State Eligibility Test (SET) -
The last date for submitting application is extended to 07.11.2008, and
Examination date is postponed to December 14, 2008.
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Showing posts with label SET 2008. Show all posts
Showing posts with label SET 2008. Show all posts
Monday, October 20, 2008
Monday, September 1, 2008
SET EXAM- Mathematics Paper III
Mathematics Paper III (Unit 16 to 24)
16. Linear Integral Equations :
Linear integral Equations of the first and second kind of Fredholm and Volterra type, solution by successive substitutions and successive approximations ; Solution of equations with separable kernels ; The Fredholm Alternative ; Holbert – Schmidt theory for symmetric kernels.
17. Numerical analysis :
Finite differences, interpolation ; Numerical solution of algebric equation ; Iteration ; Newton – Rephson method ; Solution on linear system ; Direct method ; Gauss elimination method ; Matrix – Inversion, elgenvalue problems ; Numerical differentiation and integration.Numerical solution of ordinary differential equation; iteration method, Picard’s method, Euler’s method and improved Euler’s method.
18. Integral Transform :
Laplace transform ; Transform of elementary functions, Transform of Derivatives, inverse Transform, Comrolution Theorem, Applications, Ordinary and Partial differential equations ; Fourier transforms ; sine and cosine transform, Inverse Fourier Transform, Application to ordinary and partial differential equations. 19. Mathematical Programming :
Revised simplex method, Dual simplex method, Sensitivity analysis and parametric linear programming. Kuhn – Tucker conditions of optimality. Quadratic programming ; methods due to Beale, Wofle and Vandepanne, Duality in quadranic programming, self duality, integer programming.
20. Measure Theory :
Measurable and measure spaces ; Extension of measures, signed measures, Jordan – Hahn decomposition theorems. Integration, monotone convergence theorem, Fatou’s lemma, dominated convergence theorem. Absolute continuity. Radon Nikodym theorem, Product measures, Fubini’s theorem.
21. Probability :
Sequences of events and random variables ; Zero – one laws of Borel and Kolmogorov. Almost sure convergence, convergence in mean square, Khintchine’s weak law of large numbers ; Kolmogorov’s inequality, strong law of large numbers.Convergence of series of random variables, three – series criterion. Central limit theorems of Liapounov and Lindeberg – Feller. Conditional expectation, martingales.
22. Distribution Theory :
Properties of distribution functions and characteristic functions ; continuity theorem, inversion formula, Representation of distribution function as a mixture of discrete and continuous distribution functions ; Convolutions, marginal and conditional distributions of bivariate discrete and continuous distributions.Relations between characteristic functions and moments ; Moment inequalities of Holder and Minkowski.
23. Statistical Inference and Decision Theory :
Statistical decision problem : non – randomized, mixed and randomized decision rules ; risk function, admissibility, Bayes’ rules, minimax rules, least favourable distributions, complete class, and minimal complete class. Decision problem for finite parameter class. Convex loss function. Role of sufficiency.Admissible, Bayes and minimax estimators ; Illustrations, Unbiasedness. UMVU estimators. Families of distributions with monotone likelihood property, exponential family of distributions. Test of a simple hypothesis against a simple alternative from decision – theoretic view point. Tests with Neymen structure. Uniformly most powerful unbiased tests. Locally most powerful tests. Inference on location and scale parameters estimation and tests. Equivariant estimators invariance in hypothesis testing.
24. Large sample statistical methods :
Various modes of convergence, Op and op, CLT Sheffe’s theorem, Polya’s theorem and Stutsky’s theorem. Transformation and variance stabilizing formula. Asymptotic distribution of function of sample moments. Sample quantities. Order statistics and their functions. Tests on correlations, coefficients of variation, skewness and kurtosis. Pearson Chi-square, contingency. Chi – square and likelihood ratio statistics. U – statistics. Consistency of Tests. Asymptotic relative efficiency.
Mathematics Paper III (Unit 31 to 34)
31. Demography and Vital Statistics :
Measures of fertility and mortality, period and Cohort measures.Life tables and its applications ; Methods of construction of abridged life tables. Application of stable population theory to estimate vital rates. Population projections, Stochastic models of fertility and reproduction.
