Keychains and preferential fuzzy sets with applications
- Authors: Mahlasela, Zuko
- Date: 2024-04-05
- Subjects: Uncatalogued
- Language: English
- Type: Academic theses , Doctoral theses , text
- Identifier: http://hdl.handle.net/10962/435933 , vital:73213 , DOI 10.21504/10962/435935
- Description: In this thesis, we study the preferentiality behaviour of choices under uncertainties using keychains, where a keychain is defined as an (n+ 1)-tuple of non-increasing real numbers in the unit interval, I= [0, 1]. We look at the representations of uncertainties or sets defined by vague properties using the idea of keychains, pins and pinned flags. We then apply the ideas of preferential fuzzy sets to voting patterns, economics and decision making. For voting patterns, we simulate mock trials to investigate the behaviours of choices of different individuals, the outcomes of such voting and make specific conclusions about voting strategies. It can be argued that preferentiality in voting can enhance the democratic processes in national elections. This thesis contains various representations of keychains such as binary digits, weight order, lattice and simplex representations. Another useful aspect of keychains and preferential fuzzy sets is to study the outcomes of decision making linking it to the study of keychains and finite fuzzy sets. We envisage that this study will throw light on computational aspects of any countable situations. , Thesis (PhD) -- Faculty of Science, Mathematics, 2024
- Full Text:
- Date Issued: 2024-04-05
- Authors: Mahlasela, Zuko
- Date: 2024-04-05
- Subjects: Uncatalogued
- Language: English
- Type: Academic theses , Doctoral theses , text
- Identifier: http://hdl.handle.net/10962/435933 , vital:73213 , DOI 10.21504/10962/435935
- Description: In this thesis, we study the preferentiality behaviour of choices under uncertainties using keychains, where a keychain is defined as an (n+ 1)-tuple of non-increasing real numbers in the unit interval, I= [0, 1]. We look at the representations of uncertainties or sets defined by vague properties using the idea of keychains, pins and pinned flags. We then apply the ideas of preferential fuzzy sets to voting patterns, economics and decision making. For voting patterns, we simulate mock trials to investigate the behaviours of choices of different individuals, the outcomes of such voting and make specific conclusions about voting strategies. It can be argued that preferentiality in voting can enhance the democratic processes in national elections. This thesis contains various representations of keychains such as binary digits, weight order, lattice and simplex representations. Another useful aspect of keychains and preferential fuzzy sets is to study the outcomes of decision making linking it to the study of keychains and finite fuzzy sets. We envisage that this study will throw light on computational aspects of any countable situations. , Thesis (PhD) -- Faculty of Science, Mathematics, 2024
- Full Text:
- Date Issued: 2024-04-05
Performance evaluation of baseline-dependent window functions with several weighing functions
- Authors: Vanqa, Kamvulethu
- Date: 2024-04-04
- Subjects: Uncatalogued
- Language: English
- Type: Academic theses , Master's theses , text
- Identifier: http://hdl.handle.net/10962/435850 , vital:73206
- Description: Radio interferometric data volume is exponentially increasing with the potential to cause slow processing and data storage issues for radio observations recorded at high time and frequency resolutions. This necessitates that a sort of data compression is imposed. The conventional method to compress the data is averaging across time and frequency. However, this results in amplitude loss and source distortion at the edges of the field of view. To reduce amplitude loss and source distortion, baseline-dependent window functions (BDWFs) are proposed in theliterature. BDWFs are visibility data compression methods using window functions to retainthe signals within a field of interest (FoI) and to suppress signals outside this FoI. However,BDWFs are used with window functions as discussed in the signal processing field without any optimisation. This thesis evaluates the performance of BDWFs and then proposes to use machine learning with gradient descent to optimize the window functions employed in BDWFs. Results show that the convergence of the objective function is limited due to the band-limited nature of the window functions in the Fourier space. BDWFs performance is also investigated and discussed using several weighting schemes. Results show that there exists an optimal parameter tuning (not necessarily unique) that suggests an optimal combination of BDWFs and density sampling. With this, ∼ 4 % smearing is observed within the FoI, and ∼ 80 % source suppression is achieved outside the FoI using the MeerKAT telescope at 1.4 GHz, sampled at 1 s and 184.3 kHz then averaged with BDWFs to achieve a compression factor of 4 in time and 3 in frequency. , Thesis (MA) -- Faculty of Science, Mathematics, 2024
- Full Text:
- Date Issued: 2024-04-04
- Authors: Vanqa, Kamvulethu
- Date: 2024-04-04
- Subjects: Uncatalogued
- Language: English
- Type: Academic theses , Master's theses , text
- Identifier: http://hdl.handle.net/10962/435850 , vital:73206
- Description: Radio interferometric data volume is exponentially increasing with the potential to cause slow processing and data storage issues for radio observations recorded at high time and frequency resolutions. This necessitates that a sort of data compression is imposed. The conventional method to compress the data is averaging across time and frequency. However, this results in amplitude loss and source distortion at the edges of the field of view. To reduce amplitude loss and source distortion, baseline-dependent window functions (BDWFs) are proposed in theliterature. BDWFs are visibility data compression methods using window functions to retainthe signals within a field of interest (FoI) and to suppress signals outside this FoI. However,BDWFs are used with window functions as discussed in the signal processing field without any optimisation. This thesis evaluates the performance of BDWFs and then proposes to use machine learning with gradient descent to optimize the window functions employed in BDWFs. Results show that the convergence of the objective function is limited due to the band-limited nature of the window functions in the Fourier space. BDWFs performance is also investigated and discussed using several weighting schemes. Results show that there exists an optimal parameter tuning (not necessarily unique) that suggests an optimal combination of BDWFs and density sampling. With this, ∼ 4 % smearing is observed within the FoI, and ∼ 80 % source suppression is achieved outside the FoI using the MeerKAT telescope at 1.4 GHz, sampled at 1 s and 184.3 kHz then averaged with BDWFs to achieve a compression factor of 4 in time and 3 in frequency. , Thesis (MA) -- Faculty of Science, Mathematics, 2024
- Full Text:
- Date Issued: 2024-04-04
Towards an artificial intelligence-based agent for characterising the organisation of primes
- Authors: Oyetunji, Nicole Armlade
- Date: 2024-04-04
- Subjects: Uncatalogued
- Language: English
- Type: Academic theses , Master's theses , text
- Identifier: http://hdl.handle.net/10962/435389 , vital:73153
