Complex multiplication of exactly solvable Calabi–Yau varieties

We propose a conceptual framework that leads to an abstract characterization for the exact solvability of Calabi–Yau varieties in terms of abelian varieties with complex multiplication. The abelian manifolds are derived from the cohomology of the Calabi–Yau manifold, and the conformal field theoretic quantities of the underlying string emerge from the number theoretic structure induced on the varieties by the complex multiplication symmetry. The geometric structure that provides a conceptual interpretation of the relation between geometry and conformal field theory is discrete, and turns out to be given by the torsion points on the abelian varieties.     

Regular Subgraphs in Graphs and Rooted Graphs and Definability in Monadic Second - Order Logic

We investigate the definability in monadic ∑₁¹ and monadic Π₁¹ of the problems REG_k, of whether there is a regular subgraph of degree k in some given graph, and XREG_k, of whether, for a given rooted graph, there is a regular subgraph of degree k in which the root has degree k, and their restrictions to graphs in which every vertex has degree at most k, namely REG_k^k and XREG_k^k, respectively, for k ≥ 2 (all our graphs are undirected). Our motivation partly stems from the fact (which we prove here) that REG_k^k and XREG_k^k are logspace equivalent to CONN and REACH, respectively, for k ≥ 3, where CONN is the problem of whether a given graph is connected and REACH is the problem of whether a given graph has a path joining two given vertices. We use monadic first - order reductions, monadic ∑₁¹ games and a recent technique due to Fagin, Stockmeyer and Vardi to almost completely classify whether these problems are definable in monadic ∑₁¹ and monadic Π₁¹, and we compare the definability of these problems (in monadic ∑₁¹ and monadic Π₁¹ with their computational complexity (which varies from solvable using logspace to NP - complete).     

Reactions involving fluoride ion. Part 40. Amines as initiators of fluoride ion catalysed reactions

Tetrakis(dimethylamino)ethene (TDAE) and trimethylamine react with anhydrous unsaturated fluorocarbons to produce, ‘in situ’, powerful fluoride- ion sources. These are used to initiate carbon- carbon bond forming reactions eg. oligomerisation and polyfluoroalkylation, and many of these reactions occur efficiently in the absence of a solvent.     

Numerical sensitivity of trajectories across conformational energy hypersurfaces from geometry optimized molecular orbital calculations: AM1, MNDO,and MINDO/3

The monocyclic β-lactam [[4(S)-methyl-2-oxo-1-azetidinyl]thia]acetic acid was studied by the semiempirical molecular orbital methods AM1, MNDO, and MINDO/3. Using the reaction coordinate option in the program MOPAC on VAX and Cray X-MP computers, the potential energy curve was calculated for rotation of the C2-N1-S-C torsional angle in the conformationally flexible side chain while optimizing all other geometrical variables in the molecule. The trajectory taken during geometry optimization was found to be sensitive to the computer, the program version, the convergence criteria, and the degree of code optimization used in the calculation. In order to reduce the likelihood of spurious results, conformational or reaction energy hypersurfaces need to be calculated with the more precise SCF convergence and minimization criteria available in programs for MINDO/3, MNDO, and AM1 calculations. The nitrogen in the model β-lactam antibiotic is predicted to invert periodically as the dihedral angle to the exocyclic N-substituent sweeps through 360°.     

Mathematics for young people: The royal institution masterclasses

The Mathematical Intelligencer - We expected a fair number of participants to drop out as the course progressed. Indeed, we had raised the numbers to 48 anticipating this. In fact the average level...     

A procedure for obtaining lower bounds for ground state energies of atoms and molecules

Using Löwdin's partition method we have re-derived the D. Weinstein lower bound, E > H ₁₁ - . By the same method, plus the assumption that the calculated first excited state energy is lower than a certain weighted average of approximate energies of all the excited states, we have derived a moderately better lower bound.     

A Mössbauer effect study of Y(57Fe:Mn)12

The intermetallic compound, YMn₁₂, doped with 0.5at.%⁵⁷Fe has been investigated using the⁵⁷Fe Mössbauer resonance. The data show that the Fe impurities occupy preferentially the f-site at the expense of the i- and j-sites. Whereas the i- and j-sites are expected to carry identical local magnetic moments in the ordered state, low temperature hyperfine fields of 50.08(±0.05)T and 2.90(±0.03)T are measured for the i- and j-sites respectively. It is proposed that large transferred hyperfine fields are responsible for this disparity.