|
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. |
|
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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:
|
|
|