Dpto. Estadística, Informática y Matemáticas - Estatistika, Informatika eta Matematika Saila [desde mayo 2018 / 2018ko maiatzetik]
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Publication Open Access 25 years of continuous sewage sludge application vs. mineral fertilizers on a calcareous soil affected pH but not soil carbonates(Frontiers Media, 2022) De Soto García, Isabel Sonsoles; Zamanian, Kazem; Urmeneta Martín-Calero, Henar; Enrique Martín, Alberto; Virto Quecedo, Íñigo; Zientziak; Estatistika, Informatika eta Matematika; Institute on Innovation and Sustainable Development in Food Chain - ISFOOD; Ciencias; Estadística, Informática y Matemáticas; Universidad Pública de Navarra / Nafarroako Unibertsitate PublikoaInorganic and organic fertilizers have been widely used to maintain crop yields. However, several studies have demonstrated that the dissolution of carbonates in agricultural soils by the acidification induced by N-fertilizers can result in their total or partial loss in the tilled layer of some agricultural soils. The effect of inorganic and organic fertilization on carbonates in calcareous semiarid Mediterranean soils has been less studied and is still unclear. Based on a 25-year field experiment, we investigated the effects of different fertilization strategies on the soil pH, inorganic C content, and in the proportion of pedogenic carbonates in the topsoil (0-30 cm) of a carbonate-rich soil in Navarre (N Spain). Five treatments were compared: no amendments as a control (SC), mineral fertilization (MF), and the application of sewage sludge at different doses (80 Mg ha-1 every year (80-1), 40 Mg ha-1 every year (40-1) and 40 Mg ha-1 every four years (40-4). Results showed a decrease in soil pH values by increasing the amount of sewage sludge and a small alteration in the calcite structure particularly in 40-4. However, no significant differences between treatments were found in the total content of carbonates nor in the proportion of pedogenic carbonates. The high concentration of total carbonates in the soil (16.09 ± 0.48%), and of the proportion of pedogenic carbonates (40.21 ± 1.29%, assuming a δ13C of primary carbonates = 0 ‰) seemed elevated enough to compensate for the observed acidification in the studied soil. In the case of MF, no changes were observed in the pH values, nor in the carbonate content (total and pedogenic). This suggests that the use of sewage sludge could induce changes in the future at a faster rate and of greater dimension than the use of mineral fertilizers such as the ones used in this field (46% urea, superphosphate and ClK).Publication Open Access Abstract homogeneous functions and consistently influenced/disturbed multi-expert decision making(IEEE, 2021) Santiago, Regivan; Callejas Bedregal, Benjamin; Pereira Dimuro, Graçaliz; Fernández Fernández, Francisco Javier; Bustince Sola, Humberto; Fardoun, Habib; Estadística, Informática y Matemáticas; Estatistika, Informatika eta MatematikaIn this paper we propose a new generalization for the notion of homogeneous functions. We show some properties and how it appears in some scenarios. Finally we show how this generalization can be used in order to provide a new paradigm for decision making theory called consistent influenced/disturbed decision making. In order to illustrate the applicability of this new paradigm, we provide a toy example.Publication Open Access Achieving transfer from mathematics learning(MDPI, 2023) Orón, José Víctor; Lizasoain Iriso, María Inmaculada; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2The question of transfer is a special challenge in mathematics teaching because the wide range and fragmentation of the curricula have in many cases fostered an instrumental understanding, which makes transfer difficult for the students. Although promoting a relational learning has been a huge step forward in achieving transfer, understanding usually remains at the technical level of learning. Fostering critical thinking and metacognition raises learning to the psychological level, as students are encouraged to analyse their own thinking. Despite this, our hypothesis is that transfer will only be achieved when students are helped to reach a personal dimension, being encouraged to discover their own way of approaching the global reality of their lives beyond the subject. Learning, for instance, the greatest common divisor should be an opportunity to discover that, as numbers can be presented by their prime factors, people can be recognised by their features and interests. As such, looking for the greatest common divisor should not differ from discovering common interests with friends. Integrating specific and general learning will make transfer no longer unattainable. Personalising learning means discovering how one specific learning impacts on the personal way of understanding