Domain theory and differential calculus (functions of one variable)

Published versio

Geometry-driven collapses for converting a Cech complex into a triangulation of a nicely triangulable shape

Given a set of points that sample a shape, the Rips complex of the data points is often used in machine-learning to provide an approximation of the shape easily-computed. It has been proved recently that the Rips complex captures the homotopy type of the shape assuming the vertices of the complex meet some mild sampling conditions. Unfortunately, the Rips complex is generally high-dimensional. To remedy this problem, it is tempting to simplify it through a sequence of collapses. Ideally, we would like to end up with a triangulation of the shape. Experiments suggest that, as we simplify the complex by iteratively collapsing faces, it should indeed be possible to avoid entering a dead end such as the famous Bing's house with two rooms. This paper provides a theoretical justification for this empirical observation. We demonstrate that the Rips complex of a point-cloud (for a well-chosen scale parameter) can always be turned into a simplicial complex homeomorphic to the shape by a sequence of collapses, assuming the shape is nicely triangulable and well-sampled (two concepts we will explain in the paper). To establish our result, we rely on a recent work which gives conditions under which the Rips complex can be converted into a Cech complex by a sequence of collapses. We proceed in two phases. Starting from the Cech complex, we first produce a sequence of collapses that arrives to the Cech complex, restricted by the shape. We then apply a sequence of collapses that transforms the result into the nerve of some robust covering of the shape.Comment: 24 pages, 9 figure

Lexicographic optimal homologous chains and applications to point cloud triangulations

This paper considers a particular case of the Optimal Homologous Chain Problem (OHCP), where optimality is meant as a minimal lexicographic order on chains induced by a total order on simplices. The matrix reduction algorithm used for persistent homology is used to derive polynomial algorithms solving this problem instance, whereas OHCP is NP-hard in the general case. The complexity is further improved to a quasilinear algorithm by leveraging a dual graph minimum cut formulation when the simplicial complex is a strongly connected pseudomanifold. We then show how this particular instance of the problem is relevant, by providing an application in the context of point cloud triangulation

Regular triangulations as lexicographic optimal chains

We introduce a total order on n-simplices in the n-Euclidean space for which the support of the lexicographic-minimal chain with the convex hull boundary as boundary constraint is precisely the n-dimensional Delaunay triangulation, or in a more general setting, the regular triangulation of a set of weighted points. This new characterization of regular and Delaunay triangulations is motivated by its possible generalization to submanifold triangulations as well as the recent development of polynomial-time triangulation algorithms taking advantage of this order

Feeding and oviposition preferences of Monochamus galloprovincialis on its main hosts Pinus sylvestris and Pinus pinaster

Θεωρώντας τον ρόλο του Monochamus galloprovincialis κλειδί στην εξάπλωση του νηματώδη του πεύκου στην Ευρώπη, η διαλεύκανση του εύρους των ξενιστών και των προτιμήσεων του Μ. galloprovincialis σε αυτούς είναι πρωτεύουσας σημασίας για την προστασία του δάσους. Συγκριτικές μελέτες σε συνθήκες εργαστηρίου διεξήχθησαν με σκοπό την διευκρίνηση των διατροφικών και ωοθετικών προτιμήσεων των ενηλίκων αυτού του εντόμου καθώς και της ανάπτυξης των προνυμφών του στα P. sylvestris και P. pinaster. Η διατροφή και η ωοτοκία βρέθηκαν στατιστικά πιο σημαντικές στο P. sylvestris αλλά καμία διαφορά δεν βρέθηκε στην επιβίωση των προνυμφών στα δύο είδη πεύκου. Οι προνύμφες έφτασαν στην 4η ηλικία νωρίτερα στο P. sylvestris από ότι στο P. pinaster. Τα αποτελέσματα της παρούσης μελέτης ισχυροποιούν τις υποψίες για μια γρήγορη μελλοντική εξάπλωση του νηματώδη σε ολόκληρη την Ευρώπη.Considering the key role of Monochamus galloprovincialis (Olivier) (Coleoptera: Cerambycidae) in the dispersion of the pine wood nematode, Bursaphelenchus xylophilus (Steiner & Bührer) Nickle (Nematoda: Aphelenchoidea), in Europe, defining the host spectrum and preference of M. galloprovincialis is of primary importance for forest management. Therefore, comparative studies under laboratory conditions were performed in order to define the feeding and oviposition preferences of M. galloprovincialis adults as well as the larval development on Pinus sylvestris and Pinus pinaster. Both nutrition and oviposition were significantly higher on P. sylvestris. However, no difference was found when comparing larval survival on the two Pinus species even though larvae attended the fourth instar earlier when reared on P. sylvestris than on P. pinaster. The results of this study reinforce the suspicions of a future rapid propagation of the nematode, by M. galloprovincialis, into the Pine forests all over Europe

