Modifcations to gravitational waves due to matter shells
- Authors: Naidoo, Monogaran
- Date: 2021-10-29
- Subjects: Gravitational waves , General relativity (Physics) , Einstein field equations , Cosmology , Matter shells
- Language: English
- Type: Doctoral theses , text
- Identifier: http://hdl.handle.net/10962/191118 , vital:45062 , 10.21504/10962/191119
- Description: As detections of gravitational waves (GWs) mount, the need to investigate various effects on the propagation of these waves from the time of emission until detection also grows. We investigate how a thin low density dust shell surrounding a gravitational wave source affects the propagation of GWs. The Bondi-Sachs (BS) formalism for the Einstein equations is used for the problem of a gravitational wave (GW) source surrounded by a spherical dust shell. Using linearised perturbation theory, we and the geometry of the regions exterior to, interior to and within the shell. We and that the dust shell causes the gravitational wave to be modified both in magnitude and phase, but without any energy being transferred to or from the dust. This finding is novel. In the context of cosmology, apart from the gravitational redshift, the effects are too small to be measurable; but the effect would be measurable if a GW event were to occur with a source surrounded by a massive shell and with the radius of the shell and the wavelength of the GWs of the same order. We extended our investigation to astrophysical scenarios such as binary black hole (BBH) mergers, binary neutron star (BNS) mergers, and core collapse supernovae (CCSNe). In these scenarios, instead of a monochromatic GW source, as we used in our initial investigation, we consider burst-like GW sources. The thin density shell approach is modified to include thick shells by considering concentric thin shells and integrating. Solutions are then found for these burst-like GW sources using Fourier transforms. We show that GW echoes that are claimed to be present in the Laser Interferometer Gravitational-Wave Observatory (LIGO) data of certain events, could not have been caused by a matter shell. We do and, however, that matter shells surrounding BBH mergers, BNS mergers, and CCSNe could make modifications of order a few percent to a GW signal. These modifications are expected to be measurable in GW data with current detectors if the event is close enough and at a detectable frequency; or in future detectors with increased frequency range and amplitude sensitivity. Substantial use is made of computer algebra in these investigations. In setting the scene for our investigations, we trace the evolution of general relativity (GR) from Einstein's postulation in 1915 to vindication of his theory with the confirmation of the existence of GWs a century later. We discuss the implications of our results to current and future considerations. Calculations of GWs, both analytical and numerical, have normally assumed their propagation from source to a detector on Earth in a vacuum spacetime, and so discounted the effect of intervening matter. As we enter an era of precision GW measurements, it becomes important to quantify any effects due to propagation of GWs through a non-vacuum spacetime Observational confirmation of the modification effect that we and in astrophysical scenarios involving black holes (BHs), neutron stars (NSs) and CCSNe, would also enhance our understanding of the details of the physics of these bodies. , Thesis (PhD) -- Faculty of Science, Mathematics (Pure and Applied), 2021
- Full Text:
- Date Issued: 2021-10-29
- Authors: Naidoo, Monogaran
- Date: 2021-10-29
- Subjects: Gravitational waves , General relativity (Physics) , Einstein field equations , Cosmology , Matter shells
- Language: English
- Type: Doctoral theses , text
- Identifier: http://hdl.handle.net/10962/191118 , vital:45062 , 10.21504/10962/191119
- Description: As detections of gravitational waves (GWs) mount, the need to investigate various effects on the propagation of these waves from the time of emission until detection also grows. We investigate how a thin low density dust shell surrounding a gravitational wave source affects the propagation of GWs. The Bondi-Sachs (BS) formalism for the Einstein equations is used for the problem of a gravitational wave (GW) source surrounded by a spherical dust shell. Using linearised perturbation theory, we and the geometry of the regions exterior to, interior to and within the shell. We and that the dust shell causes the gravitational wave to be modified both in magnitude and phase, but without any energy being transferred to or from the dust. This finding is novel. In the context of cosmology, apart from the gravitational redshift, the effects are too small to be measurable; but the effect would be measurable if a GW event were to occur with a source surrounded by a massive shell and with the radius of the shell and the wavelength of the GWs of the same order. We extended our investigation to astrophysical scenarios such as binary black hole (BBH) mergers, binary neutron star (BNS) mergers, and core collapse supernovae (CCSNe). In these scenarios, instead of a monochromatic GW source, as we used in our initial investigation, we consider burst-like GW sources. The thin density shell approach is modified to include thick shells by considering concentric thin shells and integrating. Solutions are then found for these burst-like GW sources using Fourier transforms. We show that GW echoes that are claimed to be present in the Laser Interferometer Gravitational-Wave Observatory (LIGO) data of certain events, could not have been caused by a matter shell. We do and, however, that matter shells surrounding BBH mergers, BNS mergers, and CCSNe could make modifications of order a few percent to a GW signal. These modifications are expected to be measurable in GW data with current detectors if the event is close enough and at a detectable frequency; or in future detectors with increased frequency range and amplitude sensitivity. Substantial use is made of computer algebra in these investigations. In setting the scene for our investigations, we trace the evolution of general relativity (GR) from Einstein's postulation in 1915 to vindication of his theory with the confirmation of the existence of GWs a century later. We discuss the implications of our results to current and future considerations. Calculations of GWs, both analytical and numerical, have normally assumed their propagation from source to a detector on Earth in a vacuum spacetime, and so discounted the effect of intervening matter. As we enter an era of precision GW measurements, it becomes important to quantify any effects due to propagation of GWs through a non-vacuum spacetime Observational confirmation of the modification effect that we and in astrophysical scenarios involving black holes (BHs), neutron stars (NSs) and CCSNe, would also enhance our understanding of the details of the physics of these bodies. , Thesis (PhD) -- Faculty of Science, Mathematics (Pure and Applied), 2021
- Full Text:
- Date Issued: 2021-10-29
Best simultaneous approximation in normed linear spaces
- Authors: Johnson, Solomon Nathan
- Date: 2018
- Subjects: Normed linear spaces , Approximation theory , Mathematical analysis
- Language: English
- Type: text , Thesis , Masters , MSc
- Identifier: http://hdl.handle.net/10962/58985 , vital:27400
- Description: In this thesis we consider the problem of simultaneously approximating elements of a set B C X by a single element of a set K C X. This type of a problem arises when the element to be approximated is not known precisely but is known to belong to a set.Thus, best simultaneous approximation is a natural generalization of best approximation which has been studied extensively. The theory of best simultaneous approximation has been studied by many authors, see for example [4], [8], [25], [28], [26] and [12] to name but a few.
- Full Text:
- Date Issued: 2018
- Authors: Johnson, Solomon Nathan
- Date: 2018
- Subjects: Normed linear spaces , Approximation theory , Mathematical analysis
- Language: English
- Type: text , Thesis , Masters , MSc
- Identifier: http://hdl.handle.net/10962/58985 , vital:27400
- Description: In this thesis we consider the problem of simultaneously approximating elements of a set B C X by a single element of a set K C X. This type of a problem arises when the element to be approximated is not known precisely but is known to belong to a set.Thus, best simultaneous approximation is a natural generalization of best approximation which has been studied extensively. The theory of best simultaneous approximation has been studied by many authors, see for example [4], [8], [25], [28], [26] and [12] to name but a few.
