Exchange-Correlation Energy from Pairing Matrix Fluctuation and the Particle-Particle Random Phase Approximation

We formulate an adiabatic connection for the exchange-correlation energy in terms of pairing matrix fluctuation. This connection opens new channels for density functional approximations based on pairing interactions. Even the simplest approximation to the pairing matrix fluctuation, the particle-particle Random Phase Approximation (pp-RPA), has some highly desirable properties. It has no delocalization error with a nearly linear energy behavior for systems with fractional charges, describes van der Waals interactions similarly and thermodynamic properties significantly better than particle-hole RPA, and eliminates static correlation error for single-bond systems. Most significantly, the pp-RPA is the first known functional that has an explicit and closed-form dependence on the occupied and unoccupied orbitals and captures the energy derivative discontinuity in strongly correlated systems. These findings illlustrate the potential of including pairing interactions within a density functional framework

The effect of precursor composition and sintering additives on the formation of ß-sialon from Al, Si and Al2O3 powders

A study was performed to investigate the effect of increasing the Al or Al2O3 precursor content, above the stoichiometric amount, on the formation of β-sialon by pressureless sintering of Al, Si and Al2O3 powders in flowing nitrogen gas. The effect of adding Y2O3 or Fe to the precursor mixture, on the β-sialon formation, was also studied. The phase morphology and yield produced by the various compositions were examined using X-ray diffraction (XRD). Additional Al2O3 decreases the β-sialon phase yield and results in a greater amount of Al2O3 in the final sintered material. Additional Al improved the conversion to β-sialon up to a maximum of 4 wt% Al beyond which the β-sialon:15R sialon ratio in the sintered material decreases. 1 wt% Y2O3 was determined to be the optimum sintering additive content, as yttrium aluminium garnet (YAG) was found to be present in materials formed from higher Y2O3 containing precursors. The presence of Fe in the precursor powder retards the formation of β-sialon by preferentially forming Fe silicides at low temperatures, thus depleting the reaction system of elemental Si, favouring the formation of 15R sialon

Spontaneous Magnetization of the Integrable Chiral Potts Model

We show how $Z$-invariance in the chiral Potts model provides a strategy to calculate the pair correlation in the general integrable chiral Potts model using only the superintegrable eigenvectors. When the distance between the two spins in the correlation function becomes infinite it becomes the square of the order parameter. In this way, we show that the spontaneous magnetization can be expressed in terms of the inner products of the eigenvectors of the $N$ asymptotically degenerate maximum eigenvalues. Using our previous results on these eigenvectors, we are able to obtain the order parameter as a sum almost identical to the one given by Baxter. This gives the known spontaneous magnetization of the chiral Potts model by an entirely different approach.Comment: LaTeX 2E document, using iopart.cls with iopams packages, 22 pages, 1 eps figure. Presented at the Simons Center for Geometry and Physics Workshop on Correlation Functions for Integrable Models 2010: January 18-22, 2010. Version 2: The identity conjectured in version 1 is now proved and its proof is presented in arXiv:1108.4713; various small corrections and improvements have been made als

Model Selection for High Dimensional Quadratic Regression via Regularization

Quadratic regression (QR) models naturally extend linear models by considering interaction effects between the covariates. To conduct model selection in QR, it is important to maintain the hierarchical model structure between main effects and interaction effects. Existing regularization methods generally achieve this goal by solving complex optimization problems, which usually demands high computational cost and hence are not feasible for high dimensional data. This paper focuses on scalable regularization methods for model selection in high dimensional QR. We first consider two-stage regularization methods and establish theoretical properties of the two-stage LASSO. Then, a new regularization method, called Regularization Algorithm under Marginality Principle (RAMP), is proposed to compute a hierarchy-preserving regularization solution path efficiently. Both methods are further extended to solve generalized QR models. Numerical results are also shown to demonstrate performance of the methods.Comment: 37 pages, 1 figure with supplementary materia