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There are four normed division algebras over real numbers, namely real numbers, complex numbers, quaternions, and octonions. Lack of commutativity and associativity make it difficult to investigate algebraic and geometric properties of octonions. Eigenvalue problem of octonionic Hermitian matrices is one of the interesting studies where we can see this difficulty of extending the basic properties from complex Hermitian matrices to octonionic Hermitian matrices. This includes the notion of orthogonality and decomposition of a Hermitian matrix using its eigenvalues and eigenvectors.Liping Huang and Wasin So derived explicit formulas for computing the roots of quaternionic quadratic equations. We extend their work to octonionic case and solve octonionic left quadratic equations. We represent left spectrum of two by two octonionic Hermitian matrix using the solutions to corresponding octonionic left quadratic equation and identify the family of matrices which admit non-real left eigenvalues. For three by three case we review previous work by Tevian Dray and Corinne Manogue of real eigenvalue problem and study characteristic equations which admit non-real roots that are correspond to non-real left eigenvalues. Finally, we discuss the right spectrum using the associator method, and provide examples using "pyoctonion" python library. Interesting applications and open problems for future studies in this literature are also included.

京都大学 / 新制・論文博士 / 博士(医学) / 乙第13584号 / 論医博第2304号 / 新制||医||1070(附属図書館) / (主査)教授 江木 盛時, 教授 湊谷 謙司, 教授 江藤 浩之 / 学位規則第4条第2項該当 / Doctor of Medical Science / Kyoto University / DFAM

This study was undertaken in an effort to determine whether or not left-handed students achieve as high pro- ficiency as do right-handed students in secondary school subjects which require motor ability. This problem came to the attention of the writer after he had examined the pre- vious writings in the general field of inquiry. It was found that mich of the material contained in these earlier studies had to do with the probable causes of handedness and what the effect would be to change a sinistral indi- vidual to dextral usage. Less material was found concern- ing the relative success of the left-handed student who was permitted to continue using his left hand.

The purpose of this research was to determine if the variables included in the Mississippi Report Card 2003-2004 utilized for the calculation of AYP can be used to predict with accuracy greater than that which can be attributed to chance, whether or not Mississippi LEAs will attain adequate yearly progress in reading and math using the logistic regression technique. An additional goal of this study is to identify whether the inclusion of a variable representing the proportion of teachers in each Mississippi LEA with a one-year teaching certificate can notably enhance the explanatory power of the logistic regression models. This study addressed two research questions: Research Question 1: Can variables (included in the Mississippi Report Card 2003-2004) required for the calculation of adequate yearly progress be used to successfully predict Adequate Yearly Progress using the Logistic Regression technique with an accuracy greater than that which can be attributed to chance? Research Question 2: Could the addition of another predictor variable (Percentage of Teachers with One-Year Educator Licenses) notably add to the predictive accuracy of the model? This study demonstrated that using the variables utilized for the calculation of AYP, a predictive model can be successfully utilized to classify Mississippi LEAs that will and will not attain AYP in reading and math with an accuracy greater than that which can be attributed to chance. This study also established that the inclusion of a variable corresponding to the percentage of teachers in a LEA with one-year educator licenses does not add to the predictive accuracy of the model.

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