Exam
## Exam

**MA Mathematics**

**Section 1: 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, diagonalization, Jordan-canonical form, Hermitian, Skew-

Hermitian and unitary matrices; Finite dimensional inner product spaces, Gram-Schmidt

orthonormalization process, self-adjoint operators, definite forms.

**Section 2: Complex Analysis**

Analytic functions, conformal mappings, bilinear transformations; complex integration:

Cauchy’s integral theorem and formula; Liouville’s theorem, maximum modulus principle;

Zeros and singularities; Taylor and Laurent’s series; residue theorem and applications for

evaluating real integrals.

**Section 3: 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,

compactness, completeness, Weierstrass approximation theorem; Lebesgue measure,

measurable functions; Lebesgue integral, Fatou’s lemma, dominated convergence

theorem.

**Section 4: Ordinary Differential Equations**

First order ordinary differential equations, existence and uniqueness theorems for initial

value problems, 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 (power series, Frobenius method);

Legendre and Bessel functions and their orthogonal properties.

**Section 5: Algebra**

Groups, subgroups, normal subgroups, quotient groups and homomorphism theorems,

automorphisms; cyclic groups and permutation groups, Sylow’s theorems and their

applications; Rings, ideals, prime and maximal ideals, quotient rings, unique factorization

domains, Principle ideal domains, Euclidean domains, polynomial rings and irreducibility

criteria; Fields, finite fields, field extensions.

**Section 6: Functional Analysis**

Normed linear spaces, Banach spaces, Hahn-Banach extension theorem, open mapping

and closed graph theorems, principle of uniform boundedness; Inner-product spaces,

Hilbert spaces, orthonormal bases, Riesz representation theorem, bounded linear

operators.

**Section 7: 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; numerical solution of systems of linear

equations: direct methods (Gauss elimination, LU decomposition); iterative methods

(Jacobi and Gauss-Seidel); numerical solution of ordinary differential equations: initial

value problems: Euler’s method, Runge-Kutta methods of order 2.

**Section 8: 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 in two dimensional Cartesian

coordinates, Interior and exterior Dirichlet problems in polar coordinates; Separation of

variables method for solving wave and diffusion equations in one space variable; Fourier

series and Fourier transform and Laplace transform methods of solutions for the above

equations.

**Section 9: Topology**

Basic concepts of topology, bases, subbases, subspace topology, order topology,

product topology, connectedness, compactness, countability and separation axioms,

Urysohn’s Lemma.

**Section 10: Probability and Statistics**

Probability space, conditional probability, Bayes theorem, independence, Random

variables, joint and conditional distributions, standard probability distributions and their

properties (Discrete uniform, Binomial, Poisson, Geometric, Negative binomial, Normal,

Exponential, Gamma, Continuous uniform, Bivariate normal, Multinomial), expectation,

conditional expectation, moments; Weak and strong law of large numbers, central limit

theorem; Sampling distributions, UMVU estimators, maximum likelihood estimators; Interval

estimation; Testing of hypotheses, standard parametric tests based on normal, , ,

distributions; Simple linear regression.

**Section 11: 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, Vogel’s approximation method for solving

transportation problems; Hungarian method for solving assignment problems.

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