PhD Thesis
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Item On Generalizations of Separation properties, Compactness and Connectedness(University of Rajshahi, Rajshahi, 2019) Biswas, Sanjoy Kumar; Akhter, Nasima; Majumdar, SubrataThe thesis is concerned with generalizations of some important and interesting properties of separation, compactness and connectedness in a span of seven chapters. In the first chapter pseudo regular and pseudo normal topological spaces have been defined. Their properties have been studied and a number of important theorems regarding these spaces have been established. In the second chapter strongly pseudo-regular and strongly pseudonormal topological spaces have been introduced and their properties have been studied. A number of important theorems have been proved in this regard. This is the third chapter. In this chapter strictly pseudo-regular and strictly pseudo-normal topological spaces have been defined and their properties have been studied. In the former class a compact set can be separated from an external point by a continuous function, while in the latter, two disjoint compact sets can be separated by a continuous function. Many important properties have been proved. The fourth chapter introduces the notions of nearly regular topological spaces of the first kind and the second kind and studies their properties. A number of important theorems regarding these spaces have been established. In this fifth chapter two new generalizations of normal spaces have been defined and studied. The spaces in these classes have been termed nearly normal topological spaces of the first kind and the second kind respectively.Item A study on Extended Newton-type Methods for Variational Inclusions(University of Rajshahi, Rajshahi, 2021) Khaton, Mst. Zamilla; Rashid, Mohammed HarunorIn this work, we deal with the two types of variational inclusions. Firstly, we consider the variational inclusion problem of the form 0 ∈ ζ(¯s) + g(¯s) + ξ(¯s), (A) where S and T are Banach spaces, ζ : S → T is differentiable in a neighborhood Υ ⊆ S of a solution s∗ of (A), g : S → T is differentiable at s∗ but may not differentiable in Υ and ξ : S ⇒ 2T is a set-valued mapping with closed graph. This work consists three parts and the main works we have done in this dissertation that are organized as follows. In the first part, particularly in Chapter 3, we study the Newton-type method for solving the variational inclusion problem (A) which is introduced in [2]. Under some suitable assumptions on the Fr´echet derivative of the differentiable function and divided difference admissible function, we establish the existence of any sequence generated by the Newtontype method and prove that the sequence generated by the method (3.1.3) converges linearly, quadratically and superlinearly to a solution of the variational inclusion (A). Specifically, when the Fr´echet derivative of the differentiable function is continuous, Lipschitz continuous and H¨older continuous, divided difference admissible function admits first order divided difference and the set-valued mapping is pseudo-Lipschitz continuous, we show the linear, quadratic and superlinear convergence by the method (3.1.3). In Chapter 4, we introduce and study the extended Newton-type-----Item Compactness and Connectedness Concept in Intuitionistic Fuzzy Topological Spaces(University of Rajshahi, 2020-12) Mahbub, Md. Aman; Hossain, Md. Sahadat; Hossain, Mohd. AltabThe fundamental concept of a fuzzy set and fuzzy set operations was first introduced by L. A. Zadeh (Zadeh, 1965) in 1965 and it provides a natural foundation for treating mathematically the fuzzy phenomena, which exists pervasively in our real world and for building new branches of fuzzy mathematics. This also provides a natural frame work for generalizing various branches of mathematics such as fuzzy topology, fuzzy group, fuzzy rings, fuzzy vector spaces, fuzzy number, fuzzy system, fuzzy function, fuzzy relation, fuzzy logic and fuzzy computation. The concepts of fuzzy topology was introduced by C. L. Chang (Chang, 1968) in 1968 based on fuzzy set. Ming and Ming (Pao-Ming & Ying-Ming, 1980) (Pao-Ming & Ying-Ming, 1980), Khedr (Khedr et al., 2001), Hutton (Hutton, 1975), Azad (Azad, 1981), Ali (Ali, 1992) (Ali et al., 1990), Lowen (Lowen, 1976) etc. discussed various properties of fuzzy topology