Action of a group on a set, Conjugacy class, Centralizer, Normalizer, Centre of Groups, Sylow theorems and Simple groups, Commutator subgroup.
Rings and Ideals:
Introductory concepts, Ideals and their operations, the Classical Isomorphism theorems. Integral Domains and Fields.
Assignment: At least 2 (30%)
Examination: 4 out of 6 questions (70%)
PMS 3112 - REAL ANALYSIS
Riemann integral :
Darboux definition for a bounded function on a closed and bounded interval, necessary and sufficient condition for integrability, integrability of monotone functions and continuous functions, linearity of the integral, additivity of the integral over the interval of integration, monotonicity of the integral, integrability of composite, integrability of the modulus and the product, integral as a limit of a sum, integration and differentiation, Fundamental Theorem of Calculus, integration by parts, integration by substitution, mean value theorems, interchanging derivative and integrals.
Functions of bounded variation and rectifiable curves :
Definition of bounded variation, total variation, additive property of total variation, total variation on [ a,x ] as a function of x, functions of bounded variation expressed as the difference of increasing functions, continuous functions of bounded variation, total variation as the integral of the absolute value of the derivative.
Improper Riemann integrals :
Tests for convergence of improper integrals (analogues of Cauchy condition, absolute convergence, comparison test, ratio limit test, Dirichelet's and Abel's tests), Cauchy's integral test, Euler's constant, derivation of Stirling 's formula, Gamma & Beta functions.
Assignment: At least 2 (30%)
Examination: 4 out of 6 questions (70%)
PMS 3113 - COMPLEX ANALYSIS
Regions and mappings, limits and continuity, differentiation, analytic functions, conformal mapping; Sequences, series, power series, uniform convergence; Exponential, logarithmic, hyperbolic and trigonometric functions; Complex integration; Cauchy's inequalities, Liouvilles' theorem, Taylor's theorem, Laurent's theorem, Zeros and Singularities, the calculus of residues.
Assignment: At least 2 (30%)
Examination: 4 out of 6 questions (70%)
PMS 3114 - INTRODUCTION TO TOPOLOGY
Fundamental concepts :
Sets, set operations, functions, equivalance relations and partitions, simple order, least upper bound, greatest lower bound, integers and real numbers, well ordering property, arbitrary cartesian products, cardinality, finite sets, countable and uncountable sets, Axiom of Choice, Zorn's Lemma.
Topology of the line :
Interior points and open sets in R, limit points, closed sets, closure, interior, convergence (comparison with the e - n 0 definition), Continuous functions (comparison with e - d definition), Compactness (comparison with the maximum, minimum value theorem and uniform continuity), Connectedness (comparison with the Intermediate Value Theorem), extension of these ideas to the Topology of the plane.
Metric spaces :
Definition of a metric space, examples, distance between sets, diameters, open balls, equivalent metrics, isometric metric spaces, convergence and completeness, continuity (in metric spaces), compactness, separability, connectedness, spaces of continuous functions.
Topological Spaces :
Definition and examples, comparison of topologies, the product topology on X x Y, the subspace Topology, closed sets and limit points, boundary, interior and exterior, continuous functions, homeomorphisms, convergence, definitions and examples for connectedness and compactness .
Gamma and Beta functions, Elliptic Integrals, Fourier Analysis; Partial Differential Equations; Solutions of linear partial differential equations with homogeneous and non-homogeneous boundary conditions; Variable separable methods; Laplace transforms; Fourier transforms, Fourier sine and cosine transforms; Hankel transforms; Melin transforms; Calculus of Variation; Chebyshw polynomials; Hermite polynomials.
Assignment: At least 2 (30%)
Examination: 4 out of 6 questions (70%)
AMS 3113 - INTRODUCTION TO NUMERICAL ANALYSIS
1. Preliminaries
(i) Basic concepts and Taylor 's theorem
(ii) Orders of convergence
(iii) Difference equations
(iv) Error analysis
2. Solutions of non-linear equations
(i) Bisection method
(ii) Newton 's method
(iii) Secant method
(iv) Fixed point functional iteration
3. Solving system of linear equations
(i) Physical problems leading to linear systems
(ii) The LU and cholesky decomposition
(iii) Direct methods
(iv) Iterative methods
(v) Error and convergence analysis
4. Approximating functions
(i) Polynomial interpolation
(ii) Divided differences
(iii) Hermite interpolation
(iv) Spline interpolation
5. Numerical differentiation and integration
(i) Numerical differentiation and Richardson Extrapolation
(ii) Numerical integration based on interpolation
(iii) Gaussian quadrature
(iv) Romberg integration
(v) Adaptive quadrature
6. Numerical solution of ordinary differential equations
(i) The Taylor series methods
(ii) Runge-Kutta method
(iii) Linear Multi steps methods
(iv) Consistency, Stability and Convergence results
(v) Introduction to stiff problems
(vi) Shooting methods for boundary value problems
Assignment: At least 2 (30%)
Examination: 4 out of 6 questions (70%)
AM3109-MATHEMATICAL MODELLING -
APPLICATIONS IN ECONOMICS AND BUSINESS
Course Content :
Introduction to economics and business. Role of mathematics in economics and business. General study of demand, supply and market equilibrium.
Static and comparative-static analysis of market models, input-output models and selected macro economic models. Effect of taxation on static market models.
Dynamic analysis in continuous and discrete time of market models, input-out put models, financial models and some macro economic models. Effect of taxation on dynamic market models.
Deterministic and probatilistic inventory models.
Elasticity and other Economic concepts: Elasticity of demand and supply - point and cross elasticities. Analysis of single product and joint products cost, revenue, average cost, price, profit functions, etc. Marginal analysis, consumer's surplus and producer's surplus, Optimisation of revenue, cost and profit functions of single product and joint products.
Consumer demand theory: Derivation of utility functions. Maximizing utility functions with and without budget constraints. Derivation of demand functions.
Indifference curves. Marginal rate of substitution and contract curves (Edgeworth box).
Linear Programming and Economic applications: Introduction, graphical method for two variables, simplex algorithm, general linear programming problems (Big M / Two phase methods), Duality -dual simplex algorithm, Sensitivity analysis. Transportation and assignment models.
Introduction to Game theory: Zero-sum matrix games, single and mixed strategy games, optimal strategies, dominance. Simple applications.
STP 3103 Regression and Time Series
Correlation, Simple linear regression, least squares procedure, Gauss Markov theorem, confidence bands, multiple linear regression, lack of fit.
Decomposition of a time series, moving averages, autocorrelation, seasonality, stationary series, forecasting, use of statistical packages, MINITAB, GLIM, INSTAT, SAS and interpretation of their outputs.
SCHEME OF EXAMINATION
The Examination in the third year Special Degree in Mathematics shall consists of 9 papers of three hours duration each, from the following:
