ma 7251: theory of numbers (prerequisite: nil) · pdf filedimensional analysis,...

MA 7251: THEORY OF NUMBERS

(Prerequisite: Nil)

Total hours: 56

Module I: (14 hours)

Divisibility: properties, the Euclidean algorithm, primes, Fundamental Theorem of Arithmetic, the sum

and the number of divisors of an integer, Mersenne and Fermat numbers, perfect numbers, linear

Diophantine equations.

Module II: (14 hours)

Congruence: basic properties, divisibility criteria, linear congruences, Chinese Remainder Theorem,

polynomial congruences of higher degree, Lagrange’s Theorem for a prime modulus, Euler phi-

function, Theorems of Fermat and Euler-Fermat, Wilson’s Theorem, pseudo primes and Carmichael

numbers.

Module III: (14 hours)

Primitive Roots, Index Arithmetic, Quadratic residues and reciprocity: quadratic residues, Euler’s

criterion, Legendre and Jacobi symbols, the law of quadratic reciprocity.

Module IV: (14 hours)

Public Key Cryptography, RSA cryptosystem, Diffie-Helman Key Exchange, Rabin Cryptosystem,

Knapsack Ciphers, Digital Signature, Secret Sharing, ElGamal Cryptosystem, Elliptic Curve

Cryptography.

References

1. K. Rosen: Elementary Theory of Numbers, Fourth edition, Addison Wesley Longman, Inc 2000.

2. I. Niven, H.S.Zuckerman and H.L.Montgomery: An Introduction to the Theory of Numbers, Fifth

Edition, John Wiley &Sons, Inc 2001.

3. T.M. Apostol, Introduction to Analytic Number Theory, Narosa Publishers, 1998.

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MA7252: KNOT THEORY

(Prerequisite: Nil)

Total hours: 56

MODULE I: (14 hours)

Knots and Links: Knots and links through their diagrams, Basic Definitions- orientation,

positive and negative crossing, writhe of a diagram, linking numbers, Isotopy of Knots-

Polynomial links and Reidemeister Moves, Hopf not isotopic to Whitehead.

MODULE II: (14 hours)

Knot colouring, labelling arcs with integers, colouring mod n, splittable links, Borromean rings,

chess board structure on a diagram, reduced connected diagram and quadrilateral decomposition,

crossing equations, colouring through the determinant of a link , Invariant of Knots:- Numerical

invariants: Crossing number-Unknotting number- Bridge number,

MODULE III: (14 hours)

Mirrors and Knot coding, mirrors, reversing orientation, chiral knots, Knots and Surfaces:-

Seifert Surfaces- Polynomials-Alexander Polynomial-Jones Polynomial- bracket polynomial,

state sums, skein relations, Conway knot and Kinoshita-Terasaka knot, Types of Knots - Torus

knots- Satellite knots- Hyperbolic knots-

MODULE IV: (14 hours)

Topology of 3-manifolds Heegaard Splittings:- Lens Spaces, Seifert Manifolds, Morse

Functions- Surgery on Links. Framing, linking number, homology of knot exterior. Surgery

description of Lens Spaces and Seifert Manifolds

References

1. Colin Adams The Knot Book, American Mathematical Society Pub., 2004.

2. Dale Rolfsen , Knots and Links, AMS Chelsea Publishing, 2003.

3. Nikolai Saveliev, Lectures on Topology of 3-manifolds: An Introduction to the Casson

Invariant, Walter de Gruyter Pub. Berlin, 1999.

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MA7253: MATHEMATICAL METHODS IN FLUID DYNAMICS

(Prerequisite: Knowledge of Calculus, Differential equations)

Total hours: 56

Module I: (15hours)

Fundamental concepts about fluids, Some mathematical concepts and notation, Kinematic of fluids, The

transport theorem, The continuity equation, The equations of motion, Stress Tensor, The Navier-Stokes

equations, The Energy equations, Boundary conditions


Equations of motion in different coordinate systems, Curvilinear coordinates, Cylindrical coordinates,

Dimensional analysis, Bukingham’s theorems, An example, Low Reynolds number flows, Steady and

unsteady plane flows, Boussinesq approximation


Rotation and vorticity, Kelvin’s circulation theorem, Potential flow, Stream line, Stream functions,

Blasius’ theorem, Kutta-Joukowski theorem, Surface phenomena, Laplace’s formula, Capillarity


Boundary-Layers concept, Prandtl boundary layer equations, General properties and exact solutions of the

boundary-layer equations for plane flows

References

1. A. J. Chorin and J. E. Marsden, “A Mathematical Introduction to Fluid Mechanics”, Springer, 1993.

2. M. Feistauer, “Mathematical Methods in Fluid Dynamics”, Longman Scientific and Technical, 1993.

3. L. D. Landau and E. M. Lifshitz, “Fluid Mechanics”, Elsevier, 2004.

4. H. Schlichting, “Boundary Layer Theory”, Springer 2001.

5. G. D. Smith, “Numerical Solution of Partial Differential Equations, Finite Difference Methods”,

Oxford, 1986.

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MA7254: ADVANCED COMPLEX ANALYSIS

(Pre-requisite:Knowledge of Complex Analysis)

Total Hours: 56

Module I: (13hours)

Review of power series, Taylor and Laurent series, Entire functions: Jenson’s Formula, Blaschke

Products, Hadamard’s theorem. The Riemann zeta function, The product development, Extension of

whole plane, Functional equations, Zeros of the zeta function.


Normal Families, Equicontinuity, Normality and compactness, Arzela’s Theorem, Families of analytic

functions, The classical definition, Riemann mapping theorem: Statement and proof, Boundary Behavior,

Use of the reflection principle, Analytic arcs.


Elliptic Functions, Simply periodic functions, Representation by exponents, Fourier Development,

Functions of finite order, Doubly periodic functions, Period Module, Unimodular transformations,

Cannonical Basis, General properties of elliptic functions.


Weierstrass Theory, Germs and Sheaves, Sections of Riemann Surfaces, Analytic continuation along arcs,

Homotopic curves, The Monodromy theorem, Branch points, the Picard theorem , subharmoic functions,

spaces pH and N , factorization theorems, shift operator, conjugate functions .

References

1. L. V. Ahlfors; “Complex Analysis”, 3rd

ed . McGraw Hill international eds., 1979.

2. W.Rudin, “Real and complex analysis”, 3rd

ed. Tata Mcgraw Hill ed.2006.

3. H. Cartan; “Elementary Theory of Analytic Functions of one or several Complex

variables”, 1/e, Dover Publications, 1995.

