A quaternionic proof of the representation formula of a quaternary quadratic form

The celebrated Four Squares Theorem of Lagrange states that every positive integer is the sum of four squares of integers. Interest in this Theorem has motivated a number of different demonstrations. While some of these demonstrations prove the existence of representations of an integer as a sum of four squares, others also produce the number of such representations. In one of these demonstrations, Hurwitz was able to use a quaternion order to obtain the formula for the number of representations. Recently the author has been able to use certain quaternion orders to demonstrate the universality of other quaternary quadratic forms besides the sum of four squares. In this paper, we develop results analogous to Hurwitz's above mentioned work by delving into the number theory of one of these quaternion orders, and discover an alternate proof of the representation formula for the corresponding quadratic form.     

Four (algorithms) in one (bag): an integrative framework of knowledge for teaching the standard algorithms of the basic arithmetic operations

In this article we present an integrative framework of knowledge for teaching the standard algorithms of the four basic arithmetic operations. The framework is based on a mathematical analysis of the algorithms, a connectionist perspective on teaching mathematics and an analogy with previous frameworks of knowledge for teaching arithmetic operations with rational numbers. In order to evaluate the potential applicability of the framework to task design, it was used for the design of mathematical learning tasks for teachers. The article includes examples of the tasks, their theoretical analysis, and empirical evidence of the sensitivity of the tasks to variations in teachers’ knowledge of the subject. This evidence is based on a study of 46 primary school teachers. The article concludes with remarks on the applicability of the framework to research and practice, highlighting its potential to encourage teaching the four algorithms with an emphasis on conceptual understanding.     

Outer-sphere oxidation of the superoxide radical anion

Efforts to use the Marcus model to describe oxidations of the superoxide radical anion (O(2-)) by transition-metal complexes have failed dramatically, with discrepancies between theory and experiment spanning 13 orders of magnitude. As a result, the prevailing view is that these reactions involve some complex interactions that are not yet understood. We now show that once the familiar form of the Marcus cross relation (MCR) is modified to allow for the relatively small size of O(2-), excellent agreement is obtained between theory and experiment. This simple finding dispels the decades of uncertainty surrounding these reactions and provides a reliable method for determining whether oxidations of (O2)- occur via inner- or outer-sphere pathways. More generally, the modified MCR is applicable to any homogeneous electron-transfer process characterized by significant differences in size between electron donors and acceptors.     

Hydrophilic interaction chromatography of seized drugs and related compounds with sub 2 μm particle columns

The use of hydrophilic interaction chromatography (HILIC) with sub 2 μm particle columns for the analysis of drugs and related compounds of forensic interest is described. This technique uses a high organic/low aqueous buffered mobile phase with a polar stationary phase, and is excellent for the separation of many of the charged solutes that are found in forensic drug exhibits. In this study, HILIC is investigated for 11 solutes of forensic interest, including weak bases, weak acids, and a neutral solute. In addition, for columns containing either ethylene bridged hybrid particles with or without an amide bonded phase, the effects of acetonitrile concentration, buffer type, buffer concentration, linear velocity, and sample concentration were studied. Based on these studies, HILIC with sub 2 μm particle columns can offer highly efficient, selective, and rapid isocratic separations of drugs and related compounds of forensic interest, with excellent peak shapes and low back pressures. This is in contrast to reverse phase chromatography (RPLC), where gradient elution is usually required, which can result in extensive overlap between acidic, neutral, and basic solutes. In addition, since HILIC exhibits a much greater loading capacity than RPLC, it could be a preferred technique for drug profiling. Furthermore, because high organic content mobile phases are highly amenable to mass spectrometric detection, the use of HILIC with tandem mass spectrometric detection for the analysis of seized drugs is described.     

Role of the alkali-metal cation size in the self-assembly of polyoxometalate-monolayer shells on gold nanoparticles

Polyoxometalate (POM)-monolayer stability constants, K, for three POM anions vary with the cation size, in the same order as that for increasing ion-pair formation with α-SiW(11)O(39)(8-) (1) in the early nucleation phase of monolayer self-assembly: Li(+) < Na(+) < K(+) < Cs(+). Cryo-TEM images demonstrating the use of the cation size to rationally control monolayer formation provide definitive evidence that the POM monolayers are electrostatically stabilized (ionic) shells, analogous in that respect to the monolayer walls of "hollow" POM-macroanion vesicles.     

Experienced engineers becoming mathematics teachers: preliminary perceptions of mathematics teaching

Engineers who choose to change careers and become mathematics teachers are a specific group as far as their mathematics learning in the context of engineering and their previous work experience are concerned. Regarding mathematics, they mainly engaged in applied mathematics associated with engineering, which is a highly practical field. This research explores experienced engineers’ perceptions of mathematics teaching-related topics, before starting their studies in a pre-service mathematics teacher preparation programme. This research explores their perceptions of mathematics as a discipline, mathematics teaching and mathematical understanding. The qualitative research involves three mechanical engineers, two industrial management engineers, and an electrical engineer. Semi-structured interviews were conducted before the beginning of the programme, and analysed qualitatively. The participants view engineering as an applied and changing discipline while perceiving mathematics as closed, rigorous, accurate, systematic, theoretical and as a tool for engineering. They mostly address general features of mathematics teaching while expressing a more multifaceted view of mathematical understanding. Due to the specific characteristics of the participants, this study may contribute to planning mathematics teacher preparation programmes for engineers.     

The Egmont Master phenomenon: X-ray fluorescence spectrometric and paper studies for art history research

The aim of the research project “Typology of Dutch Drawing” was to establish an interdisciplinary approach for investigating heterogeneous drawing collections. To define a type common to a group of drawings, we determine elements that are common to them based on style and the use of identical materials. To that end, we investigated about 750 Dutch drawings from the sixteenth century at the Dresden Kupferstich-Kabinett using art historical and scientific methods. In this work, we present a detailed analysis of 30 drawings ascribed to the Egmont Master.     

On convergence of Lavrentiev regularization for semilinear parabolic optimal control problems with pointwise control and state constraints

We consider Lavrentiev regularization for a class of semilinear parabolic optimal control problems with control constraints and pointwise state constraints and review convergence results for local solutions under Slater type assumptions as well as quadratic growth conditions. Moreover, we state a local uniqueness result for local optima under the assumptions of strict separability of the active sets as well as a second order sufficient condition for the regularized solution. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)