Fonction rationnelle

Résumé

En mathématiques, une fonction rationnelle est une fonction définie par une fraction rationnelle, c'est-à-dire une dont le numérateur et le dénominateur sont des polynômes. Définition En pratique, l'ensemble de définition est généralement \R (ensemble des réels) ou \mathbb C (ensemble des complexes). Si P et Q sont deux fonctions polynomiales et si Q n'est pas une fonction nulle, la fonction f = \frac{P}{Q} est définie pour tout x tel que Q(x) ≠ 0 par :f(x) = \frac{P(x)}{Q(x)} Une fonction qui n'est pas rationnelle est dite irrationnelle. On parle de fonction rationnelle propre quand le degré du polynôme P est inférieur à celui de Q. Domaine de définition Toute fonction polynomiale non nulle Q est acceptable mais la possibilité que pour un a donné, Q(a) = 0 implique que contrairement aux fonctions poly

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

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We define and study in terms of integral IwahoriâHecke algebras a new class of geometric operators acting on the Bruhat-Tits building of connected reductive groups over p-adic fields. These operators, which we call U-operators, generalize the geometric notion of "successors" for trees with a marked end. The first main contributions of the thesis are: (i) the integrality of the U-operator over the spherical Hecke algebra using the compatibility between Bernstein and Satake homomorphisms, (ii) in the unramified case, the U-operator attached to a cocharacter is a right root of the corresponding Hecke polynomial. In the second part of the thesis, we study some arithmetic aspects of special cycles on (products of) unitary Shimura varieties, these cycles are expected to yield new results towards the BlochâBeilinson conjectures. As a global application of (ii), we obtain: (iii) the horizontal norm relations for these GGP cycles for arbitrary n, at primes where the unitary group splits. The general local theory developed in the first part of the thesis, has the potential to result in a number of global applications along the lines of (iii) (involving other Shimura varieties and also vertical norm relations) and offers new insights into topics such as the BlasiusâRogawski conjecture as well.

EPFL2019

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Given a geometrically unirational variety over an infinite base field, we show that every finite separable extension of the base field that splits the variety is the residue field of a closed point. As an application, we obtain a characterization of function fields of smooth conics in which every sum of squares is a sum of two squares.

Springer-Verlag2013

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Alignment of uncalibrated images for multi-view classification

Ömer Sercan Arik, Pascal Frossard, Elif Vural

Efficient solutions for the classification of multi-view images can be built on graph-based algorithms when little information is known about the scene or cameras. Such methods typically require a pairwise similarity measure between images, where a common choice is the Euclidean distance. However, the accuracy of the Euclidean distance as a similarity measure is restricted to cases where images are captured from nearby viewpoints. In settings with large transformations and viewpoint changes, alignment of images is necessary prior to distance computation. We propose a method for the registration of uncalibrated images that capture the same 3D scene or object. We model the depth map of the scene as an algebraic surface, which yields a warp model in the form of a rational function between image pairs. The warp model is computed by minimizing the registration error, where the registered image is a weighted combination of two images generated with two different warp functions estimated from feature matches and image intensity functions in order to provide robust registration. We demonstrate the flexibility of our alignment method by experimentation on several wide-baseline image pairs with arbitrary scene geometries and texture levels. Moreover, the results on multi-view image classification suggest that the proposed alignment method can be effectively used in graph-based classification algorithms for the computation of pairwise distances where it achieves significant improvements over distance computation without prior alignment.

2011

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