|
Course Code:
|
MA 406
|
|
Title:
|
General Topology
|
|
Credits:
|
8.0
|
|
Pre-requisite:
|
MA 403
|
|
Description:
|
Topologies through open
sets, bases, sub-bases, closure, interior, boundary, subspaces.
Continuity, open functions, homeomorphisms, embeddings, strong and weak
topologies generated by families of functions. Quotient spaces. First
and Second countable, separable, Lindeloff, compact spaces.
Separation axioms, Urysohn"s lemma. Products, embeddings into products,
Urysohn metrisation theorem, Convergence of nets and filters. Filters
and compactness, ultrafilters, Tychonoff compactness theorem. Local
compactness, Alexandroff compactification. Function spaces,
compact-open topology. Connectedness, components, local connectedness,
paths, loops. Homotopy, fundamental group. Computation of the
fundamental group of the circle. |
|
Text/References:
|
K.D. Joshi, Introduction to General Topology, Wiley Eastern, 1983.
J.L. Kelly, General Topology, Van Nostrand, 1955.
|
|
|
Course Code:
|
MA 415
|
|
Title:
|
Mathematical Methods
|
|
Credits:
|
8.0
|
|
Pre-requisite:
|
MA 436
|
|
Description:
|
Asymptotic Methods:
asymptotic expansions, methods of strained co-ordinates and matched
asymptotic expansions. Fourier Methods with Applications: Generalised
functions, eigenfunctioon expansion and Green"s functions, Fourier
transform, Convolution, Parseval"s Theorm, fundamental solutions, and
applications to heat, Laplace and wave equations. Variational Methods:
minimum of quadratic functional, Lax-milgram theorm and applications to
boundary value theorms. Linear Integral Equations: Fredholm and
Volterra Integral Equations, Hilbert-Schmidt theory, iterative methods
and Neumann series. |
|
Text/References:
|
R.K. Bose and M.C. Joshi,
Methods of Mathematical Physics, Tata-McGraw-Hill Pub. Co. Ltd., NEw
Delhi, 1984. R. Courant and D. hilbert, Methods of mathematical
Physics, Vol 1, Wiley Eastern Pvt. Ltd., New Delhi, 1975. R. Dautray
and J.L. Lions, Mathematical Analysis and Numeraical Methods for
Science and Technology, Vol 5., Springer Verlag, Berlin, 1992. J.
Kovorkian and J.D. Cole, Perturbation Methods in Appliced Mathematics,
Springer Verlag, Berlin, 1985. S.G. Mikhlin, Variational MEthods in
Mathmatical Physics, Pergamon Press, Oxford, 1964. A. Mayfeh,
Perturbation Methods, Wiley Publ., New York, 1973. J.N. Reddy, Applied
Functional Analysis and Variational Methods, McGraw-Hill Book Co., New
York, 1987. E. Zauderer, Partial Differential Equations in Applied
Mathematics, 2nd edition, John Wiley & Sons, New York, 1989. |
|
|
Course Code:
|
MA 501
|
|
Title:
|
Algebra II
|
|
Credits:
|
6.0
|
|
Pre-requisite:
|
MA 402
|
|
Description:
|
Solution by radicals of
equations of degree at most 4, solvable groups, solvability by
radicals, Abel-Ruffini theorem, symmetric functions, Newton"s
identities for symmetric functions, Galois groups of equations of
degree at most 4. Equations with symmetric and alternating groups as
Galois groups. Reduction mod p technique. Cyclotomic extensions, norm
and trace, cyclic extensions and Hilbert"s theorem 90, Artin-Schreier
theorem, transcendental extensions. Zero divisors, nilpotent
elements, nilradical and Jacobson radical, operations on ideals,
extension and contraction, examples of rings arising in Geometry,
Combinatorics, Number Theory and Topology. |
|
Text/References:
|
O. Zariski and P. Samuel, Commutative Algebra, Vol. I, Van Nostrand, 1958.
N. Jacobson, Basic Algebra Vols. I and II, Hindustan Publishing Corporation, 1984.
M. Artin, Algebra, Prentice-Hall, 1990.
M.F. Atiyah and I.G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, 1969.
|
|
|
Course Code:
|
MA 507
|
|
Title:
|
Convex Analysis and Optimization
|
|
Credits:
|
6.0
|
|
Pre-requisite:
|
MA 404
|
|
Description:
|
Convex sets, convex cones
and convex functions. Continuity and growth properties of convex
functions. Conjugate convex functions in paired spaces. Subgradients,
normal cones and subdifferential formulae. Convex optimization.