4. J. B Conway , “Function of one complex variables”, 2/e, Narosa Publishers, 1991.

5. Nehari Z.; “Introduction to Complex Analysis”, Allyn and Bacon, Inc., 1961.

6. Nehari Z.; “Conformal Mapping”, McGraw Hill Book Company, 1952.

7. R. K. Silverman; “Complex Analysis with Application”, Prentice Hall.

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MA7255: STOCHASTIC MODELS IN OPERATIONS RESEARCH

(Pre-requisite: Probability Theory )

Total hours: 56


Continuous time Markov Chains, Poisson Processes, Birth-Death processes, Queueing systems, general

concepts, transient and steady state behavior, M/M/1 queue, system size and waiting time distributions,

M/M/1/k model, M/M/ , M/M/c models, Erlang loss models.


Erlang queueing models, the system M/ /1, the system /M/1, Bulk queues, network of Markovian

queues, channels in series, Jackson networks. Non-Markovian queueing systems, the M/G/1 model and

G/M/1 model, busy period analysis.


Reliability theory: Introduction, structure functions, reliability of systems of independent components, the

series system, the parallel system, the k-out-of-n system, bounds on the reliability function, systems with

life as a function of component lives, expected system life time, systems with repair.


Inventory Theory: Introduction to basic inventory models, EOQ models, Stochastic inventory models,

single period decision models, marginal analysis, the news vendor problem- discrete demand, continuous

demand, the EOQ with uncertain demand, (s,S) models, safety stock, periodic review policy, the ABC

inventory classification system.

REFERENCES:

1. S M Ross, Introduction to Probability Models, 6th edition, Academic Press, 2000.

2. J. Medhi, Stochastic Models in Queueing Theory, 2nd

edition, Academic Press, 2003.

3. W L Winston, Introduction to Probability Models, ( OR Vol 2), 4th edition, Thomson

Brooks/Cole, 2004.

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MA7256: ADVANCED TOPICS IN GRAPH THEORY

(Pre-requisite: Nil)

Total hours: 42

Module I: (8 hours)

Graphs: review of basics in graphs - -Trees- Blocks- Matrices-Operations on graphs, Connectivity:

Vertex Connectivity and edge connectivity – n- connected graphs-Menger’s Theorem, Traversability:

Euler graphs-Hamiltonian Graphs-Planar and Nonplanar graphs.


Metric in graph: Centre, Median, eccentric vertex, Eccentric graph, boundary vertex, complete vertex,

interior vertex, Convexity: Closure Invariants-gin(G) –gn(G)-Hull number- Geodetic Graphs- Distance

Hereditary Graphs, Symmetry: Graphs and groups-- Symmetric Graphs - Distance Symmetry-Distance

transitive graphs-distance regular graphs, Distance Sequences :Degree sequence, Eccentric Sequence -

Distance Sequences - The Distance Distribution, Mean distance.

Module III : (10 hours)

Macthings :Maximum matching-Perfect matching-Matching in bipartite graphs, Factorization:

Coverings and independence-1-fcatorization-2-factorization-Arboricity, Domination: Dominating set-

Domination number-total dominating set –total domination number.


Digraphs: Digraphs and connectedness- Tournaments- directed trees-binary trees- weighted trees and

prefix codes, Networks: Flows-cuts- The Max- Flow Min-Cut Theorem, Graph Algorithms: Polynomial

Algorithms and NP completeness ,Complexity, Search algorithms, Shortest path algorithms.

References

1. Chartrand, G, and Zhang, P ‘ Introduction to Graph Theory, McGraw Hill International

Edition, 2005.

2. Bondy,J. A. and Murty, U.S.R., ‘Graph Theory’, Springer, 2008.

3. Buckley, F. and Harary, F., ‘Distance in Graphs’, Addison - Wesley (1990).

4. Foulds C. R., ‘Graph Theory Applications’, Narosa Publishing House, 1994.

5. Harary, F., ‘Graph Theory’, Addison- Wesley pub. 1972.

6. Grimaldi, R. P., ‘Discrete and Combinatorial Mathematics: An Applied Introduction’

Addison Wesley, 1994.

7. Vasudev, C., ‘Graph Theory with Applications’, New Age international publishers, 2006.

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MA7257: DIFFERENTIAL GEOMETRY


Total hours:56

Module I ( 14 hours)

Graphs and Level sets. Vector fields, The Tangent space. Curves and Surfaces, Vector fields on

surfaces, Tangent, Curvature, Principal normal, Binormal, torsion, The Frenet Formulas,

Involutes and evolutes, the tangent surface, Orientation. Developable surfaces.

Module II (14 hours)

Gauss Map, Geodesics, Parallel transport-The Weingrten Map - Curvature of plane curves,

second fundamental theorem, Equation of Gauss and Codazzi, Lines of curvature of a surface,

tangential coordinates of a surface, Parallel surfaces.

Module III (14 Hours)

Arc lengths and line integrals- Curvature of surfaces, Intrinsic equation of a curve, Linear

element, Element of area, Intrinsic geometry.

Module IV (14 Hours)

Parameterized surfaces- Local Equivalence of surfaces and parameterized surfaces, Differential

parameters, Isometric surfaces, Geodesic curvature, normal curvature, minimal surfaces.

Reference

1. Thorpe, J.A., ‘Elementary Topics in Differential Geometry’, Springer Verlag, 1979.

2. Oneill B, ‘Elementary differential Geometry’, Academic Press - New York, 2006.

3. Do Carmo M., ‘Differential Geometry of curves and surfaces’, Englewood Cliffs, N.J.,

Prentice Hall, 1976.

4. Millman R., and Parker, G., ‘Elements of differential Geometry’, Englewood Cliffs, N.J.,

Prentice Hall, 1977.

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MA7258: ADVANCED OPERATIONS RESEARCH

(Prerequisite: Knowledge of Linear programming)

Total hours: 42


Mathematical preliminaries, Maximum and Minimum-Quadratic forms-Gradient and Hessian matrices,

Unimodal functions, Convex sets, Convex and concave functions, Mathematical programming Problems,

Varieties and characteristics, Difficulties caused by nonlinearity, Role of convexity in Non linear

programming, Unconstrained optimization, Search methods, Fibonacci search, Golden section search.


Hooke and Jeeve’s Method, Optimal gradient method, Newton’s method- Constrained nonlinear

optimization, Constrained optimization with equality constraints, Lagrangian method, Sufficiency

conditions, Optimization with inequality constraints, Kuhn-Tucker conditions, Sufficiency Conditions.


Quadratic programming, Separable programming-Frank and wolfe’s method, Kelley’cutting plane

method, Rosen’s gradient projection method, Fletcher-Reeve’s method, Penalty and Barrier method.


