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Publication . Hall, A.; Pais, S.

Mathematics, considered one of the basic areas of various formations, has been the subject of concern for many authors and researchers due to its enormous academic and educational failure [1]. Its importance in day-to-day life and the formation of individuals is irrefutable [2]. One of the problems identified in the failure of mathematics teaching is the lack of motivation that students feel towards the discipline. According to [3], motivation is an essential factor in any learning since the quality of learning is not only related to the capacity to learn, but also to the level of motivation that we have to carry out this same learning. Considering that it is imperative to make the teaching and learning process of mathematics more stimulating, taking into account modern society and student’s interests [4], the authors have developed a qualitative case study to evaluate how "mathematical magic" can contribute to increase students’ motivation for learning mathematics. In order to develop this experience, the techniques of inquiry, direct observation and analysis of documents were applied and the following instruments were used: questionnaires and respective analysis grids; production of a battery of tasks of a diversified nature; field notes and interviews. Although this project is still ongoing and therefore not yet completed, a previous analysis of the collected data allows us to conclude that the use of mathematical magic tricks in the classroom, with the purpose of motivating the students to learn mathematics, was revealed effective. Students were curious about the new tricks and were positively surprised at the relative simplicity of their explanation, as if saying to themselves, "How can such mathematical concepts bring so much surprise?" The students showed that the topics gained more meaning after performing the tricks because they witnessed real applications of the concepts, with an extremely attractive purpose. They often stated that they were going to play the tricks on their friends / family outside the classroom context, which meant that they were mastering the concepts involved.

2018Conference object

Open access

Modelling integrated multi-trophic aquaculture: Optimizing a three trophic level system

Publication . Granada, Luana; Lopes, Sofia; Novais, Sara C.; Lemos, Marco F. L.

As a fast-growing food production industry, aquaculture is dealing with the need for intensification due to the global increasing demand for fish products. However, this also implies the use of more sustainable practices to reduce negative environmental impacts currently associated with this industry, including the use of wild resources, destruction of natural ecosystems, eutrophication of effluent receiving bodies, impacts due to inadequate medication practices, among others. Using multi-species systems, such as integrated multi-trophic aquaculture, allows to produce economically important species while reducing some of these aquaculture concerns, through biomitigation of aquaculture wastes and reduction of diseases outbreaks, for example. Applying mathematical models to these systems is crucial to control and understand the interactions between species, maximizing productivity, with important environmental and economic benefits. Here, the application of some equations and models available in the literature, regarding basic parameters, is discussed – population dynamics, growth, waste production, and filtering rate – when considering the description and optimization of a theoretical integrated multi-trophic aquaculture operation composed by three trophic levels.

2018-10Journal article

Restricted access

Harmonic analysis on the proper velocity gyrogroup

Publication . Ferreira, Milton

In this paper we study harmonic analysis on the Proper Velocity (PV) gyrogroup using the gyrolanguage of analytic hyperbolic geometry. PV addition is the relativistic addition of proper velocities in special relativity and it is related with the hyperboloid model of hyperbolic geometry. The generalized harmonic analysis depends on a complex parameter $z$ and on the radius $t$ of the hyperboloid and comprises the study of the generalized translation operator, the associated convolution operator, the generalized Laplace-Beltrami operator, and its eigenfunctions, the generalized Poisson transform, and its inverse, the generalized Helgason-Fourier transform, its inverse and Plancherel's Theorem. In the limit of large $t,$ $t \rightarrow +\infty,$ the generalized harmonic analysis on the hyperboloid tends to the standard Euclidean harmonic analysis on ${\mathbb R}^n,$ thus unifying hyperbolic and Euclidean harmonic analysis.

2017Journal article

Open access

First and second fundamental solutions of the time-fractional telegraph equation of order 2α

Publication . Ferreira, Milton; Rodrigues, M. Manuela; Vieira, Nelson

In this work we obtain the first and second fundamental solutions of the multidimensional time-fractional equation of order 2α, α ∈]0, 1], where the two time-fractional derivatives are in the Caputo sense. We obtain representations of the fundamental solutions in terms of Hankel transform, double Mellin-Barnes integral, and H-functions of two variables. As an application, the fundamental solutions are used to solve a Cauchy problem and to study telegraph process with Brownian time.

2018-12-04Journal article

Open access

Eigenfunctions and fundamental solutions of the fractional Laplace and Dirac operators using Caputo derivatives

Publication . Ferreira, Milton; Vieira, Nelson Felipe Loureiro

In this paper we study eigenfunctions and fundamental solutions for the three parameter fractional Laplace operator ${}^C\!\Delta_+^{(\alpha,\beta,\gamma)}:= {}^C\!D_{x_0^+}^{1+\alpha} +{}^C\!D_{y_0^+}^{1+\beta} +{}^C\!D_{z_0^+}^{1+\gamma},$ where $(\alpha, \beta, \gamma) \in \,]0,1]^3$ and the fractional derivatives ${}^C\!D_{x_0^+}^{1+\alpha}$, ${}^C\!D_{y_0^+}^{1+\beta}$, ${}^C\!D_{z_0^+}^{1+\gamma}$ are in the Caputo sense. Applying integral transform methods we describe a complete family of eigenfunctions and fundamental solutions of the operator ${}^C\!\Delta_+^{(\alpha,\beta,\gamma)}$ in classes of functions admitting a summable fractional derivative. The solutions are expressed using the Mittag-Leffler function. From the family of fundamental solutions obtained we deduce a family of fundamental solutions of the corresponding fractional Dirac operator, which factorizes the fractional Laplace operator introduced in this paper.

2017-06Journal article

Open access