5,802 research outputs found
Countable Random Sets: Uniqueness in Law and Constructiveness
The first part of this article deals with theorems on uniqueness in law for
\sigma-finite and constructive countable random sets, which in contrast to the
usual assumptions may have points of accumulation. We discuss and compare two
approaches on uniqueness theorems: First, the study of generators for
\sigma-fields used in this context and, secondly, the analysis of hitting
functions. The last section of this paper deals with the notion of
constructiveness. We will prove a measurable selection theorem and a
decomposition theorem for constructive countable random sets, and study
constructive countable random sets with independent increments.Comment: Published in Journal of Theoretical Probability
(http://www.springerlink.com/content/0894-9840/). The final publication is
available at http://www.springerlink.co
On constructions preserving the asymptotic topology of metric spaces
We prove that graph products constructed over infinite graphs with bounded
clique number preserve finite asymptotic dimension. We also study the extent to
which Dranishnikov's property C, and Dranishnikov and Zarichnyi's straight
finite decomposition complexity are preserved by constructions such as unions,
free products, and group extensions.Comment: 13 pages, accepted for publication in NC Journal of Mathematics and
Statistic
The asymptotic dimension of box spaces of virtually nilpotent groups
We show that every box space of a virtually nilpotent group has asymptotic
dimension equal to the Hirsch length of that group
Generating all subsets of a finite set with disjoint unions
If X is an n-element set, we call a family G of subsets of X a k-generator
for X if every subset of X can be expressed as a union of at most k disjoint
sets in G. Frein, Leveque and Sebo conjectured that for n > 2k, the smallest
k-generators for X are obtained by taking a partition of X into classes of
sizes as equal as possible, and taking the union of the power-sets of the
classes. We prove this conjecture for all sufficiently large n when k = 2, and
for n a sufficiently large multiple of k when k > 2.Comment: Final version, with some additional explanations added in the proof
Interpretations of Presburger Arithmetic in Itself
Presburger arithmetic PrA is the true theory of natural numbers with
addition. We study interpretations of PrA in itself. We prove that all
one-dimensional self-interpretations are definably isomorphic to the identity
self-interpretation. In order to prove the results we show that all linear
orders that are interpretable in (N,+) are scattered orders with the finite
Hausdorff rank and that the ranks are bounded in terms of the dimension of the
respective interpretations. From our result about self-interpretations of PrA
it follows that PrA isn't one-dimensionally interpretable in any of its finite
subtheories. We note that the latter was conjectured by A. Visser.Comment: Published in proceedings of LFCS 201
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