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

Slogans políticos em campanhas eleitorais no Rio Grande do Norte: a mídia na disputa ideológica.

To the observe the periods of electoral campaigns, we can realize sharply the abundance of political propaganda that announces the candidates, their campaign record, their proposed and everything that can carry them to the victory in the urns. The propaganda is today an essential tool in the votes dispute, in consequence of this, there is, in Brazil, the increasing participation of the Politics Parties in seminars of electoral marketing. They are varied strategies of the media used in the divulging of the candidacies, among them the slogan. Instigated by the explicit relation media/political, this work intends to investigate the ideologies (values) announced by the slogans of political propaganda that circulated in electoral periods. Thus, we base on the theory of the Circle of Bakhtin, who considers the statement as unit of the verbal communication, and that conceives the language as phenomenon dialogical, and, too, in the conceptions of alien words, social voices and dialogical relations formulated by the referred theory. We use, for better to comprehend, the slogan as strategy mediatic, of the classical study done by Olivier Reboul concerning the slogan; and, like starting point of a conception about ideology, studied John Thompson to comprehends the ideological phenomenon. Our corpus it constitutes of 14 political slogans announced in the campaigns for mayor of the cities in Natal and Mossoró, both located in Rio Grande do Norte s state, Brazil, in 1996, 2000 and 2004. For the analysis of these slogans, we consider the social-historical context in which were announced, being contemplated the linguistic and discursive aspect, according to the following methodological order: the identification of alien words (interdiscours); the social voices and the kinds dialogical relations established; and, finally, the ideology announced by the statement. This way, we can arrive to the conclusion that in the electoral context occurs a true ideologies war , in other words, in the political game of an electoral campaign, the slogans announces different values, which vary in several factors function, for example, the historicity, the subject that enunciates, the discursive positionings.

Homoafetividade e abertura política em contos de Caio Fernando Abreu

This text aims to study the literary representation of homoaffectivity in short stories of Caio Fernando Abreu in the context of brazilian redemocratization, specifically in the narratives published between 1982 and 1988. To understand the relation between literature and historical context in the tales, we betake, principally, to a notion of Literature and Society, as proposed for Antonio Candido. Relating the narratives of Caio Fernando Abreu with the military dictatorship and the redemocratization processes, as well as a generational project named contraculture, we aim to verify as the search for affection for men that desire others men may be conditioned for specific socio-hystorical conditions, understanding the literary work as a privileged field of representation and comprehension of reality

Medicina filosófica: as relações entre medicina e filosofia na Grécia antiga e em Kant

This work deals with the relationship between medicine and philosophy, which has existed since Antiquity, and will also be discussed here from Kant s perspective. It presents the historical context formed by reciprocal influences of common notions regarding health/disease, balance/justice, and just measure, which are present in the medical discourse as much as in the philosophical one. It considers that Hippocratic medicine emerges from concerns about dietetics, thus creating the link between philosophy and medicine, which is important for our analysis on Kant s contributions to Hippocratic legacy. Taking into account these considerations, the work distinguishes between two aspects which are associated within the dietetics presented by Kant in his work The conflict of the faculties, studied here in the light of his Doctrine of virtue, particularly the duties to oneself in regard the care of one s body and the teleological conception. In this sense, the work indicates the role of Kantian thinking not only to enrich medical dietetics, by lending to it moral value, but also to enrich philosophy by highlighting its therapeutic effects

História, memória e espaços: experiências dos ex-combatentes de Parelhas-RN na defesa do litoral brasileiro durante a Segunda Guerra Mundial

The present essay has how I aim to analyse the memories of the ex-combatants of Parelhas-RN, specially of the components of the Força de Vigilância e Segurança do Litoral - FVSL, protagonists of the Brazilian participation in the scenery of the Second World war. Along this we looked to understand in which surrounding geographicalpartner these men were living before the War and what were the consequences of a brusque change of space owing to the convocation for the Armed Brazilian Strength in that historical context. The defense of the Brazilian coast during the War was not a so simple task, I have in mind the precariedade logistics of the Armed Strength, the attacks of submarines of the Axle that killed hundreds of civilians and Brazilian soldiers and the net of espionage mounted by Germany in Brazil. Leaving from the notion of collective memory and estrangement in Maurice Halbwachs, we will use the oral history like principal methodology, with the end of rescue these underground memories what also will make possible us the vision realizes that the protagonists themselves have of the event, besides the use of documents, photos, maps and any sort of fountains that make possible us to rebuild the scenery of Parelhas in the beginning of the War and the trajectory of life of his veterans

