Latest Publications


Books


71.  The variable order fractional calculus of variations

Almeida, Ricardo and Tavares, Dina and Torres, Delfim F. M.

Springer

This book intends to deepen the study of the fractional calculus, giving special emphasis to variable-order operators

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70.  Molecular logic and computational synthetic biology: first international symposium, MLCSB 2018, Santiago, Chile, December 17–18, 2018, Revised Selected Papers

Chaves, Madalena and Martins, Manuel A.

Springer

ria.ua.pt | doi

Book Chapters


69.  Non-invasive control of the fractional Hegselmann-Krause type model

Almeida, Ricardo and Malinowska, A.B. and Odzijewicz, T.

Non-Integer Order Calculus and its Applications. Lecture Notes in Electrical Engineering

Springer

In this paper, the fractional order Hegselmann–Krause type model with leadership is studied. We seek an optimal control strategy for the system to reach a consensus in such a way that the control mechanism is included in the leader dynamics. Necessary optimality conditions are obtained by the use of a fractional counterpart of Pontryagin Maximum Principle. The effectiveness of the proposed control strategy is illustrated by numerical examples.

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68.  A survey on fractional variational calculus

Almeida, Ricardo and Torres, Delfim F. M.

Handbook of Fractional Calculus with Applications

De Gruyter

Main results and techniques of the fractional calculus of variations are surveyed. We consider variational problems containing Caputo derivatives and study them using both indirect and direct methods. In particular, we provide necessary optimality conditions of Euler-Lagrange type for the fundamental, higher-order, and isoperimetric problems, and we compute approximate solutions based on truncated Grünwald-Letnikov approximations of the Caputo derivatives.

ria.ua.pt | doi | Peer Reviewed

67.  Analysis of fractional integro-differential equations of thermistor type

Sidi Ammi, Moulay Rchid and Torres, Delfim F. M.

Handbook of Fractional Calculus with Applications. Vol 1: Basic Theory

De Gruyter

We survey methods and results of fractional differential equations in which an unknown function is under the operation of integration and/or differentiation of fractional order. As an illustrative example, we review results as regards fractional integral and differential equations of thermistor type. Several nonlocal problems are considered: problems concerned with Riemann–Liouville, Caputo, and time-scale fractional operators. The existence and uniqueness of positive solutions are obtained through suitable fixed-point theorems in proper Banach spaces. Additionally, existence and continuation theorems are given, ensuring global existence.

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66.  Variable order Mittag–Leffler fractional operators on isolated time scales and application to the calculus of variations

Abdeljawad, Thabet and Mert, Raziye and Torres, Delfim F. M.

Fractional Derivatives with Mittag-Leffler Kernel. Studies in Systems, Decision and Control

Springer

We introduce new fractional operators of variable order in isolated time scales with Mittag–Leffler kernels. This allows a general formulation of a class of fractional variational problems involving variable-order difference operators. Main results give fractional integration by parts formulas and necessary optimality conditions of Euler–Lagrange type.

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65.  Reactive models for biological regulatory networks

Figueiredo, Daniel and Barbosa, Luís Soares

Molecular Logic and Computational Synthetic Biology: First International Symposium, MLCSB 2018, Santiago, Chile, December 17–18, 2018, Revised Selected Papers

Springer

A reactive model, as studied by D. Gabbay and his collaborators, can be regarded as a graph whose set of edges may be altered whenever one of them is crossed. In this paper we show how reactive models can describe biological regulatory networks and compare them to Boolean networks and piecewise-linear models, which are some of the most common kinds of models used nowadays. In particular, we show that, with respect to the identification of steady states, reactive Boolean networks lie between piecewise linear models and the usual, plain Boolean networks. We also show this ability is preserved by a suitable notion of bisimulation, and, therefore, by network minimisation.

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64.  Verification for Everyone? An Overview of Dynamic Logic

Madeira, Alexandre

Molecular Logic and Computational Synthetic Biology. MLCSB 2018. Lecture Notes in Computer Science

Springer

This note, reporting the homonym keynote presented in the International Symposium on Molecular Logic and Computational Synthetic Biology 2018, traces an informal roadmap on Dynamic Logic (DL) field, focusing on its versatility and resilience to be adjusted and adopted in a wide class of application domains and computational paradigms. The exposition argues the room for developments on tagging DL to the analysis of synthetic biologic domain.

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63.  A Hybrid Dynamic Logic for Event/Data-Based Systems

Hennicker, Rolf and Madeira, Alexandre and Knapp, Alexander

Lecture Notes in Computer Science - 22nd International Conference on Fundamental Approaches to Software Engineering, FASE 2019

Springer

We propose E-logic as a formal foundation for the specification and development of event-based systems with local data states. The logic is intended to cover a broad range of abstraction levels from abstract requirements specifications up to constructive specifications. Our logic uses diamond and box modalities over structured actions adopted from dynamic logic. Atomic actions are pairs Open image in new window where e is an event and a state transition predicate 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; specification refinement is defined by model class inclusion. For the presentation of constructive specifications we propose operational event/data specifications allowing for familiar, diagrammatic representations by state transition graphs. We show that E-logic is powerful enough to characterise the semantics of an operational specification by a single E-sentence. Thus the whole development process can rely on E-logic and its semantics as a common basis. This includes also a variety of implementation constructors to support, among others, event refinement and parallel composition.

ria.ua.pt | doi | Peer Reviewed

62.  About the financial efficiency of the Iberian Peninsula education

Murillo, K. and Rocha, E.

INTED2019 Proceedings

IATED

This work examines the financial efficiency of private institutions of secondary and higher education in the Iberian Peninsula, during the period 2013-2016. In particular, we study the evolution of 117 Portuguese institutions and 1885 Spanish institutions. To this end, we analyze the evolution of financial statements of each institution regarding its efficiency at three distinct stages: efficiency levels; efficiency patterns; and efficiency determinants. We use a nonparametric method that allows us to investigate changes in the institutions' efficiency standards, namely a model based on the Multidirectional Efficiency Analysis (MEA) in combination with other mathematical techniques, such that principal component analysis, clustering analysis and accumulated effort. The study allows a parallel between Spain and Portugal, characterizing the behavior of the educational levels after the financial crisis that Europe suffered in mid-2008. Our results indicate which are the levels of education that are more efficient and those that are less efficient, and which improvements could be applied to propose more effective measures according to other European experiences. The results show that Spanish higher education is more efficient than Portuguese one and the same conclusion can be inferred, with a lower degree, for the secondary education, except in 2016. Looking to higher versus secondary education, secondary education presents the biggest resource management inefficiency. Cluster analysis was applied to avoid the disparity of data in Spain. In 2015, Spain cluster 1 presents the greatest difference between the accumulated efforts of the two education levels. In contrast, Spain cluster 2 attains the smallest difference.

ria.ua.pt | doi | Peer Reviewed

61.  Oscillatory behaviour on a non-autonomous hybrid SIR-Model

Rocha, Eugénio

Molecular Logic and Computational Synthetic Biology. MLCSB 2018. Lecture Notes in Computer Science

Springer

We study the impact of some abstract agent intervention on the disease spread modelled by a SIR-model with linear growth infectivity. The intervention is meant to decrease the infectivity, which are activated by a threshold on the number of infected individuals. The coupled model is represented as a nonlinear non-autonomous hybrid system. Stability and reduction results are obtained using the notions of non-autonomous attractors, Bohl exponents, and dichotomy spectrum. Numerical examples are given where the number of infected individuals can oscillate around a equilibrium point or be a succession of bump functions, which are validated with a tool based on the notion of delta-complete decision procedures for solving satisfiability modulo theories problems over the real numbers and bounded delta-reachability. These findings seem to show that hybrid SIR-models are more flexible than standard models and generate a vast set of solution profiles. It also raises questions regarding the possibility of the agent intervention been somehow responsible for the shape and intensity of future outbreaks.

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Articles


60.  Direct transcription methods based on fractional integral approximation formulas for solving nonlinear fractional optimal control problems

Salati, Abubakar Bello and Shamsi, Mostafa and Torres, Delfim F. M.

