1

Bos 26 Jun 2016

DDEEPPAARRTTMMEENNTT OOFF MMAATTHHEEMMAATTIICCSS

SCHOOL OF MATHEMATICS AND COMPUTER SCIENCE BBHHAARRAATTHHIIDDAASSAANN UUNNIIVVEERRSSIITTYY

TIRUCHIRAPPALLI – 620 024

M.Sc., PROGRAMME IN MATHEMATICS ( CBCS )

(For the candidates to be admitted from the year 2016-2017 onwards)

Semester

Courses

I

5 Core Courses

II

4 Core Courses 1 Department Elective Course

III

3 Core Courses 1 Department Elective Course 1 University Elective Course

IV

2 Core Courses 2 Department Elective Courses 1 Project

2

* CORE COURSES (CC)

Code Title of the Course LectureHours

TutorialHours

Practical Hours Credits Prerequisite

(Exposure)

16M01CC Linear Algebra 4 2 0 5 Nil

16M02CC Real Analysis I 4 2 0 5 Nil

16M03CC Ordinary Differential Equations 4 2 0 5 Nil

16M04CC Theory of Numbers 4 2 0 5 Nil

16M05CC Discrete Mathematics 4 2 0 5 Nil

16M06CC Algebra I 4 2 0 5 Nil

16M07CC Real Analysis II 4 2 0 5 16M02CC

16M08CC Partial Differential Equations 4 2 0 5 16M03CC

16M09CC Topology 4 2 0 5 16M02CC

16M10CC Algebra II 4 2 0 5 16M01CC 16M06CC

16M11CC Complex Analysis 4 2 0 5 16M02CC 16M07CC

16M12CC Measure Theory and Integration 4 2 0 5 16M02CC 16M07CC

16M13CC Functional Analysis 4 2 0 5 16M02CC 16M09CC 16M12CC

16M14CC Differential Geometry 4 2 0 5 16M12CC

16M15CC Probability Theory 4 2 0 5 Nil

16M16CC Numerical Analysis 4 2 0 5 16M01CC 16M03CC

* DEPARTMENT ELECTIVE COURSES (DEC)

Code

Title of the Course

LectureHours

TutorialHours

Practical Hours

Credits Prerequisite(Exposure)

16M01DEC Stochastic Processes 3 2 1 4 16M14CC

16M02DEC Integral Equations and Calculus of variations

3 2 0 4 16M03CC 16M08CC

16M03DEC Fluid Dynamics 3 2 0 4 16M03CC 16M08CC

16M04DEC Mathematical Statistics 3 2 0 4 16M07CC

16M05DEC Classical Dynamics 3 2 0 4 16M03CC 16M08CC

16M06DEC Optimization Techniques 3 2 0 4 Nil

16M07DEC Operator Theory 3 2 0 4 16M13CC

3

* UNIVERSITY ELECTIVE COURSES (UEC)

Code

Title of the Course

LectureHours

TutorialHours

Practical Hours

Credits Prerequisite

16M01UEC Object Oriented Programming using C++

2 0 1 2 +2 level

Mathematics

16M02UEC Resource Management Techniques

2 1 0 2

16M03UEC Mathematical Modeling

2 1 0 2

16M04UEC

Statistics

2 1 0 2

16M05UEC General Intelligence 2 1 0 2

For each Course other than the Project

Continuous Internal Assessment (CIA) – 40 Marks

End Semester Examination (ESE) – 60 Marks

Total – 100 Marks

ESE Duration – 3 Hours.

For Project

2 Reviews – 40 Marks (2 x 20)

Evaluation (Average of Internal and

external examiners marks) – 40 Marks

Viva Voce – 20 Marks

Total – 100 Marks

Total Credits should not be less than 90

4

CORE COURSES

LINEAR ALGEBRA

Prerequisite: Nil

UNIT I Vector spaces - Subspaces - Linear Combinations and Systems of Linear Equations - Linear Dependence and Linear Independence - Bases and Dimension - Maximal Linearly Independent Subsets. UNIT II

Linear Transformations, Null Spaces, and Ranges - The Matrix Representation of a Linear Transformation - Combination of Linear Transformations and Matrix Multiplication - Invertibility and Isomorphisms - The Change of Coordinate Matrix. UNIT III

Elementary Matrix Operations and Elementary Matrices - The Rank of a Matrix and Matrix Inverses - System of Linear Equations - Theoretical Aspects and Computational Aspects - Determinants of Order 2 - Determinants of Order n - Properties of Determinants - Summary - Important Facts about Determinants. UNIT IV Eigenvalues and Eigenvectors - Diagonalizability - Cayley Hamilton Theorem. UNIT V The Jordan Canonical Form 1 - The Jordan Canonical Form 2 - The Minimal polynomial. TEXT BOOK

Stephen H. Friedberg, Arnold J. Insel and Lawrence E. Spence, Linear Algebra, Fourth Edition, PHI Learning Private Limited, New Delhi, 2014. UNIT – I - Chapter 1 UNIT – II - Chapter 2: Sections 2.1 to 2.5 UNIT – III - Chapter 3 and Chapter 4: Sections 4.1 to 4.4 UNIT – IV - Chapter 5: Sections 5.1 to 5.4 UNIT – V - Chapter 7 Sections 7.1 to 7.3 REFERENCE(S) [1] S. Kumaresan, Linear Algebra, Prentice-Hall of India Ltd, 2000. [2] K. Hoffman and R. Kunze, Linear Algebra, Second Edition, PHI, New Delhi, 1975. [3] M.Artin, Algebra, Prentice Hall of India, New Delhi, 1994. [4] Jin Ho Kwak, Linear Algebra, Second Edition, Birkhäuser, 2004. [5] I.N. Herstein, Topics in Algebra, Wiley Eastern Limited, New Delhi, 1975. [6] Gilbert Strang, Linear Algebra and its applications, Cengage Learning 8th Indian edt, 2011 [7] A.R. Rao, P. Bhimashankaram, Linear Algebra, Tata McGraw Hill, 1996. [8] V. Krishnamurthy et al, Introduction to Linear Algebra, East West Press Ltd, 1985.

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5

REAL ANALYSIS I

Prerequisite: Nil

UNIT I

Sets and Functions, Mathematical Induction, Finite and Infinite sets. Real Number system: Algebraic and Order properties: Infimum, Supremum, LUB Axiom. Countable and uncountable sets. UNIT II

Metric spaces – Definition and examples - open balls and open sets UNIT III Sequences and Series of real numbers – limit theorems – monotone sequences – Cauchy criterion – limsup, liminf - Convergent sequences in metric spaces – limit and cluster points – Cauchy sequences – Bounded sets – Dense sets.

