179,816 research outputs found

    Hadamard partitioned difference families and their descendants

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    If DD is a (4u2,2u2−u,u2−u)(4u^2,2u^2-u,u^2-u) Hadamard difference set (HDS) in GG, then {G,G∖D}\{G,G\setminus D\} is clearly a (4u2,[2u2−u,2u2+u],2u2)(4u^2,[2u^2-u,2u^2+u],2u^2) partitioned difference family (PDF). Any (v,K,λ)(v,K,\lambda)-PDF will be said of Hadamard-type if v=2λv=2\lambda as the one above. We present a doubling construction which, starting from any such PDF, leads to an infinite class of PDFs. As a special consequence, we get a PDF in a group of order 4u2(2n+1)4u^2(2n+1) and three block-sizes 4u2−2u4u^2-2u, 4u24u^2 and 4u2+2u4u^2+2u, whenever we have a (4u2,2u2−u,u2−u)(4u^2,2u^2-u,u^2-u)-HDS and the maximal prime power divisors of 2n+12n+1 are all greater than 4u2+2u4u^2+2u

    Totally frustrated states in the chromatic theory of gain graphs

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    We generalize proper coloring of gain graphs to totally frustrated states, where each vertex takes a value in a set of `qualities' or `spins' that is permuted by the gain group. (An example is the Potts model.) The number of totally frustrated states satisfies the usual deletion-contraction law but is matroidal only for standard coloring, where the group action is trivial or nearly regular. One can generalize chromatic polynomials by constructing spin sets with repeated transitive components.Comment: 14 pages, 2 figure

    Multiclass Total Variation Clustering

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    Ideas from the image processing literature have recently motivated a new set of clustering algorithms that rely on the concept of total variation. While these algorithms perform well for bi-partitioning tasks, their recursive extensions yield unimpressive results for multiclass clustering tasks. This paper presents a general framework for multiclass total variation clustering that does not rely on recursion. The results greatly outperform previous total variation algorithms and compare well with state-of-the-art NMF approaches
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