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

Animal Spirits Meets Creative Destruction

We show how a Schumpeterian process of creative destruction can induce coordination in the timing of entrepreneurial activities across diverse sectors of the economy.Consequently, a multi-sector economy, in which sector-specific, productivity improvements are made by independent, profit-seeking entrepreneurs, can exhibit regular booms, slowdowns and downturns as an inherent part of the long-run growth process.The cyclical equilibrium that we study has a higher long-run growth rate but lower welfare than the corresponding acyclical one.We find that the cycles generated by our model share some features of actual business cycles, and that across cycling economies, a negative relationship emerges between volatility and growth.economic growth;entrepreneurship;innovation;business cycles