Syllabus of Mathematical Sciences for CSIRUGC
CSIRUGC National Eligibility Test (NET) for Junior Research Fellowship and Lecturership
SYLLABUS FOR
MATHEMATICAL SCIENCES
General Aptitude (GA): Common to All Papers of Part A
Logical reasoning,graphical analysis, analytical and numerical ability, quantitative
comparison, series formation, puzzle etc.
Mathematics Syllabus
(Common Syllabus for Part ‘B & 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. Metric spaces, compactness, connectedness. Normed Linear
Spaces. Spaces of Continuous functions as examples.
Linear Algebra: Vector spaces, subspaces, lineardependence, basis, dimension,
algebra of linear transformations.Algebra of matrices, rank and determinant of matrices,
linear equations. Eigenvalues and eigenvectors, CayleyHamilton theorem.Matrix representationof
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, CauchyRiemann 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, pigeonhole principle, inclusionexclusion
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, homeomorphisms, cyclic groups, permutation groups,
Cayley’s theorem, classequations, 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.
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 nonhomogeneous linear ODEs,
variation of parameters, SturmLiouville 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 NewtonRaphson
method, Rate of convergence, Solution of systems of linear algebraic equations using
Gauss elimination and GaussSeidel methods, Finite differences, Lagrange, Hermite
and spline interpolation, Numerical differentiation and integration, Numerical solutions
of ODEs using Picard, Euler, modified Euler and RungeKutta methods.

Calculus of Variations:
Variation of a functional, EulerLagrange 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, Twodimensional 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 nstep transition probabilities, stationary distribution.

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 chisquare 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.

GaussMarkov models, estimability of parameters, Best linear unbiased estimators,
tests for linear hypotheses and confidence intervals. 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 designes , randomized blocks and Latinsquare designs.
Connectedness, and orthogonal block designs, BIBD. 2^{K} factorial experiments:
confounding and construction.
Series and parallel systems, hazard function and failure rates, censoring and life
testing.

Linear programming problem. Simplex methods, duality. Elementary queuing and inventory
models. Steadystate 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.
Syllabus of Part  C
Mathematics: This section shall carry questions from Unit I, II and III.
Statistics: Apart from Unit IV, this section shall also carry questions from
the following areas. Sequences and series, convergence, continuity, uniform continuty,
differentibility. Remann integeral, improper integerals, algebra of matrices, rank
and determinant of matrices, linear equations, eigenvalues and eigenvectors, quadratic
froms.
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