title: Superintegrability and higher order integrals for quantum systems

authors: Kalnins, Ernie G.

subjects: mathematics

abstract: We refine a method for finding a canonical form of symmetry operators of arbitrary order for the Schrödinger eigenvalue equation HΨ ≡ (Δ2 + V)Ψ = EΨ on any 2D Riemannian manifold, real or complex, that admits a separation of variables in some orthogonal coordinate system. The flat space equations with potentials V = α(x + iy)k − 1/(x − iy)k + 1 in Cartesian coordinates, and V = αr² + β/r²cos ²kθ + γ/r²sin ²kθ (the Tremblay, Turbiner and Winternitz system) in polar coordinates, have each been shown to be classically superintegrable for all rational numbers k. We apply the canonical operator method to give a constructive proof that each of these systems is also quantum superintegrable for all rational k. We develop the classical analog of the quantum canonical form for a symmetry. It is clear that our methods will generalize to other Hamiltonian systems.

text: Superintegrability and higher order integrals for quantum systems We refine a method for finding a canonical form of symmetry operators of arbitrary order for the Schrödinger eigenvalue equation HΨ ≡ (Δ2 + V)Ψ = EΨ on any 2D Riemannian manifold, real or complex, that admits a separation of variables in some orthogonal coordinate system. The flat space equations with potentials V = α(x + iy)k − 1/(x − iy)k + 1 in Cartesian coordinates, and V = αr² + β/r²cos ²kθ + γ/r²sin ²kθ (the Tremblay, Turbiner and Winternitz system) in polar coordinates, have each been shown to be classically superintegrable for all rational numbers k. We apply the canonical operator method to give a constructive proof that each of these systems is also quantum superintegrable for all rational k. We develop the classical analog of the quantum canonical form for a symmetry. It is clear that our methods will generalize to other Hamiltonian systems.

year: 2010

title: An ecological study of Paranephrops planifrons white (Decapoda: Parastacidae) in Lake Rotoiti, North Island

authors: Devcich, Alan A.

subjects: Crayfish

abstract: The Paranephrops planifrons population in mesotrophic L.  Rotoiti consists of 2 bathymetrically and temporally separate breeding groups. Late autumn breeders comprise about 80% of the population, occupy depths mainly above 30 m and at night feed in the littoral zone, where food is 80% more concentrated than elsewhere. Early summer breeders exist mainly below 30 m depth. Utilisation of the whole lake bottom and seasonal changes in food available to the early summer breeders are tentatively given as explanations of this pattern.
Crayfish show increased sensitivity to light intensities greater than 150-205 lux and consequently occupy shelters above around 12 m depth. Shelters include any recesses subjected to intensities below this range. This zone accomodates almost the entire juvenile population and ca 10-20% of the adult population. This response to light intensity is believed to be an adaptation promoting avoidance of their main predator, the shag. Nocturnalism is another adaptation reducing losses to predators, including trout. Crayfish at greater depths lie unprotected and inactive by day and ca 70-80% of these adults form a high density band around the lake at a mean annual depth of 19.2±2.8 m. The band has a mean annual vertical depth range of 11.4±2.1 m and densities up to 50 crayfish m⁻².
At dusk activity begins in the deepest waters first and crayfish forming the band migrate shorewards, while those above emerge from shelters. Within the hour feeding on detritus mainly, commences amongst the weed beds which extend to 6 m depth. At dawn a downward migration preceeds reformation of the high density band. These dual diel migrations apply almost solely to late autumn breeders and are part of a circadian rhythm, the timing of which is modified by light. Locomotion associated with migratory activity appears to be effected mainly through leg proprioceptors responsive to gravity. Directional orientation at this time and also generally, is inversely related to bottom slope angle.
The diel distribution pattern varies seasonally. In summer and autumn the high density band is displaced shorewards by a vertical distance of 3.3 m, to a mean depth of 18.3 m and there is a 4 fold increase in crayfish inhabiting shallow water shelters. This closer association with the main feeding ground is probably an adaptation to meet coincident increases in energy needs. It applies especially to late autumn breeding females, whose period of ovarian development occurs between December and April. Energy requirements appear to be met through food consumed rather than from food stored. Female feeding activity is directly related to temperature but not in males, and prewinter storage is apparently of little importance in both sexes.
Hypolimnetic deoxygenation affects early summer breeders mainly and induces a mass migration from the deeper regions to above the 30 m contour by February. Complete recolonisation of these parts follows lake turnover in May. Crayfish occur on all substrate types in approximately equal numbers except very soft muds, which support densities of <0.001 crayfish m⁻².

