223 research outputs found
On a Mutation Problem for Oriented Matroids
AbstractFor uniform oriented matroids M with n elements, there is in the realizable case a sharp lower bound Lr(n) for the number mut(M) of mutations of M: Lr(n) =n≤mut(M), see Shannon [17]. Finding a sharp lower bound L(n) ≤mut(M) in the non-realizable case is an open problem for rank d≥ 4. Las Vergnas [9] conjectured that 1 ≤L(n). We study in this article the rank 4 case. Richter-Gebert [11] showed thatL (4 k) ≤ 3 k+ 1 for k≥ 2. We confirm Las Vergnas’ conjecture for n< 13. We show that L(7k+c) ≤ 5 k+c for all integersk≥ 0 and c≥ 4, and we provide a 17 element example with a mutation free element
Oriented Matroids -- Combinatorial Structures Underlying Loop Quantum Gravity
We analyze combinatorial structures which play a central role in determining
spectral properties of the volume operator in loop quantum gravity (LQG). These
structures encode geometrical information of the embedding of arbitrary valence
vertices of a graph in 3-dimensional Riemannian space, and can be represented
by sign strings containing relative orientations of embedded edges. We
demonstrate that these signature factors are a special representation of the
general mathematical concept of an oriented matroid. Moreover, we show that
oriented matroids can also be used to describe the topology (connectedness) of
directed graphs. Hence the mathematical methods developed for oriented matroids
can be applied to the difficult combinatorics of embedded graphs underlying the
construction of LQG. As a first application we revisit the analysis of [4-5],
and find that enumeration of all possible sign configurations used there is
equivalent to enumerating all realizable oriented matroids of rank 3, and thus
can be greatly simplified. We find that for 7-valent vertices having no
coplanar triples of edge tangents, the smallest non-zero eigenvalue of the
volume spectrum does not grow as one increases the maximum spin \jmax at the
vertex, for any orientation of the edge tangents. This indicates that, in
contrast to the area operator, considering large \jmax does not necessarily
imply large volume eigenvalues. In addition we give an outlook to possible
starting points for rewriting the combinatorics of LQG in terms of oriented
matroids.Comment: 43 pages, 26 figures, LaTeX. Version published in CQG. Typos
corrected, presentation slightly extende
On the conjectured upper bounds for entries of mutation count matrices
AbstractSimple expressions for the previously given conjectural upper bounds for the entries of mutation count matrices are presented
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