Communications in Nonlinear Science and Numerical Simulation

Elsevier

This paper presents three direct methods based on Grünwald–Letnikov, trapezoidal and Simpson fractional integral formulas to solve fractional optimal control problems (FOCPs). At first, the fractional integral form of FOCP is considered, then the fractional integral is approximated by Grünwald–Letnikov, trapezoidal and Simpson formulas in a matrix approach. Thereafter, the performance index is approximated either by trapezoidal or Simpson quadrature. As a result, FOCPs are reduced to nonlinear programming problems, which can be solved by many well-developed algorithms. To improve the efficiency of the presented method, the gradient of the objective function and the Jacobian of constraints are prepared in closed forms. It is pointed out that the implementation of the methods is simple and, due to the fact that there is no need to derive necessary conditions, the methods can be simply and quickly used to solve a wide class of FOCPs. The efficiency and reliability of the presented methods are assessed by ample numerical tests involving a free final time with path constraint FOCP, a bang-bang FOCP and an optimal control of a fractional-order HIV-immune system.

ria.ua.pt | doi | Peer Reviewed

59.  Solutions of systems with the Caputo-Fabrizio fractional delta derivative on time scales

Mozyrska, Dorota and Torres, Delfim F. M. and Wyrwas, Malgorzata

Nonlinear Analysis: Hybrid Systems

Elsevier

Caputo-Fabrizio fractional delta derivatives on an arbitrary time scale are presented. When the time scale is chosen to be the set of real numbers, then the Caputo-Fabrizio fractional derivative is recovered. For isolated or partly continuous and partly discrete, i.e., hybrid time scales, one gets new fractional operators. We concentrate on the behavior of solutions to initial value problems with the Caputo-Fabrizio fractional delta derivative on an arbitrary time scale. In particular, the exponential stability of linear systems is studied. A necessary and sufficient condition for the exponential stability of linear systems with the Caputo-Fabrizio fractional delta derivative on time scales is presented. By considering a suitable fractional dynamic equation and the Laplace transform on time scales, we also propose a proper definition of Caputo-Fabrizio fractional integral on time scales. Finally, by using the Banach fixed point theorem, we prove existence and uniqueness of solution to a nonlinear initial value problem with the Caputo-Fabrizio fractional delta derivative on time scales.

ria.ua.pt | doi | Peer Reviewed

58.  New bounds for the signless Laplacian spread

Andrade, Enide and Dahl, Geir and Leal, Laura and Robbiano, María

Linear Algebra and its Applications

Elsevier

Let $G$ be an undirected simple graph. The signless Laplacian spread of $G$ is defined as the maximum distance of pairs of its signless Laplacian eigenvalues. This paper establishes some new bounds, both lower and upper, for the signless Laplacian spread. Several of these bounds depend on invariant parameters of the graph. We also use a minmax principle to find several lower bounds for this spectral invariant.

ria.ua.pt | doi | Peer Reviewed

57.  Combinatorial Perron parameters for trees

Andrade, Enide and Ciardo, Lorenzo and Dahl, Geir

Linear Algebra and its Applications

Elsevier

The notion of combinatorial Perron value was introduced in [2]. We continue the study of this parameter and also introduce a new parameter πe(M) which gives a new lower bound on the spectral radius of the bottleneck matrix M of a rooted tree. We prove a bound on the approximation error for πe(M). Several properties of these two parameters are shown. These ideas are motivated by the concept of algebraic connectivity. A certain extension property for the combinatorial Perron value is shown and it allows us to define a new center concept for caterpillars. We also compare computationally this new center to the so-called characteristic set, i.e., the center obtained from algebraic connectivity.

ria.ua.pt | doi | Peer Reviewed

56.  A recurrent nova super-remnant in the Andromeda galaxy

Darnley, M. J. and Hounsell, R. and O’Brien, T. J. and Henze, M. and Rodríguez-Gil, P. and Shafter, A. W. and Shara, M. M. and Vaytet, N. M. H. and Bode, M. F. and Ciardullo, R. and Davis, B. D. and Galera-Rosillo, R. and Harman, D. J. and Harvey, E. J. and Healy, M. W. and Ness, J.-U. and Ribeiro, V. A. R. M. and Williams, S. C.

Nature

Nature Research

The accretion of hydrogen onto a white dwarf star ignites a classical nova eruption 1,2 —a thermonuclear runaway in the accumulated envelope of gas, leading to luminosities up to a million times that of the Sun and a high-velocity mass ejection that produces a remnant shell (mainly consisting of insterstellar medium). Close to the upper mass limit of a white dwarf 3 (1.4 solar masses), rapid accretion of hydrogen (about 10 −7 solar masses per year) from a stellar companion leads to frequent eruptions on timescales of years 4,5 to decades 6 . Such binary systems are known as recurrent novae. The ejecta of recurrent novae, initially moving at velocities of up to 10,000 kilometres per second 7 , must ‘sweep up’ the surrounding interstellar medium, creating cavities in space around the nova binary. No remnant larger than one parsec across from any single classical or recurrent nova eruption is known 8–10 , but thousands of successive recurrent nova eruptions should be capable of generating shells hundreds of parsecs across. Here we report that the most frequently recurring nova, M31N 2008-12a in the Andromeda galaxy (Messier 31 or NGC 224), which erupts annually 11 , is indeed surrounded by such a super-remnant with a projected size of at least 134 by 90 parsecs. Larger than almost all known remnants of even supernova explosions 12 , the existence of this shell demonstrates that the nova M31N 2008-12a has erupted with high frequency for millions of years.

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55.  Algorithmic determination of immobile indices in convex SIP problems with polyhedral index sets

Kostyukova, O.I. and Tchemisova, T.V.

INFOR, Information Systems and Operational Research

Taylor & Francis

The concepts of immobile indices and their immobility orders are objective and important characteristics of feasible sets of semi-infinite programming (SIP) problems. They can be used for the formulation of new efficient optimality conditions without constraint qualifications. Given a class of convex SIP problems with polyhedral index sets, we describe and justify a finite constructive algorithm (algorithm DIIPS) that allows to find in a finite number of steps all immobile indices and the corresponding immobility orders along the feasible directions. This algorithm is based on a representation of the cones of feasible directions in the polyhedral index sets in the form of linear combinations of extremal rays and on the approach proposed in our previous papers for the cases of immobile indices’ sets of simpler structures. A constructive procedure of determination of the extremal rays is described, and an example illustrating the application of the DIIPS algorithm is provided.

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54.  The spread of a financial virus through Europe and beyond

Kostylenko, Olena and Rodrigues, Helena Sofia and Torres, Delfim F. M.

AIMS Mathematics

AIMS Press

We analyse the importance of international relations between countries on the financial stability. The contagion effect in the network is tested by implementing an epidemiological model, comprising a number of European countries and using bilateral data on foreign claims between them. Banking statistics of consolidated foreign claims on ultimate risk bases, obtained from the Banks of International Settlements, allow us to measure the exposure of contagion spreading from a particular country to the other national banking systems. We show that the financial system of some countries, experiencing the debt crisis, is a source of global systemic risk because they threaten the stability of a larger system, being a global threat to the intoxication of the world economy and resulting in what we call a `financial virus'. Illustrative simulations were done in the NetLogo multi-agent programmable modelling environment and in MATLAB.

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53.  Partial parking functions

Duarte, Rui and Guedes de Oliveira, António

Discrete Mathematics

Elsevier

We characterise the Pak-Stanley labels of the regions of a family of hyperplane arrangements that interpolate between the Shi arrangement and the Ish arrangement.

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52.  A Dawson-like clustering of human mitochondrial DNA sequences based on protein coding region

Soares, Inês and Duarte, Rui and Guedes de Oliveira, António and Amorim, António

Discrete Applied Mathematics

Elsevier

In the present paper, our main goal is focused in developing fast algorithms for human mtDNA sequence analyses, requiring minimum and explicit assumptions on mutation models and evolutionary pathways. We propose a new approach based on a construction of Dawson, a technique based on the ordering of the variable sites. In this approach, the first step corresponds to the computation of the order of the positions according to their capacity to separate the sequences into dichotomous groups. Aiming to avoid or at least to minimize the consideration of ambiguous evolutionary events such as insertions/deletions and recurrence, which cause well-known alignment problems, in the present study we only work with the protein coding sequence, the clearly more stable region in human mitochondrial genomes. This method was tested in a small set of 99 human mtDNA comprising representatives of all major haplogroups. The developed approach showed to be a choice to automate the clustering of human mtDNA sequences into broad groups, the output being in agreement with the canonical classification into macro-haplogroups deposited in the Phylotree database.

