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 abordagem no ensino de frações baseadas em atividades para o 6º ano do ensino fundamantal

The objective of this work if constitutes in creation a proposal for activities, in the discipline of mathematics, for the 6th year of Elementary School, that stimulates the students the develop the learning of the content of fractions, from the awareness of the insufficiency of the natural numbers for solve several problems. Thus, we prepared a set with twelve activities, starting by the comparison between measures, presenting afterward some of the meanings of fractions and ending with the operations between fractions. For so much, use has been made of materials available for use in the classroom, of forma ludic, for resolution of challenges proposed. Through these activities, it becomes possible students to recognize the necessity of using fractions for solve a amount larger of problems

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

Euler e os números pentagonais

The present investigation includes a study of Leonhard Euler and the pentagonal numbers is his article Mirabilibus Proprietatibus Numerorum Pentagonalium - E524. After a brief review of the life and work of Euler, we analyze the mathematical concepts covered in that article as well as its historical context. For this purpose, we explain the concept of figurate numbers, showing its mode of generation, as well as its geometric and algebraic representations. Then, we present a brief history of the search for the Eulerian pentagonal number theorem, based on his correspondence on the subject with Daniel Bernoulli, Nikolaus Bernoulli, Christian Goldbach and Jean Le Rond d'Alembert. At first, Euler states the theorem, but admits that he doesn t know to prove it. Finally, in a letter to Goldbach in 1750, he presents a demonstration, which is published in E541, along with an alternative proof. The expansion of the concept of pentagonal number is then explained and justified by compare the geometric and algebraic representations of the new pentagonal numbers pentagonal numbers with those of traditional pentagonal numbers. Then we explain to the pentagonal number theorem, that is, the fact that the infinite product(1 x)(1 xx)(1 x3)(1 x4)(1 x5)(1 x6)(1 x7)... is equal to the infinite series 1 x1 x2+x5+x7 x12 x15+x22+x26 ..., where the exponents are given by the pentagonal numbers (expanded) and the sign is determined by whether as more or less as the exponent is pentagonal number (traditional or expanded). We also mention that Euler relates the pentagonal number theorem to other parts of mathematics, such as the concept of partitions, generating functions, the theory of infinite products and the sum of divisors. We end with an explanation of Euler s demonstration pentagonal number theorem

Os artigos de Euler sobre os números amigáveis

Among the many methodological resources that the mathematics teacher can use in the classroom, we can cite the History of Mathematics which has contributed to the development of activities that promotes students curiosity about mathematics and its history. In this regard, the present dissertation aims to translate and analyze, mathematically and historically, the three works of Euler about amicable numbers that were writed during the Eighteenth century with the same title: De numeris amicabilibus. These works, despite being written in 1747 when Euler lived in Berlin, were published in different times and places. The first, published in 1747 in Nova Acta Eruditorum and which received the number E100 in the Eneström index, summarizes the historical context of amicable numbers, mentions the formula 2nxy & 2nz used by his precursors and presents a table containing thirty pairs of amicable numbers. The second work, E152, was published in 1750 in Opuscula varii argument. It is the result of a comprehensive review of Euler s research on amicable numbers which resulted in a catalog containing 61 pairs, a quantity which had never been achieved by any mathematician before Euler. Finally, the third work, E798, which was published in 1849 at the Opera postuma, was probably the first among the three works, to be written by Euler

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

Variação no conteúdo protéico e pigmentar em variantes cromáticas de Gracilaria domingensis nas populações naturais de Rio do Fogo-RN, BRASIL

The seaweed Gracilaria domingensis is a common species in the coast of Rio Grande do Norte. This species lives in the intertidal zone, where colour strains (red, green and brown) co-occur during the whole year. Seaweeds that live in this region are exposed to daily changes and to the rhythm of the tide. During the low tide they are exposed to dissection, hiper-or hipo-osmotic shock, high temperatures and high irradiance. The aim of this study was to analyze whether the pigment and protein content of the colour strains of G. domingensis is affected by some environmental parameters in a temporal scale. The seaweeds were collected during 10 months in the seashore of Rio do Fogo (RN). The total soluble proteins and the phycobiliprotein were extracted in phosphate buffer and the carotenoids were analyzed by a standardized method through HPLC-UV. The pigments analysis showed that phycoerithrin is the most abundant pigment in the three strains. This pigment was strongly correlated with nitrogen and the photosynthetically active radiation. Chlorophyll presented higher concentrations than carotenoids during the whole, but the ratio carotenoid/chlorophyll-a was modified by incident radiation. The most abundant carotenoid was ß-carotene and zeaxanthin, which had higher concentrations in the higher radiation months. The concentration increase of zeaxanthin in this period indicated a photoprotective response of the seaweed. The three strains presented a pigment profile that indicates different radiation tolerance profile. Our results pointed that the green strain is better adapted to high irradiance levels than the red and brown strains

