title: Structure relations and darboux contractions for 2D 2nd order superintegrable systems

authors: Heinonen, Robin

abstract: Two-dimensional quadratic algebras are generalizations of Lie algebras that include the symmetry algebras of 2nd order superintegrable systems in 2 dimensions as special cases. The superintegrable systems are exactly solvable physical systems in classical and quantum mechanics. Distinct superintegrable systems and their quadratic algebras can be related by geometric contractions, induced by Inönü-Wigner type Lie algebra contractions. These geometric contractions have important physical and geometric meanings, such as obtaining classical phenomena as limits of quantum phenomena as h → 0 and nonrelativistic phenomena from special relativistic as c → ∞, and the derivation of the Askey scheme for obtaining all hypergeometric orthogonal polynomials as limits of Racah/Wilson polynomials. In this paper we show how to simplify the structure relations for abstract nondegenerate and degenerate quadratic algebras and their contractions. In earlier papers we have classif ied contractions of 2nd order superintegrable systems on constant curvature spaces and have shown that all results are derivable from free quadratic algebras contained in the enveloping algebras of the Lie algebras e(2, C) in flat space and o(3, C) on nonzero constant curvature spaces. The quadratic algebra contractions are induced by generalizations of Inönü-Wigner contractions of these Lie algebras. As a special case we obtained the Askey scheme for hypergeometric orthogonal polynomials. After constant curvature spaces, the 4 Darboux spaces are the 2D manifolds admitting the most 2nd order Killing tensors. Here we complete this theoretical development for 2D superintegrable systems by showing that the Darboux superintegrable systems are also characterized by free quadratic algebras contained in the symmetry algebras of these spaces and that their contractions are also induced by Inönü-Wigner contractions. We present tables of the contraction results.

text: Structure relations and darboux contractions for 2D 2nd order superintegrable systems Two-dimensional quadratic algebras are generalizations of Lie algebras that include the symmetry algebras of 2nd order superintegrable systems in 2 dimensions as special cases. The superintegrable systems are exactly solvable physical systems in classical and quantum mechanics. Distinct superintegrable systems and their quadratic algebras can be related by geometric contractions, induced by Inönü-Wigner type Lie algebra contractions. These geometric contractions have important physical and geometric meanings, such as obtaining classical phenomena as limits of quantum phenomena as h → 0 and nonrelativistic phenomena from special relativistic as c → ∞, and the derivation of the Askey scheme for obtaining all hypergeometric orthogonal polynomials as limits of Racah/Wilson polynomials. In this paper we show how to simplify the structure relations for abstract nondegenerate and degenerate quadratic algebras and their contractions. In earlier papers we have classif ied contractions of 2nd order superintegrable systems on constant curvature spaces and have shown that all results are derivable from free quadratic algebras contained in the enveloping algebras of the Lie algebras e(2, C) in flat space and o(3, C) on nonzero constant curvature spaces. The quadratic algebra contractions are induced by generalizations of Inönü-Wigner contractions of these Lie algebras. As a special case we obtained the Askey scheme for hypergeometric orthogonal polynomials. After constant curvature spaces, the 4 Darboux spaces are the 2D manifolds admitting the most 2nd order Killing tensors. Here we complete this theoretical development for 2D superintegrable systems by showing that the Darboux superintegrable systems are also characterized by free quadratic algebras contained in the symmetry algebras of these spaces and that their contractions are also induced by Inönü-Wigner contractions. We present tables of the contraction results.

year: 2015

title: Adaptive models of English text

authors: Teahan, W.J.

subjects: statistical language modeling

abstract: High quality models of English text with performance approaching that of humans is important for many applications including spelling correction, speech recognition, OCR, and encryption. A number of different statistical models of English are compared with each other and with previous estimates from human subjects. It is concluded that the best current models are word based with part of speech tags. Given sufficient training text, they are able to attain performance comparable to humans.

text: Adaptive models of English text High quality models of English text with performance approaching that of humans is important for many applications including spelling correction, speech recognition, OCR, and encryption. A number of different statistical models of English are compared with each other and with previous estimates from human subjects. It is concluded that the best current models are word based with part of speech tags. Given sufficient training text, they are able to attain performance comparable to humans.

year: 1997-11

title: Key concepts in Māori and iwi histories: A critique of Te Takanga o te Wā (2015)

authors: Palmer, Georgia

subjects: History

abstract: In 2019, the New Zealand government announced a “reset” of the national history curriculum, with the intention of including more Māori content and addressing ongoing petitions to make the New Zealand Wars compulsory in all schools. Two curricula - one for mainstream schools and the other for Māori immersion Kura - were developed, with each drawing on a recent Ministry of Education publication Te Takanga o te Wā (2015) to provide key insights to Māori history concepts and themes. The Māori curricula has since taken on the name of that document, and uses those themes in its new framework, while the mainstream curriculum also adopts some of these same concepts. But how relevant are these proposed themes and concepts to Māori history, and what have Māori and iwi historians and experts actually written or said about the significance of these concepts to Māori historical practice, content, and theory?
This thesis critiques the five key themes and concepts presented in Te Takanga o te Wā and evaluates their relevance as leading Māori history concepts. These are Whakapapa, Tūrangawaewae; Mana-Motuhake; Kaitiakitanga, and Whānaungatanga. This study surveys the existing work produced by Māori historians and other experts over the past century, particularly the extent to which they have used these themes or discussed their significance to Māori and iwi historical practice and thinking. This dissertation argues that aside from whakapapa, the majority of the themes presented in Te Takanga o te Wā are not reflected in the historiography of the field, and that there are in fact a range of other crucial concepts missing. These include discussions about Wā, the importance of te reo Māori, tikanga, Mātauranga-a-iwi and other important approaches like “historical trauma”, “survivance”, Kaupapa Māori and decolonisation.

text: Key concepts in Māori and iwi histories: A critique of Te Takanga o te Wā (2015) In 2019, the New Zealand government announced a “reset” of the national history curriculum, with the intention of including more Māori content and addressing ongoing petitions to make the New Zealand Wars compulsory in all schools. Two curricula - one for mainstream schools and the other for Māori immersion Kura - were developed, with each drawing on a recent Ministry of Education publication Te Takanga o te Wā (2015) to provide key insights to Māori history concepts and themes. The Māori curricula has since taken on the name of that document, and uses those themes in its new framework, while the mainstream curriculum also adopts some of these same concepts. But how relevant are these proposed themes and concepts to Māori history, and what have Māori and iwi historians and experts actually written or said about the significance of these concepts to Māori historical practice, content, and theory?
This thesis critiques the five key themes and concepts presented in Te Takanga o te Wā and evaluates their relevance as leading Māori history concepts. These are Whakapapa, Tūrangawaewae; Mana-Motuhake; Kaitiakitanga, and Whānaungatanga. This study surveys the existing work produced by Māori historians and other experts over the past century, particularly the extent to which they have used these themes or discussed their significance to Māori and iwi historical practice and thinking. This dissertation argues that aside from whakapapa, the majority of the themes presented in Te Takanga o te Wā are not reflected in the historiography of the field, and that there are in fact a range of other crucial concepts missing. These include discussions about Wā, the importance of te reo Māori, tikanga, Mātauranga-a-iwi and other important approaches like “historical trauma”, “survivance”, Kaupapa Māori and decolonisation.

year: 2022