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Unit  I (18 Hours)Elementary Data Structures : Stacks and queues, tree, binary tree, dictionaries – Binary search tree, priority queues – Heaps, heapsort, sets and disjoint set union – Union and Find operations  Graphs – graph representations.Unit  II (18 Hours)Divide And Conquer: General method  Binary Search  Finding the maximum and minimum – Merge Sort – Quick Sort.Unit – III (18 Hours)The Greedy method: General method  Knapsack problem Tree vertex splitting Job Sequencing with deadlines Optimal storage on tapes optimal merge patterns.Unit – IV (18 Hours) Dynamic programming: The General method Multistage graphs All pairs shortest paths Optimal binary search trees. Unit  V (18 Hours) Basic traversals and Backtracking techniques for binary tree and graphs  Connected components and spanning trees  Biconnected components and DFS  Backtracking general method – The 8 Queen’s problem  Graph coloring Hamiltonian cycles. Text Book : Fundamentals of Computer Algorithms Ellis Horowitz, Sartaj Sahani and Sanguthevan Rajasekaran Galgotia Publications Pvt Ltd., 2008
Reference Book: Computer Algorithms  Sare Baase and Allan Van Gelder Pearson Education Asia, 2002 THIAGARAJAR COLLEGE (AUTONOMOUS), MADURAI – 625 009 (Re Accredited with 'A' Grade by NAAC) Department of Mathematics (From 2011 – 2013 batch onwards) Course : M.Sc. Code No. : Semester : III No. of hours allotted : 2 Paper : Non Major Elective No. of credits : 2 Title of the Paper : Fourier Transforms Course objective : To introduce the Fourier transforms and its different forms Unit – I Fourier integral formula – Weierstrss M test – Functions of exponential order – Riemann – Lebesgue Theorem – Fourier integral theorem – Different forms of Fourier integral formula – Dirichelets integral – Fejer’s integral – Applications of Fourier integral formula. Unit – II Fourier sine transform of F(x)  Fourier cosine transform of F(x) – Infinite Fourier transform of F(x) – The finite Fourier sine transform of F(x) – The finite Fourier cosine transform of F(x)
Text Book: Fourier series and Fourier transforms and their applications
Reference Book: Laplace and Fourier transforms Goyal and Gupta, Pragati Prakasham, Meerut, 12^{th} edition 1998 THIAGARAJAR COLLEGE (AUTONOMOUS), MADURAI – 625 009 (Re Accredited with 'A' Grade by NAAC) Department of Mathematics (From 2011 – 2014 batch onwards) Course : M.Sc. Code No. : Semester : III No. of hours allotted : Paper : Self study No. of credits : 5 Title of the Paper : Fuzzy Graph Theory Course objective: To introduce the fuzzy concepts in Graph Theory Unit  I Fuzzy Sets and Fuzzy Relations: Introduction – Fuzzy sets and fuzzy set operators – Fuzzy relations – Composition of fuzzy relations – Properties of fuzzy relation. Unit – II Fuzzy Graph: Introduction to fuzzy graph – Operations on fuzzy graph – Complement of a fuzzy graph  Cartesian product and composition – Union and Join. Unit  III Fuzzy Tree and Fuzzy Forest: Paths and Connectedness – Fuzzy Bridges and Fuzzy cut nodes – Fuzzy forests and fuzzy trees – Geodesics – Triangle and Parallelograms laws. Unit  IV Fuzzy Bipartite Graphs: Fuzzy Independent set and Fuzzy Bipartite graph – Fuzzy Bipartite part and Maximal Bipartite part – Maximal Fuzzy Bipartite part algorithm. Unit – V Domination in Fuzzy Graphs: Dominating Set – Fuzzy Independent Set – Bounds for (G)  More Adjacency in Fuzzy graph. Text Book: A First Look at Fuzzy Graph Theory – A. Nagoorgani and V. T. Chandrasekaran, Allied Publishers Pvt. Ltd., 2010
Reference Book: Fuzzy Graphs and Fuzzy Hypergraphs  John N. Mordeson, Premchand S. Nair, PhysicaVerlag Heidelberg, 2000. M.Phil. (Mathematics) (2011 – 2012 Batch onwards) THIAGARAJAR COLLEGE (AUTONOMOUS), MADURAI – 9 (Re – Accredited with ‘A’ Grade by NAAC) DEPARTMENT OF MATHEMATICS M.Phil. MATHEMATICS COURSE STRUCTURE (w.e.f. FROM 2011 – 2012 batch onwards) Semester – I
Semester – Ii