- Description: Machine learning has experienced significant growth in recent decades, driven by advancements in computational power and data storage. One of the applications of machine learning is in the field of number theory. Prime numbers hold significant importance in mathematics and its applications, for example in cryptography, owing to their distinct properties. Therefore, it is crucial to efficiently obtain the complete list of primes below a given threshold, with low relatively computational cost. This study extensively explores a deterministic scheme, proposed by Hawing and Okouma (2016), that is centered around Consecutive Composite Odd Numbers, showing the link between these numbers and prime numbers by examining their internal structure. The main objective of this dissertation is to develop two main artificial intelligence agents capable of learning and recognizing patterns within a list of consecutive composite odd numbers. To achieve this, the mathematical foundations of the deterministic scheme are used to generate a dataset of consecutive composite odd numbers. This dataset is further transformed into a dataset of differences to simplify the prediction problem. A literature review is conducted which encompasses research from the domains of machine learning and deep learning. Two main machine learning algorithms are implemented along with their variations, Long Short-Term Memory Networks and Error Correction Neural Networks. These models are trained independently on two separate but related datasets, the dataset of consecutive composite odd numbers and the dataset of differences between those numbers. The evaluation of these models includes relevant metrics, for example, Root Mean Square Error, Mean Absolute Percentage Error, Theil U coefficient, and Directional Accuracy. Through a comparative analysis, the study identifies the top-performing 3 models, with a particular emphasis on accuracy and computational efficiency. The results indicate that the LSTM model, when trained on difference data and coupled with exponential smoothing, displays superior performance as the most accurate model overall. It achieves a RMSE of 0.08, which significantly outperforms the dataset’s standard deviation of 0.42. This model exceeds the performance of basic estimator models, implying that a data-driven approach utilizing machine learning techniques can provide valuable insights in the field of number theory. The second best model, the ECNN trained on difference data combined with exponential smoothing, achieves an RMSE of 0.28. However, it is worth mentioning that this model is the most computationally efficient, being 32 times faster than the LSTM model. , Thesis (MSc) -- Faculty of Science, Mathematics, 2024
- Full Text:
- Date Issued: 2024-04-04
- Authors: Oyetunji, Nicole Armlade
- Date: 2024-04-04
- Subjects: Uncatalogued
- Language: English
- Type: Academic theses , Master's theses , text
- Identifier: http://hdl.handle.net/10962/435389 , vital:73153
- Description: Machine learning has experienced significant growth in recent decades, driven by advancements in computational power and data storage. One of the applications of machine learning is in the field of number theory. Prime numbers hold significant importance in mathematics and its applications, for example in cryptography, owing to their distinct properties. Therefore, it is crucial to efficiently obtain the complete list of primes below a given threshold, with low relatively computational cost. This study extensively explores a deterministic scheme, proposed by Hawing and Okouma (2016), that is centered around Consecutive Composite Odd Numbers, showing the link between these numbers and prime numbers by examining their internal structure. The main objective of this dissertation is to develop two main artificial intelligence agents capable of learning and recognizing patterns within a list of consecutive composite odd numbers. To achieve this, the mathematical foundations of the deterministic scheme are used to generate a dataset of consecutive composite odd numbers. This dataset is further transformed into a dataset of differences to simplify the prediction problem. A literature review is conducted which encompasses research from the domains of machine learning and deep learning. Two main machine learning algorithms are implemented along with their variations, Long Short-Term Memory Networks and Error Correction Neural Networks. These models are trained independently on two separate but related datasets, the dataset of consecutive composite odd numbers and the dataset of differences between those numbers. The evaluation of these models includes relevant metrics, for example, Root Mean Square Error, Mean Absolute Percentage Error, Theil U coefficient, and Directional Accuracy. Through a comparative analysis, the study identifies the top-performing 3 models, with a particular emphasis on accuracy and computational efficiency. The results indicate that the LSTM model, when trained on difference data and coupled with exponential smoothing, displays superior performance as the most accurate model overall. It achieves a RMSE of 0.08, which significantly outperforms the dataset’s standard deviation of 0.42. This model exceeds the performance of basic estimator models, implying that a data-driven approach utilizing machine learning techniques can provide valuable insights in the field of number theory. The second best model, the ECNN trained on difference data combined with exponential smoothing, achieves an RMSE of 0.28. However, it is worth mentioning that this model is the most computationally efficient, being 32 times faster than the LSTM model. , Thesis (MSc) -- Faculty of Science, Mathematics, 2024
- Full Text:
- Date Issued: 2024-04-04
Selected medicinal plants leaves identification: a computer vision approach
- Authors: Deyi, Avuya
- Date: 2023-10-13
- Subjects: Deep learning (Machine learning) , Machine learning , Convolutional neural network , Computer vision in medicine , Medicinal plants
- Language: English
- Type: Academic theses , Master's theses , text
- Identifier: http://hdl.handle.net/10962/424552 , vital:72163
- Description: Identifying and classifying medicinal plants are valuable and essential skills during drug manufacturing because several active pharmaceutical ingredients (API) are sourced from medicinal plants. For many years, identifying and classifying medicinal plants have been exclusively done by experts in the domain, such as botanists, and herbarium curators. Recently, powerful computer vision technologies, using machine learning and deep convolutional neural networks, have been developed for classifying or identifying objects on images. A convolutional neural network is a deep learning architecture that outperforms previous advanced approaches in image classification and object detection based on its efficient features extraction on images. In this thesis, we investigate different convolutional neural networks and machine learning algorithms for identifying and classifying leaves of three species of the genus Brachylaena. The three species considered are Brachylaena discolor, Brachylaena ilicifolia and Brachylaena elliptica. All three species are used medicinally by people in South Africa to treat diseases like diabetes. From 1259 labelled images of those plants species (at least 400 for each species) split into training, evaluation and test sets, we trained and evaluated different deep convolutional neural networks and machine learning models. The VGG model achieved the best results with 98.26% accuracy from cross-validation. , Thesis (MSc) -- Faculty of Science, Mathematics, 2023
- Full Text:
- Date Issued: 2023-10-13
- Authors: Deyi, Avuya
- Date: 2023-10-13
- Subjects: Deep learning (Machine learning) , Machine learning , Convolutional neural network , Computer vision in medicine , Medicinal plants
- Language: English
- Type: Academic theses , Master's theses , text
- Identifier: http://hdl.handle.net/10962/424552 , vital:72163
