Chemistry [3-1-0-8]

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Chemistry Laboratory [0-0-3-3]

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Engineering Drawing [2-0-3-7]

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Physics - I [2-1-0 -6]

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MATHEMATICS I [3-1-0-8]

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Single variable calculus: Convergence of sequences and series of real numbers; Continuity of functions; Differentiability, Rolle's theorem, mean value theorem, Taylor's theorem; Power series; Riemann integration, fundamental theorem of calculus, improper integrals; Application to length, area, volume and surface area of revolution.

Multivariable calculus: Vector functions of one variable - continuity and differentiability; Scalar valued functions of several variables, continuity, partial derivatives, directional derivatives, gradient, differentiability, chain rule; Tangent planes and normals, maxima and minima, Lagrange multiplier method; Repeated and multiple integrals with applications to volume, surface area; Change of variables; Vector fields, line and surface integrals; Green’s, Gauss’ and Stokes’ theorems and their applications.

Texts:

1. G. B. Thomas, Jr. and R. L. Finney, Calculus and Analytic Geometry, Pearson India, 9_th Edition, 2006

References:

1. R. G. Bartle and D. R. Sherbert, Introduction to Real Analysis, Wiley India, 4th Edition, 2014.
2. S. R. Ghorpade and B. V. Limaye, An Introduction to Calculus and Real Analysis, Springer India, 2006.
3. T. M. Apostol, Calculus, Volume-2, Wiley India, 2003.
4. J. E. Marsden, A. J. Tromba and A. Weinstein, Basic Multivariable Calculus, Springer India, 2002.

Electrical Sciences[3-1-0-8]

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Introduction to Computing [3-0-0-6]

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Computing Laboratory [0-0-3-3]

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Basic Electronics Laboratory [0-0-3-3]

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Engineering Mechanics [3-1-0-8]

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Physics - II [2-1-0-6]

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Physics Laboratory/Workshop [0-0-3-3]

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MATHEMATICS II [3-1-0-8]

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Linear algebra: Systems of linear equations, matrices, Gaussian elimination, echelon form, column space, null space, rank of a matrix, inverse and determinant; Vector spaces (over the field of real and complex numbers), subspaces, spanning set, linear independence, basis and dimension; Linear transformations, rank-nullity theorem, matrix of a linear transformation, change of basis and similarity; Eigenvalues and eigenvectors, algebraic and geometric multiplicity, diagonalization by similarity; Inner-product spaces, Gram-Schmidt process, orthonormal basis; Orthogonal, Hermitian and symmetric matrices, spectral theorem for real symmetric matrices

Ordinary differential equations: First order differential equations – exact differential equations, integrating factors, Bernoulli equations, existence and uniqueness theorem, applications; Higher-order linear differential equations – solutions of homogeneous and nonhomogeneous equations, method of variation of parameters, operator method; Series solutions of linear differential equations, Legendre equation and Legendre polynomials, Bessel equation and Bessel functions of first and second kinds; Systems of first-order equations, phase plane, critical points, stability.

Texts:

1. D. Poole, Linear Algebra: A Modern Introduction, Cengage Learning India Private Limited, 4th Edition, 2015.
2. S. L. Ross, Differential Equations, Wiley India, 3rd Edition, 2004.

References:

1. G. Strang, Linear Algebra and Its Applications, Cengage Learning, 4th Edition, 2006.
2. J. Gilbert and L. Gilbert, Linear Algebra and Matrix Theory, Academic Press, 1995.
3. K. Hoffman and R. Kunze, Linear Algebra, Pearson India, 2nd Edition, 2015.
4. E. A. Coddington, An Introduction to Ordinary Differential Equations, Dover Publications, 1989.
5. E. L. Ince, Ordinary Differential Equations, Dover Publications, 1958.
6. T. M. Apostol, Calculus, Volume-2, Wiley India, 2003.
7. W. E. Boyce and R. C. DiPrima, Elementary Differential Equations, Wiley India, 9th Edition, 2008.

MATHEMATICS-III[3-1-0-8]

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Complex analysis: Complex numbers and elementary properties; Complex functions - limits, continuity and differentiation, Cauchy-Riemann equations, analytic and harmonic functions, elementary analytic functions, anti-derivatives and line (contour) integrals, Cauchy-Goursat theorem, Cauchy's integral formula, Morera's theorem, Liouville's theorem, Fundamental theorem of algebra and maximum modulus principle; Power series, Taylor series, zeros of analytic functions, singularities and Laurent series, Rouche's theorem and argument principle, residues, Cauchy's Residue theorem and applications, Mobius transformations and applications.

