117 research outputs found
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 and
is maximal with respect to the weak order among matroids having as a
submatroid, with complementary contraction equal to . Any minor of the free
product of and is a free product of a repeated truncation of the
corresponding minor of with a repeated Higgs lift of the corresponding
minor of . 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 of a family of matroids that
is closed under formation of minors and free products: namely, is
cofree, cogenerated by the set of irreducible matroids belonging to .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
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
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