3,330 research outputs found

    General lower bounds on maximal determinants of binary matrices

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    We give general lower bounds on the maximal determinant of n×n {+1,-1}-matrices, both with and without the assumption of the Hadamard conjecture. Our bounds improve on earlier results of de Launey and Levin (2010) and, for certain congruence classes of

    Probabilistic lower bounds on maximal determinants of binary matrices

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    Let D(n){\mathcal D}(n) be the maximal determinant for n×nn \times n {±1}\{\pm 1\}-matrices, and R(n)=D(n)/nn/2\mathcal R(n) = {\mathcal D}(n)/n^{n/2} be the ratio of D(n){\mathcal D}(n) to the Hadamard upper bound. Using the probabilistic method, we prove new lower bounds on D(n){\mathcal D}(n) and R(n)\mathcal R(n) in terms of d=n−hd = n-h, where hh is the order of a Hadamard matrix and hh is maximal subject to h≀nh \le n. For example, R(n)>(πe/2)−d/2\mathcal R(n) > (\pi e/2)^{-d/2} if 1≀d≀31 \le d \le 3, and R(n)>(πe/2)−d/2(1−d2(π/(2h))1/2)\mathcal R(n) > (\pi e/2)^{-d/2}(1 - d^2(\pi/(2h))^{1/2}) if d>3d > 3. By a recent result of Livinskyi, d2/h1/2→0d^2/h^{1/2} \to 0 as n→∞n \to \infty, so the second bound is close to (πe/2)−d/2(\pi e/2)^{-d/2} for large nn. Previous lower bounds tended to zero as n→∞n \to \infty with dd fixed, except in the cases d∈{0,1}d \in \{0,1\}. For d≄2d \ge 2, our bounds are better for all sufficiently large nn. If the Hadamard conjecture is true, then d≀3d \le 3, so the first bound above shows that R(n)\mathcal R(n) is bounded below by a positive constant (πe/2)−3/2>0.1133(\pi e/2)^{-3/2} > 0.1133.Comment: 17 pages, 2 tables, 24 references. Shorter version of arXiv:1402.6817v4. Typos corrected in v2 and v3, new Lemma 7 in v4, updated references in v5, added Remark 2.8 and a reference in v6, updated references in v

    On minors of maximal determinant matrices

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    By an old result of Cohn (1965), a Hadamard matrix of order n has no proper Hadamard submatrices of order m > n/2. We generalise this result to maximal determinant submatrices of Hadamard matrices, and show that an interval of length asymptotically equal to n/2 is excluded from the allowable orders. We make a conjecture regarding a lower bound for sums of squares of minors of maximal determinant matrices, and give evidence in support of the conjecture. We give tables of the values taken by the minors of all maximal determinant matrices of orders up to and including 21 and make some observations on the data. Finally, we describe the algorithms that were used to compute the tables.Comment: 35 pages, 43 tables, added reference to Cohn in v

    Uniform determinantal representations

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    The problem of expressing a specific polynomial as the determinant of a square matrix of affine-linear forms arises from algebraic geometry, optimisation, complexity theory, and scientific computing. Motivated by recent developments in this last area, we introduce the notion of a uniform determinantal representation, not of a single polynomial but rather of all polynomials in a given number of variables and of a given maximal degree. We derive a lower bound on the size of the matrix, and present a construction achieving that lower bound up to a constant factor as the number of variables is fixed and the degree grows. This construction marks an improvement upon a recent construction due to Plestenjak-Hochstenbach, and we investigate the performance of new representations in their root-finding technique for bivariate systems. Furthermore, we relate uniform determinantal representations to vector spaces of singular matrices, and we conclude with a number of future research directions.Comment: 23 pages, 3 figures, 4 table

    Determinants of (–1,1)-matrices of the skew-symmetric type: a cocyclic approach

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    An n by n skew-symmetric type (−1, 1)-matrix K = [ki,j ] has 1’s on the main diagonal and ±1’s elsewhere with ki,j = −kj,i. The largest possible determinant of such a matrix K is an interesting problem. The literature is extensive for n 0 mod 4 (skew- Hadamard matrices), but for n 2 mod 4 there are few results known for this question. In this paper we approach this problem constructing cocyclic matrices over the dihedral group of 2t elements, for t odd, which are equivalent to (−1, 1)-matrices of skew type. Some explicit calculations have been done up to t = 11. To our knowledge, the upper bounds on the maximal determinant in orders 18 and 22 have been improved.Junta de AndalucĂ­a FQM-01

    Maximal determinants and saturated D-optimal designs of orders 19 and 37

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    A saturated D-optimal design is a {+1,-1} square matrix of given order with maximal determinant. We search for saturated D-optimal designs of orders 19 and 37, and find that known matrices due to Smith, Cohn, Orrick and Solomon are optimal. For order 19 we find all inequivalent saturated D-optimal designs with maximal determinant, 2^30 x 7^2 x 17, and confirm that the three known designs comprise a complete set. For order 37 we prove that the maximal determinant is 2^39 x 3^36, and find a sample of inequivalent saturated D-optimal designs. Our method is an extension of that used by Orrick to resolve the previously smallest unknown order of 15; and by Chadjipantelis, Kounias and Moyssiadis to resolve orders 17 and 21. The method is a two-step computation which first searches for candidate Gram matrices and then attempts to decompose them. Using a similar method, we also find the complete spectrum of determinant values for {+1,-1} matrices of order 13.Comment: 28 pages, 4 figure

    Geometric lower bounds for generalized ranks

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    We revisit a geometric lower bound for Waring rank of polynomials (symmetric rank of symmetric tensors) of Landsberg and Teitler and generalize it to a lower bound for rank with respect to arbitrary varieties, improving the bound given by the "non-Abelian" catalecticants recently introduced by Landsberg and Ottaviani. This is applied to give lower bounds for ranks of multihomogeneous polynomials (partially symmetric tensors); a special case is the simultaneous Waring decomposition problem for a linear system of polynomials. We generalize the classical Apolarity Lemma to multihomogeneous polynomials and give some more general statements. Finally we revisit the lower bound of Ranestad and Schreyer, and again generalize it to multihomogeneous polynomials and some more general settings.Comment: 43 pages. v2: minor change
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