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Analytical Invcstlgations in Tubulent and IVIlID Tttdmlent Flow
(198) Rahman, Md. Lutfor; Sarker, M.Shamsul Alam
The 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.-------
Quantum Gravity in Curved Space-Time of General Relativity
(University of Rajshahi, 1993) Mondal, Asit Kumar; Ahmed, Mainuddin
The main contents of this thesis may be briefly summarized as follows: Chapter 1 contains a brief account of the Plebonski spacetime . We specify there the various spacial class of spacetimes covered by the Plebanski spncetime • .Among the various special cases of the Plebanski spncetime a new spacetime discovered there· is the NUT - Kerr - Newman - Kasuya - de Sitter spacetime • This chapter also includes a discussion on the modification of the Plebanski space time and presents the modified fonn of the Plebanski spacetime • Chapter 2 deals with the separation of the variables of the Dirac equation in an arbitrary curved background spacetime • We discuss there some of the special cases of the separated Dirac equationu The pertinent equation of the separated Dirac equation will be used to derive the radial decoupled Dirac equations for the concerned background spacetimes which will be useful in smdying the problems of.Hawking radiation . Since our efforts are con can tra tad on quan mm field theory in some interesting spacetimes of general rolativity which are not the black hole spacetimes but include the black hole spncetimes as special cases , we review in Chapter J , Hawking's quanmm effects near the event horizon of NUT- Kerr- Newman spo.cetime containing flat black hole spacetimes as special cases.Chapter 4 also reviews and extends the result obtained in Chapter 3. The Hawking radiation of Dirac particles is aw.died in NUT-Kerr-Newman de Sitter spacetime containing black hole spacetimes which are asymptotically flat as well as asymptotically de Sitter 6S special cases. Chapter 5 includes the investigation of Dirac particles in Kasner-type spacetime • Chapter 6 presents Hawking's thermal radiation by black hole near the horizons of the Plebanski spacctime containing a large number of spacetimes of which some are important from the physical point of view. The result obtained in this chapter not only encompasses the lmown results but also includes some new results • This chapter ends with a concluding remarks •
A Study on Distributive Nearlattices
(University of Rajshahi, 1994) Rahman, Md.Bazlar; Noor, A.S.A.
This thesis studies the nature of distributive nearlattices. By a nearlattice S we will always mean a (lower) semilattice which· has the property that any two elements possessing a common upper bound,have a supremum, Cornish and Hickman in their paper [14],referred this property as the upper bound property, and a semilattice of this nature as a semilattice with the upper bound property. Cornish and Noor in [15] preferred to call these semilattices as nearlattices as the behaviour of such a semilattice is closer to that of a lattice than an ordiary semilattice, In this thesis we give several results on near lattices which certainly extend and generalize many results in lattice theory, In chapter 1 we discuss ideals, congruences and other results which are basic to this thesis. We include some characterizations of distributive and modular nearlattices, We generalize the separation properties given by M.H.Stone for distributive lattices. We also show that the set of prime ideals of a nearlattice Sis unordered if and only if Sis semiboolean. Chapter 2 discusses the skeletal congruences of a distributive near lattice. Skeletal congruences on distributive lattices have been studied extensively by Cornish in [ 11], Here we extend several results of Cornish for nearlattices. We also introduce the notion of disjunctive nearlattices, A distributive nearlattice S with O is called disjunctive if for O 􀀅a < b there is an element x ES such that x A a= 0 and O < x 􀀅 b, Then we give several characterizations of disjunctive nearlattices and semiboolean algebras using skeletal congruences, Finally we show that a distributive naerlattice is semiboolean if and only if 8 -----> ker8 is lattice isomorphism of Sc(S) onto KSc(S) whose inverse is the map .J ---> 8(J), In chapter 3, we discuss on normal and n-normal nearlattices, Normal lattices have been studied by several authors including Cornish [8] and Monteiro [34]; while n-normal lattices have been studied by Cornish [9] and Davey [16], In proving some of the results we have used Principle of Localization, which is an extension of lecture note of Dr. Noor on localization. This technique is very interesting and quite different from those of the previous authors,
N-Ideals of a Lattice
(University of Rajshahi, 1994) Latif, Md. Abdul
This 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.
