user's guide to viscosity solutions of second order partial differential equations

The notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems, existence theorems, and theorems about continuous dependence may now be proved by very efficient and striking arguments. The range of important applications of these results is enormous. This article is a self-contained exposition of the basic theory of viscosity solutions.Comment: 67 page

Convexity criteria and uniqueness of absolutely minimizing functions

We show that absolutely minimizing functions relative to a convex Hamiltonian $H:\mathbb{R}^n \to \mathbb{R}$ are uniquely determined by their boundary values under minimal assumptions on $H.$ Along the way, we extend the known equivalences between comparison with cones, convexity criteria, and absolutely minimizing properties, to this generality. These results perfect a long development in the uniqueness/existence theory of the archetypal problem of the calculus of variations in $L^\infty.$Comment: 34 page

Uniqueness Results for Second Order Bellman-Isaacs Equations under Quadratic Growth Assumptions and Applications

In this paper, we prove a comparison result between semicontinuous viscosity sub and supersolutions growing at most quadratically of second-order degenerate parabolic Hamilton-Jacobi-Bellman and Isaacs equations. As an application, we characterize the value function of a finite horizon stochastic control problem with unbounded controls as the unique viscosity solution of the corresponding dynamic programming equation

The Effects of Home Computers on Educational Outcomes: Evidence from a Field Experiment with Schoolchildren

Are home computers are an important input in the educational production function? To address this question, we conduct a field experiment involving the provision of free computers to schoolchildren for home use. Low-income children attending middle and high schools in 15 schools in California were randomly selected to receive free computers and followed over the school year. The results indicate that the experiment substantially increased computer ownership and total computer use among the schoolchildren with no substitution away from use at school or other locations outside the home. We find no evidence that the home computers improved educational outcomes for the treatment group. From detailed administrative data provided by the schools and a follow-up survey, we find no evidence of positive effects on a comprehensive set of outcomes such as grades, test scores, credits, attendance, school enrollment, computer skills, and college aspirations. The estimates also do not indicate that the effects of home computers on educational outcomes are instead negative. Our estimates are precise enough to rule out even modestly-sized positive or negative impacts. The lack of a positive net effect on educational outcomes may be due to displacement from non-educational uses such as for games, social networking, and entertainment. We find evidence that total hours of computer use for games and social networking increases substantially with having a home computer, and increases more than total hours of computer use for schoolwork

Species Tree Estimation for the Late Blight Pathogen, Phytophthora infestans, and Close Relatives

To better understand the evolutionary history of a group of organisms, an accurate estimate of the species phylogeny must be known. Traditionally, gene trees have served as a proxy for the species tree, although it was acknowledged early on that these trees represented different evolutionary processes. Discordances among gene trees and between the gene trees and the species tree are also expected in closely related species that have rapidly diverged, due to processes such as the incomplete sorting of ancestral polymorphisms. Recently, methods have been developed for the explicit estimation of species trees, using information from multilocus gene trees while accommodating heterogeneity among them. Here we have used three distinct approaches to estimate the species tree for five Phytophthora pathogens, including P. infestans, the causal agent of late blight disease in potato and tomato. Our concatenation-based “supergene” approach was unable to resolve relationships even with data from both the nuclear and mitochondrial genomes, and from multiple isolates per species. Our multispecies coalescent approach using both Bayesian and maximum likelihood methods was able to estimate a moderately supported species tree showing a close relationship among P. infestans, P. andina, and P. ipomoeae. The topology of the species tree was also identical to the dominant phylogenetic history estimated in our third approach, Bayesian concordance analysis. Our results support previous suggestions that P. andina is a hybrid species, with P. infestans representing one parental lineage. The other parental lineage is not known, but represents an independent evolutionary lineage more closely related to P. ipomoeae. While all five species likely originated in the New World, further study is needed to determine when and under what conditions this hybridization event may have occurred

Fire as a fundamental ecological process: Research advances and frontiers

© 2020 The Authors.Fire is a powerful ecological and evolutionary force that regulates organismal traits, population sizes, species interactions, community composition, carbon and nutrient cycling and ecosystem function. It also presents a rapidly growing societal challenge, due to both increasingly destructive wildfires and fire exclusion in fire‐dependent ecosystems. As an ecological process, fire integrates complex feedbacks among biological, social and geophysical processes, requiring coordination across several fields and scales of study. Here, we describe the diversity of ways in which fire operates as a fundamental ecological and evolutionary process on Earth. We explore research priorities in six categories of fire ecology: (a) characteristics of fire regimes, (b) changing fire regimes, (c) fire effects on above‐ground ecology, (d) fire effects on below‐ground ecology, (e) fire behaviour and (f) fire ecology modelling. We identify three emergent themes: the need to study fire across temporal scales, to assess the mechanisms underlying a variety of ecological feedbacks involving fire and to improve representation of fire in a range of modelling contexts. Synthesis: As fire regimes and our relationships with fire continue to change, prioritizing these research areas will facilitate understanding of the ecological causes and consequences of future fires and rethinking fire management alternatives.Support was provided by NSF‐DEB‐1743681 to K.K.M. and A.J.T. We thank Shalin Hai‐Jew for helpful discussion of the survey and qualitative methods.Peer reviewe