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- a-fractal function with variable parameters: an explicit representationPublication . Priyanka, T. M. C.; Serpa, Cristina; Gowrisankar, A.In this paper, new results on the alpha-fractal function with variable parameters are presented. The Weyl-Marchaud variable order fractional derivative of an alpha-fractal function with variable parameters is examined by imposing certain conditions on the scaling factors. Following the investigation of fractional derivative, the definite integral of the alpha-fractal function with variable parameters is evaluated for various intervals in the prescribed domain. Finally, an explicit structure for the alpha-fractal function is provided using the base q representation of numbers.
- Análise e otimização em sistemas de comunicação ferroviáriaPublication . Batalha, Fábio Ramos; Cal, Filipe Santiago; Lopes, Nuno David de JesusO presente relatório resulta do estágio curricular do Mestrado em Matemática Aplicada para a Indústria, realizado no Instituto Superior de Engenharia de Lisboa (ISEL), em colaboração com a empresa SOLVIT – Innovation & Development on Telecommunications, Lda. O trabalho enquadrou-se na área das telecomunicações ferroviárias, com enfoque no estudo da qualidade de sinal do sistema GSM-R (Global System for Mobile Communications – Railway), tecnologia crítica para assegurar comunicações fiáveis e seguras entre maquinistas, centros de controlo e infraestruturas ferroviárias. O estágio teve como principal objetivo a aplicação de metodologias matemáticas e computacionais à análise e otimização de comunicações móveis em ambiente ferroviário, com foco no sistema GSM-R, tendo sido desenvolvidos dois casos de estudo distintos e complementares. O primeiro caso de estudo incidiu sobre a Linha do Algarve, com enfoque na deteção de padrões e anomalias na qualidade do sinal. Para tal, foi elaborado um processo sistemático de tratamento e padronização de dados recolhidos, envolvendo filtragem, normalização, interpolação e smoothing de séries temporais. Posteriormente, foram aplicadas técnicas estatísticas para identificar anomalias nos parâmetros que caracterizam a qualidade de comunicação da rede. Os resultados obtidos permitiram construir viagens modelo representativas do comportamento típico da rede e localizar segmentos críticos, constituindo um contributo para o desenvolvimento de ferramentas automáticas de monitorização e apoio à manutenção preventiva. O segundo caso de estudo centrou-se no planeamento de frequências na Linha de Cascais, onde a topografia e a proximidade com o mar potenciam fenómenos de interferência nas comunicações. O problema foi formulado como uma tarefa de otimização combinatória, incorporando restrições práticas do sistema GSM-R, no que diz respeito à alocação de portadoras no mesmo site, ou em sites vizinhos. Para a sua resolução, foi desenvolvida uma estratégia híbrida baseada em técnicas Max-Min e heurísticas greedy, visando maximizar o rácio carrier-to-interference (C/I) ao longo do percurso. A configuração final obtida demonstrou melhorias expressivas, com ganhos significativos face às configurações de referência. De forma global, o estágio permitiu combinar métodos de análise estatística e técnicas de otimização com aplicação prática no setor ferroviário, reforçando a importância da Matemática Aplicada como ferramenta de suporte à tomada de decisão em sistemas de comunicações críticas. Os resultados alcançados evidenciam não só a aplicabilidade das metodologias desenvolvidas em contextos reais, como também o seu potencial para serem adaptadas a outras linhas ferroviárias e a diferentes cenários tecnológicos.
- Automatic completion of data gaps applied to a system of water pumpsPublication . Enguiça, Ricardo; Soares, FilipaWe consider a time series with real data from a water lift station, equipped with three water pumps which are activated and deactivated depending on certain starting and halting thresholds. Given the water level and the number of active pumps, both read every 5 min, we aim to infer when each pump was activated or deactivated. To do so, we build an algorithm that sets a hierarchy of criteria based on the past and future of a given interval to identify which thresholds have been crossed during that interval. We then fill the gaps between the 5 min time steps, modeling the water level continuously with a piecewise linear function. This filling takes into account not only every water level reading and every previously identified change of status, but also the fact that activation and deactivation of a pump has no immediate effect on water level. This allows for the fulfillment of the ultimate objective of the problem in its real context, which is to provide the water management company an estimate of how long each pump has been working. Additionally, our estimates correct the errors contained in the time series regarding the number of active pumps.
