Book Chapters
25.
Statespace estimation using the behavioral approach: a simple particular case
Ntogramatzidis, Lorenzo and Pereira, Ricardo and Rocha, Paula
CONTROLO 2020. Lecture Notes in Electrical Engineering
Springer
In this paper we apply the behavioral estimation theory developed in Ntogramatzidis et al. (2020) to the particular case of statespace systems. We derive new necessary and sufficient conditions for the solvability of the estimation problem in the presence of disturbances, and provide a method to construct an estimator in case the problem is solvable. This is a first step to investigate how our previous results, derived within the more general behavioral context, compare with the results from classical state space theory.
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Peer Reviewed
24.
Torus and quadrics intersection using GeoGebra
Breda, Ana Maria Reis D'Azevedo and Trocado, Alexandre Emanuel Batista da Silva and Santos, José Manuel dos Santos dos
ICGG 2020: proceedings of the 19th International Conference on Geometry and Graphics
Springer
This paper presents the implementation in GeoGebra of algorithms for computing the intersection curve of a quadric surface with a torus surface. We present three approaches to get and visualise the intersection curve in GeoGebra. One of the approaches makes use of the geometric capabilities of GeoGebra. The second described approach makes use of CAS to obtain a parametrization and the corresponding visualisation of the intersection curve. Finally, the third one is based on computing the projection of the intersection curve, determining its singularities and structure, and its lifting to the 3D embedding space. The research carried out reveals some of the difficulties arising from the implementation in GeoGebra of a geometric algorithm based on the algebraic equations characterising the objects in consideration.
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Articles
23.
Matrix biorthogonal polynomials: eigenvalue problems and nonAbelian discrete Painlevé equations: a Riemann–Hilbert problem perspective
Branquinho, Amílcar and Moreno, Ana Foulquié and Mañas, Manuel
Journal of Mathematical Analysis and Applications
Elsevier
In this paper we use the Riemann–Hilbert problem, with jumps supported on appropriate curves in the complex plane, for matrix biorthogonal polynomials and apply it to find Sylvester systems of differential equations for the orthogonal polynomials and its second kind functions as well. For this aim, Sylvester type differential Pearson equations for the matrix of weights are shown to be instrumental. Several applications are given, in order of increasing complexity. First, a general discussion of nonAbelian Hermite biorthogonal polynomials on the real line, understood as those whose matrix of weights is a solution of a Sylvester type Pearson equation with coefficients first degree matrix polynomials, is given. All of these are applied to the discussion of possible scenarios leading to eigenvalue problems for second order linear differential operators with matrix eigenvalues. Nonlinear matrix difference equations are discussed next. Firstly, for the general Hermite situation a general non linear relation (non trivial because of the non commutativity features of the setting) for the recursion coefficients is gotten. In the next case of higher difficulty, degree two polynomials are allowed in the Pearson equation, but the discussion is simplified by considering only a left Pearson equation. In the case, the support of the measure is on an appropriate branch of a hyperbola. The recursion coefficients are shown to fulfill a nonAbelian extension of the alternate discrete Painlevé I equation. Finally, a discussion is given for the case of degree three polynomials as coefficients in the left Pearson equation characterizing the matrix of weights. However, for simplicity only odd polynomials are allowed. In this case, a new and more general matrix extension of the discrete Painlevé I equation is found.
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22.
A matrix based list decoding algorithm for linear codes over integer residue rings
Napp, Diego and Pinto, Raquel and Saçıkara, Elif and Toste, Marisa
Linear Algebra and its Applications
Elsevier
In this paper we address the problem of list decoding of linear codes over an integer residue ring Zq, where q is a power of a prime p. The proposed procedure exploits a particular matrix representation of the linear code over Zpr , called the standard form, and the padic expansion of the tobedecoded vector. In particular, we focus on the erasure channel in which the location of the errors is known. This problem then boils down to solving a system of linear equations with coefficients in Zpr . From the paritycheck matrix representations of the code we recursively select certain equations that a codeword must satisfy and have coefficients only in the field p^{r−1}Zpr .
This yields a step by step procedure obtaining a list of the closest codewords to a given received vector with some of its coordinates erased. We show that such an algorithm actually computes all possible erased coordinates, that is, the provided list is minimal.
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21.
