Elective Courses for Ph.D.

Programming and Data Structures and Algorithms Flowcharts, C Programming: basic features, arrays and pointers, recursion, records (structures), memory management, files, input/output, Asymptotic complexity, heap sort, quick sort, sorting in linear time, elementary data structures, hash tables, binary search trees, Red-Black trees.

Texts/References:

1. B. S. Gottfried, Schaum's Outline Programming with C, McGraw Hill, 1996.
2. Cormen, Leiserson and Rivest, Introduction to Algorithms, Prentice Hall of India, 2002.

Models of Computation: space and time complexity measures, lower and upper bounds; Design techniques: greedy method, divide-and-conquer, dynamic programming; Amortized analysis: basic techniques, analysis of Fibonacci heap and disjoint-set forest; Graph algorithms: connectivity, topological sort, minimum spanning trees, shortest paths, network flow; String matching; Average-case analysis; NP-completeness.

Texts:

1. T.H. Cormen, C.E. Leiserson, R. L. Rivest and C. Stein, Introduction to Algorithms, MIT Press, 2009.

References:

1. J. Kleinberg and E. Tardos, Algorithm Design, Addison Wesley, 2006.
3. S. Dasgupta, C. Papadimitriou and U. Vazirani, Algorithms, McGraw-Hill, 2007.
A. Aho, J.E. Hopcroft and J.D. Ullman, The Design and Analysis of Computer Algorithms, Pearson, 2002.

Review of sets, functions, relations; Logic: formulae, interpretations, methods of proof in propositional and predicate logic; Number theory: division algorithm, Euclid's algorithm, fundamental theorem of arithmetic, Chinese remainder theorem; Combinatorics: permutations, combinations, partitions, recurrences, generating functions; Graph Theory: isomorphism, complete graphs, bipartite graphs, matchings, colourability, planarity; Probability: conditional probability, random variables, probability distributions, tail inequalities.

Texts:

1. E. Lehman, F.T. Leighton and A.R. Meyer, Mathematics for Computer Science, Creative Commons, 2017.
2. K.H. Rosen, Discrete Mathematics and its Applications, 7th Edn., McGraw Hill Education, 2011.

References:

1. R.L. Graham, D.E. Knuth and O. Patashnik, Concrete Mathematics, 2nd Edn., Addison-Wesley, 1994.
2. D.M. Burton, Elementary Number Theory, 7th Edn., McGraw-Hill Higher Education, 2010.
3. R. Diestel, Graph Theory, 4th Edn.,, Springer, 2010.
4. W. Feller, An introduction to probability theory and its Applications Vol. 1, 3rd Edn., Wiley, 2008.
5. S.M. Ross, A First Course in Probability, 9th Edn., Pearson, 2012.

Assignments are designed to provide hands-on experience in programming the following data structures and algorithms using the C programming language. Elementary data structures: arrays, matrices, linked lists, stacks, queues, binary trees, tree traversals; Balanced binary search trees: red-black trees, B-trees; Priority queues: binary heap; Sorting and searching: bubble, insertion, merge, quick sort, heap sort, binary search; Hashing; Strings: tries, suffix arrays, suffix trees; Sets: linked-list, disjoint-set forest; Graphs: adjacency list, adjacency matrix, depth first search, breadth first search.

Texts:

1. M.A. Weiss, Data Structures and Algorithm Analysis in C, Pearson, 2002.
2. T.H. Cormen, C.E. Leiserson, R.L. Rivest and C. Stein, Introduction to Algorithms, MIT Press, 2009.

References:

1. E. Horowitz and S. Sahni, Fundamentals of Data Structures in C, University Press, 2008.
2. R. Sedgewick, Algorithms in C Parts 1-4: Fundamentals, Data Structures, Sorting, Searching, Pearson, 1997.
3. R. Sedgewick, Algorithms in C Part 5: Graph Algorithms, Pearson, 2001.
4. M.T. Goodrich, and R. Tamassia, Data Structures and Algorithms in C++, Wiley India, 2007.
5. B.W. Kernighan and D.M. Ritchie, The C Programming Language, Prentice Hall India, 1990.

Vector spaces: subspaces, sums and direct sums; Finite dimensional vector spaces: bases and dimensions; Linear maps: null-spaces and range, invertibility; Polynomials with real and complex coefficients; Eigenvalues and eigenvectors: triangularization and diagonalization of operators on finite dimensional vector spaces; Inner-product spaces: orthonormal bases, linear functional and adjoins; Operators on inner-product spaces: self-adjoint and normal operators, minimal polynomial, Jordan form; Traces and determinants of operators and matrices.

Texts/References:

1. S. Axler, Linear Algebra Done Right, 2nd Edition, UTM, Springer, Indian Edition, 2010.
2. K. Hoffman and R. Kunze, Linear Algebra, Prentice Hall of India, 1996.
3. G. Strang, Introduction to Linear Algebra, 4th Edition, Wellesley Cambridge Press, 2009.

Completeness properties of real numbers, countable and uncountable sets, cardinality. Norms and metrics: Metric spaces, convergence of sequences, completeness, connectedness and sequential compactness; Continuity and uniform continuity; sequences and series of functions, uniform convergence, equicontinuity, Ascoli's theorem, Weierstrass approximation theorem, power series. Calculus of functions of a real variable: Differentiability, Mean value theorems, Taylor's theorem. Calculus of functions of several real variables: Partial and directional derivatives, differentiability, Chain Rule, Taylor's theorem, Maxima and Minima, Lagrange multipliers, Inverse function theorem, Implicit function theorem. Multiple Integration: Fubini's Theorem, Line integrals, Surface integrals, Green, Gauss and Stokes theorems.

