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Website : www.sssu.edu.in ; Email: registrar@sssu.edu.in

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SRI SATHYA SAI UNIVERSITY

Syllabus for Two Year M.Sc. in Mathematics

(Effective From 2009-2010 batch onwards)

M.SC. MATHEMATICS

INTRODUCTION

M.Sc. Mathematics is a four semester course. The students with B.Sc.Honours degree in Mathematics are admitted for this program. The students after completing the program either join research work or some professional courses like M.Tech. in computer science or take up jobs in various field including software development. The syllabus of M.Sc. Mathematics has been prepared by taking into consideration the GATE and CSIR-UGC syllabi of mathematics.The proposed syllabus aims to achieve the following objectives: 1. To provide broad based knowledge of mathematics. This is achieved by

giving the basic Mathematics courses as core courses (12 out of 20). The areas covered include: Analysis, Algebra, Geometry, Differential equations, Mechanics, Statistics, Operations research etc.

2. To provide scope for specialization. This is achieved by keeping 8

courses out of 20 as electives. Electives are offered from various areas of specialization in mathematics such as

Pure Mathematics: Algebraic topology, Boolean algebra etc. Applied Mathematics: Bio-fluid dynamics, Mathematical Biology etc. Computer Science: Artificial Intelligence, Computer Graphics etc. Applicable Mathematics: Wavelets and wavelet transforms, Introduction to simulation etc. Students can take the electives depending on their interest and inclination.

3. Training in Computer Science. Syllabus provides one Software laboratory

course in each of the four-semesters. In these courses the students learn some programming languages, software packages, working in different platforms etc.

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DEPARTMENT OF MATHEMATICS & COMPUTER SCIENCE

SCHEME OF INSTRUCTION AND EVALUATION

M.Sc. (MATHEMATICS) (Effective from 2009 - 2010 batch onwards) Paper Code Title of the Paper Credits Mode of Total Marks

Evaluation FIRST SEMESTER: MATH 701 Advanced Real Analysis 4 IE 100 MATH 702 Advanced Algebra 4 IE 100 MATH 703 Techniques in Applied 4 IE 100

Mathematics MATH 704 Elective - 1 3 IE 100 MATH 705 Elective - 2 3 IE 100 MATH 706 Software Lab I 2 I 50 AWR 700 Awareness Course - 1: 1 I 50 ---- 21 ---- 600 SECOND SEMESTER: MATH 801 Measure Theory 4 IE 100 MATH 802 Functional Analysis 4 IE 100 MATH 803 Numerical Linear Algebra 4 IE 100 MATH 804 Elective - 3 3 IE 100 MATH 805 Elective - 4 3 IE 100 MATH 806 Software Lab II 2 I 50 AWR 800 Awareness Course - 2: 1 I 50 ----- 21 ---- 600 THIRD SEMESTER: MATH 901 Theory of Ordinary 4 IE 100

Differntial Equations MATH 902 Probability Theory 4 IE 100 MATH 903 Differential Geometry 4 IE 100 MATH 904 Elective - 5 3 IE 100 MATH 905 Elective - 6 3 IE 100 MATH 906 Software Lab III 2 I 50 AWR 900 Awareness Course - 3: 1 I 50 ---- 21 ---- 600 FOURTH SEMESTER: MATH 1001 Mathematical Modelling 4 IE 100 MATH 1002 Optimization Techniques 4 IE 100 MATH 1003 Theory Of Statistics 4 IE 100 MATH 1004 Elective - 7 3 IE 100 MATH 1005 Elective - 8 3 IE 100 MATH 1006 Software Lab IV 2 I 50 AWR 1000 Awareness Course - 4: 1 I 50 ---- 21 ---- 600 TOTAL: 84 2400 Notes: 1. In lieu of MATH 1004 and MATH 1005 a dissertation (for 6 credits) could be taken by a candidate. 2. Students taking 5 Electives from one particular stream will be entitled to specialization in that stream. 3. The Choice of electives is at the discretion of The Head of the Dep artment. IE -- Indicates Continuous Internal Evaluation (CI E) & End Semester Examination (ESE) I -- Indicates only Continuous Internal Evaluati on E --indicates only End Semester Examination

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M.SC. (MATHEMATICS) LIST OF ELECTIVE COURSES (3 CREDITS)

Stream I : PURE MATHEMATICS 1 Algebraic Topology 2 Boolean Algebras 3 Category Theory 4 Differentiable Manifolds 5 Fuzzy Mathematics Stream II : APPLIED MATHEMATICS 1 Biofluid Dynamics 2 Advanced Fluid Dynamics 3 Mathematical Biology 4 Topics in Special Functions 5 Finite Element Methods 6. Numerical Solutions of Partial Differential Equations ( PDE ) 7. Foundations of Fuzzy Systems 8. Computational Fluid Dynamics 9. Fluid Dynamics 10.Finance Theory 11.Calculus of variations and Mechanics Stream III : COMPUTER SCIENCE 1. Artificial Intelligence 9 Signals And Linear Systems 2. Computer Networks 10.Cryptography 3. Computer Organization and Architecture 11.Digital Systems 4 Database Systems 12.Microprocessors and Microcontrollers 5 Computer Graphics 13.Bioinformatics 6 Systems Programming 14.Embedded Systems 7 Formal Languages 8 Pattern Recognition Stream IV : FUNCTIONAL ANALYSIS AND APPLICATIONS 1 Spectral Theory of Linear Operators 5.Wavelet Analysis 2 Sobolev Spaces and Sobolev Functions 6.Time scale 3 Integral Equations 7. Control Theory 4 Wavelets and Wavelet Transforms 8.Dynamical Systems Stream V : DECISION THEORY 1 Decision Analysis:Bayesian Approach 2 Decision Theory:Utility Approach 3 Game Theory 4 Multicriteria Decision Making 5 Introduction To Simulation * * *

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M.Sc. (MATHEMATICS) CORE COURSES (4 CREDITS) MATH 701 ADVANCED REAL ANALYSIS Sequences Of Continuous Functions-Limits Of Functions –The Stone And Stone-Weierstrass Approximation Theorem – Polynomial Approximation Theorem-Tietze’s Extention Theorem – Arzela – Ascoli Theorem -The Riemann -Stieltjes Theorem-Existence Of The Integral-Properties Of The Integral-The Derivative In R(P) -The Chain Rule – Mean Value Theorem - Mapping Theorem And Implicit Functions TEXT BOOK

1.The Elements Of Real Analysis,by R.G.Bartle,II nd edn, John Wiley 1964. [ Sections 24, 25, 29 , 31, 39 To 41 ]. REFERENCES

1.Functions Of Several Variables, by W.Flemming,Springer Verlag,1977. 2 .Analysis , by Serge Lang, Addision - Wesley, 1978.

* * * MATH 702 ADVANCED ALGEBRA Fields :Extension Fields - Roots Of Polynomials-The Elements Of Galois Theory, Solvability By Radicals – Galois Groups Over The Rationals. Linear Transformations : Algebra Of Linear Transformations - Characteristic Roots Canonical Forms : Triangular form - Nilpotent Transformation – Jordan Form-Rational Canonical Form -Traces and Transpose - Hermitian -Unitary And Normal Transformations - Real Quadratic Forms.

TEXT BOOK

1.Topics In Algebra , by I.N.Herstein, Vikas Publications 1982. [ Chapters 5 & 6 ].

REFERENCES

1.Linear Algebra , by K Hoffman & R.Kunz , Phipub.1984. 2.Linear Algebra, by S.H. Friedberg, A.J.Insel&L.E.Spence,Phipub.1979.

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MATH 703 TECHNIQUES IN APPLIED MATHEMATICS Perturbation Methods: Regular Perturbations - Singular Perturbations - Boundaray Layer Analysis. Calculus Of Variation: Variational Problems - Necessary Condition For Extremum - The Simplex Problems - Generalisations – Hamiltonian Theory - Isoperimetric Problems

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Integral Equations: Classification And Origin - Fredholm Equation - Symmetric Kernels - Volterra Equations. Integral Transform: Fourier Transform - Fourier Cosine, Sine Transform - Henkel Transform – Mellin Transform - Laplace Transform . TEXT BOOKS 1.Applied Mathematics - A Contemporary Approach., by David . J. Logan , John Wiley ,1987. [ Chapters 2 :2.1 To 2.3 , 3 :3.1 To 3.6 , 4 : 4.4 Only ] 2. Partial Differential Equations Of Mathematical Physics,by Tyn Myint , II nd edn Unorth Holland ,1980. [ Chapter 11 ] REFERENCES 1. Perturbation Methods by A. Nayfeh, Wiley Interscience,1973. 2. Integral Equations And P.D.E , by V.I.Smirnov,Addision Wesley Pub,1964. 3. Use Of Integral Transforms by I.N.Sneddon, Tata McGraw Hill ,1972.. 4. Integral Transforms And Their Applications,by B.Davis,Springer Verlag ,1978.

