Abstracts
Algebra and Geometry Group
Coordinator: D. Hofmann
The group of Algebra and Geometry brings together researchers with interest in Geometry, Algebra, Combinatorics, Topology, Coding Theory, Logic and Category Theory. The group has also a strong interest in various aspects of education of Mathematics, in particular, it develops the interdisciplinary project Geometrix for the implementation of different adaptive computer aided learning environments for the teaching of Mathematics.
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Title: Three adventures in the factorization world
Speaker: Paulo Almeida
Abstract: In this talk we will focus on three problems involving the factorization of three types of integers. The first topic is about integers having no factor close to its square root, and we show that the Fermat numbers have this property. Secondly we present a generalization of the Euclid-Euler theorem on perfect numbers. The third topic concerns the Euler factorization of integers and the Aubry algorithm that permit us to obtain integer solutions of quadratic diophantine equations from its rational solutions.
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Title: Dynamic Logics On-demand
Speaker: Alexandre Madeira
Abstract: In the last decades, dynamic logics have been used in different domains as a suitable formalism to reason about and specify a wide range of systems. On the other hand, logics with many-valued semantics are emerging as an interesting tool to handle devices and scenarios where uncertainty is a prime concern.
In order to combine these two aspects, we introduced in [3] a method for the systematic construction of many-valued dynamic logics. Technically, the method is parameterised by an action lattice that defines both the computational paradigm and the truth space (corresponding to the underlying Kleene algebra and residuated lattices, respectively). This parametric principle pushed then other theoretical developments, including the a method to the generation of multi-valued epistemic logic, as reported in [1]. Recently, in [2], we extended the systematic generation of dynamic logics from the propositional to the equational case, in order to capture ”full-fledged” imperative programs. The generation process is parametric on a structure specifying a notion of ”weight” assigned to programs. We overview on this talk this line of research, by raising some lines of future work.
References:
[1] Mario R. F. Benevides, Alexandre Madeira, and Manuel A. Martins. A family of graded epistemic logics. Electr. Notes Theor. Comput. Sci., 338:45-59, 2018.
[2] Leandro Gomes, Alexandre Madeira, Manisha Jain, and Lus Soares Barbosa. On the generation of equational dynamic logics for weighted imperative programs. In Formal Methods and Software Engineering - ICFEM 2019, volume 11852 of Lecture Notes in Computer Science, pages 154–169. Springer, 2019.
[3] A. Madeira, R. Neves, and M. A. Martins. An exercise on the generation of many-valued dynamic logics. Journal of Logical and Algebraic Methods in Programming, 1:1–29, 2016.
Complex and Hypercomplex Analysis Group
Coordinator: U. Kähler
The research group on Complex and Hypercomplex Analysis focuses it research on applications and higher dimensional generalisations of complex analysis with the main focus on Hypercomplex analysis. It has a wide range of international collaboration and has as major research lines hypercomplex analysis, spectral theory, inverse problems, orthogonal polynomials and Riemann-Hilbert problems as well as fractional PDEs. Its research is also closely linked to applications in engineering and data sciences, including machine and deep learning algorithms.
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Title: Hypercomplex orthogonal polynomials and shifted Vietoris sequences
Speaker: Isabel Cação
Abstract: The theory of orthogonal polynomials in the real (or complex case) is a well-establish subject widely studied by several authors and continues to attract young scientists. In the hypercomplex setting much less was done, mainly because the underlying algebra is not commutative which provides an extra challenge in the construction of power-like monomials. During the last decades, some authors constructed hypercomplex orthogonal polynomials using specific techniques that overcome the non-commutativity constraint. The aim of this talk is to study some of their properties trying to establish some similarities with the real case. Moreover, we show how the construction process relies on a certain shift of the generalized Vietoris sequences of rational numbers, defined recently.
