Regulations - 2017(MA8251)

        Eigenvalues and Eigenvectors of a real matrix – Characteristic equation – Properties of eigenvalues and eigenvectors – Statement and applications of Cayley – Hamilton Theorem – Diagonalization of matrices – Reduction of a quadratic form to canonical form by orthogonal transformation – Nature of quadratic forms.

     Gradient and directional derivative – Divergence and Curl – Vector Identities – Irrotational and solenoidal vector fields – Line integral over a plane curve – Surface integral –  Area of a curved surface – Volume integral  – Green’s theorem in a plane, Gauss divergence theorem and Stokes’ theorem (excluding proofs) – Verification and application in evaluating line, surface and volume integrals.

     Analytic functions – Necessary and sufficient conditions for analyticity in Cartesian and polar coordinates – Properties – Harmonic conjugate – Construction of analytic functions – Conformal mapping : w = z + c, cz, 1/z, z2 and bilinear transformation.

     Line integral – Cauchy’s integral theorem and Cauchy’s integral formula – Taylor and Laurent’s series expansions – Singular points – Residues – Cauchy’s Residue theorem – Application of Residue theorem for Evaluation of real integrals – Use of circular contour and semicircular contour.

     Existence conditions – Transform of elementary functions – Transform of unit step function and unit impulse function – Basic properties – Shifting theorems – Transform of derivatives and integrals – Initial and final value thorems - Inverse Laplace transform – Statement of Convolution theorem – Transforms of periodic functions – Application to solution of linear ODE of second order with constant coefficients.

                                                Regulations - 2021(MA3151)

Eigenvalues and Eigenvectors of a real matrix – Characteristic equation – Properties eigenvalues and eigenvectors – Statement and applications of Cayley – Hamilton Theorem – Diagonalization of matrices – Reduction of a quadratic form to canonical form by orthogonal transformation – Nature of quadratic forms – Applications: Stretching of an elastic membrane

Representation of functions – Limit of a function – Continuity – Derivatives – Differentiation rules – Maxima and Minima of functions of one variable.

Partial differentiation – Homogeneous functions and Euler’s theorem – Total derivative – Change of variables – Jacobians – Partial differentiation of implicit functions – Taylor’s series for functions of two variables – Maxima and Minima of functions of two variables – Lagrange’s method of undetermined multipliers.

Definite and indefinite integrals – Substitution rule – Techniques of Integration – Integration by parts, Trigonometric integrals, Trigonometric substitutions, integration of rational functions by partial fraction, Integration of irrational functions – Improper integrals. Applications : Hydrostastic force and pressure, moments and centres of mass.

Double integrals – Change of order of integration – Double integrals in polar coordinates – Area enclosed by plane curves – Triple integrals – Volume of solids – Change of variables in double and triple integrals – Applications: Moments and centres of mass, moment of inertia.

                                                  Regulations - 2021(MA3351)  

        Formation of partial differential equations –Solutions of standard types of first order partial differential equations - First order partial differential equations reducible to standard types- Lagrange’s linear equation - Linear partial differential equations of second and higher order with constant coefficients of both homogeneous and non-homogeneous types.

        Dirichlet’s conditions – General Fourier series – Odd and even functions – Half range sine series and cosine series – Root mean square value – Parseval’s identity – Harmonic analysis.

        Classification of PDE – Method of separation of variables - Fourier series solutions of one dimensional wave equation – One dimensional equation of heat conduction – Steady state solution of two dimensional equation of heat conduction (Cartesian coordinates only).

        Statement of Fourier integral theorem – Fourier transform pair – Sine and Cosine transforms – Properties – Transforms of simple functions – Convolution theorem – Parseval’s identity.

        Z-Transform – Elementary properties – Convergence of Z-Transform - Initial and Final Value Theorems -  Inverse Z-Transform using partial fraction and convolution theorem – Formation of difference equations – Solution of difference equations using Z-Transform.

        Regulations - 2021(MA3355)  

Axioms of probability – Conditional probability – Baye’s theorem - Discrete and continuous random variables – Moments – Moment generating functions – Binomial, Poisson, Geometric, Uniform, Exponential and Normal distributions - Functions of a random variable.