text: An ecological study of Paranephrops planifrons white (Decapoda: Parastacidae) in Lake Rotoiti, North Island The Paranephrops planifrons population in mesotrophic L.  Rotoiti consists of 2 bathymetrically and temporally separate breeding groups. Late autumn breeders comprise about 80% of the population, occupy depths mainly above 30 m and at night feed in the littoral zone, where food is 80% more concentrated than elsewhere. Early summer breeders exist mainly below 30 m depth. Utilisation of the whole lake bottom and seasonal changes in food available to the early summer breeders are tentatively given as explanations of this pattern.
Crayfish show increased sensitivity to light intensities greater than 150-205 lux and consequently occupy shelters above around 12 m depth. Shelters include any recesses subjected to intensities below this range. This zone accomodates almost the entire juvenile population and ca 10-20% of the adult population. This response to light intensity is believed to be an adaptation promoting avoidance of their main predator, the shag. Nocturnalism is another adaptation reducing losses to predators, including trout. Crayfish at greater depths lie unprotected and inactive by day and ca 70-80% of these adults form a high density band around the lake at a mean annual depth of 19.2±2.8 m. The band has a mean annual vertical depth range of 11.4±2.1 m and densities up to 50 crayfish m⁻².
At dusk activity begins in the deepest waters first and crayfish forming the band migrate shorewards, while those above emerge from shelters. Within the hour feeding on detritus mainly, commences amongst the weed beds which extend to 6 m depth. At dawn a downward migration preceeds reformation of the high density band. These dual diel migrations apply almost solely to late autumn breeders and are part of a circadian rhythm, the timing of which is modified by light. Locomotion associated with migratory activity appears to be effected mainly through leg proprioceptors responsive to gravity. Directional orientation at this time and also generally, is inversely related to bottom slope angle.
The diel distribution pattern varies seasonally. In summer and autumn the high density band is displaced shorewards by a vertical distance of 3.3 m, to a mean depth of 18.3 m and there is a 4 fold increase in crayfish inhabiting shallow water shelters. This closer association with the main feeding ground is probably an adaptation to meet coincident increases in energy needs. It applies especially to late autumn breeding females, whose period of ovarian development occurs between December and April. Energy requirements appear to be met through food consumed rather than from food stored. Female feeding activity is directly related to temperature but not in males, and prewinter storage is apparently of little importance in both sexes.
Hypolimnetic deoxygenation affects early summer breeders mainly and induces a mass migration from the deeper regions to above the 30 m contour by February. Complete recolonisation of these parts follows lake turnover in May. Crayfish occur on all substrate types in approximately equal numbers except very soft muds, which support densities of <0.001 crayfish m⁻².

year: 1979

title: Māori and Pacific traditional caregiving practices: Voices from the community

authors: Glasgow, Ali

abstract: Increasing numbers of Māori and Pacific infants are enrolled in early childhood services in Aotearoa/New Zealand. This phenomenon is part of a larger societal trend in which mother return to the workforce within months of their infant’s birth. This development has significant impacts on early childhood service provision, particularly in relation to addressing the cultural needs of infants. A major concern is that Māori and Pacific infants are at risk of cultural deprivation within these services. According to the research reported in this chapter, there is a call within Māori and Pacific communities for Māori and Pacific infants to be raised in culturally responsive ways in early childhood services children’s home language, culture and identity are promoted and maintained. In Aotearoa/ New Zelanad, infant caregiving theory is largely based on Western European theory and principles that promote individualism and independence, and this will be elaborated upon in the chapter. This chapter outlines findings from a nationwide Aotearoa/ New Zealand online survey conducted with Māori and Polynesian Pacific “language nest” teachers on traditional cultural practices for Māori and Polynesian infants. The language nest philosophies and programmes promote culture, language and traditional practices. Respondents expressed the desire for autonomy in providing culturally authentic caregiving practices, and frustration with educational theory and government policies that did not align, or ran contrary to, modern cultural practice. Research findings revealed Māori and Polynesian Pacific traditional cultural values, beliefs and practices, and from this common base is an emerging Polynesian theoretical model of infant caregiving practices. It is anticipated that a Polynesian theory of infant caregiving will guide Aotearoa/ New Zealand’s early childhood education services and government policy, ensuring culturally responsive practices for Māori and Polynesian Pacific infants and their families.

text: Māori and Pacific traditional caregiving practices: Voices from the community Increasing numbers of Māori and Pacific infants are enrolled in early childhood services in Aotearoa/New Zealand. This phenomenon is part of a larger societal trend in which mother return to the workforce within months of their infant’s birth. This development has significant impacts on early childhood service provision, particularly in relation to addressing the cultural needs of infants. A major concern is that Māori and Pacific infants are at risk of cultural deprivation within these services. According to the research reported in this chapter, there is a call within Māori and Pacific communities for Māori and Pacific infants to be raised in culturally responsive ways in early childhood services children’s home language, culture and identity are promoted and maintained. In Aotearoa/ New Zelanad, infant caregiving theory is largely based on Western European theory and principles that promote individualism and independence, and this will be elaborated upon in the chapter. This chapter outlines findings from a nationwide Aotearoa/ New Zealand online survey conducted with Māori and Polynesian Pacific “language nest” teachers on traditional cultural practices for Māori and Polynesian infants. The language nest philosophies and programmes promote culture, language and traditional practices. Respondents expressed the desire for autonomy in providing culturally authentic caregiving practices, and frustration with educational theory and government policies that did not align, or ran contrary to, modern cultural practice. Research findings revealed Māori and Polynesian Pacific traditional cultural values, beliefs and practices, and from this common base is an emerging Polynesian theoretical model of infant caregiving practices. It is anticipated that a Polynesian theory of infant caregiving will guide Aotearoa/ New Zealand’s early childhood education services and government policy, ensuring culturally responsive practices for Māori and Polynesian Pacific infants and their families.

year: 2016