Application Deadlines 2023-2024 here

ACADEMIC TERM |

1st semester: 3/Oct – 15/Jan |

Christmas holydays: 18/Dec – 29/Dec |

2nd semester: 12/Feb – 31/May |

Easter holydays: 25/mar – 1/Apr |

**Information about the courses** here

**SEMINARS***For information about this UC, visit the*

Course Description

**Seminar #10: MAP-PDMA PhD Program 2023/2024 — December 5, 10:00**

* Place: Via zoom at

https://videoconf-colibri.zoom.us/j/86081126614?pwd=cVJDVDhCOUtReENjVHlTUHF5T0VpQT09

* Speaker: Ana Jacinta Soares, Centre of Mathematics, University of Minho

email: ajsoares@math.uminho.pt

* Title:** A network model with human-landscape interactions**

* Abstract: Mathematical models help to understand what we see and what we think about the nature [2]. They provide qualitative explanations for patterns in nature and can be used in socio and urban planning studies in view of developing management strategies of territorial transformation. One important concept in our society is the territorial resilience, because it describes the ability of a system to tolerate the impact caused by adverse circumstances and adapt progressively its configuration. The intricate structure of an environmental system can be modelled by ecological networks that describes a mosaico of interconnected non-uniform landscape regions.

In this seminar, we present a human-landscape model for a network of Landscape Units (LUs), consisting of a network of interacting dynamical systems, all sharing the same qualita- tive structure [7]. The interaction among the various dynamical systems is defined by a linear diffusivity term describing the flux of bio-energy.

We study the dynamical system, existence of equilibria and their stability, and possible occurrence of Hopf bifurcations with consequent periodic oscillations of environmental and human variables, as typical of resilient territories. Then we show some numerical simulations of different scenarios in a sample model of an environmental system.

[1] J. Pastor, *Mathematical Ecology of Populations and Ecosystems*, in S. Friedlander and D. Serre Eds, Handbook of Mathematical Fluid Dynamics, John Wiley & Sons, 2008.

[2] R.D. Marca, M. Groppi, A.J. Soares, Human-induced oscillations in a network landscape model. Communications in Nonlinear Science and Numerical Simulation, 115 (2022), 1– 15.

**Seminar #9: MAP-PDMA PhD Program 2023/2024 — November 28, 10:00**

* Place: Room FC1 108 of DMat-FCUP, and via zoom at

https://videoconf-colibri.zoom.us/j/86081126614?pwd=cVJDVDhCOUtReENjVHlTUHF5T0VpQT09

* Speaker: Manuel Delgado, FCUP & CMUP

email: mdelgado@fc.up.pt

* Title: **Problems in numerical semigroups**

* Abstract:** **A *numerical semigroup *is a cofinite submonoid S of the additive monoid **N **of non negative integers, i.e. a subset containing 0, stable under addition and with finite complement **N**\S. Equivalently, it is a set of the form S=<a1,…,an> = **N**a1+ … +**N**an where a1,…,an are positive integers with gcd(a1,..,an)=1, called *generators *of S. The size of **N**\S is called the *genus *of S.

In this seminar will present an overview on two problems that attracted the attention of many researchers in the area of numerical semigroups. One of these problems, known as counting numerical semigroups by genus, consists of the study of the sequence (ng) of the number of numerical semigroups with genus g. The other is Wilf’s conjecture; it will be stated in the seminar after introducing the necessary terminology.

I intend to accompany the overview with experimental results.

M. Delgado, S. Eliahou and J. Fromentin, *A verification of Wilf’s conjecture up to genus 100, *(2023)https://arxiv.org/abs/2310.07742

M. Delgado, P.A. García-Sánchez, *NumericalSgps, A package for numerical semigroups*, Version 1.3.1 (2022)

Manuel Delgado, *Conjecture of Wilf: A Survey*, Numerical Semigroups: IMNS 2018. 2020:39-62.

