title: Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras

authors: Escobar Ruiz, Mauricio A.

subjects: Science & Technology

abstract: Quadratic algebras are generalizations of Lie algebras which include the symmetry algebras of 2nd order superintegrable systems in 2 dimensions as special cases. The superintegrable systems are exactly solvable physical systems in classical and quantum mechanics. Distinct superintegrable systems and their quadratic algebras can be related by geometric contractions, induced by Bôcher contractions of the conformal Lie algebra ${\mathfrak{so}}(4,\mathbb {C})$ to itself. In this paper we give a precise definition of Bôcher contractions and show how they can be classified. They subsume well known contractions of ${\mathfrak{e}}(2,\mathbb {C})$ and ${\mathfrak{so}}(3,\mathbb {C})$ and have important physical and geometric meanings, such as the derivation of the Askey scheme for obtaining all hypergeometric orthogonal polynomials as limits of Racah/Wilson polynomials. We also classify abstract nondegenerate quadratic algebras in terms of an invariant that we call a canonical form. We describe an algorithm for finding the canonical form of such algebras. We calculate explicitly all canonical forms arising from quadratic algebras of 2D nondegenerate superintegrable systems on constant curvature spaces and Darboux spaces. We further discuss contraction of quadratic algebras, focusing on those coming from superintegrable systems.

text: Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras Quadratic algebras are generalizations of Lie algebras which include the symmetry algebras of 2nd order superintegrable systems in 2 dimensions as special cases. The superintegrable systems are exactly solvable physical systems in classical and quantum mechanics. Distinct superintegrable systems and their quadratic algebras can be related by geometric contractions, induced by Bôcher contractions of the conformal Lie algebra ${\mathfrak{so}}(4,\mathbb {C})$ to itself. In this paper we give a precise definition of Bôcher contractions and show how they can be classified. They subsume well known contractions of ${\mathfrak{e}}(2,\mathbb {C})$ and ${\mathfrak{so}}(3,\mathbb {C})$ and have important physical and geometric meanings, such as the derivation of the Askey scheme for obtaining all hypergeometric orthogonal polynomials as limits of Racah/Wilson polynomials. We also classify abstract nondegenerate quadratic algebras in terms of an invariant that we call a canonical form. We describe an algorithm for finding the canonical form of such algebras. We calculate explicitly all canonical forms arising from quadratic algebras of 2D nondegenerate superintegrable systems on constant curvature spaces and Darboux spaces. We further discuss contraction of quadratic algebras, focusing on those coming from superintegrable systems.

year: 2017

title: The potential for an independent source of Younger Dryas tree-ring radiocarbon data from the Lake Ontario region, North America, and its palaeoenvironmental context

authors: Griggs, Carol

abstract: Tree rings are the gold standard for the calibration of radiocarbon dates into calendar years with limited error due to their annual record of radiocarbon content in the atmosphere. Currently, the tree-ring 14C data used in the International Radiocarbon Calibration curve (IntCal13) are from multiple sources back to ~ 10 ka cal BP for the northern hemisphere. However, the source for tree-ring calibration data prior to ~10 ka cal BP, including the Younger Dryas (YD) and most of the Early Holocene (EH) chronozones, is limited to central Europe. Substantial quantities of logs found in the lowlands of Lake Ontario in North America, dating from 12.1 to 11.2 ka cal BP, have great potential for providing unique YD-EH tree-ring 14C data from a new, previously unrepresented, geographic location and for adding a new perspective to ongoing debates such as the timing of the YD/EH transition in continental North America. Preliminary fieldwork yielded over 43 logs, predominantly spruce (Picea spp.), found buried in alluvial strata over 0.2 hectares of the floodplain at Bell Creek near Fulton, NY. Dendrochronological analysis of 32 spruce logs from this collection has produced several chronologies, one 183 years in length and a second of 147 years with a possible crossdate yielding a chronology of 260 years. Ongoing survey of the site suggests that there is high preservation potential in the same stratigraphic units in an additional ~2 hectares of floodplain, and sample collection with subsequent analysis will increase the chronology’s sample depth, extend its length, and potentially provide a robust, independent Younger Dryas – Early Holocene 14C record. Initial assessments of climatic, hydrological, environmental, and isostatic variations and events over time from the tree rings and their stable isotopes, as well as pollen, macrofossil, and sediment analyses and isostatic modeling indicate substantial changes in the Bell Creek Valley and surrounding lowlands both during the YD and into the mid-Holocene. These changes include significant alterations in the rate, volume, and direction of stream flow, and transition from a boreal riparian environment on a low-order river to a wetlands, then back to a riparian environment with much less streamflow and occasional flooding, similar to the Bell Creek Valley today.

