**List of Math courses**

**MA 417 Ordinary Differential Equations**

**MA 504 Operators on Hilbet space**

**MA 510 Introduction to Algebraic Geometry **

**MA 515 Partial Differential Equations**

**MA 521 Theory of Analytic Functions **

**MA 522 Fourier Analysis and Applications**

**MA 524 Algebraic Number Theory **

**MA 533
Advanced Probability Theory**

**MA 534 Modern Theory of Partial Differential
Equations**

**MA 538 Representation Theory of Finite Groups **

**MA 539 Spline Theory and Variational Methods **

**MA 540 Numerical Methods for Partial Differential
Equations**

**MA 562 Mathematical Theory of Finite Elements**

**MA 581 Elements of Differential Topology **

**List
of ASI courses**

**SI 421 Introduction Mathematical Software**

**SI 404 Applied Stochastic Processes**

**SI 417 Introduction to Probability Theory**

**SI 418 Advanced Programming and Unix
Environment **

**SI 501 Topics in Theoretical Computer Science**

**SI 503 Categorical Data Analysis **

**SI 511 Computer-Aided Geometric Design **

**SI 512
Finite Difference Methods for Partial Differential Equations **

**SI 515 Statistical Techniques in Data Mining **

**SI 527
Introduction to Derivatives Pricing **

**SI 530 Statistical Quality Control **

**SI 532 Statistical Decision Theory **

**SI 534 Nonparametric Statistics **

**SI 540 Stochastic Programming and Applications **

**SI 542 Mathematical Theory of Reliability**

**CS 101
Computer Programming & Utilization**

**IT 640 Modern Information System **

**EE 649
Finite Fields and Their Applications **

**EE 720 An Introduction to Number Theory and Cryptography **

**MA 401 Linear Algebra **** 3 1 0 8**

Vector spaces over fields, subspaces,
bases and dimension.

Systems of linear equations, matrices,
rank, Gaussian elimination.

Linear transformations, representation
of linear transformations by matrices, rank-nullity theorem, duality and
transpose.

Determinants, Laplace expansions,
cofactors, adjoint, Cramer's Rule.

Eigenvalues and eigenvectors,
characteristic polynomials, minimal polynomials, Cayley-Hamilton Theorem,
triangulation, diagonal-lization, rational canonical form, Jordan canonical
form.

Inner product spaces, Gram-Schmidt
orthonormalization, orthogonal projections, linear functionals and adjoints,
Hermitian, self-adjoint, unitary and normal operators, Spectral Theorem for
normal operators, Rayleigh quotient, Min-Max Principle.

Bilinear forms, symmetric and
skew-symmetric bilinear forms, real quadratic forms, Sylvester's law of
inertia, positive definiteness.

Texts / References

M.
Artin, Algebra, Prentice Hall of India, 1994.

K. Hoffman and R.
Kunze, Linear Algebra, Pearson Education (India), 2003. Prentice-Hall
of India, 1991.

S. Lang, Linear Algebra,
Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1989.

P. Lax, Linear Algebra, John Wiley & Sons, New York,. Indian Ed. 1997

H.E. Rose, Linear
Algebra, Birkhauser, 2002.

S. Lang, Algebra, 3rd Ed., Springer
(India), 2004.

O. Zariski and P. Samuel, Commutative
Algebra, Vol. I, Springer, 1975.

**MA ****403
Real Analysis I ** **3 1 0 8**

Review of basic concepts of real numbers: Archimedean property,
Completeness.

Metric spaces, compactness, connectedness,
(with emphasis on R^{n}).

Continuity and uniform continuity.

Monotonic functions, Functions of bounded variation; Absolutely continuous functions. Derivatives
of functions and Taylor's theorem.

Riemann integral and its properties,
characterization of Riemann integrable functions. Improper integrals, Gamma functions.

Sequences and series of functions, uniform
convergence and its relation to continuity, differentiation and integration. Fourier series, pointwise convergence,
Fejer's theorem, Weierstrass approximation theorem.

Texts / References

T. Apostol, Mathematical Analysis, 2nd ed.,

Narosa Publishers, 2002.

K. Ross, Elementary Analysis: The Theory

of Calculus, Springer Int. Edition, 2004.

W. Rudin, Principles of Mathematical

Analysis, 3^{rd} ed., McGraw-Hill,
1983.

**MA 419 Basic Algebra****
3 1 0 8**

Review of basics: Equivalence relations and partitions,
Division algorithm for integers, primes, unique factorization, congruences,
Chinese Remainder Theorem,

Euler j-function.

Permutations, sign of a permutation, inversons, cycles and
transpositions.

Rudiments of rings and fields, elementary properties,
polynomials in one and several variables, divisibility, irreducible
polynomials, Division algorithm, Remainder Theorem, Factor Theorem, Rational
Zeros Theorem, Relation between the roots and coefficients, Newton's Theorem on
symmetric functions, Newton's identities, Fundamental Theorem of Algebra,
(statement only), Special cases: equations of degree 4, cyclic equations.

Cyclotomic polynomials, Rational
functions, partial fraction decomposition, unique factorization of polynomials
in several variables, Resultants and discriminants.

Groups, subgroups and factor groups,
Lagrange's Theorem, homomorphisms, normal subgroups. Quotients of groups, Basic examples of
groups (including symmetric groups, matrix groups, group of rigid motions of
the plane and finite groups of motions).

Cyclic groups, generators and
relations, Cayley's Theorem, group actions, Sylow Theorems.

Direct products, Structure Theorem for finite abelian
groups.

Texts / References

M. Artin, Algebra, Prentice Hall of India, 1994.

D.S.
Dummit and R. M. Foote, Abstract Algebra, 2nd Ed.,
John Wiley, 2002.

J.A. Gallian, Contemporary Abstract
Algebra, 4^{th} ed., Narosa, 1999.

K.D.
Joshi, Foundations of Discrete Mathematics, Wiley Eastern, 1989.

T.T. Moh, Algebra, World Scientific,
1992.

S. Lang,
Undergraduate Algebra, 2nd Ed., Springer, 2001.

S. Lang, Algebra, 3^{rd} ed., Springer (India),
2004.

J. Stillwell, Elements of Algebra, Springer, 1994.

**MA 406 General Topology****
3 1 0 8**

Prerequiste: MA 403 Real Analysis

Topological Spaces: open sets, closed sets, neighbourhoods, bases,
subbases, limit points, closures, interiors, continuous functions, homeomorphisms.

Examples of topological spaces: subspace topology, product
topology, metric topology, order topology.

Quotient Topology : Construction of
cylinder, cone, Moebius band, torus, etc.

Connectedness and Compactness: Connected spaces, Connected
subspaces of the real line, Components and local connectedness, Compact spaces,
Heine-Borel Theorem, Local -compactness.

Separation Axioms: Hausdorff spaces,
Regularity, Complete Regularity, Normality, Urysohn Lemma, Tychonoff embedding
and Urysohn Metrization Theorem, Tietze Extension Theorem.

Tychnoff Theorem, One-point Compacti-fication.

Complete metric spaces and function spaces, Characterization
of compact metric spaces, equicontinuity, Ascoli-Arzela Theorem, Baire Category
Theorem. Applications: space filling curve, nowhere differentiable continuous
function.

Optional Topics:

1. Topological Groups and orbit spaces.

2. Paracompactness and partition of unity.

3. Stone-Cech Compactification.

4. Nets and filters.

Texts / References

M. A. Armstrong, Basic Topology, Springer (India), 2004.

K.D. Joshi, Introduction to General Topology, New Age
International, New Delhi, 2000.

J.L. Kelley, General Topology, Van Nostrand, Princeton,
1955.

J.R. Munkres, Topology, 2nd Ed.,
Pearson Education (India), 2001.

G.F. Simmons, Introduction to Topology and Modern Analysis,
McGraw-Hill, New York, 1963.

**MA 408 Measure Theory **** 3 1 0 8**

Prerequisite: MA 403 Real Analysis

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

Outline of extension of measures from algebras to the
generated sigma-algebras: Measurable sets; Lebesgue Measure and its properties.

Measurable functions and their
properties; Integration and Convergence theorems.

Introduction to L^{p}-spaces,
Riesz-Fischer theorem; Riesz Representation theorem for L^{2} spaces. Absolute continuity of measures,
Radon-Nikodym theorem. Dual of L^{p}-spaces.

Product measure spaces, Fubini's
theorem.

Fundamental Theorem of Calculus for Lebesgue Integrals (an
outline).

Texts / Referenes

P.R. Halmos, Measure Theory, Graduate Text in Mathematics,
Springer-Verlag, 1979.

Inder K. Rana, An Introduction to Measure and Integration (2^{nd}
ed.), Narosa Publishing House, New Delhi, 2004.

H.L. Royden, Real Analysis, 3^{rd} ed., Macmillan,
1988.

**MA 410 Multivariable Calculus ****2 1 0 6**

Prerequisites: MA 403 Real Analysis,

MA 401 Linear Algebra

Functions on Euclidean spaces, continuity,
differentiability; partial and directional derivatives, Chain Rule, Inverse
Function Theorem, Implicit Function Theorem.

Riemann Integral of real-valued functions on Euclidean
spaces, measure zero sets, Fubini's Theorem.

Partition of unity, change of
variables.

Integration on chains, tensors, differential forms, Poincare
Lemma, singular chains, integration on chains, Stokes' Theorem for integrals of
differential forms on chains. (general version). Fundamental theorem of calculus.

