Formalizing a Discrete Model of the Continuum in Coq from a Discrete Geometry Perspective

International audienceThis work presents a formalization of the discrete model of the continuum introduced by Harthong and Reeb, the Harthong-Reeb line. This model was at the origin of important developments in the Discrete Geometry field. The formalization is based on previous work by Chollet, Fuchs et al. where it was shown that the Harthong-Reeb line satisfies the axioms for constructive real numbers introduced by Bridges. Laugwitz-Schmieden numbers are then introduced and their limitations with respect to being a model of the Harthong-Reeb line is investigated. In this paper, we transpose all these definitions and properties into a formal description using the Coq proof assistant. We also show that Laugwitz-Schmieden numbers can be used to actually compute continuous functions. We hope that this work could improve techniques for both implementing numeric computations and reasoning about them in geometric systems

Foundational aspects of multiscale digitization

International audienceIn this article, we describe the theoretical foundations of the Ω-arithmetization. This method provides a multi-scale discretization of a continuous function that is a solution of a differential equation. This discretization process is based on the Harthong-Reeb line HRω. The Harthong-Reeb line is a linear space that is both discrete and continuous. This strange line HRω stems from a nonstandard point of view on arithmetic based, in this paper, on the concept of Ω-numbers introduced by Laugwitz and Schmieden. After a full description of this nonstandard background and of the first properties of HRω, we introduce the Ω-arithmetization and we apply it to some significant examples. An important point is that the constructive properties of our approach leads to algorithms which can be exactly translated into functional computer programs without uncontrolled numerical error. Afterwards, we investigate to what extent HRω fits Bridges's axioms of the constructive continuum. Finally, we give an overview of a formalization of the Harthong-Reeb line with the Coq proof assistant

Formalismes non classiques pour le traitement informatique de la topologie et de la géométrie discrète

The aim of this work is to introduce new theoretical basis for the discretization of continuous objects using non classical formalisms. This is done using a discrete model of the continuum called the Harthong-Reeb line together with the related arithmetization method which is a discretisation process of continuous functions. This study stands on a nonstandard arithmetical framework. Firstly, we use an axiomatic version of nonstandard arithmetic. In order to improve the constructive content of our method, the next step is to use another approach of nonstandard arithmetic deriving from the theory of Ω-numbers by Laugwitzand Schmieden. This second approach leads to a discrete multi-resolution representation of continuous functions. Afterwards, we investigate to what extent the Harthong-Reeb line fits Bridges axioms of the constructive continuum.L’objet de ce travail est l’utilisation de certains formalismes non classiques (analyses non standard, analyses constructives) afin de proposer des bases théoriques nouvelles autour des problèmes de discrétisations d’objets continus. Ceci est fait en utilisant un modèle discret du système des nombres réels appelé droite d’Harthong-Reeb ainsi que la méthode arithmétisation associée qui est un processus de discrétisation des fonctions continues. Cette étude repose sur un cadre arithmétique non standard. Dans un premier temps, nous utilisons une version axiomatique de l’arithmétique non standard. Puis, dans le but d’améliorer le contenu constructif de notre méthode, nous utilisons une autre approche de l’arithmétique non standard découlant de la théorie des Ω-nombres de Laugwitz et Schmieden. Cette seconde approche amène à une représentation discrète et multi-résolution de fonctions continues.Finalement, nous étudions dans quelles mesures, la droite d’Harthong-Reeb satisfait les axiomes de Bridges décrivant le continu constructif

Non classical formalisms for the computing treatment of the topoligy and the discrete geometry

L’objet de ce travail est l’utilisation de certains formalismes non classiques (analyses non standard, analyses constructives) afin de proposer des bases théoriques nouvelles autour des problèmes de discrétisations d’objets continus. Ceci est fait en utilisant un modèle discret du système des nombres réels appelé droite d’Harthong-Reeb ainsi que la méthode arithmétisation associée qui est un processus de discrétisation des fonctions continues. Cette étude repose sur un cadre arithmétique non standard. Dans un premier temps, nous utilisons une version axiomatique de l’arithmétique non standard. Puis, dans le but d’améliorer le contenu constructif de notre méthode, nous utilisons une autre approche de l’arithmétique non standard découlant de la théorie des Ω-nombres de Laugwitz et Schmieden. Cette seconde approche amène à une représentation discrète et multi-résolution de fonctions continues.Finalement, nous étudions dans quelles mesures, la droite d’Harthong-Reeb satisfait les axiomes de Bridges décrivant le continu constructif.The aim of this work is to introduce new theoretical basis for the discretization of continuous objects using non classical formalisms. This is done using a discrete model of the continuum called the Harthong-Reeb line together with the related arithmetization method which is a discretisation process of continuous functions. This study stands on a nonstandard arithmetical framework. Firstly, we use an axiomatic version of nonstandard arithmetic. In order to improve the constructive content of our method, the next step is to use another approach of nonstandard arithmetic deriving from the theory of Ω-numbers by Laugwitzand Schmieden. This second approach leads to a discrete multi-resolution representation of continuous functions. Afterwards, we investigate to what extent the Harthong-Reeb line fits Bridges axioms of the constructive continuum

A first look into a formal and constructive approach for discrete geometry using nonstandard analysis

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Insight in discrete geometry and computational content of a discrete model of the continuum

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A vascularized glioblastoma multiforme within a 3D perfused microphysiological system: combining a self-organized microvasculature and a central venule in a hydrogel

International audienceThe blood-brain barrier (BBB) hampers the development of innovative drugs and nanovectors to treat neuro-pathologies, as brain cancers. The most frequent and aggressive brain tumor is the glioblastoma multiforme (GBM). After diagnostic, the median patient survival is about 12 months. Less than 5% of diagnosed patients are still alive after 3 years. The 3D-Glimpse project develops an organ-on-chip-like in-vitro physiological microsystem, as an alternative to animal testing. The microchip is perfused to mimic the blood flow, and instrumented with biosensors for an integrated detection of nanocarriers transport. The project is divided into 4 objectives, with preliminary results and prototypes for the first 3 objectives. <br&gt1) Development of the BBB-on-chip. The combination of two methodologies allows the 3D coculture of human brain cells (HBMEC: human brain microvascular endothelial cells, HP: human pericytes, HA: human astrocytes). They organize as a vascularized tissue in a hydrogel of extracellular matrix (collagen microfibers and fibrin). The barrier function is validated under a physiologically pertinent shear stress. <br&gt2) Comparison of the BBB-on-chip in a healthy versus pathological context. The microchip is adapted to fit the microenvironment of a vascularized brain tumor, with adjunction of GBM cells (U87) to create a GBM-on-chip. <br&gt3) Instrumentation of the BBB-on-chip. Biosensing integration relies on functionalized piezo-electrical substrates (lithium niobate) for detecting the transport of nanocarriers through the BBB and a micro-processed porous membrane. <br&gt4) Screening of the therapeutic efficiency. Besides the transport, the nanocarriers will be screened for their delivery specificity, and their toxicity towards targeted and off-target cells. Dosage adjustments are expected to prevent the strong side-effects of those chemotherapeutic treatments