# Mathematics

## Courses

Abstract Algebra

Units : 8
Lecturer : Dr. Nareen Sabih Muhemed

Description : Course overview:
The study of “abstract algebra” grew out of an interest in knowing how attributes of sets of mathematical objects behave when one or more properties we associate with real numbers are restricted. For example, we are familiar with the notion that real numbers are closed under multiplication and division (that is, if we add or multiply a real number, we get a real number). But if we divide one integer by another integer, we may not get an integer as a result meaning that integers are not closed under division. We also know that if we take any two integers and multiply them in either order, we get the same result—a principle known as the commutative principle of multiplication for integers. By contrast, matrix multiplication is not generally commutative. Students of abstract algebra are interested in these sorts of properties, as they want to determine which properties hold true for any set of mathematical objects under certain operations and which types of structures result when we perform certain operations. Abstract algebra has applications in a variety of diverse fields, including computation, physics, and economics and, as a result, is an important area in mathematics. We will begin this course by reviewing basic set theory, integers, and functions in order to understand how algebraic operations arise and are used. We then will proceed to the heart of the course, which is an exploration of the fundamentals of groups, rings, and fields.

Course objective:
- Students should be able to demonstrate an understanding of the basic definitions and theorems of abstract algebra.
- Students should be able to complete problems and proofs which demonstrate both an understanding of the mechanics of the topic as well as an understanding of the basic underlying theories.
- Students should be able to follow and to construct a formal mathematical proof using each of the following methods: a direct proof, a proof by contradiction and a proof by induction.
- Students should be able to communicate mathematical ideas both in written and oral form for a variety of audiences.
- Students should be able to identify some of the key historical figures in the field of abstract algebra.
- Students should be able to demonstrate an understanding of the relationship of abstract algebra to other branches of mathematics and to related fields.
-Students should be able to independently explore related topics using resources other than the text.

Student learning outcome:
- Students will demonstrate factual knowledge including the mathematical notation and terminology used in this course. Students will read, interpret, and use the vocabulary, symbolism, and basic definitions used in abstract algebra, including binary operations, relations, groups, subgroups, homomorphisms, rings, and ideals.
- The students will describe the fundamental principles including the laws and theorems arising from the concepts covered in this course.Students will develop and apply the fundamental properties of abstract algebraic structures, their substructures, their quotient structure, and their mappings. Students will also prove basic theorems such as Lagrange’s theorem, Cayley’s theorem, and the fundamental theorems for groups and rings.
- The students will apply course material along with techniques and procedures covered in this course to solve problems. Students will use the facts, formulas, and techniques learned in this course to prove theorems about the structure, size, and nature of groups, subgroups, quotient groups, rings, subrings, ideals, quotient rings,and the associated mappings. Students will also solve problems
about the size and composition of subgroups and quotient groups; the orders of elements; isomorphic groups and rings; and the composition of ideals.
- The students will develop specific skills, competencies and thought processes sufficient to support further study or work in this or related fields .Students will acquire a level of proficiency in the fundamental concepts and applications necessary for further study, including graduate work, in academic areas requiring abstract algebra as a prerequisite, or for work in occupational fields
requiring a background in abstract algebra or other highly abstract mathematics. These fields might include the physical sciences and engineering as well as mathematics.
The Topics:
Chapter one :( Group Theory)
- Definition and Examples of Groups .
- Certain Elementary Theorems on Group .
- Two Important Groups .
- Subgroups .
- Normal Subgroups and Quotient Groups .

- Abstract Algebra, by D.M. Burton.
- A first course in abstract algebra, by J.B. fraleigh .

Mathematical Analysis

Units : 8

Description : Course overview:
Mathematical analysis is a branch of mathematics that studies continuous change and
includes the theories of differentiation, integration, measure, limits, infinite series, and
analytic functions.
These theories are usually studied in the context of real and complex numbers and functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space).

