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Course Code      :  PMAT 42983

Title                    :  Graph Theory

Pre-requisites     :  PMAT 21553

 Learning Outcomes:

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

• recognize the appearance of graphs in real life,
• identify certain real life situations that can be described using graphs,
• apply the concepts of graphs to the real world problems.

 Course Contents:

Matrices Associated with graphs and their properties, Trees: spanning trees of a connected graph, Coloring: Vertex colourings of graphs: greedy algorithm, Edge colourings of graphs, Matching and covering: Konigs theorem and Halls theorem, connectivity.

Random graphs: properties, Erdos¹s theorem on the existence of graphs with large girth and large chromatic number, the countably infinite random graph.

Group project: Applications related to real world problems

 Method of Teaching and Learning: A combination of lectures, tutorial discussions.

 Recommended Reading:

1. Robin J. Wilson: Introduction to Graph Theory (5e) Longman, 2010
2. Gary Chartrand, Ping Zhang: A First Course in Graph Theory Courier Corporation, 2013
3. Balakrishnan: T&P Of Graph Theory (Schaum's outline series) Tata McGraw-Hill Education, 2004
4. Dragos M. Cvetkovic, Michael Doob, Horst Sachs. Spectra of Graphs: Theory and Applications 3e Wiley, 1999

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Course Code    : PMAT 42793

Title                   : Advanced Theory of Riemann Integration

Pre-requisites   : PMAT 12543

 Learning Outcomes:

At the end of the course the student should be able to demonstrate knowledge of the concepts and theorems of Riemann Integration and to apply them in solving advanced integration problems.

 Course Content:

Riemann Integration: The Riemann integral, Properties of the Riemann integral, Fundamental theorem of calculus.

Improper Integrals: Properties of Improper Integrals, Leibnitz’s rule.

 Method of Teaching and Learning: A combination of lectures, tutorials and presentations.

Assessment: Based on tutorials, tests, presentations and end of course examination.

 Recommended Reading:

1. Ross, K.A., (2^nd edition, 2015) Elementary Analysis. The Theory of Calculus, Springer.
2. Widder, D.V. (2^nd edition, 2012) Advanced Calculus. Courier Corporation.

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Course Code    : PMAT 42803

Title                   : Advanced Mathematical Methods

Pre-requisites   : PMAT 22583

 Learning Outcomes:

At the end of this course, the student should be able to demonstrate knowledge of solving problems involving partial differential equations.

 Course Content:

Special Functions: Legendre Polynomials, Bessel Functions

Laplace Transforms: Laplace Transform. Linearity, Shifting Theorems, Transforms of Derivatives and Integrals, Unit Step Function (Heaviside Function), Differentiation and Integration of Transforms.

Fourier Series: Fourier Series, Even and Odd Functions, Half-Range Expansions, Fourier Integral, Fourier Cosine and Sine Transforms

Partial Differential Equations: Introduction to first order and second order partial differential equations. Parabolic, elliptic and hyperbolic partial differential equations

Integral Transforms: Laplace Transforms, Fourier Transforms, Hankel Transforms, Fourier method for partial differential equations.

Applications of Boundary Value Problems

 Method of Teaching and Learning: A combination of lectures, tutorial discussions and presentations.

 Assessment: Based on tutorials, tests, presentations and end of course examination.

 Recommended Reading:

1. Kreyszig, E., (10^th edition, 2011) Advanced Engineering Mathematics, Wiley.
2. Pinsky, M.A., (2011) Partial Differential Equations and Boundary Value Problems with Application, American Mathematical Soc.
3. Raisinghania, M.D., (1995). Advanced Differential Equations, S.Chands, India.

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Course Code    : PMAT 41783

Title                   : Differential Geometry

Pre-requisites   : PMAT 22583

Learning Outcomes:

At the end of this course the student should be able to

• demonstrate the fundamental knowledge of curves and surfaces in space
• identify the importance of the two factors curvature and torsion, and their intrinsic properties.

 Course Contents:

Theory of Curves: Concept of a curve, Arc length, Curvature and torsion, Frenet-serret formulae, General helix, intrinsic equations, Fundamental existence and uniqueness theorems for space curves, Canonical representation of a curve. Involutes and Evolutes, Theory of contact.

 Theory of Surfaces: Concept of a surface, Topological properties of a surface, Surface of revolution, Ruled surfaces, Length of arc on a surface, Vector element of an area, First and second fundamental forms, Curves on a surface, Direction coefficients, Direction ratios, Family of curves on a surface, Double family of curves. Umbilical point, Intrinsic properties of a surface, Geodesies. 

Curvature: Principle curvature and directions, Gaussian and mean curvatures, Lines of curvature Rodrigues formula.

Introduction to Riemannian geometry

 Method of Teaching and Learning: A combination of lectures, tutorials and presentations.

Assessment: Based on tutorials, tests, presentations and end of course examination.

 Recommended Reading:

1. Lipschutz, L., (1969) Differential Geometry, McGraw-
2. Willmore, T.J., (2013) An Introduction to Differential Geometry, Oxford University Press.
3. Do Carmo, M.P., (2016) Differential Geometry of Curves and Surfaces, Prentice-Hall, New Jersey.