This thesis consists of studies in the separate fields of operator algebras and non-associative algebras. Two natural operator algebra structures, A ⊗_max B and A ⊗_min B, exist on the tensor product of two given unital operator algebras A and B. Because of the different properties enjoyed by the two tensor products in connection to dilation theory, it is of interest to know when they coincide (completely isometrically). Motivated by earlier work due to Paulsen and Power, we provide conditions relating an operator algebra B and another family {C_i}_i of operator algebras under which, for any operator algebra A, the equality A ⊗_max B = A ⊗_min B either implies, or is implied by, the equalities A ⊗_max C_i = A ⊗_min C_i for every i. These results can be applied to the setting of a discrete group G pre-ordered by a subsemigroup G⁺, where B ⊆ C*_r(G) is the subalgebra of the reduced group C*-algebra of G generated by G⁺, and C_i = A(Q_i) are digraph algebras defined by considering certain pre-ordered subsets Q_i of G. The 16-dimensional algebra A₄ of real sedenions is obtained by applying the Cayley-Dickson doubling process to the real division algebra of octonions. The classification of subalgebras of A₄ up to conjugacy (i.e. by the action of the automorphism group of A₄) was completed in a previous investigation, except for the collection of those subalgebras which are isomorphic to the quaternions. We present a classification of quaternion subalgebras up to conjugacy.

There exists considerable debate in the texture community about whether grain interactions are a necessary factor to explain the development of deformation textures in polycrystalline metals. Computer simulations indicate that grain interactions play a significant role, while experimental evidence shows that the material type and starting orientation are more important in the development of texture and microstructure. A balanced review of the literature on face-centered cubic metals shows that the opposing viewpoints have developed due to the lack of any complete experimental study which considers both the intrinsic (material type and starting orientation) and extrinsic (grain interaction) factors. In this study, a novel method was developed to assemble ideally orientated crystalline aggregates in 99.99\% aluminum (Al) or copper (Cu) to experimentally evaluate the effect of grain interactions on room temperature deformation texture. Ideal orientations relevant to face-centered cubic rolling textures, Cube $\{100\}\left<001\right>$, Goss $\{110\}\left<001\right>$, Brass $\{110\}\left<1\bar{1}2\right>$ and Copper $\{112\}\left<11\bar{1}\right>$ were paired in different combinations and deformed by plane strain compression to moderate strain levels of 1.0 to 1.5. Orientation dependent mechanical behavior was distinguishable from that of the neighbor-influenced behavior. In interacting crystals the constraint on the rolling direction shear strains ($\gamma_{_{XY}}, \gamma_{_{XZ}}$) was found to be most critical to show the effect of interactions via the evolution of local microstructure and microtexture. Interacting crystals with increasing deformations were observed to gradually rotate towards the S-component, $\{123\}\langle\bar{6}\bar{3}4\rangle$. Apart from the average lattice reorientations, the interacting crystals also developed strong long-range orientation gradients inside the bulk of the crystal, which were identified as accumulating misorientations across the deformation boundaries. Based on a statistical procedure using quaternions, the orientation and interaction related heterogeneous deformations were characterized by three principal component vectors and their respective eigenvalues for both the orientation and misorientation distributions. For the case of a medium stacking fault energy metal like Cu, the texture and microstructure development depends wholly on the starting orientations. Microstructural instabilities in Cu are explained through a local slip clustering process, and the possible role of grain interactions on such instabilities is proposed. In contrast, the texture and microstructure development in a high stacking fault energy metal like Al is found to be dependent on the grain interactions. In general, orientation, grain interaction and material type were found to be key factors in the development of rolling textures in face-centered cubic metals and alloys. Moreso, in the texture development not any single parameter can be held responsible, rather, the interdependency of each of the three parameters must be considered. In this frame-work polycrystalline grains can be classified into four types according to their stability and susceptibility during deformation. / Thesis (Ph.D, Mechanical and Materials Engineering) -- Queen's University, 2009-09-25 23:59:11.809

This thesis consists of studies in the separate fields of operator algebras and non-associative algebras. Two natural operator algebra structures, A ⊗_max B and A ⊗_min B, exist on the tensor product of two given unital operator algebras A and B. Because of the different properties enjoyed by the two tensor products in connection to dilation theory, it is of interest to know when they coincide (completely isometrically). Motivated by earlier work due to Paulsen and Power, we provide conditions relating an operator algebra B and another family {C_i}_i of operator algebras under which, for any operator algebra A, the equality A ⊗_max B = A ⊗_min B either implies, or is implied by, the equalities A ⊗_max C_i = A ⊗_min C_i for every i. These results can be applied to the setting of a discrete group G pre-ordered by a subsemigroup G⁺, where B ⊆ C*_r(G) is the subalgebra of the reduced group C*-algebra of G generated by G⁺, and C_i = A(Q_i) are digraph algebras defined by considering certain pre-ordered subsets Q_i of G. The 16-dimensional algebra A₄ of real sedenions is obtained by applying the Cayley-Dickson doubling process to the real division algebra of octonions. The classification of subalgebras of A₄ up to conjugacy (i.e. by the action of the automorphism group of A₄) was completed in a previous investigation, except for the collection of those subalgebras which are isomorphic to the quaternions. We present a classification of quaternion subalgebras up to conjugacy.