Course objective:
This subject is intended to:
- introduce the main notions and methods of proof in analysis through a mathematically rigorous approach.
- investigate the main concepts of analysis for real functions of a single variable, including the introduction to set theory, sequences, series, metric space, functions, integration and measure theory.
- introduce a brief overview to number theory so that students will be ready for doing further studies such as doing Master, PhD in number theory.

Student learning outcome:
A student who completes this module successfully should be able to:
- Knowledge and understanding: define and state the main concepts and theorems of analysis; apply these in the investigation of examples; prove basic propositions in analysis.
- Intellectual skills: apply complex ideas to familiar and to novel situations; work with abstract concepts and in a context of generality; reason logically and work analytically; perform with high levels of accuracy; transfer expertise between different topics in mathematics.
- Professional skills: select and apply appropriate methods and techniques to solve problems; justify conclusions using mathematical arguments with appropriate rigour; communicate results using appropriate styles, conventions and terminology.
- Transferable skills: communicate with clarity; work effectively, independently and under direction; analyze and solve complex problems accurately; make effective use of IT; apply high levels of numeracy; adopt effective strategies for study.
- Study Master, PhD because students will be obtained what is needed for them.

The Topics:
-Preliminaries: Basics of set theory and the real number system. Countability of sets in Real numbers

-Sequences of real numbers: Convergent, divergent, monotonic and bounded sequences, subsequences, algebraic operations on sequences, Caushy sequences and another types of sequences called (C,K) summable, where K=1,2,3,….
- Series: Convergent, divergent series, absolutely convergent, conditionally convergent rearrange the terms of a series, the product of two series, anther types of convergent series called (C,K) summable, where K=1,2,3,….
- Metric spaces: Some topological concepts in metric space, openness, closedness,…, complete metric space, compact subsets in a metric space.
- Functions: Continuous functions in a metric spaces, uniform continuity, sequences of functions pointwise convergent and uniformly convergent, bounded functions.
- Integrations: Riemann and Riemann Stieltje's integrations.
- Introduction to measure theory: Inner and outer measure, measurable sets, measurable functions and Lebesgue integration.

- Real analysis 3rd edition, H.L.Royden.
- Real analysis 2nd edition, Serge Lang.
- Real variables, Burill. Kundsen.
- Methods of real analysis, Richard R. Goldberg.
- Integration, Edward James McShane.
- Mathematical analysis ll, Tom M. Apostol.
- Mathematical analysis, Steven A. Douglass.
- Anything you want you can Google it.

Mathematical Statistics

Units : 6
Lecturer : Dr. Hussein Abdulrahman Hashem

Description : Course overview:
This course is an introduction to the mathematical principles of statistical estimation and inference. Topics include: point and confidence interval estimation, principles of maximum likelihood, sufficiency and completeness, tests of simple and composite hypothesis, linear models and multiple regression, and analysis of variance.

Course objective:
Mathematical statistics is the backbone of methods applied and developed in statistical practice and research. This course will provide you with the mathematical statistics and underlying statistical theory to understand and evaluate statistical routines, expand or establish new procedures, communicate statistical ideas, and read the scientific, particularly statistically-related, literature to critically judge the relevance of research results. The goal is to bring together your understanding of statistical methods into a coherent conceptual and theoretical framework within which you may gain a greater comprehension and appreciation of statistical inference and thinking. Additionally, the material will teach you how to approach statistical problems from a mathematical perspective as a complement to the computational and data analytic training you receive in more applied courses and work.
Student learning outcome:
Upon successful completion of this course, students will be familiar with basic rules of mathematical statistics and will be able to use them in modelling uncertainty in obtaining and recording data. They will be able to utilize graphical and numerical summaries of data in understanding data generating processes. They will understand the logic of statistical inference and will be able to apply common inferential procedures. Students will be exposed to the computational aspects of statistics through the use of calculators, spreadsheet programs or special purpose data analysis packages.
The Topics:
CHAPTER 1: Probability of Events
1.1 Counting Techniques
1.2 Probability Measure
1.3 Some Properties of the Probability Measure
1.4 Conditional Probability
1.5 Bayes’ Theorem
CHAPTER 2: Random Variables and Distribution Functions

