Thesis (Ph. D.)--Ohio State University, 2007. / Title from first page of PDF file. Includes bibliographical references (p. 112-115).

Depuis les 50 dernières années, de nombreux progrès ont été faits dans la compréhension du comportement asymptotique du nombre de points rationnels de hauteur bornée sur les variétés algébriques. Des conjectures précises ont été avancées par Baryrev, Manin et Peyre quant à la formule asymptotique attendue pour une variété générale.En 1962, à l'aide d'arguments issus de la méthode du cercle de Hardy et Littlewood, B. Birch a donné une estimation précise du nombre de points à coordonnées entières bornées dans une hypersurface définie par une équation homogène. Ceci revient à démontrer la conjecture de Batyrev-Manin-Peyre pour les hypersurfaces de l'espace projectif. Plus récemment, V. Blomer et J. Brüdern ont élaboré des techniques leur permettant d'établir une formule pour le comportement asymptotique du nombre de points de hauteur bornée pour des hypersurfaces d'espaces multiprojectifs définies par des équations multihomogènes diagonales. Parallèlement, D. Schindler a démontré la conjecture pour des hypersurfaces générales d'espaces biprojectifs, à l'aide de développements de la méthode de Birch.L'objet de cette thèse a été d'utiliser et de généraliser les techniques de Schindler, Blomer et Brüdern afin de démontrer la validité de la conjecture de Batyrev-Manin-Peyre pour le cas d'hypersurfaces de variétés toriques plus générales.Ce travail est composé de trois parties. La première partie concerne le cas particulier des hypersurfaces de tridegré (1,1,1) d'un espace triprojectif. Ce cas particulier constitue une première extension des techniques de Schindler à des variétés toriques dont le rang du groupe de Picard est 3. La deuxième partie est consacrée à l'étude des hypersurfaces d'une famille de variétés toriques dont le rang du groupe de Picard est 2 et contenant la famille des espaces biprojectifs. Il s'agit en effet d'étendre la méthode de Schindler afin d'obtenir une formule asymptotique pour le nombre de points de hauteur bornée sur ces variétés. Enfin, dans la dernière partie, nous généralisons les méthodes développées dans les deux parties précédentes à des hypersurfaces des variétés toriques complètes lisses de rang de groupe dont le cône effectif est supposé simplicial, ce qui nous permet de démontrer la conjecture de Batyrev-Manin-Peyre pour ces variétés. / For the last 50 years, many progresses have been made in the understanding of the asymptotic behaviour of the number of rational points of bouded height on algebraic varieties. Some precise conjectures have been advanced by Batyrev, Manin, and Peyre for the expected asymptotic formula for a general variety.In 1962, using some arguments of the Hardy-Littlewood circle method, B. Birch gave a precise estimate for the number of integral points whose coordinates are bounded on an hypersurface defined by an homogeneous equation. This amounts to demonstrating the Batyrev-Manin-Peyre conjecture for hypersurfaces of projective spaces. More recently, V. Blomer and J. Brüdern developed some methods permitting to establish a formula for the asymptotic growth of the number of points of bounded height on hypersurfaces of multiprojective spaces defined by multihomogeneous diagonal equations. In the same time, D. Schindler proved the conjecture for general hypersurfaces of biprojective spaces by using some developements of the method of Birch.The aim of this thesis was to use and generalize the methods of Schindler, blomer, and Brüdern in order to prove the Batyrev-Manin-Peyre conjecture in the case of hypersurfaces of some general toric varieties.This work contain three parts. The first one deals with the particular case of hypersurfaces of tridegree (1,1,1) of triprojective spaces. This particular case is a first extension of the method of Schindler to some toric varieties whose rank of the Picard group is 3. The second part deals with the study of hypersurfaces of a class of toric varieties whose rank of the Picard group is 2 and containing biprojective spaces. We establish a generalization of the method of Schindler method in order to find an asymptotic formula for the number of points of bounded height on these vrieties. Finally, in the last part, we generalize the methods developed in the last two part to treat the case of hypersurfaces of complete non-singular toric vareties whose effective cone is simplicial. This permits to prove the conjecture of batyrev-Manin-Peyre for these varieties.

