A study of universal algebras in fuzzy set theory

**Authors:**Murali, V**Date:**1988**Subjects:**Fuzzy sets Algebra, Universal**Language:**English**Type:**Thesis , Doctoral , PhD**Identifier:**vital:5394 , http://hdl.handle.net/10962/d1001983**Description:**This thesis attempts a synthesis of two important and fast developing branches of mathematics, namely universal algebra and fuzzy set theory. Given an abstract algebra [X,F] where X is a non-empty set and F is a set of finitary operations on X, a fuzzy algebra [I×,F] is constructed by extending operations on X to that on I×, the set of fuzzy subsets of X (I denotes the unit interval), using Zadeh's extension principle. Homomorphisms between fuzzy algebras are defined and discussed. Fuzzy subalgebras of an algebra are defined to be elements of a fuzzy algebra which respect the extended algebra operations under inclusion of fuzzy subsets. The family of fuzzy subalgebras of an algebra is an algebraic closure system in I×. Thus the set of fuzzy subalgebras is a complete lattice. A fuzzy equivalence relation on a set is defined and a partition of such a relation into a class of fuzzy subsets is derived. Using these ideas, fuzzy functions between sets, fuzzy congruence relations, and fuzzy homomorphisms are defined. The kernels of fuzzy homomorphisms are proved to be fuzzy congruence relations, paving the way for the fuzzy isomorphism theorem. Finally, we sketch some ideas on free fuzzy subalgebras and polynomial algebras. In a nutshell, we can say that this thesis treats the central ideas of universal algebras, namely subalgebras, homomorphisms, equivalence and congruence relations, isomorphism theorems and free algebra in the fuzzy set theory setting**Full Text:****Date Issued:**1988

**Authors:**Murali, V**Date:**1988**Subjects:**Fuzzy sets Algebra, Universal**Language:**English**Type:**Thesis , Doctoral , PhD**Identifier:**vital:5394 , http://hdl.handle.net/10962/d1001983**Description:**This thesis attempts a synthesis of two important and fast developing branches of mathematics, namely universal algebra and fuzzy set theory. Given an abstract algebra [X,F] where X is a non-empty set and F is a set of finitary operations on X, a fuzzy algebra [I×,F] is constructed by extending operations on X to that on I×, the set of fuzzy subsets of X (I denotes the unit interval), using Zadeh's extension principle. Homomorphisms between fuzzy algebras are defined and discussed. Fuzzy subalgebras of an algebra are defined to be elements of a fuzzy algebra which respect the extended algebra operations under inclusion of fuzzy subsets. The family of fuzzy subalgebras of an algebra is an algebraic closure system in I×. Thus the set of fuzzy subalgebras is a complete lattice. A fuzzy equivalence relation on a set is defined and a partition of such a relation into a class of fuzzy subsets is derived. Using these ideas, fuzzy functions between sets, fuzzy congruence relations, and fuzzy homomorphisms are defined. The kernels of fuzzy homomorphisms are proved to be fuzzy congruence relations, paving the way for the fuzzy isomorphism theorem. Finally, we sketch some ideas on free fuzzy subalgebras and polynomial algebras. In a nutshell, we can say that this thesis treats the central ideas of universal algebras, namely subalgebras, homomorphisms, equivalence and congruence relations, isomorphism theorems and free algebra in the fuzzy set theory setting**Full Text:****Date Issued:**1988

