PhD Thesis

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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 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,
A study on Extended Newton-type Methods for Variational Inclusions
(University of Rajshahi, Rajshahi, 2021) Khaton, Mst. Zamilla; Rashid, Mohammed Harunor
In 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-----
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.
A Study on Interval Solutions of Nonlinear Systems
(University of Rajshahi, 2011) Mollah, Md. Shirazul Hoque; Ali, M. Zulfikar
Suppose we seek a solution of the nonlinear system f(x={f;(x1,x2, ..., Xn)} = 0 i = 1,2, ..., n where f1, f2, f3, ..., fn are continuous functions on an open set D in R”. There is good many methods for iterative interval solutions of system (I) for any such methods, R. E. Moore developed a technique for finding a safe starting point from which iterates converge, with a particular iterative method in mind, Krawczyk’s operator. Minoru Urabe established an existence and uniqueness theorem (1965) which helps verify the existence and uniqueness of an exact solution and to know the error bound to an approximate solution of a system like (1). His theorem assumes that all the computations are to be carried out in real numbers exactly. Our attempts will be made to combine M. Urabe's theorem and R. E Moore's technique theoretically as well as numerically considering interval version of Newton's method.
Analysis of Viscous Incompressible Fluid Flows and Heat Transfer
(University of Rajshahi, 2010) Uddin, Md. Sharif; Pk, M. Wazcd Ali; Gorachand Layek
We first present a brief ideas and principles in "Fluid Mechanics" which may serve as the background materials of viscous, incompressible laminar Newtonian or non-Newtonian fluid flow problems considered in the present thesis. The basic equations viz, the continuity equation and the momentum equation for the motion of viscous incompressible fluid under the limits of continuum hypothesis are presented. The laws of classical mechanics apply throughout the continuous medium under consideration. The length scale of the flow is always taken to be large compared with the molecular mean-free-path. so that the fluid can be considered as a continuum. It excludes the flow of gases at very tow pressures i.e., rarefied gases. Liquid flows can usually be treated as incompressible fluid. The classification of fluids, say, Newtonian and non-Newtonian, Prandtrs boundary layer concept, boundary layer equations, concept of similarity variable for analyzing the viscous flow problems, group theoretic approach of for finding invariant solution of an incompressible viscous fluid flows, arc discussed systematically. Using the similarity variable for some specific flow problems, a set of nonlinear ordinary differential equations, known as self similar equations arc derived. Analytical or closed-form solution as well as numerical solution of these nonlinear differential equations relating to particular class of flow problems arc obtained and the corresponding flow quantities are shown graphically and discussed physically. In general, matter is found to exist in four phases or states e.g. solid. liquid, gas and plasma (ionized gases). Out of these last three states of matter are termed as fluid. Fluid mechanics is the subject in which wc deal with the flow problems pertaining to one of these phases e.g. liquid. gas and plasma or combination, mainly of the first two or last two phases. Essentially, the fluid flow problems are of widely spread interest in various fields of engineering as well as in meteorology, oceanography and other subjects of physical sciences. We live in a world which is largely a fluid. Air, oceans, rivers and so on arc all fluids whose behaviour is mostly described using the principles of continuum hypothesis. The constitutive equations are framed using some assumptions based on the material behaviors of fluid and the flow conditions. lt's study 1s important to physicists or applied mathematicians whose main interest is in understanding the related physical phenomena. On the other hand fluid dynamical engineers worked out many problems of practical interest using empirical formula. Also an understanding of this subject helps us to explain a variety fascinating natural phenomena around us.
