Compositional evolution with mass transfer in closed systems

Evolution of compositions in time, space, temperature or other covariates is frequent in practice. For instance, the radioactive decomposition of a sample changes its composition with time. Some of the involved isotopes decompose into other isotopes of the sample, thus producing a transfer of mass from some components to other ones, but preserving the total mass present in the system. This evolution is traditionally modelled as a system of ordinary di erential equations of the mass of each component. However, this kind of evolution can be decomposed into a compositional change, expressed in terms of simplicial derivatives, and a mass evolution (constant in this example). A rst result is that the simplicial system of di erential equations is non-linear, despite of some subcompositions behaving linearly. The goal is to study the characteristics of such simplicial systems of di erential equa- tions such as linearity and stability. This is performed extracting the compositional dif ferential equations from the mass equations. Then, simplicial derivatives are expressed in coordinates of the simplex, thus reducing the problem to the standard theory of systems of di erential equations, including stability. The characterisation of stability of these non-linear systems relays on the linearisation of the system of di erential equations at the stationary point, if any. The eigenvelues of the linearised matrix and the associated behaviour of the orbits are the main tools. For a three component system, these orbits can be plotted both in coordinates of the simplex or in a ternary diagram. A characterisation of processes with transfer of mass in closed systems in terms of stability is thus concluded. Two examples are presented for illustration, one of them is a radioactive decay

Hardy-Weinberg Equilibrium and the Ternary Plot

The Hardy-Weinberg law, formulated about 100 years ago, states that under certain assumptions, the three genotypes AA, AB and BB at a bi-allelic locus are expected to occur in the proportions p2, 2pq, and q2 respectively, where p is the allele frequency of A, and q = 1-p. There are many statistical tests being used to check whether empirical marker data obeys the Hardy-Weinberg principle. Among these are the classical xi-square test (with or without continuity correction), the likelihood ratio test, Fisher's Exact test, and exact tests in combination with Monte Carlo and Markov Chain algorithms. Tests for Hardy-Weinberg equilibrium (HWE) are numerical in nature, requiring the computation of a test statistic and a p-value. There is however, ample space for the use of graphics in HWE tests, in particular for the ternary plot. Nowadays, many genetical studies are using genetical markers known as Single Nucleotide Polymorphisms (SNPs). SNP data comes in the form of counts, but from the counts one typically computes genotype frequencies and allele frequencies. These frequencies satisfy the unit-sum constraint, and their analysis therefore falls within the realm of compositional data analysis (Aitchison, 1986). SNPs are usually bi-allelic, which implies that the genotype frequencies can be adequately represented in a ternary plot. Compositions that are in exact HWE describe a parabola in the ternary plot. Compositions for which HWE cannot be rejected in a statistical test are typically “close" to the parabola, whereas compositions that differ significantly from HWE are “far". By rewriting the statistics used to test for HWE in terms of heterozygote frequencies, acceptance regions for HWE can be obtained that can be depicted in the ternary plot. This way, compositions can be tested for HWE purely on the basis of their position in the ternary plot (Graffelman & Morales, 2008). This leads to nice graphical representations where large numbers of SNPs can be tested for HWE in a single graph. Several examples of graphical tests for HWE (implemented in R software), will be shown, using SNP data from different human populations

Un análisis comparativo del papel del bucle fonológico versus la agenda visoespacial en el cálculo en niños de 7-8 años = A comparative analysis of the phonological loop versus the visuo-spatial sketchpad in mental arithmetic tasks in 7-8 y.o. children

En esta investigación se ha estudiado la relación entre dos subsistemas de la memoria de trabajo (bucle fonológico y agenda viso-espacial) y el rendimiento en cálculo con una muestra de 94 niños españoles de 7-8 años. Hemos administrado dos pruebas de cálculo diseñadas para este estudio y seis medidas simples de memoria de trabajo (de contenido verbal, numérico y espacial) de la «Batería de Tests de Memoria de Treball» de Pickering, Baqués y Gathercole (1999), y dos pruebas visuales complementarias. Los resultados muestran una correlación importante entre las medidas de contenido verbal y numérico y el rendimiento en cálculo. En cambio, no hemos encontrado ninguna relación con las medidas espaciales. Se concluye, por lo tanto, que en escolares españoles existe una relación importante entre el bucle fonológico y el rendimiento en tareas de cálculo. En cambio, el rol de la agenda viso-espacial es nulo

Local map update for large scale SLAM

A technique for simultaneous localisation and mapping (SLAM) for large scale scenarios is presented. This solution is based on the use of independent submaps of a limited size to map large areas. In addition, a global stochastic map, containing the links between adjacent submaps, is built. The information in both levels is corrected every time a loop is closed: local maps are updated with the information from overlapping maps, and the global stochastic map is optimised by means of constrained minimisation