Continuous symmetries of equations on lattices

A method is presented for calculating the Lie point symmetries of difference equations with one, or several, independent variables. The equations are given on a priori specified lattices. The Lie transformations act on the lattice, as well as on the equation. The transformations take solutions into solutions and can be used to perform symmetry reduction. Presented by P. Winternitz at the DI-CRM Workshop held in Prague, 18–21 June 2000. This article was written while S.T. and P.W. were visiting the Dipartimento di Fisica, Università di Roma Tre. They thank the Dipartimento, the INFN and the Agreement Università di Roma Tre — Université de Montréal for their support. The research of P.W. was partly supported by a research grant from NSERC of Canada.     

Detection of beryllium in digested autopsy tissues by inductively coupled plasma mass spectrometry using a high matrix interface configuration

This article describes a robust methodology using the combination of instrumental design (high matrix interface—HMI), sample dilution and internal standardization for the quantification of beryllium (Be) in various digested autopsy tissues using inductively coupled plasma mass spectrometry. The applicability of rhodium as a proper internal standard for Be was demonstrated in three types of biological matrices (i.e., femur, hair, lung tissues). Using HMI, it was possible to achieve instrumental detection limits and sensitivity of 0.6 ng L⁻¹ and 157 cps L ng⁻¹, respectively. Resilience to high salt matrices of the HMI setup was also highlighted using bone mimicking solution ([Ca²⁺] = 26 to 1,400 mg L⁻¹), providing a 14-fold increase in tolerance and a 2.7-fold decrease in method detection limit compared to optimized experimental conditions obtained without the HMI configuration. Precision of the methodology to detect low levels of Be in autopsy samples was demonstrated using hair and blood certified reference materials. Be concentration ranging from 0.015 to 255 μg kg⁻¹ in autopsy samples obtained from the U.S. Transuranium and Uranium Registries were measured using the methodology presented.     

A comparison of the Jacobian-free Newton–Krylov method and the EVP model for solving the sea ice momentum equation with a viscous-plastic formulation: A serial algorithm study

Numerical convergence properties of a recently developed Jacobian-free Newton–Krylov (JFNK) solver are compared to the ones of the widely used EVP model when solving the sea ice momentum equation with a Viscous-Plastic (VP) formulation. To do so, very accurate reference solutions are produced with an independent Picard solver with an advective time step of 10 s and a tight nonlinear convergence criterion on 10, 20, 40, and 80-km grids. Approximate solutions with the JFNK and EVP solvers are obtained for advective time steps of 10, 20 and 30 min. Because of an artificial elastic term, the EVP model permits an explicit time-stepping scheme with a relatively large subcycling time step. The elastic waves excited during the subcycling are intended to damp out and almost entirely disappear such that the approximate solution should be close to the VP solution. Results show that residual elastic waves cause the EVP approximate solution to have notable differences with the reference solution and that these differences get more important as the grid is refined. Compared to the reference solution, additional shear lines and zones of strong convergence/divergence are seen in the EVP approximate solution. The approximate solution obtained with the JFNK solver is very close to the reference solution for all spatial resolutions tested.     

Complete systems of recursive integrals and Taylor series for solutions of Sturm–Liouville equations

Given a regular nonvanishing complex valued solution y₀ of the equation , x ∈ (a,b), assume that it is n times differentiable at a point x₀ ∈ [a,b]. We present explicit formulas for calculating the first n derivatives at x₀ for any solution of the equation . That is, a map transforming the Taylor expansion of y₀ into the Taylor expansion of u is constructed. The result is obtained with the aid of the representation for solutions of the Sturm‐Liouville equation in terms of spectral parameter power series. Copyright © 2012 John Wiley & Sons, Ltd.     

Bonding nature and vibrational signatures of oxirane:(water)(n=1-3). Assessment of the performance of the dispersion-corrected DFT methods compared to the ab initio results and Fourier transform infrared experimental data

Spectroscopic properties of 1:n complexes (n = 1, 2, and 3) formed between an oxirane molecule and water clusters have been evaluated using experimental techniques (FTIR spectroscopy using a new supersonic jet experiment coupled to the infrared AILES beamline of synchrotron SOLEIL and also cryogenic neon matrix device) and theoretical approaches (SAPT, ab initio, DFT, and topological analyses). From a systematic comparison between the theoretical results (obtained with both wave function based methods and several newly hydrogen bonded adapted functionals) with the available experimental results on the studied compounds, it was concluded that only the wave function based methods (particularly coupled clusters ones) are able to well describe these compounds, while the newly hydrogen bonded adapted functionals (long-range and/or dispersion-corrected ones and also double hybrids) cannot adequately describe all the spectroscopic properties in a systematic way. The MP2 method, although more expensive than DFT, still offers a reliable method to study both isolated molecules and hydrogen bonded complexes provided the contribution of the dispersion energy in total energy is properly taken into account. The nature of interaction between oxirane and water molecules has been analyzed using the symmetry adapted perturbation theory (SAPT) method. It was evidenced that the water-oxirane interaction corresponds to the hydrogen-bonded systems with a large contribution of the dispersion energy. The nature of the oxirane-water bonding has been studied using two topological methods: atoms in molecules and electron-localization function (ELF). Geometrical structures of the titled complexes were rationalized from the spatial arrangement of ELF attractors. Secondary interaction was also accounted for the bond critical points found at H(oxirane)···O(water) bond paths.