183 research outputs found
The topology of the external activity complex of a matroid
We prove that the external activity complex of a matroid
is shellable. In fact, we show that every linear extension of LasVergnas's
external/internal order on provides a shelling of
. We also show that every linear extension of LasVergnas's
internal order on provides a shelling of the independence complex
. As a corollary, and have the same -vector.
We prove that, after removing its cone points, the external activity complex is
contractible if contains as a minor, and a sphere otherwise.Comment: Comments are welcom
The topology of the external activity complex of a matroid
International audienceWe prove that the external activity complex Act<(M) of a matroid is shellable. In fact, we show that every linear extension of Las Vergnas's external/internal order <ext/int on M provides a shelling of Act<(M). We also show that every linear extension of Las Vergnas's internal order <int on M provides a shelling of the independence complex IN(M). As a corollary, Act<(M) and M have the same h-vector. We prove that, after removing its cone points, the external activity complex is contractible if M contains U3,1 as a minor, and a sphere otherwise
A Tutte polynomial for toric arrangements
We introduce a multiplicity Tutte polynomial M(x,y), with applications to
zonotopes and toric arrangements. We prove that M(x,y) satisfies a
deletion-restriction recurrence and has positive coefficients. The
characteristic polynomial and the Poincare' polynomial of a toric arrangement
are shown to be specializations of the associated polynomial M(x,y), likewise
the corresponding polynomials for a hyperplane arrangement are specializations
of the ordinary Tutte polynomial. Furthermore, M(1,y) is the Hilbert series of
the related discrete Dahmen-Micchelli space, while M(x,1) computes the volume
and the number of integral points of the associated zonotope.Comment: Final version, to appear on Transactions AMS. 28 pages, 4 picture
Semi-inverted linear spaces and an analogue of the broken circuit complex
The image of a linear space under inversion of some coordinates is an affine
variety whose structure is governed by an underlying hyperplane arrangement. In
this paper, we generalize work by Proudfoot and Speyer to show that circuit
polynomials form a universal Groebner basis for the ideal of polynomials
vanishing on this variety. The proof relies on degenerations to the
Stanley-Reisner ideal of a simplicial complex determined by the underlying
matroid. If the linear space is real, then the semi-inverted linear space is
also an example of a hyperbolic variety, meaning that all of its intersection
points with a large family of linear spaces are real.Comment: 16 pages, 1 figure, minor revisions and added connections to the
external activity complex of a matroi
Valuations for matroid polytope subdivisions
We prove that the ranks of the subsets and the activities of the bases of a
matroid define valuations for the subdivisions of a matroid polytope into
smaller matroid polytopes.Comment: 19 pages. 2 figures; added section 6 + other correction
Rigidity and flexibility of biological networks
The network approach became a widely used tool to understand the behaviour of
complex systems in the last decade. We start from a short description of
structural rigidity theory. A detailed account on the combinatorial rigidity
analysis of protein structures, as well as local flexibility measures of
proteins and their applications in explaining allostery and thermostability is
given. We also briefly discuss the network aspects of cytoskeletal tensegrity.
Finally, we show the importance of the balance between functional flexibility
and rigidity in protein-protein interaction, metabolic, gene regulatory and
neuronal networks. Our summary raises the possibility that the concepts of
flexibility and rigidity can be generalized to all networks.Comment: 21 pages, 4 figures, 1 tabl
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