On Some Infinite Convex Invariants

The present study on some infinite convex invariants. The origin of convexity can be traced back to the period of Archimedes and Euclid. At the turn of the nineteenth centaury , convexicity became an independent branch of mathematics with its own problems, methods and theories. The convexity can be sorted out into two kinds, the first type deals with generalization of particular problems such as separation of convex sets[EL], extremality[FA], [DAV] or continuous selection Michael[M1] and the second type involved with a multi- purpose system of axioms. The theory of convex invariants has grown out of the classical results of Helly, Radon and Caratheodory in Euclidean spaces. Levi gave the first general definition of the invariants Helly number and Radon number. The notation of a convex structure was introduced by Jamison[JA4] and that of generating degree was introduced by Van de Vel[VAD8]. We also prove that for a non-coarse convex structure, rank is less than or equal to the generating degree, and also generalize Tverberg’s theorem using infinite partition numbers. Compare the transfinite topological and transfinite convex dimensions

Modification of Polyethylenes by Reactive Extrusion

The main aim of the study was to optimise the reactive extrusion conditions in the conventional modification processes of polyethylenes in a single screw extruder.The optimum conditions for peroxide crosslinking of low density polyethylene (LDPE), linear low density polyethylene (LLDPE) and their blend were determined in a torque rheometer. The actual reactive extrusion was performed in a laboratory single screw extruder using the optimum parameters. The influence of the coagent, triaUyl cyanurate (TAC), on the cross linking of low density polyethylene in the presence of peroxide was also investigated. The peroxide crosslinking was found to improve the mechanical properties and the thermal stability of the polyethylenes. The efficiency of crosslinking was found to be improved by the addition of coagent such as TAC.The optimum conditions for silane grafting viz temperature, shear rate, silane and DCP concentrations were determined on a torque rheometer in the case of LDPE, LLDPE and their blend. Silane grafting of LDPE in the presence of peroxide was performed with and without addition of water. Compounding of such mixtures in the melt at high temperatures caused decomposition of the peroxide and grafting of alkoxy silyl groups to the polyethylene chains.The optimum parameters for maleic anhydride modification of LDPE, LLDPE and their blend were determined. The grafting reaction was confinned by FTIR spectroscopy. Modification of polyethylenes with maleic anhydride in the presence of dicumyl peroxide was found to be useful in improving mechanical properties. The improvement was found to be mainly due to the grafting of carboxyl group and formation of crosslinks between the chains. The cross linking initiated improvements indicate extended property profiles and new application fields for polyethylenes.On the whole the study shows that the optimum conditions for modifying polyethylenes can be determined on a torque rheometer and actual modification can be performed in a single screw extruder by employing the optimum parameters for improved mechanical! thermal behaviour without seriously affecting their processing behaviour.

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.