Computabilidade no espaço dos intervalos reais: um modelo BSS intervalar

A matemática intervalar é uma teoria matemática originada na década de 60 com o objetivo de responder questões de exatidão e eficiência que surgem na prática da computação científica e na resolução de problemas numéricos. As abordagens clássicas para teoria da computabilidade tratam com problemas discretos (por exemplo, sobre os números naturais, números inteiros, strings sobre um alfabeto finito, grafos, etc.). No entanto, campos da matemática pura e aplicada tratam com problemas envolvendo números reais e números complexos. Isto acontece, por exemplo, em análise numérica, sistemas dinâmicos, geometria computacional e teoria da otimização. Assim, uma abordagem computacional para problemas contínuos é desejável, ou ainda necessária, para tratar formalmente com computações analógicas e computações científicas em geral. Na literatura existem diferentes abordagens para a computabilidade nos números reais, mas, uma importante diferença entre estas abordagens está na maneira como é representado o número real. Existem basicamente duas linhas de estudo da computabilidade no contínuo. Na primeira delas uma aproximação da saída com precisão arbitrária é computada a partir de uma aproximação razoável da entrada [Bra95]. A outra linha de pesquisa para computabilidade real foi desenvolvida por Blum, Shub e Smale [BSS89]. Nesta aproximação, as chamadas máquinas BSS, um número real é visto como uma entidade acabada e as funções computáveis são geradas a partir de uma classe de funções básicas (numa maneira similar às funções parciais recursivas). Nesta dissertação estudaremos o modelo BSS, usado para se caracterizar uma teoria da computabilidade sobre os números reais e estenderemos este para se modelar a computabilidade no espaço dos intervalos reais. Assim, aqui veremos uma aproximação para computabilidade intervalar epistemologicamente diferente da estudada por Bedregal e Acióly [Bed96, BA97a, BA97b], na qual um intervalo real é visto como o limite de intervalos racionais, e a computabilidade de uma função intervalar real depende da computabilidade de uma função sobre os intervalos racionais

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

Probabilidades imprecisas: intervalar, fuzzy e fuzzy intuicionista

The idea of considering imprecision in probabilities is old, beginning with the Booles George work, who in 1854 wanted to reconcile the classical logic, which allows the modeling of complete ignorance, with probabilities. In 1921, John Maynard Keynes in his book made explicit use of intervals to represent the imprecision in probabilities. But only from the work ofWalley in 1991 that were established principles that should be respected by a probability theory that deals with inaccuracies. With the emergence of the theory of fuzzy sets by Lotfi Zadeh in 1965, there is another way of dealing with uncertainty and imprecision of concepts. Quickly, they began to propose several ways to consider the ideas of Zadeh in probabilities, to deal with inaccuracies, either in the events associated with the probabilities or in the values of probabilities. In particular, James Buckley, from 2003 begins to develop a probability theory in which the fuzzy values of the probabilities are fuzzy numbers. This fuzzy probability, follows analogous principles to Walley imprecise probabilities. On the other hand, the uses of real numbers between 0 and 1 as truth degrees, as originally proposed by Zadeh, has the drawback to use very precise values for dealing with uncertainties (as one can distinguish a fairly element satisfies a property with a 0.423 level of something that meets with grade 0.424?). This motivated the development of several extensions of fuzzy set theory which includes some kind of inaccuracy. This work consider the Krassimir Atanassov extension proposed in 1983, which add an extra degree of uncertainty to model the moment of hesitation to assign the membership degree, and therefore a value indicate the degree to which the object belongs to the set while the other, the degree to which it not belongs to the set. In the Zadeh fuzzy set theory, this non membership degree is, by default, the complement of the membership degree. Thus, in this approach the non-membership degree is somehow independent of the membership degree, and this difference between the non-membership degree and the complement of the membership degree reveals the hesitation at the moment to assign a membership degree. This new extension today is called of Atanassov s intuitionistic fuzzy sets theory. It is worth noting that the term intuitionistic here has no relation to the term intuitionistic as known in the context of intuitionistic logic. In this work, will be developed two proposals for interval probability: the restricted interval probability and the unrestricted interval probability, are also introduced two notions of fuzzy probability: the constrained fuzzy probability and the unconstrained fuzzy probability and will eventually be introduced two notions of intuitionistic fuzzy probability: the restricted intuitionistic fuzzy probability and the unrestricted intuitionistic fuzzy probability

Estudo de Superfícies Seletivas de Frequência com o Uso de Inteligência Computacional

