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

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 - 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

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

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

The Private Life of Environmental Treaties

The gravitational pull of environmental treaties is felt not only by states. Yet international lawyers almost exclusively focus on states to explain treaty compliance, measure treaty implementation, and assess treaty effectiveness. This essay draws attention to a phenomenon that falls outside traditional boundaries of treaty analysis: the efforts of private corporations that aim at complying with environmental treaties. Existing models of treaty implementation are inadequate to explain these direct interactions between corporations and treaties. The dominant grammar of treaty “compliance” equally fails to fit. Using a little-studied example - the UNESCO World Heritage Convention - this essay highlights the phenomenon of corporations’ aspiring to conform their behavior to environmental treaty requirements