1,370 research outputs found
On the Monotone Upper Bound Problem
The Monotone Upper Bound Problem asks for the maximal number M(d,n) of
vertices on a strictly-increasing edge-path on a simple d-polytope with n
facets. More specifically, it asks whether the upper bound M(d,n)<=M_{ubt}(d,n)
provided by McMullen's (1970) Upper Bound Theorem is tight, where M_{ubt}(d,n)
is the number of vertices of a dual-to-cyclic d-polytope with n facets.
It was recently shown that the upper bound M(d,n)<=M_{ubt}(d,n) holds with
equality for small dimensions (d<=4: Pfeifle, 2003) and for small corank
(n<=d+2: G\"artner et al., 2001). Here we prove that it is not tight in
general: In dimension d=6 a polytope with n=9 facets can have M_{ubt}(6,9)=30
vertices, but not more than 26 <= M(6,9) <= 29 vertices can lie on a
strictly-increasing edge-path.
The proof involves classification results about neighborly polytopes, Kalai's
(1988) concept of abstract objective functions, the Holt-Klee conditions
(1998), explicit enumeration, Welzl's (2001) extended Gale diagrams, randomized
generation of instances, as well as non-realizability proofs via a version of
the Farkas lemma.Comment: 15 pages; 6 figure
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
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