Separating invariants and finite reflection groups

A separating algebra is, roughly speaking, a subalgebra of the ring of invariants whose elements distinguish between any two orbits that can be distinguished using invariants. In this paper, we introduce a geometric notion of separating algebra. This allows us to prove that only groups generated by reflections may have polynomial separating algebras, and only groups generated by bireflections may have complete intersection separating algebras.     

Understanding a time reversal process in Lamb wave propagation

This study investigates the time reversal process (TRP) of Lamb wave signals which are transmitted and received by piezoelectric transducers bonded on plate-like structures. A number of previous studies have paid attention to spatial and temporal refocusing capability of an original excitation through the TRP in highly dispersive and complex media. However, when the TRP is applied to Lamb waves in a homogeneous regular waveguide, the refocusing capability is limited due to permanent residual side bands even if the duration of the time reversed signal increases. Based on the reciprocity of elastodynamics and linear piezoelectricity, theoretical interpretation is conducted for the main and residual side bands of the reconstructed signal in the time domain. In particular, the interpretation includes the temporal effect of velocity and amplitude dispersions, the existence of multimodes, and the reflections from boundaries during the TRP. Then, numerical and experimental tests are conducted to validate the theoretical findings of this paper. Practical issues for the successful implementation of the TRP of Lamb waves are briefly addressed as well.     

Numerical simulation of the reflection of a planar shock wave over a double wedge

A numerical code based on the upwind TVD scheme for simulating the various reflection processes of a planar shock wave over a concave or convex double wedge has been developed. The numerical results were compared with actual experiments and excellent agreement was obtained. The excellent agreement serves also as a validation of the shock-capturing performance of the numerical scheme.     

Involutions of a Clifford Algebra Induced by Involutions of Orthogonal Group in Characteristic 2

Among the involutions of a Clifford algebra, those induced by the involutions of the orthogonal group are the most natural ones. In this work, several basic properties of these involutions, such as the relations between their invariants, their occurrences, and their decompositions, are investigated.     

Rank 2 Nichols Algebras with Finite Arithmetic Root System

The concept of arithmetic root systems is introduced. It is shown that there is a one-to-one correspondence between arithmetic root systems and Nichols algebras of diagonal type having a finite set of (restricted) Poincaré–Birkhoff–Witt generators. This has strong consequences for both objects. As an application all rank 2 Nichols algebras of diagonal type having a finite set of (restricted) Poincaré–Birkhoff–Witt generators are determined. Supported by the European Community under a Marie Curie Intra-European Fellowship.     

Forward equations for reflected diffusions with jumps

In this paper we obtain the forward equations associated with the evolution of the density, if it exists, of reflected diffusions on the positive orthant with jumps which form a marked point process whose random jump measure possesses a stochastic intensity. These results generalize the so-called generalized Dynkin equations for piecewise deterministic jump processes due to Davis. We then consider the stationary case where the existence of a stochastic intensity is not needed. The techniques are based on local times and the use of random jump measures. We discuss the application of these results to problems arising in queuing and storage processes as well as stationary distributions of diffusions with delayed and jump reflections at the origin.This research was supported in part by the Quebec-France Cooperative Research Program and by the Natural Sciences and Engineering Research Council of Canada under Grant OGP 0042024.     

Influence of ophthalmic lenses in the illuminance originated at a peripheral glare source and reaching the eye

To study the influence of glare on the visual performance of a subject wearing an ophthalmic lens, it is useful to know how the lens affects the illuminance reaching the eye. In this paper, considering spherical standard ophthalmic lenses and defining the relative illuminance, E_r, as the quotient between the illuminance at the cornea with and without lens, a methodology to evaluate E_r in terms of easily determined parameters is developed. Three effects are considered, pupil size variation of the system with and without lens; lateral shifts of rays transmitted through the lens and reflections at the lens. Calculations are experimentally verified employing 5 organic ophthalmic lenses of ±6; ±4 and 0.12 dioptres and 2 glass plane parallel plates 1.95 and 6.6 mm thick. Using a photometer whose sensor is 12 mm apart from the lens and 740 mm apart from a glare source subtending an eccentricity angle of 9.6°, it results E_r=1.204 for the 6 dioptres lens and E_r=0.803 for the −6 dioptres one if sensor diameter is 10 mm while, for a 719 mm distance and a 10° angle, E_r=0.922 for the thin plate and a 30 mm sensor and E_r=1.006 for the thick plate and a 10 mm sensor. Experimental and theoretical results differ in less than 3%.