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 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
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
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 -v 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.