reality (oneself, others, and world), thus making transfer natural.Publication Open Access Acoustic levitation in mid-air: recent advances, challenges, and future perspectives(American Institute of Physics, 2020) Brizzotti Andrade, Marco Aurélio; Marzo Pérez, Asier; Adamowski, Julio C.; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Gobierno de Navarra / Nafarroako Gobernua, 0011-1365-2019-000086Mid-air acoustic levitation is becoming a powerful tool to suspend and manipulate millimetric objects. Because of its unique characteristics, acoustic levitation is suitable to trap a wide variety of materials such as liquids, solids, soap bubbles, and even living creatures. Acoustic levitation can also be combined with noncontact measurement systems, allowing contactless analysis and characterization of levitating samples. In this article, we review some of the advances that have been made over the last decade. We also present the technical challenges that must be overcome in order to extend the capability of current acoustic levitation devices and, finally, we point out future directions for acoustic levitation.Publication Open Access Acoustic lock: position and orientation trapping of non-spherical sub-wavelength particles in mid-air using a single-axis acoustic levitator(American Institute of Physics, 2018) Cox, L; Croxford, A; Drinkwater, Bruce W.; Marzo Pérez, Asier; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISCWe demonstrate acoustic trapping in both position and orientation of a non-spherical particle of sub-wavelength size in mid-air. To do so, we multiplex in time a pseudo-one-dimensional vertical standing wave and a twin-trap; the vertical standing wave provides converging forces that trap in position, whereas the twin-trap applies a stabilising torque that locks the orientation. The device operates at 40 kHz, and the employed multiplexing ratio of the 2 acoustic fields is 100:50 (standing:twin) periods. This ratio can be changed to provide tunability of the relative trapping strength and converging torque. The torsional spring stiffness of the trap is measured through simulations and experiments with good agreement. Cubes from k/5.56 (1.5 mm) to k/2.5 (3.4 mm) side length were stably locked. We also apply this technique to lock different non-spherical particles in midair: cubes, pyramids, cylinders, and insects such as flies and crickets. This technique adds significant functionality to mid-air acoustic levitation and will enable applications in micro-scale manufacturing as well as containment of specimens for examination and 3D-scanning.Publication Open Access Acoustic virtual vortices with tunable orbital angular momentum for trapping of Mie particles(American Physical Society, 2018) Marzo Pérez, Asier; Caleap, Mihai; Drinkwater, Bruce W.; Estadística, Informática y Matemáticas; Estatistika, Informatika eta MatematikaAcoustic vortices can transfer angular momentum and trap particles. Here, we show that particles trapped in airborne acoustic vortices orbit at high speeds, leading to dynamic instability and ejection. We demonstrate stable trapping inside acoustic vortices by generating sequences of short-pulsed vortices of equal helicity but opposite chirality. This produces a “virtual vortex” with an orbital angular momentum that can be tuned independently of the trapping force. We use this method to adjust the rotational speed of particles inside a vortex beam and, for the first time, create three-dimensional acoustics traps for particles of wavelength order (i.e., Mie particles).Publication Open Access Acoustophoretic volumetric displays using a fast-moving levitated particle(AIP Publishing, 2019) Fushimi, Tatsuki; Marzo Pérez, Asier; Drinkwater, Bruce W.; Hill, Thomas L.; Estadística, Informática y Matemáticas; Estatistika, Informatika eta MatematikaDisplays have revolutionized the way we work and learn, and thus, the development of display technologies is of paramount importance. The possibility of a free-space display in which 3D graphics can be viewed from 360° without obstructions is an active area of research - holograms or lightfield displays can realize such a display, but they suffer from clipping and a limited field of view. Here, we use a phased array of ultrasonic emitters to realize a volumetric acoustophoretic display in which a millimetric particle is held in midair using acoustic radiation forces and moved rapidly along a 3D path. Synchronously, a light source illuminates the particle with the target color at each 3D position. We show that it is possible to render simple figures in real time (10 frames per second) as well as raster images at a lower frame rate. Additionally, we explore the dynamics of a fast-moving particle inside a phased-array levitator and identify potential sources of degradation in image quality. The dynamics are nonlinear and