Analysis on Viscous Incompressible Thermal Flow on a Ventilated Enclosure by Finite Element Method
(University of Rajshahi, 2014) Ahammad, Main Uddin; Rahman, Md. Lutfor; Rahman, Md. Mustafizur
Heat transfer in enclosure in which the influence of free (natural) and forced convection (mixed convection) are of comparable magnitude occurs frequently in engineering situations. The applications include the heat transfer improvement in heat exchanger devices, design of solar collectors, thermal design of building, air conditioning, cooling of electronic circuit boards, lubrication technologies, chemical processing equipment etc. Convection in ventilated enclosures containing obstruction has gained recent research significance as a means of heat transfer enhancement. The mathematical model of the present problem is governed by the couple equations of conservation of mass, momentum and energy. Discretization of the governing equations is achieved using a finite element scheme based on the Galerkin weighted residuals method. Then Newton–Raphson iterative algorithm is used to obtain the solutions of the obtained algebraic equations. Comparisons with previously established on particular cases of the problem are performed and the results show excellent agreement. Firstly, for mixed convection flow the effect of inlet and outlet position of a square ventilated enclosure with a centered heat generating solid body has been investigated. The bottom wall of the enclosure is kept at a uniform constant temperature, while the rest three walls of the enclosure are assumed adiabatic. A transverse uniform magnetic field is imposed in the horizontal direction normal to the right vertical wall. An external flow enters the cavity through an inlet opening whereas it exits via another outlet opening. After that, the effect of pertinent parameters in the considered flow problem in this thesis was analyzed for three different types of internal cavity solid body (heat generating, heat conducting, adiabatic) for a selected BT (bottom inlet and top outlet) configuration. Obtained results from the present study are presented in the form of streamlines, isotherms, average Nusselt number along the bottom heated surface and average fluid temperature in the cavity for each of four configurations as well as three different confined blocks for the pertinent parameters namely Reynolds number, Prandtl number, Hartmann number, solid-fluid thermal conductivity ratio, solid block diameter at the three values of Richardson number, varying from 0.1 to 10. The computational findings of this thesis reveals that both the flow and the thermal fields strongly depend on the parameters Reynolds number Re, Prandtl number Pr, Hartmann number Ha at the three convective regimes (Ri = 0.1, 1, 10). The centered solid body of the enclosure influences the steamlines pattern slightly for the smaller dimension of the block D whereas it has a considerable disparity in temperature distribution inside the enclosure. It is also observed that the solid-fluid thermal conductivity ratio K have insignificant effect on the flow fields and have significant effect on the thermal fields at the three convective regimes.
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.-------
Asymptotic Method for Time Dependent Nonlinear Differential Systems with Slowly Varying Coefficients
(University of Rajshahi, 2013) Roshid, Harun-Or-; Ali, M. Zulfikar; Dey, Pinakee
Almost all perturbation methods are developed to find periodic solutions of nonlinear system where transients are not considered. 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 process. In this dissertation, we have modified and extended the KBM method to investigate some fifth order and second order nonlinear systems in both cases with constant and slowly varying coefficients. At first, a fifth order damped nonlinear autonomous differential system is considered and a perturbation solution is developed. Then a procedure is developed for the same system with damped taking three of eigenvalues are real. After then we considered fifth order systems for over damped with small nonlinearity to obtain the transient response. We also developed a formula for fifth order critically damped nonlinear systems to control micro vibration, in micro and nano-technological industries that bring the system to equilibrium as quickly as possible without oscillating. After then we presented an analytical technique based on the extended Krylov-Bogoliubov-Mitropolskii method (by Popov) to determine approximate solutions of nonlinear differential systems whose coefficients change slowly and periodically with time. Furthermore, a non-autonomous case also investigated in which an external force acts in this system. At last, Krylov-Bogoliubov-Mitropolskii (KBM) method has been extended to certain damped-oscillatory nonlinear systems with varying coefficients. The implementations of the methods are illustrated by several examples.
Bekenstein-Hawking Entropy by Energy Quantization from Different Black Holes
(University of Rajshahi, 2016) Hossain, Md. Jakir; Rahman, Md. Atiqur; Hossain, Md. Ilias
We investigate the Bekenstein-Hawking entropy from different types of non-rotating black holes in de Sitter and Anti-de Sitter spaces by using the energy quantization mechanism in analogy with Bohr’s atomic model. We quantize the energy of the particle from the quantization of angular momentum. We also investigate the change of entropy between two nearby states as well as the thermal emission rate. In the limiting case all the results coincide with one another for the black holes in de Sitter and Anti- de Sitter spaces. The thesis is organized as follows: In chapter 1 we provide a brief discussion about our work of studying Bekenstein-Hawking entropy from black hole spacetime. In chapter 2 to 7 by using the quantization method we investigate the Lagrangian and canonical momenta of test particle, Radial motion and Effective Potential, Quantization of Circular Orbit, Energy quantization and Hawking Radiation for Schwarzschild, Schwarzschild-de Sitter (SdS), Schwarzschild Anti-de Sitter (SAdS), Reissner-Nordström (RN), Reissner-Nordström-de Sitter (RNdS), Reissner-Nordström Anti-de Sitter (RNAdS), black holes. Our new process is universally robust and the entropy framework given in this work indeed support the new perspective on quantum properties of gravity beyond classical physics, however, suggests a new idea to unify gravity with quantum theory and in the limiting case, the results are in line with that obtained by Sakalli et al. and He et al.’s method of the black hole.
