Linear Algebra

Finite dimensional vector spaces. Linear transformations and their matrix representations,

rank -- systems of linear equations, eigenvalues and eigenvectors, minimal polynomial, Cayley-Hamilton theorem, diagonalisation, Hermitian, Skew-Hermitian and unitary matrices. Finite dimensional inner product spaces, self-adjoint and Normal linear operators, spectral theorem, Quadratic forms.

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, 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, Sturm Liouville system, Greeen's functions.

Algebra

Normal subgroups and homomorphisms theorems, automorphisms. Group actions, sylow's theorems and their applications groups of order less than or equal to 20, Finite p-groups. Euclidean domains, Principal ideal domains and unique factorizations domain Prime ideals and maximal ideals in commutative rings.

Functional Analysis

Banach spaces, Hahn-Banach theorems, open mapping and closed graph theorems, principle of uniform boundedness -- Hilbert spaces, orthonormal sets, Riesz representation theorem, Self-adjoint, Unitary and Normal Linear operators on Hilbert Spaces.

Numerical Analysis

Numerical solution of algebraic and transcendental equations -- bisection, secant method, Newton-Raphson method, fixed point iteration, interpolation existence and error of polynomial interpolation, Lagrange, Newton, Hermite (osculatory) interpolations -- numerical differentiation and integration, Trapezoidal and Simpson rules -- Gaussian quadrature -- (Gauss-Legendre and Gauss-Chebyshev), method of undetermined parameters, least square and orthonormal polynomial approximation -- numerical solution of systems of linear equations direct and iterative methods, (Jacobi Gauss-Seidel and SOR) with convergence -- matrix eigenvalue problems Jacobi and Given's methods, numerical solution of ordinary differential equations initial value problems, Taylor series method, Runge-Kutta methods, Predictor-corrector methods-convergence and stability.

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, Green's functions -- solutions of Laplace, wave and diffusion equations in two variables Fourier series and transform methods of solutions of the above equations and applications to physical problems.

Mechanics

Forces in three dimensions, Poinsot central axis, virtual work, Lagrange’s equations for holonomic systems, theory of small oscillations, Hamiltonian equations --

Topology

Basic concepts of topology, product topology, connectedness, compactness, countability and separation axioms, Urysohn’s Lemma, Tietze extension theorem, metrization theorems, Tychonoff theorem on compactness of product spaces.

Probability & 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, sufficiency and consistency, maximum likelihood estimators. Testing of hypotheses, Neyman-Pearson tests, monotone likelihood ratio, likelihood ratio tests, standard parametric tests based on normal, X2 ,t, F-distributions. Linear regression and test for linearity of regression. Interval estimation.

Linear Programming

Linear programming problem and its formulation, convex sets 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, interpretation of dual variables. Balanced and unbalanced transportation problems, unimodular property and u-v method for solving transportation problems. Hungarian method for solving assignment problems.

Calculus of Variations and Integral Equations

Variational problems with fixed boundaries -- Sufficient conditions for extremum, Linear integral equations of Fredholm and Volterra type, their iterative solutions. Fredholm alternative.