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.     

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.     

Hydrophobic silica aerogels prepared via rapid supercritical extraction

Hydrophobic silica aerogels have been prepared using the rapid supercritical extraction (RSCE) technique. The RSCE technique is a one-step methanol supercritical extraction method for producing aerogel monoliths in 3 to 8 h. Standard aerogels were prepared from a tetramethoxysilane (TMOS) recipe with a molar ratio of TMOS:MeOH:H₂O:NH₄OH of 1.0:12.0:4.0:7.4 × 10⁻³. Hydrophobic aerogels were prepared using the same recipe except the TMOS was replaced with a mixture of TMOS and one of the following organosilane co-precursors: methytrimethoxysilane (MTMS), ethyltrimethoxysilane (ETMS), or propyltrimeth-oxysilane (PTMS). Results show that, by increasing the amount of catalyst and increasing gelation time, monolithic aerogels can be prepared out of volume mixtures including up to 75% MTMS, 50% ETMS or 50% PTMS in 7.5–15 h. As the amount of co-precursor is increased the aerogels become more hydrophobic (sessile tests with water droplets yield contact angles up to 155°) and less transparent (transmission through a 12.2-mm thick sample decreases from 83 to 50% at 800 nm). The skeletal and bulk density decrease and the surface area increases (550–760 m²/g) when TMOS is substituted with increasing amounts of MTMS. The amount of co-precursor does not affect the thermal conductivity. SEM imaging shows significant differences in the nanostructure for the most hydrophobic surfaces.     

A Bayesian method for construction of Markov models to describe dynamics on various time-scales

The dynamics of many biological processes of interest, such as the folding of a protein, are slow and complicated enough that a single molecular dynamics simulation trajectory of the entire process is difficult to obtain in any reasonable amount of time. Moreover, one such simulation may not be sufficient to develop an understanding of the mechanism of the process, and multiple simulations may be necessary. One approach to circumvent this computational barrier is the use of Markov state models. These models are useful because they can be constructed using data from a large number of shorter simulations instead of a single long simulation. This paper presents a new Bayesian method for the construction of Markov models from simulation data. A Markov model is specified by (τ,P,T), where τ is the mesoscopic time step, P is a partition of configuration space into mesostates, and T is an N(P)×N(P) transition rate matrix for transitions between the mesostates in one mesoscopic time step, where N(P) is the number of mesostates in P. The method presented here is different from previous Bayesian methods in several ways. (1) The method uses Bayesian analysis to determine the partition as well as the transition probabilities. (2) The method allows the construction of a Markov model for any chosen mesoscopic time-scale τ. (3) It constructs Markov models for which the diagonal elements of T are all equal to or greater than 0.5. Such a model will be called a "consistent mesoscopic Markov model" (CMMM). Such models have important advantages for providing an understanding of the dynamics on a mesoscopic time-scale. The Bayesian method uses simulation data to find a posterior probability distribution for (P,T) for any chosen τ. This distribution can be regarded as the Bayesian probability that the kinetics observed in the atomistic simulation data on the mesoscopic time-scale τ was generated by the CMMM specified by (P,T). An optimization algorithm is used to find the most probable CMMM for the chosen mesoscopic time step. We applied this method of Markov model construction to several toy systems (random walks in one and two dimensions) as well as the dynamics of alanine dipeptide in water. The resulting Markov state models were indeed successful in capturing the dynamics of our test systems on a variety of mesoscopic time-scales.     

p-Type thermoelectric properties of the oxygen-deficient perovskite Ca2Fe2O5 in the brownmillerite structure

Brownmillerite calcium ferrite was synthesized in air at 1573 K and thermoelectric properties (direct current electrical conductivity σ, Seebeck coefficient α, thermal conductivity κ, thermal expansion α_L) were measured from 373 to 1050 K in air. Seebeck coefficient was positive over all temperatures indicating conduction by holes, and electrical properties were continuous through the Pnma-Imma phase transition. Based on the thermopower and conductivity activation energies as well as estimated mobility, polaron hopping conduction was found to dominate charge transport. The low electrical conductivity, <1 S/cm, limits the power factor (α²σ), and thus the figure of merit for thermoelectric applications. The thermal conductivity values of ∼2 W/mK and their similarity to Ruddlesden-Popper phase implies the potential of the alternating tetrahedral and octahedral layers to limit phonon propagation through brownmillerite structures. Bulk linear coefficient of thermal expansion (∼14×10⁻⁶ K⁻¹) was calculated from volume data based on high-temperature in situ X-ray powder diffraction, and shows the greatest expansion perpendicular to the alternating layers.