* ELECTIVE PAPER: (In  depth study papers): (One paper is to be chosen) 1. Advanced Graph Theory 2. ODE and Stability theory 3. Computational Complexity Question paper pattern: 5 Internal choice questions 5 x 20 = 100 Marks THIAGARAJAR COLLEGE (AUTONOMOUS), MADURAI – 625 009 (Re Accredited with 'A' Grade by NAAC) Department of Mathematics (From 2011 – 2012 batch onwards) Course : M.Phil. Code No. : Semester : I No. of hours allotted : 9 Paper : Core 1 No. of credits : Title of the Paper : Research Methodology Course objective: To pay due attention to designing and adhering to the appropriate methodologies throughout for improving the quality of research and to give a brief treatment of tensor products and simple properties of the formation of fractions. Unit  I Research Methodology : Meaning of Research – Objectives of Research – Motivation in Research – Types of Research – Research Approaches – Significance of Research – Research Methods versus Methodology – Research and Scientific Method – Importance of Knowing How Research is Done – Research Process – Criteria of Good Research – Problems Encountered by Researchers in India – What is a Research Problem? – Selecting the Problem – Necessity of Defining the problem – Techniques Involved in Defining a Problem – Meaning of Research Design – Need for Research Design – Features of a Good Design – Import Concepts Relating to Research Design – Different Research Designs – Basic Principles of Experimental Designs. Unit  II LATEX : The Basics – The Document – Bibliography – Bibliographic Databases – Table of contents , Index and Glossary – Displayed Text – Rows and Columns – Typesetting Mathematics. Unit III Modules : Modules and module homomorphisms  Submodules and quotient modules  Operations and submodules – Direct sum and product – Finitely generated modules – Exact sequences – Tensor product of modules –Restriction and extension of scalars – Exactness properties of the tensor product – Algebras – Tensor product of algebras . Unit IV Rings and Modules of fractions : Local properties – Extended and contracted ideals in rings of fractions  Primary Decomposition. Unit V Integral Dependence and Valuation : Integral dependence – The goingup theorem – Integrally closed integral domains – The going down theorem – Valuation rings – Chain conditions  Noetherian rings – Primary Decomposition in Noetherian rings. Text Books: 1. Research Methodology, Methods and Techniques (Second Revised Edition)  C.R. Kothari, New Age International Publishers, Reprint 2010 2. LATEX Tutorials, A Primer  Indian TEX Users Group, Trivandrum, India, 2003 3. Introduction to Commutative Algebra  M.F. Atiyah, I.G. GeMacdonald , Addison – Wesley Publishing Company 1969
Reference Books: 1. Research Methodology  R. Panneerselvam, Prentice Hall of India, New Delhi 110 001, 2007 2. Algebra  Thomas W. Hungerford, Springer International edition, 2008 THIAGARAJAR COLLEGE (AUTONOMOUS), MADURAI – 625 009 (Re Accredited with 'A' Grade by NAAC) Department of Mathematics (From 2011 – 2012 batch onwards) Course : M.Phil. Code No. : Semester : I No. of hours allotted : 9 Paper : Core 2 No. of credits : Title of the Paper : Functional Analysis Course objective : To study certain topologicalalgebraic structures and the methods by which knowledge of these structures can be applied to analytic problem. To study the basic properties of Fourier transform. Unit  I Banach spaces: Definition and Examples – Continuous linear transformations – The Hahn – Banach Theorem – The natural imbedding of N in N**  The open mapping theorem – The conjucate of an operator. Unit  II Finite – dimensional spectral theory : Matrices – Determinants and the spectrum of an operator – The spectral theorem Unit  III Banach Algebras: Definition and examples – Regular and singular elements – Topological divisors of zero – The spectrum – The formula for the spectral radius – The radial and semisimplicity – The Gelfand mapping – Application of the formula r(x) = lim x^{n}^{ 1/n } Unit  IV Integration on product spaces : Measurability on Cartesian products – Product measure – The Fubini theorem – Completion of product measures – Convolutions – Distribution functions Unit – V Fourier transforms: Formal properties – The inversion theorem – The Plancherel theorem – The Banach Algebra L^{1} Text Book: 1) Introduction to Topology and Modern Analysis G.F. Simmons, McGraw – Hill International editions, 1963 2) Real and Complex analysis  Walter Rudin, Tata McGraw – Hill International edition, Ninth Reprint, 2010.