Constraint qualifications and Kuhn-Tucker theory. Lagrangian duality.
Applications to nonlinear programming. Chebyshev approximation.
Calculus of variation and partial differential equations.
Dubovitski-Milyutin theory. |
|
Text/References:
|
R.T. Rockafellar, Convex
Analysis, Princeton University Press, 1959. R.T. Rockafellar,
Conjugate Duality and Optimization, CBMS Lecture Notes, Series No. 13
SIAM, 1974. P.J. Laurent, Approximation et Optimization, Hermann,
1973. M.S. Bazaraa and C.M. Shetty, Foundations of Optimizations,
Lecture Notes in Economics and Management Systems, Springer-Verlag,
1976. |
|
|
Course Code:
|
MA 509
|
|
Title:
|
Elementary Number Theory
|
|
Credits:
|
6.0
|
|
Pre-requisite:
|
NIL
|
|
Description:
|
Divisibility, Primes,
Unique factorization of integers, Arithmetical functions, Mobius
inversion, congruences, Chinese remainder theorem, primitive roots,
Quadratic reciprocity, binary quadratic forms, Fermat’s two square
theorem, Lagrange’s four square theorem, discussion of Waring’s
problem, Diophantine approximations: continued fractions, rational
approximations, transcendence of Liouville numbers. |
|
Text/References:
|
· W.W. Adams and L.J. Goldstein, Introduction to the Theory of Numbers, 3rd ed., Wiley Eastern, 1972.
· Baker, A Concise Introduction to the Theory of Numbers, Cambridge University Press, 1984.
· Niven and H.S. Zuckerman, Introduction to the Theory of Numbers, 4th Ed., Wiley, New York, 1980.
|
|
|
Course Code:
|
MA 511
|
|
Title:
|
Enumerative Combinatorics I
|
|
Credits:
|
6.0
|
|
Pre-requisite:
|
MA 401
|
|
Description:
|
Basic Combinatorial Objects
: Sets, multisets, partitions of sets, partitions of numbers, finite
vector spaces, permutations, graphs etc. Basic Counting Coefficients:
The twelve fold way, binomial, q-binomial and the Stirling
coefficients, permutation statistics, etc. Sieve Methods : Principle
of inclusion-exclusion, permutations with restricted positions,
Sign-reversing involutions, determinants etc. Introduction to
combinatorial reciprocity. Introduction to symmetric functions. |
|
Text/References:
|
R.P. Stanley, Enumerative Combinatorics, Vol. I, Wadsworth and Brooks/Cole, 1986.
C. Berge, Principles of Combinatorics, Academic Press, 1972.
K.D. Joshi, Foundations of Discrete Mathematics, Wiley Eastern, 1989.
|
|
|
Course Code:
|
MA 513
|
|
Title:
|
Fourier Analysis
|
|
Credits:
|
6.0
|
|
Pre-requisite:
|
MA 408
|
|
Description:
|
Revision of Fourier series.
Tests for pointwise convergence of Fourier series. Summability of
Fourier series for integrable functions. Fourier-transforms of
integrable functions. Basic properties of Fourier transforms. Inversion
theorem, Plancheral theorem, Paley-Weiner theorem. |
|
Text/References:
|
Ganapathy Iyer, Mathematical Analysis, Tata McGraw-Hill, 1977.
