Des méthodes d’intégration par arcs de sections coniques aux échelles de modules. Legendre lecteur de Landen

Dans cet article, nous analysons en détail les deux mémoires de Legendre, Sur les intégrations par arcs d’ellipse (1786a, 1786b), en comparant les deux démonstrations qu’il donne de la rectification de l’hyperbole au moyen de deux arcs d’ellipse. Entre le premier et le second mémoire, Legendre prend connaissance d’un théorème analogue de Landen, qu’il cherche à incorporer à son projet de constitution d’un calcul des arcs d’ellipse et dont il propose une interprétation originale en le déduisant de ses propres principes. En retra?ant la préhistoire de la transformation de Landen, des premières méthodes d’intégration par arcs de sections coniques chez Maclaurin, d’Alembert et Landen jusqu’à l’élaboration par Legendre des échelles de modules, nous cherchons à mettre en lumière les conditions de son identification. Nous montrons comment Legendre fut amené à discerner dans le théorème de Landen ce qui entre ses mains allait devenir l’instrument privilégié de la mise en ordre systématique des transcendantes elliptiques, grace à une lecture orientée par les exigences spécifiques liées à une méthode d’approximation fondée sur la variation continue des paramètres et à la constitution de tables numériques d’arcs d’ellipse.     

Tuning up mind's pattern to nature's own idea: Eddington's early twenties case for variational derivatives

This paper sets out to show how Eddington's early twenties case for variational derivatives significantly bears witness to a steady and consistent shift in focus from a resolute striving for objectivity towards “selective subjectivism” and structuralism. While framing his so-called “Hamiltonian derivatives” along the lines of previously available variational methods allowing to derive gravitational field equations from an action principle, Eddington assigned them a theoretical function of his own devising in The Mathematical Theory of Relativity (1923). I make clear that two stages should be marked out in Eddington's train of thought if the meaning of such variational derivatives is to be adequately assessed. As far as they were originally intended to embody the mind's collusion with nature by linking atomicity of matter with atomicity of action, variational derivatives were at first assigned a dual role requiring of them not only to express mind's craving for permanence but also to tune up mind's privileged pattern to “Nature's own idea”. Whereas at a later stage, as affine field theory would provide a framework for world-building, such “Hamiltonian differentiation” would grow out of tune through gauge-invariance and, by disregarding how mathematical theory might precisely come into contact with actual world, would be turned into a mere heuristic device for structural knowledge.     

Human endoglin as a potential new partner involved in platelet–endothelium interactions

Complex interactions between platelets and activated endothelium occur during the thrombo-inflammatory reaction at sites of vascular injuries and during vascular hemostasis. The endothelial receptor endoglin is involved in inflammation through integrin-mediated leukocyte adhesion and transmigration; and heterozygous mutations in the endoglin gene cause hereditary hemorrhagic telangiectasia type 1. This vascular disease is characterized by a bleeding tendency that is postulated to be a consequence of telangiectasia fragility rather than a platelet defect, since platelets display normal functions in vitro in this condition. Here, we hypothesize that endoglin may act as an adhesion molecule involved in the interaction between endothelial cells and platelets through integrin recognition. We find that the extracellular domain of human endoglin promotes specific platelet adhesion under static conditions and confers resistance of adherent platelets to detachment upon exposure to flow. Also, platelets adhere to confluent endothelial cells in an endoglin-mediated process. Remarkably, Chinese hamster ovary cells ectopically expressing the human αIIbβ3 integrin acquire the capacity to adhere to myoblast transfectants expressing human endoglin, whereas platelets from Glanzmann’s thrombasthenia patients lacking the αIIbβ3 integrin are defective for endoglin-dependent adhesion to endothelial cells. Furthermore, the bleeding time, but not the prothrombin time, is significantly prolonged in endoglin-haplodeficient (Eng ^+/?) mice compared to Eng ^+/+ animals. These results suggest a new role for endoglin in αIIbβ3 integrin-mediated adhesion of platelets to the endothelium, and may provide a better understanding on the basic cellular mechanisms involved in hemostasis and thrombo-inflammatory events.