HOLY CROSS COLLEGE (AUTONOMOUS), TIRUCHIRAPPALLI 620...

HOLY CROSS COLLEGE (AUTONOMOUS), TIRUCHIRAPPALLI – 620 002.

M.Sc., MATHEMATICS

(For the candidates admitted from 2008 onwards)

SEMESTER I

Major Core 1: ALGEBRA

Total Hours: 90

Hours/week: 6 Code: P08MA1MCT01

No. of Credits: 5 Marks: 100

Objectives:

1. To give a detailed knowledge about the Counting Principle, Euclidean Rings and dual spaces.

2. To develop the concept of module theory , Galois theory and algebra of linear transformations

UNIT I: (18 hrs)

Group Theory

Another Counting principle, Sylow’s theorem (2nd Proof), Direct Products, Finite

abelian Groups.

UNIT II:(18 hrs)

Euclidean Rings

A particular Euclidean Ring, Polynomial rings, Polynomials over the rational

fields,Polynomial rings over commutative rings.

UNIT III: (18 hrs)

Vector Spaces and Modules

Dual Spaces, Modules, Extension Fields.

UNIT IV: (18 hrs)

Galois Theory

Roots of polynomials, More about Roots, The Elements of Galois Theory.

UNIT V: (18 hrs)

Linear Transformations

The Algebra of Linear transformations, Characteristic Roots, Matrices, Canonical forms-

Triangular Form, Finite Fields.

TEXT BOOK:

I.N. Herstein, Topics in Algebra, Wiley Eastern Limited, New Delhi, Second Edition,

Sixteenth Reprint, April 1994 .

Unit I: Chapter 2 (Sec.2.11 - 2.14)

Unit II: Chapter 3 (Sec.3.7 – 3.11)

Unit III: Chapter 4 (Sec.4.3,4.5), Chapter 5 – Sec 5.1

Unit IV: Chapter 5 (Sec. 5.3,5.5,5.6)

Unit V: Chapter 6 (Sec.6.1 – 6.4), Chapter 7 – Sec7.1

REFERENCE BOOKS:

1. A.R. Vashistha ,ModernAlgebra,KrishknaprakashanMandir, Meerut , Fifth Edition ,1973.

2. Serge Lang ,Algebra, Addison – Wesley Publishing Company, New York. Ninth Printing,1980.

3. M.D. Raisinghania and R.S. Aggarwal , Modern Algebra

S. Chand & Company Ltd., New Delhi,Second Edition ,1980.

4. R. Balakrishanan& N. Rama Bhadran , A Text Book of Modern Algebra Vikas Publishing Hourse Pvt. Ltd., New Delhi,Third revised Edition, 1979.

5. Surjeetsingh and QaziZameeruddin , Modern Algebra, Vikas Publishing House Pvt. Ltd.,First Edition , 1972.


M.Sc., MATHEMATICS


SEMESTER I

Major Core - 2: DATA STRUCTURES USING C

Total Hours: 90



Objectives:

1. To introduce the structure of arrays, linked list structures and binary tree representation.

2. To explain briefly about Pointers and data structures. UNIT I: (18 hrs)

Primitive Data structure

Introduction - Arrays -ordered list - Representation of arrays -stacks -Mazing problem -

Evaluation of Expressions Queues -Circular queue.

UNIT II: (18 hrs)

List Structures

Singly linked lists - linked stacks and queues- Storage pool - polynomial addition -

doubly linked lists-Binary tree -representation - traversal.

UNIT III: (18 hrs)

Pointers and Arrays

The & and * operators- Pointers expressions-Char, int and float pointers- Passing

Addresses to functions- Functions returning pointers and Arrays- Passing an entire

array to a function-More than one Dimension-Pointers and Two dimensional arrays-

Arrays-Array of Pointers.

UNIT IV: (18 hrs)

Pointers and Structures

Array of structures- More about structures-Structure pointers-Dynamic Memory

allocation- Linked lists-Stacks and Queues-Doubly linked lists.

UNIT V:(18 hrs)

Pointers and Data structures.

Merging of linked lists.-Linked lists and polynomials-Circular linked list-Binary tree-

Traversal of a Binary tree.

TEXT BOOKS:

Unit I & II:

Ellis Horowitz, SartajSahni, Fundamentals of data structures, Galgotia Publications,

New Delhi ,1983.

Chapters:1, 2.2, 2.4, 3.1, 3.2, 3.3, 4.1, 4.2, 4.3, 4.4, 4.8, 5.1, 5.2, 5.3, 5.4.

Unit III, IV & V

KanetkarYashvant P, Understanding Pointers In C, BPB Publications, Third

Edition,New Delhi, 2001

Chapters:1,2,4 and 5(Omit from Threaded binary trees).

REFERENCE BOOKS :

1.DeshpandeP. S. &Kakde O. G., C & Data Structures ,

Dreamtech Press, New Delhi , 2003

RadhaGanesan P, C and Data Structurers ,Scitech Publications (India) Pvt. Ltd., Chennai, 2002.

2.Lipschutz Seymour, Data Structures , Tata Mcgraw Hill Publishing Company Limited,

New Delhi, 2006

3.James Keogh, Data Structures Principles &Fundamentals ,Wiley

Publications, New Delhi,2004.

