COURSE CURRICULA
M.Sc. in MATHEMATICS
First Year
First Semester
Code Name L T P C
MA 401 Linear Algebra 3 1 0 8
MA 403 Real Analysis I 3 1 0 8
MA 405 Complex Analysis 3 1 0 8
CS 101 Computer Programming and Utilization 2 1 0 6
MA 409 Differential Equations I 3 1 0 8
-----------------
14 5 0 38
----------------
Contact Hours : 19
Credits : 38
L = Lecture T = Tutorial P = Practical C = Credits
Second Semester
Code Name L T P C
MA 426 Numerical Analysis I 3 0 2 8
Elective 1 3 1 0 8
Elective 2 3 1 0 8
Elective 3 3 1 0 8
Elective 4 3 1 0 8
----------------
15 4 2 40
----------------
Any four electives from Group A, Group B and Group C.
Contact Hours : 21
Credits : 40
GROUP A
MA 402 Algebra I
MA 404 Real Analysis II
MA 406 General Topology
MA 408 Measure Theory
GROUP B
MA 410 Differential Equations II
MA 412 Fluid Mechanics I
MA 414 Solid Mechanics I
MA 416 Mathematical Metehods I
GROUP C
MA 418 Theory of Statistical Distributions
MA 420 Principles of Optimization
MA 422 Probability Theory
MA 424 Theory of Sampling
Second Year
First Semester
Code Name L T P C
Elective 5 2 1 0 6
Elective 6 2 1 0 6
Elective 7 2 1 0 6
Elective 8 2 1 0 6
Elective 9 2 1 0 6
MA 585 Seminar 0 0 0 4
MA 597 Project Stage I 0 0 0 5*
________________
10 5 0 39
________________
Contact Hours : 15
Credits : 39
ELECTIVES 5 - 9
MA 501 Algebra II
MA 503 Differentiable Manifolds
MA 505 Approximation Theory
MA 507 Convex Analysis and Optimization
MA 509 Elementary Number Theory
MA 511 Enumerative Combinatorics I
MA 513 Fourier Analysis
MA 515 Graph Theory
MA 517 Operators on Hilbert Spaces
MA 519 Representation Theory of Finite Groups
MA 521 Theory of Analytic Functions
MA 541 Computational Fluid Mechanics
MA 543 Finite Element Methods and Applications
MA 545 Fluid Mechanics II
MA 547 Mathematical Methods II
MA 549 Methods of Mathematical Physics
MA 551 Numerical Analysis II
MA 553 Second Order Elliptic Partial Differential Equations
MA 555 Solid Mechanics II
MA 571 Advanced Probability
MA 573 Mathematical Theory of Reliability
MA 575 Multivariate Analysis
MA 577 Statistical Inference I
MA 579 Stochastic Processes
Second Semester
Code Name L T P C
Elective 10 2 1 0 6
Elective 11 2 1 0 6
Elective 12 2 1 0 6
Elective 13 2 1 0 6
MA 598 Project Stage II 0 0 0 15*
______________
8 4 0 39
---------------
Contact Hours : 12
Credits : 39
* Grades for MA 597 and MA 598 (20 credits) will appear together
on completion of both stages.
ELECTIVES 10 - 13
MA 502 Algebraic Number Theory
MA 504 Banach Space Techniques
MA 506 Commutative Algebra
MA 508 Algebraic Topology
MA 510 Introduction to Algebraic Geometry
MA 512 Enumerative Combinatorics II
MA 514 Locally Convex Spaces and Distribution Theory
MA 516 Operator Theory
MA 518 Spectral Approximation
MA 520 Spline Theory and Variational Methods*
MA 540 Applied Functional Analysis
MA 542 Biomechanics
MA 544 Fracture Mechanics
MA 546 MHD and Plasma Physics
MA 548 Non-Newtonian Fluid Mechanics
MA 550 Semigroup Theoretic Approach to PDEs
MA 552 Tribology
MA 554 Variational Inequalities and Applications
MA 570 Design and Analysis of Experiments
MA 572 Non-Parametric Statistical Inference
MA 574 Regression Analysis
MA 576 Statistical Decision Theory
MA 578 Statistical Inference II
MA 580 Time Series Analysis
M.Sc. IN APPLIED STATISTICS AND INFORAMTAICS
First Year
First Semester
Code Name L T P C
CS 101 Computer Programming and Utilization 2 1 0 6
SI 401 Applied Linear Algebra 2 1 2 8
SI 403 Mathematical Modelling 3 0 2 8
SI 405 Mathematical Systems Theory 3 0 0 6
SI 407 Elements of Applied Probability
and Statistics 3 1 0 8
SI 409 Discrete Structures Lab. 0 0 3 3
SI 411 Computer Lab. 0 0 3 3
_______________
13 3 10 42
----------------
Contact Hours : 26
Credits : 42
L = Lecture T = Tutorial P = Practical C = Credit
Second Semester
Code Name L T P C
SI 408 Data Structures 3 0 2 8
SI 410 Programming Languages Lab. 0 0 3 3
CS 314 Business Information Systems 2 1 0 6
CS 398 BIS Lab. 0 0 3 3
SI 402 Numerical Methods 3 0 2 8
SI 404 Optimization Techniques 3 1 0 8
SI 406 Applied Stochastic Processes 3 1 0 8
________________
14 3 10 44
----------------
Contact Hours : 27
Credits : 44
Second Year
First Semester
Code Name L T P C
SI 509 Scientific Computing Lab. 0 0 3 3
SI 513 Statistical Data Analysis 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
Elective V 3 0 0 6
SI 597 Project (Stage I) 5
______________
15 0 6 41
--------------
Contact Hours : 21
Credits : 41
It is required to choose at least one and at most two
courses from each of the following Groups of Electives.
Group I (Informatics)
SI 501 Discrete Algorithms
SI 507 System Programming
SI 511 Computer-Aided Geometric Design
CS 475 Computer Graphics.
Group II (Applied Statistics)
SI 505 Categorical Data Analysis and Regression
SI 515 Applied Multivariate Analysis
SI 519 Probabilistic Techniques in Machine Learning
SI 521 Biostatistics
Group III (Mathematical Modelling and Scientific Computing)
SI 503 Finite Difference Methods for PDE's
SI 523 Mathematical Modelling and Numerical Simulations
Second Semester
Code Name L T P C
SI 520 Informatics Lab. 0 0 3 3
Elective VI 3 0 0 6
Elective VII 3 0 0 6
Elective VIII 3 0 0 6
Elective IX 3 0 0 6
SI 596 Work Visit PN/NP
SI 598 Project (Stage II) 15
________________
12 0 3 42
----------------
Contact Hours : 15
Credits : 42
It is required to choose at least two electives from
Group V and the remaining ones from Group IV.
Group IV (Informatics and Scientific Computing)
CS 328 Programming Languages
CS 470 Modelling and Simulation
SI 506 Introduction to Automata Theory and Languages
SI 508 Digital Logic and Computer Design
SI 510 Mathematical Elements for Computer Graphics
SI 522 Large Scale Scientific Computation
Group V (Applied Statistics)
SI 502 Stochastic Programming Applications
SI 504 Experimental Designs
SI 514 Computer-Oriented Statistical Techniques
SI 516 Reliability Techniques
SI 518 Statistical Quality Control
MA 580 Time Series Analysis
Ph.D.
First Semester
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
(a) The credit requirements for students having M.Sc. or
equivalent qualification admitted to the Department
shall be 34 to 46 credits.
(b) Credits acquired through PG level courses shall be 24 or
more.
(c) Students may earn upto a maximum of 8 credits through
Seminars which should be spread over two semesters.
Second Semester
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.
COURSE CONTENTS
DEPARTMENTAL COURESE
MA 001 Preparatory Mathematics I
Complex numbers as ordered pairs. Argand's diagram. Triangle
inequality. De Moivre's Theorem.
Algebra: Quadratic equations and expressions. Permutations and
combinations. Binomial 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 applications.
Matrices and Determinants: Algebra of matrices. Determinants and
their proeprties. Inverse of a matrix. Cramer's rule.
MA 002 Preparatory Mathematics II
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, trigonometric, logarithmic, exponential, hyperbolic,
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 application to the determination of areas (simple cases).
MA 101 Introductory Mathematics 3 1 0 8
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,
application to evaluation of areas and volumes. Differential
equations of first order. Linear differential equations with
constant coefficients.
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, 6th
ed., Addison-Wesley/Narosa, 1985.
MA 103 Mathematics I 2 0 2 6
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 integral and
the fundamental 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
G.B. Thomas, and R.L. Finney, Calculus and Analytic Geometry,
6th ed., Addison-Wesley/Narosa, 1985.
T.M. Apostol, Calculus, Vol. I, 2nd ed., Wiley Eastern, 1980.
MA 104 Mathematics II 3 0 2 8
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
E. Kreyszig, Advanced Engineering Mathematics, 5th ed., Wiley
Eastern, 1985.
V. Krishnamurthy, V.P. Mainra and J.L. Arora, An Introduction to
Linear Algebra, Affiliated East-West, 1976.
T.M. Apostol, Calculus, Vol. II, 2nd ed., Wiley Eastern, 1980.
MA 203 Mathematics III 3 0 2 8
Ordinary differential equations of the 1st order, exactness and
integrating factors, variation of parameters, Picard's iteration
method.
Ordinary linear differential equations of nth order, solution of
homogeneous and nonhomogeneous 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
convergence. 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 polynomials. Fourier integrals.
Transform techniques in differential equations.
Texts/References
E. Kreyszig, Advanced Engineering Mathematics, 5th ed., Wiley
Eastern, 1985.
W.E. Boyce and R.C. DiPrima, Elementary Differential Equations and
Boundary Value Problems, 3rd ed., Wiley, 1977.
G.F. Simmons, Differential Equations with Applications and
Historical Notes, Tata McGraw-Hill, 1972.
MA 204 Mathematics IV 2 1 0 6
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
E. Kreyszig, Advanced Engineering Mathematics, 5th ed., Wiley
Eastern, 1985.
P.E. Danko, A.G. Popov, T.YA. Koznevnikova, Higher Mathematics in
Problems and Exercises, Part 2, Mir Publishers, 1983.
MA 210 Introduction to Numerical Analysis 2 1 0 6
Interpolation by polynomials. Divided differences. Error of the
interpolating polynomial. 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 convergence. Finite
difference methods. Numerical solution 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, 5th ed., Wiley,
1985.
MA 212 Probability, Random Processes and
Statistical Inference 2 1 0 6
Basic definition of probability, random variables, probability
density function, probability distribution function, standard
univariate and multivariate distributions, conditional
distributions and densities, moment generating functions,
characteristic functions, limit theorems.
Point estimation, interval estimation. Hypothesis testing.
Simple linear regression, correlation.
Random Processes : Markov processes, stationary processes.
Ergodicity, autocorrelation, cross-correlation, power spectral
density.
Exposure to statistical packages like SAS and SPSS.
Texts/References :
M. O'Flynn, Probabilities, Random Variables and Random Processes,
Harper and Row, 1982.
A. Papoulis, Probability, Random variables and Stochastic
Processes, McGraw-Hill, 1985.
H. Stark and J.W. Woods, Probability, Random Processes and
Estimation Theory for Engineers, Prentice-Hall, 1986.
A.D. Allen, Probability, Statistics and Queueing Theory with
Computer Science Applications, 2nd ed., Academic Press, 1990.
H.J. Larson, Introduction to Probability Theory and Statistical
Inference, 3rd ed., Wiley, 1969.
D.M. Himmelblau, Process Analysis by Statistical Methods, Wiley,
1970.
MA 401 Linear Algebra 3 1 0 8
Prerequisite : Nil
Vector spaces, bases and dimension, direct sums, quotient
spaces.