text: The potential for an independent source of Younger Dryas tree-ring radiocarbon data from the Lake Ontario region, North America, and its palaeoenvironmental context Tree rings are the gold standard for the calibration of radiocarbon dates into calendar years with limited error due to their annual record of radiocarbon content in the atmosphere. Currently, the tree-ring 14C data used in the International Radiocarbon Calibration curve (IntCal13) are from multiple sources back to ~ 10 ka cal BP for the northern hemisphere. However, the source for tree-ring calibration data prior to ~10 ka cal BP, including the Younger Dryas (YD) and most of the Early Holocene (EH) chronozones, is limited to central Europe. Substantial quantities of logs found in the lowlands of Lake Ontario in North America, dating from 12.1 to 11.2 ka cal BP, have great potential for providing unique YD-EH tree-ring 14C data from a new, previously unrepresented, geographic location and for adding a new perspective to ongoing debates such as the timing of the YD/EH transition in continental North America. Preliminary fieldwork yielded over 43 logs, predominantly spruce (Picea spp.), found buried in alluvial strata over 0.2 hectares of the floodplain at Bell Creek near Fulton, NY. Dendrochronological analysis of 32 spruce logs from this collection has produced several chronologies, one 183 years in length and a second of 147 years with a possible crossdate yielding a chronology of 260 years. Ongoing survey of the site suggests that there is high preservation potential in the same stratigraphic units in an additional ~2 hectares of floodplain, and sample collection with subsequent analysis will increase the chronology’s sample depth, extend its length, and potentially provide a robust, independent Younger Dryas – Early Holocene 14C record. Initial assessments of climatic, hydrological, environmental, and isostatic variations and events over time from the tree rings and their stable isotopes, as well as pollen, macrofossil, and sediment analyses and isostatic modeling indicate substantial changes in the Bell Creek Valley and surrounding lowlands both during the YD and into the mid-Holocene. These changes include significant alterations in the rate, volume, and direction of stream flow, and transition from a boreal riparian environment on a low-order river to a wetlands, then back to a riparian environment with much less streamflow and occasional flooding, similar to the Bell Creek Valley today.

year: 2015

title: Caregiver Health Related Quality of Life and Burden When Caring for Children with Traumatic Brain Injury

authors: Rose, Wallis

subjects: Caregiver Burden

abstract: Often family members take on the responsibility of caregiver when another family member sustains a traumatic brain injury (TBI). The caregiving role is a stressful task which may impact negatively upon caregivers’ psychological and physical health. Variables which may contribute to caregiver burden include: caregiver age, educational attainment, income, gender, ethnicity, marital status, supports, time post-injury, and child clinical variables: TBI severity, behavioural functioning and adaptive functioning. The literature varies in its reporting of which areas of health are most affected, and the degree to which sociodemographic and child variables impact upon negative health related quality of life (HRQoL) and burden. To address this the current study explored health outcomes for TBI caregivers using a battery of measures (Medical Study Short Form-36, Hospital Anxiety and Depression Scale, DSM-IV Depression Diagnostic Scale, BAKAS Caregiving Outcomes Scale); questionnaires were also used to collect sociodemographic information and information relating to supports and services used in the rehabilitation of the TBI child. The Behavioural Assessment System for Children was used to collect behavioural and adaptability information from the caregivers about the child. Assessments were carried out at baseline, 1-month, 6-months and 12-months post TBI from a sample of 94 TBI caregivers and at baseline from a sample of 43 Control caregivers.

