The Ups and Downs of Percent (and Some Interesting Connections)

Percent is a standardized ratio comparison that is often used to describe relative amounts of increase and decrease. In this paper, the nonsymmetric relationship between percent of increase and percent of decrease is explored in several ways. Beginning with a simple graphical representation of the relationship, the functional relationship is developed, with relative decrease expressed as a function of relative increase. Once expressed in the form f(x) = 100x/(100 + x), an immediate connection is made to two other topics commonly found within the study of rates—the work problem and the harmonic mean. Finally, successive percents of increase or decrease are shown to be everyday applications of the geometric mean. This deeper look at a common mathematical concept has uncovered rich structural relationships between topics that are often isolated in the curriculum.     

What is a design? How should we classify them?

Design theory crosses the boundary between mathematics and statistics, and includes a wide range of disparate types of design. In this paper we present a classification scheme which aims to include as many important types as possible, based on a balance among concept, representation and use.     

Synthesis of perfluorinated functionalized, branched ethers

We wish to report synthesis of perfluorinated functionalized, branched ethers from their hydrocarbon analogues by direct fluorination. Yields up to 90%, with high purities, have been obtained at ambient temperature and pressure. This technique will likely develop into a new general method for producing perfluorinated, hyperbranched and dendritic polymers.     

Embedding partial Steiner triple systems so that their automorphisms extend

It is shown that there is a function g on the natural numbers such that a partial Steiner triple system U on u points can be embedded in a Steiner triple system V on ν points, in such a way that all automorphisms of U can be extended to V, for every admissible ν satisfying ν > g(u). We find exponential upper and lower bounds for g. © 2005 Wiley Periodicals, Inc. J Combin Designs.     

On the structure of a random sum-free set

Summary There is a natural probability measure on the set of all sum-free sets of natural numbers. If we represent such a set by its characteristic functions, then the zero-one random variabless(i) are far from independent, and we cannot expect a law of large numbers to hold for them. In this paper I conjecture a decomposition of into countably many more tractible pieces (up to a null set). I prove that each piece has positive measure, and show that, within each piece, a random set almost surely has a density which is a fixed rational number depending only on the piece. For example, the first such piece is made up of sets consisting entirely of odd numbers; it has probability 0.218 ..., and its members almost surely have density 1/4.     

Polymerised bicontinuous microemulsions as stationary phases for capillary electrochromatography

Summary Polymerisation of bicontinuous microemulsions yields porous monolithic structures with well defined pore sizes that are potentially suitable for use as stationary phases for capillary electrochromatography (CEC). A variety of pore sizes can be achieved by altering the composition of the microemulsion, which typically consists of butyl methacrylate (BMA) and ethylene glycol dimethacrylate (EGDMA) as the polymerisable oil phase. The aqueous phase consists of water, a surfactant (sodium dodecyl sulphate, SDS) and a co-surfactant (1-propanol). 2-acrylamido-2-methyl-1-propane sulfonic acid (AMPS) is also added to provide charges along the polymer backbone to allow electroosmotic flow (EOF) to occur. SEM analysis shows that in-situ polymerisation yields a monolithic structure with a porous topography. Investigations have shown that these monoliths are easy to prepare, robust and suitable for the separation of phthalates. They generate higher linear velocities than are achieved using the silica based HPLC packings normally used for CEC.     

Preparation of 4-hydroxy-3-substituted, 2H-1-benzopyran-2-ones and 2H-1-benzothiopyran-2-ones from carboxylic esters and methyl salicylates or methyl thiosalicylate

C(α)-Carboxylic acid esters were treated with excess lithium diisopropylamide, condensed with methyl salicylates or methyl thiosalicylate, followed by acid cyclization to either 4-hydroxy-3-substituted, 2H-1-benzopyran-2-ones (coumarins), or 2H-1-benzothiopyran-2-ones (thiocoumarins).