ESEIG - MAT - Comunicações em eventos científicos
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- Abordagens não convencionais em manuais do ensino secundário: um exemploPublication . Soares, Filomena Baptista; Pascoal, António JoséOs matemáticos do séc. XIX só ficaram plenamente tranquilizados quando o conceito de limite se viu completamente “livre” de qualquer conotação "metafísica", ou seja, quando se soube, graças à astúcia genial dos “épsilon – delta” de Weierstrass, exprimir no estilo Arquimedes a ideia intuitiva de "verdadeiro valor" de uma quantidade indeterminada sem invocar os acréscimos "infinitamente pequenos” que, no entanto, tinham tido êxito no século XVIII. Mas o preço a pagar para apenas manipular conceitos bem definidos a partir das noções algébricas sobre os números, foi a “inversão” dos raciocínios na Análise, ou seja, o facto de que é necessário raciocinar ao contrário relativamente ao caminho heurístico e adivinhar a escolha estratégica “vencedora” em cada junção ou desdobramento lógico. Perante esta dificuldade o ensino da noção de limite viu-se “arrumado” para o 12º ano (para não dizer, aí minimizado) e os conceitos que dela dependem, como o de derivada, viram-se, nos anos anteriores, esvaziados de significado formal, sendo apresentados através de noções (próximas, mas não formais) das não convencionais reduzindo-se à expressão característica de “tende para”. Esta “tendência” não possui na Análise Clássica qualquer significado formal, e apesar de se poder considerar próxima da definição Não Convencional de limite, não lhe sendo feita qualquer referência, fica assim, impossibilitada qualquer formalização da “intuição” em questão, no entanto, pretendemos alertar, através de um exemplo, para uma “pseudo” utilização das suas noções e conceitos. Constatamos, mais uma vez, que a Análise Não Convencional parece ser um caminho possível para uma abordagem da Análise num nível não universitário.
- An economical model for dumping by dumping in a Cournot modelPublication . Banik, Nilanjan; Ferreira, Fernanda A.; Martins, J.; Pinto, Alberto A.We consider an international trade economical model where two firms of different countries compete in quantities and can use three different strategies: (i) repeated collusion, (ii) deviation from the foreigner firm followed by punishment by the home country and then followed by repeated Cournot, or (iii) repeated deviation followed by punishment. In some cases (ii) and (iii) can be interpreted as dumping.We compute the profits of both firms for each strategy and we characterize the economical parameters where each strategy is adopted by the firms.
- An integer programming framework for sequencing cutting patterns based on interval graph completionPublication . Lopes, Isabel Cristina; Carvalho, J. M. Valerio deWe derived a framework in integer programming, based on the properties of a linear ordering of the vertices in interval graphs, that acts as an edge completion model for obtaining interval graphs. This model can be applied to problems of sequencing cutting patterns, namely the minimization of open stacks problem (MOSP). By making small modifications in the objective function and using only some of the inequalities, the MOSP model is applied to another pattern sequencing problem that aims to minimize, not only the number of stacks, but also the order spread (the minimization of the stack occupation problem), and the model is tested.
- An integer programming model for the minimum interval graph completion problemPublication . Lopes, Isabel Cristina; Carvalho, J. M. Valerio deThe minimum interval graph completion problem consists of, given a graph G = ( V, E ), finding a supergraph H = ( V, E ∪ F ) that is an interval graph, while adding the least number of edges |F| . We present an integer programming formulation for solving the minimum interval graph completion problem recurring to a characteri- zation of interval graphs that produces a linear ordering of the maximal cliques of the solution graph.
- Asymmetric dynamic price competition with uncertaintyPublication . Ferreira, Fernanda A.; Pinto, Alberto A.We consider a dynamic setting-price duopoly model in which a dominant (leader) firm moves first and a subordinate (follower) firm moves second. We suppose that each firm has two different technologies, and uses one of them according to a certain probability distribution. The use of either one or the other technology affects the unitary production cost. We analyse the effect of the production costs uncertainty on the profits of the firms, for different values of the intercept demand parameters.
- Asymmetric taxation in a competition between nonprofit and for-profit firmsPublication . Ferreira, Fernanda A.This paper considers a Cournot competition between a nonprofit firm and a for-profit firm in a homogeneous goods market, with uncertain demand. Given an asymmetric tax schedule, we compute explicitly the Bayesian-Nash equilibrium. Furthermore, we analize the effects of the tax rate and the degree of altruistic preference on market equilibrium outcomes.
- Behind video lectures in a MOOCPublication . Soares, Filomena Baptista; Lopes, Ana Paula; Vieira, IsabelThe year 2012 was the “boom year” in MOOC and all its outstanding growth until now, made us move forward in designing the first MOOC in our Institution (and the third in our country, Portugal). Most MOOC are video lectured based and the learning analytic process to these ones is just taking its first steps. Designing a video-lecture seems, at a first glance, very easy: one can just record a live lesson or lecture and turn it, directly, into a video-lecture (even here one may experience some “sound” and “camera” problems); but developing some engaging, appealing video-lecture, that motivates students to embrace knowledge and that really contributes to the teaching/learning process, it is not an easy task. Therefore questions like: “What kind of information can induce knowledge construction, in a video-lecture?”, “How can a professor interact in a video-lecture when he is not really there?”, “What are the video-lectures attributes that contribute the most to viewer’s engagement?”, “What seems to be the maximum “time-resistance” of a viewer?”, and many others, raised in our minds when designing video-lectures to a Mathematics MOOC from the scratch. We believe this technological resource can be a powerful tool to enhance students' learning process. Students that were born in digital/image era, respond and react slightly different to outside stimulus, than their teachers/professors ever did or do. In this article we will describe just how we have tried to overcome some of the difficulties and challenges we tackled when producing our own video-math-lectures and in what way, we feel, videos can contribute to the teaching and learning process at higher education level.
- Bertrand and Cournot oligopolies when rivals' costs are unknownPublication . Ferreira, Fernanda A.; Ferreira, FlávioWe study Bertrand and Cournot oligopoly models with incomplete information about rivals’ costs, where the uncertainty is given by a uniform distribution. We compute the Bayesian- Nash equilibrium of both games, the ex-ante expected profits and the ex-post profits of each firm. We see that, in the price competition, even though only one firm produces in equilibrium, all firms have a positive ex-ante expected profit.
- Bertrand model under incomplete informationPublication . Ferreira, Fernanda A.; Pinto, Alberto A.We consider a Bertrand duopoly model with unknown costs. The firms' aim is to choose the price of its product according to the well-known concept of Bayesian Nash equilibrium. The chooses are made simultaneously by both firms. In this paper, we suppose that each firm has two different technologies, and uses one of them according to a certain probability distribution. The use of either one or the other technology affects the unitary production cost. We show that this game has exactly one Bayesian Nash equilibrium. We analyse the advantages, for firms and for consumers, of using the technology with highest production cost versus the one with cheapest production cost. We prove that the expected profit of each firm increases with the variance of its production costs. We also show that the expected price of each good increases with both expected production costs, being the effect of the expected production costs of the rival dominated by the effect of the own expected production costs.
- Bertrand oligopoly when rivals' costs are unknownPublication . Ferreira, Fernanda A.; Ferreira, FlávioWe study a Bertrand oligopoly model with incomplete information about rivals' costs, where the uncertainty is given by a uniform distribution. We compute the Bayesian-Nash equilibrium of this game, the ex-ante expected profit and the ex-post profit of each firm. We see that, even though only one firm produces in equilibrium, all firms have a positive ex-ante expected profit.