Pit your wits against young minds!

This column is a place for those bits of contagious mathematics that travel from person to person in the community, because they are so elegant, suprising, or appealing that one has an urge to pass them on. Contributions are most welcome.     

Chemometric approaches in environmental problems concerning PCDD and PCDF. Data interpretation and source correlation. Mechanisms of formation and destruction in MSW combustion process

Summary The role of multivariate analysis methods in evaluating, rationalizing, and working out complex environmental problems is discussed. The discussion is organized in two sections; a literature analysis of the application of chemometric methods to PCDD/PCDF data interpretation and source correlation and a review of the role of chemometric methods in analysing the results obtained by the Authors studying PCDD/PCDF formation and destruction mechanisms in MSW combustion processes.     

The rotational-conformational distribution of 2,2'-bithienyl in liquid crystals

We have investigated the distribution of conformations and orientations for a simple internal rotor molecule, 2,2'-bithienyl dissolved in liquid crystal solvents, by re-analysing published proton dipolar coupling data with the maximum entropy internal order method (MEIO). We show that detailed, model independent, conformational information can be obtained when data of sufficiently high quality are available. We also propose a novel and convenient representation method for the orientational-conformational coupling.     

Working Methods Paper: Critical considerations with respect to the identification of tin species in the environment

Biogeochemical pathways of tin species in the environment are still controversial, e.g. with regard to methylation and transmethylation phenomena, owing to the fact that the identification of methylated tin-compounds is often difficult. The previous tentative identification of a mixed methylbutyltin compound in sediment and biological samples by GC/AAS after hydride generation gave an illustration of this problem. This compound was previously identified in sediments by other authors and also suspected to occur in a contaminated sediment sample from the Boyardville Marina, France. The retention time obtained by GC/AAS corresponded to the actual retention time of a mixed methylbutyltin calibrant. However, additional checks demonstrated that the compound detected was actually monophenyltin. This evidence was produced by a thorough analysis of a selected sediment sample by alternative techniques such as GC/AAS and GC/AES after pentylation, GC/FPD and GC/MS. The results presented highlight the need for a full identification of compounds to avoid mis-interpretation.     

Separation in distributive congruence lattices

We define separable sets in algebraic lattices. For a finitely generated congruence distributive variety V \mathcal{V} , we show a close connection between non-separable sets in congruence lattices of algebras in V \mathcal{V} and the structure of subdirectly irreducible algebras in V \mathcal{V} . We apply the general results to some lattice varieties.     

Mother-to-child transmission of HIV: a simulation-based approach for the evaluation of intervention strategies

Mother-to-child transmission (MTCT) of HIV is the most significant source of HIV infection in children below the age of 15 years. In 2000 alone, about 600 000 new infections occurred, the vast majority from mothers living in developing countries who were not aware of their HIV-positive status. To date, at least 4.3 million children have died from AIDS. In this paper, we discuss the development of an operational model that can be used to evaluate intervention options for the prevention of MTCT of HIV. The problem was approached by defining suitable pregnancy risk groups, prevention options and the required model outputs to evaluate different intervention strategies. The method chosen to approach the problem was a discrete-event, three-phase simulation, built in Visual Basic, with a stochastic semi-Markov structure. The developed model takes individual pregnant women through each stage of their pregnancy, labour and birth. Different intervention strategies may be modelled at any time, including short-course antiretroviral drugs and cessation of breastfeeding. The model is demonstrated with data from Botswana, which has one of the highest HIV-infection rates in the world.     

Stability of Markovian structure observed in high frequency foreign exchange data

Contrary to the common sense in economics and financial engineering, price fluctuations at very fine level of motion exhibit various evidences against the efficient market hypothesis. We attempt to investigate this issue by studying extensive amount of foreign currency exchange data for over five years at the finest level of resolution. We specifically focus on the proposed stability in binomial conditional probabilities originally found in much smaller examples of financial time series. In order to handle very large data, we have written an efficient program in C that automatically generates those conditional probabilities. It is found that the stability is maintained for extremely large time duration that covers almost the entire period. Based on the length of conditions for which the conditional probabilities are distinguishable each other, we identify the length of memory being less than 3 movements.     

On the Metrical Theory of Continued Fraction Mixing Fibred Systems and Its Application to Jacobi-Perron Algorithm

 We study the metrical theory of fibred systems, in particular, in the case of continued fraction mixing systems. We get the limit distribution of the largest value of a continued fraction mixing stationary stochastic process with infinite expectation and some related results. These are analogous to J. Galambos, W. Philipp, and H. G. Diamond–J. D. Vaaler theorems for the regular continued fractions. As an application, we see that these theorems hold for Jacobi-Perron algorithm. Received September 30, 2001; in revised form January 8, 2002