Algebra of linear transformations, rank and nullity, dual
spaces.Inner product spaces, Gram-Schmidt orthogonalization
process. Determinants, Eigenvalues and eigen-vectors.
Cayley-Hamilton Theorem. Traces, transposes and adjoints.
Normal, unitary and self-adjoint transformations, orthogonal
projections. Spectral theorem for normal operators. Triangular
forms, nilpotent transformations, Jordan and rational canonical
forms. Quadratic forms.
Texts/References
P.R. Halmos, Finite Dimensional Vector Spaces, Princeton
University Press, 1958.
I.N. Herstein, Topics in Algebra, Wiley Eastern, 1987.
K. Hoffman and R. Kunze, Linear Algebra, Prentice-Hall, 1991.
M. Artin, Algebra, Prentice-Hall, 1990.
A. Ramanchandra Rao and P. Bhimasankaram, Linear Algebra, Tata
McGraw-Hill, 1992.
MA 402 Algebra I 3 1 0 8
Prerequisite : MA 401 (Exposure)
Review of groups, subgroups, homomorphisms, finite and discrete
groups of motions, group actions, class equation, Sylow
theorems, groups of order 12, generators and relations, SL(R),
SU(2), simplicity of alternating groups and PSL(2). Rings,
ideals, quotient rings, Euclidean domains, principal ideal
domains, unique factorization domains, primes in Z[i] and
Fermat's 2-square theorem, ideal classes in imaginary quadratic
fields.
Modules, matrices, free modules and bases, diagonalization of
integer matrices, generators and relations for modules,
structure theorem for abelian groups, applications to Jordan
canonical forms and linear operators.
Extension fields, splitting fields, fundamental theorem of
Galois Theory, constructibility by ruler and compass, finite
fields.
Texts/References:
M. Artin, Algebra, Prentice-Hall, 1990.
I.N. Herstein, Topics in Algebra, Wiley Eastern, 1987.
K.D. Joshi, Foundations of Discrete Mathematics, Wiley Eastern,
1989.
N. Jacobson, Basic Algebra, Vol. I, Hindustan Publishing
Corporation, 1984.
MA 403 Real Analysis I 3 1 0 8
Prerequisite : Nil
Metric spaces, compactness, connectedness, completeness.
Continuity.
Monotonic functions. Differentiation of vector-valued functions.
Functions of bounded variation and absolutely continuous
functions.
Riemann-Stieltjes integral and its properties. Fundamental
theorem of integral calculus.
Sequences and series of functions, uniform convergence and its
relation to continuity, differentiation and integration.
Equicontinuous families of functions, Ascoli-Arzela theorem.
Weierstrass approximation theorem. Fourier series, Fejer's
theorem, pointwise convergence.
Fourier series with respect to a general orthogonal system,
Parseval's theorem.
Texts/References :
W. Rudin, Principles of Mathematical Analysis, 3rd ed.,
McGraw-Hill, 1982.
T. Apostol, Mathematical Analysis, 3rd ed., Tata McGraw-Hill,
1974.
MA 404 Real Analysis II 3 1 0 8
Prerequisites : MA 401 (Exposure), MA 403 (Exposure)
Functions of several variables: differentiation, the inverse
function theorem, the implicit function theorem, the rank
theorem, derivatives of higher order and differentiation of
integrals, integration, change of variables, Divergence and
Stokes' theorem in Euclidean spaces.
Hilbert spaces, orthonormal basis, projection and Riesz
representation theorems. Approximation and optimization in
Hilbert spaces.
Variational problems, Lax-Milgram lemma and its applications.
Texts/References:
W. Rudin, Principles of Mathematical Analysis, 3rd ed.,
McGraw-Hill, 1982.
T. Apostol, Mathematical Analysis, 3rd ed., Tata McGraw-Hill,
1974.
W. Fleming, Functions of Several Variables, Springer-Verlag, 1977.
B.V. Limaye, Functional Analysis, 2nd ed., Wiley Eastern, 1996.
J.N. Reddy, Applied Functional Analysis and Variational Methods in
Engineering, McGraw-Hill International Edition, 1986.
MA 405 Complex Analysis 3 1 0 8
Prerequisite : Nil
Complex numbers and the point at infinity. Analytic functions.
Cauchy-Riemann conditions, harmonic functions. Mappings by
elementary functions. Riemann surfaces. Conformal mapping.
Transformations of harmonic functions. Applications to physical
problems involving Laplace's equation. Contour integrals,
Cauchy-Goursat theorem, simply and multiply connected domains.
Uniform convegence of sequences and series. Taylor and Laurent
series. Isolated singularaities and residues. Evaluation of
real integrals. Calculation of inverse Laplace transforms.
Zeros and poles. The argument principle. Rouche's theorem.
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.
MA 406 General Topology 3 1 0 8
Prerequisite : MA 403 (Exposure)
Topologies through open sets, bases, sub-bases, closure,
interior, boundary, subspaces.
Continuity, open functions, homeomorphisms, embeddings, strong
and weak topologies generated by families of functions. Quotient
spaces.
First and Second countable, separable, Lindeloff, compact
spaces.
Separation axioms, Urysohn's lemma. Products, embeddings into
products, Urysohn metrisation theorem, Convergence of nets and
filters. Filters and compactness, ultrafilters, Tychonoff
compactness theorem. Local compactness, Alexandroff
compactification. Function spaces, compact-open topology.
Connectedness, components, local connectedness, paths, loops.
Homotopy, fundamental group. Computation of the fundamental
group of the circle.
Texts/References
K.D. Joshi, Introduction to General Topology, Wiley Eastern, 1983.
J.L. Kelly, General Topology, Van Nostrand, 1955.
MA 408 Measure Theory 3 1 0 8
Prerequisite : MA 403 (Exposure)
Semi-algebra, algebra, sigma-algebra, monotone classes. Measure
spaces. Extension of measures from algebras to sigma-algebras.
Lebesgue, Lebesgue-Stieltjes measures. Properties of Lebesgue
measure on R. Simple functions, Measureable functions and their
properties. Egoroff's Theorem. Convergence a.e. and convergence
in measure. Integration with respect to a measure. Monotone
convergence theorem, Fatou's lemma, Dominated convegence
theorem. Spaces of p-integrable functions, convergence in pth
mean and its relations to other modes of convergence.
Absolute continuity of measures, Radon-Nikodym Theorem. Product
measure spaces. Fubini's theorem. Lebesgue measures in Euclidan
spaces and their properties. Change of variables formula.
Texts/References
H. Baur, Probability Theory and Elements of Measure Theory,
Academic Press, 1985.
P. Billingsley, Probability and Measure, Wiley, 1985.
P.R. Halmos, Measure Theory, Graduate Texts in Maths,
Springer-Verlag, 1979.
K.R. Parthasarathy, Introduction to Measure and Probability,
Macmillan, 1977.
MA 409 Differential Equations I 3 1 0 8
Prerequisite : Nil
Review of solution methods for first order and second order
equations; power series method and properties of Bessel
functions and Legendre polynomials, existence and uniqueness of
solutions for initial value problems, systems of linear
differential equations: Principle of superposition, fundamental
solutions and their properties. Method of variation of
parameters. Critical points and stability for autonomous
systems. Sturm-Liouville problems, eigenfunction expansions with
properties. Cauchy problem for first order quasilinear partial
differential equations: method of characteristics and Charpit's
method.
Classification of second order equations in two space variables.
Separation of variable methods for Laplace, Heat and Wave
equations.
Texts/References:
G. Birkhoff and G.C. Rota, Ordinary Differential Equations, Wiley,
1978.
M. Braun, Differential Equations and Their Applications, 4th ed.,
Springer Verlag, 1993.
E.A. Coddington, An Introduction to Ordinary Differential
Equations, Prentice-Hall, 1974.
R.P. Agarwal and R. Gupta, Essentials of Ordinary Differential
Equations, Tata McGraw-Hill, 1991.
F. John, Partial Differential Equations, 3rd ed., Narosa, 1979.
I.N. Sneddon, Elements of Partial Differential Equations,
McGraw-Hill, 1957.
A.N. Tychonov and A.A. Samarski, Partial Differential Equations of
Mathematical Physics, Vol. I, Holden-Day, 1970.
H.F. Weinberger, A First Course in Partial Differential Equations,
Blaisdell, 1965.
MA 410 Differential Equations II 3 1 0 8
Prerequisite : MA 409 (Exposure)
Classification of partial differential equations in general:
second order equations in several variables, first order
systems. Stability theory, energy conservation and dispersion.
Wave equation: Uniqueness, D'Alembert's method, method of
spherical means, method of descent and method of successive
approximation.
Fourier transforms and applications to initial value problems
for heat and wave equations. Review of method of separation of
variables, construction of Green's function and properties.
Uniqueness of solution by energy method, maximum principle for
elliptic and parabolic equations.
Symmetric Hyberbolic Systems: Basic energy inequality, existence
and uniqueness of solution.
Texts/References:
F. John, Partial Differential Equations, 3rd ed., Narosa, 1979.
I.N. Sneddon, Elements of Partial Differential Equations,
McGraw-Hill, 1957.
I.N. Tychonov and A.S. Samarski, Partial Differential Equations of
Mathematical Physics, Vol. I, Holden-Day, 1970.
H.F. Weinberger, A First Course in Partial Differential Equations,
Blaisdell, 1965.
Erich Zanderer, Partial Differential Equations of Applied
Mathematics, 2nd ed., Wiley, 1989.
MA 412 Fluid Mechanics I 3 1 0 8
Prerequisite : Nil
Continuum hypothesis, velocity field, stream function, velocity
potential, stress tensor, pressure, vorticity, strain-rate,
constitutive equation (stress and strain rate relation),
classification of fluids.
Conservation laws (mass, momentum and energy), Kelvin's
circulation theorem, Bernoulli's equation and its applications,
potential flows, and D'Alembert's paradox.
Navier-Stokes equations in cartesian, cylindrical polar,
spherical polar and curvilinear coordinate systems. Simple cases
of viscous flows (channel flow, Couette flow, Poiseuille flow,
flow between rotating and non-rotating coaxial cylinders etc.)
and their applications.
Non-dimensionalization of Navier-Stokes equation and
introduction to non-dimensional numbers (Reynolds, Prandtl,
Eckert etc.) and their implications.
Texts/References:
L.M. Milne-Thomson, Theoretical Hydrodynamics, Macmillan, 1962.
H.S. Schlichting, Boundary Layer Theory, McGraw-Hill, 1979.
A.H.P. Skelland, Non-Newtonian Flow and Heat Transfer, Wiley,
1967.
F. Chorlton, Textbook of Fluidmechanics, D. Van Nostrand, 1967.
Y.C. Fung, A First Course in Continuum Mechanics, Prentice-Hall,
1969.
MA 414 Solid Mechanics I 3 1 0 8
Prerequisite : Nil
Theory of stress, the state of stress at a point, the laws of
stress transformation, geometric representations of the state of
stress at a point. Theory of strain, changes in lengths of
straight line segments and transformation of angles between
straight line segments under the assumptions of infinitesimal
deformations.
Hooke's law, homogeneous isotropic media, fundamental
boundary-value problems of elasticity, unqiueness of solution.
Plane stress and plane strain equation.
Texts/References:
A.J. Durelli, E.A. Phillips and C.H. Tsao, Introduction to the
Theoretical and Experimental Analysis of Stress and Strain,
McGraw-Hill, 1958.
I.S. Sokolnikoff, Mathematical Theory of Elasticity, Tata
McGraw-Hill, 1977.
MA 416 Mathematical Methods I 3 1 0 8
Prerequisite : MA 409 (Exposure)
Introduction to perturbation theory : Asymptotic expansions.
Method of steepest descent. Regular and singular perturbation
methods. Methods of strained coordinates, multiple scales,
matched asymptotic expansions.
