1,005 research outputs found
Foliations and 2+1 Causal Dynamical Triangulation Models
The original models of causal dynamical triangulations construct space-time
by arranging a set of simplices in layers separated by a fixed time-like
distance. The importance of the foliation structure in the 2+1 dimensional
model is studied by considering variations in which this property is relaxed.
It turns out that the fixed-lapse condition can be equivalently replaced by a
set of global constraints that have geometrical interpretation. On the other
hand, the introduction of new types of simplices that puncture the foliating
sheets in general leads to different low-energy behavior compared to the
original model.Comment: v2: 9 pages, 3 figures, published versio
Nonorientable 3-manifolds admitting coloured triangulations with at most 30 tetrahedra
We present the census of all non-orientable, closed, connected 3-manifolds
admitting a rigid crystallization with at most 30 vertices. In order to obtain
the above result, we generate, manipulate and compare, by suitable computer
procedures, all rigid non-bipartite crystallizations up to 30 vertices.Comment: 18 pages, 3 figure
Towards a Scalable Dynamic Spatial Database System
With the rise of GPS-enabled smartphones and other similar mobile devices,
massive amounts of location data are available. However, no scalable solutions
for soft real-time spatial queries on large sets of moving objects have yet
emerged. In this paper we explore and measure the limits of actual algorithms
and implementations regarding different application scenarios. And finally we
propose a novel distributed architecture to solve the scalability issues.Comment: (2012
Computing parametric rational generating functions with a primal Barvinok algorithm
Computations with Barvinok's short rational generating functions are
traditionally being performed in the dual space, to avoid the combinatorial
complexity of inclusion--exclusion formulas for the intersecting proper faces
of cones. We prove that, on the level of indicator functions of polyhedra,
there is no need for using inclusion--exclusion formulas to account for
boundary effects: All linear identities in the space of indicator functions can
be purely expressed using half-open variants of the full-dimensional polyhedra
in the identity. This gives rise to a practically efficient, parametric
Barvinok algorithm in the primal space.Comment: 16 pages, 1 figure; v2: Minor corrections, new example and summary of
algorithm; submitted to journa
Dynamically Triangulating Lorentzian Quantum Gravity
Fruitful ideas on how to quantize gravity are few and far between. In this
paper, we give a complete description of a recently introduced non-perturbative
gravitational path integral whose continuum limit has already been investigated
extensively in d less than 4, with promising results. It is based on a
simplicial regularization of Lorentzian space-times and, most importantly,
possesses a well-defined, non-perturbative Wick rotation. We present a detailed
analysis of the geometric and mathematical properties of the discretized model
in d=3,4. This includes a derivation of Lorentzian simplicial manifold
constraints, the gravitational actions and their Wick rotation. We define a
transfer matrix for the system and show that it leads to a well-defined
self-adjoint Hamiltonian. In view of numerical simulations, we also suggest
sets of Lorentzian Monte Carlo moves. We demonstrate that certain pathological
phases found previously in Euclidean models of dynamical triangulations cannot
be realized in the Lorentzian case.Comment: 41 pages, 14 figure
Shapes of polyhedra and triangulations of the sphere
The space of shapes of a polyhedron with given total angles less than 2\pi at
each of its n vertices has a Kaehler metric, locally isometric to complex
hyperbolic space CH^{n-3}. The metric is not complete: collisions between
vertices take place a finite distance from a nonsingular point. The metric
completion is a complex hyperbolic cone-manifold. In some interesting special
cases, the metric completion is an orbifold. The concrete description of these
spaces of shapes gives information about the combinatorial classification of
triangulations of the sphere with no more than 6 triangles at a vertex.Comment: 39 pages. Published copy, also available at
http://www.maths.warwick.ac.uk/gt/GTMon1/paper25.abs.htm
IST Austria Thesis
Many methods for the reconstruction of shapes from sets of points produce ordered simplicial complexes, which are collections of vertices, edges, triangles, and their higher-dimensional analogues, called simplices, in which every simplex gets assigned a real value measuring its size. This thesis studies ordered simplicial complexes, with a focus on their topology, which reflects the connectedness of the represented shapes and the presence of holes. We are interested both in understanding better the structure of these complexes, as well as in developing algorithms for applications.
For the Delaunay triangulation, the most popular measure for a simplex is the radius of the smallest empty circumsphere. Based on it, we revisit Alpha and Wrap complexes and experimentally determine their probabilistic properties for random data. Also, we prove the existence of tri-partitions, propose algorithms to open and close holes, and extend the concepts from Euclidean to Bregman geometries
Approximation and geometric modeling with simplex B-splines associated with irregular triangles
Bivariate quadratic simplical B-splines defined by their corresponding set of knots derived from a (suboptimal) constrained Delaunay triangulation of the domain are employed to obtain a C1-smooth surface. The generation of triangle vertices is adjusted to the areal distribution of the data in the domain. We emphasize here that the vertices of the triangles initially define the knots of the B-splines and do generally not coincide with the abscissae of the data. Thus, this approach is well suited to process scattered data.\ud
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With each vertex of a given triangle we associate two additional points which give rise to six configurations of five knots defining six linearly independent bivariate quadratic B-splines supported on the convex hull of the corresponding five knots.\ud
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If we consider the vertices of the triangulation as threefold knots, the bivariate quadratic B-splines turn into the well known bivariate quadratic Bernstein-Bézier-form polynomials on triangles. Thus we might be led to think of B-splines as of smoothed versions of Bernstein-Bézier polynomials with respect to the entire domain. From the degenerate Bernstein-Bézier situation we deduce rules how to locate the additional points associated with each vertex to establish knot configurations that allow the modeling of discontinuities of the function itself or any of its directional derivatives. We find that four collinear knots out of the set of five defining an individual quadratic B-spline generate a discontinuity in the surface along the line they constitute, and that analogously three collinear knots generate a discontinuity in a first derivative.\ud
Finally, the coefficients of the linear combinations of normalized simplicial B-splines are visualized as geometric control points satisfying the convex hull property.\ud
Thus, bivariate quadratic B-splines associated with irregular triangles provide a great flexibility to approximate and model fast changing or even functions with any given discontinuities from scattered data.\ud
An example for least squares approximation with simplex splines is presented
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