Book Chapters
12.
Torus and quadrics intersection using GeoGebra
Breda, Ana Maria Reis D'Azevedo and Trocado, Alexandre Emanuel Batista da Silva and Santos, José Manuel dos Santos dos
ICGG 2020: proceedings of the 19th International Conference on Geometry and Graphics
Springer
This paper presents the implementation in GeoGebra of algorithms for computing the intersection curve of a quadric surface with a torus surface. We present three approaches to get and visualise the intersection curve in GeoGebra. One of the approaches makes use of the geometric capabilities of GeoGebra. The second described approach makes use of CAS to obtain a parametrization and the corresponding visualisation of the intersection curve. Finally, the third one is based on computing the projection of the intersection curve, determining its singularities and structure, and its lifting to the 3D embedding space. The research carried out reveals some of the difficulties arising from the implementation in GeoGebra of a geometric algorithm based on the algebraic equations characterising the objects in consideration.
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11.
Graphic and multimedia design of a narrativebased math game
Afonso, R. and Breda, A. and Rocha, E.
EDULEARN21 Proceedings
IATED
In this paper, we will describe the process of creating the graphic and multimedia design of the math serious game CNME. This game designed to facilitate and “gamify” mathematical learning processes is based on the narrative of MagellanElcano’s epic circumnavigation journey around the world. CNME game was developed by an interdisciplinary team of the research and development thematic line GEOMETRIX.
In graphic and multimedia terms the main function is to illustrate and retell events around this trip in an interactive and dynamic way, captivating and maintaining the user's interest and attention, making mathematical challenges emerge as natural / plausible events within the narrative context that unfolds.
The stimuli, used to anchor the narrative, differ according to the situation/episode that is unfolding. They are employed either to capture the player's attention or to make the experience more engaging and enjoyable.
The game is simple and logical presenting the events methodically. It follows, chronologically, the historical and remarkable events of this famous odyssey, presenting each scene as a different chapter. The exploration of these "chapters" position the user in front of different mathematical, historical, playful or general knowledge challenges and problems, which he / she must solve in order to progress to more advanced levels. These problems / challenges, of such a diverse nature, aim at learning and assessing mathematical, historical and everyday knowledge.
This game can be seen as an interactive learning environment with challenges inserted and contextualized in the game's narrative, involving several game mechanics, among which: collecting items / objects / species and building and dismantling objects.
The design phase is crucial for the development of the game as it encompasses essential factors for the success of the proposed objectives, namely, capturing attention, maintaining motivation, giving reasons for wanting to learn mathematics.
The design process includes: interaction design, graphic design and sound design.
During the interaction with the game the player is visual and aurally stimulated with vibrant landscapes and surrounding sounds, while being, psychologically, challenged to solve problems contextualized in a historical, interesting and real story line, triggering the desire to learn more and more.
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10.
CNME a math narrativebased serious game
Martins, Micaela and Breda, Ana and Rocha, Eugénio and Domingues, Ricardo
EDULEARN21 Proceedings
IATED
The teaching method has varied and evolved over the years. The year 2020 is a milestone in this variability. The COVID19 pandemic unleashed the strict need for a radical adaptation of teaching processes that, worldwide, become exclusively or almost exclusively at a distance. The impact of the digital world on our lives has been and is being felt like never before. Nonformal teaching processes gain crucial importance in this scenario.
Serious games are engaging and provide a stimulating environment in which students can explore and discover in a fun and interactive way, improving student’s motivation and performance in mathematics and making them active learners. The adoption in the educational process of serious games, promoting the development of critical thinking, and its interest, as a research topic, by scientists from various areas, namely, mathematics, have gained increasing prominence.
With regard to mathematics, despite its recognized importance in the intellectual human development, children and adolescents usually believe that it is a difficult subject, both at a conceptual and procedural level, leading to a lack of motivation and high failure rates.
In this paper, based on a solid and recent literature review, we look at the role that serious games play in the learning and motivation of children and adolescents, especially the narrative educational games focused on mathematics.
A narrative interactive serious game is defined as a serious game, in which the story exists to improve the gameplay. Within this context, the Thematic Line Geometrix of the Center for Research and Development of Mathematics and Applications (CIDMA) of the University of Aveiro developed the narrative serious game CNME based on the historical event “The Magellan  Elcano circumnavigation around the world”. This game runs on every platform that has a recent browser and it also has an application for Android and iOS. In the CNME digital and interactive game there are two game modes, the generic and the academic mode, depending on the player's profile. The generic modality was designed to promote mathematical literacy and is aimed at any citizen. The academic modality was designed to promote critical and creative thinking and is aimed at young people with mathematical knowledge at the level of the 3rd Cycle of the Portuguese Basic Education. In short, CNME is an interactive mathematical narrative game, aiming at mathematical learning in an interactive, playful and motivating way, anchored in a notable and true historical event conceived under a set of scrutinized scientific evidence.
