C.Q.D. - Revista Eletrônica Paulista de Matemática
A C.Q.D. – Revista Eletrônica Paulista de Matemática, ISSN 2316-9664, é periódico acadêmico publicado em versão eletrônica pelo Departamento de Matemática, da Faculdade de Ciências, da Universidade Estadual Paulista “Julio de Mesquita Filho” – Unesp/Bauru. Direcionada à comunidade científica, a revista foi criada em 2012 e tem como objetivo incentivar a investigação e o estudo da Matemática. Para tanto, recebe e analisa artigos submetidos por professores e alunos, e publica eletronicamente aqueles que apresentam suficientes méritos científicos e editoriais. Trata-se de uma revista diferenciada por ser totalmente virtual, de acesso gratuito e de ampla abrangência, incluindo os temas de Matemática Pura, de Matemática Aplicada e de Educação Matemática.
Fractional dynamics of a charged particle in an electromagnetic field
In this work, we analyze the modeling of physical phenomena using fractional-order differential equations, specifically focusing on the force exerted on a charged particle in an electromagnetic field. The methodological approach involves replacing integer-order derivatives with fractional-order derivatives, in the Caputo sense, in the differential equations that describe the movement of a charged particle in an electromagnetic field. This allows for a more detailed and efficient description. The results demonstrate that modeling with the Caputo fractional derivative provides a more refined solution, including the presence of damping. It is concluded that the use of fractional calculus offers significant advantages over traditional calculus, particularly in its ability to more accurately capture the system's memory and incorporate factors initially neglected in classical modeling.
Copyright (c) 2025 C.Q.D. - Revista Eletrônica Paulista de Matemática https://creativecommons.org/licenses/by/4.0
Incenter: five solved problems proposed for the International Mathematics Olympiad
Among the centers of triangles, incenter, the meeting point of the bisectors, is one of the best known due to its wide use in various fields of Mathematics and Physics. Five problems proposed for the International Mathematics Olympiad are discussed in detail. The proofs involved in the solutions are complemented by the availability of the respective links to the interactive figures using GeoGebra. It is expected that the article can be appreciated both by students preparing for the final stages of national or international competitions and by teachers interested in Olympic challenges.
A Study on the Jordan Canonical Form in vector spaces over finite fields
In this work, we will present the construction of the theorem that substantiates the existence of the Jordan Canonical Form for a given linear transformation on a vector space over and algebraically closed field using the fact that, if 𝑇 is a linear transformation in a vector space 𝑉 over a field K and 𝜆 is an eigenvalue of 𝑇, then the restriction of 𝑇 − 𝜆𝑖 to ker(𝑇 − 𝜆𝐼𝑑)𝑗 (where 𝑗 is the smallest integer such that ker(𝑇 − 𝜆𝐼𝑑)𝑗 = ker(𝑇 − 𝜆𝐼𝑑)𝑗+1) is nilpotent, which allows us to find a set of vectors 𝛽={𝑣, (𝑇 − 𝜆𝐼𝑑)(𝑣), . . . , (𝑇 − 𝜆𝐼𝑑)𝑚−1(𝑣)} such that the restriction of 𝑇 − 𝜆𝐼 to the set generated by 𝛽 can be represented by a matrix of the type 𝐽𝑚(0), therefore, the restriction of 𝑇 to [𝛽] can be represented by 𝐽𝑚(𝜆𝑖). Following that, we will use Field Theory to extend such construction in order to allow it to be used in vector spaces over finite fields, since if 𝑝(𝑥) is an irreducible polynomial over a field K, then K[𝑥]\⟨𝑝(𝑥)⟩ is a field where 𝑝(𝑥) has roots.
Aplicação de equações diferenciais em modelos presa-predador com fatores de inibição
Os modelos de interação do tipo presa-predador são comumente utilizados desde o controle biológico de pragas até mesmo na captura e emissão de carbono. Um modelo amplamente difundido é o de Lotka-Volterra (1926), sendo simples e validado. No entanto, esse modelo carece de correções que reproduzam melhor a realidade. Diante disso, foi realizada a análise qualitativa dos problemas presa-predador com fatores de inibição, utilizando as equações de Verhulst e Gompertz, juntamente com a análise numérica das aproximações para tais modelos com o software MATLAB. A análise qualitativa utilizou do estudo dos pontos de equilíbrio dos sistemas localmente lineares, da estabilidade dos sistemas lineares associados e dos sinais das derivadas, construindo trajetórias no plano de fase. Na análise numérica foram aproximadas trajetórias no plano de fase com a implementação do algoritmo de Runge-Kutta de 4a ordem. Todas as informações obtidas qualitativamente foram confirmadas por meio das aproximacões numéricas.
