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.

Box Products in Fuzzy Topological Spaces and Related Topics

The topology as the product set with a base chosen as all products of open sets in the individual spaces. This topology is known as box topology. The main objective of this study is to extend the concept of box products to fuzzy box products and to obtain some results regarding them. Owing to the fact that box products have plenty of applications in uniform and covering properties, here made an attempt to explore some inter relations of fuzzy uniform properties and fuzzy covering properties in fuzzy box products. Even though the main focus is on fuzzy box products, some brief sketches regarding hereditarily fuzzy normal spaces and fuzzy nabla product is also provided. The main results obtained include characterization of fuzzy Hausdroffness and fuzzy regularity of box products of fuzzy topological spaces. The investigation of the completeness of fuzzy uniformities in fuzzy box products proved that a fuzzy box product of spaces is fuzzy topologically complete if each co-ordinate space is fuzzy topologically complete. The thesis also prove that the fuzzy box product of a family of fuzzy α-paracompact spaces is fuzzy topologically complete. In Fuzzy box product of hereditarily fuzzy normal spaces, the main result obtained is that if a fuzzy box product of spaces is hereditarily fuzzy normal ,then every countable subset of it is fuzzy closed. It also deals with the notion of fuzzy nabla product of spaces which is a quotient of fuzzy box product. Here the study deals the relation connecting fuzzy box product and fuzzy nabla product

Study on Fuzzy Ordered FuzzyTopological Spaces

In this study we combine the notions of fuzzy order and fuzzy topology of Chang and define fuzzy ordered fuzzy topological space. Its various properties are analysed. Product, quotient, union and intersection of fuzzy orders are introduced. Besides, fuzzy order preserving maps and various fuzzy completeness are investigated. Finally an attempt is made to study the notion of generalized fuzzy ordered fuzzy topological space by considering fuzzy order defined on a fuzzy subset.

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.

Investigation on key molecular targets responsible for antidiabetic properties of Symplocos cochinchinensis (Lour.) S. Moore