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Title: Laguerre Matrix Orthogonal Polynomials and the Riemann-Hilbert Problem
Speaker: Ana Foulquié Moreno
Abstract: Inspired by previous results from the literature for the Laguerre matrix orthogonal polynomials we decided to treat these measures using the tools from the Riemann-Hilbert problem. This was done before in the so-called Hermite Matrix Biorthogonal case. To state the Riemann-Hilbert problem we need to know explicitly the measure but also the behavior of this measure at the end points of the support of the measure. The difficulties we found comes from the non-commutativity of the terms that define the measure and this gives us some restrictions. To avoid that we have to consider the matrix measure as a solution of a Pearson Differential equation. Through this characterization we still have enough information for the matrix measure in order to be able to formulate the Riemann-Hilbert problem. First and second order differential systems for the fundamental matrix, solution of the mentioned Riemann–Hilbert problem are derived. An explicit and general example is presented to illustrate the theoretical results of the work. Related matrix eigenvalue problems for second order matrix differential operators and non-Abelian extensions of a family of discrete Painlevé IV equations are discussed.
This is a joint work with Amı́car Branquinho, Universidade de Coimbra e Manuel Manãs, Universidad Complutense de Madrid.
Functional Analysis and Applications Group
Coordinator: A. Caetano
The general aim of the group is to perform research in the fields of boundary value problems, differential equations and inclusions, function spaces, integral transforms and equations, operator theory, and variational problems, as well as their applications.
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Title: Compelling relations between Convexity Analysis, Functional Analysis, Differential Eqs, Emotions, Love, and Ethics
Speaker: Eugénio Rocha
Abstract: We are particularly interested in understanding the relations between ”far away” concepts and their applications in real life scenarios. In this talk, we are going to connect three works (with several co-authors):
(a) recent results for the existence of solutions for systems of PDE involving Radon measures;
(b) an ODE model for affection/love based on a psychological model with prediction accuracy above 90 per cent;
(c) a recent enterprise AI project (NLP+ML) regarding the calculation of the consumer Need Index and the prediction of success of Kickstarter crowdfunding campaigns with accuracy of 94 per cent.
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Title: Eigenfunctions of the time-fractional diffusion-wave operator
Speaker: Manuela Rodrigues
Abstract: In this talk, we present an overview of the recent results about some new integral and series representations 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 the 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. Finally, we provide some plots of the eigenfunctions to some selected eigenvalues for different particular values of the fractional derivative of order β and the spatial dimension n.
Gravitational Geometry and Dynamics Group
Coordinator: C. Herdeiro
The goal of the Gravitational Geometry and Dynamics Group is to produce internationally competitive research in Gravitational theory both at relativisitic level, with applications to cosmology, black hole physics, graviational wave physics, high energy physics and mathematical physics, and Newtonian level, with application to solar system dynamics and exo-planet detection and dynamics.
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Title: Chansing Super Massive Black Holes
Speaker: Sonia Anton
Abstract: According with hierarchical assembly of galaxies and Super Massive Black Holes (SMBH) models, merger of smaller units is a key process to grow in mass and size. Besides providing important tools to understand galaxy evolution, merging or merged SMBH systems are relevant systems for multi-messenger science as potential gravitational wave emission sources. It is therefore of great interest to detect many these systems, as there are very few known systems. I will summarise the results obtained so far, as well as I will present the impact that the new generation of telescopes will have in this domain.
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Title: No Go Theorems and Solitons in Einstein-Maxwell-scalar models
Speaker: João Oliveira
Abstract: After a brief introduction to the topic, two non-existence results are established for self-gravitating solitons in Einstein-Maxwell-scalar models, where a scalar field is non-minimally coupled to the Maxwell field via an arbitrary function f(\Phi). The first result consists in establishing that for strictly stationary, but not necessarily static, spacetimes, using a Lichnerowicz-type argument, self-gravitating solitons do not exist. The second result consists in the generalisation of the first result to Einstein-Maxwell-Axion models, through a new coupling function g(\Phi). After these two results are established, we then discuss how we can circumvent the first result to obtain self-gravitating solitonic solutions in the Einstein-Maxwell-scalar model. These solutions are possible even in flat spacetime Maxwell-scalar models where an explicit mechanism is found to de-singularise the Coulomb field.
History of Mathematics Group
Coordinator: H.R. Malonek
One of the main aims of this group is to contribute to the work of CIDMA by supporting and promoting creative capacities in the Department that do not identify themselves with only one of the disciplines or that prefer to be integrated in forms of investigation which use other specific methods (for instance, search in libraries or archives, analysis of estates, comparative or biographical studies etc.) The proposed goals and the expected results of the historical research will be used to increase the scientific culture at any level of research, particularly at the post-graduate level. Another aim is to support the realization of commemorative activities (exhibitions, conferences) on a high scientific level which may find resonance inside and outside the Department.