Manuel Delgado, *Trimming the numerical semigroups tree to probe Wilf’s conjecture to higher genus, *(2019) https://arxiv.org/abs/1910.12377

**Seminar #8: MAP-PDMA PhD Program 2023/2024 — November 21, 10:00**

* Place: Room FC1 108 of DMat-FCUP, and via zoom at

https://videoconf-colibri.zoom.us/j/86081126614?pwd=cVJDVDhCOUtReENjVHlTUHF5T0VpQT09

* Speaker: Paula Brito, Faculdade de Economia, Universidade do Porto & LIAAD INESC TEC

email: mpbrito@fep.up.pt

* Title:** Linear models for distributional data analysis**

* Abstract: In a time when increasingly larger and complex data collections are being produced, new and adaptive forms of data representation and analysis have to be conceived and implemented. However, large amounts of data may often not be analysed directly, as collected, given their size and high entropy – consider high frequency data streams, e.g. captured by sensors, internet flows or large networks – and in general this would be of no interest, as only an analysis at higher level may put in evidence patterns of interest. Therefore, some data aggregation prior to the analysis is necessary. To prevent a too important information loss when individual observations are aggregated, variability across records should be somehow kept. New data representations are considered, as the elements of each cell of the aggregated data array will no longer be single real values or categories, as in the classical case, but finite sets of values, intervals or, more generally, distributions.

In this seminar, we are interested in numerical distributional data, where units are described by histogram or interval-valued variables. Linear models for such distributional variables are proposed, which rely on the representation of histograms or intervals by the associated quantile functions, under specific assumptions. These then allow for multivariate analysis of distributional-valued data, e.g. multiple linear regression or linear discriminant analysis.

Ref1– Dias,S.,Brito,P.(2015).LinearRegressionModelwithHistogram-ValuedVariables. Statistical Analysis and Data Mining, vol. 8, issue 2, 75-113. DOI: 10.1002/sam.11260.

Ref2– Dias, S., Brito, P., Amaral, P. (2021). Discriminant analysis of distributional data via fractional programming. European Journal of Operational Research, 294(1), 206-218.

Ref3– Brito,P.,Dias,S.(2022).AnalysisofDistributionalData.ChapmanandHall/CRC.

**Seminar #7: MAP-PDMA PhD Program 2023/2024 — November 21, 9:00**

* Place: Room FC1 108 of DMat-FCUP, and via zoom at

https://videoconf-colibri.zoom.us/j/86081126614?pwd=cVJDVDhCOUtReENjVHlTUHF5T0VpQT09

* Speaker: Filipe Martins, FCUP & CMUP

email: luis.f.martins@fc.up.pt

* Title:** Bifurcations in evolutionary matrix models in population dynamics**

* Abstract: In this talk I will consider evolutionary game-theoretic versions of a general class of non-linear matrix models frequently used in population dynamics.

Matrix models first found their way in mathematical biology in the celebrated Leslie matrix model due to Patrick H. Leslie in two papers published in the 1940’s. Many developments have occurred since, with the introduction of non-linear dependencies on the entries of the matrices, or the incorporation of evolutionary effects in the dynamics, which I will talk about in this seminar. The evolutionary components of the model track the dynamics of a vector of mean phenotypic traits subject to natural selection ([1] is an excellent introduction to the topic of Darwinian dynamics modelling in mathematical biology).

One fundamental question in population dynamics and mathematical biology is the question of population extinction and persistence, i.e., in mathematical terms, the stability/instability of the extinction equilibrium and of other non-extinction equilibria, or other types of structures such as periodic orbits. I will discuss this question through the prism of dynamic bifurcations. When the model parameters (more precisely, the inherent population growth rate), dynamic bifurcations occur, opening possibility for population persistence and non-extinction, recurrence, or to possible extinction. The results present a complete answer to a general class of evolutionary matrix models often used in mathematical biology, the mathematical assumption being that the matrix is primitive.

I will present an application of the general theoretical results to an evolutionary version of a classic Ricker population model. This application illustrates the theoretical results and, in addition, several other interesting dynamic phenomena such as period-doubling bifurcations and chaos.

Most of the results and conclusions that I will talk about in this seminar are presented in the paper [2] (joint work with Jim M. Cushing, Alberto Pinto and Amy Veprauskas).

