Remarks on formalized arithmetic and subsystems thereof

- Brink, C

**Authors:**Brink, C**Date:**1975**Subjects:**Gödel, Kurt , Logic, Symbolic and mathematical , Semantics (Philosophy) , Arithmetic -- Foundations , Number theory**Language:**English**Type:**Thesis , Masters , MSc**Identifier:**vital:5424 , http://hdl.handle.net/10962/d1009752 , Gödel, Kurt , Logic, Symbolic and mathematical , Semantics (Philosophy) , Arithmetic -- Foundations , Number theory**Description:**In a famous paper of 1931, Gödel proved that any formalization of elementary Arithmetic is incomplete, in the sense that it contains statements which are neither provable nor disprovable. Some two years before this, Presburger proved that a mutilated system of Arithmetic, employing only addition but not multiplication, is complete. This essay is partly an exposition of a system such as Presburger's, and partly an attempt to gain insight into the source of the incompleteness of Arithmetic, by linking Presburger's result with Gödel's.**Full Text:****Date Issued:**1975

**Authors:**Brink, C**Date:**1975**Subjects:**Gödel, Kurt , Logic, Symbolic and mathematical , Semantics (Philosophy) , Arithmetic -- Foundations , Number theory**Language:**English**Type:**Thesis , Masters , MSc**Identifier:**vital:5424 , http://hdl.handle.net/10962/d1009752 , Gödel, Kurt , Logic, Symbolic and mathematical , Semantics (Philosophy) , Arithmetic -- Foundations , Number theory**Description:**In a famous paper of 1931, Gödel proved that any formalization of elementary Arithmetic is incomplete, in the sense that it contains statements which are neither provable nor disprovable. Some two years before this, Presburger proved that a mutilated system of Arithmetic, employing only addition but not multiplication, is complete. This essay is partly an exposition of a system such as Presburger's, and partly an attempt to gain insight into the source of the incompleteness of Arithmetic, by linking Presburger's result with Gödel's.**Full Text:****Date Issued:**1975

Fuzzy uniform spaces

**Authors:**Burton, Michael Howard**Date:**1992**Subjects:**Fuzzy sets -- Research Uniform spaces -- Research**Language:**English**Type:**Thesis , Doctoral , PhD**Identifier:**vital:5408 , http://hdl.handle.net/10962/d1005222**Description:**For a fuzzy uniform space, the notion of a Cauchy prefilter, a precompact fuzzy set, a complete fuzzy set and a bounded fuzzy set are defined in such a way that these notions are good extensions of the corresponding notions for a uniform space. A theory of fuzzy uniform spaces is developed which generalises the theory of uniform spaces.**Full Text:****Date Issued:**1992

**Authors:**Burton, Michael Howard**Date:**1992**Subjects:**Fuzzy sets -- Research Uniform spaces -- Research**Language:**English**Type:**Thesis , Doctoral , PhD**Identifier:**vital:5408 , http://hdl.handle.net/10962/d1005222**Description:**For a fuzzy uniform space, the notion of a Cauchy prefilter, a precompact fuzzy set, a complete fuzzy set and a bounded fuzzy set are defined in such a way that these notions are good extensions of the corresponding notions for a uniform space. A theory of fuzzy uniform spaces is developed which generalises the theory of uniform spaces.**Full Text:****Date Issued:**1992

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

Instability in the magnetotail

**Authors:**English, Daniel Rowe**Date:**1977**Subjects:**Magnetotails**Language:**English**Type:**Thesis , Masters , MSc**Identifier:**vital:5425 , http://hdl.handle.net/10962/d1011764 , Magnetotails**Description:**The magnetic induction field due to the Earth only would, if undisturbed by any outside agency, resemble macroscopically the field due to a magnetic dipole. Hcwever the field is disturbed by the interplanetary magnetic field, of which the most important component is that of the Sun. If the Sun's magnetic field were effectively steady, it would also be a dipole field, and approximately constant in the region within about twenty earth radii from the earth. Also, if we treat the Sun as a dipole, its dipole axis is roughly normal to the ecliptic plane. The Earth, treated as a dipole, has an axis which is inclined to the normal to the ecliptic plane at an angle which varies daily from a few degrees to nearly a third of a right angle. However, in this paper, it is proposed to treat both dipole axes as contra-parallel and effectively normal to the ecliptic plane, so that a general idea of the combined field can be obtained. Then the effect of a steady field due to the Sun, on the Earth's field would be the formation of a "neutral ring" surrounding the Earth; that is, a closed "neutral line", this being a line of points at each of which the net nagnetic induction is zero. As the point of observation passes through this line, the field changes direction. Intro. p. v.**Full Text:****Date Issued:**1977

