Reviving Pólya's “Look Back” in a Singapore school

This study is based on the stance that Pólya's “Look Back,” though understudied, remains relevant to Mathematics curricula that place emphasis on problem solving. Although the Singapore Mathematics curriculum adopts the goal of teaching Look Back, research about how it is carried out in actual classroom practice is rare. In our project, we focus on a redesign of a teacher development programme that is targeted to help teachers realize Pólya's original vision of Look Back in the classroom. We report the cases of two teachers who have participated in the teacher development programme: their thinking about Look Back (through interview records) and their interpretation of Look Back in their classroom practice (through video records). By bringing these two spheres of data together, we discuss their degree of ‘buy-in’ to Look Back in the overall problem solving enterprise.     

The Effect of the Buffering Capacity of the Supporting Electrolyte on the Electrochemical Oxidation of Dopamine and 4‐Methylcatechol in Aqueous and Nonaqueous Solvents

Dopamine was electrochemically oxidized in aqueous solutions and in the organic solvents N,N‐dimethyl‐formamide and dimethylsulfoxide containing varying amounts of supporting electrolyte and water, to form dopamine ortho‐quinone. It was found that the electrochemical oxidation mechanism in water and in organic solvents was strongly influenced by the buffering properties of the supporting electrolyte. In aqueous solutions close to pH 7, where buffers were not used, the protons released during the oxidation process were able to sufficiently change the localized pH at the electrode surface to reduce the deprotonation rate of dopamine ortho‐quinone, thereby slowing the conversion into leucoaminochrome. In N,N‐dimethylformamide and dimethylsulfoxide solutions, in the absence of buffers, dopamine was oxidized to dopamine ortho‐quinone that survived without further reaction for several minutes at 25 °C. The voltammetric data obtained in the organic solvents were made more complicated by the presence of HCl in commercial sources of dopamine, which also underwent an oxidation process.     

Pseudozeros of polynomials and pseudospectra of companion matrices

Summary. It is well known that the zeros of a polynomial are equal to the eigenvalues of the associated companion matrix . In this paper we take a geometric view of the conditioning of these two problems and of the stability of algorithms for polynomial zerofinding. The is the set of zeros of all polynomials obtained by coefficientwise perturbations of of size ; this is a subset of the complex plane considered earlier by Mosier, and is bounded by a certain generalized lemniscate. The is another subset of defined as the set of eigenvalues of matrices with ; it is bounded by a level curve of the resolvent of $A$. We find that if $A$ is first balanced in the usual EISPACK sense, then and are usually quite close to one another. It follows that the Matlab ROOTS algorithm of balancing the companion matrix, then computing its eigenvalues, is a stable algorithm for polynomial zerofinding. Experimental comparisons with the Jenkins-Traub (IMSL) and Madsen-Reid (Harwell) Fortran codes confirm that these three algorithms have roughly similar stability properties. Received June 15, 1993     

Behavioral measures and their correlation with IPM iteration counts on semi-definite programming problems

We study four measures of problem instance behavior that might account for the observed differences in interior-point method (IPM) iterations when these methods are used to solve semidefinite programming (SDP) problem instances: (i) an aggregate geometry measure related to the primal and dual feasible regions (aspect ratios) and norms of the optimal solutions, (ii) the (Renegar-) condition measure C(d) of the data instance, (iii) a measure of the near-absence of strict complementarity of the optimal solution, and (iv) the level of degeneracy of the optimal solution. We compute these measures for the SDPLIB suite problem instances and measure the sample correlation (CORR) between these measures and IPM iteration counts (solved using the software SDPT3) when these measures have finite values. Our conclusions are roughly as follows: the aggregate geometry measure is highly correlated with IPM iterations (CORR = 0.901), and provides a very good explanation of IPM iterations, particularly for problem instances with solutions of small norm and aspect ratio. The condition measure C(d) is also correlated with IPM iterations, but less so than the aggregate geometry measure (CORR = 0.630). The near-absence of strict complementarity is weakly correlated with IPM iterations (CORR = 0.423). The level of degeneracy of the optimal solution is essentially uncorrelated with IPM iterations. This research has been partially supported through the MIT-Singapore Alliance.     

Iridium‐ and Osmium‐decorated Reduced Graphenes as Promising Catalysts for Hydrogen Evolution

Renewable energy sources are highly sought after as a result of numerous worldwide problems concerning the environment and the shortage of energy. Currently, the focus in the field is on the development of catalysts that are able to provide water splitting catalysis and energy storage for the hydrogen evolution reaction (HER). While platinum is an excellent material for HER catalysis, it is costly and rare. In this work, we investigated the electrocatalytic abilities of various graphene–metal hybrids to replace platinum for the HER. The graphene materials were doped with 4f metals, namely, iridium, osmium, platinum and rhenium, as well as 3d metals, namely, cobalt, iron and manganese. We discovered that a few hybrids, in particular iridium‐ and osmium‐doped graphenes, have the potential to become competent electrocatalysts owing to their low costs and—more importantly—to their promising electrochemical performances towards the HER. One of the more noteworthy observations of this work is the superiority of these two hybrids over MoS₂, a well‐known electrocatalyst for the HER.     

A molecular mechanics approach for analyzing tensile nonlinear deformation behavior of single-walled carbon nanotubes

In this paper, by capturing the atomic information and reflecting the behaviour governed by the nonlinear potential function, an analytical molecular mechanics approach is proposed. A constitutive relation for single-walled carbon nanotubes (SWCNT’s) is established to describe the nonlinear stress-strain curve of SWCNT’s and to predict both the elastic properties and breaking strain of SWCNT’s during tensile deformation. An analysis based on the virtual internal bond (VIB) model proposed by P. Zhang et al. is also presented for comparison. The results indicate that the proposed molecular mechanics approach is indeed an acceptable analytical method for analyzing the mechanical behavior of SWCNT’s. The project supported by the National Natural Science Foundation of China (10121202, 90305015 and 10328203), the Key Grant Project of Chinese Ministry of Education (0306) and the Research Grants Council of the Hong Kong Special Administrative Region, China (HKU 7195/04E).