Partial differential equations: Fourier series, half-range Fourier series, Fourier transforms, finite sine and cosine transforms; First order partial differential equations, solutions of linear and quasilinear first order PDEs, method of characteristics; Classification of second-order PDEs, canonical form; Initial and boundary value problems involving wave equation and heat conduction equation, boundary value problems involving Laplace equation and solutions by method of separation of variables; Initial-boundary value problems in non-rectangular coordinates.

Laplace and inverse Laplace transforms, properties, convolutions; Solution of ODEs and PDEs by Laplace transform; Solution of PDEs by Fourier transform.

Texts:

1. J. W. Brown and R. V. Churchill, Complex Variables and Applications, 7th Ed., Mc-Graw Hill, 2004.
2. I. N. Sneddon, Elements of Partial Differential Equations, McGraw Hill, 1957.
3. E. Kreyszig, Advanced Engineering Mathematics, 10th Ed., Wiley, 2015.

References:

1. J. H. Mathews and R. W. Howell, Complex Analysis for Mathematics and Engineering, 3rd Ed., Narosa,1998.
2. S. J. Farlow, Partial Differential Equations for Scientists and Engineers, Dover Publications, 1993.
3. K. Sankara Rao, Introduction to Partial Differential Equations, 3rd Ed., Prentice Hall of India, 2011

DISCRETE MATHEMATICS[3-0-0-6]

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Set theory: Sets, relations, equivalence relations, partially ordered sets, functions, countability, lattices and Boolean algebras. Logic: Well-formed formula, interpretations, propositional logic, predicate logic, theory of inference for propositional logic and predicate logic. Combinatorics: Permutations, combinations, recurrences, generating functions, partitions, special numbers like Fibonacci, Stirling and Catalan numbers. Graph Theory: Graphs and digraphs, special types of graphs, isomorphism, connectedness, Euler and Hamilton paths, planar graphs, graph colouring, trees, matching.

Texts:

1. J. P. Tremblay and R. Manohar, Discrete Mathematics with Applications to Computer Science, Tata McGraw-Hill, 1997.
2. K. H. Rosen, Discrete Mathematics & its Applications, 6th Ed., Tata McGraw-Hill, 2007.

References:

1. A. Shen and N. K. Vereshchagin, Basic Set Theory, American Mathematical Society, 2002.
2. A. Kumar, S. Kumaresan and B. K. Sarma, A Foundation Course in Mathematics, Narosa, 2018.
3. M. Huth and M. Ryan, Logic in Computer Science, Cambridge University Press, 2004.
4. V. K. Balakrishnan, Theory and Problems of Combinatorics, Schaum's Series, McGraw-Hill, 1995.
5. R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, 2nd Ed., Addison-Wesley, 1994.
6. A. Tucker, Applied Combinatorics, 6th Ed., Wiley, 2012.
7. R. Balakrishnan and K. Ranganathan, A Text Book of Graph Theory, Springer, 2000.

ELEMENTARY NUMBER THEORY AND ALGEBRA[3-0-0-6]

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Number theory: Well ordering principle, principle of mathematical induction; Division algorithm, GCD and LCM, Euclidean algorithm, linear Diophantine equation; Primes, the fundamental theorem of arithmetic; Properties of congruences, linear congruences, chinese remainder theorem; Fermat's little theorem; Arithmetic functions, Mobius inversion formula, Euler's theorem; Primitive roots; Introduction to cryptography, RSA cryptosystem, distribution of primes.

Algebra: Groups, subgroups, cyclic groups, permutation groups, Cayley's theorem, cosets and Lagrange's theorem, normal subgroups, quotient groups, homomorphisms and isomorphism theorems; Rings, integral domains, ideals, quotient rings, prime and maximal ideals, ring homomorphisms, field of quotients, polynomial rings, factorization in polynomial rings, fields, characteristic of a field, field extensions, splitting fields, finite fields.

