Course Bulletin - for Batch Year 2003 *Mathematics 2 Yr M.Sc * * Specialization : None * * Second Year * * Second Semester * * Core Course* MA 598 Project Stage II 15.0 SI 414 Optimization 8.0 * Departmental Elective* MA 502 Algebraic Number Theory 6.0 MA 506 Commutative Algebra 6.0 MA 510 Introduction to Algebric Geometry 6.0 MA 512 Enumerative Cimbinatiorics II 6.0 MA 514 Locally Convex Spaces and Distribution Theory 6.0 MA 518 Spectral Approximation 6.0 MA 521 Theory of Analytic Functions 6.0 MA 530 Nonlinear Analysis 6.0 MA 534 Modern Theory of PDE's 6.0 MA 556 Introduction to Differential Geometry 6.0 MA 568 Functional Analysis II 6.0 MA 582 Basic Algebraic Topology 6.0 * Departmental Elective* EE 678 Wavelets 6.0 EE 704 Artificial Neural Network 6.0 MA 530 Nonlinear Analysis 6.0 MA 534 Modern Theory of PDE's 6.0 MA 562 Mathematical Theory of Finite Elements 6.0 MA 568 Functional Analysis II 6.0 MA 582 Basic Algebraic Topology 6.0 MA 588 Computational Finance 6.0 MA 590 Fluid Dynamics 6.0 MA 592 Non-linear Wave Phenomena 6.0 SI 511 Computer Aided Geometric Design 6.0 SI 515 Applied Multivariate Analysis 6.0 SI 522 Large Scale Scientific Computation 6.0 SI 523 Mathematical Modelling and Numerical Simulation 6.0 SI 524 Data Mining 6.0 * Departmental Elective* MA 424 Theory of Sampling 8.0 MA 568 Functional Analysis II 6.0 MA 570 Design and Analysis of Experiments 6.0 MA 572 Non-paramatic Statistical Inference 6.0 MA 576 Statistical Decision Theory 6.0 MA 580 Time Series Analysis 6.0 MA 582 Basic Algebraic Topology 6.0 MA 594 Stochastic Calculus with Applications to Finance 6.0 SI 515 Applied Multivariate Analysis 6.0 SI 524 Data Mining 6.0 Course Contents Course Bulletin - for Batch Year 2003 *Mathematics 2 Yr M.Sc * * Specialization : None * * Second Year * * Second Semester * *Course Code: * EE 678 *Title: * Wavelets *Credits: * 6.0 *Pre-requisite: * *Description: * *Text/References: * *Course Code: * EE 704 *Title: * Artificial Neural Network *Credits: * 6.0 *Pre-requisite: * *Description: * Biological memory mechanisms. Neural basis for human memory. Neuron models. The classification problem. Linear Classifiers. Training learning and generalization. Perception convergence theorem. Ho-Kashyap algorithm. Multilayer feed forward networks. Number of hidden nodes and VC-dimension. Kolmogorov"s theorem on representation of functions of several variables. The back propagation algorithms. Other algorithms. Applications. Hopfield network. Generalized convergence theorem. Computational power and capacity. Applications. Cellular neural networks. Stability. Convergence and computational power. Applications. Kohonen"s algorithm for self organizing networks. Convergence proof. Applications. Grossberg"s algorithm. Adaptive resonance theory (ART) for binary and analog input patterns. Simulated Annealing and Boltzmann machines. Principles of statistical neuro dynamics. Deductive theory of learning. Valiant"s model. Learnability and VC-dimension. *Text/References: * Minsky M.L. and Papert S.: `Perceptrons", MIT Press, 1988. Gonzalez and Tou: `Pattern Recognition Principles`, Addison Wesley, 1974. Mc Clelland J.L. and Rumelhart O.E. ed.:`parallel distributed processing : Explorations in microstructure of cognition", MIT Press, 1986. Aarts E. and Korst J.:`Simulated Annealing and Boltzmann machines", John Wiley, 1989. Kohonen T.: `Self organization and Associative memory", Springer Verlag, 1984. *Course Code: * MA 424 *Title: * Theory of Sampling *Credits: * 8.0 *Pre-requisite: * NIL *Description: * Simple random sampling. Sampling for proportions and percentages. Estimation of sample size. Stratified random