4.KanetkarYashvant ,Data Structures Throuh C,BPB Publications,

New Delhi,2003


M.Sc., MATHEMATICS


SEMESTER I

Major Core 3 : REAL ANALYSIS

Total Hours: 90



Objectives:

1. To give the students a thorough knowledge of the various aspects of Metric Spaces in general which is imperative for any advanced learning.

2. To explain continuous functions, Riemenn integrations and Lebesgue integrals in detail.

UNIT I: (18 hrs)

Basic Topology

Metric spaces – Neighbourhood – Open sets – Closed sets – Compact sets –Perfect sets-

Cantor set – Connected sets.

UNIT II: (18 hrs)

Continuity

Limits of functions – Continuous functions – continuity and compactness – continuity

and connectedness – Discontinuities- Monotonic Functions.

UNIT III: (18 hrs)

Differentiation

The derivative of a Real function- Mean value theorems- The continuity of derivatives –

L ‘Hospital’s Rule – Derivatives of higher order-Taylor’s theorem

UNIT IV: (18 hrs)

Riemenn-Stieltje’s Integral

Definition – Existence of the Integral – Properties of the Integral – Integration and

Differentiation – Rectifiable curves.

UNIT V: (18 hrs)

Sequences and Series Of Functions

Uniform convergence-Uniform convergence and continuity – Uniform convergence and

Differentiation – Uniform convergence and Integration -Equicontinuous family of

functions – The Stone Weierstrass theorem.

TEXT BOOK:

Walter Rudin ,Principles of Mathematical Analysis , Mc-Graw Hill

International Edition, Singapore, 2002

Unit 1 – Chapter 2 (Sec 2.15 to 2.47)



Unit 4 – Chapter 6 (Sec 6.1 to 6.22, 6.26 & 6.27)

Unit 5- Chapter 7 (Sec 7.1 to 7.26)

REFERENCE BOOKS:

1.DipakChatterjee,Real Analysis, Prentice Hall of India,New Delhi,2004

2.MurrayH.Protter, Basic Elements of Real Analysis,Springer-Verlag,Newyork,2006

3.Charles Chapman Pugh, Real Mathematical Analysis, Springer-Verlag,Newyork,2004

4.PawanK.Jain,ShivK.Kawshik, An Introduction to Real Analysis,

S.Chand& Company Ltd., New Delhi,2000.

5.Malik ,S.C., Real Analysis, Prentice Hall of India,New Delhi,2004


M.Sc., MATHEMATICS


SEMESTER I

Major Core 4 : DIFFERENTIAL EQUATIONS

Total Hours: 90



Objectives:

1. To give an in depth knowledge of solving differential equations using various

methods. 2. To introduce existence and uniqueness theorems in Differential equations.

UNIT I:Second order linear equations (18 hrs)

General solution of homogeneous Equations – The use of a known solution to find

another – The homogeneous equation with constant coefficient – The method of

undetermined co-efficient – The method of variation of parameters.

UNIT II:Solution in power series (18 hrs)

Illustrative examples – Frobenius method – Bessel’s equations – Properties of Bessel’s

function –differential equation satisfied by Bessel function – a particular class of

equation – Legendre function – Hypergeometric function.

UNIT III: System of first order equations (18 hrs)

General remarks on systems – Linear systems – Homogeneous linear systems with

constant co-efficient.

UNIT IV:Partial differential equations of first order (18 hrs)

Origins of first order Partial differential equations -Cauchy’s problem for first order

Partial differential equations.- linear equations of first order – Integral surfaces passing

through a given curve –Surfaces orthogonal to a given system– Compatible systems of

first order equations – Charpit’s method.

UNIT V:Partial differential equations of the second order (18 hrs)

Origin of second order equations-Linear Partial differential equations with constant

coefficients-Equations with variable coefficients-The solution of Linear Hyperbolic

equations.

TEXT BOOKS:Units I & III.

George F. Simmons, Differential Functions with Applications And Historical Notes,

Tata Mc-Graw Hill Publishing Company Limited Company, New Delhi, 17th reprint,

1995.

Unit I: Chapter 3 (Sections 14-19)

Unit III: Chapter 7 (Sections 36-38)

Hildebrand, Advanced Calculus for Applications,

Unit II: Chapter 4 (4.1 – 4.4, 4.7 – 4.10, 4.12, 4.13)

Ian Sneddon, Elements of Partial differential Equations, Tata Mc-Graw Hill Publishing

Company Limited Company, New Delhi, 1996

Unit IV: Chapter 2 (Sections 1 to 6,9,10 )

Unit V: Chapter 3 (Sections 1, 4,5& 8)

REFERENCE BOOKS:

1.Jain M.K.,Iyengar S.R.K., Jain R.K., Computatiopnal Methods forPartial

differentialequations , New age international private limited, New Delhi ,reprint 2002.

2.Raisinghania M.D., Ordinary and Partial differential equations, S. Chand & Co., Ltd., New Delhi, 9thEdition,2006. 3.Billingham.J, King A.C., & Otto S.R., Differential Equations,Linear, Non Linear, Ordinary,

Partial, Replica Press Private Ltd, India,First South Asian Edition, 2005

4.Zafar Ahsan, Differential Equations & their Applications,Prentice Hall Of India Private Ltd, New Delhi,2006. 5. Prasad .P&Ravindran .R., Partial Differential Equations,Wiley Eastern Ltd.,

New York,2005


M.Sc., MATHEMATICS


SEMESTER I

Major Core 5 : GRAPH THEORY

Total Hours: 90



Objectives:

1. To give the rigorous introduction to the basic concepts of Graph Theory.

2. To enlighten the students with many applications of this subject.

UNIT I : (18 hrs)

Trees and Connectivity:

Trees - Cut Edges and Bonds – Cut vertices – Cayley’s formula – Application-

The Connector problem- Connectivity – Blocks – Application- Construction of reliable

communication Networks.

UNIT II : (18 hrs)

Euler Tours and Hamilton Cycles& Edge colouring

Euler Tours – Hamilton Cycles – application The Chinese Postman problem –

Edge colouring – edge chromatic number – Vizing’s theorem – application – The Time

Tabling problem.

UNIT III: (18 hrs)

Vertex Colourings:

Chromatic number-Brooke’s theorem-Hajos’ Conjecture – Chromatic

Polynomials – Girth and Chromatic number – application.

UNIT IV: (18 hrs)

Planar Graphs:

Plane and Planar graphs-Dual Graphs-Euler’s formula-bridges-

Kuratowski’sTheorem the Four Colour Conjecture.

UNIT V: (18 hrs)

Directed Graphs:

Basic ideas of independent set and Cliques. Directed Graphs –Directed Path –

Directed Cycles.

TEXT BOOK:

Bondy J.A. and Murthy V.S.R.,Graph Theory with Applications ,The Macmillan

Press,New York,1976.

Unit –I Chapters 2 & 3(Sections 2.1-2.5, 3.1-3.3)

Unit –II Chapters 4 & 6(Sections 4.1-4.3, 6.1-6.3)

Unit – III Chapter 8(Sections 8.1-8.6)

Unit – IV Chapter 9 (Sections 9.1-9.6)

Unit –V Chapter 10 (Sections 10.1-10.3)

REFERENCE BOOKS:

1. NarsinghDeo – Graph Theolry with applications to Engineering and Computer

Science Prentice – Hall of India private Limited,New Delhi,2005.

2. John Clark and Derek Allan Holton, A first look at graph Theory,

Allied Publisher Limited, First Indian reprint, 1995.

3. Frank Harray ,Graph Theory ,Narosha Publishing House ,chennai, 2001

4. BelaBollobas,Modern Graph Theory,Springer-Verlag,New York,.

5. Douglas B.,Introduction to Graph Theory,Pearson Education

Private Ltd., New Delhi, Second Edition,2002


M.Sc., MATHEMATICS


SEMESTER II

Major Core 6 : MEASURE AND INTEGRATION

Total Hours: 90



Objectives:

1. To generalize the concept of integration using measures.

2. To develop the analytical thinking of the students.

UNIT I: (18 hrs)

Lebesgue Measure

Outer measure- Measurable sets and lebesgue measure – a non measurable set –

Measurable functions – Little wood’s three principles – Egoroff’s Theorem and Lusin’s

Theorem.

UNIT II: (18 hrs)

Lebesgue Integral

The Lebesgue integral of a bounded function over a set of finite measure – The integral

of non-negative function – The general Lebesgue integral.

UNIT III: (18 hrs)

Differentiation And Integration

Differentiation of monotone functions – functions of bounded variation –

Differentiation of an integral – Absolute Continuity.

UNIT IV: (18 hrs)

General Measure And Integration

Measurable functions Integration –Signed measures-The Radon Nikodym Theorem.

UNIT V: (18 hrs)

Measure And Outer Measure

Outer measure and measurability – The Extension Theorem – Product measures.

TEXT BOOKS:

Royden Collier H.L.( 2003), REAL ANALYSIS, Pearson Education Private

Ltd.,Macmillan Co., New York.,Third Edition,.

Unit I: Chapter III

Unit II: Chapter IV (Sections:4.1-4.4)

Unit III: Chapter V (Sections5.1-5.4)

Unit IV: Chapter XI (Sections 11.1 – 11.3, 11.5, 11.6)

Unit V: Chapter XII(Sections 12.1, 12.2,12.4)

2. Indev .Rana K.( 1997), AN INTRODUCTION TO MEASURE

AND INTEGRATION, Narosa Publishing House, New Delhi.

Unit I: Proof of Egroff’s Theorem:8.2.4/Pg.223 &Lusin’s Theorem 8.2.14/Pg.227.

REFERENCE BOOKS:

1 Barra G.De .(2006),MEASURE AND INTEGRATION, New age International

Ltd., New Delhi.

2. Berberian sterling K.(1999),FUNDEMANTALS OF REAL ANALYSIS,

Springer- Verlag,New York.

3. Carothers N.L.(2006),REAL ANALYSIS,Cambridge University Press.

4. Charles Chapman Pugh (2004),REAL MATHEMATICAL ANALYSIS,

Springer- Verlag,New York.

5. Munroe (1959), INTRODUCTION MEASURE AND INTEGRATION,

Addison – Wesley Publishing Company, U.S.A.


M.Sc., MATHEMATICS


SEMESTER II

Major Core 7 : GENERAL TOPOLOGY

Total Hours: 90



Objectives:

1. To generalize the concepts of Real Analysis to topological Spaces and to give

rigorous introduction. 2. To train the students to develop analytical thinking.