Texts:

1. D. M. Burton, Elementary Number Theory, 7th Ed., McGraw Hill, 2017.
2. J. A. Gallian, Contemporary Abstract Algebra, 4th Ed., Narosa, 1998.

References:

1. I. Niven, S. Zuckerman and H. L. Montgomery, An Introduction to the Theory of Numbers, 5th Ed., Wiley-India, 1991.
2. G. A. Jones and J. M. Jones, Elementary Number Theory, Springer, 1998
3. K. H. Rosen, Elementary Number Theory and its Applications, Pearson, 2015
4. I. N. Herstein, Topics in Algebra, Wiley, 2004.
5. J. B. Fraleigh, A First Course in Abstract Algebra, Addison Wesley, 2002.

MATRIX COMPUTATIONS [3-0-2-8]

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Floating point computations, IEEE floating point arithmetic, analysis of roundoff errors; Sensitivity analysis and condition numbers; Linear systems, LU decompositions, Gaussian elimination with partial pivoting; Banded systems, positive definite systems, Cholesky decomposition - sensitivity analysis; Gram-Schmidt orthonormal process, Householder transformation, Givens rotations; QR factorization, stability of QR factorization.

Solution of linear least squares problems, normal equations, singular value decomposition(SVD), polar decomposition, Moore-Penrose inverse; Rank deficient least squares problems; Sensitivity analysis of least-squares problems; Review of canonical forms of matrices; Sensitivity of eigenvalues and eigenvectors.

Reduction to Hessenberg and tridiagonal forms; Power, inverse power and Rayleigh quotient iterations; Explicit and implicit QR algorithms for symmetric and non-symmetric matrices; Reduction to bidiagonal form; Golub-Kahan algorithm for computing SVD; Sensitivity analysis of singular values and singular vectors; Overview of iterative methods: Jacobi, Gauss-Seidel and successive over relaxation methods; Krylov subspace methods, Arnoldi and Lanczos methods, conjugate gradient method.

Texts:

1. G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd Edition, John Hopkins University Press, 1996.
2. C. T. Kelley, Iterative Methods for Linear and Nonlinear equations, SIAM, 1995.
3. D. S. Watkins, Fundamentals of Matrix Computations, John Wiley, 1991.
4. L. N. Trefethen and D. Bau III, Numerical Linear Algebra, SIAM, 1997.
5. J. W. Demmel, Applied Numerical Linear Algebra, SIAM, 1997.

Object Oriented Programming and Data structures [3-0-3-9]

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REAL ANALYSIS[3-0-0-6]

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Metrics and norms - metric spaces, normed vector spaces, convergence in metric spaces, completeness, compactness; Functions of several variables - differentiability, chain rule, Taylor's theorem, inverse function theorem, implicit function theorem; Lebesgue measure and integral - sigma-algebra of sets, measure space, Lebesgue measure, measurable functions, Lebesgue integral, Fatou’s lemma, dominated convergence theorem, monotone convergence theorem, L^p spaces.

Texts:

1. J. E. Marsden and M. J. Hoffman, Elementary Classical Analysis, 2nd Ed., W. H. Freeman, 1993.
2. M. Capinski and E. Kopp, Measure, Integral and Probability, 2nd Ed., Springer, 2007.

References:

1. N. L. Carothers, Real Analysis, Cambridge University Press, 2000.
2. G. de Barra, Measure Theory and Integration, New Age International, 1981.
3. R. C. Buck, Advanced Calculus, Waveland Press Incorporated, 2003.

PROBABILITY THEORY AND RANDOM PROCESSES [3-1-0-8]

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Probability spaces, independence, conditional probability, and basic formulae; Random variables, distribution functions, probability mass/density functions, functions of random variables; Standard univariate discrete and continuous distributions and their properties; Mathematical expectations, moments, moment generating functions, characteristic functions; Random vectors, multivariate distributions, marginal and conditional distributions, conditional expectations; Modes of convergence of sequences of random variables, laws of large numbers, central limit theorem; Definition and classification of random processes, discrete-time Markov chains, classification of states, limiting and stationary distributions, Poisson process, continuous-time Markov chains.

Texts:

1. P. G. Hoel, S. C. Port and C. J. Stone, Introduction to Probability Theory, Universal Book Stall, 2000.
2. G. R. Grimmett and D. R. Stirzaker, Probability and Random Processes, 3rd Ed., Oxford University Press, 2001.

