title: Separation equations for 2D superintegrable systems on constant curvature spaces

authors: Escobar-Ruiz, M.A.

subjects: Science & Technology

abstract: Second-order conformal quantum superintegrable systems in two dimensions are Laplace equations on a manifold with an added scalar potential and three independent 2nd order conformal symmetry operators. They encode all the information about 2D Helmholtz or time-independent Schrödinger superintegrable systems in an efficient manner: each of these systems admits a quadratic symmetry algebra (not usually a Lie algebra) and is multiseparable. We study the separation equations for the systems as a family rather than separate cases. We show that the separation equations comprise all of the various types of hypergeometric and Heun equations in full generality. In particular, they yield all of the 1D Schrödinger exactly solvable (ES) and quasi-exactly solvable (QES) systems related to the Heun operator. We focus on complex constant curvature spaces and show explicitly that there are eight pairs of Laplace separation types and these types account for all separable coordinates on the 20 flat space and 9 2-sphere Helmholtz superintegrable systems, including those for the constant potential case. The different systems are related by Stäckel transforms, by the symmetry algebras and by Böcher contractions of the conformal algebra so(4, C) to itself, which enables all systems to be derived from a single one: the generic potential on the complex 2-sphere. This approach facilitates a unified view of special function theory, incorporating hypergeometric and Heun functions in full generality.

text: Separation equations for 2D superintegrable systems on constant curvature spaces Second-order conformal quantum superintegrable systems in two dimensions are Laplace equations on a manifold with an added scalar potential and three independent 2nd order conformal symmetry operators. They encode all the information about 2D Helmholtz or time-independent Schrödinger superintegrable systems in an efficient manner: each of these systems admits a quadratic symmetry algebra (not usually a Lie algebra) and is multiseparable. We study the separation equations for the systems as a family rather than separate cases. We show that the separation equations comprise all of the various types of hypergeometric and Heun equations in full generality. In particular, they yield all of the 1D Schrödinger exactly solvable (ES) and quasi-exactly solvable (QES) systems related to the Heun operator. We focus on complex constant curvature spaces and show explicitly that there are eight pairs of Laplace separation types and these types account for all separable coordinates on the 20 flat space and 9 2-sphere Helmholtz superintegrable systems, including those for the constant potential case. The different systems are related by Stäckel transforms, by the symmetry algebras and by Böcher contractions of the conformal algebra so(4, C) to itself, which enables all systems to be derived from a single one: the generic potential on the complex 2-sphere. This approach facilitates a unified view of special function theory, incorporating hypergeometric and Heun functions in full generality.

year: 2017

title: An investigation of colour discrimination with horses (Equus caballus)

authors: Blackmore, Tania Louise

subjects: chromatic discrimination

abstract: The ability of four horses (Equus caballus) to discriminate coloured (three shades of blue, green, red, and yellow) from grey (neutral density) stimuli, produced by back projected lighting filters, was investigated in a two response forced-choice procedure. Pushes of the lever in front of a coloured screen were occasionally reinforced, pushes of the lever in front of a grey screen were never reinforced. Each colour shade was randomly paired with a grey that was brighter, one that was dimmer, and one that approximately matched the colour in terms of brightness. Each horse experienced the colours in a different order, a new colour was started after 85% correct responses over five consecutive sessions or if accuracy showed no trend over sessions. All horses reached the 85% correct with blue versus grey, three horses did so with both yellow and green versus grey. All were above chance with red versus grey but none reached criterion. Further analysis showed the wavelengths of the green stimuli used overlapped with the yellow. The results are consistent with histological and behavioural studies that suggest that horses are dichromatic. They differ from some earlier data in that they indicate horses can discriminate yellow and blue, but that they may have deficiencies in discriminating red and green.