lead to distortion in the displayed images, and this distortion increases with drawing speed. The created acoustophoretic display shows promise as a future form of display technology.Publication Open Access Acuity-based rotational patient-to-physician assignment in an emergency department using electronic health records in triage(SAGE, 2023) Cildoz Esquíroz, Marta; Ibarra Bolt, Amaya; Mallor Giménez, Fermín; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISCEmergency department (ED) operational metrics generated by a new acuity-based rotational patient-to-physician assignment (ARPA) algorithm are compared with those obtained with a simple rotational patient assignment (SRPA) system aimed only at an equitable patient distribution. The new ARPA method theoretically guarantees that no two physicians’ assigned patient loads can differ by more than one, either partially (by acuity levels) or in total; whereas SRPA guarantees only the latter. The performance of the ARPA method was assessed in practice in the ED of the main public hospital (Hospital Compound of Navarra) in the region of Navarre in Spain. This ED attends over 140 000 patients every year. Data analysis was conducted on 9,063 ED patients in the SRPA cohort, and 8,892 ED patients in the ARPA cohort. The metrics of interest are related both to patient access to healthcare and physician workload distribution: patient length of stay; arrival-to-provider time; ratio of patients exceeding the APT target threshold; and range of assigned patients across physicians by priority levels. The transition from SRPA to ARPA is associated with improvements in all ED operational metrics. This research demonstrates that ARPA is a simple and useful strategy for redesigning front-end ED processes.Publication Open Access Addendum to "uniqueness of unconditional basis of infinite direct sums of quasi-Banach spaces"(Springer, 2022) Albiac Alesanco, Fernando José; Ansorena, José L.; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Universidad Pública de Navarra / Nafarroako Unibertsitate PublikoaAfter [Uniqueness of unconditional basis of infinite direct sums of quasi-Banach spaces, Positivity 26 (2022), Paper no. 35] was published, we realized that Theorem 4.2 therein, when combined with work of Casazza and Kalton (Israel J. Math. 103:141-175, 1998) , solves the long-standing problem whether there exists a quasi-Banach space with a unique unconditional basis whose Banach envelope does not have a unique unconditional basis. Here we give examples to prove that the answer is positive. We also use auxiliary results in the aforementioned paper to give a negative answer to the question of Bourgain et al. (Mem Am Math Soc 54:iv+111, 1985)*Problem 1.11 whether the infinite direct sum l(1)(X) of a Banach space X has a unique unconditional basis whenever X does.Publication Open Access Addressing the issue of trust in elementary teachers' Maths-specific education: ANFoMAM project(Charles University (Chequia), 2019) Celi, Valentina; Cogolludo, José Ignacio; Gil Clemente, Elena; Lizasoain Iriso, María Inmaculada; Millán Gasca, Ana; Moler Cuiral, José Antonio; Regoliosi, Luigi; Estadística, Informática y Matemáticas; Estatistika, Informatika eta MatematikaTo improve primary school teachers' maths-specific education at university, our project will develop a series of workshops, as ready-to-use instruments, which closely consider children's way of learning and their relationship with mathematics. Thus, the interest of participants in children is exploited in sessions which take into account both their professional work as teachers and their own childhood experiences. The aim is to help participants to evolve in the key aspect of trust. The paper describes the objectives and first results of the ANFoMAM project, supported by the Erasmus Plus Programme in the area of strategic partnerships for innovation in higher education in Europe.Publication Open Access Admissible orders on fuzzy numbers(IEEE, 2022) Zumelzu, Nicolás; Callejas Bedregal, Benjamin; Mansilla, Edmundo; Bustince Sola, Humberto; Díaz, Roberto; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC; Estadística, Informática y MatemáticasFrom the more than two hundred partial orders for fuzzy numbers proposed in the literature, only a few are total. In this paper, we introduce the notion of admissible order for fuzzy numbers equipped with a partial order, i.e. a total order which refines the partial order. In particular, it is given special attention to the partial order proposed by Klir and Yuan in 1995. Moreover, we propose a method to construct admissible orders on fuzzy numbers in terms of linear orders defined for intervals considering a strictly increasing upper dense sequence, proving that this order is admissible for a given partial order. Finally, we use admissible orders to ranking the path costs