Degree awarded (2004): PhDEc, Economics, George Washington University / This research introduces a new theoretical framework for the analysis of access pricing (the prices that networks charge each other for the completion of calls) and the modeling of network interconnection competition. Prior work on two-way access by Armstrong (1998), Laffont, Rey and Tirole (1998), and Carter and Wright (1999), et al has been built on a two-network Hotelling (1929) differentiated competition model applied to network interconnection. The current research develops an alternative approach that is based on a cross-nested logit (CNL) discrete/continuous consumer choice model with a constant elasticity of substitution (CES) calling utility specification. A principal contribution of the new modeling framework is that in addition to being able to analyze interconnection competition among multiple networks, it is designed to incorporate the element of multiple network subscription where consumers may simultaneously subscribed to more than one type of access network. By introducing multiple-network subscription and usage substitution for users subscribed to multiple networks, the analysis allows more general assessments to be made of the impact of access pricing schemes on the degree of competition between interconnected networks. The model is also not restricted to assumptions of homogeneity in calling on the differentiated networks but can incorporate call differentiation according to network type. The model is applied to evaluate the effects of dual network subscription and asymmetric network competition and to assess multi-network competition in an environment served by two mobile networks and a fixed, wireline network. While confirming the results of prior single network subscription analysis, a central finding of the research based on the developed model is that while network competition is intensified when dual network subscription occurs, negotiated access charges between connected networks continue to serve as an instrument of collusion even in cases of non-linear (two-part) consumer tariffs. / Advisory Committee: John Kwoka, Christopher Snyder (Chair), Sumit Joshi

The overall goal of the dissertation is to contribute to the growing literature on multiple discrete-continuous (MDC) choice models. In MDC choice situations, consumers often encounter two inter-related decisions at a choice instance – which alternative(s) to choose for consumption from a set of available alternatives, and the amount to consume of the chosen alternatives. In the recent literature, there is increasing attention on modeling MDC situations based on a rigorous underlying micro-economic utility maximization framework. Among these models, the multiple-discrete continuous extreme value MDCEV model (Bhat, 2005, 2008) provides a number of advantages over other models. The primary objective of this dissertation is to extend the MDCEV framework to accommodate more realistic decision-making processes from a behavioral standpoint. The dissertation has two secondary objectives. The first is to advance the current operationalization and the econometric modeling of MDC choice situations. The second is to contribute to the transportation literature by estimating MDC models that provide new insights on individuals’ travel decision processes. The proposed extensions of the MDCEV model include: (1) To formulate and estimate a latent choice set generation model within the MDCEV framework, (2) To develop a random utility-based model formulation that extends the MDCEV model to include multiple linear constraints, and (3) To extend the MDCEV model to relax the assumption of an additively separable utility function. The methodologies developed in this dissertation allow the specification and estimation of complex MDC choice models, and may be viewed as a major advance with the potential to lead to significant breakthroughs in the way MDC choices are structured and implemented. These methodologies provide a more realistic representation of the choice process. The proposed extensions are applied to different empirical contexts within the transportation field, including participation in and travel mileage allocated to non-work activities during various time periods of the day for workers, participation in recreational activities and time allocation for workers, and household expenditures in disaggregate transportation categories. The results from these exercises clearly underline the importance of relaxing some of the assumptions made, not only in the MDCEV model, but in MDC models in general. / text

