THE APPLICATION OF C++ PROGRAMMING IN THE SIMPLEX METHOD
Abstract
In
this paper, we first introduce the general linear programming problem and then
a mechanism to convert any general linear programming problem into the standard
form.
In
chapter II, we present a systematic method for iteratively moving from one
extreme point to an adjacent extreme point in the search for an optimal
solution. We first discuss the algebra of simplex method and then the simplex
method in tableau form. The aim of this paper is not only to understand the
mechanics of the tableau method, but also the theoretical foundation upon which
the mechanical operations are built.
Finally,
we solve the linear programming problem by using a computer program with C++
language.
SOME INTEGER LINEAR PROGRAMMING PROBLEMS
Abstract
In
this thesis, we introduce integer linear programming models that are linear
programming problems in which some or all of the variables are restricted to be
integers. Thesis begins by examining the graphical solution of simple integer
program. This is followed by a discussion of various techniques for formulating
integer programming models. Finally, we present cutting plane method and branch
and bound method for solving integer programming problems.
GAME THEORY AND SOME APPLICATIONS
Abstract
A
competitive situation is called a game. The term game represents a conflict
between two or more players.
In
this paper, we will consider only the simplest kinds of games, that is, games
involving:
1.
Only two players
2.
A payoff of some amount after each play such
that one player’s win is the other player’s loss. With the indicated
restrictions, we will be able to determine the best (optimal) strategies of
play for each person.
In the last section, we give the relation
between game theory and linear programming.
EULER TOURS AND ITS APPLICATIONS
Abstract
In
this thesis, we introduce some basic definitions for Euler tours and give some
examples. Next we state equivalent conditions for the eulerian graph. We
present how to solve the Postman problem for any undirected graph and directed
graph.
PHYSICAL SYMMETRIES PRESERVING TRANSITION PROBABILITY
Abstract
We
present the characterization of θ-bilinear
forms on V × V inducing the same dualities where V is an n-dimensional
vector space over a division ring D,
and θ, an anti-automorphism of D. Next, we also present the form of
anti-automorphisms of the logic. Using these results, we shall characterize the
form of physical symmetry transformations preserving transition probability.
PROJECTIONS ON INNER PRODUCT SPACES
Abstract
We
shall obtain new vector spaces from old ones, namely, direct sum. We present a
special type of linear transformations, namely, projections. We also present
connections between direct sums and projections. Last, we shall also obtain the
partial ordering relation on the set of all orthogonal projections on an inner
product space.
UNITARY OPERATORS ON INNER PRODUCT SPACES
Abstract
We
will characterize linear transformations on F(n), linear
transformations from one space to another, linear functionals and adjoints. We
also discuss linear functionals, adjoints and unitary operators on inner
product spaces, and their properties. The last of our discussion is that for
every invertible complex matrix n × n there exists a unique
lower-triangular matrix.
SPECTRAL THEORY OF OPERATORS
Abstract
In
this thesis we shall discuss the eigenvalues and eigenvectors of an operator.
We shall find that matrices represent an operator on an invariant subspace of a
finite dimensional space. We shall also find that matrices represent an
operator in a generalized null space. We finally express that if Λ be an
arbitrary square matrix and a matrix Γ whose columns are eigenvectors and
genedralized eigenvectors of Λ then Γ-1 Λ Γ is a matrix in the
Jordan canonical form.
TREES AND MINIMUM SPANNING TREES
Abstract
Trees
are central to the structural understanding of graphs, and they have a wide
range of applications. In this thesis, some basic properties and
characterizations of a trees and rooted trees are given. The two algorithms
used to construct minimum spanning trees are also presented and some examples are
discussed.
SHORTEST PATH ALGORITHMS
Abstract
We
confine our attention to two types of problems: (1) the problem of finding a
shortest path from a vertex v to another vertex w, (2) the problem of finding a
shortest path from every vertex to every other vertex. So we discuss the
Dijkstra algorithm which is to find a shortest path and the shortest distance
from a specified vertex to every other vertex. Next we discuss the
Floyd-Warshall algorithm and Floyd algorithm which are to find a shortest path
and shortest distance from every vertex to every other vertex. Finally we
discuss Dantzig algorithm that is a special case of Floyd algorithm.
GENERALIZED INVERSES FOR SOME MATRICES
Abstract
This
thesis describes a generalization of the inverse of a nonsingular matrix, as
any solution of a vector equation. We also describe a generalization of the
inverse of a nonsingular matrix, as the unique solution of system of matrix
equations. It is used for solving linear matrix equations and among other
applications for finding an expression of the principal idempotent elements of
a matrix
SOME FIXED POINT THEOREMS
Abstract
In
this thesis, the complete metric space and the complete normed space are
investigated. Fixed points of a mapping and the principle of contraction
mapping in complete metric spaces are studied. Solving the systems of linear
equations are discussed by using Banach fixed point theorem. Moreover the
Schauder fixed point theorem is also explained.
.
ASPECT OF BOUNDED LINEAR OPERATORS ON HILBERT SPACES
Abstract
The general form of bounded linear
functionals on various spaces is studied. Properties of Hilbert-adjoint
operators are discussed. Self-adjoint, unitary and normal operators and some of
their basic properties are also discussed. Moreover, bounded linear operators
and self-adjoint linear operators which are defined on a complex Hilbert space
are studied.
NONEXPANSIVE MAPPINGS AND RELATED CLASSES OF MAPPINGS
Abstract
Some
results of nonexpansive mappings are expressed. And several general properties
of nonexpansive mapping are presented. Next classes of mapping related to
nonexpansive mappings are also studied. Moreover some results about the
existence of fixed points for nonexpansive mappings or related classes of
mappings on certain classes of Banach spaces are discussed.
CHARACTERIZATION
OF CONVEX SETS AND CONVEX FUNCTIONS ON Rn
Abstract
In this thesis, basic properties of
convex sets and convex functions on Rn are studied. We discuss the
convexity and lower semi-continuity of functions can be reduced to convexity
and closure of their epigraph. Moreover, some algebraic operations that
preserve convexity of a function are also discussed.
APPLICATIONS AND ANALYSIS OF BURNSIDE’S THEOREM
Abstract
Firstly,
we discuss the result that a G-set X exists if and only if there exists a
homomorphism of G into a set of all permutations of X. Also, we discuss a
generalization of this result and the relation between orbit and stabilizer.
Finally, we discuss Burnside’s Theorem which is to calculate the number of
different arrangements of colorings of object in a set under group
permutations, and its several applications.
SOME CHARACTERIZATIONS OF REFLEXIVITY ON NORMED SPACES
Abstract
The
fundamental definitions and examples as well as the most elementary properties
of normed spaces such as the continuity of their vector space operations are
presented. The bounded linear operators and the properties of quotient spaces
formed from normed spaces are examined. The dual space of a normed space and
characterizations up to isometric isomorphism of the duals of direct sums,
quotient spaces and subspaces of normed spaces are studied. Separability and
reflexivity of normed spaces are expressed. The most important of the some
characterizations of reflexive normed spaces are discussed.
