Title: Nonlinear diffusion in image processing

Abstract: Nonlinear diffusion equations are commonly used in many recent models for problems in sciences and engineering. In this talk we give a snapshot of our research in this area considering applications in image processing, as nonlinear diffusion filtering and segmentation. We present the main theoretical ideas and discuss the appropriate numerical methods. The main focus of our work is on implicit-explicit schemes as well as on splitting schemes. These numerical methods are studied according to their quantitative and qualitative characteristics. Some numerical results are presented in the context of a collaboration with the Institute of Biomedical Research in Light and Image (IBILI), a research institution of the Faculty of Medicine of the University of Coimbra.  

Title: Multi-objective Optimization in Engineering Problems

Abstract: The majority of real world engineering problems presents multi-objective nature. In spite of this nature, they are mostly formulated with a single objective and several constraints and, therefore, losing some of the original meaning. So, it is advantageous to consider multi-objective approaches since they provide a set of efficient solutions that, simultaneously, are optimal in terms of multiple objectives. The conflicting objectives have to be optimized to achieve an efficient set of solutions which, through the multi-objective approach, permitting the decision maker to perceive and balance several criteria in engineering problems. Although there are different ways to approach a multi-objective optimization problem, most work in the area of evolutionary multi-objective optimization is concentrated on the approximation to the Pareto optimal set. In this talk, some results are presented on the use of evolutionary algorithms to solve multi-objective problems from mechanical engineering, robotics and wastewater treatment plant process.

Title: The dual risk model and capital application involving research and
development

Abstract: We consider the dual risk model, dual to the well known classical risk model for insurance applications, where premiums are regarded as costs and claims are viewed as profits. The surplus can be interpreted as a venture capital like the capital of an economic activity involved in research and development. Like most authors, we consider an upper dividend barrier so that we model the gains of the venture capital and its return to the capital holders.
Based on the classical compound Poisson process, we show and explain clearly the dividends process dynamics, the properties of the different random quantities involved as well as their relations. The connections to the classical risk model together with the different variables involved are crucial in most of our developments. Using that connection, together with an additional upper absorbing barrier and allowing the process to continue after ruin, we derive several known and unknown results for the dual. Some results about expected discounted dividends are known from the literature, several authors have addressed the problem. We go further. Based on some of the methods retrieved from the positive claims model, we address our study on different ruin and dividend probabilities. Such as the calculation of the probability of a dividend, number of dividends, expected and amount of dividends as well as the time of getting a dividend and inter-occurrence times.
We obtain some integro-differential equations for the above results and also Laplace transforms, then we can get either numerical or analytical results for cases where solutions and/or inversions are possible.

Title: (Mathematical) Optimization in Health Care

Abstract: Does the usage of mathematics in health care simply amount to the deployment of new technology so to speed up processes? Is it just manipulating statatistical data to validate treatment quality?
In this talk we will see, through examples, that mathematics is rather useful in the assistance of health care managers to take better decisions in an environment of increasingly scarce resources. 

Title: Modeling Local Immune Responses by T cells with Regulatory T cells

Abstract: We analyse the effect of the regulatory T cells (Tregs) in the local control of the immune responses by T cells. We obtain an explicit formula for the level of antigenic stimulation of T cells as a function of the concentration of T cells and the parameters of the model. The relation between the concentration of the T cells and the antigenic stimulation of T cells is an hysteresis, that is unfold for some parameter values. We study the appearance of autoimmunity from cross-reactivity between a pathogen and a self antigen or from bystander proliferation due to the increased levels of interleukine 2 (IL-2). We also study an asymmetry in the death rates. With this asymmetry we show that the antigenic stimulation of the Tregs is able to control locally the population size of Tregs. We use a linear relation between the antigenic stimulation of T cells and the antigenic stimulation of Tregs to model a positive correlation between them. The rate of variation of the levels of antigenic stimulation determines if the outcome is an immune response or if Tregs are able to maintain control. This behavior is explained by the presence of a transcritical bifurcation for some tuning between the antigenic stimuli of T cells and Tregs.