158,386 research outputs found

    A few weight systems arising from intersection graphs

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    We show that the adjacency matrices of the intersection graphs of chord diagrams satisfy the 2-term relations of Bar-Natan and Garoufalides [bg], and hence give rise to weight systems. Among these weight systems are those associated with the Conway and HOMFLYPT polynomials. We extend these ideas to looking at a space of {\it marked} chord diagrams modulo an extended set of 2-term relations, define a set of generators for this space, and again derive weight systems from the adjacency matrices of the (marked) intersection graphs. Among these weight systems are those associated with the Kauffman polynomial.Comment: 20 pages. This version has been substantially revised. The results are largely the same, but the proofs have been reconceptualized in terms of various 2-term relations on chord diagrams and graph

    Semidefinite descriptions of the convex hull of rotation matrices

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    We study the convex hull of SO(n)SO(n), thought of as the set of n×nn\times n orthogonal matrices with unit determinant, from the point of view of semidefinite programming. We show that the convex hull of SO(n)SO(n) is doubly spectrahedral, i.e. both it and its polar have a description as the intersection of a cone of positive semidefinite matrices with an affine subspace. Our spectrahedral representations are explicit, and are of minimum size, in the sense that there are no smaller spectrahedral representations of these convex bodies.Comment: 29 pages, 1 figur

    A note on generalizations of strict diagonal dominance for real matrices

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    AbstractWe investigate classes of real square matrices possessing some weakened from of strict diagonal dominance of a real matrix whose diagonal entries are all positive. The intersection of each one of these classes with the set of all real matrices, with nonpositive off-diagonal elements, coincides with the set of all nonsingular M- matrices
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