Pourquoi, pour qui enseigner les mathématiques? Une mise en perspective historique de l’évolution des programmes de mathématiques dans la société française au XXe siècle

Evolutions of mathematical curricula in French society have been marked by the successive answers intitutions have given to the following question since one century: Why and whom teaching mathematics? Here I present two of these, one given in 1908 and one in 1967. Each symbolises a breaking period of reforms in secondary mathematics teaching in France. We will see in the two first part of this paper that they belong to two different worlds, with social, institutional, ideological and mathematical specific features. In the third part, I’ll focus on geometry, showing the effects of the different answers concerning the public and the aims of mathematical teaching.     

On the possibility of a radical decrease in the strength of many-body interactions in the organic metal α-(BETS)<Subscript>2</Subscript> KHg(SCN)<Subscript>4</Subscript>

Behavior of the de Haas-van Alphen (dHvA) oscillations depending on the angle between the magnetic field direction and the perpendicular to conducting layers in the quasi-two-dimensional organic metal α-(BETS)₂ KHg(SCN)₄ was studied in detail. The angular dependence of the dHvA oscillation amplitude exhibits a series of minima (at ±43.2°, ±64.6°, and ±72.0°) related to the “zero spin” effect, through which it is possible to estimate the splitting factor. An analysis of this value suggests that many-body interactions in the compound studied are either absent or at least radically weakened.     

Spin-relaxation and magnetoresistance in FM/SC/FM tunnel junctions

The effect of spin relaxation on tunnel magnetoresistance (TMR) in a ferromagnet/superconductor/ferromagnet (FM/SC/FM) double tunnel junction is theoretically studied. The spin accumulation in SC is determined by balancing of the spin-injection rate and the spin-relaxation rate. In the superconducting state, the spin-relaxation time τ_s becomes longer with decreasing temperature, resulting in a rapid increase of TMR. The TMR of FM/SC/FM junctions provides a useful probe to extract information about spin-relaxation in superconductors.     

Closure laws for a two-fluid two-pressure modelLois de fermeture pour un modèle à deux pressions d'écoulement diphasique

Closure laws for interfacial pressure and interfacial velocity are proposed within the frame work of two-pressure two-phase flow models. These enable us to ensure positivity of void fractions, mass fractions and internal energies when investigating field by field waves in the Riemann problem. To cite this article: F. Coquel et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 927–932.     

Analysing the information flow between financial time series

Following the recently introduced concept of transfer entropy, we attempt to measure the information flow between two financial time series, the Dow Jones and DAX stock index. Being based on Shannon entropies, this model-free approach in principle allows us to detect statistical dependencies of all types, i.e. linear and nonlinear temporal correlations. However, when available data is limited and the expected effect is rather small, a straightforward implementation suffers badly from misestimation due to finite sample effects, making it basically impossible to assess the significance of the obtained values. We therefore introduce a modified estimator, called effective transfer entropy, which leads to improved results in such conditions. In the application, we then manage to confirm an information transfer on a time scale of one minute between the two financial time series. The different economic impact of the two indices is also recovered from the data. Numerical results are then interpreted on one hand as capability of one index to explain future observations of the other, and on the other hand within terms of coupling strengths in the framework of a bivariate autoregressive stochastic model. Evidence is given for a nonlinear character of the coupling between Dow Jones and DAX.     

Universal Overconvergence of Polynomial Expansions of Harmonic Functions

For each compact subset K of ^N let (K) denote the space of functions that are harmonic on some neighbourhood of K. The space (K) is equipped with the topology of uniform convergence on K. Let Ω be an open subset of ^N such that 0Ω and ^N\Ω is connected. It is shown that there exists a series ∑H_n, where H_n is a homogeneous harmonic polynomial of degree n on ^N, such that (i) ∑H_n converges on some ball of centre 0 to a function that is continuous on Ω and harmonic on Ω, (ii) the partial sums of ∑H_n are dense in (K) for every compact subset K of ^N\Ω with connected complement. Some refinements are given and our results are compared with an analogous theorem concerning overconvergence of power series.     

Climbing Mount Scalable: Physical Resource Requirements for a Scalable Quantum Computer

The primary resource for quantum computation is Hilbert-space dimension. Whereas Hilbert space itself is an abstract construction, the number of dimensions available to a system is a physical quantity that requires physical resources. Avoiding a demand for an exponential amount of these resources places a fundamental constraint on the systems that are suitable for scalable quantum computation. To be scalable, the effective number of degrees of freedom in the computer must grow nearly linearly with the number of qubits in an equivalent qubit-based quantum computer.