# M.Sc. Courses

Course Structure

Semester- I | ||||

Sr. | Course Code | Course Description | Credits | L-T-P |
---|---|---|---|---|

1. | MAL411 | Topics in Real Analysis | 4 | 3-1-0 |

2. | MAL412 | Basic Linear Algebra | 4 | 3-1-0 |

3. | MAL413 | Introduction to Computing | 4 | 3-0-2 |

4. | MAL414 | Ordinary Differential Equation | 4 | 3-1-0 |

5. | MAL415 | Algebra | 4 | 3-1-0 |

Semester- II | ||||

Sr. | Course Code | Course Description | Credits | L-T-P |
---|---|---|---|---|

1. | MAL421 | Topics in Complex Analysis | 3 | 3-0-0 |

2. | MAL422 | Partial Differential Equation | 4 | 3-1-0 |

3. | MAL423 | Stochastic Processes | 4 | 3-1-0 |

4. | MAL424 | Numerical analysis | 4 | 3-0-2 |

5. | MAL425 | Topology | 3 | 3-0-0 |

6. | MALXXX | Elective – I |

Semester- III | ||||

Sr. | Course Code | Course Description | Credits | L-T-P |
---|---|---|---|---|

1. | MAL511 | Functional Analysis | 3 | 3-0-0 |

2. | MAL512 | Mathematical Methods | 4 | 3-1-0 |

3. | MAL513 | Optimization Techniques | 4 | 3-1-0 |

5. | MAS500 | Seminar | 2 | |

4. | MAT530 | Project – I | 3 | |

6. | MALXXX | Elective – II | ||

7. | MALXXX | Elective – III |

Semester- IV | ||||

Sr. | Course Code | Course Description | Credits | L-T-P |
---|---|---|---|---|

1. | MAT540 | Project – II | 12 | |

2. | MALXXX | Elective – IV | ||

3. | MALXXX | Elective – V |

### List of Courses at a Glance

##### MAL411 Topics in REAL ANALYSIS, 4 (3-1-0)

**Course contents :**

Metric spaces, completeness, connectedness, compactness, Heine-Borel theorem, totally bounded sets, finite intersection property, completeness of R^n, Banach fixed point theorem, perfect sets, the Cantor set.

Continuous functions, relation with connectedness and compactness, discontinuity, uniform continuous functions, monotone functions, Absolutely continuous functions, total variation and functions of bounded variations.

Differentiability and its properties, mean value theorem, Taylor's theorem, Riemann integral with properties and characterization, improper integral, Gamma function, Directional derivative, Partial derivative, Derivative as a linear transformation, Inverse and Implicit function theorems, multiple integration, Change of variables.

Sequence and series of real numbers, point wise convergence, Fejer's theorem, power series and Fourier series, uniform convergence and its relation with continuity, differentiability and inerrability, Weierstrass approximation theorem, Equi-continuous family, Arzela-Ascoli theorem.

##### MAL412 BASIC LINEAR ALGEBRA, 4 (3-1-0)

**Course contents:**

Vector spaces over fields, subspaces, bases and dimension; Systems of linear equations, matrices, rank, Gaussian elimination; Linear transformations, representation of linear transformations by matrices, rank-nullity theorem, change of basis, dual spaces, transposes of linear transformations; Determinants, Laplace expansions, cofactors, adjoint, Cramer's Rule; Eigen values and Eigen vectors, characteristic polynomials, minimal polynomials, Cayley-Hamilton Theorem, triangulation, diagonal-lization, rational canonical form, Jordan canonical form; Inner product spaces, Gram-Schmidt ortho-normalization, least square approximation, linear functionals and adjoints, Hermitian, self-adjoint, unitary and normal operators, Spectral Theorem for normal operators; Bilinear forms, symmetric and skew-symmetric bilinear forms, real quadratic forms, positive definiteness.

##### MAL413 INTRODUCTION TO COMPUTING, 4 (3-0-2)

**Course contents:**

Introduction: Computers as universal computing devise, bits, datatypes and operations, digital logic structure, The von Neumann model.

Programming: Problem solving, debugging, assembly language programming, Introduction to programming in C++, Variables and operators, control structures, pointers and arrays, functions and reference variables, Introduction to classes and templates, Developing classes for scientific applications: selected examples, Introduction to parallel processing using MPI.

##### MAL414 ORDINARY DIFFERENTIAL EQUATIONS, 4 (3-1-0)

**Course contents:**

Linear second and higher order differential equations, solutions of homogeneous and non-homogeneous equations, Method of variation of parameters.

Qualitative Properties of Solutions: Existence and uniqueness theorem, Oscillations and the Sturm Separation theorem, the Sturm Comparison theorem.

System of first order ODEs: Autonomous and non-autonomous system and stability.

Series solutions: Legendre equation and Legendre polynomials, Bessel equation and Bessel functions of first and second kinds.

Boundary Value Problems: Sturm-Liouville Boundary Value Problem, Green’s Function to solve boundary value problem.

##### MAL415 ALGEBRA, 4 (3-1-0)

**Course contents :**

Review of basics, Permutations, sign of a permutation, inversions, cycles and transpositions, groups, subgroups and factor groups, Lagrange's Theorem, homomorphism, normal subgroups, Quotients of groups, Cyclic groups, generators and relations, Cayley's Theorem, group actions, Sylow Theorems. Direct products, Structure Theorem for finite abelian groups. Simple groups and solvable groups, nilpotent groups; Free groups, free abelian groups. Rings, Examples (including polynomial rings, formal power series rings, matrix rings and group rings), ideals, prime and maximal ideals, rings of fractions, Chinese Remainder Theorem for pairwise comaximal ideals. Euclidean Domains, Principal Ideal Domains and Unique Factorizations Domains. Polynomial rings over UFD'; finite field and field extensions.

