title: Wittrick-Williams algorithm proof of bracketing and convergence theorems for eigenvalues of constrained structures with positive and negative penalty parameters

authors: Ilanko, Sinniah

subjects: Wittrick-Williams algorithm

abstract: The well-established Wittrick-Williams algorithm is used to derive novel and general proofs that show that the eigenvalues of systems with constraints can be bracketed by replacing the constraints by positive and negative pairs of either ordinary or inertial penalty parameters. It is also shown that convergence occurs from both above and below when the numerical values of these parameters are increased towards infinity. The proofs are applicable in many contexts but are derived in that of structural systems, for which the eigenvalues are either buckling load factors or the squares of natural frequencies of vibration; ordinary penalty parameters are stiffnesses of translational and rotational springs; and inertial penalty parameters are either masses or rotary inertias. The penalty parameters can be used to constrain a system or to impose constraints between systems. It is shown that the use of inertial penalty parameters has several advantages compared with using ordinary ones. Then the pth eigenvalue of a system with n constraints is bounded closely from above by the (p+n)th eigenvalue of the system with very large positive inertial penalty parameters and from below by the pth eigenvalue, when large negative values are used instead. This work is expected to enhance the versatility of numerical eigenproblem methods, e.g. the Rayleigh-Ritz method.

text: Wittrick-Williams algorithm proof of bracketing and convergence theorems for eigenvalues of constrained structures with positive and negative penalty parameters The well-established Wittrick-Williams algorithm is used to derive novel and general proofs that show that the eigenvalues of systems with constraints can be bracketed by replacing the constraints by positive and negative pairs of either ordinary or inertial penalty parameters. It is also shown that convergence occurs from both above and below when the numerical values of these parameters are increased towards infinity. The proofs are applicable in many contexts but are derived in that of structural systems, for which the eigenvalues are either buckling load factors or the squares of natural frequencies of vibration; ordinary penalty parameters are stiffnesses of translational and rotational springs; and inertial penalty parameters are either masses or rotary inertias. The penalty parameters can be used to constrain a system or to impose constraints between systems. It is shown that the use of inertial penalty parameters has several advantages compared with using ordinary ones. Then the pth eigenvalue of a system with n constraints is bounded closely from above by the (p+n)th eigenvalue of the system with very large positive inertial penalty parameters and from below by the pth eigenvalue, when large negative values are used instead. This work is expected to enhance the versatility of numerical eigenproblem methods, e.g. the Rayleigh-Ritz method.

year: 2007

title: Short Note: Report of mummified leopard seal carcass in the southern Dry Valleys, McMurdo Sound, Antarctica.

authors: Banks, Jonathan C.

subjects: leopard seal

abstract: The wide spread occurrence of mummified seal and penguin carcasses tens of kilometres from the open ocean is an interesting phenomenon occurring in the Antarctic Dry Valleys. Mummified seal carcasses were first reported by Scott’s expedition in 1903 (Scott 1969), and live seals and seal carcasses have since been reported many kilometres from the nearest ice-free ocean. Seal carcasses found in the McMurdo Dry Valleys are predominantly crabeater seals (Lobodon carcinophaga (Hombron & Jacquinot)) with a smaller number of Weddell seals, (Leptonychotes weddellii (Lesson)), also reported. Here we present only the second published report of a leopard seal carcass from the McMurdo Dry Valleys.

text: Short Note: Report of mummified leopard seal carcass in the southern Dry Valleys, McMurdo Sound, Antarctica. The wide spread occurrence of mummified seal and penguin carcasses tens of kilometres from the open ocean is an interesting phenomenon occurring in the Antarctic Dry Valleys. Mummified seal carcasses were first reported by Scott’s expedition in 1903 (Scott 1969), and live seals and seal carcasses have since been reported many kilometres from the nearest ice-free ocean. Seal carcasses found in the McMurdo Dry Valleys are predominantly crabeater seals (Lobodon carcinophaga (Hombron & Jacquinot)) with a smaller number of Weddell seals, (Leptonychotes weddellii (Lesson)), also reported. Here we present only the second published report of a leopard seal carcass from the McMurdo Dry Valleys.

