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This paper aims to analyze one’s understanding of limit of a function based on Concept Image and Concept Definition schemes, which can be used as powerful tools in research that have mathematical comprehension as part of their object of study. For that, we developed and applied a set of tasks involving limits of functions to students who had already been in a Calculus course. Vinner’s study about concept image and concept definition was the theoretical framework of this paper. The results obtained let us observe the plurality of interpretations about limits and other adjacent concepts, which allowed us to trace, in this paper, a reflection upon the cognitive conflicts concerning the (lack of) understanding of these concepts.

In this paper we introduce a q relativistic wave equation of the form i∂q· ψq + mψ0 = Eψ. We present the q− spinorial solutions using the method of separated variables in the q− relativistic wave equation. Some comments are mentioned at the end of the paper.

In this paper we propose an algebraic model for a modal epistemic logic. Although it is known the existence of algebraic models for modal logics, considering that there are so many different modal logics, so it is not usual to give an algebraic model for each such system. The basic epistemic logic used in the paper is bimodal and we can show that the epistemic algebra introduced in the paper is an adequate model for it.

Calculus of variations is a fundamental mathematical discipline focused on optimizing functionals, which map sets of functions to real numbers. This field is essential for numerous applications, including the formulation and solution of partial differential equations (PDEs) and the study of differential geometry. In PDEs, calculus of variations provides methods to find functions that minimize energy functionals, leading to solutions of various physical problems. In differential geometry, it helps understand the properties of curves and surfaces, such as geodesics, by minimizing arc-length functionals. This paper explores the intrinsic connections between these areas, highlighting key principles such as the Euler-Lagrange equation, Ekeland’s variational principle, and the Mountain Pass theorem, and their applications in solving PDEs and understanding geometrical structures.

The purpose of the present paper is to improve the global convergence results established so far concerning the Augmented Lagrangian Algorithm with exponential penalty function for solving nonlinear programming problems with equality and inequality constraints. We prove global convergence for KKT points under the PAKKT-regular constraint qualifications, which results as a consequence that accumulation points generated by the algorithm are PAKKT points. This convergence result is new for the augmented Lagrangian Method based on the exponential penalty function. An interesting consequence is that the estimates of the Lagrange multipliers computed by the method remain bounded in the presence of the quasi-normality condition. Finally we give optimality and feasibility results for the convex case.

This paper deals with a continuous-time predator-prey model of Leslie-Gower type considering the use of a physical refuge by a fraction of the prey population. The fraction of hidden prey is assumed to be dependent on the presence of predators in the environment. The conditions for the existence of equilibrium points and their local stability are established. In particular, it is shown that the point (0; 0) has a great importance in the dynamics of the model, since it determines a separating curve Σ that divides the behavior of the trajectories. Those trajectories that are above this curve have as their w- limit the point (0; 0), so the extinction of both populations may be possible depending on the initial conditions.

The logic PM4N was introduced by Jean-Yves Beziau as a modal and 4-valued system. In this introductory paper, the author presented the system from a matrix logic with four values disposed in a Boolean algebra with a modal operator for the notion of necessity. From that matrix semantics, the paper shows some valid results and stresses some motivations of the matrix semantics and of the system PM4N. In this paper we develop some additional aspects of that logic and present two deductive systems for it, a simple system of tableaux and a sequent calculus.

This paper characterizes the use of qualitative data analysis software in doctoral theses about Mathematics Education developed in Brazil, from 2010 to 2019, to support the choices of these software from paths already followed by other researchers. For this, a mapping was made to identify the theses and the aspects related to the use of these software in the field. The investigation retrieved 651 doctoral theses available in the Brazilian Digital Library of Thesis and Dissertations in which only 31 of them compose the corpus of this paper. This number evidences the little use of these software in the analyzed area. In the corpus, it was observed the predominance of the qualitative approach and the identification of nine different software, being three the most used: Atlas.ti, NVivo and CHIC, in that order. The adoption of software, as described by the authors, is linked to their potential regarding categorization, organization, visualization, data handling, time optimization and analysis facilitation and the possibility of enabling exploration of large volumes and diversity of data.