* * * MATH 801 MEASURE THEORY Semi Rings And Algebra Of Sets - Measures On Semi Ring -- Outer Measures And Measurable Sets -- The Outer Measure Generated By A Measure – Mesurable Functions -- Simple Step Functions – The Lebesgue Measure - Convergence In Measure -- Upper Functions -- Integrable Functions -- The Reimann Integral As A Lebesgue Integral. TEXT BOOK

1.Principles Of Real Analysis, by Charalambos D. Aliprantis And Owen, II nd edn Burkinshow , ] Academic Press Inc. [ Sections. 9 To 19 ] REFERENCES

1.Real Analysis by H.L. Royden,Macmillan Pub.1968. 2.Measure Theory by P.R. Halmos, Dvan Pub.1968. * * *

MATH 802 FUNCTIONAL ANALYSIS Basic Inequalities - Normed Spaces And Bounded Linear Operators - Linear Functional And The Hahn-Banach Theorem - Finite Dimensional Normed Spaces. The Baire Category Theorem And The Closed Graph Theorem - Continuous Functions On Compact Spaces And The Stone-Weierstrass Theorem - The Contraction - Mapping Theorem - Weak Topologies And The Duality - Euclidean Spaces And The Hilbert Spaces - Orthonormal Systems - Adjoint Operators - The Algebra Of Bounded Linear Operators.

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TEXT BOOK

1.Linear Analysis - An Introductory Course , by Bela Bollobas, Cambridge University Press ,Cambridge ,1990. [ Chapters 1 To 12 ] REFERENCES 1.Functional Analysis , by G.Bachmann&L.Narici,Academic Press Pub.1966. 2.Functional Analysis , by Kreyszig , Johnwiley &Sons,1978.

* * * MATH 803 NUMERICAL LINEAR ALGEBRA Fundamental Concepts : Matrix-Vector operations - Orthogonal vectors and matrices -Matrix and vector norms • Singular value decomposition (SVD) QR Factorization : Projectors and QR factorization - Gram - Schmidt orthogonalization process – Householder triangularization - Least-squares problems Conditioning and Stability : Condition numbers - Floating point Arithmetic - Stability – Conditioning of Householder triangularization - Conditioning of Back substitution - Conditioning of Least-squares problems Systems of Equations : Gaussian elimination and LU factorization - Pivoting and LUP factorization - Stability of Gaussian elimination - Cholesky Factorization. Eigenvalue Problems : Overview of eigenvalue problems - Reduction to upper-Hessenberg/Tridiagonal form - Power and inverse power iteration - QR algorithm without shifts - QR algorithm with shifts - Other eigenvalue algorithms - Computing SVD. Krylov - subspace Iterative Methods : Arnoldi iteration – GMRES method - Lanczos iteration - Orthogonal polynomials and Gauss quadrature - Conjugate gradient (CG) method - Bi-Orthogonalization method. TEXT BOOKS 1. Numerical Linear Algebra, by Lloyd Trefethen and David Bau III, SIAM, 1997. [ Chapters – 1 to 6 ] REFERENCES 1.Numerical Linear Algebra , Allaire, Grégoire, Kaber, Sidi Mahmoud , Springer(2008) 2.Applied Numerical Linear Algebra , by James W. Demmel , SIAM( 1997)

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MATH 901 THEORY OF ORDINARY DIFFERENTIAL EQUATIONS Gronwall's Inequality Linear Systems With An Introduction To Phase Space Analysis Existence And Uniqueness For Linear Systems - Homogeneous And Nonhomogeneous Systems - Systems With Constant Coefficients - Asymptotic Behaviour - Autonomous Systems - Phase Space - Two Dimensional Systems - Periodic Coefficients . Existence Theory Existence In Scarlar Case - Existence Theory For Systems Of First Order Equations - Uniqueness And Continuation Of Solutions - Dependence On Initial Conditions And Parameters. Stability Of Linear And Almost Linear Systems Definitions Of Stability – Linear Systems – Almost Linear Systems . Lyapunov's Second Method Lyapunov's Theorems And Proofs. TEXT BOOK 1.Qualitative Theory Of Ordinary Differntial Equations , by F. Brauer And J.A. Nohel, Benjamin ,1967.[ Chapters 1:Sec.1.7 Only ;Chapter 2: Except Sec. 2.6 ; Chapter 3 ; Chapter 4 : Except Sec 4.6 ; Chapter. 5:Except Sec. 5.4 And 5.5 ] REFERENCES 1.Theory Of Ordinary Differential Equations , by E.A.Coddington & N.Levinson , Tatamacgrawhill pub,1972. 2.Stability Theory By Liapunov’s Direct Method , by N.Rouche&M.Loloy,Springer - Verlag Pub,1977.

* * * MATH 902 PROBABILITY THEORY Introduction To Probability Theory: Introduction - Sample Space and Events - Probabilities defined on Events – Conditional Probabilities – Independent Events – Baye’s Formula. Random Variables : Random variables – Discrete Random variables – Continuous Random variables – Expectation of a Random variable – Jointly distributed Random variables – Moment generating Functions – Limit Theorems –Stochastic Processes. Conditional Probability and Conditional Expectation : Introduction – Discrete Case – Continuous case –Computing Expectation by conditioning –Computing Probabilities by conditioning - Applications Markov Chains : Introduction – Chapman – Klomogorov Equations – Clasification of States –Limiting Probabilities – Applications – Mean Time spent in transient states –

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Branching Processes – Time reversible Markov Chains – Markov Chain Monte Carlo Methods –Markov decision Processes. Limit Theorems: Introduction – Chebyshev’s inequality and the Weak Law of large numbers –The Central Limit Theorem – Strong Law of Large Numbers – Other inequalities – Bounding the error probability. TEXT BOOKS

1.Introduction to Probability Models , by Sheldon M Ross , Seventh Edition, Academic Press ( Elsevier ), 2001. [ Chapters 1 to 4 ] 2.A First Course in Probability, by Sheldon M Ross, Sixth Edition, Prentice Hall, 2003. [ Chapter 8 ] REFERENCES

1.An Out Line Of Statistical Theory:Vol I , by A.M.Goon, M.K.Gupta And B.Das Gupta, The World Press Pvt. Ltd , 1988. 2.Modern Probability Theory , by B.R.Bhat, Wiley Eastern Pub.,1989. 3.An Introduction To Probability Theory Vol I And II, by W.Feller, Wiley Eastern

Pub,1986 4.Linear Statistics Inferences and Its Applications, by C.R.Rao, Wiley East Pub.

* * * MATH 903 DIFFERENTIAL GEOMETRY Graph And Level Sets - Vector Fields - Tangent Spaces - Surfaces - Vector Fields On Surfaces And Orientation - The Gauss Map - Geodasics - Parallel Transport - The Weingarten Map - Curvatures Of Plane Curves - Arc Length And Line Integerals - Curvature Of Surfaces - Parametrized Surfaces - Local Equivalence Of Surfaces And Parametrized Surfaces - Surface Area And Volume . TEXT BOOK

1.Elementary Topics In Differntial Geometry, by John A. Thorpe , Springer Verlag Publishers ,1979. [ .Sections.1 to 17 ]. REFERENCE

1. A Course In Differential Geometry , by W.Klingenberg, Springer- Verlag Pub. 1978.

* * * MATH 1001 MATHEMATICAL MODELLING Mathematical Modelling: Need - Techniques - Classification – Some Characteristics of Mathematical Models. Mathematical Modelling Through Ordinary Differe ntial Equations of First Order: Linear Growth and Decay Models - Non Linear Growth And Linear Models - Compartment Models – Mathematical Modelling In Dynamics And Geometrical Problems.

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Mathematical Modelling Through Systems Of Ordinary Differential Equations Of First Order: Mathematical Modelling In Population Dynamics - Epidemics, Compartment Models - Medicine And Arms Race -Mathematical Modelling In Dynamics . Mathematical Modelling Through Second Order Ordinary Differential Equations: Mathematical Modelling Of Planetary Motions -Circular Motions and Motion Of Satellites - Miscellaneous Models.

Mathematical Modelling Through Partial D ifferential Equations:

Mass - Balance Equations - Momentum - Balance Equations - Variational Principles -Probability Generating Functions -Models Of Traffic Flow on a High Way. Mathematical Modelling Through Graphs: Situations That can be Modelled Through Graphs - Mathematical Models In Terms of Directed Graphs -Signed Graphs - Weighted Digraphs - Unoriented Graphs. TEXT BOOK 1.Mathematical Modelling , by J.N. Kapur. Wiley Eastern Limited,1988. [ Chapters : 1 :1.1to1.3; 2; 3 (Except 3.4) ; 4 ; 6 (Except 6.7,6.8 ); 7 ]. REFERENCES 1. Mathematical Models And Their Analysis , by Fredric Y.M. Wan , Harper And Row Publishers , New York. 2. A Concrete Approach To Mathematical Modelling , by Michael Mesterton Gibbons,Addison - Wesley Publishing Company 1989.. 3. Concepts Of Mathematical Modelling, by Walter J Meyer Mcgraw Hill Co.1984. 5. Case Studies In Mathematical Modelling , by W .E. Boyce. , Pitman Advanced Publishing Programs, 1981.