Integer linear programming, Gomory’s cutting plane method, Branch and Bound Algorithm, Travelling

salesman’s problem, knapsack problem, Introduction to optimization software.

References:

1. Taha, H. A., ‘Operation Research-An introduction’, Prentice Hall, 8th Edition, Prentice Hall,

2007.

2. Simmons, D.M. "Nonlinear Programming for Operations Research” Prentice Hall, 1975.

3. Bazaara, M.S., Sherali, H.D, and Shetty C.M., ‘Nonlinear Programming Theory and Algorithm’

John Wiley, 2006.

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MA7259: ADVANCED TOPOLOGY

(Pre-requisite: Topology)

Total hours: 56


Topological spaces: definition, basis, sub basis, Fundamental examples, subspace - Closure: closed sets,

limit points, Hausdorff spaces. Continuity: equivalent definitions, homeomorphisms and embeddings.

Product topology: basis and subbasis, boxproduct.


Separation Axioms and Covering Properties - Separation axioms: Hausdorff, regular, Tychonoff, and

normal topological spaces, Covering properties: Compactness, Lindelofness, paracompactness,

metacompactness, Relations between covering properties and separation axiom, Normality of

paracompactspaces, paracompactness of Lindel¨of spaces, Preservation of separation and covering

properties.


Metrizibility and Connectedness - The metrization theorems of Urysohn and Bing, Smirnov, Nagata.

Connectedness and total disconnectedness: Definitions and examples of connectedness, total

disconnectedness, zero-dimensionality. Local properties: Local compactness, local connectedness


Fundamental Group - Homotopy: homotopy of paths , Fundamental group and covering spaces, simply

connected spaces, Fundamental group of the circle, Deformation retraction: fundamental group of the

punctured plane.

References

1. Munkres J. R., ‘Topology - A first course’, Prentice Hall of India, 2000.

2. Joshi K. D., ‘Introduction to general Topology’, New age International Ltd., New Delhi 1999.

3. Simmons, G. F., ‘ Introduction to Topology and Modern Analysis’, Mc Graw Hill International

Edn. 1963.

4. Dugundji, J., ‘ Topology’, Universal Book stall, New Delhi 1995.

5. Willard, S., ‘ General Topology’, Addison-Wesley, 1970.

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MA7260: SIMULATION AND MODELLING

(Pre-requisite: Knowledge of Statistical Methods)

Total hours: 42


Introduction to system simulation, Introduction: Systems and models, Computer simulation and its

applications, Continuous system simulation, Modelling continuous systems, simulation of continuous

systems, Discrete system simulation- Methodology, event scheduling and process interaction approaches,

Random number generation, testing of randomness, generation of stochastic variates, Random samples

from continuous distributions, Uniform distribution, Exponential distribution, m-Erlang distribution,

Gamma distribution, Normal distribution, Beta distribution, Random samples from discrete distributions

– Bernoulli, Discrete uniform, Binomial, Geometric and Poisson.


Evaluation of Simulation Experiments and Simulation Languages, Evaluation of simulation experiments,

verification and validation of simulation experiments, Statistical reliability in evaluating simulation

experiments, Confidence intervals for terminating simulation runs, Simulation Languages: Programming

Considerations, General features of GPSS, SIMSCRIPT and SIMULA.


Simulation of Queuing Systems – Introduction, Parameters of queue, formulation of queuing problems,

generation of arrival pattern, generation of service pattern, simulation of single server queues, simulation

of multi-server queues, simulation of tandem queues, Computer simulation of Queuing systems.


Simulation of Stochastic Network , Introduction, Simulation of PERT Network, Definition of network

diagrams, forward pass computation, simulation of forward pass, backward pass computations,

simulation of backward pass, determination of float and slack times, determination of critical path,

simulation of complete network, merits of simulation of stochastic networks, Computer simulation of

PERT network.

References

1. Deo, N., ‘System Simulation and Digital Computer’ , PHI, Delhi, 1989.

2. Gordan, G., System Simulation" , PHI, Delhi, , 1990.

3. Banks, J., Carson, J. S., and Nelson, B. L ‘Discrete –Event System Simulation’, 2nd

edn., PHI,

New Delhi.,(2000) .

4. Law, A.M. and Kelton, W.D., ‘ Simulation Modelling and Analysis’, Mc- Graw Hill, 1990).

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MA7261: MULTI-VARIATE STATISTICAL ANALYSIS


Total hours: 42


Random vectors and matrices, Random vectors and matrices, mean vectors and covariance matrices,

matrix inequalities and maximization, Sample Geometry and random sampling-geometry of the sample,

random samples and expected values of the sample mean and covariance matrix, generalized variance,

sample mean, covariance and correlation as matrix operations, sample values of linear combinations of

variables.


The multivariate normal distribution, the multivariate normal distribution: Introduction, the multivariate

normal density and its properties, sampling from a multivariate normal distribution and maximum

likelihood estimation, the sampling distribution of X and S, Large-Sample behaviour of X and S,

assessing the assumption of normality transformations to near normality.


Inferences about a mean vector: Introduction, the plausibility of 0 as a value for a normal population

mean, Hotelling’s T2 and likelihood ratio tests, confidence regions and simultaneous comparisons of

component means, Large-sample inferences about population mean vector, inferences about mean vectors

when some observations are missing.


Comparisons of several multivariate means: Introduction paired comparison and a repeated measures

design, comparing mean vectors from two populations, comparison of several multivariate population

means (one-way MANOVA), simultaneous confidence intervals for treatment effects, two-way

multivariate analysis of variance.

References

1. Johnson, R. A. and Wichern, D.W. ‘Applied multivariate statistical analysis’, Prentice – Hall of

India Pvt. Ltd., New Delhi, 1996.

2. Anderson T.W, ‘An Introduction to Multivariate Statistical Analysis’, 2nd

edition, John Wiley and

Sons, New York, 1984.

3. Rao, C.R., ‘Linear statistical inference and its applications’, second edition, Wiley Eastern Ltd.,

New Delhi,1973.

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MA7262: DECISION THEORY

(Pre-requisite: Statistical Methods)

Total hours: 56


Game theory and decision theory: Basic elements, a comparison of game theory and decision theory,

Decision functions, Optional decision rules, Baye’s rule for estimation problems.


The main theorems of decision theory: Admissibility and completeness, decision theory, admissibility of

Baye’s rules, basic assumptions, existence of Baye’s decision rules, existence of minimal complete class,

the separating hyper plane theorem, essential completeness of the class of nonrandomized decision rules,

the minimax theorem, the complete class theorem, solving for minimax rules .


