A comparison theorem for the isoperimetric profile under curve shortening flow

We prove a comparison theorem for the isoperimetric profiles of simple closed curves evolving by the normalized curve shortening flow: If the isoperimetric profile of the region enclosed by the initial curve is greater than that of some `model' convex region with exactly four vertices and with reflection symmetry in both axes, then the inequality remains true for the isoperimetric profiles of the evolved regions. We apply this using the Angenent solution as the model region to deduce sharp time-dependent upper bounds on curvature for arbitrary embedded closed curves evolving by the normalized curve shortening flow. A slightly different comparison also gives lower bounds on curvature, and the result is a simple and direct proof of Grayson's theorem without use of any blowup or compactness arguments, Harnack estimates, or classification of self-similar solutions.Comment: 32 pages, 6 figure

An empirical analysis of ‘challenge’ as a motivational factor for educational games

Since one of the most basic and important predictors of student achievement is the amount of time a student spends engaged in learning (or time-on-task; Karweit, 1984; Frederick & Walberg, 1980); and because computer games are hugely successful at motivating users to spend time-on-task (Dondlinger, 2007; Gee, 2003; Mayo, 2007), there has understandably been a great deal of recent interest in harnessing the motivational qualities of computer games in order to create powerful, engaging educational tools (i.e., Gee, 2003; Pivec, 2007; Ruben, 1999). However, to date very little empirical academic research has investigated how, exactly, games achieve these motivational qualities. If we are to create games that produce genuinely educational outcomes, we must understand what exactly it is about games that make them so good at maintaining the player’s motivation to continue playing

A behavioural framework for designing educational computer games

Research has indicated that computer games can be innovative and powerful tools for education. Indeed, combining psychological research and games design principles offers a framework for developing educational games that promote learning while maintaining high motivation of the players. If designed correctly, it appears that games can utilize the inherent motivation demonstrated by game players to teach skills that are of immediate practical benefit. The current paper explores “the edges of gaming” in terms of proposing a novel theoretical and methodological framework for the design of educational games

The motivic Donaldson-Thomas invariants of (-2) curves

In this paper we calculate the motivic Donaldson-Thomas invariants for (-2)-curves arising from 3-fold flopping contractions in the minimal model programme. We translate this geometric situation into the machinery developed by Kontsevich and Soibelman, and using the results and framework developed previously by the authors we describe the monodromy on these invariants. In particular, in contrast to all existing known Donaldson-Thomas invariants for small resolutions of Gorenstein singularities these monodromy actions are nontrivial.Comment: 30 pages, 3 figure

Whippoorwill

https://digitalcommons.library.umaine.edu/mmb-vp/4827/thumbnail.jp