Characterizations of K-Derivations and Bi-Derivations on Lie Ideals of Gamma Rings
(University of Rajshahi, 2016) Nazneen, Ayesha; Paul, Akhil Chandra
The present thesis entitled, "CHARACTERIZATIONS OF K-DERIVATIONS AND BI-DERIVATIONS ON LIE IDEALS OF GAMMA RINGS" is the outcome of researches carried out by me under the close supervisions of Dr. Akhil Chandra Paul, Professor, Department of Mathematics, Rajshahi University, Rajshahi.The main goal of this thesis is to characterize kderivations and then to generalize it. At the begining we introduce the concept of gamma rings and then we have mentioned the extention works on gamma rings, that means kderivations, jordan k-derivations, Jordan generalized k-derivations, biderivations e.t.c. We have also mentioned the mathemeticians who have worked on these fields. In the first chapter we mainly described our works on Jordan kderivations. Here we have given the definition of k-derivation, Lie ideal, Jordan k-derivation etc. Some examples are also given. Here we define cpa (u, v) for u, v E U, a E r; where U is a Lie ideal of a r-ring M.When M is a 2-torsion free prime r-ring , then we have proved that every Jordan k-derivation is a k-derivation also. In the second chapter, we have discussed about Jordan generalized kderivation. Here we have define a new additive mapping, which is a generalized form of k-derivation.We have defined it and then defined Jordan generalized k-derivation. For a generalized k-derivation we have defined it with k-derivation. We added some examples. We have defined Jordan generalized k-derivation on Lie ideals. Here we have also defined \Va (u, v), for u, v E U, a E r; where U is a Lie ideal of a r-ring M. At the end of this chapter we have proved that every Jordan generalized k- derivation is a Jordan k-derivation and so is a k-derivation on a Lie ideal of a 2-torsion free prime r-ring M also. We have worked on semi prime r -rings in the third chapter. In this chapter we have characterized semi prime r -rings and proved that every Jordan k-derivation on a Lie ideal U of a 2-torsion free r-ring Mis a kderivation on U of M if Mis semi prime. The conception of Left centralizer is presented in the forth chapter. We have denoted it by T. Here we have mentioned an additive mapping Ba( u, v) for all u, v E U; a E r.we have also defined Jordan left centralizers. Here we have proved that every Jordan Left centralizer T is a Left centralizer ifM is a 2- torsion free semiprime r-ring. In the fifth chapter, we have discussed Jordan generalized k-derivations on Lie ideals of semiprime r-rings. We have studied Jordan generalized k-derivations earlier. We have worked those on semiprime r-rings. With a special condition we have proved first that every Jordan generalized kderivation on a Lie ideal U of a 2-torsion free semiprime r-ring M is also a generalized k-derivation on U of M. Then we have proved the same result without any special condition by using the left centralizer. In the sixth chapter, we have studied bi-derivations. We have defined symmetric mapping, bi-derivation, symmetric bi-derivation etc. Here we have also defined trace, which is associated with a bi-derivation. Using different types of conditions, we have proved that either the Lie ideal U is contained in Z(M) or the trace d is zero, if M is a prime r -ring. We have developed these results for a semiprime r-ring also. We have worked on bi-additive mappings on semiprime r-rings in chapter seven. In this chapter we have studied the commutativity of a Lie ideal of a 2-torsion free semiprime r-ring. In the eighth chapter, we have discussed symmetric bi-derivations with symmetric generalized bi-derivations. Here we have worked with two symmetric bi-derivations associated with their respective traces and have found some important results. We have worked on commutativity with symmetric bi-derivations in the nineth chapter. K. K. Dey and A. C. Paul have worked on symmetric biderivations. Some of their results are extended here on Lie ideals of prime and semiprime r-rings. At the first stage of this thesis we have discussed about Nobusawa rrings. In the tenth chapter we have tried to characterize k-derivations on Lie ideals of Nobusawa r-rings. When M is a r-ring, then it is clear that r is also an M-ring. In the basis of this idea we have found ad-derivation on a Lie ideal Q of an M-ring r. In this chapter, we have tried to find out the same types of results on these two categories of derivations on respective Lie ideals of those rings. Also we have worked on d2 and d3. In the eleveth chapter, we have described Left k-derivation. We have defined Jordan left k-derivation also. In this chapter we have used Mas a completely prime r-ring. We have proved that if M is a 2-torsion free completely prime r-ring and U is a Lie ideal of M, then every Jordan left k-derivation on U ofM is also a left k-derivation on U ofM. A complete bibliography which have been helped us to finish my total research is also added at the end of this thesis.
