Futuros Matemáticos e suas concepções sobre o conhecimento matemático e seu ensino

Taking referential theoretical conceptions of mathematical knowledge present in some of the main mathematics philosophical currents and considering that the teacher´s practice is influenced by his conception of mathematical knowledge, this research aims to understand the conceptions of mathematical knowledge and its teaching and learning teaching of future mathematicians. It follows a qualitative approach (case study) in which the data were collected by semi-structured interviews and document analysis. This investigation has pointed out that Mathematics together with Mathematics Teaching (or part of this: on the didactic and pedagogical knowledge of how to teach) could be important to formation of the future mathematician, who will probably teach in a college or university.

Análise dinâmica não-linear de um sistema não-ideal, utilizando amortecedor magneto-reológico

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

User interaction widgets for interactive theorem proving

Matita (that means pencil in Italian) is a new interactive theorem prover under development at the University of Bologna. When compared with state-of-the-art proof assistants, Matita presents both traditional and innovative aspects. The underlying calculus of the system, namely the Calculus of (Co)Inductive Constructions (CIC for short), is well-known and is used as the basis of another mainstream proof assistant—Coq—with which Matita is to some extent compatible. In the same spirit of several other systems, proof authoring is conducted by the user as a goal directed proof search, using a script for storing textual commands for the system. In the tradition of LCF, the proof language of Matita is procedural and relies on tactic and tacticals to proceed toward proof completion. The interaction paradigm offered to the user is based on the script management technique at the basis of the popularity of the Proof General generic interface for interactive theorem provers: while editing a script the user can move forth the execution point to deliver commands to the system, or back to retract (or “undo”) past commands. Matita has been developed from scratch in the past 8 years by several members of the Helm research group, this thesis author is one of such members. Matita is now a full-fledged proof assistant with a library of about 1.000 concepts. Several innovative solutions spun-off from this development effort. This thesis is about the design and implementation of some of those solutions, in particular those relevant for the topic of user interaction with theorem provers, and of which this thesis author was a major contributor. Joint work with other members of the research group is pointed out where needed. The main topics discussed in this thesis are briefly summarized below. Disambiguation. Most activities connected with interactive proving require the user to input mathematical formulae. Being mathematical notation ambiguous, parsing formulae typeset as mathematicians like to write down on paper is a challenging task; a challenge neglected by several theorem provers which usually prefer to fix an unambiguous input syntax. Exploiting features of the underlying calculus, Matita offers an efficient disambiguation engine which permit to type formulae in the familiar mathematical notation. Step-by-step tacticals. Tacticals are higher-order constructs used in proof scripts to combine tactics together. With tacticals scripts can be made shorter, readable, and more resilient to changes. Unfortunately they are de facto incompatible with state-of-the-art user interfaces based on script management. Such interfaces indeed do not permit to position the execution point inside complex tacticals, thus introducing a trade-off between the usefulness of structuring scripts and a tedious big step execution behavior during script replaying. In Matita we break this trade-off with tinycals: an alternative to a subset of LCF tacticals which can be evaluated in a more fine-grained manner. Extensible yet meaningful notation. Proof assistant users often face the need of creating new mathematical notation in order to ease the use of new concepts. The framework used in Matita for dealing with extensible notation both accounts for high quality bidimensional rendering of formulae (with the expressivity of MathMLPresentation) and provides meaningful notation, where presentational fragments are kept synchronized with semantic representation of terms. Using our approach interoperability with other systems can be achieved at the content level, and direct manipulation of formulae acting on their rendered forms is possible too. Publish/subscribe hints. Automation plays an important role in interactive proving as users like to delegate tedious proving sub-tasks to decision procedures or external reasoners. Exploiting the Web-friendliness of Matita we experimented with a broker and a network of web services (called tutors) which can try independently to complete open sub-goals of a proof, currently being authored in Matita. The user receives hints from the tutors on how to complete sub-goals and can interactively or automatically apply them to the current proof. Another innovative aspect of Matita, only marginally touched by this thesis, is the embedded content-based search engine Whelp which is exploited to various ends, from automatic theorem proving to avoiding duplicate work for the user. We also discuss the (potential) reusability in other systems of the widgets presented in this thesis and how we envisage the evolution of user interfaces for interactive theorem provers in the Web 2.0 era.

