Course Bulletin - for Batch Year 2004 *Mathematics 2 Yr M.Sc * * Specialization : None * * First Year * * First Semester * * Core Course* CS 101 Computer Programming and Utilization 6.0 MA 401 Linear Algebra 8.0 MA 403 Real Analysis I 8.0 MA 411 Introduction to Probability 6.0 MA 417 Ordinary Differential Equations 8.0 SI 409 Discrete Structures Lab 3.0 Course Contents Course Bulletin - for Batch Year 2004 *Mathematics 2 Yr M.Sc * * Specialization : None * * First Year * * First Semester * *Course Code: * CS 101 *Title: * Computer Programming and Utilization *Credits: * 6.0 *Pre-requisite: * *Description: * Functional organisation of computers, algorithms, basic progamming concepts, FOR-TRAN language programming. Program testing and debugging. Modular programming subroutines: Selected examples from Numerical Analysis, Game playing, sorting/searching methods, etc. *Text/References: * N.N. Biswas, FORTRAN IV Computer Programming, Radiant Books, 1979. K.D> Sharma, Programming in Fortran IV, Affiliated EAST WEST, 1976. *Course Code: * MA 401 *Title: * Linear Algebra *Credits: * 8.0 *Pre-requisite: * *Description: * Systems of linear equations: matrices and elementary row operations, Gaussian elimination, LU decomposition. Vector Spaces: subspaces, bases and dimension, coordinates. Linear Transformations: representation of linear transformations by matrices, rank-nullity theorm, duality and transpose, determinants. Eigenvalues and Eigenvectors: minimal and characteristic polynomials, Diagonalization, Schur"s theorem, Cayley Hamilton theorem, Jordan Canonical form. Inner Product spaces: Gram-Schmidt orthonormalization, reflectors, QR decomposition using reflectors, least squares problem, adjoint of an operator, unitary operators, rigid motions, positive(semi) definite matrices, minimum principles and Rayleigh quotients, matrix norms, condition numbers. Eigenvalue computation: power and inverse power methods, QR method. *Text/References: * M. Artin, Algebra, Prentice Hall of India, 1994. K. Hoffman and R. Kunze, Linear Algebra, Prentice Hall of India, 1991. B. Noble and J.W. Daniel, Applied Linear Algebra, 2nd ed., Prentice Hall, 1977. G. Strang, Linear Algebra and Its Applications, 3rd ed., Harcourt Brace Jovanovich, San Diego, 1988. D.S. Watkins, Fundamentals of Matrix Computations, Wiley, 1991. *Course Code: * MA 403 *Title: * Real Analysis I *Credits: * 8.0 *Pre-requisite: * NIL *Description: * Metric spaces, compactness, connectedness, completeness. Continuity. Monotonic functions. Differentiation of vector-valued functions. Functions of bounded variation and absolutely continuous functions. Riemann-Stieltjes integral and its properties. Fundamental theorem of integral calculus. Sequences and series of functions, uniform convergence and its relation to continuity, differentiation and integration. Equicontinuous families of functions, Ascoli-Arzela theorem. Weierstrass approximation theorem. Fourier series, Fejer"s theorem, pointwise convergence. *Text/References: * T. Apostol, Mathematical Analysis, 2nd ed., Addison-Wesley, 1974. Ganapati Iyer, Mathematical Analysis, Tata McGraw-Hill, 1977. W. Rudin, Principles of Mathematical Analysis, 3rd ed., McGraw-Hill, 1983. *Course Code: * MA 411 *Title: * Introduction to Probability *Credits: * 6.0 *Pre-requisite: * *Description: * Events, sigma-fields, specifying probabilities on sigma-fields, extension theorem(without proof), independence of classes of events, random variables. Distribution functions, discrete, continuous and mixed. Moments - moment generating functions, Characteristic functions. Random vectors and joint distributions, conditional distribution, conditional expectation. Convergence of random variables - the four modes of convergence and their relationships. Laws of large numbers and Central Limits Theorm. *Text/References: * P. Billingsley, Probability and Measure, John Wiley & Sons, NY, 1986. K.L. Chung, Elementary Probability Theory with Stochastics Processes, 3rd Ed., Narosa Pub. Co., 1979. J.Pitman, Probability, Narosa Pub. Co., 1993. M. Woodroofe, Probability with Application, McGraw-Hill, NY, 1975. *Course Code: * MA 417 *Title: * Ordinary Differential Equations *Credits: * 8.0 *Pre-requisite: * *Description: * Review of solution methods for first order as well as second order equations, Power series methods with properties of Bessel functions and Legendre polynomials. Existance and uniqueness of Initial Value problems; Picards and Peanos theorems, Gronwall"s inequality, continuation of solutions and maximal interval of existance, continuous dependence. Higher order linear equations and linear systems; fundamendal solutions, Wronskian, variation of constants, matrix exponential solution behaviour of solutions. Two dimensional Autonomous Systems and Phase Space Analysis: Critical points, proper and improper nodes, spiral points, and saddle points. Asymptotic Behavior: Stability ( linearized stability and Lyapunov methods.) Boundary Value Problems for second order equations: Green"s function, Sturm comparison theorms and oscillations, eigenvalue problems. *Text/References: * R.P. Agarwal and R. Gupta, Essentials of Ordinary Differential Equations, Tata McGraw-Hill Pub. Co., New Delhi, 1991. M. Braun, Differential Equations and their Applications, 4th Edition, Springer Verlag, Berlin, 1993. E.A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, Tata McGraw-Hill Publ. Co., New Delhi, 1990. L. Perko, Differential Equations and Dynamical Systems, 2nd Edition, Texts in Applied Mathematics, Vol.7, Springer Verlag, New York, 1998. M. Rama Mohana Rao, Ordinary Differential Equations: Theory and Applications. Affiliated East-West Press Pvt. Ltd., New Delhi, 1980. A. Sanchez, Ordinary Differential Equations and Stability Theory: An Introduction, Dover Pub. Co., New York, 1968. *Course Code: * SI 409 *Title: * Discrete Structures Lab *Credits: * 3.0 *Pre-requisite: * *Description: * Illustration of techniques and concepts from combinatorics and graph theory through programming exercises. Either an interpreted language such as Scheme or the language of Mathematica should be used for the programming exercises. *Text/References: * N. Biggs, Discrete Mathematics, Oxford/Clarendon Press, 1985. R. Dromey, How to Solve it By Computer, Prentice Hall India ,1996. Donald Knuth, Fundamental Algorithms, Narosa Publishers, 1985.