User Rating: 3 / 5

Course Code    : PMAT 41403

Title                   : Topology

Pre-requisites   : PMAT 21263

 Learning Outcomes:

By the end of this course, the student should be able to,

1. analyze and interpret basic topological concepts introduced in this course

2. discuss and work with abstract topological spaces and develop tools to characterize them at a depth suitable for someone aspiring to study higher-level mathematics

3. formulate tools to identify when two topological spaces are equivalent (homeomorphic)

4. differentiate between functions that define a metric on a set and those that do not

5. describe the hereditary of topological properties under continuous maps

6. summarize the basis of point-set topology.

 Course Contents:

Topological spaces: Definition, Open sets, closed sets, Basis (any topology/given topology), Finite-Closed topology, Euclidean topology

Limit Points: Definition, Boundary, Closure, Dense sets, Neighborhoods, Connectedness

Homeomorphisms: Definition, Subspaces, Non-Homeomorphic spaces, Hausdorff space

Continuous Mappings: Definition, Intermediate Value Theorem

Metric Spaces: Definition, Convergence (of a sequence), Completeness, Contraction Mappings, Baire Spaces

Compactness: Definition, Heine-Borel Theorem

Product Topology

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

 Assessment:  Based on problem sets, presentations and end of course examination  

Recommended Reading:

1. Munkres, J.R. (2015). Topology, a first course, Prentice-Hall, India.
2. Janich, K. (1984). Topology, Springer.
3. Armstrong, M.A. (1983). Basic Topology, Springer.
4. Morris, S. A. Topology Without Tears. https://www.topologywithouttears.net/topbook.pdf

User Rating: 5 / 5

Course Code      :  PMAT 42383

Title                    :  Graph Theory

Pre-requisites     :  PMAT 21553

 Learning Outcomes:

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

• recognize the applications of graphs in real life
• calculate the matrices and their properties associated with graphs
• find the minimum spanning tree
• apply graph coloring in various applications
• apply the concept of matching, covering and connectivity in real world applications
• determine the maximum flow and minimum cut of a network
• construct random graph models and apply them to the relevant real-life problems
• solve graph related application in the real-life situations.

 Course Contents:

Matrices Associated with graphs and their properties: Laplacian, Normalized Laplacian, Signless Laplacian

Trees: Spanning trees, Algorithms for shortest path and spanning trees

Graph Coloring: Vertex colouring, Edge colouring, Colouring algorithms, Chromatic polynomials

Matching: Perfect matching, Maximum matching, Alternate and Augmented paths, Hall’s marriage theorem, Stable matching,

Covering: Vertex cover, Edge cover, Konig’s theorem, Factors of a graph,

Connectivity: Vertex connectivity, Edge connectivity, Connectivity index, k-Connected graphs, Menger’s theorem

Introduction to graph Labelling: Prime labelling, Graceful labelling

Network flow problems: Network of a digraph, Flows and source/sink cuts, Ford-Fulkerson algorithm, Max-flow min-cut theorem

Random graphs: Basics in probability, Random variable, Moment generating functions, Types of random graphs, Properties of random graphs, Erdo ̈s theorem

 Method of Teaching and Learning:  A combination of lectures, group project and assignments

 Recommended Reading:

1. Wilson, R.J. (5 th Ed., 2010). Introduction to Graph Theory, Longman.
2. Chartrand, G. & Zhang, P. (2013). A First Course in Graph Theory, Courier Corporation.
3. Balakrishnan, V. (2004). T&P Of Graph Theory (Schaum's outline series) Tata McGraw-Hill Education.
4. Cvetkovic, D.M., Doob, M. & Sachs, H. (3 rd Ed., 1999). Spectra of Graphs: Theory and Applications, Wiley.
5. West, D.B. (2 nd Ed., 2005). Introduction to Graph Theory, Prentice Hall. 6. Balakrishan, R. & Ranganathan, K. (2 nd Ed., 2012). Textbook of Graph Theory, Springer.

