+ Department of Mathematics
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Syllabus - B.Sc. Special Degree in Mathematics

Level I

MSP311β: Group Theory (30 lecture hrs + 15 tutorial hrs) Same as MAT311β (Credit Value 2.5)

MSP312β: Real Analysis-III (30 lecture hrs + 15 tutorial hrs) Same as MAT312β (Credit Value 2.5)

MSP313β: Mathematical Statistics-II (30 lecture hrs+15 tutorial hrs) Same as MAT313β (Credit Value 2.5)

MSP3144: Mathematical Methods in Physics and Engineering II(60 Lecture hrs) (Credit Value 4)

Part A:
Same as AMT313β.
Part B:
Applications of Laplace Transforms to Differential Equations, Fourier Transforms, Legendre Functions, Hermite Equation, Functions and Polynomials, Laguerre Equation and Polynomials, Riccati’s Differential equation, The Dirac-Delta function,
Method of assessment: end of semester examination

MSP316β: Applied Statistics II (30 lecture hrs+15 tutorial hrs)- Same as MIM313β (Credit Value 2.5)

MSP3174: Topology (60 hrs) (Credit Value 4)
Topological Spaces, Basis for a Topology, The Subspace Topology, Closed Sets, Limit Points, Continuous Functions, The Product Topology, The Metric Topology, Connected Spaces, Compact Spaces
Method of assessment: end of semester examination

MSP3184: Measure Theory with Applications (60 hrs) (Credit Value 4)
Borel – Algebra, Borel subsets, Lebesgue outer Measure, Lebesgue measurable subsets, Lebesgue measure, Lebesgue measurable functions. Properties that hold almost everywhere, Lebesgue Integral, Lebesgue integrable functions, Monotone Convergence Theorem, Dominated Convergence Theorem, Fatou’s Lemma, Relation of Riemann and Lebesgue Integrals, Modes of convergence (topics are discussed with applications example probability theory),Introduction to martingales
Method of assessment: end of semester examination

MSP3193: Bayesian Inference and Decision Theory (45 hrs) (Credit Value 3)
Fundamentals of the Bayesian theory of inference, probability as a representation for degrees of belief, the likelihood principle, the use of Bayes Rule to revise beliefs based on evidence, conjugate prior distributions for common statistical models, methods for approximating the posterior distribution. Graphical models for representing complex probability and decision models by specifying modular components. Concepts in decision analysis, including influence diagrams, decision trees, and utility theory or/and special topics in advanced Bayesian inference and decision theory.
Method of assessment: end of semester examination

MSP3b9β Mathematical Computing (90 project hrs) (Credit Value 2.5) Same as IMT2b2β;
The symbolic/numerical mathematics package Mathematica is introduced to solve mathematical problems on the computer. In particular special attention will be given to functional programming aspects of symbolic and numerical computations in Mathematica. Method of assessment: Practical examination/assignments/project report and the end of semester oral examination and/or presentation

MSP321α: Advanced Group Theory (23 hrs) (Credit Value 1.5)
Isomorphism Theorems, Series of groups, sylow theorems, applications of the sylow theorem, free groups.
Method of assessment: end of semester examination

MSP322α: Real Analysis IV (23 hrs) (Credit Value 1.5)
Implicit functions: definition, derivative of implicit functions, Implicit function theorem Jacobians, stationary values under subsidiary conditions . More on Integration on R2 : Green’s theorem, Change of Variables in a double integral.Calculus of variation
Method of assessment: end of semester examination

MSP323β: Complex Variables (30 lecture hrs + 15 tutorial hrs)
- Same as MAT322β

MSP324α: Complex Analysis (23 hrs) (Credit Value 1.5)
Review of elementary complex analysis topics from vector analysis: Morera’s Theorem, Liouville’s Theorem, Rouche’s Theorem, Winding numbers, the generalized version of Cauchy’s theorem, Morera’s theorem, the fundamental theorem of algebra, the identity theorem, the Riemann sphere and Weierstrass-Casorati theorem, meromorphic functions, Rouche’s theorem, integration by residues, Conformal mappings and its Physical applications. Method of assessment: end of semester examination

MSP3254 Numerical Methods with Applications (60 hrs) (Credit Value 4)
Solving Linear Systems: Matrix notation, direct methods, Gauss, Jordan, Aitken Method etc.
Iterative methods: Jacobi, Gauss-Seidel, SOR method etc.
Numerical Solution of Ordinary Differential Equations: Euler and modified Euler methods and Runge-Kutta methods, convergence Criteria, errors and error propagation. Numerical Solution of Partial Differential Equations: parabolic type, Elliptic type and Hyperbolic type.
Method of assessment: end of semester examination

MSP3263: Regression Analysis (45 hrs) (Credit Value 3)
Introduction: Regression and model building, use of regression, role of the computer. Simple Linear Regression: simple linear regression model, least-square estimation of the parameters, hypothesis testing on the slop and intercept, interval estimation in simple linear regression, prediction of new observations, coefficient of determination, estimation by maximum likelihood.
Multiple Linear Regression: Multiple linear regression models, Estimation of the parameters, Hypothesis testing in multiple linear regression, Confidence interval in multiple regression, prediction of new observations, Multicollinearity.
Model Adequacy Checking: Residual analysis, Lack of fit of the regression model. Indicator Variables, Variable Selection and Model Building, Introduction to Nonlinear Regression, Introduction to Generalized Linear Models.
Method of assessment: end of semester examination

MSP3274 Differential Geometry and Tensor Analysis (60 hrs) (Credit Value 4)
Unit tangent vector, Principal normal, binormal vector and curvature of a curve, Serrate-ferret formula, surfaces, parametric curves, surfaces of revolution, metric, directional ratios and coefficients, Gauss and Mean curvature, Orthogonal trajectories, families of dual curves, Geodesics. Transformation of coordinates, summation convention, the Kronecker-delta, contravariant and covariant vectors, contravariant, covariant and mixed tensors, symmetric and skew- symmetric tensors, tensor algebra, metric tensor, conjugate metric tensor, Christoffel’s symbols of first and second kind, covariant derivatives, Riemann and Ricci tensors.Theory of Manifolds, Commutators or Lie Derivatives, Forms and Dual bases, The wedge product, Exterior and Covariant differentiation, First and Second Carton equations, The Ricci Tensor and the Einstein Tensor.
Method of assessment: end of semester examination

MSP3283 Special Topics in Statistics (eg. Multivariate Data Analysis) (45 hrs) (Credit Value 3)
Multivariate data and multivariate statistics: Introduction, Types of data, Basic multivariate statistics,The aims of multivariate analysis. Exploring multivariate data graphically: scatterplot, scatterplot matrix, checking distributional assumptions using probability plots.Cluster analysis, Principle component analysis, Logliner and logistic models for categorical multivariate data,Models for multivariate response models, discriminant analysis and factor analysis.
Method of assessment: end of semester examination

MSP3293:Applied Statistics III (30 hrs) (Credit Value 2)
One-Factor Experiments: Analysis of Variance Techniques,One-Way ANOVA, Completely Randomized De- sign, Two-way ANOVA, Randomized Block Design.
Chi-squared Tests: Goodness of Fit Test, Categorical Data, Test for Independence, Test for Homogeneity,
Nonparametric Tests: Sign test, Signed-Rank test, Rank-Sum test, Kruskal-Wallis test, Runs test, Rank Cor- relation Coefficient.
Applied Nonparametric Regression: Introduction, Basic idea of smoothing, Smoothing techniques, Choosing the smoothing parameter.
Method of assessment: end of semester examination

Level II

MSP4114 Rings & 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.