39,314 research outputs found
The Topology ToolKit
This system paper presents the Topology ToolKit (TTK), a software platform
designed for topological data analysis in scientific visualization. TTK
provides a unified, generic, efficient, and robust implementation of key
algorithms for the topological analysis of scalar data, including: critical
points, integral lines, persistence diagrams, persistence curves, merge trees,
contour trees, Morse-Smale complexes, fiber surfaces, continuous scatterplots,
Jacobi sets, Reeb spaces, and more. TTK is easily accessible to end users due
to a tight integration with ParaView. It is also easily accessible to
developers through a variety of bindings (Python, VTK/C++) for fast prototyping
or through direct, dependence-free, C++, to ease integration into pre-existing
complex systems. While developing TTK, we faced several algorithmic and
software engineering challenges, which we document in this paper. In
particular, we present an algorithm for the construction of a discrete gradient
that complies to the critical points extracted in the piecewise-linear setting.
This algorithm guarantees a combinatorial consistency across the topological
abstractions supported by TTK, and importantly, a unified implementation of
topological data simplification for multi-scale exploration and analysis. We
also present a cached triangulation data structure, that supports time
efficient and generic traversals, which self-adjusts its memory usage on demand
for input simplicial meshes and which implicitly emulates a triangulation for
regular grids with no memory overhead. Finally, we describe an original
software architecture, which guarantees memory efficient and direct accesses to
TTK features, while still allowing for researchers powerful and easy bindings
and extensions. TTK is open source (BSD license) and its code, online
documentation and video tutorials are available on TTK's website
The Topology of the AdS/CFT/Randall-Sundrum Complementarity
The background geometries of the AdS/CFT and the Randall-Sundrum theories are
locally similar, and there is strong evidence for some kind of
"complementarity" between them; yet the global structures of the respective
manifolds are very different. We show that this apparent problem can be
understood in the context of a more complete global formulation of AdS/CFT. In
this picture, the brane-world arises within the AdS/CFT geometry as the
inevitable consequence of recent results on the global structure of manifolds
with "infinities". We argue that the usual coordinates give a misleading
picture of this global structure, much as Schwarzschild coordinates conceal the
global form of Kruskal-Szekeres space.Comment: 18 pages, Discussion much expanded, several references added and
correcte
Cohomology in electromagnetic modeling
Electromagnetic modeling provides an interesting context to present a link
between physical phenomena and homology and cohomology theories. Over the past
twenty-five years, a considerable effort has been invested by the computational
electromagnetics community to develop fast and general techniques for potential
design. When magneto-quasi-static discrete formulations based on magnetic
scalar potential are employed in problems which involve conductive regions with
holes, \textit{cuts} are needed to make the boundary value problem well
defined. While an intimate connection with homology theory has been quickly
recognized, heuristic definitions of cuts are surprisingly still dominant in
the literature.
The aim of this paper is first to survey several definitions of cuts together
with their shortcomings. Then, cuts are defined as generators of the first
cohomology group over integers of a finite CW-complex. This provably general
definition has also the virtue of providing an automatic, general and efficient
algorithm for the computation of cuts. Some counter-examples show that
heuristic definitions of cuts should be abandoned. The use of cohomology theory
is not an option but the invaluable tool expressly needed to solve this
problem
On the Topological Characterization of Near Force-Free Magnetic Fields, and the work of late-onset visually-impaired Topologists
The Giroux correspondence and the notion of a near force-free magnetic field
are used to topologically characterize near force-free magnetic fields which
describe a variety of physical processes, including plasma equilibrium. As a
byproduct, the topological characterization of force-free magnetic fields
associated with current-carrying links, as conjectured by Crager and Kotiuga,
is shown to be necessary and conditions for sufficiency are given. Along the
way a paradox is exposed: The seemingly unintuitive mathematical tools, often
associated to higher dimensional topology, have their origins in three
dimensional contexts but in the hands of late-onset visually impaired
topologists. This paradox was previously exposed in the context of algorithms
for the visualization of three-dimensional magnetic fields. For this reason,
the paper concludes by developing connections between mathematics and cognitive
science in this specific context.Comment: 20 pages, no figures, a paper which was presented at a conference in
honor of the 60th birthdays of Alberto Valli and Paolo Secci. The current
preprint is from December 2014; it has been submitted to an AIMS journa
Classification of Higher Dimensional Spacetimes
We algebraically classify some higher dimensional spacetimes, including a
number of vacuum solutions of the Einstein field equations which can represent
higher dimensional black holes. We discuss some consequences of this work.Comment: 16 pages, 1 Tabl
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