References:

1. S. M. Ross, Introduction to Probability Models, 11th Ed., Academic Press, 2014.
2. J. Medhi, Stochastic Processes, 3rd Ed., New Age International, 2009.
3. W. Feller, An Introduction to Probability Theory and its Applications, Vol. 1, 3rd Ed., Wiley, 1968.
4. K. S. Trivedi, Probability and Statistics with Reliability, Queuing, and Computer Science Applications, 2nd Ed., Wiley, 2001.
5. C. M. Grinstead and J. L. Snell, Introduction to Probability, 2nd Ed., Universities Press India, 2009

MONTE CARLO SIMULATION[0-1-2-4]

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Principles of Monte Carlo; Generating random numbers (linear congruential generators, random vectors, Fibonacci generators, transformed random variables); Normally distributed random variables (Box, Muller and Marsaglia methods, correlated random variables); Low discrepancy numbers ( Monte Carlo integration, discrepancy, low discrepancy sequences, Halton sequence).

Texts:

1. P. Glasserman, Monte Carlo Methods in Financial Engineering, Springer, 2004
2. R. Seydel, Tools for Computational Finance, Springer, 2002.

SCIENTIFIC COMPUTING[3-0-2-8]

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Errors; Iterative methods for nonlinear equations; Polynomial interpolation, piecewise linear and cubic splines, spline interpolations; Numerical integration by interpolation, quadrature methods, Gaussian quadrature; Initial value problems for ordinary differential equations: Euler method, Runge-Kutta methods, multi-step methods, predictor-corrector method, stability and convergence analysis; Finite difference schemes for partial differential equations; Explicit and implicit schemes; Consistency, stability and convergence; Stability analysis (matrix method and von Neumann method), Lax equivalence theorem; Finite difference schemes for initial value problems, boundary value problems and free boundary value problems (FTCS, Backward Euler and Crank-Nicolson schemes, ADI methods, Lax Wendroff method, upwind scheme).

Texts:

1. K. E. Atkinson, An Introduction to Numerical Analysis, Wiley, 1989.
2. S. D. Conte and C. de Boor, Elementary Numerical Analysis - An Algorithmic Approach, McGraw-Hill, 1981.
3. J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, 2nd Edition, Texts in Applied Mathematics, Vol. 12, Springer Verlag, 1993.
4. J. D. Hoffman, Numerical Methods for Engineers and Scientists, McGraw-Hill, 1993.
5. R. Mitchell and S. D. F. Griffiths, The Finite Difference Methods in Partial Differential Equations, Wiley, 1980.
6. G. D. Smith, Numerical Solutions of Partial Differential Equations, 3rd Edition, Calrendorn Press, 1985.

Science Elective [3-0-0-6]

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OPTIMIZATION[3-0-0-6]

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Classification and general theory of optimization; Linear programming (LP) - formulation and geometric ideas, simplex and revised simplex methods, duality and sensitivity, transportation, assignment, and integer programming problems; Nonlinear optimization, method of Lagrange multipliers, Karush-Kuhn-Tucker theory, convex optimization; Numerical methods for unconstrained and constrained optimization (gradient method, Newton’s and quasi-Newton methods, penalty and barrier methods).

Texts:

1. M. S. Bazaraa, J. J. Jarvis and H. D. Sherali, Linear Programming and Network Flows, 4th Ed., Wiley, 2011.
2. N. S. Kambo, Mathematical Programming Techniques, Revised Ed., Affiliated East-West Press, 2008.

References:

1. E. K. P. Chong and S. H. Zak, An Introduction to Optimization, 4th Ed., Wiley, 2013.
2. M. S. Bazaraa, H. D. Sherali and C. M. Shetty, Nonlinear Programming: Theory and Algorithms, 3rd Ed., Wiley, 2013.
3. D. G. Luenberger and Y. Ye, Linear and Nonlinear Programming, 4th Ed., Springer, 2016.
4. K. G. Murty, Linear Programming, Wiley, 1983.
5. D. Gale, The Theory of Linear Economic Models, The University of Chicago Press, 1989

Formal Languages & Automata Theory[3-0-0-6]

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Alphabets, languages; Regular Languages: various types of finite automata and their equivalences thereof, minimization of finite automata, Myhill-Nerode theorem, regular expressions, regular grammars, closure properties of regular languages, pumping lemma, algorithmic properties of regular languages; Context-free languages: context-free grammars, derivation trees, ambiguous grammars, inherently ambiguous languages, Chomsky and Greibach normal form, nondeterministic and deterministic pushdown automata, Top-down and Bottom-up parsing LL(k) and LALR grammars, pumping lemma and Ogden's lemma, closure and algorithmic properties of context-free languages; Context sensitive languages: context sensitive grammars, linear bounded automata; Turing machines: recursively enumerable languages, unrestricted grammars, variants of Turing machines and equivalence thereof.

