The Witt ring of a Brauer–Severi variety

For a Brauer–Severi variety X over a field k of characteristic not two, every symmetric bilinear space over X up to Witt equivalence is defined over k. Received: 2 February 1998     

Study of the gas-phase parameters affecting the silicon-oxide film deposition induced by an ArF laser

A study of the gas-phase parameters involved in ArF laser induced chemical vapour deposition of silicon-oxide thin films is presented. A complete set of experiments has been performed showing the influence of the concentration of the precursor gases, N₂O and SiH₄, and their influence on total and partial pressures on film growth and properties. In this paper we demonstrate the ability of this LCVD method to deposit silicon oxide films of different compositions and densities by appropriate control of gas composition and total pressure. Moreover, a material specific calibration plot comprising data obtained using different preparation techniques is presented, allowing determination of the stoichiometry of SiO _x films by using FTIR spectroscopy independently of the deposition method. For the range of processing conditions examined, the experimental results suggest that chemical processes governing deposition take place mainly in the gas phase.     

Diffusing particles with electrostatic repulsion

Summary. We study a diffusion model of an interacting particles system with general drift and diffusion coefficients, and electrostatic inter-particles repulsion. More precisely, the finite particle system is shown to be well defined thanks to recent results on multivalued stochastic differential equations (see [2]), and then we consider the behaviour of this system when the number of particles goes to infinity (through the empirical measure process). In the particular case of affine drift and constant diffusion coefficient, we prove that a limiting measure-valued process exists and is the unique solution of a deterministic PDE. Our treatment of the convergence problem (as ) is partly similar to that of T. Chan [3] and L.C.G. Rogers - Z. Shi [5], except we consider here a more general case allowing collisions between particles, which leads to a second-order limiting PDE. Received: 5 August 1996 / In revised form: 17 October 1996     

Twistor Geometry for Hyperkähler Metrics on Complex Adjoint Orbits

We study the hyperkähler geometry of complex adjoint orbits from the point of view of twistor theory. We introduce, for complex semisimple adjoint orbits, the associated spectral curve and construct the twistor space as a union of certain regular adjoint orbits; we also exhibit the family of twistor lines. Furthermore, we show how our methods may be applied for describing hyperkähler metrics associated to more general spectral curves. In particular, we give an algebraic characterisation of the twistor lines.     

Approximation in weighted Hardy spaces

This paper is concerned with several approximation problems in the weighted Hardy spacesH ^p(Ω) of analytic functions in the open unit disc D of the complex plane ℂ. We prove that ifX is a relatively closed subset of D, the class of uniform limits onX of functions inH ^p(Ω) coincides, moduloH ^p(Ω), with the space of uniformly continuous functions on a certain hull ofX which are holomorphic on its interior. We also solve the simultaneous approximation problems of describing Farrell and Mergelyan sets forH ^p(Ω), giving geometric characterizations for them. By replacing approximating polynomials by polynomial multipliers of outer functions, our results lead to characterizations of the same sets with respect to cyclic vectors in the classical Hardy spacesH ^p(D), 1 ⪯p < ∞. Dedicated to Professor Nácere Hayek on the occasion of his 75th birthday.     

Variations of constants and red-shift in 6D cosmology

On the basis of the general class of solutions of 6D vacuum Einstein equations and using well-known results of the 6D geometric model of electroweak and gravitational interactions, the variations of fundamental physical constants and their influence on the observed red-shift of galaxies are investigated. By considering the interpretation of extra components of the 6D metric as reproducing the properties of matter in the 4D curved space-time, the effective quantities of matter are obtained. In particular, the states of dust and radiation are shown. The corresponding 5D models were also investigated.     

Unreasonable effectiveness of symmetry in physics

We prove that in some reasonable sense, every possible physical law can be reformulated in terms of symmetries. This result explains the well-known success of the group-theoretic approach in physics.     

Luttinger liquid with asymmetric dispersion

We present an extension of the Tomonaga-Luttinger model in which left and right-moving particles have different Fermi velocities. We derive expressions for one-particle Green's functions, momentum-distributions, density of states, charge compressibility and conductivity as functions of both the velocity difference ε and the strength of the interaction β. This allows us to identify a novel restricted region in the parameter space in which the system keeps the main features of a Luttinger liquid but with an unusual behavior of the density of states and the static charge compressibility κ. In particular κ diverges on the boundary of the restricted region, indicating the occurrence of a phase transition. Received 20 May 2002 / Received in final form 23 August 2002 Published online 19 November 2002     

Enumeration of three-dimensional convex polygons

A self-avoiding polygon (SAP) on a graph is an elementary cycle. Counting SAPs on the hypercubic lattice ℤ ^d withd≥2, is a well-known unsolved problem, which is studied both for its combinatorial and probabilistic interest and its connections with statistical mechanics. Of course, polygons on ℤ ^d are defined up to a translation, and the relevant statistic is their perimeter. A SAP on ℤ ^d is said to beconvex if its perimeter is “minimal”, that is, is exactly twice the sum of the side lengths of the smallest hyper-rectangle containing it. In 1984, Delest and Viennot enumerated convex SAPs on the square lattice [6], but no result was available in a higher dimension. We present an elementar approach to enumerate convex SAPs in any dimension. We first obtain a new proof of Delest and Viennot's result, which explains combinatorially the form of the generating function. We then compute the generating function for convex SAPs on the cubic lattice. In a dimension larger than 3, the details of the calculations become very cumbersome. However, our method suggests that the generating function for convex SAPs on ℤ ^d is always a quotient ofdifferentiably finite power series.