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
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
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
Case studies of equivalent fuzzy subgroups of finite abelian groups
- Authors: Ngcibi, Sakhile L
- Date: 2002
- Subjects: Abelian groups , Fuzzy sets
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5401 , http://hdl.handle.net/10962/d1005215 , Abelian groups , Fuzzy sets
- Description: The broad goal is to classify all fuzzy subgroups of a given type of finite group. P.S. Das introduced the ntion of level subgroups to characterize fuzzy subgroups of finite grouops. The notion of equivalence of fuzzy subgroups which is used in this thesis was first introduced by Murali and Makamba. We use this equivalence to charterise fuzzy subgroups of inite Abelian groups (p-groups in particular) for a specified prime p. We characterize some crisp subgroups of p-groups and investigate some cases on equi valent fuzzy subgroups.
- Full Text:
- Date Issued: 2002
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
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
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.
- Full Text:
- Date Issued: 2000
Aspects of renormalisation in some quantum field theories
- Authors: Roy, Alan A
- Date: 1998
- Subjects: Quantum field theory , Renormalization group
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5400 , http://hdl.handle.net/10962/d1005214 , Quantum field theory , Renormalization group
- Description: Renormalisation is an important aspect of Quantum Field Theory. It is used to create physically meaningful theories and some major developments took place in the 1970's and onwards. We consider Renormalisation in its application to the theories of ψ⁴ , Quantum Electrodynamics, Quantum Chromodynamics and the Background Field Method. Feynman diagrams are used to illustrate many of the concepts.
- Full Text:
- Date Issued: 1998
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
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.
- Full Text:
- Date Issued: 1995
Tangentially symplectic foliations
- Authors: Remsing, Claudiu Cristian
- Date: 1994
- Subjects: Geometry Problems, exercises, etc Geometry, Differential Symplectic manifolds Poisson manifolds Foliations (Mathematics)
- Language: English
- Type: Thesis , Doctoral , PhD
- Identifier: vital:5419 , http://hdl.handle.net/10962/d1005233
- Description: This thesis is concerned principally with tangential geometry and the applications of these concepts to tangentially symplectic foliations. The subject of tangential geometry is still at an elementary stage. The author here systematises current concepts and results and extends them, leading to the definition of vertical connections and vertical G-structures. Tangentially symplectic foliations are then characterised in terms of vertical symplectic forms. Some significant particular cases are discussed.
- Full Text:
- Date Issued: 1994
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
(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
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
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
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
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
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
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
Conductivity profiles for a horizontally uniform earth
- Authors: Murrell, Hugh Crozier
- Date: 1983
- Subjects: Coen, Shimon -- Criticism and interpretation Wang-Ho Yu, Michael -- Criticism and interpretation Algorithms
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5395 , http://hdl.handle.net/10962/d1001984
- Description: An investigation is made into the mathematics behind the noniterative inversion algorithm of Shimon Coen and Michael Wang-Ho Yu [1981]. The algorithm determines the conductivity profile of a horizontally uniform earth from surface measurements of apparent resistivity with a Schlumberger array. The algorithm is checked by performing the inversion on both artifical and raw field data
- Full Text:
- Date Issued: 1983