title: Second-order superintegrable systems in conformally flat spaces. I. Two-dimensional classical structure theory

authors: Kalnins, Ernie G.

subjects: Mathematics

abstract: This paper is the first in a series that lays the groundwork for a structure and classification theory of second-order superintegrable systems, both classical and quantum, in conformally flat spaces. Many examples of such systems are known, and lists of possible systems have been determined for constant curvature spaces in two and three dimensions, as well as few other spaces. Observed features of these systems are multiseparability, closure of the quadratic algebra of second-order symmetries at order 6, use of representation theory of the quadratic algebra to derive spectral properties of the quantum Schrödinger operator, and a close relationship with exactly solvable and quasi-exactly solvable systems. Our approach is, rather than focus on particular spaces and systems, to use a general theoretical method based on integrability conditions to derive structure common to all systems. In this first paper we consider classical superintegrable systems on a general two-dimensional Riemannian manifold and uncover their common structure. We show that for superintegrable systems with nondegenerate potentials there exists a standard structure based on the algebra of 2×2 symmetric matrices, that such systems are necessarily multiseparable and that the quadratic algebra closes at level 6. Superintegrable systems with degenerate potentials are also analyzed. This is all done without making use of lists of systems, so that generalization to higher dimensions, where relatively few examples are known, is much easier.

text: Second-order superintegrable systems in conformally flat spaces. I. Two-dimensional classical structure theory This paper is the first in a series that lays the groundwork for a structure and classification theory of second-order superintegrable systems, both classical and quantum, in conformally flat spaces. Many examples of such systems are known, and lists of possible systems have been determined for constant curvature spaces in two and three dimensions, as well as few other spaces. Observed features of these systems are multiseparability, closure of the quadratic algebra of second-order symmetries at order 6, use of representation theory of the quadratic algebra to derive spectral properties of the quantum Schrödinger operator, and a close relationship with exactly solvable and quasi-exactly solvable systems. Our approach is, rather than focus on particular spaces and systems, to use a general theoretical method based on integrability conditions to derive structure common to all systems. In this first paper we consider classical superintegrable systems on a general two-dimensional Riemannian manifold and uncover their common structure. We show that for superintegrable systems with nondegenerate potentials there exists a standard structure based on the algebra of 2×2 symmetric matrices, that such systems are necessarily multiseparable and that the quadratic algebra closes at level 6. Superintegrable systems with degenerate potentials are also analyzed. This is all done without making use of lists of systems, so that generalization to higher dimensions, where relatively few examples are known, is much easier.

year: 2005-04

title: Modern life at high temperatures

authors: Daniel, Roy M.

subjects: biology

abstract: A variety of micro-organisms are now known which grow optimally above 65°C, and are defined as extreme thermophiles. As might be expected they are found in both natural and artificial hot environments. Until comparatively recently the upper optimum temperature for the growth of any living organism was about 85°C. Then in 1982 Stetter described an organism, isolated from the hot sea floor of a submarine solfatara field, which grew optimally at 105°C. Since then several other organisms have been found with optimum growth temperatures at 100"C or above and a few are capable of growth at 110*C (e.g. Huber et al., 1987; Fiala and Stetter, 1986; Zillig et al., 1987; Stetter et al., 1990).

text: Modern life at high temperatures A variety of micro-organisms are now known which grow optimally above 65°C, and are defined as extreme thermophiles. As might be expected they are found in both natural and artificial hot environments. Until comparatively recently the upper optimum temperature for the growth of any living organism was about 85°C. Then in 1982 Stetter described an organism, isolated from the hot sea floor of a submarine solfatara field, which grew optimally at 105°C. Since then several other organisms have been found with optimum growth temperatures at 100"C or above and a few are capable of growth at 110*C (e.g. Huber et al., 1987; Fiala and Stetter, 1986; Zillig et al., 1987; Stetter et al., 1990).

year: 1992

title: Exploring te ao Māori: The role of museums

authors: Clarkin-Phillips, Jeanette

subjects: museum

abstract: Museums offer many opportunities for developing knowledge of the bicultural heritage of Aotearoa New Zealand. This article reports on research involving a kindergarten located in a national museum. It discusses children’s growing understanding of te ao Māori (the world of Māori) through their regular visits to the collections and exhibits in the museum and suggests teachers in early childhood centres may find connecting with their local museum a valuable resource for enhancing bicultural practice.

text: Exploring te ao Māori: The role of museums Museums offer many opportunities for developing knowledge of the bicultural heritage of Aotearoa New Zealand. This article reports on research involving a kindergarten located in a national museum. It discusses children’s growing understanding of te ao Māori (the world of Māori) through their regular visits to the collections and exhibits in the museum and suggests teachers in early childhood centres may find connecting with their local museum a valuable resource for enhancing bicultural practice.

year: 2012