The symmetry group of a model of hyperbolic plane geometry and some associated invariant optimal control problems
- Authors: Henninger, Helen Clare
- Date: 2012
- Subjects: Geometry , Symmetry groups , Symmetry (Mathematics)
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5432 , http://hdl.handle.net/10962/d1018232
- Description: In this thesis we study left-invariant control offine systems on the symmetry group of a. model of hyperbolic plane geometry, the matrix Lie group SO(1, 2)₀. We determine that there are 10 distinct classes of such control systems and for typical elements of two of these classes we provide solutions of the left-invariant optimal wntrol problem with quauratic costs. Under the identification of the Lie allgebra .so(l, 2) with Minkowski spacetime R¹̕'², we construct a controllabilility criterion for all left-invariant control affine systems on 50(1. 2)₀ which in the inhomogeneous case depends only on the presence or absence of an element in the image of the system's trace in R¹̕ ²which is identifiable using the inner product. For the solutions of both the optimal control problems, we provide explicit expressions in terms of Jacobi elliptic functions for the solutions of the reduced extremal equations and determine the nonlinear stability of the equilibrium points.
- Full Text:
- Date Issued: 2012
- Authors: Henninger, Helen Clare
- Date: 2012
- Subjects: Geometry , Symmetry groups , Symmetry (Mathematics)
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5432 , http://hdl.handle.net/10962/d1018232
- Description: In this thesis we study left-invariant control offine systems on the symmetry group of a. model of hyperbolic plane geometry, the matrix Lie group SO(1, 2)₀. We determine that there are 10 distinct classes of such control systems and for typical elements of two of these classes we provide solutions of the left-invariant optimal wntrol problem with quauratic costs. Under the identification of the Lie allgebra .so(l, 2) with Minkowski spacetime R¹̕'², we construct a controllabilility criterion for all left-invariant control affine systems on 50(1. 2)₀ which in the inhomogeneous case depends only on the presence or absence of an element in the image of the system's trace in R¹̕ ²which is identifiable using the inner product. For the solutions of both the optimal control problems, we provide explicit expressions in terms of Jacobi elliptic functions for the solutions of the reduced extremal equations and determine the nonlinear stability of the equilibrium points.
- Full Text:
- Date Issued: 2012
Water requirements and distribution of Ammophila arenaria and Scaevola plumieri on South African coastal dunes
- Authors: Peter, Craig Ingram
- Date: 2000
- Subjects: Scaevola plumieri , Sand dune plants , Sand dune planting , Plants -- Transpiration , Sandworts , Plant-water relationships , Evapotranspiration , Plants, Effect of heat on
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:4219 , http://hdl.handle.net/10962/d1003788 , Scaevola plumieri , Sand dune plants , Sand dune planting , Plants -- Transpiration , Sandworts , Plant-water relationships , Evapotranspiration , Plants, Effect of heat on
- Description: Phenomenological models are presented which predicts transpiration rates (E) of individual leaves of Scaevola plumieri, an indigenous dune pioneer, and Ammophila arenaria, an exotic grass species introduced to stabilise mobile sand. In both cases E is predictably related to atmospheric vapour pressure deficit (VPD). VPD is calculated from measurements of ambient temperature and humidity, hence, where these two environmental variables are known, E can be calculated. Possible physiological reasons for the relationships of E to VPD in both species are discussed. Scaling from measurements of E at the leaf level to the canopy level is achieved by summing the leaf area of the canopy in question. E is predicted for the entire canopy leaf area by extrapolation to this larger leaf area. Predicted transpiration rates of individual shoot within the canopy were tested gravimetrically and shown to be accurate in the case of S. plumieri, but less so in the case of A. arenaria. Using this model, the amount of water used by a known area of sand dune is shown to be less than the rainfall input in the case of S. plumieri in wet and dry years. The water use of A. arenaria exceeds rainfall in the low-rainfall year of 1995, while in 1998 rainfall input is slightly higher than water extraction by the plants. Using a geographic information system (GIS), regional maps (surfaces) of transpiration were calculated from surfaces of mean monthly temperature and mean monthly relative humidity. Monthly surfaces of transpiration were subtracted from the monthly median rainfall to produce a surface of mean monthly water deficit. Areas of water surpluses along the coast correspond with the recorded distribution of both species in the seasons that the plants are most actively growing and reproducing. This suggests that unfavourable water availability during these two species growth periods limit their distributions along the coast. In addition to unfavourable water deficits, additional climatic variables that may be important in limiting the distribution of these two species were investigated using a discriminant function analysis.
- Full Text:
- Date Issued: 2000
- Authors: Peter, Craig Ingram
- Date: 2000
- Subjects: Scaevola plumieri , Sand dune plants , Sand dune planting , Plants -- Transpiration , Sandworts , Plant-water relationships , Evapotranspiration , Plants, Effect of heat on
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:4219 , http://hdl.handle.net/10962/d1003788 , Scaevola plumieri , Sand dune plants , Sand dune planting , Plants -- Transpiration , Sandworts , Plant-water relationships , Evapotranspiration , Plants, Effect of heat on
- Description: Phenomenological models are presented which predicts transpiration rates (E) of individual leaves of Scaevola plumieri, an indigenous dune pioneer, and Ammophila arenaria, an exotic grass species introduced to stabilise mobile sand. In both cases E is predictably related to atmospheric vapour pressure deficit (VPD). VPD is calculated from measurements of ambient temperature and humidity, hence, where these two environmental variables are known, E can be calculated. Possible physiological reasons for the relationships of E to VPD in both species are discussed. Scaling from measurements of E at the leaf level to the canopy level is achieved by summing the leaf area of the canopy in question. E is predicted for the entire canopy leaf area by extrapolation to this larger leaf area. Predicted transpiration rates of individual shoot within the canopy were tested gravimetrically and shown to be accurate in the case of S. plumieri, but less so in the case of A. arenaria. Using this model, the amount of water used by a known area of sand dune is shown to be less than the rainfall input in the case of S. plumieri in wet and dry years. The water use of A. arenaria exceeds rainfall in the low-rainfall year of 1995, while in 1998 rainfall input is slightly higher than water extraction by the plants. Using a geographic information system (GIS), regional maps (surfaces) of transpiration were calculated from surfaces of mean monthly temperature and mean monthly relative humidity. Monthly surfaces of transpiration were subtracted from the monthly median rainfall to produce a surface of mean monthly water deficit. Areas of water surpluses along the coast correspond with the recorded distribution of both species in the seasons that the plants are most actively growing and reproducing. This suggests that unfavourable water availability during these two species growth periods limit their distributions along the coast. In addition to unfavourable water deficits, additional climatic variables that may be important in limiting the distribution of these two species were investigated using a discriminant function analysis.
- Full Text:
- Date Issued: 2000
A review of asbestos resources
- Authors: Abbott, Paul
- Date: 1983
- Subjects: Asbestos Asbestos -- Geology Silicate minerals Mineralogy Chrysotile
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:4910 , http://hdl.handle.net/10962/d1001570
- Full Text:
- Date Issued: 1983
- Authors: Abbott, Paul
- Date: 1983
- Subjects: Asbestos Asbestos -- Geology Silicate minerals Mineralogy Chrysotile
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:4910 , http://hdl.handle.net/10962/d1001570
- Full Text:
- Date Issued: 1983
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