abstract: Consumers search before making virtually any purchase. The notion that consumers engage in costly search is well-understood to have deep implications for market performance. However to date, no theoretical model allows for the observation that consumers often purchase more than a single product in an individual shopping occasion. Clothing, food, books, and music are but four important examples of goods that are purchased many items at a time. I develop a modeling approach that accounts for multi-purchase occasions in a structural way. My model shows that as preference for variety increases, so does the size of the consideration set. Search models that ignore preference for variety are, therefore, likely to under-predict the number of products searched. It is generally thought that lower search costs increase retail competition which pushes prices and assortments down. However, I show that there is an optimal number of products to offer depending on the intensity of consumer search costs. Consumers with high search costs prefer to shop at a store with a large assortment of goods and purchase multiple products, even if the prices that firm charges is higher than competing firms' prices. On the other hand, consumers with low search costs tend to purchase fewer goods and shop at the stores that have lower prices, as long as the store has a reasonable assortment offering. The implications for market performance are dramatic and pervasive. In particular, the misspecification of demand model in which search is important and/or multiple discreteness is observed will produce biased parameter estimates leading to erroneous managerial conclusions. / Dissertation/Thesis / Ph.D. Business Administration 2013

Continuous optimization problems are optimization problems where all variables<p>have a domain that typically is a subset of the real numbers; mixed discrete-continuous<p>optimization problems have additionally other types of variables, so<p>that some variables are continuous and others are on an ordinal or categorical<p>scale. Continuous and mixed discrete-continuous problems have a wide range<p>of applications in disciplines such as computer science, mechanical or electrical<p>engineering, economics and bioinformatics. These problems are also often hard to<p>solve due to their inherent difficulties such as a large number of variables, many<p>local optima or other factors making problems hard. Therefore, in this thesis our<p>focus is on the design, engineering and configuration of high-performing heuristic<p>optimization algorithms.<p>We tackle continuous and mixed discrete-continuous optimization problems<p>with two classes of population-based heuristic algorithms, ant colony optimization<p>(ACO) algorithms and evolution strategies. In a nutshell, the main contributions<p>of this thesis are that (i) we advance the design and engineering of ACO algorithms to algorithms that are competitive or superior to recent state-of-the-art<p>algorithms for continuous and mixed discrete-continuous optimization problems,<p>(ii) we improve upon a specific state-of-the-art evolution strategy, the covariance<p>matrix adaptation evolution strategy (CMA-ES), and (iii) we extend CMA-ES to<p>tackle mixed discrete-continuous optimization problems.<p>More in detail, we propose a unified ant colony optimization (ACO) framework<p>for continuous optimization (UACOR). This framework synthesizes algorithmic<p>components of two ACO algorithms that have been proposed in the literature<p>and an incremental ACO algorithm with local search for continuous optimization,<p>which we have proposed during my doctoral research. The design of UACOR<p>allows the usage of automatic algorithm configuration techniques to automatically<p>derive new, high-performing ACO algorithms for continuous optimization. We also<p>propose iCMAES-ILS, a hybrid algorithm that loosely couples IPOP-CMA-ES, a<p>CMA-ES variant that uses a restart schema coupled with an increasing population<p>size, and a new iterated local search (ILS) algorithm for continuous optimization.<p>The hybrid algorithm consists of an initial competition phase, in which IPOP-CMA-ES and the ILS algorithm compete for further deployment during a second<p>phase. A cooperative aspect of the hybrid algorithm is implemented in the form<p>of some limited information exchange from IPOP-CMA-ES to the ILS algorithm<p>during the initial phase. Experimental studies on recent benchmark functions<p>suites show that UACOR and iCMAES-ILS are competitive or superior to other<p>state-of-the-art algorithms.