Discrete Mathematics [3-1-0-8]

---

Set Theory - sets and classes, relations and functions, recursive definitions, posets, Zorn - s lemma, cardinal and ordinal numbers; Logic - propositional and predicate calculus, well-formed formulas, tautologies, equivalence, normal forms, theory of inference. Combinatorics - permutation and combinations, partitions, pigeonhole principle, inclusion-exclusion principle, generating functions, recurrence relations. Graph Theory - graphs and digraphs, Eulerian cycle and Hamiltonian cycle, adjacency and incidence matrices, vertex colouring, planarity, trees.

Texts:

1. J.P. Tremblay and R. Manohar, Discrete Mathematical Structures with Applications to Computer Science, Tata McGraw Hill, New Delhi, 2001.
2. C. L. Liu, Elements of Discrete Mathematics, 2nd Edn., Tata McGraw-Hill, 2000.

References:

1. K. H. Rosen, Discrete Mathematics & its Applications, 6th Edn., Tata McGraw-Hill, 2007.
2. V. K. Balakrishnan, Introductory Discrete Mathematics, Dover, 1996.
3. J. L. Hein, Discrete Structures, Logic, and Computability, 3rd Edn., Jones and Bartlett, 2010.
4. N. Deo, Graph Theory, Prentice Hall of India, 1974.

Computer Programming[3-0-2-8]

---

Introduction - the von Neumann architecture, machine language, assembly language, high level programming languages, compiler, interpreter, loader, linker, text editors, operating systems, flowchart; Basic features of programming (Using C) - data types, variables, operators, expressions, statements, control structures, functions; Advance programming features - arrays and pointers, recursion, records (structures), memory management, files, input/output, standard library functions, programming tools, testing and debugging; Fundamental operations on data - insert, delete, search, traverse and modify; Fundamental data structures - arrays, stacks, queues, linked lists; Searching and sorting - linear search, binary search, insertion-sort, bubble-sort, selection-sort; Introduction to object oriented programming.

Programming laboratory will be set in consonance with the material covered in lectures. This will include assignments in a programming language like C and C++ in GNU Linux environment.

Texts:

1. A. Kelly and I. Pohl, A Book on C, 4th Ed., Pearson Education, 1999.

References:

1. H. Schildt, C: The Complete Reference, 4th Ed., Tata Mcgraw Hill, 2000.
2. B. Kernighan and D. Ritchie, The C Programming Language, 2nd Ed., Prentice Hall of India, 1988.
3. B. Gottfried and J. Chhabra, Programming With C, Tata Mcgraw Hill, 2005.

Modern Algebra[3-1-0-8]

---

Groups, subgroups, homomorphism; Group actions, Sylow theorems; Solvable and nilpotent groups; Rings, ideals and quotient rings, maximal, prime and principal ideals; Euclidean and polynomial rings; Modules; Field extensions, Finite fields.

Texts:

1. D. Dummit and R. Foote, Abstract Algebra, Wiley, 2004.
2. N. McCoy and G. Janusz, Introduction to Abstract Algebra, 7th Edn.,Trustworthy Communications, Llc, 2009

References:

1. I. N. Herstein, Topics in Algebra, Wiley, 2008.
2. J. Fraleigh, A First Course in Abstract Algebra, Pearson, 2003.
3. P. B. Bhattacharya, S. K. Jain and S. R. Nagpaul, Basic Abstract Algebra, Cambridge University Press, 1995.

Linear Algebra [3-1-0-8]

---

Systems of linear equations, vector spaces, bases and dimensions, change of bases and change of coordinates, sums and direct sums; Linear transformations, matrix representations of linear transformations, the rank and nullity theorem; Dual spaces, transposes of linear transformations; trace and determinant, eigenvalues and eigenvectors, invariant subspaces, generalized eigenvectors; Cyclic subspaces and annihilators, the minimal polynomial, the Jordan canonical form; Inner product spaces, orthonormal bases, Gram-Schmidt process; Adjoint operators, normal, unitary, and self-adjoint operators, Schur's theorem, spectral theorem for normal operators.

Texts:

1. S. Axler, Linear Algebra Done Right, 2nd Edn., UTM, Springer, Indian edition, 2010.
2. K. Hoffman and R. Kunze, Linear Algebra, Prentice Hall of India, 1996.

References:

1. G. Schay, Introduction to Linear Algebra, Narosa, 1997.
2. G. Strang, Linear Algebra and Its applications, Nelson Engineering, 4th Edn., 2007.

