Departmental Courses

The Department conducts several courses at the B.Tech. and M.Tech. levels, and it runs two M.Sc. Programmes i) Mathematics and (ii) Applied Statistics and Informatics as well as the Ph.D. programme. The faculty members of the Department are actively involved in various research areas such as

1. Algebra and Combinatorics: Algebraic Geometry, Commutative Algebra, Enumerative Combinatorics.
2. Analysis: Approximation and Optimiza-tion, Functional Analysis, Harmonic Analysis, Partial Differential Equations, Systems and Control.

iii) Topology and Geometry: Algebraic and

Differential Topology, Differential

Geometry.

Collaborative research with other Science and Engineering Departments of the Institute is encouraged. Faculty members undertake projects sponsored by organizations such as National Board for Higher Mathematics, Indian National Science A cademy, Board of Research in Nuclear Sciences, Council of Scientific and Industrial Research, Department of Science and Technology, Department of Bio-Technology, Indian Council of Medical Research. The Department is keenly interested in Industrial interac tion.

In the Departmental Library, text books and reference works of current interest are accessible. Copies of M.Sc. Project Reports and Ph.D. Theses completed in the recent past are also available.

The Department has three Computer Laboratories with approximately 60 workstations. A Solaris/Linux/Novel Netware based LAN connected to the Intac. Several sophisticated tools for program development, graphics and geometric design are available. The students and faculty can use sophisticated mathematical and statistical software such as MATHEMATICA, SAS, MATLAB, LINDO, MACSYMA, MACAULAY etc. Microsoft platform software suites are available too.

The Mathematics Association of I.I.T. Bombay conducts academic and cultural activities which include the Maths Olympiad and it is being exclusively managed by the students of the Department for the last twenty years. In addition, there is "Popular Lecture Series in Mathematics" in which six lectures are given per year on topics accessible to a general audience.

THE FACULTY

D.V. Pai, Ph.D. (IIT Bombay)

B.V. Limaye, Ph.D. (Rochester)

P. Chaturani, Ph.D. (IIT Bombay)

K.S. Parihar, Ph.D. (IIT Kanpur)

M.C. Joshi, Ph.D. (Purdue)

[Head of the Dept.]

A.R. Shastri, Ph.D. (Bombay)

V.D. Sharma, Ph.D. (BHU)

Jagdish Prakash, Ph.D. (IIT Kanpur)

K.D. Joshi, Ph.D. (Indiana)

Akhil Ranjan, Ph.D. (Bombay)

A.K. Pani, Ph.D. (IIT Kanpur)

J.K. Verma, Ph.D. (Purdue)

R.R. Joshi, Ph.D.(IIT Bombay, UTC)

I.K. Rana, Ph.D. (ISI)

Prem Narain, Ph.D. (SUNY-Stony Brook)

R.P. Kulkarni, Ph.D. (IIT Bombay, Grenoble)

S.R. Ghorpade, Ph.D. (Purdue)

1. Subramanyam, Ph.D. (Poona)

M.K. Srinivasan, Ph.D. (Illinois)

P. Vellaisamy, Ph.D. (IIT Kanpur)

Sachin B. Patkar, Ph.D. (IIT Bombay)

S.V. Sabnis, Ph.D. (Old Dominion)

G.K. Srinivasan, Ph.D. (Minnesota)

Balwant Singh, Ph.D. (Bombay)

Code Name L T P C

CS 101 Computer Programming

and Utilization 2 1 0 6

MA 401 Linear Algebra 3 1 0 8

MA 403 Real Analysis I 3 1 0 8

MA 417 Ordinary Differential Equations 3 1 0 8

MA 411 Introduction to Probability 2 1 0 6

SI 409 Discrete Structures Lab 0 0 3 3

----------------

1. 5 3 39

Contact Hours : 21

Credits : 39

L=Lecture, T = Tutorial, P=Practical,

C=Credits

Code Name L T P C

MA 402 Algebra I 3 1 0 8

MA 444 Numerical Analysis 3 1 0 8

MA 404 Real Analysis II 3 1 0 8

MA 436 Partial Differential Equations 2 1 0 6

MA 438 Introduction to Mathematical Statistics 2 1 0 6

MA 446 Complex Analysis 2 1 0 6

-------------- 15 6 0 42 ---------------

Contact Hours : 21

Credits : 42

Code Name L T P C

MA 406 General Topology 3 1 0 8

MA 559 Functional Analysis I 3 1 0 8

MA 415 Mathematical Methods 3 1 0 8

Elective I 2 1 0 6

Elective II 2 1 0 6

MA 597* Project Stage I 5

--------------

13 5 0 41

---------------

Contact Hours : 18 Credits : 41

MA 561 Abstract Measure and Integration

MA 520 Spline Theory and Variational Methods

MA 573 Mathematical Theory of Reliability

Code Name L T P C

SI 414 Optimization 3 1 0 8

Elective III 2 1 0 6

Elective IV 2 1 0 6

Elective V 2 1 0 6

MA 598 Project Stage II 15

09 4 0 41

---------------

Contact Hours : 13 Credits : 41

SI 511 Computer Aided Geometric Designs

MA 424 Theory of Sampling

1^st Year

Code Name L T P C

CS 101 Computer Programming And Utilization 2 1 0 6

MA 401 Linear Algebra 3 1 0 8

SI 403 Mathematical Modelling 3 0 2 8

SI 405 Mathematical Systems Theory 3 0 0 6

MA 411 Introduction to

Probability 2 1 0 6

SI 409 Discrete Structures Lab. 0 0 3 3

SI 411 Computer Lab. 0 0 3 3

------------------

13 3 8 40

------------------

Contact Hours : 24

Credits : 40

L=Lecture, T=Tutorial, P= Practical,

C=Credits.

Code Name L T P C

CS 296 Software Systems Lab 0 1 3 5

MA 438 Introduction to Mathematical Statistics 2 1 0 6

MA 444 Numerical Analysis 3 0 2 8

SI 414 Optimization 3 1 0 8

SI 406 Applied Stochastic Processes 3 1 0 8

SI 408 Data Structures 3 0 2 8

-----------------

14 4 7 43

----------------

Contact Hours : 26

Credits : 45

Code Name L T P C

CS 317 Database and Information Systems 2 1 0 6

CS 387 Database and Information Systems Lab. 0 0 3 3

SI 531 Discrete Algorithms 3 0 2 8

SI 535 Categorial Data

Analysis & Regression 3 0 2 8

SI 533 Finite Difference

Methods for PDE's 3 0 2 8 Programming Language Lab. 0 0 3 3

SI 597 Project Stage I 5

-----------------

11 1 12 41

-----------------

Contact Hours : 24

Credits : 41

Code Name L T P C

SI 520 Informatics Lab. 0 0 3 3

Elective I 3 0 0 6

Elective II 3 0 0 6

Elective III 3 0 0 6

Elective IV 3 0 0 6

SI 596 Work Visit PN/NP 0

SI 598 Project Stage II 15

-----------------

1. 0 3 42

Contact Hours : 15 Credits : 42

PN/NP = Pass or Fail

Code Name L T P C

MA 825 Algebra 3 0 0 6

MA 827 Analysis 3 0 0 6

MA 829 Mathematical Methods 3 0 0 6

MA 831 Fluid Mechanics 3 0 0 6

MA 833 Weak Convergence and Martingale Theory 3 0 0 6

MA 835 Theory of Estimation 3 0 0 6

MA 837 Special Topics in Mathematics I 3 0 0 6

Seminar 0 0 0 4

1. The credit requirements for students having M.Sc. or equivalent qualification admitted to the Department shall be 34 to 46 credits.
2. Credits acquired through PG level courses shall be 24 or more.
3. Students may earn upto a maximum of 8

credits through Seminars which should

be spread over two semesters.

Code Name L T P C

MA 826 Topology 3 0 0 6

MA 828 Fucntional Analysis 3 0 0 6

MA 830 Numerical Analysis 3 0 0 6

MA 832 Elasticity 3 0 0 6

MA 834 Theory of Testing of Hypotheses 3 0 0 6

MA 836 Asymptotic Theory of Statistical Inference 3 0 0 6

MA 838 Special Topics in Mathematics II 3 0 0 6

Seminar 0 0 0 4

Note : Every student has to credit at least 3 of the 12 courses MA 825 - MA 836.

Complex numbers as ordered pairs. Argand's diagram. Triangle inequality. De Moivre's Theorem.

Algebra: Quadratic equations and express-ions. Permutations and combinations. Bino-mial theorem for a positive integral index.

Coordinate Geometry: Locus. Straight lines. Equations of circle, parabola, ellipse and hyperbola in standard forms. Parametric representation.

Vectors: Addition of vectors. Multiplication by a scalar. Scalar product, cross product and scalar triple product with geometrical appli-cations.

Matrices and Determinants: Algebra of matrices. Determinants and their properties. Inverse of a matrix. Cramer's rule.

Function. Inverse function. Elementary functions and their graphs. Limit. Continuity. Derivative and its geometrical significance. Differentiability. Derivatives of sum, difference, product and quotient of functions. Derivatives of polynomial, rational, trigono-metric, logarithmic, exponential, hyper- bolic, inverse trigonometric and inverse hyperbolic functions. Differentiation of composite and implicit functions.

Tangents and Normals. Increasing and decreasing functions. Maxima and Minima.

Integration as the inverse process of differentiation. Integration by parts and by substitution. Definite integral and its appli-cation to the determination of areas (simple cases).

Evaluation of limits of functions. Continuous functions. Differentiation of sum, difference, product and quotient of functions. Chain rule. Maxima and Minima. Partial differentiation. Integration by substitution and by parts, ap plication to evaluation of areas and volumes. Differential equations of first order. Linear differential equations with constant co-efficients.

Texts / References

N.S. Piskunov, Differential and Integral Calculus, Vol.I and II, Mir Publishers,

Moscow, 1979.

G.B. Thomas and R.L. Finney, Calculus and Analytic Geometry, 9^th ed., ISE

reprint, Addison-Wesley, 1998.

Review of the prerequisites such as limits of sequences and functions, continuity, uniform continuity and differentiability. Rolle's theorem, mean value theorems and Taylor's theorem. Newton's method for approximate solution. Riemann in tegral and the funda-mental theorem of integral calculus. Approximate integration. Applications to length, area, volume, surface area of revolution. Moments, centres of mass and gravity.

Review of vectors. Cylinders and quadric surfaces. Vector functions of one variable and their derivatives.

Partial derivatives. Chain rule. Gradient, directional derivative.

Tangent planes and normals. Maxima, minima, saddle points. Lagrange multipliers. Exact differentials.

Repeated and multiple integrals with applications to volume, surface area, moments of inertia etc.

Texts / References

T.M. Apostol, Calculus, Vol. I, 2nd ed.,

Wiley Eastern, 1980.

G.B. Thomas and R.L. Finney, Calculus and

Analytic Geometry, 9^th ed., ISE

reprint, Addison-Wesley, 1998.

Vector fields, surface integrals, line integrals, independence of path, conservative fields, divergence, curl. Green's theorem. Divergence theorem of Gauss, Stokes' theorem and applications of these theorems.

Transformations of coordinate systems and vector components. Invariance of divergence and curl. Curvilinear coordinates.

Vector spaces. Inner products. Matrices and determinants, linear transformations. Systems of linear equations. Gauss elimination, rank of a matrix. Inverse of a matrix. Bilinear and quadratic forms. Eigenvalues and eigenvectors. Similarity transformations. Diagonalization of Hermitian matrices.

Numerical methods for solving systems of linear equations. Ill-conditioning. Methods of Gauss and least squares. Inclusion of matrix eigenvalues. Finding eigenvalues by iteration.

Texts / References

T.M. Apostol, Calculus, Vol. II, 2nd ed.,

Wiley Eastern, 1980.

E. Kreyszig, Advanced Engineering

Mathematics, 9th ed., John Wiley & Sons

1999.

V. Krishnamurthy, V.P. Mainra and

J.L. Arora, An Introduction to

Linear Algebra, Affiliated East-West,

1976.

Ordinary differential equations of the 1st order, exactness and integrating factors, variation of parameters, Picard's iteration method.

Ordinary linear differential equations of n^th order, solution of homogeneous and nonho-mogeneous equations. Operator method. Methods of undetermined coefficients and variation of parameters.

Systems of differential equtions. Phase plane. Critical points. Stability.

Infinite sequences and series of real and complex numbers. Improper integrals. Cauchy criterion, tests of convergence, absolute and conditional convergence. Series of functions. Improper integrals depending on a parameter. Uniform conve rgence. Power series, radius of convergence. Power series methods for solutions of ordinary differential equations. Legendre equation and Legendre polynomials, Bessel equations and Bessel functions of first and second kind. Orthogonal sets of functions. Sturm-Liouville problems. Orthogonality of Bessel functions and Legendre polynomials.

Laplace transform. Inverse transform. Shifting on the s and t axes, convolutions, partial fractions.