##### MAL421 Topics in COMPLEX ANALYSIS, 3 (3-0-0)

**Course contents:**

The complex number system. Extended complex plane. Analytic functions. Cauchy-Riemann conditions. Mappings by elementary functions. Conformal mappings and Mobius Transformation. Complex integration. Cauchy-Goursat theorem. Cauchy integral formula. The Homotopic version of Cauchy's theorem and simple connectivity. Morera’s and Liouvile’s theorems. Uniform convergence of sequences and series. Taylor's and Laurent's series. Singularities, zeros and Poles. Isolated singularities and residues. Cauchy residue theorem. Evaluation of real integrals. The Argument Principle and Rouche's theorem. Maximum Modulus Theorem.

##### MAL422 PARTIAL DIFFERENTIAL EQUATIONS, 4 (3-1-0)

**Course contents:**

Introduction to PDE. First order quasi-linear equations. Nonlinear equations. Cauchy-Kowalewski’s theorem. Higher order equations and characteristics. Classification of second order equations. Riemann’s method and applications. One dimensional wave equation and De’Alembert’s method. Solution of three dimensional wave equation. Method of decent and Duhamel’s principle. Solutions of equations in bounded domains and uniqueness of solutions. BVPs for Laplace’s and Poisson’s equations. Maximum principle and applications. Green’s functions and properties. Existence theorem by Perron’s method. Heat equation, Maximum principle. Uniqueness of solutions via energy method. Uniqueness of solutions of IVPs for heat conduction equation. Green’s function for heat equation.

##### MAL423 STOCHASTIC PROCESSES, 4 (3-1-0)

**Course contents:**

Introduction to probability theory, Probability and counting, Some applications.

Limit theorems: Probability spaces, random variables, independence, Kolmogorov’s 0 − 1 law, Borel-Cantelli lemma, Integration, Expectation, Variance, The weak law of large numbers, The probability distribution function, Convergence of random variables, The strong law of large numbers, Weak convergence, The central limit theorem, Markov operators, Characteristic functions.

Discrete Stochastic Processes: Conditional Expectation, Martingales, Doob’s convergence theorem, Doob’s decomposition of a stochastic process, L^(p) inequality, Random walks, A discrete Feynman-Kac formula, Markov processes.

Continuous Stochastic Processes: Brownian motion, Stopping times, Continuous time martingales, Recurrence of Brownian motion, Feynman-Kac formula revisited, The Ito integral for Brownian motion, Processes of bounded quadratic variation, The Ito integral for martingales, Stochastic differential equations.

##### MAL424 NUMERICAL ANALYSIS, 4 (3-0-2)

**Course contents:**

Definition and sources of errors, solutions of nonlinear equations; Bisection method, Newton's method and its variants, fixed point iterations, convergence analysis; Newton's method for non-linear systems; Finite differences, polynomial interpolation, Hermite interpolation, spline interpolation; Numerical integration - Trapezoidal and Simpson's rules, Gaussian quadrature, Richardson extrapolation; Initial value problems - Taylor series method, Euler and modified Euler methods, Runge-Kutta methods, multistep methods and stability; Boundary value problems - finite difference method, collocation method.

##### MAL425 TOPOLOGY, 3 (3-0-0)

Topological spaces, Basis for a topology, Limit points and closure of a set, Continuous and open maps, Homeomorphisms, Subspace topology, Product and quotient topology.

Connected and locally connected spaces, Path connectedness, Components and path components, Compact and locally compact spaces, One point compactification.

Countability axioms, Separation axioms, Urysohn’s Lemma, Urysohn’s metrization theorem, Tietze extension theorem, Tychonoff’s theorem, Completely Regular Spaces, Stone-Cech Compactification.

##### MAL511 FUNCTIONAL ANALYSIS, 3 (3-0-0)

**Course contents:**

Normed linear spaces, C0, C, lp, Lp, 1≤ p ≤ ∞, C[a,b], dimension, linear transformations -continuity and boundedness, linear functional-continuity, compactness of unit ball of finite dementional spaces, equivalence of norms and continuity of inear transformations of finite dimensional spaces, dual spaces duals of C0, lp, Lp, 1≤ p ≤ ∞, separability, non-separability of l∞. reflexive spaces. Horn-Banach theorem for real and complex normed linear spaces, Uniform Boundedness Principle and its applications. Closed Graph Theorem, Open Mapping Theorem and their applications. Inner product spaces, Hilbert spaces. Orthonormal basis, Projection theorem and Riesz Representation Theorem.

##### MAL512 MATHEMATICAL METHODS, 4 (3-1-0)

**Course contents:**

Concept and calculation of Green's function, Approximate Green's function, Green's function method for differential equations, Fourier Series, Generalized Fourier series, Fourier Cosine series, Fourier Sine series, Fourier integrals. Fourier transform, Laplace transform, Z-transform, Hankel transform, Mellin transform. Solution of differential equation by Laplace and Fourier transform methods.

##### MAL513 OPTIMIZATION TECHNIQUES, 4 (3-1-0)

**Course contents:**

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, revised 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, convex optimization.

Numerical optimization techniques: line search methods, gradient methods, Newton's method, conjugate direction methods, quasi-Newton methods, projected gradient methods, penalty methods.