32. Industrial Statistics :
Control charts for variables and attributes ; Acceptance sampling by attributes ; single, double and sequential sampling plans ; OC and ASN functions, AOQL and ATI ; Acceptance sampling by varieties. Tolerance limits. Reliability analysis : Hazard function, distribution with DFR and IFR ; Series and parallel systems. Life testing experiments.
33. Inventory and Queueing theory :
Inventory ( S, S ) policy, periodic review models with stochastic demand. Dynamic inventory models. Probabilistic re-order point, lot size inventory system with and without lead time. Distribution free analysis. Solution of inventory problem with unknown density function. Warehousing problem. Queues ; Imbedded Markov Chain method to obtain steady state solution of M/G/1, G/M/1 AND M/D/C, Network models. Machine maintenance models. Design and control of queueing systems.
34. Dynamic Programming and Marketing :
Nature of dynamic programming, Deterministic processes, Non-sequential discrete optimization – allocation problems, assortment problems. Sequential discrete optimization long – term planning problems, multi stage production processes. Functional approximations. Marketing systems, application of dynamic programming to marketing problems. Introduction of new product, objective in setting market price and its policies, purchasing under fluctuating prices, Advertising and promotional decisions, Brands switching analysis, Distribution decisions.
Mathematics Paper III (Unit 1 to 6)
1. Real Analysis :
Riemann integrate functions ; improper integrate, their convergence and uniform convergence. Eulidean space R¯ , Boizano – Weleratrass theorem, compact. Subsets of R•, Heine – Borel theorem, Fourier series.Continuity of functions on R”, Differentiability of F : R• > Rm, Properties of differential, partial and directional derivatives, continuously differentiable functions. Taylor’s series. Inverse function theorem, implicit function theorem.Integral functions, line and surface integrals, Green’s theorem. Stoke’s theorem.
2. Complex Analysis :
Cauchy’s theorem for convex regions, Power series representation of Analytic functions. Liouville’s theorem, Fundamental theorem of algebra, Riemann’s theorem on removable singularities, maximum modulus principle. Schwarz lemma, Open Mapping theorem, Casoratti – Weierstrass – theorem, Weierstrass’s theorem on uniform convergence on compact sets, Bilinear transformations, Multivalued Analytic Functions, Riemann Surfaces.
3. Algebra :
Symmetric groups, alternating groups, Simple groups, Rings, Maximal ideals, Prime ideals, integral domains, Euclidean domains, principal ideal domains, Unique Factorisation domains, quotient fields, Finite fields, Algebra of Linear Transformations, Reduction of matrices to Canonical Forms, Inner Product Spaces, Orthogonality, quadratic Forms, Reduction of quadratic forms.
4. Advanced Analysis :
Elements of Metric Spaces, Convergence, continuity, compactness, Connectedness, Weierstrass’s approximation Theorem. Completeness, Bare category theorem, Labesgue measure, Labesgue integral, Differentiation and integration.
5. Advanced Algebra :
Conjugate elements and class equations of finite groups, Sylow theorems, solvable groups, Jordan Holder Theorem, Direct Products, Structure Theorem for finite abelian groups, Chain conditions on Rings : Characteristic of Field, Field extensions, Elements of Galois theory, solvability by Radicals, Ruler and compass construction.
6. Functional Analysis :
Banach Spaces, Hahn – Banach Theorem, Open mapping and closed Graph Theorems. Principle of Uniform boundedness, Boundedness and continuity of Linear Transformations. Dual Space, Embedding in the second dual, Hilbert Spaces, Projections. Orthonormal Basis, Riesz – representation theorem. Bessel’s inequality, parsaval’s identity, self adjoined operators, Normal Operators.
Mathematics Paper III (Unit 25 - 30)
25. Multivariate Statistical Analysis :
Singular and non – singular multivariate distributions. Characteristics functions. Multivariate normal distribution ; marginal and conditional distribution, distribution of linear forms, and quadratic forms, Cochran’s theorem.
Inference on parameters of multivariate normal distributions : one – population and two – population cases. Wishart distribution. Hotellings T2, Mahalanobis D2, Discrimination analysis, Principal components, Canonical correlations, Cluster analysis.
26. Linear Models and Regression :
Standard Gauss – Markov models ; Estimability of parameters ; best linear unbiased estimates ( BLUE ); Method of least squares and Gauss – Markov theorem ; Variance – covariance matrix of BLUES.
