Isotropy Representations, Eigenvalues of a Casimir Element, and Commutative Lie Subalgebras

Let h be a reductive subalgebra of a semisimple Lie algebrag and C_h U(h) be the Casimir element determined by the restrictionof the Killing form on g to h. The paper studies eigenvaluesof C_h on the isotropy representation mg/h. Some general estimatesconnecting the eigenvalues and the Dynkin indices of m are given.If h is a symmetric subalgebra, it is shown that describingthe maximal eigenvalue of C_h on exterior powers of m is connectedwith possible dimensions of commutative Lie subalgebras in m,thereby extending a result of Kostant. In this situation, aformula is also given for the maximal eigenvalue of C_h on m.More generally, a similar picture arises if h = g, where isan automorphism of finite order m and m is replaced by the eigenspaceof corresponding to a primitive mth root of unity.     

A new a posteriori error estimate for convection–reaction–diffusion problems

A new a posteriori error estimate is derived for the stationary convection–reaction–diffusion equation. In order to estimate the approximation error in the usual energy norm, the underlying bilinear form is decomposed into a computable integral and two other terms which can be estimated from above using elementary tools of functional analysis. Two auxiliary parameter-functions are introduced to construct such a splitting and tune the resulting bound. If these functions are chosen in an optimal way, the exact energy norm of the error is recovered, which proves that the estimate is sharp. The presented methodology is completely independent of the numerical technique used to compute the approximate solution. In particular, it is applicable to approximations which fail to satisfy the Galerkin orthogonality, e.g. due to an inconsistent stabilization, flux limiting, low-order quadrature rules, round-off and iteration errors, etc. Moreover, the only constant that appears in the proposed error estimate is global and stems from the Friedrichs–Poincaré inequality. Numerical experiments illustrate the potential of the proposed error estimation technique.     

Nilpotent fusion categories

In this paper we extend categorically the notion of a finite nilpotent group to fusion categories. To this end, we first analyze the trivial component of the universal grading of a fusion category C, and then introduce the upper central series of C. For fusion categories with commutative Grothendieck rings (e.g., braided fusion categories) we also introduce the lower central series. We study arithmetic and structural properties of nilpotent fusion categories, and apply our theory to modular categories and to semisimple Hopf algebras. In particular, we show that in the modular case the two central series are centralizers of each other in the sense of M. Müger.     

Comparison of direct sampling and emulsion analysis using a filter furnace for the determination of lead in crude oil by graphite furnace atomic absorption spectrometry

The determination of trace elements in crude oil is difficult due to the complex nature of the sample and the various different chemical forms in which the metals can occur. The advantage of graphite furnace atomic absorption spectrometry is that only a minimum of sample pretreatment is required. In this work two techniques have been compared to establish a fast and reliable method for lead determination in crude oil. In the first one the crude oil samples were weighed directly onto solid sampling (SS) platforms and introduced into the graphite tube for analysis. In the second one the samples were prepared as oil-in-water emulsions and analyzed in a filter furnace (FF). Twenty μL of a mixture of 0.5 mg L^− 1 Pd + 0.3 mg L^− 1 Mg + Triton X-100 has been used as the modifier, and calibration against aqueous solutions has been used for both methods. The sensitivity obtained with the FF was more than a factor of two better than that with SS; however, as a larger sample mass could be introduced in the latter case, so that the limits of detection for both techniques were 0.004 mg kg^− 1. Seven crude oil samples were analyzed using the two procedures, and all results were in agreement at a 95% confidence level using a paired Student's t-test. For validation purposes, three crude oil samples have been mineralized using an open-vessel acid digestion, and the results were in agreement with those found with direct sampling and with emulsion sampling using FF according to ANOVA test. Both methods are simple, fast and reliable, being appropriated for routine analysis; however, the direct method using SS technology should be preferred because of its simplicity, speed and commercial availability.     