El presente artículo interpreta teóricamente cómo la competencia real capitalista internacional presiona a que en las sociedades periféricas los balances comerciales dependan de ventajas absolutas de costo basadas en una sobreexplotación laboral-natural, lo que crea dinámicas cíclicas en dichas sociedades. Para reforzar tal interpretación, el artículo toma como caso de estudio al capitalismo periférico ecuatoriano, sobre el que se presentan tendencias histórico-cíclicas de mediano y largo plazo, ejemplos de actividades económicas que al parecer sustentan sus superávits comerciales desde la sobreexplotación laboral-natural, y evidencia estadística de tendencias cíclicas de corto plazo a nivel sectorial, estimando un modelo PVAR sobre un panel de datos compuesto por 43 actividades económicas observadas en el período 2007-2016. Todos los resultados empíricos muestran la importancia de las ventajas absolutas de costo basadas en la sobreexplotación sobre los balances comerciales y el surgimiento de ciclos que refuerzan la condición dependiente del capitalismo ecuatoriano. || This paper theoretically interprets how international real capitalist competition pressures trade balances of the peripheral countries to depend on absolute cost advantages based on the super-exploitation of labor power and nature; dependence which creates cyclical dynamics within those countries. To reinforce such an interpretation, the paper takes as a study case the Ecuadorian peripheral capitalism, about which some historical-cyclical trends are presented as well as some examples of economic activities that seem to obtain their trade surplus through super-exploitation. Also, the paper estimates a PVAR model over a panel data of 43 economic activities observed during the period 2007-2016; from that estimation, it is found evidence of short-run cyclical dynamics among Ecuadorian economic activities. All the empirical results evidence the importance of the absolute cost advantages gained through super-exploitation of labor power and nature on the determination of trade balances and on the rise of cycles that reinforce the dependent condition of Ecuadorian capitalism.

From a purely geometric perspective, this paper provides demonstrations of two properties related to the definite integral. The first one discusses the theorem of the definite integral of a function and its inverse. The second addresses the calculation of the area of a region rotated around the line y = x. The intention is to illustrate that when feasible, it is advantageous to explore the geometric aspects of the elements involved in a property for the purpose of its demonstration. In both cases, we assume only the monotonicity and continuity of the function.

Cournot duopoly is a mathematical model of an imperfectly competitive market, where two profit maximizing firms simultaneously choose the quantities of product to produce, considering each other's behavior. The main objective of this paper is to analytically obtain the equilibrium quantities, prices, and profits of a Cournot duopoly in a market with two objectively differentiated products composed of two properties, and also considering that the consumers subjectively differentiate these properties. We begin by introducing subjective properties differentiation into a Lancaster consumer's choice problem, where the consumer maximizes a symmetrical CES utility function depending on the properties consumed, subject to a budget constraint and to the Lancaster linear transformation of these properties in the final goods. Then we solve this optimization problem through the method of Lagrange multipliers and obtain the final products direct and inverse demand functions, and finally find the equilibrium of the proposed Cournot duopoly. The results show that if the properties are perceived by the consumers as highly differentiated, then a higher subjective properties differentiation implies that the Cournot duopoly has a well-defined equilibrium only for a smaller range of objective product differentiation. Furthermore, an increase in the consumers' subjective differentiation of properties and/or in the objective product differentiation increases the monopoly power of the firms, i.e. with a higher differentiation, the firms produce smaller quantities of products, charge higher prices and earn higher profits. The model proposed here may serve as a basis for the study of imperfectly competitive markets with product differentiation and advertising.

