MPhil Thesis
Browse
11 results
Search Results
Item An Analytical and Numerical Study on Fixed Point Theorems(University of Rajshahi, 2011) Asaduzzaman, Md.; Ali, M. ZulfikarFixed point theory has fascinated hundreds of researchers since 1922 with the celebrated Banach's fixed point theorem. There exists a vast literature on the topic and this is a very active field of research at present. Let X be a set and let T be a mapping from X intoX. A fixed point of Tis an element x EX such thatT(x) = x. In mathematics, a fixed-point theorem is a result saying that a function T will have at least one fixed point, under some conditions on T that can be stated in general terms. In other words, a fixed-point theorem is a theorem that asserts that every function that satisfies some given property must have a fixed point. Fixed point theorems give the conditions under which maps (single or multivalued) have solutions. Fixed point theory is a beautiful mixture, of analysis, topology, and geometry. If we have an equation whose explicit solution is not so easy to find, in that case we rewrite the equation in the form T(x) = x to find its solution by applying any suitable fixed-point theorem. This method can be applied not only to numerical equations but also to equations involving vectors or functions. In particular, fixed-point theorems are often used to prove the existence of solutions to differential equations. Fixed point theorems also play a fundamental role in demonstrating the existence of solutions to a wide variety of problems arising in social sciences, biology, chemistry, economics, engineering, physics and mathematics. For instance, the Banach [ 18, 40], Brouwer [ 18, 49] and Kakutani [ 18] fixed point theorems have been among the most-used tools in economics and game theory. Over the last 50 years the theory of fixed points has been revealed as a very powerful and important tool in the study of nonlinear phenomena.Item On Supra Fuzzy Topological Spaces(University of Rajshahi, 2011) Hoque, Md. Fazlul; Ali, Dewan MuslimThe fundamental concept of a fuzzy set was introduced by L. A. Zadeh [111] in 1965 to provide a foundation for the development of many areas of knowledge. Consequently, this provides a natural frame work for generalizing many algebraic and topological concepts in various directions such as fuzzy groups, fuzzy rings, fuzzy vector spaces, fuzzy supra topology, fuzzy infra topology, fuzzy bitopology etc. many other branches of mathematics have been developed all over the world during the last five decades. In 1968, Chang [19] introduced the concepts of a fuzzy topological space by using the fuzzy set. Wong [105], Lowen [60], Hutton [48], katsaras[52], Ali[3], Pu and Liu[72], etc., discussed various aspects of fuzzy topological spaces. Ying [74] introduced fuzzifying topology and developed this in a new direction with the semantic methods of continuous valued logic. In the frame work of fuzzifying topology, Sinha [93] introduced and studied To-, T1-, T2(Hausdorff)-, T3(regular)-, T4(normal)-, separation axioms. A.S. Mashhour et al. [ 64] introduced and studied the concepts of the family of fuzzifying semi open sets, fuzzifying neighborhood structure of a point and fuzzifying semi-closure of a fuzzy set. A.S. Mashhour et al. [64] also introduced and studied the R-0 and R1 separation axioms and studied their relations with the T1 and Ti-separation axioms respectively. Also, in fuzzifying topology they introduced and studied semi-To-, semi-Ro-, semi-Ti-, semi-R1-, semi-T2(semi Hausdorff)-, semi-T3(semi regular)-, semi-T4(semi normal)-, separation axioms. In 1983, A.S. Mashhour et al. [64] introduced supra topological spaces and studies s-continuous function and s • -continuous functions. In 1987, M.E. Abd EL-Monsef et al. [1] introduced the fuzzy supra topological spaces and studied fuzzy supra continuous functions and characterized a number of basic concepts..........Item On Ro and R1 Properties in Fuzzy Topological Spaces(University of Rajshahi, 2011) Azam, S. M. Faqruddin Ali; Ali, Dewan MuslimThe goal of this thesis is to find out some new R1 -concepts for fuzzy topological spaces. Besides some concepts of fuzzy R0, R1, T0, T1, T2 and regular topological spaces that are already existing in the literature are recalled. In this work, twelve R1 -axioms of fuzzy R1 -topological spaces are introduced and studied in detail. Interrelations among various R1 concepts of fuzzy topological spaces are discussed. In analogy with the well-known topological properties, a complete answer is given with regard to all possible (R1 ^ T0 = T2) and (R1 ⇒ R0)-type implications for fuzzy topological spaces. It is also shown that, though a regular topological space is also a R1 -topological space, this is not true for fuzzy topological spaces.Item Study On Some Problems of Non-Newtonian Fluid Mechanics(University of Rajshahi, 2010) Hossain, Md. Aslam; Pk, M. Wazed AliThis is an elaborate discussion of main theme ideas and the condition in "Fluid mechanics", which help us to realized a clear concept of the background and elements of various viscous Newtonian and non-Newtonian flow problems. The basic equation viz, the equation of a visco-elastic fluid of oldroyds model, the equation of continuity, the equation of motion and the equation for isotropic incompressible fluid of Newtonian and non-Newtonian. Here