Singular perturbation methods. Variational techniques : Ritz
method, Galerkin method, Least square method.
Texts/References
S.G. Mikhlin, Variational Methods in Mathematical Physics,
Macmillan, 1964.
Ali Nayfeh, Perturbation Methods, Wiley, 1973.
C.M. Bender and S.A. Orszag, Advanced Mathematical Methods for
Scientists and Engineer, McGraw-Hill, 1978.
J. Kevorkian and J.D. Cole, Perturbation Methods in Applied
Mathematics, Springer Verlag, 1985.
MA 418 Theory of Statistical Distributions 3 1 0 8
Prerequisite : Nil
Standard univariate and multivariate distributions. Generating
functions, Characteristic functions, inversion theorem,
continuity theorem.
The problem of moments. Distribution of functions of random
variables.
Pearson distributions, Edgeworth and other related expansions,
Order Statistics, distribution of random variables defined
through ranks.
Multiple and partial correlations. Limiting distributions of
likelihood ratio statistics.
Texts/References
M. Fisz, Probability and Mathematical Statistics, Wiley, 1961.
C.R. Rao, Linear Statistical Inference and its Applications, Wiley
Eastern, 1973.
M.G. Kendall, and A. Stuart, The Advanced Theory of Statistics,
Vol.1, Griffin, 1958.
N.L. Johnson, and S. Kotz, Discrete Distributions, Wiley, 1969.
N.L. Johnson, and S. Kotz, Distributions in Statistics :
Continuous Univariate Distributions, Vol. I & II, Wiley, 1970.
N.L. Johnson, and S. Kotz, Distributions in Statistics: Continuous
Multivariate Distributions, Wiley, 1972.
MA 420 Principles of Optimization 3 1 0 8
Prerequisite : Nil
Mathematical foundations. Linear Optimization. Simplex method.
Revised simplex method. Duality and sensitivity. Unconstrained
optimization of functions of several variables. Classical
techniques. Numerical methods for unconstrained optimization.
Constrained optimization of functions of several variables.
Lagrange multipliers. Kuhn-Tucker theory. Numerical methods for
constrained optimization. Convex optimization. Quadratic
optimization. Dynamic programming.
Texts/References
G. Hadley, Linear Programming, Addison Wesley, 1962.
G. Hadley, Non-linear and Dynamic Programming, Addison Wesley,
1964.
M. Panik, Classical Optimization : Foundations and Extensions,
North Holland/American Elsevier, 1976.
S.S. Rao, Optimization Theory and Applications, Wiley Eastern,
1978.
J.K. Sharma, Mathematical Models in Operations Research, Tata
McGraw-Hill, 1989.
D.M. Himmelblau, Applied Nonlinear Programming, McGraw-Hill, 1972.
MA 422 Probability Theory 3 1 0 8
Prerequisite : Nil
Probability space, conditional probability, independence of
events, Borel-Cantelli lemmas, zero-one laws.
Random variables, distribution functions, sequences of random
variables, expected value, convergence theorems, various modes
of convergence. Fubini's theorem (statement only). Joint
distributions, independence of random variables.
Moment generating function, characteristic function, central
limit theorems, laws of large numbers.
Radon-Nikodym Theorem (statement only), conditional expectation,
conditional distribution.
Texts/References
H. Bauer, Probability Theory and Elements of Measure Theory,
Academic Press, 1981.
P. Billingsley, Probability and Measure, Wiley, 1985.
MA 424 Theory of Sampling 3 1 0 8
Prerequisite : Nil
Simple random sampling. Sampling for proportions and
percentages.
Estimation of sample size. Stratified random sampling, ratio
estimators. Regression estimators. Systematic sampling. Type of
sampling unit, Subsampling with units of equal and unequal size.
Double sampling. Sources of errors in surveys.
Texts/References
W.G. Cochran, Sampling Techniques, 3rd ed., Wiley Eastern, 1977.
Des Raj, Sampling Theory, Tata McGraw-Hill, 1978.
A. Chaudhuri and H. Stenger, Survery Sampling: Theory and Methods,
Marcell Dekker, 1992.
MA 426 Numerical Analysis I 3 0 2 8
Prerequisite : Nil
Floating point arithmetic and rounding errors. Polynomial
interpolation, Newton divided differences, interpolation error.
Linear and cubic splines. Least squares approximation by
polynomials.
Numerical integration and differentiation.
Systems of linear equations: Gaussian elimination with scaling
and pivoting. LU and QR decomposition. Perturbation of a
solution.
Iterative improvement. Jacobi, Gauss-Seidel and SOR methods.
Matrix eigenvalue problem : Gershgorin theorem. Power and
Inverse Power Methods.
Solution of non-linear equations: Regula-falsi, Secant and
Newton methods.
Numerical solution of ODEs: Initial value problem, single-step
and multi-step methods with stability analysis.
Texts/References:
K.E. Atkinson, An Introduction to Numerical Analysis, Wiley, 1978.
C.E. Froberg, Introduction to Numerical Analysis, Addison-Wesley,
1981.
E. Isaacson and H.B. Keller, Analysis of Numerical Methods, Wiley,
1966.
MA 501 Algebra II 2 1 0 6
Prerequisite : MA 402
Solution by radicals of equations of degree atmost 4, solvable
groups, solvability by radicals, Abel-Ruffini theorem, symmetric
functions, Newton's identities for symmetric functions, Galois
groups of equations of degree at most 4. Equations with
symmetric nd alternating groups as Galois groups. Reduction mod
p technique. Cyclotomic extensions, norm and trace, cyclic
extensions and Hilbert's theorem 90, Artin-Schreier theorem,
transcendental extensions.
Zero divisors, nilpotent elements, nilradial and Jacobson
radical, operations on ideals, extension and contraction,
examples of rings arising in Geometry, Combinatorics, Number
Theory and Topology.
Texts/References
O. Zariski and P. Samuel, Commutative Algebra, Vol. I, Van
Nostrand, 1958.
N. Jacobson, Basic Algebra Vols. I and II, Hindustan Publishing
Corporation, 1984.
M. Artin, Algebra, Prentice-Hall, 1990.
M.F. Atiyah and I.G. Macdonald, Introduction to Commutative
Algebra, Addison-Wesley, 1969.
MA 502 Algebraic Number Theory 2 1 0 6
Prerequisites : MA 501, MA 509
Binary quadratic forms, Legendre-Gauss theory of genera.
Algebraic numbers and their basic properties, Kummer's work on
Fermat's last theorem. Unique factorization of ideals in
algebraic number fields, Class group and class number,
Ramification of primes. Discriminant, Norms of ideals,
Reciprocity laws, Cyclotomic fields and Kronecker-Weber theorem
(statement only). Introduction to class field theory.
Texts/References
J.W. Cassels, Local Fields, Cambridge Press, 1986.
J.W. Cassels and A. Frohiich, Algebraic Number Theory, Academic
Press, 1967.
H.M. Edwards, Fermat's Last Theorem, Springer-Verlag, 1977.
K. Ireland and M. Rosen, A Classical Introduction to Modern Number
Theory, 2nd ed., Springer-Verlag, 1990.
S. Lang, Algebraic Number Theory, Addison-Wesley, 1970.
D.A. Marcus, Number Fields, Springer-Verlag, 1977.
MA 503 Differentiable Manifolds 2 1 0 6
Prerequisite : MA 404
Integration on chains. Stokes' theorem. Differential manifolds,
tangent spaces, differentiable maps, immersions, Whitney's
embedding theorem, manifolds with boundary, transversality,
intersection theory mod 2. Differential forms, de Rham
cohomology,orientability of manifolds and wedge product
pairings.
Texts/References
W. Rudin, Principles of Mathematical Analysis, McGraw-Hill, 1976.
V. Guillemin and A. Pollack, Differential Topology, Prentice-Hall,
1974.
M.W. Hirsch, Differential Topology, Springer-Verlag, 1976.
MA 504 Banach Space Techniques 2 1 0 6
Prerequisite : MA 403
Normed spaces. Continuity of linear maps. Hahn-Banach theorems.
Banach spaces. Uniform boundedness principles. Closed graph and
open mapping theorems. Duals and transposes. Weak and weak-star
convergence. Reflexivity.
Texts/References
B.V. Limaye, Functional Analysis, 2nd ed., Wiley Eastern, 1996.
K. Yoshida, Functional Analysis, Springer-Verlag, 1965.
G. Goffman and G. Pedrick, First Course in Functional Analysis,
Prentice-Hall, 1974.
A. Taylor and D. Lay, Introduction to Functional Analysis, Wiley,
1980.
J.B. Conway, A Course in Functional Analysis, Springer-Verlag,
1985.
MA 505 Approximation Theory 2 1 0 6
Prerequisite: MA 403 (Exposure)
Positive operators and Korovkin's theorem, Bernstein
polynomials, Fejer's theorem, Stone-Weierstrass theorem.
Classical Chebyshev theory, discretization and discrete best
approximation, the algorithms of Remes. Degree of approximation,
moduli of continuity and K-functionals, direct and converse
theorems. Hermite-Birkhoff interpolation, piecewise polynomial
interpolation. Best approximation in normed linear spaces.
Texts/References
G.G. Lorentz, Approximation of Functions, Holt, Rinehart and
Winston,
E.W. Cheney, Introduction to Approximation Theory, McGraw-Hill,
1966.
P.J. Laurent, Approximation et Optimisation, Hermann, 1972.
I. Singer, The Theory of Best Approximation and Functional
Analysis, CBMS Lecture Notes, No. 13, SIAM, 1974.
MA 506 Commutative Algebra 2 1 0 6
Prerequisite : MA 501
Rings and modules, localization, Noetherian rings, primary
decomposition, Artinian rings, integral extensions, Hilbert's
Nullstellensatz, Noether's normalization, valuation rings,
Dedekind domains, Dimensions Theorem, Completions.
Texts/References
O. Zariski and P. Samuel, Commutative Algebra, Vols. I and II, Van
Nostrand, 1958 and 1960.
M.F. Atiyah and I.G. Macdonald, Introduction to Commutative
Algebra, Addison-Wesley, 1969.
N.S. Gopalakrishnan, Commutative Algebra, Oxonian Press, 1984.
N. Jacobson, Basic Algebra, Vol. II, Hindustan Publishing
Corporation, 1984.
D. Eisenbud, Commutative Algebra : With a View Towards Algebraic
Geometry, Springer-Verlag, 1995.
MA 507 Convex Analysis and Optimization 2 1 0 6
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
R.T. Rockafellar, Convex Analysis, Princeton University Press,
1959.
R.T. Rockafellar, Conjugate Duality and Optimization, CBMS Lecture
Notes, Series No. 13 SIAM, 1974.
P.J. Laurent, Approximation et Optimization, Hermann, 1973.
M.S. Bazaraa and C.M. Shetty, Foundations of Optimizations,
Lecture Notes in Economics and Management Systems,
Springer-Verlag, 1976.
MA 508 Algebraic Topology 2 1 0 6
Prerequisites : MA 402, MA 406 (Exposure)
Category theory : Categories and functors, natural
transformations, adjoints, universal objects. Review of homotopy
and fundamental groups. Covering spaces : path lifting, homotopy
lifting property, universal covering spaces; relation with
fundamental group.
Computation of fundamental group : Van-Kampen's theorem.
Simplicial complexes : Homology of chain complexes, Simplicial
approximation theorem; edge path groupoid.
Homology: Simplicial and singular homology, Brouwer's fixed
point theorem and invariance of domain.
Cohomology ring : Structure of mod 2 cohomology of projective
spaces.
Statement of the de Rham theorem.
Texts/References
S. Greenberg, Lectures on Algebraic Topology, Benjamin, 1967.
P. Hilton and S. Wylie, Homology Theory, An Introduction to
Algebraic Topology, Cambridge University Press, 1967.
