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.     

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).     

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.     

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.     

An electron diffraction determination of the molecular structures of silyl- and germyl-manganese pentacarbonyl in the gas phase

The molecular structures of the title compounds have been determined by gas phase electron diffraction methods. The SiMn and GeMn bond lengths are 240.7 ? 0.5 and 248.7 ? 0.2 pm respectively and the CMnC angles in the silyl and germyl cases are 94.5 ± 2° and 97 ± 2° respectively. Comparisons are made with the reported structure of CH₃Mn(CO)₅ and He^I photoelectron spectra of these compounds in an attempt to determine the extent of d → d π-bonding in the SiMn or GeMn bonds.     

Ramsey partitions of integers and pair divisions

Ifk ₁ andk ₂ are positive integers, the partitionP = (₁,₂,..., _n ) ofk ₁+k ₂ is said to be a Ramsey partition for the pairk ₁,k ₂ if for any sublistL ofP, either there is a sublist ofL which sums tok ₁ or a sublist ofP –L which sums tok ₂. Properties of Ramsey partitions are discussed. In particular it is shown that there is a unique Ramsey partition fork ₁,k ₂ having the smallest numbern of terms, and in this casen is one more than the sum of the quotients in the Euclidean algorithm fork ₁ andk ₂.An application of Ramsey partitions to the following fair division problem is also discussed: Suppose two persons are to divide a cake fairly in the ratiok ₁k ₂. This can be done trivially usingk ₁+k ₂-1 cuts. However, every Ramsey partition ofk ₁+k ₂ also yields a fair division algorithm. This method yields fewer cuts except whenk ₁=1 andk ₂=1, 2 or 4.     

Molecular Separation Barriers and Their Application to Catalytic Reactor Design

The use of semipermeable membranes for multicomponent separations based on molecular size has long been recognized. In certain applications, however, it is often desirable not to effect a separation of chemical constituents, but to sustain a separation which already exists. As an example, the efficient and economical design of a. chemical reactor using an enzyme as a catalyst depends on the accessibility of the reactant to the catalyst as well as on the degree to which a physical separation between the enzyme and the reactor product stream is maintained. A particularly simple and attractive means of achieving this is through the use of semipermeable asymmetric hollow fiber membranes. For example, by sequestering an enzyme solution within the annular macroporous support regions of an asymmetric hollow fiber, a physical separation between enzyme and a reactant solution flowing through the fiber lumen is achieved. In this way, small reactant molecules are free to diffuse across the ultrathin membrane skin into the opencell support structure where reaction will occur. Product molecules will diffuse back into the lumen, and a compact chemical reactor results. The operating behavior of this type of catalytic reactor will be described and its application to the hydrolysis of o-nitro-phenyl-B-d-galactopyranoside and of lactose is discussed.     

HPLC Method for the Simultaneous Determination of the UV-Filters Butyl Methoxy Dibenzoylmethane and Octocrylene in the Presence of Their Photodegradants

Butyl methoxy dibenzoylmethane (BMDM) and octocrylene (OC), common UV-filters in sunscreen products are often used in combination. Together they provide broad spectrum photoprotection from exposure to both UVA- and UVB-light. These UV-filters may, however, undergo photodegradation and generate photodegradants, resulting in a potential loss of photoprotection. It is thus a concern that the photostability testing as described by the ICH Guideline Q1B is not a requirement for sunscreen products in Australia, Europe or the USA. UV-filter photodegradants have in addition been shown to be toxic, highlighting the importance of their separation from the parent UV-filters. An HPLC method was developed and validated to quantitatively determine a combination of these UV-filters in the presence of their photodegradants. Reverse-phase chromatography was employed, using a C18 column and an isocratic mobile phase consisting of methanol/water/acetic acid (89/10/1 v/v). Validation according to the ICH guidelines for linearity, accuracy, precision, sensitivity, specificity and robustness was confirmed. The developed and validated method was then successfully applied to the determination of BMDM and OC in an aqueous cream base, typically used in sunscreens, after photostability testing, according to the ICH Guideline Q1B. In addition, the diketo-enol ratio of BMDM in methanol-d ₄ was determined by NMR and the two major photodegradants were identified by FTMS and LC–MS.