Structural decomposition of the chemical shielding tensor: Contributions to the asymmetry, anisotropy, and orientation

The nine elements of chemical shielding tensors contain important information about local structure, but the extraction of that information is difficult. Here we explore a semiempirical method that has the potential for providing relatively accessible structural correlations. The approach entails approximating the field-induced electron current density as entirely perpendicular to the applied field. This has two interesting consequences. ͑1͒ The resulting shielding tensor is perfectly symmetric. Thus, asymmetry in a shielding tensor is an indication of current density that is not orthogonal to the applied field. ͑2͒ The orientation dependence of the chemical shielding at a point of interest is related explicitly to the isotropic average of the chemical shielding at every point in the surrounding region. This suggests a relatively simple relationship between the orientation dependence of the chemical shielding and the molecular structure. Good correlation with experimental tensors is obtained with just one or two adjustable parameters in several series of compounds, including silicates, phosphates, hydrogen bonds, carboxyls, and amides. As expected, the results indicate that for a given center, the contribution to the shielding anisotropy that is associated with each bonded neighbor increases as the number of electrons at either the center or the neighbors increases

Axial symmetries in lattice QCD with Kaplan fermions

This paper develops in detail a lattice definition of QCD based on the chiral defect fermions recently introduced by Kaplan. The revised version provides missing technical details in the proof that non-singlet axial symmetries become exact in the limit of an infinite fifth dimension. Also several minor errors are corrected.Comment: LaTeX, 29 p

Confinement, chiral symmetry, and the lattice

Two crucial properties of QCD, confinement and chiral symmetry breaking, cannot be understood within the context of conventional Feynman perturbation theory. Non-perturbative phenomena enter the theory in a fundamental way at both the classical and quantum level. Over the years a coherent qualitative picture of the interplay between chiral symmetry, quantum mechanical anomalies, and the lattice has emerged and is reviewed here.Comment: 126 pages, 36 figures. Revision corrects additional typos and renumbers equations to be more consistent with the published versio

On a Multiplicative Multivariate Gamma Distribution with Applications in Insurance

One way to formulate a multivariate probability distribution with dependent univariate margins distributed gamma is by using the closure under convolutions property. This direction yields an additive background risk model, and it has been very well-studied. An alternative way to accomplish the same task is via an application of the Bernstein–Widder theorem with respect to a shifted inverse Beta probability density function. This way, which leads to an arguably equally popular multiplicative background risk model (MBRM), has been by far less investigated. In this paper, we reintroduce the multiplicative multivariate gamma (MMG) distribution in the most general form, and we explore its various properties thoroughly. Specifically, we study the links to the MBRM, employ the machinery of divided differences to derive the distribution of the aggregate risk random variable explicitly, look into the corresponding copula function and the measures of nonlinear correlation associated with it, and, last but not least, determine the measures of maximal tail dependence. Our main message is that the MMG distribution is (1) very intuitive and easy to communicate, (2) remarkably tractable, and (3) possesses rich dependence and tail dependence characteristics. Hence, the MMG distribution should be given serious considerations when modelling dependent risks