272,652 research outputs found
Few Cuts Meet Many Point Sets
We study the problem of how to breakup many point sets in into
smaller parts using a few splitting (shared) hyperplanes. This problem is
related to the classical Ham-Sandwich Theorem. We provide a logarithmic
approximation to the optimal solution using the greedy algorithm for submodular
optimization
Categorification of Hopf algebras of rooted trees
We exhibit a monoidal structure on the category of finite sets indexed by
P-trees for a finitary polynomial endofunctor P. This structure categorifies
the monoid scheme (over Spec N) whose semiring of functions is (a P-version of)
the Connes--Kreimer bialgebra H of rooted trees (a Hopf algebra after base
change to Z and collapsing H_0). The monoidal structure is itself given by a
polynomial functor, represented by three easily described set maps; we show
that these maps are the same as those occurring in the polynomial
representation of the free monad on P.Comment: 29 pages. Does not compile with pdflatex due to dependency on the
texdraw package. v2: expository improvements, following suggestions from the
referees; final version to appear in Centr. Eur. J. Mat
Convexity in partial cubes: the hull number
We prove that the combinatorial optimization problem of determining the hull
number of a partial cube is NP-complete. This makes partial cubes the minimal
graph class for which NP-completeness of this problem is known and improves
some earlier results in the literature.
On the other hand we provide a polynomial-time algorithm to determine the
hull number of planar partial cube quadrangulations.
Instances of the hull number problem for partial cubes described include
poset dimension and hitting sets for interiors of curves in the plane.
To obtain the above results, we investigate convexity in partial cubes and
characterize these graphs in terms of their lattice of convex subgraphs,
improving a theorem of Handa. Furthermore we provide a topological
representation theorem for planar partial cubes, generalizing a result of
Fukuda and Handa about rank three oriented matroids.Comment: 19 pages, 4 figure
BPS Open Strings and A-D-E-singularities in F-theory on K3
We improve on a recently constructed graphical representation of the
supergravity 7-brane solution and apply this refined representation to re-study
the open string description of the A-D-E-singularities in F-theory on K3. A
noteworthy feature of the graphical representation is that it provides the
complete global branch cut structure of the 7-brane solution which plays an
important role in our analysis. We first identify those groups of branes which
when made to coincide lead to the A-D-E-gauge groups. We next show that there
is always a sufficient number of open BPS strings to account for all the
generators of the gauge group. However, as we will show, there is in general no
one-to-one relation between BPS strings and gauge group generators.
For the D_{n+4}- and E-singularities, in order to relate BPS strings with
gauge group generators, we make an SU(n+4), respectively SU(5) subgroup of the
D_{n+4}- and E-gauge groups manifest. We find that only for the D-series (and
for the standard A-series) this is sufficient to identify, in a one-to-one
manner, which BPS strings correspond to which gauge group generators.Comment: 37 pages, 15 figure
Infinite graphic matroids Part I
An infinite matroid is graphic if all of its finite minors are graphic and
the intersection of any circuit with any cocircuit is finite. We show that a
matroid is graphic if and only if it can be represented by a graph-like
topological space: that is, a graph-like space in the sense of Thomassen and
Vella. This extends Tutte's characterization of finite graphic matroids.
The representation we construct has many pleasant topological properties.
Working in the representing space, we prove that any circuit in a 3-connected
graphic matroid is countable
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