Use of the median in Physics and Astronomy

The sample median is often used in statistical analyses of physical or astronomical data wherein a central value must be found from samples polluted by elements which do not belong to the population of interest or when the underlying probability law is such that the sample mean is useless for the stated purpose. Although it does not generally possesses the nice linearity properties of the mean, the median has advantages of its own, some of which are explored in this paper which elucidates analogies and differences between these two central value descriptors. Some elementary results are shown, most of which are certainly not new but not widely known either. It is observed that the moment and the quantile approaches to the description of a probability distribution are difficult to relate, but that when the quantile description is used, the sample median can be characterized very much in the same way as the sample mean; this opens the possibility of using it for estimation purposes beyond what is usually done. In order to relate the two approaches, a derivation is given of the asymptotic joint distribution of the mean and the median for a general continuous probability law.Comment: 22 pages, no figur

A simple derivation of the Lorentz transformation and of the related velocity and acceleration formulae

The Lorentz transformation is entirely derived from length contraction, itself established through the known light-clock thought experiment . This makes the derivation accessible to beginning students once Eintein's two postulates have been admitted. The formula derived for the space part of the general rotation-free Lorentz transformation is very compact and allows for an easy derivation of the velocity and acceleration transformations. A possibly new and very simple relation is found between proper acceleration and acceleration in an inertial frame.Comment: Update of physics/0606103 4 pages, 1 figure, a slightly modified version is to appear in the American Journal of Physic

On ultra-relativistic approximations, unobservable phases and other hand-waving in the derivation of the neutrino oscillation length

A wrong derivation of the phase of a propagating massive particle which has repeatedly appeared during the last years has the advantage of leading once more to the important question of the phase difference between the mass eigenstates constitutive of an oscillating neutrino described in the plane wave formalism. Serious errors of principle are pointed at in a number of simple calculations and the oscillation length is derived in a way which allows to show that the standard result can suffer but minute variations in the ultrarelativistic case.Comment: 11 pages, no figur

Kinematics of an off axis neutrino beam

We review the kinematics of a neutrino beam in the idealized case where the parent mesons momenta are parallel, but without other approximation. This reveals several interesting features, in particular in the off-axis case, which are hidden by the approximations made in a previous treatment.Comment: 14 pages, 4 figures Some typos corrected. Argument in last section streamline

Synthesis of New Chiral N-Heterocyclic Carbenes and Abnormal Carbenes

Chapter 1 Carbenes have fascinated organic chemists ever since the first evidence of their existence. The isolation and first crystallographic analysis of a stable NHC by Arduengo, in 1991, has had a major impact on the application of NHCs in chemistry. NHCs are widely used as ligands for transition metals, but also as catalysts in their own right. The aim of this project is to synthesise novel monodentate chiral NHCs 1 in order to induce a high level of stereocontrol in a selection of asymmetric reactions. The rigid ring structure, C2 symmetry and the presence of bulky groups (R) that project directly toward the reactive centre are important concepts in the design of our molecules. During the first part of my thesis, we developed a one-pot synthesis of allylic chlorides from benzaldehyde. A mechanistic study was carried out, as well as studying the scope of the reaction. We then achieved the synthesis of the desired NHCs precursors in 13 or 14 steps and carried out studies of their coordination and application in asymmetric α-arylation and allylic alkylation. Chapter 2 Recently novel coordination modes have been noted for NHCs. The so-called “abnormal” NHCs, due to their isomeric relationship between C2- and C4-bound imidazolylidene ligands are one type of relatively unexplored ligand. We have therefore developed a strategy to access chiral abnormal NHCs. Two different chiral abnormal NHC precursors: 1,2,3-triazolidinenes 2 and isomeric Hartwig’s NHCs 3 were synthesised via a short synthesis (2 or 3 steps) and metallated with a transition metal