This paper aims to identify how historical information can contribute to teacher education regarding to the Golden Ratio content. Therefore, in the theoretical framework, the relevance of the teacher's investigative stance and the contributions that the use of the History of Mathematics provides for the training of this professional are discussed, as well as about some aspects of the Golden Ratio. For this purpose, an implementation was carried out that involved the use of the History of Mathematics, in order to work on the Golden Ratio content with 4th year students of the Mathematics Degree course at a public educational institution. In this paper, the focus was directed to the study of myths and truths concerning the Golden Ratio during the planning of the implementation, as well as to the discussions of the students later. The research is qualitative in nature, based on participant observation for data collection and mixed categorization, for analysis of what had been constituted. We concluded that the use of historical information about the Golden Ratio content led to critical manifestations, so that the research teachers and students in initial training had the opportunity to deconstruct certain beliefs impregnated and disseminated in history.

In this paper we address CLOSEST STRING problem that arises in web searching, coding theory and computational molecular biology. To solve it is to find a string that minimizes the maximum Hamming distance from a given set of strings. CLOSEST STRING is an NP-hard problem. This paper proposes two linear-time algorithms, one for the general case, a kernelization algorithm, and the other for three-strings, a linear-time algorithm called Minimization First Algorithm (MFA). A formal proof of the correctness and the computational complexity of the proposed algorithms are given.

In the present paper we study episodes of the life and work of the German-Danish mathematician Werner Fenchel from the perspective of the significance of the Second World War. On the one hand, we will see how the mathematical society, in particularly the mathematician Harald Bohr, helped Fenchel to set up an academic life in Copenhagen, Denmark. On the other hand, we will see how the organisation of scientists’ contribution to the war effort in the USA during the war, and the post war military financing of academic research in the post war period interfered with some of Fenchel’s mathematical work, especially his contribution to duality theory in nonlinear programming. As such, the study presented is this paper contributes to our understanding of how mathematics and conditions of a particular place and time interfere in the history of mathematics

The generalized Kronecker graph product was introduced by Figueroa-Centeno et al. (2008). Later, Baca et al. (2018) used it for obtaining solutions of the n-queens problem on larger boards from solutions on smaller boards. In this paper, we generalize the graph product and the recent results by Baca et al. (2018), obtaining a larger class of solutions by knowing solutions on lower size boards in advance. We finalize the paper stating a couple of conjectures regarding conditions for obtaining composite solutions via graph product.

This paper aims to identify and analyze the theme of Financial Education (EF) in academic research in the latest editions of ENEM, CIBEM and SIPEM. The methodological approach is qualitative of the bibliographical. We believe that the SBEM electronic database enabled a bibliographic survey of academic production. The results pointed to the need to promote discussions about EF in the school environment, recognizing its contribution to the integral and critical formation of the subject, its articulation with social, political, economic, environmental and cultural problems, making education a progressively active social force. They also highlighted the urgent need to implement continuing education programs that enable the planning and implementation of more assertive teaching practices in this area.

Since the publication of the Base Nacional Comum Curricular (National Common Curricular Base), Financial Education has become a topic to be developed in the classroom, mainly related to the mathematics curriculum. In this sense, teachers are in charge of addressing it. In view of this, in this paper we aim to discuss understandings of Financial Education aimed at Basic Education and produced by mathematics undergraduates, mobilized in two training meetings. Based on Critical Mathematics Education and the qualitative research paradigm, we identified five topics using content analysis procedures. The results of the analysis of these topics indicate that the future teachers have knowledge of School Financial Education, although they relate the subject to economic aspects and Financial Mathematics.

Let X be a continuum, and let C(X) denote the hyperspace of all subcontinua of X. It is known that there exist monotone maps μ from C(X) into [0, ∞) such that μ({x}) = 0 for each x ∈ X, and if A is a proper subcontinuum of B, then μ(A) < μ(B). The subcontinua μ−1(t) of C(X) are called Whitney levels of C(X). In this paper, a class of closed subsets of X is employed to characterize the Whitney levels of C(X) possessing one of the following properties: irreducibility, decomposability, being a Wilder continuum, aposyndesis, semiaposyndesis, n-aposyndesis, finite aposyndesis, and connectedness colocal.

