**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.

**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.**

Real Analysis:

Real Analysis:

**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.**

Ordinary Differential Equations:

Ordinary Differential Equations:

**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.**

Algebra:

Algebra:

**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.**

Functional Analysis:

Functional 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 Legendrequadrature, 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.**

Numerical Analysis:

Numerical Analysis:

**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.**

Partial Differential Equations:

Partial Differential Equations:

**Virtual work, Lagrange’s equations for holonomic systems, Hamiltonian equations.**

Mechanics:

Mechanics:

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

Topology:

Topology:

**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, X**

Probability and Statistics:

Probability and Statistics:

^{2}, t, F – distributions; Linear regression; Interval estimation.

**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.**

Linear programming:

Linear programming:

**Variation problems with fixed boundaries; sufficient conditions for extremum, linear integral equations of Fredholm and Volterra type, their iterative solutions.**

Calculus of Variation and Integral Equations:

Calculus of Variation and Integral Equations: