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.     

Parallelization of the integral equation formulation of the polarizable continuum model for higher-order response functions

We present a parallel implementation of the integral equation formalism of the polarizable continuum model for Hartree-Fock and density functional theory calculations of energies and linear, quadratic, and cubic response functions. The contributions to the free energy of the solute due to the polarizable continuum have been implemented using a master-slave approach with load balancing to ensure good scalability also on parallel machines with a slow interconnect. We demonstrate the good scaling behavior of the code through calculations of Hartree-Fock energies and linear, quadratic, and cubic response function for a modest-sized sample molecule. We also explore the behavior of the parallelization of the integral equation formulation of the polarizable continuum model code when used in conjunction with a recent scheme for the storage of two-electron integrals in the memory of the different slaves in order to achieve superlinear scaling in the parallel calculations.     

Yukawa Couplings in Heterotic Compactification

We present a practical, algebraic method for efficiently calculating the Yukawa couplings of a large class of heterotic compactifications on Calabi-Yau three-folds with non-standard embeddings. Our methodology covers all of, though is not restricted to, the recently classified positive monads over favourable complete intersection Calabi-Yau three-folds. Since the algorithm is based on manipulating polynomials it can be easily implemented on a computer. This makes the automated investigation of Yukawa couplings for large classes of smooth heterotic compactifications a viable possibility.     

Rayleigh–Schrödinger series and Birkhoff decomposition

We derive new expressions for the Rayleigh–Schrödinger series describing the perturbation of eigenvalues of quantum Hamiltonians. The method, somehow close to the so-called dimensional renormalization in quantum field theory, involves the Birkhoff decomposition of some Laurent series built up out of explicit fully non-resonant terms present in the usual expression of the Rayleigh–Schrödinger series. Our results provide new combinatorial formulae and a new way of deriving perturbation series in quantum mechanics. More generally we prove that such a decomposition provides solutions of general normal form problems in Lie algebras.     

Interesting Reactivity in the Thermolysis and Photolysis of 2,3-Diphenyl-1-naphthol

Abstract  2,3-Diphenyl-1-naphthol (1) undergoes two unexpected reactions under different conditions. Compound (1) was heated in DMSO-d₆ and underwent a Pummerer type thermal reaction to give two isomeric products, 1-(methylthio)methoxy-2,3-diphenyl naphthol-d₅ which crystallized in the space group with a = 7.1610(9) ?, b = 11.2795(15) ?, c = 12.8905(17) ?, α = 114.049(2)°, β = 96.589(2)°, and γ = 102.945(2)°, and 2-(methylthio)methyl-2,3-diphenyl 1(2H)-naphthalenone-d₅ which crystallized in the space group with a = 8.5981(5) ?, b = 10.4374(6) ?, c = 11.1078(6) ?, α = 78.748(2)°, β = 67.709(2)°, and γ = 83.184(2)°. Photolysis (254 nm) of (1) resulted in 2,2′,3,3′-tetraphenyl-1,1′-bi-2-naphthol which crystallized in the space group P2₁/c with a = 26.3616(11) ?, b = 10.1707(4) ?, c = 23.3376(9) ?, and β = 99.034(2)°. Graphical Abstract  When 2,3-diphenyl-1-naphthol was heated in DMSO-d₆ two unexpected isomers, 1-(methylthio)methoxy-2,3-diphenyl naphthol-d₅ and the racemate 2-(methylthio)methyl-2,3-diphenyl 1(2H)-naphthalenone-d₅ were produced. Photolysis in THF at 254 nm led to the unexpected product 2,2′,3,3′-tetraphenyl-1,1′-bi-2-naphthol. All structures were elucidated by X-ray crystallography.      

Phase mixture and anti-reflection window in visible of annealed beryllium-nitride thin films on silicon crystal

Beryllium-nitride (Be₃N₂) thin films were grown on silicon Si(1 1 1) substrates by pulsed laser deposition in a RIBER LDM-32 system, and characterized with in/ex situ XPS and SIMS. The structure of the films was analyzed with XRD. The films were further analyzed for surface topographic information with SEM and profilometry, and for optical properties with optical spectroscopy. It was observed that the material, prepared at room temperature and annealed at 700 °C for 2 h, had undergone a partial phase transition to a mixture of amorphous and crystalline phases, and the thin films showed a large anti-reflection window in the visible. Therefore, the annealed Be₃N₂ thin films would be potentially useful for stable electronic packaging with desired photonic features.     

Copper-catalyzed sulfenylation of pyrroles with disulfides or thiols: directly synthesis of sulfenyl pyrroles

We present here the synthesis of sulfenyl pyrroles by copper-catalyzed sulfenylation of pyrroles with organic disulfides or thiols. The direct sulfenylation of pyrroles with organic disulfides has been accomplished in the presence of 3 mol % of CuI in DMSO at 110 °C under air atmosphere. In the other hand, sulfenylation of pyrroles with thiols were performed in the same solvent and temperature, however it is necessary 5 mol % of CuI and nitrogen atmosphere. Using these protocols we were able to produce 2-sulfenyl pyrroles in good to excellent yields and with high selectivity without use of any ligand or additive.     

Oleana‐12(13),15(16)‐diene‐3α,28‐diyl di­acetate

The title compound, C₃₄H₅₂O₄, consists of five six‐membered rings. Barring the two rings, with double bonds, all other rings are in chair conformations. Mean‐plane and ring‐puckering calculations indicate these two rings to be in distorted‐chair conformations, with distortion towards the boat conformation. There are no strong hydrogen bonds and the structure is stabilized by van der Waals interactions only. The structure is compared with those reported for other triterpenes.