Texts/References:

1. J. E. Marsden and M. J. Hoffman, Elementary Classical Analysis, 2nd Edition, W. H. Freeman, 1993.
2. P. M. Fitzpatrick, Advanced Calculus, 2nd Edition, AMS, Indian Edition, 2010.
3. N. L. Carothers, Real Analysis, Cambridge University Press, Indian Edition, 2009.
4. W. Rudin, Principles of Mathematical Analysis, 3rd Edition, McGraw Hill, 1976.

Existence and Uniqueness of Initial Value Problems: Picard's and Peano's Theorems, Gronwall's inequality, continuous dependence, maximal interval of existence. Second and Higher Order Linear Equations: Fundamental solutions, Wronskian, variation of constants, behaviour of solutions. Power series method with properties of Legendre polynomials and Bessel functions. Linear Systems: Autonomous Systems and Phase Space Analysis, matrix exponential solution, critical points, proper and improper nodes, spiral points and saddle points.

First Order Partial Differential Equations: Classification, Method of characteristics for quasi-linear and nonlinear equations, Cauchy's problem, Cauchy-Kowalewski's Theorem. Second-Order Partial Differential Equations: Classification, normal forms and characteristics, Well-posed problem, Stability theory, energy conservation, and dispersion, Adjoint differential operators. Laplace Equation: Maximum and Minimum principle, Green's identity and uniqueness by energy methods, Fundamental solution, Poisson's integral formula, Mean value property, Green's function. Heat Equation: Maximum and Minimum Principle, Duhamel's principle. Wave equation: D'Alembert solution, method of spherical means and Duhamel's principle. The Method of separation of variables for heat, Laplace and wave equations.

Texts/References:

1. G. F. Simmons and S. G. Krantz, Differential Equations: Theory, Technique, and Practice, McGraw Hill, 2006.
2. E. A. Coddington, An Introduction to Ordinary Differential Equations, Prentice Hall India, 1995.
3. E. L. Ince, Ordinary Differential Equations, Dover Publications, 1958.
4. S. L. Ross, Differential Equations, 3rd edition, Wiley India, 1984.
5. I. N. Sneddon, Elements of Partial Differential Equations, McGraw Hill, 1957.
6. S. J. Farlow, Partial Differential Equations for Scientists and Engineers, Dover Publications, 1993.
7. R. Haberman, Elementary Applied Partial Differential equations with Fourier Series and Boundary Value Problem, 4th Edition, Prentice Hall, 1998.
8. Fritz John, Partial Differential Equations, Springer-Verlag, Berlin, 1982.

Theory of Sobolev spaces: Motivation, weak derivative, definition and basic properties of Sobolev spaces, extension theorem, embedding and compactness theorems, Dual spaces, fractional order Sobolev spaces, Trace theory. Methods for solving Elliptic boundary value problems: Weak solution of elliptic boundary value problem, regularity of weak solutions, maximum principle, eigenvalue problems. Solving elliptic boundary value problem using fixed point method, Galerkin method, monotone iteration method and variational method.

Texts/References:

1. R. A. Adams and J.J.F. Fournier, Sobolev Spaces, Academic Press, 2003.
2. H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2011.
3. S. Kesavan, Topics in Functional Analysis and Applications, New Age International Private Limited, 2015.

Probability spaces, probability measures on countable and uncountable spaces, conditional probability, independence; Random variables and vectors, distribution functions, functions of random vectors, standard univariate and multivariate discrete and continuous distributions and their properties; Mathematical expectations, moments, moment generating functions, characteristic functions, inequalities, conditional expectations, covariance, correlation; 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, Markov chains, Poisson processes, Brownian motion.

Texts/References:

1. J. Jacod and P. Protter, Probability Essentials, Springer, 2004.
2. S. Ross, A First Course in Probability, 6th edition, Pearson, 2002.
3. G. R. Grimmett and D. R. Stirzaker, Probability and Random Processes, 3rd edition, Oxford University Press, 2001.
4. J. Rosenthal, A First Look at Rigorous Probability Theory, 2nd edition, World Scientific, 2006.
5. P. G. Hoel, S. C. Port and C. J. Stone, Introduction to Probability Theory, Universal Book Stall, 2000.
6. V. K. Rohatgi and A. K. Md. E. Saleh, An Introduction to Probability and Statistics, 2nd edition, Wiley, 2001.
7. W. Feller, An Introduction to Probability Theory and its Applications, Vol. 1, 3rd Edition, Wiley, 1968.

Measure theoretic formulation of probability; Random variables, Distributions, Expectations, Modes of convergence; Laws of large numbers:the weak and strong laws; Characteristic functions, Convergence in distribution, Central limit theorems; Markov chains, Random walks, Poisson process; Conditional expectations; Discrete parameter martingales,Convergence theorems for martingales; Brownian motion, Strong Markov property, Introduction to sample path properties.

Texts:

1. J.B. Walsh, Knowing the Odds: An Introduction to Probability, American Mathematical Society, 2012.
2. K. B. Athreya and S. N. Lahiri, Probability Theory, TRIM series, 2006.

References:

1. P. Billingsley, Probability and Measure, Wiley India, 3rd Edn., 2008.
2. R. Durrett, Probability: Theory and Examples, Cambridge University Press, 4th Edn.,2010.
3. O. Kallenberg, Foundations of Modern Probability, Springer, 2nd Edn., 2002.