Avaliação da toxicidade de florações naturais e de cultura de cianobactérias: efeitos sobre Ceriodaphnia silvestri (Crustacea, Cladocera)

The incidence of toxic cyanobacterial blooms is one of the important consequences of eutrophication in aquatic ecosystems. It is a very common phenomenon in reservoirs and shrimp ponds in the State of Rio Grande do Norte (RN), Brazil. Cyanobacterias produce toxins which can affect aquatic organisms and men trough the food chain. Aiming to contribute to the studies of cyanobacterias in RN, we propose: a) to evaluate the toxicity of isolated cyanobacterias in important fresh-water environments; and b) to verify the effects of both natural and cultured blooms occurred in reservoirs for human supply and in the cladoceran Ceriodaphnia silvestrii. This study was carried out using samples of natural blooms occurred between March and October of 2004 in Gargalheiras Dam (08º L e 39º W), in July of 2004 in Armando Ribeiro Gonçalves Dam (06o S e 37o W) and in commercial shrimp ponds (Litopenaeus vannamei) located in fresh-water environments. The samples were collected with plankton net (20µm.) for identification, isolation and obtaining of phytoplanktonic biomass for liophilization and later toxicity bioassays. The toxicity of cultured samples and natural blooms was investigated through bioassays in Swiss mice. Quantification of cyanobacteria in samples was conducted following the Ütermol method, with 300mL samples fixed with lugol. The toxicity test with Ceriodaphnia silvestrii followed ABNT, 2001 recommendations, and were accomplished with natural hepatotoxic bloom s samples and cultured samples of both non-toxic and neurotoxic C. raciborskii. In this test, five newborns, aged between 6 and 24 hours, were exposed to different concentrations (0 a 800 mg.L-1) of crude cyanobacterial extracts during 24 and 48 hours. Three replicates were used per treatment. The pH, temperature and dissolved oxygen at the beginning and after 24 and 48hours from the test were measured. We estimated the CL50 through the Trimmed Spearman-Karber method. The blooms were constituted by Microcystis panniformis, M. aeruginosa, Anabaena circinalis, Cylindrospermopsis raciborskii and Planktothrix agardhii, producers of mycrocistin-LR confirmed with HPLC analysis. Samples of hepatotoxic blooms registered toxinogenic potential for C. silvestrii, with CL50-24h value of 47.48 mg.L-1 and CL5048h of 38.15 mg.L-1 for GARG samples in march/2005; CL50-24h of 113,13 mg.L-1 and CL5048h of 88,24 mg.L-1 for ARG July/2004; CL50-24h of 300.39 mg.L-1 and CL50-48h of 149.89 mg.L-1 for GARG October/2005. For cultured samples, values of CL50-24h and CL50-48h for C. raciborskii toxic strains were 228.05 and 120.28 mg.L-1, respectively. There was no mortality of C. silvestrii during the tests with non-toxic C. raciborskii strain. The toxicity test with C. silvestrii presented good sensitivity degree to cyanotoxins. The toxicity of natural hepatotoxic blooms samples (microcystins) and cultured neurotoxic saxitoxins producer samples analyzed in this study give us strong indications of that toxin s influence on the zooplanktonic community structure in tropical aquatic environments. Eleven cyanobacteria strains were isolated, representing 6 species: Anabaenopsis sp., Cylindrospermopsis raciborskii, Chroococcus sp., Microcystis panniformis, Geitlerinema unigranulatum e Planktothrix agardhii. None presented toxicity in Swiss mice. The strains were catalogued and deposited in the Laboratório de Ecologia e Toxicologia de Organismos Aquáticos (LETMA), in UFRN, and will be utilized in ecotoxicológical and ecophysiological studies, aiming to clarify the causes and control of cyanobacterial blooms in aquatic environments in RN. This state s reservoirs must receive broader attention from the authorities, considering the constant blooms occurring in waters used for human consumption

Características limnológicas e estrutura trófica das comunidades de peixes de alguns lagos naturais e artificiais do Rio Grande do Norte