Mundo de medo: inquisição e cristãos-novos nos espaços coloniais capitanias de Pernambuco, Itamaracá e Paraíba (1593-1595)

Founded in 1536, the Court of the Holy Office of the Portuguese Inquisition was established as an ecclesiastical institution, but at the same time subordinate to the real powers. Among the main victims of persecution effected by the Holy Office, were the New Christians - Jews forcibly converted in 1497 or their descendants - that due to their socio-religious were repeatedly accused of heresy. This paper conducted a survey that sought to understand the historical performance of the Inquisition in Brazil in the sixteenth century on the New Christians, especially those accused of secretly retaining the religious customs of the Mosaic law, given the investigative and punitive procedures employed by the Inquisition as part of a set of actions that produce social insecurities and producers / broadcasters of fear in the populations under scrutiny. In this sense, the approach was based on the analysis of documents produced on the first visit of inspection performed in Brazil inquisitorial sixteenth century, concerning the captaincy of Pernambuco and Paraiba Itamaracá (1593-1595), not excluding, however, the sources of the first stage of visitation that occurred between 1591-1593 in the province of Bahia de Todos os Santos, even though its use is ancillary and punctual. The objective of this research was to understand the consequences of inquisitorial procedures generated on the imaginary, and the Inquisition, using the expressions and signs of fears relating to individuals contained in the New Christian complaints to the Holy Office as documentary evidence of the fear caused by the Holy Tribunal. The adoption of specific behaviors by the New Christians in the home - these spaces are appropriate and adapted to the detriment of the religious practices of Judaism features - characterizes the spatial perspective of the study, thus indicating a further objective of the study: to understand how the New Christians experienced domestic spaces in a historical context marked by behavioral surveillance generally considered morally condemned and suspected of heresy. The research was conducted to analyze the complaints and quantitative survey of some indices of documentation for the understanding of overall charges and how individuals New Christians were concerned with the domestic space, using them to maintain criptojudaica religiosity, transformed places housing often esnogas, makeshift synagogue for meetings and celebrations of Judaizing New Christians. The formulations of Michel de Certeau on appropriations and meanings of space - presented by the author in the metaphor of "practice areas" - were integrated into the workforce in order to understand the ways in which the New Christians appropriated the colonial houses, designed these spaces a very specific language in the Crypto, in which women are prominent figures. The works of Jean Delumeau and Bartholomé Benassar integrate the discussion of the Inquisition and the sensibilities of fear in the work performed. The analysis allowed the documentation to understand the meaning and the extent related to the general fear that the Inquisition represented. Some complaints are indicative of fears that can be perceived implicitly based on behaviors and attitudes adopted by the New Christians, others, however, are direct expressions of fear caused by allusion or initiative of the actions of the Inquisition in colonial Brazil in the sixteenth century

A coleção mossoroense e a construção dos mitos: Dix-Sept Rosado, o herói imolado

This research comprises a study on the values and ideals attributed to Dix-sept Rosado, by the Coleção Mossoroense journal. Local researchers demonstrate how the Coleção Mossoroense constructs an image of Mossoró, forming in the members of the city an identity with the place, a social cohesion or creating one forms to see and to say about the Mossoró city. The Coleção Mossoroense, through Vingt-un Rosado together with its family, established a form of seeing and identifying of the Mossoró s population with the city, just because it associated and it was, at the same time, a movement at the same time social (civic parties for example), political (the domination of the Rosado s family) and cultural (the books and theatrical parties about the city). The conjunction of the Coleção Mossoroense as and with a social movement, cultural and political was capable to establish a social identity, a subjective constituent . With this background, it was possible to make a study of the image attributed to Dix-sept Rosado, by the Coleção Mossoroense, through an analysis of the inherent discourse in the works by Bakhtin (1989) and Bourdieu (2000), of and on Dix-sept Rosado in the Coleção Mossoroense based on. The image of Dix-sept Rosado was constructed by the Coleção Mossoroense after its death, and the responsible for this was Vingt-un Rosado, which was the chief-editor of the Coleção Mossoroense up to 2006, year of its death. The Coleção Mossoroense built an image of Dix-sept Rosado as an enterprising man, progressive and innovator, of great capacity of leadership and oratory (a conductor of the crowds ), popular and charismatic, an honest and dynamic politician. This image made by the Coleção Mossoroense, although when enhancing in surplus the qualities of Dix-sept Rosado have as objective to transmit an image of a heroic personage, it has some relation with the reality. Dix-sept Rosado was a man identified with its locality. This can resemble a regional nationalism. The attitudes of Dix-sept Rosado must be understood inside of a broader social context, of a time marked by the populist politics. Qualities also appreciated and developed inside of a specific historical context and determined both national and locally. The Coleção Mossoroense enhances the qualities of Dix-sept Rosado so that the current politicians of the Rosado s family can establish a link of continuity in its contemporary political practice