in fuzzy weighted graphs. IEEEPublication Open Access Admissible OWA operators for fuzzy numbers(Elsevier, 2024) García-Zamora, Diego; Cruz, Anderson; Neres, Fernando; Santiago, Regivan; Roldán López de Hierro, Antonio Francisco; Paiva, Rui; Dimuro, Graçaliz P.; Martínez López, Luis; Callejas Bedregal, Benjamin; Bustince Sola, Humberto; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISCOrdered Weighted Averaging (OWA) operators are some of the most widely used aggregation functions in classic literature, but their application to fuzzy numbers has been limited due to the complexity of defining a total order in fuzzy contexts. However, the recent notion of admissible order for fuzzy numbers provides an effective method to totally order them by refining a given partial order. Therefore, this paper is devoted to defining OWA operators for fuzzy numbers with respect to admissible orders and investigating their properties. Firstly, we define the OWA operators associated with such admissible orders and then we show their main properties. Afterward, an example is presented to illustrate the applicability of these AOWA operators in linguistic decision-making. In this regard, we also develop an admissible order for trapezoidal fuzzy numbers that can be efficiently applied in practice.Publication Open Access Advances in the estimation of fixed effects in spatial models with random effects(2024) Goicoa Mangado, Tomás; Ugarte Martínez, María Dolores; Estadística, Informática y Matemáticas; Estatistika, Informatika eta MatematikaLa representación cartográfica de enfermedades permite estimar indicadores de salud específicos para áreas geográficas dentro de una región de estudio. Aunque el objetivo principal suele ser proporcionar las tasas/riesgos de incidencia o mortalidad de enfermedades como el cáncer, existen otras aplicaciones. Por ejemplo, el análisis de crímenes contra las mujeres en India. La mayor parte de la investigación en la representación cartográfica de enfermedades usa modelos mixtos de Poisson jerárquicos bayesianos que incorporan la dependencia espacial o temporal para suavizar los riesgos y reducir la variabilidad de los estimadores clásicos de los riesgos como las razones de incidencia/mortalidad estandarizadas (RIE/RME). Sin embargo, los modelos de representación cartográfica de enfermedades tienen algunos inconvenientes. Aquí nos centramos en dos de estas limitaciones. En primer lugar, estos modelos en general no son identificables y se requieren restricciones en el proceso de estimación para obtener resultados razonables. El segundo problema es la confusión espacial y está relacionado con la inclusión de covariables en los modelos. Si las covariables tienen estructura espacial, su asociación con la respuesta puede no estimarse bien debido al sesgo y la inflación de la varianza. El objetivo principal de esta tesis es doble. Por un lado, abordaremos la complejidad de incorporar restricciones de suma cero para resolver los problemas de identificación al ajustar modelos espacio-temporales ampliamente utilizados en la representación cartográfica de enfermedades utilizando NIMBLE (de Valpine et al., 2017), un sistema para crear modelos estadísticos en R que permite ajustar modelos jerárquicos bayesianos utilizando un sistema configurable de algoritmos MCMC. Por otro lado, nos centraremos en la confusión espacial, con el objetivo de proponer un método que garantice estimaciones adecuadas de efectos fijos. La presente tesis está dividida en cuatro capítulos diferentes. El primer capítulo proporciona una introducción general sobre los problemas que se van a bordar en esta tesis y el resto de los capítulos profundizan en esos problemas. Esta tesis se cierra con una sección final que resume los principales resultados e introduce algunas ideas para futuras investigaciones.Publication Open Access Affine construction methodology of aggregation functions(Elsevier, 2020) Roldán López de Hierro, Antonio Francisco; Roldán, Concepción; Bustince Sola, Humberto; Fernández Fernández, Francisco Javier; Rodríguez Martínez, Iosu; Fardoun, Habib; Lafuente López, Julio; Estadística, Informática y Matemáticas; Estatistika, Informatika eta MatematikaAggregation functions have attracted much attention in recent times because of its potential use in many areas such us data fusion and decision making. In practice, most of the aggregation functions that scientists use in their studies are constructed from very simple (usually affine or polynomial) functions. However, these are distinct in nature. In this paper, we develop a systematic study of these two classes of functions from a common point of view. To do this, we introduce the class of affine aggregation functions, which cover both the aforementioned families and most of examples of aggregation functions that are used in practice, including, by its great