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MATH 1002 OPTIMIZATION TECHNIQUES

LP-Duality And Sensitivity Analysis: Definition Of Dual- Primal-Dual Relationships- Dual Simplex Sensitivity Or Post Optimal Analysis. LP-Networks: Shortest Routs Problem- Flow Network Problems- LP-Representations Of Networks. LP-Advanced Topics: :Theoretical Development Of Simplex And The Related Theorems- Primal-Dual Algorithm And Related Theorems-Revised Simplex-*Bounded Variables-*Decomposition Algorithm-Parametric Linear Programming-Karmarkar Interior Point Algorithm-Goal Programming. Integer Programming: Formulation And Applications-Cutting Plane Algorithm-Branch And Bound Method-Zero One Implicite Enumeration. Dynamic Programming: Dynamic Programming Models-The Capital Budgeting Problems-Definition Of State-Forward And Backward Recursion Equations-Dynamic Programming Models And Competitions-Problems Of Dimensionality.

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Classical Optimization: Optimization Of Functions Of Several Variables-Unconstrainted Optimization- Jacobi’s Method- Language Method-Constrianted Optimization- Khun Tucker’s Conditions. Nonlinear Optimization: Unconstrainted Optimization- Univariate Search And Gradient Methods- Constrainted Optimization- Linear Approximation Methods- Separable, Quadratic, Geometric And Convex Programming. (*) These Topics Should Be Covered As Part Of Seminar And Not To Be Included In The Written Examination. TEXT BOOK 1.Operations Research- An Introduction, by Hamdy A.Taha , 6 th edn ,Phi- 1997. [ Chap.4,Chapters. 6 to 10, 20 to 21 ] REFERENCE 1.Optimization Techniques , by L.R.Foulds, Springer ,Utm , 1981.

* * * MATH 1003 THEORY OF STATISTICS Sampling And Sampling Distributions: Sampling - Sample Mean From Normal Distributions. Parametric Point Estimation : Methods Of Estimation-Properties Of Point Estimation - Sufficiency - Unbiased - Location Of Scale Invariance - Bayes Estimators-Vector Of Parameters- Optimum Properties Of Maximum Likelihood Estimation. Parametric Interval Estimation : Confidence Intervals- Sampling From Normal Distribution Methods Of Finding Confidence Interval- Large Sample Confidence Interval-Bayesian Interval Estimation. Tests Of Hypothesis: Simple Hypothesis - Alternative Hypothesis - Composite Hypothesis -Tests Of Hypothesis - Chi Square Tests-Tests Of Hypothesis And Confidence Intervals-Sequential Tests Of Hypothesis. TEXT BOOK

1. Introduction To Theory Of Statistics, by Mood A.M., Graybill F.A. And Boes D.C., 3 rd edn McGrawHill International Book Company, 1974. [ Chapters :. VI (Sec. 1 to 4),VII, VIII , IX ] REFERENCES 1.The AdvancedTheoryOf Statitics, by M.Kendall & A.Stuart,Charles Griffin & Co. 2.IntroductionToMathematical Statistics,by R.V.Hogg & A.T.Craig,Macmillan Pub. 3.An Introduction To Probability Theory And Mathematical Statistics , by V.K.Rohatgi, Wiley Eastern Pub.

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M.Sc. (MATHEMATICS)

ELECTIVE COURSES (3 CREDITS) Stream I : PURE MATHEMATICS 1, ALGEBRAIC TOPOLOGY Geometric Complexes and Polyhedra-Orientation Of Geometric Complexes. Simplicial Homolgy Groups-Chains-Cycles -Boundaries-Euler Poincare Theorem. Simplicial Approximation - Homomorphism Of Homolgy Groups-The Brouwer Fixed Point Theorem And Related Results. Fundamental Groups-Homototpic Paths-Covering Homotopy.Covering Spaces-Basic Properties Of Covering Spaces-Classification Of Covering Spaces-Universal Covering Spaces. Higher Homotopy Groups. TEXT BOOK

1.Basic Concepts Of Algebraic Topology, by Fred.H.Croom, Springer-Verlag, 1978. [ Chapters : 1 to 6 ] REFERENCE 1. Elements Of Algebraic Toplogy , by James R. Munkres,Benjamin / Cummins, 1984 .

* * * 2. BOOLEAN ALGEBRAS Boolean Rings - Boolean Algebras -Fields Of Sets - Order - Infinite Operations – Sub - Algebras – Homomorphisms - Ideals And Filters - The Homomorphism Theorem - Boolean Spaces – The Representation Theorems - Duality For Ideals - Duality For Homomorphisms - C ompletion – Product Of Algebras - Sum Of Algebras. TEXT BOOK

1. Lectures On Boolean Algebras , by Paul R. Halmos, Springer - Verlag Publication , 1974. [Sec 1 – 3, 6 – 9, 11 – 12, 17 – 21, 26, 27.] REFERENCE 1. Boolean Algebras , by R.Sikorski , Springer Verlag Pub ,1969. * * *

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3. CATEGORY THEORY Intoducation-Particular Objects And Morphisms-Universal Constructions-Factorization Of Morphisms-Structuring Of The Morphism Sets-Functors-Equivalent Categories-Representable And Adjoint Functions. TEXT BOOK 1. Categories , by T.S.Blyth , Longman ,1986 . [ Chapters : 1 to 8 ] REFERENCES 1) Theory Of Categories , by Barry Michell, Academic Press ,1965. 2) Categories , by Horst Schubert, Springer - Verlag Pub ,1972.

* * * 4. DIFFERENTIABLE MANIFOLDS Functions Of Several Variables-Jacobian-Inverse Function Theorem-The Rank Theorem - Differentiable ManifoldsAnd Submanifolds-Immersion-Lie Groups-The Action Of Lie Groups On A Manifold-Transformation Groups-Covering Manifolds. Vector Fields On Manifolds-Tangent Space-Vector Fields-One Parameter Groups Acting On A Manifold-One Parameter Subgroups Of Lie Groups-Lie Algebras Of Vector Fields On Manifolds.

TEXT BOOK

1)An Introduction To Differentiable Manifolds, by William.M.Boothby , Academic Press , 1975. [ Chapters : 1 to 4 ] REFERENCES 1. Manifolds,Tensor Analysis And Applications, by Ralph Abraham, Jerrold.E. Marsden , Tudor Ratice ,Addison-Wesley,1983. 2. Introduction to Differentiable Manifolds, by Auslander, and Mackenzie, McGrawhill, 1963.

* * * 5. FUZZY MATHEMATICS Fuzzy Sets And Operations On Fuzzy Sets (Book 1) Fuzzy Numbers And Their Arithmetic(Book 2). Fuzzy Relations And Functions(Book 1). Fuzzy Logic(Book 1). Fuzzy Measures And Possibility Theory(Book 3) TEXT BOOKS

1.Fuzzy Mathematical Techniques With A pplications , by Abraham Kandel, Addison-Wesley Publications ,1986 . 2.Fuzzy Arithmetic, by Arnold Kaufmann And Madan. M.Guptavan Nostrand, 1984. 3.Possibility Theory, by D.Dubois & H.Prade ,Plenum Press,1986.

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Stream II : APPLIED MATHEMATICS 1. BIOFLUID DYNAMICS Introduction - Viscosity - Laminar And Turbulent Flow - Compressible And Incompressible Flow - Basic Equations Of Fluid Mechanics - Circulatory Biofluid Mechanics - Blood Rheology - Blood Vessel Structure - Diseases Related To Obstruction Of Blood Flow. Models Of Biofluid Flows - Flows In Pipes And Ducts - Models Of Blood Flows - Poiseuille's Flow - Consequence Of Poiseuille's Flow - Applications Of Poiseuille's Law For The Study Of Blood Flow - Pulsatile Flow - Discussion On Pulsatile Flow - The Pulse Wave - Mones-Korteweg Expression For Wave Velocity In An Inviscid Fluid - Filled Elastic Cylinderical Tube - Applications In The Cardiovascular Systems - Wave Propagation Accounting For Viscosity And Its Applications To Cardiac Output Determination -Flow Through A Converging- Diverging Duct. Non-Newtonian Fluids - Classification - Time Independent Fluids - Time Dependent Fluids - Viscoelastic Fluids - Laminar Flow Of Non-Newtonian Fluids -Cassin Model - Flow Of Non-Newtonian Fluids In Elastic Tubes. The Krogh Model Of Oxygen Diffusion From Blood Vessel To Tissue - Cappilary Blood Vessel Region - Boundary Conditions - Krogh's Steady-State Model - Blum's Steady-State Model - Fluid Flow In Kidneys - Diffusion Process In The Haemodialyser - Flow In The Renal Tubule - Flow Measurement By Indicator Dilution Method - Peristaltic Flows - Peristaltic Motion In A Cylinderical Tube - Long-Wavelength Analysis. TEXT BOOK 1.Biofluid Mechanics,by Jagan N. Mazumdar, World Scientific,1992, [ Chapters : 1 To 6 ] REFERENCE 1. Biodynamics-Circulation , by Y.C.Fung, Springer-Verlag,1984. 2. Mechanics of Circulation,by C.G.Cars,T.J.Pedley,R.C.Schofer and W.A.Seed , Oxford Univ. Press,1976. * * * 2. ADVANCED FLUID DYNAMICS Flows At High Reynold's Number: Prandtl's Boundary-Layer Concept-Method Of Matched Asymptotic Expansions-Location And Nature Of Boundary Layers-Incompressible Flow Past A Flat Plate-Outer Expansion-Inner Expansion-Displacement Thickness-Separation Of Flow In A Boundary Layer-Landau's Theory-Flow In A Mixing Layer Between Two Paralle Streams-Narrow Jets-Wakes-Periodic Boundary Layer Flows. Hydrodynamic Stability: Introduction-Thermal Instability Of A Layer Of Fluidheated From Below-Stability Of Couette Flow-Rayleigh-Taylor Instabilty Of Superposed Fluids-Kelvin-Helmholt Instability-Capillary Instability Of A Liquid Jet-Stability Of Parallel Flows. TEXT BOOK

1.Theoretical Fluid Dynamics , by Bhimsen .K.Shivamoggi , Martinus Nijhoff Publishers ,1985

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REFERENCES 1.Boundary Layer Theory , by H..Schilichting ,Mcgrawhill ,1972 . 2.Hydrodynamic And Hydromagnetic Stability , by S.Chandrasekhar, Clarendon Press , 1961. 3.Perturbation Methods In Fluid Mechanics, by M.D.Vandyke ,Parabolic Press, Stanford ,1975.

