Applied

In the network externality literature, little, if any attention has been paid to the process through which consumers coordinate their adoption decisions. The primary objective of this paper is to discover how effectively rational individuals manage to coordinate their choices in a sequential choice framework. Since individuals make their choices with minimal information in this setting, perfect coordination will rarely be achieved, and it is therefore of some interest to discern both the extent to which coordination may be achieved, and the expected cost of the failure to achieve perfect coordination. We discover that when it counts, that is when the network externality is large, a substantial amount of coordination is achieved, and although perfect coordination is never guaranteed, expected relative efficiency is large.

A fast, explicit wavefield extrapolator based on the GPSPI formula is presented.

The central problem of extrapolator stability is presented and addressed by designing two half-step operators with opposing instability.

Spatial resampling is described as a very useful imaging tool.

Gabor methods can be used to approximate pseudodifferential operators.

Gabor wavefield extrapolators, based on an adaptive POU, give promising wavefield extrapolation results.

We look at the mathematical theory of partial differential equations as applied to the wave equation. In particular, we examine questions about existence and uniqueness of solutions, and various solution techniques.

An explicit part of my agenda with these lectures is to encourage more mathematicians to seriously consider issues coming from the social sciences; let me assure you that you will find different and new mathematical issues. In the other direction, I also hope to encourage social scientists to appreciate the important gains that can result by using serious mathematics; I want to encourage the social scientists to seriously consider using this powerful approach. Because of these twin goals, my lectures, and these notes, are explicitly designed to address both audiences. For instance, the beginning of each section consists of examples which are intended to help develop intuition about the issues at hand. Then, toward the end of each section, there is a slightly stronger mathematical emphasis which is intended for the mathematicians. Nevertheless, I encourage the social scientists reading these notes to push on through this somewhat more technical material.

Table of Contents:

1. Mathematical Physical vs. Social Sciences

2. Symmetry galore!

3. Singularity theory and departmental meetings

4. Evolutionary game theory

 5. Adam Smith’s “Invisible hand” — and continuous foliations

I will present a survey of modelling, computational, and analytical work on thin liquid films of viscous fluids. I will particularly focus on films that are being acted on by more than one force. For example, if you've painted the ceiling, how do you model the effects of surface tension and gravity? How do you study the dynamics of the air/liquid interface? How do things change if you're considering a freshly painted wall? Or floor?