- Full Text:
- Date Issued: 2018
Geometry of deformed special relativity
- Authors: Sixaba, Vuyile
- Date: 2018
- Subjects: Special relativity (Physics) , Quantum gravity , Quantum theory , Geometry , Heisenberg uncertainty principle
- Language: English
- Type: text , Thesis , Masters , MSc
- Identifier: http://hdl.handle.net/10962/59478 , vital:27615
- Description: We undertake a study of the classical regime in which Planck's constant and Newton's gravitational constant are negligible, but not their ratio, the Planck mass, in hopes that this could possibly lead to testable quantum gravity (QG) effects in a classical regime. In this quest for QG phenomenology we consider modifications of the standard dispersion relation of a free particle known as deformed special relativity (DSR). We try to geometrize DSR to find the geometric origin of the spacetime and momentum space. In particular, we adopt the framework of Hamilton geometry which is set up on phase space, as the cotangent bundle of configuration space in order to derive a purely phase space formulation of DSR. This is necessary when one wants to understand potential links of DSR with modifications of quantum mechanics such as Generalised Uncertainty Principles. It is subsequently observed that space-time and momentum space emerge naturally as curved and intertwined spaces. In conclusion we mention examples and applications of this framework as well as potential future developments.
- Full Text:
- Date Issued: 2018
- Authors: Sixaba, Vuyile
- Date: 2018
- Subjects: Special relativity (Physics) , Quantum gravity , Quantum theory , Geometry , Heisenberg uncertainty principle
- Language: English
- Type: text , Thesis , Masters , MSc
- Identifier: http://hdl.handle.net/10962/59478 , vital:27615
- Description: We undertake a study of the classical regime in which Planck's constant and Newton's gravitational constant are negligible, but not their ratio, the Planck mass, in hopes that this could possibly lead to testable quantum gravity (QG) effects in a classical regime. In this quest for QG phenomenology we consider modifications of the standard dispersion relation of a free particle known as deformed special relativity (DSR). We try to geometrize DSR to find the geometric origin of the spacetime and momentum space. In particular, we adopt the framework of Hamilton geometry which is set up on phase space, as the cotangent bundle of configuration space in order to derive a purely phase space formulation of DSR. This is necessary when one wants to understand potential links of DSR with modifications of quantum mechanics such as Generalised Uncertainty Principles. It is subsequently observed that space-time and momentum space emerge naturally as curved and intertwined spaces. In conclusion we mention examples and applications of this framework as well as potential future developments.
- Full Text:
- Date Issued: 2018
Contributions to the study of nonholonomic Riemannian manifolds
- Authors: Barrett, Dennis Ian
- Date: 2017
- Subjects: Riemannian manifolds , Curvature , Lie groups , Geometry, Riemannian , Tensor fields
- Language: English
- Type: Doctoral theses , text
- Identifier: http://hdl.handle.net/10962/7554 , vital:21272 , DOI https://doi.org/10.21504/10962/7554
- Description: In this thesis we consider nonholonomic Riemannian manifolds, and in particular, left- invariant nonholonomic Riemannian structures on Lie groups. These structures are closely related to mechanical systems with (positive definite) quadratic Lagrangians and nonholo- nomic constraints linear in velocities. In the first chapter, we review basic concepts of non- holonomic Riemannian geometry, including the left-invariant structures. We also examine the class of left-invariant structures with so-called Cartan-Schouten connections. The second chapter investigates the curvature of nonholonomic Riemannian manifolds and the Schouten and Wagner curvature tensors. The Schouten tensor is canonically associated to every non- holonomic Riemannian structure (in particular, we use it to define isometric invariants for structures on three-dimensional manifolds). By contrast, the Wagner tensor is not generally intrinsic, but can be used to characterise flat structures (i.e., those whose associated parallel transport is path-independent). The third chapter considers equivalence of nonholonomic Rie- mannian manifolds, particularly up to nonholonomic isometry. We also introduce the notion of a nonholonomic Riemannian submanifold, and investigate the conditions under which such a submanifold inherits its geometry from the enveloping space. The latter problem involves the concept of a geodesically invariant distribution, and we show it is also related to the curvature. In the last chapter we specialise to three-dimensional nonholonomic Riemannian manifolds. We consider the equivalence of such structures up to nonholonomic isometry and rescaling, and classify the left-invariant structures on the (three-dimensional) simply connected Lie groups. We also characterise the flat structures in three dimensions, and then classify the flat structures on the simply connected Lie groups. Lastly, we consider three typical examples of (left-invariant) nonholonomic Riemannian structures on three-dimensional Lie groups, two of which arise from problems in classical mechanics (viz., the Chaplygin problem and the Suslov problem). , Thesis (PhD) -- Faculty of Science, Mathematics (Pure and Applied), 2017
- Full Text:
- Date Issued: 2017
- Authors: Barrett, Dennis Ian
- Date: 2017
- Subjects: Riemannian manifolds , Curvature , Lie groups , Geometry, Riemannian , Tensor fields
- Language: English
- Type: Doctoral theses , text
- Identifier: http://hdl.handle.net/10962/7554 , vital:21272 , DOI https://doi.org/10.21504/10962/7554
- Description: In this thesis we consider nonholonomic Riemannian manifolds, and in particular, left- invariant nonholonomic Riemannian structures on Lie groups. These structures are closely related to mechanical systems with (positive definite) quadratic Lagrangians and nonholo- nomic constraints linear in velocities. In the first chapter, we review basic concepts of non- holonomic Riemannian geometry, including the left-invariant structures. We also examine the class of left-invariant structures with so-called Cartan-Schouten connections. The second chapter investigates the curvature of nonholonomic Riemannian manifolds and the Schouten and Wagner curvature tensors. The Schouten tensor is canonically associated to every non- holonomic Riemannian structure (in particular, we use it to define isometric invariants for structures on three-dimensional manifolds). By contrast, the Wagner tensor is not generally intrinsic, but can be used to characterise flat structures (i.e., those whose associated parallel transport is path-independent). The third chapter considers equivalence of nonholonomic Rie- mannian manifolds, particularly up to nonholonomic isometry. We also introduce the notion of a nonholonomic Riemannian submanifold, and investigate the conditions under which such a submanifold inherits its geometry from the enveloping space. The latter problem involves the concept of a geodesically invariant distribution, and we show it is also related to the curvature. In the last chapter we specialise to three-dimensional nonholonomic Riemannian manifolds. We consider the equivalence of such structures up to nonholonomic isometry and rescaling, and classify the left-invariant structures on the (three-dimensional) simply connected Lie groups. We also characterise the flat structures in three dimensions, and then classify the flat structures on the simply connected Lie groups. Lastly, we consider three typical examples of (left-invariant) nonholonomic Riemannian structures on three-dimensional Lie groups, two of which arise from problems in classical mechanics (viz., the Chaplygin problem and the Suslov problem). , Thesis (PhD) -- Faculty of Science, Mathematics (Pure and Applied), 2017
- Full Text:
- Date Issued: 2017
A combinatorial analysis of barred preferential arrangements
- Authors: Nkonkobe, Sithembele
- Date: 2016
- Language: English
- Type: Thesis , Doctoral , PhD
- Identifier: http://hdl.handle.net/10962/36228 , vital:24530
- Description: For a non-negative integer n an ordered partition of a set Xn with n distinct elements is called a preferential arrangement (PA). A barred preferential arrangement (BPA) is a preferential arrangement with bars in between the blocks of the partition. An integer sequence an associated with the counting PA's of Xn has been intensely studied over a century and a half in many different contexts. In this thesis we develop a unified combinatorial framework to study the enumeration of BPAs and a special subclass of BPAs. The results of the study lead to a positive settlement of an open problem and a conjecture by Nelsen. We derive few important identities pertaining to the number of BPAs and restricted BPAs of an n element set using generating- functionology. Later we show that the number of restricted BPAs of Xn are intricately related to well-known numbers such as Eulerian numbers, Bell numbers, Poly-Bernoulli numbers and the number of equivalence classes of fuzzy subsets of Xn under some equivalent relation.
