DEPARTMENT OF MATHEMATICS

NATIONAL INSTITUTE OF TECHNOLOGY WARANGAL 506 004

SCHEME OF INSTRUCTION AND SYLLABI OF

M.Sc. (APPLIED MATHEMATICS)

&

M.Sc. (MATHEMATICS & SCIENTIFIC COMPUTING)

(Effective from July, 2017 )

PROGRAM OBJECTIVES Master of Science (Applied Mathematics) PO1: To train the students for understanding and apply the basic concepts of sciences in real life

problems

PO2: To pursue successful careers as professional mathematicians working in industry and education

PO3: To motivate the students for research in mathematical modeling and analysis

PO4: To inspire the students who can work in frontline areas and to prepare them as well-trained

teaching faculty and scientists

Scheme of Instruction For M. Sc. (Applied Mathemat ics)

I – Semester

S.No Course code

Course Name L T P Cr

1 MA 5101 Real Analysis 3 1 0 4 2 MA 5102 Advanced Modern Algebra 3 1 0 4 3 MA 5103 Ordinary Differential Equations 3 1 0 4 4 MA 5104 Linear Algebra 3 1 0 4 5 MA 5105 Computer Programming & Data

Structures (CPDS) 3 1 0 4

6 MA 5106 Numerical Analysis 3 1 0 4 7 MA 5107 CPDS Lab 0 1 2 2

Total 18 7 2 26 II – Semester

SNo Course code

Course Name L T P Cr

1 MA 5151 Probability & Statistics 3 1 0 4 2 MA 5152 Partial Differential Equations 3 1 0 4 3 MA 5153 Integral & Discrete Transforms 3 1 0 4 4 MA 5154 Topology 3 1 0 4 5 MA 5155 Mechanics 3 1 0 4 6 ELECTIVE – I 3 0 0 3 7 MA 5156 Computational Laboratory 0 1 2 2 8 MA 5191 Seminar 0 0 2 1 Total 18 6 4 26

III – Semester SNo Course

code Course Name L T P Cr

1 MA 6101 Complex Analysis 3 1 0 4 2 MA 6102 Numer. Solution of Diff. Eqns (NSDE) 3 1 0 4 3 MA 6103 Mathematical Programming 3 1 0 4 4 MA 6104 Functional Analysis 3 1 0 4 5 ELECTIVE – II 3 0 0 3 6 ELECTIVE – III 3 0 0 3 7 MA 6105 Mathematical Programming Lab. 0 1 2 2 8 MA 6141 Seminar 0 0 2 1 Total 18 5 4 25

IV – Semester SNo Course

code Course Name L T P Cr

1 MA 6151 Finite Element Method (FEM) 3 1 0 4 2 ELECTIVE--IV 3 1 0 4 3 ELECTIVE – V 3 0 0 3 4 ELECTIVE – VI 3 0 0 3 5 MA 6152 NSDE and FEM Laboratory 0 1 2 2 6 MA 6192 Comprehensive Viva 0 0 0 2 7 MA 6199 Project Work 0 0 8 4 Total 12 3 10 21

Total number of credits for the course is 98

List of Course Titles for Electives – MSc (Applied Mathematics):

Course Code ELECTIVE - I

MA 5161 Integral Equations and Tensor Analysis

MA 5162 Mathematical Modeling

MA 5163 Differential Geometry

Course Code ELECTIVE – II AND III

MA 6111 Multivariate Data Analysis

MA 6112 Fluid Dynamics

MA 6113 Lie group Methods for Differential Equations

MA 6114 Finite Volume Method

MA 6115 Wavelet Analysis

MA 6116 Dynamical Systems

Course Code ELECTIVE – IV , V & VI

MA 6161 Measure and Integration

MA 6162 Heat and Mass Transfer

MA 6163 Perturbation Methods

MA 6164 Theory of Elasticity

MA 6165 Computational Fluid Dynamics

MA 6166 Bio-Fluid Mechanics

MA6167 Inventory, Queueing Theory & NLPP

MA6168 Hydrodynamic Stability

MA6169 Spectral Methods

PROGRAM OBJECTIVES

Master of Science (Mathematics and Scientific Computing)

PO1: To train the students for understanding and apply the basic concepts of scientific

computing

PO2: To pursue successful careers as professional mathematicians working in information

technology and business sectors

PO3: To train the students as computational scientists who can work on real life challenging

problems

PO4: To inspire the students who can work in frontline areas and to prepare them as well

trained teaching faculty and scientists

Scheme of Instruction for M. Sc. (Mathematics & Sci entific Computing)

I – Semester

SNo Course code

Course Name L T P Cr

1 MA 5101 Real Analysis 3 1 0 4 2 MA 5201 Discrete Mathematics 3 1 0 4 3 MA 5103 Ordinary Differential Equations 3 1 0 4 4 MA 5202 Numerical Linear Algebra 3 1 0 4 5 MA 5203 Problem Solving & Computer

Programming(PSCP) 3 1 0 4

6 MA 5204 Algorithmic Approach to Computational Methods 3 1 0 4 7 MA 5205 PSCP Lab 0 1 2 2

Total 18 7 2 26 II – Semester SNo Course

code Course Name L T P Cr

1 MA 5151 Probability & Statistics 3 1 0 4 2 MA 5152 Partial Differential Equations 3 1 0 4 3 MA 5251 Data Structures & Algorithms 3 1 0 4 4 MA 5252 Optimization Techniques 3 1 0 4 5 MA 5253 Object Oriented Programming (OOP) 3 1 0 4 6 ELECTIVE – I 3 0 0 3 7 MA 5254 OOP lab 0 1 2 2 8 MA 5291 Seminar 0 0 2 1 Total 18 6 4 26 III – Semester SNo Course

Code Course Name L T P Cr

1 MA 6101 Complex Analysis 3 1 0 4 2 MA 6201 Elementary Number Theory 3 1 0 4 3 MA 6202 Design and Analysis of Algorithms 3 1 0 4 4 MA 6203 Data Base Management Systems (DBMS) 3 1 0 4 5 ELECTIVE – II 3 0 0 3 6 ELECTIVE – III 3 0 0 3 7 MA 6204 DBMS Lab 0 1 2 2 8 MA 6241 Seminar 0 0 2 1 Total 18 5 4 25 IV – Semester

SNo Course code

Course Name L T P Cr

1 MA 6251 Java Programming 3 1 0 4 2 ELECTIVE—IV 3 1 0 4 3 ELECTIVE – V 3 0 0 3 4 ELECTIVE – VI 3 0 0 3 5 MA 6252 Java Programming Lab 0 1 2 2 6 MA 6292 Comprehensive Viva 0 0 0 2 7 MA 6299 Project Work 0 0 8 4 Total 12 3 10 22

Total number of credits for the course is 98

List of Course Titles for Electives – M.Sc(Mathemat ics & Scientific Computing ) :

Course Code ELECTIVE - I

MA 5261 Graph theory and Algorithms

MA 5262 Fuzzy Mathematics and Applications

MA 5263 Advanced Abstract Algebra

Course Code ELECTIVE – II AND III

MA 6211 Computer Graphics

MA 6112 Fluid Dynamics

MA 6212 Symbolic Computing

MA 6213 Elliptic Curves

MA 6214 Neural Networks

MA 6215 Parallel Computing

Course Code ELECTIVE – IV , V & VI

MA 6261 Theory of Automata

MA 6262 Approximation Theory

MA 6263 Financial Mathematics

MA 6264 Data Mining

MA 6265 Management Information Systems

MA 6266 Cryptography

MA6267 Advanced Optimization Techniques

MA6268 Data Analysis with R

MA6269 Mathematics of Data Science

Syllabus for the M.Sc. Courses MA 5101 – Real Analysis (Common to both the streams) After studying this course, the student will be able to

CO1: Find whether a given function can be Riemann integrable CO2: Test whether a given improper integral can be convergent CO3: Examine uniform convergence of given sequence and /or series of functions. CO4: Expand a given function into Fourier series. Basic Topology, finite sets, countable and uncountable sets - metric spaces - compact sets - perfect sets - connected sets. Riemann Stieltje’s integral: Definition and existence of the integral - Properties of the integral - integration and differentiation of integral with variable limits. Improper integrals: Definitions and their convergence - Tests of convergence, beta and gamma functions. Uniform convergence: Tests for uniform convergence - theorems on limit and continuity of sum functions - term by term differentiation and integration of series of functions. Power series, convergence and their properties. Fourier series: Dirichlets’ conditions - existence - problems - half range sine and cosine series. Reading :

1. Walter Rudin, Principles of Mathematical Analysis, 3rd Edition, McGraw Hill Book Co., 2013.

2. Brian S.Thomson, Andrew M.Bruckner, Judith B.Bruner, Real Analysis, Prentice Hall International, 2008.

3. William F Trench, Introduction to Real Analysis, Pearson, 2010 MA 5102 – Advanced Modern Algebra (for Applied Mathematics only) After studying this course, the student will be able to CO1: Analyze and understand group structure CO2: Analyze and understand ring structure CO3: Get introduced to extension of fields and understand their constructions CO4: Derive proofs that arise in various algebraic structures CO5: Apply the concepts and results to solve problems of Modern Algebra Group Theory: Review of basic Group theory including normal subgroups and homomorphisms; Group Actions; Cayley's Theorem; Class Equation; Automorphisms; Sylow theorems and applications; Direct Products; Finitely generated abelian groups Ring Theory: Review of basic Ring theory including ideals and ring homomorphisms; Properties of ideals; Field of quotients of an integral domain; Euclidean domains; Principal ideal domains; Unique factorization domains; Polynomial rings; Irreducibility criteria Field Theory: Introduction to fields, field extensions, algebraic closure and finite fields Reading :

1. David S. Dummit, Richard M. Foote, Abstract Algebra, Third Edition, John Wiley & Sons, 2004

2. I. N. Herstein, Topics in Algebra, Second Edition, John Wiley & Sons, 1975 3. Joseph A. Gallian, Contemporary Abstract Algebra, Eight Edition, Cengage Learning, 2013 4. S. Lang, Algebra, Revised Third Edition, Springer, 2002 5. John B. Fraleigh, A First Course in Abstract Algebra, Seventh Edition, Pearson, 2002

MA 5103 – Ordinary Differential Equations (Common to both the streams) After studying this course, the student will be able to

CO1: Determine linearly independent solutions and general solution of a non homogeneous differential equation. CO2: Find power series solution to a differential equation containing variable coefficients CO3: Discuss the existence and uniqueness of solution for an initial value problem CO4: Use Green's function to solve a non homogeneous boundary value problem First order differential equations - linear differential equations of higher order - linear dependence and Wronskian - Basic theory for linear equations - method of variation of parameters - linear equations with variable coefficients. Solution in power series - Legendre and Bessel equations - systems of differential equations - existence and uniqueness theorems - fundamental matrix - non-homogeneous linear systems - linear systems with constant coefficients and periodic coefficients - existence and uniqueness of solutions. Gronwall inequality - successive approximation - Picard’s theorem - nonuniqueness of solutions - continuous dependence on initial conditions - existence of solutions in the large. Boundary value Problems- Sturm-Liouville problem_ Green's functions Reading :

1. S.G.Deo and V. Raghavendra, Ordinary differential equations, Tata McGraw Hill Pub. Co., New Delhi,2006

2. M. Rama Mohana Rao, Ordinary differential equations - Theory and applications. Affiliated East West Press, New Delhi, 1981

3. E.A.Coddington, Introduction to Ordinary differential equations, Prentice Hall, 1995 MA 5104 – Linear Algebra (for Applied Mathematics only) After studying this course, the student will be able to

CO1. Test the consistency of system of linear algebraic equations. CO2. Verify rank nullity theorem for a given linear transformation. CO3. Find eigenvalues and canonical forms of a linear operator. CO4. Identify the importance of orthogonal property in the spectral theory. CO5. understand a bilinear form and its nature.

Systems of linear equations - matrices and elementary row operations - uniqueness of echelon forms - Moore-Penrose Generalized inverse. Vector spaces - subspaces - bases and dimension - coordinates - linear transformations and its algebra and representation by matrices - algebra of polynomials - determinant functions - permutation and uniqueness of determinants - additional properties - elementary canonical forms-characteristic values and vectors - Cayley Hamilton’s theorem - annihilating polynomial - invariant subspaces. Simultaneous triangularisation - simultaneous diagonalisation - Jordan form - inner product spaces - unitary and normal operators - bilinear forms.

