title: Superintegrability on the two-dimensional hyperboloid

authors: Kalnins, Ernie G.

subjects: Mathematics

abstract: In this work we examine the basis functions for classical and quantum mechanical systems on the two-dimensional hyperboloid that admit separation of variables in at least two coordinate systems. We present all of these cases from a unified point of view. In particular, all of the special functions that arise via variable separation have their essential features expressed in terms of their zeros. The principal new results are the details of the polynomial bases for each of the nonsubgroup bases, not just the subgroup spherical coordinate cases, and the details of the structure of the quadratic symmetry algebras.

text: Superintegrability on the two-dimensional hyperboloid In this work we examine the basis functions for classical and quantum mechanical systems on the two-dimensional hyperboloid that admit separation of variables in at least two coordinate systems. We present all of these cases from a unified point of view. In particular, all of the special functions that arise via variable separation have their essential features expressed in terms of their zeros. The principal new results are the details of the polynomial bases for each of the nonsubgroup bases, not just the subgroup spherical coordinate cases, and the details of the structure of the quadratic symmetry algebras.

year: 1997-10

title: Geologically controlled sandy beaches: Their geomorphology, morphodynamics and classification

authors: Gallop, Shari L.

subjects: Science & Technology

abstract: Beaches that are geologically controlled by rock and coral formations are the rule, not the exception. This paper reviews the current understanding of geologically controlled beaches, bringing together a range of terminologies (including embayed beaches, shore platform beaches, relict beaches, and perched beaches, among others) and processes, with the aim of exploring the multiple ways in which geology influences beach morphology and morphodynamics. We show how in addition to sediment supply, the basement geology influences where beaches will form by providing accommodation, and in the cross-shore, aspects of rock platform morphology such as elevation and slope are also important. Geologically controlled beaches can have significant variations in sediment coverage with seasons and storms, and geological controls have fundamental influences on their contemporary morphodynamics. This includes wave shadowing by headlands and rock/coral formations inducing strong alongshore gradients in wave energy, resulting in corresponding variations in morphodynamic beach state and storm response. Geologically-induced rip currents including shadow rips, deflection rips and mega-rips that can develop on embayed beaches during storms, are an integral feature of the nearshore circulation and morphodynamics of geologically controlled beaches. We bring these processes together by presenting a conceptual model of alongshore and cross-shore levels of geological control. In the longshore dimension, this ranges from beaches that are slightly embayed, through to highly embayed beaches where headlands dominate the entire beach morphodynamic response. In the cross-shore dimension, this ranges from beaches without discernible geological controls, through to relict beaches above the influence of the contemporary littoral zone. Given the prevalence of geologically controlled beaches along the world's coasts, it is paramount for coastal management to consider how these beaches differ from unconstrained beaches and avoid applying inappropriate models and tools, especially with our uncertain future climate.

text: Geologically controlled sandy beaches: Their geomorphology, morphodynamics and classification Beaches that are geologically controlled by rock and coral formations are the rule, not the exception. This paper reviews the current understanding of geologically controlled beaches, bringing together a range of terminologies (including embayed beaches, shore platform beaches, relict beaches, and perched beaches, among others) and processes, with the aim of exploring the multiple ways in which geology influences beach morphology and morphodynamics. We show how in addition to sediment supply, the basement geology influences where beaches will form by providing accommodation, and in the cross-shore, aspects of rock platform morphology such as elevation and slope are also important. Geologically controlled beaches can have significant variations in sediment coverage with seasons and storms, and geological controls have fundamental influences on their contemporary morphodynamics. This includes wave shadowing by headlands and rock/coral formations inducing strong alongshore gradients in wave energy, resulting in corresponding variations in morphodynamic beach state and storm response. Geologically-induced rip currents including shadow rips, deflection rips and mega-rips that can develop on embayed beaches during storms, are an integral feature of the nearshore circulation and morphodynamics of geologically controlled beaches. We bring these processes together by presenting a conceptual model of alongshore and cross-shore levels of geological control. In the longshore dimension, this ranges from beaches that are slightly embayed, through to highly embayed beaches where headlands dominate the entire beach morphodynamic response. In the cross-shore dimension, this ranges from beaches without discernible geological controls, through to relict beaches above the influence of the contemporary littoral zone. Given the prevalence of geologically controlled beaches along the world's coasts, it is paramount for coastal management to consider how these beaches differ from unconstrained beaches and avoid applying inappropriate models and tools, especially with our uncertain future climate.

year: 2020

title: William Grey Nicholls and Rihitoto Mataia

authors: Hart, Philip

abstract: The son of a Pakeha Maori, Nicholls had an illustrious ancestry on his mother’s side, and his wife Rihitoto Mataia also had a distinguished whakapapa. From the 1870s Nicholls farmed in Ohinemuri, and as a licensed interpreter played an important role in land transactions at the same time as he was becoming prominent in the Pakeha community. For over 30 years he invested in mining, starting at Te Aroha in 1880. He also invested in a variety of other enterprises, becoming prosperous through these investments, his farming, and in particular by acquiring and selling land. Over a 40-year period, Nicholls conducted many cases in the land court, and assisted Pakeha to acquire Maori land, and also acquired a considerable amount of land for himself and Rihitoto, who inherited many blocks from her father (but had to pay off his massive debts).
         Whereas Rihitoto was a prominent leader in the Maori community of Hauraki, Nicholls had the same role in Pakeha society. Active in a variety of initiatives to benefit the Ohinemuri community, he was a popular member and chairman of the Ohinemuri County Council, and late in life was elevated to the Legislative Council. Reputedly he assisted Maori in various ways, but clearly he identified rather more with his Pakeha ancestry. His career was a remarkably successful one, and his wife was also successful in maximizing the benefits that could be derived from her ancestry.

text: William Grey Nicholls and Rihitoto Mataia The son of a Pakeha Maori, Nicholls had an illustrious ancestry on his mother’s side, and his wife Rihitoto Mataia also had a distinguished whakapapa. From the 1870s Nicholls farmed in Ohinemuri, and as a licensed interpreter played an important role in land transactions at the same time as he was becoming prominent in the Pakeha community. For over 30 years he invested in mining, starting at Te Aroha in 1880. He also invested in a variety of other enterprises, becoming prosperous through these investments, his farming, and in particular by acquiring and selling land. Over a 40-year period, Nicholls conducted many cases in the land court, and assisted Pakeha to acquire Maori land, and also acquired a considerable amount of land for himself and Rihitoto, who inherited many blocks from her father (but had to pay off his massive debts).
         Whereas Rihitoto was a prominent leader in the Maori community of Hauraki, Nicholls had the same role in Pakeha society. Active in a variety of initiatives to benefit the Ohinemuri community, he was a popular member and chairman of the Ohinemuri County Council, and late in life was elevated to the Legislative Council. Reputedly he assisted Maori in various ways, but clearly he identified rather more with his Pakeha ancestry. His career was a remarkably successful one, and his wife was also successful in maximizing the benefits that could be derived from her ancestry.

year: 2016