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In mathematics, an integral is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of calculus, the other being differentiation. Integration started as a method to solve problems in mathematics and physics, such as finding the area under a curve, or determining displacement from velocity. Today integration is used in a wide variety of scientific fields. The integrals enumerated here are called definite integrals, which can be interpreted as the signed area of the region in the plane that is bounded by the graph of a given function between two points in the real line. Conventionally, areas above the horizontal axis of the plane are positive while areas below are negative. Integrals also refer to the concept of an antiderivative, a function whose derivative is the given function; in this case, they are also called indefinite integrals. The fund

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Le "Fragmentum de inventionibus scientiarum" de Diego Palomino

Nouara Chekhar

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

Intégration d'un odomètre optique dans un système de mobile mapping

L'objectif de ce travail de Master fut l'intégration d'un odomètre optique dans un système de cartographie mobile développé par l'Unité de Topométrie. Ce système intègre deux récepteurs GPS, une centrale inertielle et une caméra CCD. Deux étapes ponctuent ce travail. La première consiste en l'intégration technique, comprenant l'acquisition des données issues de l'odomètre et leur synchronisation avec les mesures des autres capteurs. La deuxième étape est l'intégration mathématique des données dans le processus de compensation et de filtrage des mesures de navigation.

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Fluctuation estimates for the multi-cell formula in stochastic homogenization of partitions

Matthias Ruf

In this paper we derive quantitative estimates in the context of stochastic homogenization for integral functionals defined on finite partitions, where the random surface integrand is assumed to be stationary. Requiring the integrand to satisfy in addition a multiscale functional inequality, we control quantitatively the fluctuations of the asymptotic cell formulas defining the homogenized surface integrand. As a byproduct we obtain a simplified cell formula where we replace cubes by almost flat hyperrectangles.

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Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. It has two major br

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Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These top

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In mathematics, the derivative shows the sensitivity of change of a function's output with respect to the input. Derivatives are a fundamental tool of calculus. For example, the derivative of the p

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