# Notícias

The *external numbers* are an attempt to model orders of magnitude as numbers, in relation to a nonstandard set of real numbers, rather than functions, which give the notation *O*(.) and *o*(.). The calculation rules are either equal to the rules for real numbers, or are adaptations. In particular each external number has its own zero, called *neutrix*. We present an axiomatic system for the external numbers, in analogy with the axioms for the real numbers, which we complete with an axiom that postulates the existence of a non-trivial neutrix. We build a structure satisfying all the axioms, called a *Complete Arithmetical Solid*, showing the consistency of the axiomatic system. We show how the structure captures the intrinsic imprecisions of orders of magnitudes, the Sorites paradox and informal error analysis. Some applications in error propagation and perturbation analysis are indicated.

In this talk, I will present an ordinary differential equation (ODE) model of immune response dynamics describing the behaviour of CD4 + T cells, regulatory T cells (Tregs) and interleukine-2 cytokine (IL-2) density where regulatory T cells inhibit interleukine-2 secretion. I will succinctly describe some qualitative features of the model such as time evolution, equilibria and bifurcations. I will then fit the model to data reporting the CD4 + T cell numbers from the 28 days following the infection of mice with lymphocytic choriomeningitis virus (LCMV). The data consist of two time series of the Tcell responses to the gp61 and NP309 epitopes of the disease. We observed the proliferation of T cells and, to a lower extent, Tregs during the immune activation phase following infection and subsequently, during the contraction phase, a smooth transition from faster to slower death rates. In this way we have that the ODE model was able to be calibrated thus providing a quantitative description of the data.

A Super Massive Black Hole (SMBH) known as Sagittarius A* (Sgr A*) lurks in the centre of the Milky Way. With a mass _ 4_106M_ of that of the Sun (1 M_) it affects the local stellar systems by the capture of stars, due to its gravitational influence that dominates the surrounding InterStellar Medium (ISM), leading to their disruption through the e_ects of the gravitational tidal forces. The thermal energy release during the Tidal Disruption Events (TDEs) leads to the emission of X-rays that were observed by the XMM-Newton telescope and also have a major role in the energising of the Fermi Bubbles, which are two symmetric structures that occur in both planes of the Galactic Center. This work presents the results of a parametric study of TDEs of a solar type star captured by Sgr A* and the amount of energy released during this phenomenon. The tidal disruption is evaluated in terms of the star trajectory with di_erent penetration parameters (b) and the spatial distribution of the debris is tracked over time in order to determine the fraction of stellar mass that stays bound to the black hole. Joint work with Miguel A. Avillez, Dmat,UE.

Description of an application of iso-spectral theory, which was developed by L. Bunimovich and B. Webb in the context of finite graph theory, to the problem of existence of stationary measures for certain classes of infinite graphs; a joint work with A. Baraviera and M. J. Torres.

The seminar is devoted to decision support in the case of complicated multi-objective (multi-criteria) decision problems. Approximating the Pareto frontier and subsequent informing the decision makers about it is now a recognized method of decision support especially in environmental and other public problems in the framework of which multiple decision makers have to be informed about the objective tradeoffs.

The method used in our studies (the Interactive Decision Maps (IDM) technique) applies visualization of the tradeoff among the objectives in form of decision maps, i.e. collections of bi-objective slices of the Pareto frontier. The IDM technique is based on approximating the Edgeworth-Pareto hull of the feasible objective set. Then, bi-objective slices of the Pareto frontier can be displayed on request of a user. The IDM technique can be applied in the case of more than four objectives (in the so called many-objective decision problems).

A complicated multi-objective problem is considered as an example. It is the problem of constructing control rules for a cascade of reservoirs with criteria reflecting the reliability with which the requirements imposed on the cascade are met. The problem is described by 24 objectives, by several hundreds of decision variables and by non-linear relations between them and objectives. New efficient methods were developed for approximating the Edgeworth–Pareto hull. They combine classical gradient-based methods for scalar optimization and genetic algorithms for Pareto frontier approximation. It was experimentally shown that these methods are significantly superior to the original genetic algorithm in terms of order of convergence and approximation accuracy.

**Júri:**

PRESIDENTE: SARA MARIA DE AZEVEDO E SOUSA MARQUES PEREIRA, PROF.ª AUXILIAR NO DEPARTAMENTO DE PEDAGOGIA E EDUCAÇÃO, DA UNIVERSIDADE DE ÉVORA

VOGAL-ARGUENTE: CARLOS RAMOS, PROF. AUXILIAR NO DEPARTAMENTO DE MATEMÁTICA, DA UNIVERSIDADE DE ÉVORA

VOGAL-ORIENTADOR: FERNANDO CARAPAU, PROF. AUXILIAR NO DEPARTAMENTO DE MATEMÁTICA, DA UNIVERSIDADE DE ÉVORA

VOGAL: FÁTIMA MARIA FILIPE PEREIRA, PROF.ª AUXILIAR NO DEPARTAMENTO DE MATEMÁTICA, DA UNIVERSIDADE DE ÉVORA