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Course Code:
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MA 402
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Title:
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Algebra -I
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Credits:
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8.0
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Pre-requisite:
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MA 401
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Description:
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Review of groups,
subgroups, homomorphisms, finite and discrete groups of motions, group
actions, class equation, Sylow theorems, groups of order 12, generators
and relations, SL(R), SU(2), simplicity of alternating groups and
PSL(2). Rings, ideals, quotient rings, Euclidean domains, principal
ideal domains, unique factorization domains, primes in Z[i] and
Fermat"s 2-square theorem, ideal classes in imaginary quadratic fields.
Modules, matrices, free modules and bases, diagonalization of integer
matrices, generators and relations for modules, structure theorem for
abelian groups, applications to Jordan canonical forms and linear
operators. Extension fields, splitting fields, fundamental theorem of
Galois Theory, constructibility by ruler and compass, finite fields. |
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Text/References:
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M. Artin, Algebra, Prentice-Hall, 1990.
I.N. Herstein, Topics in Algebra, Wiley Eastern, 1987.
K.D. Joshi, Foundations of Discrete Mathematics, Wiley Eastern, 1989.
N. Jacobson, Basic Algebra, Vol. I, Hindustan Publishing Corporation, 1984.
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Course Code:
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MA 404
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Title:
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Real Analysis II
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Credits:
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8.0
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Pre-requisite:
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MA 401
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Description:
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Functions of several
variables: differentiation, the inverse function theorem, the implicit
function theorem, the rank theorem, derivatives of higher order and
differentiation of integrals, integration, change of variables,
Divergence and Stokes"theorem in Euclidean spaces. Hilbert spaces,
orthonormal basis, projection and Riesz representation theorems.
Approximation and optimization in Hilbert spaces. Variational
problems, Lax-Milgram lemma and its applications. |
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Text/References:
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W. Rudin, Principles of Mathematical Analysis, 3rd ed., McGraw-Hill, 1982.
T. Apostol, Mathematical Analysis, 3rd ed., Tata McGraw-Hill, 1974.
W. Fleming, Functions of Several Variables, Springer-Verlag, 1977.
B.V. Limaye, Functional Analysis, 2nd ed., Wiley Eastern, 1996.
J.N. Reddy, Applied Functional Analysis and Variational Methods in Engineering, McGraw-Hill International Edition, 1986.
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Course Code:
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MA 436
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Title:
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Partial Differential Equations
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Credits:
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6.0
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Pre-requisite:
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MA 417
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Description:
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Cauchy Problems for first
Order Hyperbolic Equations: Method of characteristics, Monge cone.
Classification of Second Order Partial Differential Equations: Normal
forma and Characteristics. Initial and Boundary Value Problems:
Lagrange Green"s identity and uniqueness by energy methods. Stability
theory, energy conservation and dispersion. Laplace equations: mean
value property, maximum principle, Poission"s formula, Dirichlet"s
principle, existence of solution using Perron"s method(without proof). |
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Text/References:
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E. DiBenedetto, Partial Differential Equations, 2nd Printing, Birkhauser, Boston, 1995.
Fritz John, Partial Differential Equations, 3rd edition, Narosa Publ. Co., New Delhi, 1979.
P. Prasad and R. Ravindran, Partial Differential Equations, Wiley Eastern Ltd., New Delhi, 1985.
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Course Code:
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MA 438
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Title:
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Introduction to Mathematical Statistics
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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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Distribution of functions
of random variables, order statistics. Estimation - Loss function,
risk,minimum risk unbiased estimators, maximum likelihood estimation,
methods of moments, Bayes estimation, Sufficient statistics,
completeness, Basu"s theorm, exponential families, invariance and
maximal invariant statistics. Testing of Hypotheses - parametric and
non-parametric problems, examples with data analytic applications.
Confidence Intervals. |
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Text/References:
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G. Casella and B. L. Berger, Statistical Inference,
Pacific Grove, Wadsworth and Brooks, 1990.
M. H. DeGroot, Probability and Statistics, Addison-Wesley, 1986.
E.L. Lehmann and G. Casella, Theory of Point Estimation, New York, Springer-Verlag, 1998.
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Course Code:
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MA 444
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Title:
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Numerical Analysis
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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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Principles of floating point computationa and rounding errors.
System of linear Equations: Factorization methods, pivoting and scaling, residual erroe correction method.
Iteractive methods: Jacobi, Gauss-Seidel methods with convergence analysis, conjugate gradient methods.
Eigenvalue problems: Only implementation issues.
Nonlinear Systems: Newton and Newton like methods and unconstrained optimization.
Interpolation: review of Legrange 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.
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Text/References:
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K. E. Atkinson, An Introduction to Numerical Analysis, Wiley, 1989.
S.D. Conte and C. De Boor, Elementary numerical Analysis - An alogrithmic Approach, McGraw-Hill, 1981.
K. Erikson, D. Estep, P. Hansbo and C. Johnson, Computational Differential Equations, Cambridge Universal Press,
1996.
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Course Code:
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MA 446
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Title:
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Complex Analysis
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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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Complex numbers and the
point at infinity. Analytic functions. Cauchy-Teimann conditions,
harmonicity, Mappings by elementary functions. Reimann surfaces.
Conformal mappings. Contour integrals, Cauchy-goursat theorem, simply
and multiply connected domains. Uniform convergence of sequence and
series. Taylor and Laurent series. Isolated singularities and residues.
Evaluation of real integrals. Calculation of inverse Laplace
transforms. Zeros and Poles, Argument principles, Rouche"s theorm.
Winding numbers. |
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Text/References:
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R.V. Churchill and J. W. Brown, Complex Variables and Applications, International Student Edition,Mc-Graw Hill, 4th ed., 1984.
P. Henrici, Applied and Computational Complex Analysis, Vol.1, Wiley, 1974.
B.R. Palka, An Introduction to Complex Function Theory, UTM Springer-Verlag, 1991.
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