Reference book: 1) Functional Analysis  Walter Rudin, Tata McGrawHill, 1989. 2) 2) Functional Analysis – George Bachman Lawrence Narici Academic Press International, New York, 1996 THIAGARAJAR COLLEGE (AUTONOMOUS), MADURAI – 625 009 (Re Accredited with 'A' Grade by NAAC) Department of Mathematics (From 2011 – 2012 batch onwards) Course : M.Phil. Code No. : Semester : I No. of hours allotted : 9 Paper : Indepth study No. of credits : Title of the Paper : Advanced Graph Theory Course objective:
Unit  I Domination in Graphs: Dominating sets in graphs – Bounds on the domination number in terms of order, size, degree, diameter and girth – Product graphs and Vizing’s conjecture. Unit – II Conditions on dominating sets: Introduction – Independent dominating sets – Total dominating sets – Connected dominating sets. Unit  III Preliminaries: Magic Squares – Latin Squares – Magic Rectangles – Labelings – Magic Labeling – Some Applications of Magic Labelings Unit  IV Edgemagic Total Labelings: Basic ideas – Cycles – Complete bipartite graphs – Trees Unit  V Vertexmagic Total Labelings: Basic ideas – Regular graphs – Cycles and Paths – Graphs with vertices of degree one Text Books: 1. Fundamentals of domination in graphs  T. W. Haynes, S. T. Hedetniemi and Peter J. Slater Marcel Dekker Inc, New York, 1998. 2. Magic Graphs  W. D. Wallis, Birkhauser Boston, 2001
Reference Books : 1) Theory of domination in graphs  V. R. Kulli, Vishwa International Publications, Gulbarga, 2010 2) Super EdgeAntimagic GraphsA Wealth of Problems and Some Solutions  Martin Baca and Mirka Miller, Brown Walker Press, USA, 2008 THIAGARAJAR COLLEGE (AUTONOMOUS), MADURAI – 625 009 (Re Accredited with 'A' Grade by NAAC) Department of Mathematics (From 2011 – 2012 batch onwards) Course : M.Phil. Code No. : Semester : I No. of hours allotted : 9 Paper : Elective (Indepth study) No. of credits : Title of the Paper : Ordinary Differential Equations and Stability Theory Course objective: To bring together the qualitative theory of differential equations systematically at an introductory level. Unit  I Systems of Linear Differential Equations : Introduction – Systems of first order equations – Existence and Uniqueness theorem – Fundamental matrix – Nonhomogeneous linear systems – Linear systems with constant coefficients – Linear systems with Periodic coefficients. Unit  II Existence and Uniqueness of Solutions : Introduction – Preliminaries – Successive approximations – Picard’s theorem – Nonuniqueness of solutions – Continuation and dependence on initial conditions – Existence of Solutions in the large – Existence and uniqueness of solutions of systems. Unit  III Oscillations of Second Order Equations: Fundamental results – Sturm’s comparison theorem – Elementary linear oscillations – Comparison theorem HilleWintner – Oscillations of x’’ + a(t) x = 0  Elementary nonlinear oscillations. Unit  IV Boundary Value Problems: Introduction – SturmLiouville problem – Green’s functions – Nonexistence of solutions – Picard’s theorem. Unit  V Behaviour of Solutions of Linear Differential Equations : Introduction – nth order equations – Elementary critical points – Critical points of nonlinear systems – Linear systems with constant coefficients – Linear systems with variable coefficients – Second order linear differential equations. Text Book: Ordinary Differential Equations and Stability theory  S. G. Deo and V. Raghavendra, Tata McGrawHill Publishing Company Limited, New Delhi, First Reprint, 1987
Reference Books : 1. Stability theory of Differential Equations – Richard Bellman, McGrawHill Book Company, Inc, 1953. 2. Ordinary Differential Equations and Stability Theory: An Introduction  David A. Sanchez, Dover Publications, Inc., New York, 1968. THIAGARAJAR COLLEGE (AUTONOMOUS), MADURAI – 625 009 (Re Accredited with 'A' Grade by NAAC) Department of Mathematics (From 2011 – 2012 batch onwards) Course : M.Phil. Code No. : Semester : I No. of hours allotted : 9 Paper : Elective (Indepth study) No. of credits : Title of the Paper : Computational Complexity Course objective: To understand complexity in computation Unit  I Turing Machines: The standard Turing Machine – Combining Turing Machines for Complicated Tasks  Turing’s Thesis. Unit  II Other Models of Turing Machines: Monor Variations on the Turing Machine Theme – Turing Machines with More Complex Storage. Unit  III Hierarchy of Formal Languages and Automata: Recursive and Recursively Enumerable Languages – Unrestricted Grammars – ContextSensitive Grammars