Differentiable manifolds (as subspaces of Euclidean spaces),
differentiable functions on manifolds, tangent spaces, vector fields,
differential forms on manifolds, orientations, integration on manifolds,
Stokes' Theorem on manifolds.

Texts / References

V. Guillemin and A. Pollack, Differential Topology,
Prentice-Hall Inc., Englewood Cliffe, New Jersey,
1974.

W. Fleming, Functions of Several Variables, 2nd Ed., Springer-Verlag, 1977.

J.R. Munkres, Analysis on Manifolds, Addison-Wesley, 1991.

W. Rudin, Principles of Mathematical Analysis, 3rd ed.,
McGraw-Hill, 1984.

M. Spivak, Calculus on Manifolds, A Modern Approach to
Classical Theorems of Advanced Calculus, W. A. Benjamin, Inc.,
1965.

**MA 412 Complex Analysis**** ****3 1 0 8**** **

Complex numbers and the point at infinity. Analytic functions.

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

Contour integrals, Cauchy-Goursat Theorem.

Uniform convegence of sequences and series. Taylor and Laurent series. Isolated singularities and residues. Evaluation
of real integrals.

Zeroes and poles, Maximum Modulus
Principle, Argument Principle, Rouche's theorem.

Texts / References

J.B. Conway,
Functions of One Complex Variable, 2^{nd} ed., Narosa, New Delhi, 1978.

T.W. Gamelin,
Complex Analysis, Springer International Edition, 2001.

R. Remmert,
Theory of Complex Functions, Springer Verlag, 1991.

A.R. Shastri, An Introduction to Complex

Analysis, Macmilan India, New Delhi,

1999.

**MA 414 Algebra - I****
3 1 0 8**

Prerequiste: MA 401 Linear Algebra, MA
419

Basic Algebra

Simple groups and solvable groups,
nilpotent groups, simplicity of alternating groups, composition series,
Jordan-Holder Theorem. Semidirect products.
Free groups, free abelian groups.

Rings, Examples (including polynomial
rings, formal power series rings, matrix rings and group rings), ideals, prime
and maximal ideals, rings of fractions, Chinese Remainder Theorem for pairwise
comaximal ideals.

Euclidean Domains, Principal Ideal Domains and Unique
Factorizations Domains. Poly-nomial rings over UFD's.

Fields, Characteristic and prime subfields, Field
extensions, Finite, algebraic and finitely generated field extensions, Classical
ruler and compass constructions, Splitting fields and normal extensions,
algebraic closures. Finite fields, Cyclotomic fields,
Separable and inseparable extensions.

Galois groups, Fundamental Theorem of Galois Theory,
Composite extensions, Examples (including cyclotomic extensions and extensions
of finite fields).

Norm,
trace and discriminant.

Solvability
by radicals, Galois' Theorem on solvability.

Cyclic extensions, Abelian extensions,
Trans-cendental extensions.

Texts / References

M. Artin, Algebra, Prentice Hall of India, 1994.

D.S. Dummit and R. M. Foote, Abstract
Algebra, 2nd Ed., John Wiley, 2002.

J.A. Gallian, Contemporary Abstract
Algebra, 4th Ed., Narosa, 1999.

N. Jacobson, Basic Algebra I, 2nd Ed., Hindustan Publishing Co., 1984, W.H. Freeman, 1985.

**Equations **** 3 1 0 8**

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

Existence and Uniqueness of Initial Value Problems: Picard's and
Peano's Theorems, Gronwall's inequality, continuation of solutions and maximal
interval of existence, continuous dependence.

Higher Order Linear Equations and linear Systems: fundamental
solutions, Wronskian, variation of constants, matrix exponential solution,
behaviour of solutions.

Two Dimensional Autonomous Systems and Phase Space Analysis:
critical points, proper and improper nodes, spiral points and saddle points.

Asymptotic Behavior: stability (linearized stability and
Lyapunov methods).

Boundary Value Problems for Second Order Equations: Green's
function, Sturm comparision theorems and oscillations, eigenvalue problems.

Texts / References

M. Hirsch, S. Smale and R. Deveney, Differential Equations,
Dynamical Systems and Introduction to Chaos, Academic Press, 2004

L. Perko, Differential Equations and Dynamical Systems, Texts in
Applied Mathematics, Vol. 7, 2^{nd} ed., Springer
Verlag, New York, 1998.

M. Rama Mohana Rao, Ordinary Differential Equations: Theory and
Applications. Affiliated East-West Press Pvt. Ltd., New Delhi, 1980.

D. A. Sanchez, Ordinary Differential Equations and Stability
Theory: An Introduction, Dover Publ. Inc., New York, 1968.

**MA 503 Functional Analysis **** 3 1 0 8**

Prerequisites: MA 401 Linear Algebra,

MA 408 Measure Theory

Normed spaces. Continuity of linear maps.
Hahn-Banach Extension and Separation Theorems. Banach spaces. Dual spaces and transposes.

Uniform Boundedness Principle and its applications. Closed Graph
Theorem, Open Mapping Theorem and their applications. Spectrum
of a bounded operator. Examples of compact operators
on normed spaces.

Inner product spaces, Hilbert spaces. Orthonormal basis. Projection theorem and
Riesz Representation Theorem.

Texts / References

J.B. Conway, A Course in Functional Analysis, 2^{nd}
ed., Springer, Berlin, 1990.

C. Goffman and G. Pedrick, A First Course in Functional
Analysis, Prentice-Hall, 1974.

E. Kreyzig, Introduction to Functional Analysis with
Applications, John Wiley & Sons, New
York, 1978.

B.V. Limaye, Functional Analysis, 2^{nd} ed., New Age
International, New Delhi,
1996.

A. Taylor and D. Lay, Introduction to Functional Analysis,
Wiley, New York, 1980.

**Hilbert Spaces**** 2 1 0 6**

Prerequisite: MA 503 Functional Analysis

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

Compact operators on Hilbert spaces. Spectral
theorem for compact self-adjoint operators.

Application to Sturm-Liouville Problems.

Texts / References

J.B. Conway, A Course in Functional Analysis, 2^{nd}
ed., Springer, Berlin, 1990.

C. Goffman and G. Pedrick, First Course in Functional Analysis,
Prentice Hall, 1974.

I. Gohberg and S. Goldberg, Basic Operator Theory, Birkhauser,
1981.

E. Kreyzig, Introduction to Functional Analysis with
Applications, John Wiley & Sons, New
York, 1978.

B.V. Limaye, Functional Analysis, 2nd ed., New Age International,
New Delhi, 1996.

**MA 505 Algebra- II****
3 1 0 8**

Prerequisite: MA 414 Algebra I

Modules,
submodules, quotient modules and module homomorphisms.

Generation of modules, direct sums and
free modules. Tensor products of
modules. Exact sequences, projective modules.

Tensor algebras, symmetric and exterior
algebras.

Finitely generated modules over principal ideal domains,
invariant factors, elementary divisors, rational
canonical forms. Applications to finitely generated abelian groups and linear
trans-formations.

Noetherian rings and modules, Hilbert basis theorem, Primary
decomposition of ideals in noetherian rings.

Integral extensions, Going-up and Going-down theorems,
Extension and contraction of prime ideals, Noether's Normalization Lemma,
Hilbert's Nullstellensatz.

Localization of rings and modules. Primary decompositions of modules.

Texts / References

M.F. Atiyah and I. G. Macdonald,
Introduction to Commutative Algebra, Addison Wesley, 1969.

D.S. Dummit and R. M. Foote, Abstract
Algebra, 2nd Ed., John Wiley, 2002.

N. Jacobson, Basic Algebra I and II,
2nd Ed., W. H. Freeman, 1985 and 1989.

S. Lang, Algebra, 3rd Ed., Springer
(India), 2004.

O. Zariski and P. Samuel, Commutative Algebra,
Vol. I, Springer, 1975.

**MA 508 Mathematical Methods****
3 1 0 6**

Prerequisite: MA 515 Partial Differential

Equations

Asymptotic expansions, Watson's lemma,
method of stationary phase and saddle point method. Applications to differential equations.
Behaviour of solutions near an irregular singular point,
Stoke's phenomenon. Method of strained coordinates and matched
asymptotic expansions.

Variational principles, Lax-Milgram theorem and applications
to boundary value problems.

Calculus of variations and integral
equations. Volterra integral
equations of first and second kind. Iterative methods
and Neumann series.

Texts / References

C.M. Bender and S.A. Orszag, Advanced Mathematical Methods
for Scientists and Engineers, McGraw-Hill Book Co., 1978.

R. Courant & D.Hilbert, Methods of Mathe-matical
Physics, Vol. I & II, Wiley Eastern Pvt. Ltd. New Delhi, 1975.

J. Kevorkian and J.D. Cole, Perturbation Methods in Applied
Mathematics, Springer Verlag, Berlin, 1985.

S.G. Mikhlin, Variation Methods in Mathe-matical Physics,
Pergaman Press, Oxford 1964.

**MA 510 Introduction to Algebraic**

**Geometry 2 1 0 6**

Prerequisite
: MA
414

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

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

Texts / References

S.S. Abhyankar, Algebraic Geometry for
Scientists and Engineers, American Mathe-matical Society, 1990.