Real Analysis [3-1-0-8]

---

Convergence of sequence of real numbers, real valued functions of real variables, differentiability, Taylor's theorem; Functions of several variables - limit, continuity, partial and directional derivatives, differentiability, chain rule, Taylor's theorem, inverse function theorem, implicit function theorem, maxima and minima, multiple integral, change of variables, Fubini's theorem; Metrics and norms - metric spaces, convergence in metric spaces, completeness, compactness, contraction mapping, Banach fixed point theorem; Sequences and series of functions, uniform convergence, equicontinuity, Ascoli's theorem, Weierstrass approximation theorem.

Texts:

1. P. M. Fitzpatrick, Advanced Calculus, 2nd Edn., AMS, Indian Edition, 2010.
2. N. L. Carothers, Real Analysis, Cambridge University Press, Indian Edition, 2009.

References:

1. J. E. Marsden and M. J. Hoffman, Elementary Classical Analysis, 2nd Edn., W. H. Freeman, 1993.
2. W. Rudin, Principles of Mathematical Analysis, 3rd Edn., McGraw Hill, 1976.

Data structures and Algorithms[3-0-2-8]

---

Prerequistes: MA 511 Computer Programming.

Asymptotic notation; Sorting - merge sort, heap sort, priortiy queue, quick sort, sorting in linear time, order statistics; Data structures - heap, hash tables, binary search tree, balanced trees (red-black tree, AVL tree); Algorithm design techniques - divide and conquer, dynamic programming, greedy algorithm, amortized analysis; Elementary graph algorithms, minimum spanning tree, shortest path algorithms.

Programming laboratory will be set in consonance with the material covered in lectures. This will include assignments in a programming language like C and C++ in GNU Linux environment.

Texts:

1. T. H. Cormen, C. E. Leiserson, R. L. Rivest and C. Stein, Introduction to Algorithms, 2nd Ed.,Prentice-Hall of India, 2007.

References:

1. M. T. Goodrich and R. Tamassia, Data Structures and Algorithms in Java, Wiley, 2006.
2. A.V. Aho and J. E. Hopcroft, Data Structures and Algorithms, Addison-Wesley, 1983.
3. S. Sahni, Data Structures, Algorithms and Applications in C++, 2nd Ed., Universities Press, 2005.

Differential Equations [4-0-0-8]

---

Review of fundamentals of Differential equations (ODEs); Existence and uniqueness theorems, Power series solutions, Systems of Linear ODEs, Stability of linear systems.

First order linear and quasi-linear partial differential equations (PDEs), Cauchy problem, Classification of second order PDEs, characteristics, Well-posed problems, Solutions of hyperbolic, parabolic and elliptic equations, Dirichlet and Neumann problems, Maximum principles, Green's functions.

Texts:

1. E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, Tata McGraw Hill, 1990.
2. S. L. Ross, Differential Equations, 3rd Edn., Wiley India, 1984.
3. I. N. Sneddon, Elements of Partial Differential Equations, Dover Publications, 2006.
4. F. John, Partial Differential Equations, Springer, 1982.

References:

1. S. J. Farlow, Partial Differential Equations for Scientists and Engineers, Dover Publications, 1993.
2. E. L. Ince, Ordinary Differential Equations, Dover Publications, 1958.
3. F. Brauer and J. A. Nohel, The Qualitative Theory of Ordinary Differential Equations: An Introduction, Dover Publications, 1969.

Complex Analysis [3-1-0-8]

---

Review of complex numbers; Analytic functions, harmonic functions, elementary functions, branches of multiple-valued functions, conformal mappings; Complex integration, Cauchy's integral theorem, Cauchy's integral formula, theorems of Morera and Liouville, maximum-modulus theorem; Power series, Taylor's theorem and analytic continuation, zeros of analytic functions, open mapping theorem; Singularities, Laurent's theorem, Casorati-Weierstrass theorem, argument principle, Rouche's theorem, residue theorem and its applications in evaluating real integrals.

Texts:

1. R.V. Churchill and J.W. Brown, Complex Variables and Applications, 5th edition, McGraw Hill, 1990.
2. J. H. Mathews and R. W. Howell, Complex Analysis for Mathematics and Engineering, 3rd edition, Narosa, 1998.

