De solutione problematum diophanteorum per números integros : o primeiro trabalho de Euler sobre equações diofantinas

The present dissertation analyses Leonhard Euler´s early mathematical work as Diophantine Equations, De solutione problematum diophanteorum per números íntegros (On the solution of Diophantine problems in integers). It was published in 1738, although it had been presented to the St Petersburg Academy of Science five years earlier. Euler solves the problem of making the general second degree expression a perfect square, i.e., he seeks the whole number solutions to the equation ax2+bx+c = y2. For this purpose, he shows how to generate new solutions from those already obtained. Accordingly, he makes a succession of substitutions equating terms and eliminating variables until the problem reduces to finding the solution of the Pell Equation. Euler erroneously assigns this type of equation to Pell. He also makes a number of restrictions to the equation ax2+bx+c = y and works on several subthemes, from incomplete equations to polygonal numbers

A representação social de disciplinas didático-pedagógicas no contexto da formação inicial docente: um entremear de saberes

We seek, through this study, to analyze about social representation that the students of licentiate degree course of the Federal Institute of Education, Science and Technology of Rio Grande do Norte - IFRN - have about didactic-pedagogical subjects. We utilize the Social Representations Theory (MOSCOVICI, 2009; 2012; JODELET, 2001) as a theoretical and methodological contribution and as aim, we identify this social representation and understand how it is influenced by the formation of these undergraduates. So we developed the research under the seven undergraduate classroom courses offered by IFRN, namely: Biology, Spanish, Physics, Geography, Computer Science, Mathematics and Chemistry, covering units located both in the capital and in the countryside. While methodological approach we used the Procedure of Multiple Classification (PMC) - (ROAZZI, 1995), whose realization requires a set of words achieved through of Free Technique of Words Association - FTWA - (ABRIC, 1998). For this realization we have a total of one hundred twenty (120) participants, with thirty (30) in FTWA and other stage in the realization of free classification and directed that correspond to the MPR. Achieved the empirical data, we use the analysis of content (BARDIN, 2011; FRANCO, 2007) and multidimensional (ROAZZI, 1995) for the course of his interpretation. Finally, we identify the social representation of didactic and pedagogical subjects centered around the idea that it is through these subjects that can achieve the profile of "good professional" as one who gathers knowledge and attributes required for the full development of teaching involving capabilities it and characteristics that mark the sense of professionalism. Furthermore, we found that the anchoring social representation on the understanding that these disciplines "teach the teacher to be" in the image and its objectification of the "good teacher", ie, one that reaches through the training process and experiences, qualities that make it unique and able to carry full professional. Still see that the actions of the teacher trainer affects the way students perceive, assume and engage in the study of teaching and pedagogical subjects and it reflects significantly the social representation then created

Formação continuada e representação social: implicações para a educação inclusiva

This study is part of the line of research on Teacher Education, circumscribed to the Graduate Program in Education of Federal University Rio Grande do Norte. In it, we have a qualitative research, whose object returned to the influence of continued education in social representation of inclusive education of Elementary School teachers. The same was done in an Elementary School of Cruzeiro do Sul city (Brasil/Acre) and performed with the participation of 11 teachers. We assume that thinking a continuing education inclusive presupposes reflect on the redefinition and resignification of practices undertaken by the teacher. Therefore, the teacher education is conceived as a continuum in which the teacher will be graduating in forming, shaping their way of being in social situations. We argue that in the context of interactions, we are going to reconstruct social representations that guide actions and attitudes about the objects that have meaning for the group. It was with the aim of identify the influence of these social representations and at the same time submit a proposal for continuing training in structured formative needs teachers who made use of collaborative action research. Since we believe that this can either meet the assumptions of a research and vocational training to enable that researcher and other teachers, collectively, problematizing situations of practice and, at the same time able to intervene in the same it. As procedures for obtaining information and analysis of the data, we performed, respectively, semi-structured interviews (individual and collective); reflective studies sessions and the analysis of content. The path of research revealed two social representations integrated one of inclusive education and another for student with disabilities. We noticed that although the teachers criticize the character sealed and limited of continued education courses that have frequented, reveal social representations of inclusive education and students with disabilities based on theories and concepts originated from such courses. As a result, represent inclusive education as an "education for all" and student with disabilities as someone "abnormal, different, but able to learn something . It is possible to state that from the moment the teachers approched theories about the principles of inclusive education and on the possibilities of student with disabilities took new discourses, and showed signs of change in the social representations before presented. In addition, we evidenced in this production that a proposal for training within the school, fomented by the needs teachers and having as methodology to collaborative action research, provides the (re)construction of collective knowledges and practices, sharing of experiences, hopes, discoveries and concerns and the development of cooperation and attitude for the (re)development of strategies to overcome the difficulties present in everyday teaching

