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1 SATHYABAMA INSTITUTE OF SCIENCE AND TECHNOLOGY FACULTY OF SCIENCE AND HUMANITIES Sl. No. COURSE CODE 1 SMT SMT SMT PROGRAMME : M.PHIL Methodology Algebra and Analysis M A T H E M A T I C S C U R R I C U L U M Advanced Numerical Analysis Elective Project Work (Phase 1) SEMESTER 1 COURSE TITLE Research L T P C PAGE No TOTAL CREDITS : 16 SEMESTER 2 Sl. No. COURSE CODE COURSE TITLE L T P C PAGE No. 1 S96PROJ Project Work (Phase 1 & 2) TOTAL CREDITS : 20 TOTAL CREDITS FOR THE PROGRAMME : 36 ELECTIVES 1 SMT5118 Fuzzy sets and Systems 4 2 SMT5119 Advanced Topics in Graph Theory 5 3 SMT5120 Algebraic Number Theory 6 4 SMT5121 Fixed Point Theory 7 5 SMT5122 Algebraic Coding Theory 8 6 SMT5123 Computational fluid Dynamics 9 L - LECTURE HOURS, T TUTORIAL HOURS, P PRACTICAL HOURS, C CREDITS M.Phil. (MATHEMATICS) REGULAR x REGULATIONS 2015

2 SMT5115 RESEARCH METHODOLOGY L T P Credits Total Marks _ The ability to identify, reflect upon, evaluate and apply different types of information and knowledge to form independent judgements. Analytical, logical thinking and conclusions based on quantitative information will be the main objective of learning this subject. LOGIC: Propositions and Logical Operations Conditional Statements Methods of Proof Mathematical Induction. MATLAB : Programming in Matlab Polynomials, Curve Fitting and Interpolation - Applications in Numerical Analysis. TOPOLOGY : Homotopy The Fundamental Group Covering Spaces. TOPOLOGY : Geometry of Simplicial complexes Barycentric subdivisions Simplicial approximation theorem Fundamental group of a simplicial complex. DIFFERENTIAL EQUATIONS: Uncoupled Linear systems Diagonalization Exponentials of operators Fundamental theorem for Linear systems Linear Systems in R2 Complex eigen values Multiple eigen values Jordan forms Stability theory Non-homogeneous linear systems. 1. B. Kolman, R.C. Busby and S.C. Ross, Discrete Mathematical Structures, Fourth Indian reprint, Pearson Education Pvt Ltd, New Delhi, Unit I Chapter 2 2. Amos Gilat, MATLAB An Introduction with Applications, John wiley & sons, Unit II Chapters 7, 8 and I.M. Singer and J.A. Thorpe, Lecture Notes on Elementary Topology and Geometry, Springer Verlag, Unit III - Chapter 3, Unit IV - Chapter 4 4. L. Perko, Differential Equations and Dynamical systems, Springer-Verlag, First Indian Reprint, Unit V Chapter to J.P.Tremblay and R. Manohar, Discrete Mathematical Structures with Applications to Computer Science, Tata McGraw-Hill, New Delhi, James R. Munkres, Topology (2nd Edition), Prentice Hall of India, Pvt. Ltd., New Delhi, E.A Coddington and N. Levinson, Theory of Ordinary differential equations, Tata McGraw Hill, New Delhi, M.Phil. (MATHEMATICS) REGULAR x REGULATIONS 2015

3 SMT5116 ALGEBRA AND ANALYSIS L T P Credits Total Marks _ The ability to identify, reflect upon, evaluate and apply different types of information and knowledge to form independent judgements. Analytical, logical thinking and conclusions based on quantitative information will be the main objective of learning this subject. MODULES: Basic definitions Group of homomorphisms Direct products and sums of modules Free modules Vector spaces The dual space and dual module. NOETHERIAN RINGS: Basic criteria Associated primes Primary decomposition - Nakayama s lemma. RIESZ REPRESENTATION THEOREM: Topological preliminaries - Riesz representation theorem Regularity properties of Borel measures Lebegue measure continuity properties of measurable functions. FOURIER TRANSFORSMS: Formal properties Inversion theorem The Plancherel theorem Banach Algebra L1. RIEMANN MAPPING THEOREM: Preservation of angles Linear fractional transformations Normal families - Riemann Mapping Theorem. 1. Serge Lang, Algebra, Springer - Verlag, Revised Third Edition, Unit I - Chapter III: Sections 1 to 6 Unit II - Chapter X: Sections 1 to W. Rudin, Real and Complex Analysis, 3rd edition, McGraw Hill International, Unit III Chapter 2, Unit IV Chapter 9 Unit V - Chapter 14 Pages C. Musili, Rings and Modules, 2nd edition, Narosa, P.B. Bhattacharya et al., Basic Abstract Algebra, 2nd edition, Cambridge University Press, Serge Lang, Complex Analysis, Addison Wesley, V. Karunakaran, Complex Analysis 2nd edition, Narosa, New Delhi, C.D. Aliprantis and O.Burkinshaw, Priniciples of Real Analysis 2nd edition, Acade M.Phil.,(MATHEMATICS) REGULAR syllabus