A experiência de adolescentes portadores de diabetes Mellitus Tipo 1: seus significados e sentido

Worldwide, the diabetes mellitus is considered a serious problem of public health; it also involves high costs for its treatment and its complications. Even though the onset of diabetes mellitus type 1 is on 5% to 10% of the diabetic population, it is the most aggressive type of anifestation; furthermore, it is the commonest chronic disease in childhood and adolescence. Such data show the importance of understanding the development of this disease from the moment the individual perceives it, according to their experience, observing the complexity of the phenomenon thoroughly. This study aimed to comprehend the experience of the adolescent with onset of diabetes mellitus type 1 in order to learn how these young individuals perceive and reason their place in the world when they experience this illness. The methodological strategy was based on qualitative research of phenomenological inspiration. We applied the main ideas of heideggerian ontology and some concepts of existential phenomenological psychology to guide our reflexive path. The participants were 10 young individuals (06 female and 04 male), ages between 15 and 18, under treatment in the city of Recife, Pernambuco. The methodological resource was the narrative, which allowed us access to the adolescents experience. The comprehension of the reports was based on the heideggerian hermeneutics, whose priority is the interpretation of the meaning expressed by the participants speech. The results showed that factors such as social, family, economic and cultural context greatly influence the perception and the way the individuals deal with, perceive and reason their experience. We noticed that the diabetes type 1 may often represent a limitation, but it does not make it impossible to be part of the world with others. The data show a significant affective ambivalence regarding the content more directly related to the disease. This demonstrates the unique way which each one gives meaning and reasons their condition of existence. This research points out the need to understand the complexity of this phenomenon in a more complete manner, considering the adolescent in their timely and historical context. We believe we can encourage thinking that may become actions which are more fit to the reality researched regarding all the parties involved with the subject, such as health professionals, adolescents, family, support programs and public health policies

Análise dos conteúdos termodinâmicos em livros de Química e Física do ensino médio

PNLEM came out in 2004 and modified the view about schoolbook in Brazil. New perspectives pointed towards constitution of a more critical citizen as of an interdisciplinary and more contextualized teaching, made schoolbook understand these changes.But not always these alterations are checked up on by all means,because what we notice are adaptation of works have already known on market, where texts that are placed reveal environmental, socioeconomic, historical facts or daily phenomenon, or then scientific relevancy examples.We thus offer to analyse contents of chemical thermodynamics and physics for high school which are shown on schoolbook of chemistry and physics validated by PNLEM 2012, now called PNLD, studying the facts if these works bring these scientific sociocultural standpoint as well as their historical context,considering as analysis categories the following criteria: presentation of introduction, content contextualizing,presentation of concepts, thermodynamics historical context, representation in microscopic level and proposal of experiments.Eleven books were analysed,being five of chemistry and six of physics.So was applied a questionnaire with physics and chemistry teachers of high school to have an impression of their opinion about their teaching strategies and about schoolbook.This work grounded in orientations of legal documents about high school in Brazil (LDB,PCN,PCN+,OCEM) and in research on science teaching that regards specifically schoolbook and physics and chemical thermodynamics teaching.Were eleven books in number, only four denoted much coherence with new perspectives for high school, being two of chemistry and two of physics,whose concern about contextualizing and with a much integrated view of science to student s reality,indicating much social and critical character.The rest, when don t overlook,regard cultural and historical,socioeconomic contexts in isolated manner from presented concepts,aside from content

