# UG Courses Mathematics

**MAL111 Mathematics Laboratory, 2 (1-0-2)**

Rank of a matrix, consistent linear system of equations, row reduced echelon matrices, inverse of a matrix, Gauss- Jordan method of finding an inverse of a matrix.

##### Math software Tools Practice Sheet No.1

Eigen values and eigen vectors, diagonalisation of matrices, Caley-Hamilton theorem.

##### Math software Tools Practice Sheet No.2

Hermitian, Unitary and Normal Matrices, bilinear and quadratic forms.

##### Math software Tools Practice Sheet No.3

Roots of a polynomial; numerical solution of a System of algebraic equations:Newton-Raphsonand iterative methods;interpolation:Lagrange interpolation formula, interpolation formula by use of differences.

##### Math software Tools Practice Sheet No.4

Numerical differentiation; numerical integration: Trapezoidal rule and Simson’s formula; error estimates in numerical differentiation and integration.

##### Math software Tools Practice Sheet No.5

Computer graphics:plotting of line, triangle and circle; plotting of cylinder, cube and sphere; projections; rotations.

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**MAL112 Advanced Calculus, 3 (2-1-0)**

Calculus of functions of several variables,implicit functions,partial derivatives and total differentials, equality of mixed derivatives of composite functions, Taylor’s Theorem, Maxima and Minima,constrained extrema, Lagrange multipliers.Definite integrals, differentiation under integral sign, differentiation of integral swith variable limits, improper integral, Beta and Gamma functions. Multiple integrals:definitions, properties and evaluation of multiple integrals, application of double integrals (in Cartesian and polar coordinates), change of coordinates, Jacobian, line integrals, Green’s theorem, proof, first and second forms. Solution of first order differential equations. Existence and uniqueness of solution, Picard’s method of successive approximations.

**MAL114 Linear Algebra, 3 (2-0-2)**

Vector spaces, bases and dimensions, linear transformations, matrix of linear transformations, change of bases, inner product space, Graham- Schmidt orthogonalization. Triangular form, matrix norms Conditioning of linear systems, Singular Value decomposition. Direct methods:Gauss, Cholesky and Householder’s methods. Matrix iterative methods: Jacobi, Gauss-Siedel and relaxation methods, conjugate gradient methods and its pre- conditioning. Computation of eigen values and eigen vectors: Jacobi, Givens, Householder, QR and inverse methods.

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**MAL115 Real Analysis, 2 (2-0-0)**

Product of sets, mappings and their compositions, Denumerable sets,upper and lower bounds, supremum and infimum. Metric spaces Definition and examples, open closed and bounded set, interior boundary, convergence and limit of a sequence. Cauchy sequence, completeness, Bolzano-Weierstrass theorem, continuity, intermediate value theorem, and uniform continuity, connectedness, compactness and separability. Integration Riemann sums, Riemann integral of a function, integrability of a function on a closed interval, mean value theorem, improper integrals. Fourier Series, Fourier Integrals and Fourier Transforms.

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**MAL116 Introduction to Ordinary Differential Equations, 3 (3-0-0)**

Second order differential equations with constant Coefficients homogeneous and non-homogeneous differential equations, method of undetermined coefficients, annihilation method, method of variation of parameters. Wronskian and linear independence of solutions, solution of ODE by Laplace transform. Second order equations with Variable coefficients, Euler equation, linearly Independent solutions, solution of second order Equation with one known solution, application of Variations of parameters method to second order Equations with variable coefficients, Series solutions, Frobenius method, Legendere and Bessel equations, orthogonal properties of Legendre polynomials. Higher order differential equations. Boundary Value Problems and Strum-Liouville Theory: Two point boundary value problems, Strum- Liouville boundary value problems, non- homogeneous boundary value problems; series of orthogonal functions, mean convergence.

**MAL213 Introduction to Probability Theory and Stochastic Processes, 3 (3-0-0)**

Axioms of probability, conditional probability, probability space, random variable, distribution functions, standard probability distribution functions. Multidimensional random variables, marginal and conditional probability distribution, independence of random variables, bivariate, normal and multinomial distributions. Functions of several random variables, expectation, moments and moment generation functions, correlation, moment inequalities. Conditional expectation and regression, random sums, convergencein probability, weak law of large number and central limit theorem. Markov chains and random processes: Markov and other stochastic processes, stationary distributions and limit theorem, reversibility, branching processes and birth- death processes, Markov chains Monte Carlo. Queues: Single-server queues, M/M1, M/G/1, G/M/1, and G/G/1 queues.

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**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.

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**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.

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**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, Convexset, 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 & Boundand 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.