Representation systems and undergraduate proof production: A comment on Weber

Weber (2009) suggested that counterexamples can be generated by a syntactic proof production, apparently contradicting our earlier assertion (Alcock & Inglis, 2008). Here we point out that this ostensible difference is the result of Weber working with theoretical definitions that differ slightly from ours. We defend our approach by arguing that Weber’s relies upon an as yet unspecific metric for gauging the amount of work conducted in each representation system, and that it does not recognize an important asymmetry between the status of representation systems in the context of undergraduate mathematics.     

Fierz bilinear formulation of the Maxwell–Dirac equations and symmetry reductions

We study the Maxwell–Dirac equations in a manifestly gauge invariant presentation using only the spinor bilinear scalar and pseudoscalar densities, and the vector and pseudovector currents, together with their quadratic Fierz relations. The internally produced vector potential is expressed via algebraic manipulation of the Dirac equation, as a rational function of the Fierz bilinears and first derivatives (valid on the support of the scalar density), which allows a gauge invariant vector potential to be defined. This leads to a Fierz bilinear formulation of the Maxwell tensor and of the Maxwell–Dirac equations, without any reference to gauge dependent quantities. We show how demanding invariance of tensor fields under the action of a fixed (but arbitrary) Lie subgroup of the Poincaré group leads to symmetry reduced equations. The procedure is illustrated, and the reduced equations worked out explicitly for standard spherical and cylindrical cases, which are coupled third order nonlinear PDEs. Spherical symmetry necessitates the existence of magnetic monopoles, which do not affect the coupled Maxwell–Dirac system due to magnetic terms cancelling. In this paper we do not take up numerical computations. As a demonstration of the power of our approach, we also work out the symmetry reduced equations for two distinct classes of dimension 4 one-parameter families of Poincaré subgroups, one splitting and one non-splitting. The splitting class yields no solutions, whereas for the non-splitting class we find a family of formal exact solutions in closed form.     

Synthesis of a Type-VIβ-Turn Peptide Mimetic and Its Incorporation into a High-Affinity Somatostatin Receptor Ligand

The synthesis of a cis-Phe-Pro dipeptide mimetic is described, which adopts a type-VIβ-turn conformation. In this mimetic, the α-positions of Phe and Pro are joined by a CH₂CH₂ bridge, thereby forming a fused bicyclic system, and fixing a geometry similar to that seen in cis-Phe-Pro units in protein crystal structures. The dipeptide mimetic 20 was synthesized in optically pure form starting from (R)-α-allylproline ( 6 ; Schemes 1, 3, and 4), with a free carboxylic acid and an Fmoc-protected N-terminus, thereby allowing its incorporation into linear and cyclic peptides using standard solid-phase methods. The mimetic 20 was incorporated into the cyclic somatostatin analogue cyclo(-Phe = Pro-Phe-D -Trp-Lys-Thr-), where Phe = Pro represents the mimetic. This analogue shows a high affinity (pIC₅₀ 8.6) for somatostatin receptors on rat-brain cortex membranes. Based on NMR studies in aqueous solution, likely low-energy conformations for this analogue were deduced using restrained dynamic simulated annealing. The conformations found, which include a distorted type-II′ turn at D -Trp-Lys, are similar to low-energy conformations deduced elsewhere for cyclo(-Phe-Pro-Phe-D -Trp-Lys-Thr-), as well as to those seen in crystal structures of the somatostatin analogue octreotide.     

Galois groups of some vectorial polynomials

Previously nice vectorial equations were constructed having various finite classical groups as Galois groups. Here such equations are constructed for the remaining classical groups. The previous equations were genus zero equations. The present equations are strong genus zero.

     

A family of hexanuclear Mn(III) single-molecule magnets

In an attempt to employ salicylic acid (HOsalH), 2,6-dihydroxy benzoic acid {(HO)₂PhCO₂H}, and naphthalene-1,8-dicarboxylic acid {1,8-naph(CO₂H)₂} in Mn(III) salicylaldoximate chemistry as a means to alter the structural identity of the hexanucluear clusters usually obtained from this reaction system, we have isolated a family of hexanuclear Mn(III) complexes based on salicyladloxime (saoH₂) and 2-hydroxy-1-naphthaldehyde oxime (naphthsaoH₂). Five hexanuclear clusters, [Mn₆O₂(sao)₆(HOsal)₂(EtOH)₄]·EtOH (1·EtOH), [Mn₆O₂(sao)₆{1,8-naph(CO₂Me)(CO₂)}₂(MeOH)₆]·3MeOH (2·3MeOH), [Mn₆O₂(naphthsao)₆{1,8-naph(CO₂Et)(CO₂)}₂(EtOH)₆] (3·2MeOH), [Mn₆O₂(naphthsao)₆(MeCO₂)₂(EtOH)₄]·2H₂O (4·2H₂O), and [Mn₆O₂(naphthsao)₆{(HO)₂PhCO₂}₂(EtOH)₄]·4EtOH (5·4EtOH), have been synthesized and characterized by single-crystal X-ray crystallography. The magnetic properties of 3, 4, and 5 are discussed.     

A new way to rapidly create functional, fluorescent fusion proteins: random insertion of GFP with an in vitro transposition reaction

Background  

The jellyfish green fluorescent protein (GFP) can be inserted into the middle of another protein to produce a functional, fluorescent fusion protein. Finding permissive sites for insertion, however, can be difficult. Here we describe a transposon-based approach for rapidly creating libraries of GFP fusion proteins.