7,532 research outputs found
Uncountable sets of unit vectors that are separated by more than 1
Let be a Banach space. We study the circumstances under which there
exists an uncountable set of unit vectors such that
for distinct . We prove that such a set exists
if is quasi-reflexive and non-separable; if is additionally
super-reflexive then one can have for some
that depends only on . If is a non-metrisable compact,
Hausdorff space, then the unit sphere of also contains such a subset;
if moreover is perfectly normal, then one can find such a set with
cardinality equal to the density of ; this solves a problem left open by S.
K. Mercourakis and G. Vassiliadis.Comment: to appear in Studia Mat
Robust randomized matchings
The following game is played on a weighted graph: Alice selects a matching
and Bob selects a number . Alice's payoff is the ratio of the weight of
the heaviest edges of to the maximum weight of a matching of size at
most . If guarantees a payoff of at least then it is called
-robust. In 2002, Hassin and Rubinstein gave an algorithm that returns
a -robust matching, which is best possible.
We show that Alice can improve her payoff to 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
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 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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