Code | Course Name | L–T-P | Credits |

MA511 | Computer Programming | 3-0-2 | 8 |

MA501 | Discrete Mathematics | 3-1-0 | 8 |

MA521 | Modern Algebra | 3-1-0 | 8 |

MA522 | Linear Algebra | 3-1-0 | 8 |

MA541 | Real Analysis | 3-1-0 | 8 |

Code | Course Name | L–T-P | Credits |

MA547 | Complex Analysis | 3-1-0 | 8 |

MA548 | Ordinary Differential Equations | 3-1-0 | 8 |

MA549 | Topology | 3-1-0 | 8 |

MA591 | Optimization Techniques | 3-1-0 | 8 |

MA590 | PROBABILITY THEORY | 3-1-0 | 8 |

Code | Course Name | L–T-P | Credits |

MA543 | Functional Analysis | 3-1-0 | 8 |

MA567 | Partial Differential Equations | 3-1-0 | 8 |

MA572 | Numerical Analysis | 3-0-2 | 8 |

MA550 | Measure Theory | 3-1-0 | 8 |

MA---- | Elective | 3-0-0 | 6 |

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:

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

References:

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

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:

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

References:

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

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:

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

References:

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

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:

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

References:

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

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:

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

References:

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

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:

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

References:

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

Prerequisites: Nil

First order non-linear differential equations: Existence and Uniqueness problem, Gronwall’s inequality, Peano existence theorem, Picard existence and uniqueness theorem, interval of definition. Second order linear differential equations: general solution for homogeneous equations, superposition of solutions, methods of solution for non-homogeneous problems, undetermined coefficients, variation of parameters, series solutions for ODEs, types of singularity, solution at an ordinary point, solution at a singular point. nth order linear differential equations: system of equations, homogeneous system of equations, fundamental matrix, Abel-Liouville formula, system of non-homogeneous equations, stability of linear systems. Theory of two-point BVP: Green’s functions, properties of Green’s functions, Adjoint and self-adjoint BVP. Strum-Liouville’s problem, orthogonal functions, eigenvalues and eigen functions, completeness of the eigen functions.

Texts:

- Boyce, W. E. and DiPrima, R. C., Elementary Differential Equation and Boundary Value Problems, 7th Edition, John Wiley & Sons (Asia), 2001.
- Ross, S. L., Differential Equations, 3rd edition, Wiley 1984.

References:

- Simmons, G. F., Differential Equations with Applications and Historical Notes, McGraw Hill, 1991
- Coddington, E. A., An Introduction to Ordinary Differential Equations, Prentice- Hall, 1974.
- Farlow, S. J., An Introduction to Differential Equations and Their Applications, McGraw-Hill International Editions, 1994.

Prerequisites: MA541 Real Analysis

Topological spaces, Bases and sub-bases for a topology, Limit point, closure, interior, boundary of a set, dense and nowhere dense sets, Continuity, Homeomorphism, Subspace, Product and Quotient topologies. Countability axioms, Separation axioms. Connectedness; Components, path connectedness, locally connected spaces, totally disconnected spaces. Compactness; Tychonoff’s theorem, locally compact spaces, one-point compactification. Urysohn’s lemma, Tietze’s extension theorem, Urysohn’s metrization theorem.

Texts:

- J. R. Munkres: Topology, Pearson India, 2015.

References:

- C. W. Patty, Foundations of Topology, Second Edition, Jones & Barlett Student Edition, 2010.
- G. F. Simmons: Introduction to Topology and Modern Analysis, McGraw-Hill India, 2017
- S. Willard: General Topology, Dover, 2004.

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:

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

References:

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

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:

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

References:

- P. G. Hoel, S. C. Port and C. J. Stone, Introduction to Probability Theory, Universal Book Stall, 2000.
- G. R. Grimmett and D. R. Stirzaker, Probability and Random Processes, 3rd Edn., Oxford University Press, 2001.
- S. Ross, A First Course in Probability, 6th Edn., Pearson, 2002.
- W. Feller, An Introduction to Probability Theory and its Applications, Vol. 1, 3rd Edn., Wiley, 1968.
- J. Rosenthal, A First Look at Rigorous Probability Theory, 2nd Edn., World Scientific, 2006.

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:

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

References:

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

Prerequisites: MA541 Real Analysis MA548 ODE

First-Order PDE: Cauchy problem, Method of Characteristics, Lagrange Method, Compatible systems of first-order PDEs, Charpit’s method. Classification of second order PDEs, Canonical Transformations, Characteristics, Canonical forms, characteristics, Propagation of Singularities. Wave Equation: Wave equation in one dimension; Infinite, Semi-infinite, and finite String Problem, D'Alembert's formula, Wave equation in higher dimension; Solutions by spherical means, Inhomogeneous Problems; Duhamel’s principle. Laplace Equation: Fundamental solution, Mean value property, maximum principle, Green's function, Poisson's integral formula, Harnack's inequality, Dirichlet's principle. Heat equation: One dimensional homogeneous heat equation; infinite and semi-infinite rod, Inhomogeneous problems; Duhamel's Principle, Fundamental solution, Maximum principle. Separation of variables: Fourier series expansions, Solutions by separation of Variables method for Wave Equation, Heat Equation and Laplace Equation, BVP in different coordinate systems. Transform methods: Fourier Transforms, Properties of Fourier Transforms, Convolution, Fourier Sine and Cosine Transforms, Application to Initial BVP; Laplace Transforms: Properties of Laplace Transforms, Convolution, Application to initial BVP. Uniqueness ofsolutions of hyperbolic, elliptic, and parabolic equations by Energy Methods.

Texts:

- I. N. Sneddon, Elements of Partial Differential Equations, Dover Publications, 2006.
- R. McOwen, Partial Differential Equations: Methods and Applications, Pearson, 2002.

References:

- Phoolan Prasad and R. Ravindran, Partial Differential equations, New Age International, 2011.
- Qing Han, A Basic Course in Partial Differential Equations, AMS, 2011.
- L.C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, Vol. 19, American Mathematical Society, 1998
- Tyn Myint-U and Lokenath Debnath, Linear Partial Differential Equations for Scientists and Engineers, Birkhauser, 2007.

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:

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

References:

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

Prerequisites: MA541 Real Analysis

Lebesgue outer measure, Lebesgue measurable sets, Lebesgue measure. Algebra and sigma-algebra of sets, Borel sets, Outer measures and measures, Caratheodory construction. Measurable functions, Lusin’s theorem, Egoroff’s theorem. Integration of measurable functions, Monotone convergence theorem, Fatou’s lemma, Dominated convergence theorem. Lp-spaces. Product measure, Fubini’s theorem. Absolutely continuous functions, Fundamental theorem of calculus for Lebesgue integral. Radon-Nikodym theorem. Riesz representation theorem

Texts:

- G. de Barra: Measure Theory and Integration, New Age Publishers, 1st ed, 2013.
- H. L. Royden & P. M. Fitzpatrick: Real Analysis, Pearson India, 2015

References:

- D. L. Cohn: Measure Theory, Birkhauser, 1994
- W. Rudin: Real and Complex Analysis, McGraw-Hill India, 2017
- G. B. Folland: Real Analysis (2nd ed.), Wiley, 1999