Book Chapters
11.
A new mathematical model for the efficiency calculation
Galindro, Aníbal and Santos, Micael and Torres, Delfim F. M. and MartaCosta, Ana
Recent Advances in Modeling, Analysis and Systems Control: Theoretical Aspects and Applications. Studies in Systems, Decision and Control
Springer Nature
During the past sixty years, a lot of effort has been made regarding the
productive efficiency. Such endeavours provided an extensive bibliography on this
subject, culminating in two main methods, named the Stochastic Frontier Analysis
(parametric) and Data Envelopment Analysis (nonparametric). The literature states
this methodology also as the benchmark approach, since the techniques compare
the sample upon a chosen “moreefficient” reference. This article intends to disrupt
such premise, suggesting a mathematical model that relies on the optimal input
combination, provided by a differential equation system instead of an observable
sample. A numerical example is given, illustrating the application of our model’s
features.
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10.
Parametric identification of the dynamics of intersectoral balance: modelling and forecasting
Kostylenko, Olena and Rodrigues, Helena Sofia and Torres, Delfim F. M.
Recent Advances in Modeling, Analysis and Systems Control: Theoretical Aspects and Applications. Studies in Systems, Decision and Control
Springer
This work is devoted to modelling and identification of the dynamics of the intersectoral balance of a macroeconomic system. An approach to the problem of specification and identification of a weakly formalized dynamical system is developed. A matching procedure for parameters of a linear stationary Cauchy problem with a decomposition of its upshot trend and a periodic component, is proposed. Moreover, an approach for detection of significant harmonic waves, which are inherent to real macroeconomic dynamical systems, is developed.
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Articles
9.
Extremal graphs for Estrada indices
Andrade, Enide and Lenes, Eber and MalleaZepeda, Exequiel and Robbiano, María and Rodríguez Z., Jonnathan
Linear Algebra and its Applications
Elsevier
Let $mathcal{G}$ be a simple undirected connected graph. The signless Laplacian Estrada, Laplacian Estrada and Estrada indices of a graph $mathcal{G}$ is the sum of the exponentials of the signless Laplacian eigenvalues, Laplacian eigenvalues and eigenvalues of $mathcal{G}$, respectively.
The present work derives an upper bound for the Estrada index of a graph as a function of its chromatic number, in the family of graphs whose color classes have order not less than a fixed positive integer. The graphs for which the upper bound is tight is obtained.
Additionally, an upper bound for the Estrada Index of the complement of a graph in the previous family of graphs with two color classes is given. A NordhausGaddum type inequality for the Laplacian Estrada index when {$mathcal{G}$ is a bipartite} graph with color classes of order not less than $2$, is presented. Moreover, a sharp upper bound for the Estrada index of the line graph and for the signless Laplacian index of a graph in terms of connectivity is obtained.
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8.
Matrix Toda and Volterra lattices
Moreno, Ana Foulquié and Branquinho, Amílcar and GarcíaArdila, Juan C.
Applied Mathematics and Computation
Elsevier
We consider matrix Toda and Volterra lattice equations and their relation with matrix biorthogonal polynomials. From that relation, we give a method for constructing a new solution of these systems from another given one. An illustrative example is presented.
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7.
Maximal commutators and commutators of potential operators in new vanishing Morrey spaces
Almeida, Alexandre
Nonlinear Analysis
Elsevier
We study mapping properties of commutators in certain vanishing subspaces of Morrey spaces, which were recently used to solve the delicate problem of describing the closure of nice functions in Morrey norm. We show that the vanishing properties defining those subspaces are preserved under the action of maximal commutators and commutators of fractional integral operators.
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6.
Fundamental solution for natural powers of the fractional Laplace and Dirac operators in the RiemannLiouville sense
Teodoro, A. Di and Ferreira, M. and Vieira, N.
Advances in Applied Clifford Algebras
Springer
In this paper, we study the fundamental solution for natural powers of the $n$parameter fractional Laplace and Dirac operators defined via RiemannLiouville fractional derivatives. To do this we use iteration through the fractional Poisson equation starting from the fundamental solutions of the fractional Laplace $Delta_{a^+}^alpha$ and Dirac $D_{a^+}^alpha$ operators, admitting a summable fractional derivative. The family of fundamental solutions of the corresponding natural powers of fractional Laplace and Dirac operators are expressed in operator form using the MittagLeffler function.
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5.
Variable exponent TriebelLizorkinMorrey spaces
Caetano, António and Kempka, Henning
Journal of Mathematical Analysis and Applications
Elsevier
We introduce variable exponent versions of Morreyﬁed TriebelLizorkin spaces. To that end, we prove an important convolution inequality which is a replacement for the HardyLittlewood maximal inequality in the fully variable setting. Using it we obtain characterizations by means of Peetre maximal functions and use them to show the independence of the introduced spaces from the admissible system used.
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4.