Numerical simulation of some viscoelastic fluids

Numerical simulation of the Oldroyd-B type viscoelastic fluids is a very challenging problem. rnThe well-known High Weissenberg Number Problem" has haunted the mathematicians, computer scientists, and rnengineers for more than 40 years. rnWhen the Weissenberg number, which represents the ratio of elasticity to viscosity, rnexceeds some limits, simulations done by standard methods break down exponentially fast in time. rnHowever, some approaches, such as the logarithm transformation technique can significantly improve rnthe limits of the Weissenberg number until which the simulations stay stable. rnrnWe should point out that the global existence of weak solutions for the Oldroyd-B model is still open. rnLet us note that in the evolution equation of the elastic stress tensor the terms describing diffusive rneffects are typically neglected in the modelling due to their smallness. However, when keeping rnthese diffusive terms in the constitutive law the global existence of weak solutions in two-space dimension rncan been shown. rnrnThis main part of the thesis is devoted to the stability study of the Oldroyd-B viscoelastic model. rnFirstly, we show that the free energy of the diffusive Oldroyd-B model as well as its rnlogarithm transformation are dissipative in time. rnFurther, we have developed free energy dissipative schemes based on the characteristic finite element and finite difference framework. rnIn addition, the global linear stability analysis of the diffusive Oldroyd-B model has also be discussed. rnThe next part of the thesis deals with the error estimates of the combined finite element rnand finite volume discretization of a special Oldroyd-B model which covers the limiting rncase of Weissenberg number going to infinity. Theoretical results are confirmed by a series of numerical rnexperiments, which are presented in the thesis, too.

Distance education in transition

In his most recent publication, Against the Tide: Critics of Digitalisation, Otto Peters brings together some of the most formidable and critical voices and compelling perspectives on the potential hazards of digitalization. The viewpoints presented range from personal, anthropological, and pedagogical to more scientific and technical, and arise from multiple disciplines. Peters has long been a respected scholar in the field of distance education, and while Peters’ earlier work has advocated the affordances of digitalization, this latest book is an abrupt shift to the darker side of digitalization. The assembly of critics Peters has gathered come from around the world and different walks of life: journalists, educators, scientists, philosophers, lawyers, mathematicians, and computer scientists, to name a few. Their one shared bond is a deep-seated belief that digitalization will have a profound and lasting impact on humankind – and not only in positive ways. ...

Against the Tide

In his most recent publication, Against the Tide: Critics of Digitalisation, Otto Peters brings together some of the most formidable and critical voices and compelling perspectives on the potential hazards of digitalization. The viewpoints presented range from personal, anthropological, and pedagogical to more scientific and technical, and arise from multiple disciplines. Peters has long been a respected scholar in the field of distance education, and while Peters’ earlier work has advocated the affordances of digitalization, this latest book is an abrupt shift to the darker side of digitalization. The assembly of critics Peters has gathered come from around the world and different walks of life: journalists, educators, scientists, philosophers, lawyers, mathematicians, and computer scientists, to name a few. Their one shared bond is a deep-seated belief that digitalization will have a profound and lasting impact on humankind – and not only in positive ways. ...

Regulation of phyllotaxis by polar auxin transport

The regular arrangement of leaves around a plant's stem, called phyllotaxis, has for centuries attracted the attention of philosophers, mathematicians and natural scientists; however, to date, studies of phyllotaxis have been largely theoretical. Leaves and flowers are formed from the shoot apical meristem, triggered by the plant hormone auxin. Auxin is transported through plant tissues by specific cellular influx and efflux carrier proteins. Here we show that proteins involved in auxin transport regulate phyllotaxis. Our data indicate that auxin is transported upwards into the meristem through the epidermis and the outermost meristem cell layer. Existing leaf primordia act as sinks, redistributing auxin and creating its heterogeneous distribution in the meristem. Auxin accumulation occurs only at certain minimal distances from existing primordia, defining the position of future primordia. This model for phyllotaxis accounts for its reiterative nature, as well as its regularity and stability.