W. Fulton,
Algebraic Curves, Benjamin, 1969.

J. Harris, Algebraic Geometry: A First
Course, Springer-Verlag, 1992.

M. Reid, Undergraduate Algebraic Geometry, Cambridge
University Press, Cambridge, 1990.

I.R. Shafarevich, Basic Algebraic Geometry,
Springer-Verlag,
Berlin, 1974.

R.J. Walker,
Algebraic Curves, Springer- Verlag, Berlin,
1950.

**Equations ****3 1 0 8**

Prerequisites :

MA 417 Ordinary Differential Equations.

MA
410 Multivariable Calculus

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

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

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

Stability
theory, energy conservation and dispersion.

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

Heat equation: initial value problem,
fundamental solution, weak and strong maximum principle
and uniqueness results.

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

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

Texts
/ References

E. DiBenedetto, Partial Differential
Equations, Birkhauser, Boston, 1995.

L.C. Evans, Partial Differrential
Equations, Graduate Studies in Mathematics, Vol. 19, AMS,
Providence, 1998.

F. John, Partial Differential
Equations, 3^{rd} ed., Narosa Publ. Co., New Delhi,1979.

E. Zauderer, Partial Differential
Equations of Applied Mathematics, 2^{nd} ed., John Wiley and Sons, New
York, 1989.

**MA 516 Algebraic Topology****
3 1 0 8**

Prerequiste: MA 406 General Topology

Paths and homotopy, homotopy
equivalence, contractibility, deformation retracts.

Basic constructions: cones, mapping
cones, mapping cylinders, suspension.

Cell complexes, subcomplexes, CW pairs.

Fundamental groups. Examples
(including the fundamental group of the circle) and applications (including
Fundamental Theorem of Algebra, Brouwer Fixed Point Theorem and Borsuk-Ulam
Theorem, both in dimension two). Van Kampen's Theorem, Covering spaces,
lifting properties, deck transformations. universal
coverings (existence theorem optional).

Simplicial
complexes, barycentric subdivision, stars and links, simplicial approximation. Simplicial Homology. Singular
Homology. Mayer-Vietoris Sequences. Long exact sequence of
pairs and triples. Homotopy invariance and excision
(without proof).

Degree. Cellular Homology.

Applications of homology:
Jordan-Brouwer separation theorem, Invariance of dimension, Hopf's Theorem for
commutative division algebras with identity, Borsuk-Ulam Theorem, Lefschetz
Fixed Point Theorem.

Optional Topics:

Outline of the theory of: cohomology
groups, cup products, Kunneth formulas, Poincare
duality.

Texts / References

M.J. Greenberg and J. R. Harper,
Algebraic Topology, Benjamin, 1981.

W. Fulton, Algebraic topology: A First
Course, Springer-Verlag, 1995.

A. Hatcher, Algebraic Topology,
Cambridge Univ. Press, Cambridge, 2002.

W. Massey, A Basic Course in Algebraic
Topology, Springer-Verlag, Berlin, 1991.

J.R.
Munkres, Elements of Algebraic Topology, Addison Wesley, 1984.

J.J. Rotman, An Introduction to
Algebraic Topology, Springer (India), 2004.

H. Seifert and W. Threlfall, A Textbook
of Topology, translated by M. A. Goldman, Academic Press, 1980.

J.W. Vick, Homology Theory,
Springer-Verlag, 1994.

**MA 518 Spectral Approximation**** 2 1 0 6**

Prerequisite: MA 503 Functional Analysis

Spectral decomposition. Spectral sets of
finite type. Adjoint and product spaces.

Convergence of operators: norm,
collectively compact and n convergence. Error estimates.

Finite rank approximations based on
projections and approximations for integral operators.

A posteriori error estimates.

Matrix
formulations for finite rank operators.

Iterative
refinement of a simple eigenvalue.

Numerical examples.

Texts / References

M. Ahues, A. Largillier and B. V. Limaye,
Spectral Computations for Bounded Operators, Chapman and Hall/CRC, 2000.

F. Chatelin, Spectral Approximation of
Linear Operators, Academic Press, 1983.

T. Kato, Perturbation Theory of Linear Operators,
2^{nd} ed., Springer-Verlag, Berlin, 1980.

**Functions ****2 1 0 6**

Prerequisites : MA 403 Real Analysis,

MA 412
Complex Analysis.

Maximum Modulus Theorem. Schwarz Lemma. Phragmen-Lindelof Theorem.

Riemann
Mapping Theorem. Weierstrass Factor-ization Theorem.

Runge's Theorem. Simple connectedness. Mittag-Leffler Theorem.

Schwarz Reflection Principle.

Basic properties of harmonic functions.

Picard
Theorems.

Texts
/ References

L. Ahlfors, Complex Analysis, McGraw-Hill,
3^{rd} ed., New York, 1979.

J.B. Conway,
Functions of One Complex Varliable, 2^{nd} ed., Narosa, New Delhi
1978.

T.W. Gamelin,
Complex Analysis, Springer International, 2001.

R.
Narasimhan, Theory of Functions of One Complex Variable, Springer (India),
2001.

W. Rudin, Real and
Complex Analysis, 3^{rd} ed., Tata McGraw-Hill, 1987.

**MA 522 Fourier ****Analysis**** and **

**Applications ****3 1 0 8**

Prerequisite: MA 403 Real Analysis

Basic
Properties of Fourier Series: Uniqueness of Fourier
Series, Convolutions, Cesaro and Abel Summability, Fejer's theorem, Poisson
Kernel and Dirichlet problem in the unit disc. Mean square Convergence, Example
of Continuous functions with divergent Fourier series.

Distributions
and Fourier Transforms: Calculus of Distributions, Schwartz class of rapidly
decreasing functions, Fourier transforms of rapidly decreasing functions,
Riemann Lebesgue lemma, Fourier Inversion Theorem, Fourier transforms of
Gaussians.

Tempered
Distributions: Fourier transforms of tempered distributions, Convolutions,
Applications to PDEs (Laplace, Heat and Wave Equations), Schrodinger-Equation
and Uncertainty principle.

Paley-Wienner
Theorems, Poisson Summ-ation Formula: Radial Fourier transforms and Bessel's
functions. Hermite functions.

Optional
Topics:

Applications to PDEs, Wavelets and X-ray
tomography.

Applications to Number Theory.

Texts
/ References:

R.
Strichartz, A Guide to Distributions and Fourier Transforms, CRC Press.

E.M.
Stein and R. Shakarchi, Fourier Analysis: An
Introduction, Princeton University Press, Princeton 2003.

I.
Richards and H. Youn, Theory of Distributions and Non-technical Approach,
Cambridge University Press, Cambridge, 1990.

**MA 523 Basic Number Theory**** 2 1 0 6**

Prerequisites:
MA 419 Basic Algebra

Infinitude of primes, discussion of the
Prime Number Theorem, infinitude of primes in specific arithmetic progressions,
Dirichlet's theorem (without proof).

Arithmetic
functions, Mobius inversion formula. Structure of units modulo n, Euler's
phi function

Congruences,
theorems of Fermat and Euler, Wilson's theorem, linear congruences, quadratic
residues, law of quadratic reciprocity.

Binary quadratics forms, equivalence,
reduction, Fermat's two square theorem, Lagrange's
four square theorem.

Continued
fractions, rational approximations, Liouville's theorem, discussion of Roth's
theorem, transcendental numbers, transcendence of "e" and
"pi".

Diophantine
equations: Brahmagupta's equation (also known as Pell's equation), the Thue
equation, Fermat's method of descent, discussion of
the Mordell equation.

Optional
Topics:

Discussion of Waring's problem.

Discussion of the Bhargava-Conway "fifteen
theorem" for positive definite quadratic forms.

The RSA algorithm and public key encryption.

Primality testing, discussion of the Agrawal-Kayal-Saxena
theorem.

Catalan's
equation, discussion of the Gelfond-Schneider theorem, discussion of Baker's
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, Cambridge, 1984.

I. Niven and H.S. Zuckerman, An
Introduction to the Theory of Numbers, 4th Ed., Wiley,
New York, 1980.

**MA 524 Algebraic Number Theory ****2 1 0 6 **

Prerequisites: MA 505 Algebra - II
(Exposure)

Algebraic number fields.

Localisation, discrete valuation rings.

Integral
ring extensions, Dedekind domains, unique factorisation of ideals. Action of the galois
group on prime ideals.

Valuations and completions of number
fields, discussion of Ostrowski's theorem, Hensel's lemma, unramified, totally
ramified and tamely ramified extensions of p-adic fields.

Discriminants
and Ramification.

Cyclotomic fields, Gauss sums,
quadratic reciprocity revisited.

The ideal class group, finiteness of
the ideal class group, Dirichlet units theorem.

Texts / References

K. Ireland and M. Rosen, A Classical
Introduction to Modern Number Theory, 2nd ed., Springer-Verlag, Berlin, 1990.

S. Lang, Algebraic Number Theory,
Addison- Wesley, 1970.

D.A. Marcus, Number Fields,
Springer-Verlag, Berlin, 1977.

**MA 525 Dynamical Systems ****2 1 0 6**

Prerequisite:
MA 417 Ordinary Differential

Equations

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

Limit
sets and attractors. Poincare map, Poincare Benedixson theory and Poincare
index.

Texts
/ References

V.I. Arnold, Ordinary Differential
Equations, rentice Hall of India, New Delhi, 1998.

