# Mathematics

## Courses

**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:

- - solve transportation problems during the allocation of trucks to excavators
- - formulate operation research models to solve real life problem
- - proficiently allocating scarce resources to optimise and maximise profit
- - 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

**Course Reading List and References:**

- 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.

**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 .

**Course Reading List and References:****-** Abstract Algebra, by D.M. Burton.

- A first course in abstract algebra, by J.B. fraleigh .

**Units**: 8

**Lecturer**: Alaa Luqman Ibrahim

**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.

**Main 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.
- Series: Convergent, divergent series, absolutely convergent, conditionally convergent rearrange the terms of a series, the product of two series.
- 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.

**Units**: 6

**Lecturer**: Bewar Ahmed Mahmood

**Description**:

**Course overview:**

Numerical analysis is the branch of mathematics dealing with methods for obtaining approximate numerical solutions of mathematical problems.

This course presents numerical methods for solving mathematical problems. It deals with the theory and application of numerical approximation techniques as well as their computer implementation. It covers computer arithmetic, solution of nonlinear equations, interpolation and approximation, numerical integration and differentiation, solution of differential equations, and matrix computation.

**Course objective:**

The primary objective of the course is:

- To understand the number representation and errors.
- To provide the student with numerical methods of solving the non-linear equations, interpolation, differentiation, integration, differential equations, and system of linear equations.
- To improve the student’s skills in numerical methods by using the numerical analysis software and computer facilities.

**Student learning outcome:**

At the end of the course, Students will be able to:

- Estimate an error of numerical calculations.
- Apply the fundamentals of classical iteration methods to find the roots of Equations

- Apply the methods of interpolation to construct new data points within the range of a discrete set of known data points.
- Evaluate a derivative at a value using an appropriate numerical method.
- Calculate a definite integral using an appropriate numerical method
- Solve a differential equation using an appropriate numerical method.
- Solve a linear system of equations using an appropriate numerical method.
- Write computer programs to solve mathematical problems with MATLAB

**Main Topics**

**Chapter One: (Number Representation and Errors)**

1.1. Representation of numbers in different bases.

1.2. Floating-point representation.

1.3. Roundoff error.

1.4. Truncation error.

1.5. Absolute error and relative error.

1.6. Finite-Digit arithmetic.

**Chapter Two: (Solution of Nonlinear Equations)**

2.1. Bisection method.

2.2. False position method.

2.3. Newton-Raphson method.

2.4. Secant Method

2.5. Fixed Point Iteration Method.

**Chapter Three: (Interpolation and Polynomial Approximation)**

3.1. Interpolating Polynomial: Lagrange form.

3.2. Interpolating Polynomial: Newton’s divided difference formula.

**Chapter Four: (Numerical Differentiation and Integration)**

4.1. Numerical differentiation.

4.2. Numerical integration.

4.3. Newton-Cots formula.

4.4. Trapezoidal and Composite trapezoidal rule.

4.5. Simpson’s and Composite Simpson’s rule.

4.6. Romberg integration.

**Chapter Five: (Initial-Value Problems for Ordinary Differential Equations)**

5.1. Euler’s method.

5.2. Heun's method.

5.3. Runge-Kutta method of order four.

**Chapter Six: (System of Linear Equations)**

6.1. Direction Method: Gaussian elimination.

6.2. Direction Method: Triangular factorization.

6.3. Iterative Method: Jacobi iterative method.

Iterative Method: Gauss-Seidel iteration method.

**Units**: 6

**Lecturer**: Dr. Hussein Abdulrahman Hashem

**Description**:

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. 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.

**Units**: 6

**Lecturer**: Ahmed Muhammed Hussien

**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;

**Main 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*

Method of Separation of Variables

Boundary Conditions and Initial Value, Homogeneous Equations, Non-homogeneous Boundary Conditions, Inhomogeneous Equations , Heat Equations, The Wave Equation.

*Week 18, 19, 20, 21, 22*

Integral Transformation

The Laplace Transform, Properties of Laplace Transform, Inverse Laplace Transforms, Applications of Laplace Transform to Boundary Value Problems.