Platón, Aristóteles y la narrativa histórica

Una de las cuestiones más llamativas de las obras de Platón y Aristóteles estriba en su silencio acerca de la historia. Leyeron y criticaron a físicos, filósofos, matemáticos, biólogos, poetas, retóricos, políticos, etc. Sin embargo, sus citas a propósito de los historiadores de su momento cabrían en una cuartilla. En este breve artículo tratamos de ofrecer una explicación a propósito de tal omisión. Así mismo, aprovechamos para ofrecer una razón del Menéxeno y aportar una confirmación de las razones que condujeron a L. Edelstein a cuestionar la autenticidad de las cartas séptima y octava atribuidas a Platón

Platón, Aristóteles y la narrativa histórica

Una de las cuestiones más llamativas de las obras de Platón y Aristóteles estriba en su silencio acerca de la historia. Leyeron y criticaron a físicos, filósofos, matemáticos, biólogos, poetas, retóricos, políticos, etc. Sin embargo, sus citas a propósito de los historiadores de su momento cabrían en una cuartilla. En este breve artículo tratamos de ofrecer una explicación a propósito de tal omisión. Así mismo, aprovechamos para ofrecer una razón del Menéxeno y aportar una confirmación de las razones que condujeron a L. Edelstein a cuestionar la autenticidad de las cartas séptima y octava atribuidas a Platón

Platón, Aristóteles y la narrativa histórica

Una de las cuestiones más llamativas de las obras de Platón y Aristóteles estriba en su silencio acerca de la historia. Leyeron y criticaron a físicos, filósofos, matemáticos, biólogos, poetas, retóricos, políticos, etc. Sin embargo, sus citas a propósito de los historiadores de su momento cabrían en una cuartilla. En este breve artículo tratamos de ofrecer una explicación a propósito de tal omisión. Así mismo, aprovechamos para ofrecer una razón del Menéxeno y aportar una confirmación de las razones que condujeron a L. Edelstein a cuestionar la autenticidad de las cartas séptima y octava atribuidas a Platón

Arquitectura óptima: Cuatro medios prácticos para optimizar proyectos de arquitectura