- Full Text:
- Date Issued: 2016
- Authors: Nkonkobe, Sithembele
- Date: 2016
- Language: English
- Type: Thesis , Doctoral , PhD
- Identifier: http://hdl.handle.net/10962/36228 , vital:24530
- Description: For a non-negative integer n an ordered partition of a set Xn with n distinct elements is called a preferential arrangement (PA). A barred preferential arrangement (BPA) is a preferential arrangement with bars in between the blocks of the partition. An integer sequence an associated with the counting PA's of Xn has been intensely studied over a century and a half in many different contexts. In this thesis we develop a unified combinatorial framework to study the enumeration of BPAs and a special subclass of BPAs. The results of the study lead to a positive settlement of an open problem and a conjecture by Nelsen. We derive few important identities pertaining to the number of BPAs and restricted BPAs of an n element set using generating- functionology. Later we show that the number of restricted BPAs of Xn are intricately related to well-known numbers such as Eulerian numbers, Bell numbers, Poly-Bernoulli numbers and the number of equivalence classes of fuzzy subsets of Xn under some equivalent relation.
- Full Text:
- Date Issued: 2016
Cosmological structure formation using spectral methods
- Authors: Funcke, Michelle
- Date: 2016
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: http://hdl.handle.net/10962/2969 , vital:20348
- Description: Numerical simulations are becoming an increasingly important tool for understanding the growth and development of structure in the universe. Common practice is to discretize the space-time using physical variables. The discreteness is embodied by considering the dynamical variables as fields on a fixed spatial and time resolution, or by constructing the matter fields by a large number of particles which interact gravitationally (N-body methods). Recognizing that the physical quantities of interest are related to the spectrum of perturbations, we propose an alternate discretization in the frequency domain, using standard spectral methods. This approach is further aided by periodic boundary conditions which allows a straightforward decomposition of variables in a Fourier basis. Fixed resources require a high-frequency cut-off which lead to aliasing effects in non-linear equations, such as the ones considered here. This thesis describes the implementation of a 3D cosmological model based on Newtonian hydrodynamic equations in an expanding background. Initial data is constructed as a spectrum of perturbations, and evolved in the frequency domain using a pseudo-spectral evolution scheme and an explicit Runge-Kutta time integrator. The code is found to converge for both linear and non-linear evolutions, and the convergence rate is determined. The correct growth rates expected from analytical calculations are recovered in the linear case. In the non-linear model, we observe close correspondence with linear growth and are able to monitor the growth on features associated with the non-linearity. High-frequency aliasing effects were evident in the non-linear evolutions, leading to a study of two potential resolutions to this problem: a boxcar filter which adheres to“Orszag’s two thirds rule” and an exponential window function, the exponential filter suggested by Hou and Li [1], and a shifted version of the exponential filter suggested, which has the potential to alleviate high frequency- ripples resulting from the Gibbs’ phenomenon. We found that the filters were somewhat successful at reducing aliasing effects but that the Gibbs’ phenomenon could not be entirely removed by the choice of filters.
- Full Text:
- Date Issued: 2016
- Authors: Funcke, Michelle
- Date: 2016
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: http://hdl.handle.net/10962/2969 , vital:20348
- Description: Numerical simulations are becoming an increasingly important tool for understanding the growth and development of structure in the universe. Common practice is to discretize the space-time using physical variables. The discreteness is embodied by considering the dynamical variables as fields on a fixed spatial and time resolution, or by constructing the matter fields by a large number of particles which interact gravitationally (N-body methods). Recognizing that the physical quantities of interest are related to the spectrum of perturbations, we propose an alternate discretization in the frequency domain, using standard spectral methods. This approach is further aided by periodic boundary conditions which allows a straightforward decomposition of variables in a Fourier basis. Fixed resources require a high-frequency cut-off which lead to aliasing effects in non-linear equations, such as the ones considered here. This thesis describes the implementation of a 3D cosmological model based on Newtonian hydrodynamic equations in an expanding background. Initial data is constructed as a spectrum of perturbations, and evolved in the frequency domain using a pseudo-spectral evolution scheme and an explicit Runge-Kutta time integrator. The code is found to converge for both linear and non-linear evolutions, and the convergence rate is determined. The correct growth rates expected from analytical calculations are recovered in the linear case. In the non-linear model, we observe close correspondence with linear growth and are able to monitor the growth on features associated with the non-linearity. High-frequency aliasing effects were evident in the non-linear evolutions, leading to a study of two potential resolutions to this problem: a boxcar filter which adheres to“Orszag’s two thirds rule” and an exponential window function, the exponential filter suggested by Hou and Li [1], and a shifted version of the exponential filter suggested, which has the potential to alleviate high frequency- ripples resulting from the Gibbs’ phenomenon. We found that the filters were somewhat successful at reducing aliasing effects but that the Gibbs’ phenomenon could not be entirely removed by the choice of filters.
- Full Text:
- Date Issued: 2016
Observational cosmology with imperfect data
- Authors: Bester, Hertzog Landman
- Date: 2016
- Language: English
- Type: Thesis , Doctoral , PhD
- Identifier: http://hdl.handle.net/10962/463 , vital:19961
- Description: We develop a formalism suitable to infer the background geometry of a general spherically symmetric dust universe directly from data on the past lightcone. This direct observational approach makes minimal assumptions about inaccessible parts of the Universe. The non-parametric and Bayesian framework we propose provides a very direct way to test one of the most fundamental underlying assumptions of concordance cosmology viz. the Copernican principle. We present the Copernicus algorithm for this purpose. By applying the algorithm to currently available data, we demonstrate that it is not yet possible to confirm or refute the validity of the Copernican principle within the proposed framework. This is followed by an investigation which aims to determine which future data will best be able to test the Copernican principle. Our results on simulated data suggest that, besides the need to improve the current data, it will be important to identify additional model independent observables for this purpose. The main difficulty with current data is their inability to constrain the value of the cosmological constant. We show how redshift drift data could be used to infer its value with minimal assumptions about the nature of the early Universe. We also discuss some alternative applications of the algorithm.