A Tessitura de sentidos: representação social de licenciados do curso deletras da UFRN sobre o ensinar

En esta investigación, se busca comprender los procesos de formación de pregrado del Curso de Cartas da UFRN, de estas representaciones sociales acerca de los estudiantes de enseñanza. Por lo tanto, el objetivo fue conocer y analizar las representaciones sociales de curso de pregrado UFRN Cartas sobre la enseñanza. Por lo tanto, se aplica la Técnica de Libre de Palabras Asociación (TALP) con 20 participantes, después de haber sido probados individualmente, utilizando como estímulo palabra "enseñar" y limitar el número de invocaciones a 03 palabras por participante. Esta técnica fue elegida porque es una técnica proyectiva que permite la reducción y racionalización permite acceder al campo semántico de las representaciones sociales, en nuestro caso, en la enseñanza del de pregrado do Curso de Cartas da UFRN. Por lo tanto, llegamos a total de 60 llamadas, análisis de frecuencia después de 41 y 24 después de un análisis del significado que se utilizaron en otro procedimiento aplicado en esta investigación, las calificaciones de Procedimiento de Múltiples Calificaciones (PCM). Estas palabras, sobre la base de las justificaciones de las evocaciones y significados que atribuyen participantes en la investigación, se dividieron en 4 campos semánticos: Campo 1: Aspectos técnicos de la enseñanza, Campo 2: aspectos volitivos, Campo 3: aspectos relacionales de la enseñanza y de Campo 4: Elementos del campo. El PCM se aplicó de forma individual, con 50 estudiantes. Para el análisis de los datos generados, utilizamos Análisis Multidimensional (MDS) para la Escala Multidimensional de Evaluación Análisis gratuito (MSA: Multidimensional Scalogram Analysis) y el análisis de clasificación selectiva de los espacios menores (SSA: Smallest Space Analysis). El mapa generado por el análisis de MSA fue dividido en tres regiones: Aspectos de los aspectos de enseñanza, capacitación y asistencia técnica de los promotores de la profesionalización, la primera región que tiene una sub-regionalización llamada dimensión negativa. El mapa de la SSA análisis se dividió en dos partes y un subfaceta, llamado el aspecto faceta técnica de la enseñanza y la formación, la segunda faceta llamado Vínculo entre el campo y la enseñanza específica y su nombre subfaceta aspecto relacional. Del análisis del contenido de las justificaciones de la PCM, llegó a 5 categorías, a saber: Hacer la enseñanza, la faceta negativa, Escuela, Formación y faceta afectiva del profesor. El TALP aporta pruebas de que algunos de estos estudiantes ve la oportunidad de adquirir unos conocimientos en esta área, la formación en un nivel superior, como quieran, simplemente tienen un título universitario para realizar licitación abierta. El PCM nos ha hecho comprender que el contenido representacional en cuestión se configura desde el universo consensual conectado a la profesión y la formación. Por lo tanto, podemos decir que los titulares de licencias con respecto al tema estudiado anclar la representación social de "enseñar" a la imagen de ser un maestro y objetivar la imagen de la profesión

Desempenho e qualidade de frutos de tomateiro em cultivo protegido com diferentes números de hastes

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Uma fundamentação matemática para processamento digital de sinais intervalares

This work deals with a mathematical fundament for digital signal processing under point view of interval mathematics. Intend treat the open problem of precision and repesention of data in digital systems, with a intertval version of signals representation. Signals processing is a rich and complex area, therefore, this work makes a cutting with focus in systems linear invariant in the time. A vast literature in the area exists, but, some concepts in interval mathematics need to be redefined or to be elaborated for the construction of a solid theory of interval signal processing. We will construct a basic fundaments for signal processing in the interval version, such as basic properties linearity, stability, causality, a version to intervalar of linear systems e its properties. They will be presented interval versions of the convolution and the Z-transform. Will be made analysis of convergences of systems using interval Z-transform , a essentially interval distance, interval complex numbers , application in a interval filter.