Contributions to the theory of group rings

- Groenewald, Nicolas Johannes

**Authors:**Groenewald, Nicolas Johannes**Date:**1979**Subjects:**Group rings Group theory -- Mathematics**Language:**English**Type:**Thesis , Doctoral , PhD**Identifier:**vital:5391 , http://hdl.handle.net/10962/d1001980**Description:**Chapter 1 is a short review of the main results in some areas of the theory of group rings. In the first half of Chapter 2 we determine the ideal theoretic structure of the group ring RG where G is the direct product of a finite Abelian group and an ordered group with R a completely primary ring. Our choice of rings and groups entails that the study centres mainly on zero divisor ideals of group rings and hence it contributes in a small way to the zero divisor problem. We show that if R is a completely primary ring, then there exists a one-one correspondence of the prime zero divisor ideals in RG and RG¯, G finite cyclic of order n. If R is a ring with the property α, β € R, then αβ = 0 implies βα = 0, and S is an ordered semigroup, we show that if ∑α¡s¡ ∈ RS is a divisor of zero, then the coefficients α¡ belong to a zero divisor ideal in R. The converse is proved in the case where R is a commutative Noetherian ring. These results are applied to give an account of the zero divisors in the group ring over the direct product of a finite Abelian group and an ordered group with coefficients in a completely primary ring. In the second half of Chapter 2 we determine the units of the group ring RG where R is not necessarily commutative and G is an ordered group. If R is a ring such that if α, β € R and αβ = 0, then βα = 0, and if G is an ordered group, then we show that ∑αg(subscript)g is a unit in RG if and only if there exists ∑βh(subscript)h in RG such that∑αg(subscript)βg(subscript)-1 = 1 and αg(subscriptβh is nilpotent whenever GH≠1. We also show that if R is a ring with no nilpotent elements ≠0 and no idempotents ≠0,1, then RG has only trivial units. In this chapter we also consider strongly prime rings. We prove that RG is strongly prime if R is strongly prime and G is an unique product (u.p.) group. If H ⊲ G such that G/H is right ordered, then it is shown that RG is strongly prime if RH is strongly prime. In Chapter 3 results are derived to indicate the relations between certain classes of ideals in R and RG. If δ is a property of ideals defined for ideals in R and RG, then the "going up" condition holds for δ-ideals if Q being a δ-ideal in R implies that QG is a δ-ideal in RG. The "going down" condition is satisfied if P being a δ-ideal in RG implies that P∩ R is a δ-ideal in R. We proved that the "going up" and "going down" conditions are satisfied for prime ideals, ℓ-prime ideals, q-semiprime ideals and strongly prime ideals. These results are then applied to obtain certain relations between different radicals of the ring R and the group ring (semigroup ring) RG (RS). Similarly, results about the relation between the ideals and the radicals of the group rings RH and RG, where H is a central subgroup of G, are obtained. For the upper nil radical we prove that ⋃(RG) (RH) ⊆ RG, H a central subgroup of G, if G/H is an ordered group . If S is an ordered semigroup, however, then ⋃(RS) ⊆ ⋃(R)S for any ring R. In Chapter 4 we determine relations between various radicals in certain classes of group rings. In Section 4.3, as an extension of a result of Tan, we prove that P(R)G = P(RG) , R a ring with identity , if and only if the order of no finite normal subgroup of G is a zero divisor in R/P(R). If R is any ring with identity and H a normal subgroup of G such that G/H is an ordered group, we show that ⊓(RH)·RG = ⋃(RG) = ⊓(RG) , if ⋃(RH) is nilpotent. Similar results are obtained for the semigroup ring RS, S ordered. It is also shown if R is commutative and G finite of order n, then J(R)G = J(RG) if and only if n is not a zero divisor in R/J(R), J(R) being the Jacobson radical of R. For the Brown HcCoy radical we determine the following: If R is Brown McCoy semisimple or if R is a simple ring with identity, then B(RG) = (0), where G is a finitely generated torsion free Abelian group. In the last section we determine further relations between some of the previously defined radicals, in particular between P(R), U(R) and J(R). Among other results, the following relations between the abovementioned radicals are obtained: U(RS) = U(R)S = P(RS) = J(RS) where R is a left Goldie ring and S an ordered semigroup with unity**Full Text:****Date Issued:**1979

