Symmetry-breaking substitutions of [Re(CN)8]3- into the centered, face-capped octahedral clusters (CH3OH)24M9M'6(CN)48 (M = Mn, Co; M = Mo, W)

The diamagnetic complex [Re(CN)8]3- is shown to react with Mn2+ ions in methanol to generate the centered, face-capped octahedral cluster (CH3OH)24Mn9Re6(CN)48, which is structurally analogous to (CH3OH)24Mn9Mo6(CN)48. Related reactions involving stoichiometric mixtures of octacyanometalate complexes generate the substituted species (CH3OH)24Mn9Mo5Re(CN)48, (CH3OH)24Co9Mo5Re(CN)48, (CH3OH)24Mn9Mo3Re3(CN)48, (CH3OH)24Mn9W5Re(CN)48 and (CH3OH)24Co9W5Re(CN)48, in which the O(h) symmetry of the cluster core is broken. Reassessment of the magnetic properties of the Mn9Mo6(CN)48 cluster confirm that it possesses a ground state spin of S = 39/2, but does not exhibit single-molecule-magnet behavior. Lowering the symmetry of the molecule by substitutions of ReV at one or three of the MoV sites does not lead to an overall increase in the magnetic anisotropy, as probed by ac magnetic susceptibility measurements. A similar result occurs for the other substituted species, with the important exception of the new single-molecule magnet (CH3OH)24Co9W5Re(CN)48, for which the spin reversal barrier is significantly reduced relative to that observed previously in (CH3OH)24Co9W6(CN)48.     

Synthesis, structures and reactions of cyclometallated gold complexes containing (2-diphenylarsino-n-methyl)phenyl (n = 5, 6)

Reaction of (C6H3-2-AsPh2-n-Me)Li (n = 5 or 6) with [AuBr(AsPh3)] at -78 degrees C gives the corresponding cyclometallated gold(I) complexes [Au2[(mu-C6H3-n-Me)AsPh2]2] [n = 5, (1); n = 6, (9)]. 1 undergoes oxidative addition with halogens and with dibenzoyl peroxide to give digold(II) complexes [Au2X2[(mu-C6H3-5-Me)AsPh2]2] [X = Cl (2a), Br (2b), I (2c) and O2CPh (3)] containing a metal-metal bond between the 5d9 metal centres. Reaction of 2a with AgO2CMe or of 3 with C6F5Li gives the corresponding digold(II) complexes in which X = O2CMe (4) and C6F5 (6), respectively. The Au-Au distances increase in the order 4 < 2a < 2b < 2c < 6, following the covalent binding tendency of the axial ligand. Like the analogous phosphine complexes, 2a-2c and 6 in solution rearrange to form C-C coupled digold(I) complexes [Au2X2[mu-2,2-Ph2As(5,5-Me2C6H3C6H3)AsPh2]] [X = Cl (5a), X = Br (5b), X = I (5c) and C6F5 (7)] in which the gold atoms are linearly coordinated by As and X. In contrast, the products of oxidative additions to 9 depend markedly on the halogens. Reaction of 9 with chlorine gives the gold(I)-gold(III) complex, [ClAu[mu-2-Ph2As(C6H3-6-Me)]AuCl[(6-MeC6H3)-2-AsPh2]-kappa2As,C] (10), which contains a four-membered chelate ring, Ph2As(C6H3-6-Me), in the coordination sphere of the gold(III) atom. When 10 is heated, the ring is cleaved, the product being the digold(I) complex [ClAu[mu-2-Ph2As(C6H3-6-Me)]Au[AsPh2(2-Cl-3-Me-C6H3)]] (11). Reaction of 9 with bromine at 50 degrees C gives a monobromo digold(I) complex (12), which is similar to 11 except that the 2-position of the substituted aromatic ring bears hydrogen instead halogen. Reaction of 9 with iodine gives a mixture of a free tertiary arsine, (2-I-3-MeC6H3)AsPh2 (13), a digold diiodo compound (14) analogous to 11, and a gold(I)-gold(III) zwitterionic complex [I2Au(III)[(mu-C6H3-2-AsPh2-6-Me)]2Au(I)] (15) in which the bridging units are arranged head-to-head between the metal atoms. The structures of 2a-2c and 4-15 have been determined by single-crystal X-ray diffraction analysis. The different behaviour of 1 and 9 toward halogens mirrors that of their phosphine analogues; the 6-methyl substituent blocks C-C coupling of the aryl residues in the initially formed oxidative addition product. In the case of 9, the greater lability of the Au-As bond in the initial oxidative addition product may account for the more complex behaviour of this system compared with that of its phosphine analogue.     

