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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, SkewHermitian
and unitary matrices; Finite dimensional inner product spaces, Gram-Schmidt
orthonormalization process, self-adjoint operators, definite forms.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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Section 9: Topology

Basic concepts of topology, bases, subbases, subspace topology, order topology,
product topology, connectedness, compactness, countability and separation axioms,
Urysohn’s Lemma.

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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.

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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.