Este trabalho tem o objetivo de verificar como o logaritmo e a função logarítmica são abordados nos livros didáticos de Matemática do século XIX ao XXI, indicando a inserção desses conteúdos na Matemática Escolar brasileira. Baseamo-nos, dentre outros autores, em Choppin (2004), Valente (1999, 2004, 2005 e 2008) e Bittencourt (2004 e 2008) para identificarmos o livro didático como fonte de pesquisa da Matemática Escolar, a partir de suas formas conceituais, de enfoque didático ou referente à organização do saber. A fim de verificar relações entre o histórico dos logaritmos e da função logarítmica na Matemática e na Matemática Escolar, apresentamos a ideia dos logaritmos de Napier como facilitadores de cálculos, interpretamos sua definição geométrica por meio de um sistema de equações diferenciais ordinárias e, assim, mostramos um breve histórico sobre a inserção do logaritmo como função no Cálculo Integral. Os livros didáticos de Matemática analisados foram publicados entre os anos de 1879 e 2013. Nesta análise, consideramos o modo como os autores abordam ou tratam a Matemática, os tipos de atividades que são propostas, os tipos de explicações, definições, exemplos, gráficos, exercícios, aplicações e problemas associados aos logaritmos e à função logarítmica. Os resultados apontam que no século XIX os logaritmos eram tratados nos livros didáticos, quase sempre, por meio da Aritmética. A partir da década de 1890, eles são abordados, tanto no campo aritmético, pela associação à teoria das progressões, como no campo algébrico, sendo expoentes numa equação ou função. A partir da década de 1930 até os dias de hoje verificamos uma predominância da concepção algébrico-funcional dos logaritmos. Com base nas Orientações Curriculares para o Ensino Médio (2006), nos Parâmetros Curriculares Nacionais Ensino Médio (PCNs - 2000), nos Parâmetros Curriculares Nacionais+ Ensino Médio (PCNs+ Ensino Médio - 2002), no Exame Nacional do Ensino Médio e em livros didáticos recentes, verificamos que os logaritmos e a função logarítmica se inserem, atualmente, na Matemática Escolar brasileira, pela sua associação às aplicações, como por exemplo, juros compostos, dinâmica populacional, desintegração radioativa, potencial hidrogeniônico, Escala Richter, Escala de Magnitude de Momento (MMS) e nível de intensidade sonora. / This work aims to verify how the logarithm and logarithmic function are approached in the textbooks of Mathematics from the 19th to the 21st century, indicating the insertion of these contents in Brazilian School Mathematics. We are based, among other authors, in Choppin (2004), Valente (1999, 2004, 2005 and 2008) and Bittencourt (2004 and 2008) to identify the didactic book as a research source of School Mathematics, from its conceptual forms, of didactic approach or referring to the organization of knowledge. In order to verify relations between the logarithmic and logarithmic functions in Mathematics and School Mathematics, we present the idea of the logarithms of Napier as facilitators of calculations, we interpret their geometric definition by means of a system of ordinary differential equations and we show a brief history about the insertion of the logarithm as a function in the Integral Calculus. The mathematical textbooks analyzed were published between the years 1879 and 2013. In this analysis, we consider how authors approach or treat mathematics, the types of activities that are proposed, the types of explanations, definitions, examples, graphs, exercises, applications and problems associated with logarithms and logarithmic function. The results show that in the 19th century, logarithms were treated in textbooks, almost always, through Arithmetic. From the 1890s, they are approached, both in the arithmetic field, by association with the theory of progressions, and in the algebraic field, being exponents in an equation or function. From the 1930s to the present day we have found a predominance of the algebraic-functional conception of logarithms. Based on the Curriculum Guidelines for Secondary Education (2006), the National Curriculum Parameters - Secondary Education (PCN\'s - 2000), the National Curriculum Parameters + High School (PCN\'s + High School - 2002), the National High School Examination and textbooks recent, we have verified that the logarithms and the logarithmic function are currently inserted in Brazilian School Mathematics, by their association to the applications, such as compound interest, population dynamics, radioactive disintegration, hydrogen ionic potential, Richter Scale, Momentum Magnitude Scale (MMS) and level of sound intensity.

A section of the total tangent space of a scheme X of finite type over a field k, i.e. a vector field on X, corresponds to an X-valued 1-jet on X. In the language of jets the notion of a vector field becomes functorial, and the total tangent space constitutes one of an infinite family of jet schemes Jm(X) for m ≥ 0. We prove that there exist families of “logarithmic” jet schemes JDm(X) for m ≥ 0, in the category of k-schemes of finite type, associated to any given X and its family of divisors D = (D₁, . . . ,Dr). The sections of JD₁(X) correspond to so-called vector fields on X with logarithmic poles along the family of divisors D = (D₁, . . . ,Dr). To prove this, we first introduce the categories of pairs (X,D) where D is as mentioned, an r-tuple of (effective Cartier) divisors on the scheme X. The categories of pairs provide a convenient framework for working with only those jets that pull back families of divisors. / Science, Faculty of / Mathematics, Department of / Graduate

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We describe the space of polynomial fields tangent to a given an algebraic curve. / Se da una descripción del espacio de campos polinomiales tangentes a una curva algebraica dada.

In this thesis, we extend results of Emerald Stacy to compute an upper bound on the minimal height of a totally p-adic algebraic integer of degree 4 independent of p. We also compute actual values of the minimal height of a totally p-adic algebraic integer of degree 4 for small primes p.

This thesis aims to fit and compare different time series models namely the ARIMA-model, conditional heteroscedastic models and lastly a dynamic regression model with ARIMA error to Bitcoin closing price data that spans over 5 consecutive years. The purpose is to evaluate if the sanction on Russia had any effect on the cryptocurrency Bitcoin. After giving a very brief introduction to time series models and the nature of the error term, we describe the models that we want to compare. Quite early in on, autocorrelation was detected and that the time series were nonstationary. Additionally, as we are dealing with financial data, we found that the best alternative was to transform the data into logarithmic return and we then took the first difference. As we then detected a very large outlier, we decided to replace the extreme value with the mean of the two adjacent observations as we suspected it would affect the forecast interval. The dataset with first differenced log-returns was used in the ARIMA model but it turned out that there was no autocorrelation which indicated that returns in financial assets are uncorrelated across time and therefore unpredictable. The conditional heteroscedastic models, the ARCH and the GARCH models turned out to be best suitable for our data, as there was an ARCH-effect present. We could conclude that the GARCH(1,1) model using student t-distribution had the best fit, which had the lowest AIC and the highest log likelihood. In order to study the effect of the sanctions on Bitcoin volatility a dynamic regression model was used by allowing the error term to contain autocorrelation and include an independent dummy variable. The model showed that the Russian invasion of Ukraine did not, surprisingly, have any effect on the Bitcoin closing price.

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京都大学 / 新制・課程博士 / 博士(理学) / 甲第23685号 / 理博第4775号 / 新制||理||1684(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 荒川 知幸, 教授 玉川 安騎男, 教授 並河 良典 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

This thesis introduces a new effective method in statistical modeling and probabilistic decision making problems. The method is based on maximizing the Shannon Logarithmic Entropy Function for information, subject to the given prior information to serve as constraints, to generate a probability distribution. The method is known as the Maximum Entropy Principle or "Jaynes Principle". Tribus used it earlier, but in a limited case, without general application to either statistical modeling or probablistic decision making. In this thesis, a new method which generalizes the above principle is introduced. This permits practical applications, some of which are illustrated. / Thesis / Master of Engineering (ME)

No description available.