<p>To tackle mixed discrete-continuous optimization problems, we extend ACOMV <p>and propose CESMV, an ant colony optimization algorithm and a covariance matrix adaptation evolution strategy, respectively. In ACOMV and CESMV ,the decision variables of an optimization problem can be declared as continuous, ordinal, or categorical, which allows the algorithm to treat them adequately. ACOMV and<p>CESMV include three solution generation mechanisms: a continuous optimization<p>mechanism, a continuous relaxation mechanism for ordinal variables, and a categorical optimization mechanism for categorical variables. Together, these mechanisms allow ACOMV and CESMV to tackle mixed variable optimization problems.<p>We also propose a set of artificial, mixed-variable benchmark functions, which can<p>simulate discrete variables as ordered or categorical. We use them to automatically tune ACOMV and CESMV's parameters and benchmark their performance.<p>Finally we test ACOMV and CESMV on various real-world continuous and mixed-variable engineering optimization problems. Comparisons with results from the<p>literature demonstrate the effectiveness and robustness of ACOMV and CESMV<p>on mixed-variable optimization problems.<p>Apart from these main contributions, during my doctoral research I have accomplished a number of additional contributions, which concern (i) a note on the<p>bound constraints handling for the CEC'05 benchmark set, (ii) computational results for an automatically tuned IPOP-CMA-ES on the CEC'05 benchmark set and<p>(iii) a study of artificial bee colonies for continuous optimization. These additional<p>contributions are to be found in the appendix to this thesis.<p> / Doctorat en Sciences de l'ingénieur / info:eu-repo/semantics/nonPublished

Understanding how car travel and ownership respond to income and fuel prices, and how that response varies between households is crucial for car use policies and forecasts. This thesis, consisting of two papers, aims to investigate this by estimating the intemporal income and fuel price elasticities of car use using micro registry panel data on all Swedish households from 1998 to 2018. In Paper I, the income and fuel price elasticities of vehicle kilometres travelled (VKT) is estimated for all Swedish households using a linear fixed effects model. In order to investigate how different groups respond, the elasticities are estimated by income group and municipality type. The effect of income and fuel prices on VKT is largest in the middle of the income distribution but is relatively stable across municipality types. The effect of fuel prices on VKT is largest in densely populated municipalities compared to rural municipalities. Moreover, it is shown that the income elasticity is underestimated if income variable is misspecified. Paper II utilises a discrete-continuous model accounting for the effect of income and fuel prices on car ownership. It is shown that income impacts car ownership and VKT conditional on car ownership of similar magnitude, while fuel prices primarily impact VKT conditional on car ownership. Furthermore, we also estimate the model on six partially overlapping sample periods and find that the income elasticity has decreased over time, while the absolute fuel price elasticity increased up until the early 2010s and decreased thereafter. / <p>2022-10-13: ISBN (PDF) has been added in the E-version.</p>

Understanding how car travel and ownership respond to income and fuel prices, and how that response varies between households is crucial for car use policies and forecasts. This thesis, consisting of two papers, aims to investigate this by estimating the intemporal income and fuel price elasticities of car use using micro registry panel data on all Swedish households from 1998 to 2018. In Paper I, the income and fuel price elasticities of vehicle kilometres travelled (VKT) is estimated for all Swedish households using a linear fixed effects model. In order to investigate how different groups respond, the elasticities are estimated by income group and municipality type. The effect of income and fuel prices on VKT is largest in the middle of the income distribution but is relatively stable across municipality types. The effect of fuel prices on VKT is largest in densely populated municipalities compared to rural municipalities. Moreover, it is shown that the income elasticity is underestimated if income variable is misspecified. Paper II utilises a discrete-continuous model accounting for the effect of income and fuel prices on car ownership. It is shown that income impacts car ownership and VKT conditional on car ownership of similar magnitude, while fuel prices primarily impact VKT conditional on car ownership. Furthermore, we also estimate the model on six partially overlapping sample periods and find that the income elasticity has decreased over time, while the absolute fuel price elasticity increased up until the early 2010s and decreased thereafter.