REPRESENTATION THEORY OF FINITE GROUPS
AND BURNSIDE'S THEOREM
Abstract
In this thesis, we study the part of
group theory, especially: If 
is a group of
order 
where 
is a prime and 
is a natural
number, then the center of is not equal to
the set of unit element and all -groups are solvable. It is also found that a
representation of a group can be decomposed into irreducible representations;
especially: If 
is a
representation of a group 
, then for all 
, the character of , 
is a sum of 
roots of unity.
As a consequence of this, it is given that the number of irreducible
representations of is equal to the number of conjugacy classes of . Finally, by using them, Burnside's Theorem is shown.
is a group of
order where is a prime and is a natural
number, then the center of is not equal to
the set of unit element and all -groups are solvable. It is also found that a
representation of a group can be decomposed into irreducible representations;
especially: If is a
representation of a group , then for all , the character of , is a sum of roots of unity.
As a consequence of this, it is given that the number of irreducible
representations of is equal to the number of conjugacy classes of . Finally, by using them, Burnside's Theorem is shown.
PROPERTIES OF IDEALS AND EUCLIDEAN DOMAIN
Abstract
In this thesis, we discuss rings, subrings and
subfields, which are mathematically interesting and the necessary definitions
and theorems of division ring and etc: And also discuss properties of
ideals.Finally, we explain the important properties of Euclidean domain.
FINITE DIFFERENCE METHODS FOR DIFFERENTIAL
EQUATIONS
Abstract
In this thesis, we study numerical solutions
to differential equations (ordinary or partial) using the finite difference
method. The explicit and Crank-Nicolson schemes are developed, and applied to a
simple problem involving the heat equation. And then we discuss numerical
solutions to the wave equation and Laplace’s equation. Consistency,
compatibility, convergence and stability for numerical solutions to
differential equations are also investigated.
EUCLIDEAN AND FACTORIZATION DOMAINS
Abstract
In this thesis we discuss Rings and Ideals,
Euclidean Domains and Principal Ideal Domains and Unique Factorization Domains.
Firstly we discuss the useful basic definitions and theorems of rings, ideals
and the prime elements and the irreducible elements in a commutative ring. Next
we present the Euclidean domain and Principal Ideal Domain of integral domain
with unity. Finally we discuss the Unique Factorization Domain of an integral
domain with unity which include the relation between prime elements and
irreducible elements.
CHARACTERIZATIONS OF EIGENVALUES AND EIGENVECTORS
Abstract
First, we show that a square matrix of size n has at least one eigenvalue. Second,
we describe algebraic and geometric multiplicities of eigenvalues and also
dicuss some results concerning with them. Finally, we obtain the result that a
matrix is diagonalizable if and only if its algebraic and geometric
multiplicities are equal for each eigenvalue of this matrix.
ELEMENTS OF FUNCTIONAL ANALYSIS IN BANACH SPACES
Abstract
The theory of normed spaces in particular
Banach spaces and the theory of linear operators defined on them are the most
highly developed part of functional analysis; This thesis is the devoted to the
basic ideas of these theories. Then the two important theorems in this thesis
are open mapping theorem and the closed graph theorem which are the
cornerstones of the theory of Banach spaces.
BOUNDED SELF-ADJOINT LINEAR OPERATORS AND THEIR PROPERTIES
Abstract
In this thesis, we present the bounded
self-adjoint linear operators on Hilbert spaces which are mathematically
interesting and practically important. And we discuss spectral properties of
this operators. Finally, we also discuss the projection operators and their
properties.
CHARACTERIZATION OF LINEAR
TRANSFORMATIONS
Abstract
First, we show that a linear transformation
from 
into 
is represented by 
matrix with complex entries. Second, we
obtain new linear transformations from old linear transformations and also
express some results concerning injective and surjective linear transformation.
Finally, we have shown that if T is a
linear transformation on a finite dimensional vector space then the sum of the
dimensions of the range of T and the
kernel of T is equal to the dimension
of this vector space.
into
is represented by matrix with complex entries. Second, we
obtain new linear transformations from old linear transformations and also
express some results concerning injective and surjective linear transformation.
Finally, we have shown that if T is a
linear transformation on a finite dimensional vector space then the sum of the
dimensions of the range of T and the
kernel of T is equal to the dimension
of this vector space.
SOLVING THE INTEGER PROGRAMMING PROBLEMS
Abstract
In this thesis, the branch-and-bound
algorithm and the Gomory algorithm for integer programming problems are
discussed. In addition, some examples of the branch-and-bound algorithm and the
Gomory algorithm are also mentioned.
VIBRATIONS OF STRINGS AND MEMBRANES
Abstract
This thesis is mainly concerned with the
vibrations of a string and of a stretched membrane. At first, we give some
simple solutions of the wave equation, including solutions known as plane wave.
Then we study an important inequality which must be satisfied by every solution
of the wave equation. It follows from the inequality that the vaue of the
solution of the initial value problem at any given point of space-time depends
only on the values of the initial data. Finally, we discuss the vibrations of a
taut string of length L with both
fixed and of a rectangular frame of length a
and width b.
HOMOMORPHISMS AND IMBEDDING OF RINGS
Abstract
In this thesis we study the rings, integral
domain, field, characteristic of a ring, idempotent and nilpotent. Again we
also present homomorphism, endomorphism and imbedding of rings.
LINEAR TRANSFORMATIONS
Abstract
In this thesis we study the linear
transformations. Firstly, we are going to discuss vector spaces, addition of
vectors and scalar multiplication, basis and dimension. Again we also study
injective and surjective of linear transformations and matrix of a linear
transformation relative to the ordered basis and examples.
SOME PROPERTIES OF MATRICES
Abstract
First, we obtain new matrices by discussing matrix addition and scalar
multiplication. Second, we treat new operation on matrices “matrix multiplication”
and its properties. We also obtain a fundamental result about Hermitian
matrices, matrix-vector products and inner products. Finally, we express
substitution methods for inverse of some matrices.
NONLINEAR SYSTEMS OF
DIFFERENTIAL EQUATIONS
Abstract
This thesis is mainly concerned with solutions of plannar nonlinear
systems near equilibrium points resemble those of the linearized system, at
least in certain cases. At first we compute the general solution of any planar
system using the eigenvalues and eigenvectors. There is a seemingly endless
number of distinct cases. We will express in the simplest possible form nearly
all of the types of solutions. Finally we discuss the phase plane for nonlinear
system and determine the behavior of solutions near equilibrium points.
QUOTIENT TOPOLOGICAL VECTOR SPACES AND LINEAR MAPPINGHS
Abstract
This thesis carries a detailed study of
Hausdorff spaces, locally convex spaces and quotient topological vector spaces.