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9.
Towards a specification theory for fuzzy modal logic
Jain, Manisha and Gomes, Leandro and Madeira, Alexandre and Barbosa, Luis S.
2021 International Symposium on Theoretical Aspects of Software Engineering (TASE)
IEEE
Fuzziness, as a way to express imprecision, or uncertainty, in computation is an important feature in a number of current application scenarios: from hybrid systems interfacing with sensor networks with error boundaries, to knowledge bases collecting data from often noncoincident human experts. Their abstraction in e.g. fuzzy transition systems led to a number of mathematical structures to model this sort of systems and reason about them. This paper adds two more elements to this family: two modal logics, framed as institutions, to reason about fuzzy transition systems and the corresponding processes. This paves the way to the development, in the second part of the paper, of an associated theory of structured specification for fuzzy computational systems.
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Articles
8.
A new rank metric for convolutional codes
Almeida, P. and Napp, D.
Designs, Codes and Cryptography
Springer
Let F[D] be the polynomial ring with entries in a finite field F. Convolutional codes are submodules of F[D]n that can be described by left prime polynomial matrices. In the last decade there has been a great interest in convolutional codes equipped with a rank metric, called sum rank metric, due to their wide range of applications in reliable linear network coding. However, this metric suits only for delay free networks. In this work we continue this thread of research and introduce a new metric that overcomes this restriction and therefore is suitable to handle more general networks. We study this metric and provide characterizations of the distance properties in terms of the polynomial matrix representations of the convolutional code. Convolutional codes that are optimal with respect to this new metric are investigated and concrete constructions are presented. These codes are the analogs of Maximum Distance Profile convolutional codes in the context of network coding. Moreover, we show that they can be built upon a class of superregular matrices, with entries in an extension field, that preserve their superregularity properties even after multiplication with some matrices with entries in the ground field.
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7.
A semantics and a logic for Fuzzy Arden Syntax
Gomes, Leandro and Madeira, Alexandre and Barbosa, Luís Soares
Soft Computing
Springer
Fuzzy programming languages, such as the Fuzzy Arden Syntax (FAS), are used to describe behaviours which evolve in a fuzzy way and thus cannot be characterized neither by a Boolean outcome nor by a probability distribution. This paper introduces a semantics for FAS, focusing on the weighted parallel interpretation of its conditional statement. The proposed construction is based on the notion of a fuzzy multirelation which associates with each state in a program a fuzzy set of weighted possible evolutions. The latter is parametric on a residuated lattice which models the underlying semantic ‘truth space’. Finally, a family of dynamic logics, equally parametric on the residuated lattice, is introduced to reason about FAS programs.
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6.
Introducing fuzzy reactive graphs: a simple application on biology
Santiago, Regivan and Martins, Manuel A. and Figueiredo, Daniel
Soft Computing
Springer
In this paper, we propose a generalization for fuzzy graphs in order to model reactive systems with fuzziness. As we will show, the resulting fuzzy structure, called fuzzy reactive graphs (FRG), is able to model dynamical aspects of some entities which generally appear in: biology, computer science and some other fields. The dynamical aspect is captured by a transition function which updates the values of the graph after an edge has been crossed. The update process takes into account aggregation functions. The paper proposes a notion for bisimulation for such graphs and briefly shows how modal logic can be used to verify properties of systems modeled with FSGs. The paper closes with a toy example in the field of Biology.
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5.
Nondual modal operators as a basis for 4valued accessibility relations in Hybrid logic
Costa, Diana and Martins, Manuel A.
Journal of Logical and Algebraic Methods in Programming
Elsevier
The modal operators usually associated with the notions of possibility and necessity are classically duals. This paper aims to defy that duality in a paraconsistent environment, namely in a Belnapian Hybrid logic where both propositional variables and accessibility relations are fourvalued. Hybrid logic, which is an extension of Modal logic, incorporates extra machinery such as nominals – for uniquely naming states – and a satisfaction operator – so that the formula under its scope is evaluated in the state whose name the satisfaction operator indicates.
In classical Hybrid logic the semantics of negation, when it appears before compound formulas, is carried towards subformulas, meaning that eventual inconsistencies can be found at the level of nominals or propositional variables but appear unrelated to the accessibility relations. In this paper we allow inconsistencies in propositional variables and, by breaking the duality between modal operators, inconsistencies at the level of accessibility relations arise. We introduce a sound and complete tableau system and a decision procedure to check if a formula is a consequence of a set of formulas. Tableaux will be used to extract syntactic models for databases, which will then be compared using different inconsistency measures. We conclude with a discussion about bisimulation.
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4.