Localização e mapeamento simultâneos de ambientes planos usando visão monocular e representação híbrida do ambiente

The goal of this work is to propose a SLAM (Simultaneous Localization and Mapping) solution based on Extended Kalman Filter (EKF) in order to make possible a robot navigates along the environment using information from odometry and pre-existing lines on the floor. Initially, a segmentation step is necessary to classify parts of the image in floor or non floor . Then the image processing identifies floor lines and the parameters of these lines are mapped to world using a homography matrix. Finally, the identified lines are used in SLAM as landmarks in order to build a feature map. In parallel, using the corrected robot pose, the uncertainty about the pose and also the part non floor of the image, it is possible to build an occupancy grid map and generate a metric map with the obstacle s description. A greater autonomy for the robot is attained by using the two types of obtained map (the metric map and the features map). Thus, it is possible to run path planning tasks in parallel with localization and mapping. Practical results are presented to validate the proposal

Uma fundamentação para sinais e sistemas intervalares

In this work we use Interval Mathematics to establish interval counterparts for the main tools used in digital signal processing. More specifically, the approach developed here is oriented to signals, systems, sampling, quantization, coding and Fourier transforms. A detailed study for some interval arithmetics which handle with complex numbers is provided; they are: complex interval arithmetic (or rectangular), circular complex arithmetic, and interval arithmetic for polar sectors. This lead us to investigate some properties that are relevant for the development of a theory of interval digital signal processing. It is shown that the sets IR and R(C) endowed with any correct arithmetic is not an algebraic field, meaning that those sets do not behave like real and complex numbers. An alternative to the notion of interval complex width is also provided and the Kulisch- Miranker order is used in order to write complex numbers in the interval form enabling operations on endpoints. The use of interval signals and systems is possible thanks to the representation of complex values into floating point systems. That is, if a number x 2 R is not representable in a floating point system F then it is mapped to an interval [x;x], such that x is the largest number in F which is smaller than x and x is the smallest one in F which is greater than x. This interval representation is the starting point for definitions like interval signals and systems which take real or complex values. It provides the extension for notions like: causality, stability, time invariance, homogeneity, additivity and linearity to interval systems. The process of quantization is extended to its interval counterpart. Thereafter the interval versions for: quantization levels, quantization error and encoded signal are provided. It is shown that the interval levels of quantization represent complex quantization levels and the classical quantization error ranges over the interval quantization error. An estimation for the interval quantization error and an interval version for Z-transform (and hence Fourier transform) is provided. Finally, the results of an Matlab implementation is given

Mapeamento robótico 2,5-D com representação em grade de ocupação-elevação

This work introduces a new method for environment mapping with three-dimensional information from visual information for robotic accurate navigation. Many approaches of 3D mapping using occupancy grid typically requires high computacional effort to both build and store the map. We introduce an 2.5-D occupancy-elevation grid mapping, which is a discrete mapping approach, where each cell stores the occupancy probability, the height of the terrain at current place in the environment and the variance of this height. This 2.5-dimensional representation allows that a mobile robot to know whether a place in the environment is occupied by an obstacle and the height of this obstacle, thus, it can decide if is possible to traverse the obstacle. Sensorial informations necessary to construct the map is provided by a stereo vision system, which has been modeled with a robust probabilistic approach, considering the noise present in the stereo processing. The resulting maps favors the execution of tasks like decision making in the autonomous navigation, exploration, localization and path planning. Experiments carried out with a real mobile robots demonstrates that this proposed approach yields useful maps for robot autonomous navigation

Extração e Representação de Conhecimento de Séries Temporais de Demanda de Energia Elétrica Usando TSKR

The opening of the Brazilian market of electricity and competitiveness between companies in the energy sector make the search for useful information and tools that will assist in decision making activities, increase by the concessionaires. An important source of knowledge for these utilities is the time series of energy demand. The identification of behavior patterns and description of events become important for the planning execution, seeking improvements in service quality and financial benefits. This dissertation presents a methodology based on mining and representation tools of time series, in order to extract knowledge that relate series of electricity demand in various substations connected of a electric utility. The method exploits the relationship of duration, coincidence and partial order of events in multi-dimensionals time series. To represent the knowledge is used the language proposed by Mörchen (2005) called Time Series Knowledge Representation (TSKR). We conducted a case study using time series of energy demand of 8 substations interconnected by a ring system, which feeds the metropolitan area of Goiânia-GO, provided by CELG (Companhia Energética de Goiás), responsible for the service of power distribution in the state of Goiás (Brazil). Using the proposed methodology were extracted three levels of knowledge that describe the behavior of the system studied, representing clearly the system dynamics, becoming a tool to assist planning activities

Números complexos: uma proposta de mudança metodológica para uma aprendizagem significativa no ensino médio