* * * 3. MATHEMATICAL BIOLOGY Reaction Kinetics: Enzyme Kinetics-Michaelis-Menton Theory-Cooperative Phenomena-Auto Catalysis-Activation-Inhibition-Multiple Steady States-Mushrooms-Isolas. Biological Oscillators And Switches: Feedback Control Mechanisms-Oscillations And Switches Involving Two Or More Species-Simple Two Species Oscillators-Hodgkin-Huxley Theory Of Nerve Membranes-Fitzhugh-Nagumo Model-Modelling The Control Of Testosterone Secretion. Belousov-Zhabotinskii Reactions: Belousov Reaction-Field Nayes(FN) Model-Linear Stability Analysis Of FN Model-Non Local Stability Of FN Model-Relaxation Oscillators. TEXT BOOK

1.Mathematical Biology, by J.D.Murray ,Springer-Verlag,1989. REFERENCES 1.Introduction To Mathematical Biology , by S.I.Rubinov , Wiley ,1975. 2.Mathematical Models In Molecular And Cellular Biology , by L.A.Segal , Cambridge University Press,1980 . 3.The Geometry Of Bological Time , by A.T.Winfree ,Springer-Verlag,1980. 4.Oscillations And Travelling Waves In Chemical Systems,by R.J.Field And M.Burger ,Wiley ,1985..

* * * 4. TOPICS IN SPECIAL FUNCTIONS Bessel’s Functions: Bessel’s Equation-Bessel’s Functions Of The First Kind-Neumann Functions-Spherical And Modified Bessel’s Functions-Integral Representations-Integrals Involving Bessel’s Functions-Zeros Of Bessel’s Functions. Lagender Functions: The Lagender Polynomial-The Hermit Polynomial-Cheychev Polynomial-Mathieu Function. Similarity Methods: Invariant Variational Problems - Invariant Pdes- General Similarity Method. TEXT BOOKS

1.An Introduction To Applicable Mathematics - Part I : Elementry Analysis , by F.A.Hinchey,Wiley Eastern Pub.1980, [ Chapters : .5 & 6 ]. 2.AppliedMathematics-A Contemporary Approach,by J David Logan ,John Wiley, 1987..

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REFERENCES 1.Mathematical Physics , by Butkov,Addison Wesley Pub. 2.Invariant Variational Principles , by J David Logan , Academic Press,1977. 3.Tensors, Differential Forms And Variational Principles , by D Lovelock And H Rund ,Wiley Interscience , 1975 4.Similarity Solutions Of Non - Linear Partial Differential Equations , by L Dresner ,Pitman Publishing 1983.

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5. FINITE ELEMENT METHODS Continuum Boundary Value Problems And The Need For Numerical Discretisation-Finite Difference Methhods-Weighted Residual Methods-Use Of Continuous Trial Functions And The Finite Element Method-Higher Order Finite Element Approximation-Concept Of Mapping-Variational Methods-Partial Discretisation And Time Dependent Problems TEXT BOOK

1.Finite Elements And Approximation , by O.C.Zienkiewicz & K.Morgan , John Wiley And Sons(1983) [ Chapters 1,2:2.1to 2.7,3:3.1 To 3.8,4:4.1 To 4.7,5:5.1,6:6.1 To 6.8,7:7.1 To 7.4] REFERENCES 1. An Introduction To Finite Element Methods, by David.K.Brown , Surrey University Press , 1984. 2.Finite Element Analysis And Applications , by R.Wait & A.R.Mitchell , John Wiley And Sons ,1985. * * * 6. NUMERICAL SOLUTIONS OF PARTIAL DIFF ERENTIAL EQUATIONS ( PDE ) Basic Concepts in the finite difference methods, finite difference approximation in two dimensions. Basic Concepts in the finite element methods : Introduction to Finite element approximations, Method of weighted residual : Galerkin method, Sub domain method, Collocation method , finite elements on irregular subspaces. Study of solution methods based on finite difference and finite element for Parabolic Partial Differential equations

TEXT BOOK

1..Numerical Solution of Partial Differential Equations, by Leon Lapidus and George F Pinder ,( Wiley Interscience ) John Wiely and Sons 1982.

REFERENCES

1.Computational methods for Partial Differential Equations, by M.K.Jain,

S.R.K.Iyenger, R.K.Jain , Wiley Eastern Limited ,1992

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7. FOUNDATIONS OF FUZZY SYSTEMS Introduction : Fuzzy Systems - Modeling Vague , Imprecise and Uncertain Information - Elements of Fuzzy set theory:- Representation - Basic operations- Extension principle- semantics of Fuzzy Sets -Fuzzy Logic. Possibilistic Reasoning : Possibility Distributions and Uncertainty Measure, Possibilistic Inference Rules, Knowledge Representation and Hyper Graphs, Logic Based Inference Mechanisms. Fuzzy Control : Knowledge Based vs Classical Models, Mamdani and Takagi and Sogeno approaches to fuzzy control, Fuzzy controller parameters design, Understanding Fuzzy control as interpolation in the presence of Imprecision.

TEXT BOOK

1.. Foundations of Fuzzy Systems, by R .Kruse, J. Gebhardt and F. Klawonn , John Wiley & Sons. 1994. [ Chapters : 1 . 2: 2.1 to 2.7 , 3 : 3.1 to 3.5 , 4 : 4.1 to 4.4 ]

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8. COMPUTATIONAL FLUID DYNAMICS Introduction: Historical background – Basic elements of Computational Fluid Dynamics ( CFD ) – Review of pde’s for fluid flow – Classification and boundary conditions – Characteristics of second order equations – Initial Value and Boundary Value Problems-Analytical aspects of pde. Numerical Solution Procedures: Weighted Residual Approach for solution of pde’s – Stationary Convection Diffusion Equation : Finite Difference – Finite Element – Finite Volume Discretization – Schemes of Positive Type – Concept of Upwind and Central Differencing – Order of Accuracy – Explicit and Implicit schemes – Examples On Parabolic ( Unsteady Heat Diffusion ) – Elliptic ( Laplace and Poisson ) - Hyperbolic ( Wave ) Equations – Stability Non Stationary Convection - Diffusion Equation : Stability – Discrete Maximum Principle – Definition and Von Neuman Analysis for Linear Problems. Basic Equations of Fluid Dynamics : Conservation Principles – Navier Stokes ( NS ) System of Equations : Compressible and Incompressible Flow – Boundary Conditions – Reynolds Averaged NS Equations for Turbulent Flow – Spatial and Temporal Discretization on Collocated and On staggered Grids - Iterative methods – Stationary methods – Krylov Subspace methods – Multi Grid Methods – Fast Poisson Solvers. Euler Equation in one space dimension – Analytic aspects – Approximate Reimann Solver - Osher Scheme -Flux splitting schemes – Stability – Euler Equations for Inviscid Fow – Full potential Equations for Inviscid Irrotational Flow – Numerical Solution of Euler Equation in General co ordinates – Numerical solution of NS Equations in General Domains. Grid Generation for CFD Analysis: Coordinate Transformation - Discretization in General Domain - Boundary fitted Grid – Structured Grid – Equations of Motion in General Co ordinates – Algebraic

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Methods – Elliptic Poisson Solver – Transpolite Inter Polation – Unstructured Grid –Adaptive Grid CFD for Incompressible Flows: Method of Singularity (Panel) for Inviscid flow - Stream Function – Vorticity Approach – Artificial Compressibilty approach – Primitive Variables Approach – MAC and SIMPLE Algorithm – High Order Upwind Schemes – Eddy Viscosity Hypothesis - Turbulence Modeling – Turbulence Scalar Transport equations – Application Examples. CFD for Compressible Flows : Quasilinearisation of Euler equations - Godunov methods – Riemann Solvers – Jameson’s Runge– Kutta method and Beam Warming Implicit method for Euler and NS Equations – Application examples - Unified Methods for Computing Compressible and Incompressible Flow – Post Processors and Numerical Flow Visualizaton. TEXT BOOK 1.Principles of Computational Fluid Dynamics , by P.Wesseling , Springer – Vorlog , 2000. [ Chapters : 1 to 7 , 10 to 14 ] 2. Computational Fluid Dynamics , by T. J. Chung , Cambridge University Press , 2002. [ Chapters : 10,14,17,18,19,20 ] REFERENCE