**Authors:**English, Daniel Rowe**Date:**1977**Subjects:**Magnetotails**Language:**English**Type:**Thesis , Masters , MSc**Identifier:**vital:5425 , http://hdl.handle.net/10962/d1011764 , Magnetotails**Description:**The magnetic induction field due to the Earth only would, if undisturbed by any outside agency, resemble macroscopically the field due to a magnetic dipole. Hcwever the field is disturbed by the interplanetary magnetic field, of which the most important component is that of the Sun. If the Sun's magnetic field were effectively steady, it would also be a dipole field, and approximately constant in the region within about twenty earth radii from the earth. Also, if we treat the Sun as a dipole, its dipole axis is roughly normal to the ecliptic plane. The Earth, treated as a dipole, has an axis which is inclined to the normal to the ecliptic plane at an angle which varies daily from a few degrees to nearly a third of a right angle. However, in this paper, it is proposed to treat both dipole axes as contra-parallel and effectively normal to the ecliptic plane, so that a general idea of the combined field can be obtained. Then the effect of a steady field due to the Sun, on the Earth's field would be the formation of a "neutral ring" surrounding the Earth; that is, a closed "neutral line", this being a line of points at each of which the net nagnetic induction is zero. As the point of observation passes through this line, the field changes direction. Intro. p. v.**Full Text:****Date Issued:**1977

A study of fuzzy sets and systems with applications to group theory and decision making

**Authors:**Gideon, Frednard**Date:**2006**Subjects:**Fuzzy sets , Fuzzy systems , Abelian groups , Decision making**Language:**English**Type:**Thesis , Masters , MSc**Identifier:**vital:5417 , http://hdl.handle.net/10962/d1005231 , Fuzzy sets , Fuzzy systems , Abelian groups , Decision making**Description:**In this study we apply the knowledge of fuzzy sets to group structures and also to decision-making implications. We study fuzzy subgroups of finite abelian groups. We set G = Z[subscript p[superscript n]] + Z[subscript q[superscript m]]. The classification of fuzzy subgroups of G using equivalence classes is introduced. First, we present equivalence relations on fuzzy subsets of X, and then extend it to the study of equivalence relations of fuzzy subgroups of a group G. This is then followed by the notion of flags and keychains projected as tools for enumerating fuzzy subgroups of G. In addition to this, we use linear ordering of the lattice of subgroups to characterize the maximal chains of G. Then we narrow the gap between group theory and decision-making using relations. Finally, a theory of the decision-making process in a fuzzy environment leads to a fuzzy version of capital budgeting. We define the goal, constraints and decision and show how they conflict with each other using membership function implications. We establish sets of intervals for projecting decision boundaries in general. We use the knowledge of triangular fuzzy numbers which are restricted field of fuzzy logic to evaluate investment projections.**Full Text:****Date Issued:**2006