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51.  Column distances of convolutional codes over Z_p^r

Napp, Diego and Pinto, Raquel and Toste, Marisa

IEEE Transactions on Information Theory

Institute of Electrical and Electronics Engineers

Maximum distance profile codes over finite nonbinary fields have been introduced and thoroughly studied in the last decade. These codes have the property that their column distances are maximal among all codes of the same rate and degree. In this paper, we aim at studying this fundamental concept in the context of convolutional codes over a finite ring. We extensively use the concept of p-encoder to establish the theoretical framework and derive several bounds on the column distances. In particular, a method for constructing (not necessarily free) maximum distance profile convolutional codes over Zpr is presented.

ria.ua.pt | doi | Peer Reviewed

50.  Existence of solution to a nonlocal conformable fractional thermistor problem

Moulay Rchid Sidi Ammi and Torres, Delfim F. M.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics

Faculty of Sciences University of Ankara

We study a nonlocal thermistor problem for fractional derivatives in the conformable sense. Classical Schauder's fixed point theorem is used to derive the existence of a tube solution.

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49.  A space-time pseudospectral discretization method for solving diffusion optimal control problems with two-sided fractional derivatives

Mushtaq Salh Ali and Mostafa Shamsi and Hassan Khosravian-Arab and Torres, Delfim F. M. and Farid Bozorgnia

Journal of Vibration and Control

SAGE Publications

We propose a direct numerical method for the solution of an optimal control problem governed by a two-side space-fractional diffusion equation. The presented method contains two main steps. In the first step, the space variable is discretized by using the Jacobi–Gauss pseudospectral discretization and, in this way, the original problem is transformed into a classical integer–order optimal control problem. The main challenge, which we faced in this step, is to derive the left and right fractional differentiation matrices. In this respect, novel techniques for derivation of these matrices are presented. In the second step, the Legendre–Gauss–Radau pseudospectral method is employed. With these two steps, the original problem is converted into a convex quadratic optimization problem, which can be solved efficiently by available methods. Our approach can be easily implemented and extended to cover fractional optimal control problems with state constraints. Five test examples are provided to demonstrate the efficiency and validity of the presented method. The results show that our method reaches the solutions with good accuracy and a low central processing unit time.

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48.  Fractional order version of the HJB equation

Razminia, Abolhassan and AsadiZadehShiraz, Mehdi and Torres, Delfim F. M.

Journal of Computational and Nonlinear Dynamics

American Society of Mechanical Engineers

We consider an extension of the well-known Hamilton-Jacobi-Bellman (HJB) equation for fractional order dynamical systems in which a generalized performance index is considered for the related optimal control problem. Owing to the nonlocality of the fractional order operators, the classical HJB equation, in the usual form, does not hold true for fractional problems. Effectiveness of the proposed technique is illustrated through a numerical example.

ria.ua.pt | doi | Peer Reviewed

47.  Structural derivatives on time scales

Bayour, Benaoumeur and Torres, Delfim F. M.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics

Faculty of Sciences University of Ankara

We introduce the notion of structural derivative on time scales. The new operator of differentiation unifies the concepts of fractal and fractional order derivative and is motivated by lack of classical differentiability of some self-similar functions. Some properties of the new operator are proved and illustrated with examples.

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46.  Functional characterizations of trace spaces in Lipschitz domains

Touhami, Soumia and Chaira, Abdellatif and Torres, Delfim F. M.

Banach Journal of Mathematical Analysis

Duke University Press

Using a factorization theorem of Douglas, we prove functional characterizations of trace spaces Hs (∂Ω) involving a family of positive selfadjoint operators. Our method is based on the use of a suitable operator by taking the trace on the boundary ∂Ω of a bounded Lipschitz domain Ω ⊂ R d and applying Moore–Penrose pseudoinverse properties together with a special inner product on H1 (Ω). We also establish generalized results of the Moore– Penrose pseudoinverse.

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45.  Exact solution to a dynamic SIR model

Bohner, Martin and Streipert, Sabrina and Torres, Delfim F. M.

Nonlinear Analysis: Hybrid Systems

Elsevier

We investigate an epidemic model based on Bailey's continuous differential system. In the continuous time domain, we extend the classical model to time-dependent coefficients and present an alternative solution method to Gleissner's approach. If the coefficients are constant, both solution methods yield the same result. After a brief introduction to time scales, we formulate the SIR (susceptible–infected–removed) model in the general time domain and derive its solution. In the discrete case, this provides the solution to a new discrete epidemic system, which exhibits the same behavior as the continuous model. The last part is dedicated to the analysis of the limiting behavior of susceptible, infected, and removed, which contains biological relevance.

ria.ua.pt | doi | Peer Reviewed

44.  Efficiency analysis of optimal inspection management for reinforced concrete structures under carbonation-induced corrosion risk

Benítez, Pablo and Rocha, Eugénio and Talukdar, Sudip and Varum, Humberto and Rodrigues, Fernanda

Construction and Building Materials

Elsevier

Reinforced concrete is one of the most predominant materials used in constructed structures and infrastructures throughout the world. For this reason, the maintenance management of these structures is a frequent subject of study in material science and civil engineering. Thereby, several studies conclude that corrosion of reinforcement in concrete structures may become an expensive degradation mechanism whether a suitable intervention strategy throughout its service life is not applied. In this context, maintenance management is crucial to slow down the degradation process and to extend the lifespan, which can comprise two stages: inspection planning, and an assessment of repair probabilities according to the inspection results. The paper herein presented focuses on a decision-making model for the optimal inspection planning for concrete structures subject to carbonation-induced corrosion. A structured approach to optimise inspection times considering the uncertainty inherent in the corrosion-induced degradation and a trade-off between inspection costs and the structure serviceability is performed.The main objective of this work is to elaborate an efficiency analysis for a set of optimal inspection strategies that is conducted through two approaches: the Stochastic Frontier Analysis (SFA) and the Multi-direction Enveloping Analysis (MEA). This analysis provides an appraisal that reduces the number of inspection techniques necessary and the time gap between inspections to provide not only an optimal solution but the most efficient as well. Therefore, the outcome of this paper provides an inspection strategy that establishes the optimal solution for both the inspection times and the most suitable inspection technique to be applied considering the efficiency of the solution.

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43.  A sufficient optimality condition for delayed state-linear optimal control problems

Lemos-Paião, Ana Pedro and Silva, Cristiana J. and Torres, Delfim F. M.

Discrete and Continuous Dynamical Systems - Series B

American Institute of Mathematical Sciences

We give answer to an open question by proving a sufficient optimality condition for state-linear optimal control problems with time delays in state and control variables. In the proof of our main result, we transform a delayed state-linear optimal control problem to an equivalent non-delayed problem. This allows us to use a well-known theorem that ensures a sufficient optimality condition for non-delayed state-linear optimal control problems. An example is given in order to illustrate the obtained result.

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42.  On interval dynamic logic: Introducing quasi-action lattices

Santiago, Regivan and Bedregal, Benjamín and Madeira, Alexandre and Martins, Manuel A.