Eigenvalues and eigenvectors, Schur's theorem - real and complex versions, Spectral theorems for normal and Hermitian matrices - real and complex versions. Gerschgorin discs with associated perturbation theorems and inclusion results. Jordan canonical forms with application, minimal polynomials, companion matrices. Functions of matrices via spectral decompositions. Variational characterizations of eigenvalues of Hermitian matrices, Rayleigh-Ritz theorem, Courant-Fischer theorem, Weyl theorem, Cauchy interlacing theorem, Inertia and congruence, Sylvester's law of inertia. Matrix norms, spectral radius formula, relationships between matrix norms. Singular value decomposition, polar decomposition. Positive definite matrices, characterizations of definiteness, polar form and singular value decompositions, congruence and simultaneous diagonalization.

Texts:

1. R. A. Horn and C. R. Johnson, Matrix Analysis, CUP, 1985.
2. S. Axler, Linear Algebra Done Right, 2nd Edition, UTM, Springer, Indian Edition, 2010.
3. P. Lancaster and M. Tismenetsky, The Theory of Matrices, Second edition, Academic Press, 1985.
4. F. R. Gantmacher, The Theory of Matrices, Vol-I, Chelsea, 1959.

Arithmetic functions, Elementary theorems on distribution of prime numbers, Dirichlet's Theorem on primes in arithmetic progressions, Dirichlet series and Euler products, Zeta functions, Prime number theorem.

Number fields and Number rings, Prime decomposition in number rings, The ideal class group and the unit group, Dedekind zeta function and the class number formula, The distribution of primes and class field theory.

Texts:

1. Apostol, T.M., Introduction to Analytic Number Theory, UTM, Springer, 1976.
2. Marcus, D. A., Number Fields, Springer Verlag, 1977.
3. Janusz G.J. , Algebraic Number Fields, GSM, Vol-7 (2nd Ed.) AMS, 1996.

Basic notions from point-set-topology, Direct product and inverse limits, Topological groups, completions, inverse system of finite groups, Profinite groups.

Nilpotent and Solvable groups, Frattini subgroups, Chief factors, Free groups, Presentations, Matrix groups and their properties, Profinite and Pro-p-groups.

The topology of pointwise convergence.

Texts:

1. I. D. Macdonalds, Theory of Groups, Oxford University Press, 1968
2. Dixon, Linear Groups, Dover
3. D. L. Johnson, Presentation of Groups, LMS series, CUP, 1990.
4. Dixon, Man, Segal, Analytic Pro-p Groups, LMS Series, Cambridge University Press
5. James R. Munkres, Topology: a first courses, Prentice-Hall, 1992.
6. P.J. Higgins, Introduction to topological groups, LMS Lecture series 15, CUP, 1974

Group actions, Orbits and Stabilizer, Transitivity, Primitivity, Suborbits and Orbits, Linear groups, Wreath products, Structure of primitive groups and symmetric groups, Groups acting on trees, Jordan groups.

Texts:

1. Dixon and Motimer, Permutation Groups, Springer GTM, 1996.
2. Bhattacharjee et-al, Notes on Infinite Permutation Groups, Springer, 1997.
3. Cameron, Permutation Groups, LMS Series, CUP 1999.

Generators and Relations, Free groups, Subgroups of a free group, Presentation of groups, Dehn's fundamental problems, Tietze transformations, Cayley graph of a group, Free products, Generalized free products, HNN extensions, Wreath Products, Commutator calculus, Residual Properties.

Texts:

1. Baumslag, G., Topics in Combinatorial Group Theory, Lecture Notes.
2. Magnus, Karrass and Solitar, Combinatorial Group Theory, Dover, 1976.
3. Lyndon and Schupp, Combinatorial Group Theory, Springer 1977.
4. Johnson D.L., Presentation of Groups, LMS Series, CUP 1990.

Cones, Spectral properties of matrices which leave a cone invariant, Cone primitivity, Irreducible matrices, Reducible matrices, Primitive matrices, Stochastic matrices, Algebraic semigroups, Nonnegative idempotents, The semigroup $N_n$, The semigroup $D_n$, Inverse eigenvalue problems, Nonnegative matrices with given sums.

Texts:

1. Berman and Plemmons, Nonnegative Matrices in the Mathematical Sciences, SIAM, 1994.
2. Bapat and Raghavan, Nonnegative Matrices and Applications, CUP, 1997.

Congruences: linear and polynomial congruences; prime numbers: counting primes, numbers of special forms, pseudo-primes and primality testing; factorization: factorization algorithms; arithmetic functions: multiplicative and additive functions, Euler's phi function, sum and number of divisors functions, the Mobius function and other important arithmetic functions, Dirichlet products; primitive roots and quadratic residues: primitive roots, index arithmetic, quadratic residues, modular square roots; Diophantine equations: linear Diophantine equations, Pythagorean triples, Fermat's last theorem, Tell's, Bachet's and Catalan's equations, sums of squares and Waring's problem; Diophantine approximations: continued fractions, convergent, approximation theorems; quadratic fields: primes and unique factorization.

References:

1. Kenneth H. Rosen, J.G. Michaels, J.L. Gross, J.W. Grossman, D.R. Shier, Handbook of Discrete and Combinatorial Mathematics, CRC Press, 1999
2. I. Niven, H.S. Zuckerman, H.L. Montgomery, An Introduction to the Theory of Numbers, Wiley, 1991.
3. K. Chandrasekaran, An Introduction to Analytic Number Theory, Springer, 1968.

Modern Algebra: Group theory - normal series, solvable groups, nilpotent groups; Ring theory - rings and modules, prime ideals, nil and Jacobson radicals, finitely generated modules, exact sequences, tensor products, primary decomposition, Noetherian rings, Artin rings; Field theory - field extensions, automorphism groups, Galois theory.

Number Theory: congruences, residue systems, Chinese remainder theorem; quadratic residues, reciprocity law; arithmetic functions-Euler function, Mobius function; continued fractions; quadratic forms; Diophantine equations; partitions; Riemann zeta function.