1.Introduction to Computational Fluid Dynamics,by P. Niyogi, S.K.Chakrabartty and M.K.Laha , Pearson Education , 2005. 2.Numerical Fluid Flow and Heat Transfer,by S.V.Patankar,McGraw – Hill,1980 3.Computational Fluid Dynamics , by John D Anderson , Mc Graw Hill , 1995 * * *

9. FLUID DYNAMICS Part I: Governing Equations Statistical and Continuum Methods - Eulerian and Lagrangian Coordinates - Material Derivative - Control Volumes - Reynolds' Transport Theorem - Conservation of Mass - Momentum and Energy. Rotation and Rate of Shear- Constitutive Equations- Navier-Stokes Equations- Energy Equation- Governing Equations for Newtonian Fluids - Boundary Conditions, Flow Kinematics - Special Forms of the Governing Equations: Kelvin's Theorem- Bernoulli Equation - Crocco's Equation - Vorticity Equation. Part II: Ideal-Fluid Flow Governing Equations and Boundary Conditions - Potential Flows- Two – Dimensional Potential Flows: Stream Function- Complex Potential and Complex Velocity- Uniform Flows – Source- Sink - and Vortex Flows - Flow in a Sector- Flow around a Sharp Edge- Flow Due to a Doublet- Circular Cylinder without a Circulation- Circular Cylinder with Circulation- Blasius' Integral Laws- Force and Moment on a Circular Cylinder- Conformal Transformations - Joukowski Transformation- Flow around Ellipses- Kutta Condition and the Flat-Plate Airfoil - Symmetrical Joukowski Airfoil. Three - Dimensional Potential Flows: Velocity Potential - Stokes' Stream Function- Solution of the Potential Equation - Uniform Flow- Source and Sink- Flow Due to a Doublet- Flow near a Blunt Nose- Flow around a Sphere- Line-Distributed Source- Sphere in the Flow Field of a Source- Rankine Solids- d'Alembert's Paradox.

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Part III Viscous Flows of Incompressible Fluids Exact Solutions: Couette Flow - Poiseuille Flow - Flow between Rotating Cylinders - Stokes' First and Second Problems – Stagnation Point Flow – Flow in Convergent and Divergent channels. Boundary Layers: Boundary-LayerThicknesses,The Boundary-Layer Equations, Blasius' Solution – Falkner - Skan Solutions – Flow over a Wedge – Stagnation – Point Flow – Flow in a Convergent Channel – Approximate Solution for a Flat Surface – General Momentum Integral – Karman – Pohlhausen Approximation TEXT BOOK 1.Fundamentals of Mechanics of Fluids, by I. G. Currie, Mc Graw-Hill , 1993. [ Chapters 1-3; 4.1-16; 5.1-12; 7.1-5 , 7.7,7.8; 9.1-9.10 ]

REFERENCES 1. Theoretical Fluid Dynamics , by Bhimsen K.S.,Martinus , Nijhoff pub.1985. 2. An Introduction To Fluid Dynamics, by G.K.Batchelor, Cambridge Univ.Press, 1981. 3. Boundary Layer Theory , by H Schichting, McGraw Hill,1955. 4. A Text Of Fluid Dynamics , by F.Chorlton,Cbs Pub , 1985.

* * *

10. FINANCE THEORY

I. Basics Financial markets, institutions and services – Market makers and Margin accounts – Market efficiency; Yield curves; Yield to maturity - short sales – spot and forward rates. Financial markets and Economic Development

II. Risk and Uncertainty Random variables – Expectation and Variance – Expected utility Hypothesis - Risk premium- Portfolio construction - Feasibility and optimality Capital Market line and Separation Theorem.

III. Pricing Theories Capital Asset Pricing Model - Security market line - Arbitrage pricing theory – Multiple Factor Models.

IV. Options and Derivatives Derivatives - Put and Call options - Valuation of options - Binomial option pricing- Brownian Motion- Black Scholes Formula.

V. Corporate Finance Earnings-Pay out Ratio - Capital gains - Marginal Cost of Funds and Investment- Linter model- Leverage/Gearing ratio - Modigliani – Miller Theorem - Value of the Firm - Corporate Takeover

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TEXT BOOKS 1. William Sharpe, Gordon Alexander and Jeffery Bailey, Investments 6/e, Prentice Hall of India, 2006 2. John Eatwell, Murray Milgate and Peter Newman (eds.), Finance: The New Palgrave, Norton 3. Sheldon M. Ross, Mathematical Finance, 2/e, Cambridge, Cambridge University Press, 2006. 4. Harry Markowitz, Portfolio Selection, Blackwell. 5. R. E Bailey, The Economics of Financial Markets, Cambridge, Cambridge University Press, 2005

* * *

11. CALCULUS OF VARIATIONS AND MECHANICS

Calculus of variations: Concept of variation. fundamental lemma of calculus of variations. Euler's equation, Functional dependent on several independent variables, Variational problems with moving boundaries, Sufficient conditions for extrema. Mechanics: Dynamics of a particle and a system of particles in three dimensions, Principle of Linear Momentum angular momentum and energy, conservation laws, Motion of rigid body, D'Alembert's principle, Eulerian angles, Rotating frame of reference motion of top, motion under constraints, Euler's dynamic equations, Lagrange's equation, Lagrange's equation for a simple dynamical system. Hamiltonian and Canonical equations of motion. TEXT BOOKS 1. D.T Greenwood: Principle of Dynamics, Prentice- Hall of India (1988). 2. H.Goldstein, C.P Poole and J. L. Safko: Classical Mechanics, Prentice- Hall of India(2002) 3. A.S. Gupta: Calculus of Variation with applications, Prentice- Hall of India(2003). 4. N.C. Rana and P.S. Joag : Classical Mechanics, Tata McGraw Hill, New Delhi(1991).

* * *

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Stream III : COMPUTER SCIENCE 1. ARTIFICIAL INTELLIGENCE Introduction – what is AI ? – Intelligent Agents : agents , environments – Solving problems by searching : problem solving agents –Example problems – Uninformed search strategies – Informed search and exploration : Informed search strategies –Heuristic functions– Local search algorithms – Optimization problems. Logical Agents :Knowledge Based Agents – Logic – Propositional logic – Reasoning patterns – Propositional Inference – Agents based on propositional logic – First Order Logic :Representation–Using FOL–Knowledge Engineering–Inference in FOL : Unification And Lifting – Forward Chaining – Backward Chaining – Resolution – Examples. Knowledge Representation : Ontological Engineering – Categories and objects – Actions situations and events – Mental events and mental objects – Reasoning Systems – Truth maintenance systems Learning from Observations :Forms of learning – Inductive learning – Learning Decision trees - Knowledge in Learning :A logical formulation of learning – Knowledge in Learning – Explanation based learning – Learning using relevance information TEXT BOOK

1.Artificial Intelligence – A Modern Approach , by Stuart J. Russel and Peter Norvig, Prentice Hall , Pearson Education , 2003. [ Chapters & Sections : 1: 1.1 ; 2: 2.1 to 2.4 ; 3:3.1 to 3.6; 4: 4.1 to 4.3 ; 7: 7.1 to 7.7; 8 : 8.1 to 8.4; 9: 9.1 to 9.5 ; 10: 10.1 to 10.8; 18: 18.1 to 18.3 ; 19: 19.1 to 19.4 ] REFERENCES 1.Artificial Intelligence,Structures and Strategies for Complex Problem Solving,by George F.Luger and william A.Stubblefield,The Benjamin / Cummings Publishing Co, 1993. 2. Artificial Intelligence and Soft Computing, by Amit Konar , CRC Press, 2000.

* * *

2. COMPUTER NETWORKS Introduction -Network edge, Network core, ISPs and Internet, Protocol Layers and service models, OSI, TCP/IP reference models. Application Layer: Principles , Web and HTTP ,FTP ,SMPT ,DNS. Transport Layer: Services, Multiplexing and demultiplexing, principles of reliable data transfer, connection oriented transport,TCP, Connectionless support,UDP, Congestion control. Network Layer and Routing : Service models, datagram service virtual circuit service, routing principles, routing algorithms, IP protocol, routing in internet, IPV6, Multicast routing. Link Layer and LAN: The Data Link Layer services, Error Detection and Correction, Multiple Access protocols, LAN addresses and ARP, Bridges, Routers.

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Wireless and Mobile Networks: Wireless links, characteristics, CDMA, IEEE 802.11 wireless LANs, Cellular Internet Access, addressing and routing to mobile users, Mobile IP, Handling mobility in cellular networks. Network Security: Principles of Cryptography, Authentication Protocols, Digital Signatures, Message Digests, Key distribution and certification, Packet Sniffing, Secure email, secure sockets, IPsec TEXT BOOK

1. Computer Networks : A Top Down Approach Featuring the Internet , by Jim Kurose, Keith Ross, 3rd Ed, Pearson Education 2004 [Chapters 1, 2.1 - 2.5, 3, 4.1 - 4.7,5.1 – 5.6, 6.1 – 6.7, 8 ] REFERENCES 1.Data And Computer Communications, by William Stallings, VII th edn , Pearson Education , 2005.

