title: Families of Orthogonal and Biorthogonal Polynomials on the N-Sphere

authors: Kalnins, Ernie G.

subjects: Science & Technology

abstract: The Laplace–Beltrami eigenvalue equation $H\Phi = \lambda \Phi $ on the n-sphere is studied, with an added vector potential term motivated by the differential equations for the polynomial Lauricella functions $F_A $. The operator H is self adjoint with respect to the natural inner product induced on the sphere and, in certain special coordinates, it admits a spectral decomposition with eigenspaces composed entirely of polynomials. The eigenvalues are degenerate but the degeneracy can be broken through use of the possible separable coordinate systems on the n-sphere. Then a basis for each eigenspace can be selected in terms of the simultaneous eigenfunctions of a family of commuting second-order differential operators that also commute with H. The results provide a multiplicity of n-variable orthogonal and biorthogonal families of polynomials that generalize classical results for one and two variable families of Jacobi polynomials on intervals, disks, and paraboloids.

text: Families of Orthogonal and Biorthogonal Polynomials on the N-Sphere The Laplace–Beltrami eigenvalue equation $H\Phi = \lambda \Phi $ on the n-sphere is studied, with an added vector potential term motivated by the differential equations for the polynomial Lauricella functions $F_A $. The operator H is self adjoint with respect to the natural inner product induced on the sphere and, in certain special coordinates, it admits a spectral decomposition with eigenspaces composed entirely of polynomials. The eigenvalues are degenerate but the degeneracy can be broken through use of the possible separable coordinate systems on the n-sphere. Then a basis for each eigenspace can be selected in terms of the simultaneous eigenfunctions of a family of commuting second-order differential operators that also commute with H. The results provide a multiplicity of n-variable orthogonal and biorthogonal families of polynomials that generalize classical results for one and two variable families of Jacobi polynomials on intervals, disks, and paraboloids.

year: 1991

title: Type brighter

authors: Soo, Chin-En Keith

subjects: colour

abstract: Type Brighter is intended as a new way of reading the alphabet. Shape, colour and pattern create memorable sequences based on characteristics of the letterform. By utilizing colour and repetition, readability is promoted. Each letter of the English alphabet is assigned a colour and positions, resulting in a full set of unique patterns. The user types using the keyboard, and the corresponding lights are shown on the light board. The simplicity of colour makes Type Brighter an alternative to more complicated communications such as mores code, and the use of pattern creates memorable sequences of colour. User can also experience the change of ambience, while the moving colour type projects an abstract story using light.

Type Brighter aims to create a new visual language through light. Colour, shape and pattern are strong visual elements, and when combined create a memorable experience. Colour serves an important part of our everyday lives and is easily distinguishable in all situations, making Type Brighter effective in a range of applications.

text: Type brighter Type Brighter is intended as a new way of reading the alphabet. Shape, colour and pattern create memorable sequences based on characteristics of the letterform. By utilizing colour and repetition, readability is promoted. Each letter of the English alphabet is assigned a colour and positions, resulting in a full set of unique patterns. The user types using the keyboard, and the corresponding lights are shown on the light board. The simplicity of colour makes Type Brighter an alternative to more complicated communications such as mores code, and the use of pattern creates memorable sequences of colour. User can also experience the change of ambience, while the moving colour type projects an abstract story using light.

Type Brighter aims to create a new visual language through light. Colour, shape and pattern are strong visual elements, and when combined create a memorable experience. Colour serves an important part of our everyday lives and is easily distinguishable in all situations, making Type Brighter effective in a range of applications.

year: 2013

title: Ngā wāhine ariki tapairu o Te Arawa: Te Kuraimonoa, Kearoa, Whakaotirangi

authors: Retikaukau, Georgia-Louise

subjects: Te Arawa

abstract: Similar to many other iwi (tribes) in Aotearoa, the Arawa people have a rich ancestral heritage that traces back to our origins in Hawaiki through our esteemed canoe traditions. However, within these narratives, the prominence of our heroic male ancestors tends to overshadow the stories of their wives, who are often mentioned merely as secondary characters. Te Kuraimonoa, Kearoa, and Whakaotirangi represent the first three women to be named in Te Arawa traditions, yet their narratives remain vague, existing only from a male perspective. Although their names can be found scattered throughout European and Māori literature, their stories are fragmented and incomplete, varying from tribe to tribe.

The objective of this research is to delve into these three ancestresses' lives from a Te Arawa female perspective, aiming to provide a more comprehensive and nuanced understanding. Employing a mixed-method research approach, this study will investigate Māori and European literature, supplemented by thematical analysis to study and compare data gathered from interviews with female Te Arawa descendants. The insights gained from this endeavour will contribute to the creation of substantial knowledge that can benefit both present and future generations.

Comprising of seven chapters, this thesis unfolds as follows: Chapter one introduces the research purpose and provides an overview of the subsequent chapters. Chapter two outlines the research methodology employed. Chapter three presents an in-depth exploration of Te Arawa iwi, elucidating their significance and cultural context. Chapter four encompasses a comprehensive literature review, while chapter five is solely dedicated to the insights derived from the interview dialogues. Chapter six delves into the findings and engages in detailed discussions, offering a comprehensive analysis. Lastly, chapter seven serves as the conclusion, summarizing the key findings and their implications.

text: Ngā wāhine ariki tapairu o Te Arawa: Te Kuraimonoa, Kearoa, Whakaotirangi Similar to many other iwi (tribes) in Aotearoa, the Arawa people have a rich ancestral heritage that traces back to our origins in Hawaiki through our esteemed canoe traditions. However, within these narratives, the prominence of our heroic male ancestors tends to overshadow the stories of their wives, who are often mentioned merely as secondary characters. Te Kuraimonoa, Kearoa, and Whakaotirangi represent the first three women to be named in Te Arawa traditions, yet their narratives remain vague, existing only from a male perspective. Although their names can be found scattered throughout European and Māori literature, their stories are fragmented and incomplete, varying from tribe to tribe.

The objective of this research is to delve into these three ancestresses' lives from a Te Arawa female perspective, aiming to provide a more comprehensive and nuanced understanding. Employing a mixed-method research approach, this study will investigate Māori and European literature, supplemented by thematical analysis to study and compare data gathered from interviews with female Te Arawa descendants. The insights gained from this endeavour will contribute to the creation of substantial knowledge that can benefit both present and future generations.

Comprising of seven chapters, this thesis unfolds as follows: Chapter one introduces the research purpose and provides an overview of the subsequent chapters. Chapter two outlines the research methodology employed. Chapter three presents an in-depth exploration of Te Arawa iwi, elucidating their significance and cultural context. Chapter four encompasses a comprehensive literature review, while chapter five is solely dedicated to the insights derived from the interview dialogues. Chapter six delves into the findings and engages in detailed discussions, offering a comprehensive analysis. Lastly, chapter seven serves as the conclusion, summarizing the key findings and their implications.

year: 2023