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.

Geometric Algebra and Einstein’s Electron

This thesis entitled Geometric algebra and einsteins electron: Deterministic field theories .The work in this thesis clarifies an important part of Koga’s theory.Koga also developed a theory of the electron incorporating its gravitational field, using his substitutes for Einstein’s equation.The third chapter deals with the application of geometric algebra to Koga’s approach of the Dirac equation. In chapter 4 we study some aspects of the work of mendel sachs (35,36,37,).Sachs stated aim is to show how quantum mechanics is a limiting case of a general relativistic unified field theory.Chapter 5 contains a critical study and comparison of the work of Koga and Sachs. In particular, we conclude that the incorporation of Mach’s principle is not necessary in Sachs’s treatment of the Dirac equation.

Some Problems In Algebra And Topology A Study Of Fuzzy Convexity With Special Reference To Separation Properties

This thesis is a study of abstract fuzzy convexity spaces and fuzzy topology fuzzy convexity spaces No attempt seems to have been made to develop a fuzzy convexity theoryin abstract situations. The purpose of this thesis is to introduce fuzzy convexity theory in abstract situations

Spectral Analysis of Bounded Self-adjoint operators - A Linear Algebraic Approach

This thesis Entitled Spectral theory of bounded self-adjoint operators -A linear algebraic approach.The main results of the thesis can be classified as three different approaches to the spectral approximation problems. The truncation method and its perturbed versions are part of the classical linear algebraic approach to the subject. The usage of block Toeplitz-Laurent operators and the matrix valued symbols is considered as a particular example where the linear algebraic techniques are effective in simplifying problems in inverse spectral theory. The abstract approach to the spectral approximation problems via pre-conditioners and Korovkin-type theorems is an attempt to make the computations involved, well conditioned. However, in all these approaches, linear algebra comes as the central object. The objective of this study is to discuss the linear algebraic techniques in the spectral theory of bounded self-adjoint operators on a separable Hilbert space. The usage of truncation method in approximating the bounds of essential spectrum and the discrete spectral values outside these bounds is well known. The spectral gap prediction and related results was proved in the second chapter. The discrete versions of Borg-type theorems, proved in the third chapter, partly overlap with some known results in operator theory. The pure linear algebraic approach is the main novelty of the results proved here.

Gill Nets of Kerala

The fishing industry the world over is passing through a critical situation.The landings of marine fishes seemed to have reached saturation in major fishing areas of the world.As a general rule fisheries employing fishing gear and techniques used by small scale and artisanal fishermen either from shore or from onboard small fishing craft come under small-scale fisheries.This study on gill nets of Kerala, the fishing method depended upon by maximum fishermen of the state focuses on the importance of this selective and low energy fishing method in the marine fishing sector of the state.The study opens with the conceptual framework by briefly reviewing the crisis in the marine fisheries sector. Maximum fishermen depend upon gill net, which is, an important selective and low energy fishing gear. A review of relevant literature on aspects such as material, selectivity and techno-economic efficiency together with scope and main objectives of the study form the major part of the compass of the introductory chapter.This survey provided the inputs for selection of centres. The chapter presents the basis for selection of sample centres, sample units and methodology for field and experimental study.The subject matter of the fourth chapter is a basic study on gear aterials. The weathering resistance, which is an important criterion to assess the material performance, was studied for polyamide monofilament in comparison to polyamide multifilament and polyethylene twisted monofilament.The study provides supporting evidence of oxidation and characteristic C-O stretching in polyethylene and cyclic lactam .formation and presence of OH in polyamide.The study indicates that small mesh gill netting can be encouraged as a selective fishing method in the inshore waters with restrained use of 30 and 32 mm mesh sizes. The economic efficiency was assessed using standard indices such as rate of return, internal rate of return, pay back period, fishery income, energy efficiency and factor productivity. The effect of size and cost of capital and cost of production on the economics of operation is also discussed in this chapter. It was observed that level of technology did not have direct effect on economic performance.

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

Mini-trawls for Estuarine Fishing Kasargod District

Mi ni -trawls are operated by the artisanal fishermen from small wooden non-motorised canoes. Shrimp, fish and crab trawls wi th head rope length rang ing from 3.5-8 m, made of Po lyethy lene mon ofila ment (PE) twisted and Polyamide mullifilament (PA) rigged to 6-7 kll fla t rectangular wooden otter boards are common in the lower reaches of Kariango de and Chandrag iri rive rs. Since the trawling speed is less, ca tch is do minated by crus taceans. Less scope ratio also may be affecting the catching efficiency of the gear. This pape r deals with the design, operation and economics of mini traw ling carried out by a group of fisherme n in the above rivers of Kasargod district Kerala state.