A singular function and its relation with the number systems involved in its definition

Minkowski's ?(x) function can be seen as the confrontation of two number systems: regular continued fractions and the alternated dyadic system. This way of looking at it permits us to prove that its derivative, as it also happens for many other non-decreasing singular functions from [0,1] to [0,1], when it exists can only attain two values: zero and infinity. It is also proved that if the average of the partial quotients in the continued fraction expansion of x is greater than k* =5.31972, and ?'(x) exists then ?'(x)=0. In the same way, if the same average is less than k**=2 log2(F), where F is the golden ratio, then ?'(x)=infinity. Finally some results are presented concerning metric properties of continued fraction and alternated dyadic expansions.Singular function, number systems, metric number theory

Fermat's treatise on quadrature: A new reading

The Treatise on Quadrature of Fermat (c. 1659), besides containing the first known proof of the computation of the area under a higher parabola, R x+m/n dx, or under a higher hyperbola, R x-m/n dx— with the appropriate limits of integration in each case—, has a second part which was not understood by Fermat’s contemporaries. This second part of the Treatise is obscure and difficult to read and even the great Huygens described it as 'published with many mistakes and it is so obscure (with proofs redolent of error) that I have been unable to make any sense of it'. Far from the confusion that Huygens attributes to it, in this paper we try to prove that Fermat, in writing the Treatise, had a very clear goal in mind and he managed to attain it by means of a simple and original method. Fermat reduced the quadrature of a great number of algebraic curves to the quadrature of known curves: the higher parabolas and hyperbolas of the first part of the paper. Others, he reduced to the quadrature of the circle. We shall see how the clever use of two procedures, quite novel at the time: the change of variables and a particular case of the formula of integration by parts, provide Fermat with the necessary tools to square very easily curves as well-known as the folium of Descartes, the cissoid of Diocles or the witch of Agnesi.History of mathematics, quadratures, integration methods

Approximation of quadratic irrationals and their pierce expansions

In this article two aims are pursued: on the one hand, to present a rapidly converging algorithm for the approximation of square roots; on the other hand and based on the previous algorithm, to find the Pierce expansions of a certain class of quadratic irrationals as an alternative way to the method presented in 1984 by J.O. Shallit; we extend the method to find also the Pierce expansions of quadratic irrationals of the form $2 (p-1) (p - \sqrt{p^2 - 1})$ which are not covered in Shallit's work.Quadratic irrationals, Pierce series

A total order in [0,1] defined through a 'next' operator

A `next' operator, s, is built on the set R1=(0,1]-{ 1-1/e} defining a partial order that, with the help of the axiom of choice, can be extended to a total order in R1. Besides, the orbits {sn(a)}n are all dense in R1 and are constituted by elements of the same arithmetical character: if a is an algebraic irrational of degree k all the elements in a's orbit are algebraic of degree k; if a is transcendental, all are transcendental. Moreover, the asymptotic distribution function of the sequence formed by the elements in any of the half-orbits is a continuous, strictly increasing, singular function very similar to the well-known Minkowski's ?(×) function.Total orders, pierce series, singular functions

Dynamics of motivation and cohesion in a basketball team

Treball Final de Grau en Psicologia. Codi: PS1048. Curs: 2018/2019.This study is within the scope of sports psychology, specifically it has focused on the constructs of motivation and cohesion. The hypothesis is to see if the application of a program of 7 dynamics causes an increase in the variables of motivation and cohesion. To evaluate these two constructs, the Cuestionario Características Psicológicas del Rendimiento Deportivo (CPRD) Questionnaire was used, specifically the two factors of motivation and cohesion. The sample used for this research were two basketball teams in the cadet category between 14 and 16 years old, being 18 at first, although due to mortality of the sample it was reduced to 13 subjects. The design of this research is an A-B-A design in which prior to the preintervention evaluation, consent was sought from the tutors of the minors and the players were informed of the work to be done. The program with 7 dynamics was divided into two weeks, each of them lasting between 20-30 minutes and the data collection was made to coincide with the last one. The results obtained show that there are no significant differences in any of the two variables, although in one of the items that were critical in the results of the first measure, we see a significant increase after dedicating one of the dynamics. The limitations found in this research are the lack of control group, the limited time for the application of the dynamics and the small size of the sampleEste estudio se encuentra dentro del ámbito de la psicología deportiva, concretamente se ha centrado en los constructos de motivación y cohesión. La hipótesis planteada es ver si la aplicación de un programa de 7 dinámicas provoca un aumento en las variables de motivación y cohesión. Para evaluar estos dos constructos se ha utilizado el Cuestionario Características Psicológicas del Rendimiento Deportivo (CPRD) en concreto los dos factores de motivación y cohesión. La muestra utilizada para esta investigación han sido dos equipos de baloncesto de la categoría cadete entre 14 y 16 años, siendo en un inicio 18 aunque por mortalidad de la muestra se vio reducido a 13 sujetos. El diseño de esta investigación es un diseño A-B-A en el que antes de empezar la evaluación pre intervención se pidió consentimiento a los tutores de los menores y se informó a los jugadores del trabajo a realizar. El programa de las 7 dinámicas se repartieron en dos semanas, cada una de ellas tiene una duración entre 20-30 minutos y se hizo coincidir la recogida de datos con la última de ellas. Los resultados obtenidos muestran que no hay diferencias significativas en ninguna de las dos variables, aunque en uno de los ítems que resultó crítico en los resultados en la primera medida si vemos un aumento significativo después de dedicarle una de las dinámicas. Las limitaciones encontradas en esta investigación son la falta de grupo control, el tiempo limitado para la aplicación de las dinámicas y el pequeño tamaño de la muestra