Reading : 1. K.Hoffman and R.Kunze, Linear Algebra, Prentice Hall of India, New Delhi, 2003 2. Sheldon Axler, Linear algebra done right, , Springer nature, 3rd Edition, 2015 3. P.G. Bhattacharya, S.K. Jain and S.R. Nagpaul, First course in Linear Algebra, Wiley

Eastern Ltd., New Delhi, 1991 4. K.B.Datta, Matrix and Linear Algebra, Prentice Hall of India, New Delhi, 2006

MA 5105 – Computer Programming & Data Structures(CPDS) (for Applied Mathematics only) After studying this course, the student will be able to

CO1: Understand the use of input, output functions and structure of C program

CO2: Choose data types and structures to solve mathematical and scientific problem

CO3: Develop modular programs using control structures

CO4: Write programs to solve real world problems using arrays and functions

CO5: Implement sorting and searching algorithms

CO6: Develop and analyze algorithms for stacks and queues

Introduction: Keywords, identifiers, constants, variables, data types, type conversion, Types of operators and expressions, Input and output functions in C, Structure of C program, simple programs Decision and loop control Statements – IF-ELSE statement, break, continue, switch() case and nested IF statement, For loop, While loop, Do-while loop and nested loops. Arrays – Definition, Initialization, characteristics, One, Two, Three and Multidimensional Arrays Working with Strings & Standard Functions. Pointers – Introduction, features, Declaration, Arithmetic operations, pointers and Arrays, Array of pointers, pointers and strings. Functions – Declaration, Prototype, Types of functions, call by value and reference, Function with operators, Function with Arrays and Pointers. Files – Streams and file types, file operations, File I/O, Read, Write and Other file function. Structure and Union – Declaration, Initialization, structure within structure, Array of structure. Union Searching and Sorting: Sequentail search– Binary search – Selection sort - Bubble sort – Insertion sort, interchange sort, Shell sort, quick sort, merge sort, radix sort Stacks and queues- specification and implementing of stacks and queues Linked lists: Specification and implementation of lists, singly linked lists, doubly linked lists, circular list, Evaluation of a post fix expression, Convert an infix expression to post fix expression Reading: 1. Lipschitz, Programming in C (Scaum's Series), McGraw Hill, 2003. 2. Horowitz and Sahni, Fundamentals of Data Structures, Galgotia, New Delhi, 1995. 3. Tennenbam, Data Structures using C, Prentice Hall International, 2005. MA 5106 – Numerical Analysis (for Applied Mathematics only)

After studying this course, the student will be able to

CO1: Interpolate the given data and approximate the function by a polynomial CO2: Determine the numerical differentiation of a function CO3: Evaluate the Integrals numerically CO4: Solve Initial value problems numerically CO5: Find the roots of nonlinear equations Interpolation : Existence, Uniqueness of interpolating polynomial, error of interpolation - unequally spaced data; Lagrange’s, Newton’s divided difference formulae. Equally spaced data : finite difference operators and their properties, Gauss’s forward and backward formulae - Inverse interpolation - Hermite interpolation. Differentiation: Finite difference approximations for first and second order derivatives. Integration: Newton-cotes closed type methods; particular cases, error terms - Newton cotes open type methods - Romberg integration, Gaussian quadrature; Legendre, Chebyshev formulae. Solution of nonlinear and transcendental equations: Regula-Falsi, Newton-Raphson method, Chebyshev’s, method, Muller’s method, Birge-Vita method, solution of system of nonlinear equations. Approximation : Norms, Least square (using monomials and orthogonal polynomials), uniform and Chebyshev approximations Solution of linear algebraic system of equations: LU Decomposition, Gauss-Seidal methods; solution of tridiagonal system. Ill conditioned equations. Eigen values and Eigen vectors: Power and Jacobi methods. Solution of Ordinary differential equations: Initial value problems: Single step methods; Taylor’s, Euler’s, Runge-Kutta methods, error analysis; Multi-step methods: Adam-Bashforth, Nystorm’s, Adams- Moulton methods, Milne’s predictor-corrector methods. System of IVP’s and higher orders IVP’s. Reading :

1. MK Jain, SRK Iyengar and RK Jain, Numerical Methods for Engineers and Scientists, New Age International, 2008.

2. C.F.Gerald and P.O.Wheatley, Applied Numerical Analysis, Addison-Wesley, 1984. 3. K. Atkinson, Numerical Analysis, John Wiley, Singapore, 1978.

MA 5107 – CPDS Lab (for Applied Mathematics only) Programs to be written in C on 1. Branching and Loops 2. User defined functions and Library functions 3. Arrays and Pointers 4. Structures 5. Sorting 6. Searching 7. Linked Lists 8. Queue MA 5201 – Discrete Mathematics (for Mathematics & Scientific Computing only) After studying this course, the student will be able to

CO1: Apply Propositional logic and First order logic to solve problems

CO2: Determine if a logical argument is valid or invalid.

CO3: Construct induction proofs involving summations, inequalities, and divisibility

arguments.

CO4: Implement the principles of counting, permutations and combinatory to solve real

world problems

CO5: Formulate and solve recurrence relations

CO6: Prove whether a given relation is an equivalence relation/poset and will be able to draw

a Hasse diagram

CO7: Develop and analyze the concepts of Boolean algebra

Sets and propositions: Combinations of sets, Finite and Infinite sets, uncountable infinite sets, principle of inclusion and exclusion, mathematical induction. Propositions, fundamentals of logic, first order logic, ordered sets. Counting: Basics of counting, Pigeonhole principle- Permutations and combinations – Pascal’s Identity- Vandermonde’s Identity- Generalized Permutations and combinations. Generating functions, coefficients of generating functions – applications of generating functions Recurrence relations: Solving Recurrence Relations- Linear Homogeneous and Non-Homogeneous Recurrence relations.solution by the method of generating functions, sorting algorithm. Relations and functions: properties of binary relations, equivalence relations and partitions, partial and total ordering relations, Transitive closure and Warshal’s algorithm. Boolean algebra : Chains, Lattices and algebraic systems, principle of duality, basic properties of algebraic systems, distributive and complemented lattices, Boolean lattices and algebras, uniqueness of finite Boolean algebras, Boolean expressions and functions. Reading :

1. J.R.Mott, A.Kandel and Baker, Discrete Mathematics for Computer Scientists, PHI, 2006. 2. C.L.Liu, Elements of Discrete Mathematics, McGraw Hill, 1985. 3. J.P.Tremblay and R.Manohar, Discrete Mathematical Structures with applications to

Computer Science, McGraw Hill Book Co., 2004. MA 5202 – Numerical Linear Algebra (for Mathematics & Scientific Computing only) After studying this course, the student will be able to

CO1: Understand the basic concepts of linear algebra related to stability, accuracy, etc. CO2: Find QR factorization of a matrix using Householder transformation and study its applications CO3: Write various algorithms to solve system of linear equations & understand computational issues CO4: Describe the numerical procedure of eigenvalue problem CO5: Determine the SVD of a matrix and apply it to a real life problem Review of linear algebra concepts: Vector and Matrix norms and norm-preserving properties of

orthogonal matrices; Floating-point numbers and errors in computations.

Transformations in numerical linear algebra and their applications: Elementary matrices,

Householder transformation with applications to LU, QR factorizations and Hessenberg-reduction.

Orthonormal bases and orthogonal projections.

Numerical Solutions of Linear Systems: Direct methods; solutions via Gaussian eliminations and QR

factorizations with growth factors and stability; inverses and determinants; estimation of the

condition numbers; iterative methods: Jacobi and Gauss-Seidel methods and their convergence.

Least-squares Solutions to Linear Systems: existence and uniqueness, Normal equations; Pseudo

Inverse ; QR factorization methods for overdetermined systems.

Numerical Matrix Eigenvalue Problems: Gersgorin disk theorem, the Power method, the Inverse

Power methods & Rayleigh Quotient Iteration; Basic and Hessenberg QR iterations.

The Singular Value Decomposition: properties and applications of SVD

Reading:

1.Biswa Nath Datta, Numerical Linear Algebra and Applications, Second Edition, PHI, 2010.

2. Carl D. Meyer, Matrix Analysis and Applied Linear Algebra, SIAM, 2000.

3. David Kincaid and Ward Cheney, Numerical Analysis : Mathematics of Scientific Computing,

Brooks/Cole Publishing Company, California, 1991.

4. V. Sundarapandian, Numerical Linear Algebra, PHI, 2014.

MA 5203 – Problem Solving & Computer Programming(PSCP) (for Mathematics & Scientific computing only) After studying this course, the student will be able to

CO1: Develop algorithms for mathematical and scientific problems

CO2: Understand the components of computing systems

CO3: Choose data types and structures to solve mathematical and scientific problems

CO4: Develop modular programs using control structures

CO5: Able to understand the string operations

Problem solving techniques _ Algorithms_ Number representations Basics of C++: Tokens, Expressions and Operators, loops and controlling the loop execution, logic, bitwise and arithmetic operators Functions in C++ : Function Declarations, different methods of passing parameters and their purpose, Default Arguments, inline functions, Function overloading – Friend and virtual Functions Pointers, Arrays Pointers into Arrays, Constants, Pointer to Function, References Pointers to void Structures.

Strings : converting values of different types, strings: declarations, initializations, assignments, string as the example of an object: introducing methods and properties, namespaces: using and declaring, exception handling Reading

1. John R. Hubbard, Programming with C++, Second edition, McGraw-Hill, 2006 2. Stanley B. Lippman, Josee Lajoie, and Barbara E. Moo, C++ Primer (Fifth Edition) 3. Walter Savitch, Problem solving with C++ , Pearson, 2014

MA 5204 – Algorithmic Approach to Computational Methods (for Mathematics & Scientific computing only) After studying this course, the student will be able to

CO1: Understand representation of numbers in computer CO2 : Find the roots of nonlinear equations numerically CO3: Solve System of equations numerically CO4 : Interpolate the given data CO5: Evaluate integrals numerically Representation of integers and fractions, Fixed point and floating point arithmetic, error propagation, loss of significance, condition and instability, computational method of error propagation. Root finding: bisection method, secant method, regula-falsi method, Newton-Raphson method, LU decomposition, Gauss elimination with and without pivoting, Ill-conditioned equations, Gauss-Jacobi method, Gauss-Seidel method, Power method, Jacobi method to find eigenvalues. Interpolation: Lagrange's interpolation, Newton's divided difference interpolation (forward, backward), Newton-Gregory formulae, Sterling's formula. Hermite Interpolation Numerical integration: Newton-Cotes (closed type formulae)-trapezoidal rule, Simpson's 1/3 rule, Simpson's 3/8 rule. Gaussian Quadrature( Legndre and Chebyshev cases) Solution of Ordinary differential equations: Initial value problems: Single step methods; Taylor’s, Euler’s, Runge-Kutta methods, error analysis; Multi-step methods: Adam-Bashforth, Nystorm’s, Adams- Moulton methods, Milne’s predictor-corrector methods. System of IVP’s and higher order IVP’s. Reading :

1. V Rajaraman, Computer Oriented Numerical Methods, PHI, 1993. 2. M.K.Jain, SRK Iyengar and R.K Jain, Numerical Methods for Engineers and Scientists,

New Age International, 2008. MA 5205 – PSCP Lab (for Mathematics & Scientific computing only) Programs to be written in C++ on 1. Conditional Control constructs 2. Loops 3. User defined functions and Library functions 4. Arrays (Single and Multi-dimensional) 5. Pointers 6. Structures

7. String functions MA 5151 – Probability and Statistics (Common to both the streams) After studying this course, the student will be able to

CO1: Determine the mean, standard deviation and mth moment of a probability distribution CO2: Find a curve of best fit for given data CO3: Use the method of testing of hypothesis for examining the validity of a hypothesis CO4: Estimate the parameters of a population from knowledge of statistics of a sample CO5: Use the method of ANOVA for one factor experiments Random variable and sample space - notion of probability - axioms of probability - empirical approach to probability - conditional probability - independent events - Bayes’ Theorem - probability distributions with discrete and continuous random variables - joint probability mass function, marginal distribution function, joint density function. Mathematical expectation - moment generating function - Chebyshev’s inequality - weak law of large numbers - Bernoulli trials - the Binomial, negative binomial, geometric, Poisson, normal, rectangular, exponential, Gaussian, beta and gamma distributions and their moment generating functions - fit of a given theoretical model to an empirical data. Sampling and large sample tests - Introduction to testing of hypothesis - tests of significance for large samples - chi-square test - SQC - analysis of variance - t and F tests - theory of estimation - characteristics of estimation - minimum variance unbiased estimator - method of maximum likelihood estimation. Scatter diagram - linear and polynomial fitting by the method of least squares - linear correlation and linear regression - rank correlation - correlation of bivariate frequency distribution. Reading :

1. S.C.Gupta and V.K.Kapur, Fundamentals of Mathematical Statistics, S.Chand & Sons, New Delhi, 2008

2. V.K.Rohatgi and A.K.Md.Ehsanes Saleh, An Introduction to Probability theory and Mathematical Sciences, Wiley, 2001

3. Richard A Johnson, Probability and Statistics for Engineers, Pearson Education, 2005 MA 5152 – Partial Differential Equations (Common to both the streams) After studying this course, the student will be able to

CO1: Solve linear and nonlinear partial differential equations of both first and second order with different methods CO2: Introduce characteristic curves and characteristic strips CO3: Solve higher order partial differential equations CO4: Classify partial differential equations and transform into canonical form CO5: Apply partial differential equations to model and solve various physical problems Formulation - linear and quasi-linear first order partial differential equations - Paffian equation - condition for integrability - Lagrange’s method for linear equations. First order non-linear equations - method of Charpit - method of characteristics. Equations of higher order : Method of solution for the case of constant coefficients - equations of second order reduction to canonical forms - characteristic curves and the Cauchy problem- Riemann’s method for the solution of linear hyperbolic equations - Monge’s method for the solution of non-linear second order equations - method of solution by seperation of

variables. Laplace’s equation: Elementary solutions - boundary value problems - Green’s functions for Laplace’s equation - Solution using orthogonal functions. Wave equation: One dimensional equation and its solution in trigonometric series - Riemann-Volterra solution - Vibrating membrane. Diffusion equations: Elementary solution - solution in terms of orthogonal functions. Reading :

1. I. Sneddon, Elements of Partial Differential Equations, McGraw Hill, New York, 1984. 2. P.Prasad and R.Ravindran, Partial Differential Equations, Wiley Eastern, New Delhi, 1987

3. Tyn Myint-U & Lokenath Debnath: Linear Partial Differential Equations for Scientists and Engineers, Fourth Edition, Birkhauser, Bostan, 2007 4. T. Amaranath, An Elementary Course in Partial Differential Equations, Second Edition, US Edition published by Jones and Bartlett Learning, Massachusetts, 2009.