applicability, the symmetric case. Our study allows us to characterize when a function constructed from affine or polynomial functions is, in fact, a new aggregation function. We also study when sums or products of this kind of functions are again an aggregation function.Publication Open Access Agent-based simulation improves e-grocery deliveries using horizontal cooperation(IEEE, 2020) Serrano Hernández, Adrián; Faulín Fajardo, Javier; De la Torre Martinez, Rocío de la; Cadarso, Luis; Estatistika, Informatika eta Matematika; Enpresen Kudeaketa; Institute of Smart Cities - ISC; Institute for Advanced Research in Business and Economics - INARBE; Estadística, Informática y Matemáticas; Gestión de EmpresasE-commerce has increased tremendously in recent decades because of improvements in the information and telecommunications technology along with changes in societal lifestyles. More recently, e-grocery (groceries purchased online) including fresh vegetables and fruit, is gaining importance as the most-efficient delivery system in terms of cost and time. In this respect, we evaluate the effect of cooperation-based policies on service quality among different supermarkets in Pamplona, Spain. Concerning the methodology, we deploy, firstly, a detailed survey in Pamplona in order to model e-grocery demand patterns. Secondly, we develop an agent-based simulation model for generating scenarios in cooperative and non-cooperative settings, considering the real data obtained from the survey analysis. Thus, a Vehicle Routing Problem is dynamically generated and solved within the simulation framework using a biased-randomization algorithm. Finally, the results show significant reductions in lead times and better customer satisfaction when employing horizontal cooperation in e-grocery distribution.Publication Open Access Aggregation functions based on the Choquet integral applied to image resizing(Atlantis Press, 2019) Bueno, Jéssica C. S.; Dias, Camila A.; Pereira Dimuro, Graçaliz; Santos, Helida; Bustince Sola, Humberto; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC; Estadística, Informática y MatemáticasThe rising volume of data and its high complexity has brought the need of developing increasingly efficient knowledge extraction techniques, which demands efficiency both in computational cost and in accuracy. Most of problems that are handled by these techniques has complex information to be identified. So, machine learning methods are frequently used, where a variety of functions can be applied in the different steps that are employed in their architecture. One of them is the use of aggregation functions aiming at resizing images. In this context, we introduce a study of aggregation functions based on the Choquet integral, whose main characteristic in comparison with other aggregation functions is that it considers, through fuzzy measure, the interaction between the elements to be aggregated. Thus, our main goal is to present an evaluation study of the performance of the standard Choquet integral the and copula-based generalization of the Choquet integral in relation to the maximum and mean functions, looking for results that may be better than the aggregation functions commonly applied. The results of such comparisons are promising, when evaluated through image quality metrics.Publication Open Access Aggregation of individual rankings through fusion functions: criticism and optimality analysis(IEEE, 2020) Bustince Sola, Humberto; Callejas Bedregal, Benjamin; Campión Arrastia, María Jesús; Silva, Ivanoska da; Fernández Fernández, Francisco Javier; Induráin Eraso, Esteban; Raventós Pujol, Armajac; Santiago, Regivan; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Estadística, Informática y MatemáticasThroughout this paper, our main idea is to analyze from a theoretical and normative point of view different methods to aggregate individual rankings. To do so, first we introduce the concept of a general mean on an abstract set. This new concept conciliates the social choice where well-known impossibility results as the Arrovian ones are encountered and the decision-making approaches where the necessity of fusing rankings is unavoidable. Moreover it gives rise to a reasonable definition of the concept of a ranking fusion function that does indeed satisfy the axioms of a general mean. Then we will introduce some methods to build ranking fusion functions, paying a special attention to the use of score functions, and pointing out the equivalence between ranking and scoring. To conclude, we prove that any ranking fusion function introduces a partial order on rankings implemented on a finite set of alternatives. Therefore, this allows us to compare rankings and different methods of aggregation, so that in practice one should look for the maximal elements with respect to such orders defined on