title: Examples of complete solvability of 2D classical superintegrable systems

authors: Chen, Yuxan

abstract: Classical (maximal) superintegrable systems in n dimensions are Hamiltonian systems with 2n - 1 independent constants of the motion, globally defined, the maximum number possible. They are very special because they can be solved algebraically. In this paper we show explicitly, mostly through examples of 2nd order superintegrable systems in 2 dimensions, how the trajectories can be determined in detail using rather elementary algebraic, geometric and analytic methods applied to the closed quadratic algebra of symmetries of the system, without resorting to separation of variables techniques or trying to integrate Hamilton’s equations. We treat a family of 2nd order degenerate systems: oscillator analogies on Darboux, nonzero constant curvature, and flat spaces, related to one another via contractions, and obeying Kepler’s laws. Then we treat two 2nd order nondegenerate systems, an analogy of a caged Coulomb problem on the 2-sphere and its contraction to a Euclidean space caged Coulomb problem. In all cases the symmetry algebra structure provides detailed information about the trajectories, some of which are rather complicated. An interesting example is the occurrence of “metronome orbits”, trajectories confined to an arc rather than a loop, which are indicated clearly from the structure equations but might be overlooked using more traditional methods. We also treat the Post-Winternitz system, an example of a classical 4th order superintegrable system that cannot be solved using separation of variables. Finally we treat a superintegrable system, related to the addition theorem for elliptic functions, whose constants of the motion are only rational in the momenta. It is a system of special interest because its constants of the motion generate a closed polynomial algebra. This paper contains many new results but we have tried to present most of the materials in a fashion that is easily accessible to nonexperts, in order to provide entrée to superintegrablity theory.

text: Examples of complete solvability of 2D classical superintegrable systems Classical (maximal) superintegrable systems in n dimensions are Hamiltonian systems with 2n - 1 independent constants of the motion, globally defined, the maximum number possible. They are very special because they can be solved algebraically. In this paper we show explicitly, mostly through examples of 2nd order superintegrable systems in 2 dimensions, how the trajectories can be determined in detail using rather elementary algebraic, geometric and analytic methods applied to the closed quadratic algebra of symmetries of the system, without resorting to separation of variables techniques or trying to integrate Hamilton’s equations. We treat a family of 2nd order degenerate systems: oscillator analogies on Darboux, nonzero constant curvature, and flat spaces, related to one another via contractions, and obeying Kepler’s laws. Then we treat two 2nd order nondegenerate systems, an analogy of a caged Coulomb problem on the 2-sphere and its contraction to a Euclidean space caged Coulomb problem. In all cases the symmetry algebra structure provides detailed information about the trajectories, some of which are rather complicated. An interesting example is the occurrence of “metronome orbits”, trajectories confined to an arc rather than a loop, which are indicated clearly from the structure equations but might be overlooked using more traditional methods. We also treat the Post-Winternitz system, an example of a classical 4th order superintegrable system that cannot be solved using separation of variables. Finally we treat a superintegrable system, related to the addition theorem for elliptic functions, whose constants of the motion are only rational in the momenta. It is a system of special interest because its constants of the motion generate a closed polynomial algebra. This paper contains many new results but we have tried to present most of the materials in a fashion that is easily accessible to nonexperts, in order to provide entrée to superintegrablity theory.

year: 2015

title: Genomic data is missing for many highly invasive species, restricting our preparedness for escalating incursion rates.

authors: Matheson, Paige

subjects: Biodiversity

abstract: Biological invasions drive environmental change, potentially threatening native biodiversity, human health, and global economies. Population genomics is an increasingly popular tool in invasion biology, improving accuracy and providing new insights into the genetic factors that underpin invasion success compared to research based on a small number of genetic loci. We examine the extent to which population genomic resources, including reference genomes, have been used or are available for invasive species research. We find that 82% of species on the International Union for Conservation of Nature "100 Worst Invasive Alien Species" list have been studied using some form of population genetic data, but just 32% of these species have been studied using population genomic data. Further, 55% of the list's species lack a reference genome. With incursion rates escalating globally, understanding how genome-driven processes facilitate invasion is critical, but despite a promising trend of increasing uptake, "invasion genomics" is still in its infancy. We discuss how population genomic data can enhance our understanding of biological invasion and inform proactive detection and management of invasive species, and we call for more research that specifically targets this area.

text: Genomic data is missing for many highly invasive species, restricting our preparedness for escalating incursion rates. Biological invasions drive environmental change, potentially threatening native biodiversity, human health, and global economies. Population genomics is an increasingly popular tool in invasion biology, improving accuracy and providing new insights into the genetic factors that underpin invasion success compared to research based on a small number of genetic loci. We examine the extent to which population genomic resources, including reference genomes, have been used or are available for invasive species research. We find that 82% of species on the International Union for Conservation of Nature "100 Worst Invasive Alien Species" list have been studied using some form of population genetic data, but just 32% of these species have been studied using population genomic data. Further, 55% of the list's species lack a reference genome. With incursion rates escalating globally, understanding how genome-driven processes facilitate invasion is critical, but despite a promising trend of increasing uptake, "invasion genomics" is still in its infancy. We discuss how population genomic data can enhance our understanding of biological invasion and inform proactive detection and management of invasive species, and we call for more research that specifically targets this area.

year: 2022-08-17

title: Listening to the Winds of Change: Assessing culturally responsive and relational pedagogy

authors: Thin-Rabb, Polly

subjects: Culturally responsive and relational pedagogy

abstract: This thesis examines how the Rongohia te Hau tools were used to capture a snapshot of the extent that culturally responsive and relational pedagogies were occurring within classrooms in two English-medium secondary schools in Aotearoa, New Zealand. In this study, school leaders used Rongohia te Hau to measure changes in the educational experiences of Māori learners over twelve to eighteen-months, to prioritise next steps in addressing the educational inequities that were evident. 

With the contribution of teachers, leaders and kaitoro (Kia Eke Panuku facilitators), this research investigates the purpose and place of Rongohia te Hau within the Kia Eke Panuku professional learning and development context. It also presents and discusses the pedagogical shifts that occurred in these two schools over the period of this research, as evidenced by quantitative and qualitative Rongohia te Hau data. Furthermore, the learning implications that arose out of this research are analysed to help generate new questions and considerations, to enable continuing conversations for these schools, other schools, professional learning and development providers and researchers.

text: Listening to the Winds of Change: Assessing culturally responsive and relational pedagogy This thesis examines how the Rongohia te Hau tools were used to capture a snapshot of the extent that culturally responsive and relational pedagogies were occurring within classrooms in two English-medium secondary schools in Aotearoa, New Zealand. In this study, school leaders used Rongohia te Hau to measure changes in the educational experiences of Māori learners over twelve to eighteen-months, to prioritise next steps in addressing the educational inequities that were evident. 

With the contribution of teachers, leaders and kaitoro (Kia Eke Panuku facilitators), this research investigates the purpose and place of Rongohia te Hau within the Kia Eke Panuku professional learning and development context. It also presents and discusses the pedagogical shifts that occurred in these two schools over the period of this research, as evidenced by quantitative and qualitative Rongohia te Hau data. Furthermore, the learning implications that arose out of this research are analysed to help generate new questions and considerations, to enable continuing conversations for these schools, other schools, professional learning and development providers and researchers.

year: 2017