754 research outputs found
Volumetric Untrimming: Precise decomposition of trimmed trivariates into tensor products
3D objects, modeled using Computer Aided Geometric Design tools, are
traditionally represented using a boundary representation (B-rep), and
typically use spline functions to parameterize these boundary surfaces.
However, recent development in physical analysis, in isogeometric analysis
(IGA) in specific, necessitates a volumetric parametrization of the interior of
the object. IGA is performed directly by integrating over the spline spaces of
the volumetric spline representation of the object. Typically, tensor-product
B-spline trivariates are used to parameterize the volumetric domain. A general
3D object, that can be modeled in contemporary B-rep CAD tools, is typically
represented using trimmed B-spline surfaces. In order to capture the generality
of the contemporary B-rep modeling space, while supporting IGA needs, Massarwi
and Elber (2016) proposed the use of trimmed trivariates volumetric elements.
However, the use of trimmed geometry makes the integration process more
difficult since integration over trimmed B-spline basis functions is a highly
challenging task. In this work, we propose an algorithm that precisely
decomposes a trimmed B-spline trivariate into a set of (singular only on the
boundary) tensor-product B-spline trivariates, that can be utilized to simplify
the integration process in IGA. The trimmed B-spline trivariate is first
subdivided into a set of trimmed B\'ezier trivariates, at all its internal
knots. Then, each trimmed B\'ezier trivariate, is decomposed into a set of
mutually exclusive tensor-product B-spline trivariates, that precisely cover
the entire trimmed domain. This process, denoted untrimming, can be performed
in either the Euclidean space or the parametric space of the trivariate. We
present examples on complex trimmed trivariates' based geometry, and we
demonstrate the effectiveness of the method by applying IGA over the
(untrimmed) results.Comment: 18 pages, 32 figures. Contribution accepted in International
Conference on Geometric Modeling and Processing (GMP 2019
A generalization of Voronoi's reduction theory and its application
We consider Voronoi's reduction theory of positive definite quadratic forms
which is based on Delone subdivision. We extend it to forms and Delone
subdivisions having a prescribed symmetry group. Even more general, the theory
is developed for forms which are restricted to a linear subspace in the space
of quadratic forms. We apply the new theory to complete the classification of
totally real thin algebraic number fields which was recently initiated by
Bayer-Fluckiger and Nebe. Moreover, we apply it to construct new best known
sphere coverings in dimensions 9,..., 15.Comment: 31 pages, 2 figures, 2 tables, (v4) minor changes, to appear in Duke
Math.
The number of Reidemeister Moves Needed for Unknotting
There is a positive constant such that for any diagram representing
the unknot, there is a sequence of at most Reidemeister moves that
will convert it to a trivial knot diagram, is the number of crossings in
. A similar result holds for elementary moves on a polygonal knot
embedded in the 1-skeleton of the interior of a compact, orientable,
triangulated 3-manifold . There is a positive constant such that
for each , if consists of tetrahedra, and is unknotted,
then there is a sequence of at most elementary moves in which
transforms to a triangle contained inside one tetrahedron of . We obtain
explicit values for and .Comment: 48 pages, 14 figure
DeepSphere: Efficient spherical Convolutional Neural Network with HEALPix sampling for cosmological applications
Convolutional Neural Networks (CNNs) are a cornerstone of the Deep Learning
toolbox and have led to many breakthroughs in Artificial Intelligence. These
networks have mostly been developed for regular Euclidean domains such as those
supporting images, audio, or video. Because of their success, CNN-based methods
are becoming increasingly popular in Cosmology. Cosmological data often comes
as spherical maps, which make the use of the traditional CNNs more complicated.
The commonly used pixelization scheme for spherical maps is the Hierarchical
Equal Area isoLatitude Pixelisation (HEALPix). We present a spherical CNN for
analysis of full and partial HEALPix maps, which we call DeepSphere. The
spherical CNN is constructed by representing the sphere as a graph. Graphs are
versatile data structures that can act as a discrete representation of a
continuous manifold. Using the graph-based representation, we define many of
the standard CNN operations, such as convolution and pooling. With filters
restricted to being radial, our convolutions are equivariant to rotation on the
sphere, and DeepSphere can be made invariant or equivariant to rotation. This
way, DeepSphere is a special case of a graph CNN, tailored to the HEALPix
sampling of the sphere. This approach is computationally more efficient than
using spherical harmonics to perform convolutions. We demonstrate the method on
a classification problem of weak lensing mass maps from two cosmological models
and compare the performance of the CNN with that of two baseline classifiers.
The results show that the performance of DeepSphere is always superior or equal
to both of these baselines. For high noise levels and for data covering only a
smaller fraction of the sphere, DeepSphere achieves typically 10% better
classification accuracy than those baselines. Finally, we show how learned
filters can be visualized to introspect the neural network.Comment: arXiv admin note: text overlap with arXiv:astro-ph/0409513 by other
author
Tropical cycles and Chow polytopes
The Chow polytope of an algebraic cycle in a torus depends only on its
tropicalisation. Generalising this, we associate a Chow polytope to any
abstract tropical variety in a tropicalised toric variety. Several significant
polyhedra associated to tropical varieties are special cases of our Chow
polytope. The Chow polytope of a tropical variety is given by a simple
combinatorial construction: its normal subdivision is the Minkowski sum of
and a reflected skeleton of the fan of the ambient toric variety.Comment: 22 pp, 3 figs. Added discussion of arbitrary ambient toric varieties;
several improvements suggested by Eric Katz; some rearrangemen
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