Thesis (Ph. D.)--Brandeis University, 2008. / "UMI:3306459." MICROFILM COPY ALSO AVAILABLE IN THE UNIVERSITY ARCHIVES. Includes bibliographical references.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第22230号 / 理博第4544号 / 新制||理||1653(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 池田 保, 教授 雪江 明彦, 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 is concerned with attitude estimation of small flying robots using low cost, small-sized inertial and magnetic sensors. A quaternion-based extended Kalman filter is used, which adaptively compensates external acceleration. External acceleration is the main source of estimation error. If external acceleration is detected, the accelerometer measurement covariance matrix of the Kalman filter is adjusted. The proposed algorithms are verified through experiments. Selected algorithm is implemented on STM32F4DISCOVERY development board.

The aim of this work is to study the Hamiltonian quaternions H and quaternion alge- bras. The first two chapters are based on the article Quaternion algebras by K. Conrad and the rest on the book Quaternion algebras by J. Voight. In the beginning, we mainly develop the theory about quaternions, quaternion algebras and study equivalent condi- tions for being a split or non-split quaternion algebra. After that, we also characterize, up to isomorphism, quaternion algebras over several fields such as R, C or Fp. In the third chapter, the thesis deals with orders in quaternion algebras, especially Lipschitz and Hurwitz order. The fourth chapter is dedicated to the relationship between unit quaternions and rotations in R3 , thanks to which we can characterize finite subgroups of H1 , or equivalently H× . This result will be used in the last chapter, where we are mainly focused on the problem of characterization of the group of units in orders in Hamiltonian quaternions. 1

京都大学 / 新制・課程博士 / 博士(理学) / 甲第22968号 / 理博第4645号 / 新制||理||1668(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)准教授 市野 篤史, 教授 池田 保, 教授 加藤 周 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

The miniature tailsitter is a unique aircraft with inherent advantages over typical unmanned aerial vehicles. With the capabilities of both hover and level flight, these small, portable systems can produce efficient maneuvers for enhanced surveillance and autonomy with little threat to surroundings and the system itself. Such vehicles are accompanied with control challenges due to the two different flight regimes. Problems with the conventional attitude representation arise in estimation and control as the system departs from level flight conditions. Furthermore, changing dynamics and limitations in modeling and sensing give rise to significant attitude control design challenges. Restrictions in computation also result from the limited size and weight capacity of the miniature airframe. In this research, the inherent control challenges discussed above are addressed with a computationally efficient adaptive quaternion control algorithm. A backstepping method for model cancellation and consistent tracking of reference model attitude dynamics is derived. This is used in conjunction with two different algorithms designed for the identification of system parameters. For a metric of baseline performance, gain-scheduled quaternion feedback control is developed. With a regularized data-weighting recursive least-squares parameter estimation algorithm, the adaptive quaternion controller is shown to be better than the baseline method in simulation and hardware results. This method is also shown to produce universal performance for all aircraft with the three conventional control surface actuators (aileron, elevator, and rudder) barring saturation and assuming accurate system identification. Testing of attitude control algorithms requires development in quaternion-based navigational control and attitude estimation. A novel technique for hover north/east position control is derived. Also, altitude tracking in hover, given an inconsistent thrust system, is addressed with an original method of on-line throttle system identification. Means for quaternion-based level flight control are produced from adaptations made to existing techniques employed in the Brigham Young University Multi-Agent Coordination and Control Lab. Also generated are simple trajectories for transitions between flight modes. A method for the estimation of quaternion attitude is developed, which uses multiple sensors combined in a filtering technique similar to the fixed-gain Kalman filter. Simulation and hardware results of these methods are presented for concept validation. A discussion of the development and production of these testing means (a simulation environment and hardware flight test system) is provided. In culmination, a fully autonomous miniature tailsitter system is produced with results demonstrating its various capabilities.

UAVs have become an essential tool in many market segments, particularly the military where critical intelligence can be gathered by them. A tailsitter aircraft is a platform whose purpose is to efficiently merge the range and endurance of fixed-wing aircraft with the VTOL capabilities of rotorcraft and is of significant value in applications where launch and recovery area is limited or the use of launch and recovery equipment is not desirable. Developing autopilot software for a tailsitter UAV is unique in that the aircraft must be autonomously controlled over a much wider range of attitudes than conventional UAVs. Assumptions made in conventional estimation and control algorithms are not valid for tailsitter aircraft because of routine operation around gimbal lock. Quaternions are generally employed to overcome the limitations Euler angles; however, adapting the attitude representation to work at a full range of attitudes is only part of the solution. Kalman filter measurement updates and control algorithms must also work at any orientation. This research presents several methods of incorporating a magnetometer measurement into an extended Kalman filter. One method combines magnetometer and accelerometer sensor data using the solution to Wahba's problem to calculate an overall attitude measurement. Other methods correct only heading error and include using two sets of Euler angles to update the estimate, using quaternions to determine heading error and Euler angles to update the estimate, and using only quaternions to update the estimate. Quaternion feedback attitude control is widely used in tailsitter aircraft. This research also shows that in spite of its effective use in spacecraft, using the attitude error calculated via quaternions to drive flight control surfaces may not be optimal for tailsitters. It is shown that during hover when heading error is present, quaternion feedback can cause undesired behavior, particularly when the heading error is large. An alternative method for calculating attitude error called resolved tilt-twist is validated, improved, and shown to perform better than quaternion feedback. Algorithms are implemented on a commercially available autopilot and validation is performed using hardware in loop simulation. A custom interface is used to receive autopilot commands and send the autopilot simulated sensor information. The final topic covered deals with the tailsitter hovering in wind. As the tailsitter hovers, wind can cause the tailsitter to turn such that the wind is perpendicular to the wings. Wind tunnel data is taken and analyzed to explain this behavior.