Characterizations of Prime and Semiprime Gamma Rings with Derivations and Lie Ideals
(University of Rajshahi, 2014) Rahman, Md. Mizanor; Paul, Akhil Chandra
Gamma ring was first introduced by N. Nobusawa as a generalization of a classical ring. W. E. Barnes generalized the definition of a gamma ring due to Nobusawa. Presently the gamma ring due to Barnes is known as a gamma ring and gamma ring due to Nobusawa is known as gamma ring in the sense of Nobusawa and is denoted by 􀀀N-ring. It is clear that every ring is a gamma ring and every 􀀀N-ring is also a 􀀀-ring. Actually, W.E.Barnes, J.Luh and S.Kyuno studied the structures of 􀀀-rings and obtained various generalizations analogous to the corresponding parts in ring theory. Afterwards, a number of algebraists have determined a lot of fundamental properties of 􀀀-rings to classify and extend numerous significant results in classical ring theory to 􀀀-ring theory. This thesis, entitled “Characterizations of Prime and Semiprime Gamma Rings with Derivations and Lie Ideals”, aims to characterize prime and semiprime 􀀀-rings with various types of left derivations, derivations, generalized derivations, higher derivations, derivations on Lie ideals and (U,M)-derivations. All the necessary introductory definitions and examples of 􀀀-rings are discussed in considerable details in the introduction chapter. The notions of derivation and Jordan derivation in 􀀀-rings have been introduced by M. Sapanci and A. Nakajima. Afterwards, in the light of some significant results due to Jordan left derivation of a classical ring obtained by K.W.Jun and B.D.Kim, some extensive results of left derivation and Jordan left derivation of a 􀀀-ring were determined by Y.Ceven. In classical ring theory, Joso Vukman proved that if d is a Jordan left derivation of a 2-torsion free semiprime ring R and if there exists a positive integer n such that D(x)n = 0 for all x 2 R, then D = 0. He also proved that for a 2-torsion free and 3-torsion free semiprime ring R admits Jordan derivation D and G : R ! R such that D2(x) = G(x) for all x 2 R, then D = 0. In chapter 1, we extend this result to the 􀀀-ring theory in the case of Jordan left derivations. Then we construct some relevant results to prove that under a suitable condition every nonzero Jordan left derivation d of a 2-torsion free prime 􀀀-ring M induces the commutativity of M, and consequently, d is a left derivation of M. Developing a number of important results on Jordan derivations of semiprime 􀀀- rings, we then prove under a suitable condition, every Jordan derivation of a 􀀀-ring M is a derivation of M, if we consider M as a 2-torsion free (i) semiprime, and (ii) completely semiprime 􀀀-ring, respectively. We examine all these statements in chapter 2 for the clear understanding of the concepts. M. Asci and S. Ceran obtained some commutativity results of prime 􀀀-rings with left derivation. Some commutativity results in prime rings with Jordan higher left derivations were obtained by Kyuoo-Hong Park on Lie ideals and obtained some fruitful results relating this. We work on Jordan higher left derivation on a 2-torsion free prime 􀀀-ring and we show that under a suitable condition, the existence of a nonzero Jordan higher left derivation on a 2-torsion free prime 􀀀-ring M forces M commutative. For the classical ring theories, Herstein, proved a well known result that every Jordan derivation in a 2-torsion free prime ring is a derivation. Bresar proved this result in semiprime rings. Sapanci and Nakajima proved the same result in completely prime 􀀀-rings. C. Haetinger worked on higher derivations in prime rings and extended this result to Lie ideals in a prime ring. We introduce a higher derivation and a Jordan higher derivation in 􀀀-rings. Then we determine some immediate consequences due to Jordan higher derivation of 􀀀-rings to prove under a suitable condition every Jordan higher derivation of a 2-torsion free prime 􀀀-ring M is a higher derivation of M. Y. Ceven and M. A. Ozturk worked on Jordan generalized derivations in 􀀀-rings and they proved that