Publications 2021

Book Chapters

7.  Sustainable development in education: a non-parametric analysis

Murillo, Kelly and Rocha, Eugénio

INTED2021 Proceedings


The SDGs (Sustainable Development Goals) are a universal call to action to end poverty, protect the planet and improve the lives and prospects of everyone, everywhere. In 2015 all UN Member States, adopted the 2030 Agenda for the SDG, which comprises an action plan for people, the planet and prosperity with 17 objectives covering the economic, social and environmental dimensions, [1]. SDG 4 is the goal of quality education with made up of 10 targets to ensure an inclusive and equitable quality education and to promote lifelong learning opportunities for all. In this sense, it is expected that all countries increasing the number of young people and adults with relevant professional skills, decent jobs, entrepreneurship, eliminating gender and income disparities in access to education. This article examines the quality of education in 17 European countries using a model nonparametric deterministic for measuring efficiency based on MEA (Multidirectional Efficiency Analysis) [2], in combination with other mathematical techniques (such as accumulated effort and group indicator), during seven years (every three years from 2000-2018). To this end, we analyze the countries evolution at three distinct efficiency stages: levels, patterns and determinants. The study is based on the EU's set of indicators to monitor progress towards the UN SDGs: basic education (early leavers from education and training, participation in early childhood education and achievement in reading, mathematic or science), tertiary education (tertiary education attainment and employment rates of recent graduates) and adult learning (adult participation in learning). This study allows us to address questions such as: To what extent are European countries improving education quality? Which European countries have significant advances / setbacks over time? What factors are intervening in the process of the countries that are most efficient and least efficient? In other words, our results clarify which are the profiles of the countries that are most efficient, giving some insight about the improvements which could be applied in the less efficient to raise their efficiency, in view of reaching the proposed objectives for the year 2030. References: [1] Report of the Inter-Agency and Expert Group on Sustainable Development Goal Indicators (E/CN.3/2016/2/Rev.1), Economic and Social Council, United Nations, 1-39, 2016. [2] P. Bogetoft and J. L. Hougaard, Efficiency evaluations based on potential (Non-proportional) improvements, J. Productivity Analysis, 12(3), 233-247, 1999. | doi | Peer Reviewed


6.  Decompositions with atoms and molecules for variable exponent Triebel-Lizorkin-Morrey spaces

Caetano, António and Kempka, Henning

Constructive Approximation

Springer Verlag

We continue the study of the variable exponent Morreyfied Triebel-Lizorkin spaces introduced in a previous paper. Here we give characterizations by means of atoms and molecules. We also show that in some cases the number of zero moments needed for molecules, in order that an infinite linear combination of them (with coefficients in a natural sequence space) converges in the space of tempered distributions, is much smaller than what is usually required. We also establish a Sobolev type theorem for related sequence spaces, which might have independent interest. | doi | Peer Reviewed

5.  Eigenfunctions of the time‐fractional diffusion‐wave operator

Ferreira, Milton and Luchko, Yury and Rodrigues, M. Manuela and Vieira, Nelson

Mathematical Methods in the Applied Sciences


In this paper, we present some new integral and series representations for the eigenfunctions of the multidimensional time‐fractional diffusion‐wave operator with the time‐fractional derivative of order β ∈]1, 2[ defined in the Caputo sense. The integral representations are obtained in form of the inverse Fourier–Bessel transform and as a double contour integrals of the Mellin–Barnes type. Concerning series expansions, the eigenfunctions are expressed as the double generalized hypergeometric series for any β ∈]1, 2[ and as Kampé de Fériet and Lauricella series in two variables for the rational values of β. The limit cases β=1 (diffusion operator) and β=2 (wave operator) as well as an intermediate case β=32 are studied in detail. Finally, we provide several plots of the eigenfunctions to some selected eigenvalues for different particular values of the fractional derivative order β and the spatial dimension n. | doi | Peer Reviewed

4.  A new convolution operator for the linear canonical transform with applications

Castro, Luís P. and Goel, Navdeep and Silva, Anabela S.

Computational and Applied Mathematics


The linear canonical transform plays an important role in engineering and many applied fields, as it is the case of optics and signal processing. In this paper, a new convolution for the linear canonical transform is proposed and a corresponding product theorem is deduced. It is also proved a generalized Young's inequality for the introduced convolution operator. Moreover, necessary and sufficient conditions are obtained for the solvability of a class of convolution type integral equations associated with the linear canonical transform. Finally, the obtained results are implemented in multiplicative filters design, through the product in both the linear canonical transform domain and the time domain, where specific computations and comparisons are exposed. | doi | Peer Reviewed

3.  New convolutions with Hermite weight functions

Castro, Luís Pinheiro and Silva, Anabela Sousa and Tuan, Nguyen Minh

Bulletin of the Iranian Mathematical Society


In this paper, we are working with convolutions on the positive half-line, for Lebesgue integrable functions. Six new convolutions are introduced. Factorization identities for these convolutions are derived, upon the use of Fourier sine and cosine transforms and Hermite functions. Such convolutions allowus to consider systems of convolution type equations on the half-line. Using two different methods, such systems of convolution integral equations will be analyzed. Conditions for their solvability will be considered and, under such conditions, their solutions are obtained. | doi | Peer Reviewed

2.  Application of the fractional Sturm–Liouville theory to a fractional Sturm–Liouville telegraph equation

Ferreira, M. and Rodrigues, M. M. and Vieira, N.

Complex Analysis and Operator Theory


In this paper, we consider a non-homogeneous time-space-fractional telegraph equation in $n$-dimensions, which is obtained from the standard telegraph equation by replacing the first- and second-order time derivatives by Caputo fractional derivatives of corresponding fractional orders, and the Laplacian operator by a fractional Sturm-Liouville operator defined in terms of right and left fractional Riemann-Liouville derivatives. Using the method of separation of variables, we derive series representations of the solution in terms of Wright functions, for the homogeneous and non-homogeneous cases. The convergence of the series solutions is studied by using well known properties of the Wright function. We show also that our series can be written using the bivariate Mittag-Leffler function. In the end of the paper some illustrative examples are presented. | doi | Peer Reviewed

1.  Density results for Sobolev, Besov and Triebel–Lizorkin spaces on rough sets

Caetano, A. M. and Hewett, D. P. and Moiola, A.

Journal of Functional Analysis


We investigate two density questions for Sobolev, Besov and Triebel--Lizorkin spaces on rough sets. Our main results, stated in the simplest Sobolev space setting, are that: (i) for an open set $Omegasubsetmathbb R^n$, $mathcal{D}(Omega)$ is dense in ${uin H^s(mathbb R^n):{rm supp}, usubset overline{Omega}}$ whenever $partialOmega$ has zero Lebesgue measure and $Omega$ is "thick" (in the sense of Triebel); and (ii) for a $d$-set $Gammasubsetmathbb R^n$ ($0 | doi | Peer Reviewed
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