Formal proofs of operator identities by a single formal computation

A formal computation proving a new operator identity from known ones is, in principle, restricted by domains and codomains of linear operators involved, since not any two operators can be added or composed. Algebraically, identities can be modelled by noncommutative polynomials and such a formal computation proves that the polynomial corresponding to the new identity lies in the ideal generated by the polynomials corresponding to the known identities. In order to prove an operator identity, however, just proving membership of the polynomial in the ideal is not enough, since the ring of noncommutative polynomials ignores domains and codomains. We show that it suffices to additionally verify compatibility of this polynomial and of the generators of the ideal with the labelled quiver that encodes which polynomials can be realized as linear operators. Then, for every consistent representation of such a quiver in a linear category, there exists a computation in the category that proves the corresponding instance of the identity. Moreover, by assigning the same label to several edges of the quiver, the algebraic framework developed allows to model different versions of an operator by the same indeterminate in the noncommutative polynomials.     

Ein neues nichtmetallisches Element

Ohne Zusammenfassung     

Minimal expansions in redundant number systems: Fibonacci bases and Greedy algorithms

We study digit expansions with arbitrary integer digits in base q (q integer) and the Fibonacci base such that the sum of the absolute values of the digits is minimal. For the Fibonacci case, we describe a unique minimal expansion and give a greedy algorithm to compute it. Additionally, transducers to calculate minimal expansions from other expansions are given. For the case of even integer bases q, similar results are given which complement those given in [6].     

Discrepancy operators and numerical integration on compact groups

In this paper we establish an inequality of Koksma-Hlawka-type for compact groups. We first define a discrepancy for compact groups based on discrepancy operators introduced by W. Fleischer and show the relation to the classicalL ²-discrepancy. Then we prove the inequality for functions in a weightedL ²-space.     

Low-temperature hydride furnace modified for the atomic absorption spectrometric determination of metals with low appearance temperatures

The sample is vaporized from tungsten filament coils (150 W) and transported by an argon stream to the cell of a modified hydride furnace for atomic absorption spectrometry (a.a.s.). The system provides almost the same sensitivity for elements with low appearance temperatures (e.g., Bi, Cd, Pb, Tl, Zn) as graphite-furnace a.a.s. The detection limits are between 0.1 and 5 ng ml^?1, depending on the element.     

Product formulas and convolution structure for Fourier-Bessel series

One of the most far-reaching qualities of an orthogonal system is the presence of an explicit product formula. It can be utilized to establish a convolution structure and hence is essential for the harmonic analysis of the corresponding orthogonal expansion. As yet a convolution structure for Fourier-Bessel series is unknown, maybe in view of the unpractical nature of the corresponding expanding functions called Fourier-Bessel functions. It is shown in this paper that for the half-integral values of the parameter ,n=0, 1, 2,, the Fourier-Bessel functions possess a product formula, the kernel of which splits up into two different parts. While the first part is still the well-known kernel of Sonine's product formula of Bessel functions, the second part is new and reflects the boundary constraints of the Fourier-Bessel differential equation. It is given, essentially, as a finite sum over triple products of Bessel polynomials. The representation is explicit up to coefficients which are calculated here for the first two nontrivial cases and . As a consequence, a positive convolution structure is established for . The method of proof is based on solving a hyperbolic initial boundary value problem.Communicated by Tom H. Koornwinder.     

Lavandulifolioside: A New Phenylpropanoid Glycoside from Stachys lavandulifolia

A new phenlypropanoid glycoside has been isolated from the methanolic extract of the aerial parts of Stachys lavandulifolia (Lamiaceae), lavandulifolioside (1) . On the basis of chemical and spectral data the structure of the new compound 1 has been elucidated as β-(3,4-dihydroxyphenyl)ethyl O-α-L -arabinopyranosyl-(1→2)-α-L -rhamnopyranosyl-(1→3)-4-O-caffeoyl-β-D -glucopyranoside.     

What to Learn from a Comparative Genomic Sequence Analysis of <Emphasis Type="Italic">L</Emphasis>-Carnitine Dehydrogenase

Summary. In contrast to eukaryotic cells certain eubacterial strains have acquired the ability to utilize L-carnitine (R-(–)-3-hydroxy-4-(trimethylamino)butyrate) as sole source of energy, carbon and nitrogen. The first step of the L-carnitine degradation to glycine betaine is catalysed by L-carnitine dehydrogenase (L-CDH, EC 1.1.1.108) and results in the formation of the dehydrocarnitine. During the oxidation of L-carnitine a simultaneous conversion of the cofactor NAD⁺ to NADH takes place. This catabolic reaction has always been of keen interest, because it can be exploited for spectroscopic L-carnitine determination in biological fluids – a quantification method, which is developed in our lab – as well as L-carnitine production.Based on a cloned L-CDH sequence an expedition through the currently available prokaryotic genomic sequence space began to mine relevant information about bacterial L-carnitine metabolism hidden in the enormous amount of data stored in public sequence databases. Thus by means of homology-based and context-based protein function prediction is revealed that L-CDH exists in certain eubacterial genomes either as a protein of approximately 35 kDa or as a homologous fusion protein of approximately 54 kDa with an additional putative domain, which is predicted to possess a thioesterase activity. These two variants of the enzyme are found on one hand in the genome sequence of bacterial species, which were previously reported to decompose L-carnitine, and on the other hand in gram-positive bacteria, which were not known to express L-CDH. Furthermore we could not only discover that L-CDH is located in a conserved genetic entity, which genes are very likely involved in this L-carnitine catabolic pathway, but also pinpoint the exact genomic sequence position of several other enzymes, which play an essential role in the bacterial metabolism of L-carnitine precursors.     

Solution- and bound-state conformational study of N,N',N"-triacetyl chitotriose and other analogous potential inhibitors of hevamine: application of trNOESY and STD NMR spectroscopy

The solution-state conformations of N,N',N"-triacetyl chitotriose (1) and other potential chitinase inhibitors 2-4 were studied using a combination of NMR spectroscopy (NOESY) and molecular mechanics calculations. Determination solely of the global energy minimum conformation was found to be insufficient for an agreement with the NMR results. An appropriate consistency between the NMR experimental data and theoretical calculations was only reached by assessing the structures as population-weighted average conformers based on Boltzmann distributions derived from the calculated relative energies. Analogies, but also particular differences, between the synthetic compounds 2-4 and the naturally-occurring N,N',N"-triacetyl chitotriose were found. Furthermore, the conformation of compounds 1 and 2 when bound to hevamine was also studied using transferred NOESY experiments and the binding process was found to impart a level of conformational restriction on the ligands. The preferred conformation as determined for 1 in the bound state to hevamine belonged to one of the conformational families found for the compound when free in solution, although full characterisation of the bound-state conformations was impeded due to severe signal overlap. Saturation transfer difference NMR experiments were also employed to analyse the binding epitopes of the bound compounds. We thus determined that it is mainly the acetyl amido groups of the trisaccharide and the heterocyclic moiety which are in close contact with hevamine.