For over twenty years, Frege tried to find the foundations of arithmetic through logic, and by doing this, he attempted to establish the truth and certainty of the knowledge. However, when he believed his work wasdone, Bertrand Russell sent him a letter pointing out a paradox, known as Russell‟s paradox. It is often considered that Russell identified the paradox in Frege‟s theories. However, as shown in this paper, Russell, Frege and also George Cantor contributedsignificantly to the identification of the paradox. In 1902, Russell encouraged Frege to reconsider a portion of his work based in a paradox built from Cantor‟s theories. Previously, in 1885, Cantor had warned Frege about taking extensions of concepts in the construction of his system. With these considerations, Frege managed to identify the precise law and definitions that allowed the generation of the paradox in his system. The objective of this paper is to present a historical reconstruction of the paradox in Frege‟s publications and discuss it considering the correspondences exchanged between him and Russell. We shall take special attention to the role played by each of these mathematicians in the identification of the paradox and its developments. We also will show how Frege anticipated the solutions and new theories that would arise when dealing with logico-mathematical paradoxes, including but not limited to Russell‟s paradox.

In this paper we discuss the professional activity of Leopoldo Nachbin (1922-1993) in the period in which he worked at National Faculty of Philosophy of the University of Brazil (FNFi/UB, in Portuguese) until his retirement by Institute of Mathematics of the Federal University of Rio de Janeiro (IM/UFRJ, in Portuguese). We identify the influence played by some mathematicians such as G. Mammana (1893-1980), L. Schwartz (1915-2002), A. Monteiro (1907-1980), A. Weil (1906-1998), J. Dieudonné (1906-1992), M. Stone (1903-1989), and A. A. Albert (1905-1972) in the professional training of L. Nachbin. We also address the serious difficulties he encountered in the Mathematics Department of FNFi when he was entered in the competition, opened in 1950, for the post of full Professor (Professor Catedrático) of Mathematics and Superior Analysis of FNFi. We emphasize in the paper, that the period from 1947 to the 1960s marks the most creative and productive phase in terms of scientific research in the academic life of L. Nachbin.

The purpose of the present paper is to show how some current researches – published in papers and books – approach the difficulties in Mathematics learning. We intend to reflect on teaching and learning Mathematics by analysing some aspects that have been pointed out, especially about learning difficulties some students have in Mathematics area. We used the research qualitative approach with procedures of document analysis. At first, we described the meaning of the word ‘difficulty’, passing through learning difficulties and learning difficulties in Mathematics. The latest researches point the urgent need of a didactic overhaul in Mathematics teaching and maturation of all school communitiy, regardless the teaching level, concerning learning difficulties.

In this paper, we explore the interconnection and existing relationships between the Sovereign Credit Default Swaps (henceforth, CDS) and the stock markets of the main European countries. Thus, the goal of this paper is to test if the CDS premia can predict the stock market returns of the most relevant economies within the Eurozone, so that, they serve as advanced indicators like mechanisms of price transmission. For this purpose, we apply the Granger Causality test to analyze ten main European stock markets from 2004 to 2016 by using daily data. Our hypothesis is proved to work for the largest economies with liquid CDS markets, whereas the transmission mechanism between CDS and stock prices is not so evident for the smallest ones. || En este documento, exploramos la interconexión y las relaciones existentes entre los Soberanos Credit Default Swaps (en adelante, CDS) y los mercados bursátiles de los principales países europeos. Por lo tanto, el objetivo de este documento es comprobar si las primas de CDS pueden predecir los rendimientos del mercado de valores de las economías más relevantes dentro de la zona euro, de modo que sirvan como indicadores avanzados como los mecanismos de transmisión de precios. Para este propósito, aplicamos la prueba de Causalidad de Granger para analizar los diez principales mercados bursátiles europeos desde 2004 a 2016 mediante el uso de datos diarios. Se ha demostrado que nuestra hipótesis funciona para las economías más grandes con mercados de CDS líquidos, mientras que el mecanismo de transmisión entre los CDS y los precios de las acciones no es tan evidente para los más pequeños.