W. Rudin, Real and Complex Analysis, Tata McGraw-Hill, 3rd ed., 1987.
W. Rudin, Functional Analysis, Tata McGraw-Hill, 1973.
|
|
|
Course Code:
|
MA 519
|
|
Title:
|
Representation Theory of Finite Groups
|
|
Credits:
|
6.0
|
|
Pre-requisite:
|
MA 402
|
|
Description:
|
Representations,
Subrepresentations, Tensor products, Symmetric and Alternating Squares.
Characters, Schur"s lemma, Orthogonality relations, Decomposition of
regular representation, Number of irreducible representations,
canonical decomposition and explicit decompositions. Subgroups, Product
groups, Abelian groups. Induced representations. Examples : Cyclic
groups, alternating and symmetric groups. Integrality properties of
characters, Burnside p q theorem. The character of induced
reporesentation, Frobenius Reciprocity Theory, Meckey"s irreducibility
criterion, Examples of induced representations, Representations of
supersolvable groups. |
|
Text/References:
|
J.P. Serre, Linear Representation of Groups, Springer-Verlag, 1977.
N. Jacobson, Basic Algebra II, Hindustan Publishing Corproation, 1983.
M. Burrow, Representation Theory of Finite Groups, Academic Press, 1965.
S. Lang, Algebra, Addison-Wesley, 1965.
|
|
|
Course Code:
|
MA 520
|
|
Title:
|
Spline Theory and Variational Methods
|
|
Credits:
|
6.0
|
|
Pre-requisite:
|
MA 403
|
|
Description:
|
Piecewise linear
approximation. Piecewise cubic interpolation. Cubic spline
interpolation and its errors. Representation of piecewise polynomial
diminishing splines. Interpolating and smoothing splines. Approximate
representation of linear functions. Optimal quadratures. Variational
formulation of generalized splines. Surface approximation by tensor
product splines. The Rayleigh-Ritz-Galerkin procedures of elliptic
problems, Semi-discrete Galerkin procedure for parabolic problems. |
|
Text/References:
|
C. de Boor, A Practical Guide to Splines, Springer-Verlag, 1978.
M.H. Schultz, Spline Analysis, Prentice-Hall, 1973.
P.J. Laurent, Approximation et Optimization, Hermann, 1972.
|
|
|
Course Code:
|
MA 525
|
|
Title:
|
Dynamical Systems
|
|
Credits:
|
6.0
|
|
Pre-requisite:
|
MA 417
|
|
Description:
|
Review of stability of
linear systems, FLow defined by nonlinear systems of ODEs,
Linearizatin and stable manifold theorm. Hartman Grobman theorm.
Stability and Lyanpunov functions. Planar flows: saddle point, nodes,
foci, centers and nonhyperbolic critical points. Gradient and
Hamiltonian systems. Limits sets and attractors. Poincare may,
Poincare BEnedixson theory and Poincare index. |
|
Text/References:
|
P. Hartman, Ordinary Differential Equations, John Wiley & Sons, NY, 1964.
M.W. Hirsch and S. Smale, Differential Equations, Dynamical Systems and Linear Algebra, Academic Press, NY, 1974.
L. Perko, Differential Equations and Dynamical Systems, Springer Verlag, Ny, 1991.
S. Wiggins, introduction to applied Nonlinear Dynamical Systems and Chaos, TAM vol2., Springer-verlag, NY, 1990.
|
|
|
Course Code:
|
MA 527
|
|
Title:
|
Topics in Approximation Theory
|
|
Credits:
|
6.0
|
|
Pre-requisite:
|
MA 403
|
|
Description:
|
Review of density theorms:
the theorms of Korovkin, fejer and Stone-Weierstrass. The classical
Chebyshev theory, discretization and dirscrete best approximation, the
second algorithm of REmes. Degree of approximation, moduli of
continuity and K-functionals, direct and converse theorms.
Interpolation, Legrange form, extended Haar subspaces and Hermite
interpolation, Hermite Fejer interpolation, Hermite-Birkhoff
interpolation. Piecewise polynomial interpolation. |
|
Text/References:
|
R. Devore. and G.G. Lorentz, Constructive Approximatin, Springer-verlag, Berlin, 1993.
G.G. Lorentz, Approximation of functins, Holt, Rinehart and Winston, NY, 1966.