2. Computer Networks,by Andrew S.Tanenbaum,IV Ed,Pearson Education 2003 * * * 3. COMPUTER ORGANIZATION AND ARCHITECTURE Casche : Cache memory, Design principles, Pentium 4 and Power PC Cache organization. Operating System : Operating system support, Scheduling Memory management PII and Power Pc memory management. Computer Arithmetic : Computer arithmetic representation of numbers, floating point arithmetic, integer arithmetic, IEEE754 standard, booths algorithm. Instruction Characteristics: Instruction set characteristics and functions, types of operations, basic instructions in 8085. Addressing Modes and Formats : Addressing modes and formats – Pentium and Power PC Instruction Formats. CPU Structure and Functions : CPU structure and functions – Instruction cycle and Pipelining, RISC : RISC Characteristics – Pipelining – Comparison of RISC and SISC. Instruction Level Parallelism : Instruction level parallelism and super scalar processors, design issues. Control Unit : Control unit operations, microprogrammed control Parallel Organization : Parallel processing, multiple processor organization, symmetric multiprocessors , cache coherence, MESI protocol, clusters , non uniform memory access.

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TEXT BOOK

1.Computer Organization And Architecture - Designing for performance, by Williams Stalling, 6th Edn, Pearson Education2003 2.Structured Computer Organization,by Andrew Tanenbaum,2 nd edn , 1984. [ Book 1 - Chapters 4,8,9,10.1-10.4,11,12.1-12, 13.1-13.5,13.8, 14.1-14.2, 18. Book 2 – Sections 4.1 and 4.2 ]

REFERENCE BOOKS: 1..Structured Computer Organization,by Andrew Tanenbaum, IV th edn ,Pearson education. 2.Computer Architecture And Organization – John P.Hayes, McGraw Hill edn.

* * * 4. DATA BASE SYSTEMS Introduction to Database Management Systems - Relational Model – SQL – Other Relational Languages – Database Design and ER Model - Relational Database Design - Object Based Databases – Transactions TEXT BOOK

1.Database System Concepts,by, Abraham Silberschatz, Henry F. Korth, S. Sudarshan, V th edn ,Tata McGraw Hill, 2005 [ Chapters 1,2,3,5,6,7,9,15 ]

REFERENCES 1.Database Management Systems, by , Raghu Ramakrishnan , Johannes Gehrke, III rd edn , Tata McGraw Hill. 2002. 2.Database Design, Applications and Administration with ER assistant, by Micahael V. Mannino, II nd edn Tata McGraw Hill. 2002 3.Database Systems: A practical approach to design, implementation, and management, by , Connolly and Begg,IV th edn , Pearson Education ,2005 .

* * * 5. COMPUTER GRAPHICS Introduction : Lines , Frame buffers , Bresenhams algorithm for line and circle, Antialiasing , Frame buffer Display Devices : Normalized device co - ordinates , display file structure, Display controlPolygons, filling polygons, filling with pattern Transformations : Matrices , scaling , rotation, homogenous coordinates and translation, coordinate transformations, rotation about an arbitrary point

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Windowing and Clipping : Viewing transformation, Clipping,Cohen Sutherland algorithm , clipping of polygons, Generalized clipping 3 D Geometry : 3D geometry, 3D primitives, 3D transformations , rotation about an arbitrary axis, parallel projection, perspective projection Hidden Surface and Lines : Back face removal, Z – buffers, Scan line algorithm, Painters algorithm Light Color and Shading : Introduction and overview of terms like, diffuse illumination, point source illumination, transparency, Reflections, shadows, Ray Tracing, Half tone, color models Curves and Fractals : Curve generation, Interpolating Algorithms, B -splines, Bezier curves, Fractal lines and Fractal surfaces

TEXT BOOK

1. Computer Graphics: A Programming Approach , by Steven Harrington, II nd edn ,McGraw - Hill , 1987 . [Chapters 1-4,6,8,9,10,11 ]

REFERENCES

1.Mathematical Elements of Computer Graphics, by David F Rogers and J Allen Adams, Mc Graw-Hill, II nd edn 1990 2. Procedural Elements of Computer Graphics, David F Rogers,II nd edn , McGraw-Hill Int , 1998. 3.Introduction to Computer Graphics , N.Krishnamurthy, Tata McGraw-Hill Pub. Co., New Delhi, 2002

* * * 6. SYSTEMS PROGRAMMING Introduction – Simplified Instructional Computer, Assemblers , Loaders and Linkers, Macro Processors, Compilers, Operating Systems. TEXT BOOK

1.Systems Programming , by Leland Beck, III Ed, Pearson Education, 1997. [ Ch 1 to Ch. 6 ] * * *

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7. FORMAL LANGUAGES Notion Of Formal Language - Concepts And Notation- Chomsky Hierarchy Of Languages- Operations On Languages – Definition And Closure Properties- Context Free Languages - Chomsky Normal Form- Derivation Tree- Linear Grammars And Regular Languages- Greibach Normal Form- Regular Expressions. Context-Sensitive Language- Length-Increasing Grammars- Kuroda Normal Form- One-Sided Context-Sensitive Grammars. Unrestricted Phase-Structure Languages - Derivation Graph. Automata And Their Languages - Finite, Pushdown, Two-Pushdown Automata, Turing Machines. Decidability - Recursive And Enumerable Languages- Church Turing Thesis-Undecidable Problem- Complexity Of Computations- Deterministic And Non- Deterministic Automata. TEXT BOOK 1.Introduction To Formal Languages , by Gyorgy E Revesz, Mcgraw-Hill Book Co.,1985. [ Chapters: 1 To 7 ] REFERENCES 1.Formal Languages , by A.Salomaa , Academic Pub.1973. 2.Intoduction To Formal Languages And Automata , by P.Linz,Narosa Pub. 1997.

* * * 8. PATTERN RECOGNITION Introduction : Machine Perception , Pattern Recognition Systems, The Design Cycle, Learning and Adaptation Baye’s Decision Theory : Bayes Decision Theory , Minimum Error rate Classification , Classifiers , Discriminant functions and Decision Surfaces, Normal Density , Discriminant functions for the Normal Density, Bayes Decision Theory for Discrete features Maximum Likelihood and Bayesian Parameter Es timation : Maximum - Likelihood Estimation, Bayesian Estimation, Bayesian Parameter Estimation :Gaussian Case and General Theory , Problems of Dimensionality , Component Analysis and Discriminants . Non Parametric Techniques: Density Estimation, Parzen Windows , K- Nearest Neighbor Estimation,NN rule,Metrics and NN Classification, Fuzzy Classification Linear Descriminant Functions : Linear Discriminant Functions and Decision Surfaces, Generalized Discriminant Functions, The two - category linearly separable case , Minimizing the perceptron criterion function , relaxation procedures , non - separable behavior , Minimum Squared- Error procedures

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Multi Layer Neural Networks : Feed - forward Operation, and Classification, Back – propagation Algorithm , Error Surfaces , Back - propagation as Feature mapping. Unsupervised Learning and Clustering : Mixture Densities and Identifiability , Maximum - Likelihood Estimates, Application to Normal Mixtures , Unsupervised Bayesian Learning , Data Description and Clustering , Criterion functions for Clustering , Iterative Optimization , Hierarchical Clustering. TEXT BOOK

1. Pattern Classification , by Richard. O. Duda., Peter . E . Hart and David. G. Stork, John-Wiley & Sons Inc., II nd edn , NY 2004

[Chapters 1, 2.1 to 2.9, 3.1- 3.8, 4.1- 4.7, 5.1 to 5.8, 6.1- 6.5 ,10.1 to 10.9. ] REFERENCES:

1. Pattern Recognition Principles, by Tou and Gonzalez, Wesley Publishing Co, 1974 * * *

9. SIGNALS AND LINEAR SYSTEMS

Introduction

Classification of signals, basic operation on signals, properties of systems.

Time-Domain Representations of Linear Time-Invarian t Systems

Convolution, Impulse Response Representation for LTI Systems, Properties of IRR for LTI Systems, Differential and Difference Equation representation for LTI Systems. Fourier Representations for Signals

Discrete Time Fourier Series, Discrete Time Fourier Transform.

Applications of Fourier Representations

LTI Systems, Convolution and Modulation,Sampling, Reconstruction of Continuous Time Signals from samples

Representing Signals by Using Continuous-Time Compl ex Exponentials The Laplace Transform , Unilateral Laplace Transform, Inversion of the Laplace Transform, Bilateral Laplace Transform. Representing Signals by Using Discrete - T ime Complex Exponentials

The z-Transform - Properties of z-Transform, Inversion of z-Transform, Unilateral z-Transform. Application to Filters and Equalizers

Ideal Lowpass Filters , Approximating functions, Frequency Transformations, Passive Filters , Digital Filters, FIR Digital Filters, IIR Digital Filters.