UNIT I: (18 hrs)

Topology-Introduction

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: (18 hrs)

Product and Metric Topology

Continuous Functions – The Product Topology – The Metric Topology – The Metric

Topology (Continued)

UNIT III: (18 hrs)

Connectedness & Compactness

Connected Spaces – Connected Subspace of the real Line-Compact Spaces.

UNIT IV: (18 hrs)

Compactness (continued)

Compact Spaces of the Real Line – Limit point Compactness, The Count ability axioms.

UNIT V: (18 hrs)

Seperation Axioms

The separation axioms-Urysohn’s Lemma – Tietiz’s Extension theorem – The

Urysohn’sMetrization Theorem.

TEXT BOOK:

James R.Munkres (2005), TOPOLOGY, 2nd Edition, Prentice- Hall Of India,

New Delhi.

Unit -I : Chapter 2( Sections 12-17)

Unit -II: Chapter 2( Sections 18-21)

Unit -III: Chapter 3 (Sections 23,24,26)

Unit - IV: Chapter 3( Sections 27,28)

Chapter 4( Section 30)

Unit - V: Chapter 4( Sections 31 to 35)

REFERENCE BOOKS:

1. Aggarwal R.S. (2002), A TEXT BOOK ON TOPOLOGY,

Second Edition, S. Chand & Company Ltd., New Delhi.

2. James Dugundji(1993) ,TOPOLOGY, Universal Book Stall,

Third Indian reprint, New Delhi.

3. Joshi D.K.(2006), INTRODUCTION TO GENERAL TOPOLOGY,

New age International Pvt. Ltd. New Delhi.

4. Klaus Janich (2006), TOPOLOGY, Springer-Verlag,New York .

5. Sharma J.N.(1996), TOPOLOGY, Krishna Prakashan Media (P)

Ltd., 20th Revised Edition, New Delhi.


M.Sc., MATHEMATICS


SEMESTER II

Major Core 8 : COMPLEX ANALYSIS

Total Hours: 90



Objectives:

1. To introduce the students to the fascinating world of complex variable theory.

This is entirely different from analysis of real variable. 2. To introduce concept of harmonic functions and infinite products.

UNIT I: (18 hrs)

Conformality

Arcs and closed curves – Analytic functions in regions – Conformal mapping – Linear

Transformation – The cross ratio – Symmetry – Oriented circles – Elementary conformal

mappings.

UNIT II: (18 hrs)

Complex Integration

Fundamental Theorems in complex Integration – Cauchy’s Integral formula – Local

Properties of analytical Functions – Removable singularities- zeros and poles – local

mapping – maximum principle.

UNIT III: (18 hrs)

Connectivity And Residues

The general form of Cauchy’s Theorem-chains and cycles -simple connectivity-

Homology -General statement of Cauchy’s theorem-Locally exact differentials - the

calculus of residues -Residue theorem - Argument Principle - Evaluation of definite

integrals.

UNIT IV: (18 hrs)

Harmonic functions

The Mean value property - Poisson’s formula -Schwarz’s theorem - reflection principle –

power series expansion - Weierstrass’s theorem - Taylor’s series.

Unit V: (18 hrs)

Infinite Products

Infinite Products of complex numbers-Canonical Products -Entire Functions - Jensen’s

formula - Hadamard’s theorem.

TEXT BOOK

AhlforsL.V.(1979), COMPLEX ANALYSIS, Third Edition, McGraw-Hill Book

Company,New Delhi.

Unit – I: Chapter 3: sections 2,3,4 (4.1,4.2) Unit – II: Chapter 4: sections 1,2& 3 Unit – III: Chapter 4: sections 4 & 5 Unit – IV: Chapter 4: section 6, Chapter 5, section 1 Unit – V: Chapter 5: section 2 (2.2, 2.3) & section 3.

REFERENCE BOOKS:

1. Ablowitz, Athanassios.S, Fokas, Mark.J.(2005), COMPLEX VARIABLES-

INTRODUCTION AND APPLICATION, 2nd edition, , Cambridge

University Press, UK.

2. Churchill, James Ward Brown, Reul .V.( 2004), COMPLEX VARIABLES AND

APPLICATION, 7th edition , McGrawHill,New York.

3. John. M. Howie (2005), COMPLEX ANALYSIS, Springer-Verlag, New York.

4. KassanaH.S.( 2005), COMPLEX VARIABLES THEORY

AND APPLICATION, 2ndedition , Prentice Hall Of India, New Delhi.

5. Theodore (2004),W.Gamelin, COMPLEX ANALYSIS, Springer-Verlag,

New York


M.Sc., MATHEMATICS


SEMESTER II

Major Core 9 : DISCRETE MATHEMATICS

Total Hours: 90



Objectives:

1. To introduce Computability and types of grammars and languages.

2. To explain briefly about finite state machine, discrete numeric functions and

Recurrence relations

Unit I : Computability and formal languages: (18 hrs)

Russell’s paradox and noncomputability – ordered sets – Languages – Phrase structure

grammars – Types of grammars and languages.

Unit II : Finite State Machines:(18 hrs)

Introduction – finite state machines – finite state machines as models of physical

systems-Equivalent machines – Finite state machines as language recognizers – Finite

state language and type – 3 languages.