Joint distributions - Marginal and conditional distributions – Covariance - Correlation and Linear Regression - Transformation of random variables – Central limit theorem (for independent and identically distributed random variables).

Classification – Stationary process – Markov process - Poisson process - Discrete parameter - Markov chain – Chapman Kolmogorov equations (Statement only) - Limiting distributions.

Vector spaces – Subspaces – Linear combinations and linear system of equations – Linear independence and linear dependence – Bases and dimensions.

Linear transformation - Null spaces and ranges - Dimension theorem - Matrix representation of a linear transformations - Inner product - Norms - Gram Schmidt orthogonalization process - Adjoint of linear operations - Least square approximation.

        Regulations - 2017(MA8402)  

Probability – Axioms of probability – Conditional probability – Baye‘s theorem - Discrete and continuous random variables – Moments - Moment generating functions – Binomial, Poisson, Geometric, Uniform, Exponential, Gamma and Normal distributions.

Joint distributions - Marginal and conditional distributions – Covariance - Correlation and Linear Regression - Transformation of random variables – Central limit theorem (for independent and identically distributed random variables).

Classification – Stationary process – Markov process –Poisson process – Discrete parameter Markov chain – Chapman Kolmogorov equations – Limiting distributions.

Markovian queues – Birth and Death processes – Single and multiple server queueing models – Little’s formula – Queues with finite waiting rooms – Queues with impatient customers: Balking and reneging.

Finite source models – M/G/1 queue – Pollaczek Khinchin formula – M/D/1 and M/EK/1 as special cases – Series queues – Open Jackson networks.

       Regulations - 2017(MA8491)

     Solution of algebraic and transcendental equations – Fixed point iteration method – Newton Raphson method – Solution of linear system of equations – Gauss elimination method – Pivoting - Gauss Jordan method – Iterative methods of Gauss Jacobi and Gauss   Seidel – Eigenvalues of a matrix by Power method and Jacobi’s method for symmetric matrices.

     Interpolation with unequal intervals – Lagrange’s interpolation – Newton’s divided difference interpolation – Cubic Splines – Difference operators and relations - Interpolation with equal intervals – Newton’s forward and backward difference formulae.

     Approximation of derivatives using interpolation polynomials – Numerical integration using Trapezoidal, Simpson’s 1/3 rule – Romberg’s method – Two point and three point Gaussian quadrature formulae – Evaluation of double integrals by Trapezoidal and Simpson’s 1/3 rules.

     Single Step methods – Taylor’s series method – Euler’s method – Modified Euler’s method – Fourth order Runge-Kutta method for solving first order equations – Multi step methods – Milne’s and Adams-Bash forth predictor corrector methods for solving first order equations.

     Finite difference methods for solving second order two - point linear boundary value problems – Finite difference techniques for the solution of two dimensional Laplace’s and Poisson’s equations on rectangular domain – One dimensional heat flow equation by explicit and implicit (Crank Nicholson) methods – One dimensional wave equation by explicit method.

                                                Regulations - 2021(MA3303)  

     Axioms of probability – Conditional probability – Baye’s theorem - Discrete and continuous random variables – Moments – Moment generating functions – Binomial, Poisson, Geometric, Uniform, Exponential and Normal distributions – Functions of a random variable.

     Joint distributions – Marginal and conditional distributions – Covariance – Correlation and linear regression – Transformation of random variables – Central limit theorem (for independent and identically distributed random variables).

     Analytic functions – Necessary and sufficient conditions for analyticity in Cartesian and polar coordinates - Properties – Harmonic conjugates – Construction of analytic function - Conformal mapping – Mapping by functions w = z+c, cz, 1 / z, z² - Bilinear transformation.

     Line integral - Cauchy’s integral theorem – Cauchy’s integral formula – Taylor’s and Laurent’s series – Singularities – Residues – Residue theorem – Application of residue theorem for evaluation of real integrals – Applications of circular contour and semicircular contour (with poles NOT on real axis).

     Higher order linear differential equations with constant coefficients - Method of variation of parameters – Homogenous equation of Euler’s and Legendre’s type – System of simultaneous linear first order differential equations with constant coefficients - Method of undetermined coefficients.

                                                          Regulations - 2021(MA3354)  

      Propositional Logic – Propositional equivalences-Predicates and quantifiers – Nested Quantifiers – Rules of inference – introduction  to Proofs-Proof Methods and strategy.