This work presents a proposal of a methodological change to the teaching and learning of the complex numbers in the Secondary education. It is based on the inquiries and difficulties of students detected in the classrooms about the teaching of complex numbers and a questioning of the context of the mathematics teaching - that is the reason of the inquiry of this dissertation. In the searching for an efficient learning and placing the work as a research, it is presented a historical reflection of the evolution of the concept of complex numbers pointing out their more relevant focuses, such as: symbolic, numeric, geometrical and algebraic ones. Then, it shows the description of the ways of the research based on the methodology of the didactic engineering. This one is developed from the utilization of its four stages, where in the preliminary analysis stage, two data surveys are presented: the first one is concerning with the way of presenting the contents of the complex numbers in math textbooks, and the second one is concerning to the interview carried out with High school teachers who work with complex numbers in the practice of their professions. At first, in the analysis stage, it is presented the prepared and organized material to be used in the following stage. In the experimentation one, it is presented the carrying out process that was made with the second year High school students in the Centro Federal de Educação tecnológica do Rio Grande do Norte CEFET-RN. At the end, it presents, in the subsequent and validation stages, the revelation of the obtained results from the observations made in classrooms in the carrying out of the didactic sequence, the students talking and the data collection

Uma análise dos questionamentos dos alunos nas aulas de números complexos

This work presents a contribution for the studies reffering to the use of the History of Mathematics focusing on the improvement of the Teaching and Learning Process. It considers that the History of Matematics, as a way of giving meaning to the discipline and improve the quality of the Teaching and Learning Process. This research focuses on the questions of the students, classified in three categories of whys: the chronological, the logical and the pedagogical ones. Therefore, it is investigated the teaching of the Complex Numbers, from the questions of the students of the Centro Federal de Educação Tecnológica do Rio Grande do Norte (Educational Institution of Professional and Technology Education from Rio Grande do Norte). The work has the following goals: To classify and to analyse the questions of the students about the Complex Numbers in the classes of second grade of the High School, and to collate with the pointed categories used by Jones; To disccus what are the possible guidings that teachers of Mathematics can give to these questions; To present the resources needed to give support to the teacher in all things involving the History of Mathematics. Finally, to present a bibliographic research, trying to reveal supporting material to the teacher, with contents that articulate the Teaching of Mathematics with the History of Mathematics. It was found that the questionings of the pupils reffers more to the pedagogical whys, and the didatic books little contemplate other aspects of the history and little say about the sprouting and the evolution of methods of calculations used by us as well

A difícil aceitação dos números negativos: um estudo da teoria dos números de Peter Barlow (1776-1862)

The present study seeks to present a historico-epistemological analysis of the development of the mathematical concept of negative number. In order to do so, we analyzed the different forms and conditions of the construction of mathematical knowledge in different mathematical communities and, thus, identified the characteristics in the establishment of this concept. By understanding the historically constructed barriers, especially, the ones having ontologicas significant, that made the concept of negative number incompatible with that of natural number, thereby hindering the development of the concept of negative, we were able to sketch the reasons for the rejection of negative numbers by the English author Peter Barlow (1776 -1862) in his An Elementary Investigation of the Theory of Numbers, published in 1811. We also show the continuity of his difficulties with the treatment of negative numbers in the middle of the nineteenth century

O uso pedagógico de uma seqüência didática para a aquisição de algumas idéias relacionadas ao conceito de números complexos

The aim of the present work is to contribute to the teaching-learning process in Mathematics through an alternative which tries to motivate the student so that he/she will learn the basic concepts of Complex Numbers and realize that they are not pointless. Therefore, this work s general objective is to construct a didactic sequence which contains structured activities that intends to build up, in each student s thought, the concept of Complex Numbers. The didactic sequence is initially based on a review of the main historical aspects which begot the construction of those numbers. Based on these aspects, and the theories of Richard Skemp, was elaborated a sequence of structured activities linked with Maths history, having the solution of quadratic equations as a main starting point. This should make learning more accessible, because this concept permeates the students previous work and, thus, they should be more familiar with it. The methodological intervention began with the application of that sequence of activities with grade students in public schools who did not yet know the concept of Complex Numbers. It was performed in three phases: a draft study, a draft study II and the final study. Each phase was applied in a different institution, where the classes were randomly divided into groups and each group would discuss and write down the concepts they had developed about Complex Numbers. We also use of another instrument of analysis which consisted of a recorded interview of a semi-structured type, trying to find out the ways the students thought in order to construct their own concepts, i.e. the solutions of the previous activity. Their ideas about Complex Numbers were categorized according to their similarities and then analyzed. The results of the analysis show that the concepts constructed by the students were pertinent and that they complemented each other this supports the conclusion that the use of structured activities is an efficient alternative for the teaching of mathematics