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,
Eigenvalues and eigenvectors, characteristic
polynomials, minimal polynomials, Cayley-Hamilton Theorem, triangulation, diagonal-lization,
rational 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 (
S. Lang,
Linear Algebra, Undergraduate Texts in Mathematics,
P. Lax, Linear Algebra, John Wiley &
Sons,
H.E.
Rose, Linear Algebra, Birkhauser, 2002.
S. Lang, Algebra, 3rd
Ed., Springer (
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 Rn).
Continuity and uniform continuity.
Monotonic functions, Functions of bounded variation; Absolutely
continuous functions. Derivatives of functions and
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, 3rd ed., McGraw-Hill, 1983.
MA 405 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, 4th 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, 3rd ed., Springer (
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 (
K.D. Joshi, Introduction to General Topology,
New Age International, New Delhi, 2000.
J.L. Kelley, General Topology, Van Nostrand,
J.R. Munkres, Topology, 2nd Ed., Pearson
Education (
G.F. Simmons, Introduction to Topology and
Modern Analysis,
MA
408 Measure and Integration 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 Lp-spaces,
Riesz-Fischer theorem; Riesz Representation theorem for L2
spaces. Absolute continuity of measures,
Radon-Nikodym theorem. Dual of Lp-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 (2nd ed.), Narosa Publishing House,
H.L. Royden, Real Analysis, 3rd 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.,
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, 2nd ed., Narosa,
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
1999.
MA 414 Algebra - I 3 1
0 8
Prerequiste:
MA 401 Linear Algebra, MA405
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.,
MA 417 Ordinary Differential
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, 2nd
ed., Springer Verlag, New York,
1998.
M. Rama Mohana Rao, Ordinary Differential Equations: Theory and
Applications. Affiliated East-West Press Pvt. Ltd.,
D. A. Sanchez, Ordinary Differential Equations and Stability
Theory: An Introduction, Dover Publ. Inc.,
MA 503 Functional Analysis 3
1 0 8
Prerequisites: MA 401 Linear Algebra,
MA
408 Measure and
Integration
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, 2nd ed., Springer,
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,
B.V. Limaye, Functional Analysis, 2nd ed., New Age
International, New
A. Taylor and D. Lay, Introduction to Functional Analysis,
Wiley,
MA 504 Operators on
Hilbert Spaces 2
1 0 6
Prerequisite: MA 503
Functional Analysis
Adjoints of bounded operators on a Hilbert space,
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, 2nd
ed., Springer,
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,
B.V. Limaye, Functional Analysis, 2nd ed., New Age
International, New
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
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 (
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
R. Courant & D.Hilbert, Methods of
Mathe-matical Physics, Vol. I & II, Wiley Eastern Pvt. Ltd.
J. Kevorkian and J.D. Cole, Perturbation Methods
in Applied Mathematics, Springer Verlag,
S.G. Mikhlin, Variation Methods in Mathe-matical
Physics, Pergaman Press,
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.
J. Harris, Algebraic Geometry: A First Course, Springer-Verlag, 1992.
M. Reid, Undergraduate Algebraic Geometry,
I.R. Shafarevich, Basic Algebraic Geometry,
R.J. Walker, Algebraic Curves, Springer- Verlag,
MA 515 Partial Differential
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.
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,
Texts
/ References
E. DiBenedetto, Partial Differential Equations, Birkhauser,
L.C. Evans, Partial Differrential Equations,
Graduate Studies in Mathematics, Vol. 19, AMS,
F. John, Partial Differential Equations, 3rd ed.,
Narosa Publ. Co., New Delhi,1979.
E. Zauderer, Partial Differential Equations of Applied
Mathematics, 2nd ed.,
John Wiley and Sons,
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.
A. Hatcher, Algebraic Topology,
W. Massey, A Basic Course in Algebraic Topology,
J.R. Munkres,
Elements of Algebraic Topology, Addison Wesley, 1984.
J.J. Rotman,
An Introduction to Algebraic Topology, Springer (
H. Seifert and
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, 2nd ed.,
MA 521 Theory of Analytic
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, 3rd ed.,
J.B. Conway, Functions of One Complex Varliable, 2nd ed., Narosa,
T.W. Gamelin, Complex Analysis, Springer
International, 2001.
R. Narasimhan, Theory of Functions of One Complex
Variable, Springer (
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 (
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,
I. Richards and H. Youn, Theory of Distributions
and Non-technical Approach,
MA 523 Basic
Number Theory 2 1
0 6
Prerequisites:
MA 405 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,
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,
I. Niven and H.S. Zuckerman, An Introduction to the
Theory of Numbers, 4th Ed., Wiley,
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.,
S. Lang, Algebraic Number Theory, Addison-
Wesley, 1970.