      Mathematical inductions-Strong induction and well ordering – The basics of counting – The pigeonhole principle –Permutations and combinations – Recurrence relations – Solving Linear recurrence relations – generating functions – inclusion and exclusion and applications.

      Graphs and graph models-Graph terminology and special types of graphs – Representing graphs and graph isomorphism –connectivity – Euler and Hamilton paths.

      Algebraic systems-Semi groups and monoids – Groups – Subgroups and homomorphisms – Cosets  and Lagrange’s theorem – Ring & Fields (Definitions and examples).

Partial ordering – Posets – Lattices as Posets – Properties of lattices – Lattices as Algebraic systems – Sub lattices – direct product and Homomorphism – Some Special lattices – Boolean Algebra - Boolean Homomorphism.

     Graphs – Introduction – Isomorphism – Sub graphs – Walks, Paths, Circuits – Connectedness – Components – Euler graphs – Hamiltonian paths and circuits – Trees – Properties of trees – Distance and centers in tree – Rooted and binary trees.

     Spanning trees – Fundamental Circuits – Spanning trees in a weighted graph – Cut sets – Properties of cut set – All cut sets – Fundamental circuits and cut sets – Connectivity and separability – Network flows – 1-Isomorphism – 2-Isomorphism – Combinational and geometric graphs – Planer graphs – Different representation of a planar graph.

     Chromatic number – Chromatic partitioning – Chromatic polynomial – Matching – Covering – Four colour problem – Directed graphs – Types of directed graphs – Digraphs and binary relations – Directed paths and connectedness – Euler graphs.

     Fundamental principles of counting – Permutations and combinations – Binomial theorem – Combinations with repetition – Combinational numbers – Principle of Inclusion and Exclusion – Derangements – Arrangements with forbidden positions.

     Generating functions – Partitions of Integers – Exponential generating function – Summation operator – Recurrence relations – First order and second order – Non-homogeneous recurrence relations – Method of generating functions.

        Regulations - 2017(MA8352)  

     Vector spaces – Subspaces – Linear combinations and linear system of equations – Linear independence and linear dependence – Bases and dimensions.

      Linear transformation - Null spaces and ranges - Dimension theorem - Matrix representation of a linear transformations - Eigenvalues and eigenvectors - Diagonalizability.

     Inner product, norms - Gram Schmidt orthogonalization process - Adjoint of linear operations - Least square approximation.

     Formation - Solutions of first order equations - Standard types and equations reducible to standard types - Singular solutions - Lagrange‘s linear equation - Integral surface passing through a given curve - Classification of partial differential equations - Solution of linear equations of higher order with constant coefficients – Linear non-homogeneous partial differential equations.

    Dirichlet‘s conditions - General Fourier series - Half range sine and cosine series - Method of separation of variables - Solutions of one dimensional wave equation and one - dimensional heat equation - Steady state solution of two-dimensional heat equation - Fourier series solutions in Cartesian coordinates.

                                                Regulations - 2021(MA3251)  

     Sampling distributions - Estimation of parameters - Statistical hypothesis - Large sample test based on Normal distribution for single mean and difference of means – Tests based on t, Chi-square and F distributions for mean, variance and proportion – contingency table (Test for Independent) – Goodness of fit.

     One way and two way classifications – Completely randomized design – Randomized block design – Latin square design – 2² factorial design.

    Solution of algebraic and transcendental equations - Fixed point iteration method – Newton Raphson method – Solution of linear system of equations - Gauss elimination method – Pivoting –Gauss Jordan methods – Iterative methods of Gauss Jacobi and Gauss Seidel – Eigen values of a matrix by power method and Jacobi’s method for symmetric matrices.

     Lagrange’s and Newton’s divided difference interpolations – Newton’s forward and backward difference interpolation – Approximation of derivates using interpolation polynomials – Numerical single and double integrations using Trapezoidal and Simpson’s 1/3 rule.

     Single step methods: Taylor’s series method – Euler’s method – Modified Euler’s method – Fourth Order Runge-Kutta method for solving first order equations – Multi step methods: Milne’s predictor corrector methods for solving first order equations – Finite difference methods for solving second order equations.