684 research outputs found
The Computational Complexity of Knot and Link Problems
We consider the problem of deciding whether a polygonal knot in 3-dimensional
Euclidean space is unknotted, capable of being continuously deformed without
self-intersection so that it lies in a plane. We show that this problem, {\sc
unknotting problem} is in {\bf NP}. We also consider the problem, {\sc
unknotting problem} of determining whether two or more such polygons can be
split, or continuously deformed without self-intersection so that they occupy
both sides of a plane without intersecting it. We show that it also is in NP.
Finally, we show that the problem of determining the genus of a polygonal knot
(a generalization of the problem of determining whether it is unknotted) is in
{\bf PSPACE}. We also give exponential worst-case running time bounds for
deterministic algorithms to solve each of these problems. These algorithms are
based on the use of normal surfaces and decision procedures due to W. Haken,
with recent extensions by W. Jaco and J. L. Tollefson.Comment: 32 pages, 1 figur
Affine Wa(A4), Quaternions, and Decagonal Quasicrystals
We introduce a technique of projection onto the Coxeter plane of an arbitrary
higher dimensional lattice described by the affine Coxeter group. The Coxeter
plane is determined by the simple roots of the Coxeter graph I2 (h) where h is
the Coxeter number of the Coxeter group W(G) which embeds the dihedral group Dh
of order 2h as a maximal subgroup. As a simple application we demonstrate
projections of the root and weight lattices of A4 onto the Coxeter plane using
the strip (canonical) projection method. We show that the crystal spaces of the
affine Wa(A4) can be decomposed into two orthogonal spaces whose point groups
is the dihedral group D5 which acts in both spaces faithfully. The strip
projections of the root and weight lattices can be taken as models for the
decagonal quasicrystals. The paper also revises the quaternionic descriptions
of the root and weight lattices, described by the affine Coxeter group Wa(A3),
which correspond to the face centered cubic (fcc) lattice and body centered
cubic (bcc) lattice respectively. Extensions of these lattices to higher
dimensions lead to the root and weight lattices of the group Wa(An), n>=4 . We
also note that the projection of the Voronoi cell of the root lattice of Wa(A4)
describes a framework of nested decagram growing with the power of the golden
ratio recently discovered in the Islamic arts.Comment: 26 pages, 17 figure
Counting a black hole in Lorentzian product triangulations
We take a step toward a nonperturbative gravitational path integral for
black-hole geometries by deriving an expression for the expansion rate of null
geodesic congruences in the approach of causal dynamical triangulations. We
propose to use the integrated expansion rate in building a quantum horizon
finder in the sum over spacetime geometries. It takes the form of a counting
formula for various types of discrete building blocks which differ in how they
focus and defocus light rays. In the course of the derivation, we introduce the
concept of a Lorentzian dynamical triangulation of product type, whose
applicability goes beyond that of describing black-hole configurations.Comment: 42 pages, 11 figure
Classical 6j-symbols and the tetrahedron
A classical 6j-symbol is a real number which can be associated to a labelling
of the six edges of a tetrahedron by irreducible representations of SU(2). This
abstract association is traditionally used simply to express the symmetry of
the 6j-symbol, which is a purely algebraic object; however, it has a deeper
geometric significance. Ponzano and Regge, expanding on work of Wigner, gave a
striking (but unproved) asymptotic formula relating the value of the 6j-symbol,
when the dimensions of the representations are large, to the volume of an
honest Euclidean tetrahedron whose edge lengths are these dimensions. The goal
of this paper is to prove and explain this formula by using geometric
quantization. A surprising spin-off is that a generic Euclidean tetrahedron
gives rise to a family of twelve scissors-congruent but non-congruent
tetrahedra.Comment: 46 pages. Published copy, also available at
http://www.maths.warwick.ac.uk/gt/GTVol3/paper2.abs.htm
ColDICE: a parallel Vlasov-Poisson solver using moving adaptive simplicial tessellation
Resolving numerically Vlasov-Poisson equations for initially cold systems can
be reduced to following the evolution of a three-dimensional sheet evolving in
six-dimensional phase-space. We describe a public parallel numerical algorithm
consisting in representing the phase-space sheet with a conforming,
self-adaptive simplicial tessellation of which the vertices follow the
Lagrangian equations of motion. The algorithm is implemented both in six- and
four-dimensional phase-space. Refinement of the tessellation mesh is performed
using the bisection method and a local representation of the phase-space sheet
at second order relying on additional tracers created when needed at runtime.
In order to preserve in the best way the Hamiltonian nature of the system,
refinement is anisotropic and constrained by measurements of local Poincar\'e
invariants. Resolution of Poisson equation is performed using the fast Fourier
method on a regular rectangular grid, similarly to particle in cells codes. To
compute the density projected onto this grid, the intersection of the
tessellation and the grid is calculated using the method of Franklin and
Kankanhalli (1993) generalised to linear order. As preliminary tests of the
code, we study in four dimensional phase-space the evolution of an initially
small patch in a chaotic potential and the cosmological collapse of a
fluctuation composed of two sinusoidal waves. We also perform a "warm" dark
matter simulation in six-dimensional phase-space that we use to check the
parallel scaling of the code.Comment: Code and illustration movies available at:
http://www.vlasix.org/index.php?n=Main.ColDICE - Article submitted to Journal
of Computational Physic
Optimal Point Placement for Mesh Smoothing
We study the problem of moving a vertex in an unstructured mesh of
triangular, quadrilateral, or tetrahedral elements to optimize the shapes of
adjacent elements. We show that many such problems can be solved in linear time
using generalized linear programming. We also give efficient algorithms for
some mesh smoothing problems that do not fit into the generalized linear
programming paradigm.Comment: 12 pages, 3 figures. A preliminary version of this paper was
presented at the 8th ACM/SIAM Symp. on Discrete Algorithms (SODA '97). This
is the final version, and will appear in a special issue of J. Algorithms for
papers from SODA '9
Fast Complementary Dynamics via Skinning Eigenmodes
We propose a reduced-space elasto-dynamic solver that is well suited for
augmenting rigged character animations with secondary motion. At the core of
our method is a novel deformation subspace based on Linear Blend Skinning that
overcomes many of the shortcomings prior subspace methods face. Our skinning
subspace is parameterized entirely by a set of scalar weights, which we can
obtain through a small, material-aware and rig-sensitive generalized eigenvalue
problem. The resulting subspace can easily capture rotational motion and
guarantees that the resulting simulation is rotation equivariant. We further
propose a simple local-global solver for linear co-rotational elasticity and
propose a clustering method to aggregate per-tetrahedra non-linear energetic
quantities. The result is a compact simulation that is fully decoupled from the
complexity of the mesh.Comment: 20 pages, 24 figure
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