Texts:

1. J. E. Hopcroft, R. Motwani and J. D. Ullman, Introduction to Automata Theory, Languages and Computation, Pearson Education India, 2001.
2. H. R. Lewis and C. H. Papadimitriou, Elements of the Theory of Computation, Prentice Hall of India, 1997.

References:

1. M. Sipser, Introduction to the Theory of Computation, Thomson Asia, 1997.
2. D. C. Kozen, Automata and Computability, Springer-Verlag, 1997.
3. D. I. A. Cohen, Introduction to Computer Theory, John Wiley, 1997.

DESIGN AND ANALYSIS OF ALGORITHMS[3-0-0-6]

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Prerequisites: MA 221, CS 201

Models of Computation: Turing machines and random access machines, space and time complexity measures, lower and upper bounds; Design and analysis techniques: the greedy method, divide-and-conquer, dynamic programming, backtracking, branch and bound. Priority Queues: heaps, binomial heaps, Fibonacci heaps. Sorting and order statistics: sorting algorithms (insertion-sort, bubble-sort, shell-sort, quick-sort, merge-sort, heap-sort and external-sort) and their analyses, selection. Graph Algorithms: connectivity, bi-connectivity, topological sort, shortest paths, minimum spanning trees, maximum flow. Advanced topics: the disjoint set union problem; NP-completeness; geometric, approximation, parallel, and randomized algorithms.

Texts:

1. T. H. Cormen, C. E. Leiserson, R. L. Rivest and C. Stein, Introduction to Algorithms, MIT Press, 2001.

References:

1. Aho, J. E. Hopcroft and J. D. Ullman, The Design and Analysis of Computer Algorithms, Addison-Wesley, 1974.
2. E. Horowitz and S. Sahni, Fundamentals of Computer Algorithms, Galgotia Publishers, 1984.
3. M. T. Goodrich and R. Tamassia, Algorithm Design: Foundations, Analysis and Internet Examples, John Wiley, 2001.
4. H. Papadimitriou and K. Steiglitz, Combinatorial Optimization: Algorithms and Complexity, Prentice Hall of India, 1992.

FINANCIAL ENGINEERING [4-0-0-8]

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Financial markets; Cash flow, time value of money, net present value, net future value; Fixed income securities: Bonds and bonds pricing, yield curves, duration and convexity. Term structure of interest rates, spot and forward rates; Equities, risk-reward analysis, asset pricing models, mean variance portfolio optimization, Markowitz model and efficient frontier, CAPM and APT; Discrete time market models: Assumptions, portfolios and trading strategies, replicating portfolios, risk neutral probability measures, valuation of contingent claims, fundamental theorem of asset pricing; The Cox-Ross-Rubinstein (CRR) model, pricing in CRR model, Black-Scholes formula derived as a limit of the CRR pricing formula; Derivative securities: futures and forward contracts, hedging strategies using futures, pricing of futures and forward contracts, interest rate futures; Properties of options, contingent claims, trading strategies and binomial trees, pricing of stock options, options on stock indices, currencies and futures, European and American options; Greeks, delta hedging and risk management, volatility smiles; Interest rate derivatives (basic term structure model, swaps and swaptions, caps and floors).

Texts:

1. S. R. Pliska, Introduction to Mathematical Finance: Discrete Time Models, Blackwell, 1997.
2. M. Capinski and T. Zastawniak, Mathematics for Finance: An Introduction to Financial Engineering, Springer, 2005.
3. S. Roman, Introduction to the Mathematics of Finance: From Risk Management to Options Pricing, Springer, 2004.
4. J. C. Hull, Options, Futures and Other Derivatives, 6th Edition, Prentice Hall of India, 2006.