Texts/References:

1. D. S. Dummit and R. M. Foot, Abstract Algebra, John Wiley & Sons, Inc., II Edition, 1999.
2. S. Lang, Algebra, III edition, Springer, 2004.
3. C. Musili, Introduction to Rings and Modules, Narosa Publishing Company, 1997.
4. I. Stewart, Galois Theory, Academic Press, 1989.5. M. F. Atiyah and I. G. MacDonald, Introduction to Commutative Algebra, Addison Wesley, 1969.
5. D M. Burton, Elementary Number Theory, WC Brown Publishers, 1994.
6. T. M. Apostol, Introduction to Analytic Number Theory, Narosa Publishing House, 1995 (or Springer, 1976).
7. A. Baker, A Concise Introduction to the Theory of Numbers, Cambridge University Press, 1984.

Complete lattices, equivalence relations, algebras, subalgebras, homomorphism theorems, products, congruences, free algebras, term algebras, identities, subdirect products, subdirectly irreducible algebras, class operators, varieties, Birkhoff's variety theorem, congruence varieties, equational logic, equational theories, bases of varieties, many-sorted algebras.

Texts:

1. S. Burris and H. P. Sanppanavar, A Course in Universal Algebra, Springer, 1982.
2. G. Gratzer, Universal Algebra, II ed., Springer, 1979.
3. W. Wechler, Universal Algebra for Computer Scientists, Springer, 1992.
4. H. Ehrig and B. Mahr, Fundamentals of Algebraic Specification 1:Equations and Initial Semantics, Springer, 1985

Prerequisites: Nil

Review of normed spaces and Banach spaces; Bounded linear operators, Hahn-Banach theorem, Dual spaces, Open mapping theorem, Closed graph theorem, Uniform boundedness principle; Weak and weak* topologies, Alaoglu theorem; Compact operators, Riesz theory for compact operators; Spectra of bounded linear operators, Gelfand-Mazur Theorem.

Review of inner product spaces and Hilbert spaces; Orthonormal bases, Riesz representation theorem; Adjoint of a bounded linear operator, Self-adjoint, normal and unitary operators; Spectral theorem of compact self-adjoint/normal operators.

Texts/References:

1. R. Bhatia: Notes on Functional Analysis (Hindustan Book Agency)
2. J. B. Conway: A Course in Functional Analysis, 2nd edition (Springer low price edition)
3. B. V. Limaye: Functional Analysis, 3rd edition (New Age Publishers)
4. M. Schechter: Principles of Functional Analysis, 2nd edition (Universities Press)

Prerequisites: Nil

Banach space valued analytic functions, bounded linear operators in Banach spaces, adjoint operators, projection operators. Resolvent and spectra of bounded linear operators, isolated spectral values, spectral projections and spectral decompositions, spectra of adjoint operators. Linear and analytic perturbation theory for discrete spectra, continuity and analyticity of the resolvent operators, perturbation of the spectral projections, perturbation series for spectral projections and eigenvalues, a theorem of Motzkin-Taussky.

Texts/References:

1. B. V. Limaye, Spectral Perturbation and Approximation with Numerical Experiments, Proceedings of CMA, Australian National University, Vol.13, 1986.
2. T. Kato, Perturbation Theory for Linear operators, Springer, 1980.
3. G.W. Stewart and J. G. Sun, Matrix Perturbation Theory, Academic Press, 1990.

Prerequisites: Nil

Matrix valued analytic functions, projections. Invariant subspaces and spectral decompositions, singular value decompositions, pairs of projections. Matrix norms, unitarily invariant norms, gap between subspaces. General perturbation theory for matrices, Bauer-Fike and Henrici theorems, Hoffman-Wielandt theorem, analytic perturbation theory for eigenvalues. Invariant subspaces, Sylvester operator, perturbation of spectral subspaces. Matrix pencils, eigenvalues and eigenvectors of regular pencils, triangular and Weierstrass forms, Kronecker canonical form, deflating subspaces, definite pencils, perturbation of eigenvalues of regular pencils.

Texts/References:

1. G.W. Stewart and J. G. Sun, Matrix Perturbation Theory, Academic Press, 1990
2. B. V. Limaye, Spectral Perturbation and Approximation with Numerical Experiments, Proceedings of CMA, Australian National University, Vol.13, 1986.
3. T. Kato, Perturbation Theory for Linear operators, Springer, 1980.
4. F. R. Gantmacher, Applications of the Theory of Matrices, Dover, 2005.

Prerequisites: Nil

Analytic functions, properties of elementary analytic functions. Complex integration, Cauchy's theorem, Liouville's theorem, power series representation, open mapping theorem, calculus of residues. Harmonic functions, Poisson integral, Harnack's theorem, Schwarz reflection principle. Maximum modulus principle, Schwarz lemma, Phragmen-Lindelof method. Runge's theorem, Mittag-Leffler theorem, Weierstrass theorem, Jensen's formula, Hadamard's theorem. Analytic continuation, monodromy theorem. Equicontinuous, Normal families, Arzela's theorem, Riemann mapping theorem and its consequences.

Texts/References:

1. L. V. Ahlfors, Complex Analysis, McGraw-Hill, 1979.
2. J. B. Conway, Functions of One Complex Variable, 2nd Edition, Springer/Narosa, 1978.
3. S. Lang, Complex Analysis, 4th Edition, Springer, 1999.
4. R. Narasimhan and Y. Nievergelt, Complex Analysis in One Variable, Birkhauser, 2001.
5. R. Remmert, Theory of Complex Functions, Springer (India), 2005.
6. T. W. Gamelin, Complex Analysis, UTM, Springer, 2003.