Proportional fractional operators applied to the Logistic model
Com o objetivo de tentar descrever um fenômeno mais próximo da realidade, a Modelagem Matemática se apresenta como uma ferramenta essencial para a compreensão de fenômenos de diversas naturezas. Em uma mesma perspectiva, com o avanço do cálculo de ordem fracionária, diversos trabalhos foram publicados e esta Modelagem Matemática, se mostrou muito promissora. Desta forma, amparados por esta nova ótica e aliados a esta teoria generalizada de modelos de crescimento, este trabalho tem como objetivo estudar operadores fracionários proporcionais, para a avaliação das soluções analíticas e numéricas do modelo Logístico.
Algebraic estimation of apothems in regular polygons using linear regression and bootstrap optimization.
Estimating the apothem of regular polygons may be required in academical and real-life situations such as for calculating the area of regular polygons and the volume of prisms and pyramids. The calculus of apothem requires the use and knowledge of the tangent function. In the present article we describe an algebraic approximation procedure for estimation of the apothem of any regular polygon given only the length of one side and the number of sides. An initial approximation was obtained based on nearly linear correlation between the product of polygon sides number and length with the squared root of the sum of the squared apothem and squared hypotenuse of one of the inner right-angled triangles of the polygon. In a second step the bootstrap method was to use to obtain the coefficients of a second-degree polynomial. The resulting equation predicts the apothem with a minimum accuracy of 0.9997. In polygons of 3, 4, 6 and with more than 25 sides the accuracy was higher than 0.9999.
Study of Bacterial Growth Population Dynamics in Laboratory Using Interval Differential Equations
This work focused on studying the population dynamics of Escherichia coli growth in a laboratory setting using Interval Differential Equations (IDEs). Specifically, the Malthus and Verhulst models were considered to describe population growth dynamics, using parameters collected from laboratory experiments. These parameters, as reported in the article, have an interval-valued characteristic. Supported by analytical and numerical methods—Euler and fourth-order Runge-Kutta, modified for the interval theory context—interval numerical solutions were produced to study these models. The work demonstrated that IDEs, particularly the Verhulst model, faithfully represent biological data due to the intrinsic uncertainties of the system.
Characterization of Equilibrium Points and Stability Analysis of the SMIB Using the First and Second Lyapunov Methods
Stability analysis in dynamic systems plays a crucial role in ensuring the safe and reliable operation of power systems. This work aims to investigate the characteristics of equilibrium points and conduct a stability analysis in a Single Machine Infinite Bus (SMIB) power system. To achieve this, a detailed approach to nonlinear dynamic systems analysis will be considered to describe the behavior of the aforementioned power system, focusing on rotor angle stability analysis under large disturbances, exploring the First and Second Lyapunov Methods, as well as LaSalle's Invariance Principle. The simulations will be conducted using an algorithm implemented in the MATLAB environment, which will enable spectral analysis of the SMIB to investigate its stability conditions.
Towards an algebrization of a linear temporal logic
The subject of this paper is the Propositional Neighbourhood Logic (PNL), a case of linear temporal logic and an extension of Classical Propositional Logic that results from the inclusion of modal operators that deal with certain temporal aspects. In this context, the validity of a proposition is strictly related to some temporal interval. We highlight some formal aspects of this logic and its formalization in some deductive systems (or proof systems), which naturally reflect the formal aspects of time. We aim to investigate algebraic models for Propositional Neighbourhood Logic. However, in this work, we focus exclusively on presenting the Soundness result, one of the necessary steps towards obtaining an adequate (sound and complete) model, between the axiomatic system of PNL and the planned algebraic model.
Interesting properties relating areas of two triangles
This article explores an interesting geometric relationship between the areas of two triangles, in which each vertex of the smaller triangle is located on distinct sides of the larger triangle. We present and demonstrate three unusual properties, using practical examples with notable points, such as the barycenter, incenter, circumcenter, orthocenter and Gergonne point. The results obtained open paths for future geometric investigations, highlighting classical and modern concepts, such as Euler's theorem applied to pedal triangles, which relates the area of ​​triangles to orthogonal projections of a point on the sides of a triangle.
Topological structure of the phase portrait for linear vector fields in R3 and applications
This paper presents an analysis of the sign of the real part of the roots of a polynomial of degree 3 as a function of its coefficients. The problem is treated mainly from an analytical point of view, knowing that from an algebraic point of view the problem has already been treated in the literature without significant advances. As an application of the results obtained, we will fully characterize the dynamics and topological structure of the phase portrait for linear fields in R3. In addition, we present applications in the local qualitative study of equilibrium points for vector fields in R3 defined with one or more parameters.