- Description: Identifying and classifying medicinal plants are valuable and essential skills during drug manufacturing because several active pharmaceutical ingredients (API) are sourced from medicinal plants. For many years, identifying and classifying medicinal plants have been exclusively done by experts in the domain, such as botanists, and herbarium curators. Recently, powerful computer vision technologies, using machine learning and deep convolutional neural networks, have been developed for classifying or identifying objects on images. A convolutional neural network is a deep learning architecture that outperforms previous advanced approaches in image classification and object detection based on its efficient features extraction on images. In this thesis, we investigate different convolutional neural networks and machine learning algorithms for identifying and classifying leaves of three species of the genus Brachylaena. The three species considered are Brachylaena discolor, Brachylaena ilicifolia and Brachylaena elliptica. All three species are used medicinally by people in South Africa to treat diseases like diabetes. From 1259 labelled images of those plants species (at least 400 for each species) split into training, evaluation and test sets, we trained and evaluated different deep convolutional neural networks and machine learning models. The VGG model achieved the best results with 98.26% accuracy from cross-validation. , Thesis (MSc) -- Faculty of Science, Mathematics, 2023
- Full Text:
- Date Issued: 2023-10-13
Wildlife-vehicle collisions mitigation measures using road ecological data and deep learning
- Authors: Nandutu, Irene
- Date: 2023-10-13
- Subjects: Uncatalogued
- Language: English
- Type: Academic theses , Doctoral theses , text
- Identifier: http://hdl.handle.net/10962/431907 , vital:72814
- Description: Access restricted. Expected release in 2025. , Thesis (PhD) -- Faculty of Science, Mathematics, 2023
- Full Text:
- Date Issued: 2023-10-13
- Authors: Nandutu, Irene
- Date: 2023-10-13
- Subjects: Uncatalogued
- Language: English
- Type: Academic theses , Doctoral theses , text
- Identifier: http://hdl.handle.net/10962/431907 , vital:72814
- Description: Access restricted. Expected release in 2025. , Thesis (PhD) -- Faculty of Science, Mathematics, 2023
- Full Text:
- Date Issued: 2023-10-13
On the Wiener index of bicyclic graphs and graphs with fixed segment sequence
- Authors: Xhanti, Sinoxolo
- Date: 2021-10-29
- Subjects: Graph theory , Chemistry Mathematics , Chemistry Graphic methods , Wiener index , Bicyclic graphs , Fixed segment sequence , Degree sequence , Circumference , Core
- Language: English
- Type: Master's theses , text
- Identifier: http://hdl.handle.net/10962/190700 , vital:45019
- Description: Wiener index is defined as the sum of the distances between all unordered pairs of vertices in a graph. The study of the Wiener index is motivated by its application in chemistry. This thesis focuses on finding extremal bicyclic graphs relative to Wiener index under various conditions such as fixed circumference (length of the longest cycle) or fixed size of the core (maximal subgraph with no degree less than 2). A segment of a graph G is either a path whose end vertices have degree 1 or at least 3 in G and all the internal vertices have degree 2 in G, or a cycle where all the vertices have degree 2 in G except possibly one. The lengths of all the segments of G form it segment sequence. We also discuss extremal graphs with given segment sequence. , Thesis (MSc) -- Faculty of Science, Mathematics, 2021
- Full Text:
- Date Issued: 2021-10-29
- Authors: Xhanti, Sinoxolo
- Date: 2021-10-29
- Subjects: Graph theory , Chemistry Mathematics , Chemistry Graphic methods , Wiener index , Bicyclic graphs , Fixed segment sequence , Degree sequence , Circumference , Core
- Language: English
- Type: Master's theses , text
- Identifier: http://hdl.handle.net/10962/190700 , vital:45019
- Description: Wiener index is defined as the sum of the distances between all unordered pairs of vertices in a graph. The study of the Wiener index is motivated by its application in chemistry. This thesis focuses on finding extremal bicyclic graphs relative to Wiener index under various conditions such as fixed circumference (length of the longest cycle) or fixed size of the core (maximal subgraph with no degree less than 2). A segment of a graph G is either a path whose end vertices have degree 1 or at least 3 in G and all the internal vertices have degree 2 in G, or a cycle where all the vertices have degree 2 in G except possibly one. The lengths of all the segments of G form it segment sequence. We also discuss extremal graphs with given segment sequence. , Thesis (MSc) -- Faculty of Science, Mathematics, 2021
- Full Text:
- Date Issued: 2021-10-29
Numerical evolution of plane gravitational waves
- Authors: Hakata, Jonathan
- Date: 2021-10
- Subjects: Gravitational waves , Space and time , Einstein field equations , de Sitter metric , Perturbed spacetime
- Language: English
- Type: Master's theses , text
- Identifier: http://hdl.handle.net/10962/190248 , vital:44977
- Description: Unlike electromagnetic waves, gravitational waves self interact. This interaction is non-linear and can have very interesting properties which effect the curvature of space-time. A gravitational plane wave collider, implemented in the Python package COFFEE [20] that been developed in recent years by the Otago relativity group and implements the method of lines, can be reliably used to study this self-interaction. This was shown to work well numerically as profounded by Frauendiener, Stevens and Whale in 2014 [24]. For this reason, COFFEE will be used to study these gravitational wave propagations and subsequently collisions. The Einstein field equations are formulated as a well-posed initial boundary value problem (IBVP) in the Friedrich-Nagy gauge [26] and due to the large class of boundary conditions admitted by this framework, a variety of investigations into the propagation of plane gravitational waves could be carried out. This study focuses on the propagation of plane gravitational waves in the de Sitter (dS) space-time, which is the maximally symmetric solution of the Einstein’s vacuum field equations with a positive cosmological constant λ. There is substantial cosmological evidence that our universe is asymptotically de Sitter, yet no work, analytical nor numerical, has been done on gravitational plane waves propagating on such a space-time, mainly due to the increased complexity from the non-vanishing λ. Firstly, it is found analytically that with an arbitrary cosmological constant λ and a non-vanishing energy momentum tensor, the constraints will propagate. This means that we still have a wellposed IBVP, which is nontrivial since the Friedrich-Nagy gauge has only been shown to lead to a wellposed IBVP without matter [26]. Using this system, we consider one ingoing wave propagating on said space-time in vacuum. The area of the ingoing wave profile is varied and inferences are made about the different phenomena that arise in the curvature of space-time during the evolution. It is found that there exists a critical value of the wave’s area, ac, whereby taking the area below this value the system asymptotes to its initial state, and above the system diverges, indicating the presence of a singularity. Furthermore, we define an expansion parameter H to measure how the gravitational waves influence the accelerated expansion, generalising (numerically) results of Tsamis and Woodard. , Thesis (MSc) -- Faculty of Science, Mathematics, 2021
- Full Text:
- Date Issued: 2021-10
- Authors: Hakata, Jonathan
- Date: 2021-10
- Subjects: Gravitational waves , Space and time , Einstein field equations , de Sitter metric , Perturbed spacetime
- Language: English
- Type: Master's theses , text
- Identifier: http://hdl.handle.net/10962/190248 , vital:44977