every Jordan generalized derivation on some 􀀀-rings is a generalized derivation. A. Nakajima defined the notion of generalized higher derivations and investigated some elementary relations between generalized higher derivations and higher derivations in the usual sense. They also discussed Jordan generalized higher derivations and Lie derivations on rings. W. Cortes and C. Haetinger proved that every Jordan generalized higher derivations on a ring R is a generalized higher derivation. M. Ferrero and C. Haetinger proved that every Jordan higher derivation of a 2-torsion free semiprime ring is a higher derivation. C. Haetinger extended the above results of prime rings in Lie ideals. By the motivations of above works, we introduce a Jordan generalized higher derivations in 􀀀-rings. We prove that every Jordan generalized higher derivation in a 2-torsion free prime 􀀀-ring with the condition a b c = a b c for all a, b, c 2 M and , 2 􀀀, is a generalized higher derivation of M. Chapter 3 deals with all these important results elaborately. The relationship between usual derivations and Lie ideals of prime rings has been extensively studied in the last 40 years. In particular, when this relationship involves the action of the derivations on Lie ideals. In 1984, R. Awtar extended a well known result proved by I. N. Herstein to Lie ideals which states that, “every Jordan derivation on a 2-torsion free prime ring is a derivation”. In fact, R. Awtar proved that if U * Z is a square closed Lie ideal of a 2-torsion free prime ring R and d : R ! R is an additive mapping such that d(u2) = d(u)u + ud(u) for all u 2 U, then d(uv) = d(u)v + ud(v) for all u, v 2 U. M. Ashraf and N. Rehman studied on Lie ideals and Jordan left derivations of prime rings. They proved that if d : R ! R is an additive mapping on a 2-torsion free prime ring R satisfying d(u2) = 2ud(u) for all u 2 U, where U is a Lie ideal of R such that u2 2 U for all u 2 U then d(uv) = d(u)v + ud(v) for all u, v 2 U. A. K. Halder and A. C. Paul extended the results of Y. Ceven in Lie ideals. We generalize the Awtar’s result in 􀀀-rings by establishing some necessary results relating to them at the beginning of the chapter 4, we then prove if U is an admissible Lie ideal of a 2-torsion free prime 􀀀-ring M satisfying the condition a b c = a b c for all a, b, c 2 M; , 2 􀀀 and d : M ! M is a Jordan derivation on U of M, then d(u v) = d(u) v + u d(v) for all u, v 2 U; 2 􀀀 and if U is a commutative square closed Lie ideal of M, then d(u v) = d(u) v + u d(v) for all u, v 2 U and 2 􀀀. Accordingly, we then define Jordan higher derivation and higher derivation on Lie ideals of 􀀀-rings and construct some relevant results to prove the previous results analogously in case of Jordan higher derivation on Lie ideal of a 􀀀- ring. Chapter 4 is devoted to a study of these materials in order to bring out the concepts defined clearly. M. Ashraf and N. Rehman considered the question of I. N. Herstein for a Jordan generalized derivation. They showed that in a 2-torsion free ring R which has a commutator right nonzero divisor, every Jordan generalized derivation on R is a generalized derivation on R. In 2000, Nakajima defined a generalized higher derivation and gave some categorical properties. He also treated generalized higher Jordan and Lie derivations. Later, Cortes and Haetinger extended Ashraf’s theorem to generalized higher derivations. They proved that if R is 2-torsion free ring which has a commutator right nonzero divisor, then every Jordan generalized higher derivation on R is a generalized higher derivation on R. Following the notions of Jordan derivation and derivation on Lie ideals of a 􀀀-ring in the previous chapter we then introduce the concepts of a Jordan generalized derivation and generalized derivation on Lie ideals of a 􀀀-ring and we extend and generalized the above mentioned result by these newly introduced concepts. Accordingly, we then define Jordan generalized higher derivation and generalized higher derivation on Lie ideals of a 􀀀-ring and generalized the same result by