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.     

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.     

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.     

N(nPr)4[B5O6(OH)4][B(OH)3]2 and N(nBu)4[B5O6(OH)4][B(OH)3]2: Clathrates with a diamondoid arrangement of hydrogen-bonded pentaborate anions

Single-crystal X-ray structure analyses of N(nPr)₄[B₅O₆(OH)₄][B(OH)₃]₂,1, and N(nBu)₄ [B₅O₆(OH)₄][B(OH)₃]₂,2, reveal that these materials are novel clathrates, the isotypic host structures of which are three-dimensional assemblies of hydrogen-bonded [B₅O₆(OH)₄]^– ionsand B(OH)₃ molecules. The assembly of only the pentaborate anions is a distorted (i.e., along [102] elongated) fourconnected diamond-related network. The N(nPr) ₄ ⁺ and N(nBu) ₄ ⁺ ions are trapped within the complex three-dimensional channel systems of the host frameworks. Both1 and2 crystallize monoclinically with space groupP2₁/c andZ=4. The cell constants are:1:a=13.592(5),b=12.082(2),c=17.355(6) Å, =106.60(2)° (298K);2:a=13.874(3),b=12.585(1),c=17.588(4) Å, =107.04(1)° (238 K). The results obtained by both¹¹B and¹³C MAS NMR spectroscopy are discussed. Thermogravimetric studies under a flowing inert-gas atmosphere suggest that water, stemming from polycondensation of the hydrous borate species, is released from the clathrates at ca. 443 K (1) and 398 K (2) before the decomposition of the organic cations starts at ca. 603 K (1) and 603 K (2).Author for correspondence. Supplementary Data relating to this article are deposited with the British Library as supplementary publication No. SUP 82172 (82 pages).     

Tantalum Clusters in Oxidic Matrices — Synthesis,Crystal Structure and Magnetic Properties of Eu1.83Ta15O32

The mixed‐valent oxotantalate Eu_1.83Ta₁₅O₃₂ was prepared from a compressed mixture of Ta₂O₅ and the metals in a sealed Ta ampoule at 1400 °C. The crystal structure was determined by means of single crystal X‐ray diffraction: space group R3¯, a = 777.2(6) pm and c = 3523.5(3) pm, Z = 3, 984 symmetrically independent reflections, 83 variables, R_F = 0.027 for I > 2σ (I). The structure is isotypic to Ba₂Nb₁₅O₃₂. The salient feature is a [Ta(+8/3)₆O₁₂ⁱO₆^a] cluster consisting of an octahedral Ta₆ core bonded to 12 edge‐bridging inner and six outer oxygen atoms. The clusters are arranged to slabs which are sandwiched by layers of [Ta(+5)₃O₁₃] triple octahedra. Additional Ta(+5) and Eu(+2) atoms provide the cohesion of these structural units. Twelve‐fold coordinated Eu(+2) atoms are situated on a triply degenerate position 33 pm displaced from the threefold axis of symmetry. A depletion of the Eu(+2) site from 6 to 5.5 atoms per unit cell reduces the number of electrons available for Ta‐Ta bonding from 15 to 14.67 electrons per cluster. Between 125 and 320 K Eu_1.83Ta₁₅O₃₂ is semi‐conducting with a band gap of 0.23 eV. The course of the magnetization is consistently described with the Brillouin function in terms of a M_mol/(N_Aμ_B) versus B/T plot in the temperature range 5 K — 320 K and at magnetic flux densities 0.1 T — 5 T. At moderate flux densities (< 1 T) the magnetic moment agrees fairly well with the expected value of 7.94 μ_B for free Eu (2+) ions with 4f⁷ configuration in ⁸S_7/2 ground state. Below 5 K, anisotropic magnetization measurements at flux densities B < 1 T point to an onset of an antiferromagnetic ordering of Eu spins within the layers and an incipient ferromagnetic ordering perpendicular to the layers.     