[1] Joel S. Brown and Thomas L. Vincent, Evolutionary Game Theory, Natural Selection and Darwinian Dynamics, Cambridge University Press, 2005.

[2] “A bifurcation theorem for evolutionary matrix models with multiple traits”, Journal of Mathematical Biology, Vol. 75, Issue 2, pp. 491–520, 2017.

**Seminar #6: MAP-PDMA PhD Program 2023/2024 — November 14, 10:00**

https://videoconf-colibri.zoom.us/j/86081126614?pwd=cVJDVDhCOUtReENjVHlTUHF5T0VpQT09

* Speaker: Pedro Aguiar, FEUP & SYSTEC

email: pedro.aguiar@fe.up.pt

* Title:** Model based control design combining Lyapunov and optimization tools to empower trusted autonomy of robotic vehicles**

* Abstract: The past few decades have witnessed a significant research effort in the field of Lyapunov model based control design. In parallel, optimal control and optimization model based design have also expanded their range of applications, and nowadays, receding horizon approaches can be considered a mature field for particular classes of control systems.

In this talk, I will argue that Lyapunov based techniques play an important role for analysis of model based optimization methodologies and moreover, both approaches can be combined for control design resulting in powerful frameworks with formal guarantees of robustness, stability, performance, and safety. Illustrative examples in the area of motion control of autonomous robotic vehicles will be presented for Autonomous Underwater Vehicles (AUVs), Autonomous Surface Vehicles (ASVs) and Unmanned Aerial Vehicles (UAVs).

**References**

Ref 1 – Matheus Reis, A. Pedro Aguiar, Paulo Tabuada, “*Control Barrier Function based Quadratic Programs Introduce Undesirable Asymptotically Stable Equilibria”.* IEEE Control Systems Letters (L-CSS), vol. 5, no. 2, pp. 731-736, April 2021

Ref 2 – Andrea Alessandretti, A. Pedro Aguiar, and Colin N. Jones, “*An Input-to-State-Stability approach to Economic Optimization in Model Predictive Control”*. IEEE Transactions on Automatic Control, Vol. 62, No. 12, pp. 6081-6093, Dec. 2017.

Ref 3 – Andrea Alessandretti, A. Pedro Aguiar, “*An optimization-based cooperative path-following framework for multiple robotic vehicles”. *IEEE Transactions on Control of Network Systems, Vol. 7, No. 2, pp. 1002-1014, 2020

**Seminar #5: MAP-PDMA PhD Program 2023/2024 — November 14, 9:00**

https://videoconf-colibri.zoom.us/j/86081126614?pwd=cVJDVDhCOUtReENjVHlTUHF5T0VpQT09

* Speaker: Tinatin Davitashvili, Dept. Mathematics, Tbilisi State University, Georgia

email: tinatin.davitashvili@tsu.ge

* Title:** Multipoint Nonlocal Contact Problem for Elliptic Equation in Rectangular Area**

* Abstract: Nonlocal contact problem for two-dimensional linear elliptic equations is stated and investigated. The method of separation of variables is used to find the solution of a stated problem in the case of Poisson’s equation. Then, the more general problem with nonlocal multipoint contact conditions for elliptic equation with variable coefficients is considered, and the iterative method to solve the problem numerically is constructed and investigated. The uniqueness and existence of the regular solution are proved. The iterative method allows reducing the solution of a nonlocal contact problem to the solution of a sequence of classical boundary value problems. The numerical experiment is conducted. The results fully agree with the theoretical conclusions and show the efficiency of the proposed iterative procedure.

**Seminar #4: MAP-PDMA PhD Program 2023/2024 — November 7, 10:00**

https://videoconf-colibri.zoom.us/j/86081126614?pwd=cVJDVDhCOUtReENjVHlTUHF5T0VpQT09

* Speaker: Paulo Beleza Vasconcelos, Faculdade de Economia, Universidade do Porto & CMUP

email: pjv@fep.up.pt

* Title:** Low rank approximation in the computation of first kind integral equations in Tau Toolbox**

* Abstract: Tautoolbox is a mathematical library for solving integro-differential problems. Over the past few years, a class within Tautoolbox, called polynomial, has been developed for approximating functions by classical orthogonal polynomials and it is intended to be an easy-to-use yet efficient object-oriented framework.

