11,698 research outputs found
Chains, Antichains, and Complements in Infinite Partition Lattices
We consider the partition lattice on any set of transfinite
cardinality and properties of whose analogues do not hold
for finite cardinalities. Assuming the Axiom of Choice we prove: (I) the
cardinality of any maximal well-ordered chain is always exactly ; (II)
there are maximal chains in of cardinality ; (III) if,
for every cardinal , we have , there
exists a maximal chain of cardinality (but ) in
; (IV) every non-trivial maximal antichain in has
cardinality between and , and these bounds are realized.
Moreover we can construct maximal antichains of cardinality for any ; (V) all cardinals of the form
with occur as the number of
complements to some partition , and only these
cardinalities appear. Moreover, we give a direct formula for the number of
complements to a given partition; (VI) Under the Generalized Continuum
Hypothesis, the cardinalities of maximal chains, maximal antichains, and
numbers of complements are fully determined, and we provide a complete
characterization.Comment: 24 pages, 2 figures. Submitted to Algebra Universalis on 27/11/201
An upper bound on the size of diamond-free families of sets
Let be the maximum size of a family of subsets of
not containing as a (weak) subposet. The diamond poset,
denoted , is defined on four elements with the relations
and . has been studied for many posets; one of the
major open problems is determining .
Studying the average number of sets from a family of subsets of on a
maximal chain in the Boolean lattice has been a fruitful method. We
use a partitioning of the maximal chains and introduce an induction method to
show that , improving on the earlier bound of
by Kramer,
Martin and Young.Comment: Accepted by JCTA. Writing is improved based on the suggestions of
referee
On the Combinatorics of Locally Repairable Codes via Matroid Theory
This paper provides a link between matroid theory and locally repairable
codes (LRCs) that are either linear or more generally almost affine. Using this
link, new results on both LRCs and matroid theory are derived. The parameters
of LRCs are generalized to matroids, and the matroid
analogue of the generalized Singleton bound in [P. Gopalan et al., "On the
locality of codeword symbols," IEEE Trans. Inf. Theory] for linear LRCs is
given for matroids. It is shown that the given bound is not tight for certain
classes of parameters, implying a nonexistence result for the corresponding
locally repairable almost affine codes, that are coined perfect in this paper.
Constructions of classes of matroids with a large span of the parameters
and the corresponding local repair sets are given. Using
these matroid constructions, new LRCs are constructed with prescribed
parameters. The existence results on linear LRCs and the nonexistence results
on almost affine LRCs given in this paper strengthen the nonexistence and
existence results on perfect linear LRCs given in [W. Song et al., "Optimal
locally repairable codes," IEEE J. Sel. Areas Comm.].Comment: 48 pages. Submitted for publication. In this version: The text has
been edited to improve the readability. Parameter d for matroids is now
defined by the use of the rank function instead of the dual matroid. Typos
are corrected. Section III is divided into two parts, and some numberings of
theorems etc. have been change
Small Mass Implies Uniqueness of Gibbs States of a Quantum Crystal
A model of interacting quantum particles performing one-dimensional anharmonic oscillations around their equilibrium positions which form a lattice ℤ d is considered. For this model, it is proved that the set of tempered Euclidean Gibbs measures is a singleton provided the particle mass is less than a certain bound m *, which is independent of the temperature β−1. This settles a problem that was open for a long time and is an essential improvement of a similar result proved before by the same authors [5], where the bound m * depended on β in such a way that as ${{\beta \rightarrow +\infty}}
Bounds on Binary Locally Repairable Codes Tolerating Multiple Erasures
Recently, locally repairable codes has gained significant interest for their
potential applications in distributed storage systems. However, most
constructions in existence are over fields with size that grows with the number
of servers, which makes the systems computationally expensive and difficult to
maintain. Here, we study linear locally repairable codes over the binary field,
tolerating multiple local erasures. We derive bounds on the minimum distance on
such codes, and give examples of LRCs achieving these bounds. Our main
technical tools come from matroid theory, and as a byproduct of our proofs, we
show that the lattice of cyclic flats of a simple binary matroid is atomic.Comment: 9 pages, 1 figure. Parts of this paper were presented at IZS 2018.
This extended arxiv version includes corrected versions of Theorem 1.4 and
Proposition 6 that appeared in the IZS 2018 proceeding
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