using fuzzy sets and fuzzy topology. Fuzzy compactness occupies a very important place in fuzzy topological spaces and so does some of its forms. Fuzzy compactness first discussed by C. L. Chang [C. L. Chang, Fuzzy Topological Spaces, J. Math. Anal. Appl., 24(1968), 182–190], T. E. Gantner et al. [T. E. Gantner, R. C. Steinlage and R. H. Warren, Compactness in Fuzzy Topological Spaces, J. Math. Anal. Appl., 62(1978), 547–562] introduced 𝛼 -compactness, A. D. Concilio and G. Gerla [A. D. Concilio and G. Gerla, Almost Compactness in Fuzzy Topological Spaces, Fuzzy Sets and Systems, 13(1984), 187–192] discussed almost compact spaces and M. N. Mukherjee and A. Bhattacharyya [M. N. Mukherjee and A. Bhattacharyya, 𝛼 -Almost Compactness for Crisp Subsets in a Fuzzy Topological Spaces, J. Fuzzy Math, 11(1) (2003), 105–113] discussed almost 𝛼 -compact spaces. After two decades, in 1983, Atanassov (K. T. Atanassov, “Intuitionistic Fuzzy Sets,” VII ITKR`s Session, (V, Sgurev, Ed.), Sofia (1983), Bulgaria) introduced the concept of intuitionistic fuzzy sets as a generalization of fuzzy sets which looks more accurately to uncertainty quantification and provides the opportunity to precisely model the problem based on the existing knowledge and observations. An intuitionistic fuzzy set (A-IFS), developed by Atanassov (K. T. Atanassov, “Intuitionistic Fuzzy Sets,” Theory and Applications, Springer-Verlag (1999), Heidelberg, New York & K. T. Atanassov, “Intuitionistic Fuzzy Sets,” Fuzzy Sets and Systems (1986), vol. 20, 87 – 96) is a powerful tool to deal with vagueness. A prominent characteristic of A-IFS is that it assigns to each element a membership degree and a non-membership degree, and thus, A-IFS constitutes an extension of Zadeh’s fuzzy set. He added a new component (which determines the degree of non-membership) in the definition of fuzzy set. The fuzzy sets give the degree of membership of an element in a given set (and the non-membership degree equals one minus the degree of membership), while intuitionistic fuzzy sets give both a degree of membership and a degree of non-membership which are more-or-less independent from each other, the only requirement is that the sum of these two degrees is not greater than 1. In the last few years various concepts in fuzzy sets were extended to intuitionistic fuzzy sets. Intuitionistic fuzzy sets have been applied in a wide variety of fields including computer science, engineering, mathematics, medicine, chemistry and economics (K. P. Huber and M. R. Berthold, “Application of Fuzzy Graphs for Metamodeling”, Proceedings of the 2002 IEEE Conference, 640 644). In 1997, Coker [D. Coker, “An Introduction to Intuitionistic Fuzzy Topological Space,” Fuzzy Sets and Systems (1997), vol. 88, 81 –89] introduced the concept of intuitionistic fuzzy topological spaces. S. Bayhan and D. Coker, “On Fuzzy Separation Axioms in Intuitionistic Fuzzy Topological Space,” BUSEFAL (1996), vol. 67, 77 –87, D. Coker and A. Es. Hyder, “On Fuzzy Compactness in Intuitionistic Fuzzy Topological Spaces,” The Journal of Fuzzy Mathematics (1995), vol. 3, no. 4, 899 –909, S. Ozcag and D. Coker, “On Connectedness in Intuitionistic Fuzzy Special Topological Spaces,” Int. J. Math. Math. Sciences (1998), vol. 21, no. 1, 33 –40] gave some other concepts of intuitionistic fuzzy topological spaces, such as fuzzy continuity, fuzzy compactness, fuzzy connectedness, fuzzy Hausdorff space and separation axioms in intuitionistic fuzzy topological spaces. After this, many concepts in fuzzy topological spaces are being extended to intuitionistic fuzzy topological spaces. Recently many fuzzy topological concepts such as fuzzy compactness [D. Coker and A. Es. Hyder, “On Fuzzy Compactness in Intuitionistic Fuzzy Topological Spaces,” The Journal of Fuzzy Mathematics (1995), vol. 3, no. 4, 899 –909 ], fuzzy connectedness [N. Turanli and D. Coker, “Fuzzy Connectedness in Intuitionistic Fuzzy Topological Spaces,” Fuzzy Sets and Systems (2000), vol. 116, no. 3, 369 –375], fuzzy separation axioms[S. Bayhan and D. Coker, “On Separation Axioms in Intuitionistic Topological Space,” Int. J. of Math. Sci. (2001), vol. 27, no. 10, 621 –630], fuzzy continuity [H. Gurcay, D. Coker and A. Es. Hayder, “On Fuzzy Continuity in Intuitionistic Fuzzy Topological Spaces,” The Journal of Mathematics of Fuzzy Mathematics (1997), vol. 5, 365 –378], fuzzy