L'étude de la distribution des points rationnels sur les variétés algébriques est un sujet classique de la géométrie diophantienne. Le programme proposé par V. Batyrev et Y. Manin dans des années 90 donne une prédiction sur l'ordre de croissance tandis que sa version ultérieure dûe à E. Peyre conjecture l'existence d'une distribution globale. Dans cette thèse nous nous proposons une étude de la distribution locale des points rationnels de hauteur bornée sur les variétés algébriques. Ceci envisage une description plus fine que celle globale en dénombrant les points le plus proche d'un point fixé. Nous nous plaçons sur le cadre récent du travail de D. McKinnon et M. Roth qui met en évidence que la géométrie de la variété gouverne l'approximation diophantienne sur elle et nous reprenons les résultats de S. Pagelot. L'ordre de croissance espéré et l'existence d'une mesure asymptotique sur certaines surfaces toriques sont démontrés, alors que démontrons-nous un résultat totalement différent pour une autre surface sur laquelle il n'y pas de mesure asymptotique et les meilleurs approximants génériques s'obtiennent sur des courbes rationnelles nodales. Ces deux phénomènes sont de nature radicalement différente au point de vu de l'approximation diophantienne. / The study of the distribution of rational points on algebraic varieties is a classic subject of Diophantine geometry. The program proposed by V. Batyrev and Y. Manin in the 1990s gives a prediction on the order of growth whereas its later version due to E. Peyre conjectures the existence of a global distribution. In this thesis we propose a study of the local distribution of rational points of bounded height on algebraic manifolds. This aims at giving a description finer than the global one by counting the points closest to a fixed point. We set ourselves on the recent framework of the work of D. McKinnon and M. Roth who prefers that the geometry of the variety governs the Diophantine approximation on it and we take up the results of S. Pagelot. The expected order of growth and the existence of an asymptotic measure on some toric surfaces are demonstrated, while we demonstrate a totally different result for another surface on which there is no asymptotic measure and the best generic approximates are obtained on nodal rational curves. These two phenomena are of a radically different nature from the point of view of the Diophantine approximation.

This small scale exploratory research project investigated the grounding of rational number concepts in informal, everyday life situations. A qualitative approach was taken to allow for the identification and then in depth investigation, of issues of importance for such a grounding of rational number understanding. The methodology followed could be seen as a combination of grounded theory and developmental research. And the data was generated through in-depth and clinical interviews structured around a number of grounded tasks related to rational numbers. The research comprised three cycles of interviews that were transcribed and then analysed in detail, interspersed with periods of reading and reflection. The pilot cycle involved a single grade three teacher, the second cycle involved 2 grade three teachers and the third cycle involved 2 grade three children. The research identified a number of different perspectives that were all important for the development of a fundamental intuitive understanding that could be considered personally meaningful to the individual concerned and relevant to the development of rational number concepts. Firstly in order to motivate and engage the child on a personal level the grounding situation needed to be seen as personally significant by the child. Secondly, coordinating operations provided a means of developing a fundamental intuitive understanding, through coordination with affording structures of the situation that are relevant to rational numbers. Finally, goal directed actions that are deliberately structured to achieve explicit goals in a situation are important for the development of more explicit concepts and skills fundamental for rational number understanding. Different explicit structures give rise to different interpretations of rational numbers in grounding situations. In addition to these perspectives, it became evident that building and learning representations was important for developing a more particularly mathematical understanding, based on the fundamental understanding derived from the child's grounded experience. The conclusion drawn in this research as a result of this complexity, is that to achieve a comprehensive and meaningful grounding, children's learning of rational numbers will not follow a simple linear trajectory. Rather this process forms a web of learning, threading coordinating operations for intuitive development, interpretations for explicit grounding and representations to develop more formal mathematical conceptions.