Based upon the self-regulated learning theoretical framework this study examined to what extent students' Math school achievement (fifth to ninth graders from compulsory education) can be explained by different cognitive-motivational, social, educational, and contextual variables. A sample of 571 students (10 to 15 year old) enrolled in the study. Findings suggest that Math achievement can be predicted by self-efficacy in Math, school success and self-regulated learning and that these same variables can be explained by other motivational (ej., achievement goals) and contextual variables (school disruption) stressing this way the main importance of self-regulated learning processes and the role context can play in the promotion of school success. The educational implications of the results to the school levels taken are also discussed in the present paper.

Physical activity provides numerous benefits to physical and mental health, even affecting learning. Despite this, one of the main modern problems is represented by sedentariness. Children and adolescents spend most of their time at school. Therefore, a valid teaching methodology to benefit from the positive effects of physical activity and reduce sedentary moments could be interdisciplinary learning, which integrates physical education into various curricular subjects. Specifically, this paper proposes a systematic literature review of some protocols (n = 11) to investigate whether an elementary school’s integrated physical education and mathematics curriculum can improve learning in the logic-cognitive domain. From all the studies reviewed, the strong importance of interdisciplinary teaching, linking mathematics to physical education, was confirmed to reduce sedentary moments and enable a better attitude toward this discipline, breaking down the many stereotypes that often paint it as a complex and abstract subject.

In this paper we study codimension 1 Hopf bifurcation for a two dimensional autonomous nonlinear ordinary differential equations system, modeling a predator-prey interaction with Holling type II functional response and additive Allee effect in the prey equation. Positivity, dissipation, boundedness and permanence of the solutions are analyzed. Furthermore, stability and bifurcation analysis are carried out to show the existence of periodic orbits due to the occurrence of codimension 1 Hopf bifurcation, involving weak Allee effect as well as strong Allee effect. In the case of strong Allee effect, through computer simulations carried in MAPLE 13, we conjecture that this model may admit a heteroclinic bifurcation. We present some simulations which allow one to verify the analytical results.

In this paper we introduce the derivatives for non-monogenic functions. We establish the derivative for non-monogenic functions on the Dirac operator. We also propose a new type of difference operator for non-monogenic function and new type of derivative.

In this paper, we investigate the classical identities of the repunit sequence with integer indices in light of the properties of Horadan-type sequences. We highlight particularly the Tagiuri-Vajda Identity and Gelin-Cesàro Identity. Additionally, we prove that no repunit is a perfect power, either even or odd. Finally, we address a divisibility criterion for the terms of repunit rn by a prime p and its powers.

[Objective] The aim of this paper is to verify the factorial structure of the School Climate Scale in University Environments (ECEAU) through structural equation models (SEM). [Methodology] A non-experimental cross-sectional study was carried out with a random sample of 381 students of the Universidad Nacional de Costa Rica (UNA), to whom the ECEAU was applied. A SEM model was proposed to confirm the factor structure of the scale and an exploratory factor analysis (EFA) was used to provide the basis for an alternative SEM model with a different structure. The reliability of scales was determined using McDonald’s omega, and the goodness of fit indexes used for the SEM models were the RMSEA, the CFI, the TLI and the GFI. The analysis was carried out using R software. [Results] The reliability of the scales in both models was adequate (w >0.70), although the metrics in the proposed model produced better results. The AFE suggested a different structure for the ECEAU, keeping some dimensions of the original structure and modifying and eliminating others. The goodness of fit indexes in both models were almost identical, with RMSEA=0.02, CFI=0.99, TLI=0.99 and GFI=0.98, which provides supporting evidence for the factorial structures proposed. [Conclusions] The original structure of the ECEAU was confirmed, while the differences in the results of the estimation and rotation methods generated another equally valid factorial structure for measuring the school climate construct in this university, and could be applied to other similar contexts.

