Summer School 2019
Joint organization of the UC|UP, MAP-PDMA and PDMat-UA PhD Programs
University of Minho, Braga
September 9 – 13, 2019
Summer School 2018
Joint organization of the UC|UP, MAP-PDMA and PDMat-UA PhD Programs
University of Aveiro, Aveiro
September 3-7 and 10-14, 2018
MAP-PDMA Workshop Participants
PROGRAM:
June 29
J.C. Espirito Santo [UM] – 14:30 | 15:20
Rita Guerra [UA] – 15:25 | 15:45
Abstract: The main aim of this work is to obtain Heisenberg uncertainty principles for a specific oscillatory integral operator with different trigonometric phases. Additionally, invertibility theorems, Parseval type identities, Plancherel type theorems and convolutions associated with that operator are also obtained.
Manuel Baptista [CST AG] – 15:50 | 16:10
Over the years, simulations have become more and more important in the industrial context. Improvements in geometric representation, in the algorithms simulating physical processes, in results post-processing, analysis and visualisation, together with the huge development and wide availability of computational power, make simulations a valuable technique for the design and testing of products and services. One of the most cited examples is the simulation of car crashes which save huge amounts of money that physical prototypes would cost. In Electromagnetics there are numerous examples across many industries: electrical machines, particle accelerators, medical devices, home electronics, to mention a few. Antenna placement, optimisation of electronic layouts, electromagnetic interference/compatibility are among the problems to solve during the conception and development of new products containing electronics. In my talk I will try and describe the general aspects required to set up and run an EM (Electromagnetic) Simulation and what branches of Mathematics come into play, trying to raise awareness for the differences, similarities and possible synergies between academy and industry.
coffee break – 16:15 | 16:45
Weronica Wojtak [UM] – 16:45 | 17:10
Abstract: Dynamic Neural Fields (DNFs) formalized by nonlinear integro-differential equations have been introduced as models of the large-scale dynamics of spatially structured networks of neurons, in which the associated integral kernel represents the spatial distribution of synaptic connections between neurons. Neural fields have been used in the past to explain a wide range of neurobiological phenomena, including persistent population activity in higher brain areas associated with cognitive functions. For instance, the memory of sensory events is commonly believed to be stored in persistent activity patterns (“bumps”) that are initially triggered by brief inputs, and subsequently become self-sustained by recurrent interactions within the neural population. In classical neural field models, the fixed amplitude of the bump attractor is solely defined by the strength of the recurrent interactions. In this talk, I will present a novel field model consisting of two reciprocally coupled equations of Amari-type which supports a continuum of bump amplitudes. The model has been applied to explain persistent population activity that increases monotonically with the time integral of the input (parametric working memory). It robustly memorizes the strength and/or duration of inputs,and allows for changing the memory strength at any time by new relevant information. I will show how numerical simulations and analytical tools can be applied to better understand the pattern formation process.
Pedro Soares [UP] – 17:10 | 17:30
Abstract:A coupled cell network is a directed graph where we label cells (vertices) and inputs (edges) with types. To each cell, we associate a dynamical system depending on the cell’s type.Moreover, the dynamical systems are coupled according the connections between the cells in the graph and their types. The resulting dynamical system respects the network structure and it is called a coupled cell system associated to the network. We explore some examples and results about networks and coupled cell systems. One striking example is the relation between balanced colorings in the network and the existence of flow invariant subspaces, given by the equality of some cell coordinates, for any coupled cell system. We also see how coupled cell systems can support non-generic phenomena, like heteroclinic networks and bifurcations of high degeneracy.
Delfim Torres [UA] – 17:35 | 18:25
Abstract: We propose a model for the human immunodeficiency virus type 1 (HIV-1) infection with intracellular delay and prove the local asymptotical stability of the equilibrium points. Then we introduce a control function, representing the efficiency of reverse transcriptase inhibitors, and consider the pharmacological delay associated with the control. Finally, we propose and analyze an optimal control problem with state and control delays. Through numerical simulations, extremal solutions are proposed for minimization of the virus concentration and treatment costs.