these concepts. We examine all these statements in chapter 5 for the clear understanding of the concepts. (U,R)-derivations in rings have been introduced by A. K. Faraj, C. Haetinger and A. H. Majeed as a generalization of Jordan derivations on a Lie ideal of a ring. We introduce (U,M)-derivations in 􀀀-rings as a generalization of Jordan derivations on a Lie ideal of a 􀀀-ring. We construct some useful consequences of (U,M)-derivation of a prime 􀀀-ring to prove first that, d(u v) = d(u) v + u d(v) for all u, v 2 U, 2 􀀀, where U is an admissible Lie ideal of M and d is a (U,M)-derivation of M. We also prove that, if u u 2 U for all u 2 U and 2 􀀀 then d(u m) = d(u) m + u d(m) for all u 2 U,m 2 M and 2 􀀀. After introducing (U,M)-derivation in 􀀀-rings, we then introduce the concept of higher (U,M)-derivation in 􀀀-rings. We conclude the chapter 6 by proving the analogous results corresponding to the previous results considering higher (U,M)-derivations of prime 􀀀-rings almost similar way after developing a number of results regarding this newly introduced concept. Following the notion of (U,M)-derivation and higher (U,M)-derivation of a 􀀀- ring in the previous chapter here we introduce the concept of generalized (U,M)- derivation and generalized higher (U,M)-derivation in 􀀀-rings. By establishing some necessary results with generalized (U,M)-derivation, we then prove the analogous results considering generalized (U,M)-derivation of prime 􀀀-rings corresponding to the results of (U,M)-derivation of the previous chapter. A. K. Faraj, C. Haetinger and A. H. Majeed extended Awtar’s theorem to generalized higher (U,R)-derivations by proving that if R is a prime ring, char.(R) 6= 2, U is an admissible Lie ideal of R and F = (fi)i2N is a generalized higher (U,R)-derivations of R, then fn(ur) = P i+j=n fi(u)dj(r) for all u 2 U, r 2 R and n 2 N. Chapter 7 also extends this result in the case of generalized higher (U,M)-derivation of prime 􀀀-rings. Actually, it aims to prove that if U is an admissible Lie ideal of a prime 􀀀-ring M and F = (fi)i2N is a generalized higher (U,M)-derivation of M, then (i) fn(u v) = P i+j=n fi(u) dj(v) for all u, v 2 U, 2 􀀀 and n 2 N; and also, that (ii) fn(u m) = P i+j=n fi(u) dj(m) for all u 2 U,m 2 M, 2 􀀀 and n 2 N.
Characterizations of Some Radical Rings and Gamma Rings
(University of Rajshahi, 2006) Rashid, Md. Mahbubur; Paul, Akhil Chandra
The present thesis entitled "Characterizations of Some Radical Rings and Gamma Rings" is the outcome of the research by me under the supervision of Dr. A. C. Paul, Professor, Department of Mathematics, University of Rajshahi. The main emphasis of this thesis is to find radical rings. In introduction, we introduce the concepts of the complete thesis. The thesis is of eight chapters. In the first chapter, we discuss the preliminaries of the complete thesis. This chapter is the basic concept of our work. Here we discuss the definitions and some results of /:rings. We develop some characterizations of p-rings in the second chapter and in the third chapter, we generalize the p-rings. We define p-F-ring and prove the analogous properties of p-rings for p-I'-rings in the fourth chapter and the fifth chapter J-I'-rings are the generalizations of p-F-rings. In the sixth chapter, we define the regular /:rings that are more general than that of S. Kyuno, N. Nobusawa and B. Smith [14] and we also develop sufficient conditions for F-rings to be regular. In the seventh chapter, we define an abelian regular F-rings and prove that an abelian regular I'-ring is equivalent to "strongly regular" F-rings. We also develop some other properties. We define unit-regular F-rings in chapter eight and develop a number of equivalent characterizations and also develop a lattice theoretic characterization.
Compactness and Connectedness Concept in Intuitionistic Fuzzy Topological Spaces
(University of Rajshahi, 2020-12) Mahbub, Md. Aman; Hossain, Md. Sahadat; Hossain, Mohd. Altab
The 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.
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.