4 SMT5117 ADVANCED NUMERICAL ANALYSIS L T P Credits Total Marks _ The ability to identify, reflect upon, evaluate and apply different types of information and knowledge to form independent judgements. Analytical, logical thinking and conclusions based on quantitative information will be the main objective of learning this subject. Transcendental and polynomial equations: Iteration methods based on second degree equation - Rate of convergence - iterative methods Methods for finding complex roots iterative methods : Birge-Vieta method, Bairstow s method, Graeffe s root squaring method numerical solution of simultaneous linear algebraic equation - Direct methods - Gauss Jordan Method Triangularization method Cholesky method partition method. Error Analysis Iteration methods : Jacobi iteration method Gauss - Seidal iteration method SOR method. Jacobi method for symmetric matrices Power method Inverse power method. Interpolation and Approximation : Hermite Interpolations Piecewise and Spline Interpolation Appproximation Least Square Approximation. Differentiation and Integration : Numerical Differentiation Numerical Integration Methods based on Interpolation. Ordinary Differential Equations :- Multi step method Predictor Corrector method Boundary value problem Initial value methods Shooting method Finite Difference method. MATLAB : Introduction to MATLAB. 1. M.K. Jain, S.R.K. Iyengar and R.K. Jain, Numerical Methods for Scientific and Engineering Computation, III Edn. Wiley Eastern Ltd., Amos Gilat, MATLAB An Introduction with Applications, John wiley & sons, Unit I Chapter to 2.8 and Chapter of [1] Unit II Chapter , 4.8 to 4.9 of [1] Unit III Chapter 5 5.2, 5.6, 5.7 of [1] Unit IV Chapter 6 6.4, 6.5, 6.8, 6.9, 6.10 of [1] Unit V Chapters 1 4, 6 and 7 of [2] 3. Kendall E. Atkinson, An Introduction to Numerical Analysis, II Edn., John Wiley & Sons, M.K. Jain, Numerical Solution of Differential Equations, II Edn., New Age International Pvt Ltd., Samuel. D. Conte, Carl. De Boor, Elementary Numerical Analysis, McGraw-Hill International Edn., 1983 M.Phil. (MATHEMATICS) REGULAR x REGULATIONS 2015

5 RECENT SELECTED TRENDS IN MATHEMATICS [GUIDE PAPER] SMT5118 FUZZY SETS AND SYSTEMS L T P Credits Total Marks 100 COURSE IV Analytical, logical thinking and conclusions based on quantitative information will be the main objective of learning this subject. Fuzzy sets Basic types Basic concepts á-cuts Additional properties of á-cuts Extension principle for Fuzzy sets. Operations on Fuzzy sets Types of operations Fuzzy complements t-norms Fuzzy Unions Combinations of operations. Fuzzy Arithmetic Fuzzy numbers Arithmetic operations on intervals Arithmetic operations on Fuzzy numbers. Fuzzy relations Binary fuzzy relations Fuzzy equivalence relations Fuzzy compatibility relations Fuzzy ordering relations Fuzzy morphisms. Fuzzy Relation Equations General discussion Problem partitioning Solution method Fuzzy Relation Equations based on Sup-i Compositions - Fuzzy Relation Equations based on inf- ùi Compositions. 1. George J.Klir and B. Yuan, Fuzzy Sets and Fuzzy Logic, Prentice Hall of India, New Delhi, Unit I Chapter 1 and 2 Unit II Chapter 3 Unit III Chapter 4 Unit IV Chapter 5 Unit V Chapter H.J. Zimmermann, Fuzzy Set Theory M.Phil.,(MATHEMATICS) REGULAR syllabus