year: 2010

title: Toi Tuku Iho: An exploration of oranga and mana motuhake through toi

authors: Waititi, Kahurangi Rora

subjects: Toi tuku iho

abstract: This research examines how mahi toi as process, theory, and output can contribute to oranga and mana motuhake for whānau, hapū, and iwi which is explored through the tribal storytelling practices of Te Whānau a Apanui. In this thesis, I argue that kōrero tuku iho is the umbilical link to our ancestors that sustains us with teachings; cultivating a richness and depth of self and connection through story. I demonstrate that the vehicles of sharing are as varied as the stories themselves and are known as “toi” or our creative expressions of kōrero tuku iho. 
The key focus of this thesis is to reveal how toi can be used to decolonise and re-indigenise research methodologies and facilitate transformation through praxis. Iwi storytellers were interviewed, sharing that within our toi practices lie a myriad of ways to challenge colonial structures and ideologies, ground us in tīpuna knowledge systems, and provide a space to innovate solutions to the challenges of te ao hurihuri (a changing world). The kaikōrero provided a collective and relational lens that fostered the development of a Toi Tuku Iho methodology. This was processed through a creative output that I created of a digital story that was filmed to reveal how knowledge through stories can become taonga tuku iho. A Toi Tuku Iho methodology is proffered to understand the complex processes of iwi storytelling and reveal the relationality between innovation and what this thesis refers to as the tipuna continuum.
Themes explored in the thesis are an ecosystem of learning through people, place, story, whakapapa and time; how iwi oranga is viewed and contributed to by toi; and how mana motuhake is embraced and enacted through kōrero tuku iho. Furthermore, the application of kaupapa Māori creative practices to this research expands the academic confines of knowledge production to include Indigenous stories and toi. The thesis demonstrates that the research process, potential products, and outcomes, are transformed by using these practices.  Storyscapes, timescapes, landscapes and generations past and future merge together, in this research, to facilitate dreams and aspirations for future generations. The collapsing of timescapes is explored through audio-visual digital storytelling, a medium known to manoeuvre time to enhance a story. I seek to amplify the power of tribal storytelling through toi to transform the spaces we occupy both within and outside of our iwi. Mahi toi ignites our creative selves to dream and can activate a belief to explore beyond what we know. I argue that Toi Tuku Iho has the potential to engage communities, on their own terms, in their own ways and for their own means.

text: Toi Tuku Iho: An exploration of oranga and mana motuhake through toi This research examines how mahi toi as process, theory, and output can contribute to oranga and mana motuhake for whānau, hapū, and iwi which is explored through the tribal storytelling practices of Te Whānau a Apanui. In this thesis, I argue that kōrero tuku iho is the umbilical link to our ancestors that sustains us with teachings; cultivating a richness and depth of self and connection through story. I demonstrate that the vehicles of sharing are as varied as the stories themselves and are known as “toi” or our creative expressions of kōrero tuku iho. 
The key focus of this thesis is to reveal how toi can be used to decolonise and re-indigenise research methodologies and facilitate transformation through praxis. Iwi storytellers were interviewed, sharing that within our toi practices lie a myriad of ways to challenge colonial structures and ideologies, ground us in tīpuna knowledge systems, and provide a space to innovate solutions to the challenges of te ao hurihuri (a changing world). The kaikōrero provided a collective and relational lens that fostered the development of a Toi Tuku Iho methodology. This was processed through a creative output that I created of a digital story that was filmed to reveal how knowledge through stories can become taonga tuku iho. A Toi Tuku Iho methodology is proffered to understand the complex processes of iwi storytelling and reveal the relationality between innovation and what this thesis refers to as the tipuna continuum.
Themes explored in the thesis are an ecosystem of learning through people, place, story, whakapapa and time; how iwi oranga is viewed and contributed to by toi; and how mana motuhake is embraced and enacted through kōrero tuku iho. Furthermore, the application of kaupapa Māori creative practices to this research expands the academic confines of knowledge production to include Indigenous stories and toi. The thesis demonstrates that the research process, potential products, and outcomes, are transformed by using these practices.  Storyscapes, timescapes, landscapes and generations past and future merge together, in this research, to facilitate dreams and aspirations for future generations. The collapsing of timescapes is explored through audio-visual digital storytelling, a medium known to manoeuvre time to enhance a story. I seek to amplify the power of tribal storytelling through toi to transform the spaces we occupy both within and outside of our iwi. Mahi toi ignites our creative selves to dream and can activate a belief to explore beyond what we know. I argue that Toi Tuku Iho has the potential to engage communities, on their own terms, in their own ways and for their own means.

year: 2023