References:

1. N. H. Bingham and R. Kiesel, Risk Neutral Valuation, 2nd Edition, Springer, 2004.
2. K. Back, A Course in Derivative Securities, Springer, 2005.
3. S. Shreve, Stochastic Calculus for Finance, Vol 1, Springer, 2004.

Operating Systems [3-0-0-6]

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Operating Systems Laboratory [0-1-3-5]

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Data Communications [3-0-0-6]

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Theory of Computation[3-0-0-6]

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Prerequisites: MA 252 MA 351

Computability theory: Turing machines, combination of Turing machines, Turing acceptable and decidable languages, computable functions, algorithms, Church-Turing thesis, grammatically computable functions, µ-recursive functions, lambda calculus, decidability, decidable problems concerning regular and context-free languages, halting problem, Rice's theorem, Post's correspondence problem, undecidable problems from language theory via reducibility, the recursion theorem, Turing reducibility; Computational theory: computational complexity, time and space bounded computations, time and space hierarchies, relations among complexity measures, classes P and NP; reducibilities, NP-compleness.

Texts:

1. H. R. Lewis and C. H. Papadimitriou, Elements of the Theory of Computation, Prentice Hall of India, 1997.
2. J. E. Hopcroft and J. D. Ullman, Introduction to Automata Theory, Languages and Computation, Narosa, 1979.
3. M. Sipser, Introduction to the Theory of Computation, Thomson Asia, 1997.

References:

1. D. Kozen, Theory of Computation, Springer, 2006.
2. D. S. Garey and G. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, Freeman, 1979.
3. C. H. Papadimitriou, Computational Complexity, Addison-Wesley, 1994.
4. J. L. Balcazar, J. Diaz and J. Gabarro, Structural Complexity, Vol 1, 2nd Edition, Springer-Verlag, 1995.
5. E. Mendelson, Introduction to Mathematical Logic, Wadsworth & Brooks, 1987.

STOCHASTIC CALCULUS FOR FINANCE[3-0-0-6]

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Prerequisites: MA224 or equivalent and MA271 or equivalent

General probability spaces, filtrations, conditional expectations, martingales and stopping times, Markov processes; Random walks, Brownian motion and its properties; Itô integral and its properties, Itô processes, Itô-Doeblin formula; Derivation of the Black-Scholes-Merton equation, Black-Scholes-Merton formula, multi-variable stochastic calculus; Risk-neutral valuation, risk-neutral measure, Girsanov's theorem for change of measure, martingale representation theorem, fundamental theorems of asset pricing; Stochastic differential equations and their solutions, Feynman-Kac theorem and its applications

Texts:

1. S. Shreve, Stochastic Calculus for Finance, Vol. II, Springer, 2004.

References:

1. F. C. Klebaner, Introduction to Stochastic Calculus with Applications, 3rd Ed., Imperial College Press, 2012.
2. S. Shreve, Stochastic Calculus for Finance, Vol. I, Springer, 2004.
3. M. Baxter and A. Rennie, Financial Calculus, Cambridge University Press, 1996.
4. A. Etheridge, A Course in Financial Calculus, Cambridge University Press, 2003.
5. R. J. Elliott and P. E. Kopp, Mathematics of Financial Markets, Springer, 1999.

FINANCIAL ENGINEERING LAB [0-0-3-3]

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This course will focus on implementation of the financial models such as CAPM, Binomial models, Black-Scholes model, Interest rate models and asset pricing based on above models studied in Financial Engineering-I and Financial Engineering -II. The implementation will be done using S-PLUS/MALAB/C++.

Texts:

1. Y. Lyuu, Financial Engineering and Computation, Cambridge Univ. Press, 2002.
2. P. Glasserman, Monte Carlo Methods in Financial Engineering, Springer, 2004.

References:

1. D. Higham, Introduction to Financial Option Valuation: Mathematics, Stochastics and Computation, Cambridge Univ. Press, 2004.

Project - I [0-0-6-6]

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Databases Lab[0-0-3-3]

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Open Elective - II [3-0-0-6]

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COMPUTATIONAL FINANCE[3-0-2-8]

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Prerequisites: MA373 or equivalent

Review of financial market models for derivative pricing, interest rate modelling and Black-Scholes PDE; Solutions of pricing PDEs using finite difference methods, American option as free boundary problem, computation of price of American options, pricing of exotic options, upwind scheme and other methods; Monte-Carlo simulation, generating sample paths, discretization of SDE, Monte-Carlo for option valuation and Greeks, Monte-Carlo for American and exotic options; Variance reduction techniques; Monte-Carlo implementation of short rate models, forward rate models and LIBOR market model, volatility structure and calibration.

