Partially finite convex programming,Part I: Quasi relative interiors and duality theory

We study convex programs that involve the minimization of a convex function over a convex subset of a topological vector space, subject to a finite number of linear inequalities. We develop the notion of the quasi relative interior of a convex set, an extension of the relative interior in finite dimensions. We use this idea in a constraint qualification for a fundamental Fenchel duality result, and then deduce duality results for these problems despite the almost invariable failure of the standard Slater condition. Part II of this work studies applications to more concrete models, whose dual problems are often finite-dimensional and computationally tractable.     

Optimally Dense Packings of Hyperbolic Space

In previous work, a probabilistic approach to controlling difficulties of density in hyperbolic space led to a workable notion of optimal density for packings of bodies. In this paper we extend an ergodic theorem of Nevo to provide an appropriate definition of those packings to be considered optimally dense. Examples are given to illustrate various aspects of the density problem, in particular the shift in emphasis from the analysis of individual packings to spaces of packings.     

Efficient N- and C-functionalisation of cyclam macrocycles utilising bisaminal methodology

An efficient synthesis of C-functionalised cyclam macrocycles that employs bisaminal intermediates and allows N-substitution to be controlled is reported.     

Chaotic resonance: A simulation

Stochastic resonance is a statistical phenomenon that has been observed in periodically modulated, noise-driven, bistable systems. The characteristic signatures of the effect include an increase in the signal-to-noise of the output as noise is added to the system, and exponentially decreasing peaks in the probability density as a function of residence times in one state. Presented are the results of a numerical simulation where these same signatures were observed by adding achaotic driving term instead of a white noise term. Although the probability distributions of the noise and chaos inputs were significantly different, the stochastic and chaotic resonances were equal within the experimental error.     

Geometric Knowledge of Middle School Students in a Reform-based Mathematics Curriculum

National and international studies have found U.S. elementary students to be weak in their understandings and applications of geometric concepts. The University of Chicago School Mathematics Project's (UCMSP) Everyday Mathematics Program is one of the current reform-based elementary curricula incorporating geometry throughout the K-6 curriculum, with an emphasis on hands-on and problem-solving activities. In this study, the geometric knowledge of fifth and sixth graders using the UCSMP curriculum is compared to the knowledge of students using more traditional curricula. Because UCSMP students had been in the program since kindergarten, this research attempts to measure the longitudinal effects of such an approach. Along with an overall score, a subset of test items was used to assign each student a van Hiele level for geometric thinking, as well as a reasoning score. On all measures, UCSMP students substantially outperformed their counterparts, and nearly all differences were significant. Aspects of the UCSMP curriculum and the van Hiele model for learning geometry are discussed relative to these results.