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: Knowing our Neighbour: A Study of Islam for Christians in Malaysia [Book review]

authors: Pratt, Douglas

subjects: religion

abstract: Review of the book, "Knowing our Neighbour: A Study of Islam for Christians in Malaysia" by Albert Sundararaj Walters.

text: Knowing our Neighbour: A Study of Islam for Christians in Malaysia [Book review] Review of the book, "Knowing our Neighbour: A Study of Islam for Christians in Malaysia" by Albert Sundararaj Walters.

year: 2009

title: Lavinia and Henry Dunbar Johnson

authors: Hart, Philip

abstract: Rawinia Manukau, of Ngati Tamatera, married Henry Dunbar Johnson in 1868, when aged 21. Johnson had been a storekeeper at Coromandel from 1863 onwards and after 1866 had the first store at the site of the future Paeroa. In both places, but particularly in the latter, there was always the fear of conflict with the local Maori population, despite his being protected by a rangatira. After spending time at the new Thames goldfield, from 1871 onwards he was a partner in another Paeroa store, being able to erect a house because a rangatira related to his wife wished her to settle there. Lavinia (as she was known to Pakeha) obtained interests in several blocks of land in Ohinemuri, and her husband also acquired some land for a farm. He prospected Karangahake mountain from 1866 onwards, despite Maori opposition, and in 1875 and the following year actively mined there, unprofitably; Lavinia was not involved with this field, but did acquire interests in two claims when the Te Aroha one opened.
A petty squabble with a Thames neighbour resulted in Lavinia telling the latter to go back to England – she was feisty in defending her heritage. Through his marriage and close contact with Maori, Johnson understood and admired the Maori language, leading to his being appointed a licensed interpreter in 1872. After being a leading pioneer of Paeroa, in 1879 he went to Wellington to work in the Native Office, leaving his wife and family behind for a while; after then she was employed by Pakeha as a nurse and midwife. 
In 1885 Johnson was appointed to oversee the development of Rotorua, where he attempted to have Maori children educated (as his own were) and had to cope with the aftermath of the 1886 Tarawera eruption. Retrenched in 1888, he farmed for a while on his wife’s land at Te Aroha West, becoming involved in local issues and local politics. During the 1890 he obtained more official positions, and from 1896 to 1906 was a land court judge.
The Johnson family was well integrated into the dominant culture, all his daughters marrying Pakeha apart from one who married a ‘half-caste’ who had been brought up as a Pakeha. Johnson was not a Pakeha Maori in the original sense, but was accepted by Maori when living in Paeroa and as a judge tried to be kind to poverty-stricken Maori, although in time he viewed Maori as becoming lazy compared with those he had lived amongst during his first two decades in New Zealand.

text: Lavinia and Henry Dunbar Johnson Rawinia Manukau, of Ngati Tamatera, married Henry Dunbar Johnson in 1868, when aged 21. Johnson had been a storekeeper at Coromandel from 1863 onwards and after 1866 had the first store at the site of the future Paeroa. In both places, but particularly in the latter, there was always the fear of conflict with the local Maori population, despite his being protected by a rangatira. After spending time at the new Thames goldfield, from 1871 onwards he was a partner in another Paeroa store, being able to erect a house because a rangatira related to his wife wished her to settle there. Lavinia (as she was known to Pakeha) obtained interests in several blocks of land in Ohinemuri, and her husband also acquired some land for a farm. He prospected Karangahake mountain from 1866 onwards, despite Maori opposition, and in 1875 and the following year actively mined there, unprofitably; Lavinia was not involved with this field, but did acquire interests in two claims when the Te Aroha one opened.
A petty squabble with a Thames neighbour resulted in Lavinia telling the latter to go back to England – she was feisty in defending her heritage. Through his marriage and close contact with Maori, Johnson understood and admired the Maori language, leading to his being appointed a licensed interpreter in 1872. After being a leading pioneer of Paeroa, in 1879 he went to Wellington to work in the Native Office, leaving his wife and family behind for a while; after then she was employed by Pakeha as a nurse and midwife. 
In 1885 Johnson was appointed to oversee the development of Rotorua, where he attempted to have Maori children educated (as his own were) and had to cope with the aftermath of the 1886 Tarawera eruption. Retrenched in 1888, he farmed for a while on his wife’s land at Te Aroha West, becoming involved in local issues and local politics. During the 1890 he obtained more official positions, and from 1896 to 1906 was a land court judge.
The Johnson family was well integrated into the dominant culture, all his daughters marrying Pakeha apart from one who married a ‘half-caste’ who had been brought up as a Pakeha. Johnson was not a Pakeha Maori in the original sense, but was accepted by Maori when living in Paeroa and as a judge tried to be kind to poverty-stricken Maori, although in time he viewed Maori as becoming lazy compared with those he had lived amongst during his first two decades in New Zealand.

year: 2016