**Authors:**Gideon, Frednard**Date:**2006**Subjects:**Fuzzy sets , Fuzzy systems , Abelian groups , Decision making**Language:**English**Type:**Thesis , Masters , MSc**Identifier:**vital:5417 , http://hdl.handle.net/10962/d1005231 , Fuzzy sets , Fuzzy systems , Abelian groups , Decision making**Description:**In this study we apply the knowledge of fuzzy sets to group structures and also to decision-making implications. We study fuzzy subgroups of finite abelian groups. We set G = Z[subscript p[superscript n]] + Z[subscript q[superscript m]]. The classification of fuzzy subgroups of G using equivalence classes is introduced. First, we present equivalence relations on fuzzy subsets of X, and then extend it to the study of equivalence relations of fuzzy subgroups of a group G. This is then followed by the notion of flags and keychains projected as tools for enumerating fuzzy subgroups of G. In addition to this, we use linear ordering of the lattice of subgroups to characterize the maximal chains of G. Then we narrow the gap between group theory and decision-making using relations. Finally, a theory of the decision-making process in a fuzzy environment leads to a fuzzy version of capital budgeting. We define the goal, constraints and decision and show how they conflict with each other using membership function implications. We establish sets of intervals for projecting decision boundaries in general. We use the knowledge of triangular fuzzy numbers which are restricted field of fuzzy logic to evaluate investment projections.**Full Text:****Date Issued:**2006

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

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

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

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

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

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

Studies in fuzzy groups

**Authors:**Makamba, B B**Date:**1993**Subjects:**Mathematics Fuzzy sets Fuzzy systems**Language:**English**Type:**Thesis , Doctoral , PhD**Identifier:**vital:5415 , http://hdl.handle.net/10962/d1005229**Description:**In this thesis we first extend the notion of fuzzy normality to the notion of normality of a fuzzy subgroup in another fuzzy group. This leads to the study of normal series of fuzzy subgroups, and this study includes solvable and nilpotent fuzzy groups, and the fuzzy version of the Jordan-Hõlder Theorem. Furthermore we use the notion of normality to study products and direct products of fuzzy subgroups. We present a notion of fuzzy isomorphism which enables us to state and prove the three well-known isomorphism theorems and the fact that the internal direct product of two normal fuzzy subgroups is isomorphic to the external direct product of the same fuzzy subgroups. A brief discussion on fuzzy subgroups generated by fuzzy subsets is also presented, and this leads to the fuzzy version of the Basis Theorem. Finally, the notion of direct product enables us to study decomposable and indecomposable fuzzy subgroups, and this study includes the fuzzy version of the Remak-Krull-Schmidt Theorem.**Full Text:****Date Issued:**1993

**Authors:**Makamba, B B**Date:**1993**Subjects:**Mathematics Fuzzy sets Fuzzy systems**Language:**English**Type:**Thesis , Doctoral , PhD**Identifier:**vital:5415 , http://hdl.handle.net/10962/d1005229**Description:**In this thesis we first extend the notion of fuzzy normality to the notion of normality of a fuzzy subgroup in another fuzzy group. This leads to the study of normal series of fuzzy subgroups, and this study includes solvable and nilpotent fuzzy groups, and the fuzzy version of the Jordan-Hõlder Theorem. Furthermore we use the notion of normality to study products and direct products of fuzzy subgroups. We present a notion of fuzzy isomorphism which enables us to state and prove the three well-known isomorphism theorems and the fact that the internal direct product of two normal fuzzy subgroups is isomorphic to the external direct product of the same fuzzy subgroups. A brief discussion on fuzzy subgroups generated by fuzzy subsets is also presented, and this leads to the fuzzy version of the Basis Theorem. Finally, the notion of direct product enables us to study decomposable and indecomposable fuzzy subgroups, and this study includes the fuzzy version of the Remak-Krull-Schmidt Theorem.**Full Text:****Date Issued:**1993