A escassez de água é um dos principais problemas que o homem terá que enfrentar no século XXI. O grande desafio atual é atender à crescente demanda de água e ao mesmo tempo preservar o recurso que tende à escassez. A alternativa mais viável é trabalhar no sentido da conservação e uso racional, ou seja, na gestão da demanda. Um dos principais objetivos dos programas de uso racional de água (PURA) é a gestão do uso da água. Diversas instituições nacionais e estrangeiras têm desenvolvido PURA em edifícios contemplando a redução de perdas e a implementação de ações tecnológicas para a economia de água. As principais ações tecnológicas para o uso racional de água são os equipamentos economizadores. Neste trabalho foi feita uma pesquisa para identificar os principais equipamentos disponíveis no mercado e qual a sua melhor utilização. Foram pesquisadas também diversas instituições de ensino superior que estão implementando esses Programas. Foram identificadas como suas principais motivações a redução do valor pago à concessionária de água, a redução da demanda para a expansão de seus campi, a necessidade ecológica de economizar água e a preocupação com a escassez de água. Paralelamente, este trabalho buscou inicialmente a implementação de um programa de uso racional de água em um edifício da UFRGS. No entanto, devido a diversos fatores foi necessário alterar o foco do desenvolvimento do trabalho e passar para o planejamento da pré-implantação de um PURA no Anel Viário do Campus do Vale da UFRGS. As ações de pré-implantação foram dificultadas pois parte das informações fundamentais para o reconhecimento do sistema não existem ou se encontram dispersas na estrutura da Universidade. Uma das principais ações de pré-implantação seria a construção de um banco de dados que centralizasse as informações e que auxiliasse na elaboração de uma estratégia de intervenção, baseando-se, também, no histórico dos sistemas. Para a concretização da implantação de um PURA é de fundamental importância a vontade política da universidade, pois somente dessa maneira é possível fazer com que todos os usuários se tornem participantes do programa e trabalhem para a sua implantação e manutenção. / Water scarcity is one of the major problems facing man in the 21st Century. The current challenge is to meet the increasing water demand and at the same time preserve this resource that tends to become scarce. The most viable alternative is to work towards its conservation and rational use, i.e. in the management of the demand. One of the main goals of the Programs of Rational Water Use (PURA) is the reduction of the consumed water. Several national and foreign institutions have been implementing PURA on buildings, seeking a reduction in losses and making use of engineering actions to save water. The main engineering actions aimed to achieve a rational water use are the water-saving equipments. In the present study a research was conducted to identify the most common equipments available in the market and its best form of utilization. A number of institutions of higher education that are making use of this kind of programs were also researched and the main reasons for them to be doing so were identified. They are as it follows: reduction of the amount paid to the water concessionaire; reduction of the demand in order to expand their campus; the ecological necessity of saving water; and a concern with the issue of water scarcity. Parallel to that, this study initially intended to perform an implementation of a program of rational water use at a building of UFRGS. However, due to many reasons, it has become necessary to shift the focus of the work to the planning of a pre-implementation of PURA at the Anel Viário do Campus do Vale da UFRGS. The actual actions of preimplementation were proved hard to put in practice because part of the fundamental information needed for the characterization of the system does not exist or is scattered throughout the University structure. Therefore one of the main actions of a preimplementation would be the creation of a database that gathered this information together and thus helped the outlining of an intervention strategy. This database should also take into consideration the historical aspects of these systems. To accomplish the implementation of a PURA it is of the uttermost importance the political will of the university. It can act as a facilitator, turning users into conscious agents of the implementation and maintenance of a program.

Submitted by Yara de Almeida Campos Cordeiro (yarinha.a@gmail.com) on 2015-06-03T18:26:36Z No. of bitstreams: 1 Diss_Biblioteca.pdf: 649363 bytes, checksum: 3edcafbc40c828ea9dfa7b5ecbf4892b (MD5) / Approved for entry into archive by BRUNA BARROS (bruna.barros@fgv.br) on 2015-06-12T14:12:17Z (GMT) No. of bitstreams: 1 Diss_Biblioteca.pdf: 649363 bytes, checksum: 3edcafbc40c828ea9dfa7b5ecbf4892b (MD5) / Approved for entry into archive by Marcia Bacha (marcia.bacha@fgv.br) on 2015-06-12T18:08:06Z (GMT) No. of bitstreams: 1 Diss_Biblioteca.pdf: 649363 bytes, checksum: 3edcafbc40c828ea9dfa7b5ecbf4892b (MD5) / Made available in DSpace on 2015-06-12T18:08:24Z (GMT). No. of bitstreams: 1 Diss_Biblioteca.pdf: 649363 bytes, checksum: 3edcafbc40c828ea9dfa7b5ecbf4892b (MD5) Previous issue date: 2015-04 / This paper investigates the expectations formation process of economic agents about inflation rate. Using the Market Expectations System of Central Bank of Brazil, we perceive that agents do not update their forecasts every period and that even agents who update disagree in their predictions. We then focus on the two most popular types of inattention models that have been discussed in the recent literature: sticky-information and noisy-information models. Estimating a hybrid model we find that, although formally fitting the Brazilian data, it happens at the cost of a much higher degree of information rigidity than observed.

TORRES, Máro Règis Rebouças. Números algébricos e transcendentes. 66 f. Dissertação (Mestrado Profissional em Matemática) - Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2017. / Submitted by Jessyca Silva (jessyca@mat.ufc.br) on 2017-09-15T05:05:08Z No. of bitstreams: 1 2017_dis_mrrtorres.pdf: 1191154 bytes, checksum: bcb31593bd1a02e84caee6bd47906dab (MD5) / Approved for entry into archive by Rocilda Sales (rocilda@ufc.br) on 2017-09-15T11:00:00Z (GMT) No. of bitstreams: 1 2017_dis_mrrtorres.pdf: 1191154 bytes, checksum: bcb31593bd1a02e84caee6bd47906dab (MD5) / Made available in DSpace on 2017-09-15T11:00:00Z (GMT). No. of bitstreams: 1 2017_dis_mrrtorres.pdf: 1191154 bytes, checksum: bcb31593bd1a02e84caee6bd47906dab (MD5) Previous issue date: 2017 / The present work deals with algebraic and transcendent numbers characterizing them under different aspects. In particular we bring some demonstrations of the irrationality of the number π and the number of Euler, base of the natural logarithm. We will also present a demonstration of the transcendence of the number and based on the script of exercises proposed by D.G. de Figueiredo, in addition to a small historical survey on π, and, algebraic and transcendent numbers. / O presente trabalho trata sobre números algébricos e transcendentes caracterizando-os sob diferentes aspectos. Em particular trazemos algumas demonstrações da irracionalidade do número π e do número de Euler, base do logaritmo natural. Também apresentaremos uma demonstração da transcendência do número e baseada no roteiro de exercícios propostos por D.G. de Figueiredo em [4], além de um pequeno apanhado histórico sobre π, e, números algébricos e transcendentes.

Existem vários métodos para medir tipo e grau de uso de medicamentos. Esse estudo teve como objetivo caracterizar a prescrição médica para idosos, por meio de indicadores, em diferentes ambientes de atendimento à saúde em 4 municípios do Rio Grande do Sul. Buscou-se caracterizar a prescrição em locais de formação e atuação universitária, estabelecendo a prevalência de intervenções nãofarmacológicas e farmacológicas em prescrições para idosos, avaliando diferentes indicadores quanto a sua aplicabilidade em idosos e realizando inferências em relação ao uso racional de medicamentos. Fizeram parte do estudo serviços de atenção básica em saúde, além daqueles de média e alta complexidade, vinculados a ambientes de formação universitária na área da saúde, no sul do Brasil. A amostra foi constituída por prescrições médicas, obtidas diretamente dos pacientes, em nível de atenção primária em saúde e ambientes de média complexidade, ou por meio de prontuários hospitalares, durante o período de um ano. Para caracterização dos idosos, foram considerados aqueles com idade igual ou superior a 65 anos. A coleta de dados foi realizada ao longo de doze meses, em semanas, dias e turnos definidos por meio de sorteio. sorteio. Como resultados, observou-se que, nos ambientes de atenção básica em saúde e nos ambientes de média complexidade estudados, os idosos do Rio Grande do Sul apresentaram maior média de medicamentos prescritos e prevalência considerável de medicamentos inapropriados, em comparação com outros estudos que envolveram dados da população em geral em outros estados do país. Em contrapartida, esses idosos receberam menor percentual de prescrições com antimicrobianos e baixo percentual de prescrições com agentes injetáveis. As classes de medicamentos mais comumente prescritas no ambiente ambulatorial foram aquelas de uso contínuo, provavelmente em função das enfermidades crônicas