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)

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