J.R. Munkres, Elements of Algebriac Topology, Addison-Wesley,
1984.
MA 509 Elementary Number Theory 2 1 0 6
Prerequisite : Nil
Divisibility, Primes, Unique factorization of integers,
Congruence, Chinese remainder theorem, Arithmetical functions,
Mobius inversion, Quadratic reciprocity, Diophantine equations,
Fermat's two square theorem, Lagrange's four square theorem,
Waring's problem. Dirichlet's theorem on primes in arithmetic
progression. Geometry of numbers, Minkowski's theorem. Prime
number theorem.
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.
G.H. Hardy and E.M. Wright, An Introduction to the Theory of
Numbers, 4th ed., Clarendon Press, 1960.
K. Ireland and M. Rosen, A Classicial Introduction to Modern
Number Theory, Springer-Verlag, 1982.
I. Niven and H.S. Zuckerman, Introduction to the Theory of
Numbers, 3rd ed., Springer-Verlag, 1982.
J.P. Serre, A Course in Arithmetic, Narosa, 1979.
MA 510 Introduction to Algebraic Geometry 2 1 0 6
Prerequisite : MA 501
Affine and projective varieties, coordinate rings, Rational
functions and local rings, singular points and tangent lines,
Rational parametrization, Branches and valuations, Intersection
multiplicity, Bezout's theorem for plane curves, Max Noether's
theorem. Varieties, morphisms and rational maps. Resolution of
singularities of curves.
Texts/References
S.S. Abhyankar, Algebraic Geometry for Scientists and Engineers,
American Mathematical Society, 1990.
W. Fulton, Algebraic Curves, Benjamin, 1969.
M. Reid, Undergraduate Algebraic Geometry, Cambridge University
Press, 1990.
I.R. Shafarevich, Basic Algebraic Geometry, Springer-Verlag, 1974.
R.J. Walker, Algebraic Curves, Springer-Verlag, 1950.
J. Harris, Algebraic Geometry : A First Course, Springer-Verlag,
1992.
MA 511 Enumerative Combinatorics - I 2 1 0 6
Prerequisites : MA 401, 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,
determinants etc.
Introduction to combinatorial reciprocity. Introduction to
symmetric functions.
Texts/References
R.P. Stanley, Enumerative Combinatorics, Vol. I, Wadsworth and
Brooks/Cole, 1986.
C. Berge, Principles of Combinatorics, Academic Press, 1972.
K.D. Joshi, Foundations of Discrete Mathematics, Wiley Eastern,
1989.
MA 512 Enumerative Combinatorics - II 2 1 0 6
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 multivariable Lagrange
inversion.
Texts/References
R.P. Stanley, Enumerative Combinatorics, Vol. I, Wadsworth and
Brooks/Cole, 1986.
B.E. Sagan, The Symmetric Group: Representations,Combinatorial
Algorithms and Symmetric Functions, Wadsworth & Brooks/Cole, 1991.
M. Aigner, Combinatorial Theory, Springer-Verlag, 1979.
MA 513 Fourier Analysis 2 1 0 6
Prerequisite : MA 408 (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-Weiner 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.
MA 514 Locally Convex Spaces and
Distribution Theory 2 1 0 6
Prerequisites : MA 406 (Exposure), MA 408 (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
W. Rudin, Functional Analysis, McGraw-Hill, 1973.
K. Yoshida, Functional Analysis, Academic Press, 1965.
L. Hormander, The Analysis of Linear PDE, Vols. I and II,
Springer-Verlag, 1983.
MA 515 Graph Theory 2 1 0 6
Prerequisite : Nil
Graphs, subgraphs and contractions. Menger's Theorem.
2-connection.
Alternating paths. Algebraic Duality. Polynomials associated
with graphs. Planarity. Matroids.
Texts/References
W.T. Tutte, Graph Theory, Cambrdige University Press, 1985.
K.R. Parthasarathy, Basic Graph Theory, Tata McGraw-Hill, 1994.
MA 516 Operator Theory 2 1 0 6
Prerequisite : MA 517 (Exposure)
Symmetric and self-adjoint operators. Unitary operators and
Cayley transforms. Banach algebras. Gelfand-Naimark theory for
commutative B-star algebras.
Spectral representation of bounded/unbounded self-adjoint
oprators and the associated functional calculus. One parameter
group of operators and Stone's theorem. Semigroups of
self-adjoint operators.
Collectively compact sets of operators, their properties and
application to spectral theory in Banach spaces. Totally bounded
sets of compact operators.
Texts/References
F. Riesz and B.Z. Nagy, Functional Analysis, Blackie, 1956.
A. Taylor and D. Lay, Introduction to Functional Analysis, 2nd
ed., Wiley, 1980.
P.M. Anselone, Collectively Compact Operator Approximation Theory,
Prentice-Hall, 1971.
MA 517 Operators on Hilbert Spaces 2 1 0 6
Prerequisite : MA 404
Bounded operators on Hilbert spaces : adjoint operators, normal,
unitary and self-adjoint operators and their spectra, numerical
ranges. Spectral theorem for compact self-adjoint operators and
its application to Sturm-Liouville problems. Unbounded
operators.
Texts/References
B.V. Limaye, Functional Analysis, 2nd ed., Wiley Eastern, 1996.
K.Y. Yoshida, Functional Analysis, Springer-Verlag, 1965.
G. Goffman and G. Pedrick, First Course in Functional Analysis,
Prentive Hall, 1974.
D.H. Griffel, Applied Functional Analysis, Ellis Horwood Ltd.,
Wiley, 1981.
MA 518 Spectral Approximation 2 1 0 6
Prerequisite : MA 517 (Exposure)
Resolvent sets and spectra of bounded and compact operators in
Banach spaces. Spectral projection, reduced resolvent and the
nilpotent operator. Neumann expansion and the analyticity of
spectral projctions. Rayleigh-Schrodinger series and the
iterative computation of eigenelements. Numerical approximation
by methods related to projections and by quadrature methods.
Algorithms for computing eigenelements and their computational
feasibility.
Texts/References
T. Kato, Perturbation Theory of Linear Operators, 2nd
ed.,Springer-Verlag, 1980.
F. Chatelin, Spectral Approximation of Linear Operators, Academic
Press, 1983.
B.V. Limaye, Spectral Perturbation and Approximation with
Numerical Experiments, Proc. Centre Math. Anal. Vol. 13,
Australian National Univ., 1987.
MA 519 Representation Theory of Finite Groups 2 1 0 6
Prerequisite : MA 402
Representations, Subrepresentations, Tensor products, Symmetric
and Alternating Squares.
Characters, Schur's lemma, Orthogonality relations,
Decomposition of regular representation, 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 p q theorem. The
character of induced reporesentation, Frobenius Reciprocity
Theory, Meckey's irreducibility criterion, Examples of induced
representations, Representations of supersolvable groups.
Texts/References
J.P. Serre, Linear Representation of Groups, Springer-Verlag,
1977.
N. Jacobson, Basic Algebra II, Hindustan Publishing Corproation,
1983.
M. Burrow, Representation Theory of Finite Groups, Academic Press,
1965.
S. Lang, Algebra, Addison-Wesley, 1965.
MA 520 Spline Theory and Variational Methods 2 1 0 6
Prerequisite : MA 403 (Exposure)
Piecewise linear approximation. Piecewise cubic interpolation.
Cubic spline interpolation and its errors. Representation of
piecewise polynomial diminishing splines. Interpolating and
smoothing splines.
Approximate representation of linear functions. Optimal
quadratures.
Variational formulation of generalized splines. Surface
approximation by tensor product splines. The
Rayleigh-Ritz-Galerkin procedures of elliptic problems,
Semi-discrete Galerkin procedure for parabolic problems.
Texts/References
C. de Boor, A Practical Guide to Splines, Springer-Verlag, 1978.
M.H. Schultz, Spline Analysis, Prentice-Hall, 1973.
P.J. Laurent, Approximation et Optimization, Hermann, 1972.
MA 521 Theory of Analytic Functions 2 1 0 6
Prerequisites : MA 403, MA 405
Open mapping property of analytic functions,mean value property
of harmonic functions, Poisson integral representation of
harmonic functions, Schwarz lemma and Phragmen-Lindelof method.
Approximation by rational functions. Riemann mapping theorem,
simply and doubly connected domains.
Texts/References
W. Rudin, Real and Complex Analysis, Tata McGraw-Hill, 3rd ed.,
1987.
E. Hille, Analytic Function Theory, I and II, Blaisdell, 1959.
MA 540 Applied Functional Analysis 2 1 0 6
Prerequisite : MA 404 (Exposure)
Monotone operators. Fixed point theorems. Degree theory and
condensing operators. Sobolov spaces. Elliptic Boundary value
problems.
Semi-group theory. Applications to integral and
integro-differential equations.
Texts/References
M.C. Joshi and R.K. Bose, Some Topics in Nonlinear Functional
Analysis, Wiley Eastern, 1985.
S. Kesavan, Topics in Functional Analysis and Applications, Wiley
Eastern, 1989.
E. Zeilder, Nonlinear Functional Analysis and its Application,
Springer-Verlag.
MA 541 Computational Fluid Mechanics 2 1 0 6
Prerequisites : MA 407 (Exposure), MA 410 (Exposure), MA 412
General Introduction to computational methods, similarity
transformations and applications to boundary layer flows.
Computational methods for boundary value problems involving
O.D.E., explicit and implicit methods for solving unsteady
flows, method of characteristics for hyperbolic equations.
Upwind differencing and artificial viscosity, numerical
solutions of biharmonic equations, numerical methods for solving
problems involving subsonic, supersonic and transonic flows.
Texts/References
C.Y. Chow, An Introduction to Computational Fluid Mechanics,
Wiley, 1979.
M. Holt, Numerical Methods in Fluid Mechanics, Springer-Verlag,
1977.
MA 542 Biomechanics 2 1 0 6
Prerequisites : MA 412 (Exposure), MA 416 (Exposure)
The history of biomedicine - a brief review. Overall description
of the human body. Physical, chemical and rheological properties
of blood. The dynamics of the circulatory system. The human
thermal sytems. Modelling the body as compartments, sources, and
streams.
Transport through cell membranes. Artificial kidney devices.
Artificial heart-lung devices.
Texts/References
David O. Cooney, Biomedical Engineering Principles : An
Introduction to Fluid Heat and Mass Transport Processes,
Marcel Dekker, 1976.
Y.C. Fung, Biomechanics: Mechanical Properties of Living Tissues,
Springer-Verlag, 1981.
E.N. Lightfoot, Transport Phenomenon in Living Systems, Wiley,
1974.
MA 543 Finite Element Method and Applications 2 1 0 6
Prerequisite : Nil
The fundamentals of finite element method. The shape functions,
Ritz and Galerkin finite element formulations. Finite element
formulation for Laplace, wave and diffusion equations.
Texts/References
J.N. Reddy, Finite Element Method, 2nd ed., McGraw-Hill, 1993.
D.H. Norrie and G. DeVries, Introduction to Finite Element Method
Analysis, Academic Press, 1957.
MA 544 Fracture Mechanics 2 1 0 6
Prerequisites : MA 414, MA 416 (Exposure), MA 547 (Exposure)
Mathematical theories of brittle fracture. Linear elastic
fracture mechanics. Crack-border stress fields and stress
intensity factors.
Two-dimensional crack problems.
Texts/References
H. Liebowitz, Fracture, Vol. 2, Mathematical Fundamentals,
Academic Press, 1968.
I.N. Sneddon and H. Lowengrub, Crack Problems in the Classical
Theory of Elasticity, Wiley, 1969
MA 545 Fluid Mechanics II 2 1 0 6
Prerequisites : MA 412, MA 410 (Exposure), MA 416 (Exposure)
Compressible fluid flow: Thermodynamics and physical properties
of gases. Sound waves. One dimensional flows. Sub-sonic,
transonic, super-sonic and hypersonic flows. Shock conditions
and propagation of shock waves. Self-similar flows.
