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Gaussian elimination with partial pivoting (GEPP) is a widely used method to solve dense linear systems. Each GEPP step uses a row transposition pivot movement if needed to ensure the leading pivot entry is maximal in magnitude for the leading column of the remaining untriangularized subsystem. We will use theoretical and numerical approaches to study how often this pivot movement is needed. We provide full distributional descriptions for the number of pivot movements needed using GEPP using particular Haar random ensembles, as well as compare these models to other common transformations from randomized numerical linear algebra. Additionally, we introduce new random ensembles with fixed pivot movement counts and fixed sparsity, $\alpha$. Experiments estimating the empirical spectral density (ESD) of these random ensembles leads to a new conjecture on a universality class of random matrices with fixed sparsity whose scaled ESD converges to a measure on the complex unit disk that depends on $\alpha$ and is an interpolation of the uniform measure on the unit disk and the Dirac measure at the origin.

Gaussian elimination (GE) is the most used dense linear solver. Error analysis of GE with selected pivoting strategies on well-conditioned systems can focus on studying the behavior of growth factors. Although exponential growth is possible with GE with partial pivoting (GEPP), growth tends to stay much smaller in practice. Support for this behavior was provided recently by Huang and Tikhomirov's average-case analysis of GEPP, which showed GEPP growth factors for Gaussian matrices stay at most polynomial with very high probability. GE with complete pivoting (GECP) has also seen a lot of recent interest, with improvements to both lower and upper bounds on worst-case GECP growth provided by Bisain, Edelman and Urschel in 2023. We are interested in studying how GEPP and GECP behave on the same linear systems as well as studying large growth on particular subclasses of matrices, including orthogonal matrices. Moreover, as a means to better address the question of why large growth is rarely encountered, we further study matrices with a large difference in growth between using GEPP and GECP, and we explore how the smaller growth strategy dominates behavior in a small neighborhood of the initial matrix.

The recursive structure of butterfly matrices has been exploited to accelerate common methods in computational linear algebra. This was first developed by D. Stott Parker \cite{Pa95}. Recently, the machine learning community has taken particular interest in these applications. Butterfly structures can now be found integrated into architectures for software used in learning fast solvers for large linear systems and in image recognition, covering tasks such as early cancer identification or smart vehicle navigation \cite{uwBFT, butterflynet, fast_alg}. These new advances have enabled less powerful computing systems, such as in mobile devices or portable smart devices, to effectively utilize computationally heavy tools that were previously unavailable. Although empirical evidence supports the use of butterfly matrices in these newer technologies, the literature on the mathematical theory that justified these results is lacking. Building on research started in \cite{Tr19}, I will give a fuller picture of the numerical, spectral, and group properties of particular ensembles of random butterfly matrices. This document will provide a stronger mathematical foundation to further support the approaches already found in practice and can inform future applications not yet explored.

The recursive structure of butterfly matrices has been exploited to accelerate common methods in computational linear algebra. This was first developed by D. Stott Parker \cite{Pa95}. Recently, the machine learning community has taken particular interest in these applications. Butterfly structures can now be found integrated into architectures for software used in learning fast solvers for large linear systems and in image recognition, covering tasks such as early cancer identification or smart vehicle navigation \cite{uwBFT, butterflynet, fast_alg}. These new advances have enabled less powerful computing systems, such as in mobile devices or portable smart devices, to effectively utilize computationally heavy tools that were previously unavailable. Although empirical evidence supports the use of butterfly matrices in these newer technologies, the literature on the mathematical theory that justified these results is lacking. Building on research started in \cite{Tr19}, I will give a fuller picture of the numerical, spectral, and group properties of particular ensembles of random butterfly matrices. This document will provide a stronger mathematical foundation to further support the approaches already found in practice and can inform future applications not yet explored.

The recursive structure of butterfly matrices has been exploited to accelerate common methods in computational linear algebra. This was first developed by D. Stott Parker \cite{Pa95}. Recently, the machine learning community has taken particular interest in these applications. Butterfly structures can now be found integrated into architectures for software used in learning fast solvers for large linear systems and in image recognition, covering tasks such as early cancer identification or smart vehicle navigation \cite{uwBFT, butterflynet, fast_alg}. These new advances have enabled less powerful computing systems, such as in mobile devices or portable smart devices, to effectively utilize computationally heavy tools that were previously unavailable. Although empirical evidence supports the use of butterfly matrices in these newer technologies, the literature on the mathematical theory that justified these results is lacking. Building on research started in \cite{Tr19}, I will give a fuller picture of the numerical, spectral, and group properties of particular ensembles of random butterfly matrices. This document will provide a stronger mathematical foundation to further support the approaches already found in practice and can inform future applications not yet explored.