Fourier series, half-range expansions. Approximation by trigonometric poly-nomials. Fourier integrals.

Transform techniques in differential equations.

Texts / References

W.E. Boyce and R.C. DiPrima, Elementary

Differential Equations and Boundary

Value Problems, 3rd ed., Wiley, 1977.

E. Kreyszig, Advanced Engineering

Mathematics, 9th ed., John Wiley & Sons

1999.

G.F. Simmons, Differential Equations with

Applications and Historical Notes,

McGraw-Hill, New York, 1991.

Analytic functions. Cauchy-Riemann equations, Laplace equation. Elementary functions. Cauchy's integal theorem (proof by using Green's theorem), Cauchy's integral formula. Taylor series and Laurent series.

Residues and applications to evaluating real improper integrals and inverse Laplace transforms. Conformal mapping. Linear fractional transformations.

Boundary value problems involving partial differential equations such as the wave equation, the heat equation, the Laplace equation. Solutions by the method of separation of variables and by Fourier and Laplace transforms.

Texts / References

P.E. Danko, A.G. Popov,

T.YA. Koznevnikova, Higher

Mathematics in Problems and

Exercises, Part 2, Mir Publishers, 1983.

E. Kreyszig, Advanced Engineering

Mathematics, 9th ed., John Wiley & Sons

1999.

Interpolation by polynomials. Divided differences. Error of the interpolating poly-nomial. Piecewise linear and cubic spline interpolation.

Numerical integration. Composite rules. Error formulae.

Solution of a system of linear equations. Implementation of Gaussian elimination and Gauss-Seidel methods. Solution of a nonlinear equation. Bisection and secant methods.

Newton's method. Rate of convergence. Solution of a system of nonlinear equations. Numerical solution of ordinary differential equations. Euler and Runge-Kutta methods. Multi-step methods. Predictor-corrector methods. Order of convergen ce. Finite difference methods. Numerical solutions of elliptic, parabolic and hyperbolic partial differential equations.

Exposure to software packages like IMSL Subroutines, MATLAB.

Texts / References

S.D. Conte and Carl de Boor, Elementary

Numerical Analysis – An Algorithmic

Approach, 3rd ed., McGraw-Hill, 1980.

C.E. Froberg, Introduction to Numerical

Analysis, 2nd ed., Addison- Wesley,

1981.

E. Kreyszig, Advanced Engineering

Mathematics, 9th ed., John Wiley & Sons

1999.

Basic definition of probability, random variables, probability density function, probability distribution function, standard univariate and multivariate distributions, conditional distributions and densities, moment generating fu nctions, characteristic functions, limit theorems.

Point estimation, interval estimation. Hypo-thesis testing. Simple linear regression, correlation.

Random Processes : Markov processes, stationary processes. Ergodicity, auto-correlation, cross-correlation, power spectral

density.

Exposure to statistical packages like SAS and SPSS.

Texts / References

T.K. Chandra and D. Chatterjee, A First

Course in Probablity, Narosa Publishing

House, New Delhi, 2001.

D.M. Himmelblau, Process Analysis by

Statistical Methods, Wiley, 1970.

P.G. Hoel, S.C. Port and C.J. Stone,

Introduction to Probability Theory,

Universal Book Stall, New Delhi, 1998.

P.G. Hoel, S.C. Port and C.J. Stone,

Introduction to Stochastic Processes,

Universal Book Stall, New Delhi, 1994.

P.G. Hoel, S.C. Port and C.J. Stone,

Introduction to Statistical Theory,

Universal Book Stall, New Delhi, 1996.

H.J. Larson, Introduction to Probability

Theory and Statistical Inference, 3rd

ed., Wiley, 1982.

M. O'Flynn, Probabilities, Random Variables

and Random Processes, Harper and

Row, 1982.

1. Papoulis, Probability, Random variables

and Stochastic Processes, McGraw-

Hill, 1991.

R.D. Yates and D.J. Goodman, Probability and

Stochastic Processes: A Friendly Intro-

duction for Electrical and Computer

Engineers, John Wiley, New York, 1999.

Systems of linear equations: matrices and elementary row operations, Gaussian elimi-nation, LU decomposition.

Vector Spaces: subspaces, bases and dimen-sion, coordinates.

Linear Transformation: representation of linear transformations by matrices, rank-nullity theorem, 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 ortho-normalization, reflectors, QR decomposition using reflectors, least squares problem, adjoint of an operator, unitary operators, rigid motions, positive (semi) definite matrices, minimum pr inciples and Rayleigh quotients, matrix norms, condition numbers.

Eigenvalue computation: power and inverse power methods, QR method.

Texts / 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.

Groups and Symmetry: review of groups, subgroups, homomorphisms, cosets, normal subgroups, products and quotients of groups. Permutation groups, simplicity of some alternating groups and PSL(2). Introduction to matrix groups such as GL( n), SL(n), SO(n). Longitudes and latitudes of the 3-sphere and the conjugacy classes in SU(2). The group of motions of the plane, finite groups of motions, group actions and examples, orbits and stabilizers, class equation, groups of symmetries of Platoni c solids, finite subgroups of the rotation group of space. The class equation of the Icosahedral group, Sylow theorems and their application in identifying groups of order 2p and pq. Structure theorem for finitely generated abelian groups.

Rings and Factorization: Rings, homo-morphisms and ideals, quotient rings, integral domains and quotient fields, polynomials in one and several variables, roots and multiplicities, elementary symmetric functions and the fundament al theorem on symmetric functions, Resultants and discriminants, Hilbert's Nullstellensatz in the complex case. Factorization of integers and polynomials, Eisenstein's criterion, Gauss Lemma, unique factorization domains, principal ideal domains, euclidea n domains, algebraic integers in quadratic fields, Gaussian integers and Fermat's two square theorem , examples of unique factorization domains among rings of integers of imaginary quadratic fields. Application to solving certain Diophantine equations.

Fields and ruler-compass constructions: Examples of fields, algebraic and tran-scendental extensions, degree of a field extension, constructions with ruler and compass.

Texts / References

M. Artin, Algebra, Prentice Hall of India,

1994.

J. A. Gallian, Contemporary Abstract Algebra,

IV Edition Narosa Publishing House,

1999.

I. N. Herstein,Topics in Algebra, John Wiley

and Sons, 1999.

N. Jacobson, Basic Algebra, Vol I, Hindustan

Publishing Corporation, 1984.

K. D. Joshi, Foundations of Discrete

Mathematics, New Age International,

New Delhi, 2000.

S. Lang, Undergraduate Algebra, Second

Edition, Springer Verlag, 1990.

I. S. Luther and I. B. S. Passi, Algebra Vol. I

and II, Narosa Publishing House, 1996.

MA 403 Real Analysis I 3 1 0 8

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. Equiconti-nuous families of functions, Ascoli-Arzela theorem. Weierstrass approximation theorem. Fourier ser ies, Fejer's theorem, pointwise convergence.

Texts / 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.

Prerequisite: MA 403 (Exposure)

The course will consist of two parts each spanning roughly half a semester.

Part 1: Lebesgue Measure and Integration

Length function and its properties, Lebesgue outer measure, measurable sets, Lebesgue measure and its properties.

Measurable functions, Lebesgue integral, monotone convergence theorem, Fatou's Lemma, dominated convergence theorem.

Part 2: Functions of Several Variables.

Differentiation, the inverse function theorem and the implicit function theorem, Riemann Integration and change of variables.

Vector fields, Green's theorem, Divergence theorem and Stokes' theorem in R³.

Texts / References

T. Apostol, Mathematical Analysis, 1st & 2nd

ed., Addison-Wesley/Narosa, 1963/1974.

W. Fleming, Functions of Several Variables,

Springer-Verlag, 1977.

I.K. Rana, An Introduction to Measure and

Integration, Narosa Publishing House,

1997.

H.L. Royden, Real Analysis, 2nd Ed.,

Macmillan, 1971.

W. Rudin, Principles of Mathematical

Analysis, 3rd ed., McGraw-Hill, 1983.

Prerequisite : MA 403

Separation axioms, Urysohn’s lemma, Tietze extension theorem.

Products and quotient spaces, embedding into products, Urysohn metrisation.

Nets and filters. Compactness in various forms, Tychnoff compactness theorem, local compactness, one-point compactification.

Function spaces, compact-open topology. Applications to quotient maps.

Connectedness, local connectedness, paths, homotopy, fundamental group, computation of the fundamental group of the circle.

Texts / References

J. Dugundji, Topology, Prenctice-Hall, New

Delhi 1975.

K.D. Joshi, Introduction to General Topology,

New Age International, New Delhi, 2000.

J.L. Kelley, General Topology, Van

Nostrand, Princeton, 1955.

J.R. Munkres, Topology - A First Course,

Prentice-Hall, New Delhi, 1983.

Events, s -fields, specifying probabilities on s -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, con-ditional distribution, conditional cxpectation.

Convergence of random variables - the four modes of convergence and their relationships.

Laws of large numbers and Central Limit Theorem.

Texts / References

P. Billingsley, Probability and Measure, John

Wiley and Sons, New York, 1986.

K.L. Chung, Elementary Probability Theory

with Stochastics Processes, 3rd Edn.,

Narosa Publishing Co., 1979.

J. Pitman, Probability, Narosa Publishing Co.,

1993.

M. Woodroofe, Probability with Application,

McGraw-Hill, New York, 1975.

Prerequisite : MA 436 (Exposure)

Asymptotic Methods: asymptotic expansions, methods of strained co-ordinates and matched asymptotic expansions.

Fourier Methods with Applications: Gene-ralised functions, eigenfunction expansions and Green's functions, Fourier transform, Convolution, Parseval's relation, fundamental solutions, and applications to heat, Laplace and wave equ ations.

Variational Methods: minimum of quadratic functional, Lax-Milgram theorem and applications to boundary value problems.

Linear Integral Equations: Fredholm and Volterra Integral Equations, Hilbert-Schmidt theory, iterative methods and Neumann series.

Texts / References

R. K. Bose and M. C. Joshi, Methods of

Mathematical Physics, Tata-McGraw-Hill

Publ. Co. Ltd., New Delhi, 1984.

R. Courant and D. Hilbert, Methods of

Mathematical Physics, Vol.1, Wiley

Eastern Pvt. Ltd., New Delhi, 1975.

R. Dautray and J. L. Lions, Mathematical

Analysis and Numerical Methods for

Science and Technology, Vol. 5, Springer

Verlag, Berlin, 1992.

J. Kovorkian and J. D. Cole, Perturbation

Methods in Applied Mathematics,

Springer Verlag, Berlin, 1985

S. G. Mikhlin, Variational Methods in

Mathmatical Physics, Pergamon Press,

Oxford, 1964.

A. Nayfeh, Perturbation Methods, Wiley

Publ., New York, 1973.

J.N. Reddy, Applied Functional Analysis and

Variational Methods, McGraw-Hill Book

Co., New York, 1987

E. Zauderer, Partial Differential Equations in

Applied Mathematics, 2nd edition, John

Wiley and Sons, New York, 1989.

Review of solution methods for first order as well as second order equations, Power Series methods with properties of Bessel functions and Legendre polynomials.

Existence and Uniqueness of Initial Value Problems: Picards and Peanos Theorems, Gronwall’s inequality, continuation of solutions and maximal interval of existence, continuous dependence.

Higher Order Linear Equations and linear Systems: fundamental 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 compari-sion theorems and oscillations, eigenvalue problems.

Texts / References

R. P. Agarwal and R. Gupta, Essentials of

Ordinary Differential Equations, Tata

McGraw-Hill Publ. 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.

D. A. Sanchez, Ordinary Differential

Equations and Stability Theory: An

Introduction, Dover Publ. Inc., New

York, 1968.

Simple random sampling. Sampling for proportions and percentages.

Estimation of sample size. Stratified random sampling, Ratio estimators. Regression esti-mators. Systematic sampling. Type of sampling unit, Subsampling with units of equal and unequal size. Double sampling. Sources of errors in surveys .

Texts / References

1. Chaudhuri and H. Stenger, Survery

Sampling: Theory and Methods, Marcell

Dekker, 1992.

W.G. Cochran, Sampling Techniques, 3rd ed.,

Wiley Eastern, 1977.

P. Mukhopadhyay, Theory and Methods

of Survey Sampling, Prentice-Hall

of India, New Delhi, 1998.

Des Raj, Sampling Theory, Tata McGraw-

Hill, 1978.

Prerequisite : MA 417 (Exposure)

Cauchy Problems for First Order Hyperbolic Equations: method of characteristics, Monge cone.

Classification of Second Order Partial Differential Equations: normal forms and characteristics.

Initial and Boundary Value Problems: Lagrange-Green's identity and uniqueness by energy methods.

Stability theory, energy conservation and dispersion.

Laplace equation: mean value property, maximum principle, Poisson's formula, Dirichlet's principle, existence of solution using Perron's method (without proof).

Heat equation: initial value problem, maximum principle and uniqueness results.

