115 research outputs found

    An efficient sum of squares nonnegativity certificate for quaternary quartic

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    We show that for any non-negative 4-ary quartic form ff there exists a product of two non-negative quadrics qq and q′q' so that qq′fqq'f is a sum of squares (s.o.s.) of quartics. As a step towards deciding whether just one qq always suffices to make qfqf a s.o.s, we show that there exist non-s.o.s. non-negative 3-ary sextics ac−b2ac-b^2, with aa, bb, cc of degrees 2, 3, 4, respectively.Comment: LaTeX, 8 pages (significantly expanded w.r.t. version 1

    Implementing Brouwer's database of strongly regular graphs

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    Andries Brouwer maintains a public database of existence results for strongly regular graphs on n≤1300n\leq 1300 vertices. We implemented most of the infinite families of graphs listed there in the open-source software Sagemath, as well as provided constructions of the "sporadic" cases, to obtain a graph for each set of parameters with known examples. Besides providing a convenient way to verify these existence results from the actual graphs, it also extends the database to higher values of nn.Comment: 18 pages, LaTe

    The Extensions of the Generalized Quadrangle of Order (3, 9)

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    AbstractIt is shown that there is only one extension of GQ(3, 9) namely the one admitting the sporadic simple groupMcLas a flag-transitive automorphism group. The proof depends on a computer calculation

    The isometries of the cut, metric and hypermetric cones

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    We show that the symmetry groups of the cut cone Cut(n) and the metric cone Met(n) both consist of the isometries induced by the permutations on {1,...,n}; that is, Is(Cut(n))=Is(Met(n))=Sym(n) for n>4. For n=4 we have Is(Cut(4))=Is(Met(4))=Sym(3)xSym(4). This is then extended to cones containing the cuts as extreme rays and for which the triangle inequalities are facet-inducing. For instance, Is(Hyp(n))=Sym(n) for n>4, where Hyp(n) denotes the hypermetric cone.Comment: 8 pages, LaTeX, 2 postscript figure

    Implementing Hadamard Matrices in SageMath

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    Hadamard matrices are (−1,+1)(-1, +1) square matrices with mutually orthogonal rows. The Hadamard conjecture states that Hadamard matrices of order nn exist whenever nn is 11, 22, or a multiple of 44. However, no construction is known that works for all values of nn, and for some orders no Hadamard matrix has yet been found. Given the many practical applications of these matrices, it would be useful to have a way to easily check if a construction for a Hadamard matrix of order nn exists, and in case to create it. This project aimed to address this, by implementing constructions of Hadamard and skew Hadamard matrices to cover all known orders less than or equal to 10001000 in SageMath, an open-source mathematical software. Furthermore, we implemented some additional mathematical objects, such as complementary difference sets and T-sequences, which were not present in SageMath but are needed to construct Hadamard matrices. This also allows to verify the correctness of the results given in the literature; within the n≤1000n\leq 1000 range, just one order, 292292, of a skew Hadamard matrix claimed to have a known construction, required a fix.Comment: pdflatex+biber, 32 page
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