Commuting extensions and cubature formulae

Based on a novel point of view on 1-dimensional Gaussian quadrature, we present a new approach to d-dimensional cubature formulae. It is well known that the nodes of 1-dimensional Gaussian quadrature can be computed as eigenvalues of the so-called Jacobi matrix. The d-dimensional analog is that cubature nodes can be obtained from the eigenvalues of certain mutually commuting matrices. These are obtained by extending (adding rows and columns to) certain noncommuting matrices A₁,...,A_d, related to the coordinate operators x₁,...,x_d, in R^d. We prove a correspondence between cubature formulae and “commuting extensions” of A₁,...,A_d, satisfying a compatibility condition which, in appropriate coordinates, constrains certain blocks in the extended matrices to be zero. Thus, the problem of finding cubature formulae can be transformed to the problem of computing (and then simultaneously diagonalizing) commuting extensions. We give a general discussion of existence and of the expected size of commuting extensions and briefly describe our attempts at computing them.     

Calculating multidimensional discrete variable representations from cubature formulas

Finding multidimensional nondirect product discrete variable representations (DVRs) of Hamiltonian operators is one of the long standing challenges in computational quantum mechanics. The concept of a "DVR set" was introduced as a general framework for treating this problem by R. G. Littlejohn, M. Cargo, T. Carrington, Jr., K. A. Mitchell, and B. Poirier (J. Chem. Phys. 2002, 116, 8691). We present a general solution of the problem of calculating multidimensional DVR sets whose points are those of a known cubature formula. As an illustration, we calculate several new nondirect product cubature DVRs on the plane and on the sphere with up to 110 points. We also discuss simple and potentially very useful finite basis representations (FBRs), based on general (nonproduct) cubatures. Connections are drawn to a novel view on cubature presented by I. Degani, J. Schiff, and D. J. Tannor (Num. Math. 2005, 101, 479), in which commuting extensions of coordinate matrices play a central role. Our construction of DVR sets answers a problem left unresolved in the latter paper, namely, the problem of interpreting as function spaces the vector spaces on which commuting extensions act.     

Beyond Inverse Ising Model: Structure of the Analytical Solution

I consider the problem of deriving couplings of a statistical model from measured correlations, a task which generalizes the well-known inverse Ising problem. After reminding that such problem can be mapped on the one of expressing the entropy of a system as a function of its corresponding observables, I show the conditions under which this can be done without resorting to iterative algorithms. I find that inverse problems are local (the inverse Fisher information is sparse) whenever the corresponding models have a factorized form, and the entropy can be split in a sum of small cluster contributions. I illustrate these ideas through two examples (the Ising model on a tree and the one-dimensional periodic chain with arbitrary order interaction) and support the results with numerical simulations. The extension of these methods to more general scenarios is finally discussed.     

Optical control in coupled two-electron quantum dots

The dynamics of two electrons in a 2-dimensional quantum dot molecule in the presence of a time-dependent electromagnetic field is calculated from first principles. We show that carefully selected microwave pulses can exclusively populate a single state of the first excitation band and that the transition time can be further decreased by optimal pulse control. Finally we demonstrate that an oscillating charge localized state may be created by multiple transitions using a sequence of pulses.     

Development of a CZE method for the determination of mizolastine and its impurities in pharmaceutical preparations using response surface methodology

A fast and selective CZE method for the determination of mizolastine and related impurities is described. Response surface methodology was applied to study the influence of phosphate/triethanolamine (TEA) buffer concentration, heptakis(2,3,6-tri-O-methyl)-beta-CD (TMbetaCD) concentration, voltage and temperature. The optimum conditions were: 105 mM phosphate/TEA buffer (pH 3.0) containing 10 mM TMbetaCD, temperature 19 degrees C and voltage 30 kV. Validation of the method was performed in drug substance and drug product. Robustness was evaluated using a Plackett-Burman design, including pH among the considered factors. Applying the optimal conditions, the nine peaks were baseline separated in about 10 min. The method was applied to the quality control of mizolastine in controlled-release tablets.     

Assessment of the ageing of triterpenoid paint varnishes using fluorescence,Raman and FTIR spectroscopy

The assessment of the influence of natural and artificial ageing on the spectrofluorescence of triterpenoid varnishes dammar and mastic is the focus of this work. Both Fourier transform infrared (FTIR) microscopy using attenuated total reflectance and Raman spectroscopy have been employed for complementary molecular analysis of samples. Synchronous fluorescence spectroscopy, excitation emission spectroscopy, and statistical analysis of data have been used to monitor changes in the optical properties of varnish samples. Assessment of naturally and artificially aged samples using excitation emission spectroscopy suggests that extensive exposure to visible light does not lead to easily appreciable differences in the fluorescence of mastic and dammar; cluster analysis has been used to assess changes, which occur with artificial ageing under visible light, indicating that differences in the fluorescence spectra of aged triterpenoids may be insufficient for their discrimination. The results highlight significant differences between the initial fluorescence of films of dammar and mastic and the fluorescence, which develops with ageing and oxidation, and specific markers, which change with ageing in FTIR and Raman spectra, have been identified.     

Enhanced downconversion of UV light by resonant scattering of aluminum nanoparticles

Metallic nanoparticles are known to enhance nonlinear optical processes due to a local enhancement of the optical field. This strategy has been proposed to enhance downconversion in thin film solar cells, but has various disadvantages, among which is the fact that the enhancement occurs only in a tiny volume close to the particles. We report on a very different physical mechanism that can lead to significant downconversion enhancement, namely, that of resonant light scattering, and which is a large volume effect. We show that only a tiny amount of resonantly scattering metallic (aluminum) nanoparticles is enough to create a significant enhancement of the fluorescence of dye molecules in the visible wavelength range. The strategy can be applied in general to increase the emission of UV-absorbing constituents, and is of particular use for solar energy.