Wave equation: uniqueness, D'Alembert's method, method of spherical means and Duhamel's principle.

Methods of separation of variables for heat, Lapalce and wave equations.

Texts / References

E. DiBenedetto, Partial Differential

Equations,2nd printing, Birkhauser,

Boston, 1995.

Fritz John, Partial Differential Equations, 3rd.

edition, Narosa Publ. Co., New Delhi,

1979.

P. Prasad and R. Ravindran, Partial

Differential Equations, Wiley Eastern

Ltd., New Delhi, 1985.

E. Zauderer, Partial Differential Equations of

Applied Mathematics, 2nd edition, John

Wiley and Sons, New York, 1989.

Distribution of functions of random variables, order Statistics. Estimation - loss function, risk, minimum risk unbiased estimators, maximum likelihood estimation, method of moments, Bayes estimation. Sufficient Statistics, completeness , Basu's Theorem, exponential families, invariance and maximal invariant statistics.

Testing of Hypotheses - parametric and non-parametric problems, examples with data analytic applications.

Confidence Intervals.

G. Casella and B.L. Berger, Statistical

Inference, Pacific Grove, Wadsworth and

Brooks, 1990.

M.H. DeGroot, Probability and Statistics,

Addison-Wesley, 1986

E.L Lehmann and G. Casella, Theory of Point

Estimation, New York, Springer-Verlag,

1998.

Principles of floating point computations and rounding errors.

Systems of Linear Equations: factorization methods, pivoting and scaling, residual error correction method.

Iterative methods: Jacobi, Gauss-Seidel methods with convergence analysis, conjugate gradient methods.

Eigenvalue problems: only implementation issues.

Nonlinear systems: Newton and Newton like methods and unconstrained optimization.

Interpolation: review of Lagrange inter-polation techniques, piecewise linear and cubic splines, error estimates.

Approximation : uniform approximation by polynomials, data fitting and least squares approximation.

Numerical Integration: integration by interpolation, adaptive quadratures and Gauss methods

Initial Value Problems for Ordinary Differential Equations: Runge-Kutta methods, multi-step methods, predictor and corrector scheme, stability and convergence analysis.

Two Point Boundary Value Problems : finite difference methods with convergence results.

Lab. Component: Implementation of algorithms and exposure to public domain packages like LINPACK and ODEPACK.

K.E. Atkinson, An Introduction to Numerical

Analysis, Wiley, 1989.

S.D. Conte and C. De Boor, Elementary

Numerical Analysis – An Algorithmic

Approach, McGraw-Hill, 1981.

K. Eriksson, D. Estep, P. Hansbo and C.

Johnson, Computational Differential

Equations, Cambridge Univ. Press, 1996.

G.H. Golub and J.M. Ortega, Scientific

Computing and Differential Equations:

An Introduction to Numerical Methods,

Academic Press, 1992.

J. Stoer and R. Bulirsch, Introduction to

Numerical Analysis, 2nd ed., Texts in

Applied Mathematics, Vol. 12, Springer

Verlag, New York, 1993.

Complex numbers and the point at infinity. Analytic functions.

Cauchy-Riemann conditions, harmonicity, Mappings by elementary functions. Riemann surfaces. Conformal mappings.

Contour integrals, Cauchy-Goursat theorem, simply and multiply connected domains.

Uniform convegence of sequences and series. Taylor and Laurent series. Isolated singul-arities and residues. Evaluation of real integrals. Calculation of inverse Laplace transforms.

Zeroes and poles, Argument principle, Rouche’s theorem.

Winding numbers.

Texts / References

R.V. Churchill and J.W. Brown, Complex

Variables and Applications, International

Student Edition, McGraw-Hill, 4th ed.,

1984.

P. Henrici, Applied and Computational

Complex Analysis, Vol.1, Wiley, 1974.

B.R. Palka, An Introduction to Complex

Function Theory, UTM Springer-Verlag,

1991.

R. Remmert, Theory of Complex Functions,

Springer-Verlag, New York, 1991.

A.R. Shastri, An Introduction to Complex

Analysis, Macmillan India Ltd., New

Delhi, 1999.

Prerequisite : MA 402

Field Theory: Cardano's and Ferrari's method for solving cubic and quartic equations, splitting field of a polynomial, existence and uniqueness of finite fields of a given order, group of automorphisms of a finite field, proof of the fu ndamental theorem of algebra using symmetric functions, algebraic closure of a field and its uniqueness, connjugates in a field extension, separable and normal extensions, perfect fields, primitive element theorem, norm, trace and discriminant, transcende ntal extensions.

Galois Theory: Fundamental theorem of Galois theory, Galois group as a permutation group, transitivity of the Galois group, transitive subgroups of the symmetric groups of degree at most four, Galois groups of equations of degre e at most four, solvability of equations by radicals, cyclotomic extensions, constructible regular polygons, Abel-Ruffini theorem.

Modules over a PID: Modules over a commutative ring, direct sums, free modules, rank of a free module, torsion modules, structure theorem for finitely generated modules over a PID, applications to finitely generated abelian grou ps and linear transformations.

Texts / References

M. Artin, Algebra, Prentice-Hall, 1994.

I. N. Herstein, Topics in Algebra, Wiley

Eastern, 1987.

N. Jacobson, Basic Algebra, Vol. I, Hindustan

Pub. Co., 1984.

S. Lang, Algebra, Third edition, Addison-

Wesley, 1993.

Prerequisites : MA 501, MA 509

Algebraic numbers and their basic properties, Algebraic number fields and rings of integers, Discriminant of a number field, Unique factorization of ideals in algebraic number fields, Class group and class number, Ramification of primes, Kummer's Theorem, Dedekind's Discriminant Theorem, Cyclo-tomic fields and Kronecker-Weber theorem (statement only). Introduction to class field theory.

Texts / References

J.W. Cassels, Local Fields, Cambridge

University Press, 1986.

H.M. Edwards, Fermat's Last Theorem,

Springer-Verlag, 1977.

1. Frohlich and M. J. Taylor, Algebraic

Number Theory, Cambridge Studies in

Advanced Mathematics, 27, Cambridge

University Press, 1993.

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.

MA 506 Commutative Algebra 2 1 0 6Prerequisite : MA 501

Rings and modules, localisation of rings and modules, Noetherian rings and modules, primary decomposition, Artinian rings, inte-gral extensions, going up, lying over and going down theorem, Hilbert's Nullstellensatz, Noether's normalisa tion, dimension theorem, Krull's principal ideal theorem, Dedekind domains.

Texts / References

M.F. Atiyah and I.G. Macdonald, Introduction

to Commutative Algebra, Addison-

Wesley, 1969.D. Eisenbud, Commutative Algebra with a

View Toward Algebraic Geometry,

Springer-Verlag, 1995.N.S. Gopalakrishnan, Commutative Algebra,

Oxonian Press, 1984.

S. Raghavan, B. Singh and R. Sridharan,

Homological Methods in Commutative

Algebra, TIFR Mathematical Pamphlet

Number 5, Oxford University Press,

1977.M. Reid, Undergraduate Commutative

Algebra. London Mathematical Society

Student Texts, 29. Cambridge University

Press, 1995.

J.-P. Serre, Local Algebra (Translated from

French), Springer Monographs in

Mathematics, Springer-Verlag, 2000. R. Y. Sharp, Steps in Commutative Algebra,

2nd edition, Cambridge University Press,

2001. O. Zariski and P. Samuel, Commutative

Algebra, Vols. I & II, Van Nostrand, 1958

and 1960.

Prerequisite : MA 404 (Exposure)

Convex sets, convex cones and convex functions. Continuity and growth properties of convex functions. Conjugate convex functions in paired spaces. Subgradients, normal cones and subdifferential formulae.

Convex optimization. Constraint qualifications and Kuhn-Tucker theory. Lagrangian duality. Applications to nonlinear programming.

Chebyshev approximation. Calculus of variation and partial differential equations. Dubovitski-Milyutin theory.

Texts / References

M.S. Bazaraa and C.M. Shetty, Nonlinear

Programming, Theory and Algorithms,

Wiley, New York, 1979.

F.H. Clarke, Optimization and Nonsmooth

Analysis, Wiley Interscience, 1983.

J.-B. Hiriart-Urruty, Optimisation et analyse

Convexe (in French), Presses

Universitaires de France, Paris, 1998.

P.J. Laurent, Approximation et Optimization,

Hermann, 1973.

R.T. Rockafellar, Convex Analysis, Princeton

University Press, 1970.

R.T. Rockafellar, Conjugate Duality and

Optimization, CBMS Lecture Notes,

Series No. 13 SIAM, 1974.

Divisibility, Primes, Unique factorization of integers, Arithmetical functions, Mobius inversion, congruences, Chinese remainder theorem, primitive roots, Quadratic reci-procity, binary quadratic forms, Fermat's two square theore m, Lagrange's four square theorem, discussion of Waring's problem, Diophantine approximations: continued fractions, rational approximations, tran-scendence of Liouville numbers.

Texts / References

W.W. Adams and L.J. Goldstein, Introduction

to the Theory of Numbers, 3rd ed.,

Wiley Eastern, 1972.A. Baker, A Concise Introduction to the

Theory of Numbers, Cambridge

University Press, 1984.I. Niven and H.S. Zuckerman, Introduction to

the Theory of Numbers, 4^th Ed., Wiley,

New York, 1980.

Prerequisite : MA 501

Varieties: Affine and projective varieties, coordinate rings, morphisms and rational maps, local ring of a point, function fields, dimension of a variety.

Curves: Singular points and tangent lines, multiplicities and local rings, intersection multiplicities, Bezout's theorem for plane curves, Max Noether's theorem and some of its applications, group law on a nonsingular cubic, rati onal parametrization, branches and valuations.

Texts / References

S.S. Abhyankar, Algebraic Geometry for

Scientists and Engineers, American

Mathematical Society, 1990.W. Fulton, Algebraic Curves, Benjamin, 1969.J. Harris, Algebraic Geometry: A First Course,

Springer-Verlag, 1992.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.

Prerequisites : MA 401, MA 402

Basic Combinatorial Objects : Sets, multisets, partitions of sets, partitions of numbers, finite vector spaces, permutations, graphs etc.

Basic Counting Coefficients: The twelve fold way, binomial, q-binomial and the Stirling coefficients, permutation statistics, etc.

Sieve Methods : Principle of inclusion-exclusion, permutations with restricted positions, Sign-reversing involutions, deter-minants etc.

Introduction to combinatorial reciprocity. Introduction to symmetric functions.

Texts / References

C. Berge, Principles of Combinatorics,

Academic Press, 1972.

K.D. Joshi, Foundations of Discrete

Mathematics, Wiley Eastern, 2000.

R.P. Stanley, Enumerative Combinatorics,

Vol. I, Wadsworth and Brooks/Cole,

1986.

Prerequisite : MA 511

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 multi-variable Lagrange inversion.

Texts / References

M. Aigner, Combinatorial Theory, Springer-

Verlag, 1979.

B.E. Sagan, The Symmetric Group:

Representations,Combinatorial

Algorithms and Symmetric Functions,

Wadsworth & Brooks/Cole, 1991.

R.P. Stanley, Enumerative Combinatorics,

Vol. I, Wadsworth and Brooks/Cole,

1986.

Prerequisite : MA 404 (Exposure)

Revision of Fourier series. Tests for pointwise convergence of Fourier series. Summability of Fourier series for integrable functions.

Fourier-transforms of integrable functions. Basic properties of Fourier transforms. Inversion theorem, Plancheral theorem, Paley-Wiener theorem.

Texts / References

Ganapathy Iyer, Mathematical Analysis, Tata

McGraw-Hill, 1977.

W. Rudin, Real and Complex Analysis, Tata

McGraw-Hill, 3rd ed., 1987.

W. Rudin, Functional Analysis, Tata

McGraw-Hill, 1973.

Prerequisites : MA 406 (Exposure),

MA 561 (Exposure)

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.

Texts / References

L. Hormander, The Analysis of Linear PDE's,

Vols. I and II, Springer-Verlag, 1983.

W. Rudin, Functional Analysis, McGraw-Hill,

1973.

K. Yosida, Functional Analysis, 4^th edn.

Springer ISE, Narosa, New Delhi, 1974.

Prerequsite: MA 559 (Exposure)

Analyticity of Resolvent Operators. Spectral Projection and Reduced Resolvent Operators. Spectral Decomposition Theorem. Spectral sets of Finite Type.

Various modes of convergence of a sequence of operators. Upper and lower semicontinuity of the spectrum.

Error bounds for approximate eigenvalues and bases of spectral subspaces.

Finite rank approximations: Galerkin, Projection, Sloan, Nystrom and Fredholm Methods.

Solution of the Eigenvalue Problem for a finite rank operator by reducing it to a Matrix Eigenvalue Problem.

Iterative Refinement. Acceleration.

Texts / References

M. Ahues, A. Largillier and B. V. Limaye,

Spectral Computations for Bounded

Operators, Chapman and Hall/CRC.