Considerando-se como fonte primária os manuais escolares brasileiros de Matemática, o saber a ser ensinado ainda situa uma apresentação dual, polarizado no viés pragmático ou teórico, ao que se segue um procedimento temático padrão que privilegia o desenvolvimento operatório envolvendo contextos exatos, finitos e determinísticos. Em particular, essas características se acentuam gravemente no momento de introdução dos números irracionais no ensino básico, o que ocasiona uma abordagem restritiva. Para superar este quadro, Bruner (1987) fundamenta que não devemos adiar o ensino de assuntos essenciais com base na crença de que são difíceis demais, pois as ideias fundamentais de qualquer assunto podem ser ensinadas na escolaridade básica, porém demanda um trabalho para além dos aspectos técnicos, o que equivale a retomada de características ligadas à compreensão. Neste trabalho, tivemos por hipótese que os pares discreto/contínuo; exato/aproximado; finito/infinito, presentes na análise da evolução epistemológica dos números reais e descritos em Machado (2009), se constituem em pilares conceituais essenciais para fundamentar um panorama favorável a uma abordagem significativa do tema dos números irracionais, de modo a compor um amálgama entre os aspectos técnicos e semânticos. Em face da necessária reflexão, em nível educacional, em torno de tal tema, delimitamos inicialmente um contexto investigativo pautado em um estudo qualitativo orientado pela questão Como são abordados os números irracionais no ensino básico, considerando-se como fonte o livro didático de Matemática?, a fim de mapear a apresentação deste assunto no Ensino Fundamental II e no Ensino Médio. O fundamento metodológico se inspirou nos núcleos de significação, descritos em Aguiar&Ozella (2006), que buscou apreender os sentidos que constituem o conteúdo do discurso expresso nos textos dos livros didáticos. O percurso dos núcleos de significação confirmou que, nos livros didáticos analisados, a apresentação dos números irracionais ocorre de modo polarizado: alguns optam por um viés empírico e outros pela definição formal. Verificou-se que, após uma abordagem inicial, não ocorre intercâmbio destas opções, o que acarreta um rápido esgotamento das ferramentas para se desenvolver as temáticas, limitando a compreensão da complexidade dos números irracionais no ensino básico. A partir das hipóteses e da pesquisa empírica, nos propusemos a delinear as contribuições presentes no movimento dialético entre os pares discreto/contínuo, finito/infinito e exato/aproximado, cujas mútuas conexões permeiam um espaço de significações, um campo que possibilita organizar, tecer e ampliar a rede de significados, conforme Machado (1995), favorecendo um quadro de maior compreensão à apresentação dos números irracionais. O enfoque epistemológico realizado revelou uma multiplicidade de relações envolvendo os números irracionais e diversos assuntos do currículo de Matemática, não devidamente caracterizadas e exploradas no ensino básico, o que serviu de mote para a apresentação de algumas situações de ensino para ilustrar os aportes orientadores sugeridos. Acreditamos que o caminho epistemológico trilhado viabilizou uma abertura para ampliar o quadro de significados em relação a outros tópicos presentes na Matemática Elementar, considerando-se como suporte a potencialidade presente nos eixos discreto/contínuo; exato/aproximado; finito/infinito, assim como no par determinístico/aleatório. / Considering Brazilian mathematics school textbooks as a primary research source, the knowledge to be taught still has a dual presentation, polarized in a pragmatic or theoretical way, what follows a thematic procedure pattern that favors an operational development involving exact, finite and deterministic contexts. In particular, these characteristics are seriously accentuated by the time of irrational numbers introduction at basic education, which leads to a restrictive approach. To overcome this situation, Bruner (1987) states that we should not postpone teaching key issues based on the belief that they are too hard, because the fundamental ideas of any subject can be taught at basic education, but it demands a work that overcome technical aspects, considerations that are equivalent to the resumption with aspects related to understanding. In this work, we had by hypothesis that the tension inherent on discrete/continuous, exact/approximate, finite/infinite pairs, extracted from analyses on real numbers epistemological evolution and described at Machado (2009), constitutes an essential conceptual pillar to establish a helpful framework to enable a significant irrational numbers approach, in order to compose an amalgam between technical and semantic aspects. Considering the necessary educational discussion involving this theme, we initially delimited an investigative context based on a qualitative study guided by the question How irrational numbers are approached in basic education, considering mathematics textbook as a source?