The Emergence of Mathematical Modeling Competencies: An Investigation of Prospective Secondary Mathematics Teachers

A study with prospective teachers without prior mathematical modeling experience sheds light on how their newly developed conceptual understanding of modeling manifested itself in their work on the final task of a modeling module within a pedagogy course in secondary mathematics curriculum and assessment. The main purpose of the module was to provide opportunity for the prospective teachers to experience the Common Core Mathematical Practice Model with Mathematics and begin to develop competency in modeling. Their work and reflections displayed a range of proficiency in several competencies associated with the modeling process. Examples of their work illustrating these ranges are provided. The prospective teachers expressed both struggle and rewards during the process, and reflected on challenges for teaching modeling. The results suggest that infusing modules in existing courses can be an effective way to elevate prospective teachers from unfamiliarity with modeling to noticeable levels of proficiency in various modeling sub-competencies.     

Stochastic control of SDEs associated with Lévy generators and application to financial optimization

This paper is concerned with the optimal control of jump type stochastic differential equations associated with (general) Lévy generators. The maximum principle is formulated for the solutions of the equations, which is inspired by N. C. Framstad, B. Øksendal and A. Sulem [J. Optim. Theory Appl., 2004, 121: 77–98] (and a continuation, J. Bennett and J. -L. Wu [Front. Math. China, 2007, 2(4): 539–558]). The result is then applied to optimization problems in financial models driven by Lévy-type processes.     

A note on some piecewise-linear difference equations with Mersenne-type periodic solutions

This paper studies solutions of some piecewise-linear difference equations. In two particular cases, a descent argument is used to show that all solutions are periodic with either prime period 3(2 ^k  ? 1) or 6(2 ^k  ? 1) for some k ≥ 1. The existence of solutions with such periods is also considered.     

Exponentiation of infinite dimensional Z-graded lie algebras

In this article we give a new technique for exponentiating infinite dimensional graded representations of graded Lie algebras that allows for the exponentiation of some non-locally nilpotent elements. Our technique is to naturally extend the representation of the Lie algebra g on the space V naturally to a representation on a subspace £ of the dual space V ^*. After introducing the technique, we prove that it enables the exponentiation of all elements of free Lie Algebras and afhne Kac-Moody Lie algebras.     

Transition Metal‐Free Reduction of Activated Alkenes Using a Living Microorganism

Microorganisms can be programmed to perform chemical synthesis via metabolic engineering. However, despite an increasing interest in the use of de novo metabolic pathways and designer whole‐cells for small molecule synthesis, the inherent synthetic capabilities of native microorganisms remain underexplored. Herein, we report the use of unmodified E. coli BL21(DE3) cells for the reduction of keto‐acrylic compounds and apply this whole‐cell biotransformation to the synthesis of aminolevulinic acid from a lignin‐derived feedstock. The reduction reaction is rapid, chemo‐, and enantioselective, occurs under mild conditions (37 °C, aqueous media), and requires no toxic transition metals or external reductants. This study demonstrates the remarkable promiscuity of central metabolism in bacterial cells and how these processes can be leveraged for synthetic chemistry without the need for genetic manipulation.     

Postselective two-photon interference from a continuous nonclassical stream of photons emitted by a quantum dot

We report an electrically driven semiconductor single-photon source capable of emitting photons with a coherence time of up to 400 ps under fixed bias. It is shown that increasing the injection current causes the coherence time to reduce, and this effect is well explained by the fast modulation of a fluctuating environment. Hong-Ou-Mandel-type two-photon interference using a Mach-Zehnder interferometer is demonstrated using this source to test the indistinguishability of individual photons by postselecting events where two photons collide at a beam splitter. Finally, we consider how improvements in our detection system can be used to achieve a higher interference visibility.