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

Freelancer

Follow Captain Travers and the crew of the Tireless Nomad, a surplus military exploration ship, on one leg of their voyage through a particularly perilous sector of pirate patrolled space. Everything starts out normally enough, but before long chaos ensues

Race, Immigration Reform, and Heteropatriarchal Masculinity: Reframing the Obama Presidency

In this article, I argue that the macroscopic nature and complexity of race, hegemonic masculinity, and immigration issues in the United States put President Obama in a double bind for any attempt to secure reforms—situations which produce limited options and exposure to penalties in reaching solutions

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.

A Delphi Study of Additive Manufacturing Applicability for United States Air Force Civil Engineer Contingency Operations

Additive manufacturing is a relatively new technique that is gaining popularity in many applications. This research examines the possibilities for the integration of additive manufacturing (AM) machines in United States Air Force civil engineer (CE) contingency operations. A Delphi study was conducted that combined the knowledge and experience of experts in both the AM industry and the Air Force CE community to forecast the possible benefits and drawbacks of this novel AM application. The results of this Delphi study indicate that including an AM machine would be beneficial in meeting deployed Air Force CE requirements. Further, AM technology has reached a point that a pilot study would be beneficial to validate the benefits of including an AM machine in CE operations. Proposed goals of, and a design for this study are presented. Further, the results indicate that within the next five years, AM technology will have progressed far enough that a full-scale deployment of AM machines to meet Air Force CE contingency requirements will be beneficial