Dirac cohomology of the Dunkl-Opdam subalgebra via inherited Drinfeld properties

Abstract

In this paper, we define a new presentation for the Dunkl-Opdam subalgebra of the rational Cherednik algebra. This presentation uncovers the Dunkl-Opdam subalgebra as a Drinfeld algebra. We use this fact to define Dirac cohomology for the DO subalgebra. We also formalize generalized graded Hecke algebras and extend a Langlands classification to generalized graded Hecke algebras.     

New copper(II) 2-(alkylamino)troponates

The copper aminotropones Cu[ON(R′)C₇H₄R-4]₂ [R = H, R′ = Me (13), Et (14), n-Pr (15), n-Bu (16), Bz (17), MenOCH₂CH₂ (20); R = i-Pr, R′ = Me (18), n-Pr (19), MenOCH₂CH₂ (21)] have been prepared from the corresponding aminotropones HN(R′)OC₇H₄R-4 (1–7) by reacting with copper(II) acetate in aqueous ethanol. 20, 21 contain the flavourant, menthol, as part of the ligand. The structures of 5 (R = H, R′ = Bz), a hydrogen-bonded dimer, 14 and 20, both incorporating square-planar, four-coordinate copper centres, have been determined by X-ray crystallography. The antibacterial activities of complexes 13, 17, 20 and 21 have been assayed against Staphylococcus waneri, an in vitro model of plaque inhibition effects, and found to be more active than a commercial toothpaste formulation, but less active than the O,O-chelated copper(II) complex of ethylmaltol.     

Mass conservative finite volume discretization of the continuity equations in multi-component mixtures

The Stefan–Maxwell equations for multi-component diffusion result in a system of coupled continuity equations for all species in the mixture. We use a generalization of the exponential scheme to discretize this system of continuity equations with the finite volume method. The system of continuity equations in this work is obtained from a non-singular formulation of the Stefan–Maxwell equations, where the mass constraint is not applied explicitly. Instead, all mass fractions are treated as independent unknowns and the constraint is a result of the continuity equations, the boundary conditions, the diffusion algorithm and the discretization scheme. We prove that with the generalized exponential scheme, the mass constraint can be satisfied exactly, although it is not explicitly applied. A test model from the literature is used to verify the correct behavior of the scheme.     

Phosphorus-based functional groups as hydrogen bonding templates for rotaxane formation

We report on the use of the hydrogen bond acceptor properties of some phosphorus-containing functional groups for the assembly of a series of [2]rotaxanes. Phosphinamides, and the homologous thio- and selenophosphinamides, act as hydrogen bond acceptors that, in conjunction with an appropriately positioned amide group on the thread, direct the assembly of amide-based macrocycles around the axle to form rotaxanes in up to 60% yields. Employing solely phosphorus-based functional groups as the hydrogen bond accepting groups on the thread, a bis(phosphinamide) template and a phosphine oxide-phosphinamide template afforded the corresponding rotaxanes in 18 and 15% yields, respectively. X-ray crystallography of the rotaxanes shows the presence of up to four intercomponent hydrogen bonds between the amide groups of the macrocycle and various hydrogen bond accepting groups on the thread, including rare examples of amide-to-phosphinamide, -thiophosphinamide, and -selenophosphinamide groups. With a phosphine oxide-phosphinamide thread, the solid-state structure of the rotaxane is remarkable, featuring no direct intercomponent hydrogen bonds but rather a hydrogen bond network involving water molecules that bridge the H-bonding groups of the macrocycle and thread through bifurcated hydrogen bonds. The incorporation of phosphorus-based functional groups into rotaxanes may prove useful for the development of molecular shuttles in which the macrocycle can be used to hinder or expose binding ligating sites for metal-based catalysts.     

Geometric, variational discretization of continuum theories

This study derives geometric, variational discretization of continuum theories arising in fluid dynamics, magnetohydrodynamics (MHD), and the dynamics of complex fluids. A central role in these discretizations is played by the geometric formulation of fluid dynamics, which views solutions to the governing equations for perfect fluid flow as geodesics on the group of volume-preserving diffeomorphisms of the fluid domain. Inspired by this framework, we construct a finite-dimensional approximation to the diffeomorphism group and its Lie algebra, thereby permitting a variational temporal discretization of geodesics on the spatially discretized diffeomorphism group. The extension to MHD and complex fluid flow is then made through an appeal to the theory of Euler-Poincaré systems with advection, which provides a generalization of the variational formulation of ideal fluid flow to fluids with one or more advected parameters. Upon deriving a family of structured integrators for these systems, we test their performance via a numerical implementation of the update schemes on a cartesian grid. Among the hallmarks of these new numerical methods are exact preservation of momenta arising from symmetries, automatic satisfaction of solenoidal constraints on vector fields, good long-term energy behavior, robustness with respect to the spatial and temporal resolution of the discretization, and applicability to irregular meshes.     

Self-Reciprocal Irreducible Polynomials Over Finite Fields

We classify monic self-reciprocal irreducible polynomials over finite fields in terms of their orders. We also study the weights of these polynomials.     

Analytical representations of the Mössbauer recoilless fraction,Debye-Waller factors,lattice energies,and heat capacities

Simple analytical expansions are given for the recoilless fraction in Mössbauer spectroscopy, the Debye-Waller factor in X-ray scattering, and the lattice energy and heat capacity of solids. While this problem has been discussed in an earlier paper [1], computer technology has now advanced to the point that direct evaluations of the simple expansions of these quantities are useful for quick curve fitting to experimental data at any desired temperature, and these expansions are easier to evaluate than using graphs to estimate recoilless fractions and Debye temperatures. We compare this approach with a polynomial expansion in terms of Bernoulli numbers, which has only a limited domain of convergence. We explicitly evaluate the convergence of these Debye integral expansions as a function of the number of terms used and the time required.This work was prepared with the support of the U.S. Department of Energy, Grant No. DE-FG02-85 ER 45199.     

Phase shifting algorithms for non-sinusoidal waveforms with phase shift errors

In phase shifting interferometry, phase errors due to harmonic components of a fringe signal can be minimized by applying synchronous phase shifting algorithms with more than four samples. However, when the phase shift calibration is inaccurate, these algorithms cannot eliminate the effects of a non-sinusoidal waveform. It is shown that by taking a number of samples beyond one period of the fringe pattern, phase errors due to the harmonic components of the fringe signal can be eliminated, even when there exists a constant error in the phase shift interval. A general procedure for constructing phase shifting algorithms that eliminate these errors is derived. A seven-sample phase shifting algorithm is derived as an example, in which the effect of the second harmonic component can be eliminated in the presence of a constant error in the phase shift interval.