Chordal and spherical metrics, Normal families, Iteration of polynomials and rational functions, Periodic points and orbits, Singular values, Julia and Fatou sets and their characterizations, Mandelbrot set, Dynamics of transcendental entire functions, Bifurcation and Chaotic burst in the dynamics, Iteration of certain meromorphic functions, Julia sets and Fractals, Self-similarity and Fractal dimensions.

Texts:

1. J. L. Schiff, Normal Families, Springer Verlag, 1993.
2. L. Carleson and T.W. Gamelin, Complex Dynamics, Springer Verlag, 1993.
3. F. Beardon, Iteration of Rational Functions, Springer Verlag, 1991.
4. W. Bergweiler, Iteration of Meromorphic Functions, Bulletin of American Mathematical Society, Volume 29, No. 2, Pages 151-188, 1993.
5. M. F. Barnsley, Fractals Everywhere, 2nd edition, Academic Press, 1995.

Test Function and distribution: Definition, operations with distributions, convolution of distributions, Fourier transform of tempered distributions.

Sobolev spaces: Definition and properties, extension theorem, imbedding and completeness theorem, fractional order Sobolev spaces, trace theory. Application to Elliptic Problems: Weak solution of elliptic boundary value problem (BVP), regularity of weak solutions, maximum principle, eigenvalue problems, fixed point theorems and their application in semilinear elliptic BVP.

Texts:

1. R. A. Adams, Sobolev Spaces, Academic Press, 1975.
2. J. T. Oden and J.N. Reddy, An Introduction to Mathematical Theory of Finite Elements, Wiley Interscience, 1976.
3. S. Kesavan, Topics in Functional Analysis and Applications, Wiley Eastern Ltd. , New Delhi, 1989.
4. K. E. Brennan and R. Scott, The Mathematical Theory of Finite Element Methods, Springer-Verlag, Berlin, 1994.

Orthogonal systems, Trigonometric system, Fourier series in these systems, Uniqueness and convergence, Fourier series of continuous and smooth functions, $L^2$ theory of Fourier series - inversion formula and the Parseval identity, Fourier analysis and complex function theory, Paley Wiener's theorem, Tauberian theorem, Dirichlet problem, Bessel functions, Orthogonal polynomials, Fourier analysis and filters, Fourier transforms and distributions.

Texts:

1. Dym, I. H. and Mc Kean, H. P., Fourier Series and Integrals, Academic Press, 1985.
2. Folland G.B., Fourier Analysis and Applications, Brooks/ Cole Mathematics Series, 1972.
3. Katznelson, Y., An Introduction to Harmonic Analysis, Dover, New York, 1976.
4. Korner, T., Fourier Analysis, Cambridge, 1989.
5. Rudin, W., Functional Analysis, Tata Mc. Graw Hill, 1974.

Fields and a-fields, generators; Borel a-field on Euclidean, metric and general topological spaces. Monotone classes, monotone class theorem. Finitely additive measures. Measures, finite and a-finite measures. Borel measures, regularity. Outer measures; Caratheodory's extension theorem. Lebesgue measure in Euclidean spaces. Distribution functions. Measurable functions and their properties. Induced measures. a-fields generated by classes of functions; Monotone class theorem for functions. Integrability of functions. Lebesgue integrals and their properties. Fatou's lemma, monotone convergence theorem, dominated convergence theorem. Finite-dimensional product measurable spaces and measures on them. Product measures. Fubini's Theorem. Holder's. Minkowski's and Jensen 's inequalities. LP spaces; Characterizations of compact, precompact sets in LP spaces. Complex-valued measurable and integrable functions. Fourier transforms of finite measures on the real line and inversion formulae. Signed and complex-valued measures. Absolute continuity and singularity of measures. Lebesgue's differentiation theorem. Hahn decomposition theorem. Radon-Nikodym Theorem. Lebesgue decomposition theorem. Spaces of measures. Weak convergence. Helly's Theorem. Measures on locally compact spaces. Radon measures; Riesz representation theorem.

Texts/References:

1. Donald L. Cohn. Measure Theory. Birkhauser Boston, 1993.
2. M. M. Rao, Measure Theory and Integration. Marcel Dekker (Monographs and Textbooks in Pure and Applied Mathematics. 265), 2004.
3. Walter R udin. Real and Complex Analysis. 3rd ed.. McGraw-Hill, 1987.

Complex Differential Equations, Special functions, Normal families and applications, Reimann Mapping theorem, Fundementals of Univalent functions and Entire functions, Phragmen-Lindelof theorems, Gamma, Riemann-zeta functions, Harmonic functions, Dirichlet problem for disc, Analytic continuation, Runge's theorem.

Texts:

1. La, Laine, Nevanlinna theory and complex Differential Equations, Water de Gruyter, 1993.
2. R. Askey, Special Functions, Springer Verlag, 2000.
3. J.L. Schiff, Normal Families, Springer Verlag, 1993.
4. P.L. Duren, Univalent Functions, Springer Verlag, 1983.
5. R.E. Greene and S.G. Krantz, Function Theory of one Complex Variable, John Wiley and Sons, 1997.

Existence and uniqueness of solutions of ODEs, power series solution, singular points, some special functions. Nonlinear system of ODE : Preliminary concepts and definitions, the fundamental existence-uniqueness results, dependence on initial conditions and parameters, the maximum interval of existence, linearlization, stability and Liapunov functions, saddle, nodes, foci and centers, normal form theory and Hamiltonian systems. Boundary value problems : Green's function method, Sturm-Liouville problem.