Comparative analysis of the efficacy of a playful-narrative program to teach mathematics at pre-school level. In this paper, the effectiveness of a programme comprising several components that are meant to consolidate mathematical concepts and abilities at the pre-school level is analyzed. The instructional methodology of this programme is compared to other methodologies. One-hundred 5-6 year-old children made up the sample that was distributed in the following conditions: (1) traditional methodology; (2) methodology with perceptual and manipulative components, and (3) methodology with language and playful components. Mathematical competence was assessed with the Mathematical Criterial Pre-school Test and the subtest of quantitative-numeric concepts of BADyG. Participants were evaluated before and after the academic course during which they followed one of these methodologies. The results show that the programme with language and playful components is more effective than the traditional methodology (p<.000) and also more effective than the perceptual and manipulative methodology (p<.000). Implications of the results for instructional practices are analyzed.

In this paper, we used the data of Mathematics achievements from the sample of 13- year-old Spanish boys and girl who took part in the Second National Assessment of Educational Progress, carried out by the Educational Testing Service. With the aim of identifying variables related to high achievement, our goals are: 1) to analyse whether there are significant differences in 13-year-old Spanish boys and girls in their attitudes towards mathematics, their study strategies, and their performance; 2) to analyse the influence of their attitudes and study strategies on their performance, and to examine whether these relations are the same in boys and girls; 3) to propose a theoretical model that explains the relationship among the variables defined by means of structural equation modelling. The results show that: a) there are significant differences between boys and girls in the variables studied; b) it is not possible to reject the proposed model that relates study strategies and attitudes towards performance in mathematics; and c) the model tends to be stable both among boys and girls.

Following the recent work of Gómez-Déniz and Pérez-Rodríguez (2014), this paper extends the results obtained there to the normal-exponential distribution with dependence. Accordingly, the main aim of the present paper is to enhance stochastic production frontier and stochastic cost frontier modelling by proposing a bivariate distribution for dependent errors which allows us to nest the classical models. Closed-form expressions for the error term and technical efficiency are provided. An illustration using real data from the econometric literature is provided to show the applicability of the model proposed. || Continuando el reciente trabajo de Gómez-Déniz y Pérez-Rodríguez (2014), el presente artículo extiende los resultados obtenidos a la distribución normal-exponencial con dependencia. En consecuencia, el principal propósito de este artículo es mejorar el modelado de la frontera estocástica tanto de producción como de coste proponiendo para ello una distribución bivariante para errores dependientes que nos permitan encajar los modelos clásicos. Se obtienen las expresiones en forma cerrada para el término de error y la eficiencia técnica. Se ilustra la aplicabilidad del modelo propouesto usando datos reales existentes en la literatura econométrica.

La normativa contable internacional ofrece con la Norma Internacional de Contabilidad 40 (NIC 40) "Inversiones inmobiliarias" un caso referente para investigar la decisión que toman las empresas cuando se les ofrece el valor razonable o el coste histórico como criterios alternativos de valoración. En este trabajo aprovechamos la oportunidad que ofrece esta norma para aportar evidencia adicional en un contexto multinacional y multisectorial sobre cuáles son los motivos que explican la elección contable. Además, en este trabajo introducimos y comparamos el uso de las redes neuronales artificiales y los árboles de decisión, con el objetivo de evaluar la capacidad predictiva de estas metodologías, frente a la tradicionalmente utilizada regresión logística para la resolución de problemas de clasificación en este área. Los resultados de la clasificación indican que tanto las redes neuronales como los árboles de decisión pueden ser una alternativa interesante a los métodos clásicos estadísticos como la regresión logística. En particular, las dos metodologías mostraron una mayor capacidad predictiva frente a la regresión logística aunque no se encontraron diferencias significativas entre ambas. || International Accounting Standard 40 (IAS 40 - Investment properties) offers an ideal setting for research on accounting choice as it represents a paradigmatic case choosing between the fair value and the historical cost as the measurement criteria. In this paper, we take the opportunity of this standard to provide additional evidence in a multinational and multi-context on the determinants that explain the accounting choice. Furthermore, in this paper, we introduce and compare the use of artificial neural networks and decision trees in order to assess the predictive capability of these methodologies, compared to other techniques commonly used to solve classification problems in this area such as the logistic regression. The classification results indicate that both neural networks and decision trees can be an interesting alternative to classical statistical methods such as the logistic regression. In particular, both methods outperformed the logistic regression in terms of predictive ability, although no significant differences were found between both.

