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REPRESENTATIONS OF QUANTUM PERMUTATION ALGEBRAS
, 2009
"... We develop a combinatorial approach to the quantum permutation algebras, as Hopf images of representations of type π: As(n) → B(H). We discuss several general problems, including the commutativity and cocommutativity ones, the existence of tensor product or free wreath product decompositions, and ..."
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Cited by 11 (8 self)
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We develop a combinatorial approach to the quantum permutation algebras, as Hopf images of representations of type π: As(n) → B(H). We discuss several general problems, including the commutativity and cocommutativity ones, the existence of tensor product or free wreath product decompositions, and the Tannakian aspects of the construction. The main motivation comes from the quantum invariants of the complex Hadamard matrices: we show here that, under suitable regularity assumptions, the computations can be performed up to n = 6.
ON ORTHOGONAL MATRICES MAXIMIZING THE 1NORM
, 901
"... Abstract. For U ∈ O(N) we have U1 ≤ N √ N, with equality if and only if U = H / √ N, with H Hadamard matrix. Motivated by this remark, we discuss in this paper the algebraic and analytic aspects of the computation of the maximum of the 1norm on O(N). The main problem is to compute the kth mom ..."
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Cited by 9 (9 self)
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Abstract. For U ∈ O(N) we have U1 ≤ N √ N, with equality if and only if U = H / √ N, with H Hadamard matrix. Motivated by this remark, we discuss in this paper the algebraic and analytic aspects of the computation of the maximum of the 1norm on O(N). The main problem is to compute the kth moment of the 1norm, with k → ∞, and we present a number of general comments in this direction.
On the energy of (0, 1)matrices
, 2008
"... The energy of a matrix is the sum of its singular values. We study the energy of (0, 1)matrices and present two methods for constructing balanced incomplete block designs whose incidence matrices have the maximum possible energy amongst the family of all (0, 1)matrices of given order and total numb ..."
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Cited by 2 (0 self)
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The energy of a matrix is the sum of its singular values. We study the energy of (0, 1)matrices and present two methods for constructing balanced incomplete block designs whose incidence matrices have the maximum possible energy amongst the family of all (0, 1)matrices of given order and total number of ones. We also find a new upper bound for the energy of (p, q)bipartite graphs.
Almost Hadamard matrices: the case of arbitrary exponents
 Discrete Appl. Math
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The Impact of Number Theory and Computer Aided Mathematics on Solving the Hadamard Matrix Conjecture
"... In memory of Alf van der Poorten The Hadamard Conjecture has been studied since the pioneering paper of J. J. Sylvester, “Thoughts on inverse orthogonal matrices, simultaneous sign successions, tessellated pavements in two or more colours, with applications to Newtons rule, ornamental tile work and ..."
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In memory of Alf van der Poorten The Hadamard Conjecture has been studied since the pioneering paper of J. J. Sylvester, “Thoughts on inverse orthogonal matrices, simultaneous sign successions, tessellated pavements in two or more colours, with applications to Newtons rule, ornamental tile work and the theory of numbers, Phil Mag, (1867) 461–475 first appeared. We review the importance of primes on those occasions that the conjecture is confirmed. We survey the results of some computer aided construction algorithms for Hadamard matrices.
Abstract
, 2001
"... We investigate the computational complexity of the task of detecting dense regions of an unknown distribution from unlabeled samples of this distribution. We introduce a formal learning model for this task that uses a hypothesis class as its ‘antioverfitting ’ mechanism. The learning task in our m ..."
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We investigate the computational complexity of the task of detecting dense regions of an unknown distribution from unlabeled samples of this distribution. We introduce a formal learning model for this task that uses a hypothesis class as its ‘antioverfitting ’ mechanism. The learning task in our model can be reduced to a combinatorial optimization problem. We can show that for some constants, depending on the hypothesis class, these problems are N Phard to approximate to within these constant factors. We go on and introduce a new criterion for the success of approximate optimization geometric problems. The new criterion requires that the algorithm competes with hypotheses only on the points that are separated by some margin µ from their boundaries. Quite surprisingly, we discover that for each of the two hypothesis classes that we investigate, there is a ‘critical value ’ of the margin parameter µ. For any value below the critical value the problems are N Phard to approximate, while, once this value is exceeded, the problems become polytime solvable. 1