Robust formulations for economic lotsizing problem with remanufacturing
Attila, Öykü Naz and Agra, Agostinho and Akartunalı, Kerem and Arulselvan, Ashwin
European Journal of Operational Research
Elsevier
In this paper, we consider a lotsizing problem with the remanufacturing option under parameter uncertainties imposed on demands and returns. Remanufacturing has recently been a fast growing area of interest for many researchers due to increasing awareness on reducing waste in production environments, and in particular studies involving remanufacturing and parameter uncertainties simultaneously are very scarce in the literature. We first present a minmax decomposition approach for this problem, where decision maker’s problem and adversarial problem are treated iteratively. Then, we propose two novel extended reformulations for the decision maker’s problem, addressing some of the computational challenges. An original aspect of the reformulations is that they are applied only to the latest scenario added to the decision maker’s problem. Then, we present an extensive computational analysis, which provides a detailed comparison of the three formulations and evaluates the impact of key problem parameters. We conclude that the proposed extended reformulations outperform the standard formulation for a majority of the instances. We also provide insights on the impact of the problem parameters on the computational performance.
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20.
Discrete Hardy spaces for bounded domains in Rn
Cerejeiras, Paula and Kähler, Uwe and Legatiuk, Anastasiia and Legatiuk, Dmitrii
Complex Analysis and Operator Theory
Springer; Birkhäuser
Discrete function theory in higherdimensional setting has been in active development since many years. However, available results focus on studying discrete setting for such canonical domains as halfspace, while the case of bounded domains generally remained unconsidered. Therefore, this paper presents the extension of the higherdimensional function theory to the case of arbitrary bounded domains in R^n. On this way, discrete Stokes’ formula, discrete Borel–Pompeiu formula, as well as discrete Hardy spaces for general bounded domains are constructed. Finally, several discrete Hilbert problems are considered.
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19.
Eigenfunctions of the time‐fractional diffusion‐wave operator
Ferreira, Milton and Luchko, Yury and Rodrigues, M. Manuela and Vieira, Nelson
Mathematical Methods in the Applied Sciences
Wiley
In this paper, we present some new integral and series representations for the eigenfunctions of the multidimensional time‐fractional diffusion‐wave operator with the time‐fractional derivative of order β ∈]1, 2[ defined in the Caputo sense. The integral representations are obtained in form of the inverse Fourier–Bessel transform and as a double contour integrals of the Mellin–Barnes type. Concerning series expansions, the eigenfunctions are expressed as the double generalized hypergeometric series for any β ∈]1, 2[ and as Kampé de Fériet and Lauricella series in two variables for the rational values of β. The limit cases β=1 (diffusion operator) and β=2 (wave operator) as well as an intermediate case β=32 are studied in detail. Finally, we provide several plots of the eigenfunctions to some selected eigenvalues for different particular values of the fractional derivative order β and the spatial dimension n.
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18.
Fractional model of COVID19 applied to Galicia, Spain and Portugal
Ndaïrou, Faïçal and Area, Iván and Nieto, Juan J. and Silva, Cristiana J. and Torres, Delfim F. M.
Chaos, Solitons & Fractals
Elsevier
A fractional compartmental mathematical model for the spread of the COVID19
disease is proposed. Special focus has been done on the transmissibility of
superspreaders individuals. Numerical simulations are shown for data of
Galicia, Spain, and Portugal. For each region, the order of the Caputo
derivative takes a different value, that is not close to one, showing the
relevance of considering fractional models.
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17.
Spininduced scalarized black holes
Herdeiro, Carlos A. R. and Radu, Eugen and Silva, Hector O. and Sotiriou, Thomas P. and Yunes, Nicolás
Physical Review Letters
American Physical Society
It was recently shown that a scalar field suitably coupled to the
GaussBonnet invariant $mathcal{G}$ can undergo a spininduced linear
tachyonic instability near a Kerr black hole. This instability appears only
once the dimensionless spin $j$ is sufficiently large, that is, $j gtrsim
0.5$. A tachyonic instability is the hallmark of spontaneous scalarization.
Focusing, for illustrative purposes, on a class of theories that do exhibit
this instability, we show that stationary, rotating black hole solutions do
indeed have scalar hair once the spininduced instability threshold is
exceeded, while black holes that lie below the threshold are described by the
Kerr solution. Our results provide strong support for spininduced black hole
scalarization.
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16.
