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,

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

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

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.

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.

Isomorphism Theorems, Series of groups, sylow theorems, applications of the sylow theorem, free groups.

Implicit functions: definition, derivative of implicit functions, Implicit function theorem Jacobians, stationary values under subsidiary conditions . More on Integration on R

- Same as MAT322β

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

Analysing data with Computers using ’R’ software package.

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

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.

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.

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.

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.

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

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.

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.

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.

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.

© 2020- Department of Mathematics, University of Ruhuna, Sri Lanka