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

Forró eletrônico: uma análise sobre a representação da figura feminina

Understanding music as a social manifestation, able to represent the time and culture along with their uses and mores, we look at the female figure in the electronic forró genre. Through a reading of the two genres musical and feminine we seek to identify the discourses present in the compositions, which refer to the feminine with thematic and/or terms referring to women. The study focuses on the compositions between 2009 to 2010 which became hits, ie.: those which have attained the highest number of listeners to 96 FM in a program named Paradão 96, broadcasted daily in the city of Natal/RN. Grounded in such concepts as cultural industry, mass culture, folkcommunication and gender, we analyze the processes of meaning production as a result of using the female figure and more accurately, about the representation of women through the discourse of this specific genre. We have chose the descriptive research with a qualitative approach and Discourse Analysis as a technique to identify the terms that refer to women, reflecting on what would be the possible meanings found in these speeches. We point out that in our investigation the female figure turned into a sexual object, erotic and submissive, in their turn, attributes usually given to women, was not a predominant result in the speech of electronic forró gender. With a variant thematic some playful and others about the betrayal of women we have identified that, to the surprise of many, including ourselves, romantic love was the one who led the audience requests during our time frame and that the female figure has several facets within the studied genre

A Representação discursiva da figura feminina no jornal O Porvir (Currais Novos/Rio Grande do Norte - 1926-1929)

The objective of this paper is to analyze how the female figure was portrayed in discursive narratives in the bi-weekly newspaper O PORVIR , which circulated in Currais Novos, in the state of Rio Grande do Norte, Brazil, between May 2, 1926 and January 20, 1929. Our analysis was based on the semantics of the text, by applying one of the primary concepts used in Textual Analysis of Speeches, Discursive Representation (ADAM, 2011; RODRIGUES; PASSEGGI; SILVA NETO, 2010). Textual Analysis of Speeches has its roots in Textual Linguistics and consists of a theoretical and descriptive approach (ADAM, 2011). This research is characterized as documentary (SEVERINO, 2007) and its corpus includes a review of 39 editions of the aforementioned newspaper, of which we extracted 292 articles that dealt with the female figure and 396 statements for analysis. We used the following semantic categories: referencing (CASTILHO, 2010; NEVES, 2007; KOCH; MARCUSCHI, 1998; MARCUSCHI, 2008; KOCH, 2009); predication (NEVES, 2007; RODRIGUES; PASSEGGI; SILVA NETO, 2010); modification (ADAM, 2011); and spacial and temporal localization (RODRIGUES; PASSEGGI; SILVA NETO, 2010). Our analysis revealed that lady, mother, wife and homemaker were most represented, and always combined with questions about motherhood, marriage and devotion to the home. Such representation does not diverge from pre-colonial models in the early twentieth century, which served as a standard for the female figure

Produção em revista : Representação do moderno e do regional na experiência potiguar anos 1920

This work brings some reading about a few Norte-rio-grandenses writers collaborations published in literary papers which circulated during the third decade of the twentieth century in Brazil, in a way of enlarging the understanding of how the cultural debate was carried out in the years of the definition of the Brazilian Modernism, based on the relationships between literature and culture. Under the genres of prose and poetry, the selected textual collaborations for analysis have brought the traces of a small group of articulated authors in the Brazilian cultural scenery from the 1920 s, concerning the issues of the modern (linked to the modernist renovation) and the regional (under the aspects of the tradition). The set of the analyzed collaborations is representative of a modern-regional cultural practice marked by a tendency either local or cosmopolitan, thus, sharing a large dimension of the formation of the modern literary tradition, marked by different kinds of practices which give form to the project of a collective national awareness. The characterization of the existence of an intellectual field with big links, however with its own characteristics and a certain autonomy, which also gets integrated to the cultural debate of the so-called modernist years through the literary vehicles, gives to these intellectual people a co-participation in the renewing movement in the formation context of the brazilian cultural modernity of the early twentieth century

Forró eletrônico: uma análise sobre a representação da figura feminina

Entendiendo la música como una manifestación social, capaz de representar el tiempo y la cultura, junto con sus usos y costumbres, dirigimos nuestra mirada en la figura femenina presente en el género forró electrónico. A través de la lectura de los dos géneros musical y femenino buscamos identificar los discursos, presentes en las composiciones, que se refieren a lo femenino con temática y/o términos que se refieren a las mujeres. El estudio se centra en las composiciones en el período 2009 a 2010 y que se convirtieron en éxitos, es decir, aquellos que han alcanzado el mayor número de oyentes de Radio 96 FM a través del programa Paradão 96, transmitido diariamente en la ciudad de Natal/RN. Basado en conceptos tales como industria cultural, cultura de masas, folkcomunicación y género, se analizan los procesos de producción de sentido como resultado del uso de la figura femenina y con mayor precisión, sobre la representación de las mujeres a través de discursos del forró electrónico. Elegimos el enfoque descriptivo cualitativo y análisis del discurso como una técnica para identificarlos términos que se refieren a las mujeres, reflexionando sobre los posibles significados que se encuentran en estos discursos. Se señala que en nuestra investigación no predominó una figura femenina convertida en un objeto sexual, erótico y sumisa, atributos dados para las mujeres, en los discursos de forró electrónico. Con variaciones en las temáticas algunas juguetones y otras acerca de la traición de la mujer identificamos que, para sorpresa de muchos, incluidos nosotros, el amor romántico fue el que llevó las solicitudes de audiencia durante nuestro período de tiempo y que la figura femenina tiene varias facetas dentro del forró electrónico