Multipolar boson stars: macroscopic BoseEinstein condensates akin to hydrogen orbitals
Herdeiro, C. A. R. and Kunz, J. and Perapechka, I. and Radu, E. and Shnir, Ya.
Physics Letters B
Elsevier
Boson stars are often described as macroscopic BoseEinstein condensates. By
accommodating large numbers of bosons in the same quantum state, they
materialize macroscopically the intangible probability density cloud of a
single particle in the quantum world. We take this interpretation of boson
stars one step further. We show, by explicitly constructing the fully
nonlinear solutions, that static (in terms of their spacetime metric,
$g_{munu}$) boson stars, composed of a single complex scalar field, $Phi$,
can have a nontrivial multipolar structure, yielding the same morphologies for
their energy density as those that elementary hydrogen atomic orbitals have for
their probability density. This provides a close analogy between the elementary
solutions of the nonlinear EinsteinKleinGordon theory, denoted
$Phi_{(N,ell,m)}$, which could be realized in the macrocosmos, and those of
the linear Schr"odinger equation in a Coulomb potential, denoted
$Psi_{(N,ell,m)}$, that describe the microcosmos. In both cases, the
solutions are classified by a triplet of quantum numbers $(N,ell,m)$. In the
gravitational theory, multipolar boson stars can be interpreted as individual
bosonic lumps in equilibrium; remarkably, the (generic) solutions with $mneq
0$ describe gravitating solitons $[g_{munu},Phi_{(N,ell,m)}]$ without any
continuous symmetries. Multipolar boson stars analogue to hybrid orbitals are
also constructed.
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15.
Stability analysis and optimal control of a fractional HIVAIDS epidemic model with memory and general incidence rate
Boukhouima, Adnane and Lotfi, El Mehdi and Mahrouf, Marouane and Rosa, Silvério and Torres, Delfim F. M. and Yousfi, Noura
The European Physical Journal Plus
Springer Verlag; EDP Sciences; Società Italiana di Fisica
We investigate the celebrated mathematical SICA model but using fractional differential equations in order to better describe the dynamics of HIVAIDS infection. The infection process is modelled by a general functional response, and the memory effect is described by the Caputo fractional derivative. Stability and instability of equilibrium points are determined in terms of the basic reproduction number. Furthermore, a fractional optimal control system is formulated and the best strategy for minimizing the spread of the disease into the population is determined through numerical simulations based on the derived necessary optimality conditions.
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14.
Phenomenology of vectorlike leptons with Deep Learning at the Large Hadron Collider
Freitas, Felipe F. and Gonçalves, João and Morais, António P. and Pasechnik, Roman
Journal of High Energy Physics
Springer Verlag
In this paper, a model inspired by Grand Unification principles featuring three generations of vectorlike fermions, new Higgs doublets and a rich neutrino sector at the low scale is presented. Using the stateoftheart Deep Learning techniques we perform the first phenomenological analysis of this model focusing on the study of new charged vectorlike leptons (VLLs) and their possible signatures at CERN’s Large Hadron Collider (LHC). In our numerical analysis we consider signal events for vectorboson fusion and VLL pair production topologies, both involving a final state containing a pair of charged leptons of different flavor and two sterile neutrinos that provide a missing energy. We also consider the case of VLL single production where, in addition to a pair of sterile neutrinos, the final state contains only one charged lepton. We propose a novel method to identify missing transverse energy vectors by comparing the detector response with MonteCarlo simulated data. All calculated observables are provided as data sets for Deep Learning analysis, where a neural network is constructed, based on results obtained via an evolutive algorithm, whose objective is to maximise either the accuracy metric or the Asimov significance for different masses of the VLL. Taking into account the effect of the three analysed topologies, we have found that the combined significance for the observation of new VLLs at the highluminosity LHC can range from 5.7σ, for a mass of 1.25 TeV, all the way up to 28σ if the VLL mass is 200 GeV. We have also shown that by the end of the LHC RunIII a 200 GeV VLL can be excluded with a confidence of 8.8 standard deviations. The results obtained show that our model can be probed well before the end of the LHC operations and, in particular, providing important phenomenological information to constrain the energy scale at which new gauge symmetries emergent from the considered Grand Unification picture can be manifest.
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13.
Numerical solution of a class of thirdkind Volterra integral equations using Jacobi wavelets
Nemati, S. and Lima, Pedro M. and Torres, Delfim F. M.