g-closed sets[S. S. Thakur and Rekha Chaturvedi, “Generalized Closed Set in Intuitionistic Fuzzy Topology,” The Journal of Fuzzy Mathematics (2008), vol. 16, no. 3, 559 –572] and fuzzy g-continuity[S. S. Thakur and Rekha Chaturvedi, “Generalized Continuity in Intuitionistic Fuzzy Topological Spaces,” Notes on Intuitionistic Fuzzy Set (2006), vol. 12 no. 1, 38 –44] have been generalized for intuitionistic fuzzy topological spaces. Deschrijver and Kerre [G. Deschrijver and E. E. Kerre, “On the Relationship between Some Extensions of Fuzzy Set Theory,” Fuzzy Sets and Systems (2003), vol. 133, 227–235], Goguen [J. Goguen, “L-fuzzy Sets,” J. Math. Anal. Applicat. (1967), vol. 18, 145– 174] established the relationships between IFSs, L-fuzzy sets, interval-valued fuzzy sets, and interval-valued IFSs. Hausdorffness in an intuitionistic fuzzy topological space has been introduced earlier by Coker[D. Coker, “An Introduction to Intuitionistic Fuzzy Topological Space,” Fuzzy Sets and Systems (1997), vol. 88, 81 –89]. Lupianez [F. G. Lupianez. “Hausdorffness in Intuitionistic Fuzzy Topological Spaces,” Mathware and Soft Computing (2003), vol. 10, 17 –22] has also defined new notions of Hausdorffness in the intuitionistic fuzzy sense and obtained some new properties in particular in convergence. Separation axioms is very impotent in any kind of topological space. Bayhan and Coker (Bayhan & Coker, 1996) introduced fuzzy separation axioms in intuitionistic fuzzy topological spaces. Singh and Srivastava (Singh & Srivastava, 2012), Yue and Fang (Yue & Fang, 2006), Bhattacharjee and Bhaumik (Bhattacharjee & Bhaumik, 2012) also studied separation axioms in intuitionistic fuzzy topological spaces. The purpose of this thesis is to suggest new definitions of compactness and connectedness axioms in intuitionistic fuzzy topological spaces. We have studied several features of these definitions and the relations among them. We have also shown ‘good extension’ properties of all these spaces. Our criteria for definitions have been preserved as much as possible the relations between the corresponding separation properties for intuitionistic fuzzy topological spaces. The materials of this thesis have been divided into six chapters. A brief scenario of which we have presented as follows: Chapter one incorporates some of the basic definitions and results of general sets, fuzzy sets, intuitionistic sets, intuitionistic fuzzy sets and topologies based on such sets. In this chapter, subspace of topological space, product space and mapping in topological spaces x has been discussed, which are to be used as references for understanding the next chapters. Most of the results are quoted from various research papers and books. Our main works start from chapter two. In this chapter, we give seven new notions of intuitionistic fuzzy compact (in short, IF-Compact) space and investigate some relationship among them. At first we show that all these notions satisfy ‘good extension’ property. Furthermore, it proves that these intuitionistic fuzzy compact spaces are hereditary and productive. Finally, we observe that all concepts are preserved under one-one, onto and continuous mapping. In chapter three, we have introduced Q-compactness in intuitionistic fuzzy compact topological spaces. Furthermore, we have established some theorems and examples of Q-compactness in intuitionistic fuzzy topological spaces and discussed different characterizations of Q-compactness. Also we have defined 𝛿−𝑄 compactness, 𝑄−𝜎 compactness and 𝛿−𝑄−𝜎 compactness in intuitionistic fuzzy topological spaces and found different properties between Q-compactness and 𝛿−𝑄 compactness, 𝑄−𝜎 compactness and 𝛿−𝑄−𝜎 compactness in intuitionistic fuzzy topological spaces. In fourth chapter, we discusses various type of compactness in intuitionistic fuzzy topological spaces. Almost compact fuzzy sets was first constructed by Concilio and Gerla which is local property. Here we give wo new possible notions of almost compactness in intuitionistic fuzzy topological spaces are studied and investigated some of their properties. We show that these notions satisfy hereditary and productive property of intuitionistic fuzzy topological spaces. Under some conditions it is shown that image and preimage preserve intuitionistic fuzzy topological spaces. Also we give three new notions of 𝐼-compactness, 