Texts:

1. R. U. Seydel, Tools for Computational Finance, 5th Ed., Springer, 2012.
2. P. Glasserman, Monte Carlo Methods in Financial Engineering, Springer, 2004.

References:

1. Y.-l. Zhu, X. Wu, I-L. Chern and Z.-z. Sun, Derivative Securities and Difference Methods, 2nd Ed., Springer, 2013.
2. D. Higham, Introduction to Financial Option Valuation: Mathematics, Stochastics and Computation, Cambridge University Press, 2004.
3. P. Wilmott, S. Howison and J. Dewynne, The Mathematics of Financial Derivatives: A Student Introduction, Cambridge University Press, 1997.
4. Y. Lyuu, Financial Engineering and Computation, Cambridge University Press, 2002.

Project - II[0-0-10-10]

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Elective I [3-0-0-6]

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Elective II [3-0-0 -6]

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Elective III [3-0-0 -6]

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Databases[3-0-0-6]

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HSS Elective [3-0-0-6]

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HSS Elective [3-0-0-6]

---

HSS Elective [3-0-0-6]

---

HSS Elective [3-0-0-6]

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Workshop - I [0-0-3-3]

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Modern Biology [3-1-0-8]

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MODERN ALGEBRA[3-0-0-6]

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Sets, relations, mappings, equivalence relations, binary operations, monoids, semi-groups; Groups, subgroups, cyclic groups, order of an element, cosets, Lagrange's theorem, normal subgroups, quotient groups, homomorphisms, automorphsims, permutation groups, Cayley's theorem, conjugacy, class equation, Cauchy's theorem, Sylow's theorems.

Rings, ideals, quotient rings, integral domains, fields, maximal ideals, prime ideals, irreducible elements, prime elements, Euclidean domains, principal ideal domains, unique factorization domains.

Fields, characteristic of a field, prime subfield, field extensions, algebraic extensions, separable extensions, finite fields, algebraically closed field, algebraic closure of a field.

Texts:

1. J. A. Gallian, Contemporary Abstract Algebra, 4th Edition, Narosa, 1998.
2. I. N. Herstein, Topics in Algebra, Wiley, 2004.
3. P. B. Bhattacharya, S. K. Jain and S. R. Nagpaul, Basic Abstract Algebra, Cambridge University Press, 1995.

References:

1. D. S. Dummit and R. M. Foot, Abstract Algebra, 2nd Edition, John Wiley, 1999.
2. J. B. Fraleigh, A First Course in Abstract Algebra, Addison Wesley, 2002.
3. M. Artin, Algebra, Prentice Hall of India, 1991.

Science Elective [3-0-0-6]

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COMPUTER ORGANIZATION AND ARCHITECTURE[3-0-0-6]

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Number Systems: representations of numbers (binary, octal, decimal, and hexadecimal), arithmetic of signed and unsigned numbers; Boolean algebra and logic gates: gate level minimization of Boolean functions; Combinational logic circuits: design and analysis, some standard combinational circuits (encoders, decoders, multiplexers); Sample and hold Circuits, analog-to-digital converter, digital-to-analog converter.

Synchronous sequential logic circuits: design and analysis; flip-flops, registers, counters; Finite state model: state tables and state diagram, state minimization; Memory organization: hierarchical memory systems, cache memories, cache coherence, virtual memory; System buses: interconnection structures and bus interconnection, Arithmetic Logic Unit.

Study of an existing CPU: architecture, instruction set and the addressing modes supported; assembly language programming; Control unit design: instruction interpretation, hardwired and microprogrammed methods of design; RISC and CISC paradigms. I/O transfer techniques: programmed, interrupt-driven and DMA; I/O processors and channels, mapping of I/O addresses.

Texts:

1. A. S. Tenenbaum, Structured Computer Organization, 5th Edition, Prentice-Hall of India, 2005.
2. W. Stallings, Computer Organization and Architecture: Designing for Performance, 6th Edition, Prentice Hall of India, 2005.