Numerical Algorithms
Springer
We propose a spectral collocation method, based on the generalized Jacobi wavelets along with the Gauss–Jacobi quadrature formula, for solving a class of thirdkind Volterra integral equations. To do this, the interval of integration is first transformed into the interval [− 1, 1], by considering a suitable change of variable. Then, by introducing special Jacobi parameters, the integral part is approximated using the Gauss–Jacobi quadrature rule. An approximation of the unknown function is considered in terms of Jacobi wavelets functions with unknown coefficients, which must be determined. By substituting this approximation into the equation, and collocating the resulting equation at a set of collocation points, a system of linear algebraic equations is obtained. Then, we suggest a method to determine the number of basis functions necessary to attain a certain precision. Finally, some examples are included to illustrate the applicability, efficiency, and accuracy of the new scheme.
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12.
A new rank metric for convolutional codes
Almeida, P. and Napp, D.
Designs, Codes and Cryptography
Springer
Let F[D] be the polynomial ring with entries in a finite field F. Convolutional codes are submodules of F[D]n that can be described by left prime polynomial matrices. In the last decade there has been a great interest in convolutional codes equipped with a rank metric, called sum rank metric, due to their wide range of applications in reliable linear network coding. However, this metric suits only for delay free networks. In this work we continue this thread of research and introduce a new metric that overcomes this restriction and therefore is suitable to handle more general networks. We study this metric and provide characterizations of the distance properties in terms of the polynomial matrix representations of the convolutional code. Convolutional codes that are optimal with respect to this new metric are investigated and concrete constructions are presented. These codes are the analogs of Maximum Distance Profile convolutional codes in the context of network coding. Moreover, we show that they can be built upon a class of superregular matrices, with entries in an extension field, that preserve their superregularity properties even after multiplication with some matrices with entries in the ground field.
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11.
The degrees of toroidal regular proper hypermaps
Fernandes, Maria Elisa and Piedade, Claudio Alexandre
The Art of Discrete and Applied Mathematics
Slovenian Discrete and Applied Mathematics Society; University of Primorska, FAMNI
Recently the classification of all possible faithful transitive permutation representations of the group of symmetries of a regular toroidal map was accomplished.
In this paper we complete this investigation on a surface of genus 1 considering the group of a regular toroidal hypermap of type (3,3,3).
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10.
The Hjoin of arbitrary families of graphs
Cardoso, Domingos M. and Gomes, Helena and Pinheiro, Sofia J
arXiv
The Hjoin of a family of graphs G = {G1, . . . , Gp}, also called the generalized
composition, H[G1, . . . , Gp], where all graphs are undirected, simple and finite, is
the graph obtained from the graph H replacing each vertex i of H by Gi and adding
to the edges of all graphs in G the edges of the join Gi ∨ Gj , for every edge ij of H.
Some well known graph operations are particular cases of the Hjoin of a family of
graphs G as it is the case of the lexicographic product (also called composition) of
two graphs H and G, H[G], which coincides with the Hjoin of family of graphs G
where all the graphs in G are isomorphic to a fixed graph G.
So far, the known expressions for the determination of the entire spectrum of the
Hjoin in terms of the spectra of its components and an associated matrix are
limited to families of regular graphs. In this paper, we extend such a determination
to families of arbitrary graphs.
ria.ua.pt
9.
Spontaneous vectorization of electrically charged black holes
Oliveira, João M. S. and Pombo, Alexandre M.
Physical Review D
In this work, we generalise the spontaneous scalarization phenomena in EinsteinMaxwellScalar
models to a higher spin field. The result is an EinsteinMaxwellVector model wherein a vector field
is nonminimally coupled to the Maxwell invariant by an exponential coupling function. We show that
the latter guarantees the circumvention of an associated nohair theorem when the vector field has the
form of an electric field. Different than its scalar counterpart, the new spontaneously vectorized ReissnerNordstr¨om (RN) black holes are, always, undercharged while being entropically preferable. The solution
profile and domain of existence are presented and analysed.
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8.
Focus point: cancer and HIV/AIDS dynamics: from optimality to modelling
Debbouche, Amar and Nieto, Juan J. and Torres, Delfim F. M.