First-order PDEs, Cauchy problem, method of characteristics, Second-order PDEs, classification, characteristics and canonical forms. Elliptic boundary value problems : Maximum principle, Green's function, Sobolev spaces, variational formulations, weak solutions, Lax-Milgram theorem, trace theorem, Poincarénequality, energy estimates, Fredholm alternative, regularity estimates, system of conservation laws, entropy criteria.

Texts/References:

1. L. Perko, Differential Equations and Dynamical Systems, Springer, 2001.
2. J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983.
3. S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, New York, 1990
4. Lawrence C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, Vol. 19, American Mathematical Society, Providence, 1998
5. Robert C. McOwen, Partial Differential Equations - Methods and Applications, Pearson Education Inc., Indian Reprint 2004.
6. S.J. Farlow, Partial Differential Equations for Scientists and Engineers, Dover Publications,New York, 1982.

Review of Sobolev spaces. weak solutions, eigenvalues and eigenfunctions of symmetric and non-symmetric elliptic operators. evolution equations, existence of weak solutions, maximum principle, interior and boundary regularities. Nonlinear elliptic equations: Nonlinear variational problems. first and second variations, existence of minimizers, nonlinear eigenvalue problems. Nonvariational techniques: monotonicity methods, fixed point methods, Nemytskii and pseudo-nRinotone operators. geometric properties of solutions. radial symmetry. Hamilton Jacobi equations: viscosity solutions, uniqueness, control theory, Hamilton-Jacobi-Bellman equations. Semigroup methods: Strongly continuous semigroups, infinitesimal generator, Hille-Yosida theorem, applications to wave and Schrodinger equations, analytic semigroups and their generators. Energy methods for evolution problems. System of conservation laws: Riemann's problem: simple waves, rarefraction waves, shock waves, contact discontinuities, local solution of Riemann's problem, vanishing viscosity, traveling waves, entropy/entropy-flux pairs.

Texts/References:

1. Lawrence C. Evans, Partial Differential Equations, Graduate Studies in Mathmatics, Vol.19. American Mathematical Society. Providence, 1998.
2. D. Gilberg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, New York. 1983.
3. A. Pazy, Sentigroup,s of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.
4. M. Renardy, B.C. Rogers. An Introduction to Partial Differential Equations, Springer, New York, 1993.
5. O.A. Ladyzhenskaya. N.N. Uraltseva, Linear and Quasilinear' Elliptic Equations, Academic Press, 1968.
6. P.-L. Lions. Generalized Solutions of Hamilton-Jacobi Equations, Research Notes in Mathematics 69, Pitman, 1982.
7. P. Lax, Hyperbolic Systems of Conservation Laws and Mathematical Theory of Shock Waves, SIAM, 1973.

Asymptotic expansion and approximation, asymptotic solution of algebraic and transcendental equations, regular and singular perturbations for first and second-order ordinary differential equations, physical examples, initial-value problems, multiple scales, two-scale asymptotic approximation, averaging technique, composite asymptotic expansions, initial layers - matching by Van Dyke rules. Two-point boundary-value problems: Boundary layers -exponential and cusp layers, matched asymptotic expansions, composite asymptotic expansions, WKB (Wentzel, Kramers, Brillouin) expansion method, conditions for validity of the WKB approximation, patched asymptotic approximations, WKB solution of inhomogeneous ordinary differential equations. Perturbation methods for linear eigenvalue problems, Rayleigh-Schrodinger theory, singularity structure of eigenvalues as functions of complex perturbing parameter, level crossing. Nonlinear eigenvalue problems, direct error estimation, oscillatory phenomena - free conservative and free self-sustained oscillations, harmonic resonance, shock and transition layers.

Texts/References:

1. C.M. Bender, S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, Springer, New York, 1999.
2. W. Eckhaus, Asymptotic Analysis of Singular Perturbations, North-Holland, Amsterdam, 1979.
3. J. Kevorkian, J.D. Cole, Perturbation Methods in Applied Mathematics, Springer- Verlag, New York, 1981.
4. P.A. Lagerstrom, Matched Asymptotic Expansions, Springer-Verlag, New York, 1988.
5. J. A. Murdock, Perturbations -Theory and Methods, SIAM -Classics in Applied Mathematics, Vol. 27, SIAM, Philadelphia, 1999.
6. A.H. Nayfeh, Introduction to Perturbation Techniques, John Wiley & Sons, New York, 1981.
7. R.E. O'Malley, Singular Perturbation Methods for Ordinary Differential Equations, Springer-Verlag, New York, 1991.

Elliptic operators, Dirichlet and periodic boundary conditions, asymptotic expansion using multiple scales, energy proof of the homogenization formula (Tartar's method of oscillating test functions), classical correctors, Bloch waves, Bloch expansion theorem, Bloch approximation function, homogenization of elliptic systems, composite materials containing high-modulus reinforcement, boundary layer theory in composite materials, asymptotic methods and spectral problems in fluid-solid structures, multiphase flows (solid-fluid mixture).

Texts/References:

1. A. Bensoussan, J.L. Lions, G. Papanicolaou, Asymptotic Analysis of Periodic Structure, North-Holland, Amsterdam, 1978.
2. N. Bakhvalov, G. Panasenko, Homogenisation: Averaging Process in Periodic Media, Kluwer Academic Publishers, Dordrecht, 1989.
3. D.Cioranescu, P. Donato, An Introduction to Homogenization, Oxford University Press, New York, 1999.
4. E.Sanchez-Palencia, Non-Homogeneous Media and Vibration Theory, Lecture Notes in Physics, Vol. 127, Springer-Verlag, Berlin, 1980.
5. C. Conca, J.Planchard, M. Vanninathan, Fluid and Periodic Structures, John Wiley and Sons/Masson, Chichester, 1995.
6. G. Milton, The Theory of Composites, Cambridge University Press, Cambridge, UK, 2002.