- Full Text:
- Date Issued: 2016
- Authors: Bester, Hertzog Landman
- Date: 2016
- Language: English
- Type: Thesis , Doctoral , PhD
- Identifier: http://hdl.handle.net/10962/463 , vital:19961
- Description: We develop a formalism suitable to infer the background geometry of a general spherically symmetric dust universe directly from data on the past lightcone. This direct observational approach makes minimal assumptions about inaccessible parts of the Universe. The non-parametric and Bayesian framework we propose provides a very direct way to test one of the most fundamental underlying assumptions of concordance cosmology viz. the Copernican principle. We present the Copernicus algorithm for this purpose. By applying the algorithm to currently available data, we demonstrate that it is not yet possible to confirm or refute the validity of the Copernican principle within the proposed framework. This is followed by an investigation which aims to determine which future data will best be able to test the Copernican principle. Our results on simulated data suggest that, besides the need to improve the current data, it will be important to identify additional model independent observables for this purpose. The main difficulty with current data is their inability to constrain the value of the cosmological constant. We show how redshift drift data could be used to infer its value with minimal assumptions about the nature of the early Universe. We also discuss some alternative applications of the algorithm.
- Full Text:
- Date Issued: 2016
A study of barred preferential arrangements with applications to numerical approximation in electric circuits
- Authors: Nkonkobe, Sithembele
- Date: 2015
- Subjects: Electric circuits , Numerical calculations , Sequences (Mathematics)
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5433 , http://hdl.handle.net/10962/d1020394
- Description: In 1854 Cayley proposed an interesting sequence 1,1,3,13,75,541,... in connection with analytical forms called trees. Since then there has been various combinatorial interpretations of the sequence. The sequence has been interpreted as the number of preferential arrangements of members of a set with n elements. Alternatively the sequence has been interpreted as the number of ordered partitions; the outcomes in races in which ties are allowed or geometrically the number of vertices, edges and faces of simplicial objects. An interesting application of the sequence is found in combination locks. The idea of a preferential arrangement has been extended to a wider combinatorial object called barred preferential arrangement with multiple bars. In this thesis we study barred preferential arrangements combinatorially with application to resistance of certain electrical circuits. In the process we derive some results on cyclic properties of the last digit of the number of barred preferential arrangements. An algorithm in python has been developed to find the number of barred preferential arrangements.
- Full Text:
- Date Issued: 2015
- Authors: Nkonkobe, Sithembele
- Date: 2015
- Subjects: Electric circuits , Numerical calculations , Sequences (Mathematics)
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5433 , http://hdl.handle.net/10962/d1020394
- Description: In 1854 Cayley proposed an interesting sequence 1,1,3,13,75,541,... in connection with analytical forms called trees. Since then there has been various combinatorial interpretations of the sequence. The sequence has been interpreted as the number of preferential arrangements of members of a set with n elements. Alternatively the sequence has been interpreted as the number of ordered partitions; the outcomes in races in which ties are allowed or geometrically the number of vertices, edges and faces of simplicial objects. An interesting application of the sequence is found in combination locks. The idea of a preferential arrangement has been extended to a wider combinatorial object called barred preferential arrangement with multiple bars. In this thesis we study barred preferential arrangements combinatorially with application to resistance of certain electrical circuits. In the process we derive some results on cyclic properties of the last digit of the number of barred preferential arrangements. An algorithm in python has been developed to find the number of barred preferential arrangements.
- Full Text:
- Date Issued: 2015
Contributions to the study of a class of optimal control problems on the orthogonal groups SO(3) and SO(4)
- Authors: Adams, Ross Montague
- Date: 2015
- Language: English
- Type: text , Thesis , Doctoral , PhD
- Identifier: http://hdl.handle.net/10962/64826 , vital:28608
- Description: In this thesis we investigate a class of invariant optimal control problems, and their associated quadratic Hamilton-Poisson systems, on the orthogonal groups SO(3) and SO(4). Specifically, we are concerned with the class of left-invariant control affine systems. We begin by classifying all cost-extended systems on SO(3) under cost equivalence. (Cost-extended systems are closely related to optimal control problems.) A classification of all quadratic Hamilton-Poisson systems on the (minus) Lie-Poisson space so(3)*, under affine equivalence, is also obtained. For the normal forms obtained in our classification (of Hamilton-Poisson systems) we investigate the (Lyapunov) stability nature of the equilibria using spectral and energy-Casimir methods. For a subclass of these systems, we obtain analytic expressions for the integral curves of the associated Hamiltonian vector fields in terms of (basic) Jacobi elliptic functions. The explicit relationship between the classification of cost-extended systems on SO(3) and the classification of quadratic Hamilton- Poisson systems on so(3)* is provided. On SO(4), a classification of all left-invariant control affine systems under L-equivalence is obtained. We then determine which of these representatives are controllable, thus obtaining a classification under detached feedback equivalence. We also obtain a partial classification of quadratic Hamilton-Poisson systems on the Lie-Poisson space so(4)*. An investigation of the stability nature of the equilibria for a subclass of these systems is also done. Several illustrative examples of optimal control problems on the orthogonal group SO(3) are provided. More specifically, we consider an optimal control problem corresponding to a representative of our classification (of cost-extended system) for each possible number of control inputs. For each of these problems, we obtain explicit expressions for the extremal trajectories on the homogeneous space S2 by projecting the extremal trajectories on the group SO(3). The examples provided show how our classifications of cost-extended systems and Hamilton-Poisson systems can be used to obtain the optimal controls and the extremal trajectories corresponding to a large class of optimal control problems on SO(3). An example of a four-input optimal control problem on SO(4) is also provided. This example is provided to show how the solutions of certain problems on SO(4) can be related to the solutions of certain optimal control problems on SO(3).
- Full Text:
- Date Issued: 2015
- Authors: Adams, Ross Montague
- Date: 2015
- Language: English
- Type: text , Thesis , Doctoral , PhD
- Identifier: http://hdl.handle.net/10962/64826 , vital:28608
- Description: In this thesis we investigate a class of invariant optimal control problems, and their associated quadratic Hamilton-Poisson systems, on the orthogonal groups SO(3) and SO(4). Specifically, we are concerned with the class of left-invariant control affine systems. We begin by classifying all cost-extended systems on SO(3) under cost equivalence. (Cost-extended systems are closely related to optimal control problems.) A classification of all quadratic Hamilton-Poisson systems on the (minus) Lie-Poisson space so(3)*, under affine equivalence, is also obtained. For the normal forms obtained in our classification (of Hamilton-Poisson systems) we investigate the (Lyapunov) stability nature of the equilibria using spectral and energy-Casimir methods. For a subclass of these systems, we obtain analytic expressions for the integral curves of the associated Hamiltonian vector fields in terms of (basic) Jacobi elliptic functions. The explicit relationship between the classification of cost-extended systems on SO(3) and the classification of quadratic Hamilton- Poisson systems on so(3)* is provided. On SO(4), a classification of all left-invariant control affine systems under L-equivalence is obtained. We then determine which of these representatives are controllable, thus obtaining a classification under detached feedback equivalence. We also obtain a partial classification of quadratic Hamilton-Poisson systems on the Lie-Poisson space so(4)*. An investigation of the stability nature of the equilibria for a subclass of these systems is also done. Several illustrative examples of optimal control problems on the orthogonal group SO(3) are provided. More specifically, we consider an optimal control problem corresponding to a representative of our classification (of cost-extended system) for each possible number of control inputs. For each of these problems, we obtain explicit expressions for the extremal trajectories on the homogeneous space S2 by projecting the extremal trajectories on the group SO(3). The examples provided show how our classifications of cost-extended systems and Hamilton-Poisson systems can be used to obtain the optimal controls and the extremal trajectories corresponding to a large class of optimal control problems on SO(3). An example of a four-input optimal control problem on SO(4) is also provided. This example is provided to show how the solutions of certain problems on SO(4) can be related to the solutions of certain optimal control problems on SO(3).