References:

1. M. M. Mano, Digital Design, 3rd Edition, Pearson Education Asia, 2002.
2. J. Hennessy and D. Patterson, Computer Architecture A Quantitative Approach, 3rd Edition, Morgan Kaufmann, 2002.
3. C. Hamacher, Z. Vranesic, and S. Zaky, Computer Organization, 5th Edition, McGraw-Hill, 2002.
4. P. Pal Chaudhuri, Computer Organization and Design, 2nd Edition, Prentice Hall of India, 1999.
5. J. P. Hayes, Computer Architecture and Organization, 3rd Edition, McGraw-Hill, 1997.

Seminar[0-0-2-2]

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STATISTICAL ANALYSIS OF FINANCIAL DATA[3-0-2-8]

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Prerequisites: MA371

Introduction to S-Plus and data analysis: Financial data, random variables and their distributions, exploratory data analysis tools, kernel density estimation; Quantiles and Q-Q plots; Random generators and Monte Carlo samples; Continuous time processes: Maximum likelihood estimation (MLE) for common diffusion processes (Brownian Motion, Ornstein-Uhlenbeck, Cox-Ingersoll-Ross), approximate MLE of general diffusions, simulation of exact method for common diffusions, Euler and Milstein discretization schemes for general diffusions. Time series analysis: AR, MA, ARMA, ARCH and GARCH models; Identification, estimation and forecasting; Stochastic volatility time series models for term structure of interest rates. Multivariate Data Analysis: Multivariate normal samples, estimation, hypothesis testing, and simulation; Copulas and random simulations, examples of copulas families, fitting Copulas, Monte Carlo simulations with Copulas; Dimension reduction techniques, Principal Component Analysis. Elements of extreme value theory: Generalized extreme value (GEV) and generalized Pareto distribution (GPD); Block maxima, GPD and Hill methods; Quantile estimation with the Cornish-Fisher expansion.

Texts:

1. R. A. Carmona, Statistical Analysis of Financial Data in S-PLUS, Springer, 2004.
2. E. Zivot and J. Wang, Modeling Financial Time Series with S-PLUS, 2nd Edition, Springer, 2006.
3. P. Glasserman, Monte Carlo Methods in Financial Engineering, Springer, 2004.

Open Elective - I [3-0-0-6]

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FINANCIAL ENGINEERING[3-0-0-6]

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Prerequistes: MA 372 Stochastic Calculus for Finance

Continuous time financial market models, Black- Scholes model, Black-Scholes PDE and formulas, Risk neutral valuation, change of numeraire, pricing and hedging of contingent claims, Greeks, Implied volatility; Options on futures, European, American and Exotic options. Incomplete markets, market models with stochastic volatility, pricing and hedging in incomplete markets.

nd markets, term-structures of interest rates, bond pricing; Short rate models, martingale models for short rate (Vasicek, Ho-Lee, Cox-Ingersoll-Ross and Hull-White models), multifactor models; Forward rate models, Heath-Jarrow-Morton framework, pricing and hedging under short rate and forward rate models, swaps and caps; LIBOR and Swap market models, caps, swaps, swaptions, calibration and simulation. Introduction to credit risk modeling, credit derivatives, CDS and CDO.

Texts:

1. T. Bjork, Arbitrage Theory in Continuous Time, 2nd ed., Oxford Univ. Press, 2003.
2. J. C. Hull, Options, Futures and Other Derivatives, 7th Edition, Pearson Education/Prentice-Hall of India, 2008.

References:

1. S. Shreve, Stochastic Calculus for Finance, Vol. 2, Springer, 2004.
2. D. Brigo and F. Mercurio, Interest rate models:Theory and Practice, Springer, 2006.
3. N. H. Bingham and R. Kiesel, Risk Neutral Valuation, 2nd ed., Springer, 2004.
4. J.Cvitanic and F. Zapatero, Introduction to the Economics and Mathematics of Financial Markets, Prentice-Hall of India, 2007.
5. M. Musiela and M. Rutkwoski, Martingale Method in Financial Modelling, 2nd ed., Springer, 2005.
6. P. Wilmott, Derivatives: The Theory and Practice of Financial Engineering, Wiley, 1998.

HSS Elective [3-0-0-6]

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Computer Networks[3-0-0 -6]

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