M.W. Hirsch and S. Smale, Differential
Equations, Dynamical Systems and inear Algebra, Academic Press, NY, 174.

L. Perko, Differential Equations and
Dynamical Systems, Springer Verlag, NY, 1991.

S. Wiggins, Introduction to Applied
Nonlinear Dynamical Systems and Chaos, TAM Vol.2, Springer-Verlag, NY, 1990.

**MA 526 Commutative Algebra ****2 1 0 6**

Prerequisites:
MA 505 Algebra - II

Dimension theory of affine algebras:
Principal ideal theorem, Noether normalization lemma, dimension and
transcendence degree, catenary property of affine rings, dimension and degree
of the Hilbert polynomial of a graded ring, Nagata's altitude formula,
Hilbert's Nullstellensatz, finiteness of integral closure.

Hilbert-Samuel polynomials of modules :

Associated primes of modules, degree of
the Hilbert polynomial of a graded module, Hilbert series and dimension,
Dimension theorem, Hilbert-Samuel multiplicity, associativity for-mula for
multiplicity,

Complete
local rings:

Basics of completions, Artin-Rees
lemma, associated graded rings of filtrations, completions of modules, regular
local rings

Basic
Homological algebra:

Categories and functors, derived
functors, Hom and tensor products, long exact sequence of homology modules,
free resolutions, Tor and Ext, Koszul complexes.

Cohen-Macaulay
rings:

Regular sequences, quasi-regular
sequences, Ext and depth, grade of a module, Ischebeck's theorem, Basic
properties of Cohen-Macaulay rings, Macaulay's unmixed theorem, Hilbert-Samuel
multiplicity and Cohen-Macaulay rings, rings of invariants of finite groups.

Optional
Topics:

1.
Face rings of simplicial complexes, shellable simplicial complexes and their
face rings.

2.
Dedekind Domains and Valuation Theory.

Text/References

D.
Eisenbud, Commutative Algebra (with a view toward algebraic geometry) Graduate
Texts in Mathematics 150, Springer-Verlag, Berlin, 2003.

H. Matsumura, Commutative ring theory,
Cambridge Studies in Advanced Mathematics No. 8, Cambridge
University Press, Cambridge, 1980.

W. Bruns and J. Herzog, Cohen-Macaulay
Rings, (Revised edition) Cambridge Studies in Advanced Mathematics No. 39, Cambridge University Press, Cambridge, 1998.

**MA 530 Nonlinear Analysis ****2 1 0 6**

Prerequisites:
MA 503 Functional Analysis.

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

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

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

Degree theory and condensing operators with applications.

Texts
/ References

M.C. Joshi and R.K. Bose, Some Topics
in Nonlinear Functional Analysis, Wiley Eastern Ltd., New Delhi, 1985.

E. Zeilder, Nonlinear Functional
Analysis and Its Applications, Vol. I (Fixed Point Theory),
Springer Verlag, Berlin, 1985.

**MA 532 Analytic Number Theory ****2 1 0 6**

Prerequisites: MA 414 Algebra - I

MA
412 Complex Analysis

The
Wiener-Ikehara Tauberian theorem, the Prime Number Theorem.

Dirichlet's
theorem for primes in an Arithmetic Progression.

Zero free regions for the Riemann-zeta
function and other L-functions.

Euler
products and the functional equations for the Riemann zeta function and
Dirichlet L-functions.

Modular forms for the full modular
group, Eisenstein series, cusp forms, structure of the ring of modular forms.

Hecke
operators and Euler product for modular forms.

The
L-function of a modular form, functional equations.

Modular forms and the sums of four
squares.

Optional topics:

1.
Discussion of L-functions of number
fields and the Chebotarev Density Theorem.

2.
Phragmen-Lindelof Principle, Mellin
inversion formula, Hamburger's theorem.

3.
Discussion of Modular forms for
congruence subgroups.

4.
Discussion of Artin's holomorphy
conjecture and higher reciprocity laws.

5.
Discussion of elliptic curves and the
Shimura-Taniyama conjecture (Wiles' Theorem)

Text / References:

S. Lang, Algebraic Number Theory,
Addison-Wesley, 1970.

J.P. Serre, A Course in Arithmetic,
Springer-Verlag, 1973.

T. Apostol, Introduction to Analytic
Number Theory, Springer-Verlag, 1976

**Theory**** 2 1 0 6**

Probability measure, probability space,
construction of Lebesgue measure, extension theorems, limit of events,
Borel-Cantelli lemma.

Random
variables, Random vectors, distributions,
multidimensional distributions, independence.

Expectation, change of variable theorem,
convergence theorems.

Sequence of random variables, modes of
convergence. Moment
generating function and characteristics functions, inversion and uniqueness
theorems, continuity theorems, Weak and strong laws of large number, central
limit theorem.

Radon Nikodym theorem, definition and properties
of conditional expectation, conditional distributions and expectations.

Texts
/ References

P.
Billingsley, Probability and Measure, 3^{rd} ed., John Wiley &
Sons, New York, 1995.

J.
Rosenthal, A First Look at Rigorous Probability, World Scientific, Singapore,
2000.

A.N.
Shiryayev, Probability, 2^{nd} ed., Springer, New York, 1995.

K.L.
Chung, A Course in Probability Theory, Academic Press, New York, 1974.

**MA 534 Modern Theory of Partial**

**Differential Equations ****2 1 0 6**

Prerequisites:
MA 503 Functional Analysis

MA
515 Partial Differential

Equations.

Theory of distributions: supports, test
functions, regular and singular distributions, generalised derivatives.

Sobolev Spaces: definition and basic
properties, approximation by smooth functions, dual spaces, trace and imbedding
results (without proof).

Elliptic Boundary Value Problems:
abstract variational problems, Lax-Milgram Lemma, weak solutions and
wellposedness with examples, regularity result, maximum principles, eigenvalue problems.

Semigroup Theory and Applications:
exponential map, C_{0}-semigroups, Hille-Yosida and Lummer-Phillips
theorems, applications to heat and wave equations.

Texts
/ References

S.
Kesavan, Topics in Functional Analysis Wiley Eastern Ltd., New Delhi, 1989.

M. Renardy and R.C. Rogers, An
Introduction to Partial Differential Equations,2^{nd}
ed., Springer Verlag International Edition, New York, 2004.

L.C.
Evans, Partial Differential Equations, AMS, Providence, 1998.

**MA 538 Representation Theory of **

**Finite Groups **** 2 1 0 6**

Prerequisite : MA 414 Algebra I

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

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

Examples: Cyclic groups, alternating and
symmetric groups.

Integrality properties of characters,
Burnside's

p^{a}q^{b}^{ }theorem. The character of induced
representation, Frobenius Reciprocity Theorem, Meckey's irreducibility
criterion, Examples of induced representations, Representations of
supersolvable groups.

Texts
/ References

M. Burrow, Representation Theory of Finite
Groups, Academic Press, 1965.

N. Jacobson, Basic Algebra II, Hindustan
Publishing Corproation, 1983.

S.
Lang, Algebra, 3^{rd} ed. Springer (India)
2004.

J.P. Serre, Linear Representation of
Groups, Springer-Verlag, 1977.

**MA 539 Spline Theory and Variational **

**Methods **** 2 0 2 6**

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

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

Texts
/ References

C. De Boor, A Practical Guide to
Splines, Springer-Verlag, Berlin, 1978.

H.N. Mhaskar and D.V. Pai, Fundamentals
of Approximation Theory, Narosa Publishing House, New Delhi, 2000.

P.M. Prenter, Splines and Variational
Methods, Wiley-Interscience, 1989.

**MA 540 Numerical Methods for Partial**

**Differential Equations ^{
}**

Prerequisite: MA 515 Partial Differential

Equations

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

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

Texts / References

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

G.D. Smith, Numerical Solutions of Partial

Differential Equations, 3rd Edition,

Calrendorn Press, Oxford,
1985.

J.C. Strikwerda, Finite difference
Schemes and Partial Differential Equations, Wadsworth and Brooks/ Cole Advanced
Books and Software, Pacific Grove, California, 1989.

J.W. Thomas, Numerical Partial Differential
Equations : Finite Difference Methods, Texts in
Applied Mathematics, Vol. 22, Springer Verlag, NY, 1999.

J.W. Thomas, Numerical Partial Differential
Equations: Conservation Laws and Elliptic Equations, Texts in Applied
Mathematics, Vol. 33, Springer Verlag, NY, 1999.

**MA 556 Differential Geometry****
2 1 0 6**

Prerequiste: MA 410 Multivariable
Calculus

Graphs and level sets of functions on
Euclidean spaces, vector fields, integral curves of
vector fields, tangent spaces.

Surfaces in Euclidean spaces, vector
fields on surfaces, orientation, Gauss map.

Geodesics,
parallel transport, Weingarten map.

Curvature of plane curves, arc length
and line integrals.

Curvature
of surfaces.

Parametrized surfaces, local
equivalence of surfaces.

Gauss-Bonnet
Theorem, Poincare-Hopf Index Theorem.

Texts / References

M. doCarmo,
Differential Geometry of Curves and Surfaces, Prentice Hall, 1976.

B. O'Neill, Elementary Differential
Geo-metry, Academic Press, New York, 1966.

J.J. Stoker, Differential Geometry,
Wiley-Interscience, 1969.

J.A. Thorpe, Elementary Topics in
Differential Geometry, Springer (India), 2004.

