7,532 research outputs found

    Uncountable sets of unit vectors that are separated by more than 1

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    Let XX be a Banach space. We study the circumstances under which there exists an uncountable set AX\mathcal A\subset X of unit vectors such that xy>1\|x-y\|>1 for distinct x,yAx,y\in \mathcal A. We prove that such a set exists if XX is quasi-reflexive and non-separable; if XX is additionally super-reflexive then one can have xy1+ε\|x-y\|\geqslant 1+\varepsilon for some ε>0\varepsilon>0 that depends only on XX. If KK is a non-metrisable compact, Hausdorff space, then the unit sphere of X=C(K)X=C(K) also contains such a subset; if moreover KK is perfectly normal, then one can find such a set with cardinality equal to the density of XX; this solves a problem left open by S. K. Mercourakis and G. Vassiliadis.Comment: to appear in Studia Mat

    Robust randomized matchings

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    The following game is played on a weighted graph: Alice selects a matching MM and Bob selects a number kk. Alice's payoff is the ratio of the weight of the kk heaviest edges of MM to the maximum weight of a matching of size at most kk. If MM guarantees a payoff of at least α\alpha then it is called α\alpha-robust. In 2002, Hassin and Rubinstein gave an algorithm that returns a 1/21/\sqrt{2}-robust matching, which is best possible. We show that Alice can improve her payoff to 1/ln(4)1/\ln(4) by playing a randomized strategy. This result extends to a very general class of independence systems that includes matroid intersection, b-matchings, and strong 2-exchange systems. It also implies an improved approximation factor for a stochastic optimization variant known as the maximum priority matching problem and translates to an asymptotic robustness guarantee for deterministic matchings, in which Bob can only select numbers larger than a given constant. Moreover, we give a new LP-based proof of Hassin and Rubinstein's bound

    A Fourier analytic approach to the problem of mutually unbiased bases

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    We give an entirely new approach to the problem of mutually unbiased bases (MUBs), based on a Fourier analytic technique in additive combinatorics. The method provides a short and elegant generalization of the fact that there are at most d+1d+1 MUBs in \Co^d. It may also yield a proof that no complete system of MUBs exists in some composite dimensions -- a long standing open problem.Comment: 11 page
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