Interdisciplinary Courses (Off Campus)

- Probability and Statistics
- Graph and Networks
- Optimization
- Engineering Mathematics I
- Discrete Structures for Computer Science
- Engineering Mathematics II

Courses List

**MATH F111 Mathematics I**

Functions and graphs; limit and continuity; applications of derivative and integral. Conics; polar coordinates; convergence of sequences and series. Maclaurin and Taylor series. Partial derivatives. Vector calculus in Rn; vector analysis; theorems of Green, Gauss and Stokes

**MATH F112 Mathematics II**

Complex numbers, analytic functions, Cauchy's theorems; elementary functions; series expansions; calculus of residues and applications. Vector space; basis and dimension; linear transformation; range and kernel of a linear transformation; row reduction method and its application to linear system of equations.

**MATH F113 Probability & Statistics**

Probability spaces; conditional probability and independence; random variables and probability distributions;marginal and conditional distributions; independent random variables; mathematical expectation; mean and variance; binomial, Poisson and normal distributions; sum of independent random variables; law of large numbers; central limit theorem (without proof);sampling distribution and test for mean using normal and student's t distribution; test of hypothesis; correlation and linear regression.

**MATH F211 Mathematics III**

Eigen-values and eigen-vectors. Inner product space and orthonormal bases. Elementary differential equations, Hypergeometric equations, Lengendre polynomials, Bessel functions; Fourier series; Sturm-Liouville problem, series solution for differential equation, systems of first order equations; Laplace transformation and application to differential equations; one dimensional wave equation, one dimensional heat equation & Laplace equation in rectangular form.

**MATH F212 Optimization**

Introduction to optimization; linear programming; simplex methods; duality and sensitivity analysis; transportation model and its variants; integer linear programming nonlinear programming; multi-objective optimization;evolutionary computation techniques.

**MATH F213 Discrete Mathematics**

Logic and methods of proof, Elementary Combinatorics, recurrence relations, Relations and digraphs, orderings, Boolean algebra and Boolean functions.

**MATH F214 Elementary Real Analysis**

Countability and uncountability of sets; real numbers; limits and continuity; compactness and connectedness in a metric space; Riemann integration; uniform convergence.

**MATH F215 Algebra-I**

Groups, subgroups, a counting principle, normal subgroups and quotient groups, Cayley’s theorem, automorphisms, permutation groups, and Sylow’s theorems. Rings, ring of real quaternions, ideals and quotient rings, homorphisms, Eculidean rings, polynomial rings, and polynomials over the rational field.

**MATH F241 Mathematical Methods**

Integral Transforms: Fourier, Fourier sine/cosine and their inverse transforms (properties, convolution theorem and application to solve differential equation), Discrete Fourier Series, Fast Fourier transform, Calculus of Variation: Introduction, Variational problem with functionals containing first order derivatives and Euler equations, Variational problem with moving boundaries. Integral equations: Classification of integral equations, Voltera equations, Fredholm equations, Greens functions.

**MATH F242 Operations Research**

Introduction to operations research; dynamic programming; network models - including CPM and PERT; probability distributions; inventory models; queuing systems; decision making- under certainty, risk, and uncertainty; game theory; simulation techniques, systems reliability.

**MATH F243 Graphs and Networks**

Basic concepts of graphs and digraphs behind electrical communication and other networks behind social, economic and empirical structures; connectivity, reachability and vulnerability; trees, tournaments and matroids; planarity; routing and matching problems; representations; various algorithms; applications.

**MATH F244 Measure and Integration**

Lebesgue measure and integration in real numbers, Convergence and Convergence theorems, absolutely continuous functions, differentiability and integrability, theory of square integrable functions, and abstract spaces.

**MATH F311 Introduction to Topology**

Metric Spaces; Topological Spaces - subspaces, Continuity and homoeomorphism, Quotient spaces and product spaces; separation Axioms; Urysohn’s Lemma and Tietze extension Theorem; Connectedness; Compactness, Tychonoff’s Theorem, Locally Compact Spaces; Homohtopy and the fundamental group.

**MATH F341 Introduction to Functional Analysis**

Banach spaces; fundamental theorems of functional analysis; Hilbert space; elementary operator theory; spectral theory for self-adjoint operators.

