Data-Discriminants of Likelihood Equations

Maximum likelihood estimation (MLE) is a fundamental computational problem in statistics. The problem is to maximize the likelihood function with respect to given data on a statistical model. An algebraic approach to this problem is to solve a very structured parameterized polynomial system called likelihood equations. For general choices of data, the number of complex solutions to the likelihood equations is finite and called the ML-degree of the model. The only solutions to the likelihood equations that are statistically meaningful are the real/positive solutions. However, the number of real/positive solutions is not characterized by the ML-degree. We use discriminants to classify data according to the number of real/positive solutions of the likelihood equations. We call these discriminants data-discriminants (DD). We develop a probabilistic algorithm for computing DDs. Experimental results show that, for the benchmarks we have tried, the probabilistic algorithm is more efficient than the standard elimination algorithm. Based on the computational results, we discuss the real root classification problem for the 3 by 3 symmetric matrix~model.Comment: 2 table

A topological Dirac insulator in a quantum spin Hall phase : Experimental observation of first strong topological insulator

When electrons are subject to a large external magnetic field, the conventional charge quantum Hall effect \cite{Klitzing,Tsui} dictates that an electronic excitation gap is generated in the sample bulk, but metallic conduction is permitted at the boundary. Recent theoretical models suggest that certain bulk insulators with large spin-orbit interactions may also naturally support conducting topological boundary states in the extreme quantum limit, which opens up the possibility for studying unusual quantum Hall-like phenomena in zero external magnetic field. Bulk Bi$_{1-x}$Sb$_x$ single crystals are expected to be prime candidates for one such unusual Hall phase of matter known as the topological insulator. The hallmark of a topological insulator is the existence of metallic surface states that are higher dimensional analogues of the edge states that characterize a spin Hall insulator. In addition to its interesting boundary states, the bulk of Bi$_{1-x}$Sb$_x$ is predicted to exhibit three-dimensional Dirac particles, another topic of heightened current interest. Here, using incident-photon-energy-modulated (IPEM-ARPES), we report the first direct observation of massive Dirac particles in the bulk of Bi$_{0.9}$Sb$_{0.1}$, locate the Kramers' points at the sample's boundary and provide a comprehensive mapping of the topological Dirac insulator's gapless surface modes. These findings taken together suggest that the observed surface state on the boundary of the bulk insulator is a realization of the much sought exotic "topological metal". They also suggest that this material has potential application in developing next-generation quantum computing devices.Comment: 16 pages, 3 Figures. Submitted to NATURE on 25th November(2007

Prediction of Anisotropic Single-Dirac-Cones in Bi1−x{}_{1-x}Sbx{}_{x} Thin Films

The electronic band structures of Bi${}_{1-x}$Sb${}_{x}$ thin films can be varied as a function of temperature, pressure, stoichiometry, film thickness and growth orientation. We here show how different anisotropic single-Dirac-cones can be constructed in a Bi${}_{1-x}$Sb${}_{x}$ thin film for different applications or research purposes. For predicting anisotropic single-Dirac-cones, we have developed an iterative-two-dimensional-two-band model to get a consistent inverse-effective-mass-tensor and band-gap, which can be used in a general two-dimensional system that has a non-parabolic dispersion relation as in a Bi${}_{1-x}$Sb${}_{x}$ thin film system

Self-Similar Interpolation in Quantum Mechanics

An approach is developed for constructing simple analytical formulae accurately approximating solutions to eigenvalue problems of quantum mechanics. This approach is based on self-similar approximation theory. In order to derive interpolation formulae valid in the whole range of parameters of considered physical quantities, the self-similar renormalization procedure is complimented here by boundary conditions which define control functions guaranteeing correct asymptotic behaviour in the vicinity of boundary points. To emphasize the generality of the approach, it is illustrated by different problems that are typical for quantum mechanics, such as anharmonic oscillators, double-well potentials, and quasiresonance models with quasistationary states. In addition, the nonlinear Schr\"odinger equation is considered, for which both eigenvalues and wave functions are constructed.Comment: 1 file, 30 pages, RevTex, no figure

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Beyond Race and Place: Distal Sociological Determinants of HIV Disparities

Informed behavior change as an HIV prevention tool has yielded unequal successes across populations. Despite decades of HIV education, some individuals remain at high risk. The mainstream media often portrays these risk factors as products of race and national borders; however, a rich body of recent literature proposes a host of complex social factors that influence behavior, including, but not limited to: poverty, income inequality, stigmatizing social institutions and health care access. We examined the relationship between numerous social indicators and HIV incidence across eighty large U.S. cities in 1990 and 2000. During this time, major correlating factors included income inequality, poverty, educational attainment, residential segregation and marriage rates. However, these ecological factors were weighted differentially across risk groups (e.g. heterosexual, intravenous drug use, men who have sex with men (MSM)). Heterosexual risk rose significantly with poor economic indicators, while MSM risk depended more heavily on anti-homosexual stigma (as measured by same-sex marriage laws). HIV incidence among black individuals correlated significantly with numerous economic factors but also with segregation and imbalances in the male:female ratio (often an effect of mass incarceration). Our results support an overall model of HIV ecology where poverty, income inequality and social inequality (in the form of institutionalized racism and anti-homosexual stigma) have over time developed into synergistic drivers of disease transmission in the U.S., inhibiting information-based prevention efforts. The relative weights of these distal factors vary over time and by HIV risk group. Our testable model may be more generally applicable within the U.S. and beyond