sampling, ratio estimators. Regression estimators. Systematic sampling. Type of sampling unit, Subsampling with units of equal and unequal size. Double sampling. Sources of errors in surveys. *Text/References: * W.G. Cochran, Sampling Techniques, 3rd ed., Wiley Eastern, 1977. Des Raj, Sampling Theory, Tata McGraw-Hill, 1978. A. Chaudhuri and H. Stenger, Survery Sampling: Theory and Methods, Marcell Dekker, 1992. *Course Code: * MA 502 *Title: * Algebraic Number Theory *Credits: * 6.0 *Pre-requisite: * MA 501 *Description: * Binary quadratic forms, Legendre-Gauss theory of genera. Algebraic numbers and their basic properties, Kummer"s work on Fermat"s last theorem. Unique factorization of ideals in algebraic number fields, Class group and class number, Ramification of primes. Discriminant, Norms of ideals, Reciprocity laws, Cyclotomic fields and Kronecker-Weber theorem (statement only). Introduction to class field theory. *Text/References: * J.W. Cassels, Local Fields, Cambridge Press, 1986. J.W. Cassels and A. Frohiich, Algebraic Number Theory, Academic Press, 1967. H.M. Edwards, Fermat"s Last Theorem, Springer-Verlag, 1977. K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, 2nd ed., Springer-Verlag, 1990. S. Lang, Algebraic Number Theory, Addison-Wesley, 1970. D.A. Marcus, Number Fields, Springer-Verlag, 1977. *Course Code: * MA 506 *Title: * Commutative Algebra *Credits: * 6.0 *Pre-requisite: * MA 501 *Description: * Rings and modules, localization, Noetherian rings, primary decomposition, Artinian rings, integral extensions, Hilbert"s Nullstellensatz, Noether"s normalization, valuation rings, Dedekind domains, Dimensions Theorem, Completions. *Text/References: * O. Zariski and P. Samuel, Commutative Algebra, Vols. I and II, Van Nostrand, 1958 and 1960. M.F. Atiyah and I.G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, 1969. N.S. Gopalakrishnan, Commutative Algebra, Oxonian Press, 1984. N. Jacobson, Basic Algebra, Vol. II, Hindustan Publishing Corporation, 1984. D. Eisenbud, Commutative Algebra : With a View Towards Algebraic Geometry, Springer-Verlag, 1995. *Course Code: * MA 510 *Title: * Introduction to Algebric Geometry *Credits: * 6.0 *Pre-requisite: * MA 501 *Description: * Affine and projective varieties, coordinate rings, Rational functions and local rings, singular points and tangent lines, Rational parametrization, Branches and valuations, Intersection multiplicity, Bezout"s theorem for plane curves, Max Noether"s theorem. Varieties, morphisms and rational maps. Resolution of singularities of curves. *Text/References: * S.S. Abhyankar, Algebraic Geometry for Scientists and Engineers, American Mathematical Society, 1990. W. Fulton, Algebraic Curves, Benjamin, 1969. M. Reid, Undergraduate Algebraic Geometry, Cambridge University Press, 1990. I.R. Shafarevich, Basic Algebraic Geometry, Springer-Verlag, 1974. R.J. Walker, Algebraic Curves, Springer-Verlag, 1950. J. Harris, Algebraic Geometry : A First Course, Springer-Verlag, 1992. *Course Code: * MA 512 *Title: * Enumerative Cimbinatiorics II *Credits: * 6.0 *Pre-requisite: * MA 511 *Description: * Partially ordered sets, Mobius inversion. Rational generating functions: P-partitions and linear Diophantine equations. Polya theory and representation theory of the symmetric group: Combinatorial algorithms, and symmetric functions. Generating functions : Single and multivariable Lagrange inversion. *Text/References: * R.P. Stanley, Enumerative Combinatorics, Vol. I, Wadsworth and Brooks/Cole, 1986. B.E. Sagan, The Symmetric Group: Representations,Combinatorial Algorithms and Symmetric Functions, Wadsworth & Brooks/Cole, 