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

Geometrias não-eucludianas como anomalias: implicações para o ensino de geometria e medidas

This present research the aim to show to the reader the Geometry non-Euclidean while anomaly indicating the pedagogical implications and then propose a sequence of activities, divided into three blocks which show the relationship of Euclidean geometry with non-Euclidean, taking the Euclidean with respect to analysis of the anomaly in non-Euclidean. PPGECNM is tied to the line of research of History, Philosophy and Sociology of Science in the Teaching of Natural Sciences and Mathematics. Treat so on Euclid of Alexandria, his most famous work The Elements and moreover, emphasize the Fifth Postulate of Euclid, particularly the difficulties (which lasted several centuries) that mathematicians have to understand him. Until the eighteenth century, three mathematicians: Lobachevsky (1793 - 1856), Bolyai (1775 - 1856) and Gauss (1777-1855) was convinced that this axiom was correct and that there was another geometry (anomalous) as consistent as the Euclid, but that did not adapt into their parameters. It is attributed to the emergence of these three non-Euclidean geometry. For the course methodology we started with some bibliographical definitions about anomalies, after we ve featured so that our definition are better understood by the readers and then only deal geometries non-Euclidean (Hyperbolic Geometry, Spherical Geometry and Taxicab Geometry) confronting them with the Euclidean to analyze the anomalies existing in non-Euclidean geometries and observe its importance to the teaching. After this characterization follows the empirical part of the proposal which consisted the application of three blocks of activities in search of pedagogical implications of anomaly. The first on parallel lines, the second on study of triangles and the third on the shortest distance between two points. These blocks offer a work with basic elements of geometry from a historical and investigative study of geometries non-Euclidean while anomaly so the concept is understood along with it s properties without necessarily be linked to the image of the geometric elements and thus expanding or adapting to other references. For example, the block applied on the second day of activities that provides extend the result of the sum of the internal angles of any triangle, to realize that is not always 180° (only when Euclid is a reference that this conclusion can be drawn)

Modelos de condutividade térmica em rochas silicáticas, com ênfase em rochas da província Borborema, NE do Brasil

A practical approach to estimate rock thermal conductivities is to use rock models based just on the observed or expected rock mineral content. In this study, we evaluate the performances of the Krischer and Esdorn (KE), Hashin and Shtrikman (HS), classic Maxwell (CM), Maxwell-Wiener (MW), and geometric mean (GM) models in reproducing the measures of thermal conductivity of crystalline rocks.We used 1,105 samples of igneous and metamorphic rocks collected in outcroppings of the Borborema Province, Northeastern Brazil. Both thermal conductivity and petrographic modal analysis (percent volumes of quartz, K-feldspar, plagioclase, and sum of mafic minerals) were done. We divided the rocks into two groups: (a) igneous and ortho-derived (or meta-igneous) rocks and (b) metasedimentary rocks. The group of igneous and ortho-derived rocks (939 samples) covers most the lithologies de_ned in the Streckeisen diagram, with higher concentrations in the fields of granite, granodiorite, and tonalite. In the group of metasedimentary rocks (166 samples), it were sampled representative lithologies, usually of low to medium metamorphic grade. We treat the problem of reproducing the measured values of rock conductivity as an inverse problem where, besides the conductivity measurements, the volume fractions of the constituent minerals are known and the effective conductivities of the constituent minerals and model parameters are unknown. The key idea was to identify the model (and its associated estimates of effective mineral conductivities and parameters) that better reproduces the measures of rock conductivity. We evaluate the model performances by the quantity  that is equal to the percentage of number of rock samples which estimated conductivities honor the measured conductivities within the tolerance of 15%. In general, for all models, the performances were quite inferior for the metasedimentary rocks (34% <  < 65%) as compared with the igneous and ortho-derived rocks (51% <  < 70%). For igneous and ortho-derived rocks, all model performances were very similar ( = 70%), except the GM-model that presented a poor performance (51% <  < 65%); the KE and HS-models ( = 70%) were slightly superior than the CM and MW-models ( = 67%). The quartz content is the dominant factor in explaining the rock conductivity for igneous and ortho-derived rocks; in particular, using the MW-model the solution is in practice vi UFRN/CCET– Dissertação de mestrado the series association of the quartz content. On the other hand, for metasedimentary rocks, model performances were different and the performance of the KEmodel ( = 65%) was quite superior than the HS ( = 53%), CM (34% <  < 42%), MW ( = 40%), and GM (35% <  < 42%). The estimated effective mineral conductivities are stable for perturbations both in the rock conductivity measures and in the quartz volume fraction. The fact that the metasedimentary rocks are richer in platy-minerals explains partially the poor model performances, because both the high thermal anisotropy of biotite (one of the most common platy-mineral) and the difficulty in obtaining polished surfaces for measurement coupling when platyminerals are present. Independently of the rock type, both very low and very high values of rock conductivities are hardly explained by rock models based just on rock mineral content.