Review of Equations of Motion, Fundamentals of compressible flow, Boundary layer theory, Stability of laminar flows, Introduction to transition and turbulent flows.

Texts:

1. F. M. White, Viscous Fluid Flows, McGraw Hill, 1986.
2. L. D. Landau and E.M. Lifshitz, Fluid Mechanics, Pergamon Press, 1989.
3. H. Schlichting, Boundary Layer Theory, McGraw Hill, 1979.
4. J. D. Anderson, Modern Compressible Flows, McGraw Hill, 1989.
5. M. Lesieur, Turbulence in Fluids, Kluwer Academic Publishers, 1995.

Equations of Motion. Two dimensional flow. Navier-Stokes equation of motion. Velocity potential and Laplace equation. Simple irrotational flows. Separation of variables for an axisymmetric flow. Bernoulli equation for unsteady irrotational flow. Deep water wave. Shallow water wave. Theory of surface wave. Finite amplitude wave. One dimensional tidal dynamics. Linear and non-linear diffraction theory. Permutation methods. Water wave interaction with submerged spherical structures and floating cylinders. Solitary waves. Cnoidal wave. Schrodinger equation.

Texts:

1. G.K. Batchelor, An Introduction to Fluid Dynamics, CUP.
2. O.M. Phillips, The Dynamics of Upper Ocean, CUP
3. J.J. Stoker, Water Waves, Interscience.

Fick's law. Diffusion of finite size cloud. Reflection at boundary. Diffusion through random movements, diffusion with stationary velocities, dispersion of Brownian particles, reflecting and absorbing barriers. Statistical approach environmental diffusion. Lagrangian properties of turbulence. Apparent eddy diffusivity. Application to atmospheric diffusion. Experimental basis, probability distributions of particle displacements. History of a concentrated sewage plumes. Properties of planetary boundary layer, particle displacements in a wall layer, continuous line and point sources at ground level. Elevated sources, dispersion natural streams.

Texts:

1. C. T. Scanady, Turbulent Diffusion in the Environment, D. Reidel Publications .
2. F. Pasquill, Atmospheric Diffusion, Von Nostrand.
3. A. Pekalski, Diffusion Process: Experimental, Theory and Simulations, Springer-Verlag.

Introduction to the Navier-Stokes equations, Spectral approximation of Navier-Stokes equations, Simple proofs of bifurcation theorems, Steady transport equation, Existence and uniqueness theory for steady compressible flow, Finite elements methods for the incompressible Navier-Stokes equations.

Texts:

1. G. P. Galdi, J.G. Heywood and R. Rannacher, Fundamental directions in Mathematical Fluid Mechanics, Birkhauser Verlag, 2000.

Wave forces on structures, Morison equation, Froude-Krylov theory, Diffraction Theory, Perturbation method of solutions. Radiation. Dynamics of floating structures, damping and added-mass.

Interaction of water waves with spherical and cylindrical objects, hydrodynamic coefficients.

Nonlinear long waves in shallow water, KdV equation. Inverse scattering transforms and the theory of solitons. The Scrodinger equation. Soliton solution of the KdV equation.

Texts:

1. M. Rahman, Water Waves: Relating Modern Theory to Advanced Engineering Applications, OUP, 1994.
2. G. B. Whitham, Linear and Nonlinear Waves, Wiley Interscience,.
3. M. Isaacson and T. Sarpkaya, Mechanics of Wave Forces on Offshore Structures, Van Nostrand.
4. C. C. Mei, The Applied Dynamics of Ocean Surface Waves, World Scientific.

Classification of second order PDE's. Dirichlet Neumann and Robin's problem. Finite Differences schemes. Convergence analysis. Lax's equivalce theorem. Finite difference schemes for elliptic, parabolic and hyperbolic equations. Finite volume method for first and second order PDE's.

Texts:

1. G.D. Smith, Numerical solution of partial Differential Equations: Finite difference methods, Clarendon Press, Oxford.
2. C.A.J. Fletcher, Computational Techniques for Fluid Dynamics I, Springer.

Iterative methods for linear systems : Classical iterative methods (Jacobi, Gauss-Seidel and successive overrelaxation (SOR) methods), Krylov subspace methods; GMRES, Conjugate-gradient, biconjugate-gradient (BiCG), BiCGStab methods, preconditioning techniques, parallel implementations.

Finite difference method : Explicit and implicit schemes, consistence, stability and convergence, Lax equivalence theorem, numerical solutions to elliptic, parabolic and hyperbolic partial differential equations.

Approximate method of solution : Galerkin method, properties of Galerkin approximations, Petrov-Galerkin method, generalized Galerkin method.

The finite element method(FEM) : FEM for second order problems, one and two dimensional problems, finite elements(elements with a triangular mesh and a rectangular mesh and three dimensional finite elements), fourth-order problems, Hermite families of elements, isoparametric elements, numerical integration.

Texts/References:

1. D.S. Watkins, Fundamentals of Matrix Computations, second Edition, Wiley-interscience, New York, 2002.
2. L.N. Trefethen and David Bau, Numerical Linear Algebra, SIAM, 1997.
3. Joe D. Hoffman, Numerical methods for Engineers and Scientist, McGrow-Hill, 1993.
4. A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, Springer, 1994.
5. K. Atkinson and W. Han, Theoretical Numerical Analysis : A Functional Analysis Frame-work, Springer-Verlag, New York, 2001.
6. P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.
7. S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, Springer-Verlag, New York, 1994.
8. C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge University Press, 1987.

