#### MAT429-Research Project II

Research Project II

#### MAT427-Introduction to Measure and Probability

Introduction to Measure and Probability

#### MAT426-Linear Models

Linear Models

#### MAT425 - Systems Theory copy 1

**COVENANT UNIVERSITY**

**College of Science and Technology **

**Department**: Mathematics

**Course Lecturers**: Mr. O.O. Agboola & Mr. O.F. Imaga

**Programme**: Industrial Mathematics

**Course Code**: MAT425

**Course Title**: Systems Theory

**Course Units**: 3

**Semester**: Omega

**Time**: 12:00 Noon – 2:00 p.m. (Tuesdays) & 10.00 a.m. – 11:00 a.m. (Thursdays)

**Location: **CST Halls 204 and 306

a) **Course Overview**

This course requires some preliminary knowledge of calculus, linear algebra, probability and statistics. The course focuses on the theory of time-varying linear systems, in both continuous- and discrete-time, with frequent specialization to the time invariant case. Linear systems theory is the cornerstone of control theory and a prerequisite for essentially all graduate courses in this area. It is a well-established discipline that focuses on linear differential equations from the perspective of control and estimation. Topics include Lyapunov theorems, Solution of Lyapunov stability equation A^{T} P + PA = Q; Controllability and observability; Theorem on existence of solution of linear systems of differential equations with constant coefficients; Uniqueness Theorems; Theory of Stability; Elementary type of rest points. Stability under constantly operating perturbation; Test for Stability. **Prerequisites**: MAT111, MAT112, MAT121, MAT122, MAT214.

b) **Course Objectives**

At the end of the course students will be able to:

- Define and determine the critical points of a system and its stability;
- Determine the stability of a system using Lyapunov theory;
- State and prove the existence and uniqueness theorems;
- Determine the region where the solution of a system of equation exists;
- Find the region where the solution is unique and
- State and describe the observability and controllability of a system.

c) **Method of Delivery /Teaching Aids**

The course has an **in-class **component and an **out-of-class **component. The in-class component will be a combination of lectures, problem solving demonstrations, discussions, questions/answers and short problem solving activities. In the out-of-class component, students are expected to read and review their notes and textbooks, and complete homework problems.

Classroom Powerpoint presentations as well as smartboard will be used to reinforce concepts. Many sample problems will be presented on overhead transparencies. Students will be led systematically through various thinking and problem solving strategies to solve many kinds of problems. Students will be given many opportunities to practice solving problems through in-class quizzes as well as through homework assignments.

d) **Course Outline**

**Module 1 (Week 1-3)**

Stability Theorem, Lyapunov’s second method for stability & The Lyapunov Theorems

**Module 2 (Week 4-6) **

Lyapunov Equations, Solving the Lyapunov equation; **Test #1**; Tutorial for Modules 1 and 2

**Module 3 (Week 7-8)**

Existence Theorem, Uniqueness Theorem

**Module 4 (Week 9-10)**

Controllability of linear Time Invariant Systems; **Test #2**

**Module 5 (Week 11-13)**

Observability of linear Time Invariant Systems; Tutorial for Modules 3, 4 and 5.

**Tutorial/Revision Week (Week 14)**

**Final exam (Week 15-17)**

e) **Structure of the Programme / Method of Grading**

Quizzes, Homework, Class Participation, Tests and Final Exam will count in the final grade as follows:

- Attendance at class meetings, In-class work / group work (periodically), quizzes (some quizzes may be unannounced), homework, collected and graded and solutions provided (counting for 10% of the total course marks);
- Two tests, 1-hour duration for each (counting for 20% total of the course marks) and
- One (1) End-of-semester examination, not more than 3 hours’ duration counting for 70% of the total course marks.

f) **Ground Rules & Regulations**

Students would be required to maintain high level of discipline (which is the soul of an army) in the following areas:

- Regularity and punctuality at class meetings – Because regular participation enhances the learning process, students are expected to adhere to the attendance policy set forth by the University. Therefore, students are strongly encouraged to attend all classes to better prepare them for assignments, tests and other course-related activities. It should be noted that at least 75% lecture attendance is required to qualify to sit for examination.
- Regardless of the cause of absences, a student who is absent six or more days in a semester is excessively absent, and will not receive credit unless there are verified extenuating circumstances
- Students will be given assignments periodically. Students may work together to understand these assignments, but all work submitted must be the student’s original work. There is a distinct difference between providing guidance and instruction to a fellow student and allowing the direct copying of another’s answers or work.
- Late homework assignments will NOT be accepted.
- Modest dressing; and
- Good composure.

g) **Term Papers and Assignments**:

Group projects will be assigned at the discretion of the course facilitators.

h) **Alignment with Covenant University Vision and Goals**:

Prayers are to be offered at the beginning of lectures. Presentation of the learning material will be done in such a way that the knowledge acquired is useful and applicable. Efforts would be made to address students on godliness, integrity and visionary leadership.

i) **Contemporary issues/Industry Relevance**:

The course is relevant for planning a research, allocation of resources, predictions and quality control. It will provide the background needed for advanced control design.

techniques

j) **Recommended Reading/Text**:

Textbooks:

- Lecture notes for Nonlinear Systems Theory by Prof.D. Jeltsema, Prof J.M.A.Scherpen and Prof. A.J.Van der Schaft.
- Theory of Ordinary Differential Equations, Existence, Uniqueness and Stability. Jishan Hu and Wei-Ping Li. Department of Mathematics. The Hong Kong University of science and Technology.
- Jan Polderman, Jan Willems (1998). Introduction to Mathematical Systems Theory: A behavioural approach (1st ed.). New York: Springer Verlag. ISBN 0-387-982666-3
- Brian D.O. Anderson; John B. Moore(1990). Optimal Control: Linear Quadratic Methods. Englewood Cliffs, NJ: Prentice Hall. ISBN 978-013-638560-8

**Electronic Sources: NIL**

#### MAT425 - Systems Theory

**COVENANT UNIVERSITY**

**College of Science and Technology **

**Department**: Mathematics

**Course Lecturers**: Mr. O.O. Agboola & Mr. O.F. Imaga

**Programme**: Industrial Mathematics

**Course Code**: MAT425

**Course Title**: Systems Theory

**Course Units**: 3

**Semester**: Omega

**Time**: 12:00 Noon – 2:00 p.m. (Tuesdays) & 10.00 a.m. – 11:00 a.m. (Thursdays)

**Location: **CST Halls 204 and 306

a) **Course Overview**

This course requires some preliminary knowledge of calculus, linear algebra, probability and statistics. The course focuses on the theory of time-varying linear systems, in both continuous- and discrete-time, with frequent specialization to the time invariant case. Linear systems theory is the cornerstone of control theory and a prerequisite for essentially all graduate courses in this area. It is a well-established discipline that focuses on linear differential equations from the perspective of control and estimation. Topics include Lyapunov theorems, Solution of Lyapunov stability equation A^{T} P + PA = Q; Controllability and observability; Theorem on existence of solution of linear systems of differential equations with constant coefficients; Uniqueness Theorems; Theory of Stability; Elementary type of rest points. Stability under constantly operating perturbation; Test for Stability. **Prerequisites**: MAT111, MAT112, MAT121, MAT122, MAT214.

b) **Course Objectives**

At the end of the course students will be able to:

- Define and determine the critical points of a system and its stability;
- Determine the stability of a system using Lyapunov theory;
- State and prove the existence and uniqueness theorems;
- Determine the region where the solution of a system of equation exists;
- Find the region where the solution is unique and
- State and describe the observability and controllability of a system.

c) **Method of Delivery /Teaching Aids**

The course has an **in-class **component and an **out-of-class **component. The in-class component will be a combination of lectures, problem solving demonstrations, discussions, questions/answers and short problem solving activities. In the out-of-class component, students are expected to read and review their notes and textbooks, and complete homework problems.

Classroom Powerpoint presentations as well as smartboard will be used to reinforce concepts. Many sample problems will be presented on overhead transparencies. Students will be led systematically through various thinking and problem solving strategies to solve many kinds of problems. Students will be given many opportunities to practice solving problems through in-class quizzes as well as through homework assignments.

d) **Course Outline**

**Module 1 (Week 1-3)**

Stability Theorem, Lyapunov’s second method for stability & The Lyapunov Theorems

**Module 2 (Week 4-6) **

Lyapunov Equations, Solving the Lyapunov equation; **Test #1**; Tutorial for Modules 1 and 2

**Module 3 (Week 7-8)**

Existence Theorem, Uniqueness Theorem

**Module 4 (Week 9-10)**

Controllability of linear Time Invariant Systems; **Test #2**

**Module 5 (Week 11-13)**

Observability of linear Time Invariant Systems; Tutorial for Modules 3, 4 and 5.

