On the spectrum of the sum of generators for a finitely generated group

Let Γ be a finitely generated group. In the group algebra ℂ[Γ], form the averageh of a finite setS of generators of Γ. Given a unitary representation π of Γ, we relate spectral properties of the operator π(h) to properties of Γ and π. For the universal representationπ _un of Γ, we prove in particular the following results. First, the spectrum Sp(π _un (h)) contains the complex numberz of modulus one iff Sp(π _un (h)) is invariant under multiplication byz, iff there exists a character such that η(S)={z}. Second, forS ⁻¹=S, the group Γ has Kazhdan’s property (T) if and only if 1 is isolated in Sp(π _un (h)); in this case, the distance between 1 and other points of the spectrum gives a lower bound on the Kazhdan constants. Numerous examples illustrate the results.     

Interplanar torsion in the S1<--S0 electronic spectrum of jet cooled 1-phenylimidazole

The S(1)<--S(0) transition of 1-phenylimidazole (1PI) has been studied in a supersonic jet expansion by resonant two-photon ionization. The origin band at 36 075 cm(-1) is accompanied by a low frequency progression associated with torsion about the bond connecting phenyl and imidazole groups. Torsional potentials have been determined for both states. In S(0), phi(min)=37.2+/-0.5 degrees and the planar barrier is 339+/-20 cm(-1), while in S(1), phi(min)=17.6+/-0.5 degrees and the planar barrier is 57+/-2 cm(-1). The transition moment alignment is observed to be consistent with an excited state of L(b) character, in spite of the "off-axis" conjugation provided by the imidazole ring. These results are compared with ab initio calculations on both states, performed using Hartree-Fock, M?ller-Plesset second-order perturbation, density functional theory with the Becke3-Lee-Yang-Parr functional, time-dependent density functional theory, configuration interaction singles, and complete active space self-consistent field methods. Solution-phase UV spectra of neutral and protonated 1PI are also reported.     

Closest packing in dense oxides: The structure of a polymorph of Co3(AsO4)2

Crystals of Co₃(AsO₄)₂ were grown from the melt of a mixture of Co₂As₂O₇ and As₂O₅. The crystals are isostructural with Mg₃(AsO₄)₂ and are tetragonal with a = 6.858(2), c = 18.872(5) Å, Z = 6, and space group $\text{I}\text{4}\text{2d}$. A total of 1048 independent reflections were measured by diffractometer and used in the full-matrix refinement to a final R value of 0.069. The structure contains two distinct AsO₄ groups. Two of the cobalt ions are octahedrally coordinated and a third occupies a $\text{4}$ site with four short and four long CoO distances. The crystal structure of Co₃ (AsO₄)₂ is not based on the continuous three-dimensional closest packing of oxygen atoms. Nevertheless the number of oxygen atoms per cubic centimeter is 5.4 × 10²², which falls in the range of values for hexagonal and cubic closest packed structures. A better measure of the degree to which closest packing is achieved by a structure is suggested. It is based on an analysis of the polyhedra of oxygen atoms which surround each of the oxygen atoms in a structure and their relation to the polyhedra in ideally closest packed structures. In order to facilitate the analysis, polytopes of 11- and 12-vertex polyhedra were studied. A new decahexahedral 11-vertex polyhedron was found.     

The photochemical reactions of α,β,γ,δ -unsaturated diazo compounds at low temperature

The α,β:γ,δ-unsaturated tosylhydrazone lithium salts (4) undergo photolysis at ?60°C to give vinylcyclopropenes (10) and [l,2]diazeto[l,4-a] pyrroles (11) and/or (12). The formation of (11, X=H) establishes the intermediacy of (8, X=H) in the formation of the 3$\text{H}$-1,2-diazepine (9). The diazoalkene (5, X=Me) showed opposite electrocyclisation periselectivity to its thermal cyclisation and gave (11, X=Me) $\text{via}$ (8, X=Me), rather than the pyrazoles (2) and (3).     