**MATH F342 Differential Geometry**

Curve in the plane and 3D-space; Curvature of curves; Surfaces in 3D-space; First Fundamental form; Curvature of Surfaces; Gaussian and mean Curvatures; Theorema Egreguim; Geodesics; Gauss-Bonnet Theorem.

**MATH F343 Partial Differential Equations**

Nonlinear equations of first order, Charpits Method, Method of Characteristics; Elliptic, parabolic and hyperbolic partial differential equations of order 2, maximum principle, Duhamel’s principle, Greens function, Laplace transform & Fourier transform technique, solutions satisfying given conditions, partial differential equations in engineering & science.

**MATH F312 Ordinary Differential Equations**

Existence and uniqueness theorems; properties of linear systems; behaviour of solutions of nth order equations; asymptotic behaviour of linear systems; stability of linear and weakly nonlinear systems; conditions for boundedness and the number of zeros of the nontrivial solutions of second order equations; stability by Liapunov's direct method; autonomous and non-autonomous systems.

**MATH F313 Numerical Analysis**

Solution of non-linear algebraic equation; interpolation and approximation; numerical differentiation and quadrature; solution of ordinary differential equations; systems of linear equations; matrix inversion; eigenvalue and eigenvector problems; round off and conditioning.

Common Courses

**MATH F111 Mathematics I**

Functions and graphs; limit and continuity; applications of derivative and integral. Conics; polar coordinates; convergence of sequences and series. Maclaurin and Taylor series. Partial derivatives. Vector calculus in Rn; vector analysis; theorems of Green, Gauss and Stokes

**MATH F112 Mathematics I**

Complex numbers, analytic functions, Cauchy's theorems; elementary functions; series expansions; calculus of residues and applications. Vector space; basis and dimension; linear transformation; range and kernel of a linear transformation; row reduction method and its application to linear system of equations.

**MATH F113 Probability & Statistics**

Probability spaces; conditional probability and independence; random variables and probability distributions;marginal and conditional distributions; independent random variables; mathematical expectation; mean and variance; binomial, Poisson and normal distributions; sum of independent random variables; law of large numbers; central limit theorem (without proof);sampling distribution and test for mean using normal and student's t distribution; test of hypothesis; correlation and linear regression.

**MATH F211 Mathematics III**

Eigen-values and eigen-vectors. Inner product space and orthonormal bases. Elementary differential equations, Hypergeometric equations, Lengendre polynomials, Bessel functions; Fourier series; Sturm-Liouville problem, series solution for differential equation, systems of first order equations; Laplace transformation and application to differential equations; one dimensional wave equation, one dimensional heat equation & Laplace equation in rectangular form.

Core Courses

**MATH F212 Optimization**

Introduction to optimization; linear programming; simplex methods; duality and sensitivity analysis; transportation model and its variants; integer linear programming nonlinear programming; multi-objective optimization;evolutionary computation techniques.

**MATH F213 Discrete Mathematics**

Logic and methods of proof, Elementary Combinatorics, recurrence relations, Relations and digraphs, orderings, Boolean algebra and Boolean functions.

**MATH F214 Elementary Real Analysis**

Countability and uncountability of sets; real numbers; limits and continuity; compactness and connectedness in a metric space;

Riemann integration; uniform convergence.

**MATH F215 Algebra-I**

Groups, subgroups, a counting principle, normal subgroups and quotient groups, Cayley’s theorem, automorphisms, permutation groups, and Sylow’s theorems. Rings, ring of real quaternions, ideals and quotient rings, homorphisms, Eculidean rings, polynomial rings, and polynomials over the rational field.

**MATH F241 Mathematical Methods**

Integral Transforms: Fourier, Fourier sine/cosine and their inverse transforms (properties, convolution theorem and application to solve differential equation), Discrete Fourier Series, Fast Fourier transform, Calculus of Variation: Introduction, Variational problem with functionals containing first order derivatives and Euler equations, Variational problem with moving boundaries. Integral equations: Classification of integral equations, Voltera equations, Fredholm equations, Greens functions.

**MATH F242 Operations Research**

Introduction to operations research; dynamic programming; network models - including CPM and PERT; probability distributions; inventory models; queuing systems; decision making- under certainty, risk, and uncertainty; game theory; simulation techniques, systems reliability.

**MATH F243 Graphs and Networks**

Basic concepts of graphs and digraphs behind electrical communication and other networks behind social, economic and empirical structures; connectivity, reachability and vulnerability; trees, tournaments and matroids; planarity; routing and matching problems; representations; various algorithms; applications.