- Description: Unlike electromagnetic waves, gravitational waves self interact. This interaction is non-linear and can have very interesting properties which effect the curvature of space-time. A gravitational plane wave collider, implemented in the Python package COFFEE [20] that been developed in recent years by the Otago relativity group and implements the method of lines, can be reliably used to study this self-interaction. This was shown to work well numerically as profounded by Frauendiener, Stevens and Whale in 2014 [24]. For this reason, COFFEE will be used to study these gravitational wave propagations and subsequently collisions. The Einstein field equations are formulated as a well-posed initial boundary value problem (IBVP) in the Friedrich-Nagy gauge [26] and due to the large class of boundary conditions admitted by this framework, a variety of investigations into the propagation of plane gravitational waves could be carried out. This study focuses on the propagation of plane gravitational waves in the de Sitter (dS) space-time, which is the maximally symmetric solution of the Einstein’s vacuum field equations with a positive cosmological constant λ. There is substantial cosmological evidence that our universe is asymptotically de Sitter, yet no work, analytical nor numerical, has been done on gravitational plane waves propagating on such a space-time, mainly due to the increased complexity from the non-vanishing λ. Firstly, it is found analytically that with an arbitrary cosmological constant λ and a non-vanishing energy momentum tensor, the constraints will propagate. This means that we still have a wellposed IBVP, which is nontrivial since the Friedrich-Nagy gauge has only been shown to lead to a wellposed IBVP without matter [26]. Using this system, we consider one ingoing wave propagating on said space-time in vacuum. The area of the ingoing wave profile is varied and inferences are made about the different phenomena that arise in the curvature of space-time during the evolution. It is found that there exists a critical value of the wave’s area, ac, whereby taking the area below this value the system asymptotes to its initial state, and above the system diverges, indicating the presence of a singularity. Furthermore, we define an expansion parameter H to measure how the gravitational waves influence the accelerated expansion, generalising (numerically) results of Tsamis and Woodard. , Thesis (MSc) -- Faculty of Science, Mathematics, 2021
- Full Text:
- Date Issued: 2021-10
Left-invariant optimal control problems of the Engel group: classification, stability, and integration
- Authors: McLean, Catherine Eve
- Date: 2018
- Subjects: Uncatalogued
- Language: English
- Type: text , Thesis , Doctoral , PhD
- Identifier: http://hdl.handle.net/10962/62949 , vital:28323 , http://doi.org/10.21504/10962/62949
- Description: Expected release date-April 2020
- Full Text:
- Date Issued: 2018
- Authors: McLean, Catherine Eve
- Date: 2018
- Subjects: Uncatalogued
- Language: English
- Type: text , Thesis , Doctoral , PhD
- Identifier: http://hdl.handle.net/10962/62949 , vital:28323 , http://doi.org/10.21504/10962/62949
- Description: Expected release date-April 2020
- Full Text:
- Date Issued: 2018
A study of spherical solutions in chameleon scalar-tensor theories
- Authors: Mohapi, Neo
- Date: 2014
- Subjects: Scalar field theory , Equivalence principle (Physics) , General relativity (Physics) , Bosons , Dark energy (Astronomy) , Galactic dynamics
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5428 , http://hdl.handle.net/10962/d1013315
- Description: The equivalence principle has proven to be central to theories of gravity, with General Relativity being the simplest and most elegant theory to embody the principle. Most alternative theories of gravity struggle to satisfy the principle and still be distinct from GR. Extensions of cosmological and quantum theories question the irrefutably of the equivalence at every scale. The possibility of an equivalence principle violation at galactic scales would be an exciting prospect. In this thesis, we will carefully examine the equivalence principle through the study of chameleon scalar-tensor theories, this will include solutions for hypothetical stars known as boson stars. Such theories find varied application, especially in cosmology, where they model dark energy and inflation. The AWE hypothesis, is an instance of this. It is a nonuniversally coupled model in which violations of the equivalence principle on galactic scales may be apparent. We investigate spherically symmetric and static solutions within the framework of this theory. The constraints obtained from galactic rotation curves results in values of the couplings that show no significant violation of the equivalence principle or values consistent with a theory of dark energy
- Full Text:
- Date Issued: 2014
- Authors: Mohapi, Neo
- Date: 2014
- Subjects: Scalar field theory , Equivalence principle (Physics) , General relativity (Physics) , Bosons , Dark energy (Astronomy) , Galactic dynamics
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5428 , http://hdl.handle.net/10962/d1013315
- Description: The equivalence principle has proven to be central to theories of gravity, with General Relativity being the simplest and most elegant theory to embody the principle. Most alternative theories of gravity struggle to satisfy the principle and still be distinct from GR. Extensions of cosmological and quantum theories question the irrefutably of the equivalence at every scale. The possibility of an equivalence principle violation at galactic scales would be an exciting prospect. In this thesis, we will carefully examine the equivalence principle through the study of chameleon scalar-tensor theories, this will include solutions for hypothetical stars known as boson stars. Such theories find varied application, especially in cosmology, where they model dark energy and inflation. The AWE hypothesis, is an instance of this. It is a nonuniversally coupled model in which violations of the equivalence principle on galactic scales may be apparent. We investigate spherically symmetric and static solutions within the framework of this theory. The constraints obtained from galactic rotation curves results in values of the couplings that show no significant violation of the equivalence principle or values consistent with a theory of dark energy
- Full Text:
- Date Issued: 2014
Some general convergence theorems on fixed points
- Authors: Panicker, Rekha Manoj
- Date: 2014
- Subjects: Fixed point theory , Convergence , Coincidence theory (Mathematics)
- Language: English
- Type: Thesis , Doctoral , PhD
- Identifier: vital:5426 , http://hdl.handle.net/10962/d1013112
- Description: In this thesis, we first obtain coincidence and common fixed point theorems for a pair of generalized non-expansive type mappings in a normed space. Then we discuss two types of convergence theorems, namely, the convergence of Mann iteration procedures and the convergence and stability of fixed points. In addition, we discuss the viscosity approximations generated by (ψ ,ϕ)-weakly contractive mappings and a sequence of non-expansive mappings and then establish Browder and Halpern type convergence theorems on Banach spaces. With regard to iteration procedures, we obtain a result on the convergence of Mann iteration for generalized non-expansive type mappings in a Banach space which satisfies Opial's condition. And, in the case of stability of fixed points, we obtain a number of stability results for the sequence of (ψ,ϕ)- weakly contractive mappings and the sequence of their corresponding fixed points in metric and 2-metric spaces. We also present a generalization of Fraser and Nadler type stability theorems in 2-metric spaces involving a sequence of metrics.