First-order Initial-Value Problems: Continuous problem, classical difference schemes, necessary conditions, uniformly convergent exponentially fitted schemes, artificial viscosity, constant fitting factors, optimal error estimates, system of IVPs.

Texts/References:

1. R.B. Ash, Probability and Measure Theory, Second Edition, Academic Press, 2000.
2. Patrick Billingsley, Probability and Measure, Third Edition,John Wiley and Sons, 1995.
3. Y.S. Chow and H. Teicher, Probability Theory: Independence, Interchangeability,Martingales, Third Edition, Springer Verlag, 1997.

Basic concept of the finite element method, Integral formulations and variational methods, The Lax-Milgram theorem, The abstract Galerkin method, Piecewise polynomial approximation in Sobolev spaces, Finite elements, Numerical quadrature, Applications to autonomous and non-autonomous problems, Optical error bounds in energy norms, Variational crimes, Apriori error estimates.

The discontinuous Gaterkin methods, Adaptive finite element, The Autin-Nitscte duality argument, A posteriori error analysis.

Texts:

1. C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge Universityh Press, 1987.
2. P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.
3. J. N. Reddy, An Introduction to Finite Element Method, McGraw Hill, 1993.
4. K. Erikssen et al., Computational Differential Equations, Cambridge University Press, 1996.
5. C. A. J. Fletcher, Computational Galerkin Methods, Springer-Verlag, New-York inc, 1984.

Review of measure-theoretic preliminaries: fields and a-fields, Borel a-field on R" ,measures, distribution functions, measurable functions, Lebesgue integration and related results, product measures, L^P spaces, convergence forms, Radon-Nikodym theorem; Probability measures, independence of a-fields, random variables and vectors, and their distributions, functions of random vectors, independence of random vectors, convolutions of distributions; Integrability and expectation, uniform integrability, generating functions, characteristic functions, Bochner's and Levy's theorems; Infinite-dimensional products, transition functions, Tulcea's and Kolmogorov's consistency theorems; Sequences of random variables, convergence in probability, almost surely, and in L^P , weak and strong laws of large numbers, central limit theorems; Conditional expectations and their properties, regular conditional distributions and probabilities; Introduction to discrete-time martingales and Markov processes.

Texts/References:

1. R.B. Ash, Probability and Measure Theory, Second Edition, Academic Press, 2000.
2. Patrick Billingsley, Probability and Measure, Third Edition,John Wiley & Sons, 1995.
3. Y.S. Chow and H. Teicher, Probability Theory: Independence, Interchangeability, Martingales, Third Edition, Springer Verlag, 1997.

Poisson random measures and their use in discontinuous martingales and LÉY processes; Brownian motion: Hitting times, maximum process, local times and excursions; Structure of local times and excursions in terms of Poisson random measures; LÉY processes: Structure and general properties, role of Brownian motion and Poisson random measures in the structure of LÉY processes, subordinators, hitting time of subordinators.

Texts:

1. E. Cinlar, Probability and Stochastics, Graduate Texts in Mathematics 261, Springer, 2011.
2. J. Bertoin, Lé Processes, Cambridge tracts in mathematics, Cambridge University Press, 1996.
3. K. Sato, Lé Processes and Infinitely Divisible Distributions, Revised edition, Cambridge studies in advanced mathematics, Cambridge University Press, 2013.

Review of probability, random variables and distributions, generating functions and transforms; Stochastic processes, discrete and continuous-time Markov chains, renewal processes, Brownian motion; Characteristics of queueing systems, Little's formula, Markovian and non-Markovian queueing systems, embedded Markov chain applications to M/G/1, G/M/1, and related queueing systems, queues with vacations, priority queues, queues with modulated arrival process, discrete-time queues, and matrix-geometric methods in queues; Networks of queues, open and closed queueing networks, algorithms to compute the performance metrics; Simulation of queues and queueing networks; Application to manufacturing, computer and communication systems and networks.

Texts/References:

1. L. Kleinrock, Queueing Systems, Vol. 1: Theory, 1975, Vol. 2: Computer Applications, 1976, John Wiley and Sons.
2. J. Medhi, Stochastic Models in Queueing Theory, 2nd Edition, Academic Press, 2002.
3. S. Asmussen, Applied Probability and Queues, 2nd Edition, Springer, 2003.
4. D. Gross, and C.Harris, Fundamentals of Queueing Theory, 3rd Edition, John Wiley and Sons, 1998.
5. R.B. Cooper, Introduction to Queueing Theory, 2nd Edition, North-Holland, 1981.
6. R. Nelson, Probability, Stochastic Processes, and Queueing Theory: The Mathematics of Computer Performance Modelling, Springer-Verlag, 1995.
7. E. Gelenbe, and G. Pujolle, Introduction to Queueing Networks, 2nd Edition, John Wiley, 1998.

Review of linear and non-linear data structures, dynamic storage allocation techniques; Advanced data structures - B-Trees, Binomial and Fibonacci, set representation and operations; Algorithm analysis - time and space complexities; Sorting and searching algorithms; Algorithm design techniques: divide and conquer, dynalii;c programming, search and traversals, backtracking, branch and bound; Review of graph algorithms; Flow networks, sorting networks, arithmetic circuits, matrix operations, polynomials and FFT, number theoretic algorithms, string matching.

Texts:

1. T. H. Cormen, C. E. Leiserson and R. L. Rivest, Introduction to Algorithms, Prentice Hall of India Private Limited, 2002.
2. D. C. Kozen, The Design and Analysis of Algorithms, Springer Verlag, 1992.
3. A. V. Aho, J. E. Hoperoft and J. D. Ullman, The Design and Analysis of Computer Algorithms, Addison-Wesley, 1974.