Sufficient statistics and risk functions: The multivariate Normal distribution, sufficient statistics,

essentially complete classes of rules based sufficient statistics, complete sufficient statistics, and

continuity of the risk functions.


Decision problems: Invariant statistical decision problems, admissible and minimax invariant rules,

location and scale parameters, minimax estimates for the parameters of a distribution function. Multiple

decision problem- monotone multiple decision problem, Baye’s rules in multiple decision problems.

References

1. Ferguson, T. S., ‘Mathematical Statistics- Decision theoretic approach’, Academic Press, New

York, 1967.

2. White, D. J., ‘Fundamentals of Decision Theory’, North Holland, New York , 1976.

3. Schlaifer, R., ‘Analysis of Decision under uncertainty’, McGraw-Hill Book Company, 1969.

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MA7263: REGRESSION ANALYSIS


Total hours: 56


Simple regression with one independent variable(X), assumptions, estimation of parameters, standard

error of estimator, testing of hypothesis about regression parameters, standard error of prediction, Testing

of hypotheses about parallelism, equality of intercepts, congruence. Extrapolation, optimal choice of X.

Diagnostic checks and correction: graphical techniques, tests for normality, uncorrelatedness,

homoscedasticity, lack of fit, modifications like polynomial regression, transformations on Y or X, WLS,

inverse regression X(Y).


Multiple regression: Standard Gauss Markov Setup, Least square(LS) estimation, Error and estimation

spaces, Variance- Covariance of LS estimators, estimation of error variance, case with correlated

observations, LS estimation with restriction on parameters, Simultaneous estimation of linear parametric

functions, Test of Hypotheses for one and more than one linear parametric functions, confidence

intervals and regions, ANOVA.


Non Linear regression (NLS) : Linearization transforms, their use & limitations, examination of non

linearity, initial estimates, iterative procedures for NLS, grid search, Newton- Raphson , steepest descent,

Marquardt’s methods.


Logistic Regression: Logit transform, ML estimation. Tests of hypotheses, Wald test, LR test, score test,

test for overall regression, multiple logistic regression, forward and backward method, interpretation of

parameters relation with categorical data analysis, generalized linear model: link functions such as

Poisson, binomial, inverse binomial, inverse Gaussian, gamma.

References

1. Draper, N. R. and Smith, H., ‘Applied Regression Analysis’, 3rd Ed., John Wiley, 1998.

2. McCullagh, P and Nelder, J. A., ‘Generalized, Linear Models’, Chapman & Hall, 1998.

3. Ratkowsky, D. A., ‘Nonlinear Regression Modelling’, Marcel Dekker, 1983.

4. Hosmer, D.W. and Lemeshow, S., ‘Applied Logistic Regression’, John Wiley, 1989.

5. Seber, G.E.F. and Wild, C.J., ‘Nonlinear Regression’, Wiley, 1989.

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MA7264: ALGEBRAIC TOPOLOGY

(Pre-requisite: Knowledge of Topology)

Total hours: 56


Review of basic Topology: Topological spaces, subspaces, Limit points, closure, frontier, Continuous

mapping, Compactness, Arcwise connected spaces.

Module II: (14hours)

Fundamental group: Homotopy, homotopy classes, Fundamental group, Change of base point,

Topological invariance.

Module III: (14hours)

Homology groups: Geometrical motivation, Euclidean simplexes, Linear mappings, singular simplexes,

chains, Boundary of a simplex, Boundaries and cycles on any space, Homologous cycles and homology

groups, Relative homology.


Induced Homomorphisms, Topological invariance of homology groups, Homotopic mappings and the

homology groups, Prisms,Homology sequences, Simplical complexes.

References

1. Wallace, A. H., ‘An Introduction to Algebraic Topology’, Pergamon Press, 1957.

2. Maunder CRF., ‘Algebraic Topology’, Cambridge University Press, 1996.

3. Deo, S., ‘Algebraic Topology A primer’, Hindustan Book agency, 2003.

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3 1 0 3

MA7265: STATISTICAL DIGITAL SIGNAL PROCESSING

(Pre-requisite: Knowledge of Statistical methods)

Total hours: 56


Discrete-Time Random Processes: Random Variables, Random Processes, Filtering Random Processes,

Spectral Factorization, Special Types of Random Processes.


Signal Modeling: The Least Squares Method, The Pade Approximtion, Prony’s Method, Finite Data

Records, Stochastic Models.


Lattice Filters and Wiener Filtering: The FIR Lattice Filter, Split Lattice Filter, IIR Lattice Filters,

Stochastic Modeling, The FIR Wiener Filter, IIR Wiener Filter, Discrete Kalman Filter.


Spectrum Estimation: Nonparametric Methods, Minimum Variance Spectrum Estimation, The Maximum

Entropy Method, Parametric Methods, Frequency Estimation, Principal Components Spectrum

Estimation.

References

1. Hayes M. H., ‘Statistical Digital Signal Processing and Modeling’, John Wiley & Sons, 2004.

2. Miao G. J. and Clements M. A., ‘Digital Signal Processing and Statistical Classification’, Artech

House, London, 2002.

3. Gray R. M. and Davisson L. D., ‘An Introduction to Statistical Signal Processing’, Cambridge

University Press, 2004.

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MA7266: STATISTICAL METHODS FOR QUALITY MANAGEMENT


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Total Hours: 42


Design and analysis of experiments : Introduction to design and analysis of experiments, Single Factor

design and Analysis of Variance, Randomized Blocks, Latin Squares and Related Designs, introduction to

Factorial Designs : Basic Definitions and Principles, The Two-Factor Factorial Design, The General

Factorial Design, Fitting Response Curves and Surfaces, Blocking in a Factorial Design.


Statistical process control: Chance and assignable causes of quality variation, setting up of operating

control charts for RandX , Control charts for X and S, Control charts for individual measurements,

Applications of variables control charts. Control charts for Attributes- control charts for Fraction

nonconforming, control charts for nonconformities (defects).


Cumulative sum and exponentially weighted moving average control charts- The cumulative-sum control

charts, The exponentially weighted moving-average control charts, the moving average moving control

charts. Statistical process control techniques, process capability analysis, acceptance sampling for

attributes.


Reliability Statistics: Reliability definition, availability, reliability bathtub curve, estimating MTBF,

reliability prediction, confidence interval for MTBF, testing, system reliability, series systems, parallel

systems, Baye’s theorem applications, non-parametric and related test designs, hazard function, Weibul

distribution, Log-normal distribution, stress- strength inference, Binomial confidence intervals, Arrhenius

model, sequential testing.

References:

1. Grant E. L., and Leavenworth R. S., ‘Statistical Quality control’, 7th Ed.n; McGraw-

Hill Companies Inc. 1996.