Unit III : Permutations, Combinations discrete and Conditional Probabilities :(18 hrs)

Introduction – The rules of sum and product permutation – combinations – Generation

of permutations and combinations – Discrete probability-conditional probability.

Unit IV : Discrete Numeric functions and Generating functions:: (18 hrs)

Introduction – Manipulation of numeric functions – Asymptotic behavior of numeric

functions – generating functions – Combinatorial problems.

Unit V : Recurrence relations and recursive algorithms: (18 hrs)

Introduction – recurrence relations – linear recurrence relation with constant coefficients

– Homogeneous solutions – Particular solutions- total solutions – Solution by the

method of generating functions sorting algorithms – Matrix multiplication algorithms.

TEXTBOOK:

Liu C.L.(2002), ELEMENTS OF DISCRETE MATHEMATICS ,2nd Edition , Tata

McGraw Hill, New York.

Unit I: Chapter 2

Unit II: Chapter 7

Unit III: Chapter 3

Unit IV: Chapter 9

Unit V: Chapter 10

REFERENCE BOOKS:

1. Busby Robert C, Kolman Bernard, Ross Sharon Culter, Nadeem –

Ur- Rehman(2006), DISCRETE MATHEMATICAL STRUCTURES,

5th edition, , Prentice Hall of India Private Limited, New Delhi.

2. Edger G.Goodaire, Michel M.Parmenter(2002),

DISCRETE MATHEMATICS WITHGRAPH THEORY,

2ndedition, Prentice hall of India Private Limited, New Delhi.

3. Dr. Gourdu N.G. (2003), DISCRETE MATHEMATICAL STRUCTURES,

1st edition Himalaya Publishing House, Mumbai.

4. Kenneth H. Rosen (2006), DISCRETE MATHEMATICS AND ITS

APPLICATIONS , 11th Reprint , Tata McGraw Hill Company Ltd,

New York .

5. Veerarajan T. (2007), DISCRETE MATHEMATICS WITH

GRAPH THEORY COMBINATORICS, Tata McGraw Hill

Publishing Company Limited,Newyork.


M.Sc., MATHEMATICS


SEMESTER II

Non Major Elective 1 : STATISTICAL METHODS

Total Hours: 75

Hours/week: 5 Code: P08MA2NMT01


Objectives:

To understand the various statistical methods by giving real life examples.

Unit I: (15 hrs)

Measures of Central Tendency, Dispersion Skewness and Kurtosis:

Measures of Central Tendency – Mean – Median – Mode – Measures of Dispersion –

Range – Quartile deviation – Mean deviation – Standard deviation – Skewness –

Kurtosis.

Unit II: (15 hrs)

Correlation and Regression:

Correlation – Karl Pearson’s Co-efficient of correlation –Rank correlation (Correlation of

Bivariate frequency distribution to be excluded) – Regression.

Unit III: (15 hrs)

Binomial and Poisson distributions:

Discrete distribution – Mean,Variance only- Binomial distribution – Poisson distribution

-Mean & variance and simple problems only.

Unit IV: (15 hrs)

Normal distribution:

Continuous distribution – Normal distribution – Mean & variance; Moments, Properties

of Normal distribution.

Unit V: (15 hrs)

Testing of hypothesis for large samples-Test for means:

Test for difference between proportions- Test for difference between standard

deviations (problems only) &anova table.

TEXT BOOKS:

1. Pillai R.S.N and Bagavathi V. (2008), STATISTICS, S.Chand& Co limited,

New Delhi.

2. Vital P.R (2002), MATHEMATICAL STATISTICS, Margham

publications, Chennai.

REFERENCE BOOKS:

1. Dr.S.Arumugam and A.ThangapandiIssac (2004),STATISTICS , New Gamma

publishing house

2. Gupta .S.P (2006) ,STATISTICAL METHODS, Sultan Chand & Sons ,

New Delhi.

3. Navaneetham P.A. (2005),BUSINESS MATHEMATICS AND STATISTICS,

Jai Publishers

4. Sharma J.K, (2006) BUSINESS STATISTICS, Dorling Kindersley, (India)

PvtLtd,Licensees of Pearson Education in South Asia.

5. Vital P.R. (2004), BUSINESS STATISTICS, 2nd edition, Margham

publications, Chennai.


M.Sc., MATHEMATICS


SEMESTER III

Major Core 10 : FUNCTIONAL ANALYSIS

Total Hours: 90



Objectives:

1. To study three structure theorems of Functional Analysis viz., Hahn-Banach theorem,

Open mapping theorem and uniform boundedness theorem.

2. To introduce Hilbert spaces and operator theory and spectral theory of operators on a Hilbert space.

UNIT I: (18 hrs) Metric Spaces

Definition -Open sets - Closed sets –Convergence- completeness and Baire’s theorem -

Continuous mappings - Spaces of continuous functions - Euclidean and Unitary spaces.

UNIT II: (18 hrs) Banach Spaces

The definition and some examples – Continuous linear transformation – The Hahn

Banach theorem.

UNIT III: (18 hrs) Theorems under Banach spaces

The natural imbedding of N into N** - The open mapping theorem – The closed graph

theorem – The Conjugate of an operator.

UNIT IV: (18 hrs) Hilbert Spaces

The definition and some simple properties – Orthogonal complements – Orthonormal

sets – The Conjugate space H*- Adjoint of an operator – Self adjoint, normal and unitary

operators.