- Full Text:
- Date Issued: 2014
- Authors: Panicker, Rekha Manoj
- Date: 2014
- Subjects: Fixed point theory , Convergence , Coincidence theory (Mathematics)
- Language: English
- Type: Thesis , Doctoral , PhD
- Identifier: vital:5426 , http://hdl.handle.net/10962/d1013112
- Description: In this thesis, we first obtain coincidence and common fixed point theorems for a pair of generalized non-expansive type mappings in a normed space. Then we discuss two types of convergence theorems, namely, the convergence of Mann iteration procedures and the convergence and stability of fixed points. In addition, we discuss the viscosity approximations generated by (ψ ,ϕ)-weakly contractive mappings and a sequence of non-expansive mappings and then establish Browder and Halpern type convergence theorems on Banach spaces. With regard to iteration procedures, we obtain a result on the convergence of Mann iteration for generalized non-expansive type mappings in a Banach space which satisfies Opial's condition. And, in the case of stability of fixed points, we obtain a number of stability results for the sequence of (ψ,ϕ)- weakly contractive mappings and the sequence of their corresponding fixed points in metric and 2-metric spaces. We also present a generalization of Fraser and Nadler type stability theorems in 2-metric spaces involving a sequence of metrics.
- Full Text:
- Date Issued: 2014
Fixed points of single-valued and multi-valued mappings with applications
- Authors: Stofile, Simfumene
- Date: 2013
- Subjects: Fixed point theory Mappings (Mathematics) Coincidence theory (Mathematics) Metric spaces Uniform spaces Set-valued maps
- Language: English
- Type: Thesis , Doctoral , PhD
- Identifier: vital:5397 , http://hdl.handle.net/10962/d1002960
- Description: The relationship between the convergence of a sequence of self mappings of a metric space and their fixed points, known as the stability (or continuity) of fixed points has been of continuing interest and widely studied in fixed point theory. In this thesis we study the stability of common fixed points in a Hausdorff uniform space whose uniformity is generated by a family of pseudometrics, by using some general notations of convergence. These results are then extended to 2-metric spaces due to S. Gähler. In addition, a well-known theorem of T. Suzuki that generalized the Banach Contraction Principle is also extended to 2-metric spaces and applied to obtain a coincidence theorem for a pair of mappings on an arbitrary set with values in a 2-metric space. Further, we prove the existence of coincidence and fixed points of Ćirić type weakly generalized contractions in metric spaces. Subsequently, the above result is utilized to discuss applications to the convergence of modified Mann and Ishikawa iterations in a convex metric space. Finally, we obtain coincidence, fixed and stationary point results for multi-valued and hybrid pairs of mappings on a metric space.
- Full Text:
- Date Issued: 2013
- Authors: Stofile, Simfumene
- Date: 2013
- Subjects: Fixed point theory Mappings (Mathematics) Coincidence theory (Mathematics) Metric spaces Uniform spaces Set-valued maps
- Language: English
- Type: Thesis , Doctoral , PhD
- Identifier: vital:5397 , http://hdl.handle.net/10962/d1002960
- Description: The relationship between the convergence of a sequence of self mappings of a metric space and their fixed points, known as the stability (or continuity) of fixed points has been of continuing interest and widely studied in fixed point theory. In this thesis we study the stability of common fixed points in a Hausdorff uniform space whose uniformity is generated by a family of pseudometrics, by using some general notations of convergence. These results are then extended to 2-metric spaces due to S. Gähler. In addition, a well-known theorem of T. Suzuki that generalized the Banach Contraction Principle is also extended to 2-metric spaces and applied to obtain a coincidence theorem for a pair of mappings on an arbitrary set with values in a 2-metric space. Further, we prove the existence of coincidence and fixed points of Ćirić type weakly generalized contractions in metric spaces. Subsequently, the above result is utilized to discuss applications to the convergence of modified Mann and Ishikawa iterations in a convex metric space. Finally, we obtain coincidence, fixed and stationary point results for multi-valued and hybrid pairs of mappings on a metric space.
- Full Text:
- Date Issued: 2013
Numerical relativity on cosmological past null cones
- Van der Walt, Petrus Johannes
- Authors: Van der Walt, Petrus Johannes
- Date: 2013
- Subjects: Cosmology Numerical calculations General relativity (Physics) -- Mathematics Einstein field equations Gravity Gravitational waves Horizon
- Language: English
- Type: Thesis , Doctoral , PhD
- Identifier: vital:5396 , http://hdl.handle.net/10962/d1002959
- Description: The observational approach to cosmology is the endeavour to reconstruct the geometry of the Universe using only data that is theoretically verifiable within the causal boundaries of a cosmological observer. Using this approach, it was shown in [36] that given ideal cosmological observations, the only essential assumption necessary to determine the geometry of the Universe is a theory of gravity. Assuming General Relativity, the full set of Einstein field equations (EFEs) can be used to reconstruct the geometry of the Universe using direct observations on the past null cone (PNC) as initial conditions. Observationally and theoretically this is a very ambitious task and therefore, current developments have been restricted to spherically symmetric dust models while only relaxing the usual assumption of homogeneity in the radial direction. These restricted models are important for the development of theoretical foundations and also useful as verification models since they avoid the circularity of verifying what has already been assumed. The work presented in this thesis is the development of such a model where numerical relativity (NR) is used to simulate the observable universe. Similar to the work of Ellis and co-workers [36], a reference frame based on the PNC is used. The reference frame used here, however, is based on that of the characteristic formalism of NR, which has developed for calculating the propagation of gravitational waves. This provides a formalism that is well established in NR, making the use of existing algorithms possible. The Bondi-Sachs coordinates of the characteristic formalism is, however, not suitable for calculations beyond the observer apparent horizon (AH) since the diameter distance used as a radial coordinate becomes multi-valued when the cosmological PNC reconverges in the history of a universe, smaller in the past. With this taken into consideration, the Bondi-Sachs characteristic formalism is implemented for cosmology and the problem approaching the AH is investigated. Further developments address the limitations approaching the AH by introducing a metric based on the Bondi-Sachs metric where the radial coordinate is replaced with an affine parameter. The model is derived with a cosmological constant Λ incorporated into the EFEs where Λ is taken as a parameter of the theory of gravity rather than as a matter source term. Similar to the conventional characteristic formalism, this model consists of a system of differential equations for numerically evolving the EFEs as a characteristic initial value problem (CIVP). A numerical code implemented for the method has been found to be second order convergent. This code enables simulations of different models given identical data on the initial null cone and provides a method to investigate their physical consistency within the causally connected region of our current PNC. These developments closely follow existing 3D schemes developed for gravitational wave simulations, which should make it natural to extend the affine CIVP beyond spherical symmetric simulations. The developments presented in this thesis is an extended version of two papers published earlier.
