REGULATIONS AND SYLLABUS

Size: px

Start display at page:

Download "REGULATIONS AND SYLLABUS"

Noreen Morgan
5 years ago
Views:

1 REGULATIONS AND SYLLABUS Master of Science in Mathematics Effective from the Academic Year TAMIL NADU OPEN UNIVERSITY CHENNAI

2 MASTER OF MATHEMATICS (M.Sc) REGULATIONS 1. ELIGIBILITY : B.Sc. in Mathematics from any recognized University. 2. DURATION : Two Years (Academic/Calendar Year). 3. SCHEME OF EXAMINATIONS: First Year Marks Course Code Course Title Assign ments Theory Exam Total MMS 15 Algebra MMS 16 Real Analysis MMS 17 Complex Analysis and Numerical Analysis MMS18 Mathematical Statistics Second Year Marks Course Code Course Title Assign ments Theory Exam Total MMS 25 Topology and Functional Analysis MMS 26 Operations Research MMS 27 Graph Theory and Algorithms MMS 28 Differential Equations PATTERN OF THE QUESTION PAPER: Part A : Five out of eight questions 25 Marks 5 X 5 = 25 Part B : Five out of eight questions 50 Marks 5 X 10 = Total 75 Marks

3 Structure for Master of Science in Mathematics (MMS) First Year SUBJECT CODE SUBJECT NUMBER OF CREDITS MMS 15 Algebra 8 MMS 16 Real Analysis 8 MMS 17 Complex Analysis and Numerical Analysis 8 MMS 18 Mathematical Statistics 8 Second Year SUBJECT CODE SUBJECT NUMBER OF CREDITS MMS 25 Topology and Functional Analysis 8 MMS 26 Operations Research 8 MMS 27 Graph Theory and Algorithms 8 MMS 28 Differential Equations 8

4 SYLLABUS MMS 15 Algebra Block I Group Theory: Definition of a Group Some examples of Groups Preliminary Results Subgroups Order and Product of Subgroups Normal Subgroups and Quotient Groups Homomorphisms Automorphisms Cayley s Theorem Permutation Groups Conjucate of an element Cauchy s Theorem Sylow s Theorem Direct Products Finite Abelian Groups. Block II Ring Theory: Definition and Examples of rings Special Classes of Rings Homomorphisms Ideals and Quotient Rings More Ideals and Quotient Rings Field of Quotients of an Integral Domain Euclidean Rings Fermat s Theorem Polynomial Rings Block III Vector Spaces and Modules: Basic Concepts Linear Independence and Bases Dual Spaces Inner Product Spaces Modules. Block IV Fields: Extension Fields The Transcendence of e Roots of Polynomials Construction with Straightedge and Compass More About Roots The Elements of Galois Theory Solvability by Radicals Block V Linear Transformations: The Algebra of Linear Transformations Characteristic Roots Matrices Canonical Forms: Triangular Form Nilpotent Transformations

5 Reference Books 1. I.N. Herstein, Topics in Algebra, Vikas Publishing house, J.B. Fraleigh, A first course in Mordern algebra, Addition Wesley publishing house, Surjeet Singh and Quazi Zameeruddin, Modern algebra, Vikas Publishing house, 1990.

6 MMS 16 Real Analysis Block I The Real and Complex Number System: Ordered Sets Fields The Real Field The Extended Real Number System The Complex Field Euclidean Spaces. Basic Topology: Finite, Countable and Uncountable Sets Metric Spaces Compact Sets Perfect Sets Connected Sets. Numerical Sequences and Series: Convergent Sequences Subsequences Cauchy Sequences Upper and Lower Limits Some Special Sequences Series Series of Nonnegative Terms The Number e The Root and Ratio Tests Power Series Summation by Parts Absolute Convergence Addition and Multiplication of Series Rearrangements. Block II Continuity: Limits of Functions Continuous Functions Continuity and Compactness Continuity and Connectedness Discontinuities Monotonic Functions Infinite Limits and Limits at Infinity. Block III Differentiation: The Derivative of a real Function Mean Value Theorems The Continuity of Derivatives L Hospital s Rule Derivatives of Higher Order Taylor s Theorem Differentiation of Vector valued Functions. The Riemann Stieltjes Integral: Definition and Existence of the Integral Properties of the Integral Integration and Differentiation Integration of Vector-valued Functions Rectifiable Curves. Block IV Sequences and series of Functions: Discussion of Main Problem Uniform Convergence Uniform Convergence and continuity Uniform Convergence and Integration Uniform Convergence and Differentiation Equicontinuous Families of Functions The Stone Weierstrass Theorem.

7 Some Special Functions: Power Series The Exponential and logarithmic Functions The Trigonometric Functions The Algebraic Completeness of the Complex Field Fourier series The Gamma Function. Block V Functions of Several Variables: Linear Transformations Differentiation The Contraction Principle The Inverse Function Theorem The Implicit Function Theorem The Rank Theorem. Reference Books 1. W. Rudin, Principles of Real analysis, McGraw Hill, T.M. Apostol, Mathematical analysis, Addision Wesley publishing House, V.Ganapathy Iyer, Mathematical analysis, Tata Mcgraw Hill,1985.

8 MMS 17 Complex Analysis and Numerical Analysis Block I Complex Analysis Algebra of Complex numbers Geometric representation of complex numbers stereographic projection Various types of differentiation in the complex field, Cauchy Riemann equation, one point compactification and the Riemann sphere. Analytic function power series linear fractional transformation exponential logarithmic and trigonometric functions. Conformal mapping, definition and properties, elementary conformal mappings. Block II Complex integration Cauchy theorem general form of Cauchy theorem Cauchy integral formula, Morera s theorem, Liouville theorem fundamental theorem of algebra Taylor s theorem open mapping theorem maximum modulus theorem Schwartz lemma. Singularities Taylor and Laurent series expansion Weierstrass theorem Residu theorem argument principle Rouche s theorem, evaluation of standard type of integrals using residues. Block III Numerical Analysis Solution of algebraic and transcendental equations, bisection method, iteration method, method of false position, Newton Raphson method. Solution of linear system of equations, matrix inversion method, gauss Jordan elimination method, Gauss Seidel iteration method, Cholesky LU decomposition method, power method for eigen values.

9 Block IV Numerical interpolation, Newton forward and backward formula, Lagrange interpolation formula, Hermite interpolation formula, Newton divided difference formula, central difference formula, Numerical differentiation and numerical integration, traphezoid rule, Simpson rule, double integration. Block V Numerical solution of differential equations, Taylor series method, Picard method, Euler method, Runge kutta method, Predictor corrector method, Adam and Milne method. Reference Books 1. L.V. Ahlfors, Complex analysis, Mcgraw Hill, V.Karunakaran, Complex analysis, Narosa publishing house, S.Ponnusamy, Foundations of complex Analysis Narosa Publishing House, C.F. Gerald and P.O. Wheatley, Applied, Numerical Analysis, Addition Wesley, M.K. Jain, S.R.K. Iyengar and R.K. Jain, Numerical methods, Problems and Solutions, Wiley Eastern, 2002.

10 MMS 18 Mathematical Statistics Block I Probability Set function Conditional Probability and Independence Random variables of Discrete type and Continuous type distribution function its properties Expectation of a random variable moment generating function Chebeshev s inequality. Two Random variable joint density marginal probability density conditional distribution, Expectation and variance. Independence of two random variables mutual independence and pair wise independence. Distributions binomial trinomial multinomial negative binomial poisson gamma chi square normal and bivariate normal distributions t and F distributions. Block II Sampling theory transformations of the variables of discrete and continuous type distribution of order statistics Moment generating functions and expectation of functions of random variable. Limiting distribution Stochastic convergence Limiting moment generating function law of large numbers central limit theorem. Block III Point estimation measures of quality of estimators like unbiased consistent efficient and sufficient confidence intervals for proportions means variances and difference of proportions means and variances Bayesian estimates. Block IV Introduction to statistical hypotheses certain best test uniformly most powerful test likelihood ratio test Neyman Pearson theorem.

