Earnshaw (1805–1888) and Passive Magnetic Levitation

S. Earnshaw, a mathematical physicist born in England two centuries ago, was the author of studies on the transmission of sound and light. His theory on the transmission of light by “detached particles” has also influenced some studies on the stability of bodies in potential fields, particularly studies on the levitation of permanent magnets, i.e. on passive magnetic levitation. This theorem was so well known that even Maxwell mentioned it, and is today cited almost as an axiom in the ambit of passive magnetization. However, what is less well known is that the theory actually only relates to magnetostatics. In this paper, the mathematic model of the theory that refers to stability is outlined, along with some works that refer to it, in particular those on passive magnetic levitation. It is then pointed out that spaces of low instability exist in magnetostatics, and that stability may exist in the magneto-stationary and in magneto-dynamic ambit.     

One‐step synthesis of aminopyrimidines from 5‐Oxo‐4H‐benzopyrans

Novel 4‐amino‐6‐aryl‐2‐phenylpyrimidine‐5‐carbonitriles have been prepared in one step procedure from the readily available 4‐aryl‐2‐amino‐3‐cyano‐5,6,7,8‐tetrahydro‐7,7‐dimethyl‐5‐oxo‐4H‐benzopyrans. The mass spectroscopy study under EI conditions shows molecular peaks with high intensity corresponding to the loss of benzonitrile from the C2 position of the pyrimidine ring. Semiempirical (AMI and PM3) and ab initio HF/6–31G* calculations reveal a favored distorted geometry where the three rings are not in the same plane.     

Cetyltrimethylammonium hydroxide (CTAOH) as a general, ecofriendly catalyst for the formation of carbon-carbon bond through nitroalkanes

Nitroalkanes have been found to give good yields in Michael and nitroaldol (Henry) reactions by the use of a catalytic amount (10 mol%) of CTAOH, at room temperature and under solvent free conditions and in very short reaction times. The methods do not need a large excess of the nitroalkanes and show good chemoselectivity toward further functionalities.     

Lower Bounds for the Height and Size of the Ideal Class Group in CM-Fields

 We prove that, under the assumption of the Generalized Riemann Hypothesis, the exponent of the ideal class group of a CM-field goes to infinity with its absolute discriminant. This gives a positive answer to a question raised by Louboutin and Okazaki [4]. Received September 10, 2001; in revised form April 5, 2002     

Inhomogeneous Gevrey classes and ultradistributions

The inhomogeneous Gevrey classes, defined in terms of Fourier transform, are a natural extension of the standard Gevrey classes. We find equivalent characterizations and discuss algebraic and topological properties. We therefore introduce the dual spaces, the inhomogeneous ultradistributions, giving some equivalent definitions and corresponding algebraic and topological properties; in particular, a version of the Paley-Wiener-Schwartz theorem is proved in our framework. Finally, as an important example, the multianisotropic Gevrey classes and ultradistributions are considered.     

Simultaneous determination of cloramphenicol, salicylic acid and resorcinol by capillary zone electrophoresis and its application to pharmaceutical dosage forms

This study demonstrates the separation of active ingredients in acne formulations (salicylic acid, cloramphenicol and resorcinol in presence of azulene) by capillary zone electrophoresis. Factors affecting their separations were the buffer pH and concentration, applied voltage, sample preparation, and presence of additives. Optimun results were obtained with a 50 mM sodium tetraborate-50 mM sodium phosphate, pH 9.0. The carrier electrolyte gave baseline separation with good resolution, short migration times (<6 min), great reproducibility and accuracy. Calibration plots were linear over at least three orders of magnitude of analyte concentrations, the lower limits of detection being within the range 0.39-1.25 μg ml⁻¹. The procedure was fast and reliable and commercial pharmaceuticals could be analysed without prior sample clean-up procedure.     

On the Solution of Discontinuous IVPs by Adaptive Runge–Kutta Codes

In this paper an automatic technique for handling discontinuous IVPs when they are solved by means of adaptive Runge–Kutta codes is proposed. This technique detects, accurately locates and passes the discontinuities in the solution of IVPs by using the information generated by the code along the numerical integration together with a continuous interpolant of the discrete solution. A remarkable feature is that it does not require additional information on the location of the discontinuities. Some numerical experiments are presented to illustrate the reliability and efficiency of the proposed algorithms.     

Computation of the Maslov index and the spectral flow via partial signatures

Given a smooth Lagrangian path, both in the finite and in the infinite dimensional (Fredholm) case, we introduce the notion of partial signatures at each isolated intersection of the path with the Maslov cycle. For real-analytic paths, we give a formula for the computation of the Maslov index using the partial signatures; a similar formula holds for the spectral flow of real-analytic paths of Fredholm self-adjoint operators on real separable Hilbert spaces. As applications of the theory, we obtain a semi-Riemannian version of the Morse index theorem for geodesics with possibly conjugate endpoints, and we prove a bifurcation result at conjugate points along semi-Riemannian geodesics. To cite this article: R. Giambò et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

A Reality Proof in PT-Symmetric Quantum Mechanics

We review the proof of a conjecture concerning the reality of the spectra of certain PT-symmetric quantum mechanical systems, obtained via a connection between the theories of ordinary differential equations and integrable models. Spectral equivalences inspired by the correspondence are also discussed.     

Automorphisms of the q-oscillator algebra and basic orthogonal polynomials

Certain automorphisms of the quantum oscillator algebra are shown to provide an algebraic interpretation for q-generalizations of the Charlier polynomials.