Royaume-Uni : exercices d'équilibristes

Exercices de statistiques mathématiques

Par le théorème de la limite centrale, cette variable converge en loi vers la loi normale centrée réduite, c’est-à-dire, pour tout t ∈ R, on a limn→+∞ E[eitZn ] = e− t2 2 . L’objet de cet exercice est de montrer que la suite Zn ne peut pas converger en probabilité. 1. Calculer la fonction caractéristique de Z2n − Zn et montrer que cette différence converge en loi. 2. En étudiant P(|Z2n − Zn| ≥ ...

Introduction aux variétés différentiables Solutions et Exercices

Solution 12.1. Let ∑ α∈A ηα be a partition of unity supported on the coordinate charts of the orientable manifold. Let φα : Uα → Rn be the orientable coordinate charts, and let xα, . . . , xα. Then for each α define an n form on M by ωα = ηαdxα∧ · · · ∧dxα. This is smooth in Uα because it is a product of a smooth form and function. Outside of Uα the form is zero in a neighbourhood of every poin...

Exercices sur le problème d’acquisition comprimée

Exercices with the neutrino oscillation length formula

Forsaking the traditionnal hand-waving in the treatment of the motion allows to show that the ultra-relativistic approximation and the equality of kinematical variables are unnecessary ingredients in the derivation of the oscillation length using plane waves, at least in a two flavor world. It ensues that the formula is valid as it is in the non relativistic regime, provided one uses the correc...

Vital Signs Recorder for Medical Exercices and Sports

A portable device designed for in-field recording of biosignals is presented. The recorder is provided with three bipolar channels digitising at 200 Hz and lacks of interpretive intelligence. The captured data is send in real time via serial communication interface using AT commands, or stored in the internal memory for delayed transmission. Two operating modes are implemented: scheduled acquis...