2.1 Distribution Functions of Discrete Variables
2.2 Distribution Functions of Continuous Variables
CHAPTER 3: Moments of Random Variables and Chebychev Inequality
3.1 Moments of Random Variables
3.2 Expected Value of Random Variables
3.3 Variance of Random Variables
3.4 Chebychev Inequality
3.5 Moment Generating Functions
CHAPTER 4: Some Special Discrete Distributions
4.1 Bernoulli Distribution
4.2 Binomial Distribution
4.3 Geometric Distribution
4.4 Poisson Distribution
CHAPTER 5: Some Special Continuous Distributions
5.1 Uniform Distribution
5.2 Gamma Distribution
5.3 Beta Distribution
5.4 Normal Distribution
CHAPTER 6: Two Random Variables
6.1 Bivariate Discrete Random Variables
6.2 Bivariate Continuous Random Variables
6.3Conditional Distributions
6.4 Independence of Random Variables
CHAPTER 7: Product Moments of Bivariate Random Variables
7.1 Covariance of Bivariate Random Variables
7.2 Independence of Random Variables
7.3 Variance of the Linear Combination of Random Variables
7.4 Correlation and Independence
7.5 Moment Generating Functions
CHAPTER 8: Conditional Expectations of Bivariate Random Variables
8.1 Conditional Expected Values
8.2 Conditional Variance
8.3 Regression Curve
CHAPTER 9: Functions of Random Variables and Their Distribution
9.1 Distribution Function Method
9.2 Transformation Method for Univariate Case
9.3 Transformation Method for Bivariate Case
9.4 Moment Method for Sums of Random Variables
CHAPTER 10: Some Special Discrete Bivariate Distributions
10.1 Bivariate Bernoulli Distribution
10.2 Bivariate Binomial Distribution
10.3 Bivariate Geometric Distribution
10.4 Bivariate Poisson Distribution
CHAPTER 11: Some Special Continuous Bivariate Distributions
10.1 Bivariate Uniform Distribution
10.2 Bivariate Gamma Distribution
10.3 Bivariate Beta Distribution
10.4 Bivariate Normal Distribution
CHAPTER 12: Sampling Distributions Associated with the Normal Population
12.1 Chi-square distribution
12.2 Student’s t-distribution
12.3 F-distribution
CHAPTER 13: Some Techniques for Finding Point Estimators of Parameters
13.1 Moment Method
13.2 Maximum Likelihood Method
CHAPTER 14: Criteria for Evaluating the Goodness of Estimators
14.1 The Unbiased Estimator
14.2 The Relatively Efficient Estimator
14.3 The Minimum Variance Unbiased Estimator
14.4 Sufficient Estimator
14.5 Consistent Estimator
CHAPTER 15: Some Techniques for Finding Interval Estimators of Parameters
15.1 Interval Estimators and Confidence Intervals for Parameters
15.2 Confidence Interval for Population Mean
15.3 Confidence Interval for Population Variance
CHAPTER 16: Hypothesis Testing
16.1 Hypotheses for a Single Parameter
16.2 Testing of Hypotheses for Two Samples
16.3 Contingency Table: Test for Independence
CHAPTER 17: Linear Regression Models
17.1 The Simple Linear Regression Model

1 7.2 The Method of Least Squares.
17. 3 Derivation of 𝛽 and 𝛽
17.4 Quality of the Regression
17.5 Properties of the Least-Squares Estimators for the Model
17. 6 Estimation of Error Variance 𝜎
17.7 Inferences on the Least Squares Estimators

Course Reading List and References: Books Recommended:
-Freund, J. E. and Walpole, R. E. (1987). Mathematical Statistics. Englewood Cliffs:Prantice-Hall.
-Hogg, R. V. and Craig, A. T. (1978). Introduction to Mathematical Statistics. New York: Macmillan.

Numerical Analysis

Units : 6
Lecturer : Bewar Ahmed Mahmood

Description :

Course overview:
This course introduces students to classical numerical methods for approximating the solutions of problems in science, engineering, and mathematics through the use of computers. It allows students to deal with numerical methods both at a theoretical level and for programming purposes.
Numerical methods are an efficient for learning to use computers. Because numerical methods are expressly designed for computer implementation they are ideal for illustrating the computer’s powers and limitations.
When you successfully implement numerical methods on a computer, and then apply them to solve otherwise intractable problems, you will be provided with a dramatic demonstration of how computers can serve your professional development At the same time, you will also learn to acknowledge and control the errors of approximation that are part and parcel of large-scale numerical calculation.
The course content two semesters, the first semester covers error analysis, numerical methods for solving nonlinear algebraic equations, interpolation.
The second semester will be in particular on numerical integration and numerical methods for solving ordinary differential equations, direct and iterative methods for solving linear systems of algebraic equations

Course objective:
Teach students to recognize the type of problems that require numerical techniques for their solution, see some examples of the error propagation that can occur when numerical methods are applied, and have opportunity to approximate the solution to some problems that cannot be solved exactly.
To provide the student with numerical methods for solving the nonlinear equations, linear systems of equations, interpolation, differentiation, and integration. The course will also develop an understanding of the numerical solutions of ordinary differential equations.
To improve the student’s skills in numerical methods by using the numerical analysis software and computer facilities, and students will taught to use the numerical analysis methods in Matlab.

Student learning outcome:
On completion of this course, students will:
- Be able to estimate an error of numerical calculations.
- Learn fundamental numerical methods for root finding, Solving differentiation, ordinary differential equations, and linear systems of equations.
- Construct an interpolating polynomial using either the Lagrange or Newton formula.
- Derive the trapezoidal and Simpson's rules for approximating an integral.
- Know basic numerical methods and be able to implement them in Matlab.
- Choose the appropriate method to solve the problem.
- Understand how to use numerical methods to solve problems arising in a wide variety of practical applications in physics, chemistry, biosciences, engineering, etc.

The Topics:
-Errors Analysis: To introduce some kind of error we need it in the subject.

- Solution of Nonlinear Equations:
- The bisection method: How to apply it and to compute an error bound for the approximate solutions derived by the method.
- The false position: How to apply it.

- The fixed point iterative method: How to formulate the function g(x) which will satisfies the conditions on g(x), then apply the iterative scheme and the analysis of the error.
- The Newton’s method: How to apply it and the analysis of its error.
-Interpolation:
- Many times, data is given only at discrete points
- Newton’s divided difference polynomial method.

- Integration
- How to derive the Trapezoidal, the Simpson rules, and Romberg, how to apply them.
- Ordinary Differential Equations
- How to use the Euler’s method, Heun's methods (modified Euler’s method), and the Runge Kutta method for solving.
- Systems of Linear Equations
- Introduction of vectors and matrices.
- How apply the Gaussian elimination method (algorithmic approach). Give examples showing that the system has infinite number of solutions or no solution at all (singular matrix).
- How to apply the iterative methods (Jacobi and Gauss- Seidel) to solve a linear system.
- How to apply LU decomposition to solve a system of linear equations, and how to find the inverse of the square matrix using LU decomposition.

Course Reading List and References :
- Numerical Analysis by Richard L. Burden and J. Douglas Faires (ninth edition), 2011.
- Numerical methods using Matlab by John H. Mathews and Kurtis O. Fink (fourth edition).
- Numerical Methods and Analysis by James L. Buchanan and peter R. Ture.
- Applied numerical analysis by Curtis F. Geraled and Patrick O. Wheatly (third edition).
- Introduction to Numerical Methods and Matlab Programming for Engineers by Todd Young and Martin J. Mohlenkamp, 2015.
- Anything you want you can Google it.

