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Princípios básicos e avançados sobre análise combinatória e uma proposta didática com o software SageMath

Fri, 22 Aug 2025 00:00:00 GMT

Título: Princípios básicos e avançados sobre análise combinatória e uma proposta didática com o software SageMath Autor(es): Santana, Amintas Vieira de Abstract: Combinatorial Analysis is frequently regarded by both students and teachers as a chal-lenging subject. This difficulty often stems from the use of inadequate methods and approaches that, rather than fostering logical reasoning, rely almost exclusively on the mechanical application of formulas. In this study, we present fundamental counting princi-ples such as the Bijective, Additive, and Multiplicative Principles, as well as Permutations and Combinations (both simple and with repetition). We also explore advanced count-ing techniques, including the Inclusion-Exclusion Principle, Derangements, Kaplansky’s Lemmas, Recurrence-Based Counting, and the Pigeonhole Principle. Furthermore, we provide a section on counting strategies and another addressing the most common erros encountered in solving combinatorial problems. As an application of these concepts, we include solved problems from the Brazilian Mathematics Olympiad for Public and Private Schools (OBMEP) and the National High School Exam (ENEM). Finally, we propose the integration of a technological tool to support the teaching of Combinatorial Analysis in secondary education. To this end, we developed a didactic sequence of activities using the open-source software SageMath, which displays all possible grouping configurations and calculates their quantities. The activities were designed with a progressive level of difficulty, enabling students to gradually build the reasoning required to understand the underlying concepts and thereby achieve consistent learning outcomes.

Estudo sistemático da transferência de energia e de espectros de absorção em complexos de európio(III): aplicações da TDDFT e análise quimiométrica

Thu, 05 Feb 2026 00:00:00 GMT

Título: Estudo sistemático da transferência de energia e de espectros de absorção em complexos de európio(III): aplicações da TDDFT e análise quimiométrica Autor(es): Santos, Gabriel Silva Abstract: In this work, the performance of different TDDFT(Time-dependent density-functional theory) functionals in describing absorption spectra and energy transfer (ET) in Eu3+ complexes was evaluated. Initially, three β-diketonate complexes were studied: [Eu(DPTB)3(H2O)2] (Eu-1), [Eu(DPTB)3(BathPhen)] (Eu-2), and [Eu(hfaa)3(Nap-TerPyr)] (Eu-3), modeled using the DFT PBE0/TZVP/MWB52 method. TDDFT CAM-B3LYP results satisfactorily reproduced the experimental spectra of Eu-1 and Eu-2, with absorption dominated by β-diketonate ligands. ET occurred mainly through the T1→5D0 and T1→5D1 channels, with rates on the order of 107 s -1 . The higher sensitization efficiency of Eu-1 (η = 52.6%; exp.: 56.3%) compared to Eu-2 (η = 45.0%; exp.: 44.5%) was attributed to a lower T1→S0 deactivation rate. For Eu-3, Principal Component Analysis (PCA) grouped the functionals into three distinct clusters. The spectra predicted by PBE0, B1P, X3LYP, B1LYP, REVPBE0, TPSS0, PW1PW, mPW1PW, and mPW1LYP showed better agreement with experiment. The INDO/S-CIS method reproduced the spectrum well but significantly underestimated the T1 energy. TDDFT results indicated a relevant contribution of the neutral ligand to electronic absorption and of the β-diketonate ligands to ET via T1→Eu3+, with a calculated yield of 52.8% (exp.: 48.8%). Subsequently, the influence of methodological parameters on TDDFT spectra was investigated. PCA clustering of 122 experimental spectra revealed three distinct groups. The projection of theoretical spectra in PCA space, combined with Mahalanobis distances, indicated that the B3LYP functional is the most globally consistent among those studied. Therefore, it was adopted for the factorial design parameter study. The fractional factorial design (25-2 ) showed that the effects of basis set, number of excited states, use of TDA, implicit solvent, and dispersion depend on the spectral cluster: in group 1, effects were minimally relevant; in group 2, the absence of TDA improved agreement; and in group 3, dispersion corrections and absence of TDA were decisive for better reproduction of experimental profiles.

Ensino de estatística descritiva com Python : uma proposta aplicada ao ensino médio

Fri, 10 Jul 2026 00:00:00 GMT

Título: Ensino de estatística descritiva com Python : uma proposta aplicada ao ensino médio Autor(es): Santana, José Eugênio Andrade Abstract: The teaching of Descriptive Statistics in High School faces challenges related to data interpretation, tables, graphs, and the understanding of measures of central tendency and dispersion, difficulties frequently observed in school practice. Considering the importance attributed by the Brazilian National Common Core Curriculum to the reading, interpretation, and critical analysis of statistical information, this study aimed to propose, apply, describe, analyze, and reflect on a sequence of activities for Descriptive Statistics mediated by the Python programming language, aiming to integrate calculations, graphical representations, and data interpretation. This study is an experience report with a qualitative approach, developed in High School Mathematics classes at Colégio Estadual de Tempo Integral Castro Alves, located in Adustina, Bahia, Brazil. The methodological proposal consisted of integrating conventional instruction of statistical content with computational practices in Python, in which students simulated data, organized information into tables, constructed graphs, and calculated statistical measures, relating these results to the analyzed data sets. The data were analyzed through triangulation involving teacher observations, students' productions, and a feedback questionnaire, aiming to understand the perceptions and reflections emerging from the developed experience. As an educational product, a sequence of activities mediated by Python was developed, aiming at the development of statistical interpretation. The results indicate that the proposed approach has the potential to foster the understanding of Descriptive Statistics concepts and expand the possibilities for data interpretation, in line with the demands of Mathematics Education. It is not intended to claim that the use of Python, by itself, guarantees better learning outcomes, but rather that its use as a mediating tool showed potential to foster statistical interpretation processes, expand possibilities for data analysis, and stimulate students' engagement in the proposed activities.

Uma equação do calor semilinear com uma reação localizada e dados iniciais não contínuos

Tue, 19 May 2026 00:00:00 GMT

Título: Uma equação do calor semilinear com uma reação localizada e dados iniciais não contínuos Autor(es): Santos, Poliano Fagundes dos Abstract: This dissertation studies a semilinear heat equation with a nonlinear term repre senting a concentrated reaction in a closed region of Ω ⊂ Rn, a smooth and bounded domain, or the entire space Rn, with n ≥ 2. The work is based on the article [9], which allows for the consideration of non-continuous initial data. We first present the preliminary concepts and analytical tools required for the development of the theory, including special functions, results from measure theory, and Lebesgue spaces Lr(Ω), with 1 < r < ∞. We then study the linear heat equation and the associated Green’s function, deriving fundamental linear estimates in both the interior of the domain and on the boundary, which are essential for handling the localized nonlinear term. Finally, we establish the main result of the dissertation, proving local-in time existence, uniqueness of solutions, continuous dependence on the initial data, and preservation of positivity by means of Banach’s Fixed Point Theorem.