`A Study on Ultra L-topologies and Lattices of L-topologies'

The thesis is divided into nine chapters including introduction. Mainly we determine ultra L-topologies in the lattice of L- topologies and study their properties. We nd some sublattices in the lattice of L-topologies and study their properties. Also we study the lattice structure of the set of all L-closure operators on a set X.

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

Some Lattice Problems In Fuzzy Set Theory And Fuzzy Topology

It is believed that every fuzzy generalization should be formulated in such a way that it contain the ordinary set theoretic notion as a special case. Therefore the definition of fuzzy topology in the line of C.L.CHANG E9] with an arbitrary complete and distributive lattice as the membership set is taken. Almost all the results proved and presented in this thesis can, in a sense, be called generalizations of corresponding results in ordinary set theory and set topology. However the tools and the methods have to be in many of the cases, new. Here an attempt is made to solve the problem of complementation in the lattice of fuzzy topologies on a set. It is proved that in general, the lattice of fuzzy topologies is not complemented. Complements of some fuzzy topologies are found out. It is observed that (L,X) is not uniquely complemented. However, a complete analysis of the problem of complementation in the lattice of fuzzy topologies is yet to be found out

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.

Novel Reactions of TRIS(2-Thienyl)Methane and TRIS(2-Furyal)Methane

A systematic investigation of the reactivity and functionalization of two heterocyclic analogs of triphenylmethane , namely tris(2-thienyl)methane and tris(2-furyl)methane have been carried out and the results are presented in this thesis entitled "NOVEL REACTIONS OF TRIS(2-THIENYL)METHANE AND TRIS(2-FURYL)METHANE.". The history of organic free radicals dates back to Gomberg's monumental discovery of the triphenylmethyl radical in 1900. The heterocyclic analogs of triarylmethane are also interesting from the vantage point of their transformation to the corresponding radicals akin to Gomberg ' s triphenylmethyl radical and also they are prone to further transformation leading to three dimensionally elongated molecules such as dendrimers. Dendritic architectures are one of the most pervasive topologies observed in nature at the macro- and microdimensional length devices. Because of their ability to combine both organic and inorganic compounds and their propensity to either encapsulate or be engineered into unimolecular functional devices , dendrimers are versatile amongst existing nanoscale building blocks and materials.

Tridentate Schiff Bases of Some Aroylhydrazides and their Metal Complexes: Spectral and Structural Studies

The present work deals with the complexation of Schiff bases of aroylhydrazides with various transition metal ions. The hydrazone systems selected for study are capable of forming bridged polymeric structures which is one of the fascinating subjects in the crystal engineering of coordination polymers owing to their attractive new topologies and intriguing structural features. Complexation with metal ions like copper, manganese, vanadium, nickel, palladium, zinc and cadmium are tried. Various spectral techniques are employed for characterization. The structures of some complexes have been well established by single crystal X-ray diffraction studies.The work is presented in seven chapters and the last section deals with summary and conclusion. The studies reveal that the aroylhydrazone systems vary in their geometrical configuration depending on the substituents. The coordination modes of the ligands also differ upon chelating with metal ions. One of the hydrazone system selected for study proved that it could give rise to polymeric metal complexes.