CSIR-UGC National Eligibility Test (NET) for Junior Research Fellowship
and Lecturer-ship
COMMON SYLLABUS FOR PART ‘B’ AND ‘C’
MATHEMATICAL SCIENCES
|
section
|
Total no. of Q
|
Required to Answer
|
Marks for each question
|
|
Part A
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20
|
15
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2(2*15=30)
|
|
Part B
|
40
|
25
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3(3*25=75)
|
|
Part C
|
60
|
20
|
4.75(20*4.75=95)
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|
total
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145
|
75
|
200
|
|
In Part C, more than 1 option is correct,
mark will b given who give all correct answer. No partial marking in Part C
|
|||
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 as
examples.
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 ransformations. 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 mapping theorem. Taylor series, Laurent series, calculus of
residues. Conformal mappings, Mobius transformations. \
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.
Classical
Mechanics:
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 birth-and-death processes. 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
regression. 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 methods. 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 mathematics
are
expected to answer additional question from Unit II and III. Students with in
statistics
are expected to answer additional question from Unit IV.
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