Council of Scientific and Industrial Research
Human Resource Development Group
CSIR-UGC National Eligibility Test (NET) for Junior Research
UNIT – 1
Analysis: Elementary set theory, finite, countable and
uncountable sets, Real number system as a complete ordered field, Archimedean
property, supremum, infimum. Sequences and series, convergence, limsup, liminf.
Bolzano Weierstrass theorem, Heine Borel theorem. Continuity, uniform
continuity, differentiability, mean value theorem. Sequences and series of
functions, uniform convergence. Riemann sums and Riemann integral, Improper
Integrals. Monotonic functions, types of discontinuity, functions of bounded
variation, Lebesgue measure, Lebesgue integral. Functions of several variables,
directional derivative, partial derivative, derivative as a linear
transformation, inverse and implicit function theorems. Metric spaces,
compactness, connectedness. Normed linear Spaces. Spaces of continuous functions
Linear Algebra: Vector spaces, subspaces, linear
dependence, basis, dimension, algebra of linear transformations. Algebra of
matrices, rank and determinant of matrices, linear equations. Eigenvalues and
eigenvectors, Cayley-Hamilton theorem. Matrix representation of linear
transformations. Change of basis, canonical forms, diagonal forms, triangular
forms, Jordan forms. Inner product spaces, orthonormal basis. Quadratic forms,
reduction and classification of quadratic forms
UNIT – 2
Complex Analysis: Algebra of complex numbers, the
complex plane, polynomials, power series, transcendental functions such as
exponential, trigonometric and hyperbolic functions. Analytic functions,
Cauchy-Riemann equations. Contour integral, Cauchy’s theorem, Cauchy’s integral
formula, Liouville’s theorem, Maximum modulus principle, Schwarz lemma, Open
Taylor series, Laurent series, calculus of residues. Conformal mappings, Mobius
Algebra: Permutations, combinations, pigeon-hole
principle, inclusion-exclusion principle, derangements. Fundamental theorem of
arithmetic, divisibility in Z, congruences, Chinese Remainder Theorem, Euler’s
Ø- function, primitive roots. Groups, subgroups, normal subgroups, quotient
groups, homomorphisms, cyclic groups, permutation groups, Cayley’s theorem,
class equations, Sylow theorems. Rings, ideals, prime and maximal ideals,
quotient rings, unique factorization domain, principal ideal domain, Euclidean
domain. Polynomial rings and irreducibility criteria. Fields, finite fields,
field extensions, Galois Theory.
Topology: basis, dense sets, subspace and product topology, separation
axioms, connectedness and compactness.
UNIT – 3
Ordinary Differential Equations (ODEs):
Existence and uniqueness of solutions of initial value
problems for first order ordinary differential equations, singular solutions of
first order ODEs, system of first order ODEs. General theory of homogenous and
non-homogeneous linear ODEs, variation of parameters, Sturm-Liouville boundary
value problem, Green’s function.
Partial Differential Equations (PDEs):
Lagrange and Charpit methods for solving first order PDEs,
Cauchy problem for first order PDEs. Classification of second order PDEs,
General solution of higher order PDEs with constant coefficients, Method of
separation of variables for Laplace, Heat and Wave equations.
Numerical Analysis :
Numerical solutions of algebraic equations, Method of
iteration and Newton-Raphson method, Rate of convergence, Solution of systems of
linear algebraic equations using Gauss elimination and Gauss-Seidel methods,
Finite differences, Lagrange, Hermite and spline interpolation, Numerical
differentiation and integration, Numerical solutions of ODEs using Picard,
Euler, modified Euler and Runge-Kutta methods.
Calculus of Variations:
Variation of a functional, Euler-Lagrange equation, Necessary
and sufficient conditions for extrema. Variational methods for boundary value
problems in ordinary and partial differential equations.
Linear Integral Equations:
Linear integral equation of the first and second kind of
Fredholm and Volterra type, Solutions with separable kernels. Characteristic
numbers and eigenfunctions, resolvent kernel.
Generalized coordinates, Lagrange’s equations, Hamilton’s
canonical equations, Hamilton’s principle and principle of least action,
Two-dimensional motion of rigid bodies, Euler’s dynamical equations for the
motion of a rigid body about an axis, theory of small oscillations.
UNIT – 4
Descriptive statistics, exploratory data analysis:
Sample space, discrete probability, independent events, Bayes
theorem. Random variables and distribution functions (univariate and
multivariate); expectation and moments. Independent random variables, marginal
and conditional distributions. Characteristic functions. Probability
inequalities (Tchebyshef, Markov, Jensen). Modes of convergence, weak and strong
laws of large numbers, Central Limit theorems (i.i.d. case). Markov chains with
finite and countable state space, classification of states, limiting behaviour
of n-step transition probabilities, stationary distribution, Poisson and
Standard discrete and continuous univariate distributions.
sampling distributions, standard errors and asymptotic distributions,
distribution of order statistics and range.
Methods of estimation, properties of estimators, confidence
intervals. Tests of hypotheses: most powerful and uniformly most powerful tests,
likelihood ratio tests. Analysis of discrete data and chi-square test of
goodness of fit. Large sample tests.
Simple nonparametric tests for one and two sample problems,
rank correlation and test for independence. Elementary Bayesian inference.
Gauss-Markov models, estimability of parameters, best linear
unbiased estimators, confidence intervals, tests for linear hypotheses. Analysis
of variance and covariance. Fixed, random and mixed effects models. Simple and
multiple linear regression. Elementary regression diagnostics. Logistic
Multivariate normal distribution, Wishart distribution and
their properties. Distribution of quadratic forms. Inference for parameters,
partial and multiple correlation coefficients and related tests. Data reduction
techniques: Principle component analysis, Discriminant analysis, Cluster
analysis, Canonical correlation.
Simple random sampling, stratified sampling and systematic
sampling. Probability proportional to size sampling. Ratio and regression
Completely randomized designs, randomized block designs and
Latin-square designs. Connectedness and orthogonality of block designs, BIBD. 2K
factorial experiments: confounding and construction.
Hazard function and failure rates, censoring and life
testing, series and parallel systems.
Linear programming problem, simplex methods, duality.
Elementary queuing and inventory models. Steady-state solutions of Markovian
queuing models: M/M/1, M/M/1 with limited waiting space, M/M/C, M/M/C with
limited waiting space, M/G/1.
All students are expected to answer questions
from Unit I. Students in mathematicsare expected to answer additional question
from Unit II and III. Students with in statistics are expected to answer
additional question from Unit IV.