Dumb holes (also known as acoustic black holes) are fluid flows which include an "acoustic horizon:" a surface, analogous to a gravitational horizon, beyond which sound may pass but never classically return. Soundwaves in these flows will therefore experience "effective geometries" which are identical to black hole spacetimes up to a conformal factor. By adjusting the parameters of the fluid flow, it is possible to create an effective geometry which is conformal to the Anti-de Sitter black hole spacetime- a geometry which has recieved a great deal of attention in recent years due to its conjectured holographic duality to Conformal Field Theories. While we would not expect an acoustic analogue of the AdS-CFT correspondence to exist, this dumb hole provides a means, at least in principle, of experimentally testing the theoretical properties of the AdS spacetime. In particular, I have calculated the quasinormal mode spectrum of this acoustic geometry. / Graduate / 0986 / 0753 / jpbabb@yahoo.ca

GaugThe first half of the thesis concerns Abelian vortices and Yang-Mills theory. It is proved that the 5 types of vortices recently proposed by Manton are actually symmetry reductions of (anti-)self-dual Yang-Mills equations with suitable gauge groups and symmetry groups acting as isometries in a 4-manifold. As a consequence, the twistor integrability results of such vortices can be derived. It is presented a natural definition of their kinetic energy and thus the metric of the moduli space was calculated by the Samols' localisation method. Then, a modified version of the Abelian–Higgs model is proposed in such a way that spontaneous symmetry breaking and the Bogomolny argument still hold. The Painlevé test, when applied to its soliton equations, reveals a complete list of its integrable cases. The corresponding solutions are given in terms of third Painlevé transcendents and can be interpreted as original vortices on surfaces with conical singularity. The last two chapters present the following results in projective differential geometry and Hamiltonians of hydrodynamic-type systems. It is shown that the projective structures defined by the Painlevé equations are not metrisable unless either the corresponding equations admit first integrals quadratic in first derivatives or they define projectively flat structures. The corresponding first integrals can be derived from Killing vectors associated to the metrics that solve the metrisability problem. Secondly, it is given a complete set of necessary and sufficient conditions for an arbitrary affine connection in 2D to admit, locally, 0, 1, 2 or 3 Killing forms. These conditions are tensorial and simpler than the ones in previous literature. By defining suitable affine connections, it is shown that the problem of existence of Killing forms is equivalent to the conditions of the existence of Hamiltonian structures for hydrodynamic-type systems of two components.

In Markov chain models in finance and healthcare a transition matrix over a certain time interval is needed but only a transition matrix over a longer time interval may be available. The problem arises of determining a stochastic $p$th root of astochastic matrix (the given transition matrix). By exploiting the theory of functions of matrices, we develop results on the existence and characterization of stochastic $p$th roots. Our contributions include characterization of when a real matrix hasa real $p$th root, a classification of $p$th roots of a possibly singular matrix,a sufficient condition for a $p$th root of a stochastic matrix to have unit row sums,and the identification of two classes of stochastic matrices that have stochastic $p$th roots for all $p$. We also delineate a wide variety of possible configurationsas regards existence, nature (primary or nonprimary), and number of stochastic roots,and develop a necessary condition for existence of a stochastic root in terms of the spectrum of the given matrix. On the computational side, we emphasize finding an approximate stochastic root: perturb the principal root $A^{1/p}$ or the principal logarithm $\log(A)$ to the nearest stochastic matrix or the nearest intensity matrix, respectively, if they are not valid ones;minimize the residual $\normF{X^p-A}$ over all stochastic matrices $X$ and also over stochastic matrices that are primary functions of $A$. For the first two nearness problems, the global minimizers are found in the Frobenius norm. For the last two nonlinear programming problems, we derive explicit formulae for the gradient and Hessian of the objective function $\normF{X^p-A}^2$ and investigate Newton's method, a spectral projected gradient method (SPGM) and the sequential quadratic programming method to solve the problem as well as various matrices to start the iteration. Numerical experiments show that SPGM starting with the perturbed $A^{1/p}$to minimize $\normF{X^p-A}$ over all stochastic matrices is method of choice.Finally, a new algorithm is developed for computing arbitrary real powers $A^\a$ of a matrix $A\in\mathbb{C}^{n\times n}$. The algorithm starts with a Schur decomposition,takes $k$ square roots of the triangular factor $T$, evaluates an $[m/m]$ Pad\'e approximant of $(1-x)^\a$ at $I - T^$, and squares the result $k$ times. The parameters $k$ and $m$ are chosen to minimize the cost subject to achieving double precision accuracy in the evaluation of the Pad\'e approximant, making use of a result that bounds the error in the matrix Pad\'e approximant by the error in the scalar Pad\'e approximant with argument the norm of the matrix. The Pad\'e approximant is evaluated from the continued fraction representation in bottom-up fashion, which is shown to be numerically stable. In the squaring phase the diagonal and first superdiagonal are computed from explicit formulae for $T^$, yielding increased accuracy. Since the basic algorithm is designed for $\a\in(-1,1)$, a criterion for reducing an arbitrary real $\a$ to this range is developed, making use of bounds for the condition number of the $A^\a$ problem. How best to compute $A^k$ for a negative integer $k$ is also investigated. In numerical experiments the new algorithm is found to be superior in accuracy and stability to several alternatives,including the use of an eigendecomposition, a method based on the Schur--Parlett\alg\ with our new algorithm applied to the diagonal blocks and approaches based on the formula $A^\a = \exp(\a\log(A))$.

