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
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
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
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
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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