\' in order to map this subject presentation at Middle and High School. The methodological foundation was inspired in meaning core, described in Aguiar and Ozella (2006), which aims to capture the sense that constitutes the speech content expressed inside mathematics scholar textbooks. The analysis from meaning core route reveals that, in the textbooks examined, the most known irrational numbers introduction occurs in a polarized way: some opt for a pragmatic bias and others by formal definition. However, it was found that after an initial approach, there is no further relationship between these options, which causes a rapid depletion of the tools to develop these themes, which limits the complexity understanding of irrational numbers in basic education. From the hypotheses and the empirical research, we intended to delineate contributions presented on the dialectical movement between discrete/continuous, finite/infinite and exact/approximate pairs, whose mutual connections permeate a \'space of meanings\', a field that allows to organize, to weave and to expand a network of meanings, as Machado (1995), favoring a framework for better understanding the irrational numbers development in basic school. The epistemological approach performed revealed a multiplicity of relationships involving irrational numbers and various subjects of mathematics curriculum, not properly characterized and exploited in basic education, references which served as contexts for the presentation of some teaching situations to illustrate the contributions guidance suggested. We believe that the epistemological path trodden enables an opening to increase possibilities of meanings in relation to other topics of Elementary Mathematics, considering as support the capability constituents presented in discrete/continuous, exact/approximate, finite/infinity axis, as well as in deterministic/random pair.

Cette thèse est consacrée au développement de modèles mathématiques de l'hématopoïèse et de maladies du sang. Elle traite du développement de modèles hybrides discrets continus et de leurs applications à la production de cellules sanguines (l'hématopoïèse) et de maladies sanguines telles que le lymphome et le myélome. La première partie de ce travail est consacrée à la formation de cellules sanguines à partir des cellules souches de la moelle osseuse. Nous allons principalement étudier la production des globules rouges, les érythrocytes. Chez les mammifères, l'érythropoïèse se produit dans des structures particulières, les îlots érythroblastiques. Leur fonctionnement est régi par de complexes régulations intra et extracellulaire mettant en jeux différents types de cellules, d'hormones et de facteurs de croissance. Les résultats ainsi obtenus sont comparés avec des données expérimentales biologiques ou médicales chez l'humain et la souris. Le propos de la deuxième partie de cette thèse est de modéliser deux maladies du sang, le lymphome lymphoblastique à cellules T (T-LBL) et le myélome multiple (MM), ainsi que leur traitement. Le T-LBL se développe dans le thymus et affecte la production des cellules du système immunitaire. Dans le MM, les cellules malignes envahissent la moelle osseuse et détruisent les îlots érythroblastiques empêchant l'érythropoïèse. Nous développons des modèles multi-échelles de ces maladies prenant en compte la régulation intracellulaire, le niveau cellulaire et la régulation extracellulaire. La réponse au traitement dépend des caractéristiques propres à chaque patient. Plusieurs scénarios de traitements efficaces, de rechutes et une résistance au traitement sont considérés. La dernière partie porte sur un modèle d'équation de réaction diffusion qui peut être utilisé pour décrire l'évolution darwinienne des cellules cancéreuses. L'existence de “pulse solutions”, pouvant décrire localement les populations de cellules et leurs évolutions, est prouvée / The thesis is devoted to mathematical modeling of hematopoiesis and blood diseases. It is based on the development of hybrid discrete continuous models and to their applications to investigate production of blood cell (hematopoiesis) and blood diseases such as lymphoma and myeloma. The first part of the thesis concerns production of blood cells in the bone marrow. We will mainly study production of red blood cells, erythropoiesis. In mammals erythropoiesis occurs in special structures, erythroblastic islands. Their functioning is determined by complex intracellular and extracellular regulations which include various cell types, hormones and growth factors. The results of modeling are compared with biological and medical data for humans