# UG Courses

Submitted by admin on Thu, 12/29/2016 - 16:54## MA101 Calculus, 3 (3-1-0-5)

Single Variable Calculus: Limits and continuity of single variable functions, differentiation and applications of derivatives, Definite integrals, fundamental theorem of calculus, Applications to length, moments and center of mass, surfaces of revolutions, improper integrals, Sequences, series and their convergence, absolute and conditional convergence, power series. Taylor’s and Maclaurin's series.

Multi-variable Calculus: Functions of several variables-limits and continuity, partial derivatives, chain rule, gradient, directional derivatives, tangent planes, normals, extreme values, saddle points, Lagrange multipliers. Taylor’s formula. Double and triple integrals with applications, Jacobians, change of variables, line integrals, divergence, curl, conservative fields, Green’s theorem, surface integrals, Stokes’s Gauss Divergence theorem.

## MA102 Linear Algebra, Integral Transforms and Special Functions, 3 (3-1-0-5)

Linear Algebra : Vector spaces over R and C, Subspaces, Basis and Dimension, Matrices and determinants, Rank of a matrix, System of linear equations, Gauss elimination method, Linear transformations, Rank-nullity theorem, Change of basis, Eigen values, Eigen vectors, Diagonalization of a linear operator, Inner product spaces. Spectral theorem for real symmetric matrices, application to quadratic forms.

Integral Transforms: Laplace transforms of elementary functions, Inverse Laplace transforms and applications, Fourier series, Fourier transforms, Fourier cosine and sine integrals, Dirichlet integral, Inverse Fourier transforms, Special Functions: Gamma and Beta functions, Error functions.

## MA103 Differential Equations, 3 ( 3-1-0-5)

Ordinary Differential Equations: First Order Equation, Exact equations, integrating factors and Bernoulli equations. Lipschitz condition, examples on non-uniqueness. Second order differential equations with constant coefficients: homogeneous and non-homogeneous differential equations. Wronskian and linear independence of solutions, method of variation of parameters. Cauchy-Euler equations, method to second order equations with variable coefficients, Some applications, Solution of IVP using Laplace Transform and Euler’s Method. Series solutions, Frobenius method, Legendere and Bessel equations, orthogonal properties of Legendre polynomials.

Partial Differential Equations: Linear second order partial differential equations and their classification, heat equation, vibrating string, Laplace equation; method of separation of variables.

## MA104 Probability and Statistics, 3 (3-0-0-6)

Probability: Axioms of probability, conditional probability, independence of two or more events, Bayes’ theorem. Random variable, distribution functions, standard probability distributions and their properties, Simulation. Multiple random variables, marginal and conditional probability distribution, independence of random variables, bivariate normal and multinomial distributions. Functions of random variables, covariance and correlation. Conditional expectation, sum of random number of independent random variables. Convergence in probability, laws of large numbers and central limit theorem.

Statistics: Sample, population, sampling techniques, descriptive statistics, popular sampling distributions. Point estimation, parameter estimation with MLE, interval estimation, hypothesis testing. Ordinary least Squares (OLS) regression, assumptions and limitations of OLS, inference concerning regression parameters, other regressions. Analysis of variance.

## MA105 Probability and Stochastic Processes, 3 ( 3-1-0-5)

Probability: Axioms of probability, conditional probability, independence of two or more events, Bayes’ theorem. Random variable, distribution functions, standard probability distributions and their properties, Simulation. Multiple random variables, marginal and conditional probability distribution, independence of random variables, bivariate normal and multinomial distributions. Functions of random variables, covariance and correlation. Conditional expectation, sum of random number of independent random variables. Convergence in probability, laws of large numbers and central limit theorem.

Stochastic Processes: Introduction and motivation, classification of stochastic processes, Bernoulli process, Poisson process, Markov chains, single/multiple server queuing models, power spectral density.

## MAL451 Vector Field Theory, 2 (2-0-0)

Vector calculus, arc length, directional derivative, Differentiation and integration of vector valued functions, derivative of composite functions, vector equations:straightline, plane, space curves. Gradient, curl and divergence. Orthogonal Curvilinear coordinates, line, area and volume elements,expressions for gradient, curl and divergence. Line and double integrals, Green’s theorem, surface integrals, triple integrals, Stokes and divergence theorem swith applications. Conservative vector fields and path independence.

## MAL452 Complex Analysis, 3 (3-0-0)

Limit, continuity and differentiability of functions of a complex variable, analytic functions, Cauchy- Riemann equations. Definition of integral, Cauchy integral theorem, integral formula, derivatives of analytic functions, Morera’s and Liouvile’s theorems, maximum modulus principle. Poles and singularities, Taylor’s and Laurent series, isolated singular points, Cauchy residue theorem, evaluation of real integrals. Conformal and bilinear mappings.

## MAL453 Introduction to Functional Analysis, 3 (2-1-0)

Calculus of variations and applications. Normed linear spaces, Banach spaces, Hahn-Banach Theorem. Open mapping theorem, principle of uniform bounded, Hilbert Spaces. Orthogonal projections, self-adjoint, unitary and normal linear operators. Orthogonal bases,Parseval’s relation and Bassel’s inequality, Riesz representation theorem and Lax Milgram Theorem.

## MAL454 Modern Algebra, 3 (2-1-0)

Definition and example of groups, Lagrange theorem, cyclic groups, linear groups, permutation groups. Subgroups, normal subgroups and factor groups, Isomorphism theorems, Sylow theorems, and their applications. Rings and fields.

## MAL455 Operations Research, 3 (3-0-0)

Introduction to Optimization, Formulation of linear Optimization problems, Convex set, Linear Programming model, Graphical method, Simplex method, Finding a feasible basis – Big M and two phase Simplex method, Duality in Linear Program. Primal-dual relationship & economic interpretation of Duality. Dual Simplex Algorithm. Sensitivity analysis. Network analysis: Transportation & Assignment problem, Integer programming problem: Formulation, Branch & Bound and Cutting Plane methods, Dynamic Programming (DP); Non-linear Programming, Lagrange multipliers and Kuhn-Tucker conditions.

## MAL456 Fuzzy Logic And Applications, 3 (3-0-0)

Introduction: Information and Uncertainty, Classical/ Crisp set theory, Fuzzy set theory, Set theoretic operations: t-norm and t-conorm, Fuzzy relations, Fuzzy Arithmetic: Fuzzy number and fuzzy equations, Fuzzification and defuzzyfication, Propositional and predicate logic, Fuzzy rule base and approximate reasoning, Fuzzy logic, Applications , switching circuit and Boolean Algebra.