Minimization of vectors of curvilinear functionals on the second order jet bundle

Consider the multi-time multi-objective variational problem (MFP) of minimizing a vector of quotients of path independent curvilinear functionals subject to PDE and/or PDI constraints. The goal of our work is to develop an optimization theory for the second order jet bundle. While the background in Sect. 1 is introductory, the theory in Sects. 2 and 3 is new as a whole, containing our results.     

The Strauss conjecture on Kerr black hole backgrounds

We examine solutions to semilinear wave equations on black hole backgrounds and give a proof of an analog of the Strauss conjecture on the Schwarzschild and Kerr, with small angular momentum, black hole backgrounds. The key estimates are a class of weighted Strichartz estimates, which are used near infinity where the metrics can be viewed as small perturbations of the Minkowski metric, and a localized energy estimate on the black hole background, which handles the behavior in the remaining compact set.     

Singular open book structures from real mappings

We define open book structures with singular bindings. Starting with an extension of Milnor’s results on local fibrations for germs with nonisolated singularity, we find classes of genuine real analytic mappings which yield such open book structures.     

On exact quadrature formulas for harmonic functions on polyhedra

A classical quadrature result for analytic functions of a complex variable due to Motzkin and Schoenberg is extended to higher dimensions. A general scheme for integrating on polyhedra solutions of partial differential equations is discussed.

     

Solving Elliptic Equations with Brownian Motion: Bias Reduction and Temporal Difference Learning

Methodology and Computing in Applied Probability - The Feynman-Kac formula provides a way to understand solutions to elliptic partial differential equations in terms of expectations of continuous...