In underwater tracking and surveillance, the active towed array sonar presents a way of discovering and tracking adversarial submerged targets that try to stay hidden. The configuration consist of listening and emitting hydrophones towed behind a ship. Moreover, it has inherent limitations, and the characteristics of sound in the ocean are complex. By varying the pulse form emitted and the trajectory of the ship the measurement accuracy may be improved. This type of optimization constitutes a sensor management problem. In this thesis, a model of the tracking scenario has been constructed derived from Cramér-Rao bound analyses. A model predictive control approach together with information measures have been used to optimize a filter's estimated state of the target. For the simulations, the MATLAB environment has been used. Different combinations of decision horizons, information measures and variations of the Kalman filter have been studied. It has been found that the accuracy of the Extended Kalman filter is too low to give consistent results given the studied information measures. However, the Unscented Kalman filter is sufficient for this purpose.

We compute the Hilbert-Kunz functions of two-dimensional rings of type ADE by using representations of their indecomposable, maximal Cohen-Macaulay modules in terms of matrix factorizations, and as first syzygy modules of homogeneous ideals.

This dissertation is a historical biography of F.V. Engelenburg (1863-1938) and covers the period from 1889 to 1938, when Engelenburg lived and worked in South Africa. The study situates Engelenburg in the historical landscape of the Zuid-Afrikaansche Republiek during the late nineteenth and early twentieth century. The focus is mainly on Engelenburg’s journalistic career at De Volksstem, but attention is also given to his many other interests, including the development and promotion of Afrikaans and the Afrikaans academic culture, especially in the northern parts of South Africa. His work pertaining to the development of architecture, literature, aviation, the visual and performing arts, history, libraries, museums and educational institutions comes under the spotlight. His private life is considered as well in order to portray his versatility as a person. The chapters have been subdivided to highlight the variety of matters he was involved in, and a chronological approach has been followed as is customary in a biography. The study is based on archival research. In particular, Engelenburg’s private collections were used, as well as the private collections of some of his contemporaries. Engelenburg assumes a central place in the biography, with special focus on how he perceived and experienced conditions and everyday life in South Africa from the point of view of his transnational European background. His role as influential opinion-maker and political commentator on local and international politics is highlighted. His ties with political leaders and his involvement in government affairs are emphasised. The study also refers to his continued contact with his motherland, the Netherlands, and with the Dutch language. After the Anglo- Boer War, he realised that the languages of the future in South Africa would be Afrikaans (not Dutch), alongside English. His continuing support for Afrikaans as a language of instruction in schools and universities and the development of the Afrikaans literature, as well as his support for the standardization of Afrikaans helped to establish Afrikaans as an official language alongside English and Dutch in South Africa. Engelenburg’s active contribution to the work of the Zuid-Afrikaanse Akademie voor Taal, Lettere en Kuns (now the Suid- Afrikaanse Akademie vir Wetenskap en Kuns), helped to put the organization on a sound footing for future development. The Akademie can be seen as a living monument to his work in South Africa. / PhD (History)--North-West University, Vaal Triangle Campus, 2015.