UNIT IV Continuous functions – Equivalent Definitions of Continuity – Uniform Continuity -Limit of a function – Discontinuities of a Real Valued function - Compact spaces and their properties – Continuous functions on Compact spaces- Characterization of Compact Metric spaces. UNIT V

Connectedness : Connected spaces – Complete metric spaces – Examples- Baire Category Theorem – Banach Contraction Principle. TEXT BOOK(S)

[1] R.G. Bartle and D.R. Sherbert, Introduction to Real Analysis 3rd Edn, John Wiley & Sons, 2000. [2] S. Kumaresan, Topology of Metric Spaces, Narosa Publishing House, New Delhi, 2005. UNIT – I - Chapters 1 and 2 from [1] UNIT – II - Chapter 1 from [2] UNIT – III - Chapter 3 from [1] and Chapter 2 sections 2.1 to 2.5 from [2] UNIT – IV - Chapter 3, Chapter 4 from [2] (sections 3.3 and 3.6 omitted) and Chapter 5 from [1] UNIT V - Chapter 5 section 5.1 and Chapter 6 sections 6.1, 6.3 and 6.4 (section 6.2, 6.3.16 and 6.3.17 omitted) from [2] REFERENCE(S) [1] Edward D. Gaughan, Introduction to Analysis, AMS, Indian edition, 2010. [2] Kenneth A. Ross, Elementary Analysis: The Theory of Calculus, Springer Verlag, 2004. [3] M.H. Protter, C.B. Morrey, A First Course in Real Analysis, 2nd Edition, Springer Verlag, 1991 [4] S.K. Berberian, A First course in Real Analysis, Springer Verlag, 1994. [5] Charles Chapman Pugh, Real Mathematical Analysis, Springer Verlag, 2002. [6] R.P. Boas, A primer of real functions, Mathematical Association of America, 1966. [7] Tom M. Apostol, Mathematical Analysis 2 edn, Narosa, New Delhi, 1985. [8] Walter Rudin, Principles of Mathematical Analysis,Third Edition, Mcgraw Hill, 1976 [9] N.L. Carothers, Real Analysis, Cambridge University Press, South Asian Edition, 2000.

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6

ORDINARY DIFFERENTIAL EQUATIONS

Prerequisite: Nil

UNIT I The general solution of the homogeneous equation – The use of one known solution to find another – The method of variation of parameters – Power Series solutions. A review of power series – Series solutions of first order equations – Second order linear equations; Ordinary points. UNIT II Regular Singular Points – Gauss’s hypergeometric equation – The Point at infinity - Legendre Polynomials – Bessel functions – Properties of Legendre Polynomials and Bessel functions. UNIT III Linear Systems of First Order Equations – Homogeneous Equations with Constant Coefficients – The Existence and Uniqueness of Solutions of Initial Value Problem for First Order Ordinary Differential Equations – The Method of Solutions of Successive Approximations and Picard’s Theorem. UNIT IV Oscillation Theory and Boundary value problems – Qualitative Properties of Solutions – Sturm Comparison Theorems – Eigenvalues, Eigenfunctions and the Vibrating String. UNIT V Nonlinear equations: Autonomous Systems; the phase plane and its phenomena – Types of critical points; Stability – critical points and stability for linear systems – Stability by Liapunov’s direct method – Simple critical points of nonlinear systems. TEXT BOOK(S)

G.F. Simmons, Differential Equations with Applications and Historical Notes, TMH, New Delhi, 1974. UNIT – I -Chapter 3: Sections 15, 16, 19 and Chapter 5: Sections 25 to 27 UNIT – II -Chapter 5 : Sections 28 to 31 and Chapter 6: Sections 32 to 35 UNIT – III -Chapter 7: Sections 37, 38 and Chapter 11: Sections 55, 56 UNIT – IV -Chapter 4: Sections 22 to 24 UNIT – V -Chapter 8: Sections 40 to 44

REFERENCE(S) [1] M.E. Taylor, Introduction to Differential Equations, AMS Indian Edition, 2011. [2] M. Braun, Differential Equations and Their Applications, Springer, 1992. [3] Boyce and DiPrima, Elementary Differential Equations and Boundary Value Problems, 7th Edn, John

Wiley, 2001. [4] S. Deo et al, A textbook of Differential Equations, McGraw Hill, 2002. [5] Lawrence Perko, Differential Equations and Dynamical Systems, Springer, 2006. [6] E.A. Coddington and N. Levinson, Theory of Ordinary Differential Equaitons, McGraw Hill, 1955. [7] Tyn Myint-U, Ordinary Differential Equations, North-Holland, New York, 1978.

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7

THEORY OF NUMBERS

Prerequisite: Nil

UNIT I

Introduction – Divisibility – Primes – The Binomial Theorem – Congruences Euler’s totient - Fermat’s, Euler’s and Wilson’s Theorems – Solutions of congruences – The Chinese Remainder theorem. UNIT II

Prime power Moduli – Primitive roots and Power Residues – Number theory from an Algebraic Viewpoint – Groups, rings and fields. UNIT III

Quadratic Residues – Quadratic Reciprocity – The Jacobi Symbol – Binary Quadratic Forms – Equivalence and Reduction of Binary Quadratic Forms – sum of two squares. UNIT IV

Greatest integer Function – Arithmetic Functions – The Mobius Inversion Formula Recurrence Functions – Combinatorial number theory

UNIT V Diophantine Equations – The equation ax+by = c – Simultaneous Linear

Diophantine Equations – Pythagorean Triangles – Assorted examples TEXT BOOK(S) [1] Ivan Niven, Herbert S, Zuckerman and Hugh L, Montgomery, An Introduction to the

Theory of Numbers, Fifth edn., John Wiley & Sons Inc, 2004 UNIT I Chapter 1 and Chapter 2 : Sections 2.1 to 2.3 UNIT II Chapter 2 : Sections 2.6 to 2.11 UNIT III Chapter 3: Sections 3.1 to 3.6 UNIT IV Chapter 4 UNIT V Chapter 5: Sections 5.1 to 5.4 REFERENCE(S) [1] Gareth A. Jones and J. Mary Jones, Elementary Number Theory, Springer Verlag,

Indian Reprint, 2005. [2] David M. Burton, Elementary Number Theory, 6th edition, McGraw Hill, 2007. [3] George Andrews, Theory of Numbers, Saunders, 1971. [4] J. William, Fundamentals of Number Theory, Leveque, Addison-Wesley Publishing

Company, Phillipines, 1977.

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8

DISCRETE MATHEMATICS

Prerequisite: Nil

UNIT – I

Combinatorics; Sequences: Stirling Numbers: A Preview - Ordinary Generating Functions - Synthesizing Generating Functions.

UNIT II

Solving Recurrences: Types of Recurrences – Finding Generating Functions – Partial Fractions – Characterstic Roots – Simultaneous Recursions.

UNIT III

Graph Theory ; Basic results: Basic Concepts – Subgraphs – Degrees of Vertices - Paths and Connectedness

Connectivity : Vetex Cuts and Edges Cuts – Connectivity and Edge connectivity – Blocks – Cyclical Edge Connectivity of a graph – Menger’s Theorem. UNIT IV

Independent Sets and Matchings: Vertex Independent Sets and Vertex Coverings – Edge – Independent Sets – Matchings and Factors – Matchings in Bipartite Graphs. UNIT V

Planarity: Planar and Nonplanar Graphs – Euler Formula and its Consequences – K5 and K(3,3) are Nonplanar Graphs – Dual of a Plane Graph - The Four-Color theorem and the Heawood Five-Color Theorem – Kuratowski’s Theorem – Hamiltonian Plane Graphs – Tait Coloring

TEXT BOOK(S)

[1] Jonathan L. Gross, Combinatorial Methods with Computer Applications, Chapman & Hall /CRC, New York, 2008. [2] R, Balakrishnan and K.Ranganathan, A Textbook of Graph Theory, Second Edition, Springer, New York, 2012. UNIT I Chapter 1 (Section 1.6 – 1.8) of [1] UNIT II Chapter 2 (Section 2.1 - 2.5) of [1] UNIT III Chapter 1 and Chapter 3 of [2] UNIT IV Chapter 5 (Sections 5.1 -5.5) and Chapter 7 (Section 7.1 – 7.3) of [2] UNIT V Chapter 8 of [2]

REFERENCE(S) 1. C.L. Liu, Elements of Discrete Mathematics, McGraw-Hill, 1986. 2. P. Tremblay and R. Manohar, Discrete Mathematical Structures with Applications to Computer

Science, Tata McGraw-Hill, Newdelhi, 1997 3. Bondy J.A. and U.S.R. Murty, Graph Theory with Applications. North Holland, New York ,1976.