- Error estimates and generalized trial constructions for solving ODEs using physics-informed neural networksPublication . Babni, Atmane; Jamiai, Ismail; Rodrigues, José AlbertoIn this paper, we address the challenge of solving differential equations using physics-informed neural networks (PINNs), an innovative approach that integrates known physical laws into neural network training. The PINN approach involves three main steps: constructing a neural-network-based solution ansatz, defining a suitable loss function, and minimizing this loss via gradient-based optimization. We review two primary PINN formulations: the standard PINN I and an enhanced PINN II. The latter explicitly incorporates initial, final, or boundary conditions. Focusing on first-order differential equations, PINN II methods typically express the approximate solution as u˜(x,θ)=P(x)+Q(x)N(x,θ), where N(x,θ) is the neural network output with parameters θ, and P(x) and Q(x) are polynomial functions. We generalize this formulation by replacing the polynomial Q(x) with a more flexible function ϕ(x). We demonstrate that this generalized form yields a uniform approximation of the true solution, based on Cybenko’s universal approximation theorem. We further show that the approximation error diminishes as the loss function converges. Numerical experiments validate our theoretical findings and illustrate the advantages of the proposed choice of ϕ(x). Finally, we outline how this framework can be extended to higher-order or other classes of differential equations.
- Fractal Laplace transform: analyzing fractal curvesPublication . Khalili Golmankhaneh, Alireza; Welch, Kerri; Serpa, Cristina; Rodríguez-Lopez, RosanaThe concept of Laplace transform has been extended to fractal curves, enabling the solution of fractal differential equations with constant coefficients. This extension, known as the fractal Laplace transform, is particularly useful for handling inhomogeneous differential equations that involve delta Dirac functions and step functions within the realm of fractal functions. A comprehensive table of essential formulas for the fractal Laplace transform has been compiled to facilitate its application in various scenarios. By utilizing this transformative approach, researchers can now delve into the study of fractal functions and address complex problems involving non-traditional geometries. To illustrate the practicality of the fractal Laplace transform, several examples are provided, showcasing its effectiveness in solving fractal differential equations. This advancement represents a significant augmentation of the classical Laplace transform, tailored to suit the distinctive characteristics of fractal systems and functions.
- Gestão e qualidade de dados mestre no SAP IBP: sustentação do planeamento da procuraPublication . Sedas, Francisco Ribeiro; Martins, Ana Alexandra Antunes Figueiredo; Pinto, Iola Maria Silvério; Ricardo, André Henriqueto; Reis, Diogo ChambelEste relatório descreve o projeto desenvolvido no âmbito do estágio curricular do Mestrado em Matemática Aplicada para a Indústria, centrado na melhoria do planeamento da procura através de metodologias quantitativas. O trabalho enquadra-se na necessidade de reforçar a fundamentação estatística associada à seleção de variáveis explicativas e à construção de modelos preditivos aplicados a séries temporais. A investigação desenvolvida incidiu sobre a análise de dados históricos de vendas de vários produtos pertencentes a quatro famílias distintas, tendo como objetivo principal identificar variáveis preditoras relevantes e avaliar o impacto da sua inclusão em modelos de previsão. Para tal, foi implementada uma abordagem metodológica estruturada que integrou análise descritiva, três métodos complementares de seleção de variáveis, incluindo regressões lineares e a regressão Lasso, e a comparação entre diferentes modelos de previsão. O modelo principal estudado foi o SARIMAX (Seasonal AutoRegressive Integrated Moving Average with eXogenous regressors), permitindo incorporar variáveis exógenas na previsão da procura. As previsões obtidas foram comparadas com as provenientes de um modelo de Alisamento Exponencial, recorrendo à análise sistemática de métricas de erro e à verificação dos pressupostos através da avaliação dos resíduos. Os resultados evidenciaram que a integração de variáveis exógenas, aliada a um processo rigoroso de seleção de preditores, melhorou de forma consistente a precisão das previsões. O estudo reforça, assim, a importância de abordagens multivariadas e estatisticamente sustentadas na otimização do planeamento da procura e na gestão eficiente de cadeias de abastecimento.
