Generalized Hermite Polynomials and the Heat Equation for Dunkl Operators

Based on the theory of Dunkl operators, this paper presents a general concept of multivariable Hermite polynomials and Hermite functions which are associated with finite reflection groups on ℝ ^N . The definition and properties of these generalized Hermite systems extend naturally those of their classical counterparts; partial derivatives and the usual exponential kernel are here replaced by Dunkl operators and the generalized exponential kernel K of the Dunkl transform. In the case of the symmetric group S _N , our setting includes the polynomial eigenfunctions of certain Calogero-Sutherland type operators. The second part of this paper is devoted to the heat equation associated with Dunkl's Laplacian. As in the classical case, the corresponding Cauchy problem is governed by a positive one-parameter semigroup; this is assured by a maximum principle for the generalized Laplacian. The explicit solution to the Cauchy problem involves again the kernel K, which is, on the way, proven to be nonnegative for real arguments. Received: 10 March 1997 / Accepted: 7 July 1997     

The effect of the position of methyl substituents on photophysical and photochemical properties of [Ru(x,x'-dmb)(CN)4]2- complexes: experimental confirmation of the theoretical predictions

The molecular geometry, electronic structure and electronic spectra and the energy levels of the molecular orbitals responsible for the photophysical characteristics of a series of solvent tunable [Ru(x,x'-dmb)(CN)(4)](2-) complexes (where x,x'-dmb = x,x'-dimethyl-2,2'-bipyridine) were calculated by density-functional theory-based quantum chemical methods, with the purpose of proposing for experimental study the best candidate for sensitizing electron- and energy transfer processes or for light induced structural changes in the molecule. The methods applied include geometry optimization using the B3LYP functional combination and various basis sets, time-dependent density functional theory with the B3LYP and PBE0 functionals, with and without explicit inclusion of coordinated solvent H(2)O molecules and the conductor-like polarizable continuum model for solvation. The accuracy of the theoretical predictions was tested by experiments: the model compounds have been synthesized and characterized by various spectroscopic methods, such as (1)H-NMR, UV-Vis absorption and emission spectroscopy and by cyclic voltammetry. Excellent correlation was found between the theoretically calculated and the experimentally determined photophysical and photochemical characteristics. The electronic transition energies measured in water are superbly reproduced by TD-PBE0 and well by TD-B3LYP, but the performance of both functionals is worse if the solvent is acetonitrile.     

Synthesis, photochemistry, and photophysics of butadiene derivatives: influence of the methyl group on the molecular structure and photoinduced behavior

Novel butadiene derivatives display diverse photochemistry and photophysics. Excitation of 2-methyl-1-(o-vinylphenyl)-4-phenylbutadiene leads to the dihydronaphthalene derivative, whereas photolysis of the corresponding model o-methyl analogue results in the formation of the naphthalene-like derivative, deviating from the nonmethylated analogue of the prior starting compound and producing benzobi- and -tricyclic compounds. The effect of the methyl substituents is even more dramatic in the case of the dibutadienes. The parent unsubstituted compound undergoes photoinduced intramolecular cycloaddition giving benzobicyclo[3.2.1]octadiene, whereas the photochemical reaction of the corresponding dimethylated derivative shows only geometrical isomerization due to the steric effect of the substituents. Methyl groups on the butadiene backbones reduce the extent of conjugation, causing a blue-shift of the characteristic absorption band. The fluorescence efficiency is dramatically decreased, as a consequence of nonplanarity and reduced rigidity of the molecules due to the crowding by the methyl and phenyl groups together. Four molecules of very similar structures show dramatically different photoinduced behavior, revealing how changes of the nature and position of the substituents are valuable in understanding the photophysics and photochemistry of these types of compounds.     

Time-resolved study of back side ablated molybdenum thin films by ultrashort laser pulses

Ultrashort laser pulses are used to ablate a thin molybdenum layer from glass by irradiating the metal film through the transparent substrate. The trajectories of ablated molybdenum fragments are recorded using a shadowgraphic setup with a time resolution in the nanosecond range. In addition, the shape of collected molybdenum fragments is examined as a function of applied fluence. It is confirmed that in a fluence regime close to the ablation threshold one single disc is ablated as a whole and its velocity is determined in the order of 50 ms^?1. In a second fluence regime, partial melting at the center of the disc is found and small melt droplets are recorded on their flight. Mo fragments ablated in this regime feature a ring-like structure with a brittle fracture at the outer and a molten appearance at the inner edge.     