References:

1. L. V. Ahlfors, Complex Analysis, 3rd Edn., McGraw Hill, 1979.
2. J. E. Marsden and M. J. Hoffman, Basic complex analysis, 3rd Edn., W. H. Freeman, 1999.
3. D. Sarason, Complex function theory, 2nd Edn., Hindustan book agency, 2007.
4. J.B. Conway, Functions of One Complex Variable, 2nd Edn., Narosa, 1973.

PROBABILITY THEORY [3-1-0-8]

---

Axiomatic definition of probability, probability spaces, probability measures on countable and uncountable spaces, conditional probability, independence; Random variables, distribution functions, probability mass and density functions, functions of random variables, standard univariate discrete and continuous distributions and their properties; Mathematical expectations, moments, moment generating functions, characteristic functions, inequalities; Random vectors, joint, marginal and conditional distributions, conditional expectations, independence, covariance, correlation, standard multivariate distributions, functions of random vectors; Modes of convergence of sequences of random variables, weak and strong laws of large numbers, central limit theorems; Introduction to stochastic processes, definitions and examples.

Texts/References:

1. J. Jacod and P. Protter, Probability Essentials, Springer, 2004.
2. V. K. Rohatgi and A. K. Md. E. Saleh, An Introduction to Probability and Statistics, 2nd Edn., Wiley, 2001.

References:

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 Edn., Oxford University Press, 2001.
3. S. Ross, A First Course in Probability, 6th Edn., Pearson, 2002.
4. W. Feller, An Introduction to Probability Theory and its Applications, Vol. 1, 3rd Edn., Wiley, 1968.
5. J. Rosenthal, A First Look at Rigorous Probability Theory, 2nd Edn., World Scientific, 2006.

Optimization Techniques [3-1-0-8]

---

Mathematical foundations and basic definitions: concepts from linear algebra, geometry, and multivariable calculus. Linear optimization: formulation and geometrical ideas of linear programming problems, simplex method, revised simplex method, duality, sensitivity snalysis, transportation and assignment problems. Nonlinear optimization: basic theory, method of Lagrange multipliers, Karush-Kuhn-Tucker theory, convex optimization. Numerical optimization techniques: line search methods, gradient methods, Newton's method, conjugate direction methods, quasi-Newton methods, projected gradient methods, penalty methods.

Texts:

1. N. S. Kambo, Mathematical Programming Techniques, East West Press, 1997.
2. E.K.P. Chong and S.H. Zak, An Introduction to Optimization, 2nd Ed., Wiley, 2010.

References:

1. R. Fletcher, Practical Methods of Optimization, 2nd Ed., John Wiley, 2009.
2. D. G. Luenberger and Y. Ye, Linear and Nonlinear Programming, 3rd Ed., Springer India, 2010.
3. M. S. Bazarra, J.J. Jarvis, and H.D. Sherali, Linear Programming and Network Flows, 4th Ed., 2010. (3nd ed. Wiley India 2008).
4. U. Faigle, W. Kern, and G. Still, Algorithmic Principles of Mathematical Programming, Kluwe, 2002.
5. D.P. Bertsekas, Nonlinear Programming, 2nd Ed., Athena Scientific, 1999.
6. M. S. Bazarra, H.D. Sherali, and C. M. Shetty, Nonlinear Programming: Theory and Algorithms, 3rd Ed., Wiley, 2006. (2nd Edn., Wiley India, 2004).

Theory of Computation[3-1-0-8]

---

Prerequistes: MA 501 Discrete Mathematics

Alphabets, languages, grammars; Finite automata, regular languages, regular expressions; Context-free languages, pushdown automata, DCFLs; Context sensitive languages, linear bounded automata; Turing machines, recursively enumerable languages; Operations on formal languages and their properties; Chomsky hierarchy; Decision questions on languages; NP-Completeness.

Texts:

1. M. Sipser, Introduction to the Theory of Computation, Thomson, 2004.
2. H. R. Lewis and C. H. Papadimitriou, Elements of the Theory of Computation, PHI, 1981.

References:

1. J. E. Hopcroft and J. D. Ullman, Introduction to Automata Theory, Languages and Computation, Narosa, 1979.
2. Peter Linz, An Introduction to Formal Languages and Automata, Narosa, 2007.
3. D. C. Kozen, Automata and Computability, Springer-Verlag, 1997.
4. D. S. Garey and G. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, Freeman, New York, 1979.

