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

Corpos brincantes: a cultura corporal do pastoril potiguar

Pastorais, Bailes Pastoris, Festa da Lapinha, Terno de Reis, Pastor, Pastoril Religioso ou Profano... Pastoris do Norte e Nordeste brasileiro. M ário de Andrade designou-o de Pastoril, folguedo de origem ibérica, tendo sua raiz primeira nos villancico. Considerando que o Pastoril é um folguedo popular, porque nele seus participantes engajam sua vida pessoal, sua cultura e suas influências, revelando modos de ser e compreender, que são interiorizados pelos brincantes a partir da vivência de seu contexto social múltiplo e de diferentes sentidos, é que situamos esse folguedo no mundo vivido fenomenológico. A investigação é de natureza fenomenológica, elucidando os elementos citados como cenário do vivido e do sensível. O presente trabalho tem como objetivo refletir sobre a compreensão de corpo encontrado nos pastoris norteriograndenses

Entre corpos reais e virtuais: reflexões da dança contemporânea para pensar o corpo na educação física

Esta pesquisa objetiva identifi car as novas confi gurações de corpo advindas da interconexão entre Dança e Tecnologia, bem como compreender de que maneiras essas confi - gurações se aproximam e contribuem para a compreensão de corpo na Educação Física. O recurso metodológico é a análise de imagens dos vídeos de dança: Digital Brazuca e Corpo Aberto. Os vídeos analisados apontam para um repertório de saberes, que podem ser problematizados na área, e contribuem para reconfi gurar a relação homem/ambiente. Nesse sentido, promovem a plasticidade e a polissemia de relações entre o carbono e o silício, e desenham outras formas de compressão do corpo que incluem a complexidade, a incerteza, a reversibilidade e a descoberta

Corpos da Ciranda: narrativas poéticas, artístico/estética da ciranda de Lia de Itamaracá

This research is a study that deals with the language of the players bodies of Ciranda (a typical dance in circle in Brazil) – more specifically the one of Lia from Itamaracá. Our interest is to observe how this body dances, communicates, writes on time and space, establishing relations that complement and help to remain in construction. Thus, in a circular way and in an energy that is transmitted with the contact of the touch of the hands, in the power of song, in a circle that can be seen from many places, but by different angles, holding on that the particularities of its subjects players/dancers/observers, and that we propose ourselves to think: who are those players bodies and how do they build the circles of Ciranda? Therefore, during the pathway of the research, we were conducted by the phenomenological method and, from this, used the concepts of lived and sensible world. Our interest in this manifestation is, also, the body that dances and insert itself in the artistic expression, meaning and opening itself to the knowledge by the experience. Therefore, we assume a conception of the body that refers itself in the merleaupontyana fenomenologic approach, in this way, in its criativity in relation to the body as a fragmented being, as it is pointed by the Cartesiana theory. In this perspective, we understand the body in its relations with the culturals, sociais, economics and artistics issues that integrates it, in others words, in the relations that helped us to better understand the body as it is. This way, this research has as main objective to present the refletions about the players body, mainly, of Lia from Itamaracá and with this body to dance, to communicate, to write itself on time and space, estabilishing relations that complement and help to stay in construction. Such a statement leads us in this work, to inquire, for example: what mobilizes those subjects on this dance? We understand that elements as the place that is always in modification, the costumes, the musicality and the change in the movimentation of the players body in each new circle, those elements are factors that activate a permanente reconfiguration that are happening in the Ciranda dance nowadays. Finally, we assert that this investigation comes in reason to the big dimension that the Ciranda has achivied in Brazil, especially in the Northeast, as the existence of few references and registers of the reseach manifestation in the academic areas. It is possible to verify, in this research, that, in reason to this spreading, the nuances of the players bodies is even more diversified and that the missing of experiences on Itamaracá island - PE, its origin has putting away the original and community, becames, more and more, a dance of others stages and squares.

Corpos Queer e a experiência da sexualidade: notas para o conhecimento da educação física

This study had the goal of make a dialogue between queer theory and the thoughts of the French philosopher Maurice Merleau-Ponty in the categories of body and sexuality. From this dialogue, other goals were designed, namely: identify possible recurrences of the experience of bodies and queer sexualities, designed under Merleau-Ponty’s perspective, to the knowledge of Physical Education and reflect on this domain of knowledge using the notions of queer epistemology and esthesia. The study had as methodology the phenomenological attitude proposed by Merleau-Ponty and use the reduction as technic of research. Trying linking these thoughts we used the cinema of the Spanish director Pedro Almodóvar as perceptive strategy, an exercise of look as possibility of reading the world and new ways of perceiving the human being. We appreciate three films, namely: All About My Mother (1999), The Skin I Live In (2011) and Bad Education (2004), which put us in touch with bodies and queer sexualities, with the body of esthesia, of the ecstasy, sensations and lived experiences, un type of art whose contours are not fixed or determinable, postulate by Merleau-Ponty. The philosopher, provide a rich conceptual view of the body and their sexual experience, extends and opens horizons of thought and reflection about queer experience, one experience indeterminate and contingent as a singular way of inhabiting the world. Those horizons opened by the philosopher and added to the queer perspective contribute to put in question the modes of knowledge production and the knowledge about body and sexuality in Physical Education. Finally, we point that this theoretical conversation give us clues to reflect about the reverberations of a queer epistemology for Physical Education usiging one type of knowledge guided by esthesia and sensitivity as marks of another scientific rationality.

