The Pearcey Process

The extended Airy kernel describes the space-time correlation functions for the Airy process, which is the limiting process for a polynuclear growth model. The Airy functions themselves are given by integrals in which the exponents have a cubic singularity, arising from the coalescence of two saddle points in an asymptotic analysis. Pearcey functions are given by integrals in which the exponents have a quartic singularity, arising from the coalescence of three saddle points. A corresponding Pearcey kernel appears in a random matrix model and a Brownian motion model for a fixed time. This paper derives an extended Pearcey kernel by scaling the Brownian motion model at several times, and a system of partial differential equations whose solution determines associated distribution functions. We expect there to be a limiting nonstationary process consisting of infinitely many paths, which we call the Pearcey process, whose space-time correlation functions are expressible in terms of this extended kernel.     

Hydrogen peroxide promoted hydroxylation of haloarenes and heteroarenes

Addition of aqueous hydrogen peroxide significantly accelerates the substitution reactions of hydroxide salts with haloarenes bearing electron withdrawing substituents. A similar effect is observed in the reactions of hydroxide salts with halogenated heteroarenes. Reactions are carried out in water or water-THF at ambient temperature or at 50-60 °C.     

Learning to See: Making Sense of the Mathematics of Change in Middle School

In this article, we will describe the results of a study of 6th grade students learning about the mathematics of change. The students in this study worked with software environments for the computer and the graphing calculator that included a simulation of a moving elevator, linked to a graph of its velocity vs. time. We will describe how the students and their teacher negotiated the mathematical meanings of these representations, in interaction with the software and other representational tools available in the classroom. The class developed ways of selectively attending to specific features of stacks of centimeter cubes, hand-drawn graphs, and graphs (labeled velocity vs. time) on the computer screen. In addition, the class became adept at imagining the motions that corresponded to various velocity vs. time graphs. In this article, we describe this development as a process of learning to see mathematical representations of motion. The main question this article addresses is: How do students learn to see mathematical representations in ways that are consistent with the discipline of mathematics? This revised version was published online in July 2006 with corrections to the Cover Date.     

Using Inquiry to Improve Pedagogy through K‐12/University Partnerships

This article focuses on the impact a collaborative project between university graduate fellows and K‐12 classroom teachers had on improved pedagogy in the classroom and in the future at the university. Nine teams participated in a yearlong professional development project to improve pedagogy and communication skills of the participants. This study shows that the participants, the fellows and the K‐12 teacher partners, made changes in planning, implementation, and even motivation for using inquiry‐based methods in their classroom. External observations of the teams further support the individuals' claims of improved pedagogy using inquiry and impact on student conceptual understanding. The Horizon Classroom Observation instrument was used for these observations. The teams showed an overall increase in scores, as well as overall effective and exemplarily implementation of their planning. The program design, the implementation, and the results of this three‐year study will be elaborated in this article.     

A Linear Time Algorithm for Constructing Maximally Symmetric Straight Line Drawings of Triconnected Planar Graphs

Symmetry is one of the most important aesthetic criteria in graph drawing because it reveals structure in the graph. To draw graphs symmetrically, we employ two steps. The first step is to find appropriate automorphisms. The second step is to draw the graph to display the automorphisms. Our aim in this paper is to construct maximally symmetric straight line drawings of triconnected planar graphs in linear time. Previously known algorithms run in quadratic time. We show that an algorithm of Fontet can be used to find an embedding in the plane with the maximum number of symmetries, and present a new algorithm for finding a straight line drawing that achieves that maximum. Both algorithms run in linear time.     

The asymptotic numbers of regular tournaments,Eulerian digraphs and Eulerian oriented graphs

LetRT(n), ED(n) andEOG(n) be the number of labelled regular tournaments, labelled loop-free simple Eulerian digraphs, and labelled Eulerian oriented simple graphs, respectively, onn vertices. Then, asn,, for any>0. The last two families of graphs are also enumerated by their numbers of edges. The proofs use the saddle point method applied to appropriaten-dimensional integrals.     

Equilibrium studies of the sorption of Cu(II) ions onto chitosan

Chitosan is a polymer that can be obtained from the shells of seafood such as prawns, crabs, and lobsters. Chitosan has free amino groups, which can attract metal ions, and has been used as an adsorbent for the removal of metal ions from effluents. In this research, the sorption of copper ions from solution onto chitosan at two pHs has been investigated. DSC, TGA, surface area, SEM, and NMR studies have been used to report the pure physical states of chitosan and the chitosan-copper complex. The experimental isotherm data were analyzed using the Langmuir, Freundlich, and Redlich-Peterson equations. Correlation coefficients were determined for each isotherm analysis. Error functions have been used to determine the alternative single component parameters by nonlinear regression due to the inherent bias in using the correlation coefficient resulting from linearization. The error function method provided the best parameters for the isotherm equations in this system and is demonstrated for error comparison purposes.