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.

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. The 1st chapter give a brief summary of the arithmetic of fuzzy real numbers and the fuzzy normed algebra M(I). Also we explain a few preliminary definitions and results required in the later chapters. Fuzzy real numbers are introduced by Hutton,B [HU] and Rodabaugh, S.E[ROD]. Our definition slightly differs from this with an additional minor restriction. The definition of Clementina Felbin [CL1] is entirely different. The notations of [HU]and [M;Y] are retained inspite of the slight difference in the concept.the 3rd chapter In this chapter using the completion M'(I) of M(I) we give a fuzzy extension of real Hahn-Banch theorem. Some consequences of this extension are obtained. The idea of real fuzzy linear functional on fuzzy normed linear space is introduced. Some of its properties are studied. In the complex case we get only a slightly weaker analogue for the Hahn-Banch theorem, than the one [B;N] in the crisp case

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.

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.