𝐶-compactness and 𝐼−𝐶-compactness in intuitionistic fuzzy topological spaces and investigate some relations between our notionss. At last we give three new notions of paracompactness and one new notion of 𝜎-compactness in intuitionistic fuzzy topological spaces and established some properties of them. In chapter five, we give some new notions of separated, connectedness and totally connectedness and one notions of 𝑇1-space in intuitionistic fuzzy topological space and investigate some relationship among them. Also we find a relation about classical topology and intuitionistic fuzzy topology. Further, we show that connectedness in intuitionistic fuzzy topological spaces are productive. In the chapter six, we have introduced (𝑟,𝑠)-connectedness in intuitionistic fuzzy topological spaces. Furthermore, we have established some theorems and examples of (𝑟,𝑠)-connectedness in intuitionistic fuzzy topological spaces and discussed different characterizations of (𝑟,𝑠)-connectedness.Item Theoretical Investigations of Turbulence and Magneto-Hydrodynamic Turbulence in Incompressible Fluid(University of Rajshahi, 2001) Islam, Md. Anowarul; Sarker, M. Shamsul AlamThe thesis entitled "Theoretical investigations of turbulence and Magnetohydrodynamic turbulence in incompressible fluid" is being presented for the award of the degree of Doctor of Philosophy in Mathematics. It is the outcome of my researches conducted in the Department of Mathematics, Rajshahi University, Rajshahi, Bangladesh under the guidance of Dr. M. Shamsul Alam Sarker, Professor, Department of mathematics, Rajshahi University, Rajshahi-6205, Bangladesh. The thesis has been divided into six chnptcrs. The first is a general introductory chapter and gives the general idea of turbulence and Magneto-hydrodynamics turbulence and its principal concepts. Some results and theories, which arc needed in the subsequent chapter, have been included in this chapter. A brief review of the past researches related lo this thesis has also been given. In the second chapter we have discussed the decay of temperature 11uclualion in homogeneous turbulence before the final petiod for the case of multi-point and multi-time. Two-point, two-lime and three-point, three-time Fourier-lransfom1ed temperature equations is made detenninate by neglecting the fourth-order correlation tem1s. Finally, we have obtained the decay law of temperature iluctualion energy before the final period.-----Item Studies of Spinning Particles in Curved Spacetimes(University of Rajshahi, 2010) Banu, Akhtara; Ansary, Md. AbdullahWe investigate the motion of spinning particles, such as Dirac fermions, in curved spacetimes by pseudo-classical mechanics models in which the spin degrees of freedom are characterized in terms of anti-commuting Grassmann variables. The work of this thesis has been organized in 7 chapters along with an introduction at the very outset and a discussion at the end of the thesis. In the Introduction we present a short account of the work of investigating pseudo-classical spinning point particles in the four-dimensional curved spacetimes of general relativity. We review the relevant formulations and the equations of motion for the theory in chapters 1 and 2. We generalize the Killing equations in the spinning space, i.e., the configuration space of the spinning particles, spanned by the usual position coordinates and spin variables. We describe the symmetries and the corresponding constants of motion in terms of the solutions of the generalized Killing equations. Spinning space can have fermionic symmetries along with the standard kind of symmetries. The standard or generic symmetries exist for any spacetime metric gµv (x) (chapter 1 ), while the new nongeneric symmetries depend on the specific form of spacetime and are generated by the square root of bosonic constants of motion other than the Hamiltonian (chapter 2). These formalisms have been exploited in chapters 4-7 to study the pseudo-classical spinning point particles in particular types of black hole spacetimes. In chapter 3 we investigate geodesic motion of pseudo-classical spin one half point particles in the purely de Sitter spacetime and asymptotically de Sitter Schwarzschild spacetime. We apply the formalism of chapter 1 to solve the equations of motion of the pseudo-classical spinning particles in a plane. In chapter 4 we