The European Physical Journal Plus
Springer
Human cancer is a multistep process involving acquired genetic mutations, each of which imparts a particular type of growth advantage to the cell and ultimately leads to the development of a malignant phenotype. It is also a generic term for a group of diseases and figures as a leading cause of death globally; it lays a significant burden on healthcare systems and continues to be among the major health problems worldwide. The consequences of mutations in tumor cells include alterations in cell signaling pathways that result in uncontrolled cellular proliferation, insensitivity to growth inhibitory signals, resistance to apoptosis, development of cellular immortality, angiogenesis, tissue invasion and metastasis.
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7.
Modeling and forecasting of COVID19 spreading by delayed stochastic differential equations
Mahrouf, Marouane and Boukhouima, Adnane and Zine, Houssine and Lotfi, El Mehdi and Torres, Delfim F. M. and Yousfi, Noura
Axioms
MDPI
The novel coronavirus disease (COVID19) pneumonia has posed a great threat to the world recent months by causing many deaths and enormous economic damage worldwide. The first case of COVID19 in Morocco was reported on 2 March 2020, and the number of reported cases has increased day by day. In this work, we extend the wellknown SIR compartmental model to deterministic and stochastic timedelayed models in order to predict the epidemiological trend of COVID19 in Morocco and to assess the potential role of multiple preventive measures and strategies imposed by Moroccan authorities. The main features of the work include the wellposedness of the models and conditions under which the COVID19 may become extinct or persist in the population. Parameter values have been estimated from real data and numerical simulations are presented for forecasting the COVID19 spreading as well as verification of theoretical results.
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6.
Optimal control of the COVID19 pandemic: controlled sanitary deconfinement in Portugal
Silva, Cristiana J. and Cruz, Carla and Torres, Delfim F. M. and Muñuzuri, Alberto P. and Carballosa, Alejandro and Area, Iván and Nieto, Juan J. and FonsecaPinto, Rui and Passadouro, Rui and Santos, Estevão Soares dos and Abreu, Wilson and Mira, Jorge
Scientific Reports
Nature Research
The COVID19 pandemic has forced policy makers to decree urgent confinements
to stop a rapid and massive contagion. However, after that stage, societies are
being forced to find an equilibrium between the need to reduce contagion rates
and the need to reopen their economies. The experience hitherto lived has
provided data on the evolution of the pandemic, in particular the population
dynamics as a result of the public health measures enacted. This allows the
formulation of forecasting mathematical models to anticipate the consequences
of political decisions. Here we propose a model to do so and apply it to the
case of Portugal. With a mathematical deterministic model, described by a
system of ordinary differential equations, we fit the real evolution of
COVID19 in this country. After identification of the population readiness to
follow social restrictions, by analyzing the social media, we incorporate this
effect in a version of the model that allow us to check different scenarios.
This is realized by considering a Monte Carlo discrete version of the previous
model coupled via a complex network. Then, we apply optimal control theory to
maximize the number of people returning to "normal life" and minimizing the
number of active infected individuals with minimal economical costs while
warranting a low level of hospitalizations. This work allows testing various
scenarios of pandemic management (closure of sectors of the economy,
partial/total compliance with protection measures by citizens, number of beds
in intensive care units, etc.), ensuring the responsiveness of the health
system, thus being a public health decision support tool.
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5.
Spontaneous scalarization of a conducting sphere in Maxwellscalar models
Herdeiro, Carlos A. R. and Ikeda, Taishi and Minamitsuji, Masato and Nakamura, Tomohiro and Radu, Eugen
Physical Review D
American Physical Society
We study the spontaneous scalarization of a standard conducting charged sphere embedded in Maxwellscalar models in flat spacetime, wherein the scalar field
ϕ
is nonminimally coupled to the Maxwell electrodynamics. This setup serves as a toy model for the spontaneous scalarization of charged (vacuum) black holes in EinsteinMaxwellscalar (generalized scalartensor) models. In the Maxwellscalar case, unlike the black hole cases, closedform solutions exist for the scalarized configurations. We compute these configurations for three illustrations of nonminimal couplings: one that exactly linearizes the scalar field equation, and the remaining two that produce nonlinear continuations of the first one. We show that the former model leads to a runaway behavior in regions of the parameter space and neither the Coulomb nor the scalarized solutions are stable in the model; but the latter models can heal this behavior producing stable scalarized solutions that are dynamically preferred over the Coulomb one. This parallels reports on black hole scalarization in the extendedscalarGaussBonnet models. Moreover, we analyze the impact of the choice of the boundary conditions on the scalarization phenomenon. Dirichlet and Neumann boundary conditions accommodate both (linearly) stable and unstable parameter space regions, for the scalarfree conducting sphere; but radiative boundary conditions always yield an unstable scalarfree solution and preference for scalarization. Finally, we perform numerical evolution of the full Maxwellscalar system, following dynamically the scalarization process. They confirm the linear stability analysis and reveal that the scalarization phenomenon can occur in qualitatively distinct ways.
