482 research outputs found
Persistent Homology Over Directed Acyclic Graphs
We define persistent homology groups over any set of spaces which have
inclusions defined so that the corresponding directed graph between the spaces
is acyclic, as well as along any subgraph of this directed graph. This method
simultaneously generalizes standard persistent homology, zigzag persistence and
multidimensional persistence to arbitrary directed acyclic graphs, and it also
allows the study of more general families of topological spaces or point-cloud
data. We give an algorithm to compute the persistent homology groups
simultaneously for all subgraphs which contain a single source and a single
sink in arithmetic operations, where is the number of vertices in
the graph. We then demonstrate as an application of these tools a method to
overlay two distinct filtrations of the same underlying space, which allows us
to detect the most significant barcodes using considerably fewer points than
standard persistence.Comment: Revised versio
Investigations of fast neutron production by 190 GeV/c muon interactions on different targets
The production of fast neutrons (1 MeV - 1 GeV) in high energy muon-nucleus
interactions is poorly understood, yet it is fundamental to the understanding
of the background in many underground experiments. The aim of the present
experiment (CERN NA55) was to measure spallation neutrons produced by 190 GeV/c
muons scattering on carbon, copper and lead targets. We have investigated the
energy spectrum and angular distribution of spallation neutrons, and we report
the result of our measurement of the neutron production differential cross
section.Comment: 19 pages, 11 figures ep
Appeal No. 0873: Stonebridge Operating Co., LLC. v. Division of Oil & Gas Resources Management
Chief\u27s Orders 2014-39 (suspension of operations); 2014-236, 2014-238,2014-239, 2014-241 (denials of plug-back permits); 2014-253, 2014-256 thru 2014-262 & 2014-264 thru 2014-266 (plug orders
Good covers are algorithmically unrecognizable
A good cover in R^d is a collection of open contractible sets in R^d such
that the intersection of any subcollection is either contractible or empty.
Motivated by an analogy with convex sets, intersection patterns of good covers
were studied intensively. Our main result is that intersection patterns of good
covers are algorithmically unrecognizable.
More precisely, the intersection pattern of a good cover can be stored in a
simplicial complex called nerve which records which subfamilies of the good
cover intersect. A simplicial complex is topologically d-representable if it is
isomorphic to the nerve of a good cover in R^d. We prove that it is
algorithmically undecidable whether a given simplicial complex is topologically
d-representable for any fixed d \geq 5. The result remains also valid if we
replace good covers with acyclic covers or with covers by open d-balls.
As an auxiliary result we prove that if a simplicial complex is PL embeddable
into R^d, then it is topologically d-representable. We also supply this result
with showing that if a "sufficiently fine" subdivision of a k-dimensional
complex is d-representable and k \leq (2d-3)/3, then the complex is PL
embeddable into R^d.Comment: 22 pages, 5 figures; result extended also to acyclic covers in
version
Dualities in persistent (co)homology
We consider sequences of absolute and relative homology and cohomology groups
that arise naturally for a filtered cell complex. We establish algebraic
relationships between their persistence modules, and show that they contain
equivalent information. We explain how one can use the existing algorithm for
persistent homology to process any of the four modules, and relate it to a
recently introduced persistent cohomology algorithm. We present experimental
evidence for the practical efficiency of the latter algorithm.Comment: 16 pages, 3 figures, submitted to the Inverse Problems special issue
on Topological Data Analysi
The persistence landscape and some of its properties
Persistence landscapes map persistence diagrams into a function space, which
may often be taken to be a Banach space or even a Hilbert space. In the latter
case, it is a feature map and there is an associated kernel. The main advantage
of this summary is that it allows one to apply tools from statistics and
machine learning. Furthermore, the mapping from persistence diagrams to
persistence landscapes is stable and invertible. We introduce a weighted
version of the persistence landscape and define a one-parameter family of
Poisson-weighted persistence landscape kernels that may be useful for learning.
We also demonstrate some additional properties of the persistence landscape.
First, the persistence landscape may be viewed as a tropical rational function.
Second, in many cases it is possible to exactly reconstruct all of the
component persistence diagrams from an average persistence landscape. It
follows that the persistence landscape kernel is characteristic for certain
generic empirical measures. Finally, the persistence landscape distance may be
arbitrarily small compared to the interleaving distance.Comment: 18 pages, to appear in the Proceedings of the 2018 Abel Symposiu
Neutron- and muon-induced background in underground physics experiments
Background induced by neutrons in deep underground laboratories is a critical
issue for all experiments looking for rare events, such as dark matter
interactions or neutrinoless 2-beta decay. Neutrons can be produced either by
natural radioactivity, via spontaneous fission or (alpha,n) reactions, or by
interactions initiated by high-energy cosmic rays. In all underground
experiments, Monte Carlo simulations of neutron background play a crucial role
for the evaluation of the total background rate and for the optimization of
rejection strategies. The Monte Carlo methods that are commonly employed to
evaluate neutron-induced background and to optimize the experimental setup, are
reviewed and discussed. Focus is given to the issue of reliability of Monte
Carlo background estimates.Comment: 10 pages, 8 figures. Presented in the IV ILIAS Annual Meeting.
Accepted for publication on EPJ
Categorification of persistent homology
We redevelop persistent homology (topological persistence) from a categorical
point of view. The main objects of study are diagrams, indexed by the poset of
real numbers, in some target category. The set of such diagrams has an
interleaving distance, which we show generalizes the previously-studied
bottleneck distance. To illustrate the utility of this approach, we greatly
generalize previous stability results for persistence, extended persistence,
and kernel, image and cokernel persistence. We give a natural construction of a
category of interleavings of these diagrams, and show that if the target
category is abelian, so is this category of interleavings.Comment: 27 pages, v3: minor changes, to appear in Discrete & Computational
Geometr
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