Thermal Properties of Binary Mixtures of Liquid Crystals. A Review

Owing to their wide range of industrial utility, binary systems of liquid crystals are currently a topic of extensive studies. This paper is a comprehensive review of the literature from 1973, where the review of Hsu et al. ⁹⁵ ended, through mid-1981. For simplification, the organization used here is similar to that incorporated in the previous review.

Because of the volumes of literature on mesophase systems, this study includes only binary mesophase systems in which both components are thermotropic mesogens. Included here are only those systems for which phase diagrams have been constructed or systems which are similar to those whose phase diagrams have been constructed.

Much of the work on binary mixtures is done by industrial applications specialists and thus was not available for this review. Inevitably, there will be omissions from published work, although every attempt was made to include all of the available literature.     

Mesomorphic Properties of an Homologous Series of Alkoxy-Terminated Enamine-Ketone Containing Liquid Crystals

An homologous series of enamine-ketone compounds has been synthesized and their thermal properties investigated. The bis [3- p-n-alkoxyanilino)-2-butenoyl] benzenes exhibited nematic mesophases when the terminal alkoxy substituents were short and both nematic and smectic mesophases when the terminal alkoxy substituents contained twelve or more carbons. The thermal data indicate that although crystalline order is largely determined by the packing of the hydrocarbon tails, there is a low degree of interaction by the tails in the nematic mesophase.     

A Graphical Method of Exploring the Mean Structure in Longitudinal Data Analysis

Abstract

In a longitudinal study, individuals are observed over some period of time. The investigator wishes to model the responses over this time as a function of various covariates measured on these individuals. The times of measurement may be sparse and not coincident across individuals. When the covariate values are not extensively replicated, it is very difficult to propose a parametric model linking the response to the covariates because plots of the raw data are of little help. Although the response curve may only be observed at a few points, we consider the underlying curve y(t). We fit a regression model y(t) = x ^Tβ(t) + ε(t) and use the coefficient functions β(t) to suggest a suitable parametric form. Estimates of y(t) are constructed by simple interpolation, and appropriate weighting is used in the regression. We demonstrate the method on simulated data to show its ability to recover the true structure and illustrate its application to some longitudinal data from the Panel Study of Income Dynamics.     

A SCALE FOR ASSESSING PROBABILISTIC THINKING AND THE REPRESENTATIVENESS TENDENCY

We report a study of children's probability conceptions and misconceptions using a diagnostic instrument developed from the literature on the representativeness heuristic. Rasch measurement methodology was used to develop the 13-item open response instrument with a sample (n=116) of 12-15 year olds. The result is that a hierarchy of responses at two levels is confirmed for this sample, and a third level is hypothesised. Each level is characterised by the ability to overcome typical 'representativeness' effects, namely 'recency', 'random-similarity' (at level 1), 'base-rate frequency' and 'sample size' (at level 2-3). The validity of our interpretations was tested and some anomalies were identified through clinical interviews with children making the errors (n=8). Another Rasch ability measure, which we named the 'representativeness tendency', was constructed from 11 multiple-choice errors.     

CHILDREN'S PROPORTIONAL REASONING AND TENDENCY FOR AN ADDITIVE STRATEGY: THE ROLE OF MODELS

In this paper we report on 10 –14 year old children's strategies while solving two versions of ratio and proportion tasks: one ‘with models’ thought to facilitate proportional reasoning and one ‘without’. Rasch methodology was used to develop ‘with’ and ‘without models’ test versions which were given to a linked sample involving 673 children. We examine the pupils’ additive errors, their effect on ratio reasoning and how contingent on ‘model’ presentation this is. First, we provide a single scale on which pupils, item-difficulty and additive errors can be located. We then provide a new scale constructed from the error prone items, which we name the ‘tendency for additive strategy’. The measurement data is supported by qualitative data showing that the presence of ‘models’ can sometimes affect children's strategies, both positively and negatively but rarely makes a significant measurement difference on this, untutored, sample.     

Recognizing Abelian Sylow Subgroups in Finite Groups

Let p be a prime. We prove that if a finite group G has non-abelian Sylow p-subgroups, and the class size of every p-element in G is coprime to p, then G contains a simple group as a subquotient which exhibits the same property. In addition, we provide a list of all the simple groups and primes such that the Sylow p-subgroups are non-abelian and all p-elements have class size coprime to p.     

Parallel distributed-memory simplex for large-scale stochastic LP problems

We present a parallelization of the revised simplex method for large extensive forms of two-stage stochastic linear programming (LP) problems. These problems have been considered too large to solve with the simplex method; instead, decomposition approaches based on Benders decomposition or, more recently, interior-point methods are generally used. However, these approaches do not provide optimal basic solutions, which allow for efficient hot-starts (e.g., in a branch-and-bound context) and can provide important sensitivity information. Our approach exploits the dual block-angular structure of these problems inside the linear algebra of the revised simplex method in a manner suitable for high-performance distributed-memory clusters or supercomputers. While this paper focuses on stochastic LPs, the work is applicable to all problems with a dual block-angular structure. Our implementation is competitive in serial with highly efficient sparsity-exploiting simplex codes and achieves significant relative speed-ups when run in parallel. Additionally, very large problems with hundreds of millions of variables have been successfully solved to optimality. This is the largest-scale parallel sparsity-exploiting revised simplex implementation that has been developed to date and the first truly distributed solver. It is built on novel analysis of the linear algebra for dual block-angular LP problems when solved by using the revised simplex method and a novel parallel scheme for applying product-form updates.     

Precipitate assemblies formed on dislocation loops in aluminium-silver-copper alloys

The precipitation microstructure of the γ′ (AlAg₂) intermetallic phase has been examined in aluminium-silver-copper alloys. The microstructure developed in an Al-0.90at.%Ag-90at.Cu alloy was significantly different from that reported for binary Al-Ag alloys. The orientation relationship between the matrix and precipitate was unchanged; however, the γ′ phase formed assemblies with a two-dimensional, open arrangement of precipitates. Each such assembly contained two variants of the γ′ phase alternately arranged to form a faceted elliptical unit. The θ′ (Al₂Cu) phase formed on these assemblies after further ageing. Each assembly was formed via repeated precipitation of the γ′ phase on dissociated segments of a single dislocation loop. This faceted elliptical assembly has not been previously reported for the γ′ precipitate. The difference between the precipitation behaviour of the γ′ phase in Al-Ag and Al-Ag-Cu alloys was attributed to copper modifying the as-quenched defect structure of the matrix. The formation of faceted elliptical γ′ phase assemblies clarifies earlier observations on the precipitate number density and mechanical properties of aluminium-silver-copper alloys.     

Continuity of the Explosion Time in Stochastic Differential Equations

Abstract

Stochastic ordinary differential equations may have solutions that explode in finite time. In this article we prove the continuity of the explosion time with respect to the different parameters appearing in the equation, such as the initial datum, the drift, and the diffusion.