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TEXT BOOK

1. Signals and Systems, by Simon Haykin, Barry Van Veen, 2nd Ed, Wiley Pub. 2000 [ Chapters 1,2,3,4.1-4.10,6,7,8 ]

REFERENCES:

1. Signals and Systems, by Alan V. Oppenheim,Prentice hall Inc, II nd edn , 1996

2. Signals , Systems , and Transforms , by Charles L.Phillips, III rd edn ,Prentice

Hall Inc, 2002 . * * *

10. CRYPTOGRAPHY Introduction : OSI Security Architecture - Classical Encryption techniques – Cipher Principles - Data Encryption Standard - Block Cipher Design Principles and Modes of Operation - Evaluation criteria for AES - AES Cipher -Triple DES - Placement of Encryption Function - Traffic Confidentiality Public Key Cryptography : Key Management-Diffie-Hellman key Exchange - Elliptic Curve Architecture and Cryptography - Introduction to Number Theory - Confidentiality using Symmetric Encryption - Public Key Cryptography and RSA Authentication and Hash Function : Authentication requirements - Authentication functions - Message Authentication Codes - Hash Functions - Security of Hash Functions and MACs - MD5 message Digest algorithm - Secure Hash Algorithm -RIPEMD - HMAC Digital Signatures - Authentication Protocols - Digital Signature Standard Network Security : Authentication Applications: Kerberos - X.509 Authentication Service Electronic Mail Security - PGP -S/MIME - IP Security - Web Security. System Level Security : Intrusion detection - password management - Viruses and related Threats - Virus Counter measures - Firewall Design Principles - Trusted Systems. TEXT BOOK

1. Cryptography And Network Security - Principles and Practices , by William Stallings Prentice Hall of India, III rd Edn, 2003.

REFERENCES:

1.Cryptography and Network Security , by Atul Kahate, Tata McGraw -Hill, 2003. 2.Applied Cryptography, by Bruce Schneier , John Wiley & Sons Inc, 2001. * * *

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11. DIGITAL SYSTEMS Review of Digital Principles Review of number system: codes- logic gates and Boolean Algebra - Universality of NAND and NOR gates- enable/disable gate circuits - Boolean and DeMorgan’s theorems - Sum of Product forms (SOP) & Product of Sums (POS) logic circuits - simplifying Logic circuits by Karnaugh Map method. Combination Logic Functions: Binary arithmetic operations -representing positive and negative numbers - two’s compliment arithmetic -( Basic adders, Parallel Binary Adders / subtracters – Comparators – Decoders – Encoders - Code Converters - Multiplexers ( Data Selectors ) - Demultiplexers ( data distributors ) - Parity Generators / Checkers - and BCD to seven segment decoder / driver ) and their applications Sequential Logic Circuits: introduction - Latches and Flip-Flops -SR and D type latches- level triggered latches - D, JK, T, Master-Slave and Edge-Triggered Flip-Flops - Flip-Flop Operating Characteristics and timing considerations - Monostable Multi vibrator (one-shot) - Non retriggerable one shot and retriggerable one shot Flip-Flop Application. Registers : introduction - buffer registers, shift registers, types, SISO, SIPO, PISO, PIPO - bidirectional shift registers and their applications - memory registers - serial to parallel and parallel to serial conversion and arithmetic operations Counters: introduction -Asynchronous (ripple) binary up counters and frequency divider; binary down counters - IC binary Up-Down Counter, MOD counters less than 2N; propagation delay in ripple counters - synchronous counters - shift register counters/sequencers - counter application: (i) digital clock, (ii) frequency counter (iii) multiplexed display Analog and digital signal conversion and Digital Si gnal Processing: Digital to Analog conversion methods - converting analog signals to digital - analog to digital conversion methods - digital signal processing basics and the digital signal processor. Memory and Storage: Introduction - primary and secondary memories - semiconductor ROMs – types - Mask ROM, PROMs, EPROMs - Flash Memories - a basic diode ROM with internal decoding and transistor ROMs - characteristics and applications - semiconductor Read/Write memories (RAMs) - RAM types: static and dynamic, RAM applications - special type of memories: CCDs, Magnetic and Optical Storage. Programmable Logic Devices: introduction; a basic PLD; PLD arrays and classifications; PLA, PAL, CPLD and FPLD; Generic array logic : GAL 22V10 and GAL16V8; PLD Programming and software; Digital system application. 9.Integrated Circuit Logic Families: Digital IC Terminology; The TTL Logic Family; Standard TTL Series Characteristics; Improved TTL Series Characteristics; TTL Loading and Fan Out; Other TTL Characteristics; Connecting TTL Outputs Together; open-colector TTL gates; Tristate (3-state) TTL; Buffer/driver TTL gates and Schmidt trigger TTL gates; the ECL Digital Family; MOS Digital Integrated Circuits; The MOSFET; Digital MOSFET Circuit, Characteristics of MOS Logic, Complimentary MOS Logic, CMOS Series Characteristics, CMOS Open-Drain and Tristate Outputs, CMOS Transmission Gate (Bilateral Switch). TEXT BOOKS

1. Digital Fundamentals by Floyd T. L. , VIII edn , Pearson Education Asia 2004.

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2. Digital Systems: Principles and Applications; VII edn; Prentice- Hall of India 2002. 3. A first Course in Digital Electronics, by, Cook N P, edn II, Prentice Hall, 1999. 4. Digital Computer Electronics, by Malvino A. P, Tata Mc-Graw Hill, 2004. * * *

12. MICROPROCESSORS AND MICROCONTROLLERS Microprocessors : 8-bit Simple – as – possible computer ( SAP - 1, SAP -2 ) , 8085 microprocessor: Architecture, instruction sets , programming with 8085, microcomputer system Microprocessors : 16-bit system 8088 micro processor architecture,8086 software aspects , assembly language programming, procedures, macros , interrupts, interrupt routines, Peripheral Interface : I/O Interface : PPI, serial communication , programmable timers, KB & Display Controller, interrupt controller. Microcontrollers : Basic organization, 8051 CPU structure, interrupts, timers, instruction sets,timing diagram,programming,8096 architecture, instructionssets , simple programming Peripherals and Interfacing: Bus structure, memory organization, extended model and memory interfacing, polling, I /O interfacing, A / D interfacing, applications. TEXT BOOKS

1.Microprocessor Architecture , Programming and Application with 8085 , by Ramesh S Gaonkar, IV th edn, Penram International Publishing,1999. 2.MicrocomputerSystem:8086/8088 family architecture,programming and design by You Cheng Liu and Glenn A Gibson , II nd edn , Prentice Hall India , 2001 3.Programming and Customizing the 8051 Microcontroller, by Myke Predko , Tata Mcgraw Hill , 1999 REFERENCES:

1.Digital Computer Electronics, an introduction to micro – computers , by Albert Paul Malvino, Tata Mcgraw Hill , 1983

2.Design with Microcontrollers, by J.B. Peatman , Mcgrawhill International , 1989. * * *

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13. BIOINFORMATICS Introduction : The central dogma, parallel universes, watson’s definitions, top-down Vs bottom – up approach, information flow, conversance, communications. Database Networks: Definition, data management, data life cycle, database technology, interfaces, Implementation, networks: communication models, transmission technology, Protocols, topology, security. Search Engines and Data Visualization: Search process, technologies, searching and information theory, computational methods, Knowledge management, sequence visualization, structure visualization, user interfaces, animation vs simulation Statistics, Data Mining and Pattern Matching: Statistical Concepts, Micro Arrays, Imperfect data, randomness, data analysis, tools , selection, clustering , classification , data mining methods, Infrastructure pattern recognition, discovery, machine learning, text mining, pattern matching fundamentals, dynamic programming, word method, baysian method, multiple sequence alignment tools. Modeling Simulation and Colloboration : Drug discovery fundamentals, protein structure, system biology tools, collaboration and communication, case study.

TEXT BOOKS

1.Bio Informatics Computing, Bryan Bergeron, Prentice Hall , 2003

[ Chapters : 1,2,3,4,5,6,7,8,9,10 ]

REFERENCES:

1 Introduction to Bioinofmatics , by T.K.Affward , D. J .Pary Smith ,Pearson Education , 2001. 2 Bioinformatics - The machine Learning Approach , by Pierre baldi ,Soren Brunak , II nd edn,First East West Press , 2003. * * * 14. EMBEDDED SYSTEMS Embedded Architecture : Inroduction,embedded computers, design process – requirements, specifications, architectural design,components,system integration, formalism for system design. Embedded Processor and Computing Platform: ARM Processor, data operations, flow of control, SHARC Processor, memory organization, parallelism with instructions, bus configuration, memory devices, I/O devices, component interfacing. Networks : Distributed embedded architecture, networks for embedded systems , CAN bus, Ethernet, Myrinet, Internet , communication analysis , performance Analysis, allocation and scheduling.

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Real Time Characteristics: Clock driven approach , weighted round robin approach , dynamic versus static systems , release times and deadlines , earliest deadline first algorithm (EDF), scheduling System Design Techniques: Design methodologies, requirement analysis, specification, System analysis and Architecture design , Quality Assurance, examples. TEXT BOOKS

1. Computers as Components: Principles of Embedded Computing System

Design , by Wayne Wolf, Morgan Kaufman Publishers, 2001

[Chapters: 1, 4, 8, 9 ]

REFERENCES:

1. Real - Time Systems, by Jane.W.S Liu., Pearson Education Asia, 2000 2. Real-Time Systems , by C. M. Krishna and K. G. Shin , McGraw-Hill, 1997 3. Embedded System Design : Unified Hardware / Software Introduction, by Frank Vahid and Tony Givargi , John Wiley & Sons, 2000. * * *

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Stream IV:FUNCTIONAL ANALYSIS AND APPLICATIONS 1. SPECTRAL THEORY OF LINEAR OPERATORS Spectral Theory of Linear Operators In Normal Spaces-Compact Linear Operators In Normed Spaces And Their Spectrum-Spectral Thoery Of Bounded Self-Adjoint Linear Operators-Unbounded Operators In Hilbert Spaces. TEXT BOOK

1.Introductory Functional Analysis With Applications, by Erwin Kreyszig ,John Wiley, New York 1978. REFERENCES 1.Functional Analysis , by Walter Rudin, McGraw Hill Pub,1991 2.Introduction To Functional Analysis And Applications, by E. Kreyszig, John Wiley &Sons,1978.