**Finite Elements ****2 1 0 6**

Prerequisite:
MA 515 Partial Differential

Equations

MA
503 Functional Analysis

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

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

Parabolic initial and boundary value
problems: semidiscrete and completely discrete schemes with convergence
analysis.

Lab
component: Implementation of algorithms and computational experiments

using MATLAB.

Texts
/ References

K.E. Brenner and R. Scott, The Mathematical Theory of Finite
Element Methods, Springer- Verlang, Berlin, 1994.

P.G.
Ciarlet, The Finite Element Methods for Elliptic Problems, North Holland,
Amsterdam, 1978.

C. Johnson, Numerical solutions of
Partial Differential Equations by Finite Element Methods, Cambridge University
Press, Cambridge, 1987.

C. Mercier, Lectures on Topics in
Finite Element Solution of Elliptic Problems, TIFR Lectures on Mathematics and
Physics Vol. 63, Narosa Publ. House, New Delhi, 1979.

**MA 581 Elements of Differential **

**Topology **** 2 1 0 6**

Prerequisite: MA 410 Multivariable

Calculus

Differentiable Manifolds in R^{n}:
Review of inverse and implicit function theorems; tangent spaces and tangent
maps; immersions; submersions and embeddings.

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

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

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

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

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

Texts / References

1. A. Dubovin, A.T. Fomenko, S.P. Novikov, Modern Geometry Methods
and Applications - II, The Geometry and Topology of Manifolds, GTM 104,
Springer-Verlag, Berlin, 1985.

2.

V. Guillemin and A Pollack, Differential
Topology Prentice-Hall Inc., Englewood
Cliffs, New Jersey, 1974.

J. Milnor, Topology from the Differential View-point, University Press of Virginia, Charlottsville
1990.

**SI 421 Introduction to ****Mathematical Software 0 0 3 3**

Introduction to
following topics using mathematical softwares :

Linear Algebra : Solution of linear systems by elimination,
Pivoting strategy, least squares problems, eigenvalue problem.

Euler’s Method

The Second Order and
Fourth Order
Runge Kutta Methods

First
Order differential equations; IVP for ODE systems of First Order equations.

Second Order differential equations

Predictor – Correction method

Combinatorics: Making lists of combinatorial objects,
generating random combinatorial objects (like sets, permutations, partitions
etc.), Tree Search, Graph search : Breadth and depth
first search

Mathematical packages
such as MATLAB, MATHEMATICA, MAPLE etc. will be
introduced to solve problems in Linear Algebra, Differential Equations,
Numerical Analysis, Combinatorics etc.

**Texts /
References **

Chapman, MATLAB programming for
Engineers, Thomson Learning (3^{rd} Edition – 2005)

Wolfram, The MATHEMATICA book (5^{th} Edition), Wolfram Media – 2003

Heck,
Introduction to Maple, (3^{rd} Edition), Springer – 2003

**SI 402 Statistical
Inference ****3 1 0 8**

Prerequisite:SI
417 Introduction to

Probability Theory

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

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

Confidence Intervals.

**Texts / References**

G. Casella and B.L. Berger, Statistical
Inference, Wadsworth and Brooks, Pacific Grove, 1990.

M.H. DeGroot, Probability and Statistics,
Addison-Wesley, 1986

E.L.Lehmann and G. Casella, Theory of Point
Estimation, Springer-Verlag, New York,1998.

**SI
422 Regression Analysis ****2 1 0 6**

Prerequisites:
SI 417 Introduction to

Probability
Theory

Simple and multiple linear regression models
%G––%@ estimation, tests and confidence regions.
Check for normality assumption. Likelihood ratio test,
confidence intervals and hypotheses tests; tests for distributional
assumptions. Collinearity, outliers; analysis of residuals, Selecting
the $(B!H(BBest$(B!I(B regression equation,
transformation of response variables. Ridge's regression.

Texts / Reference

B.L.
Bowerman and R.T. O'Connell, Linear Statistical
Models: An Applied Approach, PWS-KENT Pub., Boston, 1990

N.R Draper. And H. Smith., Applied Regression Analysis, John Wiley and Sons
(Asia) Pvt. Ltd., Series in Probability and Statistics, 2003.

D.C.
Montgomery, E.A. Peck, G.G. Vining, Introduction to Linear Regression Analysis,
John Wiley, NY, 2003

A.A.
Sen and M. Srivastava, Regression Analysis %G–%@ Theory, Methods &
Applications,

Springer-Verlag, Berlin, 1990.

**Probability Theory**** 3 1 0 8**

Axioms of Probability, Conditional
Prob-ability and Independence, Random variables and distribution functions, Random
vectors and joint distributions, Functions of random vectors.

Expectation,
moment generating functions and characteristic functions, Conditional
expectation and distribution.

Modes of
convergence, Weak and strong laws of large numbers, Central limit theorem.

Text / References

P. Billingsley, Probability and
Measure, II Edition, John Wiley & Sons (SEA) Pvt. Ltd., 1995.

P.G. Hoel, S.C. Port and C.J. Stone,
Introduction to Probability, Universal Book Stall, New Delhi, 1998.

J.S. Rosenthal, A First Look at
Rigourous Probability Theory, World Scientific. 2000.

M. Woodroofe, Probability with
Applica-tions, McGraw-Hill Kogakusha Ltd., Tokyo, 1975.

**SI 412 Algorithms ****3 1 0 8**

Prerequisite: None. For non-Math department
students, consent of Instructor required to register.

Tools for
Analysis of Algorithms (Asymptotics, Recurrence Relations). Basic Data
Structures (Lists, Stacks, Queues, Trees, Heaps) and applications.

Sorting, Searching and Selection
(Binary Search, Insertion Sort, Merge Sort, QuickSort, Radix Sort, Counting
Sort, Heap Sort etc.. Median finding using Quick-Select, Median of Medians). Basic Graph Algorithm
(BFS, DFS, strong components etc.).

Algorithm Design Paradigms: Divide and
Conquer. Greedy Algorithms (for example, some greedy scheduling
algorithms, Dijkstra's Shortest Paths algorithm, Kruskal's Minimum Spanning
Tree Algorithm). Dynamic Programming (for example,
dynamic programming algorithms for optimal polygon triangulation, optimal
binary search tree, longest common subsequence, matrix chain multiplication,
all pairs shortest paths).

Introduction
to NP-Completeness (polynomial time reductions, verification algorithms,
classes P and NP, NP-hard and NP-complete problems).

Texts / References

R. Sedgewick, Algorithms in C++, Addison-Wesley,
1992.

T. Cormen, C. Leiserson, R. Rivest and
C. Stein, Introduction to Algorithms, MIT Press, 2001.

M.A. Weiss, Data Structures and
Algorithms Analysis in C++, Addison-Wesley, 1999.

**SI 419 Combinatorics **** 2 1 0 6**

*Prerequisites:* None.
Non-math department students need the consent of the Instructor to register.

Algorithms
and Efficiency. Graphs: Paths, Cycles, Trees, Coloring.
Trees, Spanning Trees, Graph Searching (DFS, BFS), Shortest
Paths. Bipartite Graphs and Matching problems.
Counting on Trees and Graphs. Hamiltonian and Eulerian Paths.

Groups: Cosets and Lagrange Theorem,
Cyclic Groups etc.. Permutation Groups, Orbits and
Stabilizers. Generating Functions. Symmetry and Counting: Polya Theory. Special
Topics (depending upon the instructor!)

Text / References

Normal L. Biggs, Discrete Mathematics,
Oxford University Press, Oxford, 2002.

J. Hein, Discrete Structures, Logic and
Computatibility, Jones and Barlett, 2002.

C.L.Liu, Elements of Discrete
Mathematics, McGraw Hill, 1986

**SI 416 Optimization ****2 0 2 6**

*Prerequisites:* None.
Non-math department students need the consent of the Instructor to register.

Unconstrained optimization using
calculus (Taylor's theorem, convex functions, coercive functions
).

Unconstrained optimization via
iterative methods (Newton's method, Gradient/ conjugate gradient based methods,
Quasi- Newton methods).

Constrained optimization (Penalty
methods, Lagrange multipliers, Kuhn-Tucker conditions.
Linear programming (Simplex method, Dual simplex, Duality
theory). Modeling for Optimization.

Text / Reference

M. Bazarra, C. Shetty, Nonlinear
Progra-mming, Theory and Algorithms, Wiley, 1979.

Beale, Introduction to Optimization,
John Wiley, 1988.

M.C. Joshi and K. Moudgalya,
Optimization: Theory and Practice, Narosa,
New Delhi, 2004

**SI 418 Advanced Programming and Unix **

**Environment **** 0 0 3 3**

UNIX
programming environment (file system and directory structure, and processes). Unix tools (shell
scripting, grep, tar, compress, sed, find, sort etc). Graphical User
Interface Programming using Java. Multithreaded programming
in Java. Socket programming in Java.

Text / Reference

3.
Eckel, Thinking In Java,

http://www.bruceeckel.com/javabook.html

B. Forouzan and R. Gilberg, Unix and
Shell Programming: A Textbook, 3rd ed., Brooks/Cole, 2003.

B.W. Kernighan and R. Pike, Unix
Programming Environment, Prentice Hall, 1984.

**SI 404 Applied Stochastic Processes ****2 1 0 6**

Prerequisite: SI 417 Introduction to

Probability Theory or MA 212

Stochastic processes: Description and
definition. Markov chains with finite and countably infinite
state spaces. Classification of states,
irreduci-bility, ergodicity. Basic limit theorems.