2. Montgomery D. C., ‘ Introduction to Statistical Quality Control’, 3rd

Edn., John Wiley

and sons1997

3. Montgomery D. C., ‘Design and Analysis of Experiments’, 5th Edn., John Wiley &

Sons, Inc., 2001.

4. Dovich, R. A., ‘Reliability statistics’, A S Q Quality Press., 1990.

MA7267: GENERALISED SET THEORY

(Pre-requisite:Nil)

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3 1 0 3

Total hours: 56

Module I (14 hours)

An overview of basic operations on Fuzzy sets and Multisets, Multiset relations, Compositions,

equivalence multiset relations and partitions of multisets, Multiset functions, Fuzzy Multisets.

Module II (14 hours)

Rough sets, Approximations of a set, Properties of Approximations, Rough membership function, Rough

sets and Reasoning from data: Information systems, Decision tables, Dependency of attributes, Reduction

of attributes, Indiscernibility matrices and functions.

Module III (14 hours)

Soft sets, Tabular representation of a soft set, Operations with Soft sets: soft subset, complement of a soft

set, null and absolute soft sets, AND and OR operations, Union and intersection of soft sets, DeMorgan

laws, Applications and soft analysis.

Module IV (14 hours)

Fuzzy soft sets, Operations on fuzzy soft sets, Soft fuzzy sets and its properties, Fuzzy rough sets and

rough fuzzy sets, Rough multisets, Genuine sets, Applications.

References:

1. Bing-Yuan Cao, ‘Fuzzy Information and Engineering’, Springer 2007.

2. Girish K. P., and Sunil J. J., ‘Relations and Functions in Multiset context’, Information

Sciences’ 179 (2009) 758 - 768.

3. Peters, J. F., and Skowron, A., ‘Transactions on Rough Sets I’, Springer 2004.

4. Polkowski, L., ‘ Rough Sets: Mathematical Foundations ’, Springer, 2002.

5. Demirci, M., ‘Genuine Sets’, Fuzzy Sets and Systems, 105 (1999) 377-384.

6. Zimmerman H.J., ‘Fuzzy set Theory and its Applications’, Allied Publishers Ltd, 2000.

MA7268: FOURIER ANALYSIS

(Pre-requisite:Nil)

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3 1 0 3

Total hours: 56

Module I (14 hours)

Trigonometric polynomials and trigonometric series, Fourier series in L1(T) – Riemann-Lebesgue

lemma, convolutions, Convergence, Fejer’s theorem, Fourier series of continuous functions, Fourier

series in L2(T), Bessel inequality, Parseval identity, Plancherel theorem, Fourier series of L

p functions.

Module II: I (14 hours)

LP spaces, Convolution of functions, Young’s inequality, approximate identity, Regularisation of

functions, Pointwise convergence, Fourier transform in L 1(R), Riemann-Lebesgue lemma,

Multiplication formula, Inversion, Translations and dilations, Multiplication and differentiation.

Module III: I (14 hours)

Abel means and Poisson kernel, Uniqueness theorem for Fourier transform in L1(R), Fourier transform

in L2(R), Multiplication formula, Plancherel theorem, Uniqueness theorem in L

2(R), Harmonic

functions, Dirichlet problem for the upper half plane, Point wise and norm convergence, Dirichlet

problem for the disc.

Module IV: I (14 hours)

Eigen functions of Fourier transform, Gaussian – Hermite functions, Schwartz space, Paley-Wiener

space Paley-Wiener theorem, Uncertainty principle, Hardy classes, Hardy’s theorem.

References:

1. Dym, H. and McKean H. P., ‘Fourier Series and Integrals’, Academic Press, 1985.

2. Katznelson, Y., ‘ An Introduction to Harmonic Analysis’, Cambridge University Press, 2004.

3. Helson, H., ‘Harmonic Analysis’, Hindustan Book Agency and Helson Publishing Co., 1995.

4. Stein E. M., and Shakarchi, R., ‘Fourier analysis: An Introduction’, Princeton University Press,

2003.

5. Sadosky, C., ‘Interpolation of Operators and Singular Integrals - An Introduction to

Harmonic Analysis’, Marcel Dekker, Inc., 1979.

MA7269 FUZZY SET THEORY AND APPLICATIONS


Total hours: 42


Introduction, crisp sets an overview, the notion of fuzzy sets, basic concepts of fuzzy sets, membership

functions, methods of generating membership functions, defuzzification methods, operations on fuzzy

sets, fuzzy complement, fuzzy union, fuzzy intersection, combinations of operations, General

aggregation operations.


Fuzzy numbers, arithmetic operations on intervals, arithmetic operations on fuzzy numbers, fuzzy

equations, crisp and fuzzy relations, binary relations, binary relations on a single set, equivalence and

similarity relations, compatibility or tolerance relations.

Module III : (10 hours)

Fuzzy measures, belief and plausibility measures, probability measures, possibility and necessity

measures, possibility distribution, relationship among classes of fuzzy measures.


Classical logic : an overview, fuzzy logic, approximate reasoning, other forms of implication

operations, other forms of the composition operations, fuzzy decision making, fuzzy logic in database

and information systems, fuzzy pattern recognition, fuzzy control systems.

References:

1. Klir, G. J. and Folger, T. A. ‘Fuzzy sets, Uncertainty and Information’, Prentice Hall of India,

1988.

2. Zimmerman, H. J., ‘Fuzzy Set theory and its Applications’, 4th Edition, Kluwer Academic

Publishers, 2001.

3. Goerge J Klir and Bo Yuan , Fuzzy sets and Fuzzy logic: Theory and Applications Prentice Hall

of India, 1997.

4. Nguyen H. T., and Walker, E. A., ‘ First Course in Fuzzy Logic’, 2nd

Edition , Chapman &

Hall/CRC, 1999.

5. Mendel, J. M., ‘Uncertain Rule – Based Fuzzy Logic Systems ; Introduction and New

Directions’, PH PTR, 2000.

6. Yen, J. and Langari, R., ‘ Fuzzy Logic : Intelligence Control and Information’, Pearson

Education, 1999.

7. Ross, T. J., ‘Fuzzy Logic with Engineering Applications’, McGraw Hill International Editions,

1997

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MA7270: STOCHASTIC PROCESSES

(Pre-requisite: Knowledge of Measure and Probability )

Total hours: 56


Elements of stochastic processes, Classification of general stochastic processes, Markov Chains:

Definition, examples, transition probability matrix, classification of states, basic limit theorem, limiting

distribution of Markov Chains.


Continuous time Markov Chains: General pure birth processes and Poisson processes, more about

Poisson processes, A counter model, Birth and Death processes with absorbing states, Finite state

continuous time Markov Chains.


