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Item Study of Principal n-ideals of a Lattice(Khulna University of Engineering & Technology (KUET), Khulna, Bangladesh, 2006-07) Azad, Md. Abul Kalam; Rahman, Prof. Dr. Md. BazlarThis thesis studies extensively the Principal 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 and Ayub Ali in their thesis studied thoroughly on the n-ideals and established many valuable results. For a fixed element n of a lattice L, a convex sublattice of L containing n is called an n-ideal. If L has a "0", 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 I. 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 and has provided a number of important results on them. For a lattice L, the lattice of all n-ideals of L and the lattice of all finitely generated n-ideals of L are denoted by In (L) and Fn (L) respectively, while Pn (L) represents the set of principal n-ideals of L. In this thesis, we devote ourselves in studying several properties on Pn (L) and Fn (L) which will certainly enrich many branches of lattice theory. Our results in this thesis generalize many results on normal, relatively normal, m-normal and relatively m-normal lattices. We also introduce the concept of n-annulets and α -n-ideal in studying Pn (L). In this connection it should be mentioned that if L has a 0, then putting n = 0 we find that Fn (L) is the set of all principal ideals of L which is isomorphic to L. Thus, for every result on Fn (L) in this thesis, we can obtain a result for the lattice L with 0 by substituting n = 0. Hence the result in each chapter of the thesis regarding Fn (L) are generalizations of the corresponding results in lattice theory. In chapter 2, we discuss some fundamental properties of n-ideals, which are basic to this thesis. Here we give an explicit description of Fn (L) and Pn (L) which are essential for the development of the thesis. Though Fn (L) is always a lattice, Pn (L) is not even a semilattice. But when n is a neutral element, Pn (L) becomes a meet semilattice. Moreover, we show that Pn (L) is a lattice if and only if n is a central element, and then in fact, Pn (L) = Fn (L). We also show that, for a neutral element n, the lattice L is complemented if and only if Pn (L) is so. In this chapter we also discuss on prime n-ideals. We give several properties and characterizations of prime n-ideals. We include a proof of the generalization of Stone's separation theorem. We also include a new proof of the result that for a distributive lattice L, Fn (L) is generalized Boolean if and only if prime n-ideals are unorderd. Chapter 3 discusses on minimal prime n-ideals of a lattice. We give some characterizations on minimal prime n-ideals which are essential for the further development of this chapter. Here we provide a number of results which are generalizations of the results on normal lattices. We prove that for a distributive lattice L, Fn (L) is normal if and only if each prime n-ideal of L contains a unique minimal prime n-ideal. We also show that if n is central in L, then Pn (L) is a normal lattice if and only if any two minimal prime n-ideal are comaximal which is also equivalent to < x > n ∩ n = {n} implies n* v n*=L. In chapter 4 we introduce the notion of relative n-annihilators n. We characterize distributive and modular lattices in terms of relative n-annihilators. Then we generalize several results of Mandelker on annihiltors. We use these to characterize those Fn (L) which are relatively normal lattices. Among many results we have shown that for a central element n, Pn (L) is a relatively normal lattice, if and only if any two incomparable prime n-ideal are comaximal . What is more, this is also equivalent to the condition <n,< b >n> v <n,< a >n> = L for all a,b ϵL. Pseudocomplemented distributive lattices satisfying Lee's identities form equational subclasses denoted by Bm , - 1 ≤ m ˂ w Cornish have studied distributive lattices analogues to Bm-lattices and relatively Bm-lattices. He referred then as m-normal lattices.Moreover, Beazer and Deavy have each independently obtained several characterizations of (sectionally) Bm -lattices and relatively Bm -lattices. In chapter 5 we generalize their results by studying finitely generated n-ideals which form a m-normal and a relatively m-normal lattice .We show that for a central element n ϵ L, Pn(L) is m-normal if and only if for any m+1 distinct minimal prime n-ideals P0 ............., Pn of L, P0 v ................v Pm = L. In this chapter we also show that for a central element n ϵ L, Pn (L) is relatively m-normal if and only if any m+1 pairwise incomparable prime n-ideals are comaximal. Chapter 6 introduces the concept of n-annulets and α -n-ideals of a lattice. Here we include several result on the set of n-annulets An(L) when n is a