MA 5153 – Integral and Discrete Transforms (for Applied Mathematics only) After studying this course, the student will be able to

CO1: Understand the concepts of certain integral transforms CO2: Solve differential equations using Laplace transforms CO3: Find the solution of BVP's using Fourier transforms CO4: Solve finite difference equations by using Z transforms CO5: Establish the theoretical development of FFT algorithms Laplace Transform : Definition - Functions of exponential order and examples - Transforms of elementary, transcendental and special functions - transforms of derivatives and integrals and periodic function, unit step function and impulse function - The inverse transform - Convolution theorem - solution of differential equations by the use of the transform - Laplace inverse integral - Solution of Laplace equation (in two dimensions), one dimensional heat equation and one dimensional wave equation. Fourier transform : The Fourier transform, Inverse Fourier transform, Fourier transform properties, Convolution integral, convolution theorem, correlation, correlation theorem, Parseval’s theorem, Wave from sampling, sampling theorem, frequency sampling theorem. Z transform: Z transform, inverse Z transform, Z transform properties, solution of linear difference equations by using z-transform. Discrete Fourier Transform: Fourier transform of sequences, Discrete Fourier transform, transfer function. The Fast Fourier Transform: Intuitive Development, Theoretical development of Base 2, FFT algorithm. Reading :

1. R V Churchill, Operational Mathematics, McGraw Hill, 1972. 2. F. B. Hildebrand, Methods of Applied Mathematics, PHI, New Jercy, 1960. 3. E.O.Brigham, The Fast Fourier Transforms, Prentice Hall, New Jersey, 1988. 4. E.I.Jury, Theory and applications of Z transform method, John Wiley, 1964.

MA 5154 – Topology (for Applied Mathematics only) After studying this course, the student will be able to

CO1. Define various types of topological spaces. CO2. Understand the role of connectedness and compactness in analysis. CO3. Prove a classic result namely Uryshon Lemma. CO4. Apply topological properties in obtaining Uryshon metrization theorem and Tychnoff

theorem. CO5. Understand the notion of completeness and its importance in Baire Category theorem.

Metric Spaces: Definition, open and closed sets - convergence, completeness and Baires theorem continuous mappings - spaces of continuous function - Euclidian and unitary spaces. Topological spaces: Elementary concepts - open bases and open sub-bases - weak topologies. Compactness: Tychnoff’s theorem and locally compact spaces - compactness for metric spaces - Ascoli’s theorem. Separation: T1 spaces and Hausdorff spaces - completely regular spaces and normal spaces - Urysohn’s lemma and Tietze extension theorem - Urysohn imbedding theorem - Stone-Cech copactification. Connectness : Connected spaces - components of a space - Totally disconnected spaces - locally connected spaces. Reading :

1. G.F.Simmons, Introduction to Topology and Modern Analysis, McGraw Hill, Tokyo, 2004. 2. John Kelley, General Topology, Springer, 1991. 3. J.R. Munkres, Topology, a first course, Pearson Education, 2nd Edition, 2015.

MA 5155 – Mechanics (for Applied Mathematics only) After studying Mechanics, the student will be able to

CO1: Develop equations of motion for a system of particles

CO2: Analyse the motion of a rigid body under translation.

CO3: Analyse the motion of a rigid body under rotation about a fixed point.

CO4: Develop Lagranges and Hamiltonian equations for body.

Systems of particles: Linear and Angular momentum and rate of change of angular momentum of a system of particles with respect to the fixed and moving frames of reference. Effect of impulsive forces on the systems of particles. Rigid body: Moments of inertia; Kinetic energy and angular momentum of a rigid body rotating about a fixed point and about a fixed axis; general motion of a rigid body; motion of a rigid body parallel to a fixed plane under finite and impulsive forces; Euler’s motion under no forces; effects of earth’s rotation; generalized coordinates, velocities and forces; Eulerian angles; Lagrange’s and Hamilton’s equations of motion - elementary problems; motion of a top. Reading :

1. F. Chorlton, Text book on Dynamics, CBS Pubs, New Delhi, 1985. 2. J.L. Synge and B.A.Griffith, Principles of Mechanics, McGraw Hill, 1987. 3. G R Fowles and G L Cassiday, Analytical Mechanics, Cengage , 2004.

MA 5156 – Computational Laboratory (for Applied Mathematics only) Simple programs dealing with fundamental concepts in Fortran programs using conditional statements, do loops, subscripted variables, function subprograms and subroutines. Programs Based on Numerical methods using FORTRAN. Programs for solution of quadratic equation, Solution of algebraic and transcendental equations,

Gauss-Seidel method, Inverse of a matrix/Gaussian elimination, Tridiagonal system by Thomas algorithm etc., Numerical integration , Euler’s and modified Euler’s methods, Runge-Kutta methods. MA 5251 – Data Structures & Algorithms(for Mathematics & Scientific Computing only) After studying this course, the student will be able to

CO1: Identify data structures suitable to solve problems

CO2: Develop and analyze algorithms for stacks, queues

CO3: Implement sorting and searching algorithms

CO4: Implement linked lists using pointers

Stack: Definition, Array implementation of stack (static stack): Operations PUSH, POP, And TRAVERSE. Applications of Stack: Infix, Prefix, Postfix representation and conversion using stack, Postfix expression evaluation using stack, use of stack in recursion implementation. Queue: Definition, array implementation of queue (static queue): Operations INSERT, DELETE, TRAVERSE. Applications of queue, Comparisons of array, stack and queue data structures. Introduction to Circular queue, priority queue, Double ended queue, multiple queue. Pointers: Introduction, Pointers to structures, malloc, calloc functions. Linked list: Singly and Doubly Linear link lists, Singly and doubly circular linked list : Definitions, operations INSERT , DELETE, TRAVERSE on all these list. (Insertion operation includes – insertion before a given element, insertion after a given element, insertion at given position, insertion in sorted linked list) , Implementations of Stack and Queue using linked list (Dynamic stack). Applications of linked list: String representation & string operations like string length, string reverse, string comparison, string concatenation, string copying, convert upper-case to lower and vice-versa, substring using linked list. Polynomial representation and addition of two polynomials using linked list. Josphus problem, Searching using linked list, Sorting using linked list. Reading

1. S.Lipschutz , Data Structures (Schaum’s Outlines), TMH, 1987 2. Adam Drozdek Thomson, Data Structures and Algorithm in C++, Vikas ,2013 3. Aaron M. Tenenbaum, Yedidyah Langsam, Data Structures using C & C++ , PHI, 1996

MA 5252 – Optimization Techniques (for Mathematics & Scientific Computing only) After studying this course, the student will be able to

CO1: Understand Optimization models and apply them to real life problems

CO2: Find solution to Linear and Nonlinear Optimization models

CO3: Understand to search the Optimal solution

CO4: Understand the multistage problems and derive solution

Linear Programming: Formulation - Graphical solution - Simplex method, Big M method, Two Phase method - Duality theory - Dual simplex method - Revised simplex method. Transportation problem – Finding Initial BFS and Optimal Solution. Assignment problem – Hungarian Algorithm – Travelling Salesman Problem. Sequencing problem. Non-Linear programming: Convexity- Classical methods: Lagrange’s multipliers method - Kuhn-Tucker conditions, Quadratic forms, Quadratic programming- Wolfes’ Method One dimensional search methods- Sequential search, Fibonacci search, Golden section search, Multidimensional search methods - Uni-variate search, Gradient methods, Steepest descent/ascent

methods, Conjugate gradient method, Fletcher-Reeves method, Penalty function approaches. Dynamic Programming : Principle of optimality, Recursive relations, Solution of LPP, Simple examples Reading :

1. H.A.Taha , Operations Research, An Introduction, PHI, New Delhi, 2014 2. S.S Rao, Engineering Optimization Methods & Applications, New Age, New Delhi, 2010 3. Kanti Swarup, PK Gupta and Man Mohan Operations Research, Sultan Chand & Sons,

New Delhi, 2010 4. J.C.Pant, Introduction to Operations Research, Jain Brothers, New Delhi, 2012.

MA 5253 – Object Oriented Programming(OOP) (for Mathematics & Scientific Computing only) After studying this course, the student will be able to

CO1: Implement programs using classes and objects CO2: Able to understand the overloading concept CO3: Specify the forms of inheritance and use them in programs CO4: Analyze polymorphic behavior of objects Basic concepts of object oriented programming – Benefits of oops- Object oriented languages – Applications of oops. Classes and Objects-C++ Program with class-Nesting of member functions-private member functions-Arrays within a class- memory allocation for objects-Static data members-Arrays of objects-objects as Function arguments-Friendly functions, Returning objects. Constructors and Destructors- Multiple constructors in class-Constructors with default arguments-copy constructor-Dynamic constructors. Operator overloading-overloading unary operators-overloading binary operators-overloading binary operators using Friends-Rules for overloading operators – Type conversions Inheritance-Defining derived classes-Single inheritance-multilevel inheritance-Multiple inheritance-Hierarchical inheritance-Virtual base classes – Abstract classes. Pointers, Virtual functions and Polymorphism. Reading :

1. E. Balaguruswamy, Object oriented programming with C++, 4th Edition, Tata McGraw Hill, 2008.

2. Barkakati Nabajyoti, Object-Oriented programming in C++, PHI, 1991. 3. Stroustroup Bjarne. The C++ Programming Language, 2nd Edition, Addison-Wesley,1991.

MA 5254 – OOP Lab (for Mathematics & Scientific Computing only) Simple programs in C++ languages using pointers, pointers to arrays functions. Programs with class, objects as Function arguments, Friendly functions, Constructors and Destructors, Operator overloading-overloading unary operators-overloading binary operators. Programs illustrating the implementation of various forms of inheritance (Ex. Single Hierarchical, Multilevel inheritance Etc.), Programs, which illustrates the implementation of multiple inheritance.

ELECTIVE - I (for Applied Mathematics only)

MA 5161 – Integral Equations and Tensor Analysis After studying this course, the student will be able to

CO1: Know basic steps for the application of the integral equation approach for the numerical solution of boundary value problems for elliptic equations; CO2: Apply the integral equation method to concrete elliptic problems, to build own program products. CO3: Expose students to mathematical applications of tensor algebra to handle diverse problems which occur in real life situations. CO4: Competently use tensor algebra as a tool in the field of applied sciences and related fields

Integral equations: Classification of integral equations, connection with differential equations, integral equations of the convolution type, method of successive approximations, the resolvent, Fredholm theory, Laplace and Fourier transforms with applications to integral equations. Green’s functions: Non homogeneous boundary value problems, one dimensional Green’s function, generalized functions, Green’s function in higher dimensions, problems in unbounded regions. Tensor Analysis : N-dimensional space, covariant and contravariant vectors, contraction, second & higher order tensors, quotient law, fundamental tensor, associate tensor, angle between the vectors, principal directions, Christoffel symbols, covariant and intrinsic derivatives, geodesics. Reading :

1. Barry Spain, Tensor Calculus, Dover Publications, 2003. 2. I.N. Snedden, The use of Integral Transforms, Tata Mc-Grawhill, 1979 3. John W. Dettman, Mathematical methods in Physics & Engg., McGraw Hill, NYork, 1962.

MA 5162 – Mathematical Modelling After studying this course, the student will be able to

CO1. Understand the need of modeling and various aspects of modeling process CO2. Learn population models and epidemic models

CO3. Study models in pharmacokinetics

CO4. Develop models for blood flows and other biofluid flow models

CO5. Interpret the results of mathematical models with real life problems

Introduction – Microbial population models – Single species non-age-structured population models – age structured population models – two species population models – multispecies population models – optimal exploitation models – epidemic models – models in genetics – mathematical models in pharmacokinetics – models for blood flows – models for other biofluids – diffusion and diffusion reaction models – optimization models in biology and medicine. Reading :

1. J.N.Kapur, Mathematical models in Biology and Medicine, Affiliated East-West Pvt. Ltd., 2010

2. W.J.Meyer, Concepts of Mathematical Modelling, McGraw Hill, Tokyo, 1985. 3. Frank R Giordano, William P Fox, Steven B Horton and Maurice D Weir, Mathematical

Modelling: Principles and Applications, Cengage Learning, 2009

MA 5163 - Differential Geometry After studying this course, the student will be able to

CO1: Determine the directions of tangent, normal and binormal at point on the given curve CO2: Find the geodesc curve on a given surface CO3: Find surfaces of constant curvature. CO4: Form tensor quantities and find the corresponding metric tensors. Local curve theory: Serret-Frenet formulation, fundamental existence theorem of space curves. Plane curves and their global theory: Rotation index, convex curves, isoperimetric inequality, Four vertex theorem. Local surface theory: First fundamental form and arc length, normal curvature, geodesic curvature and Gauss formulae, Geodesics, parallel vector fields along a curve and parallelism, the second fundamental form and the Weingarten map, principal, Gaussian, mean and normal curvatures, Riemannian curvature and Gauss's theorem Egregium, isometrics and fundamental theorem of surfaces. Global theory of surfaces: Geodesic coordinate patches, Gauss-Bonnet formula and Euler characteristic, index of a vector field, surfaces of constant curvature. Elements of Riemannian geometry: Concept of manifold, tensors (algebraic and analytic), covariant differentiation, symmetric properties of curvature tensor, notion of connection, Christoffel symbols; Riemannian metric and its associated affine connection, geodesic and normal coordinates. Reading

1. Millman, R. S. and Parker, G. D., Elements of Differential Geometry, Prentice Hall Inc., 1977.

2. Laugwitz, D., Differential and Riemannian Geometry, Academic Press, 1965. 3. Kumaresan, S., A course in differential geometry and Lie groups, Texts and Readings in

Mathematics, 22, Hindustan Book Agency, New Delhi, 2002.

ELECTIVE - I (for Mathematics & Scientific Comput ing only) MA 5261 - Graph Theory and Algorithms After studying this course, the student will be able to

CO1: Understand the basic concepts in graphs and isomorphism CO2: Characterize Eulerian graphs and Hamiltonian cycles CO3: Find minimal spanning tree and shortest paths CO4: Learn matching in a graph and solve assignment problem CO5: Understand planar graphs and coloring of graphs CO6: Learn various algorithms in graph theory Preliminary Concepts: Graph definition, various kinds of graphs; Incidence matrix; Isomorphism; Decomposition; Special graphs; Paths, cycles and trails - connection in graphs, bipartite graphs, Eulerian Circuits; Vertex degree and counting; Hamiltonian Cycles - necessary and sufficient conditions; Review of digraphs. Trees: Trees and distance - properties; Spanning trees; Kruskal and Prim algorithms with proofs of correctness; Shortest paths - Dijkstra's algorithm, BFS and DFS algorithms, Application to Chinese postman problem; Trees in Computer science - rooted trees, binary trees, Huffman's Algorithm. Matchings: Matching in a graph and maximum matchings; Hall's matching theorem; Maximum

bipartite matching - Augmenting path algorithm; Weighted bipartite matching - Hungarian algorithm and solving the assignment problem; Tutte's theorem. Connectivity: Connectivity; Characterizing 2-connected graphs; Menger's theorem; Network flow problems - Ford-Fulkerson labeling algorithm, Max-flow Min-cut Theorem. Coloring: Chromatic number; Greedy coloring algorithm; Brooks' theorem; Graphs with large chromatic number; Turan's theorem.