La investigación trata de mostrar cuatro acciones habitualmente empleadas al proyectar arquitectura. La tesis narra el discurso que construyen cuatro acciones como mecanismos optimizadores, fundamentales, activos y necesarios cuando creamos nuevos proyectos. En este trabajo se estudian en profundidad cuatro acciones optimizadoras a través de numerosos casos de estudio. Se estudia también la presencia de estas acciones en otros campos creativos, como la biología, el arte, la literatura, la filosofía, la matemática o la psicología de la creatividad. Se busca qué tienen en común estas cuatro acciones y se indaga sobre la posible narración que construyen entre ellas. La mayor parte de los textos que constituyen este trabajo se escriben en un formato próximo al del ensayo, empleando tiempos verbales presentes evitando los tiempos verbales pretéritos o imperfectos para potenciar la acción a través el estilo narrativo. La investigación se ha realizado a partir de fuentes bibliográficas existentes en numerosas bibliotecas. Se han llevado a cabo estudios de campo realizados a través de entrevistas personales a interlocutores expertos, no sólo de teoría arquitectónica sino también de prácticas constructivas, así como visitas a lugares íntimamente relacionados con el tema de investigación. Se ha completado el estudio de casos con ejercicios prácticos realizados por el autor de esta tesis, para profundizar con la propia investigación por empatía con los autores estudiados. La investigación bibliográfica principal se ha desarrollado en las bibliotecas de la Universidad Politécnica de Madrid, de la Universidad de Alicante, de la Universidad Autónoma de Madrid, de la Universidad Europea de Madrid, de la Universidad Camilo José Cela, de la Universidad Complutense de Madrid, de la Columbia University, de la Harvard University, de la Delft University, de Heidelberg University, de la Biblioteca Central de Madrid y de la Regional de Murcia, así como de la del COAMU. También se ha utilizado recursos bibliográficos propios. La metodología utilizada muestra desde diferentes perspectivas el problema de las acciones optimizadoras, desde obras artísticas, pasando por ejemplos de arquitectura construida hasta ejercicios puramente intelectuales. La acumulación ha sido el método de obtención de conocimiento de esta tesis. Se han acumulado conocimientos y posteriormente se ha profundizado, reflexionando sobre los datos que se han ido obteniendo. Al profundizar se enlazan unas pruebas con otras, hilando y ensamblándolas en un discurso que hace continuo y coherente la aparición de cada caso estudiado. Estas entidades probatorias son acciones que han sido aplicadas frecuentemente por distintas generaciones de autores que proyectan utilizando alguna de estas cuatro acciones. Partimos de una extensa bibliografía general y de otra específica. A través de citas e imágenes se muestra el repertorio de objetos y textos estudiados. Los casos de estudio seleccionados exponen los efectos que produce cada acción en el ejercicio del proyectar. Se ha estudiado la necesidad de cada acción en todas y cada una de las partes del ciclo creativo del proyecto, tanto en prácticas imaginadas como en construidas, de los autores que proyectan. Se citan y se interpretan las descripciones de biólogos, sociólogos, antropólogos, psicólogos, escritores, artistas, arquitectos, matemáticos, ingenieros, físicos, médicos y filósofos en los cuales estas acciones se encuentran conscientemente incorporadas en su procedimiento de proyectar y de pensar. Por último, hemos obtenido unos resultados adecuados a la metodología empleada y a los objetivos planteados gracias a la acumulación y clasificación de pruebas. Los resultados se exponen a modo de discursos conclusivos con un intencionado carácter abierto que despliega nuevas posibles nuevas vías de investigación entorno a los temas estudiados. ABSTRACT. The research seeks to show four commonly used actions in designing architecture. Thesis recounts the speech that built four actions like optimizer, fundamental, active and necessary mechanisms when we create new projects. In this work it studies in depth four optimizer actions through numerous case studies. Also, it considers the presence of these actions in other creative fields, such as biology, art, literature, philosophy, mathematics or psychology of creativity. It is intended what these four actions have in common and it explores the possible narrative constructed among them. Most of the texts that constitute this work are written in a format close to the essay, using present tenses avoiding past or imperfect tenses of enhancing the action through the narrative style. Research has been done from literature sources available in numerous libraries. Field studies have been carried out through personal interviews with expert speakers, not just of architectonic theory but also from constructive practices, as well as visits to sites closely related to the research topic. case studies with practical exercises conducted by the author of this thesis has been completed, to deepen with the own research by empathy with the studied authors. Main bibliographical investigation has been developed in the libraries of UPM, UA, UAM, UEM, the CJC, the UCM, Columbia University, Harvard University Delft University, Heidelberg University, Madrid Central Library, Regional Murcia Library and COAMU Library. Also it has been used own bibliographical resources. Methodology shows from different perspectives the problem of optimizers actions, from art, passing through examples of architecture built up to puré intellectual exercise. Accumulation has been the method of obtaining knowledge of this thesis. it has been accumulated knowledge and later it has been deepened, reflecting on the data that have been obtained. By deepening tests are linked with other, spinning and locking into a discourse that makes continuous and consistent the development of each case study. These evidentiary entities are actions that have been frequently applied by different generations of authors who project using some of these four stocks. We leave from an extensive general bibliography and another specific one. Through quotes and pictures it shows the repertoire of objects and texts studied. The selected study cases set out the effects that each action produces in the exercise of projecting. It has studied the need of each action in every parts of creative cycle of the project, both imagined as constructed practices, by the authors who project. It is quoted and interpreted the descriptions of biologists, sociologists, anthropologists, psychologists, writers, artists, architects, mathematicians, engineers, physicists, physicians and philosophers in which these actions are consciously incorporated into his projecting and thinking procedure. Finally, we have obtained adequate results to the used methodology and to the stated objectives through the accumulation and classification of evidence. The results are presented as conclusive speeches with an intentional open character that unfolds new possible research routes around the studied topics.

A new boundary condition solved with B.I.E.M.

Among the classical operators of mathematical physics the Laplacian plays an important role due to the number of different situations that can be modelled by it. Because of this a great effort has been made by mathematicians as well as by engineers to master its properties till the point that nearly everything has been said about them from a qualitative viewpoint. Quantitative results have also been obtained through the use of the new numerical techniques sustained by the computer. Finite element methods and boundary techniques have been successfully applied to engineering problems as can be seen in the technical literature (for instance [ l ] , [2], [3] . Boundary techniques are especially advantageous in those cases in which the main interest is concentrated on what is happening at the boundary. This situation is very usual in potential problems due to the properties of harmonic functions. In this paper we intend to show how a boundary condition different from the classical, but physically sound, is introduced without any violence in the discretization frame of the Boundary Integral Equation Method. The idea will be developed in the context of heat conduction in axisymmetric problems but it is hoped that its extension to other situations is straightforward. After the presentation of the method several examples will show the capabilities of modelling a physical problem.

