## Cristiana J. Silva

## Mathematical models for HIV infection and TB-HIV/AIDS co-infection

Two population mathematical models are proposed for HIV infection and TB-HIV/AIDS co-infection. The respective basic reproduction numbers are

computed, equilibria and stability are studied. A case study is presented for the HIV model and optimal control theory is applied to the TB-HIV/AIDS

co-infection model.

computed, equilibria and stability are studied. A case study is presented for the HIV model and optimal control theory is applied to the TB-HIV/AIDS

co-infection model.

Marisa L. Toste

## Convolutional Codes over Rings

We will start this talk by briefly presenting the area
of error correcting codes. This is one of the most interesting
areas of applied mathematics and is used in most of digitally
represented data, as CD players, TV, satellites, mobiles. This
is due to the fact that in real life applications errors occur and we need
to correct them. After presenting the basic ideas of this area we introduce the main
topic of this talk, namely convolutional codes over finite rings. In particular
we focus on the ring Z

^{r}_{p}. Classical convolutional codes over finite fields are a well-known class of codes widely used in mobiles and satellite communications. Convolutional codes are codes that require memory to encode the data. Convolutional codes over finite rings have been recently discovered to be the most optimal codes for code modulation. However these codes are more involved than standard codes over fields. From a mathematical point of view, they can be seen as*R*[x] - submodules of*R*[x]^{n}, where*R*is a finite ring and*R*[x] is the ring of polynomials over*R*. We will introduce this powerful class of codes and their main properties. We shall conclude the presentation by presenting some novel results on convolutional codes over finite rings which are Maximum Distance Separable.

Simão Pedro Silva Santos

## Higher-order variational problems of Herglotz type with time delay

We study, using an optimal control point of view, higher-order variational problems of Herglotz type with time delay. Namely, we deduce a higher-order Euler--Lagrange and DuBois--Reymond necessary optimality condition as well as a higher-order Noether type theorem for delayed variational problems of Herglotz type.