Students' Retention of Mathematical Knowledge and Skills in Differential Equations

This study investigates students' retention of mathematical knowledge and skills in two differential equations classes. Posttests and delayed posttests after 1 year were administered to students in inquiry‐oriented and traditional classes. The results show that students in the inquiry‐oriented class retained conceptual knowledge, as seen by their performance on modeling problems, and retained equal proficiency in procedural problems, when compared with students in the traditionally taught classes. The results of this study add additional support to the claim that teaching for conceptual understanding can lead to longer retention of mathematical knowledge.     

The Morphology of Liquid-Crystalline Polymers and the Possible Consequences for their Rheological Behaviour

Polymer liquid crystals can occur as polydomain materials where the domain size may be tens of microns. While the material within each domain may be characterized by a common order parameter, the directors of the domains can be more or less randomly distributed. Since the transition from polydomain to monodomain material only involves the removal of grain boundaries and the alignment of directors, the free energy change must necessarily be small. Such a transition can readily be achieved, therefore, by the action of any external field: electrical, magnetic, stress or surface. In this work optical photomicrographs of polymeric liquid crystals with widely varying and in some cases well controlled morphologies are presented. Probable dependence of rheological behaviour on morphology is also discussed. Such dependence is expected to be considerable under certain conditions. Due to experimental and sample limitations, however, direct correlations of rheology and morphology are sparse. Morphological consequences for the rheology of liquid-crystalline materials can be exemplified by the following possibilities. In contrast to the case of isotropic melts, wall effects can be non-negligible. Zero shear rate rheological parameters are not expected to be uniquely defined quantities since the domain sizes are large and the director may not be effectively averaged over typical sample dimensions. Non-zero shear-rate measurements of rheological parameters is effected by the propensity of: (1) individual domain directors to align under the influence of a stress field and (2) flow alignment to dominate surface-induced alignment above some critical shear rate. The effects might be manifested by a non-newtonian regime as well as yield stress behaviour and thixotropy. The kinetics of relaxation from mono- to poly-domain material has implications for the dynamic response and rheological hysterises of the material.     

On the calculation of the structure of tris-(tert-butyl)-methane

A simple potential energy function developed on unstrained molecules and containing only 16 parameters can account for the unusually long C-C bonds and high C-H stretchings in highly strained tris-(tert-butyl)methane. On average, the structural deviations from experimental data are smaller than for some recently reported potential energy functions with 20–29 parameters. The success is thought to be due to fortunate handling of nonbonded interactions which include Coulomb terms.     

Free field realizations of affine current superalgebras, screening currents and primary fields

In this paper free field realizations of affine current superalgebras are considered. Based on quantizing differential operator realizations of the corresponding basic Lie superalgebras, general and simple expressions for both the bosonic and the fermionic currents are provided. Screening currents of the first kind are also presented. Finally, explicit free field realizations of primary fields with general, possibly non-integer, weights are worked out. A formalism is used where the (generally infinite) multiplet is replaced by a generating function primary operator. The results allow setting up integral representations for correlators of primary fields corresponding to integrable representations. The results are generalizations to superalgebras of a recent work on free field realizations of affine current algebras by Petersen, Yu and the present author.     

On single and bimodal optimum buckling loads of clamped columns

We study the problem of determining the optimum shape of a thin, elastic, clamped column of given length and volume, such that the fundamental buckling load is a maximum. The column cross-sections are assumed to be geometrically similar, and a minimum allowable value is specified for the cross-sectional area.Investigating the optimization problem parametrically in terms of this minimum constraint, we reveal a significant feature. There exists a threshold value for the constraint, beyond which the optimum columns are all associated with single mode optimum buckling loads, whereas, for any value of the constraint less than the threshold value, the optimum columns are associated with bimodal fundamental buckling loads.This bimodal behaviour necessitates an extension and a mathematical reformulation of the current optimization problem, which is outlined and solved in the paper. In particular, we revise the result hitherto considered to be the optimum solution for an unconstrained column with clamped ends.     

On the influence of shell structure and matrix element fluctuations on the pairing correlation in nuclei

The uniform model for the nuclear pairing-force problem is extended to take into account the effect of fluctuations in nucleon orbital level densityρ and in pairing matrix elementsG _vv′. Simple formulas for the average dependence of level density on energy are presented, and alternative ways of estimating effects of shell bunching on the pairing correlation are derived. It is argued also that the pairing constantG might be systematically overestimated in the usual BCS numerical calculations; a semiconstant pairing force with diagonal elements larger than off-diagonal is thus suggested.