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,
Epsilon N definition of Limits of sequences
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, 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.
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 solutions of algebraic equati ons, Method of iteration and Newton-Raphson method, Rate of convergence, Solution of sy stems 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.
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 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 mo tion 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 (univa riate and multivariate); expectation and moments. Independent random variables, marginal and conditio nal 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 st ate space, classification of states, limiting behaviour of n-step 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 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, 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 distri bution and their properties. Distribution of quadratic forms. Inference for parameters, partial and multiple correlation coefficients and related tests. Data reduc tion techniques: Principle com ponent analysis, Discriminant analysis, Cluster analysis, Canonical correlation.
Simple random sampling, stratified sampling and systematic sampling. Probability proportional to size sampling. Ra tio and regression methods.
Completely randomized, randomized blocks and Latin-square designs. Connected, complete 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 met hods, 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.