F. Chatelin, Spectral Approximation of lInear

Operators, Academic Press, 1983.T. Kato, Perturbation Theory of Linear

Operators, 2nd Ed., Springer-Verlag,

1980.

Prerequisite : MA 402

Representations, Subrepresentations, Tensor products, Symmetric and Alternating Squares.

Characters, Schur's lemma, Orthogonality relations, Decomposition of regular represent-ation, Number of irreducible representations, canonical decomposition and explicit decompositions. Subgroups, Product groups, Abelian groups. Induced representations.

Examples: Cyclic groups, alternating and symmetric groups.

Integrality properties of characters, Burnside's

p^aq^b theorem. The character of induced representation, Frobenius Reciprocity Theory, Meckey's irreducibility criterion, Examples of induced representations, Representations of supersolvable groups.

Texts / References

M. Burrow, Representation Theory of Finite

Groups, Academic Press, 1965.

N. Jacobson, Basic Algebra II, Hindustan

Publishing Corproation, 1983.

S. Lang, Algebra, Addison-Wesley, 1965.

J.P. Serre, Linear Representation of Groups,

Springer-Verlag,

1977.

Even Degree and Odd Degree Spline Interpolation, end conditions, error analysis and order of convergence. Hermite inter-polation, periodic spline interpolation. B-Splines, recurrence relation for B-splines, curve fitting using splin es, optimal quadrature.

Tensor product splines, surface fitting, orthogonal spline collocation methods.

Texts / References

C. de Boor, A Practical Guide to Splines,

Springer-Verlag, 1978.

P. J. Laurent, Approximation et Optimization,

Hermann, 1972.

H.N. Mhaskar and D.V. Pai, Fundamentals of

Approximation Theory, Narosa

Publishing House, New Delhi, 2000.

P. M. Prenter, Splines and Variational

Methods, Wiley-Interscience, 1989.

Prerequisites : MA 403, MA 446

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. Rieman n mapping theorem, simply and doubly connected domains.

Texts / References

E. Hille, Analytic Function Theory, I and II,

Blaisdell, 1959.

W. Rudin, Real and Complex Analysis, Tata

McGraw-Hill, 3rd ed., 1987.

Prerequisite: MA 417 (Exposure)

Review of stability for linear systems. Flow defined by nonlinear systems of ODEs, linearization and stable manifold theorem. Hartman-Grobman theorem. Stability and Lyapunov functions. Planar flows: saddle point, nodes, foci, centers and nonhyperbolic critical points. Gradient and Hamiltonian systems. Limit sets and attractors. Poincare map, Poincare Benedixson theory and Poincare index.

Texts / References

P. Hartman, Ordinary Differential Equations,

John Wiley and Sons, NY, 1964.

M. W. Hirsch and S. Smale, Differential

Equations, Dynamical Systems and Linear

Algebra, Academic Press, NY, 1974.L. Perko, Differential Equations and

Dynamical Systems, Springer Verlag,

NY, 1991.

S. Wiggins, Introduction to Applied Nonlinear

Dynamical Systems and Chaos, TAM

Vol.2, Springer-Verlag, NY, 1990.

Prerequisite: MA 403 (Exposure)

Review of density theorems: the theorems of Korovkin, Fejer and Stone-Weierstrass.

The classical Chebyshev theory, discretization and discrete best approximation, the second algorithm of Remes.

Degree of approximation, moduli of continuity and K-functionals, direct and converse theorems.

Interpolation, Lagrange form, extended Haar subspaces and Hermite interpolation, Hermite-Fejer interpolation, Hermite-Birkhoff inter-polation. Piecewise polynomial interpolation.

Texts / References

R. DeVore and G.G. Lorentz, Constructive

Approximation, Springer-Verlag, Berlin,

1993.

G.G. Lorentz, Approximation of Functions,

Holt, Rinehart and Winston, New York,

1966.

H. N. Mhaskar and D. V. Pai, Fundamentals of

Approximation Theory, Narosa Pub-

lishing House, New Delhi, 2000.

Prerequisite: MA 436 (Exposure)

Finite differences: grids, derivation of difference equations. Elliptic equations: discrete maximum principle and stability, residual correction methods (Jacobi, Gauss-Seidel and SOR methods), LOD and ADI methods. Finite differen ce Schemes for initial and boundary value problems: Stability (matrix method, von-Neumann and energy methods), Lax-Richtmyer equivalence Theorem. Parabolic equations: explicit and implicit methods (Backward Euler and Crank-Nicolson schemes) with stability and convergence, ADI methods. Linear scalar conservation law: upwind, Lax-Wendroff and Lax-Friedrich schemes and CFL condition.

Lab Component: Exposure to MATLAB and computational experiments based on the algorithms discussed in the course.

Texts / References

1. R. Mitchell and S. D. F. Griffiths, The Finite Difference Methods in Partial Differential Equations, Wiley and Sons, NY, 1980.

R. D. Richtmyer and K. W. Morton,

Difference Methods for Initial Value

Problems, Wiley Interscience, NY, 1969.G. D. Smith, Numerical Solutions of Partial

Differential Equations, 3rd Edition,

Calrendorn Press, Oxford, 1985.J. C. Strikwerda, Finite difference Schemes

and Partial Differential Equations,

Wadsworth and Brooks/ Cole Advanced

Books and Software, Pacific Grove,

California, 1989.J. W. Thomas, Numerical Partial Differential

Equations : Finite Difference Methods,

Texts in Applied Mathematics, Vol. 22,

Springer Verlag, NY, 1999.

J. W. Thomas, Numerical Partial Differential

Equations: Conservation Laws and

Elliptic Equations, Texts in Applied

Mathematics, Vol. 33, Springer Verlag,

NY, 1999.

Prerequisites: MA 559 (Exposure).

Fixed Point Theorems with Applications: Banach contraction mapping theorem, Brouwer fixed point theorem, Leray-Schauder fixed point theorem.

Calculus in Banach spaces: Gateaux as well as Frechet derivatives, chain rule, Taylor's expansions, Implicit function theorem with applications, subdifferential.

Monotone Operators: maximal monotone operators with properties, surjectivity theorem with applications

Degree theory and condensing operators with applications.

Texts / ReferencesM. C. Joshi and R. K. Bose, Some Topics in

Nonlinear Functional Analysis, Wiley

Eastern Ltd., New Delhi, 1985.S. Kesavan, Topics in Functional Analysis,

Wiley Eastern Ltd., New Delhi, 1989.Zeilder, Nonlinear Functional Analysis and Its

Applications, Vol. 1 (Fixed Point Theory),

Springer Verlag, Berlin, 1985.

Differential Equations 2 1 0 6Prerequisites: MA 436 (Exposure),

MA 404 (Exposure)

Theory of distributions and Fourier transforms: supports, test functions, regular and singular distributions, generalised derivatives, convolution, fundamental solutions, Fourier trans-forms and inversions, Schwartz space, temper ed distributions.

Sobolev Spaces: basic properties, approxi-mation by smooth functions, trace and imbedding results (without proof).

Elliptic Boundary Value Problems: weak formulation and wellposedness with examples, regularity result, maximum principles, eigenvalue problems.

Initial Value Problems for Heat and Wave Equations: fundamental solutions, basic properties and regularity results.

Texts / References

G. B. Folland, Lectures on Partial Diferential

Equations, TIFR Lecture Notes Series,

Narosa Publ. House, New Delhi, 1983.S. Kesavan, Topics in Functional Analysis,

Wiley Eastern Ltd., New Delhi, 1989.

J. Rauch, Partial Differential Equations,

Narosa Publ. House, New Delhi, 1991.

F. Treves, Basic Linear Partial Differential

Equations, Academic Press, New York,

1975.

Geometry 2 1 0 6Prerequisite : MA 404 (Exposure)

Surfaces in Rⁿ and tangent spaces, vector-fields on surfaces, orientation, Gauss map. Geodesics and parallel transport, Weingarten map. Curvature of plane curves, arc length and line integrals.

Curvature of surfaces, parametrization of surfaces.

Rigid motions and Congruence, isometries.

Texts / References

M. doCarmo, Differential Geometry of Curves

and Surfaces, Englewood Cliffs, N.J.,

Prentice Hall, 1976.B. O'Neill, Elementary Differential Geometry,

Academic Press, N.Y., 1966.

M. Spivak, A Comprehensive Introduction to

Differential Geometry, Vols I-V, Publish or

Perish, Boston, 1970.

J. J. Stoker, Differential Geometry, Wiley-

Interscience, 1969.

J. A. Thorpe, Elementary Topics in

Differential Geometry, UTM Springer-

Verlag, 1979.

Prerequisite : MA 403, MA 404 (Exposure)

Normed linear spaces. Continuity of linear maps. Hahn-Banach theorem in extension. Banach spaces. Dual spaces. Reflexivity (definition and simple examples).

The uniform boundedness principle and its applications. The closed graph theorem, the open mapping theorem and their applications.

Spectrum of a bounded operator. Compact operators and their spectra. Fredholm alternatives.

Inner product spaces, Hilbert spaces, Orthonormal basis. Projection theorem and the Riesz representation theorem. Reflexivity of Hilbert spaces.

J.B. Conway, A Course in Functional

Analysis, Springer-Verlag, Berlin, 1985.

G. Goffman and G. Pedrick, First Course in

Functional Analysis, Prentice-Hall, 1974.

E. Kreyszig, Introductory Functional Analysis

with Applications, John Wiley & Sons,

New York, 1978.

B.V. Limaye, Functional Analysis, 2nd ed.,

New Age International, New Delhi, 1996.

A. Taylor and D. Lay, Introduction to

Functional Analysis, Wiley, New York,

1980.

Prerequisite : MA 403 (Exposure)

Semi-algebra, Algebra, Sigma-algebra, Mono-tone class, Monotone class theorem, Measure spaces.

Outline of extension of measures from algebras to the generated sigma-algebras, Measurable sets.

Measurable functions and their properties, Outline of Integration and Convergence theorems.

Introduction to L^p-spaces, Riesz-Fischer theorem.

Product measure spaces, Fubini's theorem.

Absolute continuity of measures, Radon-Nikodym theorem.

Texts / References

P. R. Halmos, Measure Theory, Graduate Text

in Mathematics, Springer-Verlag, 1979.I.K. Rana, An Introduction to Measure and

Integration, Narosa Publishing House,

New Delhi, 1997.H.L. Royden, Real Analysis. 2^nd Edn.,

Macmillan, 1968.

Finite Elements 2 1 0 6Prerequisite: MA 436 (Exposure)

MA 559 (Exposure)

Sobolev Spaces: basic elements, Poincar'e inequality. Abstract variational formulation and elliptic boundary value problem. Galerkin formulation and Cea's Lemma. Construction of finite element spaces. Polynomial approximations and interpolationa errors.

Finite element convergence analysis: Aubin-Nitsche duality argument; non-conforming elements and numerical integration; computation of finite element solutions.

Lab component: Implementation of algorithms and computational experiments with fem packages

Texts / References

K.E. Brennan and R. Scott, The Mathematical

Theory of Finite Element Methods,

Springer- Verlang, Berlin, 1994.

P. G. Ciarlet, The finite Element Methods for

Elliptic Problems, North Holland,

Amsterdam, 1978.C. Johnson, Numerical solutions of Partial

Differential Equations by Finite Element

Methods, Cambridge University Press,

Cambridge, 1987.C. Mercier, Lectures on Topics in Finite

Element Solution of Elliptic Problems,

TIFR Lectures on Mathematics and

Physics Vol. 63, Narosa Publ. House,

New Delhi, 1979.

Prerequisite: MA 559 (Exposure)

Adjoints of bounded operators on a Hilbert space, Normal, self-adjoint and unitary operators, their spectra and numerical ranges.

Spectral theorem for compact self-adjoint/ normal operators, Application to Sturm-Liouville Problems.

Hahn-Banach separation theorem, Dual spaces and transposes, Duals of L^p([a,b]) and C([a,b]).

Weak and weak* convergence, Reflexivity and Uniform Convexity.

Texts / References

C. Goffman and G. Pedrick, First Course in

Functional Analysis, Prentice Hall, 1974.

I. Gohberg and S. Goldberg, Basic Operator

Theory, Birkhauser, 1981.

B. V. Limaye, Functional Analysis, 2nd Ed.,

New Age International, New Delhi, 1996.

K. Yosida, Functional Analysis, 4^th edn.

Springer ISE, Narosa, New Delhi, 1974.

Geometry 2 1 0 6Prerequisite: MA 402 (Exposure)

Affine varieties, parametrization, ideals, orderings on monomials, Dickson's Lemma, Groebner bases, Hilbert Basis Theorem, Buchberger's algorithm, ideal membership, geometry of elimination, singular points, resultants and extension theo rem, Hilbert's Nullstellensatz, ideal variety correspondence, basic computations among ideals via Groebner bases, coordinate ring of an affine variety and algorithmic computations in it, solving equations via eigenvalues, projective varieties, projective nullstellensatz, computation of projective closure, Hilbert function and dimension of a variety and their computation, applications to graph colouring and integer programming.

