Fonction spéciale

Résumé

L'analyse mathématique regroupe sous le terme de fonctions spéciales un ensemble de fonctions analytiques non élémentaires, qui sont apparues au comme solutions d'équations de la physique mathématique, particulièrement les équations aux dérivées partielles d'ordre deux et quatre. Comme leurs propriétés ont été étudiées extensivement (et continuent de l'être), on dispose à leur sujet d'une multitude d'informations. Non seulement elles interviennent dans l'expression des solutions exactes de certaines équations aux dérivées partielles pour des conditions aux limites particulières, mais elles fournissent, par le biais des méthodes spectrales, les meilleures approximations numériques pour des conditions aux limites quelconques. Certaines d'entre elles jouent également un rôle de premier plan en théorie des nombres (fonction zêta de Riemann, logarithme intégral). Liste des fonctions spéciales Liste des fonctions spéciales

Les intégrales d'Euler : la fonction bêta, la fonc

Source officielle

https://fr.wikipedia.org/wiki/Fonction_sp%C3%A9ciale

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Bin Packing via Discrepancy of Permutations

Friedrich Eisenbrand, Doemoetoer Palvoelgyi, Thomas Rothvoss

A well-studied special case of bin packing is the 3-partition problem, where n items of size > 1/4 have to be packed in a minimum number of bins of capacity one. The famous Karmarkar-Karp algorithm transforms a fractional solution of a suitable LP relaxation for this problem into an integral solution that requires at most O(log n) additional bins. The three-permutations-problem of Beck is the following. Given any three permutations on n symbols, color the symbols red and blue, such that in any interval of any of those permutations, the number of red and blue symbols is roughly the same. The necessary difference is called the discrepancy. We establish a surprising connection between bin packing and Beck's problem: The additive integrality gap of the 3-partition linear programming relaxation can be bounded by the discrepancy of three permutations. This connection yields an alternative method to establish an O(log n) bound on the additive integrality gap of the 3-partition. Conversely, making use of a recent example of three permutations, for which a discrepancy of Omega(log n) is necessary, we prove the following: The O(log(2) n) upper bound on the additive gap for bin packing with arbitrary item sizes cannot be improved by any technique that is based on rounding up items. This lower bound holds for a large class of algorithms including the Karmarkar-Karp procedure.

Association for Computing Machinery2013

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A modular strategy for next-generation upper-limb sensory-motor neuroprostheses

Silvestro Micera, Giuseppe Schiavone, Solaiman Shokur

Neuroprosthetics is a discipline that aims at restoring lost functions to people affected by a variety of neurological disorders or neuro-traumatic lesions. It combines the expertise of computer science and electrical, mechanical, and micro/nanotechnology with cellular, molecular, and systems neuroscience. Rapid breakthroughs in the field during the past decade have brought the hope that neuroprostheses can soon become a clinical reality, in particular-as we will detail in this review-for the restoration of hand functions. We argue that any neuroprosthesis relies on a set of hardware and algorithmic building elements that we call the neurotechnological modules (NTs) used for motor decoding, movement restoration, or sensory feedback. We will show how the modular approach is already present in current neuroprosthetic solutions and how we can further exploit it to imagine the next generation of neuroprosthetics for sensory-motor restoration.

ELSEVIER2021

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Gelfand pairs admit an Iwasawa decomposition

Nicolas Monod

Every Gelfand pair (G, K) admits a decomposition G = K P, where P < G is an amenable subgroup. In particular, the Furstenberg boundary of G is homogeneous. Applications include the complete classification of non-positively curved Gelfand pairs, relying on earlier joint work with Caprace, as well as a canonical family of pure spherical functions in the sense of Gelfand-Godement for general Gelfand pairs.

SPRINGER HEIDELBERG2020

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