Fonction implicite

Résumé

En mathématiques, une équation entre différentes variables où une variable n'est pas explicitée en fonction des autres est appelée une équation implicite. Une fonction implicite est une fonction qui se déduit implicitement d'une telle équation. Plus précisément si f est une fonction de E × F dans G, où E, F et G sont des espaces vectoriels normés ou plus simplement des intervalles de R, l'équation f(x,y) = 0 définit une fonction implicite si l'on peut exprimer une des variables en fonction de l'autre pour tous les couples (x,y) vérifiant l'équation. Ou encore, l'équation f(x,y) = 0 définit une fonction implicite de E vers F, s'il existe une fonction φ dite explicite telle que , pour tout (x,y) de E × F, f(x,y) = 0 équivaut à y = φ(x). Cela revient à dire que le graphe de la relation binaire : x R y ssi f(x,y) = 0 est le graphe d'une fonction. Il est parfois possible de prouver l'existence locale d'une fonction implicite pour une équation touchant deux variables réelles, sans l'exh

Source officielle

https://fr.wikipedia.org/wiki/Fonction_implicite

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Equations de type implicite avec contraintes

Chiara Tanteri

In this thesis we will treat the Dirichlet problem for systems of implicit equations, i.e. where Ω ⊂ Rn is an open set, u : Ω → Rm, Fi : Ω × Rm × Rm×n → R, m,n ≥ 1, are continuous functions and φ, the boundary datum, is given. At first we will be interested in the study of problems of the type (1) under constraints. We will show theorems of existence of Lipschitz solutions using an approach based on Baire category theorem. As corollaries of our abstract results, we will give two theorems related to the constraints det Du > 0 and det Du = 1, the constraints most closely related to the applications. Indeed, the constraints det Du > 0 and det Du = 1 come from nonlinear elasticity and represent respectively the conditions of non interpenetration of matter and incompressibility. In the second part of this study, we will focus on the applications. We will treat many examples such as the case of singular values, potential wells under the incompressibility constraint (i.e. det Du = l), the problem of confocal ellipses, the problem of nematic elastomers, particularly related to the fields of nonlinear elasticity, the microstructure of the crystals and the optimal design, as well as the complex eikonal equation (application related to geometric optics). From the mathematical point of view, we will give sufficient conditions of solvability of the system (1). These conditions consist in characterizing the different convex hulls. Indeed, the possibility of representing these sets, in algebraic terms, gives one of the conditions which the boundary datum must satisfy so that a problem of the type (1) admits a solution. Finally we will extend to polyconvex sets properties such as the gauge, the characterization of the extreme points and the Choquet function, all of which are well-known tools within the framework of classical convex analysis.

EPFL2000

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On algebraic extra-stress models for the simulation of viscoelastic flows

Michel Deville, Gilmar Mompean

A methodology is presented for the formulation of an algebraic extra-stress model derivable from the Oldroyd-B constitutive equation for a viscoelastic fluid. An implicit algebraic tensor relation is derived directly from the differential constitutive relation by involving a slow variation condition on the evolution of the deviatoric part of the extra-stress tensor. An explicit solution to this algebraic constitutive equation is then obtained using representation bases developed previously for the turbulent flow of Newtonian fluids. A series of validation flows are then studied to assess the accuracy and range of applicability of the algebraic extra-stress model. (C) 1998 Elsevier Science B.V. All rights reserved.

1998

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, ,

The human annotations are imperfect, especially when produced by junior practitioners. Multi-expert consensus is usually regarded as golden standard, while this annotation protocol is too expensive to implement in many real-world projects. In this study, we propose a method to refine human annotation, named Neural Annotation Refinement (NeAR). It is based on a learnable implicit function, which decodes a latent vector into represented shape. By integrating the appearance as an input of implicit functions, the appearance-aware NeAR fixes the annotation artefacts. Our method is demonstrated on the application of adrenal gland analysis. We first show that the NeAR can repair distorted golden standards on a public adrenal gland segmentation dataset. Besides, we develop a new Adrenal gLand ANalysis (ALAN) dataset with the proposed NeAR, where each case consists of a 3D shape of adrenal gland and its diagnosis label (normal vs. abnormal) assigned by experts. We show that models trained on the shapes repaired by the NeAR can diagnose adrenal glands better than the original ones. The ALAN dataset will be open-source, with 1,594 shapes for adrenal gland diagnosis, which serves as a new benchmark for medical shape analysis. Code and dataset are available at https://github.com/M3DV/NeAR.

SPRINGER INTERNATIONAL PUBLISHING AG2022

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