Lab. Component: Implementation of algo-rithms developed in this course using Macaulay programming language.

Texts / References

W. W. Adams and P. Laustaunau, An

introduction to Groebner Bases, American

Mathematical Society, 1994. D. Cox, J. Little, and D. O'Shea. Ideals,

Varieties and Algorithms, 2nd edition,

Springer-Verlag, 1996.D. Cox, J. Little, and D. O'Shea, Using

Algebraic Geometry, Springer-Verlag,

1998.

Prerequisites : MA 401, MA 577 (Exposure)

Theory of linear estimation. Standard designs : CRD, RBD, LSD, BIBD and PBIBD. Factorial designs. Confounding. Missing plot technique. Analysis of covariance. Con-struction and nonexistence theory. Special designs : Split-plots, strip-plots, cross-over designs.

Texts / References

1. Dean and D. Voss, Design and Analysis

1. Dey, Theory of Block Designs, Wiley,

C.R. Hicks and K.V. Turner, Fundamental

Concepts in the Design of Experiments,

Oxford University Press, 1999.

C.F. Jeff Wu, Experiments: Planning,

Analysis, and Parameter Design Optimi-

zation, Wiley, New York, 2000.

Prerequisite : MA 577 (Exposure)

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.).

Texts / References

W.W. Daniel, Applied Nonparametric Statisti-

Stics, 2^nd Ed., Boston: PWS-KENT, 1990.

M. Hollandor, and D.A. Wolfe, Non-

parametric 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.

R.H. Randles and D.A. Wolfe, Introduction to

the Theory of Nonparametric Statistics,

Wiley, New York, 1979.

Prerequisites : MA 411

MA 438 (Exposure)

Coherent Structures, Reliability of systems of independent components, Bounds of system reliability, shape of the system reliability function, notion of ageing, parametric families of life distributions with monotone failure rate , classes of life distributions based on notions of ageing, classes of distributions in replacement policies, limit distributions for series and parallel systems. Statistical estimation and testing for popular reliability models and classes (parametric an d nonparametric).

Texsts / References

L.J. Bain, Statistical Analysis of Reliability

and Life Testing, Marcel Dekker, 1978.

R.E. Barlow and F. Proschan, Statitsical

Theory of Reliability and Life Testing,

Holt, Reinhart and Winston, 1975.

J.D. Kalbfleisch and R.L. Prentice, The

Statistical Analysis of Failure Time

Data, Wiley, 1986.

J.F. Lawless, Statistical Models and Methods

of Life Time Data, Wiley, 1982.

N.R. Mann, R.E. Shafer and N.D.

Singpurwala, Methods of Statistical

Analysis of Reliability and Life Data,

Wiley, 1974.

R.G. Miller, Survival Analysis, Wiley, 1981.

Prerequisite : MA 438

Multiple linear regression – estimation, tests and confidence regions. Check for normality assumption.

Logistic, Probit, Log-linear models for nominal and ordinal variables. Fitting of logit and log-linear models.

Texts / Reference

1. Agresti, Analysis of Categorical Data, Wiley, 1990

1. Agresti, An Introduction to Categorical

Data Analysis, John Wiley & Sons, Inc.,

New York, 1996.

E.B. Andersen, The Statistical Analysis of

Categorical Data, Springer-Verlag, 1990.

D.C. Montgomery & E.A. Peck, Introduction

to Linear Regression Analysis, 2^nd ed.,

John Wiley, 1992.

T.J. Santner & D. Duffy, The Statistical

Analysis of Discrete Data, Springer-

Verlag, 1989.

A.A. Sen & M. Srivastava, Regression

Analysis – Theory, Methods &

Applications, Springer-Verlag, 1990.

Prerequisites : MA 411, MA 438 (Exposure)

K-variate normal distribution. Estimation of the mean vector and dispersion matrix. Random sampling from multivariate normal distribution. Multivariate distribution theory. Discriminant and canonical analysis. Factor analysis. Pr incipal components.

Distribution theory associated with the analysis.

Texts / References

T.W. Anderson, An Introduction to Multi-

variate Statistical Analysis, 2^nd Ed.,

Wiley, 1984.

R. Gnanadesikan, Methods for Statistical

Data Analysis of Multivariate Obser-

vations, John Wiley, New York, 1997.

R.A. Johnson and D.W. Wicheran, Applied

Multivariate Statistical Analysis, Upper

Saddle River, Prentice Hall, 1998.

M.S. Srivastava and E.M. Carter, An Intro-

duction to Multivariate Statistics, North

Holland, 1983.

Prerequisite : MA 577

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 non-randomized rules. The minimax theorem.

Invariant statistical decision problems. Multiple decision problems.

Sequential decision problems.

Texts / References

J.O. Berger, Statistical Decision Theory :

Foundations, Concepts and Methods,

Springer-Verlag, 1980.

J.O. Berger, Statistical Design Theory and

Bayesian Analysis, 2^nd Ed., Springer-

Verlag, 1985.

T.S. Ferguson, Mathematical Statistics,

Academic Press, 1967.

S.S. Gupta and D. Huang, Multiple Statistical

Decision Theory, Springer-Verlag, New

York, 1981.

Prerequisite : MA 411

MA 438

Uniformly most powerful unbiased tests, Invariance in Estimation and Testing, Admissibility, Minimax and Bayes Estimation, Asymptotic Theory of Estimation, Asymptotic distribution of likelihood ratio statistics.

Sequential Estimation, Sequential Probability, Ratio Test.

Texts / Reference

G. Casella and R.L. Berger, Statistical

Inference, Wadsworth and Brooks, 1990.

E.L. Lehmann, Theory of Point Estimation,

John Wiley, 1983.

E.L. Lehmann, Testing Statistical Hypotheses,

2^nd ed., Wiley, 1986.

R.J. Serfling, Approximation Theorems of

Mathematical Statistics, Wiley, 1980.

Prerequisite : MA 411

Recurrent events. Renewal theory. Random walk. Markov chains and Markov processes. Stationary processes. Spectral Analysis. Branching phenomena. Semi-Markov processes. Systems with random inputs.

Texts / References

R.G. Gallager, Discrete Stochastic Processes,

Kluwer Academic Publications, Boston,

1996.

S. Karlin and H.M. Taylor, A First Course in

Stochastic Processes, Academic Press,

1975.

S. Karlin and H.M. Taylor, Second Course in

Stochastic Processes, Academic Press,

New York, 1981.

J. Medhi, Stochastic Processes, Wiley Eastern,

1982.

1. Parzen, Stochastic Processes, Holden-Day,

1972.

S.M. Ross, Stochastic Processes, Wiley, New

York, 1983.

Prerequisite : MA 577 (Exposure)

Introduction to autocorrelation function, linear stationary models like autoregressive, inte-grated moving average processes. Forecasting model identification including initial estimates of the parameters, model multiplicity etc. Model estimation, model diagnostic checking. Case studies. Computational experiments.

Texts / References

P. Bloomfield, Fourier Analysis of Time

Series: An Introduction, Wiley, New

York, 2000.

G.E.P. Box and G.M. Jenkins, Time Series

Analysis Forecasting and Control, Holden-

Day, 1976.

G.E. Box, G.M. Jenkins and G.C. Reinsel,

Forecasting and Control, Prentice-Hall,

1994.

P.J. Brockwell and R.A. Davis, Time Series,

Springer-Verlag, 1987.

C. Chatfield, The Analysis of Time Series: An

Introduction, 5^th Ed., Chapman & Hall,

Boca Raton, 1996.

J.D. Hamilton, Time Series Analysis,

Princeton University Press, 1994.

R.H. Shumway and D.S. Stoffer, Time Series

Analysis and its Applications, Springer-

Verlag, 2000.

R.A. Yaffee and M. McGee, Introduction to

Time Series Analysis and Forecasting

with Applications of SAS and SPSS,

Academic Press, San Diego, 2000.

Prerequisite: MA 404 (Exposure)

Differentiable Manifolds in Rⁿ: Review of inverse and implicit function theorems; tangent spaces and tangent maps; immersions; submersions and embeddings.

Regular Values: Regular and critical values; regular inverse image theorem; Sard's theorem; Morse lemma.

Transversality: Orientations of manifolds; oriented and mod 2 intersection numbers; degree of maps. Application to Fundamental theorem of Algebra.

*Lefschetz theory of vector fields and flows: Poincare-Hopf index theorem; Gauss-Bonnet theorem.

*Abstract manifolds: Examples such as real and complex projective spaces and Grassmannian varieties; Whitney embedding theorems.

(* indicates expository treatment intended for these parts of the syllabus.)

Texts / References

1. A. Dubovin, A. T. Fomenko, S. P.

Novikov, Modern Geometry Methods

and applications-II, The Geometry and

Topology of manifolds, Springer-

Verlag GTM 104, 1985.V. Guillemin and A Pollack, Differential

Topology Prentice-Hall Inc., Englewood

Cliffs, New Jersey, 1974.J. Milnor, Topology from the Differential

View-point, University Press of Virginia,

Charlottsville 1990.

Topology 2 1 0 6Pre-requisite: MA 406 (Exposure)Basic notions of homotopy: Homotopy equi-valence; contractibility; deformation retracts.Basic constructions: Mapping cones; mapping c ylinders; suspension.

Covering spaces and Fundamental group: Relation between subgroups of fundamental group and coverings; universal coverings (existence theorem optional).

Computation of Fundamental group: Van-Kampen's theorem and applications.

Simplicial complexes: Barycentric subd-ivision; stars and links simplicial approxi-mation; applications such as computing fundamental group of Sⁿ.

Homology Theory: Basic properties of simplicial and singular homologies; equivalence of the two theories without proof; applications to Brouwer separation theorem and invariance of domain.

*Cohomology: Singular and Cech coho-mology; cohomology of forms and De-Rham's theorem for smooth manifolds; cohomology algebra of projective spaces. Poincare duality for closed manifolds.

*Classification of Topological Surfaces.

(* indicates expository treatment intended for these parts of the syllabus)

Texts / References

W. S. Massey, Algebraic Topology: An

Introduction, GTM 56, Springer-Verlag,

1977.

J. R. Munkres, Elements of Algebraic

Topology, Addison-Wesley, 1984.

H. Seifert and W. Threlfall, A text Book of

Topology, translated by M. A. Goldman,

Academic Press 1980.

Description of continuum and kinematics. Forces, deformations, constitutive equations. Theory of motions-steady and unsteady motions. Spin vectors and tensors. Conservation laws—physical models and applications.

Texts / References

Y. C. Fung, A First Course in Continuum

Mechanics, Prentice Hall Inc., New York,

1977.

M. E. Gurtin, An Introduction to Continuum

Mechanics, Academic Press, New York,

1981.

S. Hunter, Mechanics of Continuum Media,

John Wiley and Sons, New York, 1983.

Introduction to options and markets: types of options, interest rates and present value.Black-Scholes Model: arbitrage, option values, payoffs and strategies, put call parity, Black-scholes equation, similarity solution and exact fo rmulae for European options. American options: call and put options, free boundary problem.

Binomial mehods: option valuation, divident paying stock, general formulation and implementation. Monte-Carlo simulation: valuation by simulation.

Finite Difference Methods: explicit and implicit methods with stability and convergence analysis, methods for American option--constrained matrix problem, projected SOR, time stepping algorithms with convergence and numerical exa mples.

Lab Component: Implementation of the option pricing algorithms and Evaluation for Indian companies.

Texts / References

L. Clewlow and C. Strickland, Implementing

Derivative Models, Wiley Series on

Financial Engineering, John Wiley and

Sons, Chichester, 1998.J. C. Hull, Options, Futures and Other

Derivatives, 4^th ed., Prentice Hall of India,

New Delhi, 2000.P. Wilmott, S. Howison and J. Dewynne, The

Mathematics of Financial Derivatives: A

Student Introduction, Cambridge

University Press, Cambridge, 1995.

P. Wilmott, J. Dewynne, and S. Howison,

Option Pricing: Mathematical Models and

Computation, Oxford Financial Press,

Oxford, 1993.

MA 590 Fluid Dynamics 2 1 0 6Prerequisite: MA 583 (Exposure)

Conservation laws in continuum media, similarity solutions, shallow water theory, hydrodynamic bores and breakdown of continuous solutions on a beach, dam break problem, one dimensional nozzle flows, nonlinear axisymmetric flows, flow p ast bodies and thin profiles, dissipative and dispersive flows (KdV and Burger's equations).

Texts / References

L. D. Landau and F. M. Lifscitz, Fluid

Mechanics, Pergamon Press, 2nd

Edition, 1987.

Z. Pozrikidis, Introduction to Theoretical and

Computational Fluid Dynamics, Oxford

Univ. Press, Oxford, 1997.Z. U. A. Warsi, Fluid Dynamics : Theoretical

and Computational Approach, CRC

Press, 1991.J. Zierep, Theoretical Gasdynamics, Allied

Publishing Pvt. Ltd., 1978.

