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
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.
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- 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
Methods for designing and optimizing fuzzy controllers
- Authors: Swartz, Andre Michael
- Date: 2000
- Subjects: Fuzzy sets , Fuzzy systems , Automatic control
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5412 , http://hdl.handle.net/10962/d1005226 , Fuzzy sets , Fuzzy systems , Automatic control
- Description: We start by discussing fuzzy sets and the algebra of fuzzy sets. We consider some properties of fuzzy modeling tools. This is followed by considering the Mamdani and Sugeno models for designing fuzzy controllers. Various methods for using sets of data for desining controllers are discussed. This is followed by a chapter illustrating the use of genetic algorithms in designing and optimizing fuzzy controllers.Finally we look at some previous applications of fuzzy control in telecommunication networks, and illustrate a simple application that was developed as part of the present work.
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- Date Issued: 2000
- Authors: Swartz, Andre Michael
- Date: 2000
- Subjects: Fuzzy sets , Fuzzy systems , Automatic control
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5412 , http://hdl.handle.net/10962/d1005226 , Fuzzy sets , Fuzzy systems , Automatic control
- Description: We start by discussing fuzzy sets and the algebra of fuzzy sets. We consider some properties of fuzzy modeling tools. This is followed by considering the Mamdani and Sugeno models for designing fuzzy controllers. Various methods for using sets of data for desining controllers are discussed. This is followed by a chapter illustrating the use of genetic algorithms in designing and optimizing fuzzy controllers.Finally we look at some previous applications of fuzzy control in telecommunication networks, and illustrate a simple application that was developed as part of the present work.
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- Date Issued: 2000
Fuzzy ideals in commutative rings
- Authors: Sekaran, Rajakrishnar
- Date: 1995
- Subjects: Commutative rings , Fuzzy algebra
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5407 , http://hdl.handle.net/10962/d1005221 , Commutative rings , Fuzzy algebra
- Description: In this thesis, we are concerned with various aspects of fuzzy ideals of commutative rings. The central theorem is that of primary decomposition of a fuzzy ideal as an intersection of fuzzy primary ideals in a commutative Noetherian ring. We establish the existence and the two uniqueness theorems of primary decomposition of any fuzzy ideal with membership value 1 at the zero element. In proving this central result, we build up the necessary tools such as fuzzy primary ideals and the related concept of fuzzy maximal ideals, fuzzy prime ideals and fuzzy radicals. Another approach explores various characterizations of fuzzy ideals, namely, generation and level cuts of fuzzy ideals, relation between fuzzy ideals, congruences and quotient fuzzy rings. We also tie up several authors' seemingly different definitions of fuzzy prime, primary, semiprimary and fuzzy radicals available in the literature and show some of their equivalences and implications, providing counter-examples where certain implications fail.
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- Date Issued: 1995
- Authors: Sekaran, Rajakrishnar
- Date: 1995
- Subjects: Commutative rings , Fuzzy algebra
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5407 , http://hdl.handle.net/10962/d1005221 , Commutative rings , Fuzzy algebra
- Description: In this thesis, we are concerned with various aspects of fuzzy ideals of commutative rings. The central theorem is that of primary decomposition of a fuzzy ideal as an intersection of fuzzy primary ideals in a commutative Noetherian ring. We establish the existence and the two uniqueness theorems of primary decomposition of any fuzzy ideal with membership value 1 at the zero element. In proving this central result, we build up the necessary tools such as fuzzy primary ideals and the related concept of fuzzy maximal ideals, fuzzy prime ideals and fuzzy radicals. Another approach explores various characterizations of fuzzy ideals, namely, generation and level cuts of fuzzy ideals, relation between fuzzy ideals, congruences and quotient fuzzy rings. We also tie up several authors' seemingly different definitions of fuzzy prime, primary, semiprimary and fuzzy radicals available in the literature and show some of their equivalences and implications, providing counter-examples where certain implications fail.
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- Date Issued: 1995
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.
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- 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
(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.
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- Date Issued: 1992
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
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- 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
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
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