Asymptotic Methods for some third order Nonlinear Differential Equations
(University of Rajshahi, 1995) Alam, Md. Shamsul; Sattar, Muhammad Abdus
In this thesis, we investigate the oscillations of third order nonlinear systems by the asymptotic method. The asymptotic method of Krylov-Bogoliubov-Mitropolskii (KBM) is a popular technigue for obtaining analytic solution of a second order nonlinear oscillatory system. First a third order nonlinear differential system modeling nonoscillatory process and characterized by critical damping is considered and a new perturbation technigue is developed, based on the work of Krylov-Bogoliubov-Mitropolskii, to £ind the solution of the system. Then a method is presented unifying both third order damped and overdamped systems. This method is a generalization of Bogoliubov·s asymptotic method and covers all the cases when the roots of the corresponding linear equation are real, real and complex, and real and purely imaginary.Later a third order forced nonlinear differential system modeling oscillatory process is considered and a new perturbation technique is developed to find the solution of the system. The methods are illustrated by several examples.
Cosmic String in theory of General Relativity
(1997) Biswas, Md.Elias Uddin; Ahmed, Mainuddin
This thesis is organized as follows. Chapter I contains a brief discussion of the production of various topological defects in cosmological phase transitions in the early universe. The three kinds of topological defects associated with spontaneous symmetry breaking: domain wall, cosmic string and magnetic monopole. The existence and stability of these defects is dictated by topological considerations. This chapter also includes a discussion on the standard cosmology and cosmological phase transitions. In the chapter II we review some physical properties of the cosmic string. Pair of cosmic sttings intercommuting at two points can form closed loops. A closed loop will oscillate, gradually loosing energy , until it disappear. Only significant energy loss mechanism is gravitational radiation. Gravitational radiation is indeed dominant. There is no local gravitational field due to the cosmic string. The cosmic string acts as a gravitational lens for both light rays and particles. Chapter ill addresses the topic of cosmic string evolution and structure formation. In the chapter IV we give a brief account of the Plebanski space-time. The Plebanski space-time includes many interesting space-times which are not black hole space-times but are important from the physical point of view. In the chapter V we study the equilibrium configurations of a cosmic string described by the Nambu-action in the NUT-Kerr-Newman space-time which includes as special cases the Kerr-Newman black hole space-time as well as NUT space-time which is considered as cosmological model. In this study it is interesting to note that one can obtain parallel results for Kerr-Newman black hole as well as for NUT space-time. Finally, in the chapter VI we study the equilibrium configurations of a cosmic string described by the Nambu-action in curved space-time such as the KerrNewman-Kasuya space-time which is the Kerr-Newman space-time involved with extra magnetic monopole charge. In this study it is interesting to note that the physical results remain the same whether or not the magnetic monopole exist in nature.
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.
Sectionally Pseudocomplemented Distributive Nearlattices
(University of Rajshahi, 1998) Islam, A. K. M. Sirajul; Noor, A. S. A.
