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
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
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,
Topology: Basic concepts of topology, product topology,
connectedness, compactness, countability and separation axioms, Urysohn’s
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.