Diabetes mellitus is a heterogeneous metabolic disorder characterized by hyperglycemia with disturbances in carbohydrate, protein and lipid metabolism resulting from defects in insulin secretion, insulin action or both. Currently there are 387 million people with diabetes worldwide and is expected to affect 592 million people by 2035. Insulin resistance in peripheral tissues and pancreatic beta cell dysfunction are the major challenges in the pathophysiology of diabetes. Diabetic secondary complications (like liver cirrhosis, retinopathy, microvascular and macrovascular complications) arise from persistent hyperglycemia and dyslipidemia can be disabling or even life threatening. Current medications are effective for control and management of hyperglycemia but undesirable effects, inefficiency against secondary complications and high cost are still serious issues in the present prognosis of this disorder. Hence the search for more effective and safer therapeutic agents of natural origin has been found to be highly demanding and attract attention in the present drug discovery research. The data available from Ayurveda on various medicinal plants for treatment of diabetes can efficiently yield potential new lead as antidiabetic agents. For wider acceptability and popularity of herbal remedies available in Ayurveda scientific validation by the elucidation of mechanism of action is very much essential. Modern biological techniques are available now to elucidate the biochemical basis of the effectiveness of these medicinal plants. Keeping this idea the research programme under this thesis has been planned to evaluate the molecular mechanism responsible for the antidiabetic property of Symplocos cochinchinensis, the main ingredient of Nishakathakadi Kashayam, a wellknown Ayurvedic antidiabetic preparation. A general introduction of diabetes, its pathophysiology, secondary complications and current treatment options, innovative solutions based on phytomedicine etc has been described in Chapter 1. The effect of Symplocos cochinchinensis (SC), on various in vitro biochemical targets relevant to diabetes is depicted in Chapter 2 including the preparation of plant extract. Since diabetes is a multifactorial disease, ethanolic extract of the bark of SC (SCE) and its fractions (hexane, dichloromethane, ethyl acetate and 90 % ethanol) were evaluated by in vitro methods against multiple targets such as control of postprandial hyperglycemia, insulin resistance, oxidative stress, pancreatic beta cell proliferation, inhibition of protein glycation, protein tyrosine phosphatase-1B (PTP-1B) and dipeptidyl peptidase-IV (DPPxxi IV). Among the extracts, SCE exhibited comparatively better activity like alpha glucosidase inhibition, insulin dependent glucose uptake (3 fold increase) in L6 myotubes, pancreatic beta cell regeneration in RIN-m5F and reduced triglyceride accumulation in 3T3-L1 cells, protection from hyperglycemia induced generation of reactive oxygen species in HepG2 cells with moderate antiglycation and PTP-1B inhibition. Chemical characterization by HPLC revealed the superiority of SCE over other extracts due to presence of bioactives (beta-sitosterol, phloretin 2’glucoside, oleanolic acid) in addition to minerals like magnesium, calcium, potassium, sodium, zinc and manganese. So SCE has been subjected to oral sucrose tolerance test (OGTT) to evaluate its antihyperglycemic property in mild diabetic and diabetic animal models. SCE showed significant antihyperglycemic activity in in vivo diabetic models. Chapter 3 highlights the beneficial effects of hydroethanol extract of Symplocos cochinchinensis (SCE) against hyperglycemia associated secondary complications in streptozotocin (60 mg/kg body weight) induced diabetic rat model. Proper sanction had been obtained for all the animal experiments from CSIR-CDRI institutional animal ethics committee. The experimental groups consist of normal control (NC), N + SCE 500 mg/kg bwd, diabetic control (DC), D + metformin 100 mg/kg bwd, D + SCE 250 and D + SCE 500. SCEs and metformin were administered daily for 21 days and sacrificed on day 22. Oral glucose tolerance test, plasma insulin, % HbA1c, urea, creatinine, aspartate aminotransferase (AST), alanine aminotransferase (ALT), albumin, total protein etc. were analysed. Aldose reductase (AR) activity in the eye lens was also checked. On day 21, DC rats showed significantly abnormal glucose response, HOMA-IR, % HbA1c, decreased activity of antioxidant enzymes and GSH, elevated AR activity, hepatic and renal oxidative stress markers compared to NC. DC rats also exhibited increased level of plasma urea and creatinine. Treatment with SCE protected from the deleterious alterations of biochemical parameters in a dose dependent manner including histopathological alterations in pancreas. SCE 500 exhibited significant glucose lowering effect and decreased HOMA-IR, % HbA1c, lens AR activity, and hepatic, renal oxidative stress and function markers compared to DC group. Considerable amount of liver and muscle glycogen was replenished by SCE treatment in diabetic animals. Although metformin showed better effect, the activity of SCE was very much comparable with this drug. xxii The possible molecular mechanism behind the protective property of S. cochinchinensis against the insulin resistance in peripheral tissue as well as dyslipidemia in in vivo high fructose saturated fat diet model is described in Chapter 4. Initially animal were fed a high fructose saturated fat (HFS) diet for a period of 8 weeks to develop insulin resistance and dyslipidemia. The normal diet control (ND), ND + SCE 500 mg/kg bwd, high fructose saturated fat diet control (HFS), HFS + metformin 100 mg/kg bwd, HFS + SCE 250 and HFS + SCE 500 were the experimental groups. SCEs and metformin were administered daily for the next 3 weeks and sacrificed at the end of 11th week. At the end of week 11, HFS rats showed significantly abnormal glucose and insulin tolerance, HOMA-IR, % HbA1c, adiponectin, lipid profile, liver glycolytic and gluconeogenic enzyme activities, liver and muscle triglyceride accumulation compared to ND. HFS rats also exhibited increased level of plasma inflammatory cytokines, upregulated mRNA level of gluconeogenic and lipogenic genes in liver. HFS exhibited the increased expression of GLUT-2 in liver and decreased expression of GLUT-4 in muscle and adipose. SCE treatment also preserved the architecture of pancreas, liver, and kidney tissues. Treatment with SCE reversed the alterations of biochemical parameters, improved insulin sensitivity by modifying gene expression in liver, muscle and adipose tissues. Overall results suggest that SC mediates the antidiabetic activity mainly via alpha glucosidase inhibition, improved insulin sensitivity, with antiglycation and antioxidant activities.