**MATH F244 Measure and Integration**

Lebesgue measure and integration in real numbers, Convergence and Convergence theorems, absolutely continuous functions, differentiability and integrability, theory of square integrable functions, and abstract spaces.

**MATH F311 Introduction to Topology**

Metric Spaces; Topological Spaces - subspaces, Continuity and homoeomorphism, Quotient spaces and product spaces; separation Axioms; Urysohn’s Lemma and Tietze extension Theorem; Connectedness; Compactness, Tychonoff’s Theorem, Locally Compact Spaces; Homohtopy and the fundamental group.

**MATH F341 Introduction to Functional Analysis**

Banach spaces; fundamental theorems of functional analysis; Hilbert space; elementary operator theory; spectral theory for self-adjoint operators.

**MATH F342 Differential Geometry**

Curve in the plane and 3D-space; Curvature of curves; Surfaces in 3D-space; First Fundamental form; Curvature of Surfaces; Gaussian and mean Curvatures; Theorema Egreguim; Geodesics; Gauss-Bonnet Theorem.

**MATH F343 Partial Differential Equations**

Nonlinear equations of first order, Charpits Method, Method of Characteristics; Elliptic, parabolic and hyperbolic partial differential equations of order 2, maximum principle, Duhamel’s principle, Greens function, Laplace transform & Fourier transform technique, solutions satisfying given conditions, partial differential equations in engineering & science.

**MATH F312 Ordinary Differential Equations**

Existence and uniqueness theorems; properties of linear systems; behaviour of solutions of nth order equations; asymptotic behaviour of linear systems; stability of linear and weakly nonlinear systems; conditions for boundedness and the number of zeros of the nontrivial solutions of second order equations; stability by Liapunov's direct method; autonomous and non-autonomous systems.

**MATH F313 Numerical Analysis**

Solution of non-linear algebraic equation; interpolation and approximation; numerical differentiation and quadrature; solution of ordinary differential equations; systems of linear equations; matrix inversion; eigenvalue and eigenvector problems; round off and conditioning.

Discpline Elective Courses

**BITS F314 Game Theory and Its Applications**

Strategic thinking, Rational choice, Dominance, Rationalizability, Nash equilibrium, Best response functions, Duopoly models and Nash equilibrium therein, Electoral competition, Pure strategy, Mixed strategy, Extensive forms, Sub-game perfect Nash equilibrium, Bayesian Nash equilibrium, Select Applications of Game Theory.

**BITS F343 Fuzzy Logic and Applications **

Fuzzy sets, fuzzy binary relations; fuzzy logic, fuzzy reasoning; applications in decision making, control theory, expert systems, artificial intelligence etc.

**BITS F463 Cryptography **

Objectives of cryptography; ciphers – block and stream; mathematical foundations – modular arithmetic, finite fields, discrete logarithm, primality algorithms; RSA; digital signatures; interactive proofs; zero–knowledge proofs; probabilistic algorithms; pseudo-randomness.

**CS F211 Data Structures and Algorithms **

Introduction to Abstract Data Types, Data structures and Algorithms; Analysis of Algorithms – Time and Space Complexity, Complexity Notation, Solving Recurrence Relations.; Divide-and-Conquer as a Design Technique; Recursion – Recursive Data Types, Design of Recursive Functions / Procedures, Tail Recursion, Conversion of Recursive Functions to Iterative Form. Linear data structures – Lists, Access Restricted Lists (Stacks and Queues); Searching and Order Queries. Sorting VI-47 – Sorting Algorithms (Online vs. Offline, In-memory vs. External, In-space vs. Out-of-space, Quick Sort and Randomization), Lower Bound on Complexity of Sorting Algorithms. Unordered Collections: Hash tables (Separate Chaining vs. Open Addressing, Probing, Rehashing). Binary Trees – Tree Traversals. Partially Ordered Collections: Search Trees and Height Balanced Search Trees, Heaps and Priority Queues. Probabilistic/Randomized Data Structures (such as Bloom Filters and Splay Trees). Generalized Trees – Traversals and applications. Text Processing – Basic Algorithms and Data Structures (e.g. Tries, Huffman Coding, String search / pattern matching). External Memory Data structures (B-Trees and variants). Graphs and Graph Algorithms: Representation schemes, Problems on Directed Graphs (Reachability and Strong Connectivity, Traversals, Transitive Closure. Directed Acyclic Graphs - Topological Sorting), Problems on Weighted Graphs (Shortest Paths. Spanning Trees).

