From Baking a Cake to Solving the Schrodinger Equation

The primary emphasis of this study has been to explain how modifying a cake recipe by changing either the dimensions of the cake or the amount of cake batter alters the baking time. Restricting our consideration to the genoise, one of the basic cakes of classic French cuisine, we have obtained a semi-empirical formula for its baking time as a function of oven temperature, initial temperature of the cake batter, and dimensions of the unbaked cake. The formula, which is based on the Diffusion equation, has three adjustable parameters whose values are estimated from data obtained by baking genoises in cylindrical pans of various diameters. The resulting formula for the baking time exhibits the scaling behavior typical of diffusion processes, i.e. the baking time is proportional to the (characteristic length scale)^2 of the cake. It also takes account of evaporation of moisture at the top surface of the cake, which appears to be a dominant factor affecting the baking time of a cake. In solving this problem we have obtained solutions of the Diffusion equation which are interpreted naturally and straightforwardly in the context of heat transfer; however, when interpreted in the context of the Schrodinger equation, they are somewhat peculiar. The solutions describe a system whose mass assumes different values in two different regions of space. Furthermore, the solutions exhibit characteristics similar to the evanescent modes associated with light waves propagating in a wave guide. When we consider the Schrodinger equation as a non-relativistic limit of the Klein-Gordon equation so that it includes a mass term, these are no longer solutions.Comment: 23 pages, 10 Postscript figure

Effects of cochlear implantation on binaural hearing in adults with unilateral hearing loss

A FDA clinical trial was carried out to evaluate the potential benefit of cochlear implant (CI) use for adults with unilateral moderate-to-profound sensorineural hearing loss. Subjects were 20 adults with moderate-to-profound unilateral sensorineural hearing loss and normal or near-normal hearing on the other side. A MED-EL standard electrode was implanted in the impaired ear. Outcome measures included: (a) sound localization on the horizontal plane (11 positions, −90° to 90°), (b) word recognition in quiet with the CI alone, and (c) masked sentence recognition with the target at 0° and the masker at −90°, 0°, or 90°. This battery was completed preoperatively and at 1, 3, 6, 9, and 12 months after CI activation. Normative data were also collected for 20 age-matched control subjects with normal or near-normal hearing bilaterally. The CI improved localization accuracy and reduced side bias. Word recognition with the CI alone was similar to performance of traditional CI recipients. The CI improved masked sentence recognition when the masker was presented from the front or from the side of normal or near-normal hearing. The binaural benefits observed with the CI increased between the 1- and 3-month intervals but appeared stable thereafter. In contrast to previous reports on localization and speech perception in patients with unilateral sensorineural hearing loss, CI benefits were consistently observed across individual subjects, and performance was at asymptote by the 3-month test interval. Cochlear implant settings, consistent CI use, and short duration of deafness could play a role in this result

Toward Greater Effectiveness in Community Change: Challenges and Responses for Philanthropy

Offers a model suggesting how foundations can most effectively think about, do the work of, and learn from community change. Part of the series Practice Matters: The Improving Philanthropy Project

Toward Greater Effectiveness in Community Change: Challenges and Responses for Philanthropy - Executive Summary

Philanthropies of all types seek to improve communities—for lots of reasons, and in lots of different ways. Their efforts have produced promising results and some beginning lessons about community change. But more remains to be done to ensure that philanthropic investments in community change meet expectations and that funders use the emerging lessons to move their agendas forward. Based on interviews conducted for this paper, many funders are eager to take on that challenge

Toward Greater Effectiveness in Community Change: Challenges and Responses for Philanthropy - Discussion Guide

The goal of this discussion guide is to assist foundations to engage in a dialogue about how philanthropy can become more effective in its support of community-change initiatives. The guide can be used in at least two ways. A foundation or a group of foundations intending to launch a community change initiative can use the guide as part of its planning process. Alternatively, foundations already involved in supporting a community-change initiative can use the guide as a framework to review the project's status and examine whether any changes in philanthropic practice make sense

Direct Measurement of the Radius and Density of the Transiting Exoplanet HD 189733B with the CHARA Array

We have measured the angular diameter of the transiting extrasolar planet host star HD 189733 using the CHARA O/IR interferometric array. Combining our new angular diameter of 0.377+/-0.024 mas with the Hipparcos parallax leads to a linear radius for the host star of 0.779+/-0.052 Rsol and a radius for the planet of 1.19+/-0.08 RJup. Adopting the mass of the planet as derived by its discoverers, we derive a mean density of the planet of 0.91+/-0.18 g cm-3. This is the first determination of the diameter of an extrasolar planet through purely direct means.Comment: 14 pages, 5 figures, to be published in Astrophysical Journal Letter