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9

ALGEBRA - I

Prerequisite: Nil

UNIT I Binary Operations – Groups - Subgroups – Permutations I – Permutations II – Cyclic

Groups UNIT I Isomorphisms – Direct Products – Finitely Generated Abelian groups – Groups of Cosets - Normal subgroups and factor groups- Homomorphisms UNIT III

Series of Groups – Isomorphism theorems- Proof of the Jordan Holder theorem—Group action on a set- Applications of G-sets to counting - Sylow’s theorems –Applications of Sylow theorems

UNIT IV

Rings – Integral Domains - Some non-commutative examples –The Field of quotients – Quotient rings and Ideal.

UNIT V

Homomorphism of Rings – Rings of polynomials – Factorization of Polynomials over a field – Euclidean domains- Gaussian integers and norms

TEXT BOOK

John B. Fraleigh, A First course in Abstract Algebra, Narosa Publishing House, Third edition, 1992.

UNIT –I Chapter 1, 2, 3,4,5,6 UNIT – II Chapter 7,8,9,11,12,13 UNIT – III Chapter 14,15,16,17,18,19 UNIT – IV Chapter 23,24,25,26,27,28 UNIT – V Chapter 29,30,31,33,34

REFERENCE(S)

1. J. Gallian, Contemporary Abstract Algebra, Narosa Publishing House, Fourth Edition, 1999. 2. Mark R. Sepanski, Algebra, AMS Indian Edition, 2012. 3. David S. Dummit and Richard M. Foote, Abstract Algebra, Wiley and Sons, Third Edition, 2004. 4. P.B. Bhattacharya et al., Basic Abstract Algebra, 2nd edition, Cambridge University Press, 1995. 5. C. Lanski, Concepts in Abstract Algebra, AMS Indian edition, 2010 6. M.Artin, Algebra, Prentice Hall of India, New Delhi, 1994. 7. I.N.Herstein, Topics in Algebra, John Wiley, 2nd Edition, 1975. 8. R. Solomon, Abstract Algebra, AMS Indian edition, 2010.

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10

REAL ANALYSIS II

Prerequisite: 16M02CC UNIT I Differentiation of single variable: Derivatives – The chain rule – local extrema –Rolle’s theorem – Mean Value Theorem – Taylor’s formula – Derivatives of vector – valued functions – Functions of Bounded variation and rectifiable curves – Total variation – Functions of bounded variation – Equivalence of paths – Change of parameter. UNIT II Riemann-Stieltjes integral: Definition – linear properties of the integral – Necessary conditions for the existence - First fundamental theorem of Integral calculus - Mean Value Theorems for integrals – Second fundamental theorem of Integral calculus- Change of variable in a Riemann integral – Second Mean value Theorem for Riemann integrals UNIT III

Sequence and series of functions – Point wise convergence – Uniform convergence – Uniform convergence and integration – Uniform convergence and Differentiation -Sufficient conditions for uniform convergence of a series

UNIT IV

Functions of Severable variables – Directional derivative –Total derivative – Jacobian – Chain rule –Mean Value Theorem – Taylor’s formula. UNIT V

Inverse function theorem – Implicit function theorem – Extremum problems with side conditions. TEXT BOOK(S)

Tom M. Apostol, Mathematical Analysis Second Edition, Narosa Publishing House, New Delhi, 1985. UNIT – I - Chapter 5 and 6 UNIT – II - Chapter 7 Section 7.1 -7.22 UNIT – III - Chapter 9 Section 9.1 - 9.11 and 9.14 -9.18 UNIT – IV - Chapter 12 UNIT – V -Chapter 13 REFERENCE(S) [1] M.H. Protter, C.B. Morrey, A First Course in Real Analysis, 2nd Edition, Springer Verlag International Edition, 1991. [2] Torrence Tao, Mathematical Analysis, Vol I & II, Hindustan Book Agency, 2006. [3] J.E. Marsden, A.J. Tromba, A.Weinstein, Basic multivariable calculus, Springer Verlag, 1993. [4] Robert T. Seeley, Calculas of Several Variables, Scott, Foresman and Co, 1970. [4] T.W. Korner, A Companion to Analysis, AMS Indian edition, 2011. [5] N.L. Carothers, Real Analysis, Cambridge University Press, South Asian Edition, 2000. [6] S. Kumaresan, A Course in Differential Geometry and Lie groups, Hindustan Book Agency, 2002 [7] Walter Rudin, Principles of Mathematical Analysis,Third Edition, Mcgraw Hill, 1976. [8] Tom Apostol, Calculas II, Mcgraw Hill, 1983.

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11

PARTIAL DIFFERENTIAL EQUATIONS

Prerequisite: 16M03CC UNIT I

First Order P.D.E. – Curves and Surfaces – Genesis of First Order P.D.E. – Classification of Integrals – Linear Equations of the First Order – Pfaffian Differential Equations – Compatible Systems – Charpit’s Method – Jacobi’s Method UNIT II

Integral Surfaces Through a Given Curve – Quasi-Linear Equations – Non-linear First Order P.D.E. UNIT III

Second Order P.D.E.: Genesis of Second Order P.D.E. – Classification of Second Order P.D.E. One-Dimensional Wave Equation – Vibrations of an Infinite String – Vibrations of a Semi-infinite String –Vibrations of a String of Finite Length (Method of separation of variables)

UNIT IV Laplace’s Equation: Boundary Value Problems – Maximum and Minimum

Principles – The Cauchy Problem – The Dirichlet Problem for the Upper Half Plane – The Neumann Problem for the Upper Half Plane – The Dirichlet Interior Problem for a Circle - The Dirichlet Exterior Problem for a Circle – The Neumann Problem for a Circle – The Dirichlet Problem for a Rectangle – Harnack’s Theorem . UNIT V

Heat Conduction Problem – Heat Conduction –Infinite Rod Case – Heat Conduction Finite Rod Case – Duhamel’s Principle – Wave Equation – Heat Conduction Equation - Fourier transform: - Fourier sine and cosine transforms - Properties Convolution - Solving integral equations - Finite Fourier transform - Finite Fourier sine and cosine transforms - Fourier integral theorem - Parseval's identity. TEXT BOOK(S)

[1] T.Amarnath, An Elementary Course in Partial Differential Equations , Narosa Publishing Company, 1997.

[2] Lokenath Debnath and Dambaru Bhatta , Integral Transforms and Their Applications, Chapman & Hall/CRC; 2 edition, 2006.

UNIT – I -Chapter 1: Sections 1.1 to1.8 of [1] UNIT – II -Chapter 1: Sections 1.9 to1.11 of [1] UNIT – III -Chapter 2: Sections 2.1 to 2.3.5, except 2.3.4 of [1] UNIT – IV -Chapter 2: Sections 2.4.1 to 2.4.10 of [1] UNIT – V -Chapter 2: Sections 2.4.11 to 2.6.2 of [1], Chapters 2, Sections 2.2, 2.3, 2.5, 2.11, 2.13 and 2.14 of [2] and Chapter 10, Sections 10.1 to 10.4 of [2]

REFERENCE(S)

[1] Tyn Myint-U: Partial differential equations for scientists and engineers, 3rd ed. North Holland, 1989. [2] I.C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, Vol. 19

AMS, 1998. [3] I.N. Snedden, Elements of Partial Differential Equations, McGraw Hill, 1985. [4] F. John, Partial Differential Equations, Springer Verlag, 1975. [5] Phoolan Prasad and Renuka Ravindran, Partial Differential Equations, Wiley-Eastern Ltd, 1985.