TBI caregivers experienced poorer overall health and higher levels of depression and were less likely to report positive life changes compared to Control caregivers. Older age and higher income predicted positive life changes. In addition to this higher income predicted better physical and overall health. Ethnicity was found to predict depression with the Māori/‘other ethnicity’ group suffering higher levels of depression. Children’s dysfunctional behaviour was found to predict poorer caregiver health outcomes across several domains; these include physical health, overall health, and depression; while the child’s adaptive functioning was not found to be a predictive factor in any caregiver health domains. New Zealand European and Māori TBI caregivers experienced similar health outcomes and received similar levels of support. However, Māori experienced more positive life changes than New Zealand Europeans. Caregivers’ physical and psychological well-being was found to improve over time, these changes were the greatest between the 1-month and 12-month period.
As a high percentage of children’s TBI’s occur in the home and at school, education aimed at schools and parents to inform of the effects of TBI upon children and families will bring an awareness which may encourage families to seek medical help. In doing so will offer the opportunity to receive or seek support in the initial period post TBI in the hope of reducing the burden for caregivers and producing better health outcomes. Bringing these families into contact with medical services may also help in identifying caregivers at higher risk of poorer health outcomes. Findings of more positive life changes for Māori may suggest cultural ideology is a protective factor in caregiver burden, further investigation may be necessary to understand these cultural differences and how they impact upon the caregiving role.

text: Caregiver Health Related Quality of Life and Burden When Caring for Children with Traumatic Brain Injury Often family members take on the responsibility of caregiver when another family member sustains a traumatic brain injury (TBI). The caregiving role is a stressful task which may impact negatively upon caregivers’ psychological and physical health. Variables which may contribute to caregiver burden include: caregiver age, educational attainment, income, gender, ethnicity, marital status, supports, time post-injury, and child clinical variables: TBI severity, behavioural functioning and adaptive functioning. The literature varies in its reporting of which areas of health are most affected, and the degree to which sociodemographic and child variables impact upon negative health related quality of life (HRQoL) and burden. To address this the current study explored health outcomes for TBI caregivers using a battery of measures (Medical Study Short Form-36, Hospital Anxiety and Depression Scale, DSM-IV Depression Diagnostic Scale, BAKAS Caregiving Outcomes Scale); questionnaires were also used to collect sociodemographic information and information relating to supports and services used in the rehabilitation of the TBI child. The Behavioural Assessment System for Children was used to collect behavioural and adaptability information from the caregivers about the child. Assessments were carried out at baseline, 1-month, 6-months and 12-months post TBI from a sample of 94 TBI caregivers and at baseline from a sample of 43 Control caregivers.

TBI caregivers experienced poorer overall health and higher levels of depression and were less likely to report positive life changes compared to Control caregivers. Older age and higher income predicted positive life changes. In addition to this higher income predicted better physical and overall health. Ethnicity was found to predict depression with the Māori/‘other ethnicity’ group suffering higher levels of depression. Children’s dysfunctional behaviour was found to predict poorer caregiver health outcomes across several domains; these include physical health, overall health, and depression; while the child’s adaptive functioning was not found to be a predictive factor in any caregiver health domains. New Zealand European and Māori TBI caregivers experienced similar health outcomes and received similar levels of support. However, Māori experienced more positive life changes than New Zealand Europeans. Caregivers’ physical and psychological well-being was found to improve over time, these changes were the greatest between the 1-month and 12-month period.
As a high percentage of children’s TBI’s occur in the home and at school, education aimed at schools and parents to inform of the effects of TBI upon children and families will bring an awareness which may encourage families to seek medical help. In doing so will offer the opportunity to receive or seek support in the initial period post TBI in the hope of reducing the burden for caregivers and producing better health outcomes. Bringing these families into contact with medical services may also help in identifying caregivers at higher risk of poorer health outcomes. Findings of more positive life changes for Māori may suggest cultural ideology is a protective factor in caregiver burden, further investigation may be necessary to understand these cultural differences and how they impact upon the caregiving role.

year: 2012