**Tutorial/Revision Week (Week 14)**

**Final exam (Week 15-17)**

e) **Structure of the Programme / Method of Grading**

Quizzes, Homework, Class Participation, Tests and Final Exam will count in the final grade as follows:

- Attendance at class meetings, In-class work / group work (periodically), quizzes (some quizzes may be unannounced), homework, collected and graded and solutions provided (counting for 10% of the total course marks);
- Two tests, 1-hour duration for each (counting for 20% total of the course marks) and
- One (1) End-of-semester examination, not more than 3 hours’ duration counting for 70% of the total course marks.

f) **Ground Rules & Regulations**

Students would be required to maintain high level of discipline (which is the soul of an army) in the following areas:

- Regularity and punctuality at class meetings – Because regular participation enhances the learning process, students are expected to adhere to the attendance policy set forth by the University. Therefore, students are strongly encouraged to attend all classes to better prepare them for assignments, tests and other course-related activities. It should be noted that at least 75% lecture attendance is required to qualify to sit for examination.
- Regardless of the cause of absences, a student who is absent six or more days in a semester is excessively absent, and will not receive credit unless there are verified extenuating circumstances
- Students will be given assignments periodically. Students may work together to understand these assignments, but all work submitted must be the student’s original work. There is a distinct difference between providing guidance and instruction to a fellow student and allowing the direct copying of another’s answers or work.
- Late homework assignments will NOT be accepted.
- Modest dressing; and
- Good composure.

g) **Term Papers and Assignments**:

Group projects will be assigned at the discretion of the course facilitators.

h) **Alignment with Covenant University Vision and Goals**:

Prayers are to be offered at the beginning of lectures. Presentation of the learning material will be done in such a way that the knowledge acquired is useful and applicable. Efforts would be made to address students on godliness, integrity and visionary leadership.

i) **Contemporary issues/Industry Relevance**:

The course is relevant for planning a research, allocation of resources, predictions and quality control. It will provide the background needed for advanced control design.

techniques

j) **Recommended Reading/Text**:

Textbooks:

- Lecture notes for Nonlinear Systems Theory by Prof.D. Jeltsema, Prof J.M.A.Scherpen and Prof. A.J.Van der Schaft.
- Theory of Ordinary Differential Equations, Existence, Uniqueness and Stability. Jishan Hu and Wei-Ping Li. Department of Mathematics. The Hong Kong University of science and Technology.
- Jan Polderman, Jan Willems (1998). Introduction to Mathematical Systems Theory: A behavioural approach (1st ed.). New York: Springer Verlag. ISBN 0-387-982666-3
- Brian D.O. Anderson; John B. Moore(1990). Optimal Control: Linear Quadratic Methods. Englewood Cliffs, NJ: Prentice Hall. ISBN 978-013-638560-8

**Electronic Sources: NIL**

- Teacher: OGBU FAMOUS IMAGA

#### MAT424-Survey Methodology & Quality Control

Survey Methodology & Quality Control

- Teacher: OLUWOLE AKINWUMI ODETUNMIBI
- Teacher: PELUMI EMMANUEL OGUNTUNDE

#### MAT423-Theory of Measure

** **

**Covenant University**

**College Science and Technology**

**School of Natural and Applied Sciences**

**Department Of Mathematics**

**2013/2014 Session**

** **

**PROGRAMME:** Industrial Mathematics

**COURSE CODE:** MAT 423

**COURSE TITLE:** Theory of Measure

**UNITS:** 3 Units

**COURSE LECTURER:** Dr. T. A. Anake /Dr. (Mrs.) S. A. Bishop

**SEMESTER:** Omega

**TIME: ** Monday, 12noon – 2.00p.m. and, Wednesday, 11.00a.m. -12 noon.

**LOCATION:** CST Building.

**COURSE OVERVIEW**

The course is designed to introduce the students to the extension of integration theory from Riemann integrals to Lebesgue integral. The topics covered in this course include the concept of measure, Lebesgue measure, Lebesgue outer measure, Lebesgue measurable functions and Lebesgue integrals of both bounded measurable functions and non-negative functions. These concepts have far reaching applications in the solutions of real life problems in Physical Sciences, Engineering and even Social Sciences.

**COURSE OBJECTIVES/GOALS**

At the end of the course, students should be able to:

i. Define the concept of measure and measure space

ii. Define Lebesgue measure and the Lebesgue measure space

ii. Describe the Lebesgue outer measure

iii. Define Lebesgue measurable functions

iv. Define Lebesgue Integral and prove its properties

v. State and prove some convergence theorem

vi. Show that the every Riemann Integrable function is Lebesgue Integrable

**METHOD OF DELIVERY /TEACHING AIDS**

- Guided Instructions
- Class Activity
- Assignments
- White board and marker

** **

**COURSE OUTLINE**

Module 1: Introduction to Concept of Measure

- Algebra of Sets
- Definition and Properties
- Examples of measures

Module 2: Lebesgue measure

- Introduction
- Definition and Properties
- Lebesgue Outer measure
- Sets of finite measure