apresentadas por pacientes desta faixa etária. Destacaram-se os fármacos que agem no sistema cardiovascular, no sistema nervoso e no trato gastrintestinal e metabolismo. Por sua vez, na internação de idosos nos hospitais estudados, independentemente de município, sazonalidade, gênero ou faixa etária mais avançada, identificou-se um perfil elevado de utilização de medicamentos, com polifarmácia (n=345, 85,4%) e prescrição de medicamentos inapropriados (n=325, 80,4%) em número significativo de idosos. Os medicamentos que mais apareceram nas prescrições hospitalares foram aqueles prescritos sob regime de demanda (se necessário). A prescrição inadequada aos idosos é frequentemente atribuída à falta de treinamento de uma equipe especializada em geriatria e gerontologia, além da deficiência da formação universitária. Nesse sentido, considerando-se o contexto demográfico e epidemiológico brasileiro e a caracterização da prescrição para o paciente idoso realizada no estudo, é importante priorizar ações multidisciplinares relacionadas a promoção, prevenção e recuperação de enfermidades e padronizar procedimentos, para evitar erros de prescrição, transcrição e dispensação. A educação continuada dos profissionais da área da saúde, a divulgação e a atualização de listas de medicamentos essenciais e das listas de medicamentos inapropriados para idosos podem ser ferramentas úteis para a qualificação da prescrição e a promoção do uso racional de medicamentos em idosos. A partir dos pontos vulneráveis da prescrição para idosos que foram levantados, podem ser estabelecidas mudanças nos ambientes de formação em saúde estudados, visando a construção de um perfil profissional que paute suas ações pela comunicação efetiva, interdisciplinar e compromisso social. / There are several methods to measure type and degree of drug use. This study aimed to characterize the prescription for the elderly by means of indicators in different environments of health care and 4 cities of Rio Grande do Sul. It was characterized the prescription in academic and training environments, establishing the prevalence of non-pharmacological interventions and pharmacotherapy prescriptions for the elderly, assessing the applicability of different indicators to the elderly prescriptions, and making inferences about the rational use of medicines. Participants of the study were elderly people attended in primary health care services and in medium and high complexity health care services, linked to universities located in the south of Brazil. The sample consisted of prescriptions obtained directly from patients, at the level of primary health care and environments of medium complexity, or through hospital records, during the period of one year. It was considered elderly patients those with age over 65 years. Data collection was conducted over twelve months, in weeks, days and shifts defined by lot. It was observed that in primary health care and medium complexity services the Rio Grande do Sul elderly prescriptions had a higher mean number of medications and a considerable prevalence of inappropriate medications, compared to other studies involving data from the general population. However, these elderly received a lower percentage of prescriptions with antibiotics and injectable drugs. The classes of drugs most commonly prescribed in the outpatient setting were those of continuous use, probably due to the chronic diseases that frequently occurred in the old age. Those classes involved drugs that act on the cardiovascular system, nervous system and gastrointestinal tract and metabolism. In the hospitals studied, independently of the city, seasonality, gender or older age, it was identified high profile drug utilization, with polypharmacy (n = 345, 85.4%) and inappropriate prescription of medications (n = 325, 80.4%) in great number of elderly. The drugs that most appeared in the hospital prescriptions were those prescribed under the demand scheme ("if necessary"). Inappropriate prescribing for the elderly is often attributed to the lack of training in geriatrics and gerontology and disability of university education. Considering the Brazilian epidemiological and demographic context and the prescription pattern for the elderly observed in this study, it is important to prioritize disciplinary actions related to the promotion, prevention and recovery from illness and standardize procedures to avoid errors in prescribing, transcribing, and dispensing. The continuing education of professionals in the health, distribution and update of essential drugs lists and use of lists of inappropriate drugs for elderly may be useful tools for the improvement of the prescription and promotion of rational drug use in the elderly. Based on the vulnerabilities of the prescription for the elderly that have been raised, it is possible to establish changes in the studied healthy education environments, aiming to build a professional profile that bases its actions on effective communication, interdisciplinary and social commitment.