- Full Text:
- Date Issued: 2013
- Authors: Van der Walt, Petrus Johannes
- Date: 2013
- Subjects: Cosmology Numerical calculations General relativity (Physics) -- Mathematics Einstein field equations Gravity Gravitational waves Horizon
- Language: English
- Type: Thesis , Doctoral , PhD
- Identifier: vital:5396 , http://hdl.handle.net/10962/d1002959
- Description: The observational approach to cosmology is the endeavour to reconstruct the geometry of the Universe using only data that is theoretically verifiable within the causal boundaries of a cosmological observer. Using this approach, it was shown in [36] that given ideal cosmological observations, the only essential assumption necessary to determine the geometry of the Universe is a theory of gravity. Assuming General Relativity, the full set of Einstein field equations (EFEs) can be used to reconstruct the geometry of the Universe using direct observations on the past null cone (PNC) as initial conditions. Observationally and theoretically this is a very ambitious task and therefore, current developments have been restricted to spherically symmetric dust models while only relaxing the usual assumption of homogeneity in the radial direction. These restricted models are important for the development of theoretical foundations and also useful as verification models since they avoid the circularity of verifying what has already been assumed. The work presented in this thesis is the development of such a model where numerical relativity (NR) is used to simulate the observable universe. Similar to the work of Ellis and co-workers [36], a reference frame based on the PNC is used. The reference frame used here, however, is based on that of the characteristic formalism of NR, which has developed for calculating the propagation of gravitational waves. This provides a formalism that is well established in NR, making the use of existing algorithms possible. The Bondi-Sachs coordinates of the characteristic formalism is, however, not suitable for calculations beyond the observer apparent horizon (AH) since the diameter distance used as a radial coordinate becomes multi-valued when the cosmological PNC reconverges in the history of a universe, smaller in the past. With this taken into consideration, the Bondi-Sachs characteristic formalism is implemented for cosmology and the problem approaching the AH is investigated. Further developments address the limitations approaching the AH by introducing a metric based on the Bondi-Sachs metric where the radial coordinate is replaced with an affine parameter. The model is derived with a cosmological constant Λ incorporated into the EFEs where Λ is taken as a parameter of the theory of gravity rather than as a matter source term. Similar to the conventional characteristic formalism, this model consists of a system of differential equations for numerically evolving the EFEs as a characteristic initial value problem (CIVP). A numerical code implemented for the method has been found to be second order convergent. This code enables simulations of different models given identical data on the initial null cone and provides a method to investigate their physical consistency within the causally connected region of our current PNC. These developments closely follow existing 3D schemes developed for gravitational wave simulations, which should make it natural to extend the affine CIVP beyond spherical symmetric simulations. The developments presented in this thesis is an extended version of two papers published earlier.
- Full Text:
- Date Issued: 2013
Real options valuation for South African nuclear waste management using a fuzzy mathematical approach
- Authors: Montsho, Obakeng Johannes
- Date: 2013 , 2013-06-06
- Subjects: Fuzzy mathematics , Real options (Finance) , Fuzzy sets , Business mathematics , Radioactive waste disposal -- South Africa
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5398 , http://hdl.handle.net/10962/d1003051 , Fuzzy mathematics , Real options (Finance) , Fuzzy sets , Business mathematics , Radioactive waste disposal -- South Africa
- Description: The feasibility of capital projects in an uncertain world can be determined in several ways. One of these methods is real options valuation which arose from financial option valuation theory. On the other hand fuzzy set theory was developed as a mathematical framework to capture uncertainty in project management. The valuation of real options using fuzzy numbers represents an important refinement to determining capital projects' feasibility using the real options approach. The aim of this study is to determine whether the deferral of the decommissioning time (by a decade) of an electricity-generating nuclear plant in South Africa increases decommissioning costs. Using the fuzzy binomial approach, decommissioning costs increase when decommissioning is postponed by a decade whereas use of the fuzzy Black-Scholes approach yields the opposite result. A python code was developed to assist in the computation of fuzzy binomial trees required in our study and the results of the program are incorporated in this thesis. , KMBT_363 , Adobe Acrobat 9.54 Paper Capture Plug-in
- Full Text:
- Date Issued: 2013
- Authors: Montsho, Obakeng Johannes
- Date: 2013 , 2013-06-06
- Subjects: Fuzzy mathematics , Real options (Finance) , Fuzzy sets , Business mathematics , Radioactive waste disposal -- South Africa
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5398 , http://hdl.handle.net/10962/d1003051 , Fuzzy mathematics , Real options (Finance) , Fuzzy sets , Business mathematics , Radioactive waste disposal -- South Africa
- Description: The feasibility of capital projects in an uncertain world can be determined in several ways. One of these methods is real options valuation which arose from financial option valuation theory. On the other hand fuzzy set theory was developed as a mathematical framework to capture uncertainty in project management. The valuation of real options using fuzzy numbers represents an important refinement to determining capital projects' feasibility using the real options approach. The aim of this study is to determine whether the deferral of the decommissioning time (by a decade) of an electricity-generating nuclear plant in South Africa increases decommissioning costs. Using the fuzzy binomial approach, decommissioning costs increase when decommissioning is postponed by a decade whereas use of the fuzzy Black-Scholes approach yields the opposite result. A python code was developed to assist in the computation of fuzzy binomial trees required in our study and the results of the program are incorporated in this thesis. , KMBT_363 , Adobe Acrobat 9.54 Paper Capture Plug-in
- Full Text:
- Date Issued: 2013
Characterization of stratified L-topological spaces by convergence of stratified L-filters
- Authors: Orpen, David Lisle
- Date: 2011
- Subjects: Topology , Generalized spaces , Filters (Mathematics) , Topological spaces
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5402 , http://hdl.handle.net/10962/d1005216 , Topology , Generalized spaces , Filters (Mathematics) , Topological spaces
- Description: For the case where L is an ecl-premonoid, we explore various characterizations of SL-topological spaces, in particular characterization in terms of a convergence function lim: FS L(X) ! LX. We find we have to introduce a new axiom , L on the lim function in order to completely describe SL-topological spaces, which is not required in the case where L is a frame. We generalize the classical Kowalski and Fischer axioms to the lattice context and examine their relationship to the convergence axioms. We define the category of stratified L-generalized convergence spaces, as a generalization of the classical convergence spaces and investigate conditions under which it contains the category of stratified L-topological spaces as a reflective subcategory. We investigate some subcategories of the category of stratified L-generalized convergence spaces obtained by generalizing various classical convergence axioms.