* * * 2. SOBOLEV SPACES AND SOBOLEV FUNCTIONS Sobolev Spaces and their Basic Properties-Pointwise Behaviour Of Sobolev Functions-Poincare Inequalities-A Unified Approach. TEXT BOOK

1. Weakly Differentiable Functions, by William. P. Ziemer, Springer-Verlag , 1989. New York. [ Chapterss.2 To 4 ]. REFERENCES 1.Sobolev Spaces , by R.A.Adams,Academic Press,1975 2.Introduction To Sobolev Spaces , by C.W.Clark,Univ.Columbia Pub,1968. * * * 3. INTEGRAL EQUATIONS Integral Equations,Their Origin And Classification-Modelling Of Problems As Integral Equations-Volterra Integral Equations-The Green's Function-Fredholm's Integral Equations-Existence Of Solutions-Basic Fixed Point Theorems. TEXT BOOK

1.Introduction To Integral Equations With Applications , by Abdul.J.Jerri ,Marcel Dekkes Inc.,New York ,1985. [ Chapters.1 To 6 ] REFERENCES 1.Integral Equations-A Short Course , byL.G.Chambers,International Text Book Co.1976. 2.Integral Equations - A Practical Treatment From Spectral Theory Applications, by D.Porter & D.S.G.Stirling,Cambridge Univ.Press,1990.

* * *

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4. WAVELETS AND WAVELET TRANSFORMS Continuous Wavelet Transforms (CWT), Discrete Wavelet Transform (DWT) and orthogonal wavelet decomposition – Multiresolution analysis - Construction of wavelets - Discrete wavelet transform and filter bank algorithm-Applications to digital signal processing,image compression , and boundary value problems

TEXT BOOKS

1. Fundamentals of wavelets –Theory , Algorithms , and Applications.by Jaideva C. Goswami and Andrew K. Chan. John Wiley & Sons, Inc. 1999. 2.Wavelet Transforms – Introduction to theory and applications,by Raghuveer M. Rao and Ajit S. Bopardikar. Pearson Education Asia. 2001. [ Book 1 - Chps 4 -7, 9.1- 9.8,[10], Book 2 - Chapters 1-5 ] REFERENCES 1. Wavelet Transforms and their Applications, by Lokenath Debnath, Birkhauser, Boston 2002.

* * * 5. WAVELET ANALYSIS Fourier Transforms of the square integrable functions - Riesz fisher Theorem – Poisson Summation formula _ integral wavelet transforms – Wavelet frames – Orthogonal bases of wavelets – multiresolution analysis – compactly supported wavelets – cardinal spline wavelets – fast wavelet transforms . A few recent developments in wavelet theory and applications. TEXT BOOKS 1. A first course on wavelet, by E. Hernandez and G.Weiss, CRC Press 1996. 2. Wavelet Analysis the scalable structure of information, by H.L.Resnikoff and

R.O.Wells, , Springer 1998. 3. An introduction to wavelets through linear algebra, by M.W.Frazier, springer 1999. 4. Principles of Fourier Analysis, by K.B.Howell, Chapman &Hall / CRC Press, 2001.

* * * 6. TIME SCALE Time scale calculus – First order linear equations on time scale – Hilger’s complex plane – Initial value problems - Second order linear equations on time scale – hyperbolic and trigonometric functions – Euler –Cauchy equations – laplace transforms – Self-Adjoint equations – Riccati equation – Boundary value problems and Green’s function– Eigenvalue problems. TEXT BOOK 1.Dynamic Equations On Time Scales an introduction with applications,by Martin Bohner and Allan Peterson , BIRKHAUSER BOSTON. BASEL.BERLIN 2001. [ Chp 1 to Chp 4 ]. * * *

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7. CONTROL THEORY * Introduction to Control Systems - Different types of Control Systems- Block Diagram Model – Transfer Function Model – Signal Flow Graph Model - State Variable Model - Examples. ** Basic Results of first order systems of Ordinary Differential Equations- Fixed Point Methods. ** Observability – Controllability – Stability – Stabilizability- Optimal Control.

TEXT BOOK 1. Control Systems Principles and Design, by M Gopal,Tata McGraw-Hill Publishing Company Limited. 2002. 2.Automatic Control Systems , by Benjamin C .Kuo, Prentice Hall of India. 3.Elements of Control Theory, K.Balachandran, J.P.Dauer, Narosa Publishing House, 1999. [ Book 1 & Book 2 – Topics Under * ; Book 3 – Topics Under ** ]

REFERENCES

1. Modern Control Systems, by Richard C Dorf and Robert H. Bishop, Addison- Wesley, 1998. * * * 8. DYNAMICAL SYSTEMS

First order Linear dynamical systems – Introduction to Non-Linear Dynamical System - Complex behaviour of the Nonlinear dynamical systems - Complex behaviour of the Nonlinear dynamical systems – Higher Order Linear Dynamical Systems –Dynamical Systems of several equations – Non Linear Dynamical Systems of several equations. TEXT BOOK

1.Discrete Dynamical Systems Theoryand Applications , by James T Sandefur,

Clarendon Press Oxford ,1990. [ Chapters 1 to 7 ] * * *

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Stream V: DECISION THEORY 1. DECISION ANALYSIS: BAYESIAN APPROACH Decision Trees - Utilities - Subjective Probabilities And Their Measurement- Influence Diagrams - Group Decisions - Bayesian Statistics- Bayes Estimation. TEXT BOOK 1.Decision Analysis: A Bayesian Approach , by J. Q. Smith , Chapman & Hall Publications,1988. [ Chapters 1 To 7 ]

* * * 2. DECISION THEORY:UTILITY APPROACH

Decision Problems - Decision Under Strict Uncertainty- Preference Orders And Value Functions- Multi-Attribute Value Theory- Utility Theory- Group Decision And Social Choice. TEXT BOOK 1.Decision Theory , by Simon French ,John Wiley & Sons ,1986.

[ Chapters 1 To 5 & 8 ] REFERENCE

1.Decisions With Multiple Objectives , by R.L.Keeny,H.Raiffa,John Wiley And Sons 1976.

* * * 3. GAME THEORY

Game: What Is Game - Examples Of Games - History. Two Person Zero Sum Games: Extensive Form- Normal Form- Maxmin Criterion- Mixed Strategies- Minmax Theorem- Domination- Equilibrium- Solving N X M Games. Two Person Nonzero Sum Games: Examples- Equilibrium- Proof Of Nash's Axiom- Maxmin Solution. N-Person Games: Non Cooperative Games- Characteristics Functions- Core- Stable Sets- Nucleus- Shapley Value. Market Games And Oligopoly: Edgeworth Market Games- [1,1],[M,N],[1,N] And [N,N] Market Games- Duopoloy- Oligopoly- Cournot Equilibrium. TEXT BOOK 1.Game, Theory And Application , by. C. Thomas.,Ellis Horwood Ltd Pub ,John Wiley & Sons ,1984. [ Chapters. 1 To 5. ]

REFERENCE

1. Game Theory , by N.N. Vorobev,Springer Verlag.Pub.,1977.

* * *

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4. MULTIPLE CRITERIA DECISION MAKING Binary Relation - Optimality Condition - Pareto Optimal Solution - Properties - Conditions - Goal Setting - Satisficing Solution - Compromising Solution - Value Functions - Condition - Properties. Construction Of Value Function - General And Additive - Approximation Methods - Domination And Non Dominated Solutions - Multi-Criteria And Multi-Criteria - Multi-Constraint Simplex Methods. TEXT BOOK 1.Multi - Criteria Decision Making: Concepts, Techniques & Extensions , by Po-Lung Yu, Press Publications,1985. [ Chapters 1 To 8 ]

REFERENCE 1.Multi Criteria Optimization ,by Steuer, Published By John Wiley & Sons ,1986..

* * *

5. INTRODUCTION TO SIMULATION Simulation General Principles-Random Number Generation-Random Variable Generation-Input Modelling-Verification And Validation Of Simulation Models-Basics Of Programming Languages Of Simulation-GPSS-Applications Of Simulations In Queuing And Inventory Models. TEXT BOOK 1.Discrete Event Simulation , by Jerry Banks,Carson And Nelson,Phi Pub.1996. [ Chapters.1 To 3, Sec.4.4,Chapters.8 To 11 ]. REFERENCES 1.Modeling And Analysis , by Kobayashi, Addison Wesley Pub,1981. 2.Modern Statisticalsystems And GPSS Simulation, by Karian and Dudewcz, Computer Science Press,1991. * * *