- Full Text:
- Date Issued: 2015
Counting of finite fuzzy subsets with applications to fuzzy recognition and selection strategies
- Authors: Talwanga, Matiki
- Date: 2015
- Subjects: Möbius transformations , Fuzzy sets , Functions, Zeta , Partitions (Mathematics)
- Language: English
- Type: Thesis , Doctoral , PhD
- Identifier: vital:5431 , http://hdl.handle.net/10962/d1018186
- Description: The counting of fuzzy subsets of a finite set is of great interest in both practical and theoretical contexts in Mathematics. We have used some counting techniques such as the principle of Inclusion-Exclusion and the Mõbius Inversion to enumerate the fuzzy subsets of a finite set satisfying different conditions. These two techniques are interdependent with the M¨obius inversion generalizing the principle of Inclusion-Exclusion. The enumeration is carried out each time we redefine new conditions on the set. In this study one of our aims is the recognition and identification of fuzzy subsets with same features, characteristics or conditions. To facilitate such a study, we use some ideas such as the Hamming distance, mid-point between two fuzzy subsets and cardinality of fuzzy subsets. Finally we introduce the fuzzy scanner of elements of a finite set. This is used to identify elements and fuzzy subsets of a set. The scanning process of identification and recognition facilitates the choice of entities with specified properties. We develop a procedure of selection under the fuzzy environment. This allows us a framework to resolve conflicting issues in the market place.
- Full Text:
- Date Issued: 2015
- Authors: Talwanga, Matiki
- Date: 2015
- Subjects: Möbius transformations , Fuzzy sets , Functions, Zeta , Partitions (Mathematics)
- Language: English
- Type: Thesis , Doctoral , PhD
- Identifier: vital:5431 , http://hdl.handle.net/10962/d1018186
- Description: The counting of fuzzy subsets of a finite set is of great interest in both practical and theoretical contexts in Mathematics. We have used some counting techniques such as the principle of Inclusion-Exclusion and the Mõbius Inversion to enumerate the fuzzy subsets of a finite set satisfying different conditions. These two techniques are interdependent with the M¨obius inversion generalizing the principle of Inclusion-Exclusion. The enumeration is carried out each time we redefine new conditions on the set. In this study one of our aims is the recognition and identification of fuzzy subsets with same features, characteristics or conditions. To facilitate such a study, we use some ideas such as the Hamming distance, mid-point between two fuzzy subsets and cardinality of fuzzy subsets. Finally we introduce the fuzzy scanner of elements of a finite set. This is used to identify elements and fuzzy subsets of a set. The scanning process of identification and recognition facilitates the choice of entities with specified properties. We develop a procedure of selection under the fuzzy environment. This allows us a framework to resolve conflicting issues in the market place.
- Full Text:
- Date Issued: 2015
Invariant control systems and sub-Riemannian structures on lie groups: equivalence and isometries
- Authors: Biggs, Rory
- Date: 2015
- Language: English
- Type: text , Thesis , Doctoral , PhD
- Identifier: http://hdl.handle.net/10962/64815 , vital:28607
- Description: In this thesis we consider invariant optimal control problems and invariant sub-Riemannian structures on Lie groups. Primarily, we are concerned with the equivalence and classification of problems (resp. structures). In the first chapter, both the class of invariant optimal control problems and the class of invariant sub-Riemannian structures are organised as categories. The latter category is shown to be functorially equivalent to a subcategory of the former category. Via the Pontryagin Maximum Principle, we associate to each invariant optimal control problem (resp. invariant sub-Riemannian structure) a quadratic Hamilton-Poisson system on the associated Lie-Poisson space. These Hamiltonian systems are also organised as a category and hence the Pontryagin lift is realised as a contravariant functor. Basic properties of these categories are investigated. The rest of this thesis is concerned with the classification (and investigation) of certain subclasses of structures and systems. In the second chapter, the homogeneous positive semidefinite quadratic Hamilton-Poisson systems on three-dimensional Lie-Poisson spaces are classified up to compatibility with a linear isomorphism; a list of 23 normal forms is exhibited. In the third chapter, we classify the invariant sub-Riemannian structures in three dimensions and calculate the isometry group for each normal form. By comparing our results with known results, we show that most isometries (in three dimensions) are in fact the composition of a left translation and a Lie group isomorphism. In the fourth and last chapter of this thesis, we classify the sub-Riemannian and Riemannian structures on the (2n + 1)-dimensional Heisenberg groups. Furthermore, we find the isometry group and geodesics of each normal form.
- Full Text:
- Date Issued: 2015
- Authors: Biggs, Rory
- Date: 2015
- Language: English
- Type: text , Thesis , Doctoral , PhD
- Identifier: http://hdl.handle.net/10962/64815 , vital:28607
- Description: In this thesis we consider invariant optimal control problems and invariant sub-Riemannian structures on Lie groups. Primarily, we are concerned with the equivalence and classification of problems (resp. structures). In the first chapter, both the class of invariant optimal control problems and the class of invariant sub-Riemannian structures are organised as categories. The latter category is shown to be functorially equivalent to a subcategory of the former category. Via the Pontryagin Maximum Principle, we associate to each invariant optimal control problem (resp. invariant sub-Riemannian structure) a quadratic Hamilton-Poisson system on the associated Lie-Poisson space. These Hamiltonian systems are also organised as a category and hence the Pontryagin lift is realised as a contravariant functor. Basic properties of these categories are investigated. The rest of this thesis is concerned with the classification (and investigation) of certain subclasses of structures and systems. In the second chapter, the homogeneous positive semidefinite quadratic Hamilton-Poisson systems on three-dimensional Lie-Poisson spaces are classified up to compatibility with a linear isomorphism; a list of 23 normal forms is exhibited. In the third chapter, we classify the invariant sub-Riemannian structures in three dimensions and calculate the isometry group for each normal form. By comparing our results with known results, we show that most isometries (in three dimensions) are in fact the composition of a left translation and a Lie group isomorphism. In the fourth and last chapter of this thesis, we classify the sub-Riemannian and Riemannian structures on the (2n + 1)-dimensional Heisenberg groups. Furthermore, we find the isometry group and geodesics of each normal form.
- Full Text:
- Date Issued: 2015
Invariant optimal control on the three-dimensional semi-Euclidean group: control affine and quadratic Hamilton-Poisson systems
- Authors: Barrett, Dennis Ian
- Date: 2014
- Subjects: Automorphisms , Symmetry (Mathematics) , Lyapunov stability , Geometry, Riemannian , Geometry, Affine , Elliptic functions
- Language: English
- Type: text , Thesis , Masters , MSc
- Identifier: http://hdl.handle.net/10962/64805 , vital:28605
- Description: In this thesis we consider invariant control systems and Hamilton-Poisson systems on the three dimensional semi-Euclidean group SE(1,1). We first classify the left-invariant control affine systems (under detached feedback equivalence). We provide a complete list of normal forms, as well as classifying conditions. As a corollary to this classification, we derive controllability criteria for control affine systems on SE(1,1). Secondly, we consider quadratic Hamilton-Poisson systems on the (minus) Lie-Poisson space se(1,1)*. These systems are classified up to an affine isomorphism. Six normal forms are identified for the homogeneous case, whereas sixteen representatives (including several infinite families) are obtained for the inhomogeneous systems. Thereafter we consider the stability and integration of the normal forms obtained. For all homogeneous systems, and a subclass of inhomogeneous systems, we perform a complete stability analysis and derive explicit expressions for all integral curves. (The extremal controls of a large class of optimal control problems on SE(1,1) are linearly related to these integral curves.) Lastly, we discuss the Riemannian and sub-Riemannian problems. The (left-invariant) Riemannian and sub-Riemannian structures on SE(1,1) are classified, up to isometric group automorphisms and scaling. Explicit expressions for the geodesics of the normalised structures are found.