- Full Text:
- Date Issued: 2011
- Authors: Orpen, David Lisle
- Date: 2011
- Subjects: Topology , Generalized spaces , Filters (Mathematics) , Topological spaces
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5402 , http://hdl.handle.net/10962/d1005216 , Topology , Generalized spaces , Filters (Mathematics) , Topological spaces
- Description: For the case where L is an ecl-premonoid, we explore various characterizations of SL-topological spaces, in particular characterization in terms of a convergence function lim: FS L(X) ! LX. We find we have to introduce a new axiom , L on the lim function in order to completely describe SL-topological spaces, which is not required in the case where L is a frame. We generalize the classical Kowalski and Fischer axioms to the lattice context and examine their relationship to the convergence axioms. We define the category of stratified L-generalized convergence spaces, as a generalization of the classical convergence spaces and investigate conditions under which it contains the category of stratified L-topological spaces as a reflective subcategory. We investigate some subcategories of the category of stratified L-generalized convergence spaces obtained by generalizing various classical convergence axioms.
- Full Text:
- Date Issued: 2011
Universal approximation properties of feedforward artificial neural networks.
- Authors: Redpath, Stuart Frederick
- Date: 2011
- Subjects: Neural networks (Computer science) , Artificial intelligence -- Biological applications , Functional analysis , Weierstrass-Stone Theorem , Banach-Hahn theorem
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5430 , http://hdl.handle.net/10962/d1015869
- Description: In this thesis we summarise several results in the literature which show the approximation capabilities of multilayer feedforward artificial neural networks. We show that multilayer feedforward artificial neural networks are capable of approximating continuous and measurable functions from Rn to R to any degree of accuracy under certain conditions. In particular making use of the Stone-Weierstrass and Hahn-Banach theorems, we show that a multilayer feedforward artificial neural network can approximate any continuous function to any degree of accuracy, by using either an arbitrary squashing function or any continuous sigmoidal function for activation. Making use of the Stone-Weirstrass Theorem again, we extend these approximation capabilities of multilayer feedforward artificial neural networks to the space of measurable functions under any probability measure.
- Full Text:
- Date Issued: 2011
- Authors: Redpath, Stuart Frederick
- Date: 2011
- Subjects: Neural networks (Computer science) , Artificial intelligence -- Biological applications , Functional analysis , Weierstrass-Stone Theorem , Banach-Hahn theorem
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5430 , http://hdl.handle.net/10962/d1015869
- Description: In this thesis we summarise several results in the literature which show the approximation capabilities of multilayer feedforward artificial neural networks. We show that multilayer feedforward artificial neural networks are capable of approximating continuous and measurable functions from Rn to R to any degree of accuracy under certain conditions. In particular making use of the Stone-Weierstrass and Hahn-Banach theorems, we show that a multilayer feedforward artificial neural network can approximate any continuous function to any degree of accuracy, by using either an arbitrary squashing function or any continuous sigmoidal function for activation. Making use of the Stone-Weirstrass Theorem again, we extend these approximation capabilities of multilayer feedforward artificial neural networks to the space of measurable functions under any probability measure.
- Full Text:
- Date Issued: 2011
Contributions to the study of a class of optimal control problems on the matrix lie group SO(3)
- Authors: Rodgerson, Joanne Kelly
- Date: 2009 , 2013-07-12
- Subjects: Matrix groups , Lie groups , Maximum principles (Mathematics) , Elliptic functions , Extremal problems (Mathematics)
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5421 , http://hdl.handle.net/10962/d1007199 , Matrix groups , Lie groups , Maximum principles (Mathematics) , Elliptic functions , Extremal problems (Mathematics)
- Description: The purpose of this thesis is to investigate a class of four left-invariant optimal control problems on the special orthogonal group SO(3). The set of all control-affine left-invariant control systems on SO(3) can, without loss, be reduced to a class of four typical controllable left-invariant control systems on SO(3) . The left-invariant optimal control problem on SO(3) involves finding a trajectory-control pair on SO (3), which minimizes a cost functional, and satisfies the given dynamical constraints and boundary conditions in a fixed time. The problem is lifted to the cotangent bundle T*SO(3) = SO(3) x so (3)* using the optimal Hamiltonian on so(3)*, where the maximum principle yields the optimal control. In a contribution to the study of this class of optimal control problems on SO(3), the extremal equations on so(3)* (ident ified with JR3) are integrated via elliptic functions to obtain explicit expressions for the solution curves in each typical case. The energy-Casimir method is used to give sufficient conditions for non-linear stability of the equilibrium states. , KMBT_363 , Adobe Acrobat 9.54 Paper Capture Plug-in
- Full Text:
- Date Issued: 2009
- Authors: Rodgerson, Joanne Kelly
- Date: 2009 , 2013-07-12
- Subjects: Matrix groups , Lie groups , Maximum principles (Mathematics) , Elliptic functions , Extremal problems (Mathematics)
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5421 , http://hdl.handle.net/10962/d1007199 , Matrix groups , Lie groups , Maximum principles (Mathematics) , Elliptic functions , Extremal problems (Mathematics)
- Description: The purpose of this thesis is to investigate a class of four left-invariant optimal control problems on the special orthogonal group SO(3). The set of all control-affine left-invariant control systems on SO(3) can, without loss, be reduced to a class of four typical controllable left-invariant control systems on SO(3) . The left-invariant optimal control problem on SO(3) involves finding a trajectory-control pair on SO (3), which minimizes a cost functional, and satisfies the given dynamical constraints and boundary conditions in a fixed time. The problem is lifted to the cotangent bundle T*SO(3) = SO(3) x so (3)* using the optimal Hamiltonian on so(3)*, where the maximum principle yields the optimal control. In a contribution to the study of this class of optimal control problems on SO(3), the extremal equations on so(3)* (ident ified with JR3) are integrated via elliptic functions to obtain explicit expressions for the solution curves in each typical case. The energy-Casimir method is used to give sufficient conditions for non-linear stability of the equilibrium states. , KMBT_363 , Adobe Acrobat 9.54 Paper Capture Plug-in
- Full Text:
- Date Issued: 2009
Finite fuzzy sets, keychains and their applications
- Authors: Mahlasela, Zuko
- Date: 2009
- Subjects: Fuzzy sets , Finite groups , Lattice theory , Economics -- Mathematical models
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5406 , http://hdl.handle.net/10962/d1005220 , Fuzzy sets , Finite groups , Lattice theory , Economics -- Mathematical models
- Description: The idea of keychains, an (n+1)-tuple of non-increasing real numbers in the unit interval always including 1, naturally arises in study of finite fuzzy set theory. They are a useful concept in modeling ideas of uncertainty especially those that arise in Economics, Social Sciences, Statistics and other subjects. In this thesis we define and study some basic properties of keychains with reference to Partially Ordered Sets, Lattices, Chains and Finite Fuzzy Sets. We then examine the role of keychains and their lattice diagrams in representing uncertainties that arise in such problems as in preferential voting patterns, outcomes of competitions and in Economics - Preference Relations.