**Authors:**Groenewald, Nicolas Johannes**Date:**1979**Subjects:**Group rings Group theory -- Mathematics**Language:**English**Type:**Thesis , Doctoral , PhD**Identifier:**vital:5391 , http://hdl.handle.net/10962/d1001980**Description:**Chapter 1 is a short review of the main results in some areas of the theory of group rings. In the first half of Chapter 2 we determine the ideal theoretic structure of the group ring RG where G is the direct product of a finite Abelian group and an ordered group with R a completely primary ring. Our choice of rings and groups entails that the study centres mainly on zero divisor ideals of group rings and hence it contributes in a small way to the zero divisor problem. We show that if R is a completely primary ring, then there exists a one-one correspondence of the prime zero divisor ideals in RG and RG¯, G finite cyclic of order n. If R is a ring with the property α, β € R, then αβ = 0 implies βα = 0, and S is an ordered semigroup, we show that if ∑α¡s¡ ∈ RS is a divisor of zero, then the coefficients α¡ belong to a zero divisor ideal in R. The converse is proved in the case where R is a commutative Noetherian ring. These results are applied to give an account of the zero divisors in the group ring over the direct product of a finite Abelian group and an ordered group with coefficients in a completely primary ring. In the second half of Chapter 2 we determine the units of the group ring RG where R is not necessarily commutative and G is an ordered group. If R is a ring such that if α, β € R and αβ = 0, then βα = 0, and if G is an ordered group, then we show that ∑αg(subscript)g is a unit in RG if and only if there exists ∑βh(subscript)h in RG such that∑αg(subscript)βg(subscript)-1 = 1 and αg(subscriptβh is nilpotent whenever GH≠1. We also show that if R is a ring with no nilpotent elements ≠0 and no idempotents ≠0,1, then RG has only trivial units. In this chapter we also consider strongly prime rings. We prove that RG is strongly prime if R is strongly prime and G is an unique product (u.p.) group. If H ⊲ G such that G/H is right ordered, then it is shown that RG is strongly prime if RH is strongly prime. In Chapter 3 results are derived to indicate the relations between certain classes of ideals in R and RG. If δ is a property of ideals defined for ideals in R and RG, then the "going up" condition holds for δ-ideals if Q being a δ-ideal in R implies that QG is a δ-ideal in RG. The "going down" condition is satisfied if P being a δ-ideal in RG implies that P∩ R is a δ-ideal in R. We proved that the "going up" and "going down" conditions are satisfied for prime ideals, ℓ-prime ideals, q-semiprime ideals and strongly prime ideals. These results are then applied to obtain certain relations between different radicals of the ring R and the group ring (semigroup ring) RG (RS). Similarly, results about the relation between the ideals and the radicals of the group rings RH and RG, where H is a central subgroup of G, are obtained. For the upper nil radical we prove that ⋃(RG) (RH) ⊆ RG, H a central subgroup of G, if G/H is an ordered group . If S is an ordered semigroup, however, then ⋃(RS) ⊆ ⋃(R)S for any ring R. In Chapter 4 we determine relations between various radicals in certain classes of group rings. In Section 4.3, as an extension of a result of Tan, we prove that P(R)G = P(RG) , R a ring with identity , if and only if the order of no finite normal subgroup of G is a zero divisor in R/P(R). If R is any ring with identity and H a normal subgroup of G such that G/H is an ordered group, we show that ⊓(RH)·RG = ⋃(RG) = ⊓(RG) , if ⋃(RH) is nilpotent. Similar results are obtained for the semigroup ring RS, S ordered. It is also shown if R is commutative and G finite of order n, then J(R)G = J(RG) if and only if n is not a zero divisor in R/J(R), J(R) being the Jacobson radical of R. For the Brown HcCoy radical we determine the following: If R is Brown McCoy semisimple or if R is a simple ring with identity, then B(RG) = (0), where G is a finitely generated torsion free Abelian group. In the last section we determine further relations between some of the previously defined radicals, in particular between P(R), U(R) and J(R). Among other results, the following relations between the abovementioned radicals are obtained: U(RS) = U(R)S = P(RS) = J(RS) where R is a left Goldie ring and S an ordered semigroup with unity**Full Text:****Date Issued:**1979

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

Generalisations of filters and uniform spaces

**Authors:**Muraleetharan, Murugiah**Date:**1997**Subjects:**Filters (Mathematics) , Uniform spaces**Language:**English**Type:**Thesis , Masters , MSc**Identifier:**vital:5409 , http://hdl.handle.net/10962/d1005223 , Filters (Mathematics) , Uniform spaces**Description:**The notion of a filter F ∈ 2²x has been extended to that of a : prefilter: ƒ ∈ 1²x, generalised filter ƒ ∈ 2²x x and fuzzy filter ᵩ ∈ 1¹x. A uniformity is a filter with some other conditions and the notion of a uniformity D ∈ 2²xxx has been extended to that of a : fuzzy uniformity d ∈ 1²xxx , generalised uniformity ∈ 1²xxx and super uniformity b ∈ 1¹x. We establish categorical embeddings from the category of uniform spaces into the categories of fuzzy uniform spaces, generalised uniform spaces and super uniform spaces and also categorical embeddings into the category of super uniform spaces from the categories of fuzzy uniform spaces and generalised uniform spaces.**Full Text:****Date Issued:**1997

**Authors:**Muraleetharan, Murugiah**Date:**1997**Subjects:**Filters (Mathematics) , Uniform spaces**Language:**English**Type:**Thesis , Masters , MSc**Identifier:**vital:5409 , http://hdl.handle.net/10962/d1005223 , Filters (Mathematics) , Uniform spaces**Description:**The notion of a filter F ∈ 2²x has been extended to that of a : prefilter: ƒ ∈ 1²x, generalised filter ƒ ∈ 2²x x and fuzzy filter ᵩ ∈ 1¹x. A uniformity is a filter with some other conditions and the notion of a uniformity D ∈ 2²xxx has been extended to that of a : fuzzy uniformity d ∈ 1²xxx , generalised uniformity ∈ 1²xxx and super uniformity b ∈ 1¹x. We establish categorical embeddings from the category of uniform spaces into the categories of fuzzy uniform spaces, generalised uniform spaces and super uniform spaces and also categorical embeddings into the category of super uniform spaces from the categories of fuzzy uniform spaces and generalised uniform spaces.**Full Text:****Date Issued:**1997

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

A computer analysis of some of the Harrison metrics

**Authors:**Sadler, Christopher John**Date:**1975**Subjects:**Computer science -- Mathematics , Software measurement**Language:**English**Type:**Thesis , Masters , MSc**Identifier:**vital:5427 , http://hdl.handle.net/10962/d1013151**Description:**In his paper B.K.Harrison concludes with the observation that his "solutions ... are presented as raw material for further research in General Relativity". In the same spirit, the present work started out as an attempt to process that raw material in a production-line powered by a computer. Harrison's solutions uould be fed in at one end, and the finished product, as yet undecided, would appear at the other. In the event, however, the project became more like an exercise in quality control, to continue the analogy. A search was made for algebraic criteria which would distinguish between those solutions which were acceptable for further analysis with particular regard to Gravitational radiation, and those which were not. Regrettably, no criteria could be found which characterised radiative solutions unequivocally, and, at the same time, lent themselves to a computer approach. The result is that the discussion of radiative solutions has had to be relegated to an appendix (Appendix 1), while the main body of the work is concerned with the determination of those quantities (the Newman-Penrose scalars) which would seem to be the foundation of any future computer-based analysis of gravitational radiation. Chapter 1 is an account of the underlying mathematical formulation, defining the terms, concepts and processes involved. In Chapter 2 the transformation of some of the ideas of Chapter 1 into computer software is presented. Chapter 3 is concerned with the specific metrics (the Harrison metrics) and the extent to which they have heen processed. The project has leaned heavily on papers by Harrison for the "raw material", by D' Inverno and Russell Clark, who pioneered the techniques and classified the Harrison metrics, and by Sachs for the treatment of gravitational radiation. However, the analysis of diagonal metrics, the special tetrad of Chapter 2 and the results in Appendix 2 are new.**Full Text:****Date Issued:**1975

**Authors:**Sadler, Christopher John**Date:**1975**Subjects:**Computer science -- Mathematics , Software measurement**Language:**English**Type:**Thesis , Masters , MSc**Identifier:**vital:5427 , http://hdl.handle.net/10962/d1013151**Description:**In his paper B.K.Harrison concludes with the observation that his "solutions ... are presented as raw material for further research in General Relativity". In the same spirit, the present work started out as an attempt to process that raw material in a production-line powered by a computer. Harrison's solutions uould be fed in at one end, and the finished product, as yet undecided, would appear at the other. In the event, however, the project became more like an exercise in quality control, to continue the analogy. A search was made for algebraic criteria which would distinguish between those solutions which were acceptable for further analysis with particular regard to Gravitational radiation, and those which were not. Regrettably, no criteria could be found which characterised radiative solutions unequivocally, and, at the same time, lent themselves to a computer approach. The result is that the discussion of radiative solutions has had to be relegated to an appendix (Appendix 1), while the main body of the work is concerned with the determination of those quantities (the Newman-Penrose scalars) which would seem to be the foundation of any future computer-based analysis of gravitational radiation. Chapter 1 is an account of the underlying mathematical formulation, defining the terms, concepts and processes involved. In Chapter 2 the transformation of some of the ideas of Chapter 1 into computer software is presented. Chapter 3 is concerned with the specific metrics (the Harrison metrics) and the extent to which they have heen processed. The project has leaned heavily on papers by Harrison for the "raw material", by D' Inverno and Russell Clark, who pioneered the techniques and classified the Harrison metrics, and by Sachs for the treatment of gravitational radiation. However, the analysis of diagonal metrics, the special tetrad of Chapter 2 and the results in Appendix 2 are new.**Full Text:****Date Issued:**1975

Extension theorems on L-topological spaces and L-fuzzy vector spaces

**Authors:**Pinchuck, Andrew**Date:**2002**Subjects:**Topology , Vector spaces , Generalized spaces**Language:**English**Type:**Thesis , Masters , MSc**Identifier:**vital:5405 , http://hdl.handle.net/10962/d1005219 , Topology , Vector spaces , Generalized spaces**Description:**A non-trivial example of an L-topological space, the fuzzy real line is examined. Various L-topological properties and their relationships are developed. Extension theorems on the L-fuzzy real line as well as extension theorems on more general L-topological spaces follow. Finally, a theory of L-fuzzy vector spaces leads up to a fuzzy version of the Hahn-Banach theorem.**Full Text:****Date Issued:**2002

**Authors:**Pinchuck, Andrew**Date:**2002**Subjects:**Topology , Vector spaces , Generalized spaces**Language:**English**Type:**Thesis , Masters , MSc**Identifier:**vital:5405 , http://hdl.handle.net/10962/d1005219 , Topology , Vector spaces , Generalized spaces**Description:**A non-trivial example of an L-topological space, the fuzzy real line is examined. Various L-topological properties and their relationships are developed. Extension theorems on the L-fuzzy real line as well as extension theorems on more general L-topological spaces follow. Finally, a theory of L-fuzzy vector spaces leads up to a fuzzy version of the Hahn-Banach theorem.**Full Text:****Date Issued:**2002

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

Twistors in curved space

- Ward, R S (Richard Samuel), 1951-

**Authors:**Ward, R S (Richard Samuel), 1951-**Date:**1975**Subjects:**Twistor theory , Space and time**Language:**English**Type:**Thesis , Masters , MSc**Identifier:**vital:5429 , http://hdl.handle.net/10962/d1013472**Description:**From the Introduction, p. 1. During the past decade, the theory of twistors has been introduced and developed, primarily by Professor Roger Penrose, as part of a long-term program aimed at resolving certain difficulties in present-day physical theory. These difficulties include, firstly, the problem of combining quantum mechanics and general relativity, and, secondly, the question of whether the concept of a continuum is at all relevant to physics. Most models of space-time used in general relativity employ the idea of a manifold consisting of a continuum of points. This feature of the models has often been criticised, on the grounds that physical observations are essentially discrete in nature; for reasons that are mathematical, rather than physical, the gaps between these observations are filled in a continuous fashion (see, for example, Schrodinger (I), pp.26-31). Although analysis (in its generally accepted form) demands that quantities should take on a continuous range of values, physics, as such,does not make such a demand. The situation in quantum mechanics is not all that much better since, although some quantities such as angular momentum can only take on certain discrete values, one still has to deal with the complex continuum of probability amplitudes. From this point of view it would be desirable to have all physical laws expressed in terms of combinatorial mathematics, rather than in terms of (standard) analysis.**Full Text:****Date Issued:**1975

**Authors:**Ward, R S (Richard Samuel), 1951-**Date:**1975**Subjects:**Twistor theory , Space and time**Language:**English**Type:**Thesis , Masters , MSc**Identifier:**vital:5429 , http://hdl.handle.net/10962/d1013472**Description:**From the Introduction, p. 1. During the past decade, the theory of twistors has been introduced and developed, primarily by Professor Roger Penrose, as part of a long-term program aimed at resolving certain difficulties in present-day physical theory. These difficulties include, firstly, the problem of combining quantum mechanics and general relativity, and, secondly, the question of whether the concept of a continuum is at all relevant to physics. Most models of space-time used in general relativity employ the idea of a manifold consisting of a continuum of points. This feature of the models has often been criticised, on the grounds that physical observations are essentially discrete in nature; for reasons that are mathematical, rather than physical, the gaps between these observations are filled in a continuous fashion (see, for example, Schrodinger (I), pp.26-31). Although analysis (in its generally accepted form) demands that quantities should take on a continuous range of values, physics, as such,does not make such a demand. The situation in quantum mechanics is not all that much better since, although some quantities such as angular momentum can only take on certain discrete values, one still has to deal with the complex continuum of probability amplitudes. From this point of view it would be desirable to have all physical laws expressed in terms of combinatorial mathematics, rather than in terms of (standard) analysis.**Full Text:****Date Issued:**1975