By means of semi-norm, characterization of continuous linear maps on a space
and formation of quotient topology with the help of semi-norm are discussed in
the thesis.
FINITE DIFFERENCE
APPROXIMATIONS TO THE HEAT EQUATION
Abstract
In this thesis, we study numerical solutions to the heat equation using
the finite difference method. The forward time, centered space, the
backward-time centered-space, and Crank-Nicolson schemes are developed, and
applied to a simple problem involving the one-dimensional heat equation. And
then we consider the consistency, convergence and stability for the numerical
solution to the heat equation.
EUCLIDEAN AND POLYNOMIAL RINGS
Abstract
The material that we consider in this thesis involves the notion of
polynomial and the set of all polynomials over a given field. We have some
familiarity with the notion of polynomial from our high school days and we have
seen some of the things are does with polynomial: factoring them, looking for
their roots, dividing one by another to get a remainder and so on. The emphasis
we shall give of the concept and algebraic known as a polynomial ring will be a
quite different direction from that given in high school. Furthermore we
discuss the required definitions and theorems of ring, ideals and homomorphism.
We also present the required definitions and theorems of Euclidean ring and
principal ideal domain.
MOTION OF A SYSTEM OF PARTICLES
Abstract
In this M. Sc. thesis paper, equation of
motion, interaction principle, general equation for the motion of system and
motion of the centre of mass are studied and impulse, impulsive forces, impact
of two bodies, impact of elastic bodies, the principle of linear momentum,
Newton’s experimental law and direct impact of two spheres is discussed.
Finally some suitable worked examples are presented.
STUDY ON VORTEX MOTION
Abstract
In the thesis, vortex motion, vortex filament,
rectilinear vortices with relevant examples are discussed. The images of a
vortex outside and inside of a circular cylinder are presented. Then the image
of a vortex outside and its example are mentioned. Moreover, the image of a
vortex filament in a plane is expressed.
ADJOINT OPERATOR ON INNER
PRODUCT SPACE
Abstract
In this thesis, we discuss vector spaces, which are mathematically
interesting, and the necessary definitions and theorems of subspaces and etc.
And we will discuss the concepts of length and orthogonality. Finally, we
explain the adjoint operator on inner product space.
PROPERTIES OF SUBRINGS AND
IDEALS
Abstract
In this thesis we express rings and some special classes of rings. Then
we also express the pigeonhole principle, subrings and some examples. Finally
we discuss haracteristic of a ring and ideals.
APPLICATINS OF GRAPHS AND DIGRAPHS
Abstract
In this thesis, we present some basic definitions and notations for
undirected and directed graphs and give some examples. Next we state theorems
and representation of graphs and diagraphs, and finally, we study the shortest
path problems and their related examples.
OPTIMAL SOLUTIONS FOR BALANCED TRANSPORTATION MODEL
Abstract
In this thesis, we present formulation of linear programming problem and
one example. Next, we state formulation of transportation problem as a linear
programming model and solution of a transportation problem by solving (i) The
Northwest corner rule (ii) Row minima and column minima (least-cost) method (iii)
Vogel’s approximation Method and finally, we study finding the optimal solution
using MODI method.
COMPLETE MEIRIC SPACES AND
CONTRACTION MAPPING HEOREM
Abstract
Fixed points of the contraction mappings defined on a metric space are
studied. The contraction mappings defined on complete metric space are
considered. Then the contraction mapping theorem or Banach fixed point theorem,
it states conditions sufficient for the existence and uniqueness of a fixed
point, is discussed. Moreover, two important fields of application of the
theorem, namely, ordinary differential equations, integral equations are
considered.
REPRESENTATION OF HERMITIAN
MATRICES BY ORTHOGONAL PROJECTORS
Abstract
Firstly, we consider the eigenvalues and eigenvectors of a matrix,
especially a number of results on some special matrices. Secondly, we also
express some results concerning eigenvalues and eigenvectors on similar
matrices. Lastly, we obtain the results that every square matrix is unitarily
similar to an upper triangular matrix and its consequences.
COMPACTNESS PROPERTIES IN A HAUSDORFF SPACE
Abstract
In this thesis is concerned with compactness properties in a Hausdorff
space. Some proposition of separation properties and compactness properties in
a topological space are discussed. In addition, relations between compactness
properties and separation properties in a Hausdorff space are mentioned.
MOTION OF A SPHERE IN AN INCOMPRESSIBLE FLUID
Abstract
This thesis includes the basic concepts of fluid motion. And, the
source, sink, doublet and uniform stream in three-dimension are presented. We
study the motion of a sphere in an incompressible fluid. Moreover, impulsive
motion and concentric sphere are also mentioned with some examples.
ASPECTS OF APPROXIMATION THEORY
Abstract
Important concept and properties in normed
space and Hilbert space are presented. The existence and uniqueness of best
approximation are also discussed in this thesis. It is proved that if a normed
space is strictly convex, we have uniqueness of best approximation. And it is
also discussed for Hilbert space this holds. Moreover, it is explained that for
general normed space one may need additional conditions to guarantee uniqueness
of best approximations, for instance a Haar condition in 
MAXIMAL IDEALS AND EUCLIDEAN RINGS
Abstract
First, we express rings and fields. Second, we also discuss idels,
quotient rings and homomorphisms. Finally, we obtain the properties of maxiamal
ideals and Euclidean rings.
REPRESENTATION OF UNITARY MATRICES BY REFLECTORS
Abstract
Firstly, we show that a matrix is unitary matrix if and only if the
columns or (rows) of its form a orthonormal set. We also show that if a matrix
is unitary then it is diagonalizable. Secondly, we obtain the fact that every
matrix is the product of unitary matrix and upper triangular. The main fact is
that every square unitary or orthogonal matrix is the product of elementary
reflectors.
SOLVING THE BALANCED TRANSPORTATION PROBLEMS
Abstract
In this thesis, we consider the transportation problem. Method of
finding initial basic feasible solution and optimality solution for
transportation problem are discussed. In addition, some examples of the balanced
transportation problem are also mentioned.
EXTENSION OF BOUNDED LINEAR FUNCTIONALS ON NORMED SPACES
Abstract
In this thesis we firstly express some results of bounded linear
operators and bounded linear functional on a normed space. We also express two
results concerning extension of linear functionals on a vector space namely,
Hahn-Banach Theorem on real vector space, Hahn-Banach Theorem on complex vector
space. By using these results, we obtain two theorems concening bounded linear
functionals on a normed space.
A STUDY ON BASIC FOR FLUID DYNAMICS
Abstract
In this M. Sc. thesis paper, we study the basic properties of fluid
dynamics. Firstly, we are going to discuss the density and pressure at a point,
fluid velocity and stream lines. Again we also study the equation of
continuity, stream functions for two-dimensional motion and Sources, Sinks and
Doublets in two-dimensions.