Família e subjetividade: tendências e particularidades das classes trabalhadoras no capitalismo contemporâneo

This work problematizes the inflections of the offensive of the capital on the work, having as a social-historical context the relation between the productive restructuring and the social reproduction of the working families classes. Part of the presupposition that the reproduction of the capital, to raise deep transformations in the productions, organization of the work and in the social relationships, it also produces determinations in the life and work conditions, in the affectionate relationships and in the family coexistence expressing, so, a subjective way. Thereby, the conditions of reproductions of the work in the scenery aimed by the contemporary capitalism have been demonstrating the crescent impoverishment of the workers, the alimentary insecurity, the shortage of the work, the weakness of the political organization and the regression of the State in the conduction of public policy that characterize the daily violation of human and social rights. In this approach, we seek to contemplate the several configurations of affectionate-sexual coexistence expressed by the family, articulating it to the contemporary work division, pondering about the forms of satisfaction of the needs engendered by the group for preservation of their bonds, in face to the daily adversity which translates to the growing responsibility to assist social d mands and in the impediments to the enrichment of the individuality and human diversity.

Serviço Social e Política: uma análise da participação dos assistentes sociais no Conselho Regional de Serviço Social 14ª região

This paper analyzes the political participation of Social Workers at the Social Service Regional Council Region 14th. The theoretical and methodological framework of this investigation has as its perspective the totality of social life and its determining relations within the object of study. To the production, analysis and collecting of data it was used a qualitative approach considering a bibliographical and documental research as well as a series of twelve interviews with 2005-2008 and 2008-2011 managing counselors of CRESS/RN. It was also used the data obtained from other special interviews held with the social workers in the period between 2007 and 2008. The results of this study allow and affirm the political dimension of Caseworkers and the CRESS/RN as a space of political activity with opportunities for effective and collective elaboration of strategies in order to reach the fulfillment of the ethical and professional policy of the Social Work. From a historical viewpoint, the beginning of this process is marked by the struggle for democracy, the end of military dictatorship, the establishment of the State of Human Rights. The Democratization of the so called Federal Council of Social Workers and its Regional Councils of Social Workers, CFAS / CRAS, respectively, area a result of the participation of the category in an effort to fight for democratization in Brazil. The objective of this research, so, is to understand which the socio-historical determinants are, that focus objectively and subjectively in the demobilization of social workers in CRESS Region 14th - in the contemporary and historical context. Among the results obtained we identified the ignorance of some professional workers and also of some advisors, regarding the existence and the role of the Council, as well as the commercialization of Education that compromises the quality of the professional training in its theoretical and methodological, ethical and political dimensions. According to our understanding, this shows a the presence of a non-critical professional profile based on a false reality, on the precariousness of employment contracts which undermine the political organization and submit the worker to various social exploring mechanisms such as double shift and ultimately the fragility of the management of the Regional Council -, as a consequence of the offensive capitalist system that ideologically invests to stop the political organization founded in a critical and democratic perspective. The low participation of some advisors and, in general, the category in CRESS / RN, despite its objective conditions, is a reality and it is presented to us as a challenge to future managements and policy consolidation to society. Inasmuch as the category intends to guarantee the high quality of its social workers, the demands of future counselors, their skills and abilities in dealing with regulatory issues, administrative policies that pervade the everyday life of CRESS / RN are necessary

Violência e luta por direitos no capitalismo contemporâneo: crítica à configuração do atendimento às mulheres no Rio Grande do Norte

