ISEL - Matemática
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- Activation of effector immune cells promotes tumor stochastic extinction: A homotopy analysis approachPublication . Sardanyés, Josep; Rodrigues, Carla; Januário, Cristina; Martins, Nuno; Gil-Gómez, Gabriel; Duarte, JorgeIn this article we provide homotopy solutions of a cancer nonlinear model describing the dynamics of tumor cells in interaction with healthy and effector immune cells. We apply a semi-analytic technique for solving strongly nonlinear systems – the Step Homotopy Analysis Method (SHAM). This algorithm, based on a modification of the standard homotopy analysis method (HAM), allows to obtain a one-parameter family of explicit series solutions. By using the homotopy solutions, we first investigate the dynamical effect of the activation of the effector immune cells in the deterministic dynamics, showing that an increased activation makes the system to enter into chaotic dynamics via a period-doubling bifurcation scenario. Then, by adding demographic stochasticity into the homotopy solutions, we show, as a difference from the deterministic dynamics, that an increased activation of the immune cells facilitates cancer clearance involving tumor cells extinction and healthy cells persistence. Our results highlight the importance of therapies activating the effector immune cells at early stages of cancer progression.
- Active vibration attenuation in viscoelastic laminated composite panels using multiobjective optimizationPublication . Luís, Ndilokelwa F.; Madeira, JFA; Araújo, A. L.; Ferreira, A. J. M.The optimal design of viscoelastic laminated composite panels with active piezoelectric patches is addressed in this paper. Constrained optimization is conducted to determine optimal distributions of piezoelectric patches on the top and bottom surfaces of laminated plates with viscoelastic layers. The design variables are the number and position of these patches, and the objectives are the minimization of the number of patches, the maximization of the fundamental modal loss factor and the maximization of the fundamental natural frequency. The problem is solved using the Direct MultiSearch (DMS) solver for derivative-free MultiObjective Optimization (MOO). The objective functions are evaluated by a finite element model that was developed for laminated sandwich plates incorporating piezoelectric or viscoelastic layers. Trade-off Pareto optimal fronts and the respective optimal active patch configurations are obtained and the results are analyzed and discussed.
- Algebraic structure for interaction on mixed modelsPublication . Ramos, Paulo; Fernandes, Célia; Mexia, João TiagoBinary operations on commutative Jordan algebras, CJA, can be used to study interactions between sets of factors belonging to a pair of models in which one nests the other. It should be noted that from two CJA we can, through these binary operations, build CJA. So when we nest the treatments from one model in each treatment of another model, we can study the interactions between sets of factors of the first and the second models.
- Algebraic structure for the crossing of balanced and stair nested designsPublication . Fernandes, Célia; Ramos, Paulo; Mexia, João TiagoStair nesting allows us to work with fewer observations than the most usual form of nesting, the balanced nesting. In the case of stair nesting the amount of information for the different factors is more evenly distributed. This new design leads to greater economy, because we can work with fewer observations. In this work we present the algebraic structure of the cross of balanced nested and stair nested designs, using binary operations on commutative Jordan algebras. This new cross requires fewer observations than the usual cross balanced nested designs and it is easy to carry out inference.
- Allee effect bifurcation in the γ-Ricker population model using the Lambert W functionPublication . Rocha, J. Leonel; Taha, Abdel-KaddousThe main purpose of this talk is to present the dynamical study and the bifurcation structures of the γ-Ricker population model. Resorting to the Lambert W function, the analytical solutions of the positive fixed point equation for the γ-Ricker population model are explicitly presented and conditions for the existence and stability of these fixed points are established. Another main focus of this work is the definition and characterization of the Allee effect bifurcation for the γ-Ricker population model, which is not a pitchfork bifurcation. Consequently, we prove that the phenomenon of Allee effect for the γ-Ricker population model is associated to the asymptotic behavior of the Lambert W function in a neighborhood of zero. Numerical studies are included.
