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.     

Indexing Distributions of Data: Preservice Teachers' Notions of Representativeness

The purpose of the study was to identify strategies used by preservice elementary teachers to index distributions of data. A secondary purpose of the study was to investigate whether representational form influenced the type of representative values constructed. Two hundred eighty‐three preservice teachers were administered a selection of tasks requiring them to construct representative values for a variety of distributional shapes. Participants were asked to provide justifications for their chosen representative measures. Data were analyzed from two perspectives: nature and adequacy of representative measure. Results indicated that when presented with skewed data sets, over half of the participants constructed poorly representative values. Symmetric data sets were best indexed with 67% of participants constructing highly representative values. Representational form played a significant role in determining representative values. Data presented graphically were more likely to be represented using measures of variability than were those presented nongraphically. Conversely, measures of central tendency were more frequently used to represent nongraphical data. The information arising from this study indicates that many preservice teachers construct poorly representative values to index data sets and utilize identical measures of representativeness regardless of distributional shape. The findings of the study highlight the need for exploratory data activities at the preservice level, call for provision of experiences affording insights into construction of representative values, and thus, have implications for the design of preservice mathematics education curricula.     

Constructions of economic functions and calculations of marginal rates in DEA using parametric optimization methods

The aim of the paper is to present and substantiate a technique to visualize DEA modelling results without any loss of mathematical rigour. The proposed family of parametric optimization methods allows one to construct an intersection of the multidimensional frontier with a two-dimensional plane determined by any pair of given directions. This approach reduces the efficiency analysis of production units to the investigation of well-known functions in economics. We also propose constructive methods to calculate marginal rates of substitution, marginal rates of transformation and so on.