This thesis studies the nature of a sectionally pseudocomplemented distributive nearlattice. By a nearlattice S we will always mean a meet semilattice together with the property th9.t any two elements possessing a common upper bound, have a supremum. Cornish and Hickman in their paper [14] referred this property as the upper bound property, and a semdattice of this nature as a semilattice with the upper bound property. Cornish and Noor in [15] preferred to call these semilattices as nearlattices, as the behaviour of such a semilattice is closer to that of a lattice than an ordinary semilattice. Of course a nearlattice with a largest element is a lattice. So the idea of pseudocomplementation is not possible 1n case of a general nearlattice. But for a nearlattice with a smallest element we can talk about sectionally p s eud ocom plemen ted nearlattice. Moreover, we can discuss on relatively p seudocom plemen ted nearlattices. In this thesis, we give several results on sectionally (relatively) pseudocomplemented nearlattices which certainly extend and generalize many results 1n lattice theory. ----------
KBM Asymptotic Method for third order Nonlinear Oscillations
(University of Rajshahi, 2000) Alam, Md. Shamsul; Sattar, M. Abdus
In most treatments or nonlinear oscillations hy perturbation method. only periodic oscillations are treated; transients are not considered. Krylov and Bogoliubov have used a perturbation method to discuss transients in the second order autonomous systems with small nonlinearities. The method is well known as an 'averaging method' in the theory of nonlinear oscillations. Later the method has been amplified and justified by L3ogoliubov and Mitropolskii. In this dissertation, we invt:stigatc SOllll' third order nonlinear oscillations based on the work of Krylov-13ogoliubov-Mitroplskii (KBJ\:1). First, nonlinear oscillation described by a third order ordinary autonomous differential equation is considered and a new perturbation tcclmiquc is developed. Then a method has been dcwlopcd 1(1 find asymptotic solution or a damped nonlinear system. The method is a generalization of Bogoliubov's asymptotic method and covers both under-damped and overdamped systems. Later clamped oscillations including critically damped motion have also been investigated in presence or more significant damping forces. Third order nonlinear oscillations with clamping and time delay, and with varying coenicients have been investigated separately. Moreover. a simple overdamped solution has been found for the third order weakly nonlinear systems.
Theoretical Investigations of Turbulence and Magneto-Hydrodynamic Turbulence in Incompressible Fluid
(University of Rajshahi, 2001) Islam, Md. Anowarul; Sarker, M. Shamsul Alam
The 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.-----
Spinning Particles in Curved Spacetime of General Relativity
(University of Rajshahi, 2001) Ali, M. Hossain; Ahmed, Mainuddin
In 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……….
Study of Radicals and Semisimple Classes of Rings
(University of Rajshahi, 2002) Dey, Kalyan Kumar; Majumdar, Subrata
The concept of the radical of a ring was introduced by Artin for rings with the descending chain condition with a view to obtaining a nice structure theorem for the ring. The idea was to single out the Toblerone part, of a ring, called the radical of the ring, and factor out the original ring with respect to the radical. The resulting ring, termed, semi simple has a nice description. Radica1s for rings without chain conditions were proposed by Koethe, Jacobson, Brown, McCoy, Levitzki and others for a similar purpose in an attempt to generalize Artin 's radical. All these attempts were later further generalized by Kurosh and Amitsur to define the concept of a general radical of a ring and the cones ponding semi simple ring and study these in their generality. Andrunakievic advanced these studies further. The class of rings which are radicals of themselves with respect to some radical is called a radical class, or simply, a radical, and the correponding class of the semisimple rings is called a semisimple class. A class rings may be simultaneously a radical ring with respect to some radical and a semisimple ring with respect to another radical. Such a class of rings is called a semisimple radical class. In this thesis we have studied radical classes, semisimple classes and semisimple radical classes of rings………………….
Eulerian Method for Third Order Linear Systems
(University of Rajshahi, 2003) Haque, Md. Abdul; Sattar, M. Abdus
In this thesis, we have studied the solutions of the various types of third order linear systems of ordinary differential equations. In chapter I, we have considered the third order linear homogeneous system with constant coefficients and developed Eulerian methods for all possible cases of the characteristic roots of the variational matrix. In chapter 2, we have considered the third order linear nonhomogeneous system with constant coefficients. The method covers all the cases when the roots of the characteristic equation of the corresponding homogeneous linear system are real and distinct, real and equal and complex. In finding particular solutions for the nonhomogeneous system of equations we have used the method of variation of parameters. Finally, we have obtained solutions of this system with the help of Crammer's rule. In chapter 3, we have considered the generalized form of third order linear nonhomogeneous system with constant coefficients. We have extended the Eulerian method and developed new techniques for obtaining solutions of this system. In chapter 4, we have discussed the third order linear nonhomogeneous system with variable coefficients. This problem is very difficult to solve, so we have examined a special case of this problem. By using a suitabletransformation, we have reduced it to a third order linear nonhomogeneous system with constant coefficients and have found solutions by the method of Chapter 2. We have illustrated all the methods by several suitable examples.