UNIT V: (18 hrs)

Spectral Theory:

Projections – Finite dimensional spectral theory matrices- Determinants and the

spectrum of an operator – The spectral theorem.

TEXT BOOK:

George F. Simmons (1969), INTRODUCTION TO TOPOLOGY AND MODERN

ANALYSIS, Mc-Graw-Hill International Edition, Singapore. .

Unit I: Chapter 2

Unit II: Chapter 9 (Sections 46,47,48)

Unit III: Chapter 9 (Sections 49-51)

Unit IV: Chapter 10 (Sections 52-58)

Unit V: Chapter 10 (Section 59) & 11 (Sections 60-62)

REFERENCE BOOKS:

1. Balmohan L. Limaye (1997), FUNCTIONAL ANALYSIS,

New Age International Publications, Second Edition, New Delhi,.

2. ChandrasekaraRao.K (2006), FUNCTIONAL ANALYSIS,

Narosa Publishing House, India.

3. Jain O.P. Ahuja Khalil Ahmed (1995), FUNCTIONAL ANALYSIS,

New age International Publications, New Delhi.

4. Sharma J.N., VasishthaA.R.(1988 ), FUNCTIONAL ANALYSIS,

KrisnaPrakashanMandir, Eighth Edition.

5. ThambanNair.M(2002), FUNCTIONAL ANALYSIS A FIRST COURSE,

Prentice Hall Of India Pvt. Ltd., New Delhi.. Irwin Gracy, FUNCTIONAL

ANALYSIS , Prentice Hall Of India Pvt. Ltd., New Delhi.


M.Sc., MATHEMATICS


SEMESTER III

Major Core 11 : PROBABILITY THEORY

Total Hours: 90



Objectives:

1. To make the students understand about fields, -fields and random variables.

2. To enable the students to learn about expectations, convergence in random variables

and distribution functions.

Unit I: (18 hrs) Fields and σ Fields:

Class of events – Functions and Inverse functions – Random variables – Limits of

random variables.

Unit II : (18 hrs) Probability Space:

Definition of probability – some simple properties – discrete probability space – General

probability space – Induced probability space.

Unit III : (18 hrs) Distribution functions:

Distribution functions of a random variable –Decomposition of distributive functions-

Distributive functions of vector random variables – Correspondence theorem.

Unit IV: (18 hrs) Expectation and Moments:

Definition of Expectation –Properties of expectation – Moments, Inequalities.

Unit V: (18 hrs) Convergence of Random Variables: Convergence in Probability –Convergence almost surely – Convergence in distribution

–Convergence in the rthmean -Convergence theorems for Expectations

TEXT BOOK:

B.R. Bhat (2007), MODERN PROBABILITY THEORY,3rd edition, New Age

International private ltd, New Delhi.

Unit I : Chapter 1 and 2 Omit (1.1&1.2)

Unit II : Chapter 3 (Omit 3.6)

Unit III : Chapter 4

Unit IV : Chapter 5

Unit V : Chapter 6(6.1 to 6.5)

REFERENCE BOOKS:

1. Chandra T.K and Chatterjee D. (2003), A FIRST COURSE IN PROBABILITY ,

2ndEdition,Narosa Publishing House, New Delhi.

2. Kailai Chung FaridAitsahlia(2005), ELEMENTARY PROBABILITY

THEORY WITH STOCHASTIC PROCESSES AND AN INTRODUCTIONTO

MATHEMATICAL FINANCE, 4th edition, Springer Verlag, New York.

3. MarekCapinski and ThomaszZastawniak(2003), PROBABILITY THROUGH

PROBLEMS, Springer Verlag New York Berlin Heidelberg.

4. Dr.A.Santhakumaran(2006), PROBABILITY AND TEST OF HYPOYHESES ,

1st edition, Quality CBT and course material from sonaversity.

1. Shama .T.K(2005), A TEXT BOOK OF PROBABILITY AND THEORITICAL

DISTRIBUTION, Discovery publishing house, New Delhi.


M.Sc., MATHEMATICS


SEMESTER III

Major Core 12 : CLASSICAL MECHANICS

Total Hours: 90



Objective:

To give a detailed knowledge about the mechanical system of particles, applications of

Lagrange’s equations and Hamilton’s equations as well as the theory of Hamilton

Jacobi.

Unit I: Introductory concepts (18 hrs)

The mechanical system -Equations of motion, units- Generalised co-ordinates -Degrees

of freedom , Generalised co-ordinates - configuration space - example– Constraints -

Holonomic , non holonomic and unilateral constrains- example – Virtual Work -Virtual

displacement - principle of virtual work , D’Alembert’s principle, Generalised force,

example – Energy and Momentum.

Unit II: Lagrange’s Equations: (18 hrs)

Derivation of Lagrange’s Equations –examples-Spherical and double pendulum ,

Lagrange multipliers and constraint forces – Integrals of the motion -Ignorable co-

ordinates – example- The kepler’s problem - Routhian function - conservative system -

natural systems Liouville’s System and example.

