Schramm's proof of Watts' formula

G\'{e}rard Watts predicted a formula for the probability in percolation that there is both a left--right and an up--down crossing, which was later proved by Julien Dub\'{e}dat. Here we present a simpler proof due to Oded Schramm, which builds on Cardy's formula in a conceptually appealing way: the triple derivative of Cardy's formula is the sum of two multi-arm densities. The relative sizes of the two terms are computed with Girsanov conditioning. The triple integral of one of the terms is equivalent to Watts' formula. For the relevant calculations, we present and annotate Schramm's original (and remarkably elegant) Mathematica code.Comment: Published in at http://dx.doi.org/10.1214/11-AOP652 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Wave techniques for noise modeling and measurement

The noise wave approach is applied to analysis, modeling, and measurement applications. Methods are presented for the calculation of component and network noise wave correlation matrices. Embedding calculations, relations to two-port figures-of-merit, and transformations to traditional representations are discussed. Simple expressions are derived for MESFET and HEMT noise wave parameters based on a linear equivalent circuit. A noise wave measurement technique is presented and experimentally compared with the conventional method

RinRuby: Accessing the R Interpreter from Pure Ruby

RinRuby is a Ruby library that integrates the R interpreter in Ruby, making R's statistical routines and graphics available within Ruby. The library consists of a single Ruby script that is simple to install and does not require any special compilation or installation of R. Since the library is 100% pure Ruby, it works on a variety of operating systems, Ruby implementations, and versions of R. RinRuby's methods are simple, making for readable code. This paper describes RinRuby usage, provides comprehensive documentation, gives several examples, and discusses RinRuby's implementation. The latest version of RinRuby can be found at the project website: http://rinruby.ddahl.org/.

A Coasian Approach to Efficient Water Allocation of a Transboundary River

The United States and Mexico recently resolved a decade-old water dispute that required Mexico to repay the accumulated water debt within one year. A Coasian analysis estimates the social welfare gains attainable to each country under an alternative debt repayment scheme that allows repayment over a longer time horizon and in a combination of dollars and water, instead of solely in water. Assuming average water supply conditions, under the agreed 1-year repayment contract, U.S. compensation value is 534% greater and Mexico’s compensation cost is 60% less relative to when compensation is paid exclusively in water.coase, water allocation, water compensation, water markets, Agribusiness, Consumer/Household Economics, Environmental Economics and Policy, Q1, Q2,

Tug-of-war and the infinity Laplacian

We prove that every bounded Lipschitz function F on a subset Y of a length space X admits a tautest extension to X, i.e., a unique Lipschitz extension u for which Lip_U u = Lip_{boundary of U} u for all open subsets U of X that do not intersect Y. This was previously known only for bounded domains R^n, in which case u is infinity harmonic, that is, a viscosity solution to Delta_infty u = 0. We also prove the first general uniqueness results for Delta_infty u = g on bounded subsets of R^n (when g is uniformly continuous and bounded away from zero), and analogous results for bounded length spaces. The proofs rely on a new game-theoretic description of u. Let u^epsilon(x) be the value of the following two-player zero-sum game, called tug-of-war: fix x_0=x \in X minus Y. At the kth turn, the players toss a coin and the winner chooses an x_k with d(x_k, x_{k-1})< epsilon. The game ends when x_k is in Y, and player one's payoff is F(x_k) - (epsilon^2/2) sum_{i=0}^{k-1} g(x_i) We show that the u^\epsilon converge uniformly to u as epsilon tends to zero. Even for bounded domains in R^n, the game theoretic description of infinity-harmonic functions yields new intuition and estimates; for instance, we prove power law bounds for infinity-harmonic functions in the unit disk with boundary values supported in a delta-neighborhood of a Cantor set on the unit circle.Comment: 44 pages, 4 figure