Determination of enterostatin in human cerebrospinal fluid by capillary electrophoresis with laser induced fluorescence detection

A capillary electrophoresis (CE) method with laser induced fluorescence (LIF) detection is described for quantification of enterostatin (Val-Pro-Asp-Pro-Arg), a pentapeptide involved in appetite regulation and insulin secretion. Enterostatin and two other pentapeptides belonging to the enterostatin family (i.e. Ala-Pro-Gly-Pro-Arg and Val-Pro-Gly-Pro-Arg) were well separated from each other. The peptides were fluorescently tagged with naphthalene-2,3- dicarboxaldehyde (NDA) and separated by micellar electrokinetic chromatography (MEKC) in the presence of methanol as an organic modifier. Coupled with LIF detection, the method had a detection limit of 4.8 x 10(-6) M for enterostatin. The relative standard deviation was to be 4.0% from five determinations of enterostatin at 37.2 microM in a human cerebrospinal fluid (CSF) sample. Twenty-three human CSF samples were analyzed. The level of enterostatin ranged from 24 microM to 51 microM with a mean (+/- SEM) value of 41.7 +/- 2.0 microM.     

An evaluation of a new neutron tube (Philips Type 18604) for total body in vivo activation analysis

A 14 MeV neutron generator system with a new sealed tube (Philips 18604) was evaluated for total body in vivo neutron activation analysis (TBIVNAA). The neutron output, angular distribution, rise time, reproducibility of short irradiations and leakage of tritium were investigated. The 18604 tube meets all the requirements of TBIVNAA within its normal operating specification.     

Selenium and nutrition: The accuracy and variability of the selenium content in commercial supplements

Summary Selenium is a required trace-element that has been found to be protective against serious chronic diseases such as cancer and cardiovascular disease in some, but not all, epidemiological studies using both case-control and intervention designs. As a result, the fraction of the adult U.S. population now taking a daily selenium supplement is steadily increasing. In this study, we analyzed 10 or more replicate Se supplement tablets, from each of 15 different products representing 12 different brand names with most being sampled at two different times separated by approximately 30 months. Two chemical forms, seleno-yeast and selenate were tested in 50, 100 and 200 µg/tablet dosages (seleno-yeast) and 25 and 200 µg/tablet dosages (selenate). Variations in contemporary lots were evaluated at both sampling periods. The Se content provided on the product label is generally understated. One tablet contained 2.5 times more selenium than the stated dose. Selenate supplements are less accurately labeled and more highly variable compared to yeast supplements. One popular multivitamin, labeled at 200 µg/tablet, contained tablets in excess of 300 µg. Many subjects using this supplement will exceed the 400 µg/day tolerable upper limit of intake, recently established, for Se by the Institute of Medicine’s Food and Nutrition Board.     

The Bohlmann-Rahtz route to functionalised pyridine scaffolds and their use in library synthesis

The Bohlmann-Rahtz reaction has been used to prepare 2,3,6-trisubstituted pyridines suitable for use in library synthesis. The synthesis of piperidine substituted nicotinic acid derivative 9 has been optimised and carried out on a large scale to give ca. 500 g of scaffold which was used in the generation of the pyridine library 11.     

Cationic copolymerization of tetrahydrofuran with epoxides. III. Synthesis of block copolymers

Polymerization of various cyclic ethers by BF₃·O(C₂H₅)₂ in the presence of polymeric glycol leads to the formation of hydroxyl terminated block copolymers. Where poly(oxyethylene glycol) is used as the polymeric glycol, fission of the poly(oxyethylene glycol) chain occurs, and block copolymers, containing shorter ethylene oxide unit sequences are obtained. With poly(oxypropylene glycol), on the other hand, the polymer chain remains intact. This may be due to the steric influence of the pendant methyl groups. The cyclic oligomers formed as by-products in the polymerizations are easily removed.