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     

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.     

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.     

An approach to deciding the observational equivalence of Algol-like languages

We prove that the observational equivalence of third-order finitary (i.e. recursion-free) Idealized Algol (IA) is decidable using Game Semantics. By modelling the state explicitly in our games, we show that the denotation of a term M of this fragment of IA is a compactly innocent strategy-with-state, i.e. the strategy is generated by a finite view function f_M. Given any such f_M, we construct a real-time deterministic pushdown automaton (DPDA) that recognizes the complete plays of the knowing-strategy denotation of M. Since such plays characterize observational equivalence, and there is an algorithm for deciding whether any two DPDAs recognize the same language, we obtain a procedure for deciding the observational equivalence of third-order finitary IA. Restricted to second-order terms, the DPDA representation cuts down to a deterministic finite automaton; thus our approach gives a new proof of Ghica and McCusker’s regular-expression characterization for this fragment. Our algorithmic representation of program meanings, which is compositional, provides a foundation for model-checking a wide range of behavioural properties of IA and other cognate programming languages. Another result concerns second-order IA with full recursion: we show that observational equivalence for this fragment is undecidable.     

On a class of preconditioners for solving the Helmholtz equation

In 1983, a preconditioner was proposed [J. Comput. Phys. 49 (1983) 443] based on the Laplace operator for solving the discrete Helmholtz equation efficiently with CGNR. The preconditioner is especially effective for low wavenumber cases where the linear system is slightly indefinite. Laird [Preconditioned iterative solution of the 2D Helmholtz equation, First Year's Report, St. Hugh's College, Oxford, 2001] proposed a preconditioner where an extra term is added to the Laplace operator. This term is similar to the zeroth order term in the Helmholtz equation but with reversed sign. In this paper, both approaches are further generalized to a new class of preconditioners, the so-called “shifted Laplace” preconditioners of the form Δφ−k²φ with . Numerical experiments for various wavenumbers indicate the effectiveness of the preconditioner. The preconditioner is evaluated in combination with GMRES, Bi-CGSTAB, and CGNR.     

Some non‐standard problems in viscous flow

Energy bounds are derived for Dirichlet type boundary value problems for the Navier–Stokes and Stokes equations when a combination of the solution values initially and at a later time is prescribed. The bounds are obtained by means of a differential inequality and imply uniqueness and continuous data dependence of the solutions for a range of values of the parameter in the non‐standard auxiliary condition. Copyright © 2004 John Wiley & Sons, Ltd.     

Electron transfer induced dissociations of 2- and 4-alkyl cyclohexadienones

Several 2- and 4-alkylcyclohexadienones were prepared and shown to accept electrons to produce ketyl radical anions that dissociated rapidly at room temperature to release carbon-centered radicals and an aromatic phenoxide type anion. In the PET process with benzyl-substituted cyclohexadienones, initiated with triethylamine, the benzyl radicals dimerised or abstracted an H-atom from solvent. In electrochemical reductions, and in reductions with alkali metals in liquid ammonia, the benzyl radicals were further reduced to anions.