Tests of linear hypothesis ; One – way and two – way classifications. Fixed, random and mixed effects models ( two – way classifications only ); variance components, Bivariate and multiple linear regression; Polynomial regression ; use of orthogonal polynomials. Analysis of covariance. Linear and nonlinear regression. Outliers.
27. Sample Surveys : Sampling with varying probability of selection, Hurwitz – Thompson estimator ; PPS sampling ; Double sampling, Cluster sampling. Non-sampling errors ; interpenetrating samples. Multiphase sampling. Ratio and regression methods of estimation.
28. Design of Experiments :
Factorial experiments, confounding and fractional replication. Split and strip plot designs ; Quesi – Latin square designs ; Youden square. Design for study of response surfaces ; first and second order designs. Incomplete block designs ; Balanced, connectedness and orthogonality, BIBD with recovery of inter-block information, PBIBD with 2 associate classes. Analysis of sense of experiments, estimation of residual effects. Construction of orthogonal – Latin squares, BIB designs, and confounded factorial designs. Optimality criteria for experimental designs.
29. Time – Series Analysis :
Discrete – parameter stochastic processes ; strong and weak stationarity ; autocovariance and autocorrelation, Moving average, autoregressive, autoregressive moving average and autoregressive integrated moving average processes. Box – Jenkins models. Estimation of the parameters in ARIMA models, forecasting. Perfodogram analysis.
30. Stochastic Processes :
Markov chains with finite and countable state space, classification of states, limiting behaviour of n-step transition probabilities, stationary distribution ; branching processes ; Random walk ; Gambler’s ruin. Markov processes in continuous time ; Poisson processes, birth and death processes, Wiener process.
16. Linear Integral Equations :
Linear integral Equations of the first and second kind of Fredholm and Volterra type, solution by successive substitutions and successive approximations ; Solution of equations with separable kernels ; The Fredholm Alternative ; Holbert – Schmidt theory for symmetric kernels.
17. Numerical analysis :
Finite differences, interpolation ; Numerical solution of algebric equation ; Iteration ; Newton – Rephson method ; Solution on linear system ; Direct method ; Gauss elimination method ; Matrix – Inversion, elgenvalue problems ; Numerical differentiation and integration.Numerical solution of ordinary differential equation; iteration method, Picard’s method, Euler’s method and improved Euler’s method.
18. Integral Transform :
Laplace transform ; Transform of elementary functions, Transform of Derivatives, inverse Transform, Comrolution Theorem, Applications, Ordinary and Partial differential equations ; Fourier transforms ; sine and cosine transform, Inverse Fourier Transform, Application to ordinary and partial differential equations. 19. Mathematical Programming :
Revised simplex method, Dual simplex method, Sensitivity analysis and parametric linear programming. Kuhn – Tucker conditions of optimality. Quadratic programming ; methods due to Beale, Wofle and Vandepanne, Duality in quadranic programming, self duality, integer programming.
20. Measure Theory :
Measurable and measure spaces ; Extension of measures, signed measures, Jordan – Hahn decomposition theorems. Integration, monotone convergence theorem, Fatou’s lemma, dominated convergence theorem. Absolute continuity. Radon Nikodym theorem, Product measures, Fubini’s theorem.
21. Probability :
Sequences of events and random variables ; Zero – one laws of Borel and Kolmogorov. Almost sure convergence, convergence in mean square, Khintchine’s weak law of large numbers ; Kolmogorov’s inequality, strong law of large numbers.Convergence of series of random variables, three – series criterion. Central limit theorems of Liapounov and Lindeberg – Feller. Conditional expectation, martingales.
22. Distribution Theory :
Properties of distribution functions and characteristic functions ; continuity theorem, inversion formula, Representation of distribution function as a mixture of discrete and continuous distribution functions ; Convolutions, marginal and conditional distributions of bivariate discrete and continuous distributions.Relations between characteristic functions and moments ; Moment inequalities of Holder and Minkowski.
23. Statistical Inference and Decision Theory :
Statistical decision problem : non – randomized, mixed and randomized decision rules ; risk function, admissibility, Bayes’ rules, minimax rules, least favourable distributions, complete class, and minimal complete class. Decision problem for finite parameter class. Convex loss function. Role of sufficiency.Admissible, Bayes and minimax estimators ; Illustrations, Unbiasedness. UMVU estimators. Families of distributions with monotone likelihood property, exponential family of distributions. Test of a simple hypothesis against a simple alternative from decision – theoretic view point. Tests with Neymen structure. Uniformly most powerful unbiased tests. Locally most powerful tests. Inference on location and scale parameters estimation and tests. Equivariant estimators invariance in hypothesis testing.