Methods: characteristics, hodograph, simple waves, perturbation.
Boundary layer flows in incompressible fluids: Boundary layer
approximations.
Texts/References
R. Von. Mises, Mathematical Theory of Compressible Fluid Flow,
Academic Press, 1958.
N. Curle and N.J. Davis, Modern Fluid Dynamics, Vol. 2, Van
Nostrand, 1971.
H. Schlichting, Boundary Layer Theory, McGraw-Hill, 1979.
MA 546 MHD and Plasma Physics 2 1 0 6
Prerequisites : MA 412, MA 416 (Exposure)
Motion of charged particles in electromagnetic fields, adiabatic
invariants, electric currents in an ionized gas, magnetic field
pressure, conduction and diffusion in ionized gases,
electromagnetic waves and radiation in plasmas, macroscopic
equations of plasma and fluid model, criteria for applicability
of a fluid description, hydro-magnetics, hydromagnetic flows.
Texts/References
G.W. Sutton and A. Sherman, Engineereing Magnetohydrodynamics,
McGraw-Hill, 1965.
T.J.M. Boyd and J.J. Sanderson, Plasma Dynamics, Thomas Nelson,
1969.
C.L. Longmire, Elementary Plasma Physics, Wiley Eastern, 1971.
MA 547 Mathematical Methods II 2 1 0 6
Prerequisite : MA 410 (Exposure)
Integral transforms of Fourier, Laplace, Hankel and Mellin.
Fredholm and Volterra integral equations and the iterative
solutions. Fredholm alternative, Symmetric kernels and singular
integral equations.
Texts/References
I.N. Sneddon, Fourier Transforms, McGraw-Hill, 1951.
I.N. Sneddon, The Use of Integral Transforms, Tata McGraw-Hill,
1974.
S.G. Mikhlin, Integral Equations, Pergamon Press, 1957.
A.C. Pipkin, A Course on Integral Equations, Springer-Verlag,
1991.
MA 548 Non-Newtonian Fluid Mechanics 2 1 0 6
Prerequisites : MA 412 (Exposure), MA 416 (Exposure)
Introduction to non-Newtonian Fluids. Constitutive equations.
Classification of fluids. Flow of different non-Newtonian fluids
(Casson, Power Law, Bingham, Viscoelastic, Herschel Bulkley)
through various geometries (uniform circular tubes, elliptic
tubes, non-uniform tubes, viscometric flows, etc.). Unsteady
flows, Boundary Layer flow.
Stability, Application (lubrication, blood flow etc.).
Texts/Referencess
A.H.P. Skeland, Non-Newtonian Flow and Heat Transfer, Wiley, 1967.
J.N. Kapur, B.S. Bhatt and N.C. Sacheti, Non-Newtonian Fluid
Flows, Pragati Prakashan, 1982.
G. Astarita and G. Marucci, Principles of Non-Newtonian Fluid
Mechanics, McGraw-Hill, 1974.
MA 549 Methods of Mathematical Physics 2 1 0 6
Prerequisites : MA 401 (Exposure), MA 403 (Exposure)
Theory of distributions. Boundary value problems. Spectral
theory of second-order ordinary differential operators. Linear
Integral equations. Singular integral equations and partial
differential equations.
Texts/References
R.K. Bose and M.C. Joshi, Methods of Mathematical Physics, Tata
McGraw-Hill, 1984.
R. Courant and D. Hilbert, Methods of Mathematical Physics,
Interscience, 1966.
B. Friedman, Principles and Techniques of Applied Mathematics,
Wiley, 1956.
I. Stakgold, Boundary Value Problems of Mathematical Physics, Vols
I and II, The MacMillan Co., 1968.
MA 550 Semigroup Theoretic Approach to Partial
Differential Equations 2 1 0 6
Prerequisite : MA 404 (Exposure)
Semigroup of bounded linear operators and their generators.
Hille-Yosida and Lumere-Phillips theorems, characterisation of
the generators, compact semigroups, analytic semigroups.
Abstract Cauchy problem: existence, uniqueness and regularity of
mild solutions, asymptotic behaviour. Theory of semilinear
evolution equations and applications to Schrodinger equation,
nonlinear heat and Kolv equation.
Texts/References
A. Pazy, Semigroups of Linear Operator and Applications to PDEs,
Springer-Verlag, Appl. Maths. Sci. 44, 1983.
S. Kesavan, Topics in Functional Analysis and Applications, Wiley
Eastern, 1989.
MA 551 Numerical Analysis II 2 1 0 6
Prerequisite : MA 407 (Exposure)
Boundary value problems for ordinary differential equations:
finite difference and shooting methods.
Elliptic equations: discrete maximum principle and stability,
ADI and LOD methods. Finite difference schemes for initial and
boundary value problems: Lax-Ritchmyer equivalence theorem.
Stability: matrix, Von Neumann and energy methods.
Explicit, implicit (Euler Backward and Crank-Nicolson) methods
for parabolic equations: Lax-Wendroff Scheme, Leapfrog method,
CFL conditions
Texts/References
K.E. Atkinson, An Introduction to Numerical Analysis, Wiley, 1989.
A.R. Mitchell and S.D.F. Griffiths, The Finite Difference Methods
in Partial Differential Equations, Wiley, 1980.
G.D. Smith, Numerical Solution of Partial Differential Equations.
Finite Difference Methods, 3rd ed., Calrendorn Press
Oxford, 1985.
R.D. Ritchmyer and K.W. Morton, Difference Methods for Initial
Value Problems, Interscience, Wiley, 1969.
MA 552 Tribology 2 1 0 6
Prerequisites : MA 412 (Exposure), MA 414 (Exposure),
MA 416 (Exposure) Basic differential equations. Incompressible
Lubrication. One dimensional bearings. Finite bearings.
Hydrodynamic gas bearing.
Dynamic loading. Hydrostatic bearings. Instability.
Elastohydrodynamic lubrication. Inertia and turbulence effects.
Texts/References
O. Pinkus and B. Sternlicht, Theory of Hydrodynamical Lubrication,
McGraw-Hill, 1961.
G.W. Stachowiak and A.W. Batchelor, Engineering Tribology,
Elsevier Science Publishers, 1993.
MA 553 Second Order Elliptic Partial
Differential Equations 2 1 0 6
Prerequisites : MA 403 (Exposure), MA 405 (Exposure), MA 410
(Exposure)
Test functions and distributions, Fourier transform and tempered
distributions. Elements of Sobolev spaces, approximation by
smooth functions, trace and imbedding results. Existence and
regularity of weak solutions of elliptic equations, eigenvalue
problem. Semilinear elliptic equations: monotone iteration,
Galerkin methods, Variational methods and mountain pass theorem.
Texts/References
S. Kesavan, Topics in Functional Analysis and Applications, Wiley
Eastern, 1989.
M.C. Joshi and R.K. Bose, Some Topics in Nonlinear Functional
Analysis, Wiley Eastern, 1985.
L. Nirenberg, Variational Topological Methods in Nonlinear
Problems, Bull (New Series) of AMS and No.3, 267-302, 1985.
MA 554 Variational Inequalities and Applications 2 1 0 6
Prerequisite : MA 404 (Exposure)
Minimization of convex functionals: Fundamental theorem,
variational formulation of a minimization problem, projections
on convex sets.
Variational inequalities : Fundamental theorem, minimization of
convex functionals, Gateaux derivatives and subdifferentials.
Variational problem in one dimension: The obstacle problem.
Quasivariational problems : The K-K-M lemma and the Fan lemma.
Fixed point theorems of Brouwer, Schauder and Tychonov.
Multivalued applications.
Texts/References
C. Baiocchi and A. Capelo, Variational and Quasivariational
Inequalities, Wiley, 1984.
I. Ekeland and R. Temam, Analyse Convexe et Problemes
Variationnels, Dunod/Gautheir-Villars, 1974.
MA 555 Solid Mechanics II 2 1 0 6
Prerequisites : MA 410 (Exposure), MA 414, MA 416 (Exposure)
Review of basic concepts of stress strain equilibrium equations.
Torsion of cylindrical bars, simple solutions of the torsion
problems.
Plane problems in cartesian and polar coordinates. Bending of a
plate.
Axisymmetric problems.
Texts/Referencs
I.S. Sokolnikoff, Mathematical Theory of Elasticity, McGraw-Hill,
1977.
S. Timoshenko and J.N. Goodiere, Theory of Elasticity,
McGraw-Hill, 1970.
MA 570 Design and Analysis of Experiments 2 1 0 6
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. Construction and
nonexistence theory. Special designs : Split-plots, strip-plots,
cross-over designs.
Texts/References
O. Kempthorne, Design and Analysis of Experiments, Wiley Eastern,
1967.
M.C. Chakrabarty, Mathematics of Design and Analysis of
Experiments, Asia Publishing House, 1962.
M.N. Das and N.C. Giri, Design and Analysis of Experiments, Wiley
Eastern, 1979.
A. Dey, Theory of Block Designs, Wiley, 1986.
MA 571 Advanced Probability Theory 2 1 0 6
Prerequisite : MA 422
Conditional probability, conditional expectation, Martingales,
semi-Martingales, Kolmogorov's existence theorem, stationary
processes.
Texts/References
P. Billingsley, Probability and Measure, 2nd ed., Wiley, 1986.
L. Breiman, Probability, Addison-Wesley, 1968.
J.L. Doob, Stochastic Processes, Wiley, 1953.
MA 572 Nonparametric Statistical Inference 2 1 0 6
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
M. Hollandor, and D.A. Wolfe, Nonparametric Statistical Inference,
McGraw-Hill, 1973.
E.L. Lehmann, Nonparametric Statistical Methods Based on Ranks,
McGraw-Hill, 1975.
J.W. Pratt, and J.D. Gibbons, Concepts of Nonparametric Theory,
Springer-Verlag, 1981.
MA 573 Mathematical Theory of Reliability 2 1 0 6
Prerequisites : MA 418 (Exposure), MA 422
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 and
nonparametric).
Texsts/References
R.E. Barlow and F. Proschan, Statitsical Theory of Reliability and
Life Testing, Holt, Reinhart and Winston, 1975.
J.F. Lawless, Statistical Models and Methods of Life Time Data,
Wiley, 1982.
R.G. Miller, Survival Analysis, Wiley, 1981.
L.J. Bain, Statistical Analysis of Reliability and Life Testing,
Marcel Dekker, 1978.
N.R. Mann, R.E. Shafer and N.D. Singpurwala, Methods of
Statistical Analysis of Reliability and Life Data, Wiley, 1974.
J.D. Kalbfleisch and R.L. Prentice, The Statistical Analysis of
Failure Time Data, Wiley, 1986.
MA 574 Regression Analysis 2 1 0 6
Prerequisite : MA 577
Multiple linear regression-estimation, tests and confidence
regions.
Checks for normality assumptions, outliers and influential
observations. Transformations, variable selection.
Identifiability and ill-conditioning (multi-collinearity),
transformations.
Texts/References
G.A.F. Seber and Wild, C.J., Nonlinear Regression, Wiley, 1989.
A. Sen and M. Srivastava, Regression Analysis-Theory, Methods and
Applications, Springer-Verlag, 1990.
D.C. Montgomery and E.A. Peck, Introduction to Linear Regression
Analysis, Wiley, 1982.
MA 575 Multivariate Analysis 2 1 0 6
Prerequisites : MA 418 (Exposure), MA 422
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. Principal components.
Distribution theory associated with the analysis.
Texts/References
T.W. Anderson, An Introduction to Multivariate Statistical
Analysis, Wiley, 1984.
A.M. Kshirsagar, Multivariate Analysis, Vols. I to IV, North
Holland, 1977.