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

Applications
to queueing models and relia-bility theory.

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

Texts
/ References :

V.N. Bhat, Elements of Applied Stochastic
Processes, Wiley, 1972.

V.G. Kulkarni, Modeling and Analysis of
Stochastic Systems, Chapman and Hall, London, 1995.

J.
Medhi, Stochastic Models in Queueing Theory, Academic Press, 1991.

R. Nelson, Probability, Stochastic
Processes, and Queuing Theory: The Mathematics of Computer Performance
Modelling, Springer-Verlag, New York, 1995.

S.
Ross, Stochastic Processes, 2^{nd} ed., Wiley, New York,1996.

**SI 501 Topics in Theoretical Computer **

**Science** **3 1 0 8**

Introduction
to Complexity Theory (P, NP, NP-hard, NP-complete etc.). Automata Theory and
Formal Languages (finite automata, NFA, DFA, regular languages, equivalence of
DFA and NFA, minimization of DFA, closure properties of regular languages,
regular grammars, context free grammars, parse-trees, Chomsky Normal Form,
top-down parsing).

Randomization
and Computation (Monte Carlo and Las Vegas algorithms, Role of Markov and
Chebyscheff's inequalities, Chernoff bounds in randomized algorithms,
applications of probabilistic method).

Special
Topics in Theoretical Computer Science, such as Approximation Algorithms,
Number Theoretic Algorithms, Logic and Computability.

Text / References

G. Ausiello, P. Crescenzi, G. Gambosi,
V. Kann, A. Marchetti-Spaccamela, Complexity and Approximation, Springer
Verlag, Berlin, 1999.

J. Hein, Discrete Structures, Logic and
Computatibility, Jones and Barlett, 2002.

P. Linz, An Introduction to Formal
Languages and Automata, Narosa, New Delhi, 2004.

**SI 503 Categorical Data
Analysis ****3 1 0 8**

Prerequisites: SI 422 Regression Analysis

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

Models for binary response variables.
Logistic regression-dichotomous response. Logistic regression- polytomous response. Probit and extreme
value models. Log-linear models for two and three dimensions.
Fitting of logit and log-linear models. Log-linear models for
ordinal variables.

Multi-category Logit Models.

Applications using SAS software.

Texts/ References:

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

A. Agresti, An Introduction to Categorical Data
Analysis, Wiley, New York, 1996.

E.B. Andersen, The Statistical Analysis of
Categorical Data, Springer-Verlag, Berlin, 1990.

T.J. Santner and D. Duffy, The Statistical
Analysis of Discrete Data, Springer-Verlag, Berlin, 1989.

**SI 505 Multivariate Analysis ****3 1 0 8**

Prerequisites : SI 402 statistical Inference

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, 2^{nd} Ed., Wiley, 1984.

R. Gnanadesikan, Methods for Statistical
Data Analysis of Multivariate Observations, John Wiley, New York,
1997.

R.A. Johnson and D.W. Wicheran, Applied
Multivariate Statistical Analysis, Upper Saddle River, Prentice
Hall, 1998.

M.S.
Srivastava and E.M. Carter, An Introduction to Multivariate Statistics, North
Holland, 1983.

**SI**** ****507 Numerical Analysis ****3 0 2 8**

Principles of floating point computations
and rounding errors.

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

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

Eigenvalue problems: only implementation
issues.

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

Interpolation: review of Lagrange
interpolation techniques, piecewise linear and cubic splines, error estimates.

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

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

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

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

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

Texts / References

K.E. Atkinson, An Introduction to Numerical
Analysis, Wiley, 1989.

S.D. Conte
and C. De Boor, Elementary Numerical Analysis %G–%@ An Algorithmic Approach,
McGraw-Hill, 1981.

K. Eriksson, D. Estep, P. Hansbo and C.
Johnson, Computational Differential Equations, Cambridge Univ. Press,
Cambridge, 1996.

G.H. Golub and J.M. Ortega, Scientific
Computing and Differential Equations: An Introduction to Numerical Methods,
Academic Press, 1992.

J. Stoer and R. Bulirsch, Introduction to
Numerical Analysis, 2nd ed., Texts in Applied Mathematics, Vol. 12, Springer Verlag, New York, 1993.

**SI 508 Network Models **** 2 0 2 6**

Recap of Linear Programming and
duality. Transportation and Assignment. Maximum flow
and minimum cut (duality, Ford and Fulkerson algorithm, polynomial time
algorithms).

Minimum Cost Flows (cycle cancelling
algorithms, successive path algorithms). Matching (bipartite matching, weighted
bipartite matching, cardinality general matching).

Routing algorithms (Bellman Ford
algorithm in computer networks, Dijkstra's algorithm in computer networks),
Application of network models.

Text / Reference

R.K. Ahuja, T.L. Magnanti, J.B. Orlin,
Network Flows, Prentice Hall, 1993

D. Bertsekas, Network Optimization: Continuos
and Discrete Models, Athena Scientific, 1998

M.S. Bazaraa, J.J. Jarvis, H.D.
Sherali, Linear Programming and Network Flows, Second Edition, 1990

**SI 509 Time Series
Analysis ****2 1 0 6**

Prerequisites: SI 402 Statistical Inference

Stationary processes %G–%@ strong and weak,
linear processes, estimation of mean and covariance functions. Wald decomposition Theorem.

Modeling
using ARMA processes, estimation of parameters testing model adequacy, Order
estimation.

Prediction
in stationery processes, with special reference to ARMA processes.

Frequency
domain analysis %G–%@ spectral density and its estimation, transfer functions.

ARMAX, ARIMAX models and introduction to ARCH
models.

Multivariate Time Series, State Space Models.

Texts
/ References

P.
Brockwell and R. Davis, Intoduction to Time Series and Forecasting, Springer,
Berlin, 2000.

G.E.P.
Box, G. Jenkins and G. Reinsel, Time Series Analysis-Forecasting and Control, 3^{rd}
ed., Pearson Education, 1994.

C.
Chatfield, The Analysis of Time Series %G–%@ An Introduction, Chapman and Hall
/ CRC, 4^{th} ed., 2004.

**SI 511 Computer-Aided Geometric **

**Design ****3 0 0 6**

Polynomial curves: Bezier representation,
Bernstein polynomials, Blossoming, de Castlijau algorithm. Derivatives
in terms of Bezier polygon. Degree elevation. Subdivi-sion.
Nonparametric Bezier curves.

Composite
Bezier curves.

Spline curves :
Definition and Basic properties of spline functions, B-spline curves, de Boor
algorithm. Derivatives. Insertion of new knots. Cubic spline interpolation. Inter-pretation
of parametric continuity in terms of Bezier polygon.

Geometric continuity. Frenet
frame continuity. Cubic Beta splines and significance
of the associated parameters.

Tensor
product surfaces. Bezier patches. Tri-angular patch surfaces.

Texts
/ References :

G. Frain, Curves and Surfaces for Computer
Aided Geometric Design : A Practical Guide, Academic
Press, 1988.

L. Ramshaw, Blossoming :
A Connect-the-Dots Approach to Splines, DEC systems Research
Center, Report no. 19, 1987.

**SI
512**** ****Finite Difference Methods for Partial Differential Equations**** **

Description: Review of 2nd order PDEs : Classification, separation of variaqbles and fourier
transform techniques.Automatic mesh generation techniques : Structure mesh (
transfinite interpolation), unstructured grids ( triangulation for polygonal
and non - polygonal domains).Finite difference Methods : Elliptic equations (
SOR and conjugate gradient methods, ADI schemes), parabolic equations ( explicit,
back - ward Euler and Crank - Nicolson method, LOD), hyperbolic equations ( Law
- Wendroff scheme, Leapfrod method, CFL conditions), Stability, consistency and
convergence results.Lab Component : Implementation of Algorithms developed in
this course and exposure to software packages : ODEPACK and MATLAB.

Text/References:
Gene H. Golub and James M. Ortega, Scientific Computing and Differential Equations : An Introduction to Numerical Methods, Academic
Press, 1992.P. Knupp and S. Steinberg, Fundamentals of Grid Generation, CRC
Press Inc., Boca Raton, 1994.A. R. Mitchell and D. F. Griffiths, The finite
Difference Methods in Partial Differential Equations, Wiley, 1980.G. D. Smith,
Numerical Solutions of Partial Differential Equations, Oxford Press, 1985.J. C.
Stickwards, Finite Difference Schemes and PDEs, Chapman and Hall, 1989.J. F.
Thompson, Z. U., A. Waarsi and C. W. MAstin, Numerical Grid Generations -
Foundations and Applications, North Holland, 1985.Erich Zauderer, Partial
Differential Equations of Applied Mathematics, 2nd ed., Wiley, 1989.

**SI 513 Theory of Sampling ****2 1 0 6**

Simple
random sampling. Sampling for proportions and percentages.

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

A
brief introduction to randomized response techniques and small area estimation

Texts
/ References

1.
Chaudhuri and H. Stenger, Survey Sampling:
Theory and Methods, Marcell Dekker, 1992.

W.G. Cochran, Sampling Techniques, 3rd
ed., Wiley Eastern, 1977.

P. Mukhopadhyay, Theory and Methods of
Survey Sampling, Prentice-Hall of India, New Delhi, 1998.

Des Raj, Sampling Theory, Tata McGraw-Hill,
1978.