El objetivo de este trabajo es contrastar el efecto de contagio en el fracaso empresarial como consecuencia de la proximidad geográfica entre empresas del mismo sector. Para ello, desarrollamos una aplicación empírica basada en el análisis de los estadísticos espaciales join-count sobre una muestra de empresas del sector servicios en el municipio de Murcia (España). Los resultados muestran un patrón de autocorrelación espacial significativo, de tal forma que la probabilidad de fracaso de una empresa no solo depende de sus características internas sino también de su proximidad a otras empresas fracasadas. Estos resultados pueden ser un interesante punto de partida para el desarrollo de análisis que incluyan los efectos de la proximidad geográfica entre empresas en la literatura de fracaso empresarial. || This paper aims to contrast the spillover effects in business failure derived from the geographic proximity among firms of the same sector. To get this purpose, we develop an empirical application based on the analysis of the join-count statistics on a sample of firms of service sector located in the municipality of Murcia (Spain). Our results show significant spatial autocorrelation pattern, therefore, the probability of failure in a firm not only depends on its internal characteristics, but also on the situation of failure of its vicinity peer companies. These results can be an interesting starting point in the development of papers which consider the interdependence effects among closer peer companies in business failure literature.

This paper presents the results of the management of a didactic device formulated by a Mathematics male and Biology female teacher from a secondary school in which they sought a codisciplinary study. Based on the Anthropological Theory of the Didactic, the results of an exploratory study that sought to characterize the management of a didactic device for the study of plant growth in secondary school are reported; In particular, we analyze the relevance of mathematical knowledge in its development. The main results indicate that the study enabled the protagonists to be the students: they formulated and answered questions, achieving autonomy and responsibility for the search for information in various media; however, mathematical knowledge did not occupy a prominent place during the study.

The paper discusses the problem of infinity with a special reference to mathematical truth. The roots of this problem, in a sense, go back to early efforts to provide a rational account of the phenomena of change, and the phenomenon of motion in particular, and, in connection with this, to the attempt to cast light on the complex relationship between continuity and discreteness. If mathematics is to serve as a tool of a rational description of the world, is it possible to construct a mathematical model of phenomenal continuity? This question was raised already by Zeno when the Pythagorean hypothesis of the essential ‘granularity’ of the world began to crumble as a result of the discovery of incommensurable magnitudes. The discovery of incommensurable magnitudes, and along with it the discovery of irrational numbers, produced well-known paradoxes and gave rise to the question to what extent, if at all, mathematics can deliver a ‘metaphysical’ description of the world. In more recent times, the idea of infinity was built into the infinitesimal calculus which became the foundation of the mathematical description of nature, but the metaphysical questions have not gone away. The issues surrounding actual and potential infinity continue to be the subject of intense debates. Whereas some recognise only potential infinities, others, like Georg Cantor, accept actually existent infinities. This is directly connected with the question of mathematical truth, which, according to the Platonist tradition exists as objectively given. The paper argues that the Cantorian transfinite set theory cannot withstand criticism, and that rather than providing a (metaphysical) description of reality mathematics merely offers possible models in terms of which the world might be better or more conveniently interpreted. The so-called ‘mathematical truth’ reduces to the question of ‘correctness’, viz. whether a given mathematical proposition does or does not follow from given premises.