D.A. Marcus, Number Fields,
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,
MA 526 Commutative Algebra 2
1 0 6
Prerequisites: MA 405
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,
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)
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,
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.,
E. Zeilder, Nonlinear Functional Analysis and
Its Applications, Vol. I (Fixed Point Theory), Springer Verlag,
MA 532
Analytic Number Theory 2 1 0
6
Prerequisites: MA 402 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
MA 533
Advanced Probability
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, 3rd
ed., John Wiley & Sons,
J. Rosenthal, A First Look at Rigorous
Probability, World Scientific,
A.N. Shiryayev, Probability, 2nd ed.,
Springer,
K.L. Chung, A Course in Probability Theory,
Academic Press,
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, C0-semigroups, Hille-Yosida and Lummer-Phillips theorems,
applications to heat and wave equations.
Texts
/ References
S.
Kesavan, Topics in Functional Analysis Wiley Eastern Ltd.,
M. Renardy and R.C. Rogers, An Introduction to Partial
Differential Equations,2nd ed., Springer Verlag International
Edition, New York, 2004.
L.C.
Evans, Partial Differential Equations, AMS,
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
paqb 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,
S. Lang,
Algebra, 3rd ed. Springer (
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,
H.N. Mhaskar and D.V. Pai, Fundamentals of Approximation
Theory, Narosa Publishing House,
P.M. Prenter, Splines and Variational Methods,
Wiley-Interscience, 1989.
MA 540 Numerical Methods for Partial
Differential Equations 2 1 0 6
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,
J.C. Strikwerda, Finite
difference Schemes and Partial Differential Equations,
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 3 1
0 8
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,
J.J. Stoker, Differential Geometry,
Wiley-Interscience, 1969.
J.A. Thorpe, Elementary Topics in Differential
Geometry, Springer (
MA 562 Mathematical Theory of
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,
P.G.
Ciarlet, The Finite Element Methods for Elliptic Problems,
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,
MA 581
Elements of Differential
Topology 2 1
0 6
Prerequisite: MA 410 Multivariable
Calculus
Differentiable Manifolds in Rn: Review of inverse and
implicit function theorems; tangent spaces and tangent maps; immersions;
submersions and embeddings.
Regular Values: Regular and critical values; regular inverse
image theorem; Sard's theorem; Morse lemma.
Transversality: Orientations of manifolds; oriented and mod 2
intersection numbers; degree of maps. Application to Fundamental theorem of
Algebra.
*Lefschetz theory of vector fields and flows: Poincare-Hopf
index theorem; Gauss-Bonnet theorem.
*Abstract manifolds: Examples such as real and complex
projective spaces and Grassmannian varieties; Whitney embedding theorems.
(* indicates expository treatment intended for these parts of
the syllabus.)
Texts / References
A.
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.
A.
V. Guillemin and A Pollack, Differential Topology Prentice-Hall
Inc.,
J. Milnor, Topology from the Differential View-point, University
Press of Virginia, Charlottsville 1990.
SI
401 Introduction to Computer
Architecture and Operating
Systems 0 0
3 3
Introduction to the following topics: computer
systems, CPU architecture (memory, registers, addressing, busses, instruction
set), data representation, peripheral devices, multi-processor systems,
operating systems (process, memory management, virtual storage, file systems),
basic network components and topologies.
Text / References
I. Englander, The Architecture of
Computer Hardware and System Software: An
Information Theoretic Approach, 2nd edition, John Wiley and Sons, 2002
Neil Gray, A Beginners' C++,
http://www.uow.edu.au/~nabg/ABC/ABC.
html
William Stallings, "Computer Organization
and Architecture", Prentice Hall, 5th edition, 2000
SI 402 Statistical Inference 3 1
0 8
Prerequisite:SI 407
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,
M.H. DeGroot, Probability
and Statistics, Addison-Wesley, 1986
E.L.Lehmann and G. Casella,
Theory of Point Estimation,
SI 404 Regression Analysis 2 1 0
6
Prerequisites: SI 407 Introduction to
Probability
Theory
Simple and multiple linear regression models ––
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 “Best” 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.,
N.R Draper.
And H. Smith., Applied Regression Analysis, John Wiley and Sons (
D.C.
A.A. Sen and M. Srivastava, Regression Analysis – Theory,
Methods & Applications,
SI 407 Introduction to
Probability Theory 3 1 0 8
Axioms of Probability, Conditional Prob-ability
and
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,
J.S. Rosenthal, A First Look at Rigourous
Probability Theory, World Scientific. 2000.
M. Woodroofe, Probability with Applica-tions,
McGraw-Hill Kogakusha Ltd.,
SI 412 Algorithms
3 1 0 8
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
413 Combinatorics 2 1
0 6
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,
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
Unconstrained optimization using calculus (
Unconstrained optimization via iterative 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,
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
B. 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 422 Applied Stochastic Processes 2 1 0 6
Prerequisite: SI 407 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,
J. Medhi,
Stochastic Models in Queueing Theory,
Academic Press, 1991.
R. Nelson, Probability, Stochastic Processes, and Queuing Theory: The Mathematics of Computer
Performance Modelling,
S. Ross, Stochastic Processes, 2nd
ed., Wiley,
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,
SI 503 Categorical Data Analysis 3 1 0
8
Prerequisites: SI 404 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,
E.B.
Andersen, The Statistical Analysis of Categorical Data,
T.J.
Santner and D. Duffy, The Statistical Analysis of Discrete Data,
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, 2nd Ed.,
Wiley, 1984.
R. Gnanadesikan, Methods for Statistical Data Analysis of
Multivariate Observations, John Wiley,
R.A. Johnson and D.W. Wicheran, Applied Multivariate Statistical
Analysis,
M.S. Srivastava
and E.M. Carter, An Introduction to Multivariate Statistics,
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:
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 – An Algorithmic
Approach, McGraw-Hill, 1981.
K. Eriksson, D. Estep, P. Hansbo and C. Johnson, Computational
Differential Equations, Cambridge Univ.
Press,
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 – 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 – 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,
G.E.P. Box, G. Jenkins and G. Reinsel, Time Series Analysis-Forecasting and Control,
3rd ed., Pearson Education, 1994.
C. Chatfield,
The Analysis of Time Series – An Introduction, Chapman and Hall / CRC, 4th
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
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
A. 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
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,
G.A.F. Seber, and C.J. Wild, Nonlinear Regression, John
Wiley & Sons, 1989.
W. Hardle, Applied Nonparametric
Regression,
SI
515 Statistical Techniques in
Data Mining 2 1 0 6
Pre-requisite: 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 – self organizing
maps. Genetic Algorithms, Neuro-genetic model building.
Discussion of Case
Studies.
Text Books/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,
M.H. Hassoun, Fundamentals of Artificial Neural
Networks, Prentice-Hall of
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,
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, 2nd
ed. Wiley, 1986.
T.S. Ferguson, Mathematical Statistics: A
Decision Theoretic Approach, Academic Press,
G. Casella and R.L. Berger, Statistical
Infwerence,
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.
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 407 Introduction to
Probability
Theory
Basic notions – 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 – Scholes model, Black – Scholes formula,
Risk-Neutral measure, Delta – hedging,
options on stock indices, currency options.
Texts / References
D.G. Luenberger, Investment Science,
J.C. Hull, Options, Futures and Other
Derivatives, 4th ed.,
J.C. Cox and M. Rubinstein, Options Market,
C.P Jones, Investments, Analysis and
Measurement, 5th ed., John Wiley and Sons,
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
E. Vittinghoff, D.V.
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, 6th ed.,
McGraw-Hill, 1988.
J.M. Juran and F. M. Grayna, Quality Planning and Analysis, Tata
McGraw-Hill, 1970.
D.C.
T.P. Ryan, Statistical Methods for Quality Improvement, Wiley,
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, 2nd ed., Springer-Verlag, 1985.
T.S. Ferguson, Mathematical Statistics, Academic Press,
S.S. Gupta and D. Huang, Multiple Statistical Decision
Theory,
SI
534 Nonparametric Statistics 2 1 0 6
Prerequisite:
SI-402, Statistical Inference
Kolmogorov-Smirnov Goodness-of –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
Asymptotic
Relative Efficiency of Tests. Confidence Intervals and Bounds
Texts / References
W.W. Daniel,
Applied Nonparametric Statistics, 2nd ed.,
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,
P. Sprent, Applied Nonparametric Statistical Methods,
Chapman and Hall,
B.C.
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
Texts / References
J.R. Birge, and F. Louveaux: Introduction to
Stochastic Programming. Springer,
V.V. Kolbin, Stochastic programming, D. Reidel Publications,
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,
K. Marti: Stochastic Optimization Methods.
Springer, 2005
Y. Ermoliev and R.J-B. Wets, Numerical
Techniques for Stochastic Optimization, .Springer Verlag,
Z. Michaeleawicz, General Algorithms + Data
Structures - Evolution Program.
R. J.-B. Wets and W. T. Ziemba (eds.):
Stochastic Programming: State of the Art, 1998, Annals of Oper. Res. 85, Baltzer,
SI 542
Mathematical Theory of
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,
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
CS
101 Computer Programming &
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,
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 – 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,
G. Meurant, Computer Solution of Large Linear
Systems,
Golub and C. Van Loan, Matrix Computations, John
Hopkins Press, 1996.
G.W. Stewart and J. Sun, Matrix Perturbation
Theory, Academic Press,
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.,
A. Silberschatz, H.F. Korth and S. Sudarshan,
Database System Concepts, 3rd ed., McGraw-Hill, 1997.
R.S. Pressman, Software Engineering – A
Practitioner’s Approach, 4th 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,
S. Roman, Coding and Information Theory,
Springer Verlag, 1992.
R. Lidl and H. Niederreiter, Introduction to
Finite Fields and Their Applications,
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,
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,
A. Weil, Number Theory for Beginners, Additional
references.