PakStanley labeling of the mCatalan hyperplane arrangement
Duarte, Rui and Guedes de Oliveira, António
Advances in Mathematics
Elsevier
We characterize in simple terms the PakStanley labels λ(R) of the regions
R of the mCatalan arrangement. We also propose a simple algorithm that returns R
from λ(R). Finally, we characterize in close terms the labels of the relatively bounded
regions.
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3.
Generalizações da etiquetagem de PakStanley
Duarte, Rui and Guedes de Oliveira, António
Boletim da Sociedade Portuguesa de Matemática
Sociedade Portuguesa de Matemática
Neste artigo pretendemos dar a conhecer ao leitor uma área que consideramos particularmente atraente, onde temos obtido alguns resultados que tentaremos também relatar.
O ponto de partida é uma construção (a "etiquetagem de PakStanley''), que associa um vetor a cada uma das regiões em que determinado conjunto de hiperplanos divide o espaço euclideano $R^n$. Regiões vizinhas diferem (em $1$) numa coordenada, crescendo ao afastarse de uma determinada região, etiquetada com $(1,1,dotsc,1)$.
Há cerca de vinte anos, Pak e Stanley mostraram que, no caso de os hiperplanos formarem o "arranjo de Shi'', as etiquetas de PakStanley formam um conjunto previamente estudado, o conjunto das "parking functions'', e
que a etiquetagem é bijetiva, muito embora seja difícil definir a função inversa, isto é, obter a região a partir da etiqueta.
Esta construção tem uma extensão natural a outros arranjos. Recentemente, Mazin obteve uma caracterização muito geral dos conjuntos de etiquetas assim obtidos, que implica, em particular, o resultado de Pak e Stanley. Com base neste trabalho de Mazin, estudamos as etiquetagens de outro arranjos, o arranjo de Ish recentemente definido e um conjunto de arranjos por nós introduzido, que constitui uma classe naturalmente balizada, por um lado, pelo arranjo de Shi, e por outro, pelo arranjo de Ish.
O nosso trabalho, em traços gerais, consistiu em descrever o conjunto das etiquetas respetivas e em mostrar que as etiquetagens são bijetivas, recorrendo a diferentes técnicas e resultados que aqui são explicitamente referidos. A exposição dessas técnicas e o relato desses resultados, alguns já "clássicos'' da área e outros muito recentes, foram talvez a nossa razão mais forte para a escrita deste pequeno texto.
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2.
Observational interpretations of hybrid dynamic logic with binders and silent transitions
Hennicker, Rolf and Knapp, Alexander and Madeira, Alexandre
Journal of Logical and Algebraic Methods in Programming
We extend hybrid dynamic logic with binders (for state variables) by distinguishing between observable and silent transitions. This differentiation gives rise to two kinds of observational interpretations: The first one relies on observational abstraction from the ordinary model class of a specification Sp by considering its closure under weak bisimulation. The second one uses an observational satisfaction relation for the axioms of the specification Sp, which relaxes the interpretation of state variables and the satisfaction of modal formulæ by abstracting from silent transitions. We establish a formal relationship between both approaches and show that they are equivalent under mild conditions. For the proof we instantiate the previously introduced concept of a behaviourabstractor framework to the case of dynamic logic with binders and silent transitions. As a particular outcome we provide an invariance theorem and show the HennessyMilner property for weakly bisimilar labelled transition systems and observational satisfaction. In the second part of the paper we integrate our results in a development methodology for reactive systems leading to two versions of observational refinement. We provide conditions under which both kinds of refinement are semantically equivalent, involving implementation constructors for relabelling, hiding, and parallel composition.
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1.
Hybrid dynamic logic institutions for event/databased systems
Hennicker, Rolf and Knapp, Alexander and Madeira, Alexandre
Formal Aspects of Computing
Springer
We propose EDlogic as a formal foundation for the specification and development of eventbased systems with data states. The framework is presented as an institution in the sense of Goguen and Burstall and the logic itself is parametrised by an underlying institution D whose structures are used to model data states. EDlogic is intended to cover a broad range of abstraction levels from abstract requirements specifications up to constructive specifications. It uses modal diamond and box operators over complex actions adopted from dynamic logic. Atomic actions are pairs of events and state transition predicates capturing the allowed reactions to the event. To write concrete specifications of recursive process structures we integrate (control) state variables and binders of hybrid logic. The semantic interpretation relies on event/data transition systems. For the presentation of constructive specifications we propose operational event/data specifications allowing for familiar, diagrammatic representations by state transition graphs. We show that EDlogic is powerful enough to characterise the semantics of an operational specification by a single EDsentence. Thus the whole (formal) development process for event/databased systems relies on EDlogic and its semantics as a common basis. It is supported by a variety of implementation constructors which can express, among others, event refinement and parallel composition. Due to the genericity of the approach, it is also possible to change a data state institution during system development when needed. All steps of our formal treatment are illustrated by a running example.
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