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12

TOPOLOGY

Prerequisite: 16M02CC

UNIT I

TOPOLOGICAL SPACES : Topological spaces - Basis for a topology - The order topology - The product topology on X x Y - The subspace topology - Closed sets and limit points.

UNIT II

CONTINUOUS FUNCTIONS : Continuous functions - the product topology - The metric topology.

UNIT III

CONNECTEDNESS: Connected spaces - connected subspaces of the Real line – Components and local connectedness.

UNIT IV

COMPACTNESS: Compact spaces - compact subspaces of the Real line – Arzela –

Ascoli theorem -Limit Point Compactness – Local Compactness -Tychonoff’s Theorem.

UNIT V

COUNTABILITY AND SEPARATION AXIOMS: The Countability Axioms – The separation Axioms - Normal spaces - The Urysohn Lemma - The Urysohn metrization Theorem - The Tietze extension theorem.

TEXT BOOK(S)

James R. Munkres, Topology (2nd Edition) Pearson Education Pvt. Ltd., New Delhi-2002 (Third Indian Reprint)

UNIT – I -Chapter 2: Sections 12 to 17 UNIT – II -Chapter 2: Sections 18 to 21 (Omit Section 22) UNIT – III -Chapter 3: Sections 23 to 25. UNIT – IV -Chapter 3: Sections 26 to 29 and Chapter 5: Section 37 UNIT – V -Chapter 4: Sections 30 to 35. REFERENCE(S)

1. M.A. Armstong, Basic Topology, Springer Verlag, 2005 2. J. Dugundji, Topology, Prentice Hall of India, New Delhi, 1975. 3. K.D. Joshi, Introduction to General Topology, Wiley Eastern Ltd., 1983. 4. O. Ya. Viro, Elementry Topology: Problem Textbook, AMS Indian Edition, 2012. 5. J.L. Kelly, General Topology, Van Nostrand, Reinhold Co., New York 6. L.Steen and J.Seeback, Counterexamples in Topology, Holt, Rinehart and Winston,

New York, 1970.. 7. G.F.Simmons, Introduction to Topology and Modern Analysis, McGraw-Hill, 1963.

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13

ALGEBRA – II

Prerequisites: 16M06CC

UNIT – I Prime ideals and Maximal Ideals, Irreducible polynomials. UNIT – II Classical Formulas, Splitting Fields. UNIT – III The Galois Group, Roots of Unity, Solvability by Radicals. UNIT – IV Independence of Characters, Galois Extensions, UNIT – V

The Fundamental theorem of Galois theory, Applications, Galois Great Theorem. TEXT BOOKS

Joseph Rotman, Galois Theory, 2nd edition, Springer Verlag, 1990. UNIT – I Pages 31 - 43 UNIT – II Pages 44 -58 UNIT – III Pages 59 - 75 UNIT – IV Pages 76-82 UNIT – V Pages 83-95 REFERENCE(S)

1. David S. Dummit and Richard M. Foote, Abstract Algebra, 2nd Edition, Wiley Student Edition, 2008. 2. Serge Lang. Algebra - Revised third edition - Springer – Verlag - 2002. 3. Ian Stewart, Galois Theory, Chapman and Hall, 1973. 4. R. Solomon, Abstract Algebra, AMS Indian edition, 2010. 5. C. Lanski, Concepts in Abstract Algebra, AMS Indian edition, 2010 6. John B. Fraleigh, A First course in Abstract Algebra, Narosa Publishing House, 2003. 7. M.Artin, Algebra, Prentice Hall of India, New Delhi, 1994. 8. I.N.Herstein, Topics in Algebra, John Wiley, 2nd Edition, 1975.

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14

COMPLEX ANALYSIS

Prerequisites: 16M02CC, 16M07CC, 16M09CC Unit I: The Complex Number System The real numbers - The field of complex numbers - The complex plane - Polar representation and roots of complex numbers - Lines and half planes in the complex plane - The extended plane and its spherical representation Unit II: Elementary Properties and Examples of Analytic Functions

Power series- Analytic functions - Analytic functions as mapping - Mobius transformations Unit III: Complex Integration Riemann-Stieltjes integrals - Power series representation of analytic functions - Zeros of an analytic function - The index of a closed curve - Cauchy's Theorem and Integral Formula - The homotopic version of Cauchy's Theorem and simple connectivity - Counting zeros - The Open Mapping Theorem- Goursat's Theorem Unit IV: Singularities Classification of singularities - Residues - The Argument Principle Unit V: The Maximum Modulus Theorem

The Maximum Principle - Schwarz's Lemma - Convex functions and Hadamard's Three Circles Theorem- Phragmen-Lindelof Theorem TEXT BOOK(S)

J.B. Conway, Functions of One Complex Variable, Narosa, 2 edn., 1991. UNIT – I -Chapter 1 UNIT – II -Chapter 3 UNIT – III -Chapter 4 UNIT – IV -Chapter 5 UNIT – V -Chapter 6 REFERENCE(S) [1] Bak, J., Newman, D.J., Complex Analysis, Springer-Verlag, New York, 1997. [2] L.S. Hahn and B. Epstein, Classical Complex analysis, Jones and Barlett Student Edition, 2011. [3] R. Priestely, Introduction to Complex Analysis, Oxford India, 2008. [4] Lars V. Ahlfors, Complex Analysis, Third Ed. McGraw-Hill Book Company, Tokyo, 1979. [5] Theodore W. Gamelin, Complex Analysis, Springer Verlag, 2001. [6] Donald Sarason, Notes on Complex Function theory, Hindustan Book Agency, 1994. [7] V. Karunakaran, Complex Analysis 2 edn, Narosa, New Delhi, 2005. [8] S. Ponnusamy and H. Silverman, Complex Variables with applications, Birkhauser, 2006. [9] R.V. Churchill & J.W. Brown, Complex Variables and applications, McGraw-Hill, 1990

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15

MEASURE THEORY AND INTEGRATION

Prerequisites: 16M02CC, 16M07CC

UNIT I

Measure on R: - outer measure - measurable sets – Regularity- abstract Measures – elementary properties of abstract measures. UNIT II

Integration of positive functions - Integration of complex functions – The role played by sets of measure zero. UNIT III

Measurability on cartesian products –Product Measure- The Fubini’s theorem –Completion of Product measures- convolutions. UNIT IV: Lp spaces: – convex function and inqualities - completeness of Lp spaces . UNIT V Signed Measures – Hahn Decomposition – Jordan Decomposition – Radon Nikodym theorem. Text Books: 1. G. de Barra, Measure Theory And Integration, NewAge International Pvt.Ltd. 2. W. Rudin, Real and Complex Analysis 3edn, McGraw-Hill, 1987. Unit I : Chapter 2 –Section 2.1 to2.3 from [1] and Chapter 1 - pages 5 -19 ( till the end of 1.22) from [2]. Unit II : Chapter 1 - pages 19 -31 from [2]. Unit III : Chapter 8 - pages 160-172 from [2]. Unit IV : Chapter 3 - pages 62 -70 from [2]. Unit V : Chapter 8 Sections 8.1 -8.3 from [1]. REFERENCES [1] R.G. Bartle, Elements of Integration, John Wiley, 1966. [2] C.D. Aliprantis and O.Burkinshaw, Priniciples of Real Analysis 2edn, Academic Press, Inc. New York, 1990. [3] I.K.Rana, An Introduction to Measure and Integration, 2edn ,Narosa Publishing House, NewDelhi, 2005. [4] H.L.Royden, Real Analysis, Prentice Hall of India ,Third edition, 1989. [5] R.G. Bartle, Modern theory of integration, AMS, 2000.