Renewal Processes: Definition of a renewal process and related concepts, examples of renewal processes,

special renewal processes, renewal equation and elementary renewal theorem, the renewal theorem,

generalizations and variations on renewal processes, applications of renewal theory.


Martingales: Preliminary definitions and examples. Brownian motion: Introduction and preliminaries.

Stationary Processes: Definition and examples, Mean square distance, spectral analysis of covariance

stationary processes.

References:

1. Karlin S. and Taylor, H. M., ‘A First Course in Stochastic Processes’, 2nd

Edn., Academic Press,

New York, 1975.

2. Ross, S. M., ‘Stochastic Processes’, 2nd

Edn, John Wiley and Sons, New York, 1996.

3. Medhi, J., ‘Stochastic Processes’, New Age International, New Delhi, 1991.

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MA7271: CODING THEORY


Total hours:56

MODULE 1: (14 hours)

Introduction to Coding Theory: Basic assumptions, Correcting and detecting error patterns,

Information rate, The effects of error correction and detection, Finding the most likely codeword

transmitted, Some basic algebra, Weight and distance, Maximum likelihood decoding, Reliability of

MLD, Error-detecting codes.


Linear Codes: Two important subspaces, Independence, basis, dimension, Matrices- Bases for C =

〈S〉 and C⊥- Generating matrices and encoding, Parity-check matrices, Equivalent codes, Distance

of a linear code, Cosets, MLD for linear codes, Reliability of IMLD for linear codes.


Perfect and Related Codes: Some bounds for codes, Perfect codes, Hamming codes, Extended codes,

The extended Golay code, Decoding the extended Golay code, The Golay code, Reed-Muller codes-

3Fast decoding for RM(1,m).

MODULE 1V: (14 hours)

Cyclic Linear Codes: Generating and parity check matrices for cyclic codes, Finding cyclic

codes, Dual cyclic codes, BCH Codes, Reed –Solomen Codes.

References

1. Hankerson, D. C. et. Al., ‘Coding Theory and Cryptography; The Essentials Monographs and

Textbooks in Pure and Applied Mathematics’ ; 234 CRC Press, 2013.

2. Garrett, P., ‘The Mathematics of Coding Theory: Information, Compression, Error Correction and

Finite Fields’, Pearson Education, 2004.

3. Lin, S. and Costello, D. J., ‘Error Control Coding - Fundamentals and Applications’, Prentice Hall

Inc. Englewood Cliffs

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MA7272: Reliability of Systems


Total hours:56


Introduction to reliability, Basic concepts, Cut sets, Path sets, Minimal cut and path sets, Bounds for

reliability, Reliability and Quality, Maintainability and Availability, Reliability analysis, Causes of

failures, Catastrophic and Degradation failures, Useful life of components, Component reliability and

hazard models, Mean time to failure, system reliability models, System with components in series,

parallel, k/n systems, System with mixed mode failures.


Redundancy Techniques, Component v/s unit redundancy, Weakest link techniques, Mixed

redundancy, Stand by redundancy, Redundancy optimization, Double failure and redundancy,

Maintainability and availability concepts, Two unit parallel system with repair, Signal redundancy,

Time redundancy, Software redundancy.


Hierarchical systems, Path determination method, Boolean Algebra method, Cut set approach, Logic

diagram approach, Conditional probability approach, System cost and reliability approximations,

Economics of reliability engineering, Economic cost, manufacturing cost, customers cost, Reliability

achievement cost models, Depreciation cost models, Reliability management, Management policy and

decisions.


Life testing: Introduction, hazard rate functions, Exponential distribution in life testing, Simultaneous

testing-stopping at r-th failure, Stopping by fixed time, sequential testing, Accelerated testing,

Equipment Acceptance testing, Software reliability, Software reliability models, Reliability Allocation,

A two sample problem.

References

1. Balagurusamy, E., Reliability Engineering’, Tata McGraw-Hill, 2011.

2. Shooman, M.L,. ‘ Probability Reliability An engineering Approach’, McGraw-Hill. Newyork, 1968.

3. Barlow, R.E. and Proschen, F., ‘Mathematical Theory of Reliability’, John Wiley, Newyork, 1965.

4. Aggarwal, K.K., ‘Reliability Engineering’, Springer, 2007.

5. Ross, S.M. ‘Introduction to Probability and Statistics for Engineers and Scientists’, 4/e, Elsevier, 2009.

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MA7273: OPERATOR THEORY

(Pre-requisite: -Knowledge of Functional analysis)

Total hours: 42

Module I: (10hours)

Banach algebras, Gelfand theory, C*- algebras the GNS construction, spectral theorem for normal operators,

Fredholm operators and its properties, semi-Fredhlom operators, product of operators.


Hilbert Space Operators, Parts of Spectrum, Orthogonal Projections, Invariant Subspaces, Reducing Subspaces,

Shifts, Decompositions of Operators. Compact linear operators, Spectral properties of compact bounded linear

operators, spectral theorem and functional calculus for compact normal operators .


Spectral projections, spectral decomposition theorem, spectral theorem for a bounded normal operator, Measurs

of operators. Perturbation classes,strictly singular operators, Spectral theory of integral operators: Hilbert

Schmidt theorem, Mercer,s theorem, Trace formula for integral operators, integral operators as inverse of

differential operators. Sturm- liouville systems.

Module IV: (10hours)

Unbounded operators: Basic theory of unbounded self-adjoint operators, unbounded Fredhlom operators and its

properities, essential spectrum, unbounded semi-Fredhlom operators, Spectral theorem for an unbounded self

adjoint operators.

References

1. Schechter, M.. ‘Principles of Functional Analysis’, AMS,2nd

ed., 2002.

2. Gohberg I. and Goldberg, S., ‘Basic operator Theory’, Birkhauser,1981.

3. Ahues, M., Largillier, A. and Limaye, B. V., ‘Spectral Computations for Bounded Operators’,

Chapman & Hall/CRC, 2001.

4. Conway, J. B,. ‘A course in Functional Analysis’, 2nd

ed., Springer-Verlag, 1990.