In this work we explain how this class can be useful in the solution of linear ill-posed problems, avoiding the explicit discretization for the use of finite numerical linear algebra techniques. The Tikhonov regularization method and the truncated singular value expansion are implemented. We discuss how the polynomial class has been designed to fit the needs of applications and provide a description of the available methods, including low rank approximations for bivariate functions.

Numerical experiments illustrate that this approach is capable of efficiently compute good approximations of linear discrete ill-posed problems, even facing perturbed available data function, with no programming effort. Several test problems coming from real applications are used to evaluate the performance and reliability of the solvers.

A. Alqahtani, T. Mach, L. Reichel, Solution of ill-posed problems with Chebfun, Numerical Algorithms, 92(4) 2341—2364, 2023.

S. Gazzola, P. C. Hansen, J. G. Nagy, IR tools: a MATLAB package of iterative regularization methods and large-scale test problems, Numerical Algorithms, 81(3) 773—811, 2019.

Townsend, L. N. Trefethen, An extension of Chebfun to two dimensions, SIAM Journal on Scientific Computing 35(6) C495–C518, 2013.

Townsend, L.N. Trefethen, Continuous analogues of matrix factorizations, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 471~(2173) (2015) 20140585.

P.B. Vasconcelos, J.E. Roman, J.Matos, Solving differential eigenproblems via the spectral Tau method, Numerical Algorithms, 92(3) 1789—1811, 2023.

**Seminar #3: MAP-PDMA PhD Program 2023/2024 — October 31, 10:00**

https://videoconf-colibri.zoom.us/j/86081126614?pwd=cVJDVDhCOUtReENjVHlTUHF5T0VpQT09

* Speaker: Alexandre Rodrigues, Lisbon School of Economics and Management, Center of Mathematics of University of Porto

email: arodrigues@ieg.ulisboa.pt, alexandre.rodrigues@fc.up.pt

* Title:** Bifurcations in epidemic models**

* Abstract: In this session, we will describe the SIR (modified) epidemic model with and without vaccination with the specificity that the parameter modelling the infection rate presents seasonal variations.

The tour will be based on the description of mathematical concepts linked to differential equations with periodic perturbations, bifurcation theory, chaos theory and impulsive equations.

We will also build a bridge between the mathematical results obtained and empirical descriptions already existing in the literature. Joint work with João Maurício de Carvalho (Univ. Porto).

J. Carvalho, A. Rodrigues, Strange attractors in a dynamical system inspired by a seasonally forced SIR model, Physica D, 434, 2022.

J. Carvalho, A. Rodrigues, SIR Model with Vaccination: Bifurcation Analysis, Qualitative Theory of Dynamical Systems, 2023.

J. Carvalho, A. Rodrigues, Pulse vaccination in the SIR model: global dynamics, In conclusion, 2023.

**Seminar #2: MAP-PDMA PhD Program 2023/2024 — October 24, 10:00**

https://videoconf-colibri.zoom.us/j/86081126614?pwd=cVJDVDhCOUtReENjVHlTUHF5T0VpQT09

* Speaker: Roman Chertovskih, Research Center for Systems and Technologies (SYSTEC), Advanced Production and Intelligent Systems Associated Laboratory (ARISE), Engineering Faculty, Porto University

email: roman@fe.up.pt

* Title: **How to make a magnet out of a liquid metal: dynamo theory and its applications**

* Abstract: We will discuss the dynamo problem [1] – magnetic field generation by flows of an electrically conducting fluid. We will survey the magnetic activity in the Universe: magnetic fields of planets, stars and galaxies [2], and consider an important engineering application – magnetic fields in liquid metals cooling a reactor. To simulate such magnetic phenomena, the governing equations will be introduced and the basics of magnetohydrodynamics [3] will be discussed. We also plan to overview the mathematical methods used in the analysis of such problems: ranging from the dynamical systems theory [4] and the equivariant bifurcation theory [5] to the numerical spectral methods [6]. Finally, the use of high performance computers for the considered problems will be addressed.

