Despite its large sample efficiency, the truncated flat (TF) kernel estimator of long-run covariance matrices is seldom used, because it lacks the guaranteed positive semidefiniteness and sometimes performs poorly in small samples, compared to other familiar kernel estimators. This paper proposes simple modifications to the TF estimator to enforce the positive definiteness without sacrificing the large sample efficiency and make the estimator more reliable in small samples through better utilization of the bias-variance tradeoff. We study the large sample properties of the modified TF estimators and verify their improved small-sample performances by Monte Carlo simulations.
We begin by reviewing various classical problems concerning the existence of primes or numbers with few prime factors as well as some of the key developments towards resolving these long standing questions. Then we put the theory in a natural and general geometric context of actions on affine n-space and indicate what can be established there. The methods used to develop a combinational sieve in this context involve automorphic forms, expander graphs and unexpectedly arithmetic combinatorics. Applications to classical problems such as the divisibility of the areas of Pythagorean triangles and of the curvatures of the circles in an integral Apollonian packing, are given.
The speaker introduces the formal notion of an approximately specified nonlinear regression model and investigates sequential design methodologies when the fitted model is possibly of an incorrect parametric form. He presents small-sample simulation studies which indicate that his new designs can be very successful, relative to some common competitors, in reducing mean squared error due to model misspecification and to heteroscedastic variation. His simulations also suggest that standard normal-theory inference procedures remain approximately valid under the sequential sampling schemes.
• These lectures will introduce the theory of Kirchhoff migration and imaging from an inversion perspective
• They are intended to teach some geophysics to mathematicians and some mathematics to geophysicists
• Recommended reference: Bleistein, Cohen & Stockwell, 2001, “Mathematics of Multidimensional Seismic Imaging, Migration, and Inversion”
• Geometry and Holomony
• Supersymmetry, Spinors, and Calabi-Yau
• Flux and Backreaction
• Energetics of Heterotic Flux Compactification
• Strominger System and Heterotic Flux as a Torsion
• A Supersymmetric Solution to Heterotic Flux Compactification
• Global Issues: Index Counting, Smoothness, etc
Aperiodic Order, Dynamical Systems, Operator Algebras and Topology
Abstract:
This is a slightly expanded version of a talk given at the Workshop on Aperiodic Order, held in Victoria, B.C. in August, 2002. The general subject of the talk was the densest packings of simple bodies, for instance spheres or polyhedra, in Euclidean or hyperbolic spaces, and describes recent joint work with Lewis Bowen. One of the main points was to report on our solution of the old problem of treating optimally dense packings of bodies in hyperbolic spaces. The other was to describe the general connection between aperiodicity and nonuniqueness in problems of optimal density.
In this talk we consider a random walk on a randomly colored lattice and ask what are the properties of the sequence of colors encountered by the walk.