Finite difference and finite element methods for partial differential equations on fractals

In this paper, we present numerical procedures to compute solutions of partial differential equations posed on fractals. In particular, we consider the strong form of the equation using standard graph Laplacian matrices and also weak forms of the equation derived using standard length or area measure on a discrete approximation of the fractal set. We then introduce a numerical procedure to normalize the obtained diffusions, that is, a way to compute the renormalization constant needed in the definitions of the actual partial differential equation on the fractal set. A particular case that is studied in detail is the solution of the Dirichlet problem in the Sierpinski triangle. Other examples are also presented including a non-planar Hata tree.In this paper, we present numerical procedures to compute solutions of partial differential equations posed on fractals. In particular, we consider the strong form of the equation using standard graph Laplacian matrices and also weak forms of the equation derived using standard length or área measure on a discrete approximation of the fractal set. We then introduce a numerical procedure to normalize the obtained diffusions, that is, a way to compute the renormalization constant needed in the definitions of the actual partial differential equation on the fractal set. A particular case that is studied in detail is the solution of the Dirichlet problem in the Sierpinski triangle. Other examples are also presented including a non-planar Hata tree

Nonlinear waves in a chain of magnetically coupled pendula

A motivation for the study of reduced models like one-dimensional systems in Solid State Physics is the complexity of the full problem. In recent years our group has studied theoretically, numerically and experimentally wave propagation in lattices of nonlinearly coupled oscillators. Here, we present the dynamics of magnetically coupled pendula lattices. These macroscopic systems can model the dynamical processes of matter or layered systems. We report the results obtained for harmonic wave propagation in these media, and the different regimes of mode conversion into higher harmonics strongly influenced by dispersion and discreteness, including the phenomenon of acoustic dilatation of the chain, as well as some results on the propagation of localized waves i.e., solitons and kinks.Generalitat Valenciana APOSTD/2017/042Umiversitat Politècnica de València PAID-01-14Ministerio de Economía y Competitividad (MINECO), Spain FIS2015-65998-C2-2-PJunta de Andalucía 2017/FQM-28

Evaluando la competencia de análisis epistémico de profesores de matemáticas

En el marco de un proyecto de innovación e investigación para el desarrollo de competencias de análisis didáctico del profesor de matemáticas, se ha diseñado una intervención formativa para un grupo de estudiantes de un máster para profesores de secundaria. El diseño contempla la lectura y discusión de un documento en el que se presentan nociones básicas del análisis de objetos y procesos implicados en la práctica matemática, distinguiendo las relaciones dialécticas entre objetos ostensivos y no ostensivos, así como el estudio y discusión de ejemplos ilustrativos de dicho análisis. En esta comunicación presentamos los resultados de la fase de evaluación y análisis retrospectivo de dicho diseño formativo, basada en la resolución de una tarea matemática, seguida del análisis de los objetos y significados puestos en juego por los estudiantes en la realización de la misma. Los resultados revelan algunos hechos didácticos significativos relativos a la faceta cognitiva del proceso didáctico implementado

Onto-semiotic configurations underlying diagrammatic reasoning

Diagrams and in general the use of visualization and manipulative material, play an important role in mathematics teaching and learning processes. Although several authors warn that mathematics objects should be distinguished from their possible material representations, the relations between these objects are still conflictive. In this paper, some theoretical tools from the onto-semiotic approach of mathematics knowledge are applied to analyse the diversity of objects and processes involved in mathematics activity, which is carried out using diagrammatic representations. This enables us to appreciate the synergic relations between ostensive and non-ostensive objects overlapping in mathematics practices. The onto-semiotic analysis is contextualised in a visual proof of the Pythagorean theorem

Perspectiva ontosemiótica de la visualización espacial y el razonamiento diagramático

Los diagramas, y en general el uso de visualizaciones y materiales manipulativos, desempeñan un papel importante en los procesos de enseñanza y aprendizaje de las matemáticas. En este trabajo aplicamos algunas herramientas teóricas del enfoque ontosemiótico del conocimiento matemático para analizar la diversidad de objetos y procesos implicados en la actividad matemática, que se realiza con apoyo de representaciones diagramáticas. Esto permite apreciar las relaciones sinérgicas entre los objetos ostensivos (lenguajes visuales y secuenciales) y los objetos no ostensivos (entidades abstractas y mentales) imbricados en las prácticas matemáticas