Prerequisites: MA 436

MA 415 (Exposure)

Discontinuous solutions of conservation laws, weak solutions. Evolutionary systems, shock inequalities, irreversibility. Riemann problem for a system of conservation laws (Gas dynamics). Ray methods and discontinuities. Generali zed wave front expansion for weak shock, shock stability analysis.

Texts / References

1. Anile et. al., Ray Methods for Nonlinear Waves in Fluids and Plasmas, Longman Publ. Co., 1993.

1. Jeffry, Quasilinear Hyperbolic Systems

and Waves, Pitman Publ. Co., 1976.

F. V. Shugaev and L. S. Shtemenko,

Propagation and Reflections of Shock

Waves', World Scientific Publ., 1997.

J. Smoller, Shock Waves and Reaction

Diffusion Equations, Springer Verlag, NY,

1983.

G. B. Whitham, Linear and nonlinear Waves,

John Wiley and Sons, NY, 1974.

Prerequisite : MA 579 (Exposure)

Martingales, Stochastic integrals, Ito's lemma. Stochastic Differential Equatons.

Models for Stock Prices - Black and Scholes models, Option pricing, other financial derivatives.

Texts / References

M. Baxter and A. Rennie, Financial Calculus,

Cambridge Process, Cambridge, 1998.

John C. Hull, Options, Futures and Other

Derivatives, Fourth Edition, Prentice-Hall

International Inc., 2000.Lamberton, D. and Lapeyre, B., Introduction

to Stochastic Calculus Applied to Finance,

Chapman & Hall, 1996.

Dimensional analysis and scaling, model error approximation and testing, data translation.

Fundamentals of modeling : Lagrange and Eulerian models. Basic conservation laws. Stress, strain and strain rate. Constitutive equations.

Growth and Decay models : population growth (Lofka-Volterra model), bacteria growth and decay. Radiation and crystal growth.

Interacting species and chemical reactions. Satellite problem.

Network analysis : Mechanical models, Electrical network. Fluid flow including blood flow problems, traffic flow.

Diffusion and air pollution models.

Texts / References :

M. Braun, C. S. Coleman and D. A. Draw,

Differential Equation Models, Modules in

Applied Mathematics, Vol. 1, Springer

Verlag, 1978.

C. L. Dym and E. S. Ivey, Principles of

Mathematical Modeling, Academic

Press, 1980.

1. Friedman, Mathematics in Industrial

Problems, Part 1-9, MA Series, Springer-

Verlag, 1991.

Y. C. Fung, Biomechanics, Springer Verlag,

1981.

J. Keener and J. Sneyd, Mathematical

Physiology, Springer-Verlag, 1998.

M.S. Klamkin (ed.), Mathematical Modelling.

Class Room Notes in Applied Mathem-

Atics, SIAM, 1987.

E. N. Lightfoot, Transport Phenomenon and

Living Systems, John Wiley & Sons,

1974.

H. J. White and S. Tauber, System Analysis,

W.B. Saunders Company, 1969.

System of ordinary differential equations : Reduction of nth order o.d.e. to a system of 1st order o.d.e.'s. Companion matrix. Picard's theorem for existence and uniqueness and its

implementation.

State space formulation. Concept of system. Input-output and state space. Transition matrix and its properties. Fundamental and non-fundamental solutions.

Phase Plane Analysis : Critical points and stability of a linear system. Liapunov stability method. Nonlinear system.

Transform Analysis : Laplace transform (continuous time), z-transform (discrete time). Input-output analysis.

Controllability, observability and stabiliz-ability : Controllability matrix and Gramian. Observability as a dual notion of controll-ability.

Stabilizability of a discrete system.

Texts / References

A. V. Balakrishnan, Elements of State Space

Theory of Systems, Optimization

Software Inc., 1983.

W. L. Luyben, Process Modeling, Simulation

and Control for Chemical Engineers,

McGraw-Hill International, 1990.

1. F. Simmons, Differential Equations with

Applications and Historical Notes, 2^nd

Ed., Tata McGraw-Hill, 1991.

H. J. White and S. Tauber, Systems Analysis,

W.B. Sounders, 1969.

L. A. Zadeh and E. Polak, System Theory,

McGraw-Hill, 1969.

Stochastic processes : description and definition. Markov chains with finite and countably infinite state spaces. Classification of states, irreducibility, ergodicity. Basic limit theorems. Statistical Inference. Applications to queueing models.

Markov processes with discrete and continu-ous state spaces. Poisson process, pure birth process, birth and death process. Brownian motion.

Applications to queueing models and relia-bility theory.

Basic theory and applications of renewal processes, stationary processes. Branching processes. Markov Renewal and semi-Markov

processes, regenerative processes.

Texts / References :

U. N. Bhat, Elements of Applied Stochastic

Processes, Wiley, 1972.

V.G. Kulkarni, Modeling and Analysis of

Stochastic Systems, Chapman and Hall,

London, 1995.

J. Medhi, Stochastic Models in Queueing

Theory, Academic Press, 1991.

R. Nelson, Probability, Stochastic Processes,

and Queuing Theory: The Mathematics

of Computer Performance Modelling,

Springer-Verlag, New York, 1995.

Introduction to data structures and com-plexity of algorithms. Introduction to a suitable programming language.

Arrays, lists, stacks, queues, trees, graphs, heaps, sets, hash tables.

Internal and external sorting techniques.

Tree traversals, graph traversals.

Search techniques.Tree and graph search.

Texts / References

R.L. Kruse, B.P. Leung and C.L. Tondo,

Data Strurctures and Programming in C,

Prentice Hall of India, 1991.

S. Sahni, Data Strucures, Algorithms and

Applications in C++, McGraw-Hill

International Editions, 1998.

R. Sedgewick, Algorithms in C, Parts 1-4,

Addison-Wesley, Reading, 1998.

Illustration of techniques and concepts from combinatorices 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.

Texts / 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.

Concepts in Programming Languages Paradigms and the issues such as storage management, formal translation models (BNF forms and grammers), abstractions such as subprogram (parameter passing mechanics, scopes, bindings etc .), inheritance and polymorphism etc.

Illustration of these concepts through the study of C++ Programming Language.

Texts / References

S. Lippman,`C++ Primer', Addison-Wesley

Publishing Company, 1995.

T.W. Pratt and Zelkowitz, Programming

Languages: Design and Implementation.

Prentice-Hall India, 1996.

Introduction to Unix: Introduction to C pro-gramming language, Syntax, Data Types, Pointers, Functions, Scope rules, bindings, parameters transmission mechanism, storage management.

Texts / References

B. Kernighan, R. Pike, The Unix Progra-

mming Environment, Prentice Hall India,

1995.

B. Kernighan, D. Ritchie, The C Proramming

Language, Prentice Hall India, 1996.

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 progra-mming.

Linear Programming: duality, simplex method, revised simplex method, dual simplex method, sensitivity analysis, transportation problems.

Heuristics for Combinatorial Optimization: branch and bound, hill climbing, simulated annealing, genetic algorithm, primal-dual approach.

Texts / References

E. Aarts and J.K. Lenstra, Local Search in

Combinatorial Optimization, John Wiley

and Sons, 1997.

M. Bazarra and C. Shetty, Nonlinear

Programming, Theory and Algorithms,

Wiley, New York, 1979.

Edwin K P Chong and Stanislaw H Zak, An

Introduction to Optimization, John Wiley

& Sons , 1996.

S. S. Rao, Optimization : Theory and

Applications , Wiley Eastern Ltd. 1984.

H. Taha, Operations Research : An

Introduction, Prentice Hall India, New

Delhi, 1997.

Quadratic and Nonlinear Programming solut-ion methods applied to Chance Con-strained Programming problems. Stochastic Linear and Nonlinear Programming problems arising in inventory control and other industrial applications; q ueuing models of computer networks; information processing under uncertainty. Two stage and multi-stage solution techniques. Use of Monte Carlo, probabilistic and heuristics algorithms. Genetic algorithms and neural networks for adaptive optimizatio n.

Texts / References

Yu. Ermoliev and RJB Wets, Numerical

Techniques for Stochastic Optimiation.

Springer Verlag, Berlin, 1988.

Z. Michaeleawicz, General Algorithms +

Data Structures - Evolution Program.

Springer-Verlag, 1992.

S.S. Rao, Optimization - Theory and Appli-

cations. Wiley Eastern (2nd ed.), 1987.

K. Schittkowski, More Test Examples of

Nonlinear Programming Codes.

Springer-Verlag, Berlin, 1987.

J.K. Sengupta, Stochastic Optimizations and

Economic Models. Dordrecht Reidel,

1986.

Linear Models and Estimators, Estimability of linear parametric functions. Gauss-Markoff Theorem.

Principles of Design of Experiments. General structure of analysis of designs. ANOVA, ANACOVA. Regression analysis (one independent variable).

Standard designs such as CRD, RBD, LSD, BIBD. Analysis using the missing plot technique. Factorial designs. Confounding. Analysis using Yates' algorithm.

Special designs such as split-plot, strip-plot, cross-over designs.

Orthogonal arrays, Response surface methodology. Taguchi method.

Texts / References

G.E.P. Box, W.G. Hunter and J.S. Hunter,

Statistics for Experimentors, Wiley,

1978.

N.L. Johnson and F.C. Leone, Statistics and

Experimental Design, Vol. 2, 2nd ed.,

Wiley, 1977.

D.C. Montgomery, Design and Analysis of

Experiments, 3rd ed. Wiley, 1991.

M.S. Phadke, Quality Engineering Using

Robust Design, Prentice Hall, 1989.

P.J. Ross, Taguchi Techniques for Quality

Engineering, McGraw-Hill, 1988.

H. Spaeth, Mathematical Algorithms for

Linear Regression, Academic Press,

1991.

Finite automata. Regular expressions, Regular languages and their properties. Push down automata. Context-free languages and their proepreties.

Turing machines. Turing computability. Un-decidability results. Introduction to compiler design, lexical analysis and parsing. Auto-mataic generation of lexical analyers and parsers.

Texts / References

1. Aho, R. Sethi and J.D. Ullman, Compiler

Principles,Techniques and Tools, Addison-

Wesley, 1986.

A. Holub, Compiler Design in C, Prentice-

Hall of India, 1994.

J.E. Hopcroft and J.D. Ullman, Automata,

Languages and Computation, Narosa,

1987.

Introduction to graphics hardware. Brief overview of procedural elements for computer graphics (like drawing, circle drawing, clipping, hidden line/surface removal algo-rithm).

Transformations in 2D/3D. Linear/Affine/ Projective transformations. Perspective views.

Freeform curves and surfaces. Applications from CAGD. Selected topics from Computat-ional Geometry and Applications.

Texts / References

J.D. Foley, A. van Dam, S.K. Feiner, and

J.F. Hughes, Computer Graphics:

Principles and Practice, Addison-

Wesley, 1990.

M.J. Laszlo, Computational Geometry and

Computer Graphics in C++, Prentice-

Hall India, 1999.

J. O'Rourke, Computational Geometry in C,

Cambridge Univ. Press, 1994.

Polynomial curves: Bezier representation, Bernstein polynomials, Blossoming, de Castlijau algorithm. Derivatives in terms of Bezier polygon. Degree elevation. Subdivi-sion. 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. Inter-pretation 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. Tri-angular patch surfaces.

Texts / 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.

Basic reliability models. Estimation and inferential aspects of these models. Prob-abilistic modeling of repairable systems. Statistical analysis of repairable systems and of failure data.

Texts / References :

H. Ascher and H. Feingold, Repairable system

Reliability, Marcel Dekker, 1984.

L. J. Bain and M. Engelhardt, Statistical

Analysis of Reliability and Life Testing

Models : Theory and Methods, Marcel

Dekker, 1991.

I. Gertsbakh, Reliability Theory: With Appli-

cations to Preventive Maitnenance,

Springer-Verlag, 2000.

S.E. Rigdon and A.P. Basu, Statistical

Methods for the Reliability of Repairable

Systems, Wiley, 2000.

S. K. Sinha and B. K. Kale, Life Testing and

Reliability Estimation, Wiley Eastern,

1979.

Total quality control in an industry. Quality planning, quality conformance, quality ad-herence. Quality assurance and quality management functions.

Control charts and allied techniques. Concept of quality and meaning of control. Concept of inevitability of variation-chance and assign-able causes. Pattern of variation. Principles of rational sub-grouping.

Different types of control charts. Concept of process capability and its comparison with design specifications, CUSUM charts.

Acceptance sampling. Sampling inspection versus 100 percent inspection. Basic concepts of attributes and variables inspection. OC curve, Single, double, multiple and sequential

sampling plans, Management and organisation of quality control.

Texts / References :

A. J. Duncan, Quality Control and Industrial

Statistics, 5th ed., Richard D. Irwin,

1986.

