46,014 research outputs found

### 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

### $\pi-N$ 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

### The differential graded odd nilHecke algebra

We equip the odd nilHecke algebra and its associated thick calculus category
with digrammatically local differentials. The resulting differential graded
Grothendieck groups are isomorphic to two different forms of the positive part
of quantum sl(2) at a fourth root of unity.Comment: 53 page

### The Hopf algebra of odd symmetric functions

We consider a q-analogue of the standard bilinear form on the commutative
ring of symmetric functions. The q=-1 case leads to a Z-graded Hopf
superalgebra which we call the algebra of odd symmetric functions. In the odd
setting we describe counterparts of the elementary and complete symmetric
functions, power sums, Schur functions, and combinatorial interpretations of
associated change of basis relations.Comment: 43 pages, 12 figures. v2: some correction

### 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

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