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

Corpos no espelho: um estudo antropológico sobre as construções corporais através das cirurgias plásticas na cidade de Natal

Ce travail est une contribution aux études anthropologiques conduites sur Ia beauté et le corps dans des segments moyens urbains. Il cherche à analyser le processus de construction du corps et sa relation avec le individualisme et Ia beauté et la réalisation de chirurgies plasticiennes esthétiques, à partir d'une approche de la construction sociale. En ce sens, la ligne directrice de cette discussion sont le concept de corps comme une construction sociale et culturelle spécifique de chaque société et les différentes utilisations et les représentations que les divers segments font de celui-ci. A cette fin, une recherche de caractere ethnographique a été réalisé aupres des femmes âgés de vingt à quarante et cinq ans, résidents des quartiers Tirol et Petrópolis de Ia ville de Natal-RN (Brésil), dans Ia période d'avril à novembre 2006. Le travail de terrain a mêlé observation directe et entretiens avec des femmes et des médecins chirurgiens plasticiens

Corpos diferenciados e o processo de criação da performance Kahlo em Mim Eu E(m) Kahlo

The dissertation intends to develop an investigation on the artistic existence in human beings with different bodies in society at different historical moments. In this regard and based on this scenario, the study develops a description of the stigmas production and how they are established, spread and interfere with the sociability among human beings regarded as normal and those with different bodies. Regarding the scenic arts, the text describes about the participation of artists with different bodies in the scene, specifically the freak show and postdramatic theater. The text also investigates aspects of the biography and the work of mexican artist Frida Kahlo, which underpin methodological proceedings and produce contribution to the creative process of performance Kahlo em mim Eu e(m) Kahlo , which is to investigate the practice of the scene in this dissertation

Mover/se(r) a gênese do corpo entre/corpos e seus trajetos no processo coreográfico

This paper proposes a reflection on the body between/bodies, especially in contemporary dance, in their path that starts from the choreographic construction, permeating the body of the choreographer, the dancer s body and when fulfills themselves as artistic expression, the body of the spectator. Initially discusses the body in dance as a body/space for convergence, connectedness and continuity, from the thought of the Greek philosopher Epicurus of Samos, in dialogue with the thought of Maurice Merleau-Ponty, Gilles Deleuze and José Gil. Reflect about the creation of this body/space in the relationship choreographer/dancer using as connecting thread the experiences of the author in his artistic path. Finally describes the process of creating the scenic experiment (h)áporos, which constitutes the practice scene of this dissertation, having as main objective the creation of spaces of convergence and interaction between a proponent and an affluent body that, in this move, transforms itself and the space that now cohabits / is

Formulação de massas cerâmicas para porcelanas elétricas utilizando resíduos oriundos de corpos de fusíveis

The power industry generates as waste ceramic bodies of electrical fuses that are discarded after use. The formulation of ceramic bodies for porcelain electrical insulators using waste from the bodies fuse allocation promotes environmentally appropriate, through the reuse of the material. This work evaluated the technical feasibility of using waste for use in electrical porcelains with formulations containing the residue, feldspar and kaolinite. The raw materials were processed through grinding and sieving to 200 mesh. The ceramic material obtained from the proposed formulations with 25%, 30%, 34% and 40% of the residue went through a vibratory mill for grinding and homogenization, and then were sieved at 325 mesh. The samples were shaped in a uniaxial press, with the application of 25 MPa and sintered at 1100° C, 1150°C, 1200°C, 1225°C and 1250°C, at levels of 20 and 45 minutes. Were also developed bodies of evidence with reference formulations obtained without residue, to establish a comparison on physical, mechanical and electrical. The tests were conducted and technology: linear shrinkage, porosity, water absorption, resistance to bending to three points, measuring insulation resistance electrical resistivity of the material, X-ray diffraction and X-ray fluorescence Waste characterizations pointed to the existence of two phases: mullite and quartz phases are of great importance in the microstructure of the ceramic and this fact reveals a possibility for reuse in electrical porcelains. The mullite is an important constituent because it is a phase that makes it possible to increase the mechanical strength in addition to the body allows the use at high temperatures. The use of ceramic bodies residue fuses, proved feasible for application in electrical porcelain and the most significant results were obtained by the formulations with 25% waste and sintering at 1200°C