Onto-semiotic analysis of visualization and diagrammatic reasoning tasks

The explicit recognition of the objects and processes implied in mathematic activity is a competence that teachers should develop. This competence allows teachers to understand, design, and manage the processes of mathematical learning, and assess them with suitability standards previously set. Consequently, formative processes to develop this competence should be designed. This paper describes a training design aimed at developing the teachers’ competence of epistemic and cognitive analysis of mathematics instruction processes, highlighting the role of visual and analytical languages in the constitution of mathematical objects

Análisis de la actividad matemática mediante dos herramientas teóricas: registros de representación semiótica y configuración ontosemiótica

Para comprender las dificultades y conflictos de aprendizaje, es necesario analizar las tareas matemáticas y los diversos modos de abordarlas por los estudiantes. Dicho análisis, que precisa herramientas teóricas específicas para su realización, aporta información útil para el propio diseño de las tareas y la gestión de los conocimientos en el aula. En este trabajo realizamos el análisis de una tarea que requiere la formulación de una conjetura y su demostración haciendo uso de representaciones figurales y algebraicas, aplicando dos herramientas teóricas diferentes: las nociones de registro de representación semiótica y de configuración ontosemiótica. Los resultados revelan algunas complementariedades que nos permitieron mostrar la potencial utilidad de los análisis epistémico y cognitivo realizados. Se trata de mostrar la potencial sinergia existente entre dichas herramientas y la posibilidad de avanzar en la articulación de los marcos teóricos correspondientes

Relationship between serotypes, age, and clinical presentation of invasive pneumococcal disease in Madrid, Spain, after introduction of the 7-valent pneumococcal conjugate vaccine into the vaccination calendar

To assess invasive pneumococcal disease (IPD) clinical presentations and relationships with age and serotype in hospitalized children (<15 years) after PCV7 implementation in Madrid, Spain, a prospective 2-year (May 2007 to April 2009) laboratory-confirmed (culture and/or PCR) IPD surveillance study was performed (22 hospitals). All isolates (for serotyping) and culture-negative pleural/cerebrospinal fluids were sent to the reference laboratory for pneumolysin (ply) and autolysin (lyt) gene PCR analysis. A total of 330 IPDs were identified: 263 (79.7%) confirmed by culture and 67 (20.3%) confirmed by PCR. IPD distribution by age (months) was as follows: 23.6% (<12), 15.8% (12 to 23), 15.5% (24 to 35), 22.4% (36 to 59), and 22.7% (>59). Distribution by clinical presentation was as follows: 34.5% bacteremic pneumonia, 30.3% pediatric parapneumonic empyema (PPE), 13.6% meningitis, 13.3% primary bacteremia, and 8.2% others. Meningitis and primary bacteremia were the most frequent IPDs in children <12 months old, and bacteremic pneumonia and PPE were most frequent in those >36 months old. Frequencies of IPD-associated serotypes were as follows: 1, 26.1%; 19A, 18.8%; 5, 15.5%; 7F, 8.5%; 3, 3.9%; nontypeable/ other 30 serotypes, 27.3%. Serotype 1 was linked to respiratory-associated IPD (38.6% in bacteremic pneumonia and 38.0% in PPE) and children of >36 months (51.4% for 36 to 59 months and 40.0% for >59 months), while serotype 19A was linked to nonrespiratory IPDs (31.1% in meningitis, 27.3% in primary bacteremia, and 51.9% in others) and children of <24 months (35.9% for children of <12 months and 36.5% for those 12 to 23 months old), with high nonsusceptibility rates for penicillin, cefotaxime, and erythromycin. After PCV7 implementation, non-PCV7 serotypes caused 95.5% of IPDs. The new 13-valent conjugate vaccine would provide 79.1% coverage of serotypes responsible for IPDs in this series