we also have shown the equation the velocity profile in the dimensionless form. Fluid mechanics is one of the engineering science that form the basis for all engineering like meteorology, oceanography and other subject of physical sciences. The subject branches out into various specialiies such as aerodynamics, hydraulic engineering, marine engineering, gas dynamics and rate processes. It deals with the statics, kinematics and dynamics of fluids, since the motion of a fluid is caused by unbalanced forces exerted upon it. Available methods of analysis stem from the application of the following principles concepts and laws, Newton's laws of motion, the first and second laws of thermodynamics, the principle of conservation of mass, equations of state relating fluid properties. Newton's law of viscosity, mixing-lenth concepts and restrictions caused by the presence of boundaries. In fluid flow calculations, viscosity and density are the fluid properties most generally encountered; they play the principal roles in open-and closedchannel flow and in flow around immersed bodies. Surface tension effects are of importance in the formation of droplets, in flow of small jets and in situations where liquid-gas-solid or liquid-liquid-solid interfaces occurs, as well as in the formation of capillary waves. The property of vapor pressure, accounting for changes of phase from liquid to gas, becomes important when reduced pressures are encountered. In this chapter fluid properties are discussed, as well as units and dimensions and concepts of the continuum.Item A Study on Standard n-Ideals of a Lattice(University of Rajshahi, 2014) Syeed, Ahmed, Abu Sadat; Latif, Md. AbdulThis thesis studies the nature of standard n-ideals of a lattice. The idea of n-ideals in a lattice was first introduced by Cornish and Noor. For a fixed element n of a lattice L, a convex sublattice containing n is called an n-ideals. If L has an 'O', then replacing n by 0, an n-ideal becomes an ideal. Moreover, if L has 1, an n-ideal becomes a filter by replacing n by 1. Thus, the idea of n-ideals is a kind of generalization of both ideals and filters of lattices. So, any result involving n-ideals will give a generalization of the results on ideals and filters with 0 and 1 respectively in a lattice. In this thesis we give a series of results on n-ideals of a lattice which certainly extend and generalize many works in lattice theory. Chapter-1, discusses n-ideals, finitely generated n-ideals and other results on n-ideals of a lattice which are basic to this thesis. We have shown that, a lattice L is modular (distributive) if and only if in (L), the lattice of n-ideals is modular (distributive). In chapter-2, we have discussed lattices and elements with special properties. Here we have proved the coincidence of standard and neutral elements in a wide class of lattices including modular lattices, weakly modular lattices as well as relatively complemented lattices. In modular lattices and relatively complemented lattices the proves of the results are trivial but in weakly modular lattices this prove is not so simple. In this chapter, we have proved the following results:Item On Connectedness Concepts in Fuzzy Topological Spaces(University of Rajshahi, 2013) Shayid, Md.Abu; Ali, Dewan MuslimThe goal of the thesis is to find out some new connectedness concepts in fuzzy topological spaces. Some concepts of connectedness in fuzzy topological spaces that already exist in the literature are recalled here also. In this work, various type of connectedness like Ci - connectedness (i = I, 2, 3, 4), (C3) - connectedness, stronger forms of connectedness are studied in detail. Interrelations between various connectedness concepts in fuzzy topological spaces are discussed. In each chapter of this thesis, we give several possible definitions, both existing and new, of a concept and then compare the resulting concepts and determine thereby the interrelations among them. Some other properties of these concepts have also been discussed.Item Stability Analysis of Some Mathematical Models of Epidemics(University of Rajshahi, 2013) Islam, Md. Saiful; Asaduzzaman, Md.; Mondal, Md. Nazrul IslamThe spread of communicable diseases is often described mathematically by compartmental models. Many epidemiological models have a Disease Free Equilibrium (DFE) at which the population remains in the absence of disease. The classical Susceptible Infected Removed (SIR) models are very essential as conceptual models like as predator-prey and competing species models in ecology. Some Susceptible Infected (SI) and Susceptible Infected Susceptible (SIS) type models have been considered in this study. There are two major types of control strategies available to limit the spread of infectious diseases, viz. pharmaceutical interventions (drugs, vaccines, etc.), and non-pharmaceutical interventions (social distancing, quarantine, etc.). Vaccination is important for the elimination of infectious diseases as an effective preventive strategy. Vaccination of susceptible individual has been introduced through Susceptible Infected Removed Susceptible (SIRS) models. Effective vaccines have been used successfully to control smallpox, polio and measles. Some models have been presented in this study for the transmission dynamics of infectious diseases to analyze the stability of various