Flexible and adaptive use of strategies and representations in mathematics education

The flexible and adaptive use of strategies and representations is part of a cognitive variability, which enables individuals to solve problems quickly and accurately. The development of these abilities is not simply based on growing experience; instead, we can assume that their acquisition is based on complex cognitive processes. How these processes can be described and how these can be fostered through instructional environments are research questions, which are yet to be answered satisfactorily. This special issue on flexible and adaptive use of strategies and representations in mathematics education encompasses contributions of several authors working in this particular field. They present recent research on flexible and adaptive use of strategies or representations based on theoretical and empirical perspectives. Two commentary articles discuss the presented results against the background of existing theories.     

Efficient Data Augmentation for Fitting Stochastic Epidemic Models to Prevalence Data

Stochastic epidemic models describe the dynamics of an epidemic as a disease spreads through a population. Typically, only a fraction of cases are observed at a set of discrete times. The absence of complete information about the time evolution of an epidemic gives rise to a complicated latent variable problem in which the state space size of the epidemic grows large as the population size increases. This makes analytically integrating over the missing data infeasible for populations of even moderate size. We present a data augmentation Markov chain Monte Carlo (MCMC) framework for Bayesian estimation of stochastic epidemic model parameters, in which measurements are augmented with subject-level disease histories. In our MCMC algorithm, we propose each new subject-level path, conditional on the data, using a time-inhomogenous continuous-time Markov process with rates determined by the infection histories of other individuals. The method is general, and may be applied to a broad class of epidemic models with only minimal modifications to the model dynamics and/or emission distribution. We present our algorithm in the context of multiple stochastic epidemic models in which the data are binomially sampled prevalence counts, and apply our method to data from an outbreak of influenza in a British boarding school. Supplementary material for this article is available online.     

Aryl vinyl cyclopropane Cope rearrangements

While the divinyl cyclopropane Cope rearrangement is well-known, and has been broadly applied in synthesis, examples of the aryl vinyl cyclopropane Cope rearrangement are less common and generally limited in scope or reaction yield. The aryl vinyl cyclopropane Cope rearrangement gives access to the benzocycloheptene scaffold, which is present in a variety of naturally occurring and medicinally relevant products. Herein we report a method to obtain either of two regioisomeric benzocycloheptene products via an aryl vinyl cyclopropane Cope rearrangement, featuring additive-controlled regioselectivity. Mechanistic studies indicate a dynamic equilibration of cyclopropane stereoisomers, followed by rearrangement of the cis diastereomer.     

Metal‐Free Synthesis of Fluorinated Indoles Enabled by Oxidative Dearomatization

Nitrogen heterocycles are found in a majority of approved small‐molecule pharmaceuticals, and the number of approved fluorinated drugs is increasing each decade. Therefore, new approaches for accessing fluorinated nitrogen heterocycles are of great significance. A novel, scalable, and metal‐free method for accessing a wide range of fluorinated indoles is described. This oxidative‐dearomatization‐enabled approach assembles 2‐trifluoromethyl NH‐indole products from simple commercially available anilines with hexafluoroacetylacetone in the presence of an organic oxidant. The nature of the aniline N‐capping group is critical for the success of this new reaction. Furthermore, the indole products contain a 3‐trifluoroacetyl group, which can be exploited to access a plethora of useful functional groups.     

Theoretical study of the antioxidant properties of pyridoxine

Molecules acting as antioxidants capable of scavenging reactive oxygen species (ROS) are of utmost importance in the living cell. The antioxidative properties of pyridoxine (vitamin B6) have recently been discovered. In this study, we have analyzed the reactivity of pyridoxine toward the ROS (.-)OH, (.-)OOH, and (.-)O(2)- at the density functional theory level (functionals B3LYP and MPW1B95). Two reaction types have been studied as follows: addition to the aromatic ring atoms and hydrogen/proton abstraction. Our results show that (.-)OH is the most reactive species, while (.-)OOH displays low reactivity and (.-)O2(-) does not react at all with pyridoxine. The most exergonic reactions are those where (.-)H is removed from the CH(2)OH groups or the ring-bound OH group and range from -33 to -39 kcal/mol. The most exergonic addition reactions occur by attacking the carbon atoms bonded to nitrogen but with an energy gain of only 6 kcal/mol.     

Pyrolysis mass spectrometry for distinguishing potential hoax materials from bioterror agents

Pyrolysis mass spectrometry (PyMS) was investigated as a rapid tool to distinguish potential bioterror hoax materials from samples containing pathogenic bacteria. A pyrolysis time-of-flight (TOF) mass spectrometer equipped with an alternative ionization technique, metastable atom bombardment (MAB), was used to produce sample spectra. These spectra were analyzed by principal component and discriminant analysis for pattern recognition. Materials investigated were two strains of Vibrio parahaemolyticus, one of which produced the tdh toxin, two Salmonella enterica serotypes, a biological mosquito control product containing spores of Bacillus thuringiensis, and several white to off-white powders (which could be used as hoax materials), such as flour, corn starch, methyl cellulose, and xanthan gum. PyMS distinguished bacterial samples from hoax materials. Furthermore, pattern analysis differentiated Vibrios from Salmonellae, Salmonella enterica Anatum from S. enterica Heidelberg, and the two V. parahaemolyticus strains from each other. The B. thuringiensis mixture was distinguished from other bacteria and powders, suggesting that PyMS with pattern recognition may differentiate samples containing pathogens, including Bacillus spp., from nonbiological agents and that it can be a rapid method for detection of bacteria. MS data acquisition took only 7 min for each sample.     

Derivation of class II force fields. VIII. Derivation of a general quantum mechanical force field for organic compounds

A class II valence force field covering a broad range of organic molecules has been derived employing ab initio quantum mechanical "observables." The procedure includes selecting representative molecules and molecular structures, and systematically sampling their energy surfaces as described by energies and energy first and second derivatives with respect to molecular deformations. In this article the procedure for fitting the force field parameters to these energies and energy derivatives is briefly reviewed. The application of the methodology to the derivation of a class II quantum mechanical force field (QMFF) for 32 organic functional groups is then described. A training set of 400 molecules spanning the 32 functional groups was used to parameterize the force field. The molecular families comprising the functional groups and, within each family, the torsional angles used to sample different conformers, are described. The number of stationary points (equilibria and transition states) for these molecules is given for each functional group. This set contains 1324 stationary structures, with 718 minimum energy structures and 606 transition states. The quality of the fit to the quantum data is gauged based on the deviations between the ab initio and force field energies and energy derivatives. The accuracy with which the QMFF reproduces the ab initio molecular bond lengths, bond angles, torsional angles, vibrational frequencies, and conformational energies is then given for each functional group. Consistently good accuracy is found for these computed properties for the various types of molecules. This demonstrates that the methodology is broadly applicable for the derivation of force field parameters across widely differing types of molecular structures. Copyright 2001 John Wiley & Sons, Inc. J Comput Chem 22: 1782-1800, 2001     

The stereochemistry of guggulsterol-1

The steroechemistry of guggulsterol-1, a component of the exudate of $\text{Commiphora}$$\text{mukul}$, is shown to be 20(R), 22(R)($\text{3}$).