The main objective of this work is to optimize the performance of frequency selective surfaces (FSS) composed of crossed dipole conducting patches. The optimization process is performed by determining proper values for the width of the crossed dipoles and for the FSS array periodicity, while the length of the crossed dipoles is kept constant. Particularly, the objective is to determine values that provide wide bandwidth using a search algorithm with representation in bioinspired real numbers. Typically FSS structures composed of patch elements are used for band rejection filtering applications. The FSS structures primarily act like filters depending on the type of element chosen. The region of the electromagnetic spectrum chosen for this study is the one that goes from 7 GHz to 12 GHz, which includes mostly the X-band. This frequency band was chosen to allow the use of two X-band horn antennas, in the FSS measurement setup. The design of the FSS using the developed genetic algorithm allowed increasing the structure bandwidth

Conjuntos numéricos

In this work, we present a text on the Sets Numerical using the human social needs as a tool for construction new numbers. This material is intended to present a text that reconciles the correct teaching of mathmatics and clarity needed for a good learning

A Linguagem de Especificação algébrica CASL e o Tipo de Dados Intervalos

Na computação científica é necessário que os dados sejam o mais precisos e exatos possível, porém a imprecisão dos dados de entrada desse tipo de computação pode estar associada às medidas obtidas por equipamentos que fornecem dados truncados ou arredondados, fazendo com que os cálculos com esses dados produzam resultados imprecisos. Os erros mais comuns durante a computação científica são: erros de truncamentos, que surgem em dados infinitos e que muitas vezes são truncados", ou interrompidos; erros de arredondamento que são responsáveis pela imprecisão de cálculos em seqüências finitas de operações aritméticas. Diante desse tipo de problema Moore, na década de 60, introduziu a matemática intervalar, onde foi definido um tipo de dado que permitiu trabalhar dados contínuos,possibilitando, inclusive prever o tamanho máximo do erro. A matemática intervalar é uma saída para essa questão, já que permite um controle e análise de erros de maneira automática. Porém, as propriedades algébricas dos intervalos não são as mesmas dos números reais, apesar dos números reais serem vistos como intervalos degenerados, e as propriedades algébricas dos intervalos degenerados serem exatamente as dos números reais. Partindo disso, e pensando nas técnicas de especificação algébrica, precisa-se de uma linguagem capaz de implementar uma noção auxiliar de equivalência introduzida por Santiago [6] que ``simule" as propriedades algébricas dos números reais nos intervalos. A linguagem de especificação CASL, Common Algebraic Specification Language, [1] é uma linguagem de especificação algébrica para a descrição de requisitos funcionais e projetos modulares de software, que vem sendo desenvolvida pelo CoFI, The Common Framework Initiative [2] a partir do ano de 1996. O desenvolvimento de CASL se encontra em andamento e representa um esforço conjunto de grandes expoentes da área de especificações algébricas no sentido de criar um padrão para a área. A dissertação proposta apresenta uma especificação em CASL do tipo intervalo, munido da aritmética de Moore, afim de que ele venha a estender os sistemas que manipulem dados contínuos, sendo possível não só o controle e a análise dos erros de aproximação, como também a verificação algébrica de propriedades do tipo de sistema aqui mencionado. A especificação de intervalos apresentada aqui foi feita apartir das especificações dos números racionais proposta por Mossakowaski em 2001 [3] e introduz a noção de igualdade local proposta por Santiago [6, 5, 4]

Em direção a uma representação para equações algébricas :uma lógica equacional local

The intervalar arithmetic well-known as arithmetic of Moore, doesn't possess the same properties of the real numbers, and for this reason, it is confronted with a problem of operative nature, when we want to solve intervalar equations as extension of real equations by the usual equality and of the intervalar arithmetic, for this not to possess the inverse addictive, as well as, the property of the distributivity of the multiplication for the sum doesn t be valid for any triplet of intervals. The lack of those properties disables the use of equacional logic, so much for the resolution of an intervalar equation using the same, as for a representation of a real equation, and still, for the algebraic verification of properties of a computational system, whose data are real numbers represented by intervals. However, with the notion of order of information and of approach on intervals, introduced by Acióly[6] in 1991, the idea of an intervalar equation appears to represent a real equation satisfactorily, since the terms of the intervalar equation carry the information about the solution of the real equation. In 1999, Santiago proposed the notion of simple equality and, later on, local equality for intervals [8] and [33]. Based on that idea, this dissertation extends Santiago's local groups for local algebras, following the idea of Σ-algebras according to (Hennessy[31], 1988) and (Santiago[7], 1995). One of the contributions of this dissertation, is the theorem 5.1.3.2 that it guarantees that, when deducing a local Σ-equation E t t in the proposed system SDedLoc(E), the interpretations of t and t' will be locally the same in any local Σ-algebra that satisfies the group of fixed equations local E, whenever t and t have meaning in A. This assures to a kind of safety between the local equacional logic and the local algebras

