Compatible Hamiltonian Operators for the Krichever-Novikov Equation

It has been proved by V. Sokolov that the Krichever-Novikov equation's hierarchy is hamiltonian for the non-local Hamiltonian operator H_0=u_x D^{-1} u_x and possesses twi weakly non-local recursion operatos of degree 4 and 6, L_4 and L_6. We show here that H_0, L_4H_0 and L_6H_0 are compatible Hamiltonian operators for which the Krichever-Novikov equation's hierarchy is hamiltonian

Funding in Higher Education and Economic Growth in France and the United Kingdom, 1921-2003

The 2004 Higher Education Act generated important debates about the relationships between higher education (HE), economic growth and social progress. The range of positions expressed in relation to the increase of annual tuition fees raises crucial questions about the public and private funding of HE and its individual and social economic benefits. The analysis of new historical data from the 1920s onwards shows that the expansion in university resources was not linear and may be related to long economic cycles. Moreover, private funding periodically increased in order to replace diminishing public funding, rather than taking the form of additional resources. In consequence, private funds did not provide an overall rise in the universities’ income. The considerable fluctuations of funding, combined with a more consistent growth of enrolment, led to a recurrent mismatch between resources for and access to HE, explaining the wide fluctuations of resources per student over the period. Such historical trends question whether, in the future, increased fees will be a substitute for public spending. Or will variable fees rather combine with even greater increases in public funding as part of a national project to support HE students from all social backgrounds and to boost expenditure per student

An Introduction to Topological Insulators

Electronic bands in crystals are described by an ensemble of Bloch wave functions indexed by momenta defined in the first Brillouin Zone, and their associated energies. In an insulator, an energy gap around the chemical potential separates valence bands from conduction bands. The ensemble of valence bands is then a well defined object, which can possess non-trivial or twisted topological properties. In the case of a twisted topology, the insulator is called a topological insulator. We introduce this notion of topological order in insulators as an obstruction to define the Bloch wave functions over the whole Brillouin Zone using a single phase convention. Several simple historical models displaying a topological order in dimension two are considered. Various expressions of the corresponding topological index are finally discussed.Comment: 46 pages, 29 figures. This papers aims to be a pedagogical review on topological insulators. It was written for the topical issue of "Comptes Rendus de l'Acad\'emie des Sciences - Physique" devoted to topological insulators and Dirac matte

Simple regret for infinitely many armed bandits

We consider a stochastic bandit problem with infinitely many arms. In this setting, the learner has no chance of trying all the arms even once and has to dedicate its limited number of samples only to a certain number of arms. All previous algorithms for this setting were designed for minimizing the cumulative regret of the learner. In this paper, we propose an algorithm aiming at minimizing the simple regret. As in the cumulative regret setting of infinitely many armed bandits, the rate of the simple regret will depend on a parameter $\beta$ characterizing the distribution of the near-optimal arms. We prove that depending on $\beta$, our algorithm is minimax optimal either up to a multiplicative constant or up to a $\log(n)$ factor. We also provide extensions to several important cases: when $\beta$ is unknown, in a natural setting where the near-optimal arms have a small variance, and in the case of unknown time horizon.Comment: in 32th International Conference on Machine Learning (ICML 2015

Universal metallic and insulating properties of one dimensional Anderson Localization : a numerical Landauer study

We present results on the Anderson localization in a quasi one-dimensional metallic wire in the presence of magnetic impurities. We focus within the same numerical analysis on both the universal localized and metallic regimes, and we study the evolution of these universal properties as the strength of the magnetic disorder is varied. For this purpose, we use a numerical Landauer approach, and derive the scattering matrix of the wire from electron's Green's function obtained from a recursive algorithm