24. Large sample statistical methods :
Various modes of convergence, Op and op, CLT Sheffe’s theorem, Polya’s theorem and Stutsky’s theorem. Transformation and variance stabilizing formula. Asymptotic distribution of function of sample moments. Sample quantities. Order statistics and their functions. Tests on correlations, coefficients of variation, skewness and kurtosis. Pearson Chi-square, contingency. Chi – square and likelihood ratio statistics. U – statistics. Consistency of Tests. Asymptotic relative efficiency.
Mathematics Paper III (Unit 31 to 34)
31. Demography and Vital Statistics :
Measures of fertility and mortality, period and Cohort measures.Life tables and its applications ; Methods of construction of abridged life tables. Application of stable population theory to estimate vital rates. Population projections, Stochastic models of fertility and reproduction.
32. Industrial Statistics :
Control charts for variables and attributes ; Acceptance sampling by attributes ; single, double and sequential sampling plans ; OC and ASN functions, AOQL and ATI ; Acceptance sampling by varieties. Tolerance limits. Reliability analysis : Hazard function, distribution with DFR and IFR ; Series and parallel systems. Life testing experiments.
33. Inventory and Queueing theory :
Inventory ( S, S ) policy, periodic review models with stochastic demand. Dynamic inventory models. Probabilistic re-order point, lot size inventory system with and without lead time. Distribution free analysis. Solution of inventory problem with unknown density function. Warehousing problem. Queues ; Imbedded Markov Chain method to obtain steady state solution of M/G/1, G/M/1 AND M/D/C, Network models. Machine maintenance models. Design and control of queueing systems.
34. Dynamic Programming and Marketing :
Nature of dynamic programming, Deterministic processes, Non-sequential discrete optimization – allocation problems, assortment problems. Sequential discrete optimization long – term planning problems, multi stage production processes. Functional approximations. Marketing systems, application of dynamic programming to marketing problems. Introduction of new product, objective in setting market price and its policies, purchasing under fluctuating prices, Advertising and promotional decisions, Brands switching analysis, Distribution decisions.
Mathematics Paper III (Unit 1 to 6)
1. Real Analysis :
Riemann integrate functions ; improper integrate, their convergence and uniform convergence. Eulidean space R¯ , Boizano – Weleratrass theorem, compact. Subsets of R•, Heine – Borel theorem, Fourier series.Continuity of functions on R”, Differentiability of F : R• > Rm, Properties of differential, partial and directional derivatives, continuously differentiable functions. Taylor’s series. Inverse function theorem, implicit function theorem.Integral functions, line and surface integrals, Green’s theorem. Stoke’s theorem.
2. Complex Analysis :
Cauchy’s theorem for convex regions, Power series representation of Analytic functions. Liouville’s theorem, Fundamental theorem of algebra, Riemann’s theorem on removable singularities, maximum modulus principle. Schwarz lemma, Open Mapping theorem, Casoratti – Weierstrass – theorem, Weierstrass’s theorem on uniform convergence on compact sets, Bilinear transformations, Multivalued Analytic Functions, Riemann Surfaces.
3. Algebra :
Symmetric groups, alternating groups, Simple groups, Rings, Maximal ideals, Prime ideals, integral domains, Euclidean domains, principal ideal domains, Unique Factorisation domains, quotient fields, Finite fields, Algebra of Linear Transformations, Reduction of matrices to Canonical Forms, Inner Product Spaces, Orthogonality, quadratic Forms, Reduction of quadratic forms.
4. Advanced Analysis :
Elements of Metric Spaces, Convergence, continuity, compactness, Connectedness, Weierstrass’s approximation Theorem. Completeness, Bare category theorem, Labesgue measure, Labesgue integral, Differentiation and integration.
5. Advanced Algebra :
Conjugate elements and class equations of finite groups, Sylow theorems, solvable groups, Jordan Holder Theorem, Direct Products, Structure Theorem for finite abelian groups, Chain conditions on Rings : Characteristic of Field, Field extensions, Elements of Galois theory, solvability by Radicals, Ruler and compass construction.