Especificação do tipo intervalar parametrizado em CASL

The interval datatype applications in several areas is important to construct a interval type reusable, i.e., a interval constructor can be applied to any datatype and get intervals this datatype. Since the interval is, of certain form, a set of elements limited for two bounds, left and right, with a order notions, then it s reasonable that interval constructor enclose datatypes with partial order. On the order hand, what we want is work with interval of any datatype like this we work with this datatype then. it s important to guarantee the properties of the datatype when maps to interval of this datatype. Thus, the interval constructor get a theory to parametrized interval type, i.e., a interval with generics parameters (for example rational, real, complex). Sometimes, the interval application in some algebras doesn t guarantee the mainutenance of their properties, for example, when we use interval of real, that satisfies the field properties, it doesn t guarantee the distributivity propertie. A form to surpass this problem Santiago introduced the local equality theory that weakened the notion of strong equality, and thus, allowing some properties are local keeped, what can be discard before. The interval arithmetic generalization aim to apply the interval constructor on ordered algebras weakened for local equality with the purpose of the keep their properties. How the intervals are important in applications with continuous data, it s interesting specify that theory using a specification language that supply a system development using intervals of form disciplined, trustworth and safe. Currently, the algebraic specification language, based in math models, have been use to that intention often. We choose CASL (Common Algebraic Specification Language) among others languages because CASL has several characteristics excellent to parametrized interval type, such as, provide parcialiy and parametrization

Do encantamento dos números à realidade dos fatos: conciliação entre os balanços patrimonial e social da ALUNORTE

The social balance is turning into an instrument capable of identifying the Organizational Commitment socio-environmental. The objective of the Social Balance is to present the application of company resources on socio-environmental investments internally and externally. The research was developed based on the Balance Social and Sheet from Alumina North Brazil S / A, ALUNORTE, for fiscal years 2008 and 2009, with the purpose of describing the finding of the Balance Social and Sheet from ALUNORTE about social responsibility. To validate the proposal were doing comparisons between accounting and financial datas from Alunorte and Y.Yamada, in order to highlight what they say and indicators confirm the privileges of the first against second

Adrien-Marie Legendre (1752-1833) e suas obras em Teoria dos Números

The present thesis is an analysis of Adrien-Marie Legendre s works on Number Theory, with a certain emphasis on his 1830 edition of Theory of Numbers. The role played by these works in their historical context and their influence on the development of Number Theory was investigated. A biographic study of Legendre (1752-1833) was undertaken, in which both his personal relations and his scientific productions were related to certain historical elements of the development of both his homeland, France, and the sciences in general, during the 18th and 19th centuries This study revealed notable characteristics of his personality, as well as his attitudes toward his mathematical contemporaries, especially with regard to his seemingly incessant quarrels with Gauss about the priority of various of their scientific discoveries. This is followed by a systematic study of Lagrange s work on Number Theory, including a comparative reading of certain topics, especially that of his renowned law of quadratic reciprocity, with texts of some of his contemporaries. In this way, the dynamics of the evolution of his thought in relation to his semantics, the organization of his demonstrations and his number theoretical discoveries was delimited. Finally, the impact of Legendre s work on Number Theory on the French mathematical community of the time was investigated. This investigation revealed that he not only made substantial contributions to this branch of Mathematics, but also inspired other mathematicians to advance this science even further. This indeed is a fitting legacy for his Theory of Numbers, the first modern text on Higher Arithmetic, on which he labored half his life, producing various editions. Nevertheless, Legendre also received many posthumous honors, including having his name perpetuated on the Trocadéro face of the Eiffel Tower, which contains a list of 72 eminent scientists, and having a street and an alley in Paris named after him

Obstáculos superados pelos matemáticos no passado e vivenciados pelos alunos na atualidade : a polêmica multiplicação de números inteiros