Liberdade e negação da vontade: análise do ser-livre como representação e na angústia

Common understanding about what freedom means has always been more or less related to the power to realize something intended, desired, a capability. Therefore, being free is commonly interpreted under the concept of free-will and the category of possibility to act. Although there are predecessors in History of Philosophy, Schopenhauer refuses the thesis of free will proposing otherwise the denial of willing (to live) as the ultimate possibility for human freedom, if not the only one left. The thesis that would make him famous was deeply misunderstood and so miscarried somewhat due to the way it was many times presented by the means of exotic examples wrapped in a mystical mood besides exaltations to Eastern traditions, which may satisfy anthropological curiosity instead of being capable to satisfy the reader in a philosophical way. It seems to result from Schopenhauer s thought a kind of pessimism against life. Otherwise, typical readings on the Schopenhauerian thesis are found full of inconsistencies once closely regarded, which blame does not belong to the author but to his interpreters. A new reading about the denial of willing as the ultimate possibility for human freedom demands a criticism on the inconsistencies and prejudgments deep grounded. For this, we firstly clarify the ways of understanding the willing nothing , which cannot be reduced to the mere refusal or conformism, being instead positively understood as a special manner of willing: the admission of oneself for the sake of one is. A few more than a century later The world as will and representation came to light, Heidegger proposes in his fundamental ontology that the proper being-free concerns to originary decision by which, in anguish of being suspended in nothingness, Dasein renders itself singular as the being who is in-a-world and to-death, concluding that the ultimate possibility of freedom is being-free-to-death. Developing the hypothesis that freedom, properly understood, concerns to nothingness as to indeterminate possibilities, we seek for a dialogue between Schopenhauer s thought and existential philosophy aiming to reconstitute and overcome Metaphysics tradition turning the question about freedom into a matter of Ontology. From the factual existence perspective, as we must show, every human activity (or inactivity) is ordinarily mediated by representations, in which me and world appear as distinct entities. So, each one among determininate individuals finds itself connected to the things in the world by interest, which proper concept must be sufficiently explored. Starting from this point, we may proceed to detailed analysis of usual representations of freedom aiming their destruction by Ontology and then reaching existential thesis according to Kierkegaard and Heidegger. Turning back to the analysis of Schopenhauer s work, we conclude existential understanding of freedom as will-to-be can also be found in Schopenhauer. In this way, denial of willing means ultimate freedom once the Will turns back to its own essence by suppressing the world as representation, which means the originary absolute indetermination of the extreme possibility to-be

Determinacão do período orbital de sistemas binários clipsantes

In this work is presented a new method for the determination of the orbital period (Porb) of eclipsing binary systems based on the wavelet technique. This method is applied on 18 eclipsing binary systems detected by the CoRoT (Convection Rotation and planetary transits) satellite. The periods obtained by wavelet were compared with those obtained by the conventional methods: box Fitting (EEBLS) for detached and semi-detached eclipsing binaries; and polynomial methods (ANOVA) for contact binary systems. Comparing the phase diagrams obtained by the different techniques the wavelet method determine better Porb compared with EEBLS. In the case of contact binary systems the wavelet method shows most of the times better results than the ANOVA method but when the number of data per orbital cicle is small ANOVA gives more accurate results. Thus, the wavelet technique seems to be a great tool for the analysis of data with the quality and precision given by CoRoT and the incoming photometric missions.

Evolução da excentricidade dos sistemas binários com componente evoluída

On this study we have revisited the predicted tidal circularization theory in close binary systems with a evolved component. Close binaries suffer tidal interactions that tend to synchronize periods and circularize the orbits (Zahn 1977, 1989, 1992). According to Zahn s theory we compute the integral that give us the variation of the eccentricity in a binary under the influence of tidal force and we compare the integral results with new observations for 260 binary systems with orbital solutions. Our results confirm the success of the Zahn s theory with a new data and new stellar evolutionary models, on the other hand, our results points to the need for a better description of the role of convection on this theory

