Math : Csir Ugc Net syllabus for Mathematics

CSIR-UGC National  Eligibility  Test (NET) for  Junior Research Fellowship and  Lecturer-ship   COMMON SYLLABUS FOR PART ‘B’ AND ‘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.     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 statistics  are expected  to  answer  additional  question  from  Unit  IV. 

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