A Study on Convergence of Newton's Method in Real and Interval Number
(University of Rajshahi, 2003) Rahman, Md. Majedur; Ali, Md. Zulftkar
In order to find the approximate numerical solution to a system of nonlinear equations as well as an integral and a differential operator equations, Newton's algorithm is widely used. L. V. Kantorovich [1948] and Moore [1977] studied the existence and uniqueness of solution to the system of nonlinear equations and their error bounds. M. Urabe [ 1965] also studied the existence and uniqueness of the solution to nonlinear operator equations (mainly differential operator equations). Kantorovich and Urabe's methods are two variants of Newton's method in some sense. We study the existence and uniqueness of solutions to the nonlinear systems and their error bounds. Our results will be stated in a theorem that ensures the best possible generalized error bound that is different from that given by Kantorovich and Moore. We also develop a technique that may be applied to find an approximate numerical solution to an algebraic as well as to a system of nonlinear equations both in real and interval number systems. Finally, we have treated the error estimation for the quasiperiodic solution to the Van der pol type differential operator equation based on Urabe's theorem.
A Study of Characterization of Some Regular Gamma Rings
(University of Rajshahi, 2003) Nazneen, Ayesha; Paul, Akhil Chandra
The present thesis entitled, "A Study of Characterization of Some Regular Gamma Rings " is the outcome of researches carried out by me under the close supervision of Dr. Akhil Chandra Paul, Professor, Department of Mathematics, Rajshahi University. The thesis is of six chapters. In the first chapter we have tried to introduce all types of the conceptions of the complete thesis. In the second chapter we have given the definition of r - ring due to Barnes and of the relevant things. Various types of r- rings and their examples are also presented there. Some kinds of radical and corresponding theorems are also stated and important ones are proved. The definition of k - regular r- ring is given in the third chapter. Kyuno defmed this regular r - ring. We have tried to prove that the class of all k - regular r- rings forms a radical. Some of the characterizations of this r- rings are developed. We have also shown that k - regular r- ring without zero divisors is a skew r- field…….
A Theoretical Study on Some Aspects of Turbulent Flow
(University of Rajshahi, 2003) Sultana, Mst. Shamima; Sarker, M. Shamsul Alam
The conception of turbulent flow and the accompanying transition from laminar to turbulent flow is of fundamental importance. In everyday life, we recognized three states of matter: solid, liquid and gas. Although different in many respects, liquid and gases have a common characteristic in which they differ from solids: they are fluids, lacking the ability of solids to offer permanent resistance to a deforming force. Fluids flow under the action of such forces, deforming continuously for as long as the force applied. The fluids may be classified into different types depending upon the presence of viscosity. Osborne Reynolds shows that two entirely different types of fluids flow exist. In general words, turbulent flow is a flow, in which the inertia force is dominating over the viscosity. On the other hand, Laminar flow is a flow, in which the viscosity of the fluid is dominating over the inertia force. Osborne Reynolds demonstrated this in 1883 through an experiment. Reynolds’s apparatus consist of a tank, containing water and a small tank containing dye. 0. Reynolds’s [50) was also the first to investigate in greater detail the circumstances of the transition from laminar to turbulent flow. The previously mentioned dye experiment was used by him in this connection, and he discovered the law of similarity which now bears his name, which states…………………………..
