title: Lie theory and separation of variables. 11. The EPD equation

authors: Kalnins, Ernie G.

subjects: Mathematics

abstract: We show that the Euler–Poisson–Darboux equation {∂tt -∂rr – [(2m+1)/r]∂r}Ө=0
separates in exactly nine coordinate systems corresponding to nine orbits of symmetric second-order operators in the enveloping algebra of SL(2,R), the symmetry group of this equation. We employ techniques developed in earlier papers from this series and use the representation theory of SL(2,R) to derive special function identities relating the separated solutions. We also show that the complex EPD equation separates in exactly five coordinate systems corresponding to five orbits of symmetric second-order operators in the enveloping algebra of SL(2,C).

text: Lie theory and separation of variables. 11. The EPD equation We show that the Euler–Poisson–Darboux equation {∂tt -∂rr – [(2m+1)/r]∂r}Ө=0
separates in exactly nine coordinate systems corresponding to nine orbits of symmetric second-order operators in the enveloping algebra of SL(2,R), the symmetry group of this equation. We employ techniques developed in earlier papers from this series and use the representation theory of SL(2,R) to derive special function identities relating the separated solutions. We also show that the complex EPD equation separates in exactly five coordinate systems corresponding to five orbits of symmetric second-order operators in the enveloping algebra of SL(2,C).

year: 1976-03

title: A Framework for Compositional Synthesis of Modular Nonblocking Supervisors

authors: Mohajerani, Sahar

subjects: Science & Technology

abstract: This paper describes a framework for compositional supervisor synthesis, which is applicable to all discrete event systems modeled as a set of deterministic automata. Compositional synthesis exploits the modular structure of the input model, and therefore works best for models consisting of a large number of small automata. The state-space explosion is mitigated by the use of abstraction to simplify individual components, and the property of synthesis equivalence guarantees that the final synthesis result is the same as it would have been for the non-abstracted model. The paper describes synthesis equivalent abstractions and shows their use in an algorithm to efficiently compute supervisors. The algorithm has been implemented in the DES software tool Supremica and successfully computes nonblocking modular supervisors, even for systems with more than 1014 reachable states, in less than 30 seconds.

text: A Framework for Compositional Synthesis of Modular Nonblocking Supervisors This paper describes a framework for compositional supervisor synthesis, which is applicable to all discrete event systems modeled as a set of deterministic automata. Compositional synthesis exploits the modular structure of the input model, and therefore works best for models consisting of a large number of small automata. The state-space explosion is mitigated by the use of abstraction to simplify individual components, and the property of synthesis equivalence guarantees that the final synthesis result is the same as it would have been for the non-abstracted model. The paper describes synthesis equivalent abstractions and shows their use in an algorithm to efficiently compute supervisors. The algorithm has been implemented in the DES software tool Supremica and successfully computes nonblocking modular supervisors, even for systems with more than 1014 reachable states, in less than 30 seconds.

year: 2014

title: The Environmental Justice Implications of the Planning Policy and Practice of Flood Risk Management in New Zealand

authors: Martynoga, Charlotte

subjects: environmental justce

abstract: Following an international trend, the flood defence approach historically applied in New Zealand has been superseded by a shift to flood risk management, an approach that aligns with the notion of ‘living with risk’ and devolves responsibility to risk-takers at the local level. Citizens are required to assume responsibility for assessing and minimising their own exposure, increasing their resilience and adapting to periodic flooding events. Inevitably, specific communities respond differently to flooding as their capabilities to understand, identify and manage flood risk varies. Environmental justice is the framework of inquiry within which issues of power, representation and participation in planning for flood risk management are examined to consider the injustices that are experienced by communities ‘living with risk’. 

Quantitative and qualitative methods are used to investigate the extent planning is complicit in delivering flood risk management processes that can create environmentally unjust outcomes. Flood hazard maps overlaid with contextual demographic data identify who is living in at risk spaces in three case-study communities. Primary data was collected through semi-structured interviews with local government representatives and iwi, and a questionnaire to local residents was followed by interviews. 