- Full Text:
- Date Issued: 2014
- Authors: Barrett, Dennis Ian
- Date: 2014
- Subjects: Automorphisms , Symmetry (Mathematics) , Lyapunov stability , Geometry, Riemannian , Geometry, Affine , Elliptic functions
- Language: English
- Type: text , Thesis , Masters , MSc
- Identifier: http://hdl.handle.net/10962/64805 , vital:28605
- Description: In this thesis we consider invariant control systems and Hamilton-Poisson systems on the three dimensional semi-Euclidean group SE(1,1). We first classify the left-invariant control affine systems (under detached feedback equivalence). We provide a complete list of normal forms, as well as classifying conditions. As a corollary to this classification, we derive controllability criteria for control affine systems on SE(1,1). Secondly, we consider quadratic Hamilton-Poisson systems on the (minus) Lie-Poisson space se(1,1)*. These systems are classified up to an affine isomorphism. Six normal forms are identified for the homogeneous case, whereas sixteen representatives (including several infinite families) are obtained for the inhomogeneous systems. Thereafter we consider the stability and integration of the normal forms obtained. For all homogeneous systems, and a subclass of inhomogeneous systems, we perform a complete stability analysis and derive explicit expressions for all integral curves. (The extremal controls of a large class of optimal control problems on SE(1,1) are linearly related to these integral curves.) Lastly, we discuss the Riemannian and sub-Riemannian problems. The (left-invariant) Riemannian and sub-Riemannian structures on SE(1,1) are classified, up to isometric group automorphisms and scaling. Explicit expressions for the geodesics of the normalised structures are found.
- Full Text:
- Date Issued: 2014
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
A study of a class of invariant optimal control problems on the Euclidean group SE(2)
- Authors: Adams, Ross Montague
- Date: 2011
- Subjects: Matrix groups Lie groups Extremal problems (Mathematics) Maximum principles (Mathematics) Hamilton-Jacobi equations Lyapunov stability
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5420 , http://hdl.handle.net/10962/d1006060
- Description: The aim of this thesis is to study a class of left-invariant optimal control problems on the matrix Lie group SE(2). We classify, under detached feedback equivalence, all controllable (left-invariant) control affine systems on SE(2). This result produces six types of control affine systems on SE(2). Hence, we study six associated left-invariant optimal control problems on SE(2). A left-invariant optimal control problem consists of minimizing a cost functional over the trajectory-control pairs of a left-invariant control system subject to appropriate boundary conditions. Each control problem is lifted from SE(2) to T*SE(2) ≅ SE(2) x se (2)*and then reduced to a problem on se (2)*. The maximum principle is used to obtain the optimal control and Hamiltonian corresponding to the normal extremals. Then we derive the (reduced) extremal equations on se (2)*. These equations are explicitly integrated by trigonometric and Jacobi elliptic functions. Finally, we fully classify, under Lyapunov stability, the equilibrium states of the normal extremal equations for each of the six types under consideration.
- Full Text:
- Date Issued: 2011
- Authors: Adams, Ross Montague
- Date: 2011
- Subjects: Matrix groups Lie groups Extremal problems (Mathematics) Maximum principles (Mathematics) Hamilton-Jacobi equations Lyapunov stability
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5420 , http://hdl.handle.net/10962/d1006060
- Description: The aim of this thesis is to study a class of left-invariant optimal control problems on the matrix Lie group SE(2). We classify, under detached feedback equivalence, all controllable (left-invariant) control affine systems on SE(2). This result produces six types of control affine systems on SE(2). Hence, we study six associated left-invariant optimal control problems on SE(2). A left-invariant optimal control problem consists of minimizing a cost functional over the trajectory-control pairs of a left-invariant control system subject to appropriate boundary conditions. Each control problem is lifted from SE(2) to T*SE(2) ≅ SE(2) x se (2)*and then reduced to a problem on se (2)*. The maximum principle is used to obtain the optimal control and Hamiltonian corresponding to the normal extremals. Then we derive the (reduced) extremal equations on se (2)*. These equations are explicitly integrated by trigonometric and Jacobi elliptic functions. Finally, we fully classify, under Lyapunov stability, the equilibrium states of the normal extremal equations for each of the six types under consideration.
- Full Text:
- Date Issued: 2011
Some aspects of the construction and implementation of error-correcting linear codes
- Authors: Booth, Geoffrey L
- Date: 1978
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:20967 , http://hdl.handle.net/10962/5718
- Description: From Conclusion: The study of error-correcting codes is now approximately 25 years old. The first known publication on the subject was in 1949 by M. Golay, who later did much research into the subject of perfect codes. It has been recently established that all the perfect codes are known. R.W. Hamming presented his perfect single-error correcting codes in 1950, in ~n article in the Bell System Technical Journal. These codes turned out to be a special case of the powerful Bose-Chaudhuri codes which were discovered around 1960. Various work has been done on the theory of minimal redundancy of codes for a given error-correcting performance, by Plotkin, Gilbert, Varshamov and others, between 1950 and 1960. The binary BCH codes were found to be so close to the theoretical bounds that, to date, no better codes have been discovered. Although the BCH codes are extremely efficient in terms of ratio of information to check digits, they are not easily, decoded with a minimal amount of apparatus. Petersen in 1961 described an algorithm for d e coding BCH codes, but this was cumbersome compared with the majority-logic methods of Massey and others. Thus the search began for codes which are easily decoded with comparatively simple apparatus. The finite geometry codes which were described by Rudolph in a 1964 thesis were examples of codes which are easily decoded 58 by a small number of steps of majority logic. The simplicial codes of Saltzer are even better in this respect, since they can be decoded by a single step of majority logic, but are rather inefficient . The applications of coding theory have changed over the years, as well. The first computers were huge circuits of relays, which were unreliable and prone to errors. Error correcting codes were required to minimise the possibility of incorrect results. As vacuum tubes and later transistorised circuits made computers more reliable, the need for sophisticated and powerful codes in the computer world diminished. Other used presented themselves however, for example the control systems of unmanned space craft. Because of the difficulty of sending and receiving messages in this case, · very powerful codes were required. Other uses were found in transmission lines and telephone exchanges. The codes considered in this dissertation have, for the most part, been block codes for use on the binary symmetric channel. There are, however, several other applications, such as codes for use on an erasure channel, where bits are corrupted so as to be unrecognizable, rather than changed. There are also codes for burst-error correction, where chennel noise is not randomly distributed, but occurs in "bursts" a few bits long. Certain cyclic codes are of application in these cases. The theory of error correcting codes has risen from virtual non-existence in 1950 to a major and sophisticated part of communication theory. Judging from the articles in journals, it promises to be the subject of a great deal of research for some years to come.