6. Functional Analysis :
Banach Spaces, Hahn – Banach Theorem, Open mapping and closed Graph Theorems. Principle of Uniform boundedness, Boundedness and continuity of Linear Transformations. Dual Space, Embedding in the second dual, Hilbert Spaces, Projections. Orthonormal Basis, Riesz – representation theorem. Bessel’s inequality, parsaval’s identity, self adjoined operators, Normal Operators.
Mathematics Paper III (Unit 25 - 30)
25. Multivariate Statistical Analysis :
Singular and non – singular multivariate distributions. Characteristics functions. Multivariate normal distribution ; marginal and conditional distribution, distribution of linear forms, and quadratic forms, Cochran’s theorem.
Inference on parameters of multivariate normal distributions : one – population and two – population cases. Wishart distribution. Hotellings T2, Mahalanobis D2, Discrimination analysis, Principal components, Canonical correlations, Cluster analysis.
26. Linear Models and Regression :
Standard Gauss – Markov models ; Estimability of parameters ; best linear unbiased estimates ( BLUE ); Method of least squares and Gauss – Markov theorem ; Variance – covariance matrix of BLUES.
Tests of linear hypothesis ; One – way and two – way classifications. Fixed, random and mixed effects models ( two – way classifications only ); variance components, Bivariate and multiple linear regression; Polynomial regression ; use of orthogonal polynomials. Analysis of covariance. Linear and nonlinear regression. Outliers.
27. Sample Surveys : Sampling with varying probability of selection, Hurwitz – Thompson estimator ; PPS sampling ; Double sampling, Cluster sampling. Non-sampling errors ; interpenetrating samples. Multiphase sampling. Ratio and regression methods of estimation.
28. Design of Experiments :
Factorial experiments, confounding and fractional replication. Split and strip plot designs ; Quesi – Latin square designs ; Youden square. Design for study of response surfaces ; first and second order designs. Incomplete block designs ; Balanced, connectedness and orthogonality, BIBD with recovery of inter-block information, PBIBD with 2 associate classes. Analysis of sense of experiments, estimation of residual effects. Construction of orthogonal – Latin squares, BIB designs, and confounded factorial designs. Optimality criteria for experimental designs.
29. Time – Series Analysis :
Discrete – parameter stochastic processes ; strong and weak stationarity ; autocovariance and autocorrelation, Moving average, autoregressive, autoregressive moving average and autoregressive integrated moving average processes. Box – Jenkins models. Estimation of the parameters in ARIMA models, forecasting. Perfodogram analysis.
30. Stochastic Processes :
Markov chains with finite and countable state space, classification of states, limiting behaviour of n-step transition probabilities, stationary distribution ; branching processes ; Random walk ; Gambler’s ruin. Markov processes in continuous time ; Poisson processes, birth and death processes, Wiener process.
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.
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.
SET EXAM - Mathematics Paper III (Unit 7-15)
7. Topology:
Elements of Topological Spaces, Continuity, Convergence, Homeomorphism, Compactness, Connectedness, Separation Axioms, First and Second Countability, Separability, Subspaces, Product Spaces, quotient spaces. Tychonoff's Theorem, Urysohn's Metrization theorem, Homotopy and Fundamental Group.
8.Discrete Mathematics:
Partially ordered sets, Lattices, Complete Lattices, Distributive lattices, Complements, Boolean Algebra, Boolean Expressions, Application to switching circuits, Elements of Graph Theory, Eulerian and Hamiltonian graphs, planar Graphs, Directed Graphs, Trees, Permutations and Combinations, Pigeonhole principle, principle of Inclusion and Exclusion, Derangements.
9.Ordinary and Partial Differential Equations:
Existence and Uniqueness of solution dy/dx =f(x,y) Green's function, sturm Liouville Boundary Value Problems, Cauchy Problems and Characteristics, Classification of Second Order PDE, Separation of Variables for heat equation, wave equation and Laplace equation, Special functions.
10.Number Theory:
Divisibility; Linear diophantine equations. Congruences. Quadratic residues; Sums of two squares, Arithmetic functions Mu, Tau, Phi and Sigma ( and ).
11.Mechanics:
Generalised coordinates; Lagranges equation; Hamilton's cononical equations; Variational principles - Hamilton's principles and principles of least action; Two dimensional motion of rigid bodies; Euler's dynamical equations for the motion of rigid body; Motion of a rigid body about an axis; Motion about revolving axes.
