Axiome du choix

vignette|upright=1.5|Pour tout ensemble d'ensembles non vides (les jarres), il existe une fonction qui associe à chacun de ces ensembles (ces jarres) un élément contenu dans cet ensemble (cette jarre). En mathématiques, l'axiome du choix, abrégé en « AC », est un axiome de la théorie des ensembles qui Il a été formulé pour la première fois par Ernest Zermelo en 1904 pour la démonstration du théorème de Zermelo. L'axiome du choix peut être accepté ou rejeté, selon la théorie axiomatique des ensembles choisie. Énoncé L'axiome du choix peut s'énoncer comme suit : :(0) « Pour tout ensemble X d'ensembles non vides, il existe une fonction définie sur X, appelée fonction de choix, qui à chaque ensemble A appartenant à X associe un élément de cet ensemble A. » ce qui s'écrit formellement : \forall X\left[\varnothing\notin X\Longrightarrow\exists f:X\rightarrow \bigcup X;\forall A\in X;(f(A)\in A)\right]. L'appel à cet axiome n'est pas nécessaire si X est un ensemb

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Consistent Digital Line Segments

Doemoetoer Palvoelgyi

We introduce a novel and general approach for digitalization of line segments in the plane that satisfies a set of axioms naturally arising from Euclidean axioms. In particular, we show how to derive such a system of digital segments from any total order on the integers. As a consequence, using a well-chosen total order, we manage to define a system of digital segments such that all digital segments are, in Hausdorff metric, optimally close to their corresponding Euclidean segments, thus giving an explicit construction that resolves the main question of Chun et al. (Discrete Comput. Geom. 42(3):359-378, 2009).

2012

Chargement

pyFFS: A Python Library for Fast Fourier Series Computation and Interpolation with GPU Acceleration

Eric Bezzam, Paul Hurley, Sepand Kashani, Matthieu Martin Jean-André Simeoni, Martin Vetterli

Fourier transforms are an often necessary component in many computational tasks, and can be computed efficiently through the fast Fourier transform (FFT) algorithm. However, many applications involve an underlying continuous signal, and a more natural choice would be to work with e.g. the Fourier series (FS) coefficients in order to avoid the additional overhead of translating between the analog and discrete domains. Unfortunately, there exists very little literature and tools for the manipulation of FS coefficients from discrete samples. This paper introduces a Python library called pyFFS for efficient FS coefficient computation, convolution, and interpolation. While the libraries SciPy and NumPy provide efficient functionality for discrete Fourier transform coefficients via the FFT algorithm, pyFFS addresses the computation of FS coefficients through what we call the fast Fourier series (FFS). Moreover, pyFFS includes an FS interpolation method based on the chirp Z-transform that can make it more than an order of magnitude faster than the SciPy equivalent when one wishes to perform interpolation. GPU support through the CuPy library allows for further acceleration, e.g. an order of magnitude faster for computing the 2-D FS coefficients of 1000 x 1000 samples and nearly two orders of magnitude faster for 2-D interpolation. As an application, we discuss the use of pyFFS in Fourier optics. pyFFS is available as an open source package at https://github.com/imagingofthings/pyFFS, with documentation at https://pyffs.readthedocs.io.

2022

Chargement

An overview of a new sensor calibration platform

Philipp Clausen, Stéphane Guerrier, Jan Skaloud

Inertial sensors are increasingly being employed in different types of applications. The reduced cost and the extremely small size makes them the number-one-choice in miniature embedded devices like phones, watches, and small unmanned aerial vehicles. The more complex the application, the more it is necessary to understand the structure of the error signal coming from these sensors. Indeed, their error signals are composed of deterministic and stochastic parts. The deterministic errors or faults can be compensated by proper calibration while the stochastic signal is usually ignored since its modeling is relatively difficult due to computational or statistical reasons, especially due to its complex spectral structure. However, a recently proposed approach called the Generalized Method of Wavelet Moments overcomes these limitations and this paper presents the software platform that implements this method for the analysis of the stochastic errors. As an example throughout the paper we will consider an inertial measurement unit, but the platform can be used for the stochastic calibration of any kind of sensor. The software is developed in the widely used statistical tool R using C++ language. The tools enable the user to study with ease any signal by the means of a vast range of predefined models and tools.

2017

Chargement

MATH-101(en): Analysis I (English)

We study the fundamental concepts of analysis, calculus and the integral of real-valued functions of a real variable.

MATH-604: Some Aspects of Calculus of Variations

The goal of this course is to present an overview on the solvability and the regularity of relevant models of physical, technological and economical systems, which may be formulated as minimization problems of suitable integral functionals.

CS-101: Advanced information, computation, communication I

Discrete mathematics is a discipline with applications to almost all areas of study. It provides a set of indispensable tools to computer science in particular. This course reviews (familiar) topics as diverse as mathematical reasoning, combinatorics, discrete structures & algorithmic thinking.

Théorie des ensembles

La théorie des ensembles est une branche des mathématiques, créée par le mathématicien allemand Georg Cantor à la fin du . La théorie des ensembles se donne comme primitives les notions d'ensemble e

Nombre réel

En mathématiques, un nombre réel est un nombre qui peut être représenté par une partie entière et une liste finie ou infinie de décimales. Cette définition s'applique donc aux nombres rationnels, don

Nombre cardinal

vignette|Le nombre cardinal des deux ensembles X et Y est 4 En linguistique, les nombres entiers naturels zéro, un, deux, trois, etc. s’appellent des adjectifs numéraux cardinaux. En théorie des ense