Science of Computer Programming

Elsevier

In this paper we discuss the incompatibility between the notions of validity and impreciseness in the context of Dynamic Logics. To achieve that we consider the Łukasiewicz action lattice and its interval counterpart, we show how some validities fail in the context of intervals. In order to capture the properties of action lattices that remain valid for intervals we propose a new structure called Quasi-action Lattices which generalizes action lattices and is able to model both: The Łukasiewicz action lattice, Ł, and its interval counterpart, Łˆ. The notion of graded satisfaction relation is extended to quasi-action lattices. We demonstrate that, in the case of intervals, the relation of graded satisfaction is correct (cf. Theorem 3) with respect to the graded satisfaction relation on the Łukasiewicz action lattice. Although this theorem guarantees that satisfiability is preserved on intervals, we show that validity is not. We propose, then, to weaken the notion of validity on action lattices to designated validity on quasi-action lattices. In this context, Theorem 4 guarantees that the dynamic formulæ which are valid with respect to Ł will be designated valid with respect to Łˆ

ria.ua.pt | doi | Peer Reviewed

41.  Exploring symmetry in rosettes of Truchet tiles

Hall, Andreia and Almeida, Paulo J. and Teixeira, Ricardo

Journal of Mathematics and the Arts

Taylor & Francis

In 1704 the French priest S ebastien Truchet published a paper where he explored and counted patterns made up from a square divided by a diagonal line into two colored parts,, now known as a Truchet tile. A few years later, Father Dominique Douat continued Truchet's work and published a book in 1722 containing many more patterns and further counts of con gurations. In this paper, we extend the work introduced by Truchet and Douat by considering all possible rosettes made up of an mXn array of square or non-square Truchet tiles. We then classify the rosettes according to their symmetry group and count all the distinct rosettes in each group, for all possible sizes. The results are summarized in a separate section where we further analyze the asymptotic behavior of the counts for square arrays. Finally, some applications are shown using two types of square flexagons.

ria.ua.pt | doi | Peer Reviewed

40.  On Herbrand’s Theorem for Hybrid Logic

Costa, Diana and Martins, Manuel and Marcos, João

Journal of Applied Logics – IfCoLog Journal of Logics and their Applications

Colege Publications

The original version of Herbrand’s theorem [8] for first-order logic provided the theoretical underpinning for automated theorem proving, by allowing a constructive method for associating with each first-order formula χ a sequence of quantifier-free formulas χ1, χ2, χ3, · · · so that χ has a first-order proof if and only if some χi is a tautology. Some other versions of Herbrand’s theorem have been developed for classical logic, such as the one in [6], which states that a set of quantifier-free sentences is satisfiable if and only if it is propositionally satisfiable. The literature concerning versions of Herbrand’s theorem proved in the context of non-classical logics is meager. We aim to investigate in this paper two versions of Herbrand’s theorem for hybrid logic, which is an extension of modal logic that is expressive enough so as to allow identifying specific sates of the corresponding models, as well as describing the accessibility relation that connects these states, thus being completely suitable to deal with relational structures [3]. Our main results state that a set of satisfaction statements is satisfiable in a hybrid interpretation if and only if it is propositionally satisfiable

ria.ua.pt | Peer Reviewed

39.  The number of parking functions with center of a given length

Duarte, Rui and Guedes de Oliveira, António

Advances in Applied Mathematics

Elsevier

Let 1≤r≤n and suppose that, when the Depth-first Search Algorithm is applied to a given rooted labeled tree on n+1 vertices, exactly r vertices are visited before backtracking. Let R be the set of trees with this property. We count the number of elements of R. For this purpose, we first consider a bijection, due to Perkinson, Yang and Yu, that maps R onto the set of parking function with center (defined by the authors in a previous article) of size r. A second bijection maps this set onto the set of parking functions with run r, a property that we introduce here. We then prove that the number of length n parking functions with a given run is the number of length n rook words (defined by Leven, Rhoades and Wilson) with the same run. This is done by counting related lattice paths in a ladder-shaped region. We finally count the number of length n rook words with run r, which is the answer to our initial question.

ria.ua.pt | doi | Peer Reviewed

38.  The orthogonality of the fractional circle polynomials and its application in modeling of ophthalmic surfaces

Rodrigues, Maria Manuela Fernandes and Vieira, Nelson Felipe

Journal of Physics: Conference Series

IOP Publishing

In this paper we establish some new fractional differential properties for a class of fractional circle polynomials. We apply the Zernike polynomials and a new class of fractional circle polynomials in modeling ophthalmic surfaces in visual optics and we compare the obtained results. The total RMS error is presented when addressing capability of these functions in fitting with surfaces, and it is showed that the new fractional circle polynomials can be used as an alternative to the Zernike Polynomials to represent the complete anterior corneal surface.

ria.ua.pt | doi | Peer Reviewed

37.  Modeling ophthalmic surfaces using Zernike, Bessel and Chebyshev type functions

Rodrigues, Maria Manuela Fernandes and Rosa, Andreia and Vieira, Nelson Felipe Loureiro and Murta, Joaquim

Journal of Physics: Conference Series

IOP Publishing

The visual system of the human eye is a part of the central nervous system by which the human body sees and interprets the information provided by the visible light in order to build a representation of the world around. During the propagations of the light through the eye, the retinal image can be deteriorated by diseases and disorders. For retinal images, the most important sources of images quality degradation are diffraction and optical aberrations. In order to measure and correct aberrations, there is a number of surfaces related to the anatomy and physiology of the eye. It is important to measure and mathematically model these surfaces to study their properties.

ria.ua.pt | doi | Peer Reviewed

36.  Famílias e mobilidade interna: uma análise com base nos dados do recenseamento de 2011

Gomes, Maria Cristina Sousa and Freitas, Adelaide and Pinto, Maria Luís Rocha

Análise Social

Universidade de Lisboa, Instituto de Ciências Sociais

Não é frequente considerar-se no âmbito das migrações internas a mobilidade familiar. No entanto, os dados mostram que há características dos municípios que determinam como as famílias se movimentam e se distribuem. A análise de associações entre características populacionais e familiares e variações de fluxos migratórios dos 308 municípios, desenvolvida com base na informação do recenseamento de 2011, pretende refletir sobre a mobilidade familiar em Portugal, salientando aspetos diferenciadores. As associações encontradas, sob duas escalas territoriais, permitem ressaltar dinâmicas territoriais distintas no que respeita aos fenómenos de atração e repulsão associadas à deslocação ou permanência da população nos territórios.

ria.ua.pt | doi | Peer Reviewed

35.  Invisibility in billiards is impossible in an infinite number of directions

Plakhov, Alexander and Roshchina, Vera

Journal of Dynamical and Control Systems

Springer

We show that the maximum number of directions of invisibility in a planar billiard defined in the exterior of a piecewise smooth body is at most finite.

ria.ua.pt | doi | Peer Reviewed

34.  Local properties of the surface measure of convex bodies

Plakhov, Alexander

Journal of Convex Analysis

Heldermann Verlag

It is well known that any measure in S^2 satisfying certain simple conditions is the surface measure of a bounded convex body in R^3. It is also known that a local perturbation of the surface measure may lead to a nonlocal perturbation of the corresponding convex body. We prove that, under mild conditions on the convex body, there are families of perturbations of its surface measure forming line segments, with the original measure at the midpoint, corresponding to {it local} perturbations of the body. Moreover, there is, in a sense, a huge amount of such families. We apply this result to Newton's problem of minimal resistance for convex bodies.

ria.ua.pt | Peer Reviewed

33.  On duals and parity-checks of convolutional codes over Z p r

El Oued, Mohamed and Napp, Diego and Pinto, Raquel and Toste, Marisa

Finite fields and their applications

Elsevier

A convolutional code C over Z_{p^r}((D)) is a Z_{p^r}((D))-submodule of Z_{p^r}^n((D)) that admits a polynomial set of generators, where Z_{p^r}((D)) stands for the ring of (semi-infinity) Laurent series. In this paper we study several structural properties of its dual C^{perp} . We use these results to provide a constructive algorithm to build an explicit generator matrix of C^{perp}. Moreover, we show that the transpose of such a matrix is a parity-check matrix (also called syndrome former) of C.

ria.ua.pt | doi | Peer Reviewed

32.  Warm little inflaton becomes cold dark matter

Rosa, João Pedro Trancoso Gomes and Ventura, Luís Bastos

Physical Review Letters

American Physical Society

We present a model where the inflaton can naturally account for all the dark matter in the Universe within the warm inflation paradigm. In particular, we show that the symmetries of the Warm Little Inflaton scenario (i) avoid large thermal and radiative corrections to the scalar potential, (ii) allow for sufficiently strong dissipative effects to sustain a radiation bath during inflation that becomes dominant at the end of the slow-roll regime, and (iii) enable a stable inflaton remnant in the post-inflationary epochs. The latter behaves as dark radiation until parametrically before matter-radiation equality, leading to a non-negligible contribution to the effective number of relativistic degrees of freedom during nucleosynthesis, becoming the dominant cold dark matter component in the Universe for inflaton masses in the $10^{-4}-10^{-1}$ eV range. Cold dark matter isocurvature perturbations, anti-correlated with the main adiabatic component, provide a smoking gun for this scenario that can be tested in the near future.

ria.ua.pt | doi | Peer Reviewed

31.  Boris Rufimovich Vainberg

Lakshtanov, Evgeny and Egorov, Yu. V. and A. I. Komech and Kuchment, P. A. and Maz'ya, V. G. and Molchanov, S. A. and Novikov, R. G. and Freidlin, M. I.

