17,033 research outputs found
The Szemeredi-Trotter Theorem in the Complex Plane
It is shown that points and lines in the complex Euclidean plane
determine point-line incidences. This
bound is the best possible, and it generalizes the celebrated theorem by
Szemer\'edi and Trotter about point-line incidences in the real Euclidean plane
.Comment: 24 pages, 5 figures, to appear in Combinatoric
Asymptotic Normality of Quadratic Estimators
We prove conditional asymptotic normality of a class of quadratic
U-statistics that are dominated by their degenerate second order part and have
kernels that change with the number of observations. These statistics arise in
the construction of estimators in high-dimensional semi- and non-parametric
models, and in the construction of nonparametric confidence sets. This is
illustrated by estimation of the integral of a square of a density or
regression function, and estimation of the mean response with missing data. We
show that estimators are asymptotically normal even in the case that the rate
is slower than the square root of the observations
The geometry of fractal percolation
A well studied family of random fractals called fractal percolation is
discussed. We focus on the projections of fractal percolation on the plane. Our
goal is to present stronger versions of the classical Marstrand theorem, valid
for almost every realization of fractal percolation. The extensions go in three
directions: {itemize} the statements work for all directions, not almost all,
the statements are true for more general projections, for example radial
projections onto a circle, in the case , each projection has not
only positive Lebesgue measure but also has nonempty interior. {itemize}Comment: Survey submitted for AFRT2012 conferenc
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