(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.
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- Date Issued: 1992
A computer analysis of some of the Harrison metrics
- Authors: Sadler, Christopher John
- Date: 1975
- Subjects: Computer science -- Mathematics , Software measurement
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5427 , http://hdl.handle.net/10962/d1013151
- Description: In his paper B.K.Harrison concludes with the observation that his "solutions ... are presented as raw material for further research in General Relativity". In the same spirit, the present work started out as an attempt to process that raw material in a production-line powered by a computer. Harrison's solutions uould be fed in at one end, and the finished product, as yet undecided, would appear at the other. In the event, however, the project became more like an exercise in quality control, to continue the analogy. A search was made for algebraic criteria which would distinguish between those solutions which were acceptable for further analysis with particular regard to Gravitational radiation, and those which were not. Regrettably, no criteria could be found which characterised radiative solutions unequivocally, and, at the same time, lent themselves to a computer approach. The result is that the discussion of radiative solutions has had to be relegated to an appendix (Appendix 1), while the main body of the work is concerned with the determination of those quantities (the Newman-Penrose scalars) which would seem to be the foundation of any future computer-based analysis of gravitational radiation. Chapter 1 is an account of the underlying mathematical formulation, defining the terms, concepts and processes involved. In Chapter 2 the transformation of some of the ideas of Chapter 1 into computer software is presented. Chapter 3 is concerned with the specific metrics (the Harrison metrics) and the extent to which they have heen processed. The project has leaned heavily on papers by Harrison for the "raw material", by D' Inverno and Russell Clark, who pioneered the techniques and classified the Harrison metrics, and by Sachs for the treatment of gravitational radiation. However, the analysis of diagonal metrics, the special tetrad of Chapter 2 and the results in Appendix 2 are new.
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- Date Issued: 1975
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.
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- Date Issued: 2006
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
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- Date Issued: 2014
A study of the existence of equilibrium in mathematical economics
- Authors: Xotyeni, Zukisa Gqabi
- Date: 2008
- Subjects: Equilibrium (Economics) -- Mathematical models , Macroeconomics -- Mathematical models , Economics -- Mathematical models , Welfare economics -- Mathematical models , Lattice theory , Economics, Mathematical
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5418 , http://hdl.handle.net/10962/d1005232 , Equilibrium (Economics) -- Mathematical models , Macroeconomics -- Mathematical models , Economics -- Mathematical models , Welfare economics -- Mathematical models , Lattice theory , Economics, Mathematical
- Description: In this thesis we define and study the existence of an equilibrium situation in which producers maximize their profits relative to the production vectors in their production sets, consumers satisfy their preferences in their consumption sets under certain budget constraint, and for every commodity total demand equals total supply. This competitive equilibrium situation is referred to as the Walrasian equilibrium. The existence of this equilibrium is investigated from a various mathematical points of view. These include microeconomic theory, simplicial spaces, global analysis and lattice theory.
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- Date Issued: 2008
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
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- Date Issued: 1988
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.
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- Date Issued: 1992
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.
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- Date Issued: 1998
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.
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- Date Issued: 2002
Characterization of stratified L-topological spaces by convergence of stratified L-filters
- Authors: Orpen, David Lisle
- Date: 2011
- Subjects: Topology , Generalized spaces , Filters (Mathematics) , Topological spaces
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5402 , http://hdl.handle.net/10962/d1005216 , Topology , Generalized spaces , Filters (Mathematics) , Topological spaces
- Description: For the case where L is an ecl-premonoid, we explore various characterizations of SL-topological spaces, in particular characterization in terms of a convergence function lim: FS L(X) ! LX. We find we have to introduce a new axiom , L on the lim function in order to completely describe SL-topological spaces, which is not required in the case where L is a frame. We generalize the classical Kowalski and Fischer axioms to the lattice context and examine their relationship to the convergence axioms. We define the category of stratified L-generalized convergence spaces, as a generalization of the classical convergence spaces and investigate conditions under which it contains the category of stratified L-topological spaces as a reflective subcategory. We investigate some subcategories of the category of stratified L-generalized convergence spaces obtained by generalizing various classical convergence axioms.
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- Date Issued: 2011
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
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.
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- Date Issued: 1992
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
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- Date Issued: 1983
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.
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- Date Issued: 2002
Contributions to the study of a class of optimal control problems on the matrix lie group SO(3)
- Authors: Rodgerson, Joanne Kelly
- Date: 2009 , 2013-07-12
- Subjects: Matrix groups , Lie groups , Maximum principles (Mathematics) , Elliptic functions , Extremal problems (Mathematics)
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5421 , http://hdl.handle.net/10962/d1007199 , Matrix groups , Lie groups , Maximum principles (Mathematics) , Elliptic functions , Extremal problems (Mathematics)
- Description: The purpose of this thesis is to investigate a class of four left-invariant optimal control problems on the special orthogonal group SO(3). The set of all control-affine left-invariant control systems on SO(3) can, without loss, be reduced to a class of four typical controllable left-invariant control systems on SO(3) . The left-invariant optimal control problem on SO(3) involves finding a trajectory-control pair on SO (3), which minimizes a cost functional, and satisfies the given dynamical constraints and boundary conditions in a fixed time. The problem is lifted to the cotangent bundle T*SO(3) = SO(3) x so (3)* using the optimal Hamiltonian on so(3)*, where the maximum principle yields the optimal control. In a contribution to the study of this class of optimal control problems on SO(3), the extremal equations on so(3)* (ident ified with JR3) are integrated via elliptic functions to obtain explicit expressions for the solution curves in each typical case. The energy-Casimir method is used to give sufficient conditions for non-linear stability of the equilibrium states. , KMBT_363 , Adobe Acrobat 9.54 Paper Capture Plug-in
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- Date Issued: 2009
Contributions to the theory of group rings
- 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
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- Date Issued: 1979
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
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.
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- Date Issued: 2009
Fixed points of single-valued and multi-valued mappings with applications
- Authors: Stofile, Simfumene
- Date: 2013
- Subjects: Fixed point theory Mappings (Mathematics) Coincidence theory (Mathematics) Metric spaces Uniform spaces Set-valued maps
- Language: English
- Type: Thesis , Doctoral , PhD
- Identifier: vital:5397 , http://hdl.handle.net/10962/d1002960
- Description: The relationship between the convergence of a sequence of self mappings of a metric space and their fixed points, known as the stability (or continuity) of fixed points has been of continuing interest and widely studied in fixed point theory. In this thesis we study the stability of common fixed points in a Hausdorff uniform space whose uniformity is generated by a family of pseudometrics, by using some general notations of convergence. These results are then extended to 2-metric spaces due to S. Gähler. In addition, a well-known theorem of T. Suzuki that generalized the Banach Contraction Principle is also extended to 2-metric spaces and applied to obtain a coincidence theorem for a pair of mappings on an arbitrary set with values in a 2-metric space. Further, we prove the existence of coincidence and fixed points of Ćirić type weakly generalized contractions in metric spaces. Subsequently, the above result is utilized to discuss applications to the convergence of modified Mann and Ishikawa iterations in a convex metric space. Finally, we obtain coincidence, fixed and stationary point results for multi-valued and hybrid pairs of mappings on a metric space.
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- Date Issued: 2013
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