central element of L. We proved An(L) is relatively complemented if and only if Pn(L) is sectionally quasi-complemented.Item Study of Certain Topics in Fuzzy Supra Topological Space(Khulna University of Engineering & Technology (KUET), Khulna, Bangladesh, 2013-07) Molla, Md. Yahia; Rahman, Prof. Dr. Md. BazlarAmerican Mathematician Lotfi A. Zadeh in 1965 first introduced the concept of fuzzy set. He interpreted a fuzzy set on a set as a mapping from the set into the unit interval I= [0, 1], which is a generalization of the characteristic function of the set. Many mathematicians throughout the world used this set to fuzzify different areas of mathematics. Fuzzy supra topology is one of the outcomes of such fuzzification of the usual topology. In this thesis, we have studied and have introduced several results on fuzzy supra topological spaces. At first we have discussed the standard definitions and properties of fuzzy supra R0 and R1 topological spaces, which are found in the literatures. Then we have introduced some new definitions and properties for these spaces. We have also studied the Fuzzy supra T0, T1 , T2 and Fuzzy supra regular topological spaces and obtained the following properties, such as, Good extension, Initial, Reciprocal, Productivity, Hereditary and Homeomorphism, etc. Moreover we have discussed compactness of Fuzzy Supra Topological Spaces and have proposed some new definitions, theorems and proofs.Item A study on 0-distributive nearlattice(Khulna University of Engineering & Technology (KUET), Khulna, Bangladesh, 2014-11) Rahman, Md. Zaidur; Rahman, Prof. Dr. Md. BazlarIn this thesis study of the nature of the 0-distributive nearlattices is presented. By a nearlattice S we will always mean a meet semilattice together with the property that any two elements possessing a common upper bound, have a supremum. Cornish and Hickman [14] referred this property as the upper bound property and a semilattice of this nature as a semilattice with the upperbound property. Cornish and Noor [15] preferred to call these semilattices as nearlattices, as the behaviour of such a semilattice is close to that of a lattice than an ordinary semilattice. Of course a nearlattice with a largest element is a lattice. Since any semilattice satisf'ing the descending chain condition has the upper bound property, so all finite sernilattices are nearlattices. In lattice theory, it is always very difficult to study the non-distributive and non-modular lattices. Gratzer [20] studied the non-distributive lattices by introducing the concept of distributive, standard and neutral elements in lattices. Cornish and Noor [15] extended those concepts for nearlattices to study non-distributive nearlattices. On the other hand, J.0 Varlet [66] studied another class of non-distributive lattices with 0 by introducing the concept of 0-distributivity. In fact this concept also generalizes the idea of pseudocomplement in a general lattice. This thesis extend the concept of 0-distributivity in a nearlattice to study a larger class of non-distributive nearlattices. A nearlattice S with 0 is called 0-distributive if for all x,y,z ε S with x˄y=0=X˄Z and y˅z; exists imply x˄ (y˅z) = 0. Chapter 1 gives a detailed description of nearlattices. Here we discuss ideals, congruenees, SemiBoolean algebra and many other results on nearlattice which are basic to this thesis. In Chapter 2 we introduce the concept of modular element in a nearlattice. Gratzer and Schmidt [23] introduced the notion of some special elements, e.g. distributive, standard and neutral elements, to study a larger class of non-distributive lattices. Then Cornish and Noor [15] used these concepts to nearlattices. Again Talukder and Noor [64] introduced the notion of modular elements in a join semilattice directed below. The notion of modular element is also applicable for general lattices. In this chapter, we have introduced the concept of modular and strongly distributive elements for nearlattices. i-Iere we have given several characterizations of modular and strongly distributive elements. By studying these elements and ideals we obtained many information on a class of non-distributive nearlattices. Chapter 3 and 4 are the key chapters of this thesis. In Chapter 3 we introduce the 0-distributivity in a nearlattice with 0. We include several characterizations of distributive nearlattices. We prove that a nearlattice S with 0 is 0-distributive if and only if all maximal filter of S are prime. We also show that S is 0-distributive if and only if i(s),the lattice of all ideals of S is pseudocomplemented. Then we include some prime separation properties. In this chapter we also include the notion of semi-prime ideals by extending the notion of 0-distributivity. In lattices, the notion of semi-prime ideals was given