Kinetic Models for Chemotaxis and their Drift-Diffusion Limits

Kinetic models for chemotaxis, nonlinearly coupled to a Poisson equation for the chemo-attractant density, are considered. Under suitable assumptions on the turning kernel (including models introduced by Othmer, Dunbar and Alt), convergence in the macroscopic limit to a drift-diffusion model is proven. The drift-diffusion models derived in this way include the classical Keller-Segel model. Furthermore, sufficient conditions for kinetic models are given such that finite-time-blow-up does not occur. Examples are given satisfying these conditions, whereas the macroscopic limit problem is known to exhibit finite-time-blow-up. The main analytical tools are entropy techniques for the macroscopic limit as well as results from potential theory for the control of the chemo-attractant density.Present address: Centro de Matemática e Aplicações Fundamentais, Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003, Lisboa, Portugal     

Nonlinear Free Boundary Problems with Singular Source Terms

We prove the existence of solutions to nonlinear free boundary problem with singularities at given points.     

On the Long-Time Behavior of the Quantum Fokker-Planck Equation

We analyze the long-time behavior of transport equations for a class of dissipative quantum systems with Fokker-planck type diffusion operator, subject to confining potentials of harmonic oscillator type. We establish the existence and uniqueness of a non-equilibrium steady state for the corresponding dynamics. Further, using a (classical) convex Sobolev inequality, we prove an optimal exponential rate of decay towards this state and additionally give precise dispersion estimates in those cases, where no stationary state exists.     

Carousels,Zindler curves and the floating body problem

A carousel is a dynamical system that describes the movement of an equilateral linkage in which the midpoint of each rod travels parallel to it. They are closely related to the floating body problem. We prove, using the work of Auerbach, that any figure that floats in equilibrium in every position is drawn by a carousel. Of special interest are such figures with rational perimetral density of the floating chords, which are then drawn by carousels. In particular, we prove that for some perimetral densities the only such figure is the circle, as the problem suggests.     

Beliefs and Perspectives of First‐Year,Alternative Preparation,Elementary Teachers in Urban Classrooms

Results from an earlier study ( Hart, 2002 ) suggested that a group of 14 teachers participating in an alternative preparation program for elementary teachers had developed beliefs that were consistent with current thinking in mathematics education. The current study follows 8 of those teachers into their first year of teaching in an urban classroom. Qualitative data were collected from three sources: reflection logs, mathematics case discussions, and field notes made during classroom observations. This provided a triangulation of perspectives: the teachers' views of themselves, the teachers' views of others, and the university faculty member's view of them. Also, teachers completed the Standards Belief Instrument (Zollman & Mason, 1996) at the end of Phase I and at the end of Phase II. Results from the instrument and the qualitative analysis suggest that the teachers maintained a strong reform perspective in their beliefs, but they were unable to consistently implement pedagogy that was consistent with those beliefs.     

Using Form and Function Analogy Object Boxes to Teach Human Body Systems

This study compares the use of form and function analogy object boxes to more traditional lecture and worksheet instruction during a 10th‐grade unit on human body systems. The study was conducted with two classes (N= 32) of mixed ability students at a high‐needs rural high school in central New York State. The study used a pretest/posttest design, in which the two classes alternated between conditions for the four systems (skeletal, digestive, immune, nervous). Both conditions involved students in quality instruction addressing the same concepts for the same amount of time. Additionally, all students participated in hands‐on labs. The experimental condition presented students with a set of objects analogous in form and function to parts of a human body system. Students matched objects with cards describing body system parts, mapped the analogies on a chart, generated alternative objects that could be used for the analogy, and finally, created new analogies for other body system parts. Students made significantly higher posttest and gain scores on material learned in the experimental condition, with a mean gain score average of 12.4 points out of 25, compared to 6.2 points in the control condition. Cohen's Effect Size was large, 1.36.     

Calculating scale elasticity in DEA models

In the data envelopment analysis (DEA) efficiency literature, qualitative characterizations of returns to scale (increasing, constant, or decreasing) are most common. In economics it is standard to use the scale elasticity as a quantification of scale properties for a production function representing efficient operations. Our contributions are to review DEA practices, apply the concept of scale elasticity from economic multi-output production theory to DEA piecewise linear frontier production functions, and develop formulas for scale elasticity for radial projections of inefficient observations in the relative interior of fully dimensional facets. The formulas are applied to both constructed and real data and show the differences between scale elasticities for the two valid projections (input and output orientations). Instead of getting qualitative measures of returns to scale only as was done earlier in the DEA literature, we now get a quantitative range of scale elasticity values providing more information to policy-makers.     

Polynomial Quantization on Rank One Para-Hermitian Symmetric Spaces

We construct the polynomial quantization on the space G/H where G=SL(n,R),H=GL(n–1,R). It is a variant of quantization in the spirit of Berezin. In our case covariant and contravariant symbols are polynomials on G/H. We introduce a multiplication of covariant symbols, establish the correspondence principle, study transformations of symbols (the Berezin transform) and of operators. We write a full asymptotic decomposition of the Berezin transform.     

Explicit 2-Factorisations of the Odd Graph

In this note we show how 1-factors in the middle two layers of the discrete cube can be used to construct 2-factors in the Odd graph (the Kneser graph of (k − 1)-sets from a (2k − 1)-set). In particular, we use the lexical matchings of Kierstead and Trotter, and the modular matchings of Duffus, Kierstead and Snevily, to give explicit constructions of two different 2-factorisations of the Odd graph. This revised version was published online in September 2006 with corrections to the Cover Date.