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16

FUNCTIONAL ANALYSIS

Prerequisites: 16M02CC, 16M07CC, 16M09CC, 16M12CC

UNIT I

Normed Spaces - Examples – Bounded operators – Banach spaces – Hahn Banach Theorem and Its Consequences.

UNIT II Finite dimensional Normed spaces - Uniform Boundedness principle: The theorem and its consequences - Closed Graph Theorem– Bounded Inverse Theorem– Open Mapping theorem.

UNIT III

Stone-Weierstrass Theorem – Contraction Mapping Theorem and its applications - Weak topologies and duality.

UNIT IV

Hilbert Spaces: – Orthonormal systems – Bessel’s Inequality – Parseval’s Identities – Riesz-Fischer Theorem. UNIT V Adjoint operators – Algebra of Bounded linear operators. TEXT BOOK

Bela Bollobas, Linear Analysis an introductory course, Cambridge Mathematical Textbooks, Cambridge University Press, 1990. UNIT – I -Chapter 2 and 3 UNIT – II -Chapter 4 and 5 UNIT – III -Chapters 6 , 7 and 8 UNIT – IV - Chapters 9 and 10 UNIT – V -Chapter 11 and 12 REFERENCE(S) [1] G.F.Simmons, Introduction to Topology and Modern Analysis, McGraw-Hill, 1963. [2] B.V. Limaye, Functional Analysis, Wiley Eastern Limited, Bombay, Second Print, 1985. [3] M.Thamban Nair, Functional Analysis: A first course, Prentice Hall of India, 2002.. [4] K. Yosida, Functional Analysis, Springer-Verlag, 1974. [5] E. Kreyszig, Introductory Functional Analysis with applications, John Wiley, 1978. [6] V. K. Krishan Textbook of Functional Analysis:A Problem-oriented Approach, Prentice

Hall of India, 2004

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17

DIFFERENTIAL GEOMETRY

Prerequisite: 16M07CC

Unit 1 Graphs and Level sets - Vector fields - Tangent space. Unit 2 Surfaces - Vector fields on surfaces Unit 3 Gauss map - geodesics Unit 4 Parallel Transport - Weingarten map Unit 5 Curvature of plane curves - arc length and Line integrals - Curvature of surfaces

TEXT BOOK(S)

[1] Elementary topics in Differential geometry, J.A.Thorpe ,Undergraduate texts in Mathematics, Springer- Verlag, 1979. Unit 1 - Chapters 1 to 3. Unit 2 - Chapters 4 and 5. Unit 3 - Chapters 6 and 7. Unit 4 - Chapters 8 and 9. Unit 5 - Chapters 10 to 12. REFERENCE(S)

1. S. Kumaresan, A Course in Differential Geometry and Lie groups, Texts and Readings in Mathematics 22 - Hindustan Book Agency, 2002.

2. Struik, D.T. Lectures on Classical Differential Geometry, Addison - Wesley, Mass. 1950. 3. Kobayashi S. and Nomizu. K. Foundations of Differential Geometry, Interscience

Publishers, 1963. 4. Wihelm Klingenberg: A course in Differential Geometry, Graduate Texts in Mathematics,

Springer Verlag, 1978. 5. T.J. Willmore, An Introduction to Differential Geometry, Oxford University Press,(17th

Impression) New Delhi 2002. (Indian Print).

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PROBABILITY THEORY

Prerequisites: 16M12CC UNIT I

Random Events and Random Variables - Random events – Probability axioms Combinatorial formulae – conditional probability – Bayes Theorem – Independent events Random Variables – Distribution Function – Joint Distribution – Marginal Distribution – Conditional Distribution – Independent random variables – Functions of random variables. UNIT II

Parameters of the Distribution - Expectation- Moments – The Chebyshev Inequality - Absolute moments – Order parameters – Moments of random vectors – Regression of the first and second types. UNIT III

Characteristic functions - Properties of characteristic functions – Characteristic functions and moments – semi-invariants – characteristic function of the sum of the independent random variables – Determination of distribution function by the Characteristic function – Characteristic function of multidimensional random vectors – Probability generating functions. UNIT IV

Some Probability distributions - One point , two point , Binomial – Polya – Hypergeometric – Poisson (discrete) distributions – Uniform – normal gamma – Beta – Cauchy and Laplace (continuous) distributions. UNIT V

Limit Theorems - Stochastic convergence – Bernaulli law of large numbers Convergence of sequence of distribution functions – Levy-Cramer Theorems – de Moivre- Laplace Theorem – Poisson, Chebyshev, Khintchine Weak law of large numbers – Lindberg Theorem – Lapunov Theroem – Borel-Cantelli Lemma - Kolmogorov Inequality and Kolmogorov Strong Law of large numbers. TEXT BOOKS: M. Fisz, Probability Theory and Mathematical Statistics, John Wiley and Sons, New York, 1963. UNIT – I Chapter 1: Sections 1.1 to 1.7, Chapter 2 : Sections 2.1 to 2.9 UNIT – II Chapter 3 : Sections 3.1 to 3.8 UNIT – III Chapter 4 : Sections 4.1 to 4.7 UNIT – IV Chapter 5 : Section 5.1 to 5.10 (Omit Section 5.11) UNIT – V Chapter 6 : Sections 6.1 to 6.4, 6.6 to 6.9 , 6.11 and 6.12.

(Omit Sections 6.5, 6.10,6.13 to 6.15) REFERENCES: 1. R.B. Ash, Real Analysis and Probability, Academic Press, New York, 1972 2. K.L.Chung, A course in Probability, Academic Press, New York, 1974. 3. K.R. Parthasarathy, Introduction to Probability and measure, Texts and Readings in Mathematics 22, Hindustan Book Agency, 2002. 4. R.Durrett, Probability : Theory and Examples, (2nd Edition) Duxbury Press, New York, 1996. 5. V.K.Rohatgi An Introduction to Probability Theory and Mathematical Statistics, Wiley Eastern Ltd., New Delhi, 1988(3rd Print). 6. P. Billingsley, Probability and Measure, John Wiley, 1985. 7. B.R.Bhat , Modern Probability Theory (3rd Edition), New Age International (P)Ltd, New Delhi, 1999 8. J.P. Romano and A.F. Siegel, Counter Examples in Probability and Statistics, Wadsworth and Brooks / Cole Advanced Books and Software, California, 1968.

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NUMERICAL ANALYSIS

Prerequisites: 16M01CC, 16M03CC UNIT – I

Transcendental and polynomial equations: Iteration Methods based on Second degree equation - Rate of convergence of iterative methods –Methods for finding complex roots – Polynomial equations – Birge-Vieta method, Bairstow’s method. UNIT – II

System of Linear Algebraic equations and Eigen Value Problems: Direct Methods- Gauss Jordan Elimination Method – Triangularization method – Cholesky method – Error Analysis - Iteration Methods - Jacobi iteration method – Gauss - Seidal iteration method – Eigenvalues and Eigen vectors. UNIT – III

Interpolation, Approximation and Differentiation: - Introduction –Hermite Interpolations – Piecewise and Spline Interpolation - Approximation – Least square approximation - Differentiation - Numerical Differentiation - Optimum choice of Step- length – Extrapolation methods UNIT - IV

Differentiation and Integration: - Methods based on interpolation – Methods based on undetermined coefficients – Gauss Legendre integration methods – Lobatto Integration Methods – Radau integration methods. UNIT – V

Ordinary differential equations: Introduction – Numerical Methods- Local truncation error – Euler, Backward Euler, Taylor’s Method and second order Runge-Kutta method

TEXT BOOK(S):

[1] M.K. Jain, S.R.K. Iyengar and R.K. Jain, Numerical Methods for Scientific and Engineering Computation, III Edn. Wiley Eastern Ltd., 1993.