Russian Mathematical Surveys

Turpion

Boris R. Vainberg was born on March 17, 1938, in Moscow. His father was a Lead Engineer in an aviation design institute. His mother was a homemaker. From early age, Boris was attracted to mathematics and spent much of his time at home and in school working through collections of practice problems for the Moscow Mathematical Olympiad. His first mathematical library consisted of the books he received as one of the prize-winners of these olympiads.

ria.ua.pt | doi | Peer Reviewed

30.  Scale hierarchies, symmetry breaking and particle spectra in SU(3)-family extended SUSY trinification

Camargo-Molina, José E. and Morais, António P. and Ordell, Astrid and Pasechnik, Roman and Wessén, Jonas

Physical Review D

American Physical Society

A unification of left-right $rm{SU}(3)_rm{L}times rm{SU}(3)_rm{R}$, colour $rm{SU}(3)_rm{C}$ and family $rm{SU}(3)_rm{F}$ symmetries in a maximal rank-8 subgroup of ${rm{E}}_8$ is proposed as a landmark for future explorations beyond the Standard Model (SM). We discuss the implications of this scheme in a supersymmetric (SUSY) model based on the trinification gauge $left[rm{SU}(3)right]^3$ and global $rm{SU}(3)_rm{F}$ family symmetries. Among the key properties of this model are the unification of SM Higgs and lepton sectors, a common Yukawa coupling for chiral fermions, the absence of the $mu$-problem, gauge couplings unification and proton stability to all orders in perturbation theory. The minimal field content consistent with a SM-like effective theory at low energies is composed of one $mathrm{E}_6$ $27$-plet per generation as well as three gauge and one family $rm{SU}(3)$ octets inspired by the fundamental sector of ${rm{E}}_8$. The details of the corresponding (SUSY and gauge) symmetry breaking scheme, multi-scale gauge couplings' evolution, and resulting effective low-energy scenarios are discussed.

ria.ua.pt | doi | Peer Reviewed

29.  Designing lasing and perfectly absorbing potentials

Lakshtanov, Evgeny and Vainberg, Boris and Konotop, Vladimir

Physical Review A

American Physical Society

Existence of a spectral singularity (SS) in the spectrum of a Schrödinger operator with a non-Hermitian potential requires exact matching of parameters of the potential. We provide a necessary and sufficient condition for a potential to have a SS at a given wavelength. It is shown that potentials with SSs at prescribed wavelengths can be obtained by a simple and effective procedure. In particular, the developed approach allows one to obtain potentials with several SSs and with SSs of the second order, as well as potentials obeying a given symmetry, say, PT -symmetric potentials. We illustrate all these opportunities with examples. We also describe splitting of a second-order SS under change of the potential parameters, and discuss possibilities of experimental observation of SSs of different orders.

ria.ua.pt | doi | Peer Reviewed

28.  ν-inflaton dark matter

Manso, Antonio Torres and Trancoso, João Pedro

Journal of High Energy Physics

Springer

We present a unified model where the same scalar field can drive inflation and account for the present dark matter abundance. This scenario is based on the incomplete decay of the inflaton field into right-handed neutrino pairs, which is accomplished by imposing a discrete interchange symmetry on the inflaton and on two of the right-handed neutrinos. We show that this can lead to a successful reheating of the Universe after inflation, while leaving a stable inflaton remnant at late times. This remnant may be in the form of WIMP-like inflaton particles or of an oscillating inflaton condensate, depending on whether or not the latter evaporates and reaches thermal equilibrium with the cosmic plasma. We further show that this scenario is compatible with generating light neutrino masses and mixings through the seesaw mechanism, predicting at least one massless neutrino, and also the observed baryon asymmetry via thermal leptogenesis.

ria.ua.pt | doi | Peer Reviewed

27.  The scalarised Schwarzschild-NUT spacetime

Brihaye, Yves and Herdeiro, Carlos and Radu, Eugen

Physics Letters B

Elsevier

t has recently been suggested that vacuum black holes of General Relativity (GR) can become spontaneously scalarised when appropriate non-minimal couplings to curvature invariants are considered. These models circumvent the standard black hole no scalar hair theorems of GR, allowing both the standard GR solutions and new scalarised (a.k.a. hairy) solutions, which in some cases are thermodynamically preferred. Up to now, however, only (static and spherically symmetric) scalarised Schwarzschild solutions have been considered. It would be desirable to take into account the effect of rotation; however, the higher curvature invariants introduce a considerable challenge in obtaining the corresponding scalarised rotating black holes. As a toy model for rotation, we present here the scalarised generalisation of the Schwarzschild-NUT solution, taking either the Gauss–Bonnet (GB) or the Chern–Simons (CS) curvature invariant. The NUT charge n endows spacetime with “rotation” but the angular dependence of the corresponding scalarised solutions factorises, leading to a considerable technical simplification. For GB, but not for CS, scalarisation occurs for n=0. This basic difference leads to a distinct space of solutions in the CS case, in particular exhibiting a double branch structure. In the GB case, increasing the horizon area demands a stronger non-minimal coupling for scalarisation; in the CS case, due to the double branch structure, both this and the opposite trend are found. We briefly comment also on the scalarised Reissner–Nordström-NUT solutions.

ria.ua.pt | doi | Peer Reviewed

26.  Head-on collisions and orbital mergers of Proca stars

Sanchis-Gual, Nicolas and Herdeiro, Carlos and Font, José A. and Radu, Eugen and Di Giovanni, Fabrizio

Physical Review D

American Physical Society

Proca stars, aka vector boson stars, are self-gravitating Bose-Einstein condensates obtained as numerical stationary solutions of the Einstein-(complex)-Proca system. These solitonic objects can achieve a compactness comparable to that of black holes, thus yielding an example of a black hole mimicker, which, moreover, can be both stable and form dynamically from generic initial data by the mechanism of gravitational cooling. In this paper we further explore the dynamical properties of these solitonic objects by performing both head-on collisions and orbital mergers of equal mass Proca stars, using fully nonlinear numerical evolutions. For the head-on collisions, we show that the end point and the gravitational waveform from these collisions depends on the compactness of the Proca star. Proca stars with sufficiently small compactness collide emitting gravitational radiation and leaving a stable Proca star remnant. But more compact Proca stars collide to form a transient hypermassive Proca star, which ends up decaying into a black hole, albeit temporarily surrounded by Proca quasibound states. The unstable intermediate stage can leave an imprint in the waveform, making it distinct from that of a head-on collision of black holes. The final quasinormal ringing matches that of Schwarzschild black hole, even though small deviations may occur, as a signature of sufficiently nonlinear and long-lived Proca quasibound states. For the orbital mergers, we have considered eccentric orbits and the outcome also depends on the compactness of the stars. For the binaries with the most compact stars, the binary merger forms a Kerr black hole which retains part of the initial orbital angular momentum, being surrounded by a transient Proca field remnant; in cases with lower compactness, the binary merger forms a massive Proca star with angular momentum, but out of equilibrium. As in previous studies of (scalar) boson stars, the angular momentum of such objects appears to converge to zero as a final equilibrium state is approached.

ria.ua.pt | doi | Peer Reviewed

25.  On the topological charge of S O(2) gauged Skyrmions in 2 + 1 and 3 + 1 dimensions

Navarro-Lerida, Francisco and Radu, Eugen and Tchrakian, D. H.