User Rating: 5 / 5

Course Code    : PMAT 41393

Title                   : Functional Analysis

Pre-requisites   : PMAT 21263

 Learning Outcomes:

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

1. use axioms for abstract linear spaces (over the real or complex fields) to discuss examples (and non examples) of abstract linear spaces
2. prove and apply Holder and Minkowski Inequalities and explain the general properties of metric and normed spaces and the relationships between them
3. define convergence, limit and being Cauchy sequence by using functional analysis tools
4. illustrate concepts such as completeness and complement of normed spaces and report on fundamental properties of Banach spaces
5. explain the general properties of inner product and normed spaces and the relationships between them, in particular illustrate concepts such as completeness and complement of inner product spaces.
6. analyze the fundamental properties of Hilbert spaces, the properties of linear operators defined in finite and infinite dimensions and the important applications of these properties
7. define and explain the concepts of continuity and limitation for operator, function, functional, Banach and Hilbert spaces and self-adjoint operators
8. apply the theory in the course to solve a variety of problems at an appropriate level of difficulty.

 Course Contents:

Linear spaces: Linear spaces, Subspaces and convex sets, Quotient space, Direct sums and Projections, Holder and Minkowski Inequalities

Normed Linear Spaces and Banach Spaces: Normed spaces, Open Balls, Equivalent norms, Open and closed sets, Metric spaces and Metrics Induced by norms, Translation invariant and absolute homogeneity of a norm, Quotient norms, Convergence of a sequence in a normed space, Cauchy sequence, Completeness of normed space, Banach Spaces, Bounded, Totally Bounded, and Compact Subsets of a Normed Linear Space, Finite dimensional normed linear spaces.

Inner Product Spaces and Hilbert spaces: Inner product spaces, Cauchy-Bunyakowsky-Schwarz Inequality, Inner product norm, Polarization and Parallelogram Identity, Completeness of Inner product spaces, Hilbert spaces, Orthogonality, Best approximation or nearest point, orthonormal basis.

Bounded Linear Operators and Functionals: Linear Operator, Kernal and Null spaces, Inverse Linear Operator, bounded linear operators, Operator norms, uniformly operator convergent and strongly operator convergent, Bounded linear functionals, Open Mapping Theorem, Banach’s Theorem

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

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

 Recommended Reading:

1. Teschl, G. (2020). Topics in linear and nonlinear functional analysis. American Mathematical Society.
2. Halmos, P.R. (1996). Finite Dimensional Vector Spaces, Springer.
3. Markin, M.V. (2018). Elementary Functional Analysis, de Gruyter.
4. Pinchuck, A. (2011). Functional Analysis Notes, Rhodes University.
5. Madox, I.J. (1992). Elements of Functional Analysis, Cambridge University Press.
6. Jain, P.K., Ahuja, O.P. & Ahmad, K. (2nd Ed., 2010). Functional Analysis, New Age Science Limited.
7. Kreyszig, E. (2007). Introductory Functional Analysis with Applications, John Wiley, New York.

User Rating: 5 / 5

Course Code    : PMAT 42373

Title                   : Advanced Mathematical Methods

Pre-requisites   : PMAT 22293, PMAT 22282

 Learning Outcomes:

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

1. discuss the properties of special functions as solutions of differential equations

2. classify partial differential equations using various techniques learned

3. solve hyperbolic, parabolic and elliptic equations using fundamental principles

4. apply a range of techniques to find solutions of standard PDEs

5. demonstrate accurate and efficient use of Fourier analysis techniques and their applications in the theory of PDEs

6. solve real world problems by identifying them appropriately from the perspective of partial derivative equations

 Course Content:

Special Functions: Legendre Polynomials, Bessel Functions

Partial Differential Equations: Classification of PDE, First order partial differential equations: Lagrange’s method and Charpit’s method, Second order partial differential equations: Linear Partial Differential Equations with Constant Coefficients, Partial Differential Equations of Order two with Variables Coefficients, Classification of Partial Differential Equations Reduction to Canonical or Normal Form: Parabolic, elliptic and hyperbolic partial differential equations.

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

Solutions of PDEs: Separations of variables, D’Alembert’s Solution and Characteristic solutions of the Wave Equation,

Integral Transforms: Fourier Transforms, Laplace Transforms, Hankel Transforms

 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. (10th Ed., 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.