12.Elasticity:
Analysis of strain and stress, strain and stress tensors; Geometrical representation; Compatibility conditions; Strain energy function; Constitutive relations; Elastic solids Hookes law; Saint-Venant's principle, Equations of equilibrium; Plane problems - Airy's stress function, vibrations of elastic, cylindrical and spherical media.
13.Fluid Mechanics:
Equation of continuity in fluid motion; Euler's equations of motion for perfect fluids; Two dimensional motion complex potential; Motion of sphere in perfect liquid and motion of liquid past a sphere; vorticity; Navier-Stokes's equations for viscous flows-some exact solutions.
14.Differetial Geometry:
Space curves - their curvature and torsion; Serret Frehat Formula; Fundamental theorem of space curves; Curves on surfaces; First and second fundamental form; Gaussian curvatures; Principal directions and principal curvatures; Goedesics, Fundamental equations of surface theory.
15.Calculus of Variations:
Linear functionals, minimal functional theorem, general variation of a functional, Euler- Lagrange equation; Variational methods of boundary value problems in ordinary and partial differential equations.
Elements of Topological Spaces, Continuity, Convergence, Homeomorphism, Compactness, Connectedness, Separation Axioms, First and Second Countability, Separability, Subspaces, Product Spaces, quotient spaces. Tychonoff's Theorem, Urysohn's Metrization theorem, Homotopy and Fundamental Group.
8.Discrete Mathematics:
Partially ordered sets, Lattices, Complete Lattices, Distributive lattices, Complements, Boolean Algebra, Boolean Expressions, Application to switching circuits, Elements of Graph Theory, Eulerian and Hamiltonian graphs, planar Graphs, Directed Graphs, Trees, Permutations and Combinations, Pigeonhole principle, principle of Inclusion and Exclusion, Derangements.
9.Ordinary and Partial Differential Equations:
Existence and Uniqueness of solution dy/dx =f(x,y) Green's function, sturm Liouville Boundary Value Problems, Cauchy Problems and Characteristics, Classification of Second Order PDE, Separation of Variables for heat equation, wave equation and Laplace equation, Special functions.
10.Number Theory:
Divisibility; Linear diophantine equations. Congruences. Quadratic residues; Sums of two squares, Arithmetic functions Mu, Tau, Phi and Sigma ( and ).
11.Mechanics:
Generalised coordinates; Lagranges equation; Hamilton's cononical equations; Variational principles - Hamilton's principles and principles of least action; Two dimensional motion of rigid bodies; Euler's dynamical equations for the motion of rigid body; Motion of a rigid body about an axis; Motion about revolving axes.
12.Elasticity:
Analysis of strain and stress, strain and stress tensors; Geometrical representation; Compatibility conditions; Strain energy function; Constitutive relations; Elastic solids Hookes law; Saint-Venant's principle, Equations of equilibrium; Plane problems - Airy's stress function, vibrations of elastic, cylindrical and spherical media.
13.Fluid Mechanics:
Equation of continuity in fluid motion; Euler's equations of motion for perfect fluids; Two dimensional motion complex potential; Motion of sphere in perfect liquid and motion of liquid past a sphere; vorticity; Navier-Stokes's equations for viscous flows-some exact solutions.
14.Differetial Geometry:
Space curves - their curvature and torsion; Serret Frehat Formula; Fundamental theorem of space curves; Curves on surfaces; First and second fundamental form; Gaussian curvatures; Principal directions and principal curvatures; Goedesics, Fundamental equations of surface theory.
15.Calculus of Variations:
Linear functionals, minimal functional theorem, general variation of a functional, Euler- Lagrange equation; Variational methods of boundary value problems in ordinary and partial differential equations.
Monday, July 28, 2008
UGC SET Paper I Revised
SUBJECT CODE: 00
SUB: GENERAL PAPER ON TEACHING AND RESEARCH APTITUDE
Here the main objective is to assess the teaching and research capabilities of the candidates. Hence the test is aimed at assessing the teaching and general / research aptitude as well as their awareness. They are expected to possess and exhibit cognitive abilities. Cognitive abilities include comprehension, analysis, evaluation, understanding the structure of arguments and deductive and inductive reasoning. The candidates are also expected to have a general awareness and knowledge of sources of information. They should be aware of interaction between people, environment and natural resources and their impact on quality of life. The details are given in the following sections.