Unit I – Chapter 2 – 2.4 to 2.8 of [1] Unit II – Chapter 3 – 3.2 to 3.5 of [1] Unit III – Chapter 4 – 4.1, 4.5 , 4.6, 4.8,4.9 and 5.2 – 5.4 of [1] Unit IV – Chapter 5 – 5.6 - 5.8 of [1] Unit V – Chapter 6 – 6.2 and 6.3 of [1] REFERENCES

[1] Kendall E. Atkinson, An Introduction to Numerical Analysis, II Edn., John Wiley & Sons, 1988. [2] M.K. Jain, Numerical Solution of Differential Equations, II Edn., New Age International Pvt Ltd., 1983.

[3] Samuel. D. Conte, Carl. De Boor, Elementary Numerical Analysis, Mc Graw-Hill International Edn., 1983.

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DEPARTMENT ELECTIVE PAPERS

STOCHASTIC PROCESSES

Prerequisite: 16M14CC

UNIT I

Stochastic Processes: Some notions – Specification of Stochastic processes Stationary processes – Markov Chains – Definitions and examples – Higher Transition probabilities – Generalization of Independent Bernoulli trails – Sequence of chain – Dependent trains. UNIT II

Markov chains: Classification of states and chains – determination of Higher transition probabilities – stability of a Markov system – Reducible chains – Markov chains with continuous state space. UNIT III

Markov processes with Discrete state space : Poisson processes and their extensions – Poisson process and related distribution – Generalization of Poisson process- Birth and Death process – Markov processes with discrete state space (continuous time Markov Chains). UNIT IV

Renewal processes and theory : Renewal process – Renewal processes in continuous time – Renewal equation – stopping time – Wald’s equation – Renewal theorems. UNIT V

Stochastic processes in Queuing – Queuing system – General concepts – the queuing model M/M/1 – Steady state behaviour – transient behaviour of M/M/1 Model – Non-Markovian models - the model GI/M/1.

TEXT BOOK(S)

[1] J. Medhi, Stochastic Processes, Wiley Eastern, 1982. UNIT I Ch. II : Sec 2.1 to 2.3, Ch III : Sec 3.1 to 3.3 UNIT II Ch III – Sec 3.4 to 3.6, 3.8, 3.9 and 3.11 UNIT III Ch IV : Sec 4.1 to 4.5 UNIT IV Ch VI : Sec 6.1 to 6.5 UNIT V Ch X : Sec 10.1 to 10.3, 10.7 and 10.8 (omit sec 10.2.3 & 10.2.3.1) REFERENCE(S)

1. Samuel Karlin, Howard M. Taylor, A first course in stochastic processes, 2nd edition, Academic Press, 1975.

2. Narayan Bhat , Elements of Applied Stochastic Processes, 2nd edn, John Wiley, 1984.

3. S.K. Srinivasan and K.Mehata, Stochastic Processes, Tata McGraw Hill, 1976. 4. N.U. Prabhu, Stochastic Processes. Macmillan, 1965.

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INTEGRAL EQUATIONS AND CALCULUS OF VARIATIONS

Prerequisites: 16M03CC, 16M08CC Calculus of variations Unit-I Problems with fixed boundaries

Unit-II Problems with moving boundaries – Extremals with corners – One sided variations

Unit-III Sufficient conditions for Extremum – Conditional Extremum Problems. Integral Equations UNIT IV

Linear Integral Equations - Definition, Regularity conditions – special kind of kernels – eigen values and eigen functions – convolution Integral – the inner and scalar product of two functions – Notation – reduction to a system of Algebraic equations – examples – Fredholm alternative - examples – an approximate method. UNIT V

Method of successive approximations: Iterative scheme – examples – Volterra Integral equation – examples – some results about the resolvent kernel. Classical Fredholm Theory: the method of solution of Fredholm – Fredholm’s first theorem – second theorem – third theorem. TEXT BOOK(S)

[1] Ram.P.Kanwal – Linear Integral Equations Theory and Practice, Academic Press 1971.

[2] L. Elsgolts, Differential equations and the calculus of variations, Mir publishers, Moscow 1970.

UNIT – I - Chapter 6 of [2] UNIT – II - Chapter 7,8 of [2] UNIT – III - Chapter 9,10 of [2] UNIT – IV - Chapters 1 and 2 of [1] UNIT – V - Chapters 3 and 4 of [1] REFERENCE(S) [1] S.J. Mikhlin, Linear Integral Equations (translated from Russian), Hindustan Book

Agency, 1960. [2] I.N. Snedden, Mixed Boundary Value Problems in Potential Theory, North Holland,

1966.

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FLUID DYNAMICS

Prerequisites: 16M03CC, 16M08CC

UNIT I

Real Fluids and Ideal Fluids - Velocity of a Fluid at a point - Streamlines and Path lines; Steady and Unsteady Flows - The Velocity potential – The Vorticity vector - Local and Particle Rates of Change – The Equation of continuity - Worked examples - Acceleration of a Fluid – Conditions at a rigid boundary - General analysis of fluid motion - Pressure at a point in a Fluid at Rest - Pressure at a point in Moving Fluid - Conditions at a Boundary of Two Inviscid Immiscible Fluids -Euler’s equations of motion - Bernoulli's equation - worked examples. UNIT II

Discussion of a case of steady motion under conservative body forces – Some potential theorems-Some Flows Involving Axial Symmetry - Some special two- Dimensional Flows - Impulsive Motion. Some three-dimensional Flows: Introduction - Sources, Sinks and Doublets - Images in a Rigid Infinite Plane - Axi-Symmetric Flows; Stokes stream function UNIT III

Some Two-Dimensional Flows: Meaning of a Two-Dimensional Flow - Use of cylindrical Polar coordinates – The stream function – The Complex Potential for Two- Dimensional, Irrotational, Incompressible Flow - complex velocity potentials for Standard Two-Dimensional Flows - Some worked examples – The Milne-Thomson circle theorem and applications – The Theorem of Blasius UNIT IV

The use of conformal Transformation and Hydrodynamical Aspects - Vortex rows. Viscous flow: Stress components in a Real fluid - relations between Cartesian components of stress - Translational Motion of Fluid Element - The Rate of Strain Quadric and Principal Stresses - Some Further properties of the Rate of Strain Quadric - Stress Analysis in Fluid Motion - Relations Between stress and rate of strain - The coefficient of viscosity and Laminar Flow - The Navier - Stokes equations of Motion of a Viscous Fluid . UNIT V

Some solvable problems in Viscous Flow - Steady Viscous Flow in Tubes of Uniform cross section - Diffusion of Vorticity - Energy Dissipation due to Viscosity - Steady Flow past a Fixed Sphere - Dimensional Analysis; Reynolds Number - Prandtl's Boundary Layer TEXT BOOK(S) [1] Content and Treatment as in Text Book of Fluid Dynamics by F. Chorlton (CBS Publishers &

Distributors, New Delhi-110 002) 1985. UNIT – I Chapter 2 and Chapter 3: Sections 3.1 to 3.6 UNIT – II Chapter 3: Sections 3.7 to 3.11 and Chapter 4: Sections 4.1, 4.2, 4.3, 4.5 UNIT – III Chapter5 : Sections: 5.1 to 5.9 except 5.7 UNIT – IV Chapter 5: Section 5.10 , 5.12 and Chapter 8: Sections 8.1 to 8.9 UNIT – V Chapter 8: Sections 8.10 to 8.16 REFERENCE(S) [1] J.F. Wendt, J.D. Anderson, G.Degrez and E. Dick, Computational Fluid Dynamics : An Introduction,

Springer-Verlag, 1996. [2] J.D. Anderson, Computational Fluid Dynamics, The Basics with Applications, McGraw Hill, 1995. [3] G.K. Batchelor, An Introduction to Fluid Mechanics, Foundation Books, New Delhi, 1984. [4] A.J. Chorin and A. Marsden, A Mathematical Introduction to Fluid Dynamics, Springer-Verlag, New

York, 1993. [5] S.W. Yuan, Foundations of Fluid Mechanics, Prentice Hall of India Pvt Limited, New Delhi, 1976. [6] R.K. Rathy, An Introduction to Fluid Dynamics, Oxford and IBH Publishing Company, New Delhi,

1976.