- Highly porous NiFe nanofoams synthesized by dynamic hydrogen bubble template for hydrogen evolution in alkaline mediaPublication . Carvalho, Gabriel Garcia; do Nascimento, Ricardo Espingardas; Morais Silva, Teresa; Montemor, Maria de FátimaWhen coupled with renewable energy sources, alkaline electrolysis (AEL) is a clean technology to produce hydrogen. The conventional electrodes that have been established as the commercial standard for AEL are perforated nickel plates, which have low surface area and high mass loads of active material. The use of Ni has been associated with the adsorption of protons and their recombination into H2 molecules, assisting and enabling the hydrogen evolution reactions. Due to the rising Ni price, there is a demand for less expensive electrode materials with identical, or better, performance. Thus, this work explores new alternative electrode materials and combines Ni with a cheaper and also electroactive metal, iron. For this purpose, highly porous bimetallic nanofoams are produced through the dynamic hydrogen bubble template. NiFe electrodes of different compositions are characterized by scanning electron microscopy, energy-dispersive spectroscopy, linear sweep voltammetry, and chronopotentiometry analysis. The results allow obtaining key parameters that are essential for the optimization of the electrode's response considering overpotentials, Tafel slopes, and other electrochemically relevant parameters.
- Simulating stresses and strains in solid mechanics directly from images using convolutional neural networksPublication . Vieira, Beatriz Susana; Rodrigues, José Alberto de Sousa; Bordas, Stéphane P. A.Abstract Finite Element Method (FEM) simulations provide reliable displacement, strain and stress fields, but they become costly when many geometry–load combinations must be tested. In several practical scenarios the input is an image that leads to non-rectangular, unstructured meshes, which is not ideal for strictly grid-based models. This dissertation investigates deep learning surrogates trained on FEM solutions that can deliver the mechanical response much faster. We propose a complete processing pipeline that starts from image segmentation, builds both structured and unstructured meshes, and trains two distinct models on top of them. The first one is a grid U-Net, designed for rectangular domains. The second one is MAgNET, which operates directly on the mesh and preserves the original discretization at the boundaries and at the loaded regions. Both models are trained and evaluated with exactly the same dataset, training schedule and metrics, including tests with loads above the training range and measurements of training and inference time, which enables a fair comparison between the two approaches. Results show that both surrogates reproduce displacement accurately, and that the largest strain and stress errors remain confined to the loaded boundary and to high-gradient areas. The grid U-Net is faster and very competitive on regular meshes, while MAgNET is the better option when the geometry comes from images and the mesh is unstructured.
- Velocity and energy of periodic travelling interfacial waves between two bounded fluidsPublication . Cal, Filipe; Dias, Gonçalo A. S.For a periodic travelling irrotational wave propagating at the interface between two homogeneous, incompressible and inviscid fluids bounded by horizontal planes, we generalise the Stokes definitions for the velocity of the wave propagation. Under certain conditions imposed on the horizontal velocity of the motion at the interface and supposing that the horizontal components of the velocity in each layer never reach the wave speed, we prove that the mean horizontal velocity of propagation of the wave is greater than the generalised mean horizontal velocity of the mass of the fluid. We show that, for interfacial waves of small amplitude, the excess kinetic and potential energy of the fluid have the same magnitude, but different signs, and for the nonlinear setting, we prove that the excess kinetic energy is negative.