O objetivo desta tese é explicar a redução nas taxas de sequestros políticos na Colômbia nos últimos anos a partir da análise do comportamento estratégico dos criminosos. Pontos de vista convencionais explicam a diminuição dos sequestros como o resultado exitoso da política de segurança democrática do presidente Álvaro Uribe Vélez. No entanto, até agora, tem sido desconsiderada a busca de explicações alternativas à já existente, que bem poderiam ser encontradas a partir da perspectiva de análise dos dilemas organizacionais produzidos pelos sequestros nas FARC – Forças Armadas Revolucionárias da Colômbia –, da sua interação estratégica com o governo e, em particular, de seus efeitos sobre sua atividade global e na decisão de pôr fim a essa ação criminosa. O interesse dos três ensaios que compõem esta tese é estudar as motivações deste grupo rebelde, sob o enfoque da teoria da agência, dividindo sua estrutura organizativa entre líderes (principal), que tomam as decisões estratégicas, e os combatentes (agentes), que as realizam, em um contexto de informação assimétrica, para tomar decisões racionais. Cada ensaio desenvolve a partir de diferentes perspectivas, mas tendo como base o enfoque racional de principal-agente, as razões que levaram a organização a renunciar a uma de suas atividades criminosas, considerada no princípio como uma ação estratégica eficiente que obrigaria o governo colombiano a negociar. O primeiro ensaio está focado em mostrar os custos de transação que gerou essa estratégia para os agentes e o principal. Esta análise faz uso dos mesmos instrumentos analíticos empregados para analisar os custos de qualquer transação econômica que leva a cabo uma organização legal. Mostrando que os custos dessa atividade foram altos, expressados, primeiro, em um conflito de interesses entre o líder, encarregado de esquematizar e designar tarefas, e os agentes, responsáveis por sua execução. A divergência entre estas duas partes teve origem em uma mudança nas expectativas dos agentes, que preferiam mais atividades de combate às relacionadas com o sequestro, em um contexto de perseguição constante do exército colombiano. O segundo ensaio estuda como essa mesma estratégia afetou o contexto no qual os agentes definem suas preferências. Através do uso de três enfoques diferentes da teoria econômica se expõem três interpretações diversas da mudança nas preferências dos agentes: a) uma mudança no risco; b) uma divergência entre as preferências subjacentes e induzidas; c) a presença de dimensões motivacionalmente salientes. E o terceiro ensaio apresenta um modelo formal para estabelecer um sistema de compensações eficiente que o principal oferece ao agente para atenuar o que sobre seu comportamento gerou o sequestro. Os resultados mostram que, considerando que os recursos das organizações armadas ilegais são escassos, quanto maiores são os incentivos oferecidos aos agentes para evitar que desertem, menor é a capacidade da organização para penalizar os desertores e menor a utilidade do principal. Simulando o modelo para um conjunto específico de parâmetros se conclui que a incorporação do mecanismo de autocumprimento (self-enforcing) dentro da função de utilidade do principal aumenta seus custos e propicia o baixo esforço do agente e seu comportamento oportunista. / The objective of this thesis is to explain the reduction in the rate of political kidnapping in Colombia in recent years by means of analyzing the strategic behavior of its perpetrators. This is the basic question addressed in this thesis. Conventional views interpret the fall in the kidnapping rate as an outcome of President Álvaro Uribe’s democratic security policy. I will argue, however, that this is not the whole story, since political kidnapping led Farc [for its acronym in Spanish, Fuerzas Armadas Revolucionarias de Colombia] into an unprecedented strategic situation that induced a breach between leader (principal) and combatant (agent) concerns with strong effects on its overall activity and its decision to stop that criminal action. The focus of three essays making up this thesis is on studying FARC’s motivations from the perspective of agency theory, by splitting its organizational structure into principals and agents who are acting on a setting of asymmetrical information. Each essay develops, from different perspectives, the reasons that led the organization to give up that criminal activity due to the substantial political and organizational risks involved. The first essay is focused on the transactions costs generated by the kidnapping strategy both for agents and principals. This analysis