title: Extended Kepler–Coulomb quantum superintegrable systems in three dimensions

authors: Kalnins, Ernie G.

subjects: quadratic algebras

abstract: The quantum Kepler-Coulomb system in three dimensions is well known to be second order superintegrable, with a symmetry algebra that closes polynomially under commutators. This polynomial closure is also typical for second order superintegrable systems in 2D and for second order systems in 3D with nondegenerate (four-parameter) potentials. However, the degenerate three-parameter potential for the 3D Kepler-Coulomb system (also second order superintegrable) is an exception, as its symmetry algebra does not close polynomially. The 3D four-parameter potential for the extended Kepler-Coulomb system is not even second order superintegrable, but Verrier and Evans (2008 J. Math. Phys. 49 022902) showed it was fourth order superintegrable, and Tanoudis and Daskaloyannis (2011 arXiv:11020397v1) showed that, if a second fourth order symmetry is added to the generators, the symmetry algebra closes polynomially. Here, based on the Tremblay, Turbiner and Winternitz construction, we consider an infinite class of quantum extended Kepler-Coulomb three-and four-parameter systems indexed by a pair of rational numbers (k(1), k(2)) and reducing to the usual systems when k(1) = k(2) = 1. We show these systems to be superintegrable of arbitrarily high order and determine the structure of their symmetry algebras. We demonstrate that the symmetry algebras close algebraically; only for systems admitting extra discrete symmetries is polynomial closure achieved. Underlying the structure theory is the existence of raising and lowering operators, not themselves symmetry operators or even defined independent of basis, that can be employed to construct the symmetry operators and their structure relations.

text: Extended Kepler–Coulomb quantum superintegrable systems in three dimensions The quantum Kepler-Coulomb system in three dimensions is well known to be second order superintegrable, with a symmetry algebra that closes polynomially under commutators. This polynomial closure is also typical for second order superintegrable systems in 2D and for second order systems in 3D with nondegenerate (four-parameter) potentials. However, the degenerate three-parameter potential for the 3D Kepler-Coulomb system (also second order superintegrable) is an exception, as its symmetry algebra does not close polynomially. The 3D four-parameter potential for the extended Kepler-Coulomb system is not even second order superintegrable, but Verrier and Evans (2008 J. Math. Phys. 49 022902) showed it was fourth order superintegrable, and Tanoudis and Daskaloyannis (2011 arXiv:11020397v1) showed that, if a second fourth order symmetry is added to the generators, the symmetry algebra closes polynomially. Here, based on the Tremblay, Turbiner and Winternitz construction, we consider an infinite class of quantum extended Kepler-Coulomb three-and four-parameter systems indexed by a pair of rational numbers (k(1), k(2)) and reducing to the usual systems when k(1) = k(2) = 1. We show these systems to be superintegrable of arbitrarily high order and determine the structure of their symmetry algebras. We demonstrate that the symmetry algebras close algebraically; only for systems admitting extra discrete symmetries is polynomial closure achieved. Underlying the structure theory is the existence of raising and lowering operators, not themselves symmetry operators or even defined independent of basis, that can be employed to construct the symmetry operators and their structure relations.

year: 2013

title: Questioning `Black Humour': Racial exploitation, media and health

authors: Hodgetts, Darrin

subjects: exploration

abstract: This commentary explores the relevance of media racism to health psychology. While supporting Dr Estacio's call for health psychologists to get involved in promoting social justice via the media, we propose that health psychologists should not overstate the negative influence of the media on racism in society. Media content is complex and contradictory. It contains both racist and anti-racist representations. Challenging racism requires a conceptualization of links between the representational spaces provided by the media and the everyday geographic places within which inter-personal interactions and exploitation occur.

text: Questioning `Black Humour': Racial exploitation, media and health This commentary explores the relevance of media racism to health psychology. While supporting Dr Estacio's call for health psychologists to get involved in promoting social justice via the media, we propose that health psychologists should not overstate the negative influence of the media on racism in society. Media content is complex and contradictory. It contains both racist and anti-racist representations. Challenging racism requires a conceptualization of links between the representational spaces provided by the media and the everyday geographic places within which inter-personal interactions and exploitation occur.

year: 2009

title: Positioning historical trauma theory within Aotearoa New Zealand

authors: Pihama, Leonie

abstract: This article explores the relevance of historical trauma theory for Mäori research. In exploring the impact of historical trauma upon Mäori it has become clear that the terminology associated with historical trauma theory is considered controversial in Aotearoa New Zealand. As such, this article provides an overview of key defi nitions relevant to historical trauma and explores these in relation to recent reporting related to the use of the terms “holocaust” and “genocide” in the context of colonization in Aotearoa New Zealand. It is argued that in order to engage fully with the impacts of colonization on Mäori wellbeing we must articulate fully the impact of historical trauma events and the contribution of those events to the negative health disparities experienced by many of our whänau (extended family), hapü (sub- tribes) and iwi (tribes).

text: Positioning historical trauma theory within Aotearoa New Zealand This article explores the relevance of historical trauma theory for Mäori research. In exploring the impact of historical trauma upon Mäori it has become clear that the terminology associated with historical trauma theory is considered controversial in Aotearoa New Zealand. As such, this article provides an overview of key defi nitions relevant to historical trauma and explores these in relation to recent reporting related to the use of the terms “holocaust” and “genocide” in the context of colonization in Aotearoa New Zealand. It is argued that in order to engage fully with the impacts of colonization on Mäori wellbeing we must articulate fully the impact of historical trauma events and the contribution of those events to the negative health disparities experienced by many of our whänau (extended family), hapü (sub- tribes) and iwi (tribes).

year: 2014