[1] Moffatt K., Dormy E. Self-Exciting Fluid Dynamos. Cambridge Texts in Applied Mathematics, Cambridge University Press, 2019.

[2] Rudiger G., Hollerbach R. The Magnetic Universe: Geophysical and Astrophysical Dynamo Theory. Wiley, 2004.

[3] Molokov S., Moreau R., Moffatt K. Magnetohydrodynamics: Historical Evolution and Trends. Springer, 2007.

[4] Bohr T., Jensen M.H., Paladin G., Vulpiani A. Dynamical Systems Approach to Turbulence. Cambridge Nonlinear Science Series, Cambridge University Press, 2005.

[5] Chossat P., Lauterbach R. Methods in Equivariant Bifurcations and Dynamical Systems. Advanced Series in Nonlinear Dynamics, World Scientific, 2000.

[6] Canuto C., Hussaini M.Y., Quarteroni A., Zang T.A. Spectral Methods: Fundamentals in Single Domains. Scientific Computation series, Springer, 2006.

**Seminar #1: MAP-PDMA PhD Program 2023/2024 — October 17, 10:00**

https://videoconf-colibri.zoom.us/j/86081126614?pwd=cVJDVDhCOUtReENjVHlTUHF5T0VpQT09

* Speaker: José Abílio Matos, Faculdade de Economia, Universidade do Porto & CMUP

email: jamatos@fep.up.pt

* Title: **Meta-analysis: detecting the effect size not the bias**

* Abstract: More and more computers are becoming an integral part of the Scientific process in general and of Applied Mathematics in particular. Scientific Computing, or as it is also know Compu- tational Science, is the research field that deals with the application of computers in scientific problems.

Such as any other field Scientific Computing has evolved in last decades with new tools and techniques that take advantage of the progress both in the computer power that we have available but also from the successes and failures of the methodologies when applied in practice [1]. Yet, even although computational methods are increasingly an important part of mathematical research, they are somewhat ignored in the training stage of PhD students. It was as if students were expected to learn Advanced Calculus from scratch and without any particular training.

Even with proper training there are problems that arise in practice. Some can be due just to inertia, it takes some effort to start applying new processes, and sometimes it can difficult to choose from the available options is a landscape that is continuously changing and improving. Also from a more philosophical point of view Scientific Computing is not equivalent to Computer Science but it is also not completely dissociated from it, on one hand. On the other hand we can not forget some of the insights in the Mathematical research in the past or else we will be condemned to repeat the same mistakes all over again. The challenges are many but the opportunities from the development that comes from an effective use of Scientific Computing are already known but that can go further if we apply it since the initial development of projects and not just as an after-thought.

In this talk we will discuss some of the best practices, and tools, in Scientific Computing while at the same time giving some concrete examples. Some of those are related with applica- tions to Applied Mathematics problems, for ongoing research projects ranging from Numerical Analysis[3] to Time Series Analysis[4]. Other examples are related with inner work of soft- ware release and distribution that is the equivalent of the pluming of the Scientific Computing ecosystem.

[1] Wilson G, Bryan J, Cranston K, Kitzes J, Nederbragt L, Teal TK “Good enough practices in scientific computing”. PLoS Comput Biol 13(6): e1005510. (2017) https://doi.org/ 10.1371/journal.pcbi.1005510

[2] Benureau Fabien C. Y., Rougier Nicolas P., “Re-run, Repeat, Reproduce, Reuse, Replicate: Transforming Code into Scientific Contributions. Frontiers in Neuroinformatics, vol. 11, doi: 10.3389/fninf.2017.00069, issn: 1662-5196 (2018) https://www.frontiersin. org/articles/10.3389/fninf.2017.00069

[3] Tau Toolbox – for the solution of integro-differential problems https://cmup.fc.up.pt/ tautoolbox/

[4] xdfa – a software package to compute Detrended Fluctuation Analysis (DFA) and related methods https://www.fep.up.pt/docentes/jamatos/xdfa/