Random butterfly matrices were introduced by Parker in 1995 to remove the need for pivoting when using Gaussian elimination. The growing applications of butterfly matrices have often eclipsed the mathematical understanding of how or why butterfly matrices are able to accomplish these given tasks. To help begin to close this gap using theoretical and numerical approaches, we explore the impact on the growth factor of preconditioning a linear system by butterfly matrices. These results are compared to other common methods found in randomized numerical linear algebra. In these experiments, we show preconditioning using butterfly matrices has a more significant dampening impact on large growth factors than other common preconditioners and a smaller increase to minimal growth factor systems. Moreover, we are able to determine the full distribution of the growth factors for a subclass of random butterfly matrices. Previous results by Trefethen and Schreiber relating to the distribution of random growth factors were limited to empirical estimates of the first moment for Ginibre matrices.

This paper outlines the key issues of remotely invigilated online exams (RIOEs) and presents ways to avoid and resolve the issues for educators who are considering implementing them. The purpose of this paper is to share the lessons learned during the process of implementing and evaluating RIOEs and highlight the key considerations required to conduct RIOEs more seamlessly, whilst minimising students’ cognitive load. With the continued growth, and future importance of online tertiary education, this paper provides an important contribution to the understanding of the best methods and practices by which to conduct online examinations and provides a foundation for continued research and enhancement of effective RIOEs. The paper follows an extensive Action Learning process to develop and present a case study that was conducted across nine fully online business courses in a start-up venture for the University of South Australia. Cognitive load theory underpins the case study, which enabled the researchers to gain profound understanding into the RIOE process, identify issues and offer resolutions. RIOEs require more systematic and effective design compared to traditional paper-based exams and should be supplemented by early and clear communication with students. Educators should enable and encourage students to rehearse the exam service access procedures prior to their exams and students should be provided with real-time responsive technical support for any ad hoc issues that may present during the exam. These factors play a critical role in ensuring the successful implementation of RIOEs.

This paper outlines the key issues of remotely invigilated online exams (RIOEs) and presents ways to avoid and resolve the issues for educators who are considering implementing them. The purpose of this paper is to share the lessons learned during the process of implementing and evaluating RIOEs and highlight the key considerations required to conduct RIOEs more seamlessly, whilst minimising students’ cognitive load. With the continued growth, and future importance of online tertiary education, this paper provides an important contribution to the understanding of the best methods and practices by which to conduct online examinations and provides a foundation for continued research and enhancement of effective RIOEs. The paper follows an extensive Action Learning process to develop and present a case study that was conducted across nine fully online business courses in a start-up venture for the University of South Australia. Cognitive load theory underpins the case study, which enabled the researchers to gain profound understanding into the RIOE process, identify issues and offer resolutions. RIOEs require more systematic and effective design compared to traditional paper-based exams and should be supplemented by early and clear communication with students. Educators should enable and encourage students to rehearse the exam service access procedures prior to their exams and students should be provided with real-time responsive technical support for any ad hoc issues that may present during the exam. These factors play a critical role in ensuring the successful implementation of RIOEs.

The purpose of this research was to investigate the F-16 Fault Isolation Manual from the maintenance technician's perspective. Evaluation considered the effect of various demographic factors and the maintenance technicians' perceptions on the Fault Isolation Manuals usefulness and accuracy. The findings indicate that maintenance technician's who use the manual more perceive the manual to be more useful and accurate than technicians who use the manual less. The findings also determined that technician's have a perception that the manual has several inaccurate areas.

The Chattohoochee Review was a literary journal that published new works of fiction, non-fiction, poetry and visual arts. This volume features the following entries: "A Conversation with David Kirby", "Living of Little Horse Creek", "In Lieu of Song", "Land of Cloudless Day", "Method Writing", "The Morning After the Quarrel", "Once, with a Girld who had Epilepsy", "Steven", "Jagged", "Back Then", "Met Before", "Dreaming of Rainstorms", "Four Corners", "Castaways", "A Guileless Air", "Uncle George's Lady Friend", "The Gigolo in the Gazebo", "Imagining my Daughter's Boyfriends as She Turns 13", "In Interview with Terry Kay", "Too Beholden: An Essay-review of Beet Adcock's Beholdings", "Festive Spirits, Kindred Spirits: A Review", "The Lotus Flowers: A Review".

The Chattohoochee Review was a literary journal that published new works of fiction, non-fiction, poetry and visual arts. This volume features the following entries: "A Conversation with David Kirby", "Living of Little Horse Creek", "In Lieu of Song", "Land of Cloudless Day", "Method Writing", "The Morning After the Quarrel", "Once, with a Girld who had Epilepsy", "Steven", "Jagged", "Back Then", "Met Before", "Dreaming of Rainstorms", "Four Corners", "Castaways", "A Guileless Air", "Uncle George's Lady Friend", "The Gigolo in the Gazebo", "Imagining my Daughter's Boyfriends as She Turns 13", "In Interview with Terry Kay", "Too Beholden: An Essay-review of Beet Adcock's Beholdings", "Festive Spirits, Kindred Spirits: A Review", "The Lotus Flowers: A Review".