Artificial lakes must differ from natural lakes in important structural and functional aspects that need to be understood so that these ecosystems can be properly managed. The aim of this work was to test the hypothesis that the artificial lakes (impoundments) in the semi-arid region of the Rio Grande do Norte State are more eutrophic and turbid and have different trophic structure when compared to the natural coastal lakes that occur in the humid eastern coast of the State. To test this hypothesis, 10 natural lakes and 8 artificial lakes with about 100 ha were sampled between September and November 2005 for the determination of some limnological variables and the abundance of the main fish species, which were grouped in three trophic guilds: facultative piscivores, facultative planktivores and omnivores. The results show that the artificial lakes had significantly higher concentrations of total nitrogen, total phosphorus, chlorophyll a , total and volatile suspended solids than the natural lakes. Results also show that the values of pH, total alkalinity, electric conductivity, turbidity as well as the coefficient of vertical attenuation of light were significantly higher in the artificial lakes than in the natural lakes. In the artificial lakes, the abundance of facultative planktivores was significantly higher, while the abundance of facultative piscivores significantly lower than in the natural lakes. There was no significant difference in the abundance of omnivorous fish between the two types of lakes. These results suggest that the increase in turbidity together with the other changes in the water quality of the artificial lakes, modifies the trophic structure of the fish communities reducing the importance of piscivores and the length of the food chains

Síntese de zeólitas e argilas ácidas pilarizadas a partir de matérias primas naturais

Structural changes in waste for zeolites synthesis are subject of many studies carried out in the synthesis of molecular sieves. These materials are named molecular sieves because they have well defined pore sizes and they have the capacity of select molecules by its size. In this work, it was studied the synthesis processes of two types of molecular sieves: pillared acid clays using as starting material one natural montmorillonite clay and the synthesis of zeolites from a silico-aluminous residue. This residue is a byproduct of the extraction of lithium -spodumene. The preparation of pillared acid clays was performed in two steps: 1° acid treatment of clay samples (time and temperature studies) and 2°pilarization of them with Al13 (Keggin ion). The temperature and acid concentration affect the removal of cations in the structure and porosity of the material obtained. The analysis of X-ray diffraction (XRD) and infrared spectroscopy (IR), showed that increasing the severity of the acid treatment compromises the structural material. Also the pore size distribution is approximately uniform. Despite presenting a structural disorganization, the samples were pillared. As evidenced by XRD increasing the basal spacing, specific area and uniform porosity by adsorption of N2. Regarding the microporous molecular sieves were synthesized zeolites A and NaP1 from a silico-aluminous residue, a byproduct of extracting lithium. The temperature and time of agitation during the synthesis were the most important factors for obtaining zeolite A. The aging of the gel and the highest crystallization time promoted the formation of zeolite NaP1 using a Si / Al ratio = 3.2

Mesoestruturas porosas a partir de materiais naturais

The MCM-41 mesoporous synthesis was done using rice hulls ash and chrysotile as natural alternative silica sources. For the using of these sources, chemical and thermic treatments were done in both materials. After chemical and thermic treatments, these materials were employed on the MCM-41 mesoctructures synthesis. The natural materials treated and employed in the synthesis were characterized by several techniques such as X-ray diffraction, N2 adsorption and desorption, scanning electronic microscopy and thermogravimetric analysis. MCM-41 standart samples synthetized with aerosil 200 commercial sílica were used to evaluation. The formed material from rice hulls ash showed values from BET specific area about 468 m².g-1, N2 adsorption and desorption isotherms and loss mass similar to reference materials. The silica from chrysotile calcined and leached was employed to mesoporous materials synthesis. The BET specific area showed values about 700 m².g-1, N2 adsorption and desorption isotherms type IV and loss mass similar to mesoporous materials. The formed material from calcined and leached chrysotile, without calcination, applied to phenol remotion carried high performance liquid chromatography and evaluated with organophilic clays with different treatments. By the characterization techniques were proved that mesoporous materials with lesser order that reference samples. The material formed from rice hulls ash without the calcination step achieved better adsorption results than organophilic clays

Estudo da substituição de agregados miúdos naturais por pó de pedra em concreto de cimento portland

The sharp consumption of natural resources by the construction industry has motivated numerous studies concerning the application of waste to replace partially or fully, some materials, such as aggregates, thereby reducing the environmental impact caused by the extraction of sand and crushing process. The application of stone dust from crushing process arising as an aggregate for the production of Portland cement concrete is a viable alternative in view of the high cost of natural sands, in addition to the environmental damage which causes its operation to the environment. The stone dust has reduced cost compared to natural sand because it is produced in the beds of their own quarries, which are usually located close to major urban centers. This study examined the feasibility of using stone dust from the crushing of rock gneisses in the state of Bahia, replacing natural quartz sand. In the development of scientific study was conducted to characterize physical and chemical raw materials applied and molded cylindrical specimens , using as reference values Fck 20, Fck 25 and Fck 30 MPa ( resistance characteristic of the concrete after 28 days) in following compositions stone powder: 10%, 30%, 50 %, 100% and 100% with additive. The specimens were cured and subjected to the tests of compressive strength and water absorption, then the samples were subjected to the tests of X-ray diffraction and scanning electron microscopy. The results obtained showed that the composition with 10% stone powder showed the best results regarding the physical and mechanical tests performed, confirming the reduction in compressive strength and increased water uptake increased as the content of the powder stone in the concrete composition