11 Block V Rao Cramer inequality efficient statistic sequential probability ratio test multiple comparisons sufficiency completeness and stochastic independence. Reference Books 1. R.V. Hogg and A.T. Craig, Introduction to mathematical statistics, Macmilan co, S.C. Gupta and V.K. Kapoor, Fundamentals of Mathematical statistics, Sultan chand and co, J.E. Freund, Mathematical Statistics, Prentice Hall of India, 2000.

12 Block I MMS 25 - Topology and Functional Analysis Topology Topological Spaces and Continuous Functions: Topological Spaces Definitions and examples Basis for a topology The order topology The product Topology on X Y The Subspace Topology Closed Sets and Limit Points Continuous Functions Homeomorphism Pasting lemma The Product Topology The Metric Topology Sequence lemma Uniform limit theorem The Quotient Topology. Block II Connectedness and Compactness: Connected Spaces Definition and examples Connected Sets in the real Line Intermediate value theorem Components and Path Components Local Connectedness Compact Spaces Compact Sets in the Real Line Limit Point Compactness Sequentially compact Equivalent conditions for compactness Local Compactness one point compactification. Block III Countability and Separation Axioms: The Countability Axioms The Separation Axioms The Urysohn Lemma Tietze extension theorem The Urysohn Metrization Theorem Partitions of Unity. Block IV Functional Analysis Banach Spaces: The Definition and example Continuous Linear Transfomations The Hahn Banach Theorem The Natural Imbedding of N in N** - The Open Mapping Theorem The Conjugate of an operator The uniform boundedness theorem.

13 Block V Hilbert Spaces: Definition and simple properties Orthogonal Complements Orthonormal Sets The Conjugate Space H* - The Adjoint of an operator Self Adjoin Operators Normal and Unitary Operators Projections. Reference Books 1. James R. Munkres, Topology, Prentice Hall of India Pvt Ltd, K.D. Joshi, Introduction to general topology, Wiley Estern, G.F. Simmons, Introduction to topology and modern analysis, McGraw Hill, B.V.Limaye, Functional analysis, Wiley Eastern, 1981.

14 MMS 26 Operations Research Block I Linear Programming Simplex method Duality and sensitivity analysis. Other algorithms for linear programming Dual simplex method parametric linear programming Upper bound tedhnique interior point algorithm Linear goal programming. Block II Network Analysis shortest path problem minimum spanning tree problem Maximum flow problem minimum cost flow problem Network simplex method project planning and control with PERT/CPM. Dynamic programming and Probabilistic dynamic programming. Block III Game theory: Two person Zero-sum game games with mixed strategy Graphical solution solution by Linear programming. Integer Programming cutting plane method Branch and Bound method. Block IV Queuing Theory pure Birth and Death model specialized Poisson queues m/a/l queue Pollaczek Khinthcine formula. Classical optimization theory unconstrained problems constrained problems. Block V Non linear programming algorithms unconstrained non linear algorithms constrained algorithms separable, quadratic and geometric programming.

15 Reference Books 1. H.A. Taha, Operations Research, an introduction, Prentice Hall of India, F.S. Hiller and G.J. Liebermann, Introduction to operations research, McGraw Hill, S.S. Rao, Optimization, theory and applications, Willey eastern.

