254 research outputs found
Distributed Computability in Byzantine Asynchronous Systems
In this work, we extend the topology-based approach for characterizing
computability in asynchronous crash-failure distributed systems to asynchronous
Byzantine systems. We give the first theorem with necessary and sufficient
conditions to solve arbitrary tasks in asynchronous Byzantine systems where an
adversary chooses faulty processes. In our adversarial formulation, outputs of
non-faulty processes are constrained in terms of inputs of non-faulty processes
only. For colorless tasks, an important subclass of distributed problems, the
general result reduces to an elegant model that effectively captures the
relation between the number of processes, the number of failures, as well as
the topological structure of the task's simplicial complexes.Comment: Will appear at the Proceedings of the 46th Annual Symposium on the
Theory of Computing, STOC 201
Tameness on the boundary and Ahlfors' measure conjecture
Let N be a complete hyperbolic 3-manifold that is an algebraic limit of
geometrically finite hyperbolic 3-manifolds. We show N is homeomorphic to the
interior of a compact 3-manifold, or tame, if one of the following conditions
holds:
(1) N has non-empty conformal boundary,
(2) N is not homotopy equivalent to a compression body, or
(3) N is a strong limit of geometrically finite manifolds.
The first case proves Ahlfors' measure conjecture for Kleinian groups in the
closure of the geometrically finite locus: given any algebraic limit G of
geometrically finite Kleinian groups, the limit set of G is either of Lebesgue
measure zero or all of the Riemann sphere. Thus, Ahlfors' conjecture is reduced
to the density conjecture of Bers, Sullivan, and Thurston.Comment: New revised version, 22 pages. To appear, Publ. I.H.E.S. This version
represents a fairly substantial reorganization of the logical structure of
the pape
Marden's Tameness Conjecture: history and applications
Marden's Tameness Conjecture predicts that every hyperbolic 3-manifold with
finitely generated fundamental group is homeomorphic to the interior of a
compact 3-manifold. It was recently established by Agol and Calegari-Gabai. We
will survey the history of work on this conjecture and discuss its many
applications.Comment: 30 pages, expository article based on a lecture given at the
conference on "Geometry, Topology and Analysis of Locally Symmetric Spaces
and Discrete Groups'' held in Beijing in July 2007. Article was published in
the proceedings of that conferenc
3D mesh metamorphosis from spherical parameterization for conceptual design
Engineering product design is an information intensive decision-making
process that consists of several phases including design specification
definition, design concepts generation, detailed design and analysis,
and manufacturing. Usually, generating geometry models for
visualization is a big challenge for early stage conceptual design.
Complexity of existing computer aided design packages constrains
participation of people with various backgrounds in the design
process. In addition, many design processes do not take advantage of
the rich amount of legacy information available for new concepts
creation.
The research presented here explores the use of advanced graphical
techniques to quickly and efficiently merge legacy information with
new design concepts to rapidly create new conceptual product designs.
3D mesh metamorphosis framework 3DMeshMorpher was created to
construct new models by navigating in a shape-space of registered
design models. The framework is composed of: i) a fast spherical
parameterization method to map a geometric model (genus-0) onto a unit
sphere; ii) a geometric feature identification and picking technique
based on 3D skeleton extraction; and iii) a LOD controllable 3D
remeshing scheme with spherical mesh subdivision based on the
developedspherical parameterization. This efficient software framework
enables designers to create numerous geometric concepts in real time
with a simple graphical user interface.
The spherical parameterization method is focused on closed genus-zero
meshes. It is based upon barycentric coordinates with convex boundary.
Unlike most existing similar approaches which deal with each vertex in
the mesh equally, the method developed in this research focuses
primarily on resolving overlapping areas, which helps speed the
parameterization process. The algorithm starts by normalizing the
source mesh onto a unit sphere and followed by some initial relaxation
via Gauss-Seidel iterations. Due to its emphasis on solving only
challenging overlapping regions, this parameterization process is much
faster than existing spherical mapping methods.