by Y. Ray [52]. By using these semi-prime ideals, we generalize the prime separation theorem of nearlattices in terms of annihilator ideals. Finally, we extend the concept of Glivenko congruence for 0-distributive nearlattices as well as for semi-prime ideals to establish a generalized version of prime separation theorem. In chapter 4 we discuss different properties of 0-distributive nearlattices and included several characterizations of these nearlattices Annulets and u-ideals in a distributive lattice have been studied extensively by Cornish [13]. Recently Ayub Ali, Noor and Islam [4], Noor, Ayub Ali and Islam [41] extended this concept for distributive nearlattices. In this chapter we study the annulets and a-ideals in a 0-distributive nearlattice. We give several characterizations of a -ideals. We also include a prime separation theorem for α -ideals. Finally we show that a 0-distributive nearlattice is quasicomplemented if and only if A0(S) (the dual nearlattice of annulets) is a Boolean subalgebra of A(S), where A(S) is the set of all annihilator ideals of S. Moreover, S is sectionally quasicomplemented if and only if A0(S) is relatively complemented. Chapter 5 brings the notions of 0-modular nearlattices. Ayub Ali, Hafizur Rahman and Noor [5], Jayaram [30], Noor, Ayub Ali and Islam [41] and Varlet [65] have studied different properties of 0-distributivity and 0-modularity in lattices and in semilattices. In this chapter we extend their work and include several characterizations of 0-modular nearlattices. Many mathematician including Cornish [Ii] have studied the normal lattices and p-algebras in presence of distributivity. Recently Nag, Begurn and Talukder[38] studied them in presence of 0-dirtributivity. They have generalized many results of S -algebras and D -algebras. Since the idea of pseudocomplementation is not appropriate for a nearlattice, we study the sectional pseudocomplernentation for a nearlattice. In chapter 6 we extend and generalize some results of Nag, Begum and Talukder [38] on D –algebras and S -algebras. We prove that every [0, x] , x ε S is an S -algebra if and only if it is a D -algebra where the nearlattice S is sectionally p-algebra with the condition that [o, x] for each x ε S is 1-distributive and S is 0-modular. We conclude the thesis by giving a characterization of sectionaly S-algebra whenever [0,x} for each x ε S is 1-distributive.Item CFD modeling of a modified Ahmed Car Body for Reduced Drag(Khulna University of Engineering & Technology (KUET), Khulna, Bangladesh, 2014-06) Azad, Abul Kalam; Hossain, Prof. Dr. Mohammad ArifThe wake flow behind the car presents the major contribution to a car drag. The flow over a car body is very complex. Hence MOVA consortium partners agreed to study the vehicle shape employed by Ahmed and Ramm (1984), known as Ahmed body. The aerodynamic drag of the body has great impact on the fuel consumption by a car. So for economic and for environmental reasons also drag reduction is very important. The CFD model is used to investigate and for the better understanding of the aerodynamic behavior of the flow in the surrounding area of the vehicle. The development of a good CFD model depends firstly on investigating and selecting the best grid configurations. In this simulation experiment the total element count after final refinement was within an acceptable limit 1.7 to 2 million grids. The performance of a CFD model is not only depends on the number of grids but also on the turbulence model chosen for the simulation. Sometimes on the basis of the turbulence model chosen it is also required to select the roughness height. For the above mention purpose in this simulation study it has been found that a total of 1.7 million grids are suitable and acceptable. To select this number grid dependence test is performed. To choose the turbulence model a comparison among different turbulence models is done. On the basis of Azad et al. (2012) the k-a model, which is better suited, is chosen for the simulations. As k-a model has dependence on the roughness height it was required to choose also. A further study is done to find it and the findings are published. Azad et al. (2013) has shown the results and according to them the selection of the roughness height as 0.0002m is the best choice. Out of the two strategies active and passive, a passive strategy is chosen to reduce the drag over the Ahmed car body. To manipulate the flow insertion of grooves has been considered. The grooves are placed at the top, at the slant, at the top and slant, and at the rear surface of the body. The drag, after modeling the Ahmed body, is then calculated using ANSYS-11. The results thus obtained are compared with the results obtained by Lienhart et al. (2000). It has been found that overall drag has reduced.