Volumes transverses aux feuilletages définissables dans des structures o-minimales

Let $\mathcal{F}_\lambda$ be a family of codimension $p$ foliations defined on a family $M_\lambda$ of manifolds and let $X_\lambda$ be a family of compact subsets of $M_\lambda$. Suppose that $\mathcal{F}_\lambda$, $M_\lambda$ and $X_\lambda$ are definable in an o-minimal structure and that all leaves of $\mathcal{F}_\lambda$ are closed. Given a definable family $\Omega_\lambda$ of differential $p$-forms satisfaying $i_Z\Omega_\lambda = 0$ for any vector field $Z$ tangent to $\mathcal{F}_\lambda$, we prove that there exists a constant A >0 such that the integral of $|\Omega_\lambda|$ on any transversal of $\mathcal{F}_\lambda$ intersecting each leaf in at most one point is bounded by $A$. We apply this result to prove that $p$-volumes of transverse sections of $\mathcal{F}_\lambda$ 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 $O(n^4)$ arithmetic operations, where $n$ 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

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

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 $X$, and a sphere $S$ centered at a point $p$ of \calM, the Connolly function is equal to the solid angle of the portion of $S$ containing within $X$. 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 $O(n\log n)$ 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

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