Fuzzy Absolutes and Related Topics

The study on the fuzzy absolutes and related topics. The different kinds of extensions especially compactification formed a major area of study in topology. Perfect continuous mappings always preserve certain topological properties. The concept of Fuzzy sets introduced by the American Cyberneticist L. A Zadeh started a revolution in every branch of knowledge and in particular in every branch of mathematics. Fuzziness is a kind of uncertainty and uncertainty of a symbol lies in the lack of well-defined boundaries of the set of objects to which this symbol belongs. Introduce an s-continuous mapping from a topological space to a fuzzy topological space and prove that the image of an H-closed space under an s-continuous mapping is f-H closed. Here also proved that the arbitrary product fi and sum of  fi of the s-continuous maps fi are also s-continuous. The original motivation behind the study of absolutes was the problem of characterizing the projective objects in the category of compact spaces and continuous functions.

STUDIES ON SCATTERING AND QUASI-NORMAL MODES IN BLACK HOLE SPACE-TIMES

The question of stability of black hole was first studied by Regge and Wheeler who investigated linear perturbations of the exterior Schwarzschild spacetime. Further work on this problem led to the study of quasi-normal modes which is believed as a characteristic sound of black holes. Quasi-normal modes (QNMs) describe the damped oscillations under perturbations in the surrounding geometry of a black hole with frequencies and damping times of oscillations entirely fixed by the black hole parameters.In the present work we study the influence of cosmic string on the QNMs of various black hole background spacetimes which are perturbed by a massless Dirac field.

A study on semigroup action on fuzzy subsets, inverse fuzzy automata and related topics

This thesis comprises five chapters including the introductory chapter. This includes a brief introduction and basic definitions of fuzzy set theory and its applications, semigroup action on sets, finite semigroup theory, its application in automata theory along with references which are used in this thesis. In the second chapter we defined an S-fuzzy subset of X with the extension of the notion of semigroup action of S on X to semigroup action of S on to a fuzzy subset of X using Zadeh's maximal extension principal and proved some results based on this. We also defined an S-fuzzy morphism between two S-fuzzy subsets of X and they together form a category S FSETX. Some general properties and special objects in this category are studied and finally proved that S SET and S FSET are categorically equivalent. Further we tried to generalize this concept to the action of a fuzzy semigroup on fuzzy subsets. As an application, using the above idea, we convert a _nite state automaton to a finite fuzzy state automaton. A classical automata determine whether a word is accepted by the automaton where as a _nite fuzzy state automaton determine the degree of acceptance of the word by the automaton. 1.5. Summary of the Thesis 17 In the third chapter we de_ne regular and inverse fuzzy automata, its construction, and prove that the corresponding transition monoids are regular and inverse monoids respectively. The languages accepted by an inverse fuzzy automata is an inverse fuzzy language and we give a characterization of an inverse fuzzy language. We study some of its algebraic properties and prove that the collection IFL on an alphabet does not form a variety since it is not closed under inverse homomorphic images. We also prove some results based on the fact that a semigroup is inverse if and only if idempotents commute and every L-class or R-class contains a unique idempotent. Fourth chapter includes a study of the structure of the automorphism group of a deterministic faithful inverse fuzzy automaton and prove that it is equal to a subgroup of the inverse monoid of all one-one partial fuzzy transformations on the state set. In the fifth chapter we define min-weighted and max-weighted power automata study some of its algebraic properties and prove that a fuzzy automaton and the fuzzy power automata associated with it have the same transition monoids. The thesis ends with a conclusion of the work done and the scope of further study.

Studies on scattering and thermodynamics of black holes in f(R) theory and Einstein's theory

One of the interesting consequences of Einstein's General Theory of Relativity is the black hole solutions. Until the observation made by Hawking in 1970s, it was believed that black holes are perfectly black. The General Theory of Relativity says that black holes are objects which absorb both matter and radiation crossing the event horizon. The event horizon is a surface through which even light is not able to escape. It acts as a one sided membrane that allows the passage of particles only in one direction i.e. towards the center of black holes. All the particles that are absorbed by black hole increases the mass of the black hole and thus the size of event horizon also increases. Hawking showed in 1970s that when applying quantum mechanical laws to black holes they are not perfectly black but they can emit radiation. Thus the black hole can have temperature known as Hawking temperature. In the thesis we have studied some aspects of black holes in f(R) theory of gravity and Einstein's General Theory of Relativity. The scattering of scalar field in this background space time studied in the first chapter shows that the extended black hole will scatter scalar waves and have a scattering cross section and applying tunneling mechanism we have obtained the Hawking temperature of this black hole. In the following chapter we have investigated the quasinormal properties of the extended black hole. We have studied the electromagnetic and scalar perturbations in this space-time and find that the black hole frequencies are complex and show exponential damping indicating the black hole is stable against the perturbations. In the present study we show that not only the black holes exist in modified gravities but also they have similar properties of black hole space times in General Theory of Relativity. 2 + 1 black holes or three dimensional black holes are simplified examples of more complicated four dimensional black holes. Thus these models of black holes are known as toy models of black holes in four dimensional black holes in General theory of Relativity. We have studied some properties of these types of black holes in Einstein model (General Theory of Relativity). A three dimensional black hole known as MSW is taken for our study. The thermodynamics and spectroscopy of MSW black hole are studied and obtained the area spectrum which is equispaced and different thermo dynamical properties are studied. The Dirac perturbation of this three dimensional black hole is studied and the resulting quasinormal spectrum of this three dimensional black hole is obtained. The different quasinormal frequencies are tabulated in tables and these values show an exponential damping of oscillations indicating the black hole is stable against the mass less Dirac perturbation. In General Theory of Relativity almost all solutions contain singularities. The cosmological solution and different black hole solutions of Einstein's field equation contain singularities. The regular black hole solutions are those which are solutions of Einstein's equation and have no singularity at the origin. These solutions possess event horizon but have no central singularity. Such a solution was first put forward by Bardeen. Hayward proposed a similar regular black hole solution. We have studied the thermodynamics and spectroscopy of Hay-ward regular black holes. We have also obtained the different thermodynamic properties and the area spectrum. The area spectrum is a function of the horizon radius. The entropy-heat capacity curve has a discontinuity at some value of entropy showing a phase transition.