Global time estimates for solutions to equations of dissipative type

Global time estimates of Lp-Lq norms of solutions to general strictly hyperbolic partial differential equations are considered. The case of special interest in this paper are equations exhibiting the dissipative behaviour. Results are applied to discuss time decay estimates for Fokker-Planck equations and for wave type equations with negative mass.Comment: Journees "Equations aux Derivees Partielles

Degenerate neckpinches in Ricci flow

In earlier work, we derived formal matched asymptotic profiles for families of Ricci flow solutions developing Type-II degenerate neckpinches. In the present work, we prove that there do exist Ricci flow solutions that develop singularities modeled on each such profile. In particular, we show that for each positive integer $k\geq3$, there exist compact solutions in all dimensions $m\geq3$ that become singular at the rate (T-t)^{-2+2/k}$

Structural Induction Principles for Functional Programmers

User defined recursive types are a fundamental feature of modern functional programming languages like Haskell, Clean, and the ML family of languages. Properties of programs defined by recursion on the structure of recursive types are generally proved by structural induction on the type. It is well known in the theorem proving community how to generate structural induction principles from data type declarations. These methods deserve to be better know in the functional programming community. Existing functional programming textbooks gloss over this material. And yet, if functional programmers do not know how to write down the structural induction principle for a new type - how are they supposed to reason about it? In this paper we describe an algorithm to generate structural induction principles from data type declarations. We also discuss how these methods are taught in the functional programming course at the University of Wyoming. A Haskell implementation of the algorithm is included in an appendix.Comment: In Proceedings TFPIE 2013, arXiv:1312.221

Fourier restriction to polynomial curves I: a geometric inequality

We prove a Fourier restriction result for general polynomial curves in Rd. Measuring the Fourier restriction with respect to the affine arclength measure of the curve, we obtain a universal estimate for the class of all polynomial curves of bounded degree. Our method relies on establishing a geometric inequality for general polynomial curves which is of interest in its own right. Applications of this geometric inequality to other problems in euclidean harmonic analysis have recently been established

LpL^p improving bounds for averages along curves

We establish local $(L^p,L^q)$ mapping properties for averages on curves. The exponents are sharp except for endpoints.Comment: 37 pages, simplified argument (no further need for algebraic complexity theory!), to appear, JAM

Discrete Approximations of Metric Measure Spaces of Controlled Geometry

We find a necessary and sufficient condition for a doubling metric space to carry a (1,p)-Poincare inequality. The condition involves discretizations of the metric space and Poincare inequalities on graphs.Comment: 23 Page