Module 3: Lebesgue Measurable Functions

- Lebesgue Measurable Sets
- Lebesgue Non-measurable Sets
- Lebesgue Measurable Functions

Module 4: The Lebesgue Integral

- Review of Riemann Integral
- Simple functions
- Lebesgue integral
- Lebesgue Integrable functions

Module 5: Lebesgue Integral of bounded functions over Sets of Finite Measure

- Definition and Properties
- Proof of properties

Module 6: Lebesgue Integral of Non-Negative Functions (Unbounded Functions)

- Definition and Properties
- Proof of properties
- General Lebesgue Integral

Module 7: Convergence Theorems

- Fatou’s Lemma
- Lebesgue convergence theorem
- Lebesgue Dominated Convergence Theorem
- Riemann and Lebesgue Integral

Module 8: The L^{p} Spaces

** **

**TUTORIALS:**

Tutorials will be given at the end of the course.

** **

**STRUCTURE OF PROGRAMME/METHOD OF GRADING**

- Continuous Assessment:

Test 1 10 marks

Mid semester exam 10 marks

Assignments 10 marks

- Examination 70 marks

** **

**GROUND RULES & REGUKATIONS**

- No eating in the class
- Punctuality to classes
- No use of i-pods in the class
- Dress code must be correctly adhered to
- 75% required for eligibility to semester examination.

** **

** TOPICS FOR TERM PAPERS/ASSIGNMENTS/STUDENT ACTIVITY**

Assignment and term papers will be given as the course progresses.

**ALLIGNMENT WITH COVENANT UNIVERSITY VISION/GOALS**

- Classes are conducted in such a way that the university core values are observed and respected.
- Course is delivered in a manner that the knowledge acquired is useful and applicable.

** **

**CONTEMPORARY ISSUES/INDUSTRY RELEVANCE**

Knowledge of probability and stochastic processes is applicable in banking, finance and risk management, engineering, telecommunication, biology, etc.

** **

**RECOMMENDED READING/TEXT**

Folland, G. B.. (1999). Real Analysis: Modern Techniques and Their Applications (2^{nd} Ed.). John-Wiley & Sons, New York.

Hewitt, E and Stromberg,K. (1975). *Real and Abstract Analysis*. Springer-Verlag, New York

Kingman, J.F.C. and Taylor, S.J. (1973). *Introduction to Measure and Probability. *Cambridge University Press, Great Britain.

- lecturer: KANAYO STELLA EKE

#### MAT422-Differential Equations III

**COVENANT UNIVERSITY**

**COURSE COMPACT**

**2014/2015 Academic Session**

College: College of Science and Technology

Department: Mathematics

Programme: Industrial Mathematics

Course Code: MAT422

Course Title: Differential Equations III

Units: 3

Course Facilitators: Dr. Mrs. S.A. Bishop and Mr. O.O. Agboola

Semester: Omega

Time: Wednesdays: 10.00 am – 12.00 Noon & Fridays: 10.00 am – 11.00 am

Location: Hall 201 / Hall 306

**a. Brief Overview of Course**

This course is a continuation of MAT412 - Differential Equations II. The focus of the course is the concepts and techniques for solving the partial differential equations (PDE) that permeate various scientific disciplines. The emphasis is on nonlinear PDE. Applications include problems from fluid dynamics, electrical and mechanical engineering, materials science, quantum mechanics, etc. In particular, methods of separation of variables and D’Alembert’s solution are treated.

**b. Course Objectives**

At the end of the course, student should be able to:

- identify the difference between ordinary and partial differential equations.
- identify different types of partial differential equations.
- identify P.D.E order, degree and classification of a P.D.E.
- form P.D.E by elimination of arbitrary constants and arbitrary functions
- Solve the first order linear partial differential equation
- Obtain the solution of homogeneous P.D.E. by different methods.
- use the method of separation of variables and D’Alembert’s method to solve special types of partial differential equations

**c. Methods of Lecture delivery/Teaching Aids.**

- Guided instructions

- Active student participation and interaction

- Solution of guided and related problems.

- Assignments.