In Mathematics literature some records highlight the difficulties encountered in the teaching-learning process of integers. In the past, and for a long time, many mathematicians have experienced and overcome such difficulties, which become epistemological obstacles imposed on the students and teachers nowadays. The present work comprises the results of a research conducted in the city of Natal, Brazil, in the first half of 2010, at a state school and at a federal university. It involved a total of 45 students: 20 middle high, 9 high school and 16 university students. The central aim of this study was to identify, on the one hand, which approach used for the justification of the multiplication between integers is better understood by the students and, on the other hand, the elements present in the justifications which contribute to surmount the epistemological obstacles in the processes of teaching and learning of integers. To that end, we tried to detect to which extent the epistemological obstacles faced by the students in the learning of integers get closer to the difficulties experienced by mathematicians throughout human history. Given the nature of our object of study, we have based the theoretical foundation of our research on works related to the daily life of Mathematics teaching, as well as on theorists who analyze the process of knowledge building. We conceived two research tools with the purpose of apprehending the following information about our subjects: school life; the diagnosis on the knowledge of integers and their operations, particularly the multiplication of two negative integers; the understanding of four different justifications, as elaborated by mathematicians, for the rule of signs in multiplication. Regarding the types of approach used to explain the rule of signs arithmetic, geometric, algebraic and axiomatic , we have identified in the fieldwork that, when multiplying two negative numbers, the students could better understand the arithmetic approach. Our findings indicate that the approach of the rule of signs which is considered by the majority of students to be the easiest one can be used to help understand the notion of unification of the number line, an obstacle widely known nowadays in the process of teaching-learning

Um estudo sobre a aprendizagem de números irracionais no ensino médio

The present study describes theoretical practical relationships between development and application of activities in Mathematics education. It s proposed a methodological approach to Mathematics in the first grade of Ensino Médio, supported by an experiment involving Irrational Numbers education by using constructive activities, applied obeying an educational sequence. Constructivism is used as an important theoretical reference in teaching learning process of Mathematics. The methodological intervention was done with two classes of students of the first grade of Ensino Médio, in two public schools, a state one and a federal one, located on the city of Natal, Rio Grande do Norte. The development, application and testing of the activities used on this experiment led us to think more profoundly about the value of constructivism ideas and understand that the use of activities that obey an educational sequence favors the learning. It s also discussed the research results, commented on a way to contribute to the advances of the proposal and it s more constant use. The participation and testing of the students were analyzed and judged using Skemp s Instrumental Understanding and Relational Understanding concepts. The results of the research were considered good, so we believe this methodological intervention can be used more frequently in the classes of Ensino Médio and also be applied to teachers in courses of initial education and continuous formation

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

Doenças sem doentes: Ocorrência de Distúrbios Osteomusculares Relacionados ao Trabalho DORT nos operadores de caixa de um banco

The changes incurred in the financial system with the introduction of new technologies and new forms of administration of banks has caused impact on the health of workers. These changes, which passed in the process of work, generate a combined share of the risk factors that result in numerous injuries and illnesses among banks, notably between the operators of banks tellers. The Work-Related Musculoskeletal Disordes - WRMD represent a group of occupational diseases always present among these workers. Because of its high incidence and the amount of financial resour envolved to manage the problem has been the object of constant study. This paper aims to analyze the bank teller activity; search the occurrence of WRMD in the activity, identifying the factors determining the occurrence of WRMD in the activity and determine the real number of touchs on a keyboard made by the operator and propose solutions that influence the reduction of illness in the workplace of the bank teller. Methodological tools of ergonomics are used to provide a broad knowledge of aspects of work that have been studied and influential in the generation of occupational diseases studied. It was found that activity put workers to serious risk of occupational diseases. As the main contributory factors and determinants for this illness: the requirements and control the numbers daily endorsements; evaluation system based on performance targets for productivity; management system at time of service to customers; work with stressful factors (broken box); excess of time worked; furniture of workstations with ergonomic inadequacies and policy for the prevention of occupational diseases inefficient. They have also noted cases of illness for DORT workers without fulfilling the legal requirement of the issuance of the communication of labour accident and without the removal of the employee of the workplace

Utilizando a televisão digital como um meio de comunicação para ambientes interativos reais

This Dissertation aims to provide a communication mechanism between Digital TV viewers and interaction devices, such as robots, for example, placed on the environment from which a TV program is being live broadcasted. Such communication mechanism has the objective to allow viewers controll the Interaction Devices through their TV devices, using the broadcast channel present in Interactive Digital TV systems, and receive data from the devices by the broadcast channel. This system was projected as a middlewaer system where the Interaction Devices in the TV program set are interconnected, creating a Interactive Device Network. With this approach, the system is capable of manage the devices on the network, controlling the flow of coming and leaving elements, in a transparent way for the viewers. The system yet allows the Interaction Devices communicate each other, with a integrated communication channel with no worries about the physical communication layer