Reactivity of 3-formyl- and 3-cyanothiochromones toward some N- and C-nucleophiles. Novel synthesis of 3-substituted 2-aminothiochromones

The reactions of 3-formylthiochromone with o-phenylenediamines, o-aminobenzenethiol, and indoles proceeded at the aldehyde group to give 3-(benzimidazol-2-yl)thiochromone, 3-(benzothiazol-2-yl)thiochromone, and 3-di(indol-3-yl)methylthiochromones, respectively. 3-Formyl- and 3-cyanothiochromones react with primary aromatic amines and phenylhydrazine to give the corresponding anils and phenylhydrazones of 3-formyl- and 2-amino-3-formylthiochromones. The reaction of 3-cyanochromones with o-phenylenediamines gave 2-amino-3-[(2-aminophenyl)iminomethyl]-4H-chromen-4-ones.     

Experimental measurement of the van der Waals binding energy of X-O2 clusters (X=Xe, CH3I, C3H6, C6H12)

Van der Waals binding energies for the X-O(2) complexes (X=Xe, CH(3)I, C(3)H(6), C(6)H(12)) are determined by analysis of experimental velocity map imaging data for O((3)P(2)) atoms arising from UV-photodissociation of the complex [A. V. Baklanov et al., J. Chem. Phys. 126, 124316 (2007)]. Several dissociation pathways have been observed, we focus on the channel corresponding to prompt dissociation of X-O(2) into X+2O((3)P) fragments, which is present for complexes of O(2) with all partners X. Our method is based on analysis of the kinetic energy of all three photofragments, where the O atom kinetic energy was directly measured in the experiment and the kinetic energy of the X partner was calculated using momentum conservation, along with the measured angular anisotropy for O atom recoil. We exploit the fact that the clusters are all T-shaped or nearly T-shaped, which we also confirm by ab initio calculations, along with knowledge of the transition dipole governing radiative absorption by the complex. The effect of partitioning the kinetic energy between translation along the X-O(2) and O-O coordinates on the angular anisotropy of the O atom recoil direction is discussed. Van der Waals binding energies of 110±20 cm(-1), 280±20 cm(-1), 135±30 cm(-1), and 585±20 cm(-1) are determined for Xe-O(2), CH(3)I-O(2), C(3)H(6)-O(2), and C(6)H(12)-O(2) clusters, respectively.     

A spectroscopic and computational study of Al(III) complexes in sodium cryolite melts: Ionic composition in a wide range of cryolite ratios

The structure of sodium cryolite melts was studied using Raman spectroscopy and quantum chemical calculations performed at the density functional theory level. The existence of bridged forms in the melts was argued first from the analysis of experimental Raman spectra. In the quantum chemical modelling emphasis was put on the construction of potential energy surfaces describing the formation/dissociation of certain complex species. Effects of the ionic environment were found to play a crucial role in the energetics of model processes. The structure of the simplest possible polymeric forms involving two Al centres linked through F atoms (“dimers”) was thoroughly investigated. The calculated equilibrium constants and model Raman spectra yield additional evidence in favour of the dimers. This agrees with a self-consistent analysis of a series of Raman spectra for a wide range of the melt composition.     

Observation of size-dependent thermalization in CdSe nanocrystals using time-resolved photoluminescence spectroscopy

We report heat dissipation times in semiconductor nanocrystals of CdSe. Specifically, a previously unresolved, subnanosecond decay component in the low-temperature photoluminescence decay dynamics exhibits longer decay lifetimes (tens to hundreds of picoseconds) for larger nanocrystals as well as a size-independent, ~25-meV spectral shift. We attribute the fast relaxation to transient phonon-mediated relaxation arising from nonequilibrium acoustic phonons. Following acoustic phonon dissipation, the dark exciton state recombines more slowly via LO-phonon assistance resulting in the observed spectral shift. The measured relaxation time scales agree with classical calculations of thermal diffusion, indicating that interfacial thermal conductivity does not limit thermal transport in these semiconductor nanocrystal dispersions.     

Quantum anonymous voting with anonymity check

We propose a new protocol for quantum anonymous voting having serious advantages over the existing protocols: it protects both the voters from a curious tallyman and all the participants from a dishonest voter in unconditional way. The central idea of the protocol is that the ballots are given back to the voters after the voting process, which gives a possibility for two voters to check the anonymity of the vote counting process by preparing a special entangled state of two ballots. Any attempt of cheating from the side of the tallyman results in destroying the entanglement, which can be detected by the voters.