Fonction (mathématiques)

Résumé

vignette|Diagramme de calcul pour la fonction x \mapsto \frac{2x-1}{x+3} En mathématiques, une fonction permet de définir un résultat (le plus souvent numérique) pour chaque valeur d’un ensemble appelé domaine. Ce résultat peut être obtenu par une suite de calculs arithmétiques ou par une liste de valeurs, notamment dans le cas de relevé de mesures physiques, ou encore par d’autres procédés comme les résolutions d’équations ou les passages à la limite. Le calcul effectif du résultat ou son approximation repose éventuellement sur l’élaboration de fonction informatique. Dans l’enseignement scolaire, le terme « fonction » concerne spécifiquement les fonctions réelles d’une variable réelle. De nombreuses fonctions dites usuelles sont ainsi définies comme les fonctions affines, la racine carrée ou l’exponentielle, et peuvent être combinées à l’aide des opérations arithmétiques, de la composition ou de la définition par morceaux. Ces fonctions satisfont diverses propriétés po

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https://fr.wikipedia.org/wiki/Fonction_(math%C3%A9matiques)

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Étudier les concepts fondamentaux d'analyse, et le calcul différentiel et intégral des fonctions réelles de plusieurs variables.

Étudier les concepts fondamentaux d'analyse et le calcul différentiel et intégral des fonctions réelles de plusieurs variables.

Étudier les concepts fondamentaux d'analyse et le calcul différentiel et intégral des fonctions réelles d'une variable.

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Shape and deformation measurements of large objects by fringe projection

Anne-Isabelle Desmangles

The height and/or deformation of a surface of an object (10) defined in terms of x, y and z coordinates is measured by projecting a divergent beam of light 21) onto the surface to produce a periodic pattern of fringes (30. A CCD camera (40) views the surface from another angle and images the reflected light. Phase information is extracted from at least one image to produce an optical print (36) of the object containing phase information and image-coordinates information both being related to camera pixel position, the image-coordinates being associated with phase information corresponding to the object coordinates. A processor (50) calculates the object-coordinates of the surface points through mathematical functions describing x, y and z. These mathematical functions take account of the spatial configuration and specifications of the system which are established by calibration procedures involving eg. measurements of a theodolite (60) .

2004

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The objective of this PhD thesis is the translation of, and the mathematical commentary on, a 16th-century Latin book. Its author, Diego Palomino is not well known. With a background in theology, he was a prior. In order to obtain his PhD at the University of Alcala (Madrid), he submitted a work, De mutations æris, in which he included a collection of what seems to be readings notes, entitled Fragmentum de inventionibus scientiarum. His readings have been drawn from various famous mathematicians of his time and ancient ones. The originality of his work relies mostly on his inventiveness and his style —which can be sarcastic— making the reading of it quite interesting and lively. His work consists for the main part in explaining some unclear demonstrations, or bringing new methods of solution ; he also innovates in solving pairs of indeterminate equations by providing the complete set of integral solutions. Before him, only one mathematician (Abu ̄ K ̄amil, at the end of the 10th century) did so, the other mathematicians restricting themselves to giving only one solution or a pair. Palomino did not hesitate in criticizing a well-established theory in ancient mathematics, namely Archimedes —although his critics seem to rely on a faulty edition. His book is entitled to have a significant place in the history of mathematics, for it both maintains the rigor of Greek classical mathematics and announces innovation, as did the 17th century, culminating with the discovery of the infinitesimal calculus.

EPFL2012

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Mixed-precision explicit stabilized Runge-Kutta methods for single- and multi-scale differential equations

Matteo Croci, Giacomo Rosilho De Souza

Mixed-precision algorithms combine low-and high-precision computations in order to benefit from the performance gains of reduced-precision without sacrificing accuracy. In this work, we design mixed-precision Runge-Kutta-Chebyshev (RKC) methods, where high precision is used for accuracy, and low precision for stability. Generally speaking, RKC methods are low-order explicit schemes with a stability domain growing quadratically with the number of function evaluations. For this reason, most of the computational effort is spent on stability rather than accuracy purposes. In this paper, we show that a naive mixed-precision implementation of any Runge-Kutta scheme can harm the convergence order of the method and limit its accuracy, and we introduce a new class of mixed precision RKC schemes that are instead unaffected by this limiting behavior. We present three mixed-precision schemes: a first-and a second-order RKC method, and a first order multirate RKC scheme for multiscale problems. These schemes perform only the few function evaluations needed for accuracy (1 or 2 for first-and second-order methods respectively) in high precision, while the rest are performed in low precision. We prove that while these methods are essentially as cheap as their fully low-precision equivalent, they retain the stability and convergence order of their high-precision counterpart. Indeed, numerical experiments confirm that these schemes are as accurate as the corresponding high-precision method. (C) 2022 Elsevier Inc. All rights reserved.

ACADEMIC PRESS INC ELSEVIER SCIENCE2022

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