June 30
Sofia Castro [UP] – 10:00 | 10:50
Abstract: Several problems in economics are understood by constructing models that high-light all (or, some of) the relevant questions addressed. Often these models are mathematical in nature.
This presentation will report on some models used in di fferent branches of economics and will describe some of the mathematics that proved useful in understanding them. Some of the mathematics is new and only came into being because of the economics problem. Some mathematics is standard but proved useful when applied to the particular problem.
The economics problems include topics such as industry location, oligopoly theory, growth and price setting in general equilibrium. The mathematics ranges from finding zeros of polynomials to bifurcation and singularity theory in dynamical systems, as well as game theory.
Filipe Martins [UP] – 10:55 | 11:15
The fundamental bifurcation for evolutionary matrix models with multiple traits
Abstract: One fundamental question in biology is population extinction and persistence, i.e., stability/instability of the extinction equilibrium and of non-extinction equilibria. In the case of nonlinear matrix models for structured populations, a bifurcation theorem answers this question when the projectionmatrix is primitive by showing the existence of a continuum of positive equilibria that bifurcates from the extinction equilibrium as the inherent population growth rate passes through 1. This theorem also characterizes the stability properties of the bifurcating equilibria by relating them to the direction of bifurcation, which is forward (backward) if, near the bifurcation point, the positive equilibria exist for inherent growth rates greater (less) than 1. In this paper we consider an evolutionary game theoretic version of a general nonlinear matrix model that includes the dynamics of a vector of mean phenotypic traits subject to natural selection.We extend the fundamental bifurcation theorem to this evolutionary model. We apply the results to an evolutionary version of a Ricker model with an added Allee component. This application illustrates the theoretical results and, in addition, several other interesting dynamic phenomena, such as backward bifurcation induced strong Allee effects.
coffee break – 11:20 | 11:50
Benjamim Anwasia [UM] – 11:50 | 12:10
Abstract: We consider a system of Boltzmann-type equations that describe a multi-species gaseous mixture participating in a bimolecular reversible chemical reaction under isothermal condition and perform an asymptotic analysis of a physical situation in which the dominant role in the evolution of the species is played by elastic collisions, while chemical reactions are slow, to obtain a coupled system of equation referred to as the reaction diffusion system of Maxwell-Stefan type, as the hydrodynamic limit of the simple reacting sphere kinetic model.
Liliana Silva [UP] – 12:15 | 12:30
Abstract: The stability of heteroclinic cycles may be obtained from the value of the local stability index along each connection of the cycle. We establish a way of calculating the local stability index for quasi-simple cycles : cycles whose connections are one dimensional and contained in ow-invariant spaces of equal dimension. These heteroclinic cycles exist both in symmetric and non-symmetric contexts. We make one assumption on the dynamics along the connections to ensure that the transition matrices have a convenient form. Our method applies to all simple heteroclinic cycles of type Z and to various heteroclinic cycles arising in population dynamics, namely nonsimple heteroclinic cycles, as well as to cycles that are part of a heteroclinic network. We illustrate our results with a quasi-simple (non-simple) cycle present in a heteroclinic network of the Rock-Scissors-Paper game.
lunch – 12:40 | 14:30
António Machiavelo [UP] – 14:30 | 15:20
Abstract: Modern “secret codesî, or more precisely cryptosystems, have an increasingly mathematical nature. We will explain why this is so, while attempting to give an overview of some relevant parts of modern cryptology (= cryptography + cryptanalysis). All the mathematical concepts needed to understand this talk will be explained.
Pedro Delgado [BOSCH] – 15:250 | 15:45
Nuno Paiva [NOS] – 15:50 | 16:10
coffee break – 16:15 | 16:45
Round Table – 16:35 | 18:15