376 research outputs found
Volumes transverses aux feuilletages définissables dans des structures o-minimales
Let be a family of codimension foliations defined on a family of manifolds and let be a family of compact subsets of . Suppose that , and are definable in an o-minimal structure and that all leaves of are closed. Given a definable family of differential -forms satisfaying for any vector field tangent to , we prove that there exists a constant A >0 such that the integral of on any transversal of intersecting each leaf in at most one point is bounded by . We apply this result to prove that -volumes of transverse sections of are uniformly bounded
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
Molecular Shape Analysis based uponthe Morse-Smale Complexand the Connolly Function
Docking is the process by which two or several molecules form a complex. Docking involves the geometry of the molecular surfaces, as well as chemical and energetical considerations. In the mid-eighties, Connolly proposed a docking algorithm matching surface {\em knobs} with surface {\em depressions}. Knobs and depressions refer to the extrema of the {\em Connolly} function, which is defined as follows. Given a surface \calM bounding a three-dimensional domain , and a sphere centered at a point of \calM, the Connolly function is equal to the solid angle of the portion of containing within . We recast the notions of knob and depression of the Connolly function in the framework of Morse theory for functions defined over two-dimensional manifolds. First, we study the critical points of the Connolly function for smooth surfaces. Second, we provide an efficient algorithm for computing the Connolly function over a triangulated surface. Third, we introduce a Morse-Smale decomposition based on Forman's discrete Morse theory, and provide an algorithm to construct it. This decomposition induces a partition of the surface into regions of homogeneous flow, and provides an elegant way to relate local quantities to global ones ---from critical points to Euler's characteristic of the surface. Fourth, we apply this Morse-Smale decomposition to the discrete gradient vector field induced by Connolly's function, and present experimental results for several mesh models
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
Data-Driven Analysis of Pareto Set Topology
When and why can evolutionary multi-objective optimization (EMO) algorithms
cover the entire Pareto set? That is a major concern for EMO researchers and
practitioners. A recent theoretical study revealed that (roughly speaking) if
the Pareto set forms a topological simplex (a curved line, a curved triangle, a
curved tetrahedron, etc.), then decomposition-based EMO algorithms can cover
the entire Pareto set. Usually, we cannot know the true Pareto set and have to
estimate its topology by using the population of EMO algorithms during or after
the runtime. This paper presents a data-driven approach to analyze the topology
of the Pareto set. We give a theory of how to recognize the topology of the
Pareto set from data and implement an algorithm to judge whether the true
Pareto set may form a topological simplex or not. Numerical experiments show
that the proposed method correctly recognizes the topology of high-dimensional
Pareto sets within reasonable population size.Comment: 8 pages, accepted at GECCO'18 as a full pape
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
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
The observable structure of persistence modules
In persistent topology, q-tame modules appear as a natural and large class of persistence modules indexed over the real line for which a persistence diagram is de- finable. However, unlike persistence modules indexed over a totally ordered finite set or the natural numbers, such diagrams do not provide a complete invariant of q-tame modules. The purpose of this paper is to show that the category of persistence modules can be adjusted to overcome this issue. We introduce the observable category of persistence modules: a localization of the usual category, in which the classical properties of q-tame modules still hold but where the persistence diagram is a complete isomorphism invariant and all q-tame modules admit an interval decomposition
- …