Resumo em Inglês:

ABSTRACT In this article, residual indicators were used to characterize the quality of the numerical solution of the advection-diffusion-reaction equation in a saturated porous medium. Both large and small advection regimes were considered. The small advection was exemplified by a problem with constant data, while non-constant data was considered for the large advection regime. In this case, residual quantities associated with the data must be incorporated into the residual estimates related to the spatial approximation and an auxiliary problem must be solved for the correct obtainment of the temporal estimates. The presentation of the residual indicators as a surface on the finite element mesh provides a detailed view of the regions that need refinement, allows to infer the effect of each estimate on the composition of the global estimator and, in addition, allows to follow the evolution of the residual surfaces as the contaminant front advances in the simulation process. In turn, the numerical values of the indicators allow to delimit the elements that will be refined, to compare the magnitude of the contributions among themselves, between different meshes and a better understanding of the composition of the global estimates.

Resumo em Português:

RESUMO Este trabalho propõem o uso da Matemática Intervalar em conjunto com o Método dos Mínimos Quadrados e a linguagem Python para obter a melhor solução aproximada para sistemas intervalares obtidos de experimentos físicos. Este método de resolução será aplicado para obter a melhor função aproximada que ajusta um conjunto de dados oriundos de um experimento físico, no qual um carro se desloca com aceleração constante sob um trilho de ar inclinado. Ao fazer esse tipo de abordagem usando a Matemática Intervalar, busca-se inferir como as incertezas provenientes do experimento, assim como os erros gerados pelas representações e operações numéricas em computadores, interferem no resultado obtido. Para isso, se fez necessário a utilização da biblioteca Python for Extended Scientific Computing (Python- XSC), a qual é baseada na estrutura da aritmética intervalar e fornece funções para a resolução de sistemas lineares intervalares. A aplicação do estudo feito se mostrou bastante eficiente e de fácil utilização, o que motiva sua utilização.

Resumo em Inglês:

ABSTRACT In this work, we used the Interval Mathematics, Method of Least Squares and Python language to obtain the best approximate solution for interval systems obtained from physical experiments. This resolution method will be applied to obtain the best approximate function that fits a data set from a physical experiment, in which a car was travels with constant acceleration under an inclined air trail. When doing this type of approach using Interval Mathematics, we seek to infer how the uncertainties arising from the experiment, as well as the errors generated by the representations and operations on computer, interfere with the obtained result. For this, it was necessary to use the Python for Extended Scientific Computing (Python-XSC) library, which is based on the interval arithmetic structure and provides functions for the solving of interval linear systems. The application of the study was very efficient and easy to use, which motivates its use.

Resumo em Português:

RESUMO Neste trabalho abordamos um problema real identificado no processo produtivo de indústrias de embalagens de papelão ondulado. O setor de planejamento e controle da produção emite boletins que trazem as características relativas às chapas que serão produzidas. A produção destas chapas é feita no corrugador, onde bobinas são instaladas, de acordo com as características dos boletins. O problema surge da necessidade de minimizar trocas de gramaturas e rolos corrugadores durante o processo de produção das chapas com as características contidas nos boletins, uma vez que cada boletim traz as gramaturas específicas das camadas (geralmente são 3 ou 5) da chapa, além do tipo de onda empregada no(s) miolos(s). O sequenciamento dos boletins impacta na quantidade de trocas de gramaturas e rolos corrugadores. Interpretamos este problema como um Problema de Minimização de Trocas de Ferramentas e apresentamos quatro modelos matemáticos para o caso em estudo. Resultados computacionais são apresentados comparando os resultados obtidos pelos modelos matemáticos propostos e por uma fábrica do setor, indicando que é possível reduzir em até 37% o número de trocas de gramaturas.

Resumo em Inglês:

ABSTRACT In this paper we address a real problem identified in the production process of corrugated packaging industries. The Production Planning and Control sector issues bulletins that bring the characteristics related to the sheets that will be produced. The production of these sheets is done in the corrugator, where reels are installed, according to the characteristics of the bulletins. The problem arises from the need to minimize the switches of weights and corrugator rolls during the sheets production process, since each bulletin brings the specific weight of the layers (usually 3 or 5) of the sheet, in addition to the type of wave used. The sequencing of bulletins impacts the amount of weight and corrugator rolls switches. We interpreted this problem as a Minimization of Tool Switches Problem and we present four mathematical models for the case under study. Some computational results are presented comparing the results obtained by the proposed mathematical models and by a factory of the sector, and that it is possible to reduce up to 37% the number of switches of weights.