and mice. The purpose of the second part of the thesis is to model some blood diseases, T cell Lymphoblastic lymphoma (T-LBL) and multiple myeloma (MM) and their treatment. TLBL develops in the thymus and it affects the immune system. In MM malignant cells invade the bone marrow and destroy erythroblastic islands preventing normal functioning of erythropoiesis. We developed multi-scale models of these diseases in order to take into account intracellular molecular regulation, cellular level and extracellular regulation. The response to treatment depends on the individual characteristics of the patients. Various scenarios are considered including successful treatment, relapse and development of the resistance to treatment. The last part of the thesis is devoted to a reaction-diffusion model which can be used to describe Darwinian evolution of cancer cells. Existence of pulse solutions, which can describe localized cell populations and their evolution, is proved

Understanding joint and causal relationships among multiple endogenous variables has been of much interest to researchers in the field of activity and travel behavior modeling. Structural equation models have been widely developed for modeling and analyzing the causal relationships among travel time, activity duration, car ownership, trip frequency and activity frequency. In the model, travel time and activity duration are treated as continuous variables, while car ownership, trip frequency and activity frequency as ordered discrete variables. However, many endogenous variables of interest in travel behavior are not continuous or ordered discrete but unordered discrete in nature, such as mode choice, destination choice, trip chaining pattern and time-of-day choice (it can be classified into a few categories such as AM peak, midday, PM peak and off-peak). A modeling methodology with involvement of unordered discrete variables is highly desired for better understanding the causal relationships among these variables. Under this background, the proposed dissertation study will be dedicated into seeking an appropriate modeling methodology which aids in identifying the causal relationships among activity and travel variables including unordered discrete variables. In this dissertation, the proposed modeling methodologies are applied for modeling the causal relationship between three pairs of endogenous variables: trip chaining pattern vs. mode choice, activity timing vs. duration and trip departure time vs.mode choice. The data used for modeling analysis is extracted from Swiss Travel Microcensus 2000. Such models provide us with rigorous criteria in selecting a reasonable application sequence of sub-models in the activity-based travel demand model system.

Considerando-se como fonte primária os manuais escolares brasileiros de Matemática, o saber a ser ensinado ainda situa uma apresentação dual, polarizado no viés pragmático ou teórico, ao que se segue um procedimento temático padrão que privilegia o desenvolvimento operatório envolvendo contextos exatos, finitos e determinísticos. Em particular, essas características se acentuam gravemente no momento de introdução dos números irracionais no ensino básico, o que ocasiona uma abordagem restritiva. Para superar este quadro, Bruner (1987) fundamenta que não devemos adiar o ensino de assuntos essenciais com base na crença de que são difíceis demais, pois as ideias fundamentais de qualquer assunto podem ser ensinadas na escolaridade básica, porém demanda um trabalho para além dos aspectos técnicos, o que equivale a retomada de características ligadas à compreensão. Neste trabalho, tivemos por hipótese que os pares discreto/contínuo; exato/aproximado; finito/infinito, presentes na análise da evolução epistemológica dos números reais e descritos em Machado (2009), se constituem em pilares conceituais essenciais para fundamentar um panorama favorável a uma abordagem significativa do tema dos números irracionais, de modo a compor um amálgama entre os aspectos técnicos e semânticos. Em face da necessária reflexão, em nível educacional, em torno de tal tema, delimitamos inicialmente um contexto investigativo pautado em um estudo qualitativo orientado pela questão Como são abordados os números irracionais no ensino básico, considerando-se como fonte o livro didático de Matemática?, a fim de mapear a apresentação deste assunto no Ensino Fundamental II e no Ensino Médio. O fundamento metodológico se inspirou nos núcleos de significação, descritos em Aguiar&Ozella (2006), que buscou apreender os sentidos que constituem o conteúdo do discurso expresso nos textos dos livros didáticos. O percurso dos núcleos de significação confirmou que, nos livros didáticos analisados, a apresentação dos números irracionais ocorre de modo polarizado: alguns optam por um viés empírico e