equilibrium points mathematically. Some Susceptible Vaccinated Infected Susceptible (SVIS) and Susceptible Vaccinated Infected (SVI) models have been introduced in this study by including a new compartment ‘V’ for vaccinated individual in SIS and SI type models respectively. The above models have various kinds of parameters. Mainly the stability is analyzed by bifurcation curves in the SVIS models. The basic reproductive number (R0) can be calculated due to DFE in the SVI model. Some controlling methods have been given by changing the parameters in the SVI model through R0.Item Generalizations of Some Properties of Topological and Bitopological Spaces(University of Rajshahi, 2015) Biswas, Sanjoy Kumar; Akhter, NasimaThe thesis is concerned with generalizations of some important and interesting properties of topological and bitopological spaces in a span of four chapters. The first chapter constitutes an introduction and study of (i) a weak form of strong continuity, (ii) RC- continuity, (iii) perfect continuity, (iv) contra- precontinuity and (v) contra continuity in bitopological spaces . It thus generalizes the corresponding concepts in topology introduced by Donchev, Jafari and Noiri and studied by them. In addition, generalizing the works of Ekici and Noiri, as investigation of relationships between graphs and contra δ-precontinuous functions in bitopological spaces has also been made in this chapter. In the second chapter the problems of δ-compactness of topological spaces has been generalized to the corresponding properties in bitopological spaces. Some important properties of δ-compactness in bitopological spaces have been established, which are generalizations of results of park, Srivastava and Gupta. Also, a characterization of δ-Hausdorff bitopological spaces has been made and some properties of such spaces have been established, generalizing results of Srivastava and Gupta. The third chapter introduces the notions of weakly β-continuous functions in tritopological spaces and investigates several properties of these functions, thus generalizing the corresponding works in topological spaces by Khedr, Al-Areefi and Noiri and in bitopological spaces by Tahiliani. In the fourth chapter the idea of density topology has been introduced for tritopological spaces and has been used to prove certain theorem involving some separation properties. The concept of density of sets in a tritopological spaces and the notion of its trioclosure generalizing topology have been introduced and fruitfully used for study of separation properties.Item Separation Axioms on L-Topological Spaces(University of Rajshahi, 2015) Islam, Rafiqul; Hossain, Md. SahadatThe notion of a fuzzy set, as proposed by L.A. Zadeh[90] in 1965 to provide a foundation for the evolution of many areas of knowledge. After then in quick succession, L-fuzzy sets were introduced by Goguen[24] in 1967. As a result, this provides a natural frame work for generalizing many algebraic and topological concepts in various directions such as L-fuzzy sets, fuzzy logics, fuzzy control, fuzzy groups, fuzzy rings, fuzzy vector spaces, fuzzy topology, fuzzy bitopology, L-topology etc. Many other branches of mathematics have been developed all over the world during the last five decades. Chang [12] first introduced and studied the concept of a fuzzy topological space by using the fuzzy set in 1968. Hutton[31-34] ,Reilly[33-34],Wong [83-84], Lowen[47-48], Srivastava[73-79], Dude[16-17], Cutler[14], Ying[88], Ali[2-9], Hossain[26-30], Pu and Liu[53-54], etc., discussed and developed various aspects of fuzzy topological spaces. Ying [88] introduced fuzzifying topology and developed this in a new direction with the semantic methods of continuous valued logic. With the help of fuzzifying topology, Sinha[69-70] introduced and studied T0, T1, T2-(Hausdorff), T3-(regular), T4- (normal), separation axioms. Mashhour et al. [51-52] introduced and studied the concepts of the family of fuzzifying semi-open sets, fuzzifying neighbourhood structure of a point and fuzzifying semi-closure of a fuzzy set.................Item New Approximate Solution of Non-Linear Differential Systems(University of Rajshahi, 2014) Pervin, Mst. Razia; Shanta, Shewli ShamimMost of the perturbation methods are developed to find periodic solutions of nonlinear systems; transients are not considered. At first, Krylov and Bogoliubov introduced a perturbation method which is well known as “asymptotic averaging method” to discuss the transients in the second order autonomous systems with small nonlinearities. Later, this method has been amplified and justified by Bogoliubov and Mitropolskii. Mitropolskii has extended the method for slowly varying coefficients to determine the steady state periodic motions and transient processes. In this dissertation, we have modified and extended the KBM method to investigate some second order nonlinear systems. Firstly, a second order time dependent nonlinear differential system is considered. Then a new perturbation technique is developed to find an asymptotic solution of nonlinear systems in presence of an external force. Finally, this technique is used to obtain an asymptotic solution of a time dependent nonlinear differential system with slowly varying coefficients using the extended KBM method. These methods are illustrated with several examples.