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Title: Os 500 anos do Tratado da prática d’arismetica de Gaspar Nicolas
Speaker: Teresa Costa Clain
Abstract: Comemorou-se em 15 de Novembro de 2019 os quinhentos anos do Tratado da Pratica d’Arismetica de Gaspar Nicolas. Esta obra foi o primeiro tratado de aritmética prática de autor português publicado em Portugal. Pertence a um conjunto de livros que entraram em cena numa época marcada por uma vontade de perpetuar na forma escrita o conhecimento outrora transmitido oralmente. Para esta mudana contribuı́ram a emancipação das lı́nguas vernáculas relativamente ao latim e a eclosão da imprensa. O tratado insere-se ainda numa época de grande desenvolvimento comercial, com necessidades de modelos, para programar os grandes negócios associados à expansão marı́tima e comercial, utilizando algoritmos apoiados na vulgarização do cálculo com os números indo-árabes. O Tratado da Pratica d’Arismetica aborda ainda temas clássicos, tais como, como as progresses, raı́zes quadradas e cúbicas, os problemas com números, entre outros que, embora desligados do mundo mercantil, assumiram um lugar de destaque no saber matemtico da época. Estes temas estão associados a processos calculatórios, contudo, apresentam esboços de um ideal muito próximo do pensamento matemático. Nesta sessão apresentaremos, de forma sucinta, a obra de Gaspar Nicolas destacando alguns temas e problemas abordados.
Optimization, Graph Theory, and Combinatorics Group
Coordinator: A. Plakhov
The general objectives of OGTC are to contribute for the development of new results in optimization and operations research, mathematical physics (including theory of billiards and inverse problems), spectral graph theory and its applications to combinatorial optimization, combinatorial matrix theory and combinatorics. It is part of these objectives to give advanced training in any of these areas, namely to do supervision of masters and doctoral theses. It is also among the goals of this group to apply the knowledge of its members as well as the obtained results to real world problems and in expected R& D projects as partnerships with industrial companies and other institutions.
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Title: On the 4th Moore graph with diameter 2
Speaker: Domingos M. Cardoso
Abstract: The Moore graphs had its origin on a article by Hoffman and Singleton published in 1960. According to the authors, the problem of finding the largest order n ∆,D of a graph with maximum degree ∆ and diameter not greater than D was first proposed by Moore. In the same paper they introduced the so called Moore’s upper bound for n ∆,D and designated by Moore graphs the nontrivial simple connected graphs for which this upper bound is attained. As it is well known, a More Graph which is neither a complete graph nor a cycle of order 2D + 1 > 5 has D = 2 and ∆ ∈ {2, 3, 7}, and possibly ∆ = 57. The Moore (∆, D)-graphs, with D = 2, are the strongly regular graphs C 5 , G 1 , the Petersen graph, G 2 , the Hoffman-Singleton graph, G 3 , and possibly the strongly regular graph with parameters (3250, 57, 0, 1), G 4 , herein called the 4th Moore graph with diameter 2. The existence of G 4 is an open problem with 60 years and became one of the biggest challenges in Graph Theory. From the literature, it is known that if G 4 there exists, then its stability number is not greater than 400.
In our research, assuming that G 4 there exists, we have proved that every maximum stable set of G ∈ {G j , j = 2, 3, 4} is a
(0, τ )-regular set, with τ = 2 for G 2 , τ = 3 for G 3 and τ = 8 for G 4 . As a consequence, the stability number of G 4 is equal to 400. Moreover, we have deduced a necessary and sufficient condition for the existence of Moore graphs of order n, with diameter 2 and a (0, τ )-regular set. Using this necessary and sufficient condition, an algorithm for the construction of the Moore graphs with diameter 2 and valency d ∈ {3, 7, 57} was proposed. However, so far, our algorithmic implementations are computationally effective only for G 2 and G 3 .
Joint work with António Pereira.
