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- Provas de Agregação em Matemática: lição, relatório e curriculumPublication . Costa, Fernando Pestana daNuma primeira parte faz-se uma descrição dos tipos de equações de coagulação-fragmentação mais comuns nas literaturas matemática e científica, referindo-se alguns aspectos históricos considerados relevantes, bem como várias áreas de aplicações. Na segunda parte descrevem-se resultados matemáticos relativos a existência e unicidade de soluções de alguns destes sistemas, nomeadamente os sistemas discretos de Smoluchowski e de coagulação-fragmentação: começando com uma breve apresentação dos espaços funcionais utilizados, passam- se depois em revista os resultados sobre existència de soluções fornecendo-se uma descrição breve das ideias subjacentes às demonstrações. Esta parte termina com uma secção dedicada aos problemas de unicidade. Nas terceira e quarta partes descrevem-se diversos aspectos do comportamento de soluções. Focam-se com especial atenção questões sobre a convergência para equilíbrios a tempos longos, sobre o comportamento auto-semelhante de soluções e sobre a conservação, ou não conservação, de densidade. Todas estas questões, além da óbvia relevância matemática, têm também interpretações físicas de clara importância para as aplicações.
- Dynamics of a differential system using invariant regionsPublication . Costa, Fernando Pestana daThe longtime behaviour of a two dimensional system of ordinary differential equations with singularities is studied using conveniently defined positively invariant sets and auxiliary functions. The approach uses only elementary techniques of phase plane analysis and provides a good geometric insight into the dynamical behaviour of the system. It provides dynamical information analogous to what is usually obtained via centre manifold techniques but does not require the flow to be defined at the limit point.
- Dynamics of a Non-Autonomous ODE System Occurring in Coagulation TheoryPublication . Costa, Fernando Pestana da; Sasportes, RafaelWe consider a constant coefficient coagulation equation with Becker–D¨oring type interactions and power law input of monomers J1(t)=αtω, with α > 0 and ω>−1 2 . For this infinite dimensional system we prove solutions converge to similarity profiles as t and j converge to infinity in a similarity way, namely with either j/ς or (j −ς)/√ς constants, where ς =ς(t) is a function of t only. This work generalizes to the non-autonomous case a recent result of da Costa et al. (2004). Markov Processes Relat. Fields 12, 367–398. and provides a rigorous derivation of formal results obtained by Wattis J. Phys. A: Math. Gen. 37, 7823–7841. The main part of the approach is the analysis of a bidimensional non-autonomous system obtained through an appropriate change of variables; this is achieved by the use of differential inequalities and qualitative theory methods. The results about rate of convergence of solutions of the bidimensional system thus obtained are fed into an integral formula representation for the solutions of the infinite dimensional system which is then estimated by an adaptation of methods used by da Costa et al. (2004). Markov Processes Relat. Fields 12, 367–398.
- Long-time behaviour and self-similarity in a coagulation equation with input of monomersPublication . Costa, Fernando Pestana da; Roessel, Henry J. van; Wattis, Jonathan ADFor a coagulation equation with Becker-Doring type interactions and time-independent monomer input we study the detailed long-time behaviour of nonnegative solutions and prove the convergence to a self-similar function
- Kinetics and thermodynamics of poly (9,9-dioctylfluorene) beta-phase formation in dilute solutionPublication . Costa, Fernando Pestana da; Maçanita, António L.; Morgado, Jorge; Dias, Fernando B.; Burrows, Hugh D.; Monkman, Andrew P.Poly(9,9-dioctylfluorene) (PFO) adopts a particular type of conformation in dilute solutions of the poor solvent methylcyclohexane (MCH) below 273 K, which is revealed by the appearance of a red-shifted absorption peak at 437−438 nm. The formation of this ordered conformation depends on the temperature but is independent of polymer concentration over the range studied (3−25 μg/mL). On the basis of absorption, steady-state, and time-resolved fluorescence data, the new absorption peak at 437−438 nm is assigned to a highly ordered conformation of PFO chains, analogous to the so-called β-phase first identified in PFO films. From the study of PFO solutions in MCH as a function of temperature, we conclude that these ordered segments (β-conformation) coexist with less ordered domains in the same chain. When the ordered domains are present, they act as efficient energy traps and the fluorescence from the disordered regions is quenched. The transition between the disordered and the ordered PFO conformations is adequately described by a mechanism that involves two steps: a first, essentially intramolecular, one from a relatively disordered (α) to an ordered conformation (β), followed by aggregation of chains containing β-conformation into anisotropic ordered domains. From the temperature dependence of the 437−438 nm peak intensity, the transition temperature Tβ = 261 K, enthalpy ΔHβ = −18.0 kcal mol-1, and entropy ΔSβ = −68.4 cal K-1 mol-1 were obtained. The formation of the β-conformation domains were also followed as a function of time at 260 K. The rate constants at 260 K were determined, showing an order of magnitude around 10-3 s-1 (kα→β = 5.9 × 10-4 s-1; kβ→α = 9 × 10-4 s-1; kagg = 2.3 × 10-3 M-1 s-1; kdiss = 4.4 × 10-4 s-1). This small magnitude explains the long times required for a “complete” conversion to the β-conformation.