6 SATHYABAMA INSTITUTE OF SCIENCE AND TECHNOLOGY FACULTY OF SCIENCE AND HUMANITIES SMT5119 ADVANCED TOPICS IN GRAPH THEORY T P Credits Total Marks _ Analytical, logical thinking and conclusions based on quantitative information this will be the main objective of learning subject. Perfect graphs. Other classes of Perfect graphs. Labelings of graphs. Factorizations and decompositions. Domination in graphs. 1. D.B. West, Introduction to graph theory, PHI (2002). Unit I & II Chapter H G.Chartrand and L.Lesniak, Graphs and Digraphs, Chapen & Hall/ CRC Press, 1996 Unit III Chapter 9 Section 3 Unit IV - Chapter 9 Section 2 Unit V - Chapter 10 Sections 1 and 2 3. J.Clark and D.A.Holton, A First look at Graph Theory, Allied Publishers, New Delhi, R. Gould. Graph Theory, Benjamin/Cummings, Menlo Park, A.Gibbons, Algorithmic Graph Theory, Cambridge University Press, Cambridge, R.J..Wilson. and J.J.Watkins, Graphs: An Introductory Approach, John Wiley and Sons, New York, K.R. Parthasarathy, Basic Graph Theory, Tata Mc-Graw Hill Publishing Company, New Delhi, M.Phil.,(MATHEMATICS) REGULAR syllabus

7 SMT5120 ALGEBRAIC NUMBER THEORY L T P Credits Total Marks _ Analytical, logical thinking and conclusions based on quantitative information will be the main objective of learning this subject. Algebraic background: Rings and fields - Factorization of polynomials - Field extensions - Symmetric polynomials - Modules - Free abelian groups. Algebraic numbers: conjugates and discriminants - Algebraic integers - Integral bases - Norms and traces. Quadratic fields - Cyclotomic fields - Factorization into irreducibles - examples of non-unique factorization into irreducibles. Prime factorization - Euclidean domains - Euclidean quadratic fields - consequences of unique factorization - the Ramanjuan- Nagell theorem. Fractional Ideals - Prime factorization of ideals - The norm of an ideal. 1. Ian Stewart and David Tall - Algebraic Number Theory, Chapman and Hall Mathematics Series (1979). Unit III: Chapter 3, and Sections of Chapter 4 Unit IV: Sections of Chapter 4. Unit V: Chapter 5 Unit I: Chapter 1 Unit II: Chapter 2 M.Phil. (MATHEMATICS) REGULAR x REGULATIONS 2015

8 SMT5121 FIXED POINT THEORY L T P Credits Total Marks _ Analytical, logical thinking and conclusions based on quantitative information will be the main objective of learning this subject. Banach s contraction principle Further extensions- Caristi Ekeland principle -Equivalence of Caristiprinciples. Tarsiki s Fixed point theorem - Hyperconvex spaces Properties fixed point theorems intersection of hyper convex spaces Isbell s convex hull. Uniformly convex Banach spaces Fixed point theorem of Browder, Gohde and Kirk. Reflexive Banach spaces Normal structure- Fixed point theorems. Generalized Banach Fixed-point theorem- Upper and lower semi continuity of multivalued maps Generalized Schauder Fixed point theorem Variational Inequalities and the Browder Fixed-Point theorem Extremal Principle Applications to Game Theory Michael s selection theorem. Fixed point theorem for continuous functions- Brouwer's theorem -Schauder's theorem - applications - Hairy ball theorem - pancake problems- Kyfan's best approximation theorem. 1. E. Zeidler, Nonlinear Functional Analysis and its applications, Vol. I Springer Verlag New york (1986) 2. M. A. Khamsi & W. A. Kirk, An introduction of Metric spaces and Fixed point theory, John Wiley & sons (2001) UNIT I - Chapter 3 ( ) from [2] UNIT II - Chapter 4 from [2] UNIT III - Chapter 10 ( ) from[1] and chapter 5 ( ) from [2] UNIT IV - Chapter 9 from [1] UNIT V - Chapter 2 from [1] 3. D.R. Smart, Fixed point theory, Cambridge University Press, (1974). 4. V.I. Istratescu, Fixed point theory, D. Reidel Publsihing Company, Boston (1979). M.Phil.,(MATHEMATICS) REGULAR syllabus