Feeding and oviposition preferences of Monochamus galloprovincialis on its main hosts Pinus sylvestris and Pinus pinaster

Θεωρώντας τον ρόλο του Monochamus galloprovincialis κλειδί στην εξάπλωση του νηματώδη του πεύκου στην Ευρώπη, η διαλεύκανση του εύρους των ξενιστών και των προτιμήσεων του Μ. galloprovincialis σε αυτούς είναι πρωτεύουσας σημασίας για την προστασία του δάσους. Συγκριτικές μελέτες σε συνθήκες εργαστηρίου διεξήχθησαν με σκοπό την διευκρίνηση των διατροφικών και ωοθετικών προτιμήσεων των ενηλίκων αυτού του εντόμου καθώς και της ανάπτυξης των προνυμφών του στα P. sylvestris και P. pinaster. Η διατροφή και η ωοτοκία βρέθηκαν στατιστικά πιο σημαντικές στο P. sylvestris αλλά καμία διαφορά δεν βρέθηκε στην επιβίωση των προνυμφών στα δύο είδη πεύκου. Οι προνύμφες έφτασαν στην 4η ηλικία νωρίτερα στο P. sylvestris από ότι στο P. pinaster. Τα αποτελέσματα της παρούσης μελέτης ισχυροποιούν τις υποψίες για μια γρήγορη μελλοντική εξάπλωση του νηματώδη σε ολόκληρη την Ευρώπη.Considering the key role of Monochamus galloprovincialis (Olivier) (Coleoptera: Cerambycidae) in the dispersion of the pine wood nematode, Bursaphelenchus xylophilus (Steiner & Bührer) Nickle (Nematoda: Aphelenchoidea), in Europe, defining the host spectrum and preference of M. galloprovincialis is of primary importance for forest management. Therefore, comparative studies under laboratory conditions were performed in order to define the feeding and oviposition preferences of M. galloprovincialis adults as well as the larval development on Pinus sylvestris and Pinus pinaster. Both nutrition and oviposition were significantly higher on P. sylvestris. However, no difference was found when comparing larval survival on the two Pinus species even though larvae attended the fourth instar earlier when reared on P. sylvestris than on P. pinaster. The results of this study reinforce the suspicions of a future rapid propagation of the nematode, by M. galloprovincialis, into the Pine forests all over Europe

Modélisation de la distribution géographique de deux scolytes, Tomicus destruens et Tomicus piniperda, en Europe et dans la région méditerranéenne