H.N. Mhaskar and D.V. PAi, Fundamentals of Approximation theory, Narosa Publishing House, ND, 2000.
|
|
|
Course Code:
|
MA 529
|
|
Title:
|
Numerical Methods for Partial Differential Equatio
|
|
Credits:
|
6.0
|
|
Pre-requisite:
|
MA 436
|
|
Description:
|
Finite difference: 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
theorm. Parabolic equatioons: 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 schems and CFL condition. Lab component: Exposure to
MATLAB and computational experimeents based on the algorithms discussed
in the course. |
|
Text/References:
|
A. R> Mitchell and
S.D.F. Griffiths, The finite Difference Methods in partial differential
Equations, Wiley & sons, 1980. R.D. Richtmyer and K.W. Morton,
Difference Methods for Initial Value Problems, Wiley Interscience, NY,
1969. G.D. Smith, Numerical solutions of partial Differential
Equations, 3rd edition, clarendon 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 AMthematics, Vol22, 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. |
|
|
Course Code:
|
MA 559
|
|
Title:
|
Functional Analysis
|
|
Credits:
|
8.0
|
|
Pre-requisite:
|
MA 403
|
|
Description:
|
Normed linear spaces.
Continuity of linear maps. Hahn-Banach theorm in extension. Banach
spaces. Dual spaces. Reflexivity(definiton and simple examples). The
uniform boundedness principle and its applicatioons. The closed graph
theorm, the open mapping theorm and their applications. Spectrum of a
bounded operator. Compact operators and their spectra. Fredholm
alternatives. Inner product spaces, hilbert spaces, Orthonormal basis.
Projection theorm and the Riesz representation theorm. Reflexivity of
Hilbert spaces. |
|
Text/References:
|
J.B. Conway, A course in Functional Analysis, SPringer-Verlag, Berlin, 1985.
G. Goffman and G. Pedrick, First course in functional analysis, Prentice-Hall, 1974.
E. Kreyszig, Introductory Functional Analysis with applications, John Wiley & Sons, NY, 1978.
B.V. Limaye, Functional Analysis, 2nd ed., New Age International, New Delhi, 1996.
A. Taylor and D. Lay, Introduction to functional analysis, Wiley, Ny, 1980.
|
|
|
Course Code:
|
MA 561
|
|
Title:
|
Abstract Measure and Integration
|
|
Credits:
|
6.0
|
|
Pre-requisite:
|
MA 403
|
|
Description:
|
Semi-algebra, Algebra,
Sigma-algebra, Monotone class, Monotone class theorm, Measure spaces.
Outline of extension of measures from algebras to the generated
sigma-algebras, Measurable sets. MEasurable functions and their
properties, outline of integration and convergence theorms.
Introduction to L-spaces, Riesz-fischer theorm. Product measure spaces,
Fubini"s theorm. Absolute continuity to measures,Random Nikodym theorm. |
|
Text/References:
|
P.R. Halmos, Measure Theory, Graduate Text in Mathematics, Springer-Verlag, 1979.
I.K. Rana, An Introduction to Measure and Integration, Narosa Publishing House, ND, 1997.
H.L. royden, Real Analysis, 2nd Ed., Macmillan, 1968.
|
|
|
Course Code:
|
MA 569
|
|
Title:
|
Computational Commutative Algebra and Algebraic Ge
|
|
Credits:
|
6.0
|
|
Pre-requisite:
|
MA 402
|
|
Description:
|
Affine varieties,
parametrization, ideals, orderings on monomials, Dickson"s Lemma,
Groebner bases, hilbert Basis Theorm, Buchberger"s algorithm, ideal
membership, geometry of elimination, singular points, resultants and
extension theorm, hilbert"s Nullstellensatz, ideal variety
correspondance, basic computation among ideals via Groebner bases,
Coordinate ring of an affine variety and algorithmic computations in
it, solving equations via eigenvalues, projective varieties, projective
nullstellensatz, computation of projective nullstellensatz, computation
of projective closure, hilbert function and fimension of a variety and
their computation. Applications to graph colouring and integer
programming. Lab. Component: Implementation of algorithms developed in
this course using Macaylay programming language. |
|
Text/References:
|
W.W. Adams and P. Laustaunau, An introduction to Groebner Bases, American Mathematical Society, 1994.