- A hierarchical cluster system based on Horton-Strahler rules for river networksPublication . Costa, Fernando Pestana da; Grinfeld, Michael; Wattis, Jonathan ADWe consider a cluster system in which each cluster is characterized by two parameters: an \order" i; following Horton-Strahler's rules, and a \mass" j following the usual additive rule. Denoting by ci;j (t) the concen- tration of clusters of order i and mass j at time t; we derive a coagulation- like ordinary di erential system for the time dynamics of these clusters. Results about existence and the behaviour of solutions as t ! 1 are ob- tained, in particular we prove that ci;j (t) ! 0 and Ni(c(t)) ! 0 as t ! 1; where the functional Ni( ) measures the total amount of clusters of a given xed order i: Exact and approximate equations for the time evolution of these functionals are derived. We also present numerical results that sug- gest the existence of self-similar solutions to these approximate equations and discuss its possible relevance for an interpretation of Horton's law of river numbers
- Construções geométricasPublication . Costa, Fernando Pestana daNo primeiro capítulo deste texto apresentamos algumas construções geométricas elementares com régua e compasso. No segundo capítulo fornecemos uma panorâmica elementar dos poliedros, incluindo o estudo do teorema de Descartes e da regra de Euler. Num apêndice são incluidas algumas planificações de poliedros que podem ser utilizadas para a construção de modelos tri-dimensionais. Este texto foi produzido para apoio a um curso breve para professores do ensino básico.
- Convergence to self-similarity in an addition model with power-like time-dependent input of monomersPublication . Costa, Fernando Pestana da; Sasportes, Rafael; Pinto, João TeixeiraIn this note we extend the results published in Ref. 1 to a coagulation system with Becker-Doring type interactions and time-dependent input of monomers $J_{1}(t)$ of power–like type: $J_{1}(t)/(\alpha t^{\omega }) \rightarrow 1$ as $t \rightarrow \infty$, with $\alpha > 0$ and $\omega > − \frac{1}{2}$. The general framework of the proof follows Ref. 1 but a different strategy is needed at a number of points.
- New challenges for e-learning post-graduation in exact and technological sciencesPublication . Caeiro, Sandra; Costa, Fernando Pestana da; Moura, Ana Pinto de; Martinho, Ana Paula; Nicolau, Paula Bacelar; Ramos, Maria do Rosário; Araújo, João; Azeiteiro, Ulisses; Rocio, VitorLearning and teaching in an on-line environment are, in many ways, much like teaching and learning in any other formal education context. However, the pervasive effect of the on-line medium creates a unique environment for teaching and learning (Anderson, 2004). This century will see the emergence of a post-modern era of distance education characterized by increased diversity and choice (Garrison, 2000). Such development is made possible by new communication technologies, as exemplified by the evolution of the open universities in their adoption of new models to replace/complement the traditional self-paced, independent learning model of the industrial era (Davis, 1999). The implementation of these new models to formal master courses in specific areas of life sciences, mathematics and technology is a new challenge, where face to face field and laboratory activities are often compulsory. This paper presents a preliminary reflection on the working experience of three masters courses taught at Portuguese distance learning university. In Portugal those courses are novel both in terms of the e- or b-learning regime, and in terms of curricular contents and professional competence outcomes.
- Interchanging knowledge and experiments in a learning community in a view of collaborative workPublication . Amador, Filomena; Valadares, Jorge; Coelho, José; Rocio, Vitor; Caetano, Fernando J. P.; Gaspar, Cândido Dias; Ribeiro, Carlos; Remédios, Jaime; Costa, Fernando Pestana daIn this work we describe and analyze an experiment based in a programme of permanent seminars – e-DCET organized with the objective of developing practices of collaborative work inside the Departamento de Ciências Exactas e Tecnológicas (Universidade Aberta - Portugal), valuing the active sharing of pedagogic practices and knowledge at a variety of levels. Various themes were discussed and analyzed, among them we highlighted some experiences of online teaching in formal and non-formal contexts, user management in Moodle LMS, news from international e-learning meetings, visual representations and teaching chemistry online, pseudo-code as a communication language using Notetab and interactive materials in teaching mathematics.