Physics Letters B

Elsevier

The question of the dependence of the topological charge q of a gauged Skyrmion, on the gauge field, is studied quantitatively. Two examples, both gauged with SO(2) are studied and contrasted: i) The O(3) model in 2+1 dimensions, and ii) The O(4) model in 3+1 dimensions. In case i), where the (usual) Chern–Simons (CS) term is present, the value of q changes sign, going through zero. This evolution is tracked by a parameter characterising the solutions in the given theory. In case ii), in which dimensions no CS density is available, the evolution of q is not observed.

ria.ua.pt | doi | Peer Reviewed

24.  Compact objects and the swampland

Herdeiro, Carlos A. R. and Radu, Eugen and Uzawa, Kunihito

Journal of High Energy Physics

Springer Verlag

Recently, two simple criteria were proposed to assess if vacua emerging from an effective scalar field theory are part of the string “landscape” or “swampland”. The former are the vacua that emerge from string compactifications; the latter are not obtained by any such compactification and hence may not survive in a UV completed theory of gravity. So far, these criteria have been applied to inflationary and dark energy models. Here we consider them in the context of solitonic compact objects made up of scalar fields: boson stars. Analysing several models (static, rotating, with and without self-interactions), we find that, in this context, the criteria are not independent. Furthermore, we find the universal behaviour that in the region wherein the boson stars are expected to be perturbatively stable, the compact objects may be part of the landscape. By contrast, in the region where they may be faithful black hole mimickers, in the sense they possess a light ring, the criteria fail (are obeyed) for static (rotating) ultracompact boson stars, which should thus be part of the swampland (landscape). We also consider hairy black holes interpolating between these boson stars and the Kerr solution and establish the part of the domain of existence where the swampland criteria are violated. In interpreting these results one should bear in mind, however, that the swampland criteria are not quantitatively strict.

ria.ua.pt | doi | Peer Reviewed

23.  Magnetized accretion disks around Kerr black holes with scalar hair: Constant angular momentum disks

Gimeno-Soler, Sergio and Font, José A. and Herdeiro, Carlos and Radu, Eugen

Physical Review D

American Physical Society

Testing the true nature of black holes - the no-hair hypothesis - will become increasingly more precise in the next few years as new observational data is collected in both the gravitational-wave channel and the electromagnetic channel. In this paper we consider numerically generated spacetimes of Kerr black holes with synchronized scalar hair and build stationary models of magnetized thick disks (or tori) around them. Our approach assumes that the disks are not self-gravitating, they obey a polytropic equation of state, the distribution of their specific angular momentum is constant, and they are marginally stable, i.e., the disks completely fill their Roche lobe. Moreover, contrary to existing approaches in the literature, our models are thermodynamically relativist, as the specific enthalpy of the fluid can adopt values significantly larger than unity. We study the dependence of the morphology and properties of the accretion tori on the type of black hole considered, from purely Kerr black holes with varying degrees of spin parameter, namely from a Schwarzschild black hole to a nearly extremal Kerr case, to Kerr black holes with scalar hair with different Arnowitt-Deser-Misner mass and horizon angular velocity. Comparisons between the disk properties for both types of black holes are presented. The sequences of magnetized, equilibrium disks around Kerr black holes with scalar hair discussed in this study are morphologically and thermodynamically different than their Kerr black hole counterparts, namely their vertical size is larger, the high-density central region is more extended, and the fluid is more relativistic. Therefore, we expect significant differences to appear when these sequences are used as initial data for numerical relativity codes to investigate their dynamical (nonlinear) stability and used in tandem with ray-tracing codes to obtain synthetic images of black holes (i.e., shadows) in astrophysically relevant situations where the light source is provided by an emitting accretion disk

ria.ua.pt | doi | Peer Reviewed

22.  Gravitating solitons and black holes with synchronised hair in the four dimensional O(3) sigma-model

Herdeiro, C. and Perapechka, I. and Radu, E. and Shnir, Ya

Journal of High Energy Physics

Springer Verlag

We consider the O(3) non-linear sigma-model, composed of three real scalar fields with a standard kinetic term and with a symmetry breaking potential in four space-time dimensions. We show that this simple, geometrically motivated model, admits both self-gravitating, asymptotically flat, non-topological solitons and hairy black holes, when minimally coupled to Einstein’s gravity, without the need to introduce higher order kinetic terms in the scalar fields action. Both spherically symmetric and spinning, axially symmetric solutions are studied. The solutions are obtained under a ansatz with oscillation (in the static case) or rotation (in the spinning case) in the internal space. Thus, there is symmetry non-inheritance: the matter sector is not invariant under the individual spacetime isometries. For the hairy black holes, which are necessarily spinning, the internal rotation (isorotation) must be synchronous with the rotational angular velocity of the event horizon. We explore the domain of existence of the solutions and some of their physical properties, that resemble closely those of (mini) boson stars and Kerr black holes with synchronised scalar hair in Einstein-(massive, complex)-Klein-Gordon theory

ria.ua.pt | doi | Peer Reviewed

21.  Black hole scalarisation from the breakdown of scale-invariance

Herdeiro, Carlos A. R. and Radu, Eugen

Physical Review D

American Physical Society

Electro-vacuum black holes are scale-invariant; their energy-momentum tensor is traceless. Quantum corrections of various sorts, however, can often produce a trace anomaly and a breakdown of scale-invariance. The (quantum-corrected) black hole solutions of the corresponding gravitational effective field theory (EFT) have a non-vanishing Ricci scalar. Then, the presence of a scalar field with the standard non-minimal coupling $xi phi^2 R$ naturally triggers a spontaneous scalarisation of the corresponding black holes. This scalarisation phenomenon occurs for an (infinite) discrete set of $xi$. We illustrate the occurrence of this phenomenon for two examples of static, spherically symmetric, asymptotically flat black hole solution of EFTs. In one example the trace anomaly comes from the matter sector -- a novel, closed form, generalisation of the Reissner-Nordstr"om solution with an $F^4$ correction -- whereas in the other example it comes from the geometry sector -- a noncommutative geometry generalization of the Schwarzschild black hole. For comparison, we also consider the scalarisation of a black hole surrounded by (non-conformally invariant) classical matter (Einstein-Maxwell-dilaton black holes). We find that the scalarised solutions are, generically, entropically favoured.

ria.ua.pt | doi | Peer Reviewed

20.  Boson and Dirac stars in D ≥ 4 dimensions

Blázquez-Salcedo, Jose Luis and Knoll, Christian and Radu, Eugen

Physics Letters B

Elsevier

We present a comparative study of spherically symmetric, localized, particle-like solutions for spin s=0,1/2 and 1 gravitating fields in a D-dimensional, asymptotically flat spacetime. These fields are massive, possessing a harmonic time dependence and no self-interaction. Special attention is paid to the mathematical similarities and physical differences between the bosonic and fermionic cases. We find that the generic pattern of solutions is similar for any value of the spin s, depending only on the dimensionality of spacetime, the cases D=4,5 being special.

ria.ua.pt | doi | Peer Reviewed

19.  Kerr black holes with synchronised scalar hair and higher azimuthal harmonic index

Delgado, Jorge F. M. and Herdeiro, Carlos A. R. and Radu, Eugen

Physics Letters B

Elsevier

Kerr black holes with synchronised scalar hair and azimuthal harmonic index m>1 are constructed and studied. The corresponding domain of existence has a broader frequency range than the fundamental m=1 family; moreover, larger ADM masses, M and angular momenta J are allowed. Amongst other salient features, non-uniqueness of solutions for fixed global quantities is observed: solutions with the same M and J co-exist, for consecutive values of m, and the ones with larger m are always entropically favoured. Our analysis demonstrates, moreover, the qualitative universality of various features observed for m=1 solutions, such as the shape of the domain of existence, the typology of ergo-regions, and the horizon geometry, which is studied through its isometric embedding in Euclidean 3-space.

ria.ua.pt | doi | Peer Reviewed

18.  Investigating the use and acceptance of technologies by professors in a higher education institution

Costa, Carolina and Alvelos, Helena and Teixeira, Leonor

International Journal of Online Pedagogy and Course Design

IGI Global

This article analyses the use and acceptance of technologies by professors in the teaching and learning context in a higher education institution. In the empirical study, a questionnaire based on the technology acceptance model was applied. The results indicated that the most used technologies are Moodle, Facebook and YouTube and it was concluded that in general, those technologies are well accepted. Few statistically significant differences between respondents’ gender, scientific areas or ages were found, revealing that the use of those technologies is already widespread in the studied institution. Results also showed that perceived usefulness and perceived ease of use are two important determinants of Moodle acceptance, and that the majority of respondents did not know the MOOC concept. This article is valuable for researchers in the area and for professors that want to implement the use technologies in the teaching and learning context.

ria.ua.pt | doi | Peer Reviewed

17.  Product preservation and stable units for reflections into idempotent subvarieties

Xarez, João José and Xarez, Isabel Andrade

Categories and General Algebraic Structures with Applications

Shahid Beheshti University, Tehran, Iran

We give a necessary and sufficient condition for the preservation of finite products by a reflection of a variety of universal algebras into an idempotent subvariety. It is also shown that simple and semi-left-exact reflections into subvarieties of universal algebras are the same. It then follows that a reflection of a variety of universal algebras into an idempotent subvariety has stable units if and only if it is simple and the above-mentioned condition holds.

ria.ua.pt | Peer Reviewed

16.  Analysis and numerical approximation of tempered fractional calculus of variations problems

Almeida, Ricardo and Morgado, M. Luísa

Journal of Computational and Applied Mathematics

Elsevier

In this paper, we study variational problems where the cost functional involves the tempered Caputo fractional derivative. Several important optimization conditions are derived to find the optimal solution. Sufficient and necessary conditions are presented for different variational problems. For example, the cases of integral (isoperimetric problem) and holonomic constraints are considered, as well as problems with high order derivatives. A numerical scheme is proposed to determine approximations of the solution and it is illustrated through some examples

ria.ua.pt | doi | Peer Reviewed

15.  Optimal control measures for a susceptible‐carrier‐infectious‐recovered‐susceptible malware propagation model

Gonçalves, João N.C. and Rodrigues, Helena Sofia and Monteiro, M. Teresa T.

