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ABSTRACT A fractional order mathematical model that already exists in the literature, was considered. This model was established to study the effects of medicinal treatment in people infected with the human immunodeficiency virus (HIV). The importance of this study is that the model evaluates, among other parameters, the density of healthy and HIV-infected CD4+ T cells. These data are very necessary for the subject infected by the virus given the effects that an antiretroviral treatment causes in it. The objective of this work is to consider several numerical schemes that involve fractional derivatives in order to compare their behaviors and to obtain a good approximation of the mentioned model solution. Convergence of these schemes will be studied as well as sensitivity with respect to the variation of the parameters η (drug efficacy) and α (fractional derivative order). Furthermore, through the collection of medical records of people living with HIV, it is intended to determine the optimal fractional derivative order for the model and to compare it with the classical model.Resumo em Inglês:
ABSTRACT In this work we present a numerical study of surface water waves over variable topographies for the linear Euler equations based on a conformal mapping and Fourier transform. We show that in the shallow-water limit the Jacobian of the conformal mapping brings all the topographic effects from the bottom to the free surface. Implementation of the numerical method is illustrated by a MATLAB program. The numerical results are validated by comparing them with exact solutions when the bottom topography is flat, and with theoretical results for an uneven topography.Resumo em Português:
RESUMO Neste artigo o método multigrid algébrico baseado em agregação suavizada, o método clássico de Ruge-Stuben e o método GMRES pré-condicionado por multigrid algébrico foram utilizados para a solução da equação do fluxo livre estacionário em domínio georreferenciado. A disponibilidade dos códigos computacionais permitiu avaliar a aproximação de elementos finitos sob a perspectiva dos métodos multigrid algébricos e respectiva combinação, como pré-condicionante, com o método GMRES. As diferenças máximas entre soluções por diferentes métodos, o tempo necessário para obter as soluções dos sistemas lineares associados, em cada uma das iterações de Picard, os residuais de cada um dos métodos iterativos e os residuais em cada uma das iteradas de Picard são apresentados e discutidos. Como resultado da análise, os métodos pré-condicionados são mais eficientes no sentido do menor tempo computacional aliado à estabilidade do número de iterações. A análise dos residuais das iterações de Picard permite comparar a evolução dos diferentes métodos de solução dos sistemas lineares. O detalhamento dos residuais dos métodos iterativos em cada passo das iterações de Picard permitiu uma visão mais abrangente e uma análise da convergência. Em detalhes, o método baseado em agregação suavizada necessita de um número expressivamente menor de iterações quando comparado ao método clássico de Ruge Stüben nas primeiras iterações de Picard. O pré-condicionamento reduz o número de iterações em relação às iterações iniciais e há uma persistência da redução do número de iterações do método baseado em agregação em relação ao método clássico.Resumo em Inglês:
ABSTRACT In this paper, the algebraic multigrid method based on smoothed aggregation, the classical method of Ruge-Stúben and the method GMRES preconditioned by algebraic multigrid were used to solve the steady-state free flow equation in a georeferenced domain. The availability of computer codes allowed us to evaluate the approximation of finite elements from the perspective of algebraic multigrid methods and their combination, as preconditioning, with the GMRES method. The maximum differences between solutions by different methods, the time required to obtain the solutions of the associated linear systems, in each of the Picard iterations, the residuals of each of the iterative methods and the residuals of each of the Picard iterations are presented and discussed. As a result of the analysis, the preconditioned methods are more efficient in the sense of less computational time combined with the stability of the number of iterations. The analysis of the residuals of Picard iterations allows comparing the evolution of the different methods of solving linear systems. The detailing of the residuals of the iterative methods in each step of Picard’s iterations allowed a more comprehensive view and an analysis of convergence. In detail, the method based on smoothed aggregation needs a significantly smaller number of iterations when compared to the classical method of Ruge Stüben in the first iterations of Picard. The preconditioning reduces the number of iterations in relation to the initial iterations and there is a persistence of the reduction in the number of iterations of the aggregation-based method in relation to the classical method.Resumo em Inglês:
ABSTRACT Because of the current scenario of the SARS-CoV-2 (COVID-19) pandemic in Brazil, whose vaccination campaign is in its initial stage, government authorities have pointed towards the complete reopening of the economy. And recently, for the in-personal return of classroom teaching in schools. Given the family relationship, one of the questions that remains without an answer is: what are the consequences of the schools’ reopening on the dissemination of COVID-19? The purpose of this work is to analyze a variant of the compartmental SIRD (Susceptible, Infected, Recovered, Social Distancing) model in a structured, interacting age population representing six age groups, from the basic education age to the elderly. We present a complete analysis of the well-posedness of the proposed mathematical model. We discuss distinct disease spreading scenarios based on observations of the mathematical behavior of the proposed dynamics. Moreover, we present the existence of the stationary points in terms of the parameters of the model and the number of infected age groups. Finally, we present different numerical simulations of the predicted scenarios by the model. Those numerical realizations support the conclusion that an early school reopening, resulting in the decreasing social isolation of young people, causes the infection curve to grow considerably, even for other age groups.Resumo em Inglês:
ABSTRACT In this work we obtain a variational formulation and a priori estimates for approximate solutions of a problem involving fractional diffusion equations.Resumo em Português:
RESUMO Problemas de corte e empacotamento fazem parte do processo de planejamento da produção em muitas indústrias (e.g. papel, vidro, móveis). Em algumas dessas indústrias, um objeto retangular grande deve ser cortado em itens retangulares menores e existe uma capacidade limitada para o estoque dos itens. Nesse contexto, surge o problema de corte bidimensional guilhotinado 2-estágios restrito (PCBG-2est). Alguns autores têm proposto algoritmos de programação dinâmica para resolver o problema no caso irrestrito. Para o caso restrito essa técnica ainda apresenta alguns desafios devido à dimensão do espaço de estados. Nesse artigo propõem-se duas heurísticas baseadas em programação dinâmica e no método de Gilmore e Gomory para resolver o PCBG-2est restrito. São apresentados resultados do estudo computacional realizado com três conjuntos de instâncias que mostram a eficiência da proposta. Em particular, para instâncias similares às encontradas na indústria moveleira foram obtidas soluções com gap máximo médio de 4.4%.Resumo em Inglês:
ABSTRACT Cutting and packing problems are part of the production planning process in many industries (e.g. paper, glass, furniture). In some furniture industries, large rectangular objects have to be cut into smaller rectangles and there is a limited storage space for work in process. In this case there is interest in solving the constrained two-dimensional two-stages guillotine cutting problem (PCBG-2est). Several authors applied dynamic programming algorithms for solving the unconstrained two-dimensional cutting problem. However, for the constrained case this technique still presents some challenges due to the size of the state space. We propose a heuristic based on the two-step method of Gilmore and Gomory for the constrained PCBG-2est considering special constraints associated with the cutting equipment. The results of a computational study with three sets of instances show the efficiency of the proposal. In particular, for instances that are similar to the furniture industry, solutions were obtained with an average maximum gap of 4.4%.Resumo em Inglês:
ABSTRACT The aim of this paper is to analyse the evolution of the COVID-19 pandemic in Rio Grande do Sul by applying graph-theoretical tools, particularly spectral clustering techniques, on weighted graphs defined on the set of 167 municipalities in the state with population 10,000 or more, which are based on data provided by government agencies and other sources. To respond to this outbreak, the state has adopted a system by which pre-determined regions are assigned flags on a weekly basis, and different measures go into effect according to the flag assigned. Our results suggest that considering a flexible approach to the regions themselves might be a useful additional tool to give more leeway to cities with lower incidence rates, while keeping the focus on public safety. Moreover, simulations show that the combination of pendulum migration and isolation data used in this paper leads to a coherent qualitative description of the evolution of the pandemic in Rio Grande do Sul. These simulations also confirm the dampening effect of isolation on the dissemination of the disease.Resumo em Inglês:
ABSTRACT The study of the concentration dynamics of a pollutant substance in a fluid is a classic problem of fluid mechanics given by the transport equation u t + cu x = 0, where u = u(x,t) denotes the pollutant concentration along a horizontal pipe of a fixed cross-section in the positive x direction at he time t > 0 and c represents the fluid propagation velocity. In view of that, the velocity of propagation of the fluid is a physical quantity, obtained, generally in an approximate form, which makes such quantity uncertain. In this paper, we propose to obtain the concentration when the constant c represents the fuzzy set. The concentration was obtained by using the Zadeh’s Extension Principle. Through the concentration obtained, we analyze the influence of uncertainty on the fluid propagation velocity in the concentration dynamics and explore possible practial applications in case-studies of engineering, environmental and soil sciences.Resumo em Inglês:
ABSTRACT This article analyzes the performance of combining information from Scanning Electron Microscopy (SEM) micrographs with Static Light Scattering (SLS) measurements for retrieving the so-called Particle Size Distribution (PSD) in terms of experimental features. The corresponding data fusion is implemented using a novel Monte Carlo-based method consisting in a SMF (Sampling-Mapping-Filtering) approach. This approach provides an important reference to assess the strategy of the experiment for this specific problem by means of solving an inverse problem. Furthermore, low levels of volume fraction and a PSD represented by log-normal distributions are considered in order to reduce processing and model errors due to ill-posedness. The prior statistics corresponding to the SEM micrographs have been achieved by means of the Jackknife procedure used as a resampling technique. The likelihood term considers iid normal measurements generated from the Local Monodisperse Approximation (LMA) and also makes use of the same model as forward linear model, in an inversion case known as inverse crime. However, it has been proved that the LMA performs well in practice for low fraction volume systems as considered here. The PSD retrieval is measured in terms of improvement in precision with respect to one of the log-normal parameters in SEM micrographs, i.e., the desirability. Estimates are expressed as a function of a typical system parameter such as polydispersity, as well as experimental variables, i.e., number of particles per micrograph (PPM) and noise level ε in the SLS measurements. These estimations are then analyzed by means of the Box-Behnken (BB) design and the response surface methodology (RSM) in order to generate a surrogate model from which rules for the optimization of the experiment are made when desirability is maximized. Finally, a Rule-Based System (RBS) is proposed for future use.Resumo em Inglês:
ABSTRACT The study of optimality conditions and constraint qualification is a key topic in nonlinear optimization. In this work, we present a reformulation of the well-known second-order constraint qualification described by McCormick in 17. This reformulation is based on the use of feasible arcs, but is independent of Lagrange multipliers. Using such a reformulation, we can show that a local minimizer verifies the strong second-order necessary optimality condition. We can also prove that the reformulation is weaker than the known relaxed constant rank constraint qualification in 19. Furthermore, we demonstrate that the condition is neither related to the MFCQ + WCR in 8 nor to the CCP2 condition, the companion constraint qualification associated with the second-order sequential optimality condition AKKT 2 in 5.Resumo em Inglês:
ABSTRACT The aim of this paper is to apply the diffusive metric technique defined by the spectral analysis of graph Laplacians to the set of the 41 cities belonging to AMBA, the largest urban concentration in Argentina, based on public transport and neighborhood. It could be expected that the propagation of any epidemic desease would follow the paths determined by those metrics. Our result reflects that the isolation measures decided by the health administration helped at the atenuation of the actual spread of COVID-19 in AMBA.Resumo em Português:
RESUMO O fretamento de veículos para realizar o transporte de funcionários em substituição ao transporte público é uma realidade para diversas empresas. Os benefícios obtidos com essa opção alcançam o conforto dos funcionários e o controle da entrada e saída desses nos diversos turnos de trabalho, o que ajuda no cotidiano de produção da empresa. Assim, planejar adequadamente as rotas para esses veículos também é importante. Neste contexto, este artigo estuda o problema de roteamento de veículos modelado por meio de fluxo de produtos em arcos para propor uma nova modelagem matemática que adiciona um certo controle na quantidade de passageiros entre veículos, no sentido de que cada veículo usado transporta aproximadamente a mesma quantidade de passageiros. Um estudo de caso referente ao transporte fretado de trabalhadores de uma indústria do ramo alimentício na cidade de Itumbiara-GO foi conduzido como forma de validar a viabilidade prática do modelo proposto. Uma série de experimentos computacionais e ajustes de parâmetros foram realizados com o objetivo de investigar o impacto prático de planejar rotas equilibradas em termos da quantidade de passageiros, número de visitas e da distância total percorrida. Estatística descritiva e testes de hipóteses validaram as soluções do modelo matemático proposto para uma parte significativa dos experimentos computacionais.Resumo em Inglês:
ABSTRACT Many companies have chosen private commute options over public transportation for employee displacement. The benefits obtained with these options aim at the comfort of the employees and management of the entry and exit time in the different work shifts, helping in the daily production of the companies. Thus, planning suitable routes for the vehicles is also essential. This article studies the vehicle routing problem based on a two-commodity network flow formulation to propose new mathematical modeling that explicitly controls the number of passengers between vehicles, targeting the design of routes with approximately the same number of passengers. To validate the practical feasibility of the proposed model, we solved a case study regarding the private commute transport of workers from the food industry in the city of Itumbiara-Brazil. This paper reports a series of computational experiments with parameter variation to evaluate the practical impact of planning balanced routes regarding the number of passengers, the number of visits, and the total distance traveled. Descriptive statistics and hypothesis tests validated the solutions of the proposed mathematical model for a significant part of the computational experiments.