rankings IEEE.Publication Open Access Aggregator to electric vehicle LoRaWAN based communication analysis in vehicle-to-grid systems in smart cities(IEEE, 2020) Klaina, Hicham; Picallo Guembe, Imanol; López Iturri, Peio; Astrain Escola, José Javier; Azpilicueta Fernández de las Heras, Leyre; Falcone Lanas, Francisco Javier; Ingeniaritza Elektrikoa, Elektronikoaren eta Telekomunikazio Ingeniaritzaren; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC; Ingeniería Eléctrica, Electrónica y de Comunicación; Estadística, Informática y MatemáticasRecently, there has been growing attention to the power grid management due to the increasing concerns on global warming. With the advancement in electric vehicles (EV) industry and the evolution in batteries, EVs become an important contributor to the grid with capability of bidirectional power exchange with the grid. In this context, Vehicle-to-Grid (V2G) systems enable multiple functionalities between EVs and the corresponding aggregator. Thus, reliable, long-range communication capabilities between aggregator and EVs is compulsory. In this paper, wireless channel analysis for aggregator and electrical vehicle communication using Long-Range Wide Area Network (LoRaWAN) technology in V2G is presented, in order to test a low-cost solution with large coverage and reduced power consumption profile. Wireless channel and system-level measurements have been performed in a real urban scenario between EV's charging station in Pamplona (Spain) and a vehicle in motion using LoRaWAN 868 MHz devices. Wireless channel characterization is performed by implementing a full 3D urban scenario model, including elements such as buildings, vehicles, users and urban infrastructure such as lamp posts and benches. By means of in-house developed 3D Ray Launching algorithm with hybrid simulation capabilities, estimations of received power levels, signal to noise ratio and time domain parameters have been obtained, for the complete volume of the scenario under test in dense urban conditions. V2G end to end communication has been validated by implementing an intra-vehicle Controller Area Network-BUS (CAN BUS) data gathering system connected to the vehicle LoRaWAN transceiver and subsequently, to a cloud-based web service. The results show that the accurate deterministic based radio channel analysis enables to optimize the network design of LoRaWAN networks in a vehicular environment, considering inter-vehicular and infrastructure links, enabling scalable, low cost end to end data exchange for the deployment of ancillary V2G services.Publication Open Access Alleviating confounding in spatio-temporal areal models with an application on crimes against women in India(SAGE Publications, 2021) Adin Urtasun, Aritz; Goicoa Mangado, Tomás; Hodges, James S.; Schnell, Patrick M.; Ugarte Martínez, María Dolores; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Estadística, Informática y MatemáticasAssessing associations between a response of interest and a set of covariates in spatial areal models is the leitmotiv of ecological regression. However, the presence of spatially correlated random effects can mask or even bias estimates of such associations due to confounding effects if they are not carefully handled. Though potentially harmful, confounding issues have often been ignored in practice leading to wrong conclusions about the underlying associations between the response and the covariates. In spatio-temporal areal models, the temporal dimension may emerge as a new source of confounding, and the problem may be even worse. In this work, we propose two approaches to deal with confounding of fixed effects by spatial and temporal random effects, while obtaining good model predictions. In particular, restricted regression and an apparently—though in fact not—equivalent procedure using constraints are proposed within both fully Bayes and empirical Bayes approaches. The methods are compared in terms of fixed-effect estimates and model selection criteria. The techniques are used to assess the association between dowry deaths and certain socio-demographic covariates in the districts of Uttar Pradesh, India.Publication Embargo Análisis de idoneidad en propuestas para el estudio de cónicas con GeoGebra en Educación Secundaria(2023) Abaurrea Larrayoz, Jaione; Rodríguez Wilhelmi, Miguel; Estadística, Informática y Matemáticas; Estatistika, Informatika eta MatematikaEn esta tesis doctoral se propone una manera de trabajar las cónicas en Educación Secundaria diferente a las propuestas que en general se plantean en los libros de texto. Así, se describen y analizan actividades que se alejan de la enseñanza meramente transmisiva de las matemáticas, para enfatizar la acción matemática que implica llevar a cabo los procesos de: resolución de problemas, razonamiento y prueba, conexiones y representación.