Unit III: Special applications of Lagrange’s equations: (18 hrs)

Rayleigh’s Dissipation function– Gyroscopic system – velocity dependent

potentials:Hamilton’s Principle -Stationary values of a function - constrained stationary

values , stationary value definite integral - Example - – The brachistochrone problem –

Geodesic path, case of n dependent variables- Hamilton’s Principle, Non-holonomic

systems , Multipliers rule.

Unit IV: Hamilton’s equations (18 hrs)

Hamilton‘s equations-Derivation of Hamilton’s equations-The form of Hamilton

function - Legendre transformation-examples-Other variational principles - Modified

Hamilton’s principle , Principle of least action and example - Phase space.

Unit V: Hamilton Jacobi Theory: (18 hrs)

Hamilton’s principle functions-The canonical integral- Paffian differential forms-The

Hamilton Jacobi equations -Jacobi’s theorem, conservative system and ignorable co-

ordinates and examples – Separability -Liouville’s system - Stackel’s theorem - example.

TEXT BOOK:

Greenwood D.T. (1979),CLASSICAL DYNAMICS, Prentice Hall of India Private

limited,New Delhi.

Unit I : Chapter I

Unit II : Chapter II

Unit III : Chapter III (Sections 3.1, 3.3, 3.4) &Chapter IV (Section 4.1 )

Unit IV : Chapter IV ( Sections 4.2, 4.3, 4.4 )

Unit V : Chapter V

REFERENCE BOOKS:

1 Batchelor G.K. (2005), AN INTRODUCTION TO FLUID DYNAMICS ,

ManasSaikia for foundation Books Pvt Ltd, NewDelhi.

2. DuraiPandianP.,LaxmiDuraiPandian, MuthmizJayapragasam(2003),

MECHANICS, 4th edition S.Chand and Company Ltd, New Delhi.

3. Gupta S.L, Kumar. V, Sharma.H.V(2003), CLASSICAL MECHANICS,

Ninteenth Edition 2001, Reprint 2003, K.K.Mittal for PragathiPrakashan, Meerut.

4. Irving H.Shames (2003), MECHANICS OF FLUIDS ,McGraw Hill

Company Limited, New Delhi.

6. Rana N.C., Joag P.S. (2004), CLASSICAL MECHANICS, Tata McGraw Hill Company Limited ,New Delhi.


M.Sc., MATHEMATICS


SEMESTER III

Major Elective 1 : THEORY OF NUMBERS

Total Hours: 90

Hours/week: 6 Code: P08MA3MET01


Objective:

To provide the students to know about Divisibility, congruences, Legendre

function,Binary quadratic forms and Diophantine Equations.

UNIT I: (18 hrs) Divisibility and Congruences

Introduction – Divisibility – Primes – The Binomial Theorem – Congruences –

Euler’s quotient – Fermat’s Euler’s and Wilson’s Theorems – Solutions of

congruences – The Chinese Remainder theorem.

UNIT II: (18 hrs) Congruences(Continued)

Techniques of numerical calculations – Public key cryptography – Prime power Moduli

– Primitive roots and Power Residues – Congruences of degree two.

UNIT III: (18 hrs) Quadratic Reciprocity

Number theory from an Algebraic Viewpoint – Groups, rings and fields – Quadratic

Residues – The Legendre symbol (a/r) where r is an odd prime – Quadratic Reciprocity

– The Jacobi symbol (P/q) where q is an odd positive integer.

UNIT IV : (18 hrs)

Quadratic Forms &Some functions of Number TheoryBinary Quadratic Forms –

Equivalence and Reduction of Binary Quadratic Forms –

Sums of three squares – Positive Definite Binary Quadratic forms – Greatest integer

Function – Arithmetic Functions – The Mobius Inversion Formula – Recurrence

Functions – Combinatorial number theory.

UNIT V: (18 hrs) Diophantine Equations

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

Triangles – Assorted examples.

TEXT BOOK:

Ivan Niven, Herbert S. Zuckerman ,Hugh L. Montgomery (2006),

AN INTRODUCTION TO THE THEORY OFNUMBERS, Fifth edition, John Wiley

&Sons Private Limited, Singapore.

UNIT I Chapter 1 and Chapter 2: Sections 2.1 to 2.3 UNIT II Chapter 2: Sections 2.4 to 2.9 UNIT III Chapter 2: Sections 2.10, 2.11 and Chapter 3: Sections 3.1 to 3.3 UNIT IV Chapter 3: Sections 3.4 to 3.7 and Chapter 4 UNITV: Chapter 5: Sections 5.1 to 5.4 REFERENCE BOOKS:

1. Andrews,George E. (1989), THE NUMBER THEORY,HindLaw,New Delhi.

2. David M. Burton (2007), ELEMENTARY NUMBER THEORY, Tata Mc_Graw

Hill Publishing Company Limited,New Delhi.

3. Ireland,Kenneth(2005),CLASSICAL INTRODUCTION TO MODERN

NUMBER THEORY,Wiley-Dreamtech,India Private Limited,New Delhi.

4. KumaraveluS. ,SusheelaKumaravelu (2002), ELEMENTS OF

NUMBER THEORY,S.K.V.Publishers,Nagercoil.

5. NadkarniM.G.,Dani J.S.(1999), NUMBER THEORY, Tata Mc_Graw

Hill Publishing Company Limited,New Delhi.


