45,662 research outputs found
On the Complexity of Polytope Isomorphism Problems
We show that the problem to decide whether two (convex) polytopes, given by
their vertex-facet incidences, are combinatorially isomorphic is graph
isomorphism complete, even for simple or simplicial polytopes. On the other
hand, we give a polynomial time algorithm for the combinatorial polytope
isomorphism problem in bounded dimensions. Furthermore, we derive that the
problems to decide whether two polytopes, given either by vertex or by facet
descriptions, are projectively or affinely isomorphic are graph isomorphism
hard.
The original version of the paper (June 2001, 11 pages) had the title ``On
the Complexity of Isomorphism Problems Related to Polytopes''. The main
difference between the current and the former version is a new polynomial time
algorithm for polytope isomorphism in bounded dimension that does not rely on
Luks polynomial time algorithm for checking two graphs of bounded valence for
isomorphism. Furthermore, the treatment of geometric isomorphism problems was
extended.Comment: 16 pages; to appear in: Graphs and Comb.; replaces our paper ``On the
Complexity of Isomorphism Problems Related to Polytopes'' (June 2001
On C-smooth Surfaces of Constant Width
A number of results for C-smooth surfaces of constant width in Euclidean
3-space are obtained. In particular, an integral inequality
for constant width surfaces is established. This is used to prove that the
ratio of volume to cubed width of a constant width surface is reduced by
shrinking it along its normal lines. We also give a characterization of
surfaces of constant width that have rational support function.
Our techniques, which are complex differential geometric in nature, allow us
to construct explicit smooth surfaces of constant width in ,
and their focal sets. They also allow for easy construction of tetrahedrally
symmetric surfaces of constant width.Comment: 14 pages AMS-LATEX, 5 figure
Combinatorial Space Tiling
The present article studies combinatorial tilings of Euclidean or spherical
spaces by polytopes, serving two main purposes: first, to survey some of the
main developments in combinatorial space tiling; and second, to highlight some
new and some old open problems in this area.Comment: 16 pages; to appear in "Symmetry: Culture and Science
Grid Representations and the Chromatic Number
A grid drawing of a graph maps vertices to grid points and edges to line
segments that avoid grid points representing other vertices. We show that there
is a number of grid points that some line segment of an arbitrary grid drawing
must intersect. This number is closely connected to the chromatic number.
Second, we study how many columns we need to draw a graph in the grid,
introducing some new \NP-complete problems. Finally, we show that any planar
graph has a planar grid drawing where every line segment contains exactly two
grid points. This result proves conjectures asked by David Flores-Pe\~naloza
and Francisco Javier Zaragoza Martinez.Comment: 22 pages, 8 figure
Construction of eigenvarieties in small cohomological dimensions for semi-simple, simply connected groups
We study low order terms of Emerton's spectral sequence for simply connected,
simple groups. As a result, for real rank 1 groups, we show that Emerton's
method for constructing eigenvarieties is successful in cohomological dimension
1. For real rank 2 groups, we show that a slight modification of Emerton's
method allows one to construct eigenvarieties in cohomological dimension 2
Holography, a covariant c-function, and the geometry of the renormalization group
We propose a covariant geometrical expression for the c-function for theories
which admit dual gravitational descriptions. We state a c-theorem with respect
to this quantity and prove it. We apply the expression to a class of
geometries, from domain walls in gauged supergravities, to extremal and near
extremal Dp branes, and the AdS Schwarzschild black hole. In all cases, we find
agreement with expectations.Comment: 20 pages, 1 figure; minor clarifications, a reference added, and a
few typos correcte
Nonautonomous control of stable and unstable manifolds in two-dimensional flows
We outline a method for controlling the location of stable and unstable
manifolds in the following sense. From a known location of the stable and
unstable manifolds in a steady two-dimensional flow, the primary segments of
the manifolds are to be moved to a user-specified time-varying location which
is near the steady location. We determine the nonautonomous perturbation to the
vector field required to achieve this control, and give a theoretical bound for
the error in the manifolds resulting from applying this control. The efficacy
of the control strategy is illustrated via a numerical example
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