83 research outputs found
Axioms for consensus functions on the n-cube
An elementary general result is proved that allows for simple
characterizations of well-known location/consensus functions (median, mean and
center) on the n-cube. In addition, alternate new characterizations are given
for the median and anti-median functions on the n-cube.Comment: 12 page
Purely non-atomic weak L^p spaces
Let \msp be a purely non-atomic measure space, and let . If
\weakLp\msp is isomorphic, as a Banach space, to \weakLp\mspp for some
purely atomic measure space \mspp, then there is a measurable partition
such that
is countably
generated and -finite, and that or for every
measurable . In particular, \weakLp\msp is
isomorphic to
Local Testing for Membership in Lattices
Motivated by the structural analogies between point lattices and linear error-correcting codes, and by the mature theory on locally testable codes, we initiate a systematic study of local testing for membership in lattices. Testing membership in lattices is also motivated in practice, by applications to integer programming, error detection in lattice-based communication, and cryptography. Apart from establishing the conceptual foundations of lattice testing, our results include the following: 1. We demonstrate upper and lower bounds on the query complexity of local testing for the well-known family of code formula lattices. Furthermore, we instantiate our results with code formula lattices constructed from Reed-Muller codes, and obtain nearly-tight bounds. 2. We show that in order to achieve low query complexity, it is sufficient to design one-sided non-adaptive canonical tests. This result is akin to, and based on an analogous result for error-correcting codes due to Ben-Sasson et al. (SIAM J. Computing 35(1) pp1-21)
Subspace arrangements defined by products of linear forms
We consider the vanishing ideal of an arrangement of linear subspaces in a
vector space and investigate when this ideal can be generated by products of
linear forms. We introduce a combinatorial construction (blocker duality) which
yields such generators in cases with a lot of combinatorial structure, and we
present the examples that motivated our work. We give a construction which
produces all elements of this type in the vanishing ideal of the arrangement.
This leads to an algorithm for deciding if the ideal is generated by products
of linear forms. We also consider generic arrangements of points in
and lines in Comment: 20 pages; AMSLatex; v.2: proof of Proposition 5.1.3 corrected; proof
of Proposition 5.1.6 shortened; references added, v.3: minor corrections;
final version; to appear in the Journal of the London Mathematical Societ
The space of subgroups of an abelian group
We carry out the Cantor-Bendixson analysis of the space of all subgroups of
any countable abelian group and we deduce a complete classification of such
spaces up to homeomorphism.Comment: 22 pages, no figur
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