5. Lang, S., ‘Complex Analysis’, 4th Ed., Springer, 1999.

6. Limaye, B.V., ‘Functional Analysis’ 2/e, New Age Publishers, 2006.

7. Riesz, F. and SzNagy, B., ‘Functional Analysis’, Dover Publications, 1990.

8. Rudin W., ‘Functional Analysis’, 2/e, Tata McGraw - Hill edn.,2006.

9. Yosida, K., ‘Functional Analysis’, 5th ed., Narosa, 1979.

10. Sunder, V.S., ‘Functional analysis’ spectral theory”, Birkhauser,1998.

11. Douglas, R. G.,’ Banach Algebra Techniques in Operator Theory’, Academic Press, 1972.

12. Dunford, N.,and Schwartz, J.T. ‘Linear operators, part I: General theory’, Inter science, Newyork,1958.

13. Dunford, N.,and Schwartz, J.T. ‘Linear operators, part II:Spectral theory’, Inter science, Newyork,1963.

14. Murphy, G.J., ‘ C*-Algebras and Operator Theory’, Academic Press Inc., 1990.

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MA7274: WAVELETS THEORY


Total hours: 42

ModuleI: (11hours)

Vector spaces and Bases, Linear transformation, Matrices and change of basis, Inner products, Hilbert Space,

Fourier transforms , Parsevl identity and Plancherel theorem, Basic Properties of Discrete Fourier Transforms ,

Translation invariant Linear Transforms ,The Fast Fourier Transforms.


Construction of wavelets on ZN ,The Haar system, Shannon Wavelets, Real Shannon wavelets , Daubechies’s D6

wavelets on ZN., Examples and applications.

Module III: (11hours)

Wavelets on Z: l2 (Z), Complete orthonormal sets in Hilbert spaces , ),(2 L and Fourier series ,The Fourier

Transform and convolution on l2 (Z) , First stage Wavelets on Z , Implementation and Examples.


Wavelets on R : L2 (R) and approximate identities , The Fourier transform on R , Multiresolution analysis ,

Construction of MRA .

References:

1. Frazier, Ml. W. ‘An Introduction to Wavelets through Linear Algebra’, Springer, Newyork, 1999.

2. Goswami, J. C. and Chan, A. K., ‘Fundamentals of Wavelets Theory Algorithms and

Applications’, John Wiley and Sons, Newyork. , 1999.

3. Nievergelt, Y. ‘Wavelets made easy’, Birkhauser, Boston,1999.

4. Bachman, G., Narici, L. and Beckenstein, E., ‘Fourier and wavelet analysis’, Springer, 2006

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MA7275: QUEUEING THEORY

(Pre-requisite: knowledge of Probability and Measure & Stochastic Processes)

Total hours: 42


Simple Markovian Queuing Models – Introduction to queuing theory, Characteristics of queuing processes,

Measures of effectiveness, Markovian queueing models, steady state solution of the M/M/1 model, waiting time

distributions, Little’s formula, queues with parallel channels and truncation, Erlang’s loss formula, Queues with

unlimited service, finite source queues.

Module II :(12 hours)

Transient behaviour of M/M/1 queues, transient behaviour of M/M/, Busy period analysis for M/M/1 and

M/M/c models. Advanced Markovian models, Bulk input 1//][ MM X model, Bulk service 1// ][YMM model,

Erlangian models 1//1// MEandEM kk , A brief discussion of priority queues.


Queuing networks – series queues, open Jackson networks, closed Jackson networks, Cyclic queues, Extension

of Jackson networks, Non Jackson networks.


Models with general arrival pattern, The M/G/1 queuing model, The Pollaczek-khintchine formula, Departure

point steady state system size probabilities, ergodic theory, special cases 1// kEM and M/D/1, waiting times,

busy period analysis, general input and exponential service models, arrival point steady state system size

probabilities.

References

1. Gross, D. and Harris, C. M., ‘Fundamentals of Queueing Theory’, 2/e, John Wiley and Sons, New York,

1985.

2. Kleinrock L., ‘Queueing Systems”, Vol 1 & Vol 2, John Wiley and Sons, New York, 1995.

3. S. M. Ross; Introduction to Probability Models”, 8/e, Academic Press, New York, 2002.

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MA7276: NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS


Total hours: 56

Module 1: (14 hours)

Introduction; Initial Value problem for first order ODEs, Initial Value problem for system of first order ODEs’,

Reduction of higher order ODEs to first order systems; First order system with constant coefficients, linear

difference equation with constant coefficients,


Numerical Methods for ODEs: Single Step Methods: Order and convergence of the general explicit

one-step methods, derivation of classical RungeKutta methods, error bounds and error estimate of

RungeKutta methods,RungeKutta methods of order greater than four, numerical errors ; weak stability

theory for Runge-Kutta methods, implicit Rungekutta methods.

Module 3 :(14 hours)

General Linear Multi-Step Methods, derivation through Taylor expansions, derivation through

numerical integration, derivation through interpolation, convergence, order and error constants, Local

and global truncation error, consistency and zero stability, Error bounds and local and global

truncation error, weak stability theory, interval of absolute and relative stability, comparison of implicit

and explicit Linear Multistep methods,

Module 4:(14 hours)

Predictor-Corrector methods, Local truncation error of predictor-corrector methods: Milne’s device,

weak stability of predictor corrector methods. Stiff ODEs; Implicit Stability Theory: A-stability, L-

stability, B-stability, Backward Difference Formulas Methods: formulas and stability regions , Two-

Point Boundary Value Problems; Finite-Difference Methods, Shooting Methods, Collocation Methods.

References

1. Atkinson, Han and Stewart, ‘Numerical Solution of Ordinary Differential Equations’, John

Wiley & Sons, 2009.

2. Shampine, Gladwell and Thompson, ‘Solving ODEs with MATLAB’, Cambridge University

Press, 2003.

3. Hairer, Nørsett and Wanner, ‘Solving Ordinary Differential Equations I -- Nonstiff Problems’,

Springer, 2010,

4. Hairer and Wanner, ‘Solving Ordinary Differential Equations II -- Stiff and Differential-

Algebraic Problems’, Springer, 2010,.

5. Hairer, Lubich and Wanner, ‘Geometric Numerical Integration’, Springer, 2010.

6. Lambert, J. D., ‘Numerical Methods for Ordinary Differential Systems: The Initial Value

Problems’, John-Wiley, 1991.

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MA7277: Numerical solutions for partial differential equations

(Prerequisite: Knowledge of programming in Matlab)

Total hours: 56

Module-1: (13 hours)

Classification of PDEs, finite difference approximations to derivates, truncation errors, boundary conditions:

Dirichlet, Neumann and Robin type boundary conditions. Review of iterative methods to linear system of

equations: Jacobi, Gauss-seidel, SOR. Matrix form of iterative methods and their convergence. Conjugate

gradient method. Initial value problems, Initial boundary value problems and their analysis of convergence,

consistency and stability. Lax theorem, Von Neumann criterion for stability.