On a class of pseudo-differential operators in IRⁿ

**Authors:**Matjila, D M**Date:**1988**Subjects:**Pseudodifferential operators Operator theory**Language:**English**Type:**Thesis , Masters , MSc**Identifier:**vital:5392 , http://hdl.handle.net/10962/d1001981**Description:**The class of pseudo-differential operators with symbols from Sm (superscript) po̧̧ (subscipt)(Ωx IRⁿ) has been extensively studied.The main assumption which characterises this class of symbols is that a(x,Ȩ) є Sm (superscript)po̧̧ (subscipt)(Ωx IRⁿ) should have a polynomial growth in the Ȩ variable only. The x-variable is controlled on compact subsets of Ω. A polynomial growth in both the x and Ȩ variables on a C°°(lR²ⁿ) function a(x,Ȩ) gives rise to a different class of symbols and a corresponding class of operators. In this work, such symbols and the action of the operators on the functional spaces S(lRⁿ) , S'(lRⁿ) and the Sobolev spaces Qs (superscript) (lRⁿ) (s є lRⁿ) are studied. A study of the calculus (i.e. transposes, adjoints and compositions) and the functional analysis of these operators is done with special attention to L-boundedness and compactness. The class of hypoelliptic pseudo-differential operators in IRⁿ is introduced as a subclass of those considered earlier.These operators possess the property that they allow a pseudo- inverse or parametrix. In conclusion. the spectral theory of these operators is considered. Since a general spectral theory would be beyond the scope of this work, only some special cases of the pseudo-differential operators in IRⁿ are considered. A few applications of this spectral theory are discussed**Full Text:****Date Issued:**1988

**Authors:**Matjila, D M**Date:**1988**Subjects:**Pseudodifferential operators Operator theory**Language:**English**Type:**Thesis , Masters , MSc**Identifier:**vital:5392 , http://hdl.handle.net/10962/d1001981**Description:**The class of pseudo-differential operators with symbols from Sm (superscript) po̧̧ (subscipt)(Ωx IRⁿ) has been extensively studied.The main assumption which characterises this class of symbols is that a(x,Ȩ) є Sm (superscript)po̧̧ (subscipt)(Ωx IRⁿ) should have a polynomial growth in the Ȩ variable only. The x-variable is controlled on compact subsets of Ω. A polynomial growth in both the x and Ȩ variables on a C°°(lR²ⁿ) function a(x,Ȩ) gives rise to a different class of symbols and a corresponding class of operators. In this work, such symbols and the action of the operators on the functional spaces S(lRⁿ) , S'(lRⁿ) and the Sobolev spaces Qs (superscript) (lRⁿ) (s є lRⁿ) are studied. A study of the calculus (i.e. transposes, adjoints and compositions) and the functional analysis of these operators is done with special attention to L-boundedness and compactness. The class of hypoelliptic pseudo-differential operators in IRⁿ is introduced as a subclass of those considered earlier.These operators possess the property that they allow a pseudo- inverse or parametrix. In conclusion. the spectral theory of these operators is considered. Since a general spectral theory would be beyond the scope of this work, only some special cases of the pseudo-differential operators in IRⁿ are considered. A few applications of this spectral theory are discussed**Full Text:****Date Issued:**1988

(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

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

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

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

Comparison of different notions of compactness in the fuzzy topological space

**Authors:**Morapeli, E Z**Date:**1989**Subjects:**Fuzzy mathematics Fuzzy topology**Language:**English**Type:**Thesis , Masters , MSc**Identifier:**vital:5393 , http://hdl.handle.net/10962/d1001982**Description:**Various notions of compactness in a fuzzy topological space have been introduced by different authors. The aim of this thesis is to compare them. We find that in a T₂ space (in the sense that no fuzzy net converges to two fuzzy points with different supports) all these notions are equivalent for the whole space. Furthermore, for N-compactness and f-compactness (being the only notions that are defined for an arbitrary fuzzy subset) we have equivalence under a stronger condition, namely, a T₂ space in the sense that every prime prefilter has an adherence that is non-zero in at most one point**Full Text:****Date Issued:**1989

**Authors:**Morapeli, E Z**Date:**1989**Subjects:**Fuzzy mathematics Fuzzy topology**Language:**English**Type:**Thesis , Masters , MSc**Identifier:**vital:5393 , http://hdl.handle.net/10962/d1001982**Description:**Various notions of compactness in a fuzzy topological space have been introduced by different authors. The aim of this thesis is to compare them. We find that in a T₂ space (in the sense that no fuzzy net converges to two fuzzy points with different supports) all these notions are equivalent for the whole space. Furthermore, for N-compactness and f-compactness (being the only notions that are defined for an arbitrary fuzzy subset) we have equivalence under a stronger condition, namely, a T₂ space in the sense that every prime prefilter has an adherence that is non-zero in at most one point**Full Text:****Date Issued:**1989

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