E. L. Grant and R. Levenworth, Statistical

Quality Control, 6^th ed., McGraw-Hill,

1988.

J. M. Juran and F. M. Grayna, Quality

Planning and Analysis, Tata McGraw-

Hill, 1970.

D.C. Montgomery, Introduction to Statistical

Quality Control, Wiley, 1985.

T.P. Ryan, Statistical Methods for Quality

Improvement, Wiley, New York, 2000.

Introduction to inductive probability and machine learning. Statistical pattern recog-nition and clustering techniques. Stochastic approximation and rough classification. Bayesian classification. Sequential probability and incre mental machine learning.

Nonparametric methods for leader inde-pendent sample based learning.

Random generate and test algorithms. Stochastic heuristics in guided learning by discovery.

Texts / References :

R. F. Albrecht, Artificial Neural Nets and

Genetic Algorithms, Springer-Verlag,

1993.

S. C. Choi and E. Y. Rodin, Statistical

Methods of Discrimination and

Classification : Advances in Theory and

Applications, Pergamon Press, 1986.

S. J. Hanson et. al., (Eds.) Machine Learning –

from Theory to Applications, Lecture

Notes in Computer Science, Vol. 661,

Springer-Verlag, 1993.

R. S. Michaelski et. al., (Eds.) Machine

Learning - An Artificial Intelligence

Approach, Springer, 1984.

J. Press, Bayesian Statistics : Principles,

Models and applications, Wiley, 1989.

W. D. Wayne, Applied Nonparametric

Statistics, 2nd ed., PWS-KENT, 1990.

SI 520 Informatics Lab 0 0 3 3

Intac Technologies: Advanced Java Technologies. Software Development Tools: Parser Generators, GUI Builders. Software Components: Beans, Active X controls.

Texts / References

http://java.sun.com

http://microsoft.com

Randomization and control of clinical trials. Sampling in clinical studies. Cohort analysis. Sampling distributions and hypo-thesis testing for clinical and laboratory data. Importance of type I, type II errors and s ample sizes in the design and interpretation of control trials. Inferencing with incomplete data.

Logistic and multiple regression models in

drug response analysis. Distribution free tests and nonparametric regression analysis of bio-medical measurements. Methods based on rank orders.

Vital statistics. Analysis of survival data. Sampling and testing in epidemiological studies.

Texts / References

Daniel Wayne W., Biostatistics : A Found-

ation for Analysis in the Health

Sciences (5h ed.), John Wiley, N.Y.,

1991.

Friedman, L.M., Furberg C. and Demets

D.L., Fundamentals of Clinical Trials.

Mosby-Year Book Inc., St., Louis, 1996.

Hosmer, D.W. and Lemeshow, S. Applied

Logistic Regression, John Wiley and

Sons, NY, 1989.

Peace K.E. (ed.) Statistical Issues in Drug

Research and Development. Marcel

Dekkar, NY, 1990.

Selected Papers from JASA and Biometrika.

Prerequisite: SI 503 (Exposure)

Large sparse linear systems: Storage schemes, preconditioners, GMRES algorithms, Multi-grid Algorithms with implementation.

Nonlinear Solvers: Newton's method and some of its variations, continuation methods, conjugate direction method and Davidon-Fletcher-Powell Algorithms. Non-linear Multigrid with applications.

Software Support: HOMOPACK, LAPACK, MADPACK.

Texts / References

O. Axelsson, Iterative Solution Methods

Cambridge Univ. Press, 1994.

23. Hackbusch, Multigrid Methods and

Applications. Springer-Verlag, 1985.

J.M. Ortega and W.C. Rheinboldt, Iterative

Solution of Nonlinear Equations in Several

Variables. Academic Press, NY, 1970.

C.W. Ueberrhuber, Numerical Computation :

Methods, Software and Analysis.

Springer-Verlag, Berlin, 1997.

16. Wesseling, An Introduction to Multigrid Methods. John Wiley & Sons, 1992.

Review of continuum model, Transport phenomena, Air quality modelling, (pollu-tion from chimney), Furnace reaction analysis,

De-icing helicopter blades (free and moving boundary problems), modelling microwave heating, Food contamination from the pack-aging, Electron Beam Lithography, Color negative film development, photocopy mach-ine; Selected case studies .

Software Support: MATHEMATICA, LSODE, GNUPLOT, MATLAB.

Texts / References

J. Crank, Free and Moving Boundary

Problems, Oxford Univ. Press, 1987.

A. Friedman, Mathematics in Industrial

Problems Part 1 – 9, IMA Series,

Springer-Verlag, 1991.

1. Friedman and W. Littman, Industrial

Mathematics for Under- graduates.

SIAM Publ. 1994.

Y.C. Fung, A First Course in Continuum

Mechanics, Prentice-Hall, 1969.

A. James (Ed.), An Introduction to Water

Quality Modelling, Wiley Pub. 1984.

M.S. Klamkin, (ed.), Mathematical

Modelling: Classroom Notes in Applied

Mathematics, SIAM, 1987.

Lecture Notes on Heat and Mass Transfer :

A Problem driven approach, M.Sc. in

Industrial Mathematics. Univ. Strathclyde,

U.K., 1995.

Mathematical preliminaries: assymptotic notation. Advanced Data Structures: Hash tables, Binomial Heaps, Disjoint sets.

Greedy Algorithms: Huffman coding, Minimum Spanning Tree construction, Dijkstra's Shortest Path construction. Dynamic

Programming Algorithms: Matrix-chain multi-plication, All pairs shortest path problems, Minimum weight triangulation of convex polygons. Divide and conquer: Linear time selection, Euclidean closest pair problem, Strassen's matrix multi plication algorithm.

Backtracking and Branch and Bound methods: Graph colouring, Integer programming. Approximation algorithms: Vertex cover, Euclidean travelling salesman problem, Set cover problem.

Texts / References:

T. Cormen, C. Leiserson, and R. Rivest,

Introduction to Algorithms, MIT

Press and McGraw Hill Book Company,

1991.

U. Manber, Introduction to Algorithms: A

Creative Approach, Addison-Wesley,

1989.

Review of 2nd order PDEs : Classification, separation of variables and Fourier transform techniques.

Automatic mesh generation techniques : Structured mesh (transfinite interpolation), unstructured grids (triangulation for polygonal and nonpolygonal domains).

Finite Difference Methods : Elliptic equations (SOR and conjugate gradient methods, ADI schemes), parabolic equations (explicit, backward Euler and Crank-Nicolson method, LOD), hyperbolic equations (Law-Wendroff scheme, Leapfrog method, CFL conditions) Stability, consistency and convergence results.

Lab Component: Implementation of Algo-rithms developed in this course and exposure to software packages: ODEPACK and MATLAB.

Texts / References

Gene H. Golub and James M. Ortega,

Scientific Computing and Differential

Equations : An Introduction to

Numerical Methods, Academic Press,

1992.

P. Knupp and S. Steinberg, Fundamentals of

Grid Generation, CRC Press Inc., Boca

Raton, 1994.

A.R. Mitchell and D.F. Griffiths, The Finite

Difference Methods in Partial

Differential Equations, Wiley, 1980.

G.D. Smith, Numerical Solutions of Partial

Differential Equations, Oxford Press,

1985.

J.C. Stickwards, Finite Difference Schemes

and PDEs, Chapman and Hall, 1989.

J.F. Thompson, Z.U., A. Waarsi and C.W.

Mastin, Numerical Grid Generations –

Foundations and Applications, North

Holland, 1985.

Erich Zauderer, Partial Differential Equations

of Applied Mathematics, 2^nd ed., Wiley,

1989.

Two-way contingency tables: Table structure for two dimensions. Ways of comparing proportions. Measures of associations. Sampling distributions. Goodness-of-fit tests, testing of independence. Exact and large sample inference .

Models of binary response variables. Logistic regression. Logistic models for categorical data. Probit and extreme value models. Log-linear models for two and three dimensions. Fitting of logit and log-linear models. Log-linear and logit models for ordinary variables.

Regression: Simple, multiple, non-linear regression, likelihood ratio test, confidence intervals and hypotheses tests, tests for distributional assumptions Collinearity, outliers, analysis of residuals. Model building, Principal component and ridge regression.

Lab component: Relevant real life problems to be done using statistical Software Packages such as SAS etc.

Texts / References

A. Agresti, Analysis of Categorical Data,

Wiley, 1990.

1. Agresti, An Introduction to Categorical

E.B. Andersen, The Statistical Analysis of

Categorical Data, Springer-Verlag, 1990.

R.F. Gunst and R.L. Mason, Regression

Analysis and its Applications – A Data

Oriented Approach, Marcel Dekkar, 1980.

T.J. Santner and D. Duffy, The Statistical

Analysis of Discrete Data, Springer-

Verlag, 1989.

A.A. Sen and M. Srivastava, Regression

Analysis – Theory, Methods and

Applications, Springer-Verlag, 1990.

Probability. Random variables. Standard distributions and their applications. Point and interval estimation. Testing of hypothesis. Regression and correlation analysis. ANOVA and ANACOVA. Design of experiments. Sequential analysi s.

Texts/References

A.B. Bowker and G.J. Liberman, Engineering

Statistics, Asia, 1972.

R.V. Hogg and E.A. Tanis, Probability and

Statistical Ingference, 2nd Ed.,

Macmillan, 1983.

N.L. Johnson and F.C. Xeen Leone, Statistics

and Experimental Design in Engineering

and the Physical Sciences,Vol.I and II,

2nd Ed. Wiley Interscicen, 1977.

Modules over PID with applications to the structure of finitely generated abelian groups and canonical forms of matrices. Basic theory of commulative rings: Localization, integral dependence, Noetherian and Artinian rings, Hilber t's Nullstellensatz, Hilbert series of graded algebras with applications to simplicial complexes, lattice points of convex polytopes and counting magic squares.

Texts / References

N. Jacobson, Basic Algebra, Vol. 1 and 2,

Hindustan Publishing Corporation, 1984.

T. Hibi, Algebraic Combinatorics on Convex

Polytopes, Carslaw Publications, 1993.

S. Lang, Algebra, 3rd ed., Addison-Wesley,

1993.

Zariski and Samuel, Commutative Algebra,

Van Nostrand, Princeton,Vol. I 1958, Vol.

II 1960. (New Printing by Springer-

Verlag)

Review of differentiable manifolds, tangent spaces and derivative of a map. Morse functions, local surfaces, passing critical points and attaching cells. CW complexes and CW homology. Cobordism. Isotopies, extending isotopies, gl uing manifolds, isotopies of discs. Surfaces, model surfaces, characterizations of disc, classification of compact surfaces.

Texts/References

M.W. Hirsch, Differential Topology,

Springer-Verlag, 1976.

Antoni A. Kosinski, Differential Manifolds,

Academic Press, New York, 1993.

J. Milnor, Morse Theory, Annals of Math.

Studies, # 51, PrincetonUniv. Press,

1963.

M. Morse, The Calculus of Variations in the

Large, AMS Colloquium Publication,

Vol.18, 1934.

Review of measure theory, Vitali covering theorem and its applications (Fundamental Theorem of Calculus for Lebesgue Integral). Complex measures, total variation, absolute continuity, Radon-Nikodym theorem with applications, posi tive Borel measures, Riesz representation theorem. Change of variables formula for Lebesgue integrals in Euclidean spaces.

Texts / References :

E. Hewitt and K. Stromberg, Real and

Abstract Analysis, Springer-Verlag, 1969.

S. Lang, Real and Functional Analysis, 3^rd

Ed., Springer-Verlag, Berlin, 1993.

I.K. Rana, An Introduction to Measure and

Integration, Narosa Publishing House,

1997.

W. Rudin, Real and Complex Analysis, Tata

McGraw-Hill, 3rd ed., 1985.

Topological vector spaces, separation properties, linear maps, boundedness and continuity, seminorms and local convexity, Baire category, Banach-Steinhaus theorm, open mapping and closed graph theorems, Hahn Banach theorems. Weak topologies. Distributions.

Calculus with distributions, convolutions, Fourier transforms, tempered distributions, Payley-Wiener theorems, Sobolev's lemma.

Texts / References

S. Lang, Real and Functional Analysis, 3^rd

Ed., Springer-Verlag, Berlin, 1993.

W. Rudin, Functional Analysis, Tata

McGraw-Hill, 1974.

F. Treves, Topological Vector Spaces,

Distributions and Kernels, Academic

Press, 1967.

K. Yosida, Functional Analysis, 4^th edn.

Springer ISE, Narosa, New Delhi, 1974.

Review of ordinary differential equations. Sturm-Liouville problems.

Green's functions. Phase plane analysis and stability. Review of first and second order partial differential equations, classification of equations. Asymptotic series and perturbation methods.

Survey of the integral transforms such as Fourier transforms, Laplace transforms, Hankel transforms. Application of integral transforms to the reduction of various boundary value problems. Fredholm and Voltera integral equations. Iterat ive solutions.

Singular integral equations.

