On Versal Unfoldings of Singularities for General Two-Dimensional Spatial Motions

Local models are given for singularities which can appear on the trajectories of general two-dimensional spatial motions. Versal unfoldings of these model singularities give rise to computer generated pictures describing the family of trajectories arising from small deformations of the tracing point.     

Balanced and 1-balanced graph constructions

There are several density functions for graphs which have found use in various applications. In this paper, we examine two of them, the first being given by b(G)=|E(G)|/|V(G)|, and the other being given by g(G)=|E(G)|/(|V(G)|−ω(G)), where ω(G) denotes the number of components of G. Graphs for which b(H)≤b(G) for all subgraphs H of G are called balanced graphs, and graphs for which g(H)≤g(G) for all subgraphs H of G are called 1-balanced graphs (also sometimes called strongly balanced or uniformly dense in the literature). Although the functions b and g are very similar, they distinguish classes of graphs sufficiently differently that b(G) is useful in studying random graphs, g(G) has been useful in designing networks with reduced vulnerability to attack and in studying the World Wide Web, and a similar function is useful in the study of rigidity. First we give a new characterization of balanced graphs. Then we introduce a graph construction which generalizes the Cartesian product of graphs to produce what we call a generalized Cartesian product. We show that generalized Cartesian product derived from a tree and 1-balanced graphs are 1-balanced, and we use this to prove that the generalized Cartesian products derived from 1-balanced graphs are 1-balanced.     

Linear Programming with Costs and Incomes in Two or More Separated Currencies

If the balance of payments of any country is to be kept favourable, it is essential that decision makers at all levels in that country be aware of the influences alternative policies have on this balance. The normal formulation of a linear program matrix does not reveal this information, neither does it separate costs from incomes in working out the objective function and opportunity costs.The modification to matrix layout suggested enables costs and incomes to be shown separately, and further enables each to be detailed in any number of currencies. In U.K. problems one would deal normally in pounds and use dollars to denote all foreign currencies. By examination of vectors in the solution matrix it is possible to see the total effect of each variable on costs and incomes, each split into the correct currencies, so that net pound or net dollar effects can be found.To obtain full benefit from this exercise the decision-maker has to be prepared to take note of balance of payments influences, as well as direct influences on his own profits. A major side-benefit is that the act of separating monies into local and foreign categories brings an added awareness, at many levels of an organization, of the possible effects different policies may have on the balance of payments. This system is currently being used in the Chilean steel industry in order to rationalize their import and export policies.     

Development of prototype titanate ion-exchange loaded-membranes for strontium, cesium and actinide decontamination from aqueous media

We have successfully incorporated high surface area particles of titanate ion-exchange materials (monosodium titanate and crystalline silicotitanate) into porous and inert support membrane fibrils. The resulting membrane sheets were used to evaluate the removal of surrogate radioactive materials for cesium-137 and strontium-90 from high caustic nuclear waste simulants. The membrane supports met the nominal requirement for non-chemical interaction with the embedded ion-exchange materials and were porous enough to allow sufficient liquid flow. Most of the stamped out 47-mm size titanium impregnated ion-exchange membrane discs removed more than 96% of dissolved cesium-133 and strontium-88 from caustic nuclear waste salt simulants.     

Transforming a graph into a 1-balanced graph

Let G be a non-trivial, loopless graph and for each non-trivial subgraph H of G, let . The graph G is 1-balanced if γ(G), the maximum among g(H), taken over all non-trivial subgraphs H of G, is attained when H=G. This quantity γ(G) is called the fractional arboricity of the graph G. The value γ(G) appears in a paper by Picard and Queyranne and has been studied extensively by Catlin, Grossman, Hobbs and Lai. The quantity γ(G)−g(G) measures how much a given graph G differs from being 1-balanced. In this paper, we describe a systematic method of modifying a given graph to obtain a 1-balanced graph on the same number of vertices and edges. We obtain this by a sequence of iterations; each iteration re-defining one end-vertex of an edge in the given graph. After each iteration, either the value γ of the new graph formed is less than that of the graph from the previous iteration or the size of the maximal γ-achieving subgraph of the new graph is smaller than that of the graph in the previous iteration. We show that our algorithm is polynomial in time complexity. Further ways to decrease the number of iterations are also discussed.     

The use of chemical contrast in the SEM analysis of polymer blends

A novel method for investigating the morphology of multiphase polymers is discussed. In this procedure, inherent differences in secondary and backscattered electron emission from the various components rather than topographical variations are used to produce image contrast. The combined use of various selective stains and energy-dispersive x-ray analysis to accentuate and positively identify individual phases is discussed. The effects of coating thickness and type and changes in accelerating voltage on image quality are also described.     

Mono-organotin(IV) compounds as esterification and transesterification catalysts

A series of monoorganotin(IV) compounds has been investigated as transesterification catalysts for the reaction of butyl propionate with methanol. The most active catalysts were found to be those which contain tin-halogen bonds, e.g. monobutyltin trichloride (BuSnCl₃), and the least effective were the coordinatively saturated monoorganotin derivatives. Certain of the mono(2-carboalkoxyethyl)tin compounds were found to undergo a facile autocatalysed transesterification reaction with alcohols. Coordination of the carbonyl group in the ester to the tin catalyst is an important factor influencing its activity. A study of the catalysis of the esterification of propionic acid by BuSnCl₃ is reported.     

The 1970 survey of British rheological research

Summary This paper summarizes a survey of British rheological research made during 1970 by the National Engineering Laboratory. The survey recorded 1031 research projects being carried out in 581 laboratories and included an index of 1486 active workers in the field of rheology. An analysis of this information has shown the distribution of interest in the various classes of materials, in different phenomena, in different processes and in the different methods of approach.

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Zusammenfassung Diese Veröffentlichung gibt einen Überblick über die rheologische Forschung in Großbritannien im Jahre 1970, verfaßt vom National Engineering Laboratory (NEL). Der Überblick umfaßt 1031 Forschungsprojekte, die in 581 Laboratorien ausgeführt wurden, und enthält ein Verzeichnis von 1486 auf dem Gebiet der Rheologie tätigen Forschern. Eine Analyse dieser Informationen zeigt die Verteilung des Interesses betreffend die verschiedenen Stoffklassen, Phänomene, Verfahren und Untersuchungsmethoden.