Optimal Control Applications and Methods

Wiley

Purposing to lessen malware propagation, this paper proposes optimal control measures for a susceptible-carrier-infectious-recovered-susceptible (SCIRS) epidemiological model formed by a system of ordinary differential equations. By taking advantage of real-world data related to the number of reported cybercrimes in Japan from 2012 to 2017, an optimal control problem is formulated to minimize the number of infected devices in a cost-effective way. The existence and uniqueness of the results related to the optimality system are proved. Overall, numerical simulations show the usefulness of the proposed control strategies in reducing the spread of malware infections.

ria.ua.pt | doi | Peer Reviewed

14.  The risk of contagion spreading and its optimal control in the economy

Kostylenko, Olena and Rodrigues, Helena Sofia and Torres, Delfim F. M.

Statistics, Optimization and Information Computing

International Academic Press

The global crisis of 2008 provoked a heightened interest among scientists to study the phenomenon, its propagation and negative consequences. The process of modelling the spread of a virus is commonly used in epidemiology. Conceptually, the spread of a disease among a population is similar to the contagion process in economy. This similarity allows considering the contagion in the world financial system using the same mathematical model of infection spread that is often used in epidemiology. Our research focuses on the dynamic behaviour of contagion spreading in the global financial network. The effect of infection by a systemic spread of risks in the network of national banking systems of countries is tested. An optimal control problem is then formulated to simulate a control that may avoid significant financial losses. The results show that the proposed approach describes well the reality of the world economy, and emphasizes the importance of international relations between countries on the financial stability.

ria.ua.pt | Peer Reviewed

13.  New sampling theorem and multiplicative filtering in the FRFT domain

Anh, P. K. and Castro, Luís P. and Thao, P. T. and Tuan, N. M.

Signal, Image and Video Processing

Springer Verlag

Having in consideration a fractional convolution associated with the fractional Fourier transform (FRFT), we propose a novel reconstruction formula for bandlimited signals in the FRFT domain without using the classical Shannon theorem. This may be considered the main contribution of this work, and numerical experiments are implemented to demonstrate the effectiveness of the proposed sampling theorem. As a second goal, we also look for the designing of multiplicative filters. Indeed, we also convert the multiplicative filtering in FRFT domain to the time domain, which can be realized by Fast Fourier transform. Two concrete examples are included where the use of the present results is illustrated.

ria.ua.pt | doi | Peer Reviewed

12.  Multiscale information storage of linear long-range correlated stochastic processes

Faes, Luca and Pereira, Margarida Almeida and Silva, Maria Eduarda and Pernice, Riccardo and Busacca, Alessandro and Javorka, Michal and Rocha, Ana Paula

Physical review. E

American Physical Society

Information storage, reflecting the capability of a dynamical system to keep predictable information during its evolution over time, is a key element of intrinsic distributed computation, useful for the description of the dynamical complexity of several physical and biological processes. Here we introduce a parametric approach which allows one to compute information storage across multiple timescales in stochastic processes displaying both short-term dynamics and long-range correlations (LRC). Our analysis is performed in the popular framework of multiscale entropy, whereby a time series is first "coarse grained" at the chosen timescale through low-pass filtering and downsampling, and then its complexity is evaluated in terms of conditional entropy. Within this framework, our approach makes use of linear fractionally integrated autoregressive (ARFI) models to derive analytical expressions for the information storage computed at multiple timescales. Specifically, we exploit state space models to provide the representation of lowpass filtered and downsampled ARFI processes, from which information storage is computed at any given timescale relating the process variance to the prediction error variance. This enhances the practical usability of multiscale information storage, as it enables a computationally reliable quantification of a complexity measure which incorporates the effects of LRC together with that of short-term dynamics. The proposed measure is first assessed in simulated ARFI processes reproducing different types of autoregressive dynamics and different degrees of LRC, studying both the theoretical values and the finite sample performance. We find that LRC alter substantially the complexity of ARFI processes even at short timescales, and that reliable estimation of complexity can be achieved at longer timescales only when LRC are properly modeled. Then, we assess multiscale information storage in physiological time series measured in humans during resting state and postural stress, revealing unprecedented responses to stress of the complexity of heart period and systolic arterial pressure variability, which are related to the different role played by LRC in the two conditions.

ria.ua.pt | doi | Peer Reviewed

11.  Chaos analysis and explicit series solutions to the seasonally forced SIR epidemic model

Duarte, Jorge and Martins, Nuno and Rogovchenko, Svitlana and Rogovchenko, Yuriy and Januário, Cristina

Journal of Mathematical Biology

Springer

Despite numerous studies of epidemiological systems, the role of seasonality in the recurrent epidemics is not entirely understood. During certain periods of the year incidence rates of a number of endemic infectious diseases may fluctuate dramatically. This influences the dynamics of mathematical models describing the spread of infection and often leads to chaotic oscillations. In this paper, we are concerned with a generalization of a classical Susceptible-Infected-Recovered epidemic model which accounts for seasonal effects. Combining numerical and analytic techniques, we gain new insights into the complex dynamics of a recurrent disease influenced by the seasonality. Computation of the Lyapunov spectrum allows us to identify different chaotic regimes, determine the fractal dimension and estimate the predictability of the appearance of attractors in the system. Applying the homotopy analysis method, we obtain series solutions to the original nonautonomous SIR model with a high level of accuracy and use these approximations to analyze the dynamics of the system. The efficiency of the method is guaranteed by the optimal choice of an auxiliary control parameter which ensures the rapid convergence of the series to the exact solution of the forced SIR epidemic model.

ria.ua.pt | doi | Peer Reviewed

10.  New convolutions weighted by Hermite functions and their applications

Castro, L. P. and Guerra, R. C. and Tuan, N. M.

Mathematical Inequalities and Applications

Element D.O.O.

We introduce eight new convolutions weighted by multi-dimensional Hermite functions, prove two Young-type inequalities, and exhibit their applications in different subjects. One application consists in the study of the solvability of a very general class of integral equations whose kernel depends on four different functions. Necessary and sufficient conditions for the unique solvability of such integral equations are here obtained.

ria.ua.pt | doi | Peer Reviewed

9.  An epidemiological MSEIR model described by the Caputo fractional derivative

Almeida, Ricardo and Cruz, Artur Miguel and Martins, Natália and Monteiro, Maria Teresa

International Journal of Dynamics and Control

Springer

A fractional MSEIR model is presented, involving the Caputo fractional derivative. The equilibrium points and the basic reproduction number are computed. An analysis of the local asymptotic stability at the disease free equilibrium is given. Finally a numerical simulation, using Matlab based on optimization techniques, of the varicella outbreak among Shenzhen school children, China, is carried out.

ria.ua.pt | doi | Peer Reviewed

8.  Optimal impulse control of dynamical systems

Piunovskiy, Alexey and Plakhov, Alexander and Torres, Delfim F. M. and Zhang, Yi

SIAM Journal on Control and Optimization

Using the tools of the Markov Decision Processes, we justify the dynamic programming approach to the optimal impulse control of deterministic dynamical systems. We prove the equivalence of the integral and differential forms of the optimality equation. The theory is illustrated by an example from mathematical epidemiology. The developed methods can be also useful for the study of piecewise deterministic Markov processes.

ria.ua.pt | Peer Reviewed

7.  A numerical approach for solving fractional optimal control problems using modified hat functions

Nemati, Somayeh and Lima, Pedro M. and Torres, Delfim F. M.