Operation Research

Units : 4
Lecturer : Huda Younus Najm

Description : Course overview:
This course is an introductory and practical course to the study of operations research
application in mining projects. It is designed primarily for mathematics students to replicate what is happening in classroom so as to be able to apply the knowledge and skills gained during and after course of study to real life situations they might face in the mathematics. It involves demonstration of principles and techniques of operations research using real life projects. Topics to be covered include operation research and model formulation, solution of the operation research model, phases of an operation research study, techniques of operation research or operations research solution tools such as Linear Programming (LP) (Two phase (two variables) LP, Transportation models, Network models, etc.
Operation Research is applied to problems that concern how to conduct and coordinate operations( i.e. activities) with an organization. The major phases of a typical Operation  Research (OR) study are the following:

-Define the problem and gather relevant data.
-Formulate a mathematical model to represent the problem.
-Develop a computer-based procedure for deriving a solution to the problem resulting from the model.
-Test the model and refine it as needed.
-Prepare for the ongoing application of the model as prescribed by management.
-Implement.
Then we will discuss these phases in turn starting with the process of defining the problem and gathering data which includes determining such things as: The appropriate objectives, constraints on what can be done, interrelations between the are to be studied and the
other areas of organization, possible alternative courses of action, The second phase is to formulate a mathematical model. A mathematical model is defined by a system of equations and related mathematical expressions that describe the essence of a problem.

Course objective:
The aim of the subject is to study: define the problem of life and analyses. This subject is intended to:
1. This course is intended to provide students with a knowledge that can make them appreciate the use of various research operations tools in decision making in organizations.
2. Making decisions as well as being able to formulate organizational problems into OR models for seeking optimal solutions.
3. Grasp the methodology of OR problem solving.
4. Understand and differentiate deterministic/probabilistic/ dynamic problem solving situations. Develop formulation skills in building be able to understand and interpret solutions and sensitivity/ parametric analyses.
5. Introduce a brief overview to operation research so that students will be ready for doing further studies such as doing Master, PhD in operation research.

Student learning outcome:
Upon successful completion of this course, the student will be able to:
(Knowledge based)
- explain the meaning of operations research
- know the various techniques of operations research;
- apply the techniques used in operations research to solve real life problem in mining industry
- select an optimum solution with profit maximization;
- have complete understand of the significant role operation research play in mining project completion at every stage of the mines (Skills)
- use operations research to:

1. - solve transportation problems during the allocation of trucks to excavators
2. - formulate operation research models to solve real life problem
3. - proficiently allocating scarce resources to optimise and maximise profit
4. - eliminate customers / clients waiting period for service delivery

- turn real life problems into formulation of models to be solve by linear programming etc.
- determine critical path analysis to solve real life project scheduling time and timely delivery .
- use critical path analysis and programming evaluation production and review techniques for timely project scheduling
- conduct literature search on the internet in the use of operation research techniques in mining projects execution and completion.

The Topics:
Chapter one: basics of operations research
- Introduction, Mathematical models in operation research, phases of operation research study, scientific method in operation research.
- Convex sets and linear programming
- Mathematical program.
- Problem formulation.
Chapter two: Methods for solving Linear programming model
- Graphical method, Special case of graphical solution (multiple optimal, infeasible solution, Unbound solution Degeneracy solution).
Chapter three: Methods for solving Linear programming model
- Simplex method, the steps of the simplex algorithm
- Requirements for standard form
- M-technique or Big –method
- Artificial variables techniques: (the two phase method)
- Simplex multipliers method
Chapter Four: Duality and Sensitivity analysis
- Dual problem
- Dual simplex method.
Chapter Five: Transportation problem
- Transportation problem, definition of the model.
- How to find the Basic solution: (Northwest corner method)
- How to find the Basic solution: (Least- cost method)
- How to find the Basic solution: (Vogel method).
- Development of basic solution: The stopping stone method; Modified distribution method ( multipliers method).
Chapter six: Network Models
- Network definition
- Network analysis: Critical path method ; Pert projects evaluation and review technique.
Chapter seven: Game Theory
- Definition game theory, saddle point, solution by graphical.
- Linear Programming method.
Chapter eight: Optimization
- Introduction to optimization
- Mathematical model