**CS F364 Design and Analysis of Algorithms**

Basic Design Techniques – Divide-and-Conquer, Greedy, Dynamic Programming (Examples, Analysis, General Structure of Solutions, Limitations and Applicability). Specialized Design Techniques: Network Flow, Randomization (Examples, Analysis, Limitations). Complexity Classes and Hardness of Problems – P, NP, Reductions, NP-hardness and NP-Completeness, Reduction Techniques, Basic NP-complete problems. Design Techniques for Hard Problems – Backtracking, Branch-and-Bound, and Approximation (General approaches and structure of solution, Analysis, and Limitations). Linear Programming – LP Problem and Simplex Algorithm, Approach for using LP for modeling and solving problems. Introduction to Design and Analysis of Parallel and Multi-threaded Algorithms.

**MATH F231 Number Theory**

Primes and factorization; division algorithm; congruences and modular arithmetic; Chinese remainder theorem Euler phifunction and primitive roots of unity; Gauss's quadratic reciprocity law; applications to periodic decimals and periodic continued fractions.

**MATH F314 Algebra-II**

Dual spaces, modules, fields, finite fields, extension of fields: algebraic extension, separable and inseparable extension, normal extension, sptitting fields, Galois extension, and Galois group. The algebra of linear transformations, characteristic roots and characteristic vectors, canonical forms: triangular form, nilpotent form, and Jordan form

**MATH F353 Statistical Inference and Applications**

Review of elements of probability and statistical methods, Classical Decision theory including parametric and non-parametric methods for testing of hypotheses, Analysis of Variance: One way and two way classifications, Design of experiments: Analysis of Completely randomized design, Randomized block design and Latin square design with one or more missing values, Statistical Quality control for variables and measurements.

**MATH F354 Complex Analysis**

A rigorous treatment of the theory of analytic functions of complex variables including Cauchy's theorems; maximum modulus theorem; the principles of argument; Jensen's formula; Mittag Lefler theorem; Weierstrass canonical products and analytic continuation

**MATH F378 Advanced Probability Theory**

Measure theoretic probability and probability space, Law of large numbers and independence, convergence, Central limit theorems, Higher dimensional limit theorems, Random walks and their properties, Martingale and their properties, Martingale convergence theorem, Radon-Nikodym derivative, Doob’s inequality, Backward martingales, Markov chain and their properties, finite state ergodicity, recurrence and transience.

**MATH F420 Mathematical Modeling**

Elementary mathematical models; Role of mathematics in problem solving; Concepts of mathematical modeling; Pitfalls in modelling; System approach; formulation, Analyses of models; Sensitivity analysis, Simulation approach. One or more of the interrelated topics will be covered from the following: optimal harvesting models, Environmental models, traffic flow models, bio-fluid flow models, socio-economic models, financial models, stochastic models, etc. providing a fertile ground for interdisciplinary courses. The selection of topics will depend upon the circumstances and mutual interest of the current students and faculty

**MATH F421 Combinatorial Mathematics**

Advanced theory of permutations and combinations; elementary counting functions; theory of partitions; theorems on choice including Ramsey's theorem; the mobius function; permutation groups; Polya's theorem and Debrauijn's generalisation; graphical enumeration problems.

**MATH F422 Numerical Methodology for Partial Differential Equations**

Classification of PDEs, nature of well-posed problems, interpretation of PDEs by characteristics and physical basis, appropriate boundary/initial conditions. Numerical methods for PDEs: Finite difference approximations to partial derivatives, Explicit and implicit schemes, Richardson Extrapolation Convergence, Stability and Consistency of Elliptic, Parabolic and Hyperbolic PDEs. Introduction to finite volume method, Computational experiments based on the algorithms

**MATH F423 Introduction to Algebraic Topology**

Homotopy; Fundamental group and Computation; Covering Spaces; Universal Covering Spaces; Simplicial Complexes; Simplicial Homology and Computation.

