Paracompactness of Metric Spaces and the Axiom of Multiple Choice

The axiom of multiple choice implies that metric spaces are paracompact but the reverse implication cannot be proved in set theory without the axiom of choice.     

A mild and efficient AgSbF6-catalyzed synthesis of fully substituted pyrroles through a sequential propargylation/amination/cycloisomerization reaction

Development of an efficient synthesis of fully substituted pyrroles via a sequential propargylation/amination/cycloisomerization was accomplished using AgSbF₆ as a catalyst. The one-pot three-component reaction of propargylic alcohols, 1,3-dicarbonyl compounds, and primary amines proceeds at a mild temperature, which prevents the formation of furan by-product. The reaction was also successfully applied to the more basic aliphatic amines with the addition of 1.1 equiv of acetic acid.     

Live lectures or online videos: students’ resource choices in a first-year university mathematics module

In Maths for Business, a mathematics module for non-mathematics specialists, students are given the choice of completing the module content via short online videos, live lectures or a combination of both. In this study, we identify students’ specific usage patterns with both of these resources and discuss their reasons for the preferences they exhibit. In 2015–2016, we collected quantitative data on each student's resource usage (attendance at live lectures and access of online videos) for the entire class of 522 students and employed model-based clustering which identified four distinct resource usage patterns with lectures and/or videos. We also collected qualitative data on students’ perceptions of resource usage through a survey administered at the end of the semester, to which 161 students responded. The 161 survey responses were linked to each cluster and analysed using thematic analysis. Perceived benefits of videos include flexibility of scheduling and pace, and avoidance of large, long lectures. In contrast, the main perceived advantages of lectures are the ability to engage in group tasks, to ask questions, and to learn ‘gradually’. Students in the two clusters with high lecture attendance achieved, on average, higher marks in the module.     

A continuous flow-batch hybrid reactor for commodity chemical synthesis enabled by inline NMR and temperature monitoring

Inline, real time NMR and temperature measurements have been used to optimise the continuous flow synthesis of difluoromethyltrimethylsilane (TMSCF₂H) by the reduction of the Ruppert-Prakash reagent (TMSCF₃). These measurements were used to maximise the space-time-yield, while ensuring this exothermic process remains safe. In this way, a three-fold increase in space-time-yield was achieved compared to the reported batch procedure, isolating 25?g of pure TMSCF₂H after 105?min.     

Thresholds and optimal binary comparison search trees

We present an O(n⁴)-time algorithm for the following problem: Given a set of items with known access frequencies, find the optimal binary search tree under the realistic assumption that each comparison can only result in a two-way decision: either an equality comparison or a less-than comparisons. This improves the best known result of O(n⁵) time, which is based on split tree algorithms. Our algorithm relies on establishing thresholds on the frequency of an item that can occur as an equality comparison at the root of an optimal tree.     

Solution to torsion relations in finsler-spacetime tangent bundle

In the Finsler-spacetime tangent bundle, a simple solution is determined to the torsion relations that were obtained previously to maintain (1) compatibility with Cartan's theory of Finsler space, (2) the almost complex structure, and (3) the vanishing of the covariant derivative of the almost complex structure.