Texts / References

R.K. Bose and M.C. Joshi, Methods of

Mathematical Physics, Tata-McGraw

Hill, 1984.

W.E. Boyce and R.C.Diprima, Elementary

Differential Equations and Boundary

Value Problems, Wiley, 1977.

S.G.Mikhlin, Integral Equations, Pergamon

Press, 1957.

I.N. Sneddon, Elements of Partial Differential

Equations,McGraw-Hill, 1957.

I.N. Sneddon, The use of Integral Transforms,

Tata McGraw-Hill, 1974.

1. Zauderer, Partial Differential Equations of

Applied Mathematics, Wiley, 1989.

MA 830 Numerical Analysis 3 0 0 6

Review of some requisites. Iterative methods for the solution of linear and non-linear algebraic systems of equations including Gauss-Seidel, SOR and conjugate gradient Newton methods. Derivations of the conditions of their conve rgence and stability with illustrations.

Discussion of finite difference methods (including ADI and LOD) and their convergence for the solution of elliptic, parabolic and hyperbolic partial differential equations.

Development of finite element methods with applications to various types of initial and boundary value problems.

Texts / References

K.J. Bathe, Finite Element Procedures in

Engineering Analysis, Prentice-Hall of

India, 1990.

G.F. Carey and J.T. Oden, Finite Elements :

Computational Aspects, Vol. III,

Prentice-Hall, 1984.

2. Issacson and H. B. Keller, Analysis of

Numerical Methods,Wiley, 1966.

M.K. Jain, Numerical Solution of Differential

Equations, Wiley Eastern, 1984.

A.A. Samarskii and E.S. Nikolaev, Numeical

Methods for Grid Equations, Birkhauser-

Verlag, 1989.

J.C. Strikwerda, Finite Difference Schemes

and Partial Differential Equations,

Wordsworth and Brooke/Coles

Advanced Books and Software,1989.

Basic equations of fluid flow. Constitutive equations and classification of fluids. Incom-pressible (Ideal, Newtonian, Non-Newtonian, Viscoelastic) fluid flows. Basic thermo-dynamics, compressible fluid flows. Super-sonic, transo nic and subsonic flows. Nonlinear wave propagation including shock waves of arbitrary strength.

Texts / References

D.J. Acheson, Elementary Fluid Dynamics.

Clarendon Press, 1990.

M.M. Denn, Process Fluid Mechanics,

Prentice-Hall, 1980.

Landau and Lipschitz, Fluid mechanics,

Pergamon Press, 1959.

R.V. Mises, Mathematical Theory of

Compressible Fluid Flows, Academic

Press, 1958.

A.R. Paterson, A First Course in Fluid

Dynamics, Cambridge Univ. Press, 1983.

H. Schlichting, Boundary Layer Theory,

McGraw-Hill, 1979.

A.H.P. Selland, Non-Newtonian Fluid and

Heat Transfer, Wiley, 1967.

Derivation of the basic equations of elasticity. Use of curvilinear coordinates. Solution for isotropic bodies in terms of potential functions. Problems depending on one har-monic function. Theories of plane strain and plane stre ss. Solutions of some plane problems using complex variable techniques. Axi-symmetric problems in the theory of elasticity.

Texts / References

T.M. Atanackovic and A. Guran, Theory

Of Elasticity for Scientists and Engineers,

Birkhauser, Boston, 2000.

A.E. Green and W. Zerna, Theoretical

Elasticity, Clarendon Press, 1963.

A.K. Mal and S.J. Singh, Deformation of

Elastic Solids, Prentice-Hall Inc.,

New Jersey, 1991.

I.S. Sokolnikoff, Mathematical Theory of

Elasticity, Tata McGraw-Hill, 1977.

Review of conditional expectations. Martingales in discrete and continuous time. Square integrable Martingales. Weak convergence in metric spaces with special reference to C([0,1]) space. Dependent variables. Diffusion processes and mixing. Martingale Central Limit Theorem.

Texts / References

P. Billingsley, Convergence of Probability

Measures, Wiley, 1999.

R.J. Elliot, Stochastic Calculus and

Applications, Springer-Verlag, 1982.

K.R. Parthasarathy, Probability Measures on

Metric Spaces, Academic Press, 1967.]

A.W. Van-der-Vaart and J.A. Wellner, Weak

Convergence and Empirical Processes:

With Applications to Statistics, Springer-

Verlag, 1996.

D. Williams, Probability with Martingales,

Cambridge Mathematical Textbooks,

1991.

UMP tests. Neymann-Pearson fundamental lemma. Distributions with ML ratio. Con-fidence bounds. Generalization of the fundamental lemma.

Least favourable distributions. Applications to normal distribution.

Similarity and completeness. UMP unbiased two-sided tests. Applications to exponential families. Fisher-Beherns problem. Unbaised confidence sets. Most powerful permutation and invariant tests. Admissibility of tests. Chi-square and likelihood ratio tests. Minimax tests and invariance. The Hunt-Stein theorem and its applications.

Texts / References

T.S. Ferguson, Mathematical Statistics : A

Decision Theoretic Approach, Academic

Press, 1967.

L. Le Cam, Asymptotic in Statistics, Springer-

Verlag, 1990.

E.L. Lehmann, Testing Statistical Hypotheses,

Wiley, 1986.

Elements of decision theory such as complete class theorem, admissibility of Bayes rule, minimax theorem.

Review of sufficiency, consistency, and efficiency. UMVU estimators and their properties. Application to normal and expo-nential one and two sample problems. Information inequality (multiparameter case). Equivariance. Invariance . Application to location and scale families.

MRE estimation.

Bayes and minimax estimation for exponential families. Admissibility of estimators. Blyth's ratio method. Karlin's sufficient conditions.

Pitman's estimator and its properties. Simul-taneous estimation.

Stein's phenomenon. Shrinkage estimation.

Texts / References

J. Berger, Statistical Decision Theory,

Springer-Verlag, 1980.

T.S. Ferguson, Mathematical Statistics : A

Decision Theoretic Approach, Academic

Press, 1967.

E.L. Lehmann, Theory of Point Estimation,

Wiley, 1983.

S. Zacks, The Theory of Statistical Inference,

Wiley, 1971.

Best asymptotically normal estimation. First and second order efficiency of estimators. Large deviations and measures of efficency.

Locally asymptotically normal models. Locally asymptotically best decision rules. Contiguity of measures and applications.

Texts / References

L. Le Cam, Asymptotic Methods in Statistical

Deceision Theory, Springer-Verlag, 1986.

L. Le Cam, Asymptotics in Statistics,

Springer-Verlag, 1990.

E.L. Lehmann, Theory of Point Estimation,

Wiley, 1983.

G.C. Roussas, Contiguity of Probability

Measures, Cambridge Univ. Press, 1972.

R.J. Serfling, Approximation Theorems of

Mathematical Statistics, Wiley, 1980.

This course will consist of lectures by faculty members on specialised areas in Mathematics. There will be three weekly meetings of one our each. More than one special topic can be covered in parallel under the same course. The c ourse will be coordinated by a single faculty member.

Functional organisation of computers, algo-rithms, basic programming concepts, FORTRAN language programming. Program testing and debugging. Modular programming subroutines: Selected examples from Numer-ical Analysis, Game playing , Sorting/ Searching methods, etc.

Texts / References

N.N. Biswas, FORTRAN IV Computer

Programming, Radiant Books, 1979 .

K.D. Sharma, Programming in Fortran IV,

Affiliated East West, 1976.

Introduction to the UNIX operating sytem (file system and directory structure, and processes).

Unix tools (shell programming, grep, tar, compress, sed, find, sort etc.). Programming in AWK. Introduction to Wold Wide Web (html, http, cgi).

Programming Using Java, Graphical User Interface programming using Java. Socket programming in Java.

Programming tools (make, source code control using sccs/rcs, debuggers). Document pro-cessing using Latex.

Nature of Business Systems and Data Pro-cessing.

Data Models, ER Diagrams and Data Flow diagrams. Introduction to Relational Theory, Normalisation.

SQL-92 and host language interfaces. Storage structures. Indexing and Hashing Techniques. Introduction to query processing and tran-sactions.

Web based Information Systems. Information Retrieval Systems.

Texts / References

Abraham Silberschatz, Henry F. Korth and

S. Sudarshan, Database System Concepts

3rd Ed, McGraw Hill, 1997.

James A. Senn, Analysis and Design of

Information Systems, McGraw-Hill, 1990.

Ramez Elmasri and Shamkant Navathe,

Fundamentals of Database Systems 2nd Ed,

Benjamin Cummings, 1994.

Use of database systems supporting interactive SQL and 4GLs, spread-sheets, client-server GUI development using packages like PowerBuilder or Gupta SQL, and using the Web. Design of applications and user interfaces using thes e systems.

CS 682 Software Engineering 3 0 0 6

Prerequisites: Exposure to Systems Progra-mming and Data Processing. Software life cycle - important steps and effort distribution. Aspects of estimation and scheduling. Software evaluation techniques-modular design : coupling and cohe sion, Software and complexity measures. Issues in software reliability.

System Analysis: Requirement analysis. Specification languages. Feasibility analysis. File and data structure design, Systems analysis tools.

Software design methodologies, Data flow and Data Structure oriented design strategies. Software development, coding, verification, and integration. Issues in project management-team structure, scheduling, software quality assura nce.

Object Oriented methodology: object orineted paradigm, OO analysis and design, examples of methodolgies (e.g., Rumbaugh's OMT).

Texts / References

J. Martin, Rapid Application Development,

Maxwell MacMillan, 1991.

B. Meyer, Object Oriented Software

Construction, Prentice Hall, 1988.

R. S. Pressman, Software Engineering - A

Practioner's Approach,3rd Edition,

McGrawHill,1992.

J. Rumbaugh et. al., Object Oriented Modeling

and Design, Prentice Hall,1991.

G. G. Schulmeyer, Zero Defect Software,

McGraw-Hill, 1992.

EE 678 Wavelets 3 0 0 6

Introduction to time frequency analysis; the how, what and why about wavelets.

Short-time Fourier transform, Wigner-Ville transform.

Continuous time wavelet transform, Discrete wavelet transform, tiling of the time-frequency plane and wavepacket analysis.

Construction of wavelets. Multiresolution analysis. Introduction to frames and biortho-gonal wavelets.

Multirate signal processing and filter bank theory.

Application of wavelet theory to signal denoising, image and video compression, multi-tone digital communication, transient detection.

Texts / References

A.N. Akansu and R.A. Haddad, Multi-

resolution signal Decomposition:

Transforms, Subbands and Wavelets,

Academic Press, Oranld, Florida, 1992.

B.Boashash, Time-Frequency Signal Analysis,

In editor S.Haykin,), Advanced Spectral

Analysis, pages 418--517. Prentice Hall,

New Jersey, 1991.

Y.T. Chan, Wavelet Basics, Kluwer

Publishers, Boston, 1993.

C. K. Chui, An Introduction to Wavelets,

Academic Press Inc., New York,1992.

I. Daubechies, Ten Lectures on Wavelets,

Society for Industrial and Applied

Mathematics, Philadelphia, PA, 1992.

Gerald Kaiser, A Friendly Guide to Wavelets,

Birkhauser, New York, 1995.

P. P. Vaidyanathan, Multirate Systems and

Filter Banks, Prentice Hall, New Jersey,

1993.

Introduction: Biological neurons and memory: Structure and function of a single neuron; Artificial neural networks (ANN); Typical applications of ANNs: Classification, Cluster-ing, vector quantization, pattern recognition, functi on approximation, forecasting, control, optimization; Basic approach of the working of ANN - Training, Learning and Gener-alization.

Supervised learning: single-layer networks; perceptron-linear separability, training algo-rithm, limitations; multi-layer networks-architecture, back propagation algorithm (BTA) and other training algorithms, appli-cations. Adapt ive multi-layer networks-architecture, training algorithms; recurrent networks; feed-forward networks; radial-basis-runction (RBF) networks.

Unsupervised learning: Winner-takes-all networks; Hamming networks; maxnet; simple competitive learning; vector-quantization; counter propagation networks; adaptive resonance theory; Kohonen's self-organizing maps; principal comp onent analysis.

Associated Models: Hopfield networks, brain-in-a-box network; Boltzmann machine.

Optimization methods: Hopfield networks for-TSP, Solution of simultaneous linear equations; Iterated gradient descent; simulated annealing; fenetic algorithm.

Texts / References

A. Cichocki and R. Unbehauen, Neural

Networks for Optimization and Signal

Processing, John Wiley and Sons, 1993.

K. Mehrotra, C.K. Mohan and Sanjay Ranka,

Elements of Artificial Neural Networks,

MIT Press, 1997 - [Indian Reprint Penram

International Publishing (India), 1997]

Simon Haykin, Neural Networks - A Compre-

hensive Foundation, Macmillan Publishing

Co., New York, 1994.

J. M. Zurada, Introduction to Artificial Neural

Networks, (Indian edition) Jaico Publishers,

Mumbai, 1997.