Advancing Mathematical Activity: A Practice-Oriented View of Advanced Mathematical Thinking

The purpose of this article is to contribute to the dialogue about the notion of advanced mathematical thinking by offering an alternative characterization for this idea, namely advancing mathematical activity. We use the term advancing (versus advanced) because we emphasize the progression and evolution of students' reasoning in relation to their previous activity. We also use the term activity, rather than thinking. This shift in language reflects our characterization of progression in mathematical thinking as acts of participation in a variety of different socially or culturally situated mathematical practices. For these practices, we emphasize the changing nature of students' mathematical activity and frame the process of progression in terms of multiple layers of horizontal and vertical mathematizing.     

A Runge-Kutta method for index 1 stochastic differential-algebraic equations with scalar noise

The paper deals with the numerical treatment of stochastic differential-algebraic equations of index one with a scalar driving Wiener process. Therefore, a particularly customized stochastic Runge-Kutta method is introduced. Order conditions for convergence with order 1.0 in the mean-square sense are calculated and coefficients for some schemes are presented. The proposed schemes are stiffly accurate and applicable to nonlinear stochastic differential-algebraic equations. As an advantage they do not require the calculation of any pseudo-inverses or projectors. Further, the mean-square stability of the proposed schemes is analyzed and simulation results are presented bringing out their good performance.     

Biological evaluation of Trans-African highways

The Trans-African highway network is a unique concept of integrated development of transport corridors spanning all African countries and providing landlocked countries access to seaports. The planned road system is still maturing and just partially complete, thus giving us a chance to play with different scenarios of its growth and to consider potential alternative transport networks. We study the evolving transport network in the African continent with a groundbreaking technique of imitating growing transport networks with slime mould Physarum polycephalum. We represent the major urban areas of Africa with a source of nutrients, inoculate a piece of the slime mould in Cairo and allow the mould to span all urban areas with its network of protoplasmic tubes. We then compare the slime mould networks with existing and planned highway corridors. We found that slime mould provides a good approximation of the Trans-African highway network, with some roads of Eastern Africa delineated by P. polycephalum in a larger number of laboratory experiments. We demonstrate direct matches between protoplasmic tubes and Trans-Sahelian as well as Lagos-Mombasa corridors. Finally we analyse the bio-logic of transport network development in families of generalised Physarum graphs.     

On the survival of peptide cations after electron capture: Role of internal hydrogen bonding and microsolvation

Electron capture by both bare and microsolvated small peptide dications was investigated by colliding these ions with sodium vapor in an accelerator mass spectrometer to provide insight into processes that occur on the microsecond time frame. Survival of the intact peptide monocation after electron capture depends strongly on molecular size. For dipeptides, no intact reduced species were observed; the predominant ions correspond to loss of hydrogen and ammonia. In contrast, the intact reduced species was observed for larger peptides. Calculated structures indicate that the diprotonated dipeptide ions form largely extended structures with low probability of internal ionic hydrogen bonding (i.e., charge solvation) whereas internal ionic H-bonding occurs extensively for larger peptide dications. Solvation of the peptide ions with between one to seven methanol molecules reduces the total extent of H loss even for dipeptides where intact reduced species can survive more than a microsecond after electron capture. The yield of ions corresponding to cleavage of NCalpha bonds (c+ and z+* ions) does not depend strongly on peptide size but decreases with the extent of microsolvation for the dipeptide dications. H-bonding appears to play an important role for the survival of the intact reduced ions but less so for the formation of c+ and z+* ions. Our results indicate that electron capture predominantly occurs at the ammonium groups (at least 70 to 80%), and provides important new insights into the electron capture dissociation process.     

The tandem ring-closing metathesis-isomerization approach to 6-deoxyglycals

Protected 3,6-dideoxyglycals have been synthesized de novo as single isomers starting from ethyl lactate by using the tandem RCM-isomerization reaction as the key step. Different relative configurations become accessible by addition of vinyl- or allyl-metal compounds to protected lactaldehydes under Cram-chelate or Felkin-Anh control. The concept is exemplified for glycals of L-rhodinose and L-amicetose, as well as for ring-expanded non-natural analogues thereof. This novel approach to glycals is also applicable to the synthesis of disaccharide glycals via a reiterative strategy, as exemplified for the dimer of L-rhodinose and its non-natural ring expanded analogue.     

Square versus tetrahedral iron clusters with polyoxometalate ligands

Two new insoluble transition metal substituted phosphotungstates, (C2N2H10)11[{(B-alpha-PW9O34)Fe3(OH)3}4(PO4)4Fe].38H2O(1) and K4(C2N2H10)12[(alpha-PW10Fe2O39)4].30H2O(2), have been isolated by the hydrothermal reaction of [A-alpha-PW9O34]9-, Fe(III) ions and ethylenediamine. Compound 1 has a tetrahedral symmetry and contains a Fe13 core built from the assembly of four Fe(III) trisubstituted [B-alpha-PW9O34]9- anions around a central disordered iron ion via four phosphato ligands. The anion in 2 can be described as a square of disubstituted [PW10O37]9- anions linked by Fe(III)-O-Fe(III) bridges. Magnetic measurements performed on 1 and 2 have shown the occurrence of antiferromagnetic interactions between the iron ions and have allowed the coupling constants between the magnetic centers to be determined.     

Electronic and geometric properties of ETS-10: QM/MM studies of cluster models

Hybrid DFT/MM methods have been used to investigate the electronic and geometric properties of the microporous titanosilicate ETS-10. A comparison of finite length and periodic models demonstrates that band gap energies for ETS-10 can be well represented with relatively small cluster models. Optimization of finite clusters leads to different local geometries for bulk and end sites, where the local bulk TiO6 geometry is in good agreement with recent experimental results. Geometry optimizations reveal that any asymmetry within the axial O-Ti-O chain is negligible. The band gap in the optimized model corresponds to a O(2p) --> Tibulk(3d) transition. The results suggest that the three Ti atom, single chain, symmetric, finite cluster is an effective model for the geometric and electronic properties of bulk and end TiO6 groups in ETS-10.     

Solvent-free synthesis of metal complexes

Avoiding the use of solvents in synthesis can reduce environmental contamination and even be more convenient than using solvent-based synthesis. In this tutorial review we focus on recent research into the use of mechanochemistry (grinding) to synthesise metal complexes in the absence of solvent. We include synthesis of mononuclear complexes, coordination clusters, spacious coordination cages, and 1-, 2- and 3-dimensional coordination polymers (metal organic frameworks) which can even exhibit microporosity. Remarkably, in many cases, mechanochemical synthesis is actually faster and more convenient than the original solvent-based methods. Examples of solvent-free methods other than grinding are also briefly discussed, and the positive outlook for this growing topic is emphasised.