APPLICATIONS OF CIRCLE THEOREM
Abstract
In this thesis, two-dimensioned source and sink, doublet and vortex are
presented. There is also the study of uniform stream of complex potential
included. Moreover, calculation of Blasius and Circle Theorem and relevant
examples are also presented.
APPLICATIONS OF MATRIX REPRESENTATION
Abstract
In this thesis, we present some basic definitions and theorems of linear
transformation, isomorphism and etc. Next we state theorem and examples of
vector representation. Finally, we explain the important properties of matrix
representation.
INVARIANT SUBSPACES
Abstract
In this thesis we study the invariant subspaces. Firstly, we are going
to discuss vector spaces and eigen values and eigen vectors. Again we also
study annihilating polynomials, theorems and some examples.
COMPACTNESS PROPERTIES IN A HAUSDORFF SPACE
Abstract
In this thesis is concerned with compactness properties in a Hausdorff
space. Some proposition of separation properties and compactness properties in
a topological space are discussed. In addition, relations between compactness
properties and separation properties in a Hausdorff space are mentioned.
VARIOUS TYPE OF EQUATION OF MOTION
Abstract
In this thesis, some properties of the fluid, equation of motion with
relevant examples are discussed. The equation of motion in cylindrical
coordinates and spherical polar coordinates are presented. Then impulsive
action, equation of motion under impulsive force (vector form) and it examples
are mentioned. Moreover, equation of motion under impulsive force (cartesian
form) is expressed.
SOME PROPERTIES IN METRIC SPACE AND INNER PRODUCT SPACE
Abstract
In this thesis, we present some basic definitions and notations for
vector space, metric space and inner product space and give some examples. Next
we state characterizations of this spaces and finally, we discuss some
properties of vector space, metric space and inner product spaces and their
related examples.
SOME BASIC CONCEPT FOR GRAPH AND DIGRAPH
Abstract
In this M.Sc. thesis paper, we introduce come basic definitions and
notations for graph and digraph and give some examples. Next we state theorems
and examples for connectedness and component of a graph and directed graph, and
finally, we study shortest path algorithm and their related examples.
THE NATURE OF THE FLOW IN MOTION
Abstract
In this M.Sc. thesis paper, two-dimensional motion, stream lines,
irrotational motion in two-dimensions and complex potential in two-dimensional,
irrotational, incompressible flow are studied, definition of source, sink and
doublet and complex potentials for source, sink and doublet are also discussed.
Again, velocity and acceleration of a fluid particle and the system of equation
of continuity. Finally some suitable worked examples are presented.
IDEALS, HOMOMORPHISMS, AND QUOTIENT RINGS
Abstract
In this thesis we study concept in group, some concept in ring,
properties of ideals and homomorphisms. We study isomorphism and quotient
rings.
ORTHONORMAL SET OVER THE VECTOR SPACE
Abstract
In this thesis, we study orthonormal basis the vector space. Firstly we
are going to discuss vector spaces, some concepts of vector space, linear
transformation on vector spaces and quotient space. Secondly, we also study
inner product space. Finally, we study orthonormal set over the vector space
relative to theorem and examples.
STUDY ON EQUATION OF CONSERVATION OF MASS
Abstract
In this thesis, some properties of the fluid, fluid motion with relevant
examples are discussed. The equation of continuity in cartesian coordinates and
cylindrical coordinates are presented. Then vortex line, vortex tube and vortex
filament are mentioned. Moreover, rotational and irrotational motion is
expressed.
VECTOR SPACES AND MODULES
Abstract
In this thesis we study vector spaces and subspaces, isomorphisms,
linear independence, bases and finite-dimensional. We also present dual spaces,
inner product spaces and modules.
A STUDY ON SYSTEMS OF LINEAR EQUATIONS
Abstract
In this thesis, we present some basic definitions and notations for
systems of linear equations, pivot operations and more on systems of linear
equations and give some examples. Next we discuss the characterizations of
systems of linear equations and their related examples.
PROPERTIES OF POSETS AND LATTICES
Abstract
In this thesis we discuss partial order relations which would finally
lead us to the definition of a lattice. Firstly we discuss various types of
relations that can be defined on a set. Next, we discuss uniqueness of greatest
lower bounds (least upper bounds) and each partially order set can be
represented by helping of a digram. Finally, we discuss properties of partially
order sets and lattices.
PROPERTIES OF IDEALS IN LATTICE
Abstract
In this thesis we discuss characterization of ideals. Firstly, we
express the useful basic definitions of partially ordered set and its
properties. Next, we are now well-equipped to define a lattice and study its
properties. Finally, we discuss ideals, dual ideals, principal ideals,
principal dual ideals, prime ideals, dual prime ideals and their properties.
SOME APPLICATIONS OF THE DIMENSION THEOREM TO THE SOLUTION
OF THE SYSTEM OF LINEAR EQUATIONS
Abstract
This
thesis is concerned with a study on solutions of linear equations. For this
study, some basic definitions and properties related to the vector space are
required and so these are firstly discussed. Then the dimensional theorem is
studied. Finally, the dimensional theorem is applied to the solution of the
system of linear equation.
THE
RELATION BETWEEN MATRICES AND LINEAR MAPS
Abstract
This thesis is concerned with a
study on the relation between matrices and linear maps. For this study, some
basic definitions and properties related to the vector space are required and
so these are primarily discussed. Linear Mappings and their properties are
explained. The linear map associated with a matrix and the matrix associated
with a linear map are also studied.
CHARACTERIZATION
OF MEASURABLE SETS AND MEASURABLE FUNCTIONS
Abstract
Several properties, concerning algebra of sets, are presented. Two
important properties of Lebesgue measure are also discussed. In this thesis,
certain operations performed on measurable functions lead again to measurable
functions and consequence of equality 
are explained.
are explained.
STUDY ON
EQUATIONS OF MOTION FOR INCOMPRESSIBLE AND INVISCID FLUID IN STEADY MOTION
Abstract
In this thesis, curvilinear coordinate systems are explained. Basic
hydrodynamic equations such as equation of continuity and equation of motion
are derived. Finally, we study steady motion for incompressible and inviscid
fluid in steady motion with some examples.
STUDY ON SOME
EQUATIONS OF STREAMLINES
FOR THE
INCOMPRESSIBLE FLUID
Abstract
In this thesis, basis properties of the fluid motion are first studied.
Then basic hydrodynamic equations such as the equations of continuity and
equations of motion are derived. The stream lines and Reynold's transport
theorem are also discussed. The equations of stream lines for the steady motion
of an incompressible fluid are explained with some examples.
THE EQUATIONS OF
MOTION FOR A PARTICLE MOVING IN ELLIPTIC ORBITS
Abstract
In this thesis concerned with the central
force motion which is a plane curved motion. The areal velocity is defined and
shows that it is the same as the magnitude of angular momentum or moment of
momentum for a unit mass which is a constant. Two types of coordinates system,
the reciprocal polar coordinates and pedal coordinates are introduced.