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Title: Graphs with convex quadratic stability number - results and applications
Speaker: Maria de Fátima Pacheco
Abstract: Determining the stability number of an arbitrary graph - which is the cardinality of a maximum stable set - is a NP-complete problem and, therefore, the possibility of find- ing a polynomial algorithm for the determination of a maximum stable set is highly unlikely. A graph has convex quadratic stability number if its stability number is determined by solving a convex quadratic program associated to the corresponding adjacency matrix. The aim of this talk is to present recognition procedures for graphs with convex quadratic stability number, applying results that relate the eigenvalues of the adjacency matrix and maximum stable sets. The application of such procedures to particular cases have wide implications and new approaches to well known problems such as efficient domination, the determination of graphs with perfect matchings and Hamiltonian cycles will also be described.
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Title: Maritime Inventory Routing Problem
Speaker: Filipe Rodrigues
Abstract: In this talk we present a general overview of a very important problem arising in many real-world situations: The Maritime Inventory Routing Problem (MIRP). We start by describing the problem and show its importance for the industry. Then we explain why the MIRP is naturally an optimization problem subject to uncertainty and mention some of the approaches that can be used to solve it. Moreover, we explain how such approaches can be applied and the main differences between them. A set of real instances is used to show the main benefits and drawbacks of the approaches tested.
Probabilities and Statistics Group
Coordinator: I. Pereira
The group of Probability and Statistics has two main objectives: to develop fundamental research on Probability and Mathematical Statistics, and to use the concepts and the methodologies from those fields in applied research programs, like technological, clinical, biological or social scientific projects.
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Title: Normalized entropy: a powerful information measure in regression analysis
Speaker: Pedro Macedo
Abstract: Info-metrics is a research area at the intersection of statistics, computer science and decision theory, where the maximum entropy principle established by Edwin Jaynes plays a fundamental role. Since normalized entropy can be used to measure the information content of the signal component in regression models, recent developments with normalized entropy, namely a possible solution for the Freedman’s paradox and a proposed aggregation technique for large-scale data, are discussed in this talk. Results from simulation studies suggest that normalized entropy is a powerful information measure in regression analysis.
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Title: Approximate Bayesian Computation for parameter estimation in HIV dynamic models
Speaker: Diana Rocha
Abstract: The clinical follow-up of an HIV patient allows the development of a personal treatment plan, where the status of the patient can be evaluated from the viral load (VL) values and theCD4 + T cells count along time. This work aims at contributing to the characterization of the patient follow-up from a mathematical model that describes the VL and CD4 + T temporal dynamics (formulated as a system of nonlinear ordinary differential equations) whose parameters are estimated from the VL and CD4 + T set of values observed for the patient. Model parameters are usually estimated from simulation-based approaches as the Markov Chain Monte Carlo (MCMC) despite being computationally demanding. More recently, Approximate Bayesian Computation-based (ABC) approaches became promising alternatives to overcome the MCMC computational drawback. In this work, ABC-based approaches are further explored aiming to improve the estimation process while increasing its computational efficiency. The methods are compared through simulated data that mimics VL and CD4 + T temporal trajectories of HIV patients.
This is a joint work with Sónia Gouveia (IEETA and CIDMA), Manuel Scotto (CEMAT-UL), Carla Pinto (CMUP and ISEP) and João Nuno Tavares (CMUP).
Systems and Control Group
Coordinator: D. F. M. Torres
The main goal of the group is to carry out research in theoretical and practical aspects in the area of systems and control. The group specific areas of interest are: behavioral systems and convolutional codes, adaptive control and control of drug administration, calculus of variations, optimal control, fractional calculus, time scales, and optimal control applied to epidemiological models.
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Title: A dynamical system approach for some models arising in biology
Speaker: Cristina Januário
Abstract: The complexity of biological models makes its understanding sometimes a challenge. The conceptual richness and applicability of the theory of dynamical systems, allows us to obtain some insights about the models dynamical behavior. In this talk we provide examples of this approach.
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Title: Rank metric convolutional codes
Speaker: Filipa Santana
Abstract: Rank metric convolutional codes are a generalization of the linear rank metric codes and they are better suited to be used in multi-shot network coding. In this talk we present these codes. In particular, we consider the sum rank distance and the column rank distance since they measure the error correction capability of these codes and we introduce constructions of rank metric convolutional codes with optimal sum rank distance and optimal column rank distance.
