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Syllabus for Mathematics (MA)
Linear Algebra:
Finite dimensional vector spaces; Linear transformations and their matrix
representations, rank; systems of linear equations, eigen values and eigen
vectors, minimal polynomial, Cayley-Hamilton Theroem, diagonalisation, Hermitian,
Skew-Hermitian and unitary matrices; Finite dimensional inner product spaces,
Gram-Schmidt orthonormalization process, self-adjoint operators.
Complex Analysis:
Analytic functions, conformal mappings, bilinear transformations; complex
integration: Cauchy's integral theorem and formula; Liouville's theorem, maximum
modulus principle; Taylor and Laurent's series; residue theorem and applications
for evaluating real integrals.
Real Analysis:
Sequences and series of functions, uniform convergence, power series, Fourier
series, functions of several variables, maxima, minima; Riemann integration,
multiple integrals, line, surface and volume integrals, theorems of Green,
Stokes and Gauss; metric spaces, completeness, Weierstrass approximation
theorem, compactness; Lebesgue measure, measurable functions; Lebesgue integral,
Fatou's lemma, dominated convergence theorem.
Ordinary Differential Equations:
First order ordinary differential equations, existence and uniqueness theorems,
systems of linear first order ordinary differential equations, linear ordinary
differential equations of higher order with constant coefficients; linear second
order ordinary differential equations with variable coefficients; method of
Laplace transforms for solving ordinary differential equations, series
solutions; Legendre and Bessel functions and their orthogonality.
Algebra:
Normal subgroups and homomorphism theorems, automorphisms; Group actions,
Sylow's theorems and their applications; Euclidean domains, Principle ideal
domains and unique factorization domains. Prime ideals and maximal ideals in
commutative rings; Fields, finite fields.
Functional Analysis:
Banach spaces, Hahn-Banach extension theorem, open mapping and closed graph
theorems, principle of uniform boundedness; Hilbert spaces, orthonormal bases,
Riesz representation theorem, bounded linear operators.
Numerical Analysis:
Numerical solution of algebraic and transcendental equations: bisection, secant
method, Newton-Raphson method, fixed point iteration; interpolation: error of
polynomial interpolation, Lagrange, Newton interpolations; numerical
differentiation; numerical integration: Trapezoidal and Simpson rules, Gauss
Legendre quadrature, method of undetermined parameters; least square polynomial
approximation; numerical solution of systems of linear equations: direct methods
(Gauss elimination, LU decomposition); iterative methods (Jacobi and
Gauss-Seidel); matrix eigenvalue problems: power method, numerical solution of
ordinary differential equations: initial value problems: Taylor series methods,
Euler's method, Runge-Kutta methods.
Partial Differential Equations:
Linear and quasilinear first order partial differential equations, method of
characteristics; second order linear equations in two variables and their
classification; Cauchy, Dirichlet and Neumann problems; solutions of Laplace,
wave and diffusion equations in two variables; Fourier series and Fourier
transform and Laplace transform methods of solutions for the above equations.
Mechanics:
Virtual work, Lagrange's equations for holonomic systems, Hamiltonian equations.
Topology:
Basic concepts of topology, product topology, connectedness, compactness,
countability and separation axioms, Urysohn's Lemma.
Probability and Statistics:
Probability space, conditional probability, Bayes theorem, independence, Random variables, joint
and conditional distributions, standard probability distributions and their
properties, expectation, conditional expectation, moments; Weak and strong law
of large numbers, central limit theorem; Sampling distributions, UMVU
estimators, maximum likelihood estimators, Testing of hypotheses, standard
parametric tests based on normal, X2 , t, F - distributions;
Linear regression; Interval estimation.
Linear programming:
Linear programming problem and its formulation, convex sets and their
properties, graphical method, basic feasible solution, simplex method, big-M and
two phase methods; infeasible and unbounded LPP's, alternate optima; Dual
problem and duality theorems, dual simplex method and its application in post
optimality analysis; Balanced and unbalanced transportation problems, u -u method for solving transportation problems; Hungarian method for solving
assignment problems.
Calculus of Variation and Integral Equations:
Variation problems with fixed boundaries; sufficient conditions for extremum, linear integral equations
of Fredholm and Volterra type, their iterative solutions.
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