Planar Graphs: Planar graphs; dual of a plane graph; Euler's formula; Kuratowski's Theorem;

Five Color Theorem; Four Colour Problem.

Reading : 1. Douglas B. West, Introduction to Graph Theory, Second Edition, Pearson Education, Inc., 2001 2. R. Diestel, Graph Theory, Fifth Edition, Springer, 2017 3. J. A. Bondy and U. S. R. Murty, Graph Theory, Springer, 2008 4. Narsingh Deo, Graph Theory with Applications to Engineering and Computer Science, Prentice-Hall, Inc., 1974

MA 5262 - Fuzzy Mathematics and Applications After studying this course, the student will be able to

CO1.Uunderstand the concept of crisp set and fuzzy set theory CO2. Apply set operations on fuzzy sets CO3. Solve problems related to propositional logic CO4. Apply the concept of fuzzy logic, fuzzy relations to solve simple problems CO5. Apply logic of Boolean algebra to switching circuits CO6. Understand the working of fuzzy logic controller through examples Crisp set theory (CST): Introduction, Relations between sets, Operations on sets, Characteristic functions, Cartesian products of crisp sets, crisp relations on sets. Fuzzy set theory (FST) : Introduction, concept of fuzzy set (FS), Relation between FS, operations on FS, properties of standard operations, certain numbers associated with a FS, certain crisp sets associated with FS, Certain FS associated with given FS, Extension principle. Propositional Logic (PL1) : Introduction, Syntax of PL1, Semantics of PL1, certain properties satisfied by connectives, inference rules, Derivation, Resolution. Predicate Logic (PL2) : Introduction, Syntax of PL2, Semantics of PL2, certain properties satisfied by connectives and quantifiers, inference rules, Derivation, Resolution Fuzzy Relations (FR): Introduction, Operations on FR, α-cuts of FR, Composition of FR, Projections of FR, Cylindrical extensions, Cylindrical closure, FR on a domain. Fuzzy Logic (FL): Introduction, Three-valued logics, N-valued logics and infinite valued logics, Fuzzy logics, Fuzzy propositions and their interpretations in terms of fuzzy sets, Fuzzy rules and their interpretations in terms of FR, fuzzy inference, More on fuzzy inference, Generalizations of FLSwitching functions (SF) and Switching circuits (SC): Introduction, SF, Disjunctive normal form, SC, Relation between SF and SC, Equivalence and simplification of circuits, Introduction of Boolean Algebra BA, Identification, Complete Disjunctive normal form. Applications: Introduction to fuzzy logic controller (FLC), Fuzzy expert systems, classical control theory versus fuzzy control, examples, working of FLC through examples, Details of FLC, Mathematical formulation of FLC, Introduction of fuzzy methods in decision making.

Reading :

1. G.J. Klir and B.Yuan, Fuzzy sets and Fuzzy Logic–Theory and Applications, PHI, 1997. 2. T.J.Ross , Fuzzy Logic with Engineering Applications, McGraw-Hill, 1995. 3. M. Ganesh, Introduction to Fuzzy Sets and Fuzzy Logic, PHI, 2001. 4. H. Zimmermann, Fuzzy Set Theory and its applications, Kluwer Academic, 1991

MA 5263 – Symbolic Computing After studying this course, the student will be able to

CO1: To introduce the fundamental concepts of Symbolic computing CO2: To provide a foundation to use basic building blocks of Mathematica and Matlab CO3: Learn to write Mathematica and Matlab Scripts. CO4: To explore various applications of Matlab in Mathematics CO5: To provide the basic knowledge to use Matlab for programming. Difference between Numeric computing and Symbolic computing, Introduction to Mathematica, Parts of Mathematica, Basics of programming in Mathematica, Built-in functions and constants, Numeric calculation using Mathematica, Symbolic computing with Mathematica, Buil-in function for Matrices and Linear Algebra, Solving equations, Calculus with Mathematica, Solving ordinary Differential equations with Mathematica, Graphics and built-in graphics functions, User defined functions, Conditionals and looping in Mathematica. Simple programs using Mathematica. Introduction to MATLAB, Advantages and disadvantages of MATLAB, MATLAB environment, MATLAB basics, Programming in MATLAB, Built-in functions, Application to Linear algebra, curve fitting and interpolation, numerical integration and solving Ordinary differential equations. Branching statements, loops and program design, User defined functions, Input and output functions, introduction to plotting, handling Graphics. Reading :

1. Paul R.Wellin, Mathematica, Wolfram Research Inc., 2005 2. Cleve Moler, Numerical Computing with MATLAB, SIAM, 2004

MA 6101 – Complex Analysis (Common to both the streams) After studying this course, the student will be able to

CO1. Introduce the analyticity of complex functions and study their applications CO2: Evaluate complex integrals and Expand complex functions. CO3. Determine and classify the zeros and singularities of the complex functions CO4. Evaluate improper integrals by residue theorem CO5. Learn the uniqueness of conformal transformation Complex variable - functions of a complex variable - continuity - differentiability - analytic functions - complex integration - Cauchy’s theorem. Cauchy’s integral formula - Morera’s theorem - Taylor’s theorem - Cauchy’s inequality - Liouville’s theorem - zeros of an analytic function - singularities. Laurent’s theorem - Residue - Cauchy’s residue theorem - contour integration - the fundamental theorem of algebra - Poisson’s integral formula. Analytic continuation - branches of a many-valued function - Riemann surface. The maximum modulus theorem - mean values of f(z) - Conformal representation - Bilinear transformation - Transformation by elementary functions - uniqueness of conformal transformation - representation of a polygon on a half plane - representation of any region on a circle.

Reading :

1. R.V. Churchill and Brown J.W, Complex variables and applications, McGraw Hill, Tokyo, 1974.

2. Copson, E.T., Theory of complex variables, Oxford University Press, New Delhi, 1974. 3. S. Ponnusamy & Herb Silverman: Complex Variables with Applications, Birkhauser, Boston,

2006. MA 6102 – Numerical Solution of Differential Equations (NSDE) (for Applied Mathematics only) After studying this course, the student will be able to

CO1: Solve the linear and non-linear initial value problems in ordinary differential equations using the explicit and implicit multistep methods CO2: Solve the two point boundary value problems in ordinary differential equations using the cubic splines CO3: Solve the heat equation, wave equation and the Laplace equation in one dimensional and 2- dimensional space using the finite difference methods CO4: Find the stability, convergence and the error analysis of the finite difference methods Ordinary Differential Equations: Multistep (Explicit and Implicit) Methods for Initial Value problems, Stability and convergence analysis, Linear and nonlinear boundary value problems, Quasilinearization. Shooting methods. Finite Difference Methods : Finite difference approximations for derivatives, boundary value problems with explicit boundary conditions, implicit boundary conditions, error analysis, stability analysis, convergence analysis. Cubic splines and their application for solving two point boundary value problems. Partial Differential Equations: Finite difference approximations for partial derivatives and finite difference schemes for Parabolic equations: Schmidt’s two level, multilevel explicit methods, Crank-Nicolson’s two level, multilevel implicit methods, Dirichlet’s problem, Neumann problem, mixed boundary value problem. Hyperbolic Equations: Explicit methods, implicit methods, one space dimension, two space dimensions, ADI methods. Elliptic equations: Laplace equation, Poisson equation, iterative schemes, Dirichlet’s problem, Neumann problem, mixed boundary value problem, ADI methods. Reading :

1. M.K.Jain, Numerical Solution of Differential Equations, Wiley Eastern, Delhi, 1985. 2. G.D.Smith, Numerical Solution of Partial Differential Equations, Oxford University Press,

2004. 3. M.K.Jain, S.R.K.Iyengar and R.K.Jain, Computational Methods for Partial Differential

Equations, Wiley Eastern, 2002. MA 6103 – Mathematical Programming (for Applied Mathematics only) After studying this course, the student will be able to

CO1: Formulation of a LPP and understand graphical solution CO2: Determine the solution of a LPP by simplex methods CO3: Application of post optimality analysis CO4: Solution of transportation and assignment problems CO5: Determine the solution of ILPP CO6: Determine an optimal solution by dynamic programming Linear Programming : Lines and hyperplanes - convex sets, convex hull - Formulation of a Linear

Programming Problem - Theorems dealing with vertices of feasible regions and optimality - Graphical solution - Simplex method ( including Big M method and two phase method) - Dual problem - duality theory - dual simplex method - sensitivity analysis - revised simplex method - parametric programming . Transportation problem - existence of solution - degeneracy - MODI method (including the theory). Integer Programming: Gomory’s cutting plane method for an integer linear programming problem and a mixed integer linear programming problem - Assignment problem - travelling salesman problem. Dynamic programming: Multistage decision process - concept of sub optimization - principle of optimality - computational procedure in dynamic programming -Application to problems involving discrete variables, continuous variables and constraints involving equations and inequations - application to linear programming problem. Reading :

1. H.A.Taha , Operations Research, An Introduction, PHI, 2008 2. N.S. Kambo, Mathematical Programming Techniques, East-West Pub., Delhi, 1991 3. Kanti Swarup et. al., Operations Research, Sultan Chand and Co., 2006 4. J.C.Pant, Introduction to Operations Research, Jain Brothers, 2008.

MA 6104 – Functional Analysis (for Applied Mathematics only) After studying this course, the student will be able to

CO1. Define various examples for normed linear spaces. CO2. Understand the nature of Banach spaces and Hilbert spaces. CO3. Prove the open mapping theorem and closed graph theorem. CO4. Apply results of this course in solving operator equations.

Algebraic systems: Groups, rings, structure of rings, linear spaces, dimension of a linear space, linear transformation algebras. Banach spaces: Definition and some examples - continuous linear transformation - Hahn-Banach theorem - Natural imbedding of N in N** - the open mapping theorem - the conjugate of an operator. Hilbert spaces: Definition and some simple properties - Orthogonal complements - orthonormal sets - conjugate space H*, adjoint of an operator - self-adjoint operators - Normal and unitary operators - projections. Reading :

1. G.F.Simmons, Introduction to Topology and Modern Analysis, McGraw Hill, Tokyo, 2004. 2. E. Kreyszig, Introductory functional analysis with applications, Wiley, New York, 2007. 3. I.J.Maddox, Elements of Functional Analysis, Cambridge Univ. Press, New Delhi, 1992.

MA 6105 – Mathematical Programming Laboratory (for Applied Mathematics only) Programs based on Mathematical Programming using C language. Simple programs dealing with fundamentals of C language for Simplex method, Two phase method, Big-M method, Revised simplex method, Transportation algorithm, Dual simplex method , Assignment problem MA 6201 – Elementary Number Theory (for Mathematics & Scientific Computing only)

After studying this course, the student will be able to

CO1: Understand the distribution of prime numbers CO2: Understand concepts of congruences, quadratic residues and solving equations involving them CO3: Learn number theoretic functions and their properties CO4: Solve certain types of Diophantine equations CO5: Know Farey fractions, continued fractions and multiplicative characters Divisibility - primes; fundamental theorem of arithmetic; prime number theorem Theory of congruences - Chinese remainder theorem; prime power moduli; prime modulus; primitive roots and power residues; Fermat's theorems Quadratic residues - Legendre and Jacobi symbols; quadratic reciprocity; binary quadratic forms Number theoretic functions - greatest integer function; arithmetic functions; Mobius inversion formula

Diophantine equations - basic type ax + by = c; simultaneous linear equations; Pythagorean

triangles; assorted problems Introduction to Farey fractions, continued fractions, Pell's equation, multiplicative characters. Reading :

1. I. Niven, H. S. Zuckerman and H. L. Montgomery, An Introduction to the Theory of Numbers, Fifty Edition, John Wiley & Sons, 1991

2. Kenneth Ireland, Michael Rosen, A Classical Introduction to Modern Number Theory, Second Edition, Springer, 1990

3. David M. Burton, Elementary Number Theory, Seventh Edition, McGraw-Hill, 2011

MA 6202 – Design & Analysis of Algorithms (for Mathematics & Scientific Computing only) After studying this course, the student will be able to

CO1.Uunderstand the basic concepts of algorithms and analysis CO2.Aanalyze time and space complexity CO3.Uunderstand algorithm design methodology CO4. Apply important algorithm methodology to solve problems CO5. Understand the difference between P and NP classes of problems Analyzing Algorithms : Concepts in Algorithms Analysis – asymptotic complexity of algorithms – Growth functions Recurrences. Master Theorem. Divide and Conquer Method: Binary Search, Quick Sort, Expected Running Time of Randomized Quick Sort, Merge Sort, Strassen’s Matrix Multiplication Algorithm, Data Structures for Set manipulation problems: Binary tree traversal algorithms, disjoint-set union algorithms Graph Algorithms : Representations of graphs – Breadth-first search – Depth-first search – Minimum spanning tree – The algorithms of Kruskal and Prim – Shortest paths – Dijkstra’s algorithm Greedy Method: Activity Selection Problem, Knapsack Problem, single source shortest path problem Dynamic Programming: Solution to 0-1 Knapsack Problem, multistage graphs, TSP using Dynamic Programming Backtracking: Basic examples, N-Queen’s Problem, sum of subsets problem