is based on the same theoretical tools used to study the costs held by any legal organization. I found that the costs of kidnapping were high, expressed first in a conflict of interest between the leader –responsible for designing and assigning tasks—and the agents in charge of its implementation. The divergence was due to a shift in the expectations of agents who preferred combat activities over the menial tasks associated with kidnapping, in a context of heavy pressure by the Colombian Army. In contradistinction to legal organizations in which such type of divergence can be solved, in part, by paying higher wages to agents in order to extract their best effort, this alternative is not feasible for FARC, for those who joined the organization are supposed to have an ideological and political commitment. The second essay studies how the kidnapping strategy affected the preferences of agents and their behavior by means of using three different approaches from economic theory: (a) a change in risk, (b) a divergence between underlying and induced preferences and, (c) the presence of salient motivational dimensions. The third essay examines, through a principal-agent model, the nature of the trade-off between incentives and enforcing mechanisms that the leadership of an Armed illegal organization offers to its agents. Using a MATLAB’s optimization tool-box, I computed the optimal transfer system for a given parameterization of the model, and analyzed its properties. The numerical analysis shows that the inclusion of a self-enforcing mechanism on the leader’s objective function increases the costs for the principal and could lead agents to choose low efforts and engage in opportunistic behavior.

A presente pesquisa tem como foco o ensino dos números racionais no Ensino Fundamental. Seu objetivo é analisar as potencialidades e proposta de ensino que utiliza o computador, em particular do so fltiwmaitraeç dõee sg edoem uemtriaa dinâmica GeoGebra, e a metodologia de aulas de matemática investigativa. Essa proposta de ensino foi aplicada em turmas de 8º ano do Ensino Fundamental de uma escola pública, localizada na cidade de Sapucaia do Sul/RS. Esses estudantes já traziam conhecimentos sobre números racionais e, diante disso, esse campo numérico foi retomado com a intenção de verificar e contornar possíveis déficits de aprendizagem, assim como de ampliar o conhecimento sobre esses números por meio de investigações em que pudessem ser observadas algumas de suas características. Para isso, propusemos atividades que articulavam simultaneamente diferentes representações dos números racionais. A metodologia de pesquisa empregada foi a qualitativa. A proposta de ensino apresentada no final deste texto como alternativa de trabalho para o ensino de números racionais, e para professores, diante da análise realizada, poderem refletir acerca de suas vpiostueanicsi adliod asdoefstw ea rliem GitaeçoõGese.b Drao, sq ruees uflatavdooresc oebratimdo sa,o pso daelumnooss dae sctoamcaprr eoesn rseãcou rdsooss números racionais e suas regularidades, e o cenário investigativo-tecnológico, que fez com que eles se mantivessem engajados na investigação como agentes de seu aprendizado. / This research focuses on the teaching of rational numbers in Elementary School. It's aim is to analyse The potentialities and limitations of a teaching approach which proposes the use of the computer, particularly the dynamic geometry software GeoGebra, and the methodology of investigative math classes. This teaching approach was applied to students of the 8th year of an Elementary Education public school located at Sapucaia do Sul/RS, Brazil. These students already studied rational numbers at school in previous years. Thus this numerical field has been taken into account with the intention to check and bypass possible learning deficits, as well as increase knowledge of these numbers through investigations in which it could be observed some your characteristics. For this purpose, we proposed activities that simultaneously articulated different representations of rational numbers. The research is based in a qualitative paradigm. The teaching approach is presented at the end of this text as an alternative way for teaching rational numbers and other teachers in view of the analysis can consider its potentialities and limitations. From the results, we can highlight the visuals of the GeoGebra software, which favored students the understanding of rational numbers and their regularities and the investigative - technological scenario, which caused them to remain engaged in research as their learning agents.