Resumo em Português:

RESUMO O cálculo I1 foi introduzido em 1995 por Sette e Carnielli. Este sistema possui um caráter intuicionista, no mesmo sentido do sistema lógico desenvolvido por Arend Heyting (1898-1980), o qual surgiu como a lógica subjacente a Matemática Intuicionista, ou construtivista, por exemplo, ¬¬A→A não ser uma tautologia em I1. Ademais, o cálculo I1 é uma lógica trivalorada que, ao contrário da lógica clássica, não admite apenas dois valores de verdade, mas sim três, estes são T, F* e F. Os valores T e F denotam, respectivamente, verdade e falsidade, enquanto que F* pode ser interpretado como “falsidade por falta de evidência positiva”. O ambiente semântico dessa lógica, há apenas um valor distinguido. Neste trabalho, desenvolvemos um método dedutivo alternativo ao axiomático para o sistema I1, ou seja, introduzimos um sistema de tableaux analíticos para tal lógica. Estabelecemos, por meio de teoremas, que toda dedução obtida do sistema axiomático também será deduzida pelo sistema de tableaux proposto.

Resumo em Inglês:

ABSTRACT Sette and Carnielli introduced the axiomatic system I1 in 1995, which presents an intuitionistic character, in the same regard of the logical system developed by Arend Heyting (1898-1980), that is, as a logic underlying to the constructive mathematics, for instance, ¬¬A→A is not an I1 tautology. On the other hand, this system is a 3-valued logic, in which presents in addition to the classical truth values, the truth-value F*, that can be understood as “false by lack of positive evidence”. In the semantic environment of this logic, there is only one distinguished value. In this work, we develop an alternative deductive method to the axiomatic for the system I1, namely, we introduce an analytic tableaux system for such logic. We establish, by means of a theorem, that every deduction obtained from the axiomatic system, will be also deduced by the tableaux system proposed.

Resumo em Inglês:

ABSTRACT By using the notion of Carathéodory solution to differential equations, the present work studies the boundedness of solutions of discontinuous differential equations. For these discontinuous systems determined by discontinuous differential equations, results are obtained that guarantee sufficient conditions to boundedness of solutions in terms of nonsmooth Lyapunov functions.

Resumo em Inglês:

ABSTRACT The aim of this paper is to present, in a broad context, a proposal of mathematical foundation that serves as framework for the development of the electronic structure calculation methodology called “Density Matrix Tight Binding method (DMTB)”, which was originally presented in the literature by the group led by physicist David Vanderbilt in 19939, as well as its relationship to the computational modeling that can be chosen for its implementation. The approach adopted makes it clear that the final mathematical formulation of this methodology is completely dependent on the computational strategies chosen for its effective implementation. Thus, we put the DMTB as a mathematical-computational model of variable final formulation. Finally, we propose an implementation based on the nonlinear conjugate gradient method. The final model obtained is slightly different from the DMTB that was originally presented in 1993, in agreement with the version presented by Millam and Scuseria in 199711. The approach used develops the mathematical aspects, aiming at the effective computational implementation of the methodology.

Resumo em Inglês:

ABSTRACT The present work concerns the COVID-19’s spread over The United States, Brazil and Colombia. Although countries show differences in economic development, but similarities such as continental dimension or native social interaction, the spread of COVID-19 in them has some similarities. At the moment, the countries are living the disease with temporal delay. Thus, we used a database on WHO Coronavirus, Mathematical Modeling and Numerical Simulations to describe the most recent COVID-19 development patterns in these countries, which we saw.