Complexity Classes: Example NP-complete problems Reading : 1. Cormen T.H., Leiserson C.E., Rivest R.L. and C. Stein, Introduction to Algorithms, 3rd Edition, PHI, New Delhi, 2004. 2. Horowitz E., Sahni S. and Rajasekaran S., Fundamentals of Computer Algorithms, Second Edition, Universities Press, 2011. 3. Aho A V, Hopcroft J E, and Ullman J.D., The Design and Analysis of Computer Algorithms, Pearson, 10 th Impression, New Delhi, 2012. 4. Baase S. and Gelder A.V., Computer Algorithms: Introduction to Design and Analysis, 3rd Edition, Addison and Wesley, 2000. 5. Levitin A., Introduction to the Design and Analysis of Algorithms, 2nd Impression, Pearson Education, New Delhi, 2009. MA 6203 – Data Base Management Systems (for Mathematics & Scientific Computing only) After studying this course, the student will be able to

CO1: Understand functional components of the DBMS CO2: Devise queries using Relational Algebra, Relational Calculus and SQL CO3: Design database schema CO4: Develop ER model CO5: Understand transaction processing, concurrency control and security Introduction, Purpose of Data base systems, Data models, Data Independence, Database manager, Database Administrator. Entity - Relation Model: Entities, Entities and relationships, Mapping constraints, E- R diagrams, Generalization and aggregation, Design of E - R database scheme. Relational Model: Structure of relational database, the relational algebra, the tuple relational calculus, the domain relational calculus, Modifying the database, Views. Relational Commercial languages : SQL, Query by example, Quel Integrity Constraints : Domain constraints, referential integrity, Functional dependencies, Assertions and Triggers Relational Database Design : Pitfalls in relational database design, First, Second and Third, Fourth and fifth normal forms Query Processing : Query interpretation, Equivalence of expressions, Estimation of query processing cost, Join strategies, Structure of query optimizer Transaction Processing: Strage Model, Recovery form transaction failure, Dead lock handling, High performance transaction systems, long duration transactions, Extended relational systems and Distributed databases. Security and Integrity: L Security and integrity violations, Authorization and Views, Security specification in SQL. Reading :

1. A.Silverschatz, H.F. Korth, and S. Sudarshan, Database System concepts, McGraw Hill, New York, 1997

2. Jeffery D. Ullman, Principles of Database Systems, Galgotia, 1984 3. Ramkrishnan R and J Gehrke, Database Management Systems, Third edition, McGraw

Hill,,2003

MA 6204 – DBMS Lab (for Mathematics & Scientific Computing only) DDL, DML, DCL Statements Built in functions and Aggregate functions SQL : Ordinary Query, Sub Query, Correlated Sub Query PL/SQL, Data types, Control Structures, Error handling mechanism, Subprograms (procedures and functions), Stored procedures, Data base triggers and exception RDBMS: Building forms using form designers, triggers, all types of triggers, menus and Reports. MA 6141/6241 – Seminar (Common to both the streams) Every student has to give a seminar and a committee of staff members from the department shall evaluate the same. The DAC (PG & R) of the Department shall constitute this committee.

ELECTIVES II AND III (for Applied Mathematics only) MA 6111 - Multivariate Data Analysis After studying this course, the student will be able to

CO1: Analyze Multivariate Distributions and their characteristics CO2: Perform MANOVA CO3: Analyze Conjoint analysis CO4: Analyze Cluster analysis, Multidimensional scaling Multivariate analysis of variance – Differences between MANOVA and discriminant analysis – A hypothetical illustration of MANOVA – A decision process for MANOVA – conjoint analysis – comparing conjoint analysis with other multivariate methods – designing a conjoint analysis experiment – managerial applications of conjoint analysis – alternate conjoint methodologies – an illustration of conjoint analysis – canonical correlation analysis – analyzing relationships with canonical correlation – interpreting the canonical variate – validation and diagnosis – cluster analysis – cluster analysis decision process – multidimensional scaling – comparing MDS to other interdependence techniques – a decision framework for perceptual mapping – correspondence analysis. Reading :

1. Joseph F.Hair, Rolph E. Anderson , Multivariate data analysis, Pearson Education, 2003. 2. M.G.Kendall, A course in Multivariate Analysis, Charles Griffith, 1968.

MA 6112 - Fluid Dynamics After studying this course, the student will be able to

CO1: Find stream lines and path lines of a fluid flow

CO2: Find complex velocity potential for an incompressible and irrotational flow.

CO3: Set up equations of motion with boundary conditions for a given fluid flown problem and

solve them for the velocity field

CO4: Analyse the flow in ducts of different cross sections and estimate the volumetric flow rate.

Kinematics of fluids in motion: Real fluids and ideal fluids – velocity of a fluid at a point – stream lines and path lines – steady and unsteady flows – the velocity potential – the velocity vector – local and particle rates of change – the equation of continuity – acceleration of fluid – conditions at a rigid boundary. Equations of motion of fluid: Euler’s equations of motion – Bernoulli’s equation – some flows

involving axial symmetry – some special two-dimensional flows. Some three dimensional flows: Introduction – sources, sinks and doublets – axisymmetric flows – Stokes’ stream function. The Milne-Thomson circle theorem – the theorem of Blasius – applications. Viscous flows: Stress analysis in fluid motion – relations between stress and rate of strain – the coefficient of viscosity and laminar flow – the Navier-Stokes’ equations of motion of viscous fluid – steady motion between parallel planes, through tube of uniform cross section and flow between concentric rotating cylinders. Steady viscous flow in tubes of uniform cross section – a uniqueness theorem – tube having uniform elliptic cross section – tube having equilateral triangular cross section – steady flow past a fixed sphere. Reading :

1. Frank Chorlton, Fluid Dynamics, CBS Publishers, Delhi, 2004 2. L.M.Milne Thomson, Theoretical Hydrodynamics, Dover Publications, 1960

MA 6113-- Lie group Methods for Differential Equations After studying this course, the student will be able to

CO1: Show competence in the field of ordinary and partial differential equations.

CO2: Show analytic skills and working knowledge in Lie’s integration methods.

CO3: Solve linear and non-linear differential equations.

CO4: Reduce a vast amount of nonlinear second-order ordinary equations used in applications to four

canonical forms and integrate them

CO5: Know the terminology in group analysis of differential equations.

Introduction to Lie group analysis: Lie group of transformations - Groups; Group of transformations; One parameter Lie groups of transformations - Infinitesimal transformations: First order theorem of Lie; Infinitesimal generators; Invariant functions; Canonical coordinates; Invariants of points, curves and surfaces - Extended infinitesimal - Extended transformations (Prolongations) - Symmetry reductions - Multi parameter Lie groups of transformations. Group analysis of ordinary differential equations: Invariance of ordinary differential equations - Prolongation techniques - Calculation of Lie symmetry groups - Differential equations admitting a given group - Invariant solutions - Group classification for ordinary differential equations - Symmetry analysis for systems of ordinary differential equations. Group analysis of partial differential equations: Invariance of partial differential equations - Prolongation formulae - Determining equations - Infinitesimal of partial differential equations - Invariant solutions - Group classification for partial differential equations - Lie symmetries for systems of partial differential equations. Reading

1. Bluman, G. W. and Kumei, S., Symmetries and Differential Equations, Springer-Verlag, Heidelberg, Berlin, 1989.

2 Ovsiannikov, L.V., Group Analysis of Differential Equations, Academic Press, New York, 1982 (Moscow, Nauka, 1978, in Russian). 3. P. Olver, Applications of Lie groups to Differential Equations, 2nd Edition, GTM 107, Springer-Verlag, Berlin, 1993. 4. Bluman, G. W. and Cole, J. D., Similarity Methods for Differential Equations, Applied

Mathematical Sciences, Vol. 13. Springer-Verlag, New York-Heidelberg, 1974. 5. Hydon, P.E. Symmetry methods for differential equations: a beginner’s guide, Cambridge University Press, 2000. 6. Bluman, G & Anco, S, Symmetry and Integration Methods for Differential Equations, Springer, New York, 2002. 7. Bluman, G. W., Cheviakov, A. F. and Anco, S. C., Applications of Symmetry Methods to Partial Differential Equations, Applied Mathematical Sciences, 168, Springer, 2010.

MA 6114 - Finite Volume Method After studying this course, the student will be able to

CO1: Derive the Conservation Equations CO2: Discretize linear partial differential equations (PDE) CO3: Identify source terms in PDE and their linearization CO4: Solve diffusive problems: Steady and unsteady 1D, Steady 2D and 3D problems CO5: Solve convective problems using upwind, QUICK and hybrid schemes CO6: Solve the velocity and pressure coupling using SIMPLE, SIMPLER, SIMPLEC algorithms Introduction - Obtaining the Integral Form from the Differential Form - Finite Volume Meshes - Discretising the Semi-Integral Equation – Implementation Finite Volume Schemes - FVM on a Cartesian Mesh - Finite Volume Schemes in 1D and 3D - Time Step Calculation for a Finite Volume Scheme - Finite Volume FOU 2D Scheme - Boundary Conditions - Coding a Finite Volume Solver Derivation of Equations - Conservation Laws - Control Volume Approach - Deriving the Integral Form of the 2D Linear Advection Equation - Further Finite Volume Schemes - Linear Interpolation - Quadratic Interpolation - Converting from Finite Difference to Finite Volume Systems of Equations - The Shallow Water Equations - General FVS for the SWE - FVS for the 2D SWE on a Structured Mesh - Heuristic Time Step for a 2D SWE FVS Reading :

1. D.M.Causon, C.G.Mingham, & L.Own: Introductory Finite Volume Methods for Partial Differential Equations, Springer, 2009.

2. H.Versteeg & W.Malalasekera, An introduction to CFD: The Finite Volume Method, Pearson, 2009.

MA 6115 - Wavelet Analysis After studying this course, the student will be able to

CO1: Expand a function in Haar wavelets CO2: Construct Meyer wavelets to a given function CO3: Find Daubechies wavelet series to a given function CO4: Analyze two or more dimensional problems using wavelets Heuristic treatment of the wavelet transform – wavelet transform – Haar wavelet expansion : Haar functions and Haar series, Haar sums and Dyadic projections, completeness of the Haar functions, Haar series in C0 and Lp spaces, pointwise convergence of Haar series, construction of standard Brownian motion, Haar function representation of Brownian motion – Multiresolution analysis : Orthogonal systems, scaling functions, from scaling function to MRA, Meyer wavelets, from scaling function to orthonormal wavelet – Wavelets with compact support : from scaling filter to scaling function, explicit representation of compact wavelets, Daubechies recipe, Hernandez-Weiss recipe, smoothness of wavelets - convergence properties of wavelet expansions: wavelet series in Lp spaces,

large scale analysis, almost everywhere convergence, convergence at a pre-assigned point – Wavelets in several variables: tensor product of wavelets, general formulation of MRA and wavelets in Rd, Examples of wavelets in Rd. Reading :

1. Mark A. Pinsky, Introduction to Fourier analysis and wavelets, Cenage Learning India Pvt. Ltd, 2002.

2. M.V.Altaisky , Wavelets, Theory, Applications Implementation, University Press, 2009

MA 6116 - Dynamical Systems After studying this course, the student will be able to

CO1: A one & 2-dimensional systems of ODE, find the fixed points, determine the linearisation of the system about such solutions and discuss their stability; CO2: Understand the distinction between hyperbolic and non-hyperbolic fixed points; CO3: Sketch a phase portrait of linear and nonlinear one- and two-dimensional systems of ODEs; CO4: Investigate bifurcations of one-dimensional dynamical systems in general; draw bifurcation diagrams. Linear Systems: Review of stability for linear systems of two equations. Local Theory for Nonlinear Planar Systems: Flow defined by a differential equation, Linearization and stable manifold theorem, Hartman-Grobman theorem, Stability and Lyapunov functions, Saddles, nodes, foci, centers and nonhyperbolic critical points. Gradient and Hamiltonian systems. Global Theory for Nonlinear Planar Systems: Limit sets and attractors, Poincaré map, Poincaré Benedixson theory and Poincare index theorem. Bifurcation Theory for Nonlinear Systems: Structural stability and Peixoto's theorem, Bifurcations at nonhyperbolic equilibrium points. Reading

1. L. Perko, Differential Equations and Dynamical Systems, Springer Verlag, 1991. 2. M. W. Hirsch and S. Smale, Differential Equations, Dynamical Systems and Linear Algebra

Academic Press, 2010. 3. P. Hartman, Ordinary Differential Equations, 2nd edition, SIAM 2002. 4. C. Chicone, Ordinary Differential Equations with Applications, 2nd Edition, Springer, 2006.

ELECTIVES II AND III (for Mathematics & Scientific Computing only) MA 6211 - Computer Graphics After studying this course, the student will be able to

CO1: Understand the structure of an interactive computer graphics system, and the separation

of system components.

CO2: Develop and analyses the algorithms for generation lines and polygons

CO3: Apply the geometrical transformations on objects.

CO4: Implement the techniques for segmentation

CO5: Differentiate different techniques for windowing and clipping

Introduction : Pixels and frame buffers - Coordinate systems - vector generation - line drawing and circle generation - algorithms and initializing of lines - thick line segments - character generation - display file and its structure. Polygons: Polygon representation - inside test - filling of polygon. 2D Transformations: Matrices - coordinate transformations - rotation about an arbitrary point - other transformations and inverse transformations. Segments: Segment table - operations on segments - image transformation and other display file structures. Windowing and Clipping: The viewing transformations - clipping - the cohen sutherland outcode algorithm - the sutherland Hodgman algorithm - clipping of polygons and multiple windowing. 3D Transformations: Rotation about an arbitrary axis - parallel projection - respective projection - Clipping in three dimensions - clipping planes and 3D viewing transformations. Hidden surfaces and Lines: Back face algorithms, Z buffers - scan line algorithms - the painter’s algorithms - warnock’s algorithm - Franklin algorithm and hidden line methods. Shading: Shading equations - smooth shading - Gouraud and phong shading methods - shadows. Curves and Fractals: Curve generation - interpolation - B-Splines - Benzier curves - fractal lines and fractal surfaces. Reading :

1. S.Harrington, Computer Graphics - A Programming Approach, McGraw Hill, New York, 1983.

2. D.F.Rogers & J.A.Adams, Mathematical Elements of Computer Graphics, McGraw Hill, New York, 1990.

MA 6112 - Fluid Dynamics After studying this course, the student will be able to

CO1: Find stream lines and path lines of a fluid flow

CO2: Find complex velocity potential for an incompressible and irrotational flow.