M.S. Srivastava and E.M. Carter, An Introduction to Multivariate
Statistics, North Holland, 1983.
MA 576 Statistical Decision Theory 2 1 0 6
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 nonrandomized 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.
T.S. Ferguson, Mathematical Statistics, Academic Press, 1967.
MA 577 Statistical Inference I 2 1 0 6
Prerequisites : MA 418 (Exposure), MA 422
Point estimation. Cramer-Rao inequality, Bhattacharya bounds.
Sufficient estimators, Rao-Blackwell theorem. Maximum likelihood
and other methods of estimation. Tests and statistical
hypothesis.
Critical region. Power, Neyman-Pearson lemmas. Likelihood ratio
principle. MP, UMP, LMPU tests, similar tests.
Statistical decision theory. Loss function. Risk functions.
Admissibility. Bayes and minimax solutions. Randomized decision
functions. Sequential decision rules. Sequential analysis.
Texts/References
C.R. Rao, Linear Statistical Inference and its Applications, Wiley
Eastern, 1974.
M.G. Kendall and A. Stuart, The Advanced Theory of Statistics,
Vol. II, Griffin, 1966.
E.L. Lehmann, Testing Statistical Hypotheses, 2nd ed., Wiley,
1986.
G. Casella and R.L. Berger, Statistical Inference, Wadsworth and
Brooks, 1990.
MA 578 Statistical Inference II 2 1 0 6
Prerequisite : MA 577
UMPU, Invariance, Asymptotic Theory of Estimation, efficiency,
super-efficiency, properties of m.l.e., asymptotic distribution
of likelihood ratio statistics.
Texts/References
L. LeCam, Asymptotic Methods in Statistical Decision Theory,
Springer-Verlag, 1986.
E.L. Lehmann, Theory of Point Estimation, Wiley, 1983.
E.L. Lehmann, Testing Statistical Hypothesis, 2nd ed., Wiley,
1986.
R.J. Serfling, Approximation Theorems of Mathematical Statistics,
Wiley, 1980.
MA 579 Stochastic Processes 2 1 0 6
Prerequisite : MA 422
Recurrent events. Renewal theory. Random walk. Markov chains and
Markov processes. Stationary processes. Spectral Analysis.
Stochastic calculus. Branching phenomena. Semi-Markov processes.
Systems with random inputs.
Texts/References
D.R. Cox and H.D. Miller, The Theory of Stochastic Processes,
Methuen, 1970.
E. Parzen, Stochastic Processes, Holden-Day, 1972.
R.O. Howard, Dynamic Probabilistic Systems, Vol. 1 and 2, Wiley,
1971.
S.K. Srinivasan and K. Mehta, Stochastic Processes, Tata
McGraw-Hill, 1976.
J. Medhi, Stochastic Processes, Wiley Eastern, 1982.
S. Karlin and H.M. Taylor, A First Course in Stochastic Processes,
Academic Press, 1975.
MA 580 Time Series Analysis 2 1 0 6
Prerequisite : MA 577 (Exposure)
Introduction to autocorrelation function, linear stationary
models like autoregressive, integrated moving average processes.
Forecasting model identification including initial estimates of
the parameters, model multiplicity etc. Model estimation, model
diagnostic checking. Case studies. Computational experiments.
Texts/References
C. Chatfield, The Analysis of Time Series: An Introduction,
Chapman & Hall, 1984.
G.E.P. Box and G.M. Jenkins, Time Series Analysis Forecasting
and Control, Holden-Day, 1976.
P.J. Brockwell and R.A. Davis, Time Series, Springer-Verlag, 1987.
MA 603 Statistical Methods for Analysis and Design 3 0 0 6
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 analysis.
Texts/References
A.B. Bowker and G.J. Liberman, Engineering Statistics,
Asia, 1972.
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.
R.V. Hogg and E.A. Tanis, Probability and Statistical
Ingference, 2nd Ed., Macmillan, 1983.
MA 825 Algebra 3 0 0 6
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, Hilbert'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.
S. Lang, Algebra, 3rd ed., Addison-Wesley, 1993.
T. Hibi, Algebraic Combinatorics on Convex Polytopes, Carslaw
Publications, 1993.
Zariski and Samuel, Commutative Algebra, Van Nostrand, Princeton,
Vol. I 1958, Vol. II 1960. (New Printing by Springer-Verlag)
MA 826 Topology 3 0 0 6
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, gluing
manifolds, isotopies of discs. Surfaces, model surfaces,
characterizations of disc, classification of compact surfaces.
Texts/References
M.W. Hirsch, Differential Topology, Springer-Verlag, 1976.
J. Milnor, Morse Theory, Annals of Math. Studies, # 51, Princeton
Univ. Press, 1963.
M. Morse, The Calculus of Variations in the Large, AMS Colloquium
Publication, Vol.18, 1934.
MA 827 Analysis 3 0 0 6
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, positive
Borel measures, Riesz representation theorem. Change of
variables formula for Lebesgue integrals in Euclidean spaces.
Texts/References :
W. Rudin, Real and Complex Analysis, Tata McGraw-Hill, 3rd ed.,
1985.
E. Hewitt and K. Stromberg, Real and Abstract Analysis,
Springer-Verlag, 1969.
MA 828 Functional Analysis 3 0 0 6
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
W. Rudin, Functional Analysis, Tata McGraw-Hill, 1974.
F. Treves, Topological Vector Spaces, Distributions and Kernels,
Academic Press, 1967.
K. Yosida, Functional Analysis, Springer-Verlag, 1968.
MA 829 Mathematical Methods 3 0 0 6
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. Iterative solutions.
Singular integral equations.
Texts/References
W.E. Boyce and R.C.Diprima, Elementary Differential Equations and
Boundary Value Problems, Wiley, 1977.
I.N. Sneddon, Elements of Partial Differential Equations,
McGraw-Hill, 1957.
E. Zauderer, Partial Differential Equations of Applied
Mathematics, Wiley, 1989.
I.N. Sneddon, The use of Integral Transforms, Tata McGraw-Hill,
1974.
S.G.Mikhlin, Integral Equations, Pergamon Press, 1957.
MA 830 Numerical Analysis 3 0 0 6
Review of some requisities. 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 convergence 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
E. Issacson and H. B. Keller, Analysis of Numerical Methods,
Wiley, 1966.
A.A. Samarskii and E.S. Nikolaev, Numeical Methods for Grid
Equations, Birkhauser-Verlag, 1989.
M.K. Jain, Numerical Solution of Differential Equations, Wiley
Eastern, 1984.
K.J. Bathe, Finite Element Procedures in Engineering Analysis,
Prentice-Hall of India, 1990.
J.C. Strikwerda, Finite Difference Schemes and Partial
Differential Equations, Wordsworth and Brooke/Coles
Advanced Books and Software,1989.
G.F. Carey and J.T. Oden, Finite Elements : Computational Aspects,
Vol. III, Prentice-Hall, 1984.
MA 831 Fluid Mechanics 3 0 0 6
Basic equations of fluid flow. Constitutive equations and
classification of fluids. Incompressible (Ideal, Newtonian,
Non-Newtonian, Viscoelastic) fluid flows. Basic thermodynamics,
compressible fluid flows. Supersonic, transonic and subsonic
flows.
Nonlinear wave propagation including shock waves of arbitrary
strength.
Texts/References
Landau and Lipschitz, Fluid mechanics, Pergamon Press, 1959.
H. Schlichting, Boundary Layer Theory, McGraw-Hill, 1979.
A.H.P. Selland, Non-Newtonian Fluid and Heat Transfer, Wiley,
1967.
M.M. Denn, Process Fluid Mechanics, Prentice-Hall, 1980.
D.J. Acheson, Elementary Fluid Dynamics. Clarendon Press, 1990.
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.
MA 832 Elasticity 3 0 0 6
Derivation of the basic equations of elasticity. Use of
curvilinear coordinates. Solution for isotropic bodies in terms
of potential functions. Problems depending on one harmonic
function. Theories of plane strain and plane stress. Solutions
of some plane problems using complex variable techniques.
Axisymmetric problems in the theory of elasticity.
Texts/References
A.E. Green and W. Zerna, Theoretical Elasticity, Clarendon Press,
1963.
I.S. Sokolnikoff, Mathematical Theory of Elasticity, Tata
McGraw-Hill, 1977.
MA 833 Weak Convergence and Martingale Theory 3 0 0 6
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, 1968.
R.J. Elliot, Stochastic Calculus and Applications,
Springer-Verlag, 1982.
K.R. Parthasarathy, Probability Measures on Metric Spaces,
Academic Press, 1967.
MA 834 Theory of Testing of Hypotheses 3 0 0 6
UMP tests. Neymann-Pearson fundamental lemma. Distributions with
ML ratio. Confidence 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.
E.L. Lehmann, Testing Statistical Hypotheses, Wiley, 1986.
C.R. Rao, Linear Statistical Inference and its Applications, Wiley
Eastern, 1974.
MA 835 Theory of Estimation 3 0 0 6
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
exponential 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. Simultaneous 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.
MA 836 Asymptotic Theory of Statistical Inference 3 0 0 6
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.
MA 837/MA 838 Special Topics in Mathematics I/II 3 0 0 6
This course will consist of lectures by faculty members on
specialised areas in Mathematics. There will be three weekly
meetings of one hour each. More than one special topic can be
covered in parallel under the same course. The course will be
coordinated by a single faculty member.
SI 401 Applied Linear Algebra 2 1 2 8
Floating point round-off analysis. Linear independence.
Dimension. Rank of a matrix. Vector and matrix norms.
Linear Systems : Overdetermined, underdetermined and nonsingular
systems. Condition of a system. LU decomposition and Gauss
elimination.
Pivoting and scaling strategies. QR factorization and
Householder orthogonalization. Stability. Least square solution.
Generalized inverse. Iterative improvement of a solution.
Eigenvalue Problems : Cayley-Hamilton theorem. Discussion of
canonical forms. Diagonalization of symmetric matrices.
Quadratic forms and their relation to eigenvalue problems.
Sensitivity of eigenvalues and eigenvectors to perturbation.
Location of eigenvalues. Gershgorin's theorem. Power and inverse
power methods. Rayleigh quotient iteration.
Simultaneous orthogonalization and QR method.
Exposure to software packages like Mathematica, IMSL, Matlab,
Gauss, LAPACK.
Texts/References :
G. E. Forsythe and C. B. Moler, Computer Solution of Linear
Algebraic Systems, Prentice Hall, 1967.
B. Noble and J. W. Daniel, Applied Linear Algebra, 2nd ed.,
Prentice Hall, 1977.
G. W. Stewart, Introduction to Matrix Computations, Academic
Press, 1973.
D. S. Watkins, Fundamentals of Matrix Computation, Wiley, 1991.
SI 402 Numerical Methods 3 0 2 8
Polynomial interpolation. Piecewise polynomial and cubic spline
interpolation. Least square approximation. Numerical integration
: various rules and their composite versions. Numerical
differentiation.
Methods for single non-linear equation. Bisection and secant
methods.
Newton's method: convergence and rate of convergence. System of
equations.
Numerical solution of ordinary differential equations. Euler
method.
Runge-Kutta and multi-step methods. Predictor-corrector method.
Exposure to software packages like Mathematica, Matlab, IMSL
Subroutines.
Texts/References :
S. D. Conte and Carl de Boor, Elementary Numerical Analysis - An
Algorithmic Approach, McGraw-Hill, 1981.
K. E. Atkinson, An Introduction to Numerical Analysis, Wiley,
1989.
F. B. Hildebrand, Introduction to Numerical Analysis, Tata
McGraw-Hill, 1974.
SI 403 Mathematical Modeling 3 2 0 8
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 :
C. L. Dym and E. S. Ivey, Principles of Mathematical Modeling,
Academic Press, 1980.