To ensure the correspondence of features from different models, we
introduce a skeleton based feature identification and picking method
for features alignment. Unlike traditional methods that align single
point for each feature, this method can provide alignments for
complete feature areas. This could help users to create more
reasonable intermediate morphing results with preserved topological
features. This skeleton featuring framework could potentially be
extended to automatic features alignment for geometries with similar
topologies. The skeleton extracted could also be applied for other
applications such as skeleton-based animations.
The 3D remeshing algorithm with spherical mesh subdivision is
developed to generate a common connectivity for different mesh models.
This method is derived from the concept of spherical mesh subdivision.
The local recursive subdivision can be set to match the desired LOD
(level of details) for source spherical mesh. Such LOD is controllable
and this allows various outputs with different resolutions. Such
recursive subdivision then follows by a triangular correction process
which ensures valid triangulations for the remeshing. And the final
mesh merging and reconstruction process produces the remeshing model
with desired LOD specified from user. Usually the final merged model
contains all the geometric details from each model with reasonable
amount of vertices, unlike other existing methods that result in big
amount of vertices in the merged model. Such multi-resolution outputs
with controllable LOD could also be applied in various other computer
graphics applications such as computer games
Moment polytopes for symplectic manifolds with monodromy
A natural way of generalising Hamiltonian toric manifolds is to permit the
presence of generic isolated singularities for the moment map. For a class of
such ``almost-toric 4-manifolds'' which admits a Hamiltonian -action we
show that one can associate a group of convex polygons that generalise the
celebrated moment polytopes of Atiyah, Guillemin-Sternberg. As an application,
we derive a Duistermaat-Heckman formula demonstrating a strong effect of the
possible monodromy of the underlying integrable system.Comment: finally a revision of the 2003 preprint. 29 pages, 8 figure
Unstructured un-split geometrical Volume-of-Fluid methods -- A review
Geometrical Volume-of-Fluid (VoF) methods mainly support structured meshes,
and only a small number of contributions in the scientific literature report
results with unstructured meshes and three spatial dimensions. Unstructured
meshes are traditionally used for handling geometrically complex solution
domains that are prevalent when simulating problems of industrial relevance.
However, three-dimensional geometrical operations are significantly more
complex than their two-dimensional counterparts, which is confirmed by the
ratio of publications with three-dimensional results on unstructured meshes to
publications with two-dimensional results or support for structured meshes.
Additionally, unstructured meshes present challenges in serial and parallel
computational efficiency, accuracy, implementation complexity, and robustness.
Ongoing research is still very active, focusing on different issues: interface
positioning in general polyhedra, estimation of interface normal vectors,
advection accuracy, and parallel and serial computational efficiency.
This survey tries to give a complete and critical overview of classical, as
well as contemporary geometrical VOF methods with concise explanations of the
underlying ideas and sub-algorithms, focusing primarily on unstructured meshes
and three dimensional calculations. Reviewed methods are listed in historical
order and compared in terms of accuracy and computational efficiency
Conjectures on Convergence and Scalar Curvature
Here we survey the compactness and geometric stability conjectures formulated
by the participants at the 2018 IAS Emerging Topics Workshop on {\em Scalar
Curvature and Convergence}. We have tried to survey all the progress towards
these conjectures as well as related examples, although it is impossible to
cover everything. We focus primarily on sequences of compact Riemannian
manifolds with nonnegative scalar curvature and their limit spaces. Christina
Sormani is grateful to have had the opportunity to write up our ideas and has
done her best to credit everyone involved within the paper even though she is
the only author listed above. In truth we are a team of over thirty people
working together and apart on these deep questions and we welcome everyone who
is interested in these conjectures to join us.Comment: Please email us any comments or corrections. 57 pages, 20 figures,
IAS Emerging Topics on Scalar Curvature and Convergenc
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