Looking for Difference Sets in Groups with Dihedral Images

We prove four theorems about groups with a dihedral (or cyclic) image containing a difference set. For the first two, suppose G, a group of order 2p with p an odd prime, contains a nontrivial (v, k, ) difference set D with order n = k – prime to p and self-conjugate modulo p. If G has an image of order p, then 0 2a + 2 for a unique choice of = ±1, and for a = (k – )/2p. If G has an image of order 2p, then and ( – 1)/( – 1). There are further constraints on n, a and . We give examples in which these theorems imply no difference set can exist in a group of a specified order, including filling in some entries in Smith's extension to nonabelian groups of Lander's tables. A similar theorem covers the case when p|n. Finally, we show that if G contains a nontrivial (v, k, ) difference set D and has a dihedral image D _2m with either (n, m) = 1 or m = p ^t for p an odd prime dividing n, then one of the C ₂ intersection numbers of D is divisible by m. Again, this gives some non-existence results.     

A sharp bound for the degree of proper monomial mappings between balls

The authors prove that a proper monomial holomorphic mapping from the two-ball to the N-ball has degree at most 2N-3, and that this result is sharp. The authors first show that certain group-invariant polynomials (related to Lucas polynomials) achieve the bound. To establish the bound the authors introduce a graph-theoretic approach that requires determining the number of sinks in a directed graph associated with the quotient polynomial. The proof also relies on a result of the first author that expresses all proper polynomial holomorphic mappings between balls in terms of tensor products.     

Abelian Subgroup Separability of Haken 3-Manifolds and Closed Hyperbolic n-Orbifolds

We prove that abelian subgroups of fundamental groups of Haken3-manifolds and closed hyperbolic n-orbifolds are separable.2000 Mathematical Subject Classification: 20E26, 51M10, 57M05.     

Prediction of structure-dependent charge transfer rates for a Li atom outside a Si(0 0 1) surface

We investigate the broadening of the 2s energy level of a Li atom outside a Si(0 0 1) surface using a first principles approach. The covalent nature of the Si surface produces large variations in Li energy level widths as a function of lateral position across the surface. The widths above symmetric Si dimers are predicted to be much larger than above buckled Si dimers, suggesting that charge transfer will occur primarily above symmetric dimers. We discuss the ramifications of our results on the controversy surrounding the relative abundance of the buckled vs. symmetric dimers on the Si surface.     

On pseudocyclic table algebras and applications to pseudocyclic association schemes

We effect a complete study of the thermodynamic formalism, the entropy spectrum of Birkhoff averages, and the ergodic optimization problem for a family of parabolic horseshoes. We consider a large class of potentials that are not necessarily regular, and we describe both the uniqueness of equilibrium measures and the occurrence of phase transitions for nonregular potentials in this class. Our approach consists in reducing the problems to the study of renewal shifts. We also describe applications of this approach to hyperbolic horseshoes as well as to noninvertible maps, both parabolic (with the Manneville-Pomeau map) and uniformly expanding. This allows us to recover in a unified manner several results scattered in the literature. For the family of hyperbolic horseshoes, we also describe the dimension spectrum of equilibrium measures of a class of potentials that are not necessarily regular. In particular, the dimension spectra need not be strictly convex.     

The Effect of Platinum on Diffusion Kinetics in β‐NiAl: Implications for Thermal Barrier Coating Lifetimes

Platinum is added to thermal barrier coatings (TBCs) as it is observed empirically to extend their lifetime, but the mechanism by which Pt acts is unknown. Since Pt has been proposed to alter diffusivities in NiAl, a key component of TBCs, we use first‐principles quantum mechanics calculations to investigate atomic level diffusion mechanisms. Here, we examine the effect of Pt on five previously proposed mechanisms for Ni diffusion in NiAl: next‐nearest‐neighbor jumps, the triple defect mechanism, and three variants of the six jump cycle. We predict that Pt increases the rate of Ni diffusion by stabilizing point defects and defect clusters that are diffusion intermediates. Previously, we predicted the triple defect mechanism to be a dominant Ni diffusion mechanism; it simultaneously results in long‐range Al diffusion in the opposite direction. Since Pt increases the rate of Ni diffusion, it also increases Al diffusion in NiAl, which may be key to extending the coating lifetime.     

Can density functional theory (DFT) be used as an aid to a deeper understanding of tandem mass spectrometric fragmentation pathways?

Prediction of tandem mass spectrometric (MS/MS) fragmentation for non‐peptidic molecules based on structure is of immense interest to the mass spectrometrist. If a reliable approach to MS/MS prediction could be achieved its impact within the pharmaceutical industry could be immense. Many publications have stressed that the fragmentation of a molecular ion or protonated molecule is a complex process that depends on many parameters, making prediction difficult. Commercial prediction software relies on a collection of general heuristic rules of fragmentation, which involve cleaving every bond in the structure to produce a list of ‘expected’ masses which can be compared with the experimental data. These approaches do not take into account the thermodynamic or molecular orbital effects that impact on the molecule at the point of protonation which could influence the potential sites of bond cleavage based on the structural motif. A series of compounds have been studied by examining the experimentally derived high‐resolution MS/MS data and comparing it with the in silico modelling of the neutral and protonated structures. The effect that protonation at specific sites can have on the bond lengths has also been determined. We have calculated the thermodynamically most stable protonated species and have observed how that information can help predict the cleavage site for that ion. The data have shown that this use of in silico techniques could be a possible way to predict MS/MS spectra. Copyright © 2009 John Wiley & Sons, Ltd.     

A Survey of Calculator Usage in High Schools

This survey investigated the current status of calculator use in classrooms and schools. The results indicated that the prevailing policy in the sample of high schools is to allow the use of calculators during classroom learning activities and tests. Scientific calculators are more frequently used than graphing calculators in algebra I and geometry; whereas, graphing calculators are more frequently used in algebra II and precalculus/trigonometry. At the time of this survey, school policies regarding the use of graphing calculators with symbolic algebra capabilities were still not determined.