

AbstractMöbius inversion techniques developed by Rota [1] are used to justify Ryser's calculation [2, 3] of the permanent of a matrix, and to establish an alternative method of calculation (Proposition 4)

Invariant-theoretic methods in scene analysis and structural mechanics

We discuss possible applications of invariant theory to unsolved problems in applied geometry. In particular, we discuss projective conditions for correctness of plane drawings of 3-dimensional geometric forms, and for special mechanical behavior of bar-and-joint structures

On the generic rigidity of plane frameworks

Projet ICSLANo abstrac

Chirality and the isotopy classification of skew lines in projective 3-space

This article concerns isotopy invariants of finite configurations of skew lines in projective 3-space. We develop the theory of the chiral signature and of the Kauffman polynomial of a configuration. Invariance of the Kauffman polynomial under two types of diagram moves is shown by a direct combinatorial argument. The connection between a configuration and its plane projections is established in the context of oriented projective geometry, using oriented directrices and limit isotopies. Using a map to the Klein spherical model of projective space, we arrive at a linked-circle model of the configuration, and to a convenient ball-and-string model, the temari model. Linear graphs provide codes for chiral signatures, and permit the identification of those signatures which can be realized as simple stacked configurations of lines, which we call spindles. A catalogue of all unlabeled configurations of up to six lines, together with their Kauffman polynomials, is appended

Isostatic phase transition and instability in stiff granular materials

In this letter, structural rigidity concepts are used to understand the origin of instabilities in granular aggregates. It is shown that: a) The contact network of a noncohesive granular aggregate becomes exactly isostatic in the limit of large stiffness-to-load ratio. b) Isostaticity is responsible for the anomalously large susceptibility to perturbation of these systems, and c) The load-stress response function of granular materials is critical (power-law distributed) in the isostatic limit. Thus there is a phase transition in the limit of intinitely large stiffness, and the resulting isostatic phase is characterized by huge instability to perturbation.Comment: RevTeX, 4 pages w/eps figures [psfig]. To appear in Phys. Rev. Let