title: Infinite order symmetries for two-dimensional separable Schrödinger equations

authors: Kalnins, Ernie G.

subjects: mathematics

abstract: Consider a non-relativistic Hamiltonian operator H in 2 dimensions consisting of a kinetic energy term plus a potential. We show that if the associated Schrödinger eigenvalue equation admits an orthogonal separation of variables, there is a calculus to describe the (in general) infinite-order differential operator symmetries of the Schrödinger equation. The calculus is formal but can be made rigorous when all functions in the eigenvaue equation are analytic. The infinite-order calculus exhibits structure that is not apparent when one studies only finite-order symmetries. The search for finite-order symmetries can then be reposed as one of looking for solutions of a coupled system of PDEs that are polynomial in certain parameters. We go further and extend the calculus to the situation where the Schrödinger equation admits a second-order symmetry operator, not necessarily associated with orthogonal separable coordinates.

text: Infinite order symmetries for two-dimensional separable Schrödinger equations Consider a non-relativistic Hamiltonian operator H in 2 dimensions consisting of a kinetic energy term plus a potential. We show that if the associated Schrödinger eigenvalue equation admits an orthogonal separation of variables, there is a calculus to describe the (in general) infinite-order differential operator symmetries of the Schrödinger equation. The calculus is formal but can be made rigorous when all functions in the eigenvaue equation are analytic. The infinite-order calculus exhibits structure that is not apparent when one studies only finite-order symmetries. The search for finite-order symmetries can then be reposed as one of looking for solutions of a coupled system of PDEs that are polynomial in certain parameters. We go further and extend the calculus to the situation where the Schrödinger equation admits a second-order symmetry operator, not necessarily associated with orthogonal separable coordinates.

year: 2004

title: The potential for Charophyte re-establishment in large, shallow, eutrophic lakes with special reference to Lake Waikare, New Zealand.

authors: Hopkins, Aareka

subjects: lake restoration

abstract: Lake Waikare is a large, shallow eutrophic lake devoid of submerged macrophytes. I investigated potential methods for re-establishing submerged macrophytes in the lake. Specifically, I subjected charophyte (Chara corallina) plantlets to two treatments of exposure in the lake (in areas exposed and sheltered from wind) to test for survival and growth under these conditions, and inside and outside fish exclosures to test for growth and survival in the presence of fish. 

While plantlets grew outside the exclosures in winter, their accumulated biomass over 21 days was less than protected plantlets. In winter, the accumulated biomass was lower outside than inside exclosures (by ~40%) at the sheltered site and was lower outside than inside exclosures (by 43%) at the exposed site. Overall, growth rates in winter were higher at the sheltered site (compared to the exposed site) by ~7%. In summer, charophyte accumulated biomass inside the exclosures increased by 85%, while at the sheltered site accumulated biomass increased by 58%. Outside the exclosures in summer the plantlets were completely removed at both sites. Overall, growth rates where higher at the exposed site than the sheltered site by 31%. Fish were responsible for the partial removal of plantlets in winter and total removal of plantlets in summer, and therefore affect the survival and growth of charophytes in Lake Waikare. The embayment at the sheltered site provides the best location in winter for re-establishment of charophytes from oospores because better growth rates were obtained there, and its sheltered location provides protection from severe wave action found at the exposed site. Oospores did not germinate after being submersed in the lake for 90 days due to heavy sedimentation.

To induce an improvement in the present light climate, Alum was tested to determine its effectiveness and longevity for settling lake sediments to allow charophytes to establish and grow. Examining the settling rates of Lake Waikare sediments and water treated with Alum over a range of suspended sediment concentrations and time intervals, sediments settled faster with Alum than without for at least 15 days (at 200 g l^1 suspended sediment concentration) and it remained active to 60 days but at reduced effectiveness. At the other concentrations tested (100 g l^1 and 300 g l^1 suspended sediment concentration), Alum responses were insignificant.