1991. M. Aigner, Combinatorial Theory, Springer-Verlag, 1979. *Course Code: * MA 514 *Title: * Locally Convex Spaces and Distribution Theory *Credits: * 6.0 *Pre-requisite: * MA 406 *Description: * Locally convex spaces and their metrizability. Frechet spaces. Weak topologies. Test function spaces. Calculus with distributions. Localization. Distributions as derivatives. Convolutions. Fourier transforms. Tempered distributions.Paley-Wiener theorems. Sobolev"s lemma. Fundamental solutions of partial differential equations. Elliptic equations. *Text/References: * W. Rudin, Functional Analysis, McGraw-Hill, 1973. K. Yoshida, Functional Analysis, Academic Press, 1965. L. Hormander, The Analysis of Linear PDE, Vols. I and II, Springer-Verlag, 1983. *Course Code: * MA 518 *Title: * Spectral Approximation *Credits: * 6.0 *Pre-requisite: * MA 517 *Description: * Resolvent sets and spectra of bounded and compact operators in Banach spaces. Spectral projection, reduced resolvent and the nilpotent operator. Neumann expansion and the analyticity of spectral projctions. Rayleigh-Schrodinger series and the iterative computation of eigenelements. Numerical approximation by methods related to projections and by quadrature methods. Algorithms for computing eigenelements and their computational feasibility. *Text/References: * T. Kato, Perturbation Theory of Linear Operators, 2nd ed.,Springer-Verlag, 1980. F. Chatelin, Spectral Approximation of Linear Operators, Academic Press, 1983. B.V. Limaye, Spectral Perturbation and Approximation with Numerical Experiments, Proc. Centre Math. Anal. Vol. 13, Australian National Univ., 1987. *Course Code: * MA 521 *Title: * Theory of Analytic Functions *Credits: * 6.0 *Pre-requisite: * MA 403 *Description: * Open mapping property of analytic functions,mean value property of harmonic functions, Poisson integral representation of harmonic functions, Schwarz lemma and Phragmen-Lindelof method. Approximation by rational functions. Riemann mapping theorem, simply and doubly connected domains. *Text/References: * W. Rudin, Real and Complex Analysis, Tata McGraw-Hill, 3rd ed., 1987. E. Hille, Analytic Function Theory, I and II, Blaisdell, 1959. *Course Code: * MA 530 *Title: * Nonlinear Analysis *Credits: * 6.0 *Pre-requisite: * *Description: * *Text/References: * *Course Code: * MA 534 *Title: * Modern Theory of PDE's *Credits: * 6.0 *Pre-requisite: * *Description: * *Text/References: * *Course Code: * MA 556 *Title: * Introduction to Differential Geometry *Credits: * 6.0 *Pre-requisite: * *Description: * *Text/References: * *Course Code: * MA 562 *Title: * Mathematical Theory of Finite Elements *Credits: * 6.0 *Pre-requisite: * *Description: * *Text/References: * *Course Code: * MA 568 *Title: * Functional Analysis II *Credits: * 6.0 *Pre-requisite: * *Description: * *Text/References: * *Course Code: * MA 570 *Title: * Design and Analysis of Experiments *Credits: * 6.0 *Pre-requisite: * MA 401 *Description: * Theory of linear estimation. Standard designs : CRD, RBD, LSD, BIBD and PBIBD. Factorial designs. Confounding. Missing plot technique. Analysis of covariance. Construction and nonexistence theory. Special designs : Split-plots, strip-plots, cross-over designs. *Text/References: * O. Kempthorne, Design and Analysis of Experiments, Wiley Eastern, 1967. M.C. Chakrabarty, Mathematics of Design and Analysis of Experiments, Asia Publishing House, 1962. M.N. Das and N.C. Giri, Design and Analysis of Experiments, Wiley Eastern, 1979. A. Dey, Theory of Block Designs, Wiley, 1986. *Course Code: * MA 572 *Title: * Non-paramatic Statistical Inference *Credits: * 6.0 *Pre-requisite: * MA 577 *Description: * The empirical distribution and its basic properties. Location and scale parameters. Estimation and Testing in one sample problem. Asymptotic Relative Efficiency. Testing, many sample problems (Tests for Independence, Equality of distribution function, etc.). *Text/References: * M. Hollandor, and D.A. Wolfe, Nonparametric Statistical Inference, McGraw-Hill, 1973. E.L. Lehmann, Nonparametric Statistical Methods Based on Ranks, McGraw-Hill, 1975. J.W. Pratt, and J.D. Gibbons, Concepts of Nonparametric Theory, Springer-Verlag, 1981. *Course Code: * MA 576 *Title: * Statistical Decision Theory *Credits: * 6.0 *Pre-requisite: * MA 577 *Description: * Decision functions, Risk functions, utility and subjective probability, Randomization, Optimal decision rules. Admissibility and completeness, Existence of Bayes Decision Rules, Existence of a Minimal complete class, Essential completeness of the class of nonrandomized rules. The minimax theorem. Invariant statistical decision problems. Multiple decision problems. Sequential decision problems. *Text/References: * J.O. Berger, Statistical Decision Theory : Foundations, Concepts and Methods, Springer-Verlag, 1980. T.S. Ferguson, Mathematical Statistics, Academic Press, 1967. *Course Code: * MA 580 *Title: * Time Series Analysis *Credits: * 6.0 *Pre-requisite: * MA 577 *Description: * Introduction to autocorrelation function, linear stationary models like autoregressive, integrated moving average processes. Forecasting model identification including initial estimates of the parameters, model multiplicity etc. Model estimation, model diagnostic checking. Case studies. Computational experiments. *Text/References: * C. Chatfield, The Analysis of Time Series: An Introduction, Chapman & Hall, 1984. G.E.P. Box and G.M. Jenkins, Time Series Analysis Forecasting and Control, Holden-Day, 1976. P.J. Brockwell and R.A. Davis, Time Series, Springer-Verlag, 1987. *Course Code: * MA 582 *Title: * Basic Algebraic Topology *Credits: * 6.0 *Pre-requisite: * *Description: * *Text/References: * *Course Code: * MA 588 *Title: * Computational Finance *Credits: * 6.0 *Pre-requisite: * *Description: * *Text/References: * *Course Code: * MA 590 *Title: * Fluid Dynamics *Credits: * 6.0 *Pre-requisite: * *Description: * *Text/References: * *Course Code: * MA 592 *Title: * Non-linear Wave Phenomena *Credits: * 6.0 *Pre-requisite: * *Description: * *Text/References: * *Course Code: * MA 594 *Title: * Stochastic Calculus with Applications to Finance *Credits: * 6.0 *Pre-requisite: * *Description: * *Text/References: * *Course Code: * MA 598 *Title: * Project Stage II *Credits: * 15.0 *Pre-requisite: * *Description: * *Text/References: * *Course Code: * SI 414 *Title: * Optimization *Credits: * 8.0 *Pre-requisite: * *Description: * Classical Optimization Theory: unconstrained optimization, calculus for necessary and sufficient conditions, Newton-Raphson method, unconstrained nonlinear algorithms, direct search, gradient methods. Constrained Optimization Theory: Jacobian and Lagrangian based approaches, Kuhn Tucker conditions, penalty function methods, separable programming, quadratic programming. Linear Programming: duality, simplex method revised simplex method, dual simplex method, sensitivity analysis, transportation problems. Heuristics for Combinational Optimization: Branch and Bound, hill climbing, simulated annealing, generic algoriths, primal-dual approach. *Text/References: * E. Arts and J.K. Lenstra, Local Search in Combinational Optimization, John Wiley and Sons, 1997. M. Bazarra and C. Shetty, Nonlinear Programming, Theory and Algorithms, Wiley, NY,1979. Edwin