Studies on Fuzzy Graphs

In this thesis an attempt to develop the properties of basic concepts in fuzzy graphs such as fuzzy bridges, fuzzy cutnodes, fuzzy trees and blocks in fuzzy graphs have been made. The notion of complement of a fuzzy graph is modified and some of its properties are studied. Since the notion of complement has just been initiated, several properties of G and G available for crisp graphs can be studied for fuzzy graphs also. Mainly focused on fuzzy trees defined by Rosenfeld in [10] , several other types of fuzzy trees are defined depending on the acyclicity level of a fuzzy graph. It is observed that there are selfcentered fuzzy trees. Some operations on fuzzy graphs and prove that complement of the union two fuzzy graphs is the join of their complements and complement of the join of two fuzzy graphs is union of their complements. The study of fuzzy graphs made in this thesis is far from being complete. The wide ranging applications of graph theory and the interdisciplinary nature of fuzzy set theory, if properly blended together could pave a way for a substantial growth of fuzzy graph theory.

Some Generalizations of Fuzzy Metrizability

The main purpose of the study is to extent concept of the class of spaces called ‘generalized metric spaces’ to fuzzy context and investigates its properties. Any class of spaces defined by a property possessed by all metric spaces could technically be called as a class of ‘generalized metric spaces’. But the term is meant for classes, which are ‘close’ to metrizable spaces in some under certain kinds of mappings. The theory of generalized metric spaces is closely related to ‘metrization theory’. The class of spaces likes Morita’s M- spaces, Borges’s w-spaces, Arhangelskii’s p-spaces, Okuyama’s  spaces have major roles in the theory of generalized metric spaces. The thesis introduces fuzzy metrizable spaces, fuzzy submetrizable spaces and proves some characterizations of fuzzy submetrizable spaces, and also the fuzzy generalized metric spaces like fuzzy w-spaces, fuzzy Moore spaces, fuzzy M-spaces, fuzzy k-spaces, fuzzy -spaces study of their properties, prove some equivalent conditions for fuzzy p-spaces. The concept of a network is one of the most useful tools in the theory of generalized metric spaces. The -spaces is a class of generalized metric spaces having a network.

Fuzzy Topological Games and Related Topics

The main purpose of study is to extend the concept of the topological game G(K, X) and some other kinds of games into fuzzy topological games and to obtain some results regarding them. Owing to the fact that topological games have plenty of applications in covering properties, it made an attempt to explore some inter relations of games and covering properties in fuzzy topological spaces. Even though the main focus is on fuzzy para-meta compact spaces and closure preserving shading families, some brief sketches regarding fuzzy P-spaces and Shading Dimension is also provided. In a topological game players choose some objects related to the topological structure of a space such as points, closed subsets, open covers etc. More over the condition on a play to be winning for a player may also include topological notions such as closure, convergence, etc. It turns out that topological games are related to the Baire property, Baire spaces, Completeness properties, Convergence properties, Separation properties, Covering and Base properties, Continuous images, Suslin sets, Singular spaces etc.

A Study Of Frames In The Fuzzy And Intuitionist Fuzzy Contexts

The fuzzy set theory has a wider scope of applicability than classical set theory in solving various problems. Fuzzy set theory in the last three decades as a formal theory which got formalized by generalizing the original ideas and concepts in classical mathematical areas and as a very powerful modeling language, that can cope with a large fraction of uncertainties of real life situations. In Intuitionistic Fuzzy sets a new component degree of non membership in addition to the degree of membership in the case of fuzzy sets with the requirement that their sum be less than or equal to one. The main objective of this thesis is to study frames in Fuzzy and Intuitionistic Fuzzy contexts. The thesis proved some results such as ifµ is a fuzzy subset of a frame F, then µ is a fuzzy frame of F iff each non-empty level subset µt of µ is a subframe of F, the category Fuzzfrm of fuzzy frames has products and the category Fuzzfrm of fuzzy frames is complete. It define a fuzzy-quotient frame of F to be a fuzzy partition of F, that is, a subset of IF and having a frame structure with respect to new operations and study the notion of intuitionistic fuzzy frames and obtain some results and introduce the concept of Intuitionistic fuzzy Quotient frames. Finally it establish the categorical link between frames and intuitionistic fuzzy topologies.

Consequence modelling,vulnerability assessment and fuzzy fault tree analysis of hazardous storages in an industrial area