On the energy of singular and non singular graphs
Andrade, Enide and Carmona, Juan R. and Poveda, Alex and Robbiano, María
MATCH Communications in Mathematical and in Computer Chemistry
University of Kragujevac
Let $G$ be a simple undirected graph with $n$ vertices, $m$ edges, adjacency matrix $A$, largest eigenvalue $rho$ and nullity $kappa$. The energy of $G,$ $mathcal{E}(G)$ is the sum of its singular values. In this work lower bounds for $mathcal{E}(G)$ in terms of the coefficient of $mu^{kappa}$ in the expansion of characteristic polynomial, $p(mu)=det{(mu IA)}$ are obtained.
In particular one of the bounds generalizes a lower bound obtained by K. Das, S. A. Mojallal and I. Gutman in $2013$ to the case of graphs with given nullity. The bipartite case is also studied obtaining in this case, a sufficient condition to improve the spectral lower bound $2rho.$ Considering an increasing sequence convergent to $rho$ a convergent increasing sequence of lower bounds for the energy of $G$ is constructed.
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3.
Numerical optimal control of HIV transmission in Octave/MATLAB
Campos, Carlos and Silva, Cristiana J. and Torres, Delfim F. M.
Mathematical and Computational Applications
MDPI
We provide easy and readable GNU Octave/MATLAB code for the simulation of
mathematical models described by ordinary differential equations and for the solution of optimal
control problems through Pontryagin’s maximum principle. For that, we consider a normalized
HIV/AIDS transmission dynamics model based on the one proposed in our recent contribution (Silva,
C.J.; Torres, D.F.M. A SICA compartmental model in epidemiology with application to HIV/AIDS in
Cape Verde. Ecol. Complex. 2017, 30, 70–75), given by a system of four ordinary differential equations.
An HIV initial value problem is solved numerically using the ode45 GNU Octave function and three
standard methods implemented by us in Octave/MATLAB: Euler method and secondorder and
fourthorder Runge–Kutta methods. Afterwards, a control function is introduced into the normalized
HIV model and an optimal control problem is formulated, where the goal is to find the optimal
HIV prevention strategy that maximizes the fraction of uninfected HIV individuals with the least
HIV new infections and cost associated with the control measures. The optimal control problem is
characterized analytically using the Pontryagin Maximum Principle, and the extremals are computed
numerically by implementing a forwardbackward fourthorder Runge–Kutta method. Complete
algorithms, for both uncontrolled initial value and optimal control problems, developed under the
free GNU Octave software and compatible with MATLAB are provided along the article.
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2.
Regional enlarged observability of Caputo fractional differential equations
Zouiten, Hayat and Boutoulout, Ali and Torres, Delfim F. M.
Discrete and Continuous Dynamical Systems  Series S
American Institute of Mathematical Sciences (AIMS)
We consider the regional enlarged observability problem for fractional
evolution differential equations involving Caputo derivatives. Using the
Hilbert Uniqueness Method, we show that it is possible to rebuild the initial
state between two prescribed functions only in an internal subregion of the
whole domain. Finally, an example is provided to illustrate the theory.
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1.
Graphs with clusters perturbed by regular graphs: Aαspectrum and applications
Cardoso, Domingos M. and Pastén, Germain and Rojo, Oscar
Discussiones Mathematicae Graph Theory
De Gruyter
Given a graph $G$, its adjacency matrix $A(G)$ and its diagonal matrix of vertex degrees $D(G)$, consider the matrix $A_{alpha}left( Gright) = alpha Dleft( Gright) +(1alpha)Aleft(Gright)$, where $alpha inleft[ 0,1right)$. The $A_{alpha}$ spectrum of $G$ is the multiset of eigenvalues of $A_{alpha}(G)$ and these eigenvalues are the $alpha$ eigenvalues of $G$. A cluster in $G$ is a pair of vertex subsets $(C,S)$, where $C$ is a set of cardinality $C ge 2$ of pairwise coneighbor vertices sharing the same set $S$ of $S$ neighbors. Assuming that $G$ is connected and it has a cluster $(C,S)$, $G(H)$ is obtained from $G$ and an $r$ regular graph $H$ of order $C$ by identifying its vertices with the vertices in $C$, eigenvalues of $A_{alpha}(G)$ and $A_{alpha}(G(H))$ are deduced and if $A_{alpha}(H)$ is positive semidefinite then the $i$th eigenvalue of $A_{alpha}(G(H))$ is greater than or equal to $i$th eigenvalue of $A_{alpha}(G)$. These results are
extended to graphs with several pairwise disjoint clusters $(C_1,S_1), ldots, (C_k,S_k)$. As an application, the effect on the energy, $alpha$Estrada index and $alpha$index of a graph $G$ with clusters when the edges of regular graphs are added to $G$ are analyzed. Finally, the $A_{alpha}$ spectrum of the corona product $G circ H$ of a connected graph $G$ and a regular graph $H$ is determined.
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