Fourth Stage



Units : 8
Lecturer : Prof. Dr. Alias Barakat Khalaf

Description :

1) Topological Spaces

2) Bases and Subbases

3) Continuity And Topological Equivalence

5) Separation Axioms

6) Compactness

7) Product Spaces

Complex Analysis

Units : 6
Lecturer : Jiyar Ahmed

Description :

In first year you studied real-valued functions of a real variable. Real functions vary enormously in how nicely behaved they are in terms of properties like continuity, differentiability and so forth. The library of standard functions that we study, such as rational functions, trigonometric functions, and hyperbolic functions, are all extremely well-behaved; much better even than a 'typical' infinitely differentiable real function.

  To understand why, it turns out that one should look at complex functions of a complex variable. Differentiability for these functions, although having an identical looking definition, is a much stronger property than in the real case. The behaviour of complex functions which are differentiable on a reasonable set is very restricted. In a certain sense, all such functions are almost polynomials and this explains many of the properties of our standard real functions.

  Studying these complex functions requires us to find new ways of picturing functions. The graphical methods of real variable calculus are of limited use, but geometric reasoning will be just as important.

 An important concept throughout the course will be that of a path integral. Here, instead of integrating a function over an interval [a,b], we integrate functions over a curve in the plane. Many of our main theorems will concern how one can evaluate these path integrals with doing any integration! The powerful integral formulas due to Cauchy and Goursat have lots of lovely consequences, including enabling us to do many real integrals for which our real valued methods fail.


Lecturer : Huda Younus Najm

Description :

The aim of the subject is to study define the problem of life and analyses. This subject is intended to: 

This is a course  for students  interested in solving optimization problems. Because of the wide  (and growing) use of optimization in science, engineering, economics, and industry, it is  essential for students and practitioners alike to develop an understanding of optimization  algorithms. Knowledge of the capabilities and limitations of these algorithms leads to a better  understanding of their impact on various applications, and points the way to future research  on improving and extending optimization algorithms. Our goal in this course is to give a comprehensive description of the most powerful, state-of-the-art, techniques  for solving continuous optimization problems. By presenting the motivating ideas for each  algorithm, we try to stimulate the reader’s intuition and make the technical details easier to  follow. Formal mathematical requirements are kept to a minimum.

Graph Theory

Description :

Course overview:
This course focuses on the mathematical theory of graphs; a few applications and algorithms will be discussed. Topics include trees, connectivity, Eulerian and Hamiltonian graphs, edge and vertex colorings, Wiener Polynomial , planar graphs, and directed graphs. An advanced topic completes the course. Familiarity with linear algebra and basic counting methods such as binomial coefficients is assumed. Comfort with reading and writing mathematical proofs is required. The overall goal of the course is for students to learn the main concepts of Graph Theory and to be able to be able to write rigorous proofs involving these concepts.

Course objective:
This course aims are twofold. First, to discuss some of the major results of graph theory, and to provide an introduction to the language, methods and terminology of the subject. Second, to emphasise various approaches (algorithmic, probabilistic, etc) that have proved fruitful in  odern graph theory: these modes of thinking about the subject have also proved successful in  ther areas of mathematics, and we hope that students will find the techniques learnt in this  ourse to be useful in other areas of mathematics aims, and the understanding of fundamental definitions and properties of graphs, the ability to read and write rigorous mathematical proofs involving graphs, recognition of the numerous applications of graph theory in computer science and engineering.

Student learning outcome:
A student who masters of the content of this course will be qualified to:
- Continue with the study of special topics and to apply graph theory to other fields.
- Present the various topics in the theory of graphs in a logical order.
- Indicate the historical background, and to clarify the exposition by including figures to illustrate concepts and results.
- The course can thus be used as a text in the tradition of the more method, with the student
gaining mathematical power by being encouraged to prove all theorems stated without proof.
- Study Master, PhD because students will be obtained what is needed for them.

The Topics:
CH.1: Basic Result
-The Konigsberg Bridge problem, Basic Concepts, Degree of Vertices, Subgraphs, Path and Connectedness, Distance, The Transmission, Digraphs, Line graph, The Adjacency Matrix, Operations on Graphs.
CH. 2: Degree Sequences
-Degree Sequences of Graph, Degree of Singularity of Graph.
CH.3:Connectivity of Graphs
-Cut Vertex, Bridges and Blocks, Bipartite graphs and trees, Connectivity.
CH.4:Tours and matchings
-Eulerian graphs, Hamiltonian graphs Matchings.
CH.5: Colorability
-Vertex Colorings, Edge Coloring of Graphs, Homomorphism, The Chromatic polynomial, Connectivity, Vertex Connectivity and Edge Connectivity
CH.6: Graphs and surfaces
-Planar Graphs, Coloring planar Graphs, Genus of a Graphs, Dual graphs and Kuratowiski’s theorem.
CH.7: Wiener Polynomials
CH.8: Flow in Network

Course Reading List and References:
[1] R. Balakrishnan and K. Ranganathan; A Textbook of Graph Theory, SpringerVerlag, Inc., New York, 2000.
[2] L. W. Beineke and R. J. Wilson; Selected Topics in Graph Theory, Academic Press, Inc., London, 1978.
[3] F. Harary; Graph Theory, Addison-Wesley, Reading, MA., 1969.

