Diophantine approximation by conjugate algebraic integers

Building on work of Davenport and Schmidt, we mainly prove two results. The first one is a version of Gel'fond's transcendence criterion which provides a sufficient condition for a complex or $p$-adic number $\xi$ to be algebraic in terms of the existence of polynomials of bounded degree taking small values at $\xi$ together with most of their derivatives. The second one, which follows from this criterion by an argument of duality, is a result of simultaneous approximation by conjugate algebraic integers for a fixed number $\xi$ that is either transcendental or algebraic of sufficiently large degree. We also present several constructions showing that these results are essentially optimal.Comment: The section 4 of this new version has been rewritten to simplify the proof of the main result. Other results in Sections 9 and 10 have been improved. To appear in Compositio Mat

Characteristics of good and competitive equilibrium

We analyse the consequences of a change of the characteristics of goods due to new information on the equilibrium of a pure exchange economy with n goods and m agents. Some changes of the characteristics of goods à la Lancaster have a positive effect on utility. In the general competitive equilibrium, some agents gain but others may lose. Nevertheless there is an increase of a linear combination of the utilities of the agents. The different possibilities of gains and losses are explicitly analysed in an example with two goods and two agents.Pure exchange economy, Characteristic of goods, Price determination, Pareto efficiency

Hochschild cohomology and Atiyah classes

In this paper we prove that on a smooth algebraic variety the HKR-morphism twisted by the square root of the Todd genus gives an isomorphism between the sheaf of poly-vector fields and the sheaf of poly-differential operators, both considered as derived Gerstenhaber algebras. In particular we obtain an isomorphism between Hochschild cohomology and the cohomology of poly-vector fields which is compatible with the Lie bracket and the cupproduct. The latter compatibility is an unpublished result by Kontsevich. Our proof is set in the framework of Lie algebroids and so applies without modification in much more general settings as well.Comment: Reference to work of Cattaneo, Felder and Willwacher adde

Hochschild cohomology for Lie algebroids

We define the Hochschild (co)homology of a ringed space relative to a locally free Lie algebroid. Our definitions mimic those of Swan and Caldararu for an algebraic variety. We show that our (co)homology groups can be computed using suitable standard complexes. Our formulae depend on certain natural structures on jetbundles over Lie algebroids. In an appendix we explain this by showing that such jetbundles are formal groupoids which serve as the formal exponentiation of the Lie algebroid.Comment: The authors were informed that the fact that jetbundles are formal groupoids is already contained in arXiv:0904.4736 (with a somewhat different proof

Piecework versus merit pay: a Mean Fi eld Game approach to academic behavior

This paper applies the Mean Fi eld Game approach pioneered by Lasry and Lions (2007) to the analysis of the researchers' academic productivity. It provides a theoretical motivation for the stability of the universaly observed Lotka's law. It shows that a remuneration scheme taking into account the researchers rank with respect to the academic resume can induce a larger number of researchers to overtake a minimal production standard. It thus appears as superior to piecework remuneration.Mean Field Game, Academic production, incentives, Lotka's law.