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     

Influence of relative configuration of disubstituted cyclopentanes and -hexanes on 13C shifts

(13)C shifts of disubstituted cyclopentane and cyclohexane derivatives were compared in dependence on the relative configuration of the two substituents. A diequatorial substitution correlates with deshielding compared to other substitution patterns. Some novel fluorinated cyclopentanes and -hexanes including their DFT calculation-assisted structure elucidation are described.     

Metal-metal bond or isolated metal centers? Interaction of Hg(CN)2 with square planar transition metal cyanides

Three adducts have been prepared from Hg(CN)(2) and square planar M(II)(CN)(4)(2)(-) transition metal cyanides (M = Pt, Pd, or Ni, with d(8) electron shell) as solids. The structure of the compounds K(2)PtHg(CN)(6).2H(2)O (1), Na(2)PdHg(CN)(6).2H(2)O (2), and K(2)NiHg(CN)(6).2H(2)O (3) have been studied by single-crystal X-ray diffraction, XPS, Raman spectroscopy, and luminescence spectroscopy in the solid state. The structure of K(2)PtHg(CN)(6).2H(2)O consists of one-dimensional wires. No CN(-) bridges occur between the heterometallic centers. The wires are strictly linear, and the Pt(II) and Hg(II) centers alternate. The distance d(Hg)(-)(Pt) is relatively short, 3.460 A. Time-resolved luminescence spectra indicate that Hg(CN)(2) units incorporated into the structure act as electron traps and shorten the lifetime of both the short-lived and longer-lived exited states in 1 compared to K(2)[Pt(CN)(4)].2H(2)O. The structures of Na(2)PdHg(CN)(6).2H(2)O and K(2)NiHg(CN)(6).2H(2)O can be considered as double salts; the lack of heterometallophilic interaction between the remote Hg(II) and Pd(II) atoms, d(Hg)(-)(Pd) = 4.92 A, and Hg(II) and Ni(II) atoms, d(Hg)(-)(Ni) = 4.61 A, is apparent. Electron binding energy values of the metallic centers measured by XPS show that there is no electron transfer between the metal ions in the three adducts. In solution, experimental findings clearly indicate the lack of metal-metal bond formation in all studied Hg(II)-CN(-)-M(II)(CN)(4)(2)(-) systems (M = Pt, Pd, or Ni).     

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.     

Preparation and Spectral Characterization of Sulphur-Containing Diarylazo Compounds

Abstract

By reaction of arylazo-substituted aryl halides with mercaptides, new arylazo-substituted arylsulfides are formed. These compounds are deeply coloured and exhibit an intense absorption band in the visible region, which can be reproduced quantitatively and interpreted satisfactorily by means of quantum chemical calculations in the framework of the PPP and LHM methods, respectively.     

Guest-induced conformation shift of p-sulphonatothiacalix[4]arene in the solid state and solution manipulated by [Zn(dipy)3]2+

The ¹H NMR and X-ray analysis data reveal the guest-induced cone → partial cone and cone → 1,2-alternate conformational shift in aqueous methanol solutions and solid state for a p-sulphonatothiacalix[4]arene-[Zn(dipy)₃]²⁺ host–guest system. The experimental data, together with DFT calculations, show that the guest-induced conformational shift of p-sulphonatothiacalix[4]arene is facilitated by the deprotonation of phenolic groups.     

Rigid rod polymers with regularly spaced silicon-centered phthalocyanine units in the backbone

We describe the synthesis of new rigid rod polymers possessing a backbone of silicon phthalo-cyanine units and stiff 4,4′-dioxybiphenyl or 2,6-dioxynaphthalene spacers. These modified phthalocyaninato-polysiloxane derivatives are characterised by mass, absorption and ¹³C-CPMAS spectroscopies, differential scanning calorimetry and X-ray powder diffractometry. UV/VIS spectra show only slight differences to the corresponding phthalocyanine monomers, which indicates the absence of excited state interactions between the chromophores within the polymer backbone.     

Crystalline Bacterial Cell Surface Layers (S Layers): From Supramolecular Cell Structure to Biomimetics and Nanotechnology

An astonishingly broad application potential in biotechnology, biomimetics, and nanotechnology is revealed by studies on the structure, chemistry, biosynthesis, genetics, self-assembly, and function of supramolecular surface layers (S layers). These are monomolecular, crystalline assemblies of protein or glycoprotein subunits and represent one of the most commonly observed surface structures of prokaryotic cell envelopes (see schematic representation of an archaebacterial cell envelope).     

On list critical graphs

In this paper we discuss some basic properties of k-list critical graphs. A graph G is k-list critical if there exists a list assignment L for G with |L(v)|=k−1 for all vertices v of G such that every proper subgraph of G is L-colorable, but G itself is not L-colorable. This generalizes the usual definition of a k-chromatic critical graph, where L(v)={1,…,k−1} for all vertices v of G. While the investigation of k-critical graphs is a well established part of coloring theory, not much is known about k-list critical graphs. Several unexpected phenomena occur, for instance a k-list critical graph may contain another one as a proper induced subgraph, with the same value of k. We also show that, for all 2≤p≤k, there is a minimal k-list critical graph with chromatic number p. Furthermore, we discuss the question, for which values of k and n is the complete graph K_nk-list critical. While this is the case for all 5≤k≤n, K_n is not 4-list critical if n is large.