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.     

Limited stability of the pH reference material sodium tetraborate decahydrate (“borax”)

Thermogravimetric and X-ray powder diffraction analyses revealed that the pH reference material sodium tetraborate decahydrate (“borax”) is not a stable substance but, loosing some of its water of crystallization, transforms slowly into the pentahydrate. The connected pH changes of the solutions are opposite to and larger than, those expected according to the dilution value of the decahydrate as shown by differential potentiometric measurements. The transformation rate depends on the storing time in the closed original container, the time of and after the first exposure of the material to the atmosphere, the frequency of and the relative humidity during subsequent exposures, and the temperature. The transformation reaction is not understood in detail. However, the experiments showed that sodium tetraborate decahydrate is a reliable pH reference material, whose pH is constant within ΔpH=±0.005 within at least two years after preparation, even if occasionally exposed to the atmosphere. Further work will show whether this time limit can be somewhat extended.     

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.     

Fluorescence quenching by excimer formation: quenching constant approximations for excimer formation-dissociation by classical potential models

Fluorescence quenching by excimer formation is studied on the assumption that the excimer formation and dissociation can be modeled as overdamped motion in an attractive potential (classical potential models). An approach to the zeroth-order, concentration-independent quenching constants is proposed which starts from a mean reaction-time ansatz and reduces the calculation essentially to the solution of the eigenvalue problem for the Smoluchowski operator which describes the excimer equilibration. For a square-well potential model it is shown that a quenching constant expansion in terms of relaxation modes, truncated at the kinetic level, gives a satisfactory approximation of the recently obtained exact zeroth-order result under defined conditions. It is demonstrated how this two-mode approach can be applied for a quenching constant estimation if the excimer formation and dissociation are modeled by more realistic interaction potentials, as for instance, Morse- or Gaussian-type ones.     

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.