46,695 research outputs found
Positivity, decay, and extinction for a singular diffusion equation with gradient absorption
We study qualitative properties of non-negative solutions to the Cauchy
problem for the fast diffusion equation with gradient absorption \partial_t u
-\Delta_{p}u+|\nabla u|^{q}=0\quad in\;\; (0,\infty)\times\RR^N, where , , and . Based on gradient estimates for the solutions, we
classify the behavior of the solutions for large times, obtaining either
positivity as for , optimal decay estimates as
for , or extinction in finite time for . In addition, we show how the diffusion prevents extinction in finite
time in some ranges of exponents where extinction occurs for the non-diffusive
Hamilton-Jacobi equation
Optimal testing for properties of distributions
Given samples from an unknown discrete distribution p, is it possible to distinguish whether p belongs to some class of distributions C versus p being far from every distribution in C? This fundamental question has received tremendous attention in statistics, focusing primarily on asymptotic analysis, as well as in information theory and theoretical computer science, where the emphasis has been on small sample size and computational complexity. Nevertheless, even for basic properties of discrete distributions such as monotonicity, independence, logconcavity, unimodality, and monotone-hazard rate, the optimal sample complexity
is unknown. We provide a general approach via which we obtain sample-optimal and computationally efficient testers for all these distribution families. At the core of our approach is an algorithm which solves the following problem: Given samples from an unknown distribution p, and a known distribution q, are p and q close in x[superscript 2]-distance, or far in total variation distance? The optimality of our testers is established by providing matching lower bounds, up to constant factors. Finally, a necessary building block for our testers and an important byproduct of our work are the first known computationally efficient proper learners for discrete log-concave, monotone hazard rate distributions
Impressions of convexity - An illustration for commutator bounds
We determine the sharpest constant such that for all complex
matrices and , and for Schatten -, - and -norms the inequality
is valid. The main theoretical
tool in our investigations is complex interpolation theory.Comment: 32 pages, 88 picture
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