**MATH F424 Applied Stochastic Process**

Definition and examples of Stochastic Processes (SPs), classification of random processes according to state space and parameter space, types of SPs, elementary problems; Stationary Process: Weakly stationary and strongly stationary processes, moving average and autoregressive processes; Martingales: definition and examples of martingales; Markov Chains: Transition probability, classification of states and chains, stability of Markov chains, irreducibility, stationary distribution ergodic theorem; Continuous-time Markov Chains (CTMCs): Poisson process, birth-death process and their applications; Continuous time and continuous state space: Brownian motion, Wiener process and applications; Renewal processes in discrete and continuous time; Renewal reward process; Branching Processes; Galton-Watson branching process and its properties.

**MATH F431 Distribution Theory**

C-infinity functions, distributions and their derivatives; support, convolution and regularization; distributions of finite order; multiplication of distributions; Fourier transforms of distributions; temperate distributions and their Fourier transforms; fundamental solutions.

**MATH F432 Applied Statistical Methods**

Review of estimation and testing of hypotheses; Simple and multiple regression methodology through method of least squares, Multicollinearity and residual analysis, Categorical data handling through logistic regression; Multivariate data analysis by Hoteling ??, Mahalanobis ??, discriminant analysis, cluster analysis and factor analysis; Data handling and forecasting time series data by various components time series methodology; Statistical Quality Control of variables and attributes control charts; Non parametric data handling through Kruskal walls test, Mann Whitney and KS two sample test.

**MATH F441 Discrete Mathematical Structures**

One or more of the interrelated topics will be covered from the following: graphs, designs, codes, shift register sequences, groups, fields, Boolean algebras, analysis of algorithms, Fast Fourier Transform etc. providing a fertile ground for interaction between mathematics and modern areas of computer science. The selection of the topics will depend upon the circumstance and current interest of faculty. MATH

**F444 Numerical Solutions of Ordinary Differential Equations**

Introduction to ODEs, Numerical Techniques for One Step Methods, Convergence and Absolute Stability, Numerical techniques for Linear Multi-Step Methods, Zero Stability, Consistency, Convergence, Predictor-Corrector methods, Absolute Stability of Predictor-Corrector methods, Stiff ODEs and its numerical methods, Finite Difference Methods to Linear and Nonlinear Boundary Value Problems, Stability and Convergence Analysis, Differential Algebraic Equations, Numerical techniques for Differential Algebraic Equations, Introduction to One dimensional Finite Element Methods, Comparison VI-90 between Finite Difference Methods and Finite Element Methods, Variational formulation, Finite Element Approximation, Approximation Errors, Convergence of solution, Order of Convergence. MATH

**F445 Mathematical Fluid Dynamics**

Introduction to the Fluid Dynamics and Fundamental Concepts, Lagrange and Eulerian Descriptions, Continuum hypothesis, Conservation of Mass based on different approaches, Equation of Continuity in different Coordinates, Potential Flow, Laplace Equation, one-, two- and three-dimensional flow, Conservation of Linear Momentum, Euler's Equation, Bernoulli's equation, Constitutive equations for Newtonian Fluid, Navier-Stokes Equations, First Law of Thermodynamics, Reynolds number, Exact Solution of Navier-Stokes Equation, Boundary Layer Approximations, Setting up the Boundary-Layer Equations, Limit Equation For the Flat Plate, Discussion of Blasius' Equation, Description of Flow Past a Circular Cylinder , Decay of a Laminar Shear Layer

**MATH F456 Cosmology**

History of cosmological ideas, Observational overview of the universe, Expansion of the universe, Newtonian gravity, Friedman equation, the fluid and acceleration equations, Geometry of the universe, Infinite and observable universe, Big bang, Simple cosmological models, Hubble law, redshift, Observational parameters, the cosmological constant, the age of the universe, weighing the universe, dark matter, CMB, the early universe, Nucleosynthesis, Inflationary universe, Initial singularity, standard cosmological model, general relativistic cosmology, classic cosmology, neutrino cosmology, baryogensis, structure of the universe.

**MATH F471 Nonlinear Optimization**

Introduction; convexity and cones; Kuhun Tucker theory; unconstrained and constrained optimization; gradient methods; polynomial optimization; penalty function; generalized convex functions; duality in nonlinear programming; optimality criterion for generalised convex functions; fractional programming.

**MATH F481 Commutative Algebra**

Modules; direct sums and products; finitely generated modules, exact sequences; tensor product of modules; rings and modules of fractions; localization; Noetherian modules and primary decompositions; integral dependence and valuation theory; integrally discrete valuation rings and Dedekind domains; fractional ideals.