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Title: Decoding of 2D convolutional codes over an erasure channel
Speaker: Rita Simões
Abstract: In this talk we address the problem of decoding 2D convolutional codes over an erasure channel. To this end we introduce the notion of neighbours around a set of erasures which can be considered an analogue of the notion of sliding window in the context of 1D convolutional codes. The main idea is to reduce the decoding problem of 2D convolutional codes to a problem of decoding a set of associated 1D convolutional codes. We first show how to recover sets of erasures that are distributed on vertical, horizontal and diagonal lines. Finally we outline some ideas to treat any set of erasures distributed randomly on the 2D plane.
Thematic lines
BioMath
Coordinator: C. J. Silva
The aim of the thematic line is the development of research in mathematical biology, establishing links between different groups of CIDMA. Some of the topics include mathematical epidemiology, biostatistics, molecular biology, epigenetics, hybrid systems and biological network design.
Url: https://sites.google.com/view/ltbiomath/home
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Title: Complex network of epidemic models - case study in Cape Verde
Speaker: Cristiana J. Silva
Abstract: We start by revisiting an HIV/AIDS compartmental model that fits the reality of Cape Verde from 1987 to 2014. Based on the epidemiological model, we propose an original complex network model in an heterogeneous geographical area. The complex network is constructed by coupling nonidentical instances the HIV/AIDS epidemiological model for which a disease-free equilibrium and an endemic equilibrium can coexist. After proving the existence of a positively invariant region for the solutions of the complex network problem, we investigate the effect of the coupling on the dynamics of the network, and establish the existence of a unique disease-free equilibrium for the whole network, which is globally asymptotically stable. We prove the existence of an optimal topology that minimizes the level of infected individuals, and apply the theoretical results to the case of the Cape Verde archipelago.
This talk is based on two joints works with Delfim F. M. Torres (CIDMA) and Guillaume Cantin (University of Le Havre).
GEOMETRIX
Coordinator: A. Breda
The thematic line GEOMETRIX, composed by an interdisciplinary team bringing together researchers,school teachers and university students from different scientific areas, is focused on the “conception and imple-mentation of environments promoting research, interaction and a sense of community enabling well being andformal and informal learning”. Its action is distributed by the following areas of intervention.
1. Creation of real and virtual 3D mathematical models to support the learning of mathematics;
2. Creation of environments promoting social interaction in fragile groups of the contemporary society;
3. Development of serious games;
4. Research on combinatorial geometry.
PICS - Inverse Problems and Applications in Health Sciences
Coordinator: P. Cerejeiras
The thematic line in Inverse Problems in Health Sciences was create in 2015. The line coordinates the common efforts of researchers from different groups on problems related to the field of health sciences. Currently, the line concentrates its efforts on two projects, namely:
Project 1. Diagnostic of Thyroid Cancer
Project 2. Modelation of Ophthalmic Surfaces but is also promoting methods of machine and deep learning.
URL: http://sweet.ua.pt/pceres/LtPICS
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Title: Modeling ophthalmic surfaces using Zernike, Bessel and Chebyshev type functions
Speaker: Manuela Rodrigues
Abstract: In this talk the application of Zernike, Bessel and Chebyshev functions is studied and the results are compared when modeling ophthalmic surfaces in visual optics. The total RMS error is presented when addressing the capability of these functions in fitting with different surfaces. It is shown that Chebyshev polynomials could be appropriate alternatives of the Zernike polynomials to represent complete anterior corneal surfaces.
TFC - From Theory to Computational Frameworks
Coordinator: E. Rocha
The thematic line LT-TFC intends to join together CIDMA members that want to reduce the time gap between the publication of recognized theoretical results and their application to different scientific fields, target communities, or society problems; mainly by researching and developing new Computational Frameworks, supported by their research work.
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Title: Activities of LT-TFC
Speaker: Daniel Figueiredo
Abstract: The thematic line LT-TFC (da Teoria às Ferramentas Computacionais) aims to reduce the time gap between the publication of recognized theoretical results and their application to different scientific fields, target communities, or society problems. In 2019, the main focus was:
1. The development of the tool rPrism (a software for reactive weighted state transition models);
2. A framework for fitting epidemic data into hybrid automata;
3. Participation on the development of a prototype for the efficient and adaptive control of electric water heaters.