**SI 514 Statistical Modeling**** 2 1 0 6**

Prerequisites:
SI 402 Statistical Inference

Nonlinear regression, Nonparametric
regression, generalized additive models, Bootstrap methods, kernel methods,
neural network, Artificial Intelligence, a few topics from machine learning.

Texts
/ References:

T.
Hastie, and R. Tibshirani, Generalized Additive Models, Chapman and Hall,
London, 1990.

G.A.F.
Seber, and C.J. Wild, Nonlinear Regression, John Wiley & Sons, 1989.

W. Hardle,
Applied Nonparametric Regression, Cambridge University Press, London, 1990.

**SI 515 Statistical
Techniques in **

**Data Mining ****2 1 0 6 **

Pre-requisite: p SI 402 Statistical Inference

Introduction
to Data Mining and its Virtuous Cycle.

Cluster Analysis: Hierarchical and
Non-hierarchical techniques. Classification
and Discriminant Analysis Tools: CART, Random forests, Fisher's discriminant
functions and other related rules, Bayesian
classification and learning rules.

Dimension Reduction and Visualization
Techniques: Multidimensional scaling, Principal Component Analysis, Chernoff
faces, Sun-ray charts.

Algorithms for data-mining
using multiple nonlinear and nonparametric regression.

Neural Networks: Multi-layer perceptron,
predictive ANN model building using back-propagation algorithm. Exploratory
data analysis using Neural Networks %G–%@ self organizing
maps. Genetic Algorithms, Neuro-genetic model building.

Discussion of Case
Studies.

Texts/References:

L. Breiman, J.H. Friedman,
R.A. Olschen and C.J. Stone, Classification of Regresion Trees, Wadsowrth
Publisher, Belmont, CA, 1984.

D.J. Hand, H. Mannila
and P. Smith, Principles of Data Minng, MIT Press,

Cambridge, MA 2001.

M.H. Hassoun,
Fundamentals of Artificial Neural Networks, Prentice-Hall of India,

New Delhi 1998.

T. Hastie, R. Tibshirani & J. H. Friedman,
The elements of Statistical Learning: Data Mining, Inference & Prediction,
Springer Series in Statistics, Springer-Verlag, New York 2001.

R.A. Johnson and D.W.
Wichern, Applied Multivariate Analysis, Upper Saddle River, Prentice-Hall, N.J.
1998.

S. James Press, Subjective and
Objective Bayesian Statistics: Principles, Models, and Applications, 2nd
Edition, Wiley, 2002.

**SI
525 Testing of Hypothesis ****2 1 0 6**

Prerequisites:
SI 402 Statistical Inference

Statistical hypotheses, Neyman-Pearsaon
fundmeantal lemma, Monotone likelihood ratio, confidence bounds, generalization
of fundamental lemma, two-sided hypotheses.

Unbiased
tests, UMP unbiased tests, applications to standard distributions, similarity
and completion, Pemutation tests; most powerful
permutation tests.

Symmetry and invariance, most powerful invariant
tests, unbiased and invariance.

Tests
with guaranteed power, maxi-min tests and invariance. Likelihood
ratio tests and its properties.

Texts
/ References

E.L. Lehmann, Testing Statistical Hyp-otheses, 2^{nd}
ed. Wiley, 1986.

T.S.
Ferguson, Mathematical Statistics: A Decision Theoretic Approach, Academic
Press, New York, 1967.

G.
Casella and R.L. Berger, Statistical Infwerence, Wordsworth & Brooks,
California, 1990.

**SI 526 Experimental
Designs**** 2 0 2 6**

Prerequisites: SI 402 Statistical Inference

Linear
Models and Estimators, Estimability of linear parametric functions.
Gauss-Markoff Theorem. One-way classification and two-way
classification models and their analyses. Standard
designs such as CRD, RBD, LSD, BIBD. Analysis using
the missing plot technique.

Fctorial
designs. Confounding. Analysis using Yates' algorithm.
Fractional factorial.

A brief introduction to Random Effects models
and their analyses.

A
brief introduction to special designs such as split-plot, strip-plot, cross-over designs.

Response surface
methodology.

Applications using SAS software.

Texts / References

A.M.
Kshirsagar, A First Course in Linear Models, Marcel Dekker, 1983.

D.C.
Montgomery, Design and Analysis of Experiments, 3rd Ed.,
John Wiley & Sons, 1991.

C.F.J.
Wu and M. Hamada, Experiments: Planning Analysis, and Parameter Design
Optimization, John Wiley & Sons, 2002.

**SI
527 Introduction to Derivatives **

**Pricing ****2 1 0 6**

Prerequisites:
SI 417 Introduction to

Probability
Theory

Basic
notions %G–%@ Cash flow, present value of a cash flow, securities, fixed income
securities, types of markets.

Forward
and futures contracts, options, properties of stock option prices, trading
strategies involving options, option pricing using Binomial trees, Black %G–%@
Scholes model, Black %G–%@ Scholes formula, Risk-Neutral measure, Delta %G–%@
hedging, options on stock indices, currency options.

Texts
/ References

D.G.
Luenberger, Investment Science, Oxford University Press, Oxford, 1998.

J.C.
Hull, Options, Futures and Other Derivatives, 4^{th} ed.,
Prentice-Hall, New York, 2000.

J.C.
Cox and M. Rubinstein, Options Market, Englewood Cliffs, N.J.:
Prentice Hall, 1985.

C.P
Jones, Investments, Analysis and Measurement, 5^{th} ed., John Wiley
and Sons, New York, 1996.

**SI 528 Biostatistics ****2 1 0 6**

Pre-requisite: SI 402 Statistical Inference

Introduction
to clinical trials and other types of clinical research, bias and random error
in clinical studies, overview of Phase I-IV trials, multi-center trials; randomized,
controlled clinical trials; concept of blinding/masking in clinical trials.

Design
of Phase 1-3 clinical trials: parallel vs. cross-over designs, cross-sectional
vs. longitudinal designs, review of factorial designs, objectives and endpoints
of clinical trials, formulation of appropriate hypotheses (equivalence,
non-inferiority, etc.); sample size calculation; design for bioequivalence/

bioavailability
trials, sequential stopping in clinical trials.

Analysis
of Phase 1-3 trials: Use of generalized linear models; analysis of categorical
outcomes, Bayesian and non-parametric methods;
analysis of survival data from clinical trials

Epidemiological
studies: case-control and cohort designs; odds ratio and relative risk;
logistic and multiple regression models.

Texts/ References:

S.C.
Chow and J.P. Liu, Design and Analysis of Clinical Trials - Concepts &
Methodologies, John Wiley & Sons, NY, 1998.

S.C.
Chow and J.P. Liu, Design and Analysis of Bioavailability & Bioequivalence Studies, Marcel Dekker, 2000.

W.W.
Daniel, Biostatistics: A Foundation for Analysis in the Health Sciences (6th ed.), John
Wiley, NewYork, 2002.

J.L.
Fleiss, The Design and Analysis of Clinical Experiments, John Wiley & Sons,
1986.

D.W.
Hosmer and S. Lemeshow, Applied Logistic Regression, John Wiley and Sons, NY,
1989.

E.
Vittinghoff, D.V. Glidden, S.C. Shiboski and C.E. McCulloch, Regression Methods
in Biostatistics, Springer Verlag, 2005.

J.G.
Ibrahim, M-H Chen and D. Sinha, Bayesian survival analysis, Springer, NY, 2001.

**SI 530 Statistical Quality Control ****2 1 0 6**

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

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

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

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

Texts
/ References :

A.J. Duncan, Quality Control and Industrial
Statistics, 5th ed., Richard D. Irwin, 1986.

E.L.
Grant and R. Levenworth, Statistical Quality Control, 6^{th} ed.,
McGraw-Hill, 1988.

J.M. Juran and F. M. Grayna, Quality
Planning and Analysis, Tata McGraw-Hill, 1970.

D.C. Montgomery, Introduction to Statistical
Quality Control, Wiley, 1985.

T.P. Ryan, Statistical Methods for Quality
Improvement, Wiley, New York, 2000.

**SI 532 Statistical Decision Theory**** 2 1 0 6**

Prerequisite : SI 402 Statistical Inference

Decision functions, Risk functions,
utility and subjective probability, Randomization,

Optimal
decision rules. Admissibility and completeness, Existence of Bayes Decision
Rules, Existence of a Minimal complete class, Essential completeness of the
class of non-randomized rules. The minimax theorem.

Invariant
statistical decision problems. Multiple decision problems.

Sequential decision problems.

Texts
/ References

J.O. Berger, Statistical Decision Theory : Foundations, Concepts and Methods, Springer-Verlag,
1980.

J.O. Berger, Statistical Design Theory
and Bayesian Analysis, 2^{nd} ed., Springer-Verlag, 1985.

T.S. Ferguson, Mathematical Statistics,
Academic Press, New York, 1967.

S.S. Gupta and D. Huang, Multiple
Statistical Decision Theory, Springer-Verlag, New York, 1981.

**SI 534 Nonparametric Statistics**** 2 1 0 6**

Prerequisite:
SI 402, Statistical Inference

Kolmogorov-Smirnov Goodness-of %G–%@Fit Test.

The empirical
distribution and its basic properties. Order
Statistics. Inferences concerning Location parameter
based on one-sample and two-sample problems. Inferences
concerning Scale parameters. General Distribution Tests based on Two or
More Independent Samples.