CO3: Set up equations of motion with boundary conditions for a given fluid flown problem and

solve them for the velocity field

CO4: Analyse the flow in ducts of different cross sections and estimate the volumetric flow rate.

Kinematics of fluids in motion: Real fluids and ideal fluids – velocity of a fluid at a point – stream lines and path lines – steady and unsteady flows – the velocity potential – the velocity vector – local and particle rates of change – the equation of continuity – acceleration of fluid – conditions at a rigid boundary. Equations of motion of fluid: Euler’s equations of motion – Bernoulli’s equation – some flows involving axial symmetry – some special two-dimensional flows. Some three dimensional flows: Introduction – sources, sinks and doublets – axisymmetric flows – Stokes’ stream function. The Milne-Thomson circle theorem – the theorem of Blasius – applications. Viscous flows: Stress analysis in fluid motion – relations between stress and rate of strain – the coefficient of viscosity and laminar flow – the Navier-Stokes’ equations of motion of viscous fluid – steady motion between parallel planes, through tube of uniform cross section and flow between concentric rotating cylinders. Steady viscous flow in tubes of uniform cross section – a uniqueness theorem – tube having uniform elliptic cross section – tube having equilateral triangular cross section – steady flow past a fixed sphere. Reading :

1. Frank Chorlton, Fluid Dynamics, CBS Publishers, Delhi, 2004 2. L.M.Milne Thomson, Theoretical Hydrodynamics, Macmillan Company, New York, 1960

MA6212-- Introduction to Python Programming After studying this course, the student will be able to CO1: Introduce the fundamental concepts of Python CO2: Provide a foundation to use basic building blocks of Python CO3: Learn to write Python Scripts. CO4: Explore various Exception Handling mechanisms CO5: Provide the basic knowledge to use Python with OOP Terminology. Introduction : Python Introduction, Python features, Python Environment Variables, Simple Programs, Python Identifiers, Reserved words, Assigning values to the variables, Multiple assignment, Standard data types, Type Conversion, Operators in Python, Decision Making, Looping, Loop Control statement, Mathematical functions- Random number function, Trigonometric functions, Mathematical Constants, Strings & Lists: Assigning values in strings, String manipulations, String special operators, String formatting operators, Triple Quotes, Raw String, Unicode String, Build-i n-String methods, Lists Introduction, Accessing values in list, List manipulations, List Operations, Indexing, slicing & matrices. Functions & Methods Built –in Functions and methods, Tuples- introduction, accessing values, Tuple functions, Dictionary- Introduction, Accessing values, Functions, Time tuple functions, Calendar tuple functions, time module functions, calendar module functions, and other module functions, user defined functions, Pass by value & pass by reference, function arguments & its types. Python Scripts: Import statements, Locating modules, Namespace, dir(), global(), local(), reload(), Packages in python, I/O function, Opening and closing files, file object attribute, manipulations of the files, Directories in python, File and Directory related methods. Exception: Exception, Handling Exception, example programs, try-finally, Argument of an Exception, Raising an Exception, User-defined exceptions. Reading: 1. Introduction to Python Programming, PovelSolin, Martin Novak,2012 2. Introduction to Python Programming, Jacob Fredslund, 2007 3. An Introduction to Python, John C. Lusth, 2011 4. Introduction to Python, DaveKuhlman, 2008 MA6213---Elliptic Curves After studying this course, the student will be able to

CO1: Know the basic arithmetic of elliptic curves

CO2: Understand the concept of torsion points and pairings in elliptic curves CO3: Compute the order of a point and the order of the group of points in elliptic curves over finite fields CO4: Apply the elliptic curve method for factorization and primality testing CO5: Learn about elliptic curves over rational and complex fields

Basic theory and tools: Weierstrass Equations; Group law on elliptic curves; a little of projective geometry; projective space and the point at infinity; other equations for Elliptic curves and other

coordinate systems; the -invariant; elliptic curves in characteristic 2; Endomorphisms, Frobenius

map; Singular curves; Elliptic curves mod ; Torsion points; Division polynomials; Weil pairing; Tate-Lichtenbaum pairing. Elliptic curves over Finite Fields: Frobenius endomorphism; Hasse theorem; Baby step, Giant step for finding order of a point; Schoof's algorithm for finding order of the group of points; supersingular curves. Applications: A cryptosystem based on Weil Pairing; Lenstra's factorization algorithm using elliptic curves; Primality testing using elliptic curves.

Number theoretic perspective: An introductory treatment of the topics - Elliptic curves over ,

Elliptic curves over , Divisors, Pairings, Isogenies.

Reading : 1 Lawrence C. Washington, Elliptic Curves: Number Theory and Cryptography, Second Edition, Chapman & Hall/CRC, 2008 2. Joseph H. Silverman, The Arithmetic of Elliptic Curves, Second Edition, Springer, 2009 3. Neal Koblitz, Introduction to Elliptic Curves and Modular Form, Second Edition, Springer, 1984 4. Anthony W. Knapp, Elliptic Curves, Princeton University Press, 1992

MA6214---Neural Networks After studying this course, the student will be able to

CO1: Learn different types of neural networks and different types of learning models

CO2: Determine the Mathematical foundations of neural network models.

CO3: Implement of neural networks using training algorithms such as the feed-forward, back- propagation algorithm.

CO4: Design, build and train neural networks for practical purposes

Introductory Concepts: ‘Neurons’ and their basic function- Math review- Mathematical Machinery and Review- How and Why Perceptrons Can Compute Logic Statements- Training Perceptrons Using Supervised Learning Techniques- Training Multi-layer

Neural Networks Using Supervised Learning Techniques: Recurrent Neural Networks and Unsupervised Learning: Optimization Techniques-Implementation and Performance Considerations-Variations on the Hopfield Network-A Stochastic Version of the Hopfield Network: The Boltzmann Machine-A Stochastic Version of the Binary Associative Memory: Restricted Boltzmann Machines-Competitive Learning and Self-Organizing Maps-Neural Network Modifications and Applications-Cellular Neural Networks and the Future of Massively Parallel Computation

Reading: 1. Raul Rojas, Neural Networks - A Systematic Introduction, Springer-Verlag, Berlin, New-York,

1996.

2. Koch, Christof. Biophysics of Computation: Information Processing in Single Neurons,

Oxford University Press, 2004.

MA 6215 - Parallel Computing After studying this course, the student will be able to

CO1.Uunderstand the fundamentals of parallel computing CO2. Design and analyze the parallel algorithms for solving linear system of equations CO3. Design parallel algorithms for eigenvalue problems CO4. Develop parallel algorithm code for iterative methods CO5.Aapply parallel computing method for solving simple differential equations Introduction to parallel programming – data parallelism – functional parallelism –pipelining – Flynn's taxonomy – parallel algorithm design – task/channel model –Foster's design methodology – case studies: boundary value problem – finding the maximum – n-body problem – Speedup and efficiency – Amdahl's law – Gustafson-Barsis's Law – Karp-Flatt Metric –Isoefficiency metric The message-passing model – the message-passing interface – MPI standard – basic concepts of MPI: MPI_Init, MPI_Comm_size, MPI_Comm_rank, MPI_Send, MPI_Recv, MPI_Finalize timing the MPI programs: MPI_Wtime, MPI_Wtick – collective communication: MPI_Reduce, MPI_Barrier, MPI_Bcast, MPI_Gather, MPI_Scatter – case studies: the sieve of Eratosthenes, Floyd's algorithm, Matrix-vector multiplication Monte Carlo methods – parallel random number generators – random number distributions – case studies – Matrix multiplication – row wise block-stripped algorithm – Cannon's algorithm – solving linear systems – back substitution – Gaussian elimination – iterative methods – conjugate gradient method Sorting algorithms – quicksort – parallel quick sort – hyper quick sort – sorting by regular sampling – Fast Fourier transform – combinatorial search – divide and conquer – parallel backtrack search – parallel branch and bound – parallel alpha-beta search.

Reading:

1. M. J. Quinn, Parallel Computing – Theory and Practice, Second Edition, Tata McGraw-Hill Publishing Company Ltd., 2002.

2. B. Wilkinson and M. Allen, Parallel Programming–Techniques and applications using networked workstations and parallel computers, 2 nd Edition, Pearson Education, 2005.

3. Michael J. Quinn, Parallel Programming in C with MPI and OpenMP, Tata McGraw-Hill Publishing Company Ltd., 2003

MA 6151 – Finite Element Method (FEM) (for Applied Mathematics only) After studying this course, the student will be able to

CO1: Determine an extremum by calculus of variations approach CO2: Formation of a variational problem for a boundary value problem CO3: Solution of one dimensional problems CO4: Solution of two dimensional problems by rectangular and triangular elements Introduction to Calculus of Variations.

Finite Element Method: Variational formulation - Rayleigh-Ritz minimization - weighted residuals - Galerkin method applied to boundary value problems. Global and local finite element models in one dimension - derivation of finite element equation. Finite element interpolation - polynomial elements in one dimension, two dimensional elements, natural coordinates, triangular elements, rectangular elements, Lagrangian and Hermite elements for rectangular elements - global interpolation functions. Local and global forms of finite element equations - boundary conditions - methods of solution for a steady state problem - Newton-Raphson continuation - one dimensional heat and wave equations. Reading :

1. J.N.Reddy, An introduction to the Finite Element Method, McGraw Hill, NY, 2005. 2. I.J. Chung, Finite element analysis in Fluid Dynamics, McGraw Hill Inc., 1978

MA 6152 – NSDE and FEM Lab (for Applied Mathematics only) Writing programs using Finite difference schemes for Schmidt's two level, Crank- Nicolson's two level for heat conduction problem. Explicit and Implicit methods for Wave equation, Laplace equation and Poisson equation. Writing programs using linear, quadratic and cubic elements for one dimensional general linear second order differential equation. MA 6251 – JAVA Programming (for Mathematics & Scientific Computing only) After studying this course, the student will be able to

CO1: To introduce the fundamental concepts of java CO2: To provide a foundation to use basic concepts in Java CO3: Learn to write Java Scripts. CO4: To explore various Exception Handling mechanisms CO5: To provide the basic knowledge to use Java with OOP Terminology. Java Basics - History of Java, Java buzzwords, comments, data types, variables, constants, scope and life time of variables, operators, operator hierarchy, expressions, type conversion and casting, enumerated types, control flow-block scope, conditional statements, loops, break and continue statements, simple java program, arrays, input and output, formatting output, Review of OOP concepts, encapsulation, inheritance, polymorphism, classes, objects, constructors, methods, parameter passing, static fields and methods, access control, this reference, overloading methods and constructors, recursion, garbage collection, building strings, exploring string class, Inheritance – Inheritance concept, benefits of inheritance, Super classes and Sub classes, Member access rules, Inheritance hierarchies, super uses, preventing inheritance: final classes and methods, casting, polymorphism- dynamic binding, method overriding, abstract classes and methods, the Object class and its methods. Interfaces – Interfaces vs. Abstract classes, defining an interface, implementing interfaces, accessing implementations through interface references, extending interface. Files – streams- byte streams, character streams, text Input/output, binary input/output, random access file operations, File management using File class, Using java.io.

Networking in Java – Introduction, Manipulating URLs, Ex. Client/Server Interaction with Stream Socket Connections, Connectionless Client/Server Interaction with Datagrams, Using java.net. Exception handling – Dealing with errors, benefits of exception handling, the classification of exceptions- exception hierarchy, checked exceptions and unchecked exceptions, usage of try, catch, throw, throws and finally, rethrowing exceptions, exception specification, built in exceptions, creating own exception sub classes, Guide lines for proper use of exceptions. Multithreading - Differences between multiple processes and multiple threads, thread states, creating threads, interrupting threads, thread priorities, synchronizing threads, interthread communication, thread groups, daemon threads. Reading : 1. Herbert Schildt, Java: the complete reference, 7th editon, TMH,2011 2. P.J.Deitel and H.M.Deitel, Java for Programmers, Pearson education, 2009 3. Cay S.Horstmann and Gary Cornell, Core Java, Volume 1-Fundamentals, Eighth edition, Pearson eduction, 2013 4. D.S.Malik, Java Programming, Cengage Learning, 2009 5. R.A. Johnson, An introduction to Java programming and object oriented application development, Cengage Learning, 2007 MA 6252 – JAVA Programming Lab (for Mathematics & Scientific Computing only) Software packages (with a menu driven basis) have to be developed for the topics covered in the subjects Numerical Methods, Statistical Methods, Operations Research, Finite Difference and Finite Element Methods covered in the earlier/present semesters MA 6192/6292 – Comprehensive Viva (Common to both the streams) Each student has to appear for comprehensive viva in front of the panel of examiners. MA 6199/6299 – Project Work (Common to both the streams) Each student will be allotted to a faculty member by DAC (PG & R) for this term project work. It has to be carried out on a topic chosen by the student in consultation with the supervisor. The project supervisor will periodically review the student progress over the semester. The progress of the work will be evaluated in the middle of the semester by DAC (PG & R). Finally, the student has to submit the project report before the end examinations of the IV semester and present the findings for the final evaluation.

Elective – IV , V & VI (for Applied Mathematics only) MA 6161 – Measure and Integration After studying this course, the student will be able to

CO1. Identify the class of measurable sets. CO2. Derive properties of Lebesgue measurable sets and functions. CO3. Determine whether the given function is Lebesgue integrable or not. CO4. Prove Fatou's Lemma and Lebesgue dominated convergence theorem. CO5. find the advantages of Lebesgue integration than Riemann integration

Algebras of sets - Borel subsets of R - Lebesgue outer measure and its properties - algebras of measurable sets in R – non-measurable set - example of measurable set which is not a Borel set - Lebesgue measure and its properties - Lebesgue-Stieltje’s measure - measurable functions - point wise convergence and convergence in measure - Egoroff theorem - Lebesgue integral - Lebesgue criterion of Riemann integrability - Fatou’s lemma - convergence theorem - Differentiation of an integral - absolute continuity with respect to Lebesgue measure. Lebesgue integral in the plane - Fubini’s theorem. Reading :

1. H. L. Royden, Patric K and Fitz Patrick, Real Analysis, 4th Edition, PHI, 2011. 2. Paul R. Halmos, Measure Theory, D.Vanostrand, New Jersy, 1964.