Investigación-acción: habilidades sociales en un alula de Infantil

Treball Final de Grau en Mestre o Mestra d'Educació Infantil. Codi: MI1040. Curs acadèmic: 2017/2018El trabajo que se presenta es una investigación-acción realizada en un aula de cinco años de Infantil. En esta aula se detectó un problema de habilidades sociales por parte de algunos alumnos/as, para confirmar el problema se realizó una entrevista, la observación y un Test Sociométrico. Una vez confirmado el problema se implantaron en el aula dos acciones, la reestructuración del aula y diversas dinámicas de cohesión de grupo. Esta intervención se ha realizado durante 4 semanas, utilizando dos sesiones por semana. Una vez se ha terminado la intervención se ha vuelto ha recoger información para ser analizada y comprobar si las acciones que se han realizado han sido útiles o no lo han sido. Los resultados que se han obtenido son positivos ya que se han alcanzado los objetivos que se habían marcado.This work is an action research conducted in a five-year-olds classroom. In this classroom a problem concerning social skills was identified among some students, in order to confirm the problem, a series of interviews, surveillance and a Sociometrical Test were carried out. When the problem was corroborated, two actions were developed in the classroom: the reorganization of the classroom and some dynamics of group cohesion. This intervention has taken four weeks, using two sessions per week. When the intervention finished, the information was gathered in order to carry out an analysis and to check if the carried out actions have been useful or not. The results obtained are positive because the targets set have been reached

Evaluating the impact of future memory technologies in the design of multicore processors

