**SECTION- II **

**Algebra, Real Analysis, Complex Analysis, Partial Differential equations. Mechanics, Hydrodynamics, Numerical Analysis. Statistics including probability operational research. **

**Algebra. **

Groups, sub-groups, normal sub-groups, homomorphism of groups, quotient groups. Basic isomorphism theorems. Sylow theorems. Permutation Groups, Cayley's theorem. Rings and Ideals, Principal Ideal domains, unique factorisations domains and Euclidean domains. Field Extensions. Finite fields.

**Real Analysis. **

Metric spaces, their topology with special reference to Rn sequence in a metric space, Cauchy sequence Completeness, Completion Continuous functions, Uniform Continuity, Properties of Continuous function on Compact sets. Riemann-Stieltjes integral, Improper integrals and their conditions of existence. Differentiation of functions of several variables, Implicit function theorem, maxima and minima, Absolute and conditional convergence series of real and complex terms, Re-arrangement of series. Uniform convergence, Infinite products, Continuity, differentiability and integrability for series, Multiple integrals.

**Complex Analysis**.

Analytic functions, Cauchy's theorem, Cauchy's integral formula, Power series, Taylor's, Singularities, Cauchy's Residue theorem and Contour integration.

**Partial Differential Equations. **

Formations of partial differential equations. Types of integrals of partial differential Equations of first order Charpit’s method. Partial differential equation with constant co-efficient.

**Mechanics. **

Generalised co-ordinates, Constraints, Holonomic and Non-holonomic systems, D' Alembert's principle and Lagrange equations. Moment of Inertia, Motion of rigid bodies in two dimensions.

**Hydro-dynamics. **

Equation of continuity, Momentum and energy. Inviscid Flow Theory – Two-dimensional motion, streaming motion, Sources and Sinks.

**Numerical Analysis. **

** Transcendental and Polynomial Equations –** Methods of tabulation, bisection, Regula-falsi, secant, and Newton-Raphson and order of its convergence.

**Interpolation and Numerical differentiation –** Polynomial interpolation with equal or unequal steps

**Spline interpolation –** Cubic splines. Numerical differentiation formulae with error terms.

** Numerical integration –** Problems of approximate quadratic quadrature formula with equi-spaced arguments. Gaussian quadrature convergence.

**Ordinary differential equations –** Euler’s method, Multistep-predictor corrector methods – Adam's and Milne's method, convergence and stability, Runge – Kutta methods.

**Probability and Statistics. **

**1. Statistical methods –** Concept of statistical population and random sample. Collection and presentation of data. Measure of location and dispersion. Moments and Sheppard's correction cumulants. Measures of Skewness and Kurtosis. Curve fitting by least squares regression, correlation and correlation ratio. Rank correlation, Partial correlation co-efficient and Multiple correlation co-efficient.

**2. Probability –** Discrete sample space, Events, their union and intersection, etc., Probability – Classical relative frequency and axiomatic approaches. Probability in continuum probability space conditional probability and independence, Basic laws of probability, Probability of combination of events, Bayes theorem, Random variable probability function, Probability density function. Distributions function, Mathematical expectation. Marginal and conditional distributions, Conditional expectation.

**3. Probability distributions –** Binomial, Poisson Normal Gamma, Beata. Cauchy, Multinomial,

Hypergeometric, Negative Binomial, Chebyshev's Lemma. (Weak) law of large numbers, Central limit theorem for independent and identical varieties, standard errors, Sampling distribution of T.F and Chi-square and their uses in tests of significance large sample tests for mean and proportion.

**Operational Research. **

**Mathematical Programming –** Definition and some elementary properties of convex sets, simplex methods, degeneracy, duality, sensitivity analysis rectangular games and their solutions. Transportation and assignment problems. Kuhn-Tucker condition for non-linear programming Bellman's optimality principle and some elementary applications of dynamic programming.

**Theory of Queues –** Analysis of steady-state and transient solution for queueing system with Poisson arrivals and exponential service time. Deterministic replacement models, sequencing problems with two machines, n jobs, 3 machines, n jobs (special case) and n machines, 2 jobs.