investigate geodesic motions of pseudo-classical spinning point particles in the Euclidean Taub-NUT space. Applying the formalism of chapters 1 and 2, we describe the symmetries and derive the corresponding constants of motion. The spinning space is found to admit hidden supersymmetries, generated by the mysterious Killing-Yano tensors. We use these formalisms to analyze the motion of the pseudo-classical spinning particles on a cone and plane. In chapter 5 we analyze the geodesic motion of pseudo-classical spinning particles in the NUT-Taub spacetime. We find that spinning spacetime admits new fermionic supersymmetries along with generic symmetries. The corresponding constants of motion have been obtained and the motion has been described on a cone and on a plane. In chapter 6 we study geodesic motion of pseudo-classical spinning particles in the Taub-NUT-de Sitter spacetime, which is the Taub-NUT spacetime generalized with cosmological constant. We obtain the conserved quantities for spinning space and describe the motion of the pseudo-classical Dirac fermion on a cone and plane. Geodesic motions of pseudo-classical spinning point particles are analyzed for the generalized NUT spacetime in chapter 7. Both types of symmetries, generic and nongeneric, are found to exist in the spinning space. We obtain the corresponding conserved quantities and investigate the motion of the pseudo-classical spinning particles on a cone and plane. Finally, we present the concluding remarks of this work in the discussion, at the end of the thesis.Item Analytical Invcstlgations in Tubulent and IVIlID Tttdmlent Flow(198) Rahman, Md. Lutfor; Sarker, M.Shamsul AlamThe thesis entitled "Analytical Investigations in Turbulent and MHD Turbulent Flow" is being presented for the award of the degree of Doctor of Philosophy in Mathematics. It is the outcome of my research conducted in the Department of Mathematics, Rajshahi University during the year 1994-1998 under the guidance of Dr. M. Shamsul Alam Sarker, Department of Mathematics, Rajshahi University, Rajshahi-6205, Bangladesh. The whole thesis has been divided into six chapters. The first is an introductory chapter and gives the general idea of turbulence, magnetohydrodynamic turbulence and its principal concepts. Throughout the work we have considered the flow of fluids to be isotropic and homogeneous. The notions generally adopted are those used by Batchelor, Chandrasekhar and Deissler in their research papers. Number inside brackets [ ] refer to the references which are arranged alphabetical at the end of the thesis. In the second chapter, we have derived the equation for tl1e rate of change of magnetic field covariance in MHD turbulent flow. 111e result shows that the defming scalars of the magnetic field covariance depend on the defining scalar H of two point magnetic field correlation. In tl1e third chapter, the decay of turbulence at ti.mes before the final period in presence of dust particles is studied. Two and three point correlation equation is used to obtain a relation for tl1e triple correlations and the equation is made detenninate by neglecting the quadruple correlations. Finally, we obtained tl1e energy decay law of dusty fluid turbulence before tl1e final period. In the fourth chapter, we have studied the decay of dusty fluid MHD turbulence before the final period. Tirree point correlation equation is used to obtain a relation for the triple correlations applicable at times before the final period. In this case the equation is made determinate by neglecting tl1e quadrnple correlations. Finally, we obtain the energy decay law of dusty fluid MHD turbulence at times before the final period.-------Item A Study on Finitely Generated N-Ideals of a Lattice(University of Rajshahi, 1998) Ali, Md. Ayub; Noor, A. S. A.This thesis studies extensively the finitely generated n-ideals of a lattice. The idea of n-ideals in a lattice was first introduced by Cornish and Noor in studying the kernels around a particular element n, of a skeletal congruence on a distributive lattice. Then Latif in his thesis "n-ideals of a lattice" studied thoroughly on the n-ideals and established many valuable results. For a fixed element n of a lattice L, a convex sub lattice of L containing n is called an n-ideal. If L has a "O", then replacing n by 0, an n-ideal becomes an ideal and if L has a "1" then it 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. The n-ideal generated by a finite number of elements of a lattice is called a finitely generated n-ideal, while the n-ideal generated by a single. element is known as a principal n-ideal. Latif in his thesis has given a neat description on finitely generated n-ideals of a lattice a'nd has provided a number of important results on them.Item Spinning Particles in Curved Spacetime of General Relativity(University of Rajshahi, 2001) Ali, M. Hossain; Ahmed, MainuddinIn Introduction we give a brief account of our work of studying spinning particles in curved spacetime of General Relativity. In Chapter I we discuss the relevant equations for the motion of spinning particles in curved spacetime. We present the generalized Killing equations for spinning space and describe the constants of motion. In Chapter II we derive the constants and the equations of motion for spinning particles moving in the Schwarzschild spacetime. We discuss various types of orbits and describe exact solutions in a plane. In Chapter III we extend the work of Chapter II in the Reissner-Nordstrom spacetime, which is the Schwarzschild spacetime generalized with a charged parameter and then further extend this work in the Reissner-Nordstrom spacetime generalized with a NUT (or magnetic mass) parameter. In the Reissner-Nordstrom spacetime we investigate the motion of spinning particles on a plane for bound state orbits, while in the NUT-Reissner-Nordstrom spacetime we analyze the motion on a cone and on a plane……….Item Effects of Air Bubbles on Storm Surge(University of Rajshahi, 2011) Rahman, Md. Azizur; Hoque, Md. Ashabul; Rahman, M. Zillur; Debsarma, Sujit KumarThe development of real time operational surge prediction system in the Bay of Bengal incorporating air bubble effects is still limited and is needed immediate attention. In this respects the study has been focused the wave energy dissipation, sea level rise due to air entrainment during storm surge and its various consequences. Furthermore, the thesis has also been investigated the two most severe cyclones of 1991 and AILA, 2009 that crossed the coast of Bangladesh. The energy dissipation by entrained air bubbles under breaking waves is studied theoretically both in spilling and plunging breakers. Results show that there is a correlation between the sudden reductions of wave height and entrained air bubbles into water. The study also focuses that in plunging breaking waves, the wave energy is dissipated on a very short way, whereas, in a spilling breaker this way is of the order of some wave lengths. Mathematical formulations of depth-averaged hydrodynamic equations have also been developed in terms of air bubbles. The results show that the water level increases with the increasing of air bubbles. Moreover, we have also found that wave energy increases with the increasing of void fraction. A nonlinear storm surge model in terms of air bubble effect has been developed for the Bay of Bengal along Bangladesh coast The model is included the analysis area of the head Bay of Bengal. These equations have been solved by finite difference method where the coasts and islands are included in the stair step method. The dynamic effect of pressure gradient outside the storm surge region in the model is found to be negligible in comparison with the dominant effect of wind. Using the model storm surges are simulated for the severe cyclones April, 1991 and "AILA" 2009. The simulation is made at some coastal and island locations of Bangladesh and simulated water levels are found higher in presence of air bubbles. That is, it observes that approximately 1.5%-3.5% water level increases with the increasing of void fraction from 10%-70% associated with AILA, 2009, whereas it increases up to 2%-4.5% for void fraction 10%-70% in the cyclone April, 1991. The results are in good agreement with the reported post storm survey information.Item N-Ideals of a Lattice(University of Rajshahi, 1994) Latif, Md. AbdulThis thesis studies the nature of n-ideals of a lattice. The topic arose out of a study on the kernels, around a particular element n, of a skeletal congruence on a distributive 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-ideal. 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 O 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.