Mathematics

                                                                                           Subject : Mathematics

Unit-I – Algebra

 
Groups – Examples – Cyclic Groups- Permulation Groups – Lagrange’s theorem- Cosets – Normal groups – Homomorphism – Theorems – Cayley’s theorem – Cauchy’s Theorem – Sylow’s theorem – Finitely Generated Abelian Groups – Rings- Euclidian Rings- Polynomial Rings- U.F.D. – Quotient – Fields of integral domains- Ideals- Maximal ideals – Vector Spaces – Linear independence and Bases – Dual spaces – Inner product spaces – Linear transformation – rank – Characteristic roots of matrices – Cayley Hamilton Theorem – Canonical form under equivalence – Fields – Characteristics of a field – Algebraic extensions – Roots of Polynomials – Splitting fields – Simple extensions – Elements of Galois theory- Finite fields.

Unit-II – Real Analysis

 
Cardinal numbers – Countable and uncountable cordinals – Cantor’s diagonal process – Properties of real numbers – Order – Completeness of R-Lub property in R-Cauchy sequence – Maximum and minimum limits of sequences – Topology of R.Heine Borel – Bolzano Weierstrass – Compact if and only if closed and bounded – Connected subset of R-Lindelof’s covering theorem – Continuous functions in relation to compact subsets and connected subsets- Uniformly continuous function – Derivatives – Left and right derivatives – Mean value theorem – Rolle’s theorem- Taylor’s theorem- L’ Hospital’s Rule – Riemann integral – Fundamental theorem of Calculus –Lebesgue measure and Lebesque integral on R’Lchesque integral of Bounded Measurable function – other sets of finite measure – Comparison of Riemann and Lebesque integrals – Monotone convergence theorem – Repeated integrals.

Unit-III – Fourier series and Fourier Integrals

 
Integration of Fourier series – Fejer’s theorem on (C.1) summability at a point – Fejer’s-Lebsque theorem on (C.1) summability almost everywhere – Riesz-Fisher theorem – Bessel’s inequality and Parseval’s theorem – Properties of Fourier co-efficients – Fourier transform in L (-D, D) – Fourier Integral theorem – Convolution theorem for Fourier transforms and Poisson summation formula.

Unit-IV – Differential Geometry

 
Curves in spaces – Serret-Frenet formulas – Locus of centers of curvature – Spherical curvature – Intrinsic equation – Helices – Spherical indicatrix surfaces – Envelope – Edge of regression – Developable surfaces associated to a curve – first and second fundamental forms – lines of curvature – Meusnieu’s theorem – Gaussian curvature – Euler’s theorem –

Duplin’s Indicatrix – Surface of revolution conjugate systems – Asymptritic lines – Isolmetric lines – Geodesics.

Unit-V – Operations Research

 
Linear programming – Simplex Computational procedure – Geometric interpretation of the simplex procedure – The revised simplex method – Duality problems – Degeneracy procedure – Peturbation techniques – integer programming – Transportation problem– Non-linear programming – The convex programming problem – Dyamic programming – Approximation in function space, successive approximations – Game theory – The maximum and minimum principle – Fundamental theory of games – queuing theory / single server and multi server models (M/G/I), (G/M/I), (G/G1/I) models, Erlang service distributions cost Model and optimization – Mathematical theory of inventory control – Feed back control in inventory management – Optional inventory policies in deterministic models – Storage models – Damtype models – Dams with discrete input and continuous output – Replacement theory – Deterministic Stochostic cases – Models for unbounded horizons and uncertain case – Markovian decision models in replacement theory – Reliability – Failure rates – System reliability – Reliability of growth models – Net work analysis – Directed net work – Max flowmin cut theorem – CPM- PERT – Probabilistic condition and decisional network analysis.

Unit-VI – Functional Analysis

 
Banach Spaces – Definition and example – continuous linear transformations – Banach theorem – Natural embedding of X in X – Open mapping and closed graph theorem – Properties of conjugate of an operator – Hilbert spaces – Orthonormal bases – Conjugate space H – Adjoint of an operator – Projections- l2 as a Hilbert space – lp space – Holders and Minkowski inequalities – Matrices – Basic operations of matrices – Determinant of a matrix – Determinant and spectrum of an operator – Spectral theorem for operators on a finite dimensional Hilbert space – Regular and singular elements in a Banach Algebra – Topological divisor of zero – Spectrum of an element in a Branch algebra – the formula for the spectral radius radical and semi simplicity.

Unit-VII – Complex Analysis

 
Introduction to the concept of analytic function – limits and continuity – analytic functions – Polynomials and rational functions elementary theory of power series – Maclaurin’s series – uniform convergence power series and Abel’s limit theorem – Analytic functions as mapping – conformality arcs and closed curves – Analytical functions in regions – Conformal mapping – Linear transformations – the linear group, the cross ratio and symmetry – Complex integration – Fundamental theorems – line integrals – rectifiable arcs – line integrals as functions of arcs – Cauchy’s theorem for a rectangle, Cauchy’s theorem in a Circular disc, Cauchy’s integal formula – The index of a point with respect to a closed curve, the integral formula –

higher derivatives – Local properties of Analytic functions and removable singularities- Taylor’s theorem – Zeros and Poles – the local mapping and the maximum modulus Principle.

Unit-VIII – Differential Equations

 
Linear differential equation – constant co-efficients – Existence of solutions – Wrongskian – independence of solutions – Initial value problems for second order equations – Integration in series – Bessel’s equation – Legendre and Hermite Polynomials – elementary properties – Total differential equations – first order partial differential equation – Charpits method.

Unit-IX – Statistics – I

 
Statistical Method – Concepts of Statistical population and random sample – Collections and presentation of data – Measures of location and dispersion – Moments and shepherd correction – cumulate – Measures of skewness and Kurtosis – Curve fitting by least squares – Regression – Correlation and correlation ratio – rank correlation – Partial correlation – Multiple correlation coefficient – Probability Discrete – sample space, events – their union – intersection etc. – Probability classical relative frequency and axiomatic approaches – Probability in continuous probability space – conditional probability and independence – Basic laws of probability of combination of events – Baye’s theorem – probability functions – Probability density functions – Distribution function – Mathematical Expectations – Marginal and conditional distribution – Conditional expectations.

Unit-X – Statistics-II

 
Probability distributions – Binomial, Poisson, Normal, Gama, Beta, Cauchy, Multinomial Hypergeometric, Negative Binomial – Chehychev’s lemma (weak) law of large numbers – Central limit theorem for independent identical variates, Standard Errors – sampling distributions of t, F and Chi square – and their uses in tests of significance – Large sample tests for mean and proportions – Sample surveys – Sampling frame – sampling with equal probability with or without replacement – stratified sampling – Brief study of two stage systematic and cluster sampling methods – regression and ratio estimates – Design of experiments, principles of experimentation – Analysis of variance – Completely randomized block and latin square designs.