"It’s the Memory, Stupid!" In 1996, Richard Sites, one of the fathers of Computer Architecture and lead designer of the DEC alpha, wrote a paper [36] with the title above. In that paper he realized that the only important design issue for microprocessors in the next decade would be the memory subsystems design. After more than a decade later, the community of researchers started to digest and internalize this quote. Now, after more than two decades, it can be said that a lot of progress has been done since 1996 but the expectations of the enormous data sets that software is going to handle in the followings years tells that more aggressive designs are needed. Another reason new memory technologies are needed is because of the multicore architecture which has increased the required memory bandwidth. This architecture completely extended across the main computer sectors was the result of continuing the Moore’s law in exchange of adding more difficulties for software and hardware developers. All of this has promoted this project. First, it has been decided to create a bridge of the state of the art DRAM simulator Ramulator [30] with the micro-architecture simulator TaskSim [34]. Once the bridge has been completed, the second goal of this project has been to make an evaluation of the impact of the current and future memory technologies in multicore architectures. As a first approach, this new infrastructure has been used to evaluate the behavior of several parallel applications concluding that the execution time of the applications varies significantly across different memory technologies which even increase the differences while simulating different processors. The doubtless winner among all the memory technologies evaluated has been HBM which in some cases has achieved the best expected memory cycle response time."És la Memòria, Estúpid!" El 1996, Richard Sites, un dels pares de l’Arquitectura de Computadors i dissenyador principal del DEC Alpha, va escriure un article [36] amb el títol anterior. En aquest article es va adonar que l’única qüestió important per al disseny de microprocessadors en la dècada següent seria el disseny de subsistemes de memòria. Després de més d’una dècada, la comunitat d’investigadors va començar a digerir i assimilar aquesta cita. Ara, després de més de dues dècades, es pot dir que un gran progrés s’ha fet des de 1996, però les expectatives dels enormes conjunts de dades que el software utilitzarà en els següents anys indica que es necessiten dissenys més agressius. Una altra raó per el qual es necessiten noves tecnologies de memòria és degut a les arquitectures multinucli que augmenten l’ample de banda de memòria requerit. Aquesta arquitectura completament estesa a través dels sectors principals va ser el resultat de seguir la llei de Moore a canvi d’afegir més dificultats per als desenvolupadors de software i hardware. Tot això ha promogut aquest projecte. En primer lloc, s’ha decidit crear un pont sobre el capdavanter simulador de DRAM Ramulator [30] amb un de micro-arquitectura TaskSim [34]. Una vegada que s’ha completat el pont, el segon objectiu d’aquest projecte ha estat fer una avaluació de l’impacte de les tecnologies de memòria actuals i futures en les arquitectures multinucli. Com a primera aproximació, s’ha utilitzat aquesta nova infraestructura per avaluar el comportament de diverses aplicacions paral·leles arribant a la conclusió que el temps d’execució de les aplicacions varia significativament entre diferents tecnologies de memòria, que fins i tot augmenten les diferències amb la simulació de diferents processadors. El guanyador, sens dubte, entre totes les tecnologies de memòria ha estat HBM que en alguns casos ha aconseguit el millor temps de cicle de memòria esperat."Es la Memoria, Estupido!" En 1996, Richard Sites, uno de los padres de la Arquitectura de Computadores y diseñador principal del DEC alpha, escribió un artículo [36] con el título anterior. En ese artículo se dio cuenta de que el único problema de diseño importante para los microprocesadores en la próxima década sería el diseño del subsistema de memoria. Una década más tarde, la comunidad de investigadores comenzó a digerir e interiorizar esta cita. Ahora, después de más de dos décadas, se puede decir que se han hecho muchos progresos desde 1996, pero las expectativas de los enormes conjuntos de datos que el software va a manejar en los próximos años indican que se necesitan diseños más agresivos. Otra razón por la que se necesitan nuevas tecnologías de memoria es debido a la arquitectura multinúcleo que aumenta el ancho de banda de memoria requerido. Esta arquitectura completamente extendida a través de los principales sectores fue el resultado de continuar la ley de Moore a cambio de añadir más dificultades para los desarrolladores de software y hardware. Todo esto ha promovido este proyecto. En primer lugar, se ha decidido crear un puente entre el potente simulador de DRAM Ramulator [30] con uno de micro-arquitecura TaskSim [34]. Una vez que el puente se ha completado, el segundo objetivo de este proyecto ha sido hacer una evaluación del impacto de las tecnologías de memoria actuales y futuras en arquitecturas multinúcleo. Como primera aproximación, se ha utilizado esta nueva infraestructura para evaluar varias aplicaciones paralelas llegando a la conclusión de que el tiempo de ejecución de las aplicaciones varía significativamente entre las diferentes tecnologías de memoria y que incluso aumentan las diferencias al simular diferentes procesadores. El ganador sin duda entre todas las tecnologías de memoria evaluadas ha sido HBM que en algunos casos ha logrado el mejor tiempo de ciclo de memoria de respuesta esperado

Is there a (new) bubble in the US housing market? An empirical specification to identify rational housing bubbles

Treballs Finals del Màster d'Economia, Facultat d'Economia i Empresa, Universitat de Barcelona. Curs: 2020-2022, Tutor: Sergi Basco MascaróFor the last two years (2020 and 2021), real home prices have risen enormously in the United States. Indeed, the peak prices of 2021 were even more prominent than those experienced during the last housing bubble, which burst in 2006. Thereby, one relevant question strongly arises: has it emerged a new housing bubble in the US economy? Based on previous academic rational bubble theoretical frameworks and the current account identity, this study demonstrates empirically that such a recent and extreme appreciation trend is not a consequence of a rational housing bubble. Through an OLS panel regression model it is applied an empirical test on three different periods of interest: the Dot-Com Bubble (1996-2000), the Housing Bubble (2002-2006), and the COVID-19 explored episode (2020-2021). From 2020 to 2021 and during the Dot-Com Bubble, an insignificant statistical influence of the US foreign capital inflows is detected on the real house price evolution. While at the same time, in the course of the Housing Bubble, such an effect of the international capital movements on the housing appreciation is statistically significant. Following the previous literature, this empirical evidence identifies the Housing Bubble as the only analysed period containing a rational bubble in the housing market. Consequently, as a result, the Dot-Com Bubble and the years under COVID-19 are not detected as rational housing bubble episodes. This categorization of the Dot-Com Bubble and the Housing Bubble is consistent with earlier academic research. Hence, the reported novel findings related to 2020 and 2021 are endowed with important credibility since the used empirical strategy can accurately classify prior documented rational bubble events. Moreover, these results exhibit a solid validity after applying several robustness tests. Thus, this analysis offers the first comprehensive empirical evidence postulating that a rational bubble did not emerge in the US housing market during the first two years of the COVID-19 pandemic