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.

Over the past 10 years, smartwatches have become increasingly popular for commercial use. Their ever-increasing capabilities, accuracy, and sophistication of smartwatches is making them increasingly appealing to physical activity researchers as a valuable research tool. The non-invasive nature, prevalence, and versatility of smartwatches is being utilized to track heart rate, blood-oxygen levels, activity and movement, and sleep. However, the current state of the art lacks a uniform method to extract, organize, and analyze data collected from these devices. The objective of this research was to develop a Python software library that is widely available, highly capable, and easy to use with the data collected by the Apple Watch. The library was designed to offer data science, visualization, and mining features that help physical activity research find and communicate patterns in the Apple Health data. The custom-built caching system of the library provides near-instant runtime to parse and analyze large files without trading off on memory usage. The Wanjara Smartwatch Library has significantly better performance, proven reliability and robustness, and improved usability than the alternatives discovered in the review of the literature. / Master of Science / Over the past 10 years, smartwatches have become increasingly popular for commercial use. Their ever-increasing capabilities, accuracy, and sophistication of smartwatches is making them increasingly appealing to physical activity researchers as a valuable research tool. The non-invasive nature, prevalence, and versatility of smartwatches is being utilized to track heart rate, blood-oxygen levels, activity and movement, and sleep. However, the current state of the art lacks a uniform method to extract, organize, and analyze data collected from these devices. The objective of this research was to develop a Python software library that is widely available, highly capable, and easy to use with the data collected by the Apple Watch. The library was designed to offer data science, visualization, and mining features that help physical activity research find and communicate patterns in the Apple Health data. The custom-built caching system of the library provides near-instant runtime to parse and analyze large files without trading off on memory usage. The Wanjara Smartwatch Library has significantly better performance, proven reliability and robustness, and improved usability than the alternatives discovered in the review of the literature.

This thesis elaborates the Büchi automata theory and introduces a library that enables working with them. Basics of the automata theory is explained. The main part of this work is focused on Büchi automata, which belong to finite automata. The main properties of Büchi automata are explained and proved. The knowledge of those properties is important for understanding the algorithms mentioned in this work. The next part describes those algorithms and explains their principles in details. The design of library is described in the next part of this work. Main part is aimed at the implementation of the library and auxiliary scripts. The most interesting and important parts of methods are described in detail. Closing part describes testing of library functionality.