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Provence (prəˈvɒ̃s, USalsoprəʊˈ-, UKalsoprɒˈ-, pʁɔvɑ̃s) is a geographical region and historical province of southeastern France, which extends from the left bank of the lower Rhône to the west to the Italian border to the east; it is bordered by the Mediterranean Sea to the south. It largely corresponds with the modern administrative region of Provence-Alpes-Côte d'Azur and includes the departments of Var, Bouches-du-Rhône, Alpes-de-Haute-Provence, as well as parts of Alpes-Maritimes and Vaucluse. The largest city of the region and its modern-day capital is Marseille. The Romans made the region the first Roman province beyond the Alps and called it Provincia Romana, which evolved into the present name. Until 1481 it was ruled by the Counts of Provence from their capital in Aix-en-Provence, then became a province of the Kings of France. While it has been part of France for more than 500 years, it still retains a distinct cultural and linguistic identity

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Ce cours présente différentes facettes de l'optique moderne et met à la fois l'accent sur des bases rigoureuses et des applications pratiques. Le cours inclut une partie théorique avec un cours et des exercices, ainsi qu'une partie expérimentale avec des travaux pratiques.

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On the configuration LP for maximum budgeted allocation

Christos Kalaitzis, Aleksander Madry, Lukas Polacek, Ola Nils Anders Svensson

We study the maximum budgeted allocation problem, i.e., the problem of selling a set of m indivisible goods to n players, each with a separate budget, such that we maximize the collected revenue. Since the natural assignment LP is known to have an integrality gap of , which matches the best known approximation algorithms, our main focus is to improve our understanding of the stronger configuration LP relaxation. In this direction, we prove that the integrality gap of the configuration LP is strictly better than , and provide corresponding polynomial time roundings, in the following restrictions of the problem: (i) the restricted budgeted allocation problem, in which all the players have the same budget and every item has the same value for any player it can be sold to, and (ii) the graph MBA problem, in which an item can be assigned to at most 2 players. Finally, we improve the best known upper bound on the integrality gap for the general case from to and also prove hardness of approximation results for both cases.

Springer Heidelberg2015

On the Configuration LP for Maximum Budgeted Allocation

Christos Kalaitzis, Aleksander Madry, Lukas Polacek, Ola Nils Anders Svensson

We study the Maximum Budgeted Allocation problem, i.e., the problem of selling a set of m indivisible goods to n players, each with a separate budget, such that we maximize the collected revenue. Since the natural assignment LP is known to have an integrality gap of, which matches the best known approximation algorithms, our main focus is to improve our understanding of the stronger configuration LP relaxation. In this direction, we prove that the integrality gap of the configuration LP is strictly better than, and provide corresponding polynomial time roundings, in the following restrictions of the problem: (i) the Restricted Budgeted Allocation problem, in which all the players have the same budget and every item has the same value for any player it can be sold to, and (ii) the graph MBA problem, in which an item can be assigned to at most 2 players. Finally, we improve the best known upper bound on the integrality gap for the general case from 5/6 to 2√2 2 ≈ 0.828 and also prove hardness of approximation results for both cases. © 2014 Springer International Publishing Switzerland.

Springer2014

Hybrid Bounds For Automorphic Forms On Ellipsoids Over Number Fields

Philippe Michel

We prove upper bounds for Hecke-Laplace eigenfunctions on certain Riemannian manifolds X of arithmetic type, uniformly in the eigenvalue and the volume of the manifold. The manifolds under consideration are d-fold products of 2-spheres or 3-spheres, realized as adelic quotients of quaternion algebras over totally real number fields. In the volume aspect we prove a ('Weyl-type') saving of vol (X)(-1/6+epsilon).

Cambridge Univ Press2013

Marseille

Marseille (mɑːrˈseɪ , maʁsɛj, maχˈsɛjə; also spelled in English as Marseilles) is the prefecture of the French department of Bouches-du-Rhône and capital of the Provence-Alpes-Côte d'Azur region. Situ

Middle Ages

In the history of Europe, the Middle Ages or medieval period (also spelled mediæval or mediaeval) lasted approximately from the late 5th to the late 15th centuries, aligning with the post-classical p

Aix-en-Provence

Aix-en-Provence (UKˌɛks_ɒ̃_prɒˈvɒ̃s, USˌeɪks_ɒ̃_proʊˈvɒ̃s,ˌɛks-), or simply Aix (medieval Occitan: Aics), is a city and commune in southern France, about north of Marseille. A former capital of Pro