An improved light climate achieved by fish removal or Alum treatment will likely not be sufficient to permit the re-establishment of submerged macrophytes due to the turbid, algal-dominated state of the lake. The present nutrient and sediment levels, wave climate and fish influence must be mitigated so charophyte plantlets can be established.

text: The potential for Charophyte re-establishment in large, shallow, eutrophic lakes with special reference to Lake Waikare, New Zealand. Lake Waikare is a large, shallow eutrophic lake devoid of submerged macrophytes. I investigated potential methods for re-establishing submerged macrophytes in the lake. Specifically, I subjected charophyte (Chara corallina) plantlets to two treatments of exposure in the lake (in areas exposed and sheltered from wind) to test for survival and growth under these conditions, and inside and outside fish exclosures to test for growth and survival in the presence of fish. 

While plantlets grew outside the exclosures in winter, their accumulated biomass over 21 days was less than protected plantlets. In winter, the accumulated biomass was lower outside than inside exclosures (by ~40%) at the sheltered site and was lower outside than inside exclosures (by 43%) at the exposed site. Overall, growth rates in winter were higher at the sheltered site (compared to the exposed site) by ~7%. In summer, charophyte accumulated biomass inside the exclosures increased by 85%, while at the sheltered site accumulated biomass increased by 58%. Outside the exclosures in summer the plantlets were completely removed at both sites. Overall, growth rates where higher at the exposed site than the sheltered site by 31%. Fish were responsible for the partial removal of plantlets in winter and total removal of plantlets in summer, and therefore affect the survival and growth of charophytes in Lake Waikare. The embayment at the sheltered site provides the best location in winter for re-establishment of charophytes from oospores because better growth rates were obtained there, and its sheltered location provides protection from severe wave action found at the exposed site. Oospores did not germinate after being submersed in the lake for 90 days due to heavy sedimentation.

To induce an improvement in the present light climate, Alum was tested to determine its effectiveness and longevity for settling lake sediments to allow charophytes to establish and grow. Examining the settling rates of Lake Waikare sediments and water treated with Alum over a range of suspended sediment concentrations and time intervals, sediments settled faster with Alum than without for at least 15 days (at 200 g l^1 suspended sediment concentration) and it remained active to 60 days but at reduced effectiveness. At the other concentrations tested (100 g l^1 and 300 g l^1 suspended sediment concentration), Alum responses were insignificant.

An improved light climate achieved by fish removal or Alum treatment will likely not be sufficient to permit the re-establishment of submerged macrophytes due to the turbid, algal-dominated state of the lake. The present nutrient and sediment levels, wave climate and fish influence must be mitigated so charophyte plantlets can be established.

year: 2006

title: Mormonism and the white possessive: Moving critical indigenous studies theory into the religious realm

authors: Simon, Hemopereki

abstract: This essay responds to the pressing need to pursue topics related to Mormonism from a research perspective that is at once critical, culturally affirming, and grounded in Indigenous views. The goal here is to provide a theoretical framework for Critical Indigenous Studies perspectives to emerge in Mormon and Lamanite Studies. I therefore submit the following questions as a contribution to and critique of Lamanite Studies within the broader fields of Mormon, Cultural, and Religious Studies:
1. Which theoretical frame(s) from Critical Indigenous 
Studies is/are best suited to address Lamanite and Mormon issues? 
2. How might we reframe current Critical Indigenous 
Studies Theory to undertake research on Mormon and 
Western religious topics? 