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CLASSICAL DYNAMICS

Prerequisites: 16M03CC, 16M08CC UNIT I

Introductory concepts: The mechanical system - Generalised Coordinates - constraints - virtual work - Energy and momentum. UNIT II

Lagrange's equation: Derivation and examples - Integrals of the Motion UNIT III

Hamilton's equations: Hamilton's principle - Hamilton’s equations - Other variational principles - phase space. UNIT IV

Hamilton - Jacobi Theory: Hamilton's Principal Function – The Hamilton - Jacobi equation - Separability. UNIT V

Canonical Transformations: Differential forms and Generating functions – Special Transformations – Lagrange and Poisson Brackets.

TEXT BOOK(S)

[1] Donald T. Greenwood, Classical Dynamics, PHI Pvt. Ltd., New Delhi, 1985. UNIT – I -Chapter 1: Sections 1.1 to 1.5 UNIT – II -Chapter 2: Sections 2.1 to 2.4 UNIT – III -Chapter 3: Sections 3.1 - 3.2 and 3.4 (section 3.3 omitted) UNIT – IV -Chapter 4: Sections 4.1 to 4.4 UNIT – V -Chapter 5: Sections 5.1 to 5.3 REFERENCE(S) [1] H. Goldstein, Classical Mechanics, (2nd Edition), Narosa Publishing House, New

Delhi, 1998. [2] John L Synge and Byron A Griffith, Priniciples of Mechanics, McGraw-Hill, New York, 1959. [2] Narayan Chandra Rana & Promod Sharad Chandra Joag, Classical Mechanics, Tata

McGraw Hill, 1991.

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MATHEMATICAL STATISTICS

Prerequisite: Nil

UNIT I

SAMPLE MOMENTS AND THEIR FUNCTIONS: Notion of a sample and a statistic – Distribution functions of 2, s2 and (2, s2) - 2 distribution - Student t-distribution - Fisher's Z-distribution - Snedecor's F-distribution - Distribution of sample mean from non-normal populations.

UNIT II

SIGNIFICANCE TEST: Concept of a statistical test - Parametric tests for small samples and large samples - c2 test Kolmogorov Theorem 10.11.1 - Smirnov Theorem 10.11.2 - Tests of Kolmogorov and Smirnov type - The Wald-Wolfovitz and Wilcoxon-Mann-Whitney tests - Independence Tests by contingency tables.

UNIT III

ESTIMATION: Preliminary notion - Consistency estimation - Unbiased estimates – Sufficiency - Efficiency - Asymptotically most efficient estimates - methods of finding estimates - confidence Interval.

UNIT IV

ANALYSIS OF VARIANCE: One way classification and two-way classification. Hypotheses Testing Poser functions - OC function- Most Powerful test - Uniformly most powerful test - unbiased test.

UNIT V

SEQUENTIAL ANALYSIS: SPRT - Auxiliary Theorem - Wald's fundamental identity – OC function and SPRT - E(n) and Determination of A and B - Testing a hypothesis, concerning p on 0-1 distribution and m in Normal distribution.

TEXT BOOK(S) [1] M. Fisz, Probability Theory and Mathematical Statistics, John Wiley and sons, New York,

1963. UNIT I Chapter 9 : Sections 9.1 to 9.8 UNIT II Chapter 10 : Sections 10.11, Chapters 11,12 Sections : 12.1 to 12.7. UNIT III Chapter 13 : Sections 13.1 to 13.8 (Omit Section 13.9) UNIT IV Chapter 15 : Sections 15.1 and 15.2 (Omit Section 15.3)

Chapter 16 : Sections 16.1 to 16.5 (Omit Section 16.6 and 16.7) UNIT V Chapter 17 : Sections 17.1 to 17.9 ( Omit Section 17.10) REFERENCE(S)

1. E.J.Dudewicz and S.N.Mishra, Modern Mathematical Statistics, Jobn Wiley and Sons, New York, 1988.

2. V.K.Rohatgi An Introduction to Probability Theory and Mathematical Statistics, Wiley Eastern Ltd., New Delhi, 1988(3rd Print).

3. G.G.Roussas, A First Course in Mathematical Statistics, Addison Wesley Publishing Company, 1973.

4. B.L.Vander Waerden, Mathematical Statistics, G.Allen & Unwin Ltd., London, 1968.

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OPTIMIZATION TECHNIQUES

Prerequisite: Nil

UNIT I Linear Programming Problem – Simplex method – Integer Programming – Gomory's

all I.P.P method – Fractional Cut Method – All integer L.P.P – Mixed integer L.P.P- Branch and Bound method. UNIT II

Dynamic Programming – The recursive equation approach- Solution of discrete D.P.P – Some applications- Solutions of L.P.P by Dynamic Programming. UNIT III

Queueing system – Deterministic Queueing System – Probability distributions in Queueing systems – Classification of Queueing Models – Transient and Steady States – Poisson Queueing Systems – Non-Poisson Queueing systems – Cost Models in Queueing – Other Queueing Models – Queueing Constrol – Queueing Theory and inventory Control UNIT IV

Inventory models – the concept of EOQ – deterministic inventory problems- with no shortages, with shortages – problems of EOQ with price breaks – Inventory problems with uncertain demands – One period problem – One period problem - without set-up cost , with set-up cost UNIT V Non-Linear Programming – Formulation – constrained optimization – with equaling constraints, with in-equaling constraints – saddle point problems – Methods – Graphical sign - kuhn – Tucker conditions with non- negative constrains – quadratic programming- wolfe's modified simplex method – Beale's method – separable convex programming. TEXT BOOK

Kanti Swarup, P . K. Gupta, Man Mohan, Operations Research (15th Edn.),. Sultan Chand & sons, New Delhi.

UNIT I Sections 4.1 to 4.4, 7.1 – 7.7 UNIT II Sections 13.1 to 13.7 UNIT III Sections 21.1 to 21.14 UNIT IV Sections 19.1 to 19.12, 20.4 – 20.6 UNIT V Sections 27.1 - 27.7, 28.1 – 28.8. REFERENCE(S)

[1] Hamdy A. Taha, Operations Research (7th Edn.), McGraw Hill Publications, New Delhi.2002.

[2] Bazaara, Jarvis and Sherali, Linear Programming and Network Flows, 2th ed., John Wiley

[3] O.L. Mangasarian, Non Linear Programming, McGraw Hill, New York, 1979. [4] Mokther S. Bazaraa and C.M. Shetty, Non Linear Programming, Theory and

Algorithms, Willy, New York, 1979. [5] Prem Kumar Gupta and D.S. Hira, Operations Research : An Introduction, S. Chand

and Co., Ltd. New Delhi, 1983. [6] S.S. Rao, Optimization Theory and Applications, Wiley Eastern Limited, New

Delhi, 1979. [7] G. Hadley, Linear Programming, Narosa Publishing House.

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OPERATOR THEORY

Prerequisite: 16M13CC

UNIT I Compact Operators: Characterizations – Some Properties.