Lítio e parâmetros orbitais em sistemas binários com componentes evoluídas

One of the best established properties of the single late type evolved stars is that their rotational velocity and lithium content decrease with effective temperature and age. Nevertheless, the root cause of this property, as well as the link between rotation and lithium abundance and, in particular, the effects of binarity on rotation and lithium content in binary systems with evolved component, are not yet completely established. How does the gravitational tides, in binary systems, affects rotational evolution and lithium dilution? Trying to answer these questions, we have carried out an observational survey, in the lithium region centered at the lithium I line A6707.81A, for a large sample of about 100 binary systems with evolved component along the spectral range F, G and K, with the CES spectrometer mounted at the CAT 1.44 m Telescope of the ESO, La Silla, Chile. By combining the abundances of lithium issued from these observations with rotational velocity and orbital parameters, we have found a number of important results. First of all, we confirm that in this class of binary systems rotation is effectively affected by tidal effects. Binary systems with orbital period lower than about 100 days and circular or nearly circular orbits, present rotational velocity enhanced in relation to the single giant stars and to the binary systems with an orbital period larger than 100 days. This is clearly the result of the synchonization between the rotational and orbital motions due to tidal effects. In addition, we have found that lithium abundances in binary systems with giant components present the same gradual decreasing with effective temperature, observed in the single giants of same luminosity class and spectral types. We have found no lithium-rich binary systems, in contrast with single giants. A remarkable result from the present study is the one showing that synchronized binary systems with giant component retains more of their original lithium than the unsynchronized systems. In fact, we have found a possible "inhibited zone", in which synchronized binary systems with giant component having lithium abundance lower than a threshold level should be unusual. Finally, the present study also shows that the binary systems with giant component presenting the highest lithium contents are those with the highest rotation rates

O reino encantado do sertão: uma crítica da produção e do fechamento da representação do sertão no romance de Ariano Suassuna

Based on the relation history, literature and space, this work examines the production and the closing of the representation of the hinterland in the Romance d A Pedra do Reino of the paraiban writer Ariano Suassuna. Inspired by method proposed by deconstructionist by Jacques Derrida, our reflection decomposes the writing scene of Suassuna by means of it s tracks to interpret the operation of the representation of the space-hinterland for the speech of the writer. Being thus, we present the elements chosen for Suassuna to weave an identification it s with this space-hinterland, taking a priori as a sentimental place and of familiar origin . The construction of this relation of belonging in great is measured grounded for a familiar memory entangled in the context politician who culminated in the Revolution of 1930 and that he resulted for the writer in the death of father. Suassuna launches a look on the identity and the Brazilian culture that guided in the tradition concept it searchs to deny the gift grounding its speech in the return to the past, a past order . Specifically in it s romance we notice to exist the fusing of these tracks in the production of a representation of the hinterland that it intends to close a concept of this space and its culture. Soon, the idea of kingdom sends to the space of the enchantments manufactured in infancy and retaken by the intellectual radical who in the anxiety of honor the father figure, does not save efforts in constructing a space representation that, over all, either faithful the representation that it searchs to trace of proper itself. Under one it marks of universality , the hinterland-kingdom of Suassuna if discloses eurocentric in the measure where it perpetuates discurse the filiation of the European cultural traditions.

Da Pena à Espada: Xenofonte e a Representação de Esparta em A Constituição dos Lacedemônios

We intend to analyze how, through your own views and social practices, Xenophon composed the image of Spartans and their poliad regime during the period following the end of the 5th century to the early 4th century before BCE a time of great political turbulence in the Hellenic Poleis. In order to do so, we will use the writings in The Constitution of the Lacedaemonians, to point three elements that we believe are essential understanding of the Xenophon´s narrative: who was Xenophon and in which ways his life experiences influenced his narrative style; the idea of City, i.e., Xenophon´s idea of the Polis and how he defined it as a community of Citizens; and finally, to establish through which tools Xenophon build an image of Sparta and Spartans by way of their representations in his writings

A representação da mulher e as questões de gênero na toponímia urbana de Caicó-RN

This piece wants to investigate the relations between gender identities and the toponymy in Caicó city. To do so, we will discuss the reason why places are named, the practice of naming places, so as the process of choosing the names of neighborhoods, streets and city public squares. Thereby, we will investigate the space changes in Caicó city since its formation, as a village, until the decade of 1970. This period of time was chosen in order to discuss the toponymic process of the city, besides having noticed that public places and locations in Caico with the feminine nominations only appear from the second half of the 60s decade and in the beginning of the 70s. This way, we preferred to investigate who were the first Caicó women to have their names in the urban space; why women only begin to have homage from the decade of the 60s on; what is the importance and the location of the places with feminine names; and what gender image is there on this homage

Representação de linhas de transmissão por meio de variáveis de estado levando em consideração o efeito da freqüência sobre os parâmetros longitudinais

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Uso das tecnologias na representação descritiva: o padrão de descrição bibliográfica semântica MarcOnt Initiative nos ambientes informacionais digitais

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)