Mathematical Studies for the IB Diploma.

Este práctico libro de ejercicios cubre las necesidades básicas para conseguir el Diploma de Bachillerato Internacional en estudios de matemáticas. Incluye consejos de profesores y estudiantes para realizar un examen óptimo y obtener el mayor rendimiento posible. Los temas del libro son: álgebra y números (ecuaciones de segundo grado), lógica y probabilidad, funciones (funciones exponenciales y gráficos), geometría y trigonometría (la regla del seno y el área de un triángulo, la regla del coseno), estadísticas (clasificación de datos, tablas de frecuencia y polígonos), cálculo diferencial (ecuaciones de línea tangentes), matemáticas financieras.

Improving awareness about the meaning of the principle of mathematical induction. 'Fomentar la conciencia sobre el significado del principio de inducción matemática'.

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Why children differ in their mathematical attainment at primary school?.'¿Por qué los niños difieren en su competencia matemática en la escuela primaria?'.

Resúmen tomado parcialmente de la publicación.- El artículo forma parte de un monográfico dedicado a Psicología de las Matemáticas

Developing Purposeful Mathematical Thinking : a Curious Tale of Apple Trees. 'Desarrollo de un pensamiento matemático intencionado : un relato curioso de manzanos'.

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Prospective elementary school teachers' ways of making sense of mathematical problem posing. 'Modos en que futuros profesores de primaria dan sentido a la invención de problemas matemáticos'.

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Hybrid tasks : promoting statistical thinking and critical thinking through the same mathematical activities

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Mathematics teachers' ideas about mathematical models : a diverse landscape

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Forms of generalization in students experiencing mathematical learning difficulties

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Lessons from PISA 2012 about mathematical literacy : an illustrated essay

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Construction and modification of the autonomy of school mathematical knowledge in Portugal

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Procedures and examples for examining a wide range of student outcomes from 1:1 student computing settings

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Computing Distance Functions from Generalized Sources on Weighted Polyhedral Surfaces

We present algorithms for computing approximate distance functions and shortest paths from a generalized source (point, segment, polygonal chain or polygonal region) on a weighted non-convex polyhedral surface in which obstacles (represented by polygonal chains or polygons) are allowed. We also describe an algorithm for discretizing, by using graphics hardware capabilities, distance functions. Finally, we present algorithms for computing discrete k-order Voronoi diagrams

Critical thoughts on computing atom condensed Fukui functions

Different procedures to obtain atom condensed Fukui functions are described. It is shown how the resulting values may differ depending on the exact approach to atom condensed Fukui functions. The condensed Fukui function can be computed using either the fragment of molecular response approach or the response of molecular fragment approach. The two approaches are nonequivalent; only the latter approach corresponds in general with a population difference expression. The Mulliken approach does not depend on the approach taken but has some computational drawbacks. The different resulting expressions are tested for a wide set of molecules. In practice one must make seemingly arbitrary choices about how to compute condensed Fukui functions, which suggests questioning the role of these indicators in conceptual density-functional theory

A systematic and feasible method for computing nuclear contributions to electrical properties of polyatomic molecules

An analytic method to evaluate nuclear contributions to electrical properties of polyatomic molecules is presented. Such contributions control changes induced by an electric field on equilibrium geometry (nuclear relaxation contribution) and vibrational motion (vibrational contribution) of a molecular system. Expressions to compute the nuclear contributions have been derived from a power series expansion of the potential energy. These contributions to the electrical properties are given in terms of energy derivatives with respect to normal coordinates, electric field intensity or both. Only one calculation of such derivatives at the field-free equilibrium geometry is required. To show the useful efficiency of the analytical evaluation of electrical properties (the so-called AEEP method), results for calculations on water and pyridine at the SCF/TZ2P and the MP2/TZ2P levels of theory are reported. The results obtained are compared with previous theoretical calculations and with experimental values

An approach to coded structured light to obtain three dimensional information

The human visual ability to perceive depth looks like a puzzle. We perceive three-dimensional spatial information quickly and efficiently by using the binocular stereopsis of our eyes and, what is mote important the learning of the most common objects which we achieved through living. Nowadays, modelling the behaviour of our brain is a fiction, that is why the huge problem of 3D perception and further, interpretation is split into a sequence of easier problems. A lot of research is involved in robot vision in order to obtain 3D information of the surrounded scene. Most of this research is based on modelling the stereopsis of humans by using two cameras as if they were two eyes. This method is known as stereo vision and has been widely studied in the past and is being studied at present, and a lot of work will be surely done in the future. This fact allows us to affirm that this topic is one of the most interesting ones in computer vision. The stereo vision principle is based on obtaining the three dimensional position of an object point from the position of its projective points in both camera image planes. However, before inferring 3D information, the mathematical models of both cameras have to be known. This step is known as camera calibration and is broadly describes in the thesis. Perhaps the most important problem in stereo vision is the determination of the pair of homologue points in the two images, known as the correspondence problem, and it is also one of the most difficult problems to be solved which is currently investigated by a lot of researchers. The epipolar geometry allows us to reduce the correspondence problem. An approach to the epipolar geometry is describes in the thesis. Nevertheless, it does not solve it at all as a lot of considerations have to be taken into account. As an example we have to consider points without correspondence due to a surface occlusion or simply due to a projection out of the camera scope. The interest of the thesis is focused on structured light which has been considered as one of the most frequently used techniques in order to reduce the problems related lo stereo vision. Structured light is based on the relationship between a projected light pattern its projection and an image sensor. The deformations between the pattern projected into the scene and the one captured by the camera, permits to obtain three dimensional information of the illuminated scene. This technique has been widely used in such applications as: 3D object reconstruction, robot navigation, quality control, and so on. Although the projection of regular patterns solve the problem of points without match, it does not solve the problem of multiple matching, which leads us to use hard computing algorithms in order to search the correct matches. In recent years, another structured light technique has increased in importance. This technique is based on the codification of the light projected on the scene in order to be used as a tool to obtain an unique match. Each token of light is imaged by the camera, we have to read the label (decode the pattern) in order to solve the correspondence problem. The advantages and disadvantages of stereo vision against structured light and a survey on coded structured light are related and discussed. The work carried out in the frame of this thesis has permitted to present a new coded structured light pattern which solves the correspondence problem uniquely and robust. Unique, as each token of light is coded by a different word which removes the problem of multiple matching. Robust, since the pattern has been coded using the position of each token of light with respect to both co-ordinate axis. Algorithms and experimental results are included in the thesis. The reader can see examples 3D measurement of static objects, and the more complicated measurement of moving objects. The technique can be used in both cases as the pattern is coded by a single projection shot. Then it can be used in several applications of robot vision. Our interest is focused on the mathematical study of the camera and pattern projector models. We are also interested in how these models can be obtained by calibration, and how they can be used to obtained three dimensional information from two correspondence points. Furthermore, we have studied structured light and coded structured light, and we have presented a new coded structured light pattern. However, in this thesis we started from the assumption that the correspondence points could be well-segmented from the captured image. Computer vision constitutes a huge problem and a lot of work is being done at all levels of human vision modelling, starting from a)image acquisition; b) further image enhancement, filtering and processing, c) image segmentation which involves thresholding, thinning, contour detection, texture and colour analysis, and so on. The interest of this thesis starts in the next step, usually known as depth perception or 3D measurement.