- Allee's dynamics and bifurcation structures in von Bertalanffy's population size functionsPublication . Rocha, J. Leonel; Taha, Abdel-Kaddous; Fournier-Prunaret, D.The interest and the relevance of the study of the population dynamics and the extinction phenomenon are our main motivation to investigate the induction of Allee Effect in von Bertalanffy's population size functions. The adjustment or correction factor of rational type introduced allows us to analyze simultaneously strong and weak Allee's functions and functions with no Allee effect, whose classification is dependent on the stability of the fixed point x = 0. This classification is founded on the concepts of strong and weak Allee's effects to the population growth rates associated. The transition from strong Allee effect to no Allee effect, passing through the weak Allee effect, is verified with the evolution of the rarefaction critical density or Allee's limit. The existence of cusp points on a fold bifurcation curve is related to this phenomenon of transition on Allee's dynamics. Moreover, the "foliated" structure of the parameter plane considered is also explained, with respect to the evolution of the Allee limit. The bifurcation analysis is based on the configurations of fold and flip bifurcation curves. The chaotic semistability and the nonadmissibility bifurcation curves are proposed to this family of 1D maps, which allow us to define and characterize the corresponding Allee effect region.
- Allee's dynamics and bifurcation structures in von Bertalanffy's population size functionsPublication . Rocha, J. Leonel; Taha, Abdel-Kaddous; Fournier-Prunaret, D.The interest and the relevance of the study of the population dynamics and the extinction phenomenon are our main motivation to investigate the induction of Allee Effect in von Bertalanffy's population size functions. The adjustment or correction factor of rational type introduced allows us to analyze simultaneously strong and weak Allee's functions and functions with no Allee effect, whose classification is dependent on the stability of the fixed point x = 0. This classification is founded on the concepts of strong and weak Allee's effects to the population growth rates associated. The transition from strong Allee effect to no Allee effect, passing through the weak Allee effect, is verified with the evolution of the rarefaction critical density or Allee's limit. The existence of cusp points on a fold bifurcation curve is related to this phenomenon of transition on Allee's dynamics. Moreover, the "foliated" structure of the parameter plane considered is also explained, with respect to the evolution of the Allee limit. The bifurcation analysis is based on the configurations of fold and flip bifurcation curves. The chaotic semistability and the nonadmissibility bifurcation curves are proposed to this family of 1D maps, which allow us to define and characterize the corresponding Allee effect region.
- Allee's effect bifurcation in generalized logistic mapsPublication . Rocha, J. Leonel; Taha, Abdel-KaddousThis paper concerns the study of the Allee effect on the dynamical behavior of a new class of generalized logistic maps. The fundamentals of the dynamics of this 4-parameter family of one-dimensional maps are presented. A complete classification of the nature and stability of its fixed points is provided. The main results relate to the Allee effect bifurcation: a new type of bifurcation introduced for this class of unimodal maps. A necessary and sufficient condition so that the Allee fixed point is a snap-back repeller is established. In addition, in the parameters space is defined an Allee's effect region, which determines the existence of an essential extinction for the generalized logistic maps. Local and global bifurcations of generalized logistic maps are investigated.
- An algorithm for constrained optimization with applications to the design of mechanical structuresPublication . Barbarosie, Cristian; Lopes, Sérgio; Toader, Anca-MariaWe propose an algorithm for minimizing a functional under constraints. It uses _rst order derivatives of both the objective function and the constraints. The step is computed as a sum between a steepest descent step (which minimizes the objective functional) and a correction step related to the Newton method (which aims to solve the equality constraints). The linear combination between these two steps envolves coefficients similar to Lagrange multipliers which are computed in a natural way based on the Newton method. The algorithm uses no projection and thus the iterates are not feasible; the constraints are only satis_ed in the limit (after convergence). Although the algorithm can be used as a general-purpose optimization tool, it is designed speci_cally for problems where _rst order derivatives of both objective and constraint functionals are available but not second order derivatives (as is often the case in structural optimization).
- An alternative proof on higher order derivatives of a multilinear mapPublication . Carvalho, SoniaAs a generalization of the formulas proved by Bhatia, Grover and Jain (Derivatives of tensor powers and their norms. Electron J Linear Algebra. 2013;26:604-619), in recent papers (The kth derivative of the immannant and the chi-symmetric tensor power of an operator. Electron J Linear Algebra. 2014;27:Article 18, On derivatives and norms of generalized matrix functions and respective symmetric powers. Electron J Linear Algebra. 2015;30:Article 22) Carvalho and Freitas obtained formulas for directional derivatives, of all orders, for generalized matrix functions and for every symmetric tensor power associated with a character xi of a subgroup G of the symmetric group S-m. Throughout our work, we used some well-known formulas for the derivatives of all orders of a multilinear map, since the maps that we studied are all multilinear. In this paper, we intend to present an alternative proof of these formulas, using the multilinearity argument.