outros pela definição formal. Verificou-se que, após uma abordagem inicial, não ocorre intercâmbio destas opções, o que acarreta um rápido esgotamento das ferramentas para se desenvolver as temáticas, limitando a compreensão da complexidade dos números irracionais no ensino básico. A partir das hipóteses e da pesquisa empírica, nos propusemos a delinear as contribuições presentes no movimento dialético entre os pares discreto/contínuo, finito/infinito e exato/aproximado, cujas mútuas conexões permeiam um espaço de significações, um campo que possibilita organizar, tecer e ampliar a rede de significados, conforme Machado (1995), favorecendo um quadro de maior compreensão à apresentação dos números irracionais. O enfoque epistemológico realizado revelou uma multiplicidade de relações envolvendo os números irracionais e diversos assuntos do currículo de Matemática, não devidamente caracterizadas e exploradas no ensino básico, o que serviu de mote para a apresentação de algumas situações de ensino para ilustrar os aportes orientadores sugeridos. Acreditamos que o caminho epistemológico trilhado viabilizou uma abertura para ampliar o quadro de significados em relação a outros tópicos presentes na Matemática Elementar, considerando-se como suporte a potencialidade presente nos eixos discreto/contínuo; exato/aproximado; finito/infinito, assim como no par determinístico/aleatório. / Considering Brazilian mathematics school textbooks as a primary research source, the knowledge to be taught still has a dual presentation, polarized in a pragmatic or theoretical way, what follows a thematic procedure pattern that favors an operational development involving exact, finite and deterministic contexts. In particular, these characteristics are seriously accentuated by the time of irrational numbers introduction at basic education, which leads to a restrictive approach. To overcome this situation, Bruner (1987) states that we should not postpone teaching key issues based on the belief that they are too hard, because the fundamental ideas of any subject can be taught at basic education, but it demands a work that overcome technical aspects, considerations that are equivalent to the resumption with aspects related to understanding. In this work, we had by hypothesis that the tension inherent on discrete/continuous, exact/approximate, finite/infinite pairs, extracted from analyses on real numbers epistemological evolution and described at Machado (2009), constitutes an essential conceptual pillar to establish a helpful framework to enable a significant irrational numbers approach, in order to compose an amalgam between technical and semantic aspects. Considering the necessary educational discussion involving this theme, we initially delimited an investigative context based on a qualitative study guided by the question How irrational numbers are approached in basic education, considering mathematics textbook as a source?\' in order to map this subject presentation at Middle and High School. The methodological foundation was inspired in meaning core, described in Aguiar and Ozella (2006), which aims to capture the sense that constitutes the speech content expressed inside mathematics scholar textbooks. The analysis from meaning core route reveals that, in the textbooks examined, the most known irrational numbers introduction occurs in a polarized way: some opt for a pragmatic bias and others by formal definition. However, it was found that after an initial approach, there is no further relationship between these options, which causes a rapid depletion of the tools to develop these themes, which limits the complexity understanding of irrational numbers in basic education. From the hypotheses and the empirical research, we intended to delineate contributions presented on the dialectical movement between discrete/continuous, finite/infinite and exact/approximate pairs, whose mutual connections permeate a \'space of meanings\', a field that allows to organize, to weave and to expand a network of meanings, as Machado (1995), favoring a framework for better understanding the irrational numbers development in basic school. The epistemological approach performed revealed a multiplicity of relationships involving irrational numbers and various subjects of mathematics curriculum, not properly characterized and exploited in basic education, references which served as contexts for the presentation of some teaching situations to illustrate the contributions guidance suggested. We believe that the epistemological path trodden enables an opening to increase possibilities of meanings in relation to other topics of Elementary Mathematics, considering as support the capability constituents presented in discrete/continuous, exact/approximate, finite/infinity axis, as well as in deterministic/random pair.