Parabolic equations: explicit and implicit methods for one and two dimentional parabolic equations, Crank-

Nicolson method, numerical examples, weighted average approximation, consistency, convergence and stability,

alternate direction method in two dimensions, Peaceman-Rachford scheme, Douglas-Rachford scheme.


Hyperbolic equations: Finite difference methods for first and second order wave equation, Lax-wendroff explicit

method, CFL condition for one and two dimentsions, ADI schemes fo two dimensional hyperbolic equations,

Lax-wendroff method for a system of hyperbolic equations, Wendroff’s implicit approximation, reduction of a

first order equation to a system of ordinary differential equations, numerical examples.


Elliptic equations: Numerical examples: a torsion problem, a heat conduction problem with derivative boundary

conditions. Finite differences in polar co-ordinates, techniques near a curved boundary, improvement of the

accuracy of the solutions. Analysis of the discretization error of the five-point approximation to Poisson’s

equation.

(Most of the tutorial classes will be dealt with implementation of numerical schemes for solving partial

differential equations in Matlab.)

References:

1. Morton, K.W. and Mayers, D.F., ‘Numerical solution of partial differential equations’, Cambridge,

2011.

2. Smith, G.D., ‘Numerical solution of partial differential equations’, finite difference methods, oxford,

2010.

3. Leveque, R. J., ‘Finite difference methods for ordinary and partial differential equations’, SIAM, 2007.

4. Thomas, J.W., ‘Numerical partial differential equations: Finite difference methods’, Springer, 1998.

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MA7278: SPECTRAL THEORY OF HILBERT SPACE OPERATORS

(Pre-requisite: nil)

Total hours: 56


Elements of Hilbert space theory, Bounded linear operators on Hilbert spaces, Bounded linear functionals,

projection ,Riesz representation theorem, Adjoint, Self adjoint, Unitary Normal operators.


Spectral properties of bounded linear operators, Resolvant and spectrum, spectral theory, Complex

analysis in spectral theory.

ModuleIII: (16 hours)

Compact linear operators, spectral theory of compact self adjoint operators; Formula for the inverse

operator, Minimum-maximum Properities of eigenvalues, compact normal operators, Operator equations,

Fredholm alternative.


Spectral properties of bounded self adjoint linear operators, Positive operators, Square root of an operator,

Projection operators, spectral family and spectral family of bounded self adjoint linear operor, spectral

representation of bounded self adjoint Linear operators, Extension of spectral theorem to continuous

functions.

References

1. Arveson, W., ‘An Invitation to C* algebras’, Springer, 2000.

2. Gohberg, I. and Goldberg, S., ‘ Basic operator Theory’, Birkhauser,1981

3. Reed, M. and Simon, B., ‘Methods in Mathematical Physics’, Academic Press, 1986.

4. Courant R. and Hilbert, D., ‘Methods of mathematical Physics’, Interscience, 1996

5. 5. Kreyszig, E., ‘Introductory Functional Analysis with applications’, Wiley Eastern, 2001.

6. Rudin, W., ‘Functional Analysis’, 2/e, Tata McGraw - Hill ed., 2006.

7. Limaye, B.V., ‘Functional Analysis’, 2/e, New Age Publishers, 2006.

8. Conway, J. B., ‘A course in Functional Analysis’, 2nd

ed., Springer-Verlag, 1990.

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MA7279: NUMERICAL LINEAR ALGEBRA


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Total Hours: 56


Review Linear Algebra Basic Concepts, Conditioning and Stability; Condition numbers, Floating point

arithmetic , Stability of various algorithms, Linear Equation Solving: Gaussian Elimination, Pivoting,

Stability of Gaussian Elimination, Cholesky Factorization, Jordan canonical form and applications.


Positive definite systems, LU decomposition, Orthogonal Matrices, Projectors and QR Factorization,

Gram-Schmidt Process, Householder Transformation, Least Square Problems, Numerical Computation

of Eigenvalues and Eigenvectors : Gerschgorin’s Method, Power method, Jacobi’s and Givens method,

Hessenberg form, inverse iteration, Schur factorization, QR algorithm, sensitivity of eigenvalues and

eigenvectors .


Singular Value Decomposition(SVD), Computing the SVD, applications, QR algorithm for SVD


Generalized inverses of matrices , computing the Moore- Penrose generalized inverse of a matrix,

References

1. Trefethen, L.N. and David Bau III, “Numerical Linear Algebra”, SIAM, 1997.

2. Watkins, D. S. , ‘Fundamentals of Matrix Computations’, 2nd

ed., John Wiley & sons, Inc.,

2002.

3. Golub, G. and Loan , C. V., ‘Matrix Computations’, 3rd

ed., John Hopkins University Press, 1996.

4. Hoffman K. and Kunze, R., ‘ Linear Algebra’, Prentice Hall of India, 1971.

5. 5Halmos, P. R., ‘Finite dimensional vector spaces’, Narosa Publishing House, 1974.

MA7280: FRACTAL THEORY AND APPLICATIONS


Total hours: 56


The space of fractals and Iterative function systems: The matrix space (H(X),h), the

completeness of space of fractals, transformations on the real line, affine transformations in the

Eucledian plane, Mobius transformations on the Riemann sphere, Analytic transformations, the

contractio0n mapping theorem, the deterministic algorithm, random iteration algorithm,

condensation sets, the continuous dependence of fractals on parameters.


Chaotic dynamics on fractals and fractal dimension: The addresses of points on fractals, continuous

transformations from code space to fractals, dynamical systems, dynamics on fractals, equivalent

dynamical systems, shadow of deterministic dynamics, shadowing theorem, Chaotic dynamics on fractals,

fractal dimension, theoretical and experimental determination of fractal dimension, Housdo rff-

Besicovitch dimension.

Module II1: (14 hours)

Fractal interpolation,Julia sets and Mandelbrot’ssets: Applications for fractal functions, fractal

interpolation functions, the fractal dimension of fractal interpolation functions, hidden variable fractal

interpolation, space filling curves, escape time algorithm, Julia sets, IFS for Julia sets, Application of

Julia sets to Newton’s method Invariant sets of continuous open mappings, map of fractals, Mandelbrot’s

sets, Mandelbrot’s sets for Julia sets.


Measures on fractals and application: Invariant measures on fractals, measures, integration, Elton’s

theorem, recurrent iterated function systems, applications to computer graphics, fractal compression,

fractal antennas.

References

1. Devaney, R. L., ‘An introduction to Chaotic Dynamical systems’, Secnd edition, Addison

Wesley,1989.

2. Mandelbrot, B.B., ‘The fractal geometry of Nature’, W.H. Freeman and Company, New York,

1982.

3. Peitgen, Jurgens and Saupe, ‘Chaos and Fractals’, Springer- Verlag, 1992.

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