16 MMS 27 - Graph Theory and Algorithms Block I An Introduction to graphs: Definitions and basic concepts Graph Models Vertex degrees Isomorphism and Automorphism Special class of graphs The pigeonhole principles and Turan s theorem Walk, Path and Connectedness Distance, Radius, Diameter and Girth Subgraphs and Isometric subgraphs Operations on Graphs The Adjacency, Incidence and Path matrices Algorithms Introduction to Algorithms Breadth-first search Algorithm Dijkstra s Algorithm Ford s Algorithm. Bipartite Graphs: Characterisations of bipartite graphs Trees cut edges and cut vertices Spanning trees and isometric trees Cayley s Formula Binary trees Algorithms Spanning tree Algorithm Kruskal s Algorithm Prim s Algorithm. Block II Connectivity: Connectivity and edge connectivity 2-Connected graphs Menger s Theorem Separable graphs, 1-Isomorphism and 2-Isomorphism. Graphic Sequences: Degree sequences Graphic sequences Wang and Kleitman s Theorem Algorithms Algorithm 1 Algorithm 2. Block III Eulerian and Hamiltonian Graphs: Charecterisations of Eulerian Graphs Degree Sets Randomly Eulerian Graphs Application Algorithm Fleury s Algorithm Further Readings Enumeration Hamiltonian Graphs Hamilton Cycle in Power Graphs and Line Graphs Hamiltonian Sequences Application Algorithms Two Optimal Algorithm The Closest Insertion Algorithm Albertson s Algorithm Related Parameters. Matchings: Matching System of Distinct Representatives and Marriage Problem Covering 1-Factor Stable Matchings Application Algorithm The Hungaria Algorithm Algorithm for Maximum Matching.

17 Block IV Independence: Independent Sets Edge colourings Application Vizing s Theorem Vertex Colouring Uniquely Colourable Graphs Brook s Bound and Improvements Hajos Conjecture Mycielski s Construction Line-distinguishing Colourings Chromatic Polynomials Algorithm Sequential Colouring Algorithm. Block V Planar Graphs: Planar Embedding Euler s Formula Maximum Planar Graphs Geometric dual Characterisations of Planar Graphs Algorithm DMP Planarity Algorithm Colouring in Planar Graphs Face Colouring. Reference Books 1. M.Murugan, Graph Theory and Algorithms, Muthali Publishing House, Annanagar, Chennai, J.A. Bondy and U.S.R. Murthy, Graph Theory with applications, Macmillan Co., D.B.West, Introduction to graph theory, Prentice Hall of India, 2001.

18 Block I MMS 28 - Differential Equations Ordinary differential equations linear differential equations of higher order linear independence and Wronskian, method of variation of parameters homogeneous linear differential equations with constant coefficients. Block II Solutions in power series second order linear differential equations with ordinary points Legendre equation and Legendre polynomial, Second order equation with regular singular points Bessel equations. Block III Systems of linear differential equations system of first order equation existence and uniqueness theorem fundamental matrix linear system with constant and periodic coefficients Existence and uniqueness of solutions successive approximations Picard theorem. Block IV Partial differential equations of second order linear partial differential equation with constant and variable coefficients, characteristic curves of second order equations characteristic equations in three variables the solution of linear hyperbolic equations. Block V Elementary solutions of Laplace equations, families of equi potential surfaces, boundary value problems, Kelvin s inversion theorem, the theory of Green s function for Laplace equation. Reference Books 1. E.A. Coddington, An introduction to ordinary differential equations Prentice Hall of India, S.G.Deo and Ragavendra Rao, Ordinary differential equations and stability theory, Prentice Hall of India, Jan Snedden, Elements of Partial differential equations, Mc.Graw Hill, 1985.

MATHEMATICS. Course Syllabus. Section A: Linear Algebra. Subject Code: MA. Course Structure. Ordinary Differential Equations

MATHEMATICS. Course Syllabus. Section A: Linear Algebra. Subject Code: MA. Course Structure. Ordinary Differential Equations MATHEMATICS Subject Code: MA Course Structure Sections/Units Section A Section B Section C Linear Algebra Complex Analysis Real Analysis Topics Section D Section E Section F Section G Section H Section