Gelfand-Kirillov dimension of algebras with locally nilpotent derivations

Let R be a finitely generated algebra over a field of characteristic 0 with a locally nilpotent derivation δ ≠ 0. We show that if {ie313-1}, where the invariants {ie313-2} are prime and satisfy a polynomial identity, then {ie313-3}. Furthermore, when R is a domain, the same conclusion holds without the assumption that R is finitely generated. This enables us to obtain a result on skew polynomial rings. These results extend work of Bell and Smoktunowicz on domains with GK dimension in the interval [2, 3).     

Roles of xyloglucan and pectin on the mechanical properties of bacterial cellulose composite films

Xyloglucan and pectin are major non-cellulosic components of most primary plant cell walls. It is believed that xyloglucan and perhaps pectin are functioning as tethers between cellulose microfibrils in the cell walls. In order to understand the role of xyloglucan and pectin in cell wall mechanical properties, model cell wall composites created using Gluconacetobacter xylinus cellulose or cellulose nanowhiskers (CNWs) derived there from with different amounts of xyloglucan and/or pectin have been prepared and measured under extension conditions. Compared with pure CNW films, CNW composites with lower amounts of xyloglucan or pectin did not show significant differences in mechanical behavior. Only when the additives were as high as 60 %, the films exhibited a slightly lower Young’s modulus. However, when cultured with xyloglucan or pectin, the bacterial cellulose (BC) composites produced by G. xylinus showed much lower modulus compared with that of the pure BC films. Xyloglucan was able to further reduce the modulus and extensibility of the film compared to that of pectin. It is proposed that surface coating or tethering of xyloglucan or pectin of cellulose microfibrils does not alone affect the mechanical properties of cell wall materials. The implication from this work is that xyloglucan or pectin alters the mechanical properties of cellulose networks during rather than after the cellulose biosynthesis process, which impacts the nature of the connection between these compounds.     

Rolf Huisgen's Classic Studies of Cyclic Triene Diels–Alder Reactions Elaborated by Modern Computational Analysis

Rolf Huisgen explored the Diels–Alder reactions of 1,3,5-cycloheptatriene (CHT) and cyclooctatetraene (COT) with the dienophiles maleic anhydride and 4-phenyl-1,2,4-triazoline-3,5-dione (PTAD) to determine the kinetics and mechanisms of various electrocyclizations and Diels–Alder reactions. These reactions have been examined with density functional theory. Modern computational chemistry has provided information not previously available by experiment. Transition states for all the reactions have been identified, and their Gibbs energies are used to explain the experimental reactivities. Zwitterionic intermediates were not found in the [4+2] cycloadditions of both CHT or COT with PTAD and are thus not involved in these reactions. [2+2+2] cycloadditions, as an alternative path to the Diels–Alder products, are highly disfavored. Rapid double nitrogen inversion was found for the cycloaddition products with PTAD.     

Density functionals with asymptotic-potential corrections are required for the simulation of spectroscopic properties of materials

Five effects of correction of the asymptotic potential error in density functionals are identified that significantly improve calculated properties of molecular excited states involving charge-transfer character. Newly developed materials-science computational methods are used to demonstrate how these effects manifest in materials spectroscopy. Connection is made considering chlorophyll-a as a paradigm for molecular spectroscopy, 22 iconic materials as paradigms for 3D materials spectroscopy, and the V_N⁻ defect in hexagonal boron nitride as an example of the spectroscopy of defects in 2D materials pertaining to nanophotonics. Defects can equally be thought of as being “molecular” and “materials” in nature and hence bridge the relms of molecular and materials spectroscopies. It is concluded that the density functional HSE06, currently considered as the standard for accurate calculations of materials spectroscopy, should be replaced, in most instances, by the computationally similar but asymptotically corrected CAM-B3LYP functional, with some specific functionals for materials-use only providing further improvements.

Spectroscopic transitions in materials that involve charge transfer require asymptotically corrected density functionals. As most transitions do have some charge transfer character, use of such methods are generally warranted.     