Studies on Hydromagnetic Stability of Newtonian and Non-Newtonian Fluid
(University of Rajshahi, 2004) Rahman, M. Monsur; Sarker, M. Shamsul Alam
In this thesis hydromagnetic stabilities with respect to both axisymmetric and nonaxisymmetric perturbations of an incompressible perfectly conducting viscous fluid in presence of a magnetic field has been investigated. In chapter two, we have discussed the hydromagnetic stability with respect to axisymmetric disturbance of an in-compressible viscous fluid rotating between two perfectly conducting infinite co-axial cylinders in presence of a magnetic field by inner product method. The complex characteristic equation for the growth rate has been simplified and deduce propagation conditions for unstable, oscillatory and stable modes. In chapter three, hydromagnetic stability of helical flows in viscous fluid has been discussed. In chapter four, we have studied the flow of a highly conducting viscous incompressible fluid which is flowing between two paralled non-conducting planes in a uniform transverse magnetic field perpendicular to the plane. The results obtained have been discussed with the help of tables and graphs. In chapter five, we have investigated the unsteady Ml ID now of a visco-elastic Rivlin Ericksen fluid with transient pressure gradient through a uniform circular cylinder in a uniform transverse magnetic field. Here, the velocity profile of a fluid clement has been calculated theoretically and graphically. In chapter six, an attempt has been made to investigate the unsteady flow of an incompressible visco-elastic Rivlin-Ericksen fluid between two concentric cylinders with transient pressure gradient in a uniform transverse magnetic field. Here we have calculated the velocity profile of a fluid clement theoretically and graphically. In chapter seven, we have investigated the unsteady unidirectional now of an incompressible visco-elastic oldroyd type fluid between two concentric cylinders under the action of a magnetic field with time varying body forces. Here we have calculated velocity profile of a fluid element theoretically and graphically, we also have discussed the stability of the velocity profile. ln chapter eight, Hyromagnetic Stability of viscoelastic oldroyd fluid with the time varying body force through a rectangular channel has been discussed……………………………
Some Theoretical Studies on Turbulence and Magneto-Hydrodynamic Turbulence
(University of Rajshahi, 2004) Azad, Md. Abul Kalam; Sarker, M. Shamsul Alam
The theory of turbulent motion has received considerable attention in recent developments of high-speed jet aircraft, plasma physics and chemical engineering. The formation of a turbulent boundary layer is one of the most frequently encountered pheno111ena in high-speed aerodynamics. It is common experience that the flow observed in nature such as rivers and winds usually differ from stream flow or laminar flow of a viscous fluid. The mean motion of such flow does not satisfy the Navier-Stockes equations for a viscous fluid. Such flows, which occur at high Reynolds numbers, are often termed turbulent flows. In turbulent flow, the steady motion of the fluid is steady so far as the temporal mean values of velocities and the pressures are concerned where as actually both velocities and the pressures are irregularly fluctuating. The velocity and pressure distributions in turbulent flows as well as the energy losses are determined mainly by turbulent fluctuations. The essential characteristic of turbulent !lows is that the turbulent fluctuations are random in nature. The instability of laminar flow at a high Reynolds numbers, are causes disruption of the laminar pattern of fluid motion. With sufficient disturbances the result is known as turbulence. The irregular, chaotic motion of fluid particle characterizes turbulence. It is far too complicated to be known in complete detail. This fact has been recognized by all theories extant today start with a stochastic formulation of the phenomenon. At least, the technical people understand the meaning of turbulence. The use of the word "Turbulence" to characterize a certain type of flow, namely, the counterpart of streamline motion is comparatively recent. Reynolds, 0 [I 02] made the first systematic experimental investigation of turbulent flow. The turbulent motion of fluid was described by Reynolds [I 02], one of the pioneers in the study of turbulent flows as "sinuous motion" because fluid particles in turbulent flow appeared to follow sinusoidal or irregular paths. Turbulence is rather a familiar notion; yet it is not easy to define in such a way as to cover the detailed characteristic comprehended in it and to make the definition agree with the modern view of it held by professionals in this field of applied science. The word "Turbulence" means: agitation, commotion, disturbance etc. This definition is, however, too general, and does not suffice to characterize turbulent fluid motion in the modern sense. In 1937, Taylor and Vonkarman [ 128) gave the following definition: " Turbulence is an irregular motion which in general makes its appearance in fluids, gaseous or liquid, when they flow past solid surface or even when neighboring streams of the same fluid flow past or over one another". According to this definition, the flow has to satisfy the condition of irregularity. But this irregularity is a very important feature. Because of irregularity, it is impossible to describe the motion in all details as a function of time and space co-ordinates. But fortunately turbulent motion is irregular in the sense that it is possible to describe it by laws of probability. It appears possible to indicate distinct average values of various quantities, such as velocity, pressure, temperature, etc and this is very important. If turbulent motion were entirely irregular, it would be inaccessible to any mathematical treatment. Therefore, it is not sufficient just to say that turbulence is an irregular motion yet we do not have clear-cut definition or turbulence. This is rather difficult. Hinze [41) suggested that, "Turbulent fluid motion is an irregular condition of flow in which various quantities show a random variation with time and space co-ordinates, so that statistically distinct average values can be discerned"……………………….