9 SMT5122 ALGEBRAIC CODING THEORY L T P Credits Total Marks _ Analytical, logical thinking and conclusions based on quantitative information will be the main objective of learning this subject. Linear Code: Introduction - Linear codes, Encoding, Decoding - Check Matrices and Dual Code. Linear Code continued: Classification by Isometry Semilinear Isometry Classes of Linear codes The Weight Enumerator. Bounds and Modifications: Combinatorial Bounds for the Parameters - New Codes from Old and Minimum distance Further Modifications and Constructions. Reed Muller Codes MDS Codes. Cyclic Codes: Cyclic Codes as Group Algebra Codes Polynomial Representaion of Cyclic Codes BCH Codes and Reed-Solomon Codes. 1. Anton Betten, Michael Braun, Harald Fripertinger, Adalbert Kerber, Axel Kohnert and Alfred Wassermann, Error-Correcting Linear Codes, Classification by Isometry and Applications, Springer-Verlag berlin Heidelberg 2006 UNIT I Chapter 1 (Section ) UNIT II Chapter 1 (Section ) UNIT III Chapter 2 (Section ) UNIT IV Chapter 2 (Section 2.4 & 2.5) UNIT V Chapter 4 (Section ) 2. F. J Mac Williams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North Holland Publishing Company D.G. Hoffman et al, Coding Theory: The Essentials, Marcel Dekker, Inc, 1991 M.Phil. (MATHEMATICS) REGULAR x REGULATIONS 2015

10 SATHYABAMA INSTITUTE OF SCIENCE AND TECHNOLOGY FACULTY OF SCIENCE AND HUMANITIES SMT5123 Computational Fluid Dynamics L T P Credits Total Marks 100 COURSE IX The ability to identify, reflect upon, evaluate and apply different types of information and knowledge to form independent judgements. Analytical, logical thinking and conclusions based on quantitative information will be the main objective of learning this subject. Introduction Conduction heat transfer Thermal conductivity Convection heat transfer Radiation heat transfer Dimensions and Units Governing Equations and Boundary Conditions: Basics of computational fluid dynamics Governing equations of fluid dynamics Continuity, Momentum and Energy equations Chemical species transport Physical boundary conditions Time-averaged equations for Turbulent Flow Turbulent Kinetic Energy Equations Mathematical behaviour of PDEs on CFD - Elliptic, Parabolic and Hyperbolic equations Principles of Convection Viscous flow- Inviscid flow Laminar boundary layer on a flat plate Energy equation of the boundary layer The thermal boundary layer Relation between fluid friction and heat transfer Turbulent Boundary layer Heat transfer in Laminar tube flow Turbulent flow in a tube Heat transfer in High speed flow. Natural Convection Systems Free convection heat transfer on a vertical flat plate Empirical relations for free convection Free convection from vertical planes, Cylinders, Horizontal Cylinders, Horizontal plates, Inclined surfaces Non newtonian fluids Simplified equations for Air, Free convection from spheres Free convection in enclosed spaces Combined free and forced convection. Finite Volume Method for Diffusion Finite volume formulation for steady state One, Two and Three - dimensional diffusion problems. One dimensional unsteady heat conduction through Explicit, Crank Nicolson and fully implicit schemes Finite Volume Method for Convection Diffusion Steady one-dimensional convection and diffusion Central, upwind differencing schemes- Properties of discretization schemes Conservativeness, Boundedness, Transportiveness, Hybrid, Power-law, QUICK Schemes. 1. J. P. Holman, Heat Transfer, McGraw-Hill, Singapore, Unit I-III 2. H. K. Versteeg and W. Malalasekera, An Introduction to Computational Fluid Dynamics: The finite volume Method, Longman, England, Unit I & Unit IV-V 3. F.P. Incropera, D. P. Dewitt, T. L. Bergman and A. S. Lavine, Fundamentals of Heat and Mass Transfer, John Wiley & Sons, USA, S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing Corporation, USA, 2004 M.Phil.,(MATHEMATICS) REGULAR syllabus

VEER NARMAD SOUTH GUJARAT UNIVERSITY, SURAT

VEER NARMAD SOUTH GUJARAT UNIVERSITY, SURAT Syllabus for M.Sc. (Mathematics) Scheme of Teaching and Examination Semester II Subject Code Subject Scheme Of Teaching Scheme Of Examination PGMTH L P Total