Complete regularity and related concepts in L-uniform spaces

**Authors:**Harnett, Rait Sicklen**Date:**1992**Subjects:**Uniform spaces , Mathematics -- Research**Language:**English**Type:**Thesis , Masters , MSc**Identifier:**vital:5403 , http://hdl.handle.net/10962/d1005217 , Uniform spaces , Mathematics -- Research**Description:**L will denote a completely distributive lattice with an order reversing involution. The concept of an L-uniform space is introduced. An extension theorem concerning L-uniformly continuous functions is proved. A characterisation of L-uniformizability, involving L-complete regularity is given. With respect to L--completely regular spaces it is shown that the topological modification of an L-completely regular space is completely regular. Furthermore it is shown that the topologically generated L-topology of a completely regular space is L-completely regular.**Full Text:****Date Issued:**1992

**Authors:**Harnett, Rait Sicklen**Date:**1992**Subjects:**Uniform spaces , Mathematics -- Research**Language:**English**Type:**Thesis , Masters , MSc**Identifier:**vital:5403 , http://hdl.handle.net/10962/d1005217 , Uniform spaces , Mathematics -- Research**Description:**L will denote a completely distributive lattice with an order reversing involution. The concept of an L-uniform space is introduced. An extension theorem concerning L-uniformly continuous functions is proved. A characterisation of L-uniformizability, involving L-complete regularity is given. With respect to L--completely regular spaces it is shown that the topological modification of an L-completely regular space is completely regular. Furthermore it is shown that the topologically generated L-topology of a completely regular space is L-completely regular.**Full Text:****Date Issued:**1992

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

Sobriety of crisp and fuzzy topological spaces

**Authors:**Jacot-Guillarmod, Paul**Date:**2004**Subjects:**Topological spaces , Fuzzy sets**Language:**English**Type:**Thesis , Masters , MSc**Identifier:**vital:5414 , http://hdl.handle.net/10962/d1005228 , Topological spaces , Fuzzy sets**Description:**The objective of this thesis is a survey of crisp and fuzzy sober topological spaces. We begin by examining sobriety of crisp topological spaces. We then extend this to the L- topological case and obtain analogous results and characterizations to those of the crisp case. We then brie y examine semi-sobriety of (L;M)-topological spaces.**Full Text:****Date Issued:**2004

**Authors:**Jacot-Guillarmod, Paul**Date:**2004**Subjects:**Topological spaces , Fuzzy sets**Language:**English**Type:**Thesis , Masters , MSc**Identifier:**vital:5414 , http://hdl.handle.net/10962/d1005228 , Topological spaces , Fuzzy sets**Description:**The objective of this thesis is a survey of crisp and fuzzy sober topological spaces. We begin by examining sobriety of crisp topological spaces. We then extend this to the L- topological case and obtain analogous results and characterizations to those of the crisp case. We then brie y examine semi-sobriety of (L;M)-topological spaces.**Full Text:****Date Issued:**2004

Studies of equivalent fuzzy subgroups of finite abelian p-Groups of rank two and their subgroup lattices

**Authors:**Ngcibi, Sakhile Leonard**Date:**2006**Subjects:**Abelian groups Fuzzy sets Finite groups Group theory Polynomials**Language:**English**Type:**Thesis , Doctoral , PhD**Identifier:**vital:5416 , http://hdl.handle.net/10962/d1005230**Description:**We determine the number and nature of distinct equivalence classes of fuzzy subgroups of finite Abelian p-group G of rank two under a natural equivalence relation on fuzzy subgroups. Our discussions embrace the necessary theory from groups with special emphasis on finite p-groups as a step towards the classification of crisp subgroups as well as maximal chains of subgroups. Unique naming of subgroup generators as discussed in this work facilitates counting of subgroups and chains of subgroups from subgroup lattices of the groups. We cover aspects of fuzzy theory including fuzzy (homo-) isomorphism together with operations on fuzzy subgroups. The equivalence characterization as discussed here is finer than isomorphism. We introduce the theory of keychains with a view towards the enumeration of maximal chains as well as fuzzy subgroups under the equivalence relation mentioned above. We discuss a strategy to develop subgroup lattices of the groups used in the discussion, and give examples for specific cases of prime p and positive integers n,m. We derive formulas for both the number of maximal chains as well as the number of distinct equivalence classes of fuzzy subgroups. The results are in the form of polynomials in p (known in the literature as Hall polynomials) with combinatorial coefficients. Finally we give a brief investigation of the results from a graph-theoretic point of view. We view the subgroup lattices of these groups as simple, connected, symmetric graphs.**Full Text:****Date Issued:**2006