K.P. Chong and Stanislaw H. Zak, An Introduction to Optimization, John Wiley & Sons, 1996. *Course Code: * SI 511 *Title: * Computer Aided Geometric Design *Credits: * 6.0 *Pre-requisite: * *Description: * Polynomial curves : Bezier representation, Bernstein polynomials, Blossoming, de Castlijau algorithm. Derivatives in terms of Bezier polygon. Degree elevation. Subdivision. Nonparametric Bezier curves. Composite Bezier curves. Spline curves : Definition and Basic properties of spline functions, B-spline curves, de Boor algorithm. Derivatives. Insertion of new knots. Cubic spline interpolation. Interpretation of parametric continuity in terms of Bezier polygon. Geometric continuity. Frenet frame continuity. Cubic Beta splines and significance of the associated parameters. Tensor product surfaces. Bezier patches. Triangular patch surfaces. *Text/References: * G. Frain, Curves and Surfaces for Computer Aided Geometric Design : A Practical Guide, Academic Press, 1988. L. Ramshaw, Blossoming : A Connect-the-Dots Approach to Splines, DEC systems Research Center, Report no. 19, 1987. *Course Code: * SI 515 *Title: * Applied Multivariate Analysis *Credits: * 6.0 *Pre-requisite: * *Description: * Matrix algebra and random vectors. Sample geometry and random sampling. The multivariate normal distribution. Inferences about a mean vector. Large sample inference about population mean vectors, proportions. Comparison of several multivariate population means. Two-way multivariate analysis of variance, classical linear regression model, least square estimation and inferences about the regression model. Model checking and other aspects of regression. Multivariate multiple regression. Principal component techniques. Factor analysis. Separation and classification for two populations. Fisher"s method for discrimination among several populations. Hierarchical and nonhierarchical clustering methods. *Text/References: * R. Gnanadesikan, Methods for Statistical Data Analysis of Multivariate Observations, Wiley, 1977. D. F. Morrison, Multivariate Statistical Methods, 2nd ed., McGraw-Hill, 1976. N. H. Timm, Multivariate Analysis with Applications in Education and Psychology, Brooks/Cole, 1975. *Course Code: * SI 522 *Title: * Large Scale Scientific Computation *Credits: * 6.0 *Pre-requisite: * *Description: * *Text/References: * *Course Code: * SI 523 *Title: * Mathematical Modelling and Numerical Simulation *Credits: * 6.0 *Pre-requisite: * *Description: * Review of continuum model, Transport phenomena, Air quality modelling, (pollution from chimney), Furnace reaction analysis, De-icing helicopter blades (free and moving boundary problems), modelling microwave heating, Food contamination from the packaging, Electron Beam Lithography, Color negative film development, photocopy machine; Selected case studies. Software Support: MATHEMATICA, LSODE, GNUPLOT, MATLAB. *Text/References: * A. Friedman and W. Littman, Industrial Mathematics for Under- graduates. SIAM Publ. 1994.. J. Crank, Free and Moving Boundary Problems, Oxford Univ. Press, 1987. A. James (Ed.), An Introduction to Water Quality Modelling, Wiley Pub. 1984. M.S. Klamkin, (ed.), Mathematical Moddelling: Classroom Notes in Applied Mathematics, SIAM Publications. A. Friedman, Mathematics in Industrial Problems Part 1 ? 9, IMA Series, Springer-Verlag. Lecture Notes on Heat and Mass Transfer : A Problem Driven Approach, M.Sc. in Industrial Mathematics. Univ. Strathclyde, U.K., 1995. Y.C. Fung, A First Course in Continuum Mechanics, Prentice-Hall, 1969. *Course Code: * SI 524 *Title: * Data Mining *Credits: * 6.0 *Pre-requisite: * *Description: * *Text/References: *