D. Cox, J. Little and D. O"Shea. Ideals, Varieties and Algorithms, 2nd Editions, Springer-Verlag, 1996.
D. Cox, J. Little and D. O"Shea, Using Algebraic Geometry, Springer-Verlag, 1998.
|
|
|
Course Code:
|
MA 573
|
|
Title:
|
Mathematical Theory of Reliability
|
|
Credits:
|
6.0
|
|
Pre-requisite:
|
MA 418
|
|
Description:
|
Coherent Structures,
Reliability of systems of independent components, Bounds of system
reliability, shape of the system reliability function,notion of ageing,
parametric families of life distributions with monotone failure rate,
classes of life distributions based on notions of ageing, classes of
distributions in replacement policies, limit distributions for series
and parallel systems. Statistical estimation and testing for popular
reliability models and classes (parametric and nonparametric). |
|
Text/References:
|
R.E. Barlow and F. Proschan, Statitsical Theory of Reliability and Life Testing, Holt, Reinhart and Winston, 1975.
J.F. Lawless, Statistical Models and Methods of Life Time Data, Wiley, 1982.
R.G. Miller, Survival Analysis, Wiley, 1981.
L.J. Bain, Statistical Analysis of Reliability and Life Testing, Marcel Dekker, 1978.
N.R. Mann, R.E. Shafer and N.D. Singpurwala, Methods of Statistical Analysis of Reliability and Life Data, Wiley, 1974.
J.D. Kalbfleisch and R.L. Prentice, The Statistical Analysis of Failure Time Data, Wiley, 1986.
|
|
|
Course Code:
|
MA 574
|
|
Title:
|
Generalized Linear Models
|
|
Credits:
|
6.0
|
|
Pre-requisite:
|
MA 438
|
|
Description:
|
Multiple linear regression-estimation, tests and confidence regions.
Logistic Probit, Log-linear models for nominal and ordinal variables. Fitting of logit and log-linear models.
|
|
Text/References:
|
A. Agresti, Analysis of Categorical Data Wiley, 1990
A. Agresti, An Introduction to Categorical Data Analysis, John Wiley & Sons, New York 1996
E.B. Andersen, The statistical analysis of Categorical Data, Springer-Verlag, 1990.
D C Montgomery & E.A Peck, Introduction to Linear Regression Analysis, 2nd ed., John Wiley, 1992.
T.J. Santner & D.Duffy, The Statistical Analysis of Discrete Data, Springer-Varlag, 1989.
A A Sen & M Srivastava, Regression Analysis Theory, Methods & Applications, Springer Verlag, 1990
|
|
|
Course Code:
|
MA 575
|
|
Title:
|
Multivariate Analysis
|
|
Credits:
|
6.0
|
|
Pre-requisite:
|
MA 418
|
|
Description:
|
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. |
|
Text/References:
|
T.W. Anderson, An Introduction to Multivariate Statistical Analysis, Wiley, 1984.
A.M. Kshirsagar, Multivariate Analysis, Vols. I to IV, North Holland, 1977.
M.S. Srivastava and E.M. Carter, An Introduction to Multivariate Statistics, North Holland, 1983.
|
|
|
Course Code:
|
MA 577
|
|
Title:
|
Statistical Inference I
|
|
Credits:
|
6.0
|
|
Pre-requisite:
|
MA 418
|
|
Description:
|
Point estimation.