- Hamdy A. Taha, "Operations research an introduction", second edition, (2003).
- J.C. Pant ., "Introduction to Optimization Operation Research ", ( 2004).
- Edwin K.p.Ghong and Stanislaw H.Zak ,"An introduction to Optimization." , (2001).
- James K. Strayer, " Linear Programming and Applications" , (1989) Springer
- Taiwo Owoeye: Operation Research; Olugbenga Press Publication (2001). ISBN 987-2430. 60p.
- Hiller, F.S. And L.J. Lieberman: Introduction to operation research, Holden Day, San Francisco (6th Ed.) (1995).
- Internet website.

Partial Differential Equations

Units : 6

Description : Course overview:
If calculus is the heart of modern science, then differential equations are its guts. All physical laws, from the motion of a vibrating string to the orbits of the planets to Einstein’s field equations, are expressed in terms of differential equations. Classically, ordinary differential equations described one dimensional phenomena and partial differential equations described higher dimensional phenomena.

Course objective:
On completion of the course students should be able to:
- give basic definitions;
- to identify and classify an PDEs ;
- use standard methods to solve linear and nonlinear first order PDEs;
- classify the second order linear PDEs;
- to evaluate Fourier Series of periodic function;
- to evaluate eigenvalue and eigenfunction of problem;
- to solve problem with boundary and/or initial value;

Student learning outcome:
A student who completes this module successfully should be able to:
- The student will learn to formulate partial differential equations (PDEs) and seek understanding of their solutions.
- The student will recognise basic types of partial differential equations which are solvable, and will understand the features of linear equations in particular.
- Students will be familiar to derive methods to solve ordinary differential equations.
- Study Master, PhD because students will be obtained what is needed for them.

The Topics:
Week 1, 2:
Total Differential Equations, Method of Obtaining Primitive, Solution by Inspection, Homogenous Total Differential Equations, Simultaneous, Differential Equations.

Week 3, 4
Introduction to Partial Differential Equation, Partial Differential Equation, Classification of Equations, Elimination of Arbitrary Constants, Elimination of Arbitrary Functions.

Week 5, 6, 7
First Order Partial Differential Equation, Linear First Order Partial Differential Equation and
Solving Methods, Non-linear First Order Partial Differential Equation and Solving Methods, Special Cases of First Order Partial Differential Equation.

Week 8, 9, 10, 11, 12, 13
Second Order Linear Partial Differential Equation: Linear Second Order Partial Differential Equation with Constant Coefficients, Some Cases of Linear Second Order Partial Differential Equation with Variable Coefficients, Classification Linear Second Order Partial Differential Equation with Variable Coefficients.

Week 14, 15, 16, 17
Fourier Series Fourier Series, Cosine and Sine Series, Complex Fourier Series, Change of Interval, Differentiation and Integration of Fourier Series.

Week 18, 19, 20, 21
Method of Separation of Variables Boundary Conditions and Initial Value, Sturem –
Liouville Theory, Homogeneous Equations, Nonhomogeneous Boundary Conditions, Inhomogeneous Equations , Heat Equations, The Wave Equation.

Week 22, 23, 24, 25, 26
Integral Transformation :The Laplace Transform, Properties of Laplace Transform, Inverse Laplace Transforms, Applications of Laplace Transform to Boundary Value Problems, The
Fourier Integral Transform, Properties of Fourier Integral Transform, Inverse Fourier Integral Transform, Applications of Fourier Integral Transform to Boundary Value Problems