text: An investigation of colour discrimination with horses (Equus caballus) The ability of four horses (Equus caballus) to discriminate coloured (three shades of blue, green, red, and yellow) from grey (neutral density) stimuli, produced by back projected lighting filters, was investigated in a two response forced-choice procedure. Pushes of the lever in front of a coloured screen were occasionally reinforced, pushes of the lever in front of a grey screen were never reinforced. Each colour shade was randomly paired with a grey that was brighter, one that was dimmer, and one that approximately matched the colour in terms of brightness. Each horse experienced the colours in a different order, a new colour was started after 85% correct responses over five consecutive sessions or if accuracy showed no trend over sessions. All horses reached the 85% correct with blue versus grey, three horses did so with both yellow and green versus grey. All were above chance with red versus grey but none reached criterion. Further analysis showed the wavelengths of the green stimuli used overlapped with the yellow. The results are consistent with histological and behavioural studies that suggest that horses are dichromatic. They differ from some earlier data in that they indicate horses can discriminate yellow and blue, but that they may have deficiencies in discriminating red and green.

year: 2008

title: Maori children : Conceptions of Death and Tangihanga

authors: Jacob, Juanita Emily

subjects: Maori

abstract: Research pertaining to Maori children‟s experiences and perceptions of death and tangihanga is sparse. Much of what is available, relating to children and their experiences with death, particularly death of a loved one, is generalised and stems from Western paradigms of knowledge. In contrast, this study aimed to investigate Maori children‟s experience relating to death and tangi through the eyes of Maori parents. Five areas were explored with Maori parents: (1) childhood experiences of Maori parents relating to death and tangi, (2) parental conceptualisation pertaining to ideas of an afterlife (3) how and when Maori parents talk with children about topics relating to death, tangi and an afterlife (4) how Maori children understand and conceptualise these events, and (5) how these practices will continue on in the future. Findings of this study suggest that Maori children: (a) received abundant support from kinship networks to help them cope with their grief, (b) they were included and involved in all aspects relating to death and tangi, (c) they were encouraged to express emotions openly and without restraint, (d) they understood and comprehended death through personal experience and exposure to tangi, (e) the business of grieving came secondary to the fact that tangi was more like a holiday with family, (f) Maori parents informed and talked to their children about death and tangi, and (g) they often use both Christian ideologies and cultural beliefs to explained death and afterlife to their children. From this study we learn that death was not hidden from children, that parents talked with their children in very open and age relevant ways, and considered their children‟s participation in tangi as an important way to grieve and ensure continuity with kinship networks and support. This study suggests that the challenge now is to ensure that these practices continue to persist between parents and their children, and continue on through generations.

text: Maori children : Conceptions of Death and Tangihanga Research pertaining to Maori children‟s experiences and perceptions of death and tangihanga is sparse. Much of what is available, relating to children and their experiences with death, particularly death of a loved one, is generalised and stems from Western paradigms of knowledge. In contrast, this study aimed to investigate Maori children‟s experience relating to death and tangi through the eyes of Maori parents. Five areas were explored with Maori parents: (1) childhood experiences of Maori parents relating to death and tangi, (2) parental conceptualisation pertaining to ideas of an afterlife (3) how and when Maori parents talk with children about topics relating to death, tangi and an afterlife (4) how Maori children understand and conceptualise these events, and (5) how these practices will continue on in the future. Findings of this study suggest that Maori children: (a) received abundant support from kinship networks to help them cope with their grief, (b) they were included and involved in all aspects relating to death and tangi, (c) they were encouraged to express emotions openly and without restraint, (d) they understood and comprehended death through personal experience and exposure to tangi, (e) the business of grieving came secondary to the fact that tangi was more like a holiday with family, (f) Maori parents informed and talked to their children about death and tangi, and (g) they often use both Christian ideologies and cultural beliefs to explained death and afterlife to their children. From this study we learn that death was not hidden from children, that parents talked with their children in very open and age relevant ways, and considered their children‟s participation in tangi as an important way to grieve and ensure continuity with kinship networks and support. This study suggests that the challenge now is to ensure that these practices continue to persist between parents and their children, and continue on through generations.

year: 2011