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- Trapped modes along periodic structures submerged in a three-layer fluid with a background steady flowPublication . Dias, Gonçalo; Pereira, Bruno M. M.In this study, we study the trapping of linear water waves by infinite arrays of three-dimensional fixed periodic structures in a three-layer fluid. Each layer has an independent uniform velocity field with respect to the fixed ground in addition to the internal modes along the interfaces between layers. Dynamical stability between velocity shear and gravitational pull constrains the layer velocities to a neighbourhood of the diagonal U1=U2=U3 in velocity space. A non-linear spectral problem results from the variational formulation. This problem can be linearized, resulting in a geometric condition (from energy minimization) that ensures the existence of trapped modes within the limits set by stability. These modes are solutions living the discrete spectrum that do not radiate energy to infinity. Symmetries reduce the global problem to solutions in the first octant of the three-dimensional velocity space. Examples are shown of configurations of obstacles which satisfy the stability and geometric conditions, depending on the values of the layer velocities. The robustness of the result of the vertical column from previous studies is confirmed in the new configurations. This allows for comparison principles (Cavalieri's principle, etc.) to be used in determining whether trapped modes are generated.
- Stochastic graph-based models of tumor growth and celular interationsPublication . Rodrigues, José AlbertoThe tumor microenvironment is a highly dynamic and complex system where cellular interactions evolve over time, influencing tumor growth, immune response, and treatment resistance. In this study, we develop a graph-theoretic framework to model the tumor microenvironment, where nodes represent different cell types, and edges denote their interactions. The temporal evolution of the tumor microenvironment is governed by fundamental biological processes, including proliferation, apoptosis, migration, and angiogenesis, which we model using differential equations with stochastic effects. Specifically, we describe tumor cell population dynamics using a logistic growth model incorporating both apoptosis and random fluctuations. Additionally, we construct a dynamic network to represent cellular interactions, allowing for an analysis of structural changes over time. Through numerical simulations, we investigate how key parameters such as proliferation rates, apoptosis thresholds, and stochastic fluctuations influence tumor progression and network topology. Our findings demonstrate that graph theory provides a powerful mathematical tool to analyze the spatiotemporal evolution of tumors, offering insights into potential therapeutic strategies. This approach has implications for optimizing cancer treatments by targeting critical network structures within the tumor microenvironment.
- Flow of a periodic interfacial travelling water wavePublication . Cal, Filipe; Dias, GonçaloWe consider a symmetric periodic travelling wave propagating at the interface between two homogeneous, incompressible, irrotational and inviscid fluids bounded by horizontal planes. For interfacial waves of small amplitude, we present a formula for the interface wave depending on the pressure at the rigid lid and at the flat bottom, and, for the general non-linear case, we derive a lower bound for the interfacial wave height. Under certain conditions imposed on the horizontal component of the motion at the interface and supposing that the horizontal components of the velocity in each layer never reach the wave speed, we study the monotonicity of the horizontal component of the velocity field along the streamlines and also analyze the monotonicity of the pressure along horizontal lines throughout the fluid in both layers, and along the boundary of the domain, between the crest and the trough. Finally, based on the behavior of the velocity field components, we build a pictorial description of the particle paths in both layers.
- Trapped modes along periodic structures submerged in a two-layer fluid with free surface and a background steady flowPublication . Dias, Gonçalo; Pereira, BrunoThis study examines the trapping of linear water waves by an endless structure of stationary, three-dimensional periodic obstacles within a two-layer fluid system. The setup features a lower layer of either limited or unlimited depth, overlaid by an upper layer of finite thickness bounded by a free surface, with each layer exhibiting its own constant background speed relative to the fixed reference frame. For real roots to emerge in the dispersion relation, an additional stability condition on the layer velocities is necessary. By selecting adequate choices for the background flow, a non-linear eigenvalue problem is derived from the variational formulation; its reasonable approximation yields a geometric criterion that guarantees the presence of trapped modes (subject to the aforementioned stability bounds). The selection of the eigenvalue is influenced by velocity owing to the presence of an interface and free surface. Due to inherent symmetries, the overall analysis can be confined to the positive quadrant of the velocity domain. Illustrations are provided for various obstacle setups that produce trapped modes in diverse ways.
- 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.
- 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.
- 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.
- 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.
- The modelling of urban running racesPublication . Enguiça, Ricardo; Lopes, Nuno D.In this paper, we model mass running urban races, taking into consideration several conditioning factors. The main goal is to find ideal configurations of the start of the race, splitting it into several waves, reducing the density of athletes and the overall time lost, when comparing the normal race results with a race without density constraints. This model takes into account distinct realistic runners' profiles, changes in slope and width on the race course and its influence on the running pace. Moreover, density levels, dynamics of the start of the race and time between the departure of waves are also considered.