Investigation of surface topology of printed nanoparticle layers using wide‐angle low‐Q scattering

A new small‐angle scattering technique in reflection geometry is described which enables a topological study of rough surfaces. This is achieved by using long‐wavelength soft X‐rays which are scattered at wide angles but in the low‐Q range normally associated with small‐angle scattering. The use of nanometre‐wavelength radiation restricts the penetration to a thin surface layer which follows the topology of the surface, while moving the scattered beam to wider angles preventing shadowing by the surface features. The technique is, however, only applicable to rough surfaces for which there is no specular reflection, so that only the scattered beam was detected by the detector. As an example, a study of the surfaces of rough layers of silicon produced by the deposition of nanoparticles by blade‐coating is presented. The surfaces of the blade‐coated layers have rough features of the order of several micrometers. Using 2 nm and 13 nm X‐rays scattered at angular ranges of 5°≤θ≤ 51° and 5°≤θ≤ 45°, respectively, a combined range of scattering vector of 0.00842 Å^?1≤Q≤ 0.4883 Å^?1 was obtained. Comparison with previous transmission SAXS and USAXS studies of the same materials indicates that the new method does probe the surface topology rather than the internal microstructure.     

Biomimetic sensor chip monitoring real-time food degradation: correlating chemical deterioration with microbiological status

Abstract  Conventional tests currently in use for the detection and identification of food-borne pathogens and also of microbial food deterioration are time-consuming as they are based on conventional culturing techniques or monitoring environmental conditions like the increase of temperature or change in pH. These parameters do not reflect the real quality of the meat to be tested. The aim of our approach was to create a simple and cheap sensor providing reasonable sensitivity and selectivity to indicate the bacterial infection in real-time monitoring combined with a memory effect that cannot easily be corrupted. Thus, an optical thin film sensor chip was developed able to detect bacterial decay of food through a specific colour change. The design of the sensor relates to the phenomenon of “anomalous absorption”, which can best be described as a thin film enhanced absorption. A metal nanoparticle layer positioned at a well-defined distance to a smooth metal surface shows that the minimum of spectral reflectivity strongly depends on the thickness of the distance layer. This setup represents a special kind of reflection interference filter. In such a sensor setup a biomimetic polymer is integrated, which is degradable by lytic enzymes excreted by microorganisms in food decay. Meat deterioration under controlled conditions is correlated to the amount of enzymes secreted by microorganisms and the bacterial count. Thus, after incubation of the sensor setup with standard meat preparations, the enzymes released from decaying cell material change the thickness of the polymer layer and generate an easily visible colour change. This setup would be useful for integration into meat packaging. Graphical Abstract   M. Barth, U. Bohrn and M. Ibrišimović contributed equally to this work.     

Distinction of tris(diimine)ruthenium(II) enantiomers chiral by virtue of helical chirality: temperature-dependent deuterium NMR spectroscopy in partially oriented phases

2H NMR spectra of perdeuterated tris(diimine)ruthenium(II) complexes have been recorded in lyotropic liquid crystalline phase formed by the chiral polypeptide, poly-gamma-benzyl-L-glutamate (PBLG) and co-solvents. It is demonstrated that the left- and right-rotation isomers of these octahedral metal complexes with D3 symmetry can be distinguished. The effects of temperature and ligand size on spectral resolution were also studied. Although in the case of free bipyridine ligands excellent optical resolution could be obtained at room temperature in the complexes studied, the increase in ligand size has a detrimental effect on the resolution. This can be compensated to some extent by elevating the sample temperature and decreasing the deuterium relaxation rates, but the available temperature range and therefore the applicability of the technique are limited.     

An uncertainty principle for Hankel transforms

There exists a generalized Hankel transform of order on , which is based on the eigenfunctions of the Dunkl operator

For this transform coincides with the usual Fourier transform on . In this paper the operator replaces the usual first derivative in order to obtain a sharp uncertainty principle for generalized Hankel transforms on . It generalizes the classical Weyl-Heisenberg uncertainty principle for the position and momentum operators on ; moreover, it implies a Weyl-Heisenberg inequality for the classical Hankel transform of arbitrary order on

     

Staudinger reaction at in-bridgehead positions of phosphorus macrobicyclic compounds

Reaction of in,in-phosphite 1 with thiophosphoryl azide 2 affords in,in-dithiophosphate 3, in,in-thiophosphate-imidophosphate 4, and in,in-phosphite-imidophosphate 5. Compounds 4 and 5 are the first examples of the modification of in-bridgehead positions in macrobicyclic compounds with groups larger than methyl. The benzaldehyde arms of the in-substituent in 4 and 5 jut out of the cage bars. In 4 they are trapped between the macrocyclic arms to give the NMR spectra of a Cs-symmetric solution-state structure. In contrast, in 5 the benzaldehyde arms can move between the gaps of the cage. This results in 1H and 13C NMR spectra which are consistent for a compound with C3v symmetry. In,out-diimidophosphate 7 is obtained in moderate yield by reaction of in,out-phosphite 6 with thiophosphoryl azide 2. Its in-benzaldehyde moieties are not fixed between the cage arms, but can freely move from one gap to the next as is indicated by NMR measurements.