Functional Analysis[3-1-0-8]

---

Prerequisites: MA541 Real Analysis

Normed linear spaces, Banach spaces; Continuity of linear maps, Hahn-Banach theorem, open mapping and closed graph theorems, uniform boundedness principle; Duals and transposes, weak and weak* convergence, reflexivity; Spectra of bounded linear operators, compact operators and their spectra; Hilbert spaces, bounded linear operators on Hilbert spaces; Adjoint operators, normal, unitary, self-adjoint operators and their spectra, spectral theorem for compact self-adjoint operators.

Texts:

1. B. V. Limaye, Functional Analysis, 2nd edition, Wiley Eastern, 1996.
2. E. Kreyszig, Introduction to Functional Analysis with Applications, John Wiley and Sons, 1978.

References:

1. J.B. Conway, A Course in Functional Analysis, Springer, 1990.

Numerical Analysis[3-0-2-8]

---

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

Texts:

1. D. Kincaid and W. Cheney, Numerical Analysis: Mathematics of Scientific Computing, 3rd Edn., AMS, 2002.
2. K. E. Atkinson, Introduction to Numerical Analysis, 2nd Edn., John Wiley, 1989.

References:

1. S. D. Conte and Carl de Boor, Elementary Numerical Analysis - An Algorithmic Approach, 3rd Edn., McGraw Hill, 1980

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

---

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

---

[-]

---

Numerical Linear Algebra[3-0-2-8]

---

Prerequisites: MA522 Linear Algebra

Fundamentals - overview of matrix computations, norms of vectors and matrices, singular value decomposition (SVD), IEEE floating point arithmetic, analysis of roundoff errors, stability and ill-conditioning; Linear systems - LU factorization, Gaussian eliminations, Cholesky factorization, stability and sensitivity analysis; Jacobi, Gauss-Seidel and successive overrelaxation methods; Linear least-squares - Gram- Schmidt orthonormal process, rotators and reflectors, QR factorization, stability of QR factorization; QR method linear least-squares problems, normal equations, Moore- Penrose inverse, rank deficient least-squares problems, sensitivity analysis. Eigenvalues and singular values - Schur's decomposition, reduction of matrices to Hessenberg and tridiagonal forms; Power, inverse power and Rayleigh quotient iterations; QR algorithm, implementation of implicit QR algorithm; Sensitivity analysis of eiegnvalues; Reduction of matrices to bidiagonal forms, QR algorithm for SVD.

Software Support: MATLAB.

Texts:

1. L. N. Trefethen and David Bau, Numerical Linear Algebra, SIAM, 1997.
2. D. S. Watkins, Fundamentals of Matrix Computation, 2nd Edn., Wiley, 2002.

References:

1. J.W. Demmel, Applied Numerical Linear Algebra, SIAM, 1997.
2. B. N. Datta, Numerical Linear Algebra and Applications, 2nd Edn., SIAM, 2010.
3. G. H. Golub and C.F.Van Loan, Matrix Computation, 3rd Edn., Hindustan book agency, 2007.

Numerics of Partial Differential Equations[3-0-2-8]

---

Prerequisites: MA 542 Differential Equations

Finite difference schemes for partial differential equations - explicit and implicit schemes; Consistency, stability and convergence - stability analysis by matrix method and von Neumann method, Lax's equivalence theorem; Finite difference schemes for initial and boundary value problems - FTCS, backward Euler and Crank-Nicolson schemes, ADI methods, Lax Wendroff method, upwind scheme; CFL conditions; Finite element method for ordinary differential equations - variational methods, method of weighted residuals, finite element analysis of one-dimensional problems.

Texts:

1. G. D. Smith, Numerical Solutions to Partial Differential Equations, Oxford University Press, 3rd Edn., 1986.
2. J. C. Strikwerda, Finite Difference Schemes and Partial Differential Equations, SIAM, 2004.
3. J. N. Reddy, An Introduction to Finite Element Method, 3rd Edn., McGraw Hill, 2005.

References:

1. L. Lapidus and G. F. Pinder, Numerical Solution of Partial Differential Equations in Science and Engineering, John Wiley, 1982.
2. K. W. Morton and D. F. Mayers, Numerical Solution of Partial Differential Equations, Cambridge University Press, 2nd Edn., 2005.
3. C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Dover Publications, 2009.

Elective-III [3-0-0 -6]

---

Elective-IV/Open Elective[3-0-0 -6]

---

Project[0-0-14-14]

---