Note:
1.Each section gets equal wieghtage
(5 questions and 10 marks from each section)
2.Whenever pictorial questions are set for the sighted candidates, a passage followed by equal number of questions should be set for the visually handicapped candidates.
I. Teaching Aptitude :
*Teaching : Nature,objectives, characteristics and basic requirements.
*Learner's characteristics
*Factors affecting teaching
*Methods of teaching
*Evaluation systems.
II.Research Aptitude:
Research : Meaning, characteristics and types;
Steps of research;
Methods of research;
Research Ethics;
Paper, article, workshop, seminar, conference and symposium;
Thesis writing: its characteristics and format.
III. Reading comprehension
*A passage to be set with questions to be answered
IV. Communication
Communication: Nature, characteristics, types, barriers and effective classroom communication.
V. Mathematical Reasoning
Number series; letter series; codes;
Relationships; classification.
VI.Logical Reasoning
Understanding the structure of arguments;
Evaluating and distinguishing deductive and inductive reasoning;
Verbal analogies: Word analogy – Applied analogy;
Verbal classification;
Reasoning Logical Diagrams :
Simple diagrammatic relationship, multi-diagrammatic relationship;
Venn diagram; Analytical Reasoning.
VII. Data Interpretation
Sources, acquisition and interpretation of data;
Quantitative and qualitative data;
Graphical representation and mapping of data.
VIII. Information and Communication Technology(ICT)
ICT: meaning, advantages, disadvantages and uses;
General abbreviations and terminology;
Basics of internet and e-mailing.
IX. People and Environment
People and environment interaction;
Sources of pollution;
Pollutants and their impact on human life, exploitation of natural and energy resources;
Natural hazards and mitigation.
X. Higher Education System:
Governance, Polity and Administration
Structure of the institutions for higher learning and research in India;
formal and distance education;
professional/technical and general education;
value education: governance;
polity and administration;
concept, institutions and their interactions.
SUB: GENERAL PAPER ON TEACHING AND RESEARCH APTITUDE
Here the main objective is to assess the teaching and research capabilities of the candidates. Hence the test is aimed at assessing the teaching and general / research aptitude as well as their awareness. They are expected to possess and exhibit cognitive abilities. Cognitive abilities include comprehension, analysis, evaluation, understanding the structure of arguments and deductive and inductive reasoning. The candidates are also expected to have a general awareness and knowledge of sources of information. They should be aware of interaction between people, environment and natural resources and their impact on quality of life. The details are given in the following sections.
Note:
1.Each section gets equal wieghtage
(5 questions and 10 marks from each section)
2.Whenever pictorial questions are set for the sighted candidates, a passage followed by equal number of questions should be set for the visually handicapped candidates.
I. Teaching Aptitude :
*Teaching : Nature,objectives, characteristics and basic requirements.
*Learner's characteristics
*Factors affecting teaching
*Methods of teaching
*Evaluation systems.
II.Research Aptitude:
Research : Meaning, characteristics and types;
Steps of research;
Methods of research;
Research Ethics;
Paper, article, workshop, seminar, conference and symposium;
Thesis writing: its characteristics and format.
III. Reading comprehension
*A passage to be set with questions to be answered
IV. Communication
Communication: Nature, characteristics, types, barriers and effective classroom communication.
V. Mathematical Reasoning
Number series; letter series; codes;
Relationships; classification.
VI.Logical Reasoning
Understanding the structure of arguments;
Evaluating and distinguishing deductive and inductive reasoning;
Verbal analogies: Word analogy – Applied analogy;
Verbal classification;
Reasoning Logical Diagrams :
Simple diagrammatic relationship, multi-diagrammatic relationship;
Venn diagram; Analytical Reasoning.
VII. Data Interpretation
Sources, acquisition and interpretation of data;
Quantitative and qualitative data;
Graphical representation and mapping of data.
VIII. Information and Communication Technology(ICT)
ICT: meaning, advantages, disadvantages and uses;
General abbreviations and terminology;
Basics of internet and e-mailing.
IX. People and Environment
People and environment interaction;
Sources of pollution;
Pollutants and their impact on human life, exploitation of natural and energy resources;
Natural hazards and mitigation.
X. Higher Education System:
Governance, Polity and Administration
Structure of the institutions for higher learning and research in India;
formal and distance education;
professional/technical and general education;
value education: governance;
polity and administration;
concept, institutions and their interactions.
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