Special Functions

Units : 4
Lecturer : Asst. Prof. Dr. Shayma Adil Murad

Description :

Course overview:  

Special Functions are particular mathematical functions which have more or less established names and notations due to their importance in mathematical analysisfunctional analysis, physics, or other applications.   There is no general formal definition, but the list of mathematical functions contains functions which are commonly accepted as special. Many special functions   appear as solutions of differential equations or integrals of elementary functions. In mathematics, several functions or groups of functions are  important enough to deserve their own names. There is a large theory of special functions which developed out of  statistics and mathematical physics.

Special functions are just that: specialized functions beyond the familiar trigonometric or exponential functions. The one studied (hypergeometric functions, orthogonal polynomial,  and so on) arise very naturally in areas of analysis, number theory, Lie groups, and combinatory. Very detailed information is often available. After successful completion of this course, students will be able to  develop key concepts in Special functions.

 Course objective:

 The student will be able to:

  • Understand the basic use for both the Gamma and Beta functions.
  • Solve the Legendre differential equation by series method and find the conditions necessary for a polynomial solution.
  • Derive and apply the generating function and recurrence relations for Laguerre , Hermite, Jacobie's and Bessel's Polynomials
  • Employ the orthogonality relation of Legendre polynomials to develop functions as series of such polynomials. Also Understand the basic use for modified Bessle's functions, moreover derive and apply the generating function and recurrence relations for modified Bessle's functions. The goal of the course is to provide the student with an introduction to this piece of mathematical, including the fundamentals of solution techniques of a variety equations of (Bessel’s, Laguerre, Chebyshev Polynomail,  hypergeometric  Function)  by using power series method. To achieve this goal, students will, completion of homework assignments, and exams.
Functional Analysis

Units : 4
Lecturer : Alaa Luqman Ibrahim

Description :

Course overview:

Functional analysis is generally regarded as an activity whereby functions are broken down into sub-functions, and relationships between sub-functions (e.g. sequence, concurrency, control flow, item flow, logical branching, looping, iteration, replication, etc.). Therefore, the breakdown (decomposition) of a function describes how that function is to be accomplished (performed). Functional analysis has two major applications. The first is as a tool for CAPTURE and VALIDATION of requirements. That is, functional analysis is a technique used WITHIN requirements analysis. Anybody who has ever developed a use case, or written down how something is to be used for a particular use, has used functional analysis in this application. There are, of course, much more robust ways of using functional analysis as a requirements analysis tool than the traditional IT use case approach. The second major application of functional analysis is as a design tool, a LOGICAL DESIGN tool. In this application, requirements level functions are broken down into solution level functions. Take the “Conduct the Olympic Games (a requirements level function)” system. When we define functions such as “design the stadium”, “build the stadium”, “obtain certificate of occupancy”, “conduct event trial”, etc., we are DECIDING UPON, we are CREATING, solution level functions. It is exactly the same with technology-based solutions. As soon as we decide that the function “design the stadium” is to be performed by the “stadium contractor” object, we have CREATED a functional requirement on that object. That is the design application of functional analysis.


Course objective:

We will cover a large amount of material. This should give you some warning of the fact that this will not be a course to relax at; though this does not mean that you will not be able to enjoy it. Functional Analysis is not only a beautiful subject but it is also very useful and powerful in applied mathematics and theoretical physics. It is also basic for the understanding and development of very many other mathematical theories like the Theory of Partial Differential Equations and Operator Theory. Do not be fooled by the fact things start slow. This is the kind of course where things keep on building up continuously, with new things appearing rather often. In a mathematically sounding language, this course is "locally easy" but "globally hard". That means that if you keep up to date with the lectures, read my notes regularly and do the exercises, you should not have any problem and might even be able to enjoy this course. Otherwise, you will soon find yourself in deep trouble.


Main Topics


  • Vector spaces, linear combinations, linear dependence, linear independence, span, basis, dimension, subspace of a vector space, further properties of vector spaces, theorems and examples.
  • Normed space, Banach space, convergence of sequence and series in a normed space, finite dimensional normed space, subspaces, equivalent norms, the product of Banach spaces, isomorphism of a normed space, theorems and examples.


  • Linear operators, properties of linear operators , the null of the operators, bounded and continuous linear operators with applications, inverse of operators, the graph of the operators, homeomorphism operators, linear operators on a finite dimensional normed space, theorems and examples.


  • Linear functional, bounded and continuous linear functional with applications, linear functional on a finite dimensional normed space, theorems and examples.


  • Fundamental theorems for normed and Banach spaces, Zorn's lemma, Hahn Banach theorem, convergent of sequences of operators, open mapping theorem, closed graph theorem, partially ordered set, reflexive spaces, the dual space of a normed space reflexivity on a finite dimensional normed space and in a Hilbert space, theorems and examples.


•     Inner product (pre-Hilbert) space, Hilbert space. Convergence of  a sequence in a pre-Hilbert space, subspaces of  a Hilbert space, orthogonal complement, direct sum. Orthogonal projection, orthonormal sets and sequences, Bessel inequality, Hilbert adjoint operators, self-adjoint operators, unitary operators, normal operators, self adjoint of product, sequence of self-adjoint operators, theorems and examples.    

Research Project

Units : 4