- White board and marker

- Lecture notes and textbooks

**d. Course Outlines / Weekly Lecture Schedule**

**Week 1** - Introduction to Partial Differential Equations - Definition, Examples, Types and Classification; Derivation of PDEs by elimination of arbitrary constants and by elimination of arbitrary functions

**Week 2** Partial Differential equations of the first order: Langrange method, Standard method, Charpits method

**Week 3** Partial Differential equations of the first order: Langrange method, Standard method, Charpits method continues

**Week 4** Partial Differential equations of the second order: method by inspection, Monge’s method, Monge’s method of integration

**Week 5** Partial Differential equations of the second order: method by inspection, Monge’s method, Monge’s method of integration continues

**Week 6**

**Review for Test 1 and Test 1 Week 7** Solution to general Linear Partial Differential Equations of higher order

**Week 8** Solution to general Linear Partial Differential Equations of higher order continues

**Week 9** Solutions to special types of partial differential equations: Hyperbolic, Parabolic, Elliptic equations, Diffusion equation, Wave equation. Methods of solution: Separation of variables, D’Alembert’s method

**Week 10**

**Review for Test 2 and Test 2**

**Week 11** Solutions to special types of partial differential equations: Hyperbolic, Parabolic, Elliptic equations, Diffusion equation, Wave equation. Methods of solution: Separation of variables, D’Alembert’s method continues

**Week 12**

**Revision Week** and Upload of Final Lecture Attendance

**Week 13** (Final Year Project Defence Week)

**Week 14** **& Week 15 (Omega Semester Examinations)**

**e. Structure of the Programme/Method of Grading **

Continuous Assessment:

Test 1 10%

Test 2 10%

Assignment & Home Work 10%

Examination 70%

**f. Ground Rules & Regulations**

Students are to maintain high level of discipline in the following areas.

- Punctuality
- Modest Dressing
- Quietness
- 75% lecture attendance for eligibility to semester examination.

**g. Assignment**

Students are given assignments at the end of the lecture.

**h. Alignment with Covenant University Vision/Goals**

* Prayers at the commencement of lectures and commitment to God.

* Classes are conducted with total compliance to the university core values.

* Course is delivered in a manner that the knowledge acquired is useful and applicable.

**i. Industry Relevance**

Modeled problems in various fields of engineering and some aspect of sciences require the tool of differential equation to achieve result. Thus, the relevance cannot be overemphasized.

- Modeling and solving real life problems.

**j. Recommended Reading/Text**

- A Text book of Engineering Mathematics (Vol.2) by Peter V. O‘ Neil, Cengage Learning.
- C. Prasad, Advanced Mathematics for Engineers, Prasad Mudralaya.
- E. Kreyszig: Advanced Engineering Mathematics, Wiley Eastern.
- M.D. Raisinghania: Ordinary & Partial Differential Equations, S. Chand Publication
- Engineering Mathematics by V. Sundaram, R. Balasubramanian and K. A. Lakshminarayanan. Volume 2 and Volume 3.

- Facilitator: ABIODUN ADEGBOYEGA OPANUGA

#### MAT224-Introduction to Numerical Analysis

Introduction to Numerical Analysis

- Teacher: JIMEVWO GODWIN OGHONYON

#### MAT222-Mathematical Methods II

Mathematical Methods II

Mathematical methods II, as a continuation of Mathematical methods I, is an aspect of Mathematics that equips students of Mathematics and the physical sciences with the right tools with which to tackle problems in Mathematics. However, it is expected that the students have some knowledge of Calculus and that he/she is therefore fairly familiar with such topics as simple differentiation, integration, and the use of the sine, cosine, exponential and logarithmic functions. This aspect of Mathematical methods will focus on topics such as D Operators and its applications, Series solution of Differential equations, Differentiation and integration of integrals, line integrals, multiple integrals and introduction to Fourier series and applications. The student should as a requirement master MAT212.

- Facilitator: ABIODUN ADEGBOYEGA OPANUGA

#### MAT123-Mathematics VII: Statistics II

Mathematics VII: Statistics II

- Teacher: OLUWOLE AKINWUMI ODETUNMIBI
- Teacher: HILARY IZUCHUKWU OKAGBUE

#### MAT122-Mathematics VI: Vector Algebra

MAT122, Vector Algebra is a one-semester 2-Credit foundation level course. The main objective of this course is to give students a good foundation in Vector Analysis, a course they might be taking in depth later on either in mathematics or physics, or engineering. Topics include 3-dimensional coordinate system, vectors and algebra of vectors, dot and cross products of vectors and calculus of vector functions.

- Facilitator: OLASUNMBO OLAOLUWA AGBOOLA
- Facilitator: JIMEVWO GODWIN OGHONYON

#### MAT121-Mathematics V: Calculus

All students must ensure that assignment are completed before the next class.

- Teacher: GRACE OLUWAFUNKE ALAO
- Teacher: SUNDAY ONOS EDEKI
- Teacher: OGBU FAMOUS IMAGA
- Teacher: ADEREMI SAMUEL OJO