This research analyzes the Rio Grande do Norte care services to women who face violence in the context of contemporary capitalism. To do so, we situate the patriarchy in the set of current social relations and its relationship with the corporate determinations in everyday life. The new functions of the Patriarchy in the capital sociability permeates the lives of individuals and particularizes the relationships of violence which affects women, requiring, in the immediate level, policy-making to face them. The research found an arsenal of contradictory possibilities and limitations in dealing with violence. In this process, forms of struggle and resistance predominate, which appear as possibilities and limits were identified relate to the socio-historical context of regression of the rights, historical moment in which increase the objective difficulties in everyday life to ensure the legal achievements. It is worth to emphasize the achievements and contradictions that characterize the struggle process for rights, linking services to women to the social policies and to the limits they face in opposition to the aims of the State to meet the mandatory requirements of capital, reducing its role as the main guarantor of policies and rights. In this sense, the trajectory of the achievements that have referred to the proclamation of a specific law to deal with combat violence against women, the Maria da Penha Law - 11.340/06, which provides an integrated set of measures that, if implemented, would allow the women protection from relations of violence they experience. We identified in Rio Grande do Norte precarious services that are essential to achieve the Maria da Penha Law. This situation requires a feminist organization to claim the rights that enable women to see themselves as people with rights in the process of collective struggle. This is the historical need for continuity of struggles that accumulate policies for the existence of a new model of social relations of gender. One of the possibilities that are presented in the current context is the impact on the public budget in order to ensure compliance with the budget for public policies for women - woman budget. In this perspective, feminist segments in national and state level have been organized to understand the functioning and monitoring of social policies. This is a condition and prerequisite for ensuring policies to ensure basic rights and the violence combat , which still requires an integrated set of services. The survey results allow us to consider that the struggle for rights is necessary at this historical moment, however it is not sufficient in human emancipation, which requires new forms of social relations that determine substantive equality between men and women. Thus, the feminist movement faces the challenge to organize and strengthen itself in daily life, in order to execute a project that changes the meaning of women's rights, articulated to a corporate project which wants other command in the set of social relations . This study emphasizes the need for a more and more organic connection between feminism and social struggles, to ensure the inclusion of women in anti-capitalist struggle

As Faces e os Disfaces da Informalidade no Capitalismo Conteporâneo: Um Estudo do Comércio de Rua em Pau dos Ferros/RN

This paper discusses the expression of informality in contemporary capitalism. Thematic of relevance to the analysis of the reality of work today and the logic that moves the capital, its real presence in the lives of individuals. The street trading of Pau dos Ferros town, popularly known as "street market" was chosen as the search space. The main objective is to seize and examine the articulations and logic, present in the configuration of the street trading of this city, located in the state of Rio Grande do Norte, explaining the functionality of informality for capitalist accumulation, but also for the reproduction of segments of the working class. Our analysis is based in the perspective of totality, trying to grasp the historical determinations of the phenomenon in focus. It includes the analysis of the mechanisms used by the capital to reproduce itself in the current historical context, which has been implicated in the composition of the labor markets of different countries and in various forms of exploitation to which workers in general are subject. It also means discussing the development of capitalism in Brazil, the logic that permeates its dependence, and especially the use of over-exploitation of labor, as a lever for internal accumulation. The course of investigation consisted of theoretical research to form the basis of theoretical and methodological analysis and to outline the context in which our research object is inserted, and field research conducted in two phases: systematic observation, which allowed to map traders features and the infrastructure of commerce, and the conduction of interviews with key informants. The material collected was scrutinized according to analytical scheme inspired by the content analysis. Among the main considerations developed from the research process we include: the street trading of Pau dos Ferros remains shrouded in the majority sale of agricultural products, this demonstrates the structural characteristics of the region. However, the supply of this product is no longer restricted to the excess of small local producers. The presence of the dealer changed the distribution of the product, streamlining it. In parallel, business practices are developed, practices in which traded goods (industrial) reflect the moment of capitalist restoration, a larger business network. The reflections also made it possible to show that street trading follows developing on the basis of informal work, which gains functionality to the system, as it is configured as a space commonly used to drain part of the production, of industries (clothing/shoes), especially if the distribution is considered as an essential element of the complex process that aims at capital appreciation. This activity has been functioning as a place of employment and income generation for the subjects who are away from formal employment, masking, this way, unemployment, moreover, they allow them to continue as consumers. Such expressions reflect the ability and the logic of capital to expand and aggregate into so many realities. It is underway today, the logic that has led many workers to join the project of domination of capital, by the illusory chance to become capitalists. The aim has been to turn the subject into a consumer and the worker an enterprising

Equações de segundo grau e mudança de variáveis

In this work are presented, as a review and in a historical context, the most used methods to solve quadratic equations. It is also shown the simplest type of change of variables, namely: x = Ay + B where A;B 2 R, and some changes of variables that were used to solve quadratic equations throughout history. Finally, a change of variable, which has been used by the author in the classroom as an alternative method, is presented and the result of this methodoly is illustrated by the responses of a test that was done by the students in classroom