Singularity of Data Analytic Operations

Statistical data by their very nature are indeterminate in the sense that if one repeated the process of collecting the data the new data set would be somewhat different from the original. Therefore, a statistical method, a map $\Phi$ taking a data set $x$ to a point in some space F, should be stable at $x$: Small perturbations in $x$ should result in a small change in $\Phi(x)$. Otherwise, $\Phi$ is useless at $x$ or -- and this is important -- near $x$. So one doesn't want $\Phi$ to have "singularities," data sets $x$ s.t.\ the the limit of $\Phi(y)$ as $y$ approaches $x$ doesn't exist. (Yes, the same issue arises elsewhere in applied math.) However, broad classes of statistical methods have topological obstructions of continuity: They must have singularities. We show why and give lower bounds on the Hausdorff dimension, even Hausdorff measure, of the set of singularities of such data maps. There seem to be numerous examples. We apply mainly topological methods to study the (topological) singularities of functions defined (on dense subsets of) "data spaces" and taking values in spaces with nontrivial homology. At least in this book, data spaces are usually compact manifolds. The purpose is to gain insight into the numerical conditioning of statistical description, data summarization, and inference and learning methods. We prove general results that can often be used to bound below the dimension of the singular set. We apply our topological results to develop lower bounds on Hausdorff measure of the singular set. We apply these methods to the study of plane fitting and measuring location of data on spheres. \emph{This is not a "final" version, merely another attempt.}Comment: 325 pages, 8 figure

π−N\pi-N from an Extended Effective Field Theory

Third order chiral perturbation theory accounts for the $\pi-N$ scattering phase shift data out to energies slightly below the position of the $\Delta$ resonance. The low energy constants are not accurately determined. Explicit inclusion of the $\Delta$ field is favored.Comment: 2 pages latex, working group talk, Chiral Dynamics 2000, Jefferson Lab., VA, July 2000, World Scientific, to be pu

MONOLITH: a next generation experiment for athospheric neutrinos

MONOLITH is a massive magnetized tracking calorimeter, optimized for the detection of atmospheric muon neutrinos, proposed at the Gran Sasso laboratory in Italy. The main goal is to establish (or reject) the neutrino oscillation hypothesis through an explicit observation of the full first oscillation swing (the ``L/E pattern''). Its performance, status and prospects are briefly reviewed.Comment: Talk given at Europhysics Neutrino Oscillation Workshop (NOW2000), Otranto, Italy, September 9-16, 2000 (4 pages, 3 figures

Co-movement, Capital and Contracts: 'Normal' Cycles Through Creative Destruction

We develop a unified theory of endogenous business cycles in which expansions are neoclassical growth periods driven by productivity improvements and capital accumulation, while downturns are the result of Keynesian contractions in aggregate demand below potential output. Recessions allow skilled labor to be reallocated to growth promoting activities which fuel subsequent expansions. However, rigidities in production and contractual limitations, inherent to the process of creative destruction, leave capital severely underutilized. A key feature of our equilibrium is the endogenous emergence of long term supply contracts between capitalist owners and producers.Long-term contracting;investment irreversibility;putty-clay technology;asset- specificity;Endogenous cycles and growth

VEGA Pathfinder navigation for Giotto Halley encounter

Results of the VEGA Pathfinder concept which was used to successfully target the European Space Agnecy's Giotto spacecraft to a 600 km encounter with the comet Halley are presented. Pathfinder was an international cooperative navigation activity involving USSR, European and U.S. space agencies. The final Giotto targeting maneuver was based on a comet location determined from optical data acquired by the earlier arriving Soviet VEGA spacecraft. Inertial pointing angles extracted from optical images of the comet nucleus were combined with a precise estimate of the VEGA encounter orbits determined using VLBI data acquired by NASA's Deep Space Network to predict the location of Halley at Giotto encounter. This article describes the VLBI techniques used to determine the VEGA orbits and shows that the insensitivity of the VLBI data strategy to unmodeled dynamic error sources resulted in estimates of the VEGA orbits with an accuracy of 50 km