We mainly consider finding formula solutions to quaternion equations and calculating the generalized inverses of quaternion polynomial matrices.

We mainly consider finding formula solutions to quaternion equations and calculating the generalized inverses of quaternion polynomial matrices.

Berlin, Freie University, Diss., 2005. / Dateiformat: zip, Dateien im PDF-Format.

Münster (Westfalen), Univ., Diss., 2009

One important part of inertial navigation is the estimation of the direction relative to the Earth’s geographic North Pole, the so called heading. In this project, a gyroscope and an accelerometer were used together in an Extended Kalman Filter with a quaternion as the state space variable, representing the attitude. Given the attitude of the system, measurements from a magnetometer were rotated to a horizontal coordinate frame in order to calculate the direction toward Earth’s magnetic North Pole. Comparing this direction with the angle toward the Geographic North Pole given by a GPS, the local magnetic declination was estimated with the purpose of correcting the heading in the future. Different methods for detecting disturbances on the magnetometer in order to automatically decide when it is to be trusted was discussed and evaluated. Routines for easily performing sensor calibration was created. The outcome of the project was a well working attitude estimation, simply performed calibration routines and a set of methods working together to detect magnetometer disturbances. / En viktig del av ett tröghetsnavigeringssystem är skattningen av riktningen relativt jordens geografiska nordpol, den så kallade bäringen. I detta projekt användes ett gyroskop och en accelerometer tillsammans i ett Extended Kalman filter med en quaternion som tillståndsvariabel för att representera attityden. Givet systemets attityd roterades mätningar från en magnetometer till ett horisontellt koordinatsystem för att beräkna riktningen mot magnetiska nordpolen. Genom att jämföra denna riktning med vinkeln mot geografiska nordpolen kunde den lokala magnetiska deklinationen skattas för att sedan användas i framtiden för att korrigera bäringen. Olika metoder för att detektera störningar på magnetometern för att automatiskt bestämma när den är störd diskuteras och utvärderas. Rutiner för att enkelt kalibrera sensorerna skapades. Projektets resultat var en väl fungerande attitydestimering, enkla kalibreringsmetoder samt ett par metoder för att detektera störningar på magnetometern.

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La première partie de ce mémoire relève les principaux problèmes de nature algébrique et géométrique qu'ont dû résoudre les mathématiciens avant d'accepter l'existence des nombres complexes; l'une des conséquences de cet exercice est de proposer l'esquisse d'une approche plus adéquate à l'enseignement des nombres complexes au collégial. La deuxième partie présente l'approche géométrique des quaternions, tel que formulée par leur inventeur (Hamilton), puis démontre leurs principales propriétés géométriques dans le contexte de l'algèbre linéaire. Dans la troisième partie, l'axiomatisation de l'intuition géométrique est abordée dans le contexte des fondements proposés par Hilbert en regard des géométries non euclidiennes. ______________________________________________________________________________ MOTS-CLÉS DE L’AUTEUR : Histoire des nombres complexes, quaternions, fondements de la géométrie.

The main goal of this Masters' thesis is to explore isomorphism types of quaternion algebras using the theory of quadratic forms, number theory and algebra. I would also present ways to characterize quaternion algebras, and talk about how quaternion algebras are important in Brauer groups by describing a theorem proved by Merkurjev in 1981.

The main goal of this Masters' thesis is to explore isomorphism types of quaternion algebras using the theory of quadratic forms, number theory and algebra. I would also present ways to characterize quaternion algebras, and talk about how quaternion algebras are important in Brauer groups by describing a theorem proved by Merkurjev in 1981.

In this thesis, we study Quaternionic Analysis, which is the most natural and close generalization of complex analysis. Quaternion Analysis conserves many of its important features by key reference A. Sudbery. Quaternions are a non-commutative multiplication system in which all the other field hypotheses are valid, so the investigation of their properties and structure became the basis of this study. The first chapter contains a brief history of what led to the discovery of quaternions and their construction as a fourdimensional. Chapter two develops quaternionic algebra which is become a common part of mathematics and physics culture. In the third chapter, we present some theorems on therepresentations of quaternions utilising regularity and Cauchy-Riemann-Fueter. Quaternion derivatives in the mathematical literature are typically defined only for analytic (regular) functions. Moreover, this chapter shows how regular functions can be constructed from harmonic functions. The fourth and last chapter summarises the weaknesses and strengths of this thesis and provides suggestions for further study.