## CSIR NET Mathematical Science Syllabus

CSIR NET mathematical science candidates should read the complete syllabus before starting their preparation. Syllabus helps candidates in preparing in right direction.

## CSIR Exam Pattern

Exam duration – 3 Hours

Maximum Marks – 200

Negative Marking will of 25% for part A & B and there will be no negative marking for part C

MATHEMATICAL SCIENCES

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 transformations. Change of basis, canonical forms, diagonal forms,  triangular forms, Jordan forms.

Inner product spaces, orthonormal basis.

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 homogeneous 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 behavior 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.