Tests for
Randomness and equality of distributions. Tests for Independence. The one-sample regression
problem.

Asymptotic Relative Efficiency of Tests. Confidence Intervals and Bounds

Texts
/ References

W.W.
Daniel, Applied Nonparametric Statistics, 2^{nd}
ed., Boston: PWS-KENT, 1990.

M.
Hollandor, and D.A. Wolfe, Non-parametric Statistical Inference, McGraw-Hill,
1973.

E.L.
Lehmann, Nonparametric Statistical Methods Based on Ranks, McGraw-Hill, 1975.

J.D. Gibbons, Nonparametric Statistical
Inference Marcel Dekker, NewYork, 1985

R.H. Randles and D.A. Wolfe, Introduction
to the Theory of Nonparametric Statistics,Wiley, New York,
1979.

P. Sprent, Applied
Nonparametric Statistical Methods**, **Chapman and Hall, London,
1989

B.C. Arnold, N. Balakrishnan and H.
N. Nagaraja, First Course in Order Statistics. John Wiley, NewYork, 1992

J.K. Ghosh and R.V. Ramamoorthi, Bayesian Nonparametrics,
Springer Verlag, NY, 2003.

**SI 540 Stochastic
Programming and**

**Applications 3 0 0 6**

Quadratic and Nonlinear Programming
solution methods applied to Chance Constrained Pro-gramming problems. Stochastic Linear and Non-linear Progra-mming Problems.
Applications in inventory control and other industrial systems, opti-mization
of queuing models of computer networks, information processing under
uncertainty. Two stage and multi-stage solution techniques.
Dynamic programming with Recourse. Use of Monte Carlo,
probabilistic and heuristics algorithms. Genetic algorithms and Neural networks
for adaptive optimization in random environment.

**Texts / References**

J.R. Birge, and F. Louveaux: Introduction
to Stochastic Programming. Springer, New York, 1997.

V.V. Kolbin, Stochastic programming,** **D. Reidel
Publications, Dordrecht, 1977

S.S. Rao
Engineering Optimization: Theory and Practice. 3rd Ed.,
John Wiley & Sons Inc., NY 1996/ 2002.

J.K. Sengupta, Stochastic Optimizations
and Economic Models. D. Reidel Publications, Dordrecht, 1986.

K.
Marti: Stochastic Optimization Methods. Springer, 2005

Y. Ermoliev and R.J-B. Wets, Numerical
Techniques for Stochastic Optimization, .Springer
Verlag, Berlin, 1988.

Z. Michaeleawicz, General Algorithms +
Data Structures - Evolution Program. Springer-Verlag, Berlin,
1992.

R. J.-B. Wets
and W. T. Ziemba (eds.): Stochastic Programming: State of the Art, 1998, Annals
of Oper. Res. 85, Baltzer, Amsterdam, 1999.

**Reliability ****2 1 0 6**

Pre-requisites: SI 402 Statistical Inference

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 inferential aspects for (i) standard reliability
models, (ii) parametric and non-parametric classes of aging distri-butions.

Texts
/ References

H.
Ascher and H. Feingold, Repairable Systems Reliability: Modeling, Inference,
Mis-conceptions and Their Causes, Marcel Dekker, 1984.

L.J.
Bain and M. Engelhardt, Statistical Analysis of Reliability and Life Testing
Models: Theory and Methods, Marcel Dekker, New York, 1991.

R.E.
Barlow and F. Proschan, Statistical Theory of Reliability and Life Testing,
Holt, Reinhart and Winston, 1975.

J.D.
Kalbfleisch and R.L. Prentice, The Statistical Analysis of Failure Time Data,
Wiley, 1986.

J.F. Lawless,
Statistical Models and Methods for Life Time Data, John Wiley & Sons, 1982.

S.K.
Sinha, Reliability and Life Testing, Wiley Eastern, New Delhi, 1986.

**Utilization ****2 0 2 6**

Functional organization
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, New Delhi, 1976.

**CS 206 Formal Methods in
CS**** 2 0 1 6**

Propositional Logic and First Order
Logic: Syntax and semantics. Proof systems such as Hilbert, Natural Deductions,
Sequent and Resolution, Clasual Form, Herbrand Theorem, Unification and
Resolution Theorem Proving, Applications of logic to Program Specification and
Verification: specification of Abstract Data Types, Hoare logic, assertions,
invariants, weakest preconditions, Formal models of programs: Complete partial
orders as domains, continuous functions, domain constructors, fix point. Denotational semantic of a while-do language.

Text/References:

D. Gries, The Science of Programming,
Springer-Verlag, 1977.

R.C. Backhouse, Program Construction and
Verification, Prentice Hall, 1986.

W.K. Grassman and J.P.
Tremblay, Logic and Discrete Mathematics %G–%@ a Computer Science Perspective,
Prentice Hall, 1991.

J. Loeckxm, H.D. Ehrich and M. Wolf,
Specifications of Abstract Data Types, Wiley-Teubner, 1996.

J. Gallier, Logic for Computer Science:
Foundations of Automated Theorem Proving, Wiley, 1981.

D.A. Schmidt, Denotational Semantics: A
Methodology for Language Development, Allyn and Bacon, Inc.,
1986.

M.J.C. Gordon, Programming Language
Theory and its Implementations, Prentice Hall International, 1988.

N. Francez, Program Verification, Addison
Wesley, 1992.

**EE 636 Matrix
Computations ****3 0 0 6**

Basic iterative methods
for solutions of linear systems and their rates of convergence. Generalized conjugate gradient, Krybov
space and Lanczos methods. Iterative methods for symmetric,
non-symmetric and generalized eigenvalue problems. Singular
value decompo-sitions. Fast computations for
structured matrices. Polynomial matrix computations.
Perturbation bounds for eigenvalues.

Texts/Reference:

O. Axelsson, Iterative Solution Methods,
Cambridge University Press, Cambridge 1994.

G. Meurant, Computer Solution of Large Linear
Systems, North Holland, 1999.

Golub and C. Van Loan, Matrix
Computations, John Hopkins Press, 1996.

G.W. Stewart and J. Sun,
Matrix Perturbation Theory, Academic Press, New York, 1990.

**IT 640
Modern Information System ****3
0 0 6 **

Introduction to Information Systems,
Introduction to Database Management Systems, Software Engineering, Information
Technology and basic of networking, Internet Technologies, Web and HTML,
Distributed systems, Corporate Information systems.

Texts/References:

N.L. Sada. Structured
COBOL, programming with Business Applications, Pitamber Publ. Co., New
Delhi, 1991.

A. Silberschatz, H.F. Korth and S.
Sudarshan, Database System Concepts, 3^{rd} ed., McGraw-Hill, 1997.

R.S. Pressman, Software Engineering %G–%@
A Practitioner's Approach, 4^{th} ed., McGraw-Hill, 1995.

Object Oriented Modeling and Design,
Prentice Hall, 1991.

**EE 649 Finite Fields and
Their **

**Applications ****3 0 0 6**

Basic of finite fields: Groups, rings,
fields, polynomials, field extensions, characterization of finite fields, roots
of irreducible polynomials, traces, norms, bases, roots of unity, cyclotomic
polynomials, representation of elements of finite fields. Wedderbum's theorem,
order of polynomials, primitive polynomials, construction of irreducible polynomials,
linearized polynomials, binomials, trinomials.

Applications to algebraic coding theory:
Linear codes, cyclic codes, Goppa codes.

Texts/References:

R. Lidl and H. Niederreiter, Finite
Fields, Cambridge University Press, Cambridge, 1997.

S. Roman, Coding and Information Theory,
Springer Verlag, 1992.

R. Lidl and H. Niederreiter, Introduction
to Finite Fields and Their Applications, Cambridge University Press, 1986,
Chapters 1 %G–%@ 3 and 8.

**EE 720 An
Introduction to Number **

**Theory
and Cryptography ****3
0 0 6**

Some Topics
In Elementary Number Theory: Time estimates for doing arithmetic. Divisibility and the Euclidean algorithm. Congruences. Some applications to factoring.

Finite Fields
and Quadratic Residues:

Finite
fields, Quadratic residues and reciprocity.

Cryptography:

some simple cryptosystems. Enciphering
matrices.

Public Key:

The
idea of public key cryptography. RSA. Discrete log.

Elliptic
Curves:

Basic facts. Elliptic curve cryptosystems.

Texts/References:

N. Koblitz, A Course in Number and Theory
and Cryptography, Graduate Texts in Mathematics, No.114, Springer-Verlag, New
York/Berlin/Heidelberg, 1987.

A. Baker, A Concise Introduction to the
Theory of Numbers, Cambridge University Press, New York/Port Chester/Melbourne/
Sydney, 1990.

A.N. Parshin and I.R.
Shafarevich (Eds.), Number Theory, Encyclopaedia of Mathe-matics Sciences, Vol. 49, Springer-Verlag,
New York/Berlin/Heidelberg, 1995.

J. Stillwell, Elements of Number Theory,
Undergraduate Texts in Mathematics, Springer-Verlag,NewYork/Berlin/Heidelberg,
2003.

H.C.A. van
Tilborg, An Introduction to Cryptography, Kluwer Academic Publishers, Boston/
Dordrecht/Lancaster, 1988.

A. Weil, Number Theory for Beginners,
Additional references.