Communications in Nonlinear Science and Numerical Simulation

Elsevier

We introduce a numerical method, based on modified hat functions, for solving a class of fractional optimal control problems. In our scheme, the control and the fractional derivative of the state function are considered as linear combinations of the modified hat functions. The fractional derivative is considered in the Caputo sense while the Riemann-Liouville integral operator is used to give approximations for the state function and some of its derivatives. To this aim, we use the fractional order integration operational matrix of the modified hat functions and some properties of the Caputo derivative and Riemann-Liouville integral operators. Using results of the considered basis functions, solving the fractional optimal control problem is reduced to the solution of a system of nonlinear algebraic equations. An error bound is proved for the approximate optimal value of the performance index obtained by the proposed method. The method is then generalized for solving a class of fractional optimal control problems with inequality constraints. The most important advantages of our method are easy implementation, simple operations, and elimination of numerical integration. Some illustrative examples are considered to demonstrate the effectiveness and accuracy of the proposed technique.

ria.ua.pt | doi | Peer Reviewed

6.  Ground state, bound states and bifurcation properties for a Schrodinger-Poisson system with critical exponent

Chen, Jianqing and Huang, Lirong and Rocha, Eugénio

Electronic Journal of Differential Equations

Texas State University, Department of Mathematics

This article concerns the existence of ground state and bound states, and the study of their bifurcation properties for the Schrödinger-Poisson system(Forumala Presented). Under suitable assumptions on the coefficient h(x), we prove that the ground state must bifurcate from zero, and that another bound state bifurcates from a solution, when µ = µ 1 is the first eigenvalue of −∆u + u = µh(x)u in H 1 (R 3 )

ria.ua.pt | Peer Reviewed

Proceedings


5.  Using an artistic approach to the teaching of non-euclidean geometry in a professional development course for mathematics teachers

Hall, Andreia and Brás, Isabel and Pais, Sónia

INTED 2019 Proceedings

IATED

Elliot Eisner (1933-2014), a pioneer in arts education, suggested that an artistic approach to education could improve its quality and lead to a new vision for teaching and learning [1]. This is true for any subject, including mathematics. Geometry related topics make a perfect setting for a deeper contribution of art to education and allow for a complete symbiosis between the teaching of mathematics and an artistic education. One such topic is the study of non-Euclidean geometry which is now briefly addressed in the middle school mathematics curriculum in Portugal [2]. We believe that the learning and teaching of non-Euclidean geometry can be facilitated by taking the role of an artist and creating works of art, eventually inspired by renowned artists such as M.C. Escher. In this paper, we present some results of a professional development course for mathematics teachers where the participants studied basic non-Euclidean geometry concepts and created ceramic pieces using the Poincaré disk. The course took place in a Portuguese university, from January to March 2018, and involved 20 teachers of grades 1 to 12. The authors have developed a qualitative case study to evaluate how an artistic approach to the teaching of non-Euclidean geometry is perceived, by the mathematics teachers, as a contribution to the learning process. Overall, the activities developed have proved to be successful examples of interdisciplinary methodologies that bring into the teaching of mathematics usual procedures in the teaching of the arts. Moreover, the artistic approach followed during the course helped the teachers develop their geometric competences concerning non-Euclidean geometry in a more solid appropriation and application of the geometric concepts involved.

ria.ua.pt | Peer Reviewed

4.  Mathematical magic in undergraduate mathematical classes for pre-service teachers

Hall, Andreia and Pais, Sónia

INTED 2019 Proceedings

IATED

Nowadays we live in an ever-changing society. The educational context is no exception, and requires a renewal of paradigms. Profound changes to the role and function of the teacher and the students are particularly vital [1], [2], [3]. The current University students have different motivations, different attitudes towards the teaching and learning process, and different responses to specific classroom environments and instructional practices [4]. The challenge that teachers face today is to motivate students to learn, get them to commit to and have an active role in their learning [5]. 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 [6], the authors have developed several strategies to increase the interest and improve the success of students in mathematics. Thus, they decided to use mathematical magic tricks in their classes to raise the motivation of the students, captivating them and stimulating their interest in mathematics. In this work, a case study to evaluate how mathematical magic can contribute to increase students' motivation for learning mathematics is presented. The paper describes how it is conceived as well as the main results.

ria.ua.pt | Peer Reviewed

3.  New classes of monohedral spherical tilings by non-convex spherical hexagons and non-convex spherical Pentagons with GeoGebra

Breda, Ana and Santos, José Santos dos

34th International Conference on Computers and Their Applications - CATA 2019

In previous works we have ilustrate a procedure to obtain spherical tiling with GeoGebra. We have found new classes of monohedral spherical tiling by four spherical pentagons, and new class of dihedral spherical tiling by twelve spherical pentagons. One again, we would make use of GeoGebra to show how we can do generate new classes of monohedral non-convex hexagonal spherical tilings, H(C,τ), changing the side gluing rules of the regular spherical octahedral tiling, by local action of particular subgroups of spherical isometries. In relation to one of the new classes, by hexagonal tiles, we describe some of its properties. We also show the existence of a a new family of monohedral pentagonal tiling which arises as a degenarated case associated to the family H(C ,0) . All these classes of spherical tilings have emerged as a result of an interactive construction process, only possible by the use of newly produced GeoGebra tools and the dynamic interaction capabilities of this software.

ria.ua.pt | doi | Peer Reviewed

2.  Dificuldades de alunos do 8.º ano na construção de diagramas de extremos e quartis

Carvalho, Maria José and Fernandes, José António and Freitas, Adelaide

Actas del Tercer Congreso Internacional Virtual de Educación Estadística

Universidad de Granada, Facultad de Ciencias de la Educacion

Neste estudo são analisadas as resoluções de 23 alunos portugueses, do 8.º ano, de tarefas sobre o tema diagramas de extremos e quartis (DEQ) quando os dados são fornecidos de duas formas distintas: dados não organizados e dados representados num...

ria.ua.pt | Peer Reviewed

1.  Analyzing the practices of knowledge sharing and collaboration considering faculty members and researchers: an empirical study

Chedid, Marcello and Alvelos, Helena and Teixeira, Leonor

INTED2019 Proceedings - 13th International Technology, Education and Development Conference

IATED

Knowledge sharing and collaboration among university researchers and/or teachers are impacted by: (i) life cycles associated with technological innovations; (ii) economic and cultural globalization; (iii) the educational needs of an increasingly knowledge-driven society; and (iv) the training needs for highperforming professional activities. These factors constitute major opportunities for change in higher education institutions around the world and can be found in internal and/or external collaboration relationships. A collaboration relationship is expected to benefit the related members and teams (teachers, researchers, students and professionals), the communities/organizations that establish the relationship and, consequently, the surrounding society. Several studies reveal that collaboration can be strongly influenced by knowledge sharing. However, often universities evidence an individualistic culture where knowledge is related to the experience and the competence of each researcher and/or teacher. Thus, knowledge is frequently considered as property and as a differentiation factor of each individual. Additionally, the perception of little benefits and rewards in sharing knowledge within the institution lead to the existence of a real distance among the work of the referred actors, who focus their efforts towards individual objectives rather than in common goals. In this context, the individual characteristics tend to show a major impact on knowledge sharing. Considering the limited number of studies in the university’s context regarding knowledge sharing and collaboration practices, especially within Portuguese higher education institutions, the need to explore and deepen the understanding of these questions was identified. Taking this issue into account, this work aims at analyzing the practices of knowledge sharing and collaboration among researchers and/or teachers at the University of Aveiro, Portugal. The empirical study was conducted on the basis of a questionnaire prepared taking into account the literature on the area. The sample consisted of teachers and researchers from University of Aveiro, Portugal, and the questionnaire was provided online. The data was collected anonymously and analyzed with the statistical software SPSS IBM 24. The results present some important facts that can help higher education institutions and their members to define and implement strategies and practices in order to promote higher levels of knowledge sharing and stronger collaboration relationships.

ria.ua.pt | doi | Peer Reviewed

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