**Authors:**Ngcibi, Sakhile Leonard**Date:**2006**Subjects:**Abelian groups Fuzzy sets Finite groups Group theory Polynomials**Language:**English**Type:**Thesis , Doctoral , PhD**Identifier:**vital:5416 , http://hdl.handle.net/10962/d1005230**Description:**We determine the number and nature of distinct equivalence classes of fuzzy subgroups of finite Abelian p-group G of rank two under a natural equivalence relation on fuzzy subgroups. Our discussions embrace the necessary theory from groups with special emphasis on finite p-groups as a step towards the classification of crisp subgroups as well as maximal chains of subgroups. Unique naming of subgroup generators as discussed in this work facilitates counting of subgroups and chains of subgroups from subgroup lattices of the groups. We cover aspects of fuzzy theory including fuzzy (homo-) isomorphism together with operations on fuzzy subgroups. The equivalence characterization as discussed here is finer than isomorphism. We introduce the theory of keychains with a view towards the enumeration of maximal chains as well as fuzzy subgroups under the equivalence relation mentioned above. We discuss a strategy to develop subgroup lattices of the groups used in the discussion, and give examples for specific cases of prime p and positive integers n,m. We derive formulas for both the number of maximal chains as well as the number of distinct equivalence classes of fuzzy subgroups. The results are in the form of polynomials in p (known in the literature as Hall polynomials) with combinatorial coefficients. Finally we give a brief investigation of the results from a graph-theoretic point of view. We view the subgroup lattices of these groups as simple, connected, symmetric graphs.**Full Text:****Date Issued:**2006

Aspects of fuzzy spaces with special reference to cardinality, dimension, and order-homomorphisms

**Authors:**Lubczonok, Pawel**Date:**1992**Subjects:**Fuzzy sets Topological spaces**Language:**English**Type:**Thesis , Doctoral , PhD**Identifier:**vital:5399 , http://hdl.handle.net/10962/d1005213**Description:**Aspects of fuzzy vector spaces and fuzzy groups are investigated, including linear independence, basis, dimension, group order, finitely generated groups and cyclic groups. It was necessary to consider cardinality of fuzzy sets and related issues, which included a question of ways in which to define functions between fuzzy sets. Among the results proved, are the additivity property of dimension for fuzzy vector spaces, Lagrange's Theorem for fuzzy groups ( the existing version of this theorem does not take fuzziness into account at all), a compactness property of finitely generated fuzzy groups and an extension of an earlier result on the order-homomorphisms. An open question is posed with regard to the existence of a basis for an arbitrary fuzzy vector space.**Full Text:****Date Issued:**1992

**Authors:**Lubczonok, Pawel**Date:**1992**Subjects:**Fuzzy sets Topological spaces**Language:**English**Type:**Thesis , Doctoral , PhD**Identifier:**vital:5399 , http://hdl.handle.net/10962/d1005213**Description:**Aspects of fuzzy vector spaces and fuzzy groups are investigated, including linear independence, basis, dimension, group order, finitely generated groups and cyclic groups. It was necessary to consider cardinality of fuzzy sets and related issues, which included a question of ways in which to define functions between fuzzy sets. Among the results proved, are the additivity property of dimension for fuzzy vector spaces, Lagrange's Theorem for fuzzy groups ( the existing version of this theorem does not take fuzziness into account at all), a compactness property of finitely generated fuzzy groups and an extension of an earlier result on the order-homomorphisms. An open question is posed with regard to the existence of a basis for an arbitrary fuzzy vector space.**Full Text:****Date Issued:**1992

(L, M)-fuzzy topological spaces

**Authors:**Matutu, Phethiwe Precious**Date:**1992**Subjects:**Topological spaces , Fuzzy sets**Language:**English**Type:**Thesis , Masters , MSc**Identifier:**vital:5410 , http://hdl.handle.net/10962/d1005224 , Topological spaces , Fuzzy sets**Description:**The objective of this thesis is to develop certain aspects of the theory of (L,M)-fuzzy topological spaces, where L and M are complete lattices (with additional conditions when necessary). We obtain results which are to a large extent analogous to results given in a series of papers of Šostak (where L = M = [0,1]) but not necessarily with analogous proofs. Often, our generalizations require a variety of techniques from lattice theory e.g. from continuity or complete distributive lattices.**Full Text:****Date Issued:**1992