and Languages – The Chomsky Hierarchy. Unit  IV Limits of Algorithmic Computation: Some Problems that cannot be solved by Turing Machines – Undecidable Problems for Recursively Enumerable Laguages – The Post Correspondance Problem – Undecidable Problems for ContextFree Languages – A question of efficiency. Unit – V An Overview of Computational Complexity: Efficiency of computation – Turing Machine Models and Complexity – Language Families and Complexity Classes – The Complexity Classes P and NP – Some NP Problems – PolynomialTime Reduction – NPCompleteness and an Open Question. Text Book: An Introduction to Formal Languages and Automata – Peter Linz, Jones and Bartlett Publishers, Fourth Edition, 2006
Reference book: Fundamentals of Computer Algorithms  Ellis Horowitz, Sartaj Sahani,Sanguthevar Rajasekaran, Galgotia, 2007 Certificate and Diploma Courses THIAGARAJAR COLLEGE (AUTONOMOUS), MADURAI – 625 009 (Re Accredited with 'A' Grade by NAAC) Department of Mathematics (From 2011 – 2014 batch onwards) Course : Certificate course Code No. : Semester : III No. of hours allotted :40 Paper : No. of credits : 02 Title of the Paper : Astronomy Course objective: To study the solar systems and Astronomical observations Unit  I (8 Hours) Celestial sphere – Diuranal motion, celestial axis and equator – Celestial Horizon – Zenith and Nadir – Transit – Due east and due west Unit – II (8 Hours) Zones of Earth – Trace the variation in the duration of day and night – duration of perpectual day in the place of latitude  Analytical condition for perpetual day Unit  III (8 Hours) Terrestrial latitudes and longitudes – Change of latitude and longitude –The reduction of latitude – The radius of curvature of earth at a station of Geographical latitude Unit  IV (8 Hours) Kepler’s Law of planetary motion  Longitude of Perigee  Eccentricity of the earth orbit around the Sun – Newton’s deductions from Kepler’s law – Fix the position of a planet in its elliptic orbit Unit  V (8 Hours) Equation of time – seasons – calendar – conversion of time Text Book: Astronomy  S. Kumaravelu and Susila Kumaravelu, Jeeva computers, 2009
Reference Book: Dynamic Astronomy – Robert T Dixon, Prentice Hall  Gale, 6^{th} Edition, 1992 THIAGARAJAR COLLEGE (AUTONOMOUS), MADURAI – 625 009 (Re Accredited with 'A' Grade by NAAC) Department of Mathematics (From 2011 – 2012 batch onwards) Course : Diploma Course Code No. : Semester :III(M.Sc.,) No. of hours allotted : Paper : No. of credits : Title of the Paper : Statistics Course Objective: To introduce the statistical techniques Correlation, Regression and some of the statistical tools t – test, F test and Chisquare test Note: Derivations and Proofs of the theorems are not included. Only problems are to be dealt with. Unit – I (10 Hours) Correlation: Introduction – Meaning of Correlation – Scatter Diagram – Karl Pearson’s Coefficient of correlation – Calculation of the correlation coefficient for a bivariate frequency distribution – Probable Error of correlation coefficient – Rank correlation. Unit – II (10 Hours) Linear and Curvilinear Regression: Introduction – Linear Regression – Curvilinear regression – Regression curves. Unit – III (14 Hours) Large Sample Theory: Introduction – Types of Sampling – Parameter and Statistic – Tests of Significance – Procedure for testing of hypothesis – Test of significance for large samples – Sampling attributes – Sampling variables. Unit – IV (14 Hours) Exact Sampling Distributions  I (Chi – square Distribution): Introduction – M.G.F. of Chi –square distribution – Some theorems on Chi – square distribution (Statements only) – Linear transformation – Applications of Chi – square distribution. Unit – V (12 Hours) Exact Sampling Distributions – II ( t , F and z distributions ) : Introduction Student’s ‘t’ Distribution  Applications of t – Distribution  Distribution of sample correlation coefficient when population correlation coefficient (Sawkin’s Method) Text Book: Fundamentals of Mathematical Statistics  S.C. Gupta, V.K. Kapoor, Sultan Chand & Sons, New Delhi, Eleventh Thoroughly Revised Edition, Reprint 2009
Reference Book: Research Methodology, Methods and Techniques  C.R. Kothari New Age International Publishers, Second Revised Edition, Reprint 2010 B.C.A. THIAGARAJAR COLLEGE (AUTONOMOUS) MADURAI – 9. ReAccredited with ‘A’ Grade by NAAC DEPARTMENT OF MATHEMATICS BACHELOR OF COMPUTER APPLICATION COURSE STRUCTURE (w.e.f 20112014 batch onwards) Semester – I
Semester – II
Semester – III
Semester – IV
Semester – V
