Collapse of magnetized hypermassive neutron stars in general relativity

Hypermassive neutron stars (HMNSs)--equilibrium configurations supported against collapse by rapid differential rotation--are possible transient remnants of binary neutron-star mergers. Using newly developed codes for magnetohydrodynamic simulations in dynamical spacetimes, we are able to track the evolution of a magnetized HMNS in full general relativity for the first time. We find that secular angular momentum transport due to magnetic braking and the magnetorotational instability results in the collapse of an HMNS to a rotating black hole, accompanied by a gravitational wave burst. The nascent black hole is surrounded by a hot, massive torus undergoing quasistationary accretion and a collimated magnetic field. This scenario suggests that HMNS collapse is a possible candidate for the central engine of short gamma-ray bursts.     

Conformally covariant nonlinear equations on tensor-spinors

We prove that to each conformally covariant equation on tensor-spinors on a Riemannian or pseudo-Riemannian manifold with spin structure one can add a nonlinear term without losing the property of conformal covariance. It follows in particular that, on a manifold of dimension n, the nonlinear Dirac equation, Pψ + λ|ψ|^1/(n?1)ψ = 0, where P is the Dirac operator and λ is a constant, is conformally covariant. This generalizes a result of Gürsey [1]. Some results of Ørsted [2], concerning a nonlinear equation associated with the Laplacian on function, and of Branson, concerning distinguished nonlinearities associated with his modified Laplacian on differential forms [3] are also derived as particular cases of this general result.     

Symmetries in Differential Geometry: a Computational Approach to Prolongations

The aim of this work is to develop a systematic manner to close overdetermined systems arising from conformal Killing tensors (CKT). The research performs this action for 1-tensor and 2-tensors. This research makes it possible to develop a new general method for any rank of CKT. This method can also be applied to other types of Killing equations, as well as to overdetermined systems constrained by some other conditions. The major methodological apparatus of the research is a decomposition of the section bundles where the covariant derivatives of the CKT land via generalized gradients. This decomposition generates a tree in which each row represents a higher derivative. After using the conformal Killing equation, just a few components (branches) survive, which means that most of them can be expressed in terms of lower order terms. This results in a finite number of independent jets. Thus, any higher covariant derivative can be written in terms of these jets. The findings of this work are significant methodologically and, more specifically, in the potential for the discovery of symmetries. First, this work has uncovered a new method that could be used to close overdetermined systems arising from conformal Killing tensors (CKT). Second, through an application of this method, this research finds higher symmetry operators of first and second degree, which are known by other means, for the Laplace operator. The findings also reveal the first order symmetry operators for the Yamabe case. Moreover, the research leads to conjectures about the second order symmetries of the Yamabe operator.     

Evaluation of metal and radionuclide data from neutron activation and acid-digestion-based spectrometry analyses of background soils: Significance in environmental restoration

A faster, more cost-effective, and higher-quality data acquisition for natural background-level metals and radionuclides in soils is needed for remedial investigations of contaminated sites. The advantages and disadvantages of neutron activation analysis (NAA) compared with those of acid-digestion-based spectrometry (ADS) methods were evaluated using Al, Sb, As, Cr, Co, Fe, Mg, Mn, Hg, K, Ag,²³²Th,²³⁵U,²³⁸U, V, and Zn data. The ADS methods used for this project were inductively coupled plasma (ICP), ICP-mass spectrometry (ICP-MS), and alpha spectrometry. Scatter plots showed that the NAA results for As, Co, Fe, Mn,²³²Th, and²³⁸U are reasonably correlated with the results from the other analytical methods. Compared to NAA, however, the ADS methods underestimated Al, Cr, Mg, K, V, and Zn. Because of the high detection limits of ADS methods, the scatter plots of Sb, Hg, and Ag did not show a definite relationship. The NAA results were highly correlated with the alpha spectrometry results for²³²Th and²³⁸U but poorly correlated for²³⁵U. The NAA, including the delayed neutron counting, was a far superior technique for quantifying background levels of radionuclides (²³²Th,²³⁵U, and²³⁸U) and metals (Al, Cr, Mg, K, V, and Zn) in soils.     

Option Pricing on Multiple Assets

Since 1973, the Black–Scholes formula has been used in financial markets to price financial derivatives such as options. In the classical Black–Scholes model for the market, the following type of mix is assumed or postulated: in the simplest case, it consists of an essentially riskless bond and a single risky asset. Hence, certainty mixed with uncertainty: safe vs risky! Here we consider more complex products where each component in a portfolio entails several variables with constraints. This leads to elegant models based on multivariable stochastic integration, and describing several securities simultaneously [Etheridge, A Course in Financial Calculus, Cambridge University Press, UK (2002), Jiang, Mathematical Modeling and Methods of Option Pricing, Higher Education, Beijing, China (2003)] and [Broadie, Detemple, Math. Financ. 7:241–286 (1997)]. We derive a general asymptotic solution in a short time interval using the heat kernel expansion on a Riemannian metric. We then use our formula to predict the better price of options on multiple underlying assets. We then apply our method to the case known as the two-color rainbow option, i.e., the special case of the model with two underlying assets. This asymptotic solution is important, as it explains hidden effects in a class of financial models.This paper is dedicated to the memory of the first named author, Professor Thomas P. Branson (1953–2006).     

Creating a suite of macros for meta-analysis in SAS: A case study in collaboration

A series of macros that have been created to perform fixed and random effects meta-analysis in SAS^® are described as is the motivation for their creation. These macros are being made freely available on the internet for others to use. The application of the macros is illustrated using an example of trials in pre-eclampsia.     

The conformal deformation detour complex for the obstruction tensor

On pseudo-Riemannian manifolds of even dimension , with everywhere vanishing (Fefferman-Graham) obstruction tensor, we construct a complex of conformally invariant differential operators. The complex controls the infinitesimal deformations of obstruction-flat structures, and, in the case of Riemannian signature the complex is elliptic.

     

Second order conformal covariants

We complete the classification of second order, bundle valued differential operator invariants of Riemannian and pseudo-Riemannian conformal structure, by classifying such operators which pass between bundles associated to different representations of the rotation or spin group.