**Authors:**Matutu, Phethiwe Precious**Date:**1992**Subjects:**Topological spaces , Fuzzy sets**Language:**English**Type:**Thesis , Masters , MSc**Identifier:**vital:5410 , http://hdl.handle.net/10962/d1005224 , Topological spaces , Fuzzy sets**Description:**The objective of this thesis is to develop certain aspects of the theory of (L,M)-fuzzy topological spaces, where L and M are complete lattices (with additional conditions when necessary). We obtain results which are to a large extent analogous to results given in a series of papers of Šostak (where L = M = [0,1]) but not necessarily with analogous proofs. Often, our generalizations require a variety of techniques from lattice theory e.g. from continuity or complete distributive lattices.**Full Text:****Date Issued:**1992

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

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

Continuity and generalized continuity in dynamics and other applications

**Authors:**Mimna, Roy Allan**Date:**2002**Subjects:**Topological dynamics -- Research Dynamics -- Mathematical models -- Research Perturbation (Mathematics) Attractors (Mathematics) Baire classes Mathematics -- Research**Language:**English**Type:**Thesis , Doctoral , PhD**Identifier:**vital:5404 , http://hdl.handle.net/10962/d1005218**Description:**The topological dynamics of continuous and noncontinuous dynamical systems are investigated. Various definitions of chaos are studied, as well as notions of stability. Results are obtained on asymptotically stable sets and the perturbation stability of such sets. The primary focus is on the traditional point sets of topological dynamics, including the chain recurrent set, omega-limit sets and attractors. The basic setting is that of a continuous function on a compact metric space, sometimes with additional properties on the space. The investigation includes results on the dynamical properties of typical continuous functions in the sense of Baire category. Results are also developed concerning dynamical systems involving quasi-continuous functions. An invariance property for the omega-limit sets of such functions is given. Omega-limit sets are characterized for Riemann integrable derivatives and derivatiyes which are continuous almost everywhere. Techniques used in the investigation and formulation of results include finding theorems which relate the rather disparate notions of dynamical properties and generalized continuity. In addition to dynamical systems, numerous other applications of generalized continuity are imoestigated. Techniques used include application of the Baire Category Theorem and the notion of semi-closure. For example, results are formulated concerning functions determined by dense sets, including separately continuous functions, thus generalizing the classical result for continuous functions on dense subsets of the domain. The uniform boundedness theorem is extended to functions which are not necessarily continuous, including various derivatives. The closed graph theorem is strictly generalized in two separate ways, and applications are presented using these generalizations. An invariance property of separately continuous functions is given. Cluster sets are studied in connection with separate continuity, and various results are presented concerning locally bounded functions.**Full Text:****Date Issued:**2002

**Authors:**Mimna, Roy Allan**Date:**2002**Subjects:**Topological dynamics -- Research Dynamics -- Mathematical models -- Research Perturbation (Mathematics) Attractors (Mathematics) Baire classes Mathematics -- Research**Language:**English**Type:**Thesis , Doctoral , PhD**Identifier:**vital:5404 , http://hdl.handle.net/10962/d1005218**Description:**The topological dynamics of continuous and noncontinuous dynamical systems are investigated. Various definitions of chaos are studied, as well as notions of stability. Results are obtained on asymptotically stable sets and the perturbation stability of such sets. The primary focus is on the traditional point sets of topological dynamics, including the chain recurrent set, omega-limit sets and attractors. The basic setting is that of a continuous function on a compact metric space, sometimes with additional properties on the space. The investigation includes results on the dynamical properties of typical continuous functions in the sense of Baire category. Results are also developed concerning dynamical systems involving quasi-continuous functions. An invariance property for the omega-limit sets of such functions is given. Omega-limit sets are characterized for Riemann integrable derivatives and derivatiyes which are continuous almost everywhere. Techniques used in the investigation and formulation of results include finding theorems which relate the rather disparate notions of dynamical properties and generalized continuity. In addition to dynamical systems, numerous other applications of generalized continuity are imoestigated. Techniques used include application of the Baire Category Theorem and the notion of semi-closure. For example, results are formulated concerning functions determined by dense sets, including separately continuous functions, thus generalizing the classical result for continuous functions on dense subsets of the domain. The uniform boundedness theorem is extended to functions which are not necessarily continuous, including various derivatives. The closed graph theorem is strictly generalized in two separate ways, and applications are presented using these generalizations. An invariance property of separately continuous functions is given. Cluster sets are studied in connection with separate continuity, and various results are presented concerning locally bounded functions.**Full Text:****Date Issued:**2002

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

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