On Galileo’s Tallest Column

The height at which an unloaded column will fail under its own weight was calculated for first time by Galileo for cylindrical columns. Galileo questioned himself if there exists a shape function for the cross-section of the column with which the latter can attains a greater height than the cylindrical column. The problem is not solved since then, although the definition of the so named “constant maximum strength” solids seems to give an affirmative answer to Galileo’s question, in the form of shapes than can attains infinite height, even when loaded with a useful load at the top. The main contribution of this work is to show that Galileo’s problem is (i) an important problem for structural design theory of buildings and other structures, (ii) not solved by the time being in any sense and (iii) a interesting problem for mathematicians involved in related but very different problems (as Euler’s tallest column). A contemporary formulation of the problem is included as a result of a research on the subject.

Biomimetismo entre los endemismos arquitectónicos y taxones en Cantabria

Es notoria la importancia de la naturaleza en la evolución de la arquitectura. Hasta la gran eclosión de la industria manufacturera y la mejora en las infraestructuras de comunicación, se podría decir que ambas corrían paralelas, para a partir de ahí dejar “congelada” la sabiduría popular y dar un salto a la globalidad, generando una situación de libertad arquitectónica prácticamente total, que independiza la construcción de su ubicación. "El biomimetismo es esencialmente un campo de investigación interdisciplinar, una serie de colaboraciones entre botánicos, físicos, matemáticos, ingenieros y zoólogos; donde la rígida división entre disciplinas «puras» cede lugar a un área de investigación que apunta a generar tecnología inteligente (smarttechnologies), utilizando materiales o procesos que sean de alguna manera sensibles al medio ambiente." (M. Weinstock, 1998). “La morfología de las plantas en los diferentes climas parece tener cierta analogía con la edificación, ya que algunas de las tensiones que inciden en su forma (tales como las variaciones de temperatura) corresponden de manera similar a las necesidades humanas.” (V. Olgyay 1963). En el presente trabajo se han estudiado las especies endémicas que nos rodean, para poder leer a través de ellas millones de años de supervivencia en este entorno, con el fin de mimetizar sus respuestas. También se han estudiado diferentes tipologías de arquitectura vernácula y su simulación energética, con el propósito de evaluar la demanda energética optima exigible. A partir de una ubicación específica, la orientación, compacidad, perforación y las características de la envolvente son los elementos que más influyen en la demanda energética de una edificación. Tanto la forma como los materiales pueden ser mimetizados con la naturaleza. En esta Tesis se han cuantificado los parámetros de diseño formales tomando como referencia las especies vegetales o la arquitectura vernácula, sin perder de vista los objetivos buscados por normativas o institutos en la reducción del consumo energético vinculado a la calefacción y ventilación. ABSTRACT The importance of the nature in the evolution of the architecture is well-known. Until the great burst of the manufacturing industry and the improvement in communication infrastructures, it would be possible to be said that both ran parallel, stops there from leaving “frozen” the popular wisdom and jump to the globalization, creating a situation almost complete architectural freedom, that it frees the construction of its location. "Biomimicry is essentially an interdisciplinary field of research, a series of collaborations among botanists, physicists, mathematicians, engineers and zoologists; where the rigid division between "pure" disciplines gives way to an area of research that aims to generate intelligent technology (smarttechnologies), using materials and processes that are in some environmentally sensitive manner. "(M. Weinstock, 1998). “The morphology of the plants in different climates seems to have some analogy with the building, as some of the tensions that affect their form (such as the temperature variations) are similar to the human necessities.” (V. Olgyay 1963). In the present work, the endemic species that surround to us have been studied, to be able to read through them millions of years of survival in this environment, in order to mimic their answers. Also different types or popular architecture and their energy simulation have been studied, in order to evaluate the rate of energy optimum demand. Orientation, compactness, perforation and characteristics of the envelope are the elements that influence more in the energy demand of a building. The shape and materials can be mimic with nature. Each of them has been quantified in this work by reference plant species or popular architecture, without losing sight of the objectives sought by regulations or institutes about reduction in energy consumption.

The multivariate L1-median and associated data depth

This paper gives three related results: (i) a new, simple, fast, monotonically converging algorithm for deriving the L1-median of a data cloud in ℝd, a problem that can be traced to Fermat and has fascinated applied mathematicians for over three centuries; (ii) a new general definition for depth functions, as functions of multivariate medians, so that different definitions of medians will, correspondingly, give rise to different dept functions; and (iii) a simple closed-form formula of the L1-depth function for a given data cloud in ℝd.