Aitchison Geometry for Probability and Likelihood as a new approach to mathematical statistics

The Aitchison vector space structure for the simplex is generalized to a Hilbert space structure A2(P) for distributions and likelihoods on arbitrary spaces. Central notations of statistics, such as Information or Likelihood, can be identified in the algebraical structure of A2(P) and their corresponding notions in compositional data analysis, such as Aitchison distance or centered log ratio transform. In this way very elaborated aspects of mathematical statistics can be understood easily in the light of a simple vector space structure and of compositional data analysis. E.g. combination of statistical information such as Bayesian updating, combination of likelihood and robust M-estimation functions are simple additions/ perturbations in A2(Pprior). Weighting observations corresponds to a weighted addition of the corresponding evidence. Likelihood based statistics for general exponential families turns out to have a particularly easy interpretation in terms of A2(P). Regular exponential families form finite dimensional linear subspaces of A2(P) and they correspond to finite dimensional subspaces formed by their posterior in the dual information space A2(Pprior). The Aitchison norm can identified with mean Fisher information. The closing constant itself is identified with a generalization of the cummulant function and shown to be Kullback Leiblers directed information. Fisher information is the local geometry of the manifold induced by the A2(P) derivative of the Kullback Leibler information and the space A2(P) can therefore be seen as the tangential geometry of statistical inference at the distribution P. The discussion of A2(P) valued random variables, such as estimation functions or likelihoods, give a further interpretation of Fisher information as the expected squared norm of evidence and a scale free understanding of unbiased reasoning

Vertebrates Limb Geometry in the Simplex space

A novel metric comparison of the appendicular skeleton (fore and hind limb) of different vertebrates using the Compositional Data Analysis (CDA) methodological approach it’s presented. 355 specimens belonging in various taxa of Dinosauria (Sauropodomorpha, Theropoda, Ornithischia and Aves) and Mammalia (Prothotheria, Metatheria and Eutheria) were analyzed with CDA. A special focus has been put on Sauropodomorpha dinosaurs and the Aitchinson distance has been used as a measure of disparity in limb elements proportions to infer some aspects of functional morphology

Hyperbolic Geometry

Exercises, exam questions and solutions for a fourth year hyperbolic geometry course. Diagrams for the questions are all together in the support.zip file, as .eps files

Teaching and learning geometry : issues and methods in mathematical education.

Recurso para la evaluación de la enseñanza y el aprendizaje de la geometría en la enseñanza secundaria desde la perspectiva de los nuevos docentes y de los que tienen más experiencia. Está diseñado para ampliar y profundizar el conocimiento de la materia y ofrecer consejos prácticos e ideas para el aula en el contexto de la práctica y la investigación actual. Hace especial hincapié en: comprender las ideas fundamentales del currículo de geometría; el aprendizaje de la geometría de manera efectiva; la investigación y la práctica actual; las ideas erróneas y los errores; el razonamiento de la geometría; la solución de problemas; el papel de la tecnología en el aprendizaje de la geometría.

Ontological beliefs and their impact on teaching elementary geometry. 'Creencias ontológicas y su impacto en la enseñanza de la geometría elemental'.

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Reasoning by contradiction in dynamic geometry

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The effect of flow field & geometry on the dynamic contact angle

