Semigroups, rings, and Markov chains

We analyze random walks on a class of semigroups called ``left-regular bands''. These walks include the hyperplane chamber walks of Bidigare, Hanlon, and Rockmore. Using methods of ring theory, we show that the transition matrices are diagonalizable and we calculate the eigenvalues and multiplicities. The methods lead to explicit formulas for the projections onto the eigenspaces. As examples of these semigroup walks, we construct a random walk on the maximal chains of any distributive lattice, as well as two random walks associated with any matroid. The examples include a q-analogue of the Tsetlin library. The multiplicities of the eigenvalues in the matroid walks are ``generalized derangement numbers'', which may be of independent interest.Comment: To appear in J. Theoret. Proba

The Effect of Walkthrough Observations on Teacher Perspectives in Christian Schools

This study investigated the effects on teacher perceptions of frequent, brief classroom observations in Christian schools. Teachers (N=111) responded to 13 belief and value statements prior to and after the term during which administrators conducted weekly, brief, unannounced observations in their classes. Teachers reported significant positive change regarding (a) analyzing reasons for selecting methods to assess learning, (b) being encouraged after class observations, and (c) being encouraged after receiving feedback related to the observations

A Modern Approach to Evidence

A Review of A Modern Approach to Evidence by Richard O. Lempert and Stephen A. Saltzbur

Finding the future:evolving interaction design

The main aim of this project is to design and prototype a simplified example of a mobile operating system that makes use of both edge swipe control and 'smart' graphical instructions. The research will consider how these methods can be used to design a truly inclusive and accessible interface. The effectiveness of these features will be validated through user experiments and focus groups over the course of the project, with the findings of user testing used to inform design practice

Forest diagrams for elements of Thompson's group F

We introduce forest diagrams to represent elements of Thompson's group F. These diagrams relate to a certain action of F on the real line in the same way that tree diagrams relate to the standard action of F on the unit interval. Using forest diagrams, we give a conceptually simple length formula for elements of F with respect to the {x_0,x_1} generating set, and we discuss the construction of minimum-length words for positive elements. Finally, we use forest diagrams and the length formula to examine the structure of the Cayley graph of F.Comment: 44 pages, 70 figure