Synchrotron X‐ray tests of an L‐shaped laterally graded multilayer mirror for the analyzer system of the ultra‐high‐resolution IXS spectrometer at NSLS‐II

Characterization and testing of an L‐shaped laterally graded multilayer mirror are presented. This mirror is designed as a two‐dimensional collimating optics for the analyzer system of the ultra‐high‐resolution inelastic X‐ray scattering (IXS) spectrometer at National Synchrotron Light Source II (NSLS‐II). The characterization includes point‐to‐point reflectivity measurements, lattice parameter determination and mirror metrology (figure, slope error and roughness). The synchrotron X‐ray test of the mirror was carried out reversely as a focusing device. The results show that the L‐shaped laterally graded multilayer mirror is suitable to be used, with high efficiency, for the analyzer system of the IXS spectrometer at NSLS‐II.     

Real loci of based loop groups

Let (G, K) be a Riemannian symmetric pair of maximal rank, where G is a compact simply connected Lie group and K is the fixed point set of an involutive automorphism σ. This induces an involutive automorphism τ of the based loop space Ω(G). There exists a maximal torus T ⊂ G such that the canonical action of T × S ¹ on Ω(G) is compatible with τ (in the sense of Duistermaat). This allows us to formulate and prove a version of Duistermaat’s convexity theorem. Namely, the images of Ω(G) and Ω(G) ^τ (fixed point set of τ) under the T × S ¹ moment map on Ω(G) are equal. The space Ω(G) ^τ is homotopy equivalent to the loop space Ω(G/K) of the Riemannian symmetric space G/K. We prove a stronger form of a result of Bott and Samelson which relates the cohomology rings with coefficients in \mathbbZ₂ {\mathbb{Z}_2} of Ω(G) and Ω(G/K). Namely, the two cohomology rings are isomorphic, by a degree-halving isomorphism (Bott and Samelson [BS] had proved that the Betti numbers are equal). A version of this theorem involving equivariant cohomology is also proved. The proof uses the notion of conjugation space in the sense of Hausmann, Holm, and Puppe [HHP].     

Tensors,spinors, and functions on the unit sphere

A representation of tensors and spinors at a point of space-time as spin and conformally weighted functions on the unit sphere is derived. Methods for performing algebraic operations on tensors and spinors in this representation are discussed.Supported in part by grants Nos. GP-35773X1 and MPS74-18020 from the National Science Foundation.     

Synthesis of Polymers Containing Potassium Acyltrifluoroborates (KATs) and Post-polymerization Ligation and Conjugation

We report the synthesis of monomers for atom-transfer radical polymerization (ATRP) and a reversible addition-fragmentation chain transfer (RAFT) agent bearing trifluoroborate iminiums (TIMs), which are quantitatively converted into potassium acyltrifluoroborates (KATs) after polymerization. The resulting KAT-containing polymers are suitable for rapid amide-forming ligations for both post-polymerization modification and polymer conjugation. The polymer conjugation occurs rapidly, even under dilute (micromolar) aqueous conditions at ambient temperatures, thereby enabling the synthesis of a variety of linear and star-shaped block copolymers. In addition, we applied post-polymerization modification to the covalent linking of a photocaged cyclic antibiotic (gramicidin S) to the side chains of the KAT-containing copolymer. Cellular assays revealed that the polymer–antibiotic conjugate is biocompatible and provides efficient light-controlled release of the antibiotic on demand.     

The general relativistic hydrogen atom

The general relativistic Dirac equation is formulated in an arbitrary curved space-time using differential forms. These equations are applied to spherically symmetric systems with arbitrary charge and mass. For the case of a black hole (with event horizon) it is shown that the Dirac Hamiltonian is self-adjoint, has essential spectrum the whole real line and no bound states. Although rigorous results are obtained only for a spherically symmetric system, it is argued that, in the presence of any event horizon there will be no bound states. The case of a naked singularity is investigated with the results that the Dirac Hamiltonian is not self-adjoint. The self-adjoint extensions preserving angular momentum are studied and their spectrum is found to consist of an essential spectrum corresponding to that of a free electron plus eigenvalues in the gap (–mc ², +mc ²). It is shown that, for certain boundary conditions, neutrino bound states exist.Supported in part by the National Science Foundation