20,040 research outputs found
Quadrilateral-octagon coordinates for almost normal surfaces
Normal and almost normal surfaces are essential tools for algorithmic
3-manifold topology, but to use them requires exponentially slow enumeration
algorithms in a high-dimensional vector space. The quadrilateral coordinates of
Tollefson alleviate this problem considerably for normal surfaces, by reducing
the dimension of this vector space from 7n to 3n (where n is the complexity of
the underlying triangulation). Here we develop an analogous theory for
octagonal almost normal surfaces, using quadrilateral and octagon coordinates
to reduce this dimension from 10n to 6n. As an application, we show that
quadrilateral-octagon coordinates can be used exclusively in the streamlined
3-sphere recognition algorithm of Jaco, Rubinstein and Thompson, reducing
experimental running times by factors of thousands. We also introduce joint
coordinates, a system with only 3n dimensions for octagonal almost normal
surfaces that has appealing geometric properties.Comment: 34 pages, 20 figures; v2: Simplified the proof of Theorem 4.5 using
cohomology, plus other minor changes; v3: Minor housekeepin
Scattering Amplitudes and Toric Geometry
In this paper we provide a first attempt towards a toric geometric
interpretation of scattering amplitudes. In recent investigations it has indeed
been proposed that the all-loop integrand of planar N=4 SYM can be represented
in terms of well defined finite objects called on-shell diagrams drawn on
disks. Furthermore it has been shown that the physical information of on-shell
diagrams is encoded in the geometry of auxiliary algebraic varieties called the
totally non negative Grassmannians. In this new formulation the infinite
dimensional symmetry of the theory is manifest and many results, that are quite
tricky to obtain in terms of the standard Lagrangian formulation of the theory,
are instead manifest. In this paper, elaborating on previous results, we
provide another picture of the scattering amplitudes in terms of toric
geometry. In particular we describe in detail the toric varieties associated to
an on-shell diagram, how the singularities of the amplitudes are encoded in
some subspaces of the toric variety, and how this picture maps onto the
Grassmannian description. Eventually we discuss the action of cluster
transformations on the toric varieties. The hope is to provide an alternative
description of the scattering amplitudes that could contribute in the
developing of this very interesting field of research.Comment: 58 pages, 25 figures, typos corrected, a reference added, to be
published in JHE
Converting between quadrilateral and standard solution sets in normal surface theory
The enumeration of normal surfaces is a crucial but very slow operation in
algorithmic 3-manifold topology. At the heart of this operation is a polytope
vertex enumeration in a high-dimensional space (standard coordinates).
Tollefson's Q-theory speeds up this operation by using a much smaller space
(quadrilateral coordinates), at the cost of a reduced solution set that might
not always be sufficient for our needs. In this paper we present algorithms for
converting between solution sets in quadrilateral and standard coordinates. As
a consequence we obtain a new algorithm for enumerating all standard vertex
normal surfaces, yielding both the speed of quadrilateral coordinates and the
wider applicability of standard coordinates. Experimentation with the software
package Regina shows this new algorithm to be extremely fast in practice,
improving speed for large cases by factors from thousands up to millions.Comment: 55 pages, 10 figures; v2: minor fixes only, plus a reformat for the
journal styl
Maximal admissible faces and asymptotic bounds for the normal surface solution space
The enumeration of normal surfaces is a key bottleneck in computational
three-dimensional topology. The underlying procedure is the enumeration of
admissible vertices of a high-dimensional polytope, where admissibility is a
powerful but non-linear and non-convex constraint. The main results of this
paper are significant improvements upon the best known asymptotic bounds on the
number of admissible vertices, using polytopes in both the standard normal
surface coordinate system and the streamlined quadrilateral coordinate system.
To achieve these results we examine the layout of admissible points within
these polytopes. We show that these points correspond to well-behaved
substructures of the face lattice, and we study properties of the corresponding
"admissible faces". Key lemmata include upper bounds on the number of maximal
admissible faces of each dimension, and a bijection between the maximal
admissible faces in the two coordinate systems mentioned above.Comment: 31 pages, 10 figures, 2 tables; v2: minor revisions (to appear in
Journal of Combinatorial Theory A
Stereo image processing system for robot vision
More and more applications (path planning, collision avoidance
methods) require 3D description of the surround world. This paper
describes a stereo vision system that uses 2D (grayscale or color) images
to extract simple 2D geometric entities (points, lines) applying a
low-level feature detector. The features are matched across views with a
graph matching algorithm. During the projective reconstruction the 3D
description of the scene is recovered. The developed system uses uncalibrated
cameras, therefore only projective 3D structure can be detected
defined up to a collineation. Using the Euclidean information about a
known set of predefined objects stored in database and the results of the
recognition algorithm, the description can be updated to a metric one
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