A note on congruences of semilattices with sectionally finite height

We present a bijective correspondence between congruences of semilattices with sectionally finite height (i.e., meet-semilattices whose principal downsets have finite length) and certain special subsets of their universes. We characterize these subsets from a purely order-theoretic point of view and prove that the bijection coincides with the Leibniz operator of abstract algebraic logic.     

On the effect of friend feedbacks

Given ?? an (A,B)‐invariant subspace, we prove that the set of friend feedbacks is a linear variety, which can be considered as the direct sum of the feedbacks of the restriction to ?? and the co‐restriction to ??^⊥. In particular, when the natural controllability hypothesis hold, both pole assignments are simultaneously possible by means of a convenient friend feedback. Copyright © 2010 John Wiley & Sons, Ltd.     

A networking method to compare theories: metacognition in problem solving reformulated within the Anthropological Theory of the Didactic

An important role of theory in research is to provide new ways of conceptualizing practical questions, essentially by transforming them into scientific problems that can be more easily delimited, typified and approached. In mathematics education, theoretical developments around ‘metacognition’ initially appeared in the research domain of Problem Solving closely related to the practical question of how to learn (and teach) to solve non-routine problems. This paper presents a networking method to approach a notion as ‘metacognition’ within a different theoretical perspective, as the one provided by the Anthropological Theory of the Didactic. Instead of trying to directly ‘translate’ this notion from one perspective to another, the strategy used consists in going back to the practical question that is at the origin of ‘metacognition’ and show how the new perspective relates this initial question to a very different kind of phenomena. The analysis is supported by an empirical study focused on a teaching proposal in grade 10 concerning the problem of comparing mobile phone tariffs.     

Enumeration of polycarborane isomers: especially dicarboranes

Various sorts of isomer enumeration problems are addressed in the context of polycarboranes, with special illustrative focus on the case of dicarboranes, for which then various numerical results are given. A systematic and general Pólya-theoretic methodology is used to make the computations, including some new techniques being applicable to a wide range of nano-structures built from a framework of like local subunits.     

Development of an analytical methodology to quantify melamine in milk using micellar liquid chromatography and validation according to EU Regulation 2002/654/EC

Melamine is a toxic triazine, illegally used as an additive in milk to apparently increase the amount of protein. A chromatographic procedure using a C18 column and a micellar mobile phase of sodium dodecyl sulphate (0.05 M) and propanol (7.5%), buffered at pH 3, and a detection set by absorbance at 210 nm, was reported for the resolution and quantification of melamine in liquid and powdered milk samples. In this work, samples were diluted with a SDS solution and were directly injected, thus avoiding long extraction and experimental procedures. Melamine was eluted in nearly 9.3 min without overlapping the protein band or other endogeneous compounds. The optimal mobile phase composition was taken using a chemometrical approach that considers the retention factor, efficiency and peak shape. Validation was performed following the European Commission's indications (European Decision 2002/657/EC), and the main analytical parameters studied were: linearity (0.02-100 ppm; r² = 0.999), limit of detection (5 ppb), intra- and inter-day precision (R.S.D. <7.6% and <9.7%, respectively) and robustness (R.S.D. <7.4% for retention time and <5.0% for area). Sensitivity was adequate to detect melamine under the safety limits proposed by the US FDA. Finally, recoveries for several milk samples were found in the 85-109% range.     

On polynomial digraphs

Let Φ(x,y) be a bivariate polynomial with complex coefficients. The zeroes of Φ(x,y) are given a combinatorial structure by considering them as arcs of a directed graph G(Φ). This paper studies some relationship between the polynomial Φ(x,y) and the structure of G(Φ).     

Red Queen strange attractors in host–parasite replicator gene-for-gene coevolution

We study a continuous time model describing gene-for-gene, host–parasite interactions among self-replicating macromolecules evolving in both neutral and rugged fitness landscapes. Our model considers polymorphic genotypic populations of sequences with 3 bits undergoing mutation and incorporating a “type II” non-linear functional response in the host–parasite interaction. We show, for both fitness landscapes, a wide range of chaotic coevolutionary dynamics governed by Red Queen strange attractors. The analysis of a rugged fitness landscape for parasite sequences reveals that fittest genotypes achieve lower stationary concentration values, as opposed to the flattest ones, which undergo a higher stationary concentration. Our model also shows that the increase of parasites pressure (higher self-replication and mutation rates) generically involves a simplification of the host–parasite dynamical behavior, involving the transition from a chaotic to an ordered coevolutionary phase. Moreover, the same transition can also be found when hosts “run” faster through the hypercube. Our results, in agreement with previous studies in host–parasite coevolution, suggest that chaos might be common in coevolutionary dynamics of changing self-replicating entities undergoing a host–parasite ecology.