The Small Index Property for Free Groups and Relatively Free Groups

1. Some groups with the small index property In this paper we shall use the methods of the paper [6] by Hodges,Hodkinson, Lascar and Shelah to show that the free group ofcountably infinite rank and certain relatively free groups ofcountably infinite rank have the small index property.     

Four unit linking groups IV. Liquid crystals of positive dielectric anisotropy

Three separate series of new materials of weak to strongly positive dielectric anisotropy have been prepared. Each series contains four sub-sets of materials each incorporating a different four unit linking group (i.e., C₄H₈, C₄H₆, C₃H₆O and C₃H₄O) and the same series of end groups (i.e. F, CN and OCF₃) in various substitution patterns. The synthesis and liquid crystal transition temperatures of these novel substances are described and compared with those of the corresponding materials incorporating standard central linkages (i.e.-, C₂H₄, CH₂O, COO). The effect of an additional trans carbon-carbon double bond in the terminal alkyl chain and in the central linking unit has also been studied.     

A two-echelon base-stock inventory model with Poisson demand and the sequential processing of orders at the upper echelon

We consider a two-echelon inventory system with a number of non-identical, independent ‘retailers’ at the lower echelon and a single ‘supplier’ at the upper echelon. Each retailer experiences Poisson demand and operates a base stock policy with backorders. The supplier manufactures to order and holds no stock. Orders are produced, in first-come first-served sequence, with a fixed production time. The supplier therefore functions as an M/D/1 queue. We are interested in the performance characteristics (average inventory, average backorder level) at each retailer. By finding the distribution of order lead time and hence the distribution of demand during order lead time, we find the steady state inventory and backorder levels based on the assumption that order lead times are independent of demand during order lead time at a retailer. We also propose two alternative approximation procedures based on assumed forms for the order lead time distribution. Finally we provide a derivation of the steady state inventory and backorder levels which will be exact as long as there is no transportation time on orders between the supplier and retailers. A numerical comparison is made between the exact and approximate measures. We conclude by recommending an approach which is intuitive and computationally straightforward.     

Effect of stoichiometry on liquid crystalline supramolecular polymers formed with complementary nucleobase pair interactions

We report herein studies on the liquid crystalline behavior of a series of supramolecular materials that contain different ratios of two complementary symmetrically-substituted alkoxy-bis(phenylethynyl)benzene AA- and BB-type monomers. One monomer has thymine units placed at either end of the rigid mesogenic core, while the other has N⁶-(4-methoxybenzoyl)-adenine units placed on the ends. Differential scanning calorimetric and polarized optical microscopy studies have been carried out on these systems. These studies show that the material's behavior is strongly dependent on its thermal history. As a result, the materials can exhibit, on heating, either a liquid crystalline phase, a crystalline phase, or the coexistence of crystalline and liquid crystalline regions. © 2006 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 44: 5049–5059, 2006     

Soluble models for dynamics driven by a super-diffusive noise

We explicitly discuss scalar Langevin type of equations where the deterministic part is linear, but where the integrated noise source is a non-linear diffusion process exhibiting superdiffusive behavior. We calculate transient and stationary probabilities and study the possibility of noise induced transitions from a unimodal to a bimodal probability shape. Illustrations from finance and dynamical systems are given.     

Self-inhibition or the lack of it in the pyrolyses of neopentane and n-hexane

Published data show that in its early stages (up to 3% decomposition), the pyrolysis of n-hexane in the ranges 723–823 K and 10–100 Torr is not inhibited by the olefin products, in contrast with neopentane pyrolysis which is very strongly inhibited in similar conditions. Detailed consideration of the chain mechanisms for the two pyrolyses shows that the reactivity of the chain terminating radical towards hydrogen abstraction from an allylic C? H bond in product olefin is the factor which determines whether or not observable self-inhibition occurs. Thus, n-hexane pyrolysis, whose chain decomposition is terminated by recombination and disproportionation of ethyl, is not significantly self-inhibited, whereas that of neopentane which is terminated by recombination of methyl is very strongly inhibited because methyl is 14× more reactive than ethyl. The implications for other alkanes are briefly discussed.     

Stock Control and the Undershoot of the Re-order Level

Practical stock-control systems are frequently based on a policy of reviewing the stock balance and placing an order if this has fallen to or below a set level R. In order to facilitate the calculation of the control parameters, the assumption is generally made that the stock balance is exactly R when an order is placed, whereas in practice it may have fallen below R. This paper uses a steady-state analysis to evaluate some relatively simple adjustments which may be incorporated to handle this undershoot.