- Full Text:
- Date Issued: 1978
- Authors: Booth, Geoffrey L
- Date: 1978
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:20967 , http://hdl.handle.net/10962/5718
- Description: From Conclusion: The study of error-correcting codes is now approximately 25 years old. The first known publication on the subject was in 1949 by M. Golay, who later did much research into the subject of perfect codes. It has been recently established that all the perfect codes are known. R.W. Hamming presented his perfect single-error correcting codes in 1950, in ~n article in the Bell System Technical Journal. These codes turned out to be a special case of the powerful Bose-Chaudhuri codes which were discovered around 1960. Various work has been done on the theory of minimal redundancy of codes for a given error-correcting performance, by Plotkin, Gilbert, Varshamov and others, between 1950 and 1960. The binary BCH codes were found to be so close to the theoretical bounds that, to date, no better codes have been discovered. Although the BCH codes are extremely efficient in terms of ratio of information to check digits, they are not easily, decoded with a minimal amount of apparatus. Petersen in 1961 described an algorithm for d e coding BCH codes, but this was cumbersome compared with the majority-logic methods of Massey and others. Thus the search began for codes which are easily decoded with comparatively simple apparatus. The finite geometry codes which were described by Rudolph in a 1964 thesis were examples of codes which are easily decoded 58 by a small number of steps of majority logic. The simplicial codes of Saltzer are even better in this respect, since they can be decoded by a single step of majority logic, but are rather inefficient . The applications of coding theory have changed over the years, as well. The first computers were huge circuits of relays, which were unreliable and prone to errors. Error correcting codes were required to minimise the possibility of incorrect results. As vacuum tubes and later transistorised circuits made computers more reliable, the need for sophisticated and powerful codes in the computer world diminished. Other used presented themselves however, for example the control systems of unmanned space craft. Because of the difficulty of sending and receiving messages in this case, · very powerful codes were required. Other uses were found in transmission lines and telephone exchanges. The codes considered in this dissertation have, for the most part, been block codes for use on the binary symmetric channel. There are, however, several other applications, such as codes for use on an erasure channel, where bits are corrupted so as to be unrecognizable, rather than changed. There are also codes for burst-error correction, where chennel noise is not randomly distributed, but occurs in "bursts" a few bits long. Certain cyclic codes are of application in these cases. The theory of error correcting codes has risen from virtual non-existence in 1950 to a major and sophisticated part of communication theory. Judging from the articles in journals, it promises to be the subject of a great deal of research for some years to come.
- Full Text:
- Date Issued: 1978
Some aspects of the theory, application, and computation of generalised inverses of matrices
- Authors: Cretchley, Partricia C
- Date: 1977
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: http://hdl.handle.net/10962/6212 , vital:21063
- Description: The idea of generalising the classical notion of the inverse of a non-singular matrix arose as far back as in 1920, but it was not until the late fifties that the development of the theory gained any impetus. Since then , as is the case in the development of many new concepts , work done in parallel in various parts of the world has resulted in a great deal of untidiness in the literature : confusion over terminology , and even duplication of theory. More recently, however, some attempts have been made to bring together people active in the field of generalised inverses, in order to reach consensus on some aspects of definition and terminology, and to publish more general works on the subject. Towards this purpose, a symposium on the theory and application of generalised inverses of matrices was held in Lubbock, Texas, and its proceedings published in 1968 (see [25] ). A few other works of this nature (see [4], (19a] ) have appeared , but the bulk of the literature still comprises numerous diverse papers offering further ideas on the theoretical properties which these matrices have , and drawing attention to their application in areas of statistics , numerical analysis , filtering , modern control and estimation theory, pattern recognition and many others. This essay offers a look at generalised inverses in the following way: firstly a broad basis and background is established in the first three chapters to provide greater understanding of the motivation for the remaining chapters, where the approach then changes to become far more detailed. Within this general framework, Chapter 1 offers a brief glimpse of the history and development of work in the field. In Chapter 2 some of the most significant properties of these inverses are described, while in Chapters 3 and 4 and 5 attention is given to interesting and remarkable computational algorithms relating to generalised inverses (some well suited to machine processing). The material of Chapters 4 and 5 is largely due to Decell, Stallings and Boullion, and Tanabe, in [6], [24] and [27], respectively, while the source of material for the first three chapters is the literature generally, with Penrose's two papers providing a rough framework for Chapters 1 and 2 (see [17]).
- Full Text:
- Date Issued: 1977
- Authors: Cretchley, Partricia C
- Date: 1977
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: http://hdl.handle.net/10962/6212 , vital:21063
- Description: The idea of generalising the classical notion of the inverse of a non-singular matrix arose as far back as in 1920, but it was not until the late fifties that the development of the theory gained any impetus. Since then , as is the case in the development of many new concepts , work done in parallel in various parts of the world has resulted in a great deal of untidiness in the literature : confusion over terminology , and even duplication of theory. More recently, however, some attempts have been made to bring together people active in the field of generalised inverses, in order to reach consensus on some aspects of definition and terminology, and to publish more general works on the subject. Towards this purpose, a symposium on the theory and application of generalised inverses of matrices was held in Lubbock, Texas, and its proceedings published in 1968 (see [25] ). A few other works of this nature (see [4], (19a] ) have appeared , but the bulk of the literature still comprises numerous diverse papers offering further ideas on the theoretical properties which these matrices have , and drawing attention to their application in areas of statistics , numerical analysis , filtering , modern control and estimation theory, pattern recognition and many others. This essay offers a look at generalised inverses in the following way: firstly a broad basis and background is established in the first three chapters to provide greater understanding of the motivation for the remaining chapters, where the approach then changes to become far more detailed. Within this general framework, Chapter 1 offers a brief glimpse of the history and development of work in the field. In Chapter 2 some of the most significant properties of these inverses are described, while in Chapters 3 and 4 and 5 attention is given to interesting and remarkable computational algorithms relating to generalised inverses (some well suited to machine processing). The material of Chapters 4 and 5 is largely due to Decell, Stallings and Boullion, and Tanabe, in [6], [24] and [27], respectively, while the source of material for the first three chapters is the literature generally, with Penrose's two papers providing a rough framework for Chapters 1 and 2 (see [17]).