UNIT II

Spectral Results for Banach Space Operators – Spectrum – Eigen spectrum – Resolvent set – Riesz - Schauder Theory

UNIT III Operators on Hilbert Spaces –Adjoint – Self Adjoint – Normal – Unitary – Opertors – Numerical range – Hilbert-Schmidt operators.

UNIT IV Spectral Results for Hilbert space Operators- Normal and Self Adjoint Operators- Spectral Representations

UNIT V Banach Algebras – Regular and Singular elements – Spectrum – Gelfand mapping – Gelfand- Neumark Theorem TEXT BOOKS

[1] M.Thamban Nair, Functional Analysis: A first course, Prentice Hall of India, New Delhi, 2002.. [2] G.F.Simmons, Introduction to Topology and Modern Analysis, McGraw-Hill International Ed. 1963. UNIT – I -Chapter 9 from [1] UNIT – II -Chapter 10 from [1] UNIT – III -Chapter 11 from [1] UNIT – IV - Chapters 12 and 13 from [1] UNIT – V -Chapters 12 and 13 from [2] REFERENCE(S) [1] Bela Bollobas, Linear Analysis, An introductory course, Cambridge University Press, 1990 [2] B.V. Limaye, Functional Analysis, Wiley Eastern Limited, Bombay, Second Print, 1985. [3] K. Yosida, Functional Analysis, Springer-Verlag, 1974. [4] E. Kreyszig, Introductory Functional Analysis with applications, John Wiley, 1978.

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UNIVERSITY ELECTIVE COURSES

OBJECT ORIENTED PROGRAMMING using C++

Prerequisite: Nil

UNIT I Programming Paradigms – Introduction to OOP – Advantages of OOP-Characteristics of OO languages – Overview of C++ - C++ programming basics – Functions: Simple Functions – Call by value – Call by reference – Returning values of different type – Function overloading – inline functions – Default arguments – Recursive functions. UNIT II Class – Objects – Constructors – Destructors – Objects as function arguments – Returning objects from functions – Structures and Classes – Static data – Static function – Array of objects. UNIT III Access specifiers – Friend function – Friend class – Operator overloading – Type casting – Pointers – Template. UNIT IV Inheritance – Derived class constructors – Class hierarchies – Types of inheritance – Virtual base class – Function overriding – Virtual functions – Pure virtual functions – Abstract class. UNIT V Files and Streams: I/O manipulators – Streams – Error handling during file operations String I/O – Character I/O – Object I/O – I/O with multiple objects – File pointers – Disk I/O with member functions. TEXT BOOK(S)

Robert Lafore, Object-Oriented Programming in Microsoft C++, Galgotia Publications, New Delhi, 2000.

UNIT – I -Chapters 1, 2 and 3 UNIT – II -Chapters 4 and 5 UNIT – III -Chapters 6 and 7 UNIT – IV -Chapters 9 and 10 UNIT – V -Chapters 11 and 12

REFERENCE(S) [1] E.Balagurusamy, Object-Oriented Programming with C++, Second Edition, 2002. [2] Bjarne Stroustrup, The C++ Programming Language, Addison-Wesley, New York,

1999. [3] StephenPrata,”C++ Primer Plus”, 6th Edition ,Addison-Wesley Professional, 2011.

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RESOURCE MANAGEMENT TECHNIQUES

Prerequisite: Nil

UNIT I

Linear programming problem – Mathematical formulation – Graphical solution and extension – Simplex method.

UNIT II

Duality in Linear programming.

UNIT III

Transportation problem – Assignment problem. UNIT IV

Game Theory. UNIT V

Network Scheduling by PERT/CPM. TEXT BOOK(S) [1] Kanti Swarup, P. K. Gupta, Man Mohan, Operations Research, Sultan Chand and

Sons, 2010. UNIT – I -Chapters 2, 3 and 4 UNIT – II -Chapter 5 UNIT – III -Chapters 10 and 11 UNIT – IV -Chapter 17 UNIT – V -Chapter 25 REFERENCE(S) [1] Bazaara, Jarvis and Sherali, Linear Programming and Network Flows, 2th

ed., John Wiley. [2] Hamdy Taha, Operations Research, Pearson Education.

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MATHEMATICAL MODELLING

Prerequisite: Nil

UNIT I

Mathematical Modelling through Ordinary Differential Equations of First order :

Linear Growth and Decay Models – Non-Linear Growth and Decay Models – Compartment Models

– Dynamics problems – Geometrical problems.

UNIT II

Mathematical Modelling through Systems of Ordinary Differential Equations of First

Order : Population Dynamics – Epidemics – Compartment Models – Economics – Medicine, Arms

Race, Battles and International Trade – Dynamics.

UNIT III

Mathematical Modelling through Ordinary Differential Equations of Second Order:

Planetary Motions – Circular Motion and Motion of Satellites – Mathematical Modelling through

Linear Differential Equations of Second Order – Miscellaneous Mathematical Models.

UNIT IV

Mathematical Modelling through Difference Equations : Simple Models – Basic Theory

of Linear Difference Equations with Constant Coefficients – Economics and Finance – Population

Dynamics and Genetics – Probability Theory.

UNIT V

Mathematical Modelling through Graphs : Solutions which can be Modelled through

Graphs – Mathematical Modelling in Terms of Directed Graphs, Signed Graphs, Weighted Digraphs

and Unoriented Graphs.

TEXT BOOK(S)

J.N. Kapur, Mathematical Modelling, Wiley Eastern Limited, New Delhi, 1988. REFERENCE(S) [1] J. N. Kapur, Mathematical Models in Biology and Medicine, Affiliated East –West Press Pvt

Limited, New Delhi, 1981.

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STATISTICS

Prerequisite: Nil

UNIT I

Collection, classification and tabulation of data, graphical and diagramatic representation – Bar diagrams, Pie diagram, Histogram, Frequency polygon, frequency curve and Ogives. UNIT II

Measures of central tendency – Mean, Median and Mode in series of individual observations, Discrete series, Continuous series (inclusive), More than frequency, Less than frequency, Mid-value and open-end class.

UNIT III

Measures of dispersion – Range, Quartile deviation, Mean deviation about an average, Standard deviation and co-efficient of variation for individual, discrete and continuous type data. UNIT IV

Correlation – Different types of correlation – Positive, Negative, Simple, Partial Multiple, Linear and non-Linear correlation. Methods of correlation – Karlpearson’s Spearman’s correlations and Concurrent deviation . UNIT V

Regression types and method of analysis, Regression line, Regression equations, Deviation taken from arithmetic mean of X and Y, Deviation taken from assumed mean, Partial and multiple regression coefficients – Applications TEXT BOOK(S) [1] S.C.Gupta, V.K.Kapoor, Fundamentals of Mathematical Statistics, Sultan Chand and

Sons, New Delhi, 1994. REFERENCE(S) [1] Freund J.E.(2001); Mathematical Statistics, Prentice Hall of India. [2] Goon, A.M., Gupta M.K., Dos Gupta, B, (1991), Fundamentals of Statistics, Vol.I,

World Press, Calcutta.

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GENERAL INTELLIGENCE

Prerequisite: Nil

UNIT I

Problems on Numbers- Average-Problems on Ages.

UNIT II Percentage-Profit & Loss-Simple Interest-Compound Interest.

UNIT III

Ratio & Proportion-Partnership-Calender-Clocks.

UNIT IV Time and work-Pipes & Cistern

UNIT V Time & Distance-Problems on Trains-Boats and Streams

TEXT BOOK Dinesh Khattar, The Pearson Guide To Quantitative Aptitude For Competitive Examinations, Pearson Education, 3 edition, 2015.

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