Least Squares Ranking on Graphs

Given a set of alternatives to be ranked, and some pairwise comparison data, ranking is a least squares computation on a graph. The vertices are the alternatives, and the edge values comprise the comparison data. The basic idea is very simple and old: come up with values on vertices such that their differences match the given edge data. Since an exact match will usually be impossible, one settles for matching in a least squares sense. This formulation was first described by Leake in 1976 for rankingfootball teams and appears as an example in Professor Gilbert Strang's classic linear algebra textbook. If one is willing to look into the residual a little further, then the problem really comes alive, as shown effectively by the remarkable recent paper of Jiang et al. With or without this twist, the humble least squares problem on graphs has far-reaching connections with many current areas ofresearch. These connections are to theoretical computer science (spectral graph theory, and multilevel methods for graph Laplacian systems); numerical analysis (algebraic multigrid, and finite element exterior calculus); other mathematics (Hodge decomposition, and random clique complexes); and applications (arbitrage, and ranking of sports teams). Not all of these connections are explored in this paper, but many are. The underlying ideas are easy to explain, requiring only the four fundamental subspaces from elementary linear algebra. One of our aims is to explain these basic ideas and connections, to get researchers in many fields interested in this topic. Another aim is to use our numerical experiments for guidance on selecting methods and exposing the need for further development.Comment: Added missing references, comparison of linear solvers overhauled, conclusion section added, some new figures adde

Green's Functions and the Adiabatic Hyperspherical Method

We address the few-body problem using the adiabatic hyperspherical representation. A general form for the hyperangular Green's function in $d$-dimensions is derived. The resulting Lippmann-Schwinger equation is solved for the case of three-particles with s-wave zero-range interactions. Identical particle symmetry is incorporated in a general and intuitive way. Complete semi-analytic expressions for the nonadiabatic channel couplings are derived. Finally, a model to describe the atom-loss due to three-body recombination for a three-component fermi-gas of $^{6}$Li atoms is presented.Comment: 14 pages, 8 figures, 2 table

Sodium Absorption From the Exoplanetary Atmosphere of HD189733b Detected in the Optical Transmission Spectrum

We present the first ground-based detection of sodium absorption in the transmission spectrum of an extrasolar planet. Absorption due to the atmosphere of the extrasolar planet HD189733b is detected in both lines of the NaI doublet. High spectral resolution observations were taken of eleven transits with the High Resolution Spectrograph (HRS) on the 9.2 meter Hobby-Eberly Telescope (HET). The NaI absorption in the transmission spectrum due to HD189733b is (-67.2 +/- 20.7) x 10^-5 deeper in the ``narrow'' spectral band that encompasses both lines relative to adjacent bands. The 1-sigma error includes both random and systematic errors, and the detection is >3-sigma. This amount of relative absorption in NaI for HD189733b is ~3x larger than detected for HD209458b by Charbonneau et al. (2002), and indicates these two hot-Jupiters may have significantly different atmospheric properties.Comment: 12 pages, 2 figures; Accepted for publication in ApJ Letter

The Theology and Methods of George Whitefield

George Whitefield ranks among the most influential evangelists of all time. Unfortunately, the impact of his theology and methods on Church Growth Thought has never been researched. Seth Polk, senior pastor of Cross Lanes Baptist Church in West Virginia, explores this aspect of George Whitefield’s ministry in “The Theology and Methods of George Whitefield.

Causal connectivity of evolved neural networks during behavior

To show how causal interactions in neural dynamics are modulated by behavior, it is valuable to analyze these interactions without perturbing or lesioning the neural mechanism. This paper proposes a method, based on a graph-theoretic extension of vector autoregressive modeling and 'Granger causality,' for characterizing causal interactions generated within intact neural mechanisms. This method, called 'causal connectivity analysis' is illustrated via model neural networks optimized for controlling target fixation in a simulated head-eye system, in which the structure of the environment can be experimentally varied. Causal connectivity analysis of this model yields novel insights into neural mechanisms underlying sensorimotor coordination. In contrast to networks supporting comparatively simple behavior, networks supporting rich adaptive behavior show a higher density of causal interactions, as well as a stronger causal flow from sensory inputs to motor outputs. They also show different arrangements of 'causal sources' and 'causal sinks': nodes that differentially affect, or are affected by, the remainder of the network. Finally, analysis of causal connectivity can predict the functional consequences of network lesions. These results suggest that causal connectivity analysis may have useful applications in the analysis of neural dynamics