Correspondências científicas como uma relação didática entre história e ensino de matemática: o exemplo das cartas de Euler a uma princesa da Alemanha

Lettres àune Princesse d'Allemagne sur divers sujets de physique et de philosophie (Letters to a Princess of Germany on various topics of physics and philosophy) is the work taken as an object of study of this thesis. It is a literary success written in the eighteenth century by the Swiss mathematician and physicist Leonhard Paul Euler (1707-1783) in order to meet a request from the Prussian king, Frederick II, the Great (1712-1786) to accept to guide the intellectual education of his niece, the young princess Anhalt-Dessau (1745-1808). The method of teaching and learning through letters elected to the education of the German monarch resulted in a collection of 234 matches in which Euler theory is about music, Philosophy, Mechanics, Optics, Astronomy, Theology and Ethics among others. The research seeks to point out mathematical content contained in this reference work based on the exploitation and adaptation of original historical works as an articulator of development activities for teaching mathematics in basic education and in accordance with the National Curriculum Parameters of Mathematics (NCP) work. The general objective point out the limits and didactic potential of Lettres à une Princesse d'Allemagne sur divers sujets de physique et de philosophie as a source of support for teachers of basic education in developing activities for teaching mathematics. The discussions raised point to concrete possibilities of entanglement between the extracted mathematical content of the bulge of the work with current teaching methodologies from resizing the use of letters according to Freire's pedagogical perspective of the correspondence, and especially the use of new communication channels in the century XXI, both aimed at dialogue and approximation between those who write and those who read.

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

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

Análise da expressão imuno-histoquímica da ciclofilina A, EMMPRIN e MMP-7 na doença periodontal

Periodontal disease is an infectious disease resulting from the immunoinflammatory response of the host to microorganisms present in the dental biofilm which causes tissue destruction. The objective of this study was to evaluate the immunohistochemical expression of cyclophilin A (CYPA), extracellular matrix metalloproteinase inducer (EMMPRIN) and matrix metalloproteinase 7 (MMP-7) in human specimens of clinically healthy gingiva (n=32), biofilm-induced gingivitis (n=28), and chronic periodontitis (n=30). Immunopositivity for CYPA, EMMPRIN and MMP-7 differed significantly between the three groups, with higher percentages of staining in chronic periodontitis specimens, followed by chronic gingivitis and healthy gingiva specimens (p < 0.001). Immunoexpression of CYPA and MMP-7 was higher in the intense inflammatory infiltrate observed mainly in cases of periodontitis. Analysis of possible correlations between the immunoexpression of EMMPRIN, MMP-7 and CYPA and between the expression of these proteins and clinical parameters (probing depth and clinical attachment loss) showed a positive correlation of CYPA expression with MMP-7 (r = 0.831; p < 0.001) and EMMPRIN (r = 0.289; p = 0.006). In addition, there was a significant positive correlation between probing depth and expression of MMP-7 (r = 0.726; p < 0.001), EMMPRIN (r = 0.345; p = 0.001), and CYPA (r = 0.803; p < 0.001). These results suggest that CYPA, EMMPRIN and MMP-7 are associated with the pathogenesis and progression of periodontal disease

Eficácia do tratamento não-cirúrgico em indivíduos com mucosite peri-implantar

It has been shown that the development of peri-implant mucositis is associated with biofilm accumulation. It is believed that the therapeutic approaches used in periodontal disease may have a positive effect in the cases of peri-implant disease. The aim of this study was to evaluate the effectiveness of non-surgical treatment of peri-implant mucositis, with or without the use of chlorhexidine 0,12% in subjects rehabilitated with osseointegrated implants. Thus, patients were randomly divided into test group (chlorhexidine surgical therapy) and control (non-surgical treatment). This therapy consisted of an adaptation of the (Full Mouth scalling and Root Planing) nonoperative protocol FMSRP, but without the use of ultrasound. The visible plaque index (VPI), gingival bleeding index (GBI), probing depth (PD), bleeding on probing (BOP) and keratinized mucosa clinical parameters were evaluated at baseline and at different times after treatment. The data were not normally distributed and the implant was considered the sampling unit. Data were analyzed using Fri edman and Wilcoxon chi-square (=5%), tests using the Statistical Package for Social Sciences 17.0 (SPSS). Thus, 119 implants were evaluated, 61 in the test group and 58 in the control group. The results showed statistically significant differences for the variables: average BTI implants in both groups (p<0,001), mean ISG implants both in the test group (p<0,001), and control (p= 0,006) of implants; PS for the test group (p< 0,001) and control (p = 0,015) and SS (p<0,001) in the two treatment groups. However, there was no statistically significant difference when the groups were compared. The PS and SS variables showed no statistically significant difference in any of independent interest to the study (age, sex, smoking, treatment group, keratinized mucosa at different times, peri-implant biotype, average VPI implants and GBI). Thus, it can be concluded that both the mechanical treatment isolated as its association with chlorhexidine mouthwash 0.12% can be used for the treatment of peri-implant mucositis. Moreover, the condition of oral h ygiene has improved between baseline and six months and the depth and bleeding on probing decreased after three and six months

A relação de Euler para poliedros

In this paper we analyze the Euler Relation generally using as a means to visualize the fundamental idea presented manipulation of concrete materials, so that there is greater ease of understanding of the content, expanding learning for secondary students and even fundamental. The study is an introduction to the topic and leads the reader to understand that the notorious Euler Relation if inadequately presented, is not sufficient to establish the existence of a polyhedron. For analyzing some examples, the text inserts the idea of doubt, showing cases where it is not fit enough numbers to validate the Euler Relation. The research also highlights a theorem certainly unfamiliar to many students and teachers to research the polyhedra, presenting some very simple inequalities relating the amounts of edges, vertices and faces of any convex polyhedron, which clearly specifies the conditions and sufficient necessary for us to see, without the need of viewing the existence of the solid screen. And so we can see various polyhedra and facilitate understanding of what we are exposed, we will use Geogebra, dynamic application that combines mathematical concepts of algebra and geometry and can be found through the link http://www.geogebra.org