A number of recent experiments suggest that, at a given wetting speed, the dynamic contact angle formed by an advancing liquid-gas interface with a solid substrate depends on the flow field and geometry near the moving contact line. In the present work, this effect is investigated in the framework of an earlier developed theory that was based on the fact that dynamic wetting is, by its very name, a process of formation of a new liquid-solid interface (newly “wetted” solid surface) and hence should be considered not as a singular problem but as a particular case from a general class of flows with forming or/and disappearing interfaces. The results demonstrate that, in the flow configuration of curtain coating, where a liquid sheet (“curtain”) impinges onto a moving solid substrate, the actual dynamic contact angle indeed depends not only on the wetting speed and material constants of the contacting media, as in the so-called slip models, but also on the inlet velocity of the curtain, its height, and the angle between the falling curtain and the solid surface. In other words, for the same wetting speed the dynamic contact angle can be varied by manipulating the flow field and geometry near the moving contact line. The obtained results have important experimental implications: given that the dynamic contact angle is determined by the values of the surface tensions at the contact line and hence depends on the distributions of the surface parameters along the interfaces, which can be influenced by the flow field, one can use the overall flow conditions and the contact angle as a macroscopic multiparametric signal-response pair that probes the dynamics of the liquid-solid interface. This approach would allow one to investigate experimentally such properties of the interface as, for example, its equation of state and the rheological properties involved in the interface’s response to an external torque, and would help to measure its parameters, such as the coefficient of sliding friction, the surface-tension relaxation time, and so on.

A groove-ploughing theory for the production of mega-scale glacial lineations, and implications for ice-stream mechanics

Mega-scale glacial lineations (MSGLs) are longitudinally aligned corrugations (ridge-groove structures 6-100 km long) in sediment produced subglacially. They are indicators of fast flow and a common signature of ice-stream beds. We develop a qualitative theory that accounts for their formation, and use numerical modelling, and observations of ice-stream beds to provide supporting evidence. Ice in contact with a rough (scale of 10-10(3) m) bedrock surface will mimic the form of the bed. Because of flow acceleration and convergence in ice-stream onset zones, the ice-base roughness elements experience transverse strain, transforming them from irregular bumps into longitudinally aligned keels of ice protruding downwards. Where such keels slide across a soft sedimentary bed, they plough through the sediments, carving elongate grooves, and deforming material up into intervening ridges. This explains MSGLs and has important implications for ice-stream mechanics. Groove ploughing provides the means to acquire new lubricating sediment and to transport large volumes of it downstream. Keels may provide basal drag in the force budget of ice streams, thereby playing a role in flow regulation and stability We speculate that groove ploughing permits significant ice-stream widening, thus facilitating high-magnitude ice discharge.

Surface determination of the air-earth electrical current density using co-located sensors of different geometry

A vertical conduction current flows in the atmosphere as a result of the global atmospheric electric circuit. The current at the surface consists of the conduction current and a locally generated displacement current, which are often approximately equal in magnitude. A method of separating the two currents using two collectors of different geometry is investigated. The picoammeters connected to the collectors have a RC time constant of approximately 3 s, permitting the investigation of higher frequency air-earth current changes than previously achieved. The displacement current component of the air-earth current derived from the instrument agrees with calculations using simultaneous data from a co-located fast response electric field mill. The mean value of the nondisplacement current measured over 9 h was 1.76 +/- 0.002 pA m(-2). (c) 2006 American Institute of Physics.

The expected imprint of flux rope geometry on suprathermal electrons in magnetic clouds

Magnetic clouds are a subset of interplanetary coronal mass ejections characterized by a smooth rotation in the magnetic field direction, which is interpreted as a signature of a magnetic flux rope. Suprathermal electron observations indicate that one or both ends of a magnetic cloud typically remain connected to the Sun as it moves out through the heliosphere. With distance from the axis of the flux rope, out toward its edge, the magnetic field winds more tightly about the axis and electrons must traverse longer magnetic field lines to reach the same heliocentric distance. This increased time of flight allows greater pitch-angle scattering to occur, meaning suprathermal electron pitch-angle distributions should be systematically broader at the edges of the flux rope than at the axis. We model this effect with an analytical magnetic flux rope model and a numerical scheme for suprathermal electron pitch-angle scattering and find that the signature of a magnetic flux rope should be observable with the typical pitch-angle resolution of suprathermal electron data provided ACE's SWEPAM instrument. Evidence of this signature in the observations, however, is weak, possibly because reconnection of magnetic fields within the flux rope acts to intermix flux tubes.