Article publié suite à l'événement : MEDINSECT 3 ; Hammamet-Tunis (Tunisie) - (2012-05-08 - 2012-05-11).Preliminary note – Data and results presented in that paper come from the original article: Horn A., Kerdelhué C., Lieutier F., Rossi J.P., 2012. Predicting the distribution of the two bark beetles Tomicus destruens and Tomicus piniperda in Europe and the Mediterranean region. Agricultural and Forest Entomology, 14, 358-366.Various factors such as climate and resource availability influence the geographical distributions of organisms. Species that are susceptible to small temperature variations are known to experience rapid distribution shifts as a result of current global warming, sometimes leading to new threats to agriculture and forests. Tomicus destruens and Tomicus piniperda (Coleoptera, Curculionidae, Scolytinae) cause economic damage to pines in Europe and around the Mediterranean Basin. However, their respective potential distributions have not yet been studied at a large scale, mostly because these two species have long been misidentified. The present study aimed to investigate the influence of climatic factors on the geographical distributions of both Tomicus species in Europe and around the Mediterranean Sea, and to establish maps of suitable areas. We used occurrence data from 114 published localities where the presence or absence of both species was unambiguously recorded and confirmed by molecular data, and we gathered WorldClim meteorological records to link the occurrence of insects and climatic data and to build potential distribution maps. The two studied Tomicus species presented parapatric distributions and opposite climate requirements. T. destruens occurs in locations with warmer temperatures, whereas T. piniperda occurs under colder, continental climates. Amongst the investigated climatic variables, temperature appeared to be most correlated with both species distributions. We further extended our approach to explore potential geographical distributions under climate change scenarios. This showed that the distribution of both species is expected to exhibit strong alteration in the near future (2080) corresponding to a marked expansion of T. destruens towards northern Europe and a retractation and fragmentation of the distribution of T. piniperda.Résumé - Différents facteurs, tels que le climat ou la disponibilité de certaines ressources, affectent la distribution géographique des organismes. Aujourd’hui, les espèces sensibles à de faibles variations de température connaissent des modifications de leur aire de distribution sous l’effet du changement climatique, avec parfois de nouveaux risques pour l’agriculture et les forêts. Tomicus destruens et Tomicus piniperda (Coleoptera, Curculionidae, Scolytinae) causent des dégâts importants aux pins en Europe et sur le pourtour méditerranéen mais leur aire de distribution potentielle à une large échelle est encore mal connue, en particulier parce que ces deux espèces ont longtemps été confondues. L’étude présentée ici a pour objectif de mieux comprendre l’impact des variables climatiques sur la distribution géographique des deux espèces et d’établir des cartes de distribution potentielle, actuelles et futures. Nous avons utilisé 114 données d’occurrence publiées pour lesquelles la présence et l’absence des deux espèces sont avérées par des diagnostiques moléculaires. Nous avons utilisé les données climatiques disponibles dans la base « WorldClim » afin d’ajuster des modèles linéaires généralisés pour modéliser les relations climat – occurrences de T. destruens et T. piniperda. Les résultats obtenus montrent que les deux espèces ont des distributions parapatriques et des préférences climatiques différentes. T. destruens est présent dans les zones les plus chaudes tandis que T. piniperda occupe les régions les plus froides ayant probablement des saisons marquées. Nous avons utilisé les modèles précédents pour réaliser une projection permettant d’estimer la distribution potentielle de chaque espèce dans le cadre du scénario de changement climatique IPCC 4 du CIAT (modèle CCCM avec un scénario d’émission a2a). Les résultats indiquent que les distributions des deux espèces devraient connaître d’importants changements à court terme (2080). Ainsi, T. destruens remonterait vers le nord de l’Europe tandis que l’aire de distribution de T. piniperda serait à la fois réduite et très fragmentée

Stability of Curvature Measures

We address the problem of curvature estimation from sampled compact sets. The main contribution is a stability result: we show that the gaussian, mean or anisotropic curvature measures of the offset of a compact set K with positive $\mu$-reach can be estimated by the same curvature measures of the offset of a compact set K' close to K in the Hausdorff sense. We show how these curvature measures can be computed for finite unions of balls. The curvature measures of the offset of a compact set with positive $\mu$-reach can thus be approximated by the curvature measures of the offset of a point-cloud sample. These results can also be interpreted as a framework for an effective and robust notion of curvature