The Elucidation of Peroxyl Radical Reactions in Aqueous Solution with the Help of Radiation-Chemical Methods

Whenever free radicals are formed, independent of whether this occurs thermally, is induced by UV or ionizing irradiation, or takes place in redox reactions, they are converted rapidly into the corresponding peroxyl radicals in the presence of oxygen. Peroxyl radical reactions in aqueous environments are observed not only in aquatic systems (e.g., rivers, lakes and oceans) but also in the living cell and to a considerable degree even in the atmosphere (in water droplets). The peroxyl radical chemistry occurring in this medium is often very different from that observed in the gas phase or in organic solvents. In spite of the great importance of these reactions in medicine (for example in anti-cancer irradiation therapy and ischaemia) there have been comparatively few studies of peroxyl reactions in aqueous media. Radiation-chemical techniques such as pulse radiolysis offer the best means for carrying out such studies, so that it is not surprising that the majority of the information available in this area has been obtained with the help of radiation-chemical methods. The radiation chemistry of water can be con trolled in such a manner that the main products are ^˙OH radicals (90 % yield), which react with substrate molecules to give substrate radicals and in the presence of oxygen to give substrate peroxyl radicals. The experimental conditions can also be varied in such a way that HO/O radicals can be formed in 100 % yield and caused to react with substrates. We therefore have a simple access to these intermediates, which are extremely important in biological systems. A detailed product analysis, supported by kinetic studies carried out with the help of pulse radiolysis, has been used to clarify the chemistry of a series of peroxyl radicals, so that sufficient material is now available to justify a review of the variety of the peroxyl radical reactions studied by means of radiation-chemical methods. A more general survey of the physical properties of the peroxyl radicals and their unimolecular and bimolecular reactions will be followed by a discussion of selected examples of various classes of substance. Because of the great biological importance of radical-induced DNA damage this area will also be treated briefly.     

Electron transfer initiated Diels-Alder reaction with allenes as dienophiles

The first examples of using aryl allenes as dienophiles in the electron transfer induced Diels-Alder reaction are described. According to mechanistic tests the reaction proceeds via a cycloaddition of the diene^+· to the allene.     

Raman imaging of PLGA microsphere degradation inside macrophages

Understanding the degradation behavior of polymeric microspheres is crucial for the successful application of such devices in controlled drug delivery. The degradation mechanism of poly(lactic-co-glycolic acid) (PLGA) microspheres inside phagocytic cells is not known, but different models for degradation in aqueous solution have been proposed. We have used confocal Raman spectroscopy and imaging to study the intracellular degradation of PLGA microspheres inside individual macrophages. Our results show that ingested microspheres degrade in a heterogeneous manner, with a more rapid degradation in the center. Comparison of Raman spectra from degrading beads with those of uningested beads reveals that ester hydrolysis occurs throughout the phagocytosed microspheres, with a selective loss of glycolic acid units. Furthermore, we show that PLGA degradation is a cell-mediated process, possibly caused by the low pH of the phagosome and/or the presence of hydrolytic enzymes. In conclusion, we have demonstrated that the chemical composition of degrading polymers inside cells can be probed by Raman spectral imaging. This technique will expand the capabilities of investigating biomaterial degradation in vivo.     

Rational choice based on vague preferences

This paper studies vague preferences. In contrast to the concept of fuzzy preferences, the more general notion of vague preferences does not entail the assumption that the different degrees of preference are completely ordered. It is shown that the more general concept of a vague preference order arises very naturally in the context of aggregating a set of (exact) weak orders. Furthermore, necessary and sufficient conditions for the rationalizability of an exact choice function by means of a vague preference order are given. It turns out that in the general case these conditions are weaker than in the case where the degrees of preference are completely ordered. Consequently, there exist choice functions which are rationalizable by a general vague preference order but which are not rationalizable by a fuzzy preference order, or more generally by any vague preference order which induces a complete ranking of the degrees of preference. Examples of such choice functions are provided.This is a revised version of a discussion paper which was written at the University of Karlsruhe. I am grateful to Georg Bol and an anonymous referee for valuable comments. The research has been made possible by a fellowship of the DFG (Deutsche Forschungsgemeinschaft) which is also gratefully acknowledged.