Local dynamics near to equilibrium points
The Pell sequence in the tricomplex ring
In this study we investigate a Pell-type sequence, perpetrated by a second-order linear recurrence which specifies the Horadam sequence. Here, we present, study and analyze results concerning this new class of Pell sequences.
A characterization of adjunction in a many-valued modal system
Galois connections are pairs of functions, defined over ordered sets, that preserve some particular aspects. They are studied in the context of algebraic structures. As a logic topic, we take the four-valued logic PM4N that contemplates, at least, two modal operators for the notions of necessary and possible. In this paper, we develop a particular four-valued implication for PM4N, which constitutes a Galois pair with the two modal operators. Then, we show some properties of the logic just considering the correlates algebraic developments.
Fuzzy model to check the susceptibility of young people in relation to fake news
Known as fake news, the various forms of misinformation have become a major vulnerability in today's world. They can influence, shape or distort our perception of what is real. In this work, we present a Fuzzy Rule-Based System (FRBS) with the objective of establishing an appropriate modeling that allows inferring how susceptible, in terms of believing and/or sharing; a person may be when faced with fake news. To do this, we use data collected from online forms from participants in one of the Project modules titled “Artificial Intelligence: from Logic to Humanities” supported by CNPq. The data was used to feed a fuzzy model that allows inferring the participants' degree of susceptibility to misinformation. The results showed that in a sample of 42 project participants, only 9.5% had a high degree of contamination by “fake news”. The internal validity of the model was established by comparing the results obtained through the FRBS with a manual analysis of the data by the expert.
Funções quadráticas na sala de aula: uma proposta de ensino através da modelagem matemática
We propose a didactic approach to teaching quadratic functions through mathematical modeling. The proposal is based on a problem that requires determining one side of a rectangle as a function of the height of a trapezoid formed within a triangle, under the condition that the areas of the regions formed are equal. Additionally, the triangle and the rectangle must have the same total areas and heights. We explored this problem situation through an experiment involving the transfer of liquids by gravity between triangular and rectangular reservoirs, allowing students to manipulate, investigate, and validate mathematical concepts. The use of tools such as spreadsheets and GeoGebra simulations provided a dynamic learning environment, enhancing the understanding of the properties of quadratic functions. The proposal was implemented with a first-year high school class, followed by an analysis of the results obtained.
Identification of fatty acids from açai in mistures subjected to crossflow microfiltration using Artificial Neural Networks
This article proposes the use of Artificial Neural Networks as an alternative tool to classify mixtures of fatty acids from açai subjected to crossflow microfiltration. The neural networks were trained with the Levenberg-Marquardt and Bayesian Regularization supervised learning algorithms, using the software MATLAB and datasets from literature, related to pressure, flow speed, time and transmembrane flow rate. The evaluation criteria were consists on analyzing the confusion matriz, the error histogram and the performance graphics. The neural network that used the Bayesian Regularization had 99,1\% accuracy for the mixtures with water/oleic acid and water/palmitic acid and also 96,5\% accuracy when a third class was added to the datasets, containing water and both the fatty acids, showing the validity of the tool.
In this article we examine how to approximately infer the area of ​​the region in the $xy$ plane bounded by the graph of the function $x\in[0,1]\mapsto 1+\frac{1}{n}sin(x\sqrt{n}) $, $n\in\mathbb{N}$, and along the $x$ and $y$ axes, and along the line $x=1$ using elements from Elementary Mathematics (covered in Basic Education). We also show how this analysis can be considered for the approximate calculation of areas of flat figures obtained by varying the sides of polygons via trigonometric functions; To this end, we define the notion of functions with good oscillatory behavior. We obtain formulas using geometric concepts from plane geometry aimed at activities at the Basic Education level. We use the graphic packages \LaTeX\ 'animate', 'tikz', 'tkz-euclide' and 'pgfplots' to produce figures and animations.