- Full Text:
- Date Issued: 1977
The Hamilton-Jacobi theory in general relativity theory and certain Petrov type D metrics
- Authors: Matravers, David Richard
- Date: 1973
- Subjects: Hamilton-Jacobi equations , General relativity (Physics) , Generalized spaces
- Language: English
- Type: Thesis , Doctoral , PhD
- Identifier: vital:5422 , http://hdl.handle.net/10962/d1007551 , Hamilton-Jacobi equations , General relativity (Physics) , Generalized spaces
- Description: Introduction: The discovery of new solutions to Einstein's field equations has long been a problem in General Relativity. However due to new techniques of Newman and Penrose [1], Carter [2] and others there has been a considerable proliferation of new solutions in recent times. Consequently a new problem has arisen. How are we to interpret the new solutions physically? The tools available, despite a spate of papers in the past fifteen years, remain inadequate although often sophisticated. Any attempts at physical interpretations of metrics are beset with difficulties. There is always the possibility that two entirely different physical pictures will emerge. For example a direct approach would be to attempt an "infilling" of the metric, that is, an extension of the metric into the region occupied by the gravitating matter. However even for the Kerr [1] metric the infilling is by no means unique, in fact a most natural "infilling" turns out to be unphysical (Israel [1]). Yet few people would doubt the physical significance of the Kerr metric. Viewed in this light our attempt to discuss, among other things, the physical interpretation of type D metrics is slightly ambitious. However the problems with regard to this type of metric are not as formidable as for most of the other metrics, since we have been able to integrate the geodesic equations. Nevertheless it is still not possible to produce complete answers to all the questions posed. After a chapter on Mathematical preliminaries the study divides naturally into four sections. We start with an outline of the Hamilton-Jacobi theory of Rund [1] and then go on to show how this theory can be applied to the Carter [2] metrics. In the process we lay a foundation in the calculus of variations for Carter's work. This leads us to the construction of Killing tensors for all but one of the Kinnersley [1] type D vacuum metrics and the Cartei [2] metrics which are not necessarily vacuum metrics. The geodesic equations, for these metrics, are integrated using the Hamilton-Jacobi procedure. The remaining chapters are devoted to the Kinnersley [1] type D vacuum metrics. We omit his class I metrics since these are the Schwarzschild metrics, and have been studied in detail before. Chapter three is devoted to a general study of his class II a metric, a generalisation of the Kerr [1] and NUT (Newman, Tamburino and Unti [1]) metrics. We integrate the geodesic equations and discuss certain general properties: the question of geodesic completeness, the asymptotic properties, and the existence of Killing horizons. Chapter four is concerned with the interpretation of the new parameter 'l', that arises in the class II a and NUT metrics. This parameter was interpreted by Demianski and Newman [1] as a magnetic monopole of mass. Our work centers on the possibility of obtaining observable effects from the presence of 'l'. We have been able to show that its presence is observable, at least in principle, from a study of the motion of particles in the field. In the first place, if l is comparable to the mass of the gravitating system, a comparatively large perihelion shift is to be expected. The possibility of anomalous behaviour in the orbits of test particles, quite unlike anything that occurs in a Newtonian or Schwarzschild field, also arises. In the fifth chapter the Kinnersley class IV metrics are considered. These metrics, which in their simplest form have been known for some time, present serious problems and no interpretations have been suggested. Our discussion is essentially exploratory and the information that does emerge takes the form of suggestions rather than conclusions. Intrinsically the metrics give the impression that interesting results should be obtainable since they are asymptotically flat in certain directions. However the case that we have dealt with does not appear to represent a radiation metric.
- Full Text:
- Date Issued: 1973
- Authors: Matravers, David Richard
- Date: 1973
- Subjects: Hamilton-Jacobi equations , General relativity (Physics) , Generalized spaces
- Language: English
- Type: Thesis , Doctoral , PhD
- Identifier: vital:5422 , http://hdl.handle.net/10962/d1007551 , Hamilton-Jacobi equations , General relativity (Physics) , Generalized spaces
- Description: Introduction: The discovery of new solutions to Einstein's field equations has long been a problem in General Relativity. However due to new techniques of Newman and Penrose [1], Carter [2] and others there has been a considerable proliferation of new solutions in recent times. Consequently a new problem has arisen. How are we to interpret the new solutions physically? The tools available, despite a spate of papers in the past fifteen years, remain inadequate although often sophisticated. Any attempts at physical interpretations of metrics are beset with difficulties. There is always the possibility that two entirely different physical pictures will emerge. For example a direct approach would be to attempt an "infilling" of the metric, that is, an extension of the metric into the region occupied by the gravitating matter. However even for the Kerr [1] metric the infilling is by no means unique, in fact a most natural "infilling" turns out to be unphysical (Israel [1]). Yet few people would doubt the physical significance of the Kerr metric. Viewed in this light our attempt to discuss, among other things, the physical interpretation of type D metrics is slightly ambitious. However the problems with regard to this type of metric are not as formidable as for most of the other metrics, since we have been able to integrate the geodesic equations. Nevertheless it is still not possible to produce complete answers to all the questions posed. After a chapter on Mathematical preliminaries the study divides naturally into four sections. We start with an outline of the Hamilton-Jacobi theory of Rund [1] and then go on to show how this theory can be applied to the Carter [2] metrics. In the process we lay a foundation in the calculus of variations for Carter's work. This leads us to the construction of Killing tensors for all but one of the Kinnersley [1] type D vacuum metrics and the Cartei [2] metrics which are not necessarily vacuum metrics. The geodesic equations, for these metrics, are integrated using the Hamilton-Jacobi procedure. The remaining chapters are devoted to the Kinnersley [1] type D vacuum metrics. We omit his class I metrics since these are the Schwarzschild metrics, and have been studied in detail before. Chapter three is devoted to a general study of his class II a metric, a generalisation of the Kerr [1] and NUT (Newman, Tamburino and Unti [1]) metrics. We integrate the geodesic equations and discuss certain general properties: the question of geodesic completeness, the asymptotic properties, and the existence of Killing horizons. Chapter four is concerned with the interpretation of the new parameter 'l', that arises in the class II a and NUT metrics. This parameter was interpreted by Demianski and Newman [1] as a magnetic monopole of mass. Our work centers on the possibility of obtaining observable effects from the presence of 'l'. We have been able to show that its presence is observable, at least in principle, from a study of the motion of particles in the field. In the first place, if l is comparable to the mass of the gravitating system, a comparatively large perihelion shift is to be expected. The possibility of anomalous behaviour in the orbits of test particles, quite unlike anything that occurs in a Newtonian or Schwarzschild field, also arises. In the fifth chapter the Kinnersley class IV metrics are considered. These metrics, which in their simplest form have been known for some time, present serious problems and no interpretations have been suggested. Our discussion is essentially exploratory and the information that does emerge takes the form of suggestions rather than conclusions. Intrinsically the metrics give the impression that interesting results should be obtainable since they are asymptotically flat in certain directions. However the case that we have dealt with does not appear to represent a radiation metric.
- Full Text:
- Date Issued: 1973
A probability operator
- Authors: Sinclair, Allan M
- Date: 1965
- Subjects: Mathematics , Probabilities
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5423 , http://hdl.handle.net/10962/d1007702 , Mathematics , Probabilities
- Description: From Introduction: In probability theory it is often convenient to represent laws by characteristic functions, these being particularly suited to classical analysis. Trotter has suggest ted that probability laws can also be represented by probability operators. These operators are easily handled since they are continuous, and hence bounded, positive linear operators on a normed linear space. This representation arises because distribution functions and their complete convergence correspond to probability operators and their complete convergence. Hence the relations between distribution functions and probability operators will be discussed before the introduction of probability laws.
- Full Text:
- Date Issued: 1965
- Authors: Sinclair, Allan M
- Date: 1965
- Subjects: Mathematics , Probabilities
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5423 , http://hdl.handle.net/10962/d1007702 , Mathematics , Probabilities
- Description: From Introduction: In probability theory it is often convenient to represent laws by characteristic functions, these being particularly suited to classical analysis. Trotter has suggest ted that probability laws can also be represented by probability operators. These operators are easily handled since they are continuous, and hence bounded, positive linear operators on a normed linear space. This representation arises because distribution functions and their complete convergence correspond to probability operators and their complete convergence. Hence the relations between distribution functions and probability operators will be discussed before the introduction of probability laws.
- Full Text:
- Date Issued: 1965
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