The hazards associated with major accident hazard (MAH) industries are fire, explosion and toxic gas releases. Of these, toxic gas release is the worst as it has the potential to cause extensive fatalities. Qualitative and quantitative hazard analyses are essential for the identitication and quantification of the hazards associated with chemical industries. This research work presents the results of a consequence analysis carried out to assess the damage potential of the hazardous material storages in an industrial area of central Kerala, India. A survey carried out in the major accident hazard (MAH) units in the industrial belt revealed that the major hazardous chemicals stored by the various industrial units are ammonia, chlorine, benzene, naphtha, cyclohexane, cyclohexanone and LPG. The damage potential of the above chemicals is assessed using consequence modelling. Modelling of pool fires for naphtha, cyclohexane, cyclohexanone, benzene and ammonia are carried out using TNO model. Vapor cloud explosion (VCE) modelling of LPG, cyclohexane and benzene are carried out using TNT equivalent model. Boiling liquid expanding vapor explosion (BLEVE) modelling of LPG is also carried out. Dispersion modelling of toxic chemicals like chlorine, ammonia and benzene is carried out using the ALOHA air quality model. Threat zones for different hazardous storages are estimated based on the consequence modelling. The distance covered by the threat zone was found to be maximum for chlorine release from a chlor-alkali industry located in the area. The results of consequence modelling are useful for the estimation of individual risk and societal risk in the above industrial area.Vulnerability assessment is carried out using probit functions for toxic, thermal and pressure loads. Individual and societal risks are also estimated at different locations. Mapping of threat zones due to different incident outcome cases from different MAH industries is done with the help of Are GIS.Fault Tree Analysis (FTA) is an established technique for hazard evaluation. This technique has the advantage of being both qualitative and quantitative, if the probabilities and frequencies of the basic events are known. However it is often difficult to estimate precisely the failure probability of the components due to insufficient data or vague characteristics of the basic event. It has been reported that availability of the failure probability data pertaining to local conditions is surprisingly limited in India. This thesis outlines the generation of failure probability values of the basic events that lead to the release of chlorine from the storage and filling facility of a major chlor-alkali industry located in the area using expert elicitation and proven fuzzy logic. Sensitivity analysis has been done to evaluate the percentage contribution of each basic event that could lead to chlorine release. Two dimensional fuzzy fault tree analysis (TDFFTA) has been proposed for balancing the hesitation factor invo1ved in expert elicitation .

Studies in Fuzzy Commutative Algebra

The main objective of this thesis was to extend some basic concepts and results in module theory in algebra to the fuzzy setting.The concepts like simple module, semisimple module and exact sequences of R-modules form an important area of study in crisp module theory. In this thesis generalising these concepts to the fuzzy setting we have introduced concepts of ‘simple and semisimple L-modules’ and proved some results which include results analogous to those in crisp case. Also we have defined and studied the concept of ‘exact sequences of L-modules’.Further extending the concepts in crisp theory, we have introduced the fuzzy analogues ‘projective and injective L-modules’. We have proved many results in this context. Further we have defined and explored notion of ‘essential L-submodules of an L-module’. Still there are results in crisp theory related to the topics covered in this thesis which are to be investigated in the fuzzy setting. There are a lot of ideas still left in algebra, related to the theory of modules, such as the ‘injective hull of a module’, ‘tensor product of modules’ etc. for which the fuzzy analogues are not defined and explored.

A characterization of fuzzy trees

In this paper some properties of fuzzy bridges are studied.A characterization of fuzzy trees is obtained using these concepts.

Studies on Fuzzy Matroitls and Related Topics

The doctoral thesis focuses on the Studies on fuzzy Matroids and related topics.Since the publication of the classical paper on fuzzy sets by L. A. Zadeh in 1965.the theory of fuzzy mathematics has gained more and more recognition from many researchers in a wide range of scientific fields. Among various branches of pure and applied mathematics, convexity was one of the areas where the notion of fuzzy set was applied. Many researchers have been involved in extending the notion of abstract convexity to the broader framework of fuzzy setting. As a result, a number of concepts have been formulated and explored. However. many concepts are yet to be fuzzified. The main objective of this thesis was to extend some basic concepts and results in convexity theory to the fuzzy setting. The concept like matroids, independent structures. classical convex invariants like Helly number, Caratheodoty number, Radon number and Exchange number form an important area of study in crisp convexity theory. In this thesis, we try to generalize some of these concepts to the fuzzy setting. Finally, we have defined different types of fuzzy matroids derived from vector spaces and discussed some of their properties.

Some problems in set topology relating group of homeomorphisms and order

In this thesis we investigate some problems in set theoretical topology related to the concepts of the group of homeomorphisms and order. Many problems considered are directly or indirectly related to the concept of the group of homeomorphisms of a topological space onto itself. Order theoretic methods are used extensively. Chapter-l deals with the group of homeomorphisms. This concept has been investigated by several authors for many years from different angles. It was observed that nonhomeomorphic topological spaces can have isomorphic groups of homeomorphisms. Many problems relating the topological properties of a space and the algebraic properties of its group of homeomorphisms were investigated. The group of isomorphisms of several algebraic, geometric, order theoretic and topological structures had also been investigated. A related concept of the semigroup of continuous functions of a topological space also received attention

A study on fuzzy semi inner product spaces

Mathematical models are often used to describe physical realities. However, the physical realities are imprecise while the mathematical concepts are required to be precise and perfect. Even mathematicians like H. Poincare worried about this. He observed that mathematical models are over idealizations, for instance, he said that only in Mathematics, equality is a transitive relation. A first attempt to save this situation was perhaps given by K. Menger in 1951 by introducing the concept of statistical metric space in which the distance between points is a probability distribution on the set of nonnegative real numbers rather than a mere nonnegative real number. Other attempts were made by M.J. Frank, U. Hbhle, B. Schweizer, A. Sklar and others. An aspect in common to all these approaches is that they model impreciseness in a probabilistic manner. They are not able to deal with situations in which impreciseness is not apparently of a probabilistic nature. This thesis is confined to introducing and developing a theory of fuzzy semi inner product spaces.