Monday, September 1, 2008

SET EXAM Mathematics Paper II

1. Basic concepts of Real and Complex analysis :

Sequences and series, continuity, uniform continuity, Differentiability, Mean Value Theorem, sequences and series of functions, uniform convergence, Riemann integral – definition and simple properties. Algebra of Complex numbers, Analytic functions. Cauchy’s Theorem and integral formula, Power series, Taylor’s and Laurent’s series, Residues, Contour integration.

2.Basic Concepts of Linear Algebra :

Space of n-vectors, Linear dependence, Basis, Linear transformation, Algebra of matrices, Rank of a matrix, Determinants, Linear equations, Quadratic forms, Characteristic roots and vectors.

3.Basic concepts of probability :

Sample space, discrete probability, simple theorems on probability, independence of events, Bayes Theorem. Discrete and continuous random variables, Binomial, Paisson and Normal distributions ; Expectation and moments, independence of random variables, Chebyshev’s inequality.

4. Linear Programming Basic Concepts:

Convex sets. Linear Programming Problem ( LPP ). Examples of LPP, Hyperplane, open and closed half – spaces. Feasible, basic feasible and optimal solutions. Extreme point and graphical method.

5. Real Analysis :

Finite, countable and uncountable sets, Bounded and unbounded sets. Archimedean property, ordered field, completeness of R, Extended real number system, liens up and limits of a sequence, the epsilon – delta definition of continuity and convergence, the algebra of continuous functions, monotonic functions, types of discontinuities, infinite limits and limits at infinity, functions of bounded variation, elements of metric spaces.

6.Complex Analysis:

Riemann Sphere and Stereographic projection. Lines, Circles, crossratio. Mobius transformations, Analytic functions, Cauchy - Riemann equations, line integrals, Cauchy's theorem, Morera's theorem, Liouville's theorem, integral formula, zero-sets of analytic functions, exponential, sine and cosine functions, Power series representation, Classification of singularities, Conformal Mapping.

7. Algebra:

Group, subgroups, Normal subgroups, Quotient Groups, Homomorphisms, Cyclic Groups, permutation Groups, Cayley's Theorem, Rings, Ideals, Integral Domains, Fields, Polynomial Rings.

8.Linear Algebra:

Vector spaces, subspaces, quotient spaces, Linear independence, Bases, Dimension. The algebra of linear Transformations, kernel, range, isomorphism, Matrix Representation of a linear transformation, change of bases, Linear functionals, dual space, projection, determinant function, eigenvalues and eigen vectors, Cayley-Hamilton Theorem, Invariant Sub-spaces, Canonical Forms: diagonal form, Triangular form, Jordan Form, Inner product spaces.

9. Differential Equations:

First order ODE, singular solutions, initial value Problems of First Order ODE, General theory of homogeneous and non-homogeneous Linear ODE, Variation of Parameters. Lagrange's and Charpit's methods of solving first order Partial Differential Equations. PDE's of higher order with constant coefficients.

10.Data Analysis Basic Concepts:

Graphical representation, measures of central tendency and dispersion. Bivariate data, correlation and regression. Least squares - polynomial regression, Applications of normal distribution.

11.Probability: Axiomatic definition of probability. Random variables and distribution functions (univariate and multivariate); expectation and moments; independent events and independent random variables; Bayes' theorem; marginal and conditional distribution in the multivariate case, covariance matrix and correlation coefficients (product moment, partial and multiple), regression.
Moment generating functions, characteristic functions; probability inequalities (Tchebyshef, Markov, Jensen). Convergence in probability and in distribution; weak law of large numbers and central limit theorem for independent identically distributed random variables with finite variance.
12.Probability Distribution: Bernoulli, Binomial, Multinomial, Hypergeomatric, Poisson, Geometric and Negative binomial distributions, Uniform, exponential, Cauchy, Beta, Gamma, and normal (univariate and multivariate) distributions Transformations of random variables; sampling distributions. t, F and chi-square distributions as sampling distributions, Standard errors and large sample distributions. Distribution of order statistics and range.
13. Theory of Statistics: Methods of estimation: maximum likelihood method, method of moments, minimum chi-square method, least-squares method. Unbiasedness, efficiency, consistency. Cramer-Rao inequality. Sufficient Statistics. Rao-Blackwell Theorem. Uniformly minimum variance unbiased estimators. Estimation by confidence intervals. Tests of hypotheses: Simple and composite hypotheses, two types of errors, critical region, randomized test, power function, most powerful and uniformly most powerful tests. Likelihood-ratio tests. Wald's sequential probability ratio test.
14. Statistical methods and Data Analysis: Tests for mean and variance in the normal distribution: one-population and two- population cases; related confidence intervals. Tests for product moment, partial and multiple correlation coefficients; comparison of k linear regressions. Fitting polynomial regression; related test. Analysis of discrete data: chi-square test of goodness of fit, contingency tables. Analysis of variance: one-way and two-way classification (equal number of observations per cell). Large-sample tests through normal approximation. Nonparametric tests: sign test, median test, Mann-Whitney test, Wilcoxon test for one and two-samples, rank correlation and test of independence.
15. Operational Research Modelling: Definition and scope of Operational Research. Different types of models. Replacement models and sequencing theory, Inventory problems and their analytical structure. Simple deterministic and stochastic models of inventory control. Basic characteristics of queueing system, different performance measures. Steady state solution of Markovian queueing models: M/M/1, M/M/1 with limited waiting space M/M/C, M/M/C with limited waiting space.
16.Linear Programming: Linear Programming, Simplex method, Duality in linear programming. Transformation and assignment problems. Two person-zero sum games. Equivalence of rectangular game and linear programming.
17. Finite Population: Sampling Techniques and Estimation: Simple random sampling with and without replacement. Stratified sampling; allocation problem; systematic sampling. Two stage sampling. Related estimation problems in the above cases.
18. Design of Experiments: Basic principles of experimental design. Randomisation structure and analysis of completely randomised, randomised blocks and Latin-square designs. Factorial experiments. Analysis of 2n factorial experiments in randomised blocks.


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