Particles moving under the inverse square law always described conic orbit is
proved. Finally we discussed about the elliptic orbit moving under the inverse
square law when a blow is given along tangentially and normally.
VARIOUS FORMS OF
FOURIER INTEGRAL THEOREM
Abstract
In this thesis, we study the Fourier
series of the periodic functions. The Fourier series of even and odd functions,
half range expansions are discussed. In addition, some examples of the Fourier
integral theorem are also explained.
APPLICATIONS TO
PARTIAL DIFFERENTIAL EQUATIONS OF ORDER TWO
Abstract
In this thesis, differential equations, partial differential equations
of first order with relevant examples are discussed. And then applications of
partial differential equations of order two are mentioned. Moreover, variable
separation method and Laplace's Equation and its solution are expressed.
SOME TYPES OF
MATCHING
Abstract
Let 
be a graph. A
subset M of E is called a matching of G provided no two
edges of M are adjacent. We give some conditions for the existence of a
matching in a bipartite graph and perfect matching in a simple graph. In the
last section, we study on algorithm to find maximum matching in the simple
graph.
be a graph. A
subset M of E is called a matching of G provided no two
edges of M are adjacent. We give some conditions for the existence of a
matching in a bipartite graph and perfect matching in a simple graph. In the
last section, we study on algorithm to find maximum matching in the simple
graph.
GRAPH
REPRESENTATION AND GRAPH ISOMORPHISM
Abstract
In this thesis, we study some basic definitions and notations for
graphs. We also discuss Euler tour and Hamilton cycle of a graph. We present a
necessary condition for isomorphism between two graphs and we determine pairs
of graph are isomorphic.
A BRIEF STUDY OF
OPERATORS
Abstract
We discuss the theory of operators
on Banach spaces. And we consider the operators in Hilbert space, which has
more structure than a normed space and a correspondingly richer operator
theory. Because we are interested in the eigenvalue problem, we shall consider
only operators which map a Hilbert space into itself. We also express the
eigenvalues of a self-adjoint operators and compact operators.
UNBOUNDED LINEAR
OPERATORS IN HILBERT SPACE
Abstract
In this thesis, we state the unbounded linear operators in Hilbert
space. Unbounded linear operators occur in many applications, notably in
connection with differential equations and in quantum mechanics. Their theory
is more complicated then that of bounded operators. Most unbounded linear
operators occurring in pratical problems are closed or have closed linear
extensions. And then we study the spectral properties of self-adjoint linear
operators and unitary operators.
A STUDY ON
BRANCH AND BOUND ALGORITHM
Abstract
In this thesis, we study simplex method to solve the linear programs.
Next, we discuss the integer programming problem by using branch and bound
algorithm and finally, we present the optimal solution in travelling salesman
problem by using this algorithm.
TOPOLOGICAL
VECTOR SPACES ON FUNCTIONAL ANALYSIS
Abstract
In this thesis, we present some basic definitions and notations for
topological vector space and give some illustrative examples. Next we state
characterizations of this space and finally, we discuss some properties of
topological vector spaces on functional analysis and their related examples.
UNBOUNDED LINEAR
OPERATORS IN HILBERT SPACE
Abstract
In this thesis, we state the unbounded linear operators in Hilbert
space. Unbounded linear operators occur in many applications, notably in
connection with differential equations and in quantum mechanics. Their theory
is more complicated then that of bounded operators. Most unbounded linear
operators occurring in pratical problems are closed or have closed linear
extensions. And then we study the spectral properties of self-adjoint linear
operators and unitary operators.
GROUPS OF RIGID
MOTIONS OF REGULAR N-GONS
Abstract
We first present symmetries of the geometric
figures, that is, examine the permutations of vertices of them. Next, we also
express the some results of cyclic groups, especially, cyclic groups are
abelian. The main fact of this thesis is that the group of rigid motions of a
regular n-gon (
) is a subgroup of symmetric group on n letters.
) is a subgroup of symmetric group on n letters.
GROUPS OF RIGID
MOTIONS ON THE PLANE
Abstract
Firstly, we present some illustrated example and basic facts on groups.
We also discuss groups of rigid motions; especially, groups of reflections,
translations and rotations on the plane. The main fact is that an isometry on
the plane is a linear transformation and is represented by an element in the
group of two by two orthogonal matrices.
NORMS AND INNER
PRODUCTS
Abstract
In this thesis, we study linear transformation, orthonormal basis, dual
space, orthogonal complements and linearly independents. We also present norms,
inner product spaces and isomorphism.
STUDY ON DUALITY
OF LINEAR PROGRAMING
Abstract
In this thesis, basic properties of primal and dual linear programming
problem are described. Based on this method, basic feasible solutions for the
optimal solution of maximization or minimization problems are discussed. Some
theorems and some examples of the primal and dual linear programming problem
are also explained.
GENERALIZED
GROUP MULTIPLICATION
Abstract
We discuss the basic properties of the groups of rigid motions on a
geometric figure; especially, examine the some facts on the permutation groups
on n letters. The last fact is that
the number of ways in which the vertices of a square can be colored red or blue
by using a group of rigid motions.
ACYCLIC DIRECTED
GRAPH (DIGRAPH)
Abstract
We introduce basic concepts of directed graph (digraph). Directed paths
and directed cycles are discussed. We define a tournament and directed Hamilton
path of a digraph D. We study some
properties of an important class of directed graphs, namely, the acyclic
directed graphs. Some theorems related to maximal acyclic directed graphs are
stated and proved.
MODULES OVER A
PRINCIPAL IDEAL DOMAIN
Abstract
In this thesis, we study base ring,
submodule, R-module, the rank of a
free module, Epimorphisms. We also present principal ideal domain, annihilator,
cyclic module and primary module.
TRAVELING
THROUGH A GRAPH
Abstract
In this thesis, we first express some basic
definitions, notations and some examples in graph theory. The relations between
the degrees of vertices and the number of edges are presented. We consider some
properties of connected and disconnected graphs. We study the existence of an
Euler trail in a graph is related to the degrees of the vertices. Finally, we
discuss sufficient conditions for a graph to be hamiltonian.
NAVIER-STOKES’
EQUATIONS IN TERMS OF CURVILINEAR COORDINATES
Abstract
In this thesis, we study the Navier-Stokes’
Equation in terms of orthogonal curvilinear coordinates of incompressible
viscous fluid motions.
SOME
APPLICATIONS OF BLASIUS’ THEOREM
Abstract
In this thesis, we introduce the basic concepts of fluid motion. Next we
discuss inviscid two-dimensional motion. Lastly, we proof Blasius’ theorem and
describe some applications.
GROUPS AND RINGS
ISOMORPHISMS
Abstract
In this thesis, we express basic definitions
of groups, subgroups, normal subgroups and homomorphisms. Then, we discuss the
theorems of ideals and isomorphisms. In addition, isomorphisms on groups and
rings are also presented.
NILPOTENT LINEAR
TRANSFORMATION
Abstract
In this thesis, we present some basis definitions and theorems of linear
system of equation, linear transformation, basis and etc. Next we start
theorems and examples of eigenvalues and eigenvectors of a matrix. Finally, we
explain the properties of matrix representation and nilpotent linear
transformation.
FACTORIZATION
OF POLYNOMIALS
Abstract
In this thesis, we discuss factorization of integer, which are
mathematically interesting, and definitions and theorems of ring, ideals and
etc. And also discuss properties of Euclidean ring. Finally, we explain the
definitions and theorems of factorization polynomial.
TWO-DIMENSIONAL SOURCE, SINK, DOUBLET
AND ITS IMAGE SYSTEM
Abstract
In this thesis, firstly we present equation of continuity, equation of
motion and streamlines. Secondly we express source, sink and doublet in two
dimensions. Finally, we discuss the image systems concerning with them.
STEADY FLOW
THROUGH A CURVILINEAR SQUARE
Abstract
In this thesis, analysis of stress in fluid such as stress vector,
stress components in cartesian coordinate system, symmetry in stress vector,
principal stress and stress invariants are expressed. Basis equations of a
viscous fluid are derived. Steady viscous flow in tube of uniform cross section
and the rate of discharge are discussed. Finally, the rate of discharge for the
curvilinear square is derived.
FORCES ON A
SPHERE IN A VISCOUS FLUID
Abstract
In this thesis, we introduce the
basic concepts of vectors and tensors. The Navier-Stokes equation for the fluid
motion and the Reynolds number are derived. Lastly, describe the constitutive
equation and we calculate a force exerts on a sphere in the fluid and the
couple required to maintain rotation the sphere.
STUDY ON
CREEPING FLOW PAST A FIXED SPHERE
Abstract
In this thesis, we introduce the basic concepts of vectors and tensors.
The Navier-Stokes equation for the fluid motion and the Reynolds number are
derived. Finally, we calculate axisymmetric flow in spherical coordinates,
streamlines and the drag coefficient of creeping flow past a fixed sphere.
MAXIMAL IDEALS
AND PRIME IDEALS
Abstract
Firstly, we discuss about the ring
with some definitions and theorems. Then, some definitions and theorems of
ideal, field and quotient ring are discussed. Some theorems on ring
homomorphisms are also presented. Finally, we take up the question of when a
quotient ring of a ring is a field and when it is an integral domain.
SOME SPECTRAL
THEORY OF BOUNDED SELF-ADJOINT LINEAR OPERATORS ON HILBERT SPACES
Abstract
The bounded self-adjoint linear operators on a Hilbert spaces are
studied. Which are mathematically interesting and practically important in
applications. Next, the spectral properties of bounded self-adjoint linear
operators on a complex Hilbert space are also discussed. The spectrum of a
bounded self-adjoint linear operator is real and eigenvectors corresponding to
different eigenvalues are studied. The concepts of a positive operator are
expressed.
LINEAR OPERATORS
BETWEEN NORMED SPACES
Abstract
In this thesis, firstly, the fundamental definitions and examples, as
well as the most elementary properties of normed spaces such as the continuity
of their vector space operations are presented. The basic properties of bounded
linear operators between normed spaces, including properties of normed space
isomorphisms are studied. The continuous linear operators between normed spaces
are also discussed.
EULERIAN TOUR
AND THE POSTMAN PROBLEM FOR UNDIRECTED GRAPHS
Abstract
A trail in a graph is an Euler trail if every edge of the graph appears
as an edge in the trail exactly once. A closed Euler trail is an Euler tour. A
graph is said to be an Euler graph if it has an Euler tour. The postman problem
consists of finding a minimum weight tour of a graph 
traversing all
its edges at least once. We present how to solve the postman problems for any
undirected graph.
traversing all
its edges at least once. We present how to solve the postman problems for any
undirected graph.
FINDING MINIMUM
SPANNING TREES IN UNDIRECTED GRAPHS
Abstract
Trees are central to the structural
understanding of graphs, and they have a wide range of applications. In this
thesis, some basic properties of a trees and rooted trees are given. The two
algorithms used to construct minimum spanning trees in undirected graphs are
also presented and some examples are discussed.
EDGE COLOURING
OF A GRAPH
Abstract
In this thesis, we first express some basic definitions and some
examples in graph theory. The relations between the degrees of vertices and the
number of edges are presented. We study the existence of an Euler trail in a
graph is related to the degrees of the vertices. Then we discuss sufficient
conditions for a graph to be hamiltonian. For simple graph, Vizing’s theorem
has shown that 
colours are
sufficient. We consider the question of existence of 2 edge colouring of a
graph in which 2 colours for every vertex in the graph. Finally, we determine
whether a given graph is of class one or of class two.
colours are
sufficient. We consider the question of existence of 2 edge colouring of a
graph in which 2 colours for every vertex in the graph. Finally, we determine
whether a given graph is of class one or of class two.
VERTEX COLOURING
OF A GRAPH
Abstract
In this thesis, we first express some basis definitions and some
examples in graph theory. The relations between the degrees of vertices and the
number of edges are presented. We consider some properties of connected and
disconnected graphs. We discuss basic principles for calculating chromatic
numbers. Then we study upper bounds and lower bounds for chromatic number of a
graph. Finally, we describe graphs that are critical with respect to chromatic
number.
STUDYING ON AN
EUCLIDEAN RING
Abstract
In this thesis, basic definitions of
groups and theorems are presented. Next, we study the ring theory and some
examples. Finally, we discuss about an Euclidean ring to be the ring of
Gaussian integers and a Fermat theorem.
LEVEL
ASSIGNMENTS OF AN ACYCLIC DIGRAPH
Abstract
For a subset 
of a digraph
, we define a reachable set 
of S.
Then we define a basis for. We express definitions of n-basis and transmitter.
We prove that every acyclic digraph without isolates has a unique basis
consisting of its transmitters. We also prove that every acyclic digraph has a
unique 1-basis. We discuss level assignments in an acyclic digraph. Finally, we
show that every acyclic digraph has a gradable expansion.
of a digraph, we define a reachable set of S.
Then we define a basis for. We express definitions of n-basis and transmitter.
We prove that every acyclic digraph without isolates has a unique basis
consisting of its transmitters. We also prove that every acyclic digraph has a
unique 1-basis. We discuss level assignments in an acyclic digraph. Finally, we
show that every acyclic digraph has a gradable expansion.
A STUDY ON
DIRICHLET PROBLEM INVOLVING LAPLACE EQUATION FOR A CIRCLE
Abstract
The classifications of second-order equations with two independent
variables are studied. The determinations of the general solution for a class
of relatively simple equations are illustrated.