M.Sc., MATHEMATICS


SEMESTER III

MAJOR ELECTIVE 1: AUTOMATA THEORY

Total Hours: 90



OBJECTIVE:

1. To understand the nuances of Automata and Grammar.

2. To enable the students to understand the applications of these techniques in

computerscience.

UNIT I: FINITE AUTOMATA AND REGULAR EXPRESSIONS (18HRS)

Definitions and examples - Deterministic and Nondeterministic finite Automata -

- Finite Automata with ∈ − moves.

UNIT II: CONTEXT FREE GRAMMAR (18HRS)

Regular expressions and their relationship with automation - Grammar - Ambiguous and

unambiguous grammars - Derivation trees – Chomsky Normal form.

UNIT III: PUSHDOWN AUTOMATON (18HRS)

Pushdown Automaton - Definition and examples - Relation with Context free languages.

UNIT IV: FINITE AUTOMATA AND LEXICAL ANALYSIS (18HRS)

Role of a lexical analyzer - Minimizing the number of states of a DFA - Implementation of a

lexical analyzer.

UNIT V: BASIC PARSING TECHNIQUES

(18HRS)

Parsers - Bottom up Parsers - Shift reduce - operator precedence - Top down Parsers -

Recursive descent - Predictive parsers.

TEXTBOOKS:

1. John E. Hopcroft and Jeffrey D. Ullman, Introduction to Automata theory, Languages

and Computations, Narosa Publishing House, Chennai,2000.

UNIT I: Chapter 2:Sections2.1-2.4

UNIT II: Chapter 2, Section 2.5, Chapter 4, Sections 4.1-4.3, 4.5, 4.6

UNIT III: Chapter 5: Section 5.2& 5.3

2. A.V. Aho and Jeffrey D. Ullman, Principles of Compiler Design, Narosa Publishing

House, Chennai,2002.

UNIT IV: Chapter 3: Section 3.1-3.8

UNIT V: Chapter 5: Section 5.1-5.5

REFERENCE BOOKS:

1. Harry R. Lewis and Christos H. Papadimitriou, Elements of the Theory of Computation,

Second Edition, Prentice Hall,1997.

2. A.V. Aho, Monica S. Lam, R. Sethi, J.D. Ullman, Compilers: Principles, Techniques

and Tools, Second Edition, Addison-Wesley,2007.


M.Sc., MATHEMATICS


SEMESTER III

Non Major Elective 2 : QUANTITATIVE TECHNIQUES

Total Hours: 75

Hours/week: 5 Code: P08MA3NMT02


Objective:

To give detailed knowledge aboutthe basics of Inventory models ,project scheduling

and Decision making.

Unit I: (15 hrs) PERT/CPM Methods

Network Scheduling – PERT/CPM. – Network and Basic Components – Rules of

Network construction – Time Calculations in Networks – Critical Path Method – PERT.

Unit II: (15 hrs) Inventory Management

Types of inventory – Need for Inventory control-Economic Order Quantity – E.O.Q.

with shortages– safety stock and R.O.L.

Unit III: (15 hrs)

Decision analysis

Decision under Risk – Expected Money Value Criterion – Decision Trees – Decision

under uncertainity – Minimax Criterion

Unit IV: (15 hrs)

GameTheory

Theory of games – Pure and Mixed strategies- Principles of Dominance – Algebraic

Method -Graphical Method.

Unit V: (15 hrs) Replacement problems

Introduction- Replacement of equipments that deteriorates gradually- value of money

does not change with time- value of money change with time.

NOTE:

No theory – only problems.

TEXT BOOK:

KantiSwaru P. Manmohan and Gupta(2006), OPERATIONS RESEARCH,

Sultan Chand Son Pvt. Limited.

Unit I: Chapter 21 (Sections 21.1-21.6)

Unit II: Chapter 19 (Sections19.1-19.7 & 19.8)

UnitIII: Chapter 16 (Sections 16.1-16.7)

Unit IV:Chapter 17 (Sections17.1-17.7)

Unit V: Chapter 18 (Sections 18.1-18.2)

REFERENCE BOOKS:

1. Gupta P.K., Hira S. (2005), OPERATION RESEARCH, S Chand & Co.

Limited New Delhi.

2. MariappanP.(2001), OPERATION RESEARCH METHODS &

APPLICATIONS, New Century Book House Private Limited..

3. PanneerSevvam (2003), OPERATION RESEARCH, Prentice Hall of India

Private Limited, New Delhi.

4. Sharma J.K.(2007), OPERATION RESEARCH THEORY & APPLICATIONS,

Macmillan India Limited,Chennai.

5. TahaHamadyA.( 2002), OPERATIONS RESEARCH AN INTRODUCTION,

Pearson Education Publishing Limited,New Delhi.


M.Sc., MATHEMATICS


SEMESTER IV

Major Core 13 : OPERATIONS RESEARCH

Total Hours: 90



Objectives:

1. To discuss the methods of solving Integer Programming Problems and

NLPP programming algorithms.

2. To give detailed knowledge on Dynamic programming, Decision Analysis

and Inventory models.

Unit I: (18 hrs)

Integer Programming

Introduction to integer programming – some applications of Integer Programming –

methods of Integer Programming – branch and bound method – zero-one implicit

enumeration method.

Unit II: (18 hrs)

Dynamic Programming

Elements of the DP model – the capital budgeting problem – more on the definition of

the state – examples of DP