The analysis demonstrates how the environmental justice components of distributive justice, procedural justice, justice as recognition and a capabilities approach to justice are tied together in the political and social processes of managing floods. Procedural justice is based on participatory parity so ensuring all members of the affected community are treated fairly in the deliberative and discursive decision-making is essential. Evidence revealed a community’s limited access to extensive flood risk information, unequal power sharing in decision-making and community participation, and restricted ability for disadvantaged groups to access legal processes. Distributive justice demands the use of multi-criteria analysis, rather than cost-benefit analysis, in prioritising and directing flood risk management to vulnerable communities. Using direct benefit rating to fund flood mitigation works heightens existing inequalities within communities and demands consideration of social difference in flood risk vulnerabilities. To ensure justice as recognition, local and indigenous knowledge needs to be valued and included in decision-making processes. Whilst working party arrangements are not inclusive they empower communities to be actively involved, promote trust and ownership of their local place and flood risk project. Recognition of identities and cultural practices is crucial for self-determination and minimises the marginalisation of voices. Planners need to examine social aspects of how people perceive, adapt and cope with flood risk, alongside place-based vulnerability. Policy, in embracing a capabilities approach to justice, would focus on the functionings people actually achieve rather than the opportunities. This calls for removing aggregations to look at the capabilities of individuals and communities to manage and respond to their flood risk. Judgements on planning initiatives need to be based on whether distributional outcomes enhance the capabilities of the relatively disadvantaged, thereby improving the resilience capacity of a community to manage flood risk. 

This study bridges a research gap in drawing together flood risk management, planning and environmental justice; advancing understanding of the environmental injustices within flood risk management in New Zealand.

text: The Environmental Justice Implications of the Planning Policy and Practice of Flood Risk Management in New Zealand Following an international trend, the flood defence approach historically applied in New Zealand has been superseded by a shift to flood risk management, an approach that aligns with the notion of ‘living with risk’ and devolves responsibility to risk-takers at the local level. Citizens are required to assume responsibility for assessing and minimising their own exposure, increasing their resilience and adapting to periodic flooding events. Inevitably, specific communities respond differently to flooding as their capabilities to understand, identify and manage flood risk varies. Environmental justice is the framework of inquiry within which issues of power, representation and participation in planning for flood risk management are examined to consider the injustices that are experienced by communities ‘living with risk’. 

Quantitative and qualitative methods are used to investigate the extent planning is complicit in delivering flood risk management processes that can create environmentally unjust outcomes. Flood hazard maps overlaid with contextual demographic data identify who is living in at risk spaces in three case-study communities. Primary data was collected through semi-structured interviews with local government representatives and iwi, and a questionnaire to local residents was followed by interviews. 

The analysis demonstrates how the environmental justice components of distributive justice, procedural justice, justice as recognition and a capabilities approach to justice are tied together in the political and social processes of managing floods. Procedural justice is based on participatory parity so ensuring all members of the affected community are treated fairly in the deliberative and discursive decision-making is essential. Evidence revealed a community’s limited access to extensive flood risk information, unequal power sharing in decision-making and community participation, and restricted ability for disadvantaged groups to access legal processes. Distributive justice demands the use of multi-criteria analysis, rather than cost-benefit analysis, in prioritising and directing flood risk management to vulnerable communities. Using direct benefit rating to fund flood mitigation works heightens existing inequalities within communities and demands consideration of social difference in flood risk vulnerabilities. To ensure justice as recognition, local and indigenous knowledge needs to be valued and included in decision-making processes. Whilst working party arrangements are not inclusive they empower communities to be actively involved, promote trust and ownership of their local place and flood risk project. Recognition of identities and cultural practices is crucial for self-determination and minimises the marginalisation of voices. Planners need to examine social aspects of how people perceive, adapt and cope with flood risk, alongside place-based vulnerability. Policy, in embracing a capabilities approach to justice, would focus on the functionings people actually achieve rather than the opportunities. This calls for removing aggregations to look at the capabilities of individuals and communities to manage and respond to their flood risk. Judgements on planning initiatives need to be based on whether distributional outcomes enhance the capabilities of the relatively disadvantaged, thereby improving the resilience capacity of a community to manage flood risk. 

This study bridges a research gap in drawing together flood risk management, planning and environmental justice; advancing understanding of the environmental injustices within flood risk management in New Zealand.

year: 2018