In this paper we first explore how Wittgenstein’s philosophy provides a conceptual tools to discuss the possibility of the simultaneous existence of culturally different mathematical practices. We will argue that Wittgenstein’s later work will be a fruitful framework to serve as a philosophical background to investigate ethnomathematics (Wittgenstein 1973). We will give an overview of Wittgenstein’s later work which is referred to by many researchers in the field of ethnomathematics. The central philosophical investigation concerns Wittgenstein’s shift to abandoning the essentialist concept of language and therefore denying the existence of a universal language. Languages—or ‘language games’ as Wittgenstein calls them—are immersed in a form of life, in a cultural or social formation and are embedded in the totality of communal activities. This gives rise to the idea of rationality as an invention or as a construct that emerges in specific local contexts. In the second part of the paper we introduce, analyse and compare the mathematical aspects of two activities known as string figure-making and sand drawing, to illustrate Wittgenstein’s ideas. Based on an ethnomathematical comparative analysis, we will argue that there is evidence of invariant and distinguishing features of a mathematical rationality, as expressed in both string figure-making and sand drawing practices, from one society to another. Finally, we suggest that a philosophical-anthropological approach to mathematical practices may allow us to better understand the interrelations between mathematics and cultures. Philosophical investigations may help the reflection on the possibility of culturally determined ethnomathematics, while an anthropological approach, using ethnographical methods, may afford new materials for the analysis of ethnomathematics and its links to the cultural context. This combined approach will help us to better characterize mathematical practices in both sociological and epistemological terms.

This paper describes an emerging approach to the design of task sequences and the theory that undergirds it. The approach aims at promoting particular mathematical concepts, understood as the result of reflective abstraction. Central to this approach is the identification of available student activities from which students can abstract the intended ideas. The approach differs from approaches in which learning to solve the problem posed is the intended learning. The paper illustrates the approach through data from a teaching experiment on division of fractions.

El artículo tiene por objetivo determinar las variables que pueden condicionar la decisión observable de un individuo con respecto a escuchar música grabada en Colombia. Para ello, se estiman modelos probit usando microdatos de la Encuesta de Consumo Cultural 2008. Los resultados muestran que individuos con educación superior, que toman cursos y talleres culturales y se sitúan en los niveles medios y altos de estrato socio-económico presentan la mayor probabilidad de escuchar música grabada. Igualmente, se encuentra que individuos que se auto-reconocen como afro-descendientes se asocian con una menor probabilidad de escuchar música grabada. El artículo contribuye a la escasa literatura empírica sobre los determinantes de la participación cultural en países de América Latina. || This paper purpose is to determine the variables that could influence the individual observable decision of listening to recorded music in Colombia. To do so, probit models are estimated using micro-data from the Encuesta de Consumo Cultural 2008 [Cultural Consumption Survey 2008 ]. Empirical results show that individuals with higher education that take cultural courses and workshops, and belong to medium and high socio-economic strata, are those with the highest probability of listening to recorded music. Additionally, self-identied individuals as African-Colombian-descendent present the lowest likelihood. This paper contributes to the scarce empirical literature on the determinants of cultural participation in Latin America.

This paper gives a survey over the existence of uniform L∞ a priori bounds for positive solutions of subcritical elliptic equations (P)p -\Delta_p u =f(u), in \Omega, u = 0, on \partial\Omega widening the known ranges of subcritical nonlinearities for which positive solutions are a-priori bounded. Our arguments rely on the moving planes method, a Pohozaev identity, W1,q regularity for q > N, and Morrey’s Theorem. In this part I, when p = 2, we show that there exists a-priori bounds for classical, positive solutions of (P)2 with f(u) = u2∗−1/[ln(e + u)]α, with 2∗ = 2N/(N − 2), and α > 2/(N − 2). Appealing to the Kelvin transform, we cover non-convex domains. In a forthcoming paper containing part II, we extend our results for Hamiltonian elliptic systems (see [22]), and for the p-Laplacian (see [10]). We also study the asymptotic behavior of radially symmetric solutions uα = uα(r) of (P)2 as α → 0 (see [24]).