M. Braun, C. S. Coleman and D. A. Draw, Differential Equation
Models, Modules in Applied Mathematics, Vol. 1, Springer
Verlag, 1978.
H. J. White and S. Tauber, System Analysis, W.B. Saunders Company,
1969.
Y. C. Fung, Biomechanics, Springer Verlag, 1981.
E. N. Lightfoot, Transport Phenomenon and Living Systems, Wiley,
1974.
SI 404 Optimization Techniques 3 1 0 8
Linear Programming : Problem formulation, simplex and revised
simplex methods. Duality and sensitivity. Case studies. Interior
point methods.
Nonlinear Programming : Problem formulation. Basic concepts from
calculus of several variables, linear algebra and convex
analysis.
Iterative methods for unconstrained optimization. Least square
optimization. Convex programming and Karush-Kuhn-Tucker theory.
Penalty methods. Optimization with equality constraints.
Texts/References :
V. Chva'tal, Linear Programming, W.H. Freeman, 1983.
A. L. Peressini, F. E. Sullivan and J. J. Uhl, Jr., Springer
Verlag, 1988.
S. Bradley, A. Hax and T. Magnanti, Applied Mathematical
Programming, Addison-Wesley, 1978.
A. Arbel, Exploring Interior Point Linear Programming, Algorithms
and Software, M.I.T. Press, 1993.
SI 405 Mathematical Systems Theory 3 0 0 6
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 nonfundamental 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 stabilizability :
Controllability matrix and Gramian. Observability as a dual
notion of controllability.
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.
G. F. Simmons, Differential Equations with Applications and
Historical Notes, Tata McGraw-Hill, 1972.
H. J. White and S. Tauber, Systems Analysis, W.B. Sounders, 1969.
L. A. Zadeh and E. Polak, System Theory, McGraw-Hill, 1969.
SI 406 Applied Stochastic Processes 3 1 0 8
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 continuous state spaces.
Poisson process, pure birth process, birth and death process.
Brownian motion.
Applications to queueing models and reliability 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.
P. G. Hoel, S. C. Port and C. J. Stone, Introduction to Stochastic
Processes, Houghton Mifflin, 1972.
A. O. Allen, Probability, Statistics and Queueing Theory with
Computer Science Applications, 2nd ed., Academic Press, 1990.
J. Medhi, Stochastic Models in Queueing Theory, Academic Press,
1991.
SI 407 Elements of Applied Probability and Statistics 3 1 0 8
Probability, conditional probability. Bayes theorem,
independence of events. Random variables. Distribution
functions. Random vectors, conditional distributions. Expected
values and moments. Standard univariate distributions. Weak laws
of large numbers. Central Limit Theorems. Multivariate
distributions such as multivariate normal and t distributions.
Empirical distributions, distributions of order statistics,
range, extremes.
Point estimation : Sufficiency, method of moments and maximum
likelihood. UMVU estimation.
Simple and Composite hypotheses, two sample tests.
Neyman-Pearson lemma, UMP tests, Non-parametric tests : Sign
tests, Run tests, Rank tests, Chi-square goodness of fit.
Texts/References :
M. Woodroof, Probability with Applications, McGraw-Hill, 1975.
G. G. Roussas, A First Course in Mathematical Statistics,
Addison-Wesley, 1975.
H. J. Larson, Introduction to Probability Theory and Statistical
Inference, 3rd ed., Wiley, 1992.
A. O. Allen, Probability, Statistics and Queueing Theory with
Computer Science Applications, 2nd ed., Academic Press, 1990.
SI 408 Data Structures 3 0 2 8
Introduction to data structures and complexity 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
Yedidyah Langsam,Moshe J. Augenstein and Aaron M.Tenenbaum, Data
Structures Using C and C++,
Prentice Hall, India, 1997.
R.L. Kruse, Bruse P. Leung and L.Clovis Tonda, Data Structures
and Programming Design in C.
Prentical Hall India.
E. Horowiz,S. Sahni, Fundamentals of Data Structures,
Galgotia Publishers, 1983.
SI 409 Discrete Structures Lab. 0 0 3 3
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.
Donald Knuth, Fundamental Algorithms,
Narosa Publishers, 1985.
R. Dromey, How to Solve it By Computer,
Prentice Hall India, 1996.
SI 410 Programming Languages Lab 0 0 3 3
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.
SI 411 Computer Lab. 0 0 3 3
Introduction to Unix: Introduction to C programming language,
Syntax, Data Types, Pointers, Functions, Scope rules,
bindings, parameters transmission mechanism, storage management.
Texts/References
B. Kernighan, D.Ritchie, The C Proramming Language,
Prentice Hall India, 1996.
B. Kernighan, R. Pike, The Unix Programming Environment,
Prentice Hall India, 1995.
SI 501 Discrete Algorithms 3 0 0 6
Mathematical preliminaries: assymptotic notation. Advanced
Data Structures: Hash tables, Binomial Heaps, Disjoint sets.
Greedy Algorithms: Huffman coding, Minimum Spanning Tree con-
struction, Dijkstra's Shortest Path construction. Dynamic
Programming Algorithms: Matrix-chain multiplication, All pairs
shortest path problems, Minimum weight triangulation of convex
polygons. Divide and conquer: Linear time selection, Euclidean
closest pair problem, Strassen's matrix multiplication 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.
SI 502 Stochastic Programming Applications 3 0 0 6
Quadratic and Nonlinear Programming solution methods applied to
Chance Constrained Programming problems. Stochastic Linear and
Nonlinear Programming problems arising in inventory control
and other industrial applications; queuing models of computer
networks; information processing under uncertainty. Two stage
and multistage solution techniques. Use of Monte Carlo,
probabilistic and heuristics algorithms. Genetic algorithms and
neural networks for adaptive optimization.
Texts/References
S.S. Rao, Optimization - Theory and Applications.
Wiley Eastern (2nd ed.), 1987.
J.K. Sengupta, Stochastic Optimizations and Economic Models.
Dordrecht Reidel, 1986.
Yu. Ermoliev and RJB Wets, Numerical Techniques for Stochastic
Optimiation.
Springer Verlag, Berlin, 1988.
K. Schittkowski, More Test Examples of Nonlinear Programming
Codes.
Springer-Verlag, Berlin, 1987.
Z. Michaeleawicz, General Algorithms + Data Structures -
Evolution Program.
Springer-Verlag, 1992.
SI 503 Finite Difference Methods for Partial Differential
Equations 3 0 0 6
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 (Lax-Wendroff scheme, Leapfrog method, CFL
conditions) Stability, consistency and convergence results.
Texts/References
J.C. Stickwards, Finite Difference Schemes and PDEs,
Chapman and Hall, 1989.
P. Knupp and S. Steinberg, Fundamentals of Grid Generation,
CRC Press Inc., Boca Raton, 1994.
J.F. Thompson, Z.U., A. Waarsi and C.W. Mastin, Numerical Grid
Generations - Foundations and Applications,
North Holland, 1985.
A.R. Mitchell and D.F. Griffiths, The Finite Difference Methods
in Partial Differential Equations,
Wiley, 1980.
G.D. Smith, Numerical Solutions to Partial Differential
Equations,
Oxford Press, 1985.
Erich Zauderer, Partial Differential Equations of Applied
Mathematics 2nd ed.
Wiley, 1989.
Gene H. Golub and James M. Ortega, Scientific Computing and
Differential Equations : An Introduction to Numerical
Methods.
A.P. 1992.
SI 504 Experimental Designs 3 0 0 6
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
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.
H. Spaeth, Mathematical Algorithms for Linear Regression,
Academic Press, 1991.
M.S. Phadke, Quality Engineering Using Robust Design, Prentice
Hall, 1989.
P.J. Ross, Taguchi Techniques for Quality Engineering, McGraw-
Hill, 1988.
G.E.P. Box, W.G. Hunter and J.S. Hunter, Statistics for
Experimentors, Wiley, 1978.
SI 505 Categorical Data Analysis and Regression 3 0 0 6
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 model 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.
Texts/References:
E.B. Andersen, The Statistical Analysis of Categorical Data,
Springer-Verlarg, 1990.
A. Agresti, Analysis of Categorical Data, Wiley, 1990.
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.
R. F. Gunst and R.L. Mason, Regression Analysis and its
Applications - A Data Oriented Approach, Marcel Dekkar,
1980.
SI 506 Introduction to Atuomata Theory
and Languages 3 0 0 6
Finite automata. Regular expressions, Regular languages and
their properties. Push down automata. Context-free languages and
their proepreties.
Turing machines. Turing computability. Undecidability results.
Introduction to compiler design, lexical analysis and parsing.
Automataic generation of lexical analyers and parsers.
Texts/References
J.E. Hopcroft and J.D. Ullman, Automata, Languages and
Computation, Narosa, 1987.
A. 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.
SI 507 Systems Programming 3 0 0 6
Introduction to Unix Operating Systems. File System, Process
Control. Interprocesses Communication. Memory Management.
Special topics from other operating systems.
Overview of MINIX.
Texts/References
M.J. Bach, The Design of Unix Operating System, Prentice-Hall of
India, 1986.
A.S. Tennenbaum, Operating Systems : Design and Implementation,
Prentice Hall of India, 1989.
SI 508 Digital Logic and Computer Design 3 0 0 6
Boolean alebgra. Normal forms. Minimization of switching
functions.
Logic gates. Circuits for arithmetic computation. Sequential
logic.
Introduction to microprocessor design.
Texts/References
M.M. Mano, Digital Logic and Computer Design, Prentice-Hall of
India, 1986.
Z. Kohavi, Switching Theory and Finite Automata, Tata
McGraw-Hill, 1978.
SI 509 Scientific Computing Lab 0 0 3 3
Solutions of large linear systems generated by finite difference
discretization of two point boundary value problems and Poisson
equations : banded and sparse solvers, iterative methods.
Nonlinear systems generated by discretization of system of
nonlinear ODEs and nonlinear two point boundary value problems.
Use of ODEPACK, use of GNUPLOT, Mmesh generation (transfinite
interpolation techniques), Sample numerical experiments for
partial differential equations using MATLAB.
Software Support : GNUPLOT, LSODE, MATLAB
Texts/References
Gene H. Golub and James M. Ortega: Scientific Computing and
Differential Equations : An Introduction to Numerical
Methods,
AP 1992.
G. Lindfield and J. Penny, Numerical Methods Using MATLAB,
Ellis Horwood Ltd., NY, 1995.
SI 510 Mathematical Elements for Computer
Graphics. 3 0 0 6
Introduction to graphics hardware. Brief overview of procedural
elements for computer graphics (like drawing, circle drawing,
clipping, hidden line/surface removal algorithm).
Transformations in 2D/3D. Linear/Affine/Projective
transformations. Perspective views.
Freeform curves and surfaces. Applications from CAGD. Selected
topics from Computational Geometry and Applications.
Texts/References
D.F. Rogers and Adams, Mathematical Elements for Computer
Graphics, McGraw-Hill, 1989.
D.F. Rogers, Procedural Elements for Computer Graphics.
McGRaw-Hill, 1985.
J. O'Rourke, Computational Geometry in C, Cambridge Univ. Press
1994.
SI 511 Computer-Aided Geometric Designs 3 0 0 6
Polynomial curves : Bezier representation, Bernstein
polynomials, Blossoming, de Castlijau algorithm. Derivatives in
terms of Bezier polygon. Degree elevation. Subdivision.
Nonparametric Bezier curves.
Composite Bezier curves.
Spline curves : Definition and Basic properties of spline
functions, B-spline curves, de Boor algorithm. Derivatives.
Insertion of new knots. Cubic spline interpolation.
Interpretation of parametric continuity in terms of Bezier
polygon.
Geometric continuity. Frenet frame continuity. Cubic Beta
splines and significance of the associated parameters.