Some Topics in Fuzzy Topological Spaces
(University of Rajshahi, 2004) Hossain, Md. Sahadat; Ali, Dewan Muslim
The concept of fuzzy set was introduced by the American Mathematician L. A. Zadeh for the first time in 1965. This provides a natural framework for generalizing many algebraic and topological concepts in various directions. Accordingly fuzzy groups, fuzzy ideals, fuzzy rings, fuzzy vector space, fuzzy measure, fuzzy topology, fuzzy topological groups, fuzzy topological vector space and many other branches have been developed all over the world during the last four decades. It is still developing in many directions. While reviewing the literature in fuzzy topology, we have seen the gap in separation axiom of fuzzy topological space, the counter part of which in ordinary topological space drew attention of several eminent mathematicians like Azad, K. K.; Chang, C. L.; Ali, D. M.; Wuyts, P.; Srivastava, A. K. and Lowen, R. etc. We aim to develop of theories of Fuzzy To, Fuzzy T1, Fuzzy T2, Fuzzy Ro, Fuzzy R1, Fuzzy Regular and Fuzzy Normal spaces analogous to its counterpart in ordinary topology. The material of the thesis has been divided into six chapters, a brief scenario of which we present as follows. Chapter one incorporates some of the basic definitions and results of fuzzy set, fuzzy topology and mappings. These results are ready references for the work in subsequent chapters. Results are stated without proof and can be seen in the papers referred to.
Use of Fox Derivatives for Solution of Group Theoretic Problem
(University of Rajshahi, 2004) Sultana, Quazi Selina; Majumdar, Subrnta
The thesis is an exposition of use of Fox derivatives for solution of topological and group theoretic problems. Our own contribution is in the latter area. Developed by Fox, free differential calculus emerged as a powerful tool for the solution of topological problems. In addition to reviewing application of this tool in various mathematical situations, (i) we have applied it to provide new proofs of a few very important results in combinatorial group theory, (ii) solved the conjugacy problem for torsion-free polycyclic-by-finite groups and torsion-free groups with single defining relations and (iii) determined homology and cohomology of a few classes of groups. In the first chapter the free differential calculus of Fox has been introduced and its application to classification of topological spaces, has been briefly described. In this context, Lens spaces, knots, links and braids have been defined and use of Alexander polynomials in their classification described A few useful tenns in combinatorial group theory have been defined and a number of fundamental results stated. The second chapter describes Birman's work on the relationship of the Jacobians with automorphisms of free groups. This has been used to determine the automorphism group of a free group of rank 2. The third chapter provides a new proof of Freiheitssatz, a fundamental theorem in the theory of single relator groups. This was first proved by Magnus who used it to solve the word problem for such groups. Many proofs (some of them geometric) of his theorem and its generalisations have so far been given by eminent mathematicians, viz. Lyndon, Schupp, Bums, etc. We proved this theorem using Fox derivatives following Majumdar's ideas. ---------