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4.
New convolutions with Hermite weight functions
Castro, Luís Pinheiro and Silva, Anabela Sousa and Tuan, Nguyen Minh
Bulletin of the Iranian Mathematical Society
Springer
In this paper, we are working with convolutions on the positive halfline, for Lebesgue
integrable functions. Six new convolutions are introduced. Factorization identities for
these convolutions are derived, upon the use of Fourier sine and cosine transforms and
Hermite functions. Such convolutions allowus to consider systems of convolution type
equations on the halfline. Using two different methods, such systems of convolution
integral equations will be analyzed. Conditions for their solvability will be considered
and, under such conditions, their solutions are obtained.
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3.
Black holes, stationary clouds and magnetic fields
Santos, Nuno M. and Herdeiro, Carlos A. R.
Physics Letters B
Elsevier
As the electron in the hydrogen atom, a bosonic field can bind itself to a
black hole occupying a discrete infinite set of states. When (i) the spacetime
is prone to superradiance and (ii) a confinement mechanism is present, some of
such states are infinitely longlived. These equilibrium configurations, known
as stationary clouds, are states "synchronized" with a rotating black hole's
event horizon. For most, if not all, stationary clouds studied in the
literature so far, the requirements (i)(ii) are independent of each other.
However, this is not always the case. This paper shows that massless neutral
scalar fields can form stationary clouds around a ReissnerNordstr"{o}m black
hole when both are subject to a uniform magnetic field. The latter
simultaneously enacts both requirements by creating an ergoregion (thereby
opening up the possibility of superradiance) and trapping the scalar field in
the black hole's vicinity. This leads to some novel features, in particular,
that only black holes with a subset of the possible charge to mass ratios can
support stationary clouds.
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2.
Quasinormal modes of hot, cold and bald Einstein–Maxwellscalar black holes
BlázquezSalcedo, Jose Luis and Herdeiro, Carlos A. R. and Kahlen, Sarah and Kunz, Jutta and Pombo, Alexandre M. and Radu, Eugen
The European Physical Journal C
SpringerOpen
Einstein–Maxwellscalar models allow for different classes of black hole solutions, depending on the nonminimal coupling function f(ϕ) employed, between the scalar field and the Maxwell invariant. Here, we address the linear mode stability of the black hole solutions obtained recently for a quartic coupling function, f(ϕ)=1+αϕ4 (BlázquezSalcedo et al. in Phys. Lett. B 806:135493, 2020). Besides the bald Reissner–Nordström solutions, this coupling allows for two branches of scalarized black holes, termed cold and hot, respectively. For these three branches of black holes we calculate the spectrum of quasinormal modes. It consists of polar scalarled modes, polar and axial electromagneticled modes, and polar and axial gravitationalled modes. We demonstrate that the only unstable mode present is the radial scalarled mode of the cold branch. Consequently, the bald Reissner–Nordström branch and the hot scalarized branch are both modestable. The nontrivial scalar field in the scalarized background solutions leads to the breaking of the degeneracy between axial and polar modes present for Reissner–Nordström solutions. This isospectrality is only slightly broken on the cold branch, but it is strongly broken on the hot branch.
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1.
Control of COVID19 dynamics through a fractionalorder model
Bushnaq, Samia and Saeed, Tareq and Torres, Delfim F. M. and Zeb, Anwar
Alexandria Engineering Journal
Elsevier
We investigate, through a fractional mathematical model, the effects of
physical distance on the SARSCoV2 virus transmission. Two controls are
considered in our model for eradication of the spread of COVID19: media
education, through campaigns explaining the importance of social distancing,
use of face masks, etc., towards all population, while the second one is
quarantine social isolation of the exposed individuals. A general fractional
order optimal control problem, and associated optimality conditions of
Pontryagin type, are discussed, with the goal to minimize the number of
susceptible and infected while maximizing the number of recovered. The
extremals are then numerically obtained.
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