9,005 research outputs found

    Semantics for a Quantum Programming Language by Operator Algebras

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    This paper presents a novel semantics for a quantum programming language by operator algebras, which are known to give a formulation for quantum theory that is alternative to the one by Hilbert spaces. We show that the opposite category of the category of W*-algebras and normal completely positive subunital maps is an elementary quantum flow chart category in the sense of Selinger. As a consequence, it gives a denotational semantics for Selinger's first-order functional quantum programming language QPL. The use of operator algebras allows us to accommodate infinite structures and to handle classical and quantum computations in a unified way.Comment: In Proceedings QPL 2014, arXiv:1412.810

    Set optimization - a rather short introduction

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    Recent developments in set optimization are surveyed and extended including various set relations as well as fundamental constructions of a convex analysis for set- and vector-valued functions, and duality for set optimization problems. Extensive sections with bibliographical comments summarize the state of the art. Applications to vector optimization and financial risk measures are discussed along with algorithmic approaches to set optimization problems

    Improved bounds for the crossing numbers of K_m,n and K_n

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    It has been long--conjectured that the crossing number cr(K_m,n) of the complete bipartite graph K_m,n equals the Zarankiewicz Number Z(m,n):= floor((m-1)/2) floor(m/2) floor((n-1)/2) floor(n/2). Another long--standing conjecture states that the crossing number cr(K_n) of the complete graph K_n equals Z(n):= floor(n/2) floor((n-1)/2) floor((n-2)/2) floor((n-3)/2)/4. In this paper we show the following improved bounds on the asymptotic ratios of these crossing numbers and their conjectured values: (i) for each fixed m >= 9, lim_{n->infty} cr(K_m,n)/Z(m,n) >= 0.83m/(m-1); (ii) lim_{n->infty} cr(K_n,n)/Z(n,n) >= 0.83; and (iii) lim_{n->infty} cr(K_n)/Z(n) >= 0.83. The previous best known lower bounds were 0.8m/(m-1), 0.8, and 0.8, respectively. These improved bounds are obtained as a consequence of the new bound cr(K_{7,n}) >= 2.1796n^2 - 4.5n. To obtain this improved lower bound for cr(K_{7,n}), we use some elementary topological facts on drawings of K_{2,7} to set up a quadratic program on 6! variables whose minimum p satisfies cr(K_{7,n}) >= (p/2)n^2 - 4.5n, and then use state--of--the--art quadratic optimization techniques combined with a bit of invariant theory of permutation groups to show that p >= 4.3593.Comment: LaTeX, 18 pages, 2 figure

    Computing Invariants of Simplicial Manifolds

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    This is a survey of known algorithms in algebraic topology with a focus on finite simplicial complexes and, in particular, simplicial manifolds. Wherever possible an elementary approach is chosen. This way the text may also serve as a condensed but very basic introduction to the algebraic topology of simplicial manifolds. This text will appear as a chapter in the forthcoming book "Triangulated Manifolds with Few Vertices" by Frank H. Lutz.Comment: 13 pages, 3 figure
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