Cramer-Rao inequality, Bhattacharya bounds. Sufficient estimators,
Rao-Blackwell theorem. Maximum likelihood and other methods of
estimation. Tests and statistical hypothesis. Critical region. Power,
Neyman-Pearson lemmas. Likelihood ratio principle. MP, UMP, LMPU tests,
similar tests. Statistical decision theory. Loss function. Risk
functions. Admissibility. Bayes and minimax solutions. Randomized
decision functions. Sequential decision rules. Sequential analysis. |
|
Text/References:
|
C.R. Rao, Linear Statistical Inference and its Applications, Wiley Eastern, 1974.
M.G. Kendall and A. Stuart, The Advanced Theory of Statistics, Vol. II, Griffin, 1966.
E.L. Lehmann, Testing Statistical Hypotheses, 2nd ed., Wiley, 1986.
G. Casella and R.L. Berger, Statistical Inference, Wadsworth and Brooks, 1990.
|
|
|
Course Code:
|
MA 579
|
|
Title:
|
Stochastic Processes
|
|
Credits:
|
6.0
|
|
Pre-requisite:
|
MA 422
|
|
Description:
|
Recurrent events. Renewal
theory. Random walk. Markov chains and Markov processes. Stationary
processes. Spectral Analysis. Stochastic calculus. Branching phenomena.
Semi-Markov processes. Systems with random inputs. |
|
Text/References:
|
D.R. Cox and H.D. Miller, The Theory of Stochastic Processes, Methuen, 1970.
E. Parzen, Stochastic Processes, Holden-Day, 1972.
R.O. Howard, Dynamic Probabilistic Systems, Vol. 1 and 2, Wiley, 1971.
S.K. Srinivasan and K. Mehta, Stochastic Processes, Tata McGraw-Hill, 1976.
J. Medhi, Stochastic Processes, Wiley Eastern, 1982.
S. Karlin and H.M. Taylor, A First Course in Stochastic Processes, Academic Press, 1975.
|
|
|
Course Code:
|
MA 581
|
|
Title:
|
Elements of Differential Topology
|
|
Credits:
|
6.0
|
|
Pre-requisite:
|
MA 404
|
|
Description:
|
Differentiable Manifolds in
Rn: Review of inverse and implicit function theorms, tangent spaces and
tangent maps; immersions, submersions and embeddings. Regular Values:
Regular and Critical values; reguler inverse image theorm; Sard"s
theorm; Morse Lemma. Transeversality: Orientations of manifolds;
oriented and mod2 intersection numbers; degree of maps. Application to
fundamental theorm of Algebra. Lefschetz theory of vector fields and
flows: Poincase-Hopf index theorm; Gauss-Bonnet theorm. Abstract
manifolds: Examples such as real and complex projective spaces and
Grassmannian varieties; Whitney embedding theorms. |
|
Text/References:
|
A.A. Dubovin, A.T. Fomenko,
S.P. Novikov, Modern Geometry Methods and applications-II, the Geometry
and Topology of manifolds, Springer-Verlag GTM 104, 1985. V. Guillemin
and A pollack, Differential Topology Prentice-Hall Inc., Englewood
cliffs, New Jersey, 1974. J. Milnor, Topology from the differentia View
Point, University Press of Vriginia, Charlottsville, 1990. |
|
|
Course Code:
|
MA 583
|
|
Title:
|
Introduction to Continuum Mechanics
|
|
Credits:
|
6.0
|
|
Pre-requisite:
|
|
|
Description:
|
Description of continuous
and kinematics. Forces, deformations, constitutive equations. Theory of
motions-steady and unsteady motions. Spin vectors and tensors.
Conservation laws- physical models and applications. |
|
Text/References:
|
Y. C. Fung, A first Course in Continum Mechanics, Prentice Hall Inc., New York, 1977.
M. E. Gurtin, An introduction to Continuum MEchanics, Academic Press, NY, 1981.
S. Hunter, Mechanics of Continuum Media, John Wiley & Sons, NY, 1983.
|
|
|
Course Code:
|
MA 597
|
|
Title:
|
Project Stage I
|
|
Credits:
|
5.0
|
|
Pre-requisite:
|
|
|
Description:
|
|
|
Text/References:
|
|
|