Polinômios e números primos: explorando padrões através de cálculos computacionais e a Conjectura 41
Hardy e Littlewood, ao longo de suas notáveis carreiras, propuseram várias conjecturas matemáticas. Em particular, a Conjectura F de Hardy-Littlewood, formulada em 1922, sugere que certas funções quadráticas geram infinitos números primos e estabelece uma relação assintótica para a função de contagem de números primos. Este trabalho apresenta um polinômio de segundo grau, v41(x)=x2+81x+412, derivado do n-quadrado Zeta, que representa uma aplicação específica da Conjectura F. Propomos, portanto, uma abordagem alternativa para a demonstração da Conjectura F através da Conjectura 41. "A Conjectura 41 postula que ``A função de contagem de primos-41, gerada pelo polinômio v41(x), é assintoticamente igual a x/(Alnx + B)". Adicionalmente, constatamos que, se a Conjectura 41 for verdadeira, o polinômio em questão gerará infinitos números primos.
Copyright (c) 2024 C.Q.D. - Revista Eletrônica Paulista de Matemática https://creativecommons.org/licenses/by/4.0
The paths of the roots of the quadratic function II
In Yamaoka (2023) we fixed two real coefficients and varied the remaining real coefficient of the quadratic function to obtain the paths described by the two roots of the function in the $\C$ plane. In this article, we used the inverse $\Phi$ of the stereographic projection to take the paths of both roots to the sphere $\mathbb{S}^2 \subset \R^3$: except in the subcase in which the paths of the images by $\Phi$ of the two roots are restricted to the south pole $S$ of $\mathbb{S}^2$, in each of the other subcases the paths of the images by $\Phi$ of both roots rest on one or two circumferences on $\mathbb{S}^2$. As the variable coefficient approaches $-\infty$ and $+\infty$, we observed the relations that the two roots have, via $\Phi$, with the north $N$ and/or south $S$ poles of $\mathbb{S}^2$. Let $G_j \subset \mathbb{S}^2$ be the set of points of the path of the image by $\Phi$ of the root $j$ on $\mathbb{S}^2$, $j=1,2$. We determined the common adherent points to $G_1$ and to $G_2$. We used Differential Calculus, Analytical Geometry with vector treatment, the inverse of stereographic projection and the distance between two sets.
The matRad software tutorial for radiotherapy planning
The matRad is an effective software for learning and developing advanced techniques in radiotherapy planning. It was developed by the German Cancer Research Center DKFZ to plan treatments that utilize photons, protons, and carbon ions, and it stands out for calculating dose with inverse planning and presenting treatment plans. The use of this program contributes to the enhancement of clinical outcomes by delivering a more conformal dose to the tumor and reducing side effects. This article describes the steps for obtaining, installing, and using the matRad software. An application in liver cancer treatment is presented to develop a radiotherapy plan, investigating the entire graphical interface.
Uma abordagem numérica para modelos populacionais considerando incerteza do tipo intervalar nas condições iniciais
This work presents a comparative study between Zadeh’s extension principle and Sup - $J_0$ extension principle, in uncertainty propagation, focusing on populational models described by differential equations. In this study, it is considered that the initial conditions of the models are given by intervals, in order to incorporate the intrinsic uncertainty of the phenomenon. To this end, population dynamics models are studied using Euler numerical method, which is adapted to interval arithmetic. Specifically, three models are considered, Malthus, Verhust and Gompertz. Simulations are provided for each model in order to analyze the uncertainty propagation for both Zadeh's and $J_0$ sums. The obtained results corroborate the theoretical results, showing that Zadeh’s sum propagates uncertainty over time, whereas the Sup - $J_0$ extension principle reduces the uncertainty.
Triangle areas and some inferences on plane geometry
The condition for the existence of a triangle ABC with sides a, b and c establishes that a+b>c. The present work identifies in the elements of the triangle, a certain quantity k>0 such that a+b=c+k. For this identification, the intercept points M, N and P of the triangle with its incircle of center I were obtained. The main result is the Equivalence of Areas between the Re-entrant Quadrilateral ADBC, with D being the vertex running through the incircle that minimizes its area, and the area of the Deltoid (or Kite) CNIP. As a geometric contribution, a construction is suggested to obtain the intercepts M, N and P on ABC without the prior need to trace the incircle. Finally, a dynamic tool was applied to trace coordinates defined, point-to-point, by the areas of the Re-entrant Quadrilaterals ADBC, ADCB and ACDB, while the common vertex D runs through the incircle of the triangle ABC. Elliptical trajectories of these points were revealed.
Transformando frequências em melodias: análise de ondas sonoras e da curva de Railsback
This paper investigates the principles of acoustics and musical language, with an emphasis on the transformation of frequencies into melodies through the piano. The study focuses on the analysis of the Railsback tuning curve, applied by piano technicians to optimize sound quality. Using the MATLAB programming environment, we examine in detail various waveforms, including sinusoidal, square and triangular, to analyze their characteristics and effects on sound synthesis. We implement an algorithm that incorporates the Railsback curve to adjust the frequencies of the keys of a piano, allowing the reproduction of music with harmonious acoustic balance.
