### Special Course Description – Level II

**MSP4114 Algebra (Rings and Field Theory) (60 hrs) (Credit Value 4)**

Ring and fields: rings and fields, integral domain, characteristic of a ring, subrings and subfields , Ideals , maximal ideals and prime ideals. Homomorphisms and imbedding of rings,

Isomorphism: Quotient rings, homomorphism, imbedding of rings, more on ideals, Isomorphism theorems Euclidean and factorization domains: Euclidean domains, prime and irreducible elements, polynomial rings , unique factorization domains.

Extension fields: Introduction to extension fields, algebraic extension, roots of polynomials, splitting fields, ruler

and compass constructions, prime subfields, separable extension.

Galois theory: Normal extension, automorphism of field extension, fundamental theorem of Galois theory, Galoi’s extension, finite fields.

*Method of assessment: end of semester examination*

**MSP4b26 Seminars and Research/Study Project-Mathematics/Statistics -(Credit Value 6)**

Every special degree student is required to conduct supervised investigation on a research topic assigned at the beginning of the semester and is required to submit a dissertation.

*Method of assessment: Seminars/Presentations,Dissertation and Oral Examination*

MSP4134 Functional Analysis (60 hrs) (Credit Value 4)

Metric Spaces, Limit and Continuity, Connectedness, Completeness and Compactness, Completion of Metric Spaces, Normed Vector Spaces, Normed Spaces, Finite Dimensional Normed Spaces, Linear Subspaces of Normed Spaces, Banach Spaces, Fundamental Theorems for Normed and Banach Spaces, Inner Product Spaces, Hilbert Spaces, Orthogonal Expansions, Separable Hilbert Spaces, Linear Operators and Functionals, Liner Transformations on Hilbert Spaces, Spectrum of a Linear Operators.

*Method of assessment: end of semester examination*

**MSP4144 Time Series Analysis (60 hrs) (Credit Value 4)**

Introduction to basic concepts of time series analysis such as auto-regression, moving averages, integration, ARIMA, autocorrelation, and trends and volatility.Stationarity, testing for unit roots, and structural change different formulations of lags, and causality. Time series forecasting. Time series modelling, such as multi-equation models, cointegration and error-correction models or/and special topics in advanced time series analysis.

*Method of assessment: end of semester examination*

**MSP4153: Statistical Laboratory (60 hrs) (Credit Value 3)**

Analysing data with Computers using ’R’ software package.

**MSP4164: Analytical and Numerical Methods for PDEs (60 hrs) (Credit Value 4)**

Analytical methods for Partial Differential Equations: Introduction to Elliptic, Parabolic and Hyperbolic PDEs, Initial and boundary value problems, Superposition Principle of solutions, Fourier series, Separation of variables, Homogeneous and non-homogeneous problems, Time dependent and independent non-homogeneous problems, Sturm-Liouville Systems, Eigenvalues and eigenfunctions, Finite Fourier Transforms and non-homogeneous problems, Problems in Infinite Spatial Domains, Fourier Transforms, Fourier Transforms method for PDEs, Laplace Transforms methods for PDEs. Numerical Methods for Partial Differential Equations: Approximation of partial derivatives using finite differences, Finite-difference methods for parabolic, hyperbolic and elliptic equations, Heat equation, Wave equation and Poisson equation as examples, Convergence and Stability, Finite-element methods for PDEs in one dimensional space

*Method of assessment: end of semester examination*

**MSP4214: Mathematical Foundations of Quantum Mechanics / Special Topics in Mathematical Physics (60 hrs) (Credit Value 4) (This module shall be offered as a teaching module or a reading module.)**

Physical background, Dynamics, Observables, The uncertainty principle, spectral theory, Scattering States, Special Cases (e.g. infinite well, potential well etc), Many-particle systems, density matrices, Survey of modern philosophy of quantum theory/quantum computing. Course contents of Special Topics in Applied Mathematics will depend on the availability of staff members.

*Method of assessment: end of semester examination*

**MSP4224: Introduction to Stochastic Analysis ( 60 hrs) (Credit Value 4) Prerequisites MSP3184: Measure Theory with Applications**

Basic Stochastic Processes, Brownian Motion Calculus. Stochastic Differential Equations, Diffusion Processes, Martingales, Calculus for Semimartingales, Pure Jump Processes, Change of Probability Measure , Applications in Finance, Biology, Engineering, Physics and other areas, computational solutions. Special topics in stochastic modelling.

*Method of assessment: end of semester examination*

MSP4234: Topics in Applied Mathematics I(60 hrs) (Credit Value 4) (Eg. Dynamical Systems/Control Theory)

Course contents of Special Topics in Applied Mathematics will depend on the availability of staff members and shall be announced at the beginning of the academic year.

MSP4244: Topics in Applied Mathematics II (60 hrs) (Credit Value 4) (Eg Geo-mathematics/Relativity Theory/ Electromagnetic Theory/Computational Fluid Dynamics)

Course contents of Special Topics in Applied Mathematics will depend on the availability of staff members and shall be announced at the beginning of the academic year.

**MSP4254: Special Topics in Applied Mathematics (60 hrs) (Credit Value 4)**

Course contents will depend on the availability of staff members and shall be announced at the beginning of the

academic year.

**MSP4263: Design and Analysis of Experiments/Operations Research (45 hrs) (Credit Value 3)**

Introduction to the Design of Experiments, Analysis of Variance, One Factor Experiments, Randomized Complete

Blocks, Latin Squares, Comparisons among treatments, Factorial Experiments (Two or More Factors) , The 2k

factorial Experiments design, Confounding, Fractional Factorial Experiments, Higher Fractions and Screening Designs, Taguchi’s Robust Parameter Design, Control and Noise Variables.

*Method of assessment: end of semester examination
*

**MSP4273: Special Topics in Statistics (45 hrs) (Credit Value 3)**

Introduction to Distributions and Inference for Categorical Data: Categorical response data, distributions for categorical data, statistical inference for categorical data. Describing Contingency Tables: Probability structure for contingency tables, comparing two proportions, partial association in stratified 2 × 2 tables, Extensions for I × J tables. Inference for Contingency Tables: Confidence intervals for association parameters, Testing independence in

two-way contingency tables, two-way tables with ordered classifications, small-sample tests of independence.

Logistic Regression: Interpreting parameters in logistic regression, Inference for logistic regression, Multiple Lo-

gistic Regression, Fitting logistic regression models. Building and Applying Logistic Regression Models, Log-linear

models for contingency tables and building of log-linear Models.

**MSP4283: Introduction to Stochastic Processes (45 hrs) (Credit Value 3)**

Discrete and continuous Markov chains, point processes, random walks, branching processes and the analysis of their limiting behaviour. Renewal theory, Brownian motion, Gaussian processes and martingales.

*Method of assessment: end of semester examination*

MSP4293: Medical statistics (45 hrs) (Credit Value 3)

Clinical Trials: Basic Concepts and designs: controlled and uncontrolled clinical trials, historical controls, protocol, placebo, randomization, blind and double blind trials, ethical issues. Multiplicity and meta-analysis: intern analysis, multi-center trials, combining trials. Cross over trials, Binary response data, Analysis of cohort and case-control studies

Survival Data Analysis:

Basic concepts: survival function, hazard function, censoring.

Single sample methods: life-tables, Kaplan-Meier survival curve, parametric models.

Two sample methods: log-rank test, parametric comparisons.

Regression model: inclusion of covariates, Cox’s proportional hazards model, competing risks.