MA 6162 - Heat and Mass Transfer After studying this course, the student will be able to

CO1: Understand the boundary layer concepts and derive boundary layer equations for velocity, temperature. CO2: Compute temperature distribution in steady-state and unsteady-state heat conduction. CO3: Interpret and analyze free and mixed convection heat transfer. CO4: Analyze the heat and mass transfer past vertical and horizontal bodies.

Laminar Boundary layer flow: Concept of boundary layer, velocity and thermal boundary layer, Derivation of boundary layer equations for two dimensional incompressible flow, similarity solutions, dimensional analysis, wall heating conditions and uniform wall heat flux boundary layer over a flat plate (Blasius Equations). Convective transport: Free convection, mixed convection, Free and mixed Convection on a Vertical Flat Plate with a Constant Wall Temperature, Constant Heat Flow, Variable Surface Temperature, Variable Heat Flux on a Surface, Free Convection on a Vertical Surface in Stratified Media. Free and mixed convection on a Vertical Cylinder, Horizontal Cylinder, Inclined Cylinder and cone. Mass transfer: Properties of mixtures, mass conversation, mass diffusivities, and boundary conditions, laminar forced convection, impermeable surface model, other external forced convection configurations, internal forced convection, natural convection, mass function and mass line, effect of chemical reaction. Convection in porous media: Mass conservation, Darcy flow model and the Forchheimer modification, first law of thermodynamics, second law of thermodynamics, forced convection, natural convection boundary layers, enclosed porous media heated from the side, penetrative convection, enclosed porous media heated from below, multiple flow scales distributed non-uniformly. Reading

1. H.Schlichting and K.Gersten, Boundary Layer Theory, Springer publisher, 2000. 2. Franz Durst, Fluid Mechanics (An introduction to the theory of fluid flows), Springer

publisher, 2008. 3. Adrian Bejan, Convection Heat Transfer, Wiley-India, 2012.

MA6163 --Perturbation Methods After studying this course, the student will be able to

CO1: Solve perturbation problems in differential equations

CO2: Understand boundary layer in fluid flow problems CO3: Understand regular and singular perturbation theory CO4: Use asymptotic expansions to solve perturbation problems Introduction: Parameter perturbations, Coordinate perturbations, Order Symbols and Gauge functions, Asymtotic expansions and Sequences, Convergent versus Asymtotic Series, Nonuniform Expansions, Elementary operations on Asymtotic Expansions. Straight forward expansions and sources of nonuniformity: Infinite domains, A small parameter multiplying highest derivative. Type change of partial differential equations, the presence of singularities, the role of coordinate systems. The method of strained coordinates: The method of strained parameters, Lighthill's technique, Temples technique, Renormalization technique, Limitations of the method of Strained coordinates The methods of Matched and Composite Asymptotic Expansions: The methods of Matched Asymptotic Expansions, The methods of Composite Asymptotic Expansions Variation of Parameters and Methods of Averaging: Variation of Parameters, The method of averaging, Struble's technique, The Krylov-Bogoliubov-Mitropolski technique, The method of averaging by using canonical variables, Von-Zeipel's procedure, Averaging by using the Lie series and transforms, Averaging by using Lagragians. The method of Multiple Scales: Description of method, Applications of the Derivative, Expansion method, The two variable expansion procedure, generalized method Reading: 1. A H Nayfeh, Perturbation Methods, Wiley, New York, 1973 2. A.H. Nayfeh, Introduction to Perturbation Techniques, John Wiley & Sons, 1981 MA 6164 - Theory of Elasticity After studying this course, the student will be able to

CO1: Derive the components of strain and stress tensors

CO2: Find torsion for bars of uniform cross sections

CO3: Analyse the two and three dimensional problems of elasticity

CO4: Examine different types of waves generated. Analysis Of Strain: Deformation, strain tensor in rectangular Cartesian coordinates, Geometric interpretation of infinitesimal strain, rotation, compatibility of strain components, properties of strain tensor, strain in spherical and cylindrical polar coordinates. Analysis Of Stress: Stresses, laws of motion, Cauchy’s formula, equations of equilibrium, transformation of coordinates, Plane state of stresses, Cauchy’s stress quadric, shearing stress, Mohr’s circle, stress deviation, stress tensor in general coordinates, physical components of a stress tensor in general coordinates, equation of equilibrium in curvilinear coordinates. Linear Theory Of Elasticity: Generalized Hooke’s law, Stress-Strain relationship for an isotropic elastic material, Basic equation of elasticity for homogeneous isotropic bodies, boundary value problems, the problem of equilibrium and the uniqueness of solution of elasticity, Saint-Venant’s principle. Torsion: Torsion of prismatic bars, torsion of circular, elliptic and rectangular bars, membrane analogy, torsion of rectangular section and hollow thin walled sections. Solution Of Two And Three Dimensional Problems: Bending of a cantilever beam, simply supported beam with simple loadings. Semi infinite medium subjected to simple loadings. Plane elastic waves, Rayleigh surface waves, Love waves, Vibration of an infinite isotropic solid cylinder.

READING: 1. Sokolnikoff I.S. Mathematical Theory of Elasticity, Tata-McGraw Hill, New Delhi,1974. 2. Hetnarski R.B. and Ignaczak J. Mathematical Theory of Elasticity, Taylor &Francis, London, 2004. 3. Achenbach J.D. Wave Propagation in Elastic Solids, North-Holland Pub. Co.,Amsterdam, 1973. 4. Fung Y.C., Foundations of Solid Mechanics, Prentice Hall Inc., New Jercy, 1965 5. Srinath L.S., Advanced Mechanics of Solids, Tata McGraw Hill, New Delhi, 3rd Edition, 2008. MA 6165 – Computational Fluid Dynamics After studying this course, the student will be able to

CO1.Develop mathematical models of fluid flow and derive the governing equations CO2. Solve flow problems related to inviscid flows and potential flows CO3. Learn the essential numerical background for solving partial differential equations governing the fluid flow CO4. Develop flow simulation code for simple viscous fluid flow and heat transfer problems CO5. Interpret the results of fluid flow with physics of the problem Basic Equations of Fluid Dynamics: Continuity equation; Momentum equation; Energy equation ; Main Non-dimensional groups- Reynolds number, Froude number, Prandtl number, Mach number, Specific heat ratio and others. Equations expressed in conservative form. Inviscid Flows. Incompressible potential flows; Flows due to Sources and Sinks; Inverse method-I Von Karman’s method for approximating flow past bodies of revolution; Inverse method-II : Conformal mapping; Panel method; Elliptic equations -Potential flows in ducts or around bodies –Circular cylinder inside a channel; Propagation of a finite amplitude wave and formation of a shock-Method of characteristics. Viscous Fluid flows : Governing equations for viscous flows; Structure of a plane shock wave; self similar laminar Boundary layer flows; flat plate thermometer problem; Ordinary boundary value problems involving derivative boundary conditions- pipe and open channel flows; Explicit method for solving generalized Rayleigh problem; Implicit method for solving starting flow in a channel problem; Numerical solution of a bi-harmonic equations-Stokes flows. Reading :

1. Chuen-Yen-Chow and Sedat Biringen, An introduction to computational Fluid Mechanics, Wiley, 2011.

2. Tarit Kumar Bose, Computational Fluid Dynamics, Wiley Eastern Ltd., 1988. 3. C.A.J. Fletcher, Computational Techniques for fluid Dynamics Vol. I and .II, Springer-

Verlag, Berlin, 1991. MA 6166 – Bio-Fluid Mechanics After studying this course, the student will be able to

CO1: Estimate volumetric flow rate of blood in blood vessels. CO2: Analyze pressure in stenotic regions in blood vessels CO3: Simplify governing equations for blood flow by taking small amplitude and long wave length approximations CO4: Analyze filtration process of the blood flow in Renal tubes Fundamental concepts of Biomechanics. Cardiovascular system: Basic concepts about blood, blood vessels, governing equations, models on blood flow, flow in large blood vessels, microcirculation, pulsatile flow, stenotic region flow. Peristalsis: Basic concepts, governing equations, peristaltic transport under long wave length

approximation, peristaltic flow for small amplitudes and small Reynold’s number. Flow in Renal tubules: Basic concepts, governing equations, ultra filtration, flow through proximal tubules, flow through tubes with varying cross section. Reading :

1. J.N.Kapur, Mathematical Models in Biology and Medicine, Affiliated East West Press, New Delhi, 1992.

2. C.G.Caro, T.J.Pedley, R.C.Schroter, & W.A.Seed, Mechanics of circulation, Oxford University Press., 2011.

3. Y.C.Fung, Bio-Dynamics: Circulation, Springer-Verlag., 2010. MA6167--Inventory, Queueing Theory and NLPP After studying this course, the student will be able to

CO1: Determine the characteristics of a queuing model

CO2: Determine the EOQ for an inventory model

CO3: Determine the solution of a CNLPP

CO4: Determine the solution of a QPP

Queuing theory : Characteristics of queueing systems - the birth and death process - steady state solutions - single server model (finite and infinite capacities) - single server model (with SIRO) - models with state dependent arrival and service rates- waiting time distributions. Inventory Control : Inventory control for single commodity - deterministic inventory models ( without and with shortages) - Probabilistic inventory ( both discrete and continuous) control models. Nonlinear programming problem : Constrained NLPP, Lagrange’s multipliers method - convex NLPP, Kuhn-Tucker conditioins (including the proof) - Quadratic programming problem (Wolfe’s method) - Geometric programming Reading :

1. H.A.Taha , Operations Research, An Introduction , PHI, Delhi, 2014 2. H.M.Wagner, Principles of Operations Research, PHI, Delhi, 2010. 3. J.C.Pant, Introduction to Optimization: Operations Research, Jain Brothers, Delhi, 2015.

MA6168 --Hydrodynamic Stability After studying this course, the student will be able to

CO1 : Derive linearised stability equations for a given basic state;

CO2 : Perform a normal-mode analysis leading to an eigenvalue problems;

CO3 : Use the ideas of weakly non-linear stability theory in simple systems; CO4 : Appreciate the different physical mechanisms leading to instability. Fundamental concept of fluid (in)stability; some examples and applications, Equations of motion for fluid flows; Navier-Stokes and Boussinesq equations; Some analytical solutions, Equations for perturbations; non-dimensionalization; example: Rayleigh-Benard problem , Linearization; example: Couette problem , and modal analysis, Rayleigh-Benard instability, Centrifugal instability (Taylor-Couette flow) , Numerical stability analysis, Kelvin-Helmholtz instability , Stability of parallel

flows, Stability of boundary layers, Zombie vortex instability , Crow instability and baroclinic instability,End of the modal linear stability analysis ,Non-normal growth and non-modal analysis, End of linear stability analysis, Bifurcation theory and Applications, Nonlinear stability and Transition to chaos . Reading : 1. P. G. Drazin, Introduction to hydrodynamic stability, Cambridge University Press, 2007. 2. P. K. Kundu and I. M. Cohen, Fluid mechanics, Academic Press, 2012 3. P. G. Drazin and W. H. Reid, Hydrodynamic stability, Cambridge University Press, 2004 4. P. J. Schmid and D. S. Henningson, Stability and transition in shear flows, Springer, 2001 5. W. O. Criminale, T. L. Jackson, R. D. Joslin, Theory and computation of hydrodynamic stability, Cambridge University Press, 2003 MA 6169-- Spectral Methods After studying this course, the student will be able to

CO1. Understand the basics of spectral methods and Matlab tools CO2. Evaluate Fourier spectral differentiation using differentiation matrices and FFT's CO3. Solve IVP's and BVP's using spectral methods CO4. Determine stability, convergence criterions and stifness. CO5. Understand the mathematical concepts of spectral element methods.

Introduction: Historical background. Introduction to spectral methods via orthogonal functions. Some examples of spectral methods. Spectral differentiation versus Finite differences. MATLAB as a tool in problem solving. Basic layout of spectral methods. Fourier Spectral Differentiation: Fourier approximation. Fourier spectral differentiation via differentiation matrices. Fourier spectral differentiation via FFT. Smoothness and accuracy. Aliasing and aliasing removal. MATLAB demonstrations. Chebyshev Spectral Differentiation: Polynomial approximation. Jacobi polynomials. Chebyshev spectral differentiation via Differentiation matrices. Chebyshev spectral differentiation via FFT. Smoothness and accuracy. MATLAB demonstrations. Initial Value Problems: Method of lines treatment of problems with mixed initial/boundary conditions. Semi-implicit methods. Method of integrating factors. Case studies and MATLAB demonstrations. Boundary Value Problems: Treatment of problems Dirichlet/Neumann/Robin type boundary conditions. Eigen boundary value problems. Boundary value problems in Polar coordinates. Differential eigen problems. Case studies and MATLAB demonstrations. Time Stepping: Linear multistep and multistage methods. Stability and convergence criterions. The concept of stability regions. Stiffness. Introduction to Spectral Element Method: Weak variational formulation. Elemental representation and parametric mapping. Legendre Spectral differentiation and integration. Local elemental operations. Global operations. Boundary representation. Case studies and MATLAB demonstrations. Reading :

1. L. N. Trefethen, Spectral Methods in Matlab, SIAM, 2000.

2. Canuto, C., Hussaini, M. Y., Quarteroni, A, and Zang, T. A., Spectral Methods : Fundamentals in Single Domain, Springer Verlag, 2006.

3. Gottlieb, D. and Orszag, S. A., Numerical Analysis of Spectral Methods: Theory and

Applications, CBMS-NSF 26, Philadelphia: SIAM, 1977.