Analytic conjugation between planar differential systems and potential systems
The classic Poincaré Normal Form Theorem (see [1]) states that a critical point of an analytic planar vector field is a non-degenerate center if and only if there is an analytic coordinate change such that in the new coordinates the vector field initial is of the form $f(x^2+y^2)\big(y\frac{\partial}{\partial x}-x\frac{\partial}{\partial y}\big)$, where $f$ is an analytic function defined in a neighborhood of the origin such that $f(0)>0.$ In this article it is proved that an analytical planar vector field with a non-degenerate center at $(0,0)$ is analytically conjugate, in a neighborhood of $(0,0)$, to a Hamiltonian vector field of the form $y\frac{\partial}{\partial x}-V'(x)\frac{\partial}{\partial y}$, where $V$ is an analytic function defined in a neighborhood of the origin such that $V(0)=V'(0)=0$ and $V''(0)>0.$ This result is a partial answer to a problem proposed by Chicone in 1987 (see [2]).
Viga prismática delgada e elástica: três aplicações de importância prática na engenharia
Prismatic beams subjected to forces can undergo elastic deformations. In this study, three applications involving an almost straight prismatic steel beam are analyzed. The first consists of a concentrated load P at its midpoint, where the modeling is treated in two different ways. One form uses the bending moment equation with boundary conditions applied at one of its ends and at the center point of the beam. This approach was in good agreement with the model developed in this study, where the boundary conditions are applied at the ends of the beam. In the second application, a spring is placed under the bar at the midpoint, in order to attenuate the vertical movement caused by P force. The elasticity curve of these deformations was determined here. The third application involves the placement of three equidistant loads on the bar to obtain an unforced kinematic representation of this movement.
Although the study of projectile motion after an oblique launch be a recurrent topic of consideration both in high school as well in junior disciplines of Calculus and Physics in college, its classical approach, in this context, presents it as a mere application of uniform motion and uniformly accelerated motion in two dimensions. Therefore, some of main properties as well interesting applications of this phenomenon are, in general, neglected. In this paper, we develop the basic theory of projectile motion, pointing out their main properties, and apply it to several non trivial and interesting problems that can be extended beyond the scope of this text, e.g. \cite{SS:04}.
Theorems of Menelaus, Pascal and Ceva: five solved problems proposed for the International Mathematics Olympiad
The Theorems of Menelaus, Pascal and Ceva establish necessary and sufficient conditions for the concurrence and collinearity of certain lines and points. Five problems proposed for the International Mathematics Olympiad are discussed in detail. The proofs involved in the solutions are complemented by the availability of the respective links to the interactive figures using GeoGebra. It is expected that the article can be appreciated both by students preparing for the final stages of national or international competitions and by teachers interested in Olympic challenges. Other topics that appear are: quadrilaterals, similarity, point power, median base, incenter and homothety.
In basic Physics courses, students come into contact with the second law of thermodynamics, which leads them to define a new state function called “entropy”. However, the definitions came from empirical studies. We will show in a generalized way the Ludwig Eduard Boltzmann formulation for entropy, in addition to some topological properties of the set of entropic functions.
The Jacobsthal generalized fundamental system: properties and identities
Neste trabalho estudamos o sistema fundamental generalizado de Jacobsthal, destacando suas principais propriedades e algumas identidades. São usadas as propriedades matriciais da bem conhecida matriz casoratiana para obtenção das identidades deste sistema. Também apresentamos expressões combinatórias para as propriedades e para as identidades. A metodologia que adotamos foi a revisão bibliográfica, onde procuramos responder a seguinte pergunta: Quais são as principais propriedades e identidades do sistema fundamental generalizado de Jacobsthal? Começamos falando sobre a família de números de Jacobsthal generalizados, destacando seu papel fundamental e em seguida, apresentamos propriedades e identidades através do estudo da respectiva matriz de Casoratian deste sistema. Ainda, destacamos a fórmula combinatória para os números generalizados. Esperamos que este trabalho contribua para uma divulgação deste conteúdo e a elaboração de outras pesquisas futuras.
This article presents an adaptation of the arms race model developed by Richardson, which is based on a system of differential equations, via set theory and fuzzy logic. To this end, we implemented a fuzzy inference system combined with traditional numerical methods with the aim of solving the differential equations proposed by Richardson when the field of directions is only partially known. This methodology allows the specialist to understand and modify complex mathematical models intuitively without losing mathematical rigor.