Tensor product surfaces. Bezier patches. Triangular patch
surfaces.
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.
SI 512 Combinatorial Optimization 3 0 0 6
Networks and Matroids : Maximum flow, minimum cost flow,
bipartite and nonbipartite matchings.
Matroids : Greedy algorithm, matroid intersection and union.
Integer Programming : Model formulations, properties of integral
polyhedra and computational complexity, relaxation and valid
inequalities, duality, cutting plane algorithms, branch and
bound.
Heuristics.
Texts/References :
R. K. Ahuja, T. L. Magnanti, and J. B. Orlin, Network Flows :
Theory, Applications and Algorithms, Prentice Hall, 1993.
G. L. Nemhauser and L. A. Wolsey, Integer and Combinatorial
Optimization, Wiley, 1988.
A. Schrijver, Theory of Linear and Integer Programming, Wiley,
1986.
SI 513 Statistical Data Analysis Lab. 0 0 3 3
Categorical Data Analysis and Regression: Basic of regression
analysis, Estimation, Testing (point and interval),
prediction, checking model adequacy and residual analysis.
Nonparametric Statistics: Basic tests such as Kolmogorov-
Smirnov test, sign test, median test etc.
Applied Multivariate Analysis : Discriminant Analysis,
Cluster Analysis, Principal Component Analysis, Factor Analysis.
References :
A. Agresti, Analysis of Categorical Data, Wiley, 1990.
A.A. Sen and M. Srivastava, Regression Analysis - Theory,
Methods and Applications, Springer-Verlag, 1990.
W.J. Conover, Practical Nonparametric Statistics, Wiley, New
York, 1971.
J.D. Gibbons, Nonparametric Methods in Quantitative Analysis,
Holt, New York, 1976.
M. Hollander and D.A. Wolfe, Nonparametric Statistical Methods,
Wiley, New York, 1973.
R.A. Johnson and D.W. Wichern, Applied Multivariate Statistical
Analysis, Prentice Hall, Inc.,Englewood Cliffs, New
Jersey, 1982.
SI 514 Computer-Oriented Statistical Techniques 3 0 0 6
Density estimation : Kernel estimator, bandwidth selection,
nearest-neighbour estimator, maximum penalized likelihood
method, spline smoothing, applications.
EM algorithm : Principle of data augmentation, definition and
illustration of the algorithm, convergence.
Gibbs sampling : Illustration in the bivariate case,
generalization to three or more variables, convergence,
applications.
Resampling techniques : Bootstrap method, bootstrap confidence
intervals and estimates of bias, bandwidth selection by
bootstrap, jacknife and its relation to the bootstrap, the delta
method.
Texts/References :
B. W. Silverman, Density Estimation for Statistics and Data
Analysis, Chapman and Hall, 1986.
M. A. Tanner, Tools for Statistical Inference, Springer-Verlag,
1991.
B. Efron and R. J. Tibshirani, An Introduction to the Bootstrap,
Chapman and Hall, 1993.
SI 515 Applied Multivariate Analysis 3 0 0 6
Matrix algebra and random vectors. Sample geometry and random
sampling. The multivariate normal distribution. Inferences about
a mean vector. Large sample inference about population mean
vectors, proportions. Comparison of several multivariate
population means.
Two-way multivariate analysis of variance, classical linear
regression model, least square estimation and inferences about
the regression model. Model checking and other aspects of
regression. Multivariate multiple regression. Principal
component techniques. Factor analysis.
Separation and classification for two populations. Fisher's
method for discrimination among several populations.
Hierarchical and nonhierarchical clustering methods.
Texts/References :
R. Gnanadesikan, Methods for Statistical Data Analysis of
Multivariate Observations, Wiley, 1977.
D. F. Morrison, Multivariate Statistical Methods, 2nd ed.,
McGraw-Hill, 1976.
N. H. Timm, Multivariate Analysis with Applications in Education
and Psychology, Brooks/Cole, 1975.
SI 516 Reliability Techniques 3 0 0 6
Basic reliability models. Estimation and inferential aspects of
these models. Probabilistic 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.
S. K. Sinha and B. K. Kale, Life Testing and Reliability
Estimation, Wiley Eastern, 1979.
SI 518 Statistical Quality Control 2 1 0 6
Total quality control in an industry. Quality planning, quality
conformance, quality adherence. 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 assignable 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 :
J. M. Juran and F. M. Grayna, Quality Planning and Analysis, Tata
McGraw-Hill, 1970.
A. J. Duncan, Quality Control and Industrial Statistics, 5th ed.,
Richard D. Irwin, 1986.
A. V. Feigenbaum, Total Quality Control Engineering and
Management, McGraw-Hill, 1961.
E. L. Grant and R. Levenworth, Statistical Quality Control, 6th
ed., McGraw-Hill, 1988.
SI 519 Probabilistic Techniques in Machine Learning 3 0 0 6
Introduction to inductive probability and machine learning.
Statistical pattern recognition and clustering techniques.
Stochastic approximation and rough classification. Bayesian
classification. Sequential probability and incremental machine
learning.
Nonparametric methods for leader independent 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
Web and Intac Programming : Java applications programming,
HTML, CGI Programming.
Texts/References
P. Naughton, H.Schildt, Java : The Complete Reference.
Tata McGraw-Hill, 1997.
K. Jamsa, S. Lalani,S. Weakley, Web Programming.
Frank Bros. and Co., 1996.
SI 521 Biostatistics 3 0 0 6
Randomization and control of clinical trials. Sampling in
clinical studies. Cohort analysis. Sampling distributions and
hypothesis testing for clinical and laboratory data. Importance
of type I, type II errors and sample 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 Foundation 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.
SI 522 Large Scale Scientific Computation 3 0 0 6
Exposure SI 503
Large sparse linear systems: Storage schemes, preconditioners,
GMRES algorithms, Multigrid Algorithms with implementation.
Nonlinear Solvers: Newton's method and some of its variations,
continuation methods, conjugate direction method and Davidon-
Fletcher - Powell Algorithms. Nonlinear Multigrid with
applications.
Software Support: HOMOPACK, LAPACK, MADPACK.
Texts/References
J.M. Ortega and W.C. Rheinboldt, Iterative Solution of Nonlinear
Equations in Several Variables.
Academic Press, NY, 1970.
O.Axelsson, Iterative Solution Methods
Cambridge Univ. Press, 1994.
W. Hackbusch, Multigrid Methods and Applications.
Springer-Verlag, 1985.
P. Wesseling, An Introduction to Multigrid Methods.
John Wiley & Sons, 1992.
C.W. Ueberrhuber, Numerical Computation : Methods, Software
and Analysis.
Springer-Verlag, Berlin, 1997.
SI 523 Mathematical Modelling and Numerical Simulation 3 0 0 6
Review of continuum model, Transport phenomena, Air quality
modelling, (pollution from chimney), Furnace reaction analysis,
De-icing helicopter blades (free and moving boundary
problems), modelling microwave heating, Food contamination from
the packaging, Electron Beam Lithography, Color negative film
development, photocopy machine; Selected case studies.
Software Support: MATHEMATICA, LSODE, GNUPLOT, MATLAB.
Texts/References
A. Friedman and W. Littman, Industrial Mathematics for Under-
graduates.
SIAM Publ. 1994..
J. Crank, Free and Moving Boundary Problems,
Oxford Univ. Press, 1987.
A. James (Ed.), An Introduction to Water Quality Modelling,
Wiley Pub. 1984.
M.S. Klamkin, (ed.), Mathematical Modelling : Classroom Notes
in Applied Mathematics,.
SIAM Publications.
A. Friedman, Mathematics in Industrial Problems Part 1 - 9.
IMA Series, Springer-Verlag.
Lecture Notes on Heat and Mass Transfer : A Problem driven
approach, M.Sc. in Industrial Mathematics.
Univ. Strathclyde, U.K., 1995.
Y.C. Fung, A First Course in Continuum Mechanics.
Prentice-Hall, 1969.
OTHTER COURSES OFFERED
CS 101 Computer Programming and Utilisation 2 1 0 6
Functional organisation of computers, algorithms, basic
programming concepts, FORTRAN language programming. Program
testing and debugging. Modular programming subroutines: Selected
examples from Numerical 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.
CS 314 Business Information Systems 2 1 0 6
Nature of Business Systems. Data Processing Tasks, Modern
Storage devices and other peripherals.
Data Models, ER Diagrams and Data Flow diagrams.
Introduction to Relational Theory, Normalisation, File Design.
SQL and host language interfaces, User Interface development:
screens, reports, transactions.
Use of leading product to construct a business application.
Texts/References
James A. Senn, Analysis and Design of Information Systems,
McGraw-Hill, 1990.
H.F. Korth and A. Silberschatz, Database System Concepts, McGraw-Hill,
1991.
R. Elmasri and S. Navathe, Fundamentals of Databse Systems, Benjamin
Cummings, 1993.
CS 398 Business Information Systems Lab 0 0 3 3
Experiments in COBOL on topics such as (i) indexed and relative
files, (ii) report generation, (iii) screen section (
interactive processing). Experiments in usage of tools and
packages like spread-sheets, FOXPRO etc. Laboratory experiments
in use of interactive SQL, 4GLs, client-server environments like
powerBuilder, Gupta SQL etc. Designing an application using
these packages.
CS 470 Modelling and Simulation 3 0 0 6
Selected illustrative examples of simulation applications.
Models: Structural, Process, Continuous, Discrete,
Deterministic, Random, input/output, static, dynamic,
multilevel. Simulation: Analog/Digital/Hybrid, verification,
validation. Data Modelling and Analysis : Population parameters,
hypotheses testing, confidence-intervals, goodness of fit,
estimating transient/steady-state characterstics, variance
reduction. Simulation Process : Problem formulating, model
building, data acquisition, model translation, verification,
validation, strategic and tactical planning, experimentation,
analysis of results, implementation and documentation.
Simulation Languages : Examples from SIMSCRIPT, GPSS, GASP,
SIMULA, etc.
Texts/References:
G.Gordon, `System Simulation', 2nd ed., Prentice Hall, 1978.
Narsing Deo, `System Simulation with Digital Computers', Prentice
Hall, 1976.
J.R. Leigh, `Modelling and Simulation', Peter Peregrims Ltd., 1983.
A.M.Law, W.D.Kelton, `Simulation Modelling and Analysis', Mcgraw Hill,
1982.
CS 475 Computer Graphics 3 0 0 6
Interactive graphics programming, Coordinate systems. Graphical
output primitives and segments. Concept of logical input
devices. Examples from GKS and Core.
Graphics hardware : Vector and Raster CRTs Display controllers
and processors, plotters, keyboard, Light pen, Tablet and
Stylus, Mouse.
Graphics system software: Clipping, Normalising transformation,
Display file organisation.
Raster conversion algorithms, Implementation of logical input
devices.
Geometric transformation : Homogeneous coordinate systems,
Matrix formulation and concatenation of transformations.
3-D graphics : 3-D output primitives specification and
computation of projections, 3-D clipping, Hidden line and hidden
surface elimination, Models of illumination and shading.
Texts/References:
J.D.Foley, A Van Dam, Fundamentals of Interactive Computer Graphics,
Addison Wesley, 1981.
W.M.Newman, R.Sproull, Principles of inter active Computer Graphics,
2/e, McGraw Hill, 1979.
D.F.Rogers, Procedural Elements for Computer Graphics, McGraw Hill,
1985.
D.F.Rogers, Mathematical Elements for Computer Graphics, 2/e, McGraw
Hill, 1989.
S.Harrington, Computer Graphics : A Programming Approach, 2/e, McGraw
Hill, 1987.
G.Enderle, K. Kansy, G. Pfaff, Computer Graphics Programming GKS - The
Graphics Standard, Springer Verlag, 1983.