Then Fourier series essential for the further study of partial
differential equations are discussed. Boundary value problems and maximum
principle are also discussed. Moreover, the existence of the solution of the
Dirichlet problem for interior and exterior of a circle are studied.
SOME CONTINUOUS
FUNCTIONS
Abstract
In this thesis, we present some basic definitions and notations for some
continuous functions and give some illustrative examples. Next we state
characterizations of these and finally, we discuss some properties of
continuous functions on real numbers, metric spaces and topological space and
their related examples.
SOME PROPERTIES
OF PROJECTION OPERATORS ON HILBERT SPACES
Abstract
The inner product and the associated idea of a pair of vectors being
perpendicular or orthogonal are presented. The sets of perpendicular vectors
can form a basis for an infinite-dimensional space and the geometrical aspects
of Hilbert space theory are also discussed. Moreover, the projection operators
on a Hilbert space and their basis properties are expressed.
EQUATIONS OF CONTINUITY AND EQUATIONS OF
MOTION IN ORTHOGONAL CURVILINEAR COORDINATES
Abstract
This thesis is mainly concentrated
with the equations of continuity and equations of motion in orthogonal
curvilinear coordinates. Firstly, the basic concepts of vector algebra, the
velocity and acceleration of a particle, the motion of a fluid element, the
equation of continuity, Reynold’s Transport Theorem and the equations of motion
are discussed. Secondly, the arc length and the volume element, the gradient of
, the divergence of 
the curl of 
and the
Laplacian of in curvilinear
coordinates are considered. Finally, the gradient of , the divergence of the curl of the Laplacian
of 
the equations
of continuity and the equations of motion in orthogonal curvilinear coordinates
are studied.
, the divergence of the curl of and the
Laplacian of in curvilinear
coordinates are considered. Finally, the gradient of , the divergence of the curl of the Laplacian
of the equations
of continuity and the equations of motion in orthogonal curvilinear coordinates
are studied.
IDEALS AND
SOLVING ITS PROBLEMS
Abstract
In this thesis, we study ideals and solving its problems. Firstly, we
are going to discuss rings and field, ideals, homomorphisms and quotient rings.
Again we also study prime ideals, maximal ideals, theorem and some examples.
POLYNOMIALS
AND MATRICES
Abstract
In this thesis, we study polynomials and matrices on a finite
dimensional vector space. Firstly, we are going to discuss vector spaces and
eigen values and eigen vectors. Then we also study diagonalizable on finite
dimensional vector spaces, theorems and some example.
ANALYSIS OF THE
ORDERS OF CYCLIC GROUPS
Abstract
In this thesis, we present some basic definitions and notations for some
continuous functions and give some illustrative examples. Next we state
characterizations of this and finally, we discuss some properties of continuous
functions on real numbers, metric space and topological space and their related
examples.
VARIOUS KIND OF
RINGS
Abstract
In this thesis, we express basic definitions of rings, subrings,
commutative rings and Boolean rings. Then, we discuss the properties of ideals,
maximal ideals and simple rings. In addition, some examples and properties of
the Euclidean rings and principle ideal rings are also presented.
INNER PRODUCT
AND ORTHOGONALITY
Abstract
Some examples of Hilbert spaces are
presented. We discuss every inner product space is normed space but not all
normed spaces are inner product spaces.
Properties of orthogonal elements in
inner product spaces are studied. It is also presented how to obtain an
orthonormal sequence if an arbitrary linearly independent sequence is given.
Moreover, properties of a total orthonormal set are discussed.
ANALYSIS ON
CONGRUENCE CLASSES OF INTEGERS
Abstract
We express a generalization of equality
of integers, namely, congruence. We also discuss some properties of congruence
classes, namely, ring, integral domain, etc. Finally, we will show that the
sets of congruence class Zmn
and Zm × Zn are isomorphic when m and n are relatively prime.
ANALYSIS AND
APPLICATIONS ON CHINESE REMAINDER THEOREM
Abstract
Firstly, we express the division algorithm and its applications.
Secondly, we also express Euclidean Algorithm and its applications. Finally, we
solve the two congruences by using Chinese Remainder Theorem.
DESIGN CIRCUIT
BY USING BOOLEAN FUNCTIONS
Abstract
In this thesis, it is known that the structure
of the lattice is closely like to the structure of algebra. Namely, a
complement distributive lattice, a Boolean lattice, is a Boolean algebra.
Especially, one of major applications of Boolean algebra is that circuit is
designed by using Boolean functions.
A STUDY ON
QUOTIENT RINGS AND FINITE FIELDS
Abstract
First, we express commutative rings.
Second, we also discuss fields, polynomials and homomorphisms. Finally, we
obtain the properties of quotient rings and finite fields.
SOME
REPRESENTATIONS OF A LINEAR TRANSFORMATION AND DIAGONALIZATION
Abstract
We study the properties of linear
transformations and the matrix representations of a linear transformation. Then
we present the eigenvalues and eigenvectors of square matrix. Finally, we study
a diagonalizable matrix.
FINITE
DIFFERENCE SCHEMES AND STABILITY OF THE WAVE EQUATION
Abstract
In this thesis, numerical
differentiations of a single variable and two variables are presented. The finite
difference method and boundary conditions are illustrated. Explicit and
implicit difference schemes of wave equation are discussed. And then stability
of the wave equation is studied.
INVISCID TWO-DIMENSIONAL VORTEX MOTION
Abstract
In this thesis, motion in two
dimensions is first studied. Then stream function and complex potential are
also discussed. Image system of source, sink and doublet are studied. Inviscid
two-dimensional vortex motions are explained with some examples.
FORCES ON A
RIGID BOUNDARY IN AN INVISCID INCOMPRESSIBLE FLUID
Abstract
Two-dimensional motion of an
incompressible fluid is studied by complex variables method. The complex
potential for sources, sinks, doublets and vortices in an irrotational
incompressible fluid flow are obtained first. We also present three-dimensional
sources, sinks and doublets. And then we study forces on a cylinder in an
inviscid fluid by using Blasius’ Theorem. Moreover, we find stream functions
and velocity potentials for the flows with spherical boundaries by sphere
theorem. Finally the result, often referred as D’Alembert’s paradox for a
finite body of arbitrary shape, is established.
THE STRUCTURE K-CONNECTED GRAPH
Abstract
At first, the basic definitions of graph
and some examples are expressed. Connected graph, tree, cut edge, cut vertex
and some theorems are stated and proved. Finally connectivity and
edge-connectivity and solving some theorems are stated and proved.
COMPACTNESS AND
SOME FIXED POINT THEOREMS ON NORMED SPACES
Abstract
The classical spaces with basic
definitions, examples and theorems are described. Then compact sets and
relatively compact sets are proved in normed spaces. Contradiction and fixed
point theorems are discussed. Finally sequences of contradictions and fixed
point theorems are studied.
STUDY ON
ERROR-CORRECTING CODE
Abstract
It is studied that the application of
finite group under addition of conjugacy classes over the finite field Z2, namely, binary linear
code in coding theory. It will introduce the Hamming distance and Hamming
weight and next shows that the distance between two words and the weight of the
difference of these words are same. Finally, it is shown that encoded messages
are decoded by three ways, namely, parity-check matrix decoding, coset decoding
and syndrome decoding.
HIGHER
DIMENSIONAL LINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS
Abstract
At first, higher dimensional linear
algebra and higher order linear differential equations with constant
coefficients are studied. Then we state the general solution and the Wronskian
with some examples are discussed.
Eigenvalues and eigenvectors in higher dimension are also investigated.
Finally, some examples for changing coordinates system are presented.
PHASE PORTRAITS
FOR PLANAR AUTONOMOUS SYSTEMS
Abstract
At first,
basic background from algebra and second-order differential equations are
discussed. Planar linear systems and some examples are also studied. Then
various types of eigenvalues are stated. Finally, phase portraits for planar
systems with examples are investigated.
INDEPENDENCE
NUMBERS AND COVERING NUMBERS OF A GRAPH
Abstract
Vertex-independent set, edge-independent
set of a graph G and two types of
covering of a graph namely, vertex cover and edge cover are studied. Theorems
concerning with independence number, edge-independence number, matching number
and edge covering number are expressed. We are also discussed some theorems about
the critical graphs.
BLOCK-CUT VERTEX
TREE OF A GRAPH
Abstract
In this thesis the basic definitions of
a graph, the concepts of cut vertex, cut edge and block of a graph are
introduced. Then the definitions of tree, block-cut vertex tree, block graph and
cut vertex graph are stated. Also mentioned, some theorems related to block-cut
vertex tree and block point tree are proved.
A STUDY ON
EXTENDABLE GRAPH
Abstract
At first, basic definitions and
notations are expressed. Matchings in bipartite graphs and non-bipartite graph
are consider. Then maximum matching, minimum covering and some theorems are
expressed.Especially, maximum matching and perfect matching are studied.
Finally, theorems on k-extendable
graphs are investigated.
MINIMUM SPANNING
TREES OF GRAPHS
Abstract
In this thesis, firstly some basic
definitions and examples in graph theory are expressed. Then several results
and concepts involving trees are presented. The relation between cutsets and
spanning trees and the relation between cycles and cotrees are discussed.
Finally, for constructing minimum spanning trees, Kruskal’s algorithm and
Prim’s algorithm are stated.
STRUCTURE
THEOREMS ON PERMUTATION GROUPS
Abstract
We first present the basic properties of
groups and subgroups. Then we express the examples and properties of cyclic
groups. The main fact of this thesis is the structures of the permutation
groups and their properties.
EUCLIDEAN AND
FACTORIZATION DOMAINS
Abstract
In this thesis, we study Euclidean and
Factorization Domains and solving its problems. Firstly, we are going to
discuss rings, fields and quotient rings. Again we also study prime ideals,
principal ideals and maximal ideals, theorems and some examples.
LAMINAR STEADY FLOW OF INCOMPRESSIBLE
VISCOUS FLUID IN TUBES OF CROSS-SECTION WITH CIRCULAR AND NON-CIRCULAR
Abstract
This thesis is mainly expressed as
laminar steady flow of incompressible viscous fluid in tubes of cross-section
with circular and non-circular. In the first section, we discussed the methods
of describing fluid motion, velocity and acceleration of a fluid particle,
motion of a fluid element, equation of continuity, Reynolds’ Transport Theorem
and the equations of motion. In the second section, we considered the steady
unidirectional flow, two dimensional flow through the straight channel, couette
flow and the Hagen-Poiseuille flow. In the last portion, we studied the tube of
non-circular cross-section, a uniqueness theorem and tube having uniform
elliptic, equilateral triangular, rectangular and cardiod cross-sections.
MODULAR AND
DISTRIBUTIVE LATTICES
Abstract
In this thesis, basic concepts of partially ordered sets (posets) and
lattices are introduced. Some ideals of lattices and homomorphisms between two
lattices are expressed. Modular equality becomes an inequality of lattices are
expressed. Distributive lattices which is related to modular lattices are also
explained.
HOMOMORPHISMS
AND IMBEDDING OF RINGS
Abstract
Firstly basic definitions, lemmas and
some examples of ring and subring are studied. Secondly we describe ideal and
quotient rings and we prove homomorphism theorem. Finally some theorems of
isomorphism of rings and imbedding of rings are discussed.
SOME PROPERTIES
OF BOUNDED LINEAR OPERATORS ON BANACH SPACES
Abstract
The basic definitions and examples of
vector spaces, normed spaces and Banach spaces are studied. Then the linear
operators especially, with examples and theorems of bounded linear operators on
normed spaces are expressed. Finally, we study the properties of bounded linear
operators on Banach spaces are discussed.
SOME FIXED POINT THEOREMS IN NORMED LINEAR SPACES
Abstract
Fixed points of the contraction mappings
defined on a metric space are studied. The contraction mappings defined on
complete metric space are considered. Then the contraction mapping theorem or
Banach fixed point theorem, it states conditions sufficient for the existence
and uniqueness of a fixed point, is discussed. Moreover, two important fields
of application of the theorem, namely, ordinary differential equations,
integral equations are considered.
CHARACTERIZATION
OF CONVEX SETS AND CONVEX FUNCTIONS ON Rn
Abstract
In this thesis, basic properties of
convex sets and convex functions on Rn are studied. We discuss the
convexity and lower semi-continuity of functions can be reduced to convexity
and closure of their epigraph. Moreover, some algebraic operations that preserve
convexity of a function are also discussed.
REPRESENTATION THEORY OF FINITE GROUPS
AND BURNSIDE'S THEOREM
Abstract
In this thesis, we study the part of
group theory, especially: If is a group of
order where is a prime and is a natural
number, then the center of is not equal to
the set of unit element and all -groups are solvable. It is also found that a
representation of a group can be decomposed into irreducible representations;
especially: If is a
representation of a group , then for all , the character of , is a sum of roots of unity.
As a consequence of this, it is given that the number of irreducible
representations of is equal to the number of conjugacy classes of . Finally, by using them, Burnside's Theorem is shown.
is a group of
order where is a prime and is a natural
number, then the center of is not equal to
the set of unit element and all -groups are solvable. It is also found that a
representation of a group can be decomposed into irreducible representations;
especially: If is a
representation of a group , then for all , the character of , is a sum of roots of unity.
As a consequence of this, it is given that the number of irreducible
representations of is equal to the number of conjugacy classes of . Finally, by using them, Burnside's Theorem is shown.