To address these questions, the article first outlines its 
positionality as a basis for understanding Simon’s approach to the research. The bulk of the research here focuses on describing Lamanitism in terms of its relationship to racism and whiteness. The article therefore continues with a description of the figure of 
Hagoth in The Book of Mormon, with a particular emphasis on their relationship to Te Moana-Nui-ā-Kiwa. From there the article outlines Unger’s theorem of contextuality and considers how Lamanitism might be considered an “artificial context” from a Critical Indigenous Studies perspective. A discussion of the Doctrine of Discovery, Manifest Destiny, and Moana Jackson’s idea of the culture of colonization is followed by a description of the “Racial Contract” and its speculative extension to Indigenous adherents of the Mormon Church. The article then explicates important and interrelated tikanga values from Te Ao Māori such as tapu and mana. Lastly, the article identifies the “white possessive” as a key doctrine for Critical Indigenous Studies and undertakes a reworking of the doctrine for its application to the Mormon Church. More specifically, I transmute the notion “white patriarchal sovereignty” into “white patriarchal salvation” in order to tailor it to critical Indigenous discussions of the Church. 
In my conclusion I attempt to answer the question – What is to be done? I assert that the Church is in conflict with the cultural revival of Te Ao Māori and the idea of “original instructions.” The Church’s claim to salvific truth and thinking is contradicted by my very identity and culturally informed ontology. The ensuing discussion confirms that as a colonizing entity the Church operates parallel to the government as a co-invading force in Indigenous lands. The idea of a Lamanite people as Polynesians, which includes Māori is called into question. I provide a continuum model to explain Indigenous realities in relation to religiosity and decolonization and suggest some topics as case studies for further investigation. The true purpose of this article is to facilitate the integration of critical Indigenous with Mormon and/or Lamanite Studies to bring about much-needed dialogue, discussion, and debate on the place of indigeneity in The Church of Jesus Christ of Latter-Day Saints.

text: Mormonism and the white possessive: Moving critical indigenous studies theory into the religious realm This essay responds to the pressing need to pursue topics related to Mormonism from a research perspective that is at once critical, culturally affirming, and grounded in Indigenous views. The goal here is to provide a theoretical framework for Critical Indigenous Studies perspectives to emerge in Mormon and Lamanite Studies. I therefore submit the following questions as a contribution to and critique of Lamanite Studies within the broader fields of Mormon, Cultural, and Religious Studies:
1. Which theoretical frame(s) from Critical Indigenous 
Studies is/are best suited to address Lamanite and Mormon issues? 
2. How might we reframe current Critical Indigenous 
Studies Theory to undertake research on Mormon and 
Western religious topics? 

To address these questions, the article first outlines its 
positionality as a basis for understanding Simon’s approach to the research. The bulk of the research here focuses on describing Lamanitism in terms of its relationship to racism and whiteness. The article therefore continues with a description of the figure of 
Hagoth in The Book of Mormon, with a particular emphasis on their relationship to Te Moana-Nui-ā-Kiwa. From there the article outlines Unger’s theorem of contextuality and considers how Lamanitism might be considered an “artificial context” from a Critical Indigenous Studies perspective. A discussion of the Doctrine of Discovery, Manifest Destiny, and Moana Jackson’s idea of the culture of colonization is followed by a description of the “Racial Contract” and its speculative extension to Indigenous adherents of the Mormon Church. The article then explicates important and interrelated tikanga values from Te Ao Māori such as tapu and mana. Lastly, the article identifies the “white possessive” as a key doctrine for Critical Indigenous Studies and undertakes a reworking of the doctrine for its application to the Mormon Church. More specifically, I transmute the notion “white patriarchal sovereignty” into “white patriarchal salvation” in order to tailor it to critical Indigenous discussions of the Church. 
In my conclusion I attempt to answer the question – What is to be done? I assert that the Church is in conflict with the cultural revival of Te Ao Māori and the idea of “original instructions.” The Church’s claim to salvific truth and thinking is contradicted by my very identity and culturally informed ontology. The ensuing discussion confirms that as a colonizing entity the Church operates parallel to the government as a co-invading force in Indigenous lands. The idea of a Lamanite people as Polynesians, which includes Māori is called into question. I provide a continuum model to explain Indigenous realities in relation to religiosity and decolonization and suggest some topics as case studies for further investigation. The true purpose of this article is to facilitate the integration of critical Indigenous with Mormon and/or Lamanite Studies to bring about much-needed dialogue, discussion, and debate on the place of indigeneity in The Church of Jesus Christ of Latter-Day Saints.

year: 2022