This dissertation is a historical biography of F.V. Engelenburg (1863-1938) and covers the period from 1889 to 1938, when Engelenburg lived and worked in South Africa. The study situates Engelenburg in the historical landscape of the Zuid-Afrikaansche Republiek during the late nineteenth and early twentieth century. The focus is mainly on Engelenburg’s journalistic career at De Volksstem, but attention is also given to his many other interests, including the development and promotion of Afrikaans and the Afrikaans academic culture, especially in the northern parts of South Africa. His work pertaining to the development of architecture, literature, aviation, the visual and performing arts, history, libraries, museums and educational institutions comes under the spotlight. His private life is considered as well in order to portray his versatility as a person. The chapters have been subdivided to highlight the variety of matters he was involved in, and a chronological approach has been followed as is customary in a biography. The study is based on archival research. In particular, Engelenburg’s private collections were used, as well as the private collections of some of his contemporaries. Engelenburg assumes a central place in the biography, with special focus on how he perceived and experienced conditions and everyday life in South Africa from the point of view of his transnational European background. His role as influential opinion-maker and political commentator on local and international politics is highlighted. His ties with political leaders and his involvement in government affairs are emphasised. The study also refers to his continued contact with his motherland, the Netherlands, and with the Dutch language. After the Anglo- Boer War, he realised that the languages of the future in South Africa would be Afrikaans (not Dutch), alongside English. His continuing support for Afrikaans as a language of instruction in schools and universities and the development of the Afrikaans literature, as well as his support for the standardization of Afrikaans helped to establish Afrikaans as an official language alongside English and Dutch in South Africa. Engelenburg’s active contribution to the work of the Zuid-Afrikaanse Akademie voor Taal, Lettere en Kuns (now the Suid- Afrikaanse Akademie vir Wetenskap en Kuns), helped to put the organization on a sound footing for future development. The Akademie can be seen as a living monument to his work in South Africa. / PhD (History)--North-West University, Vaal Triangle Campus, 2015.

L'histoire du théorème de Jordan est abordée sous l'angle d'une question d'identité posée sur la période qui sépare la date de 1870 et l'énoncé par Camille Jordan d'une forme canonique des substitutions linéaires des années trente du vingtième siècle au cours desquelles le théorème de Jordan de la décomposition matricielle acquiert une place centrale dans la théorie des matrices canoniques. A partir d'un moment historique de référence, la controverse entre Jordan et Kronecker de 1874, le théorème de Jordan permet de jeter un regard original sur l'histoire de la période 1870-1930 en suivant le rôle joué par des savoirs tacites, des idéaux et des pratiques propres à des réseaux et des communautés. Ce regard permet notamment de mettre en évidence la dynamique d'une tension entre formes canoniques et invariants dans l'évolution de la signification de la notion de forme en mathématiques et contribue à l'histoire de l'algèbre linéaire en décrivant le rôle joué par une méthode de décomposition indissociable d'un mode particulier de représentation : la décomposition matricielle.

Data security will play a central role in the design of future IT systems. The PC has been a major driver of the digital economy. Recently, there has been a shift towards IT applications realized as embedded systems, because they have proved to be good solutions for many applications, especially those which require data processing in real time. Examples include security for wireless phones, wireless computing, pay-TV, and copy protection schemes for audio/video consumer products and digital cinemas. Most of these embedded applications will be wireless, which makes the communication channel vulnerable. The implementation of cryptographic systems presents several requirements and challenges. For example, the performance of algorithms is often crucial, and guaranteeing security is a formidable challenge. One needs encryption algorithms to run at the transmission rates of the communication links at speeds that are achieved through custom hardware devices. Public-key cryptosystems such as RSA, DSA and DSS have traditionally been used to accomplish secure communication via insecure channels. Elliptic curves are the basis for a relatively new class of public-key schemes. It is predicted that elliptic curve cryptosystems (ECCs) will replace many existing schemes in the near future. The main reason for the attractiveness of ECC is the fact that significantly smaller parameters can be used in ECC than in other competitive system, but with equivalent levels of security. The benefits of having smaller key size include faster computations, and reduction in processing power, storage space and bandwidth. This makes ECC ideal for constrained environments where resources such as power, processing time and memory are limited. The implementation of ECC requires several choices, such as the type of the underlying finite field, algorithms for implementing the finite field arithmetic, the type of the elliptic curve, algorithms for implementing the elliptic curve group operation, and elliptic curve protocols. Many of these selections may have a major impact on overall performance. In this dissertation a finite field from a special class called the Optimal Extension Field (OEF) is chosen as the underlying finite field of implementing ECC. OEFs utilize the fast integer arithmetic available on modern microcontrollers to produce very efficient results without resorting to multiprecision operations or arithmetic using polynomials of large degree. This dissertation discusses the theoretical and implementation issues associated with the development of this finite field in a low end embedded system. It also presents various improvement techniques for OEF arithmetic. The main objectives of this dissertation are to --Implement the functions required to perform the finite field arithmetic operations. -- Implement the functions required to generate an elliptic curve and to embed data on that elliptic curve. -- Implement the functions required to perform the elliptic curve group operation. All of these functions constitute a library that could be used to implement any elliptic curve cryptosystem. In this dissertation this library is implemented in an 8-bit AVR Atmel microcontroller. / Dissertation (MEng (Computer Engineering))--University of Pretoria, 2006. / Electrical, Electronic and Computer Engineering / unrestricted

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