4. Canuto, C., Hussaini, M. Y., Quarteroni, A, and Zang, T. A., Spectral Methods : Evolution to Complex Geometries and Applications to Fluid Dynamics, Springer Series in Scientific Computation, 2007.

5. Canuto, C., Hussaini, M. Y., Quarteroni, A, and Zang, T. A., Spectral Methods in Fluid Dynamics, Springer Series in Computational Physics, 1988.

6. Guo, B-Y, Spectral methods and their applications, World Scientific, 1998

Elective – IV , V & VI (for Mathematics & Scientific Computing only)

MA 6261 – Theory Of Automata After studying this course, the student will be able to

CO1: Classify and create the Languages CO2 : Design the Automata to accept the given language CO3 : Design the Turing machine CO4: Find the equivalences among machines Preliminaries: Sets, Relations, Equivalence relation, partition, Transitive closures, Kleene’ closure *, Strings, Alphabets, Languages, Recursive definitions. Regular Languages and Finite Automata: Regular Expressions, Regular Languages, Finite State Machines, Deterministic finite automata (DFA), Non-deterministic finite automata (NFA), Non-deterministic finite automata with ε moves (NFA-ε), ε-closure, Equivalence of DFA, NFA and NFA-ε, Language accepted by Finite Automata, Kleene’s Theorem Properties of Regular Sets: Properties of the Languages accepted by finite automata, Regular and non-regular languages, Minimal finite automata, Pumping lemma, Myhill - Nerode theorem. Closure properties of Regular languages, Context Free Languages and Pushdown Automata: Context free grammars (CFG), context free languages (CFL), closure properties of context free languages, Chomsky normal form, Greibach normal form, Pumping lemma for CFL, parsing, Pushdown automata (PDA), CFG for PDA, PDA for CFG, phrase structured grammars and languages and context sensitive grammars and languages. Turing Machines: Turing machine model, example, Modification of Turing machines, Church’s hypothesis and Non-deterministic Turing machines. Reading :

1. Hopcroft J. and Ullman J.D., Introduction to Automata Theory, Languages and Computation, Narosa Publishing, 1989.

2. Martin, J.C., Introduction to Languages and the Theory of Computation, Tata McGraw Hill, 2009.

3. Carrel J. and Long D., Theory of finite automata with an introduction to formal languages, Prentice Hall, 1989.

MA 6262 - Approximation Theory

After studying this course, the student will be able to

CO1: Approximate the function by least square principle CO2: Find Generalized approximations CO3: Approximate function by special functions CO4: Understand uniform approximation Approximation in normed linear spaces: Existence- uniqueness – convexity – characterization of best uniform Approximations –uniqueness results – Haar subspaces – approximation of real Valued functions on an interval. Chebyshev polynomials: Properties – more on external properties of Chebyshev polynomials – strong Uniqueness and continuity of metric projection – discretization – discrete best Approximation. Interpolation Introduction – algebraic formulation of finite interpolation – Lagrange’s form – Extended Haar subspaces and Hermite interpolation – Hermite – Fejer interpolation. Best approximation in normed linear spaces: Introduction – approximate properties of sets – characterization and duality. Projection , Continuity of metric projections – convexity, solarity and cheyshevity of sets – best Simultaneous approximation. Reading

1. Hrushikesh N. Mhaskar and Devidas V. Pai., “Fundamentals of approximation theory”, Narosa Publishing House, New Delhi, 2000.

2. Ward Cheney and Will Light, “A course in approximation theory”, Brooks/Cole Publishing Company, New York, 2000.

3. Cheney E.W.,”Introduction to approximation theory”, McGraw Hill, New York, 1966. 4. Singer I.,”Best Approximation in Normed Linear Spaces by element of linear subspaces”,

Springer-Verlag, Berlin, 1970. MA 6263 – Financial Mathematics After studying this course, the student will be able to

CO1: Understand the concept of Brownian motion CO2: Compute option pricing for a given rate of interest CO3: Apply Black Scholes formula for option costs CO4: Learn portfolio selection problem

INTRODUCTION : Probability and Random variables-Geometric Browninan Motion as a limit of simpler models – Brownian motion. PRESENT VALUE ANALYSIS AND ARBITRAGE Interest rates – Present value analysis – Rate of Return – Continuously variable interest rates – Pricing contracts via Arbitrage – An example in options pricing. ARBITRAGE THEOREM AND BLACK-SCHOLES FORMULA The Arbitrage theorem – Multiperiod binomial model – Black-Scholes formula –Properties of Black – Scholes option cost – Delta Hedging Arbitrage Strategy –Pricing American put options. EXPECTED UTILITY Limitations of arbitrage pricing – Valuing investments by expected utility – The portfolio selection problem – Capital asset pricing model – Rates of Return – Single period and geometric Brownian motion.

EXOTIC OPTIONS Barrier options – Asian and look back options – Monte Carlo Simulation – Pricing exotic option by simulation – More efficient simulation estimators – Options with nonlinear pay offs – pricing approximations via multi period binomial models -- The Black - Scholes Formula. Reading:

1. Ross S.M., “An elementary introduction to Mathematical Finance”, 2nd Edition, Cambridge University Press, 1999

2. Marek Musiela and Marck Rutkowski,” Martingale Methods in Financial Modelling”, Springer, 2nd Edition, 2005.

3. Marek Capiński, Tomasz Zastawniak; Mathematics for Finance: An Introduction to Financial Engineering; Springer, 2003

4. Petr Zima and Robert L. Brown; Mathematics of Finance; Mc Graw Hill, Schaum’s outline series, 2011

MA 6264 - Data Mining After studying this course, the student will be able to

CO1: Understand data mining tasks and issues CO2: Apply different techniques for data mining CO3: Analyze multi-dimensional modeling techniques CO4: Implement of the clustering techniques CO5: Evaluate the performance of algorithms for Association Rules.

Introduction: Basic Data Mining Tasks, Data Mining Issues, Data Mining Metrics, Data Mining from a Database Perspective. Data Mining Techniques: A Statistical Perspective on Data Mining, Similarity Measures, Decision Trees, Neural Networks, Genetic Algorithms. Classification: Statistical-Based Algorithms, Distance-Based Algorithms, Decision Tree-Based Algorithms, Neural Network-Based Algorithms, Rule-Based Algorithms, Combining Techniques. Clustering: Similarity and Distance Measures, Hierarchical Algorithms, Partitional Algorithms, Clustering Large Databases, Clustering with Categorical Attributes. Association Rules: Basic Algorithms, Incremental Rules, Advanced Association Rule Techniques, Measuring the Quality of Rules. Advanced Techniques: Web Mining, Spatial Mining and Temporal Mining. Reading

1. J. Han and M. Kamber, Data Mining: Concepts and Techniques, 2nd Ed. Morgan Kaufman. 2006.

2. M. H. Dunham. Data Mining: Introductory and Advanced Topics, Pearson Education. 2001. 3. I. H. Witten and E. Frank. Data Mining: Practical Machine Learning Tools and Techniques,

Morgan Kaufmann. 2000. 4. D. Hand, H. Mannila and P. Smyth. Principles of Data Mining, Prentice-Hall. 2001.

MA 6265 – Management Information Systems After studying this course, the student will be able to

CO1: Determine the key terminologies and concepts of MIS

CO2: Design, develop and implement Information Technology solutions for business problems

CO3: Understand ethical issues that occur in business

CO4: Plan projects, work in team settings and deliver project outcomes in time

Meaning, Nature, Need, Role, Importance, Evolution of Management through Information system. Relatedness of MIS with management activities, Management functions and decision making. Concept of ‘Balanced MIS’ effectiveness and efficiency criteria. Development of MIS, Information System Planning – Methodology and Tools/Techniques for systematic identification, evaluation, modification of MIS. A study of major financial, production, manpower and marketing MIS. End user computing and development. Advanced MIS – concept, need and problems in achieving advanced MIS, Decision support systems, Export systems. Rationale of computer application. Reading :

1. Charles S.Parker, Management Information Systems- Strategy and Action, McGraw Hill Intl. Edition., 1989.

2. James O’Brian, Management Information Systems, McGraw Hill, 1998. MA 6266 – Cryptography After studying this course, the student will be able to

CO1: Understand the structure of various secret key cryptosystems like stream ciphers, DES, AES

CO2: Understand and analyze public key cryptosystems like RSA and ElGamal

CO3: Learn certain algorithms for prime factorization and discrete-log problems

CO4: Learn elliptic curves and cryptosystems based on these curves

CO5: Know the concepts of hash functions and digital signature schemes

Introduction : Review on basic group theory and basic number theory; Historical ciphers and their cryptanalysis; Principles of modern cryptography; perfect secrecy and one-time pad. Private-key Cryptography: Stream ciphers; Block ciphers - SPN, Feistel design, DES, AES; Introduction to differential and linear cryptanalysis. Public-key Cryptography: RSA Cryptosystem; Primality testing; Algorithms for factoring; Diffie-Hellman key-exchange protocol; Discrete-Logarithm Problem; ElGamal Cryptosystem; Algorithms for DLP. Elliptic curves: basic facts; elliptic-curve cryptosystem. Discussion on Hash functions, Digital signatures and other cryptography topics of relevance. Reading

1. Douglas R. Stinson, Cryptography: Theory and Practice, Third Edition, Chapman & Hall/CRC, 2006

2. Christof Paar and Jan Pelzl, Understanding Cryptography, Springer, 2010 3. Wade Trappe and Lawrence Washington, Introduction to Cryptography with Coding Theory,

Second Edition, Pearson, 2006 4. Jonathan Katz and Yehuda Lindell, Introduction to Modern Cryptography, Second Edition,

CRC Press, Taylor & Francis Group, 2015 MA 6267– Advanced Optimization Techniques After studying this course, the student will be able to

CO1 Differentiate and classify traditional and non-traditional optimization methods. CO2 Formulate an optimization problem to solve complex problems. CO3 Apply A*, AO*, Branch and Bound search techniques for problem solving.

CO4 Apply GA, PSO and ACO algorithms for various optimization problems

Problem Solving Methods : Problem Space, Problem solving, State space, Algorithm’s performance and Complexity, Search Algorithms, Depth first search method, Breadth first search method, Branch and Bound search method, Introduction to P type, NP complete and NP Hard problems. Classical methods versus Non-traditional methods.

Evolutionary Methods: Principles of Evolutionary Processes and genetics, Introduction to evolutionary algorithms, Evolutionary strategy, Evolutionary programming.

Genetic Algorithm : Basic concepts, working principle, procedures of GA, flow chart of GA, Genetic representations, (encoding) Initialization and selection, Genetic operators, Mutation, Generational Cycle, Genetic programming, Simple applications.

Swarm Optimization: Introduction to Swarm intelligence, Ant colony optimization (ACO), Meta-heuristic, Algorithm for Travelling Salesman Problem, Particle swarm optimization (PSO), Other variants of swarm intelligence algorithms, Simple problems and applications.

Artificial Neural Networks: Neuron, Nerve structure and synapse, Artificial Neuron and its model, activation functions, Neural network architecture: single layer and multilayer feed forward networks, recurrent networks. Back propagation algorithm, factors affecting back propagation training, Simple applications. Reading:

1. Kalyanmoy Deb, Multi-objective Optimization using Evolutionary Algorithms, John Wiley and Sons, 2012.

2. Maurice Clerc, Particle Swarm optimization, ISTE, USA, South Asian Edn, 2007 3. Marco D & Thomas S, Ant Colony optimization, MIT Press, London, 2004 4. Tettamanzi Andrea, Tomassini and Marco, Soft Computing Integrating Evolutionary, Neural

and Fuzzy Systems, Springer, 2001. 5. S S Rao, Engineering Optimization, New Age, New Delhi, 2014

MA6268 - Data Analysis with R After studying fluid dynamics, the student will be able to

CO1: manage and manipulate a given data in R

CO2: extract data from various sources and store

CO3: compare and analyse data

CO4: build statistical models for the data

Basics of R: Variables, Data types, vectors, calling functions, missing data

Advanced Data Structures: Data frames, Lists, Matrices and Arrays

Reading Data into R: Reading CSVs, Excel Data, Data from statistical tools, Data from websites.

Functions and control statements: function arguments, return values, do.call, if and else, switch,

for loops, while loops.

Group Manipulation: Apply family, aggregate, plyr, data.table

Probability and Statistics: Normal, Binomial and Poisson distributions.; dbinom(), pbinom(),

qbinom(), rbinom() for probability distributions. Correlation and covariance, t test, ANOVA

Linear Models: Simple linear regression, multiple regression, logistic regression, Poisson regression,

survival analysis

Modal diagnostics: residuals, comparing modals, cross validation, bootstrap.

Reading : 1. Jared P Lander, R for every one- Advanced analysis and graphics, Pearson Education 2014

2. Garrett Grolemund, Hands on programming with R, Oreilly, SPD(Shroff Publications and

Distributors Pvt Ltd) 2014.

MA6269--- Mathematics of Data Science After studying fluid dynamics, the student will be able to

CO1: Analyze the basics of Data Science CO2: Apply PCA, Spectral Clustering CO3: Compute dimension reduction and clustering of random graphs CO4: Apply Approximation algorithms Introduction of Data Science, Visualization of data, Resampling, Distributions, Linear Model & Baysian Model, Simple examples Principal Component Analysis (PCA), Spectral Clustering _ Cheeger’s inequality, Concentration of measure and tail bounds in probability. Dimension reduction through Johnson-Lindelstrauss Lemma and Gordon’s Escape through a Mesh Theorem. Approximation algorithms in Theoretical Computer science and the Max-cut problem. Clustering of random graphs: Stochastic Block model. Basics of duality in Optimization. Synchronization, inverse problems on graphs. Reading: 1. Joel Grus, Data science from scratch, O'Reilly Media, 2015 2. Lillian Pierson, Data science for Dummies, 2nd Edn, Wiley, 2017 3. Murtaza Haider, Getting started with data science, IBM Press, 2016 4. https://ocw.mit.edu/mathematics