- Full Text:
- Date Issued: 2009
- Authors: Mahlasela, Zuko
- Date: 2009
- Subjects: Fuzzy sets , Finite groups , Lattice theory , Economics -- Mathematical models
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5406 , http://hdl.handle.net/10962/d1005220 , Fuzzy sets , Finite groups , Lattice theory , Economics -- Mathematical models
- Description: The idea of keychains, an (n+1)-tuple of non-increasing real numbers in the unit interval always including 1, naturally arises in study of finite fuzzy set theory. They are a useful concept in modeling ideas of uncertainty especially those that arise in Economics, Social Sciences, Statistics and other subjects. In this thesis we define and study some basic properties of keychains with reference to Partially Ordered Sets, Lattices, Chains and Finite Fuzzy Sets. We then examine the role of keychains and their lattice diagrams in representing uncertainties that arise in such problems as in preferential voting patterns, outcomes of competitions and in Economics - Preference Relations.
- Full Text:
- Date Issued: 2009
The principle of inclusion-exclusion and möbius function as counting techniques in finite fuzzy subsets
- Authors: Talwanga, Matiki
- Date: 2009
- Subjects: Fuzzy logic , Fuzzy sets , Fuzzy systems , Möbius function
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5413 , http://hdl.handle.net/10962/d1005227 , Fuzzy logic , Fuzzy sets , Fuzzy systems , Möbius function
- Description: The broad goal in this thesis is to enumerate elements and fuzzy subsets of a finite set enjoying some useful properties through the well-known counting technique of the principle of inclusion-exclusion. We consider the set of membership values to be finite and uniformly spaced in the real unit interval. Further we define an equivalence relation with regards to the cardinalities of fuzzy subsets providing the Möbius function and Möbius inversion in that context.
- Full Text:
- Date Issued: 2009
- Authors: Talwanga, Matiki
- Date: 2009
- Subjects: Fuzzy logic , Fuzzy sets , Fuzzy systems , Möbius function
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5413 , http://hdl.handle.net/10962/d1005227 , Fuzzy logic , Fuzzy sets , Fuzzy systems , Möbius function
- Description: The broad goal in this thesis is to enumerate elements and fuzzy subsets of a finite set enjoying some useful properties through the well-known counting technique of the principle of inclusion-exclusion. We consider the set of membership values to be finite and uniformly spaced in the real unit interval. Further we define an equivalence relation with regards to the cardinalities of fuzzy subsets providing the Möbius function and Möbius inversion in that context.
- Full Text:
- Date Issued: 2009
A study of the existence of equilibrium in mathematical economics
- Authors: Xotyeni, Zukisa Gqabi
- Date: 2008
- Subjects: Equilibrium (Economics) -- Mathematical models , Macroeconomics -- Mathematical models , Economics -- Mathematical models , Welfare economics -- Mathematical models , Lattice theory , Economics, Mathematical
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5418 , http://hdl.handle.net/10962/d1005232 , Equilibrium (Economics) -- Mathematical models , Macroeconomics -- Mathematical models , Economics -- Mathematical models , Welfare economics -- Mathematical models , Lattice theory , Economics, Mathematical
- Description: In this thesis we define and study the existence of an equilibrium situation in which producers maximize their profits relative to the production vectors in their production sets, consumers satisfy their preferences in their consumption sets under certain budget constraint, and for every commodity total demand equals total supply. This competitive equilibrium situation is referred to as the Walrasian equilibrium. The existence of this equilibrium is investigated from a various mathematical points of view. These include microeconomic theory, simplicial spaces, global analysis and lattice theory.
- Full Text:
- Date Issued: 2008
- Authors: Xotyeni, Zukisa Gqabi
- Date: 2008
- Subjects: Equilibrium (Economics) -- Mathematical models , Macroeconomics -- Mathematical models , Economics -- Mathematical models , Welfare economics -- Mathematical models , Lattice theory , Economics, Mathematical
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5418 , http://hdl.handle.net/10962/d1005232 , Equilibrium (Economics) -- Mathematical models , Macroeconomics -- Mathematical models , Economics -- Mathematical models , Welfare economics -- Mathematical models , Lattice theory , Economics, Mathematical
- Description: In this thesis we define and study the existence of an equilibrium situation in which producers maximize their profits relative to the production vectors in their production sets, consumers satisfy their preferences in their consumption sets under certain budget constraint, and for every commodity total demand equals total supply. This competitive equilibrium situation is referred to as the Walrasian equilibrium. The existence of this equilibrium is investigated from a various mathematical points of view. These include microeconomic theory, simplicial spaces, global analysis and lattice theory.
- Full Text:
- Date Issued: 2008
Lattice-valued uniform convergence spaces the case of enriched lattices
- Authors: Craig, Andrew Philip Knott
- Date: 2008
- Subjects: Lattice theory , Uniform spaces , Convergence
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5411 , http://hdl.handle.net/10962/d1005225 , Lattice theory , Uniform spaces , Convergence
- Description: Using a pseudo-bisymmetric enriched cl-premonoid as the underlying lattice, we examine different categories of lattice-valued spaces. Lattice-valued topological spaces, uniform spaces and limit spaces are described, and we produce a new definition of stratified lattice-valued uniform convergence spaces in this generalised lattice context. We show that the category of stratified L-uniform convergence spaces is topological, and that the forgetful functor preserves initial constructions for the underlying stratified L-limit space. For the case of L a complete Heyting algebra, it is shown that the category of stratified L-uniform convergence spaces is cartesian closed.
- Full Text:
- Date Issued: 2008
- Authors: Craig, Andrew Philip Knott
- Date: 2008
- Subjects: Lattice theory , Uniform spaces , Convergence
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5411 , http://hdl.handle.net/10962/d1005225 , Lattice theory , Uniform spaces , Convergence
- Description: Using a pseudo-bisymmetric enriched cl-premonoid as the underlying lattice, we examine different categories of lattice-valued spaces. Lattice-valued topological spaces, uniform spaces and limit spaces are described, and we produce a new definition of stratified lattice-valued uniform convergence spaces in this generalised lattice context. We show that the category of stratified L-uniform convergence spaces is topological, and that the forgetful functor preserves initial constructions for the underlying stratified L-limit space. For the case of L a complete Heyting algebra, it is shown that the category of stratified L-uniform convergence spaces is cartesian closed.
- Full Text:
- Date Issued: 2008