Resumo em Português:

RESUMO O Problema das p-Medianas Capacitado (PPMC) consiste em localizar p facilidades em uma rede composta por n vértices e decidir qual facilidade atenderá cada vértice, a fim de minimizar a soma de todas as distâncias de cada facilidade para cada vértice e atender às restrições de capacidade máxima de cada facilidade. Neste trabalho, dado que o PPMC é da classe NP-difícil, quatro variantes da metaheurística General Variable Neighborhood Search (GVNS) são implementadas para resolver o PPMC: G-VND, G-RVND, GG-VND e GG-RVND. Elas diferem entre si com relação ao método usado para construir uma solução inicial e o usado para a busca local. Nas duas primeiras variantes, a solução inicial é gerada de forma aleatória e o método de busca local é feita via Variable Neighborhood Descent (VND) ou Random Variable Neighborhood Descent (RVND), respectivamente. Por sua vez, nas duas últimas, a solução inicial é feita via a fase de construção do método Greedy Randomized Adaptive Search Procedure (GRASP). Usando instâncias padrões de teste da literatura, mostramos, inicialmente, que a variante GG-VND teve melhor desempenho quando comparada com as demais. Em seguida, mostramos que, quando comparada com os algoritmos da literatura, esta variante possui desempenho equivalente ou superior.

Resumo em Inglês:

ABSTRACT The Capacitated p-Median Problem (CPMP) consists of locating p facilities in a network composed of n clients and deciding which facility will serve each client to minimize the sum of all distances from each facility to each client and meet the facility capacity constraints. Since the CPMP belongs to the NP-hard class, in this work, four variants of the General Variable Neighborhood Search (GVNS) metaheuristic, named G-VND, G-RVND, GG-VND, and GG-RVND, are implemented for treating the CPMP. They differ concerning the method used to build an initial solution and the one used for local search. In the first two variants, the initial solution is generated randomly, and the local search method is performed via Variable Neighborhood Descent (VND) or Random Variable Neighborhood Descent (RVND), respectively. In turn, in the last two, the initial solution is executed via the construction phase of the Greedy Randomized Adaptive Search Procedure (GRASP) method. Using benchmark instances from the literature, we initially showed that the GG-VND variant performed better than others. Then, we show that this variant has an equivalent or superior performance when compared with the literature algorithms.

Resumo em Inglês:

ABSTRACT Portfolio selection is undoubtedly one of the most challenging topics in the area of finance. Since Markowitz’s initial contribution in 1952, portfolio allocation strategies have been intensely discussed in the literature. With the development of online optimization techniques, dynamic learning algorithms have proven to be an effective approach to building portfolios, although they do not assess the risk related to each investment decision. In this work, we compared the performance of the Online Gradient Descent (OGD) algorithm and a modification of the method, that takes into account risk metrics controlling for the Beta of the portfolio. In order to control for the Beta, each asset was modeled using the CAPM model and a time varying Beta that follows a random walk. We compared both the traditional OGD algorithm and the OGD with Beta constraints with the Uniform Constant Rebalanced Portfolio and two different indexes for the Brazilian market, composed of small caps and the assets that belong to the Ibovespa index. Controlling the Beta proved to be an efficient strategy when the investor chooses an appropriate interval for the Beta during bull markets or bear markets. Moreover, the time varying Beta was an efficient metric to force the desired correlation with the market and also to reduce the volatility of the portfolio during bear markets.

Resumo em Inglês:

ABSTRACT Mathematical and computational modeling have been increasingly applied in many areas of cancer research, aiming to improve the understanding of tumorigenic mechanisms and to suggest more effective therapy protocols. The mathematical description of the tumor growth dynamics is often made using the exponential, logistic, and Gompertz models. However, recent literature has suggested that the Allee effect may play an important role in the early stages of tumor dynamics, including cancer relapse and metastasis. For a model to provide reliable predictions, it is necessary to have a rigorous evaluation of the uncertainty inherent in the modeling process. In this work, our main objective is to show how a model framework that integrates sensitivity analysis, model calibration, and model selection techniques can improve and systematically characterize model and data uncertainties. We investigate five distinct models with different complexities, which encompass the exponential, logistic, Gompertz, and weak and strong Allee effect dynamics. Using tumor growth data published in the literature, we perform a global sensitivity analysis, apply a Bayesian framework for parameter inference, evaluate the associated sensitivity matrices, and use different information criteria for model selection (First- and Second-Order Akaike Information Criteria and Bayesian Information Criterion). We show that such a wider methodology allows having a more detailed picture of each model assumption and uncertainty, calibration reliability, ultimately improving tumor mathematical description. The used in vivo data suggested the existence of both a competitive effect among tumor cells and a weak Allee effect in the growth dynamics. The proposed model framework highlights the need for more detailed experimental studies on the influence of the Allee effect on the analyzed cancer scenario.