Damped Forced Vibration of Some Quasi-Linear Differential Systems
(University of Rajshahi, 2008) Dey, Pinakee; Sattar, M A.; Ali, M. Zulfikar; Alam, M. Shamsul
There are many approaches for approximating solutions of nonlinear vibrating problems. The most common methods for constructing approximate analytical solutions to the nonlinear vibrating problems are the perturbation methods. These methods are developed to find only periodic vibrations of the nonlinear differential systems. In order to investigate the transients of nonlinear vibrations, Krylov and BogoLiubov introduced a perturbation method to discuss the transients in the second order autonomous systems with small nonlinearities. The method is well known as an "asymptotic averaging method" in the theory of nonlinear vibrations. Then the method was amplified and justified by BogoLiubov and Mitropolskii. These methods were applied to autonomous systems. Later, Arya and Bojadziev, Bojadziev and Hung, and Shamsul extended the Krylov-Bogo Liubov-Mitropolskii (KBM) method to sometime dependent nonlinear differential systems. In this dissertation, we extend the work of KBM and investigate some other time dependent non-linear differential 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 vibrations in presence of a slowly decaying external force. We then find an asymptotic solution of a time dependent nonlinear differential system with slowly varying coefficients using the KBM method. Later, we find the perturbation solutions of damped forced vibrations using the modified KBM method, in which the coefficients change slowly varying with time. Further, this technique is used to obtain the second approximate solution of second order forced vibrations. Finally, this technique is used to obtain the higher approximate solution of an n-th order damped forced vibrating problem in the resonance case, and the stability of the stationary regime of vibrations has also been investigated. The methods are illustrated by several examples.
Effects of Air Bubbles on Storm Surge
(University of Rajshahi, 2011) Rahman, Md. Azizur; Hoque, Md. Ashabul; Rahman, M. Zillur; Debsarma, Sujit Kumar
The 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.
Investigation of Some Aspects of Protoplanetary Model of Planetary Formation
(University of Rajshahi, 2005) Paul, Gour Chandra; Bhattacharjee, Shishir Kumar
In this thesis we have investigated some aspects of the protoplanetary theory of planetary formation, namely, the structure of a protoplanet, sedimentation of heavy elements in a protoplanet, and the effect of mass loss on the orbit of a protoplanet. The thesis contains five different chapters. The first chapter deals with a brief outline of the current view of planetary formation while in the other chapters we have investigated the problems under consideration. In chapter 2, we have determined the structure of a protoplanet by numerical method in which the protoplanet is assumed to be a sphere of solar composition, which is in a steady state of quasi-static equilibrium. It is also assumed that the only source of energy in a protoplanet is gravitational. Regarding the heat transference of heat inside the protoplanet we have considered two cases of interest i) the convective case and ii) the conductive - radiative case. In chapter 3, the distribution of thermodynamic variables in a protoplanet has been determined by polytropic method assuming that the protoplanet is a polytope of index n = .5, I, 3/2 and 3. In the fourth chapter, we have investigated the segregation time of falling grains inside a protoplanet. We have calculated the time for two possible cases of interest, namely, i) the mass of the grain remains constant during falling, ii) the grain mass increases due to its adherence with other grains, and have found that a solid core having mass roughly equal to that of a terrestrial type planet can form at the center of a protoplanet in a reasonable short period of time on astronomical scale. In chapter 5, we have investigated the effect of mass loss on the orbital distance of a protoplanet in a two-body problem as well as in a three-body problem, and have shown that the planetary spacing observed today can satisfactorily be explained in terms of mass loss from a set of identical protoplanets.
Investigation of some aspects of protoplanetary model of planetary formation
(University of Rajshahi, 2005) Paul, Gour Chandra; Bhattachaijee, Shishir Kumar
In this thesis we have investigated some aspects of the protoplanetary theory of planetary formation, namely, the structure of a protoplanet, sedimentation of heavy elements in a protoplanet, and the effect of mass loss on the orbit of a protoplanet. The thesis contains five different chapters. The first chapter deals with a brief outline of the current view of planetary formation while in the other chapters we have investigated the problems under consideration. In chapter 2, we have determined the structure of a protoplanet by numerical method in which the protoplanet is assumed to be a sphere of solar composition, which is in a steady state of quasi-static equilibrium. It is also assumed that the only source of energy in a protoplanet is gravitational. Regarding the heat transference of heat inside the protoplanet we have considered two cases of interest i) the convective case and ii) the conductive - radiative case. In chapter 3, the distribution of thermodynamic variables in a protoplanet has been determined by polytropic method assuming that the protoplanet is a polytrope of index n = .5, I, 3/2 and 3. In the fourth chapter, we have investigated the segregation time of falling grains inside a protoplanet. We have calculated the time for two possible cases of interest, namely, i) the mass of the grain remains constant during falling, ii) the grain mass increases due to its adherence with other grains, and have found that a solid core having mass roughly equal to that of a terrestrial type planet can form at the centre of a protoplanet in a reasonable short period of time on astronomical $Cale.
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.