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Course Code:
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CS 101
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Title:
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Computer Programming and Utilization
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Credits:
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6.0
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Pre-requisite:
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Description:
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Functional organisation of
computers, algorithms, basic progamming concepts, FOR-TRAN language
programming. Program testing and debugging. Modular programming
subroutines: Selected examples from Numerical Analysis, Game playing,
sorting/searching methods, etc. |
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Text/References:
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N.N. Biswas, FORTRAN IV Computer Programming, Radiant Books, 1979.
K.D> Sharma, Programming in Fortran IV, Affiliated EAST WEST, 1976.
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Course Code:
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MA 401
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Title:
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Linear Algebra
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Credits:
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8.0
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Pre-requisite:
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Description:
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Systems of linear
equations: matrices and elementary row operations, Gaussian
elimination, LU decomposition. Vector Spaces: subspaces, bases and
dimension, coordinates. Linear Transformations: representation of
linear transformations by matrices, rank-nullity theorm, duality and
transpose, determinants. Eigenvalues and Eigenvectors: minimal and
characteristic polynomials, Diagonalization, Schur"s theorem, Cayley
Hamilton theorem, Jordan Canonical form. Inner Product spaces:
Gram-Schmidt orthonormalization, reflectors, QR decomposition using
reflectors, least squares problem, adjoint of an operator, unitary
operators, rigid motions, positive(semi) definite matrices, minimum
principles and Rayleigh quotients, matrix norms, condition numbers.
Eigenvalue computation: power and inverse power methods, QR method. |
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Text/References:
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M. Artin, Algebra, Prentice Hall of India, 1994.
K. Hoffman and R. Kunze, Linear Algebra, Prentice Hall of India, 1991.
B. Noble and J.W. Daniel, Applied Linear Algebra, 2nd ed., Prentice Hall, 1977.
G. Strang, Linear Algebra and Its Applications, 3rd ed., Harcourt Brace Jovanovich, San Diego, 1988.
D.S. Watkins, Fundamentals of Matrix Computations, Wiley, 1991.
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Course Code:
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MA 403
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Title:
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Real Analysis I
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Credits:
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8.0
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Pre-requisite:
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NIL
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Description:
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Metric spaces, compactness,
connectedness, completeness. Continuity. Monotonic functions.
Differentiation of vector-valued functions. Functions of bounded
variation and absolutely continuous functions. Riemann-Stieltjes
integral and its properties. Fundamental theorem of integral calculus.
Sequences and series of functions, uniform convergence and its relation
to continuity, differentiation and integration. Equicontinuous families
of functions, Ascoli-Arzela theorem. Weierstrass approximation theorem.
Fourier series, Fejer"s theorem, pointwise convergence. |
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Text/References:
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T. Apostol, Mathematical Analysis, 2nd ed., Addison-Wesley, 1974.
Ganapati Iyer, Mathematical Analysis, Tata McGraw-Hill, 1977.
W. Rudin, Principles of Mathematical
Analysis, 3rd ed., McGraw-Hill, 1983.
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Course Code:
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MA 411
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Title:
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Introduction to Probability
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Credits:
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6.0
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Pre-requisite:
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Description:
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Events, sigma-fields,
specifying probabilities on sigma-fields, extension theorem(without
proof), independence of classes of events, random variables.
Distribution functions, discrete, continuous and mixed. Moments - moment
generating functions, Characteristic functions. Random vectors and
joint distributions, conditional distribution, conditional expectation.
Convergence of random variables - the four modes of convergence and
their relationships. Laws of large numbers and Central Limits Theorm. |
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Text/References:
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P. Billingsley, Probability and Measure, John Wiley & Sons, NY, 1986.
K.L. Chung, Elementary Probability Theory with Stochastics Processes, 3rd Ed., Narosa Pub. Co., 1979.
J.Pitman, Probability, Narosa Pub. Co., 1993.
M. Woodroofe, Probability with Application, McGraw-Hill, NY, 1975.
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Course Code:
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MA 417
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Title:
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Ordinary Differential Equations
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Credits:
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8.0
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Pre-requisite:
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Description:
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Review of solution methods
for first order as well as second order equations, Power series methods
with properties of Bessel functions and Legendre polynomials. Existance
and uniqueness of Initial Value problems; Picards and Peanos theorems,
Gronwall"s inequality, continuation of solutions and maximal interval
of existance, continuous dependence. Higher order linear equations and
linear systems; fundamendal 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 comparison theorms and oscillations, eigenvalue
problems. |
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Text/References:
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R.P. Agarwal and R. Gupta,
Essentials of Ordinary Differential Equations, Tata McGraw-Hill Pub.
Co., New Delhi, 1991. M. Braun, Differential Equations and their
Applications, 4th Edition, Springer Verlag, Berlin, 1993. E.A.
Coddington and N. Levinson, Theory of Ordinary Differential Equations,
Tata McGraw-Hill Publ. Co., New Delhi, 1990. L. Perko, Differential
Equations and Dynamical Systems, 2nd Edition, Texts in Applied
Mathematics, Vol.7, Springer Verlag, New York, 1998. M. Rama Mohana
Rao, Ordinary Differential Equations: Theory and Applications.
Affiliated East-West Press Pvt. Ltd., New Delhi, 1980. A. Sanchez,
Ordinary Differential Equations and Stability Theory: An Introduction,
Dover Pub. Co., New York, 1968. |
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Course Code:
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SI 409
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Title:
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Discrete Structures Lab
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Credits:
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3.0
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Pre-requisite:
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Description:
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Illustration of techniques and concepts from combinatorics and graph theory through programming exercises.
Either an interpreted language such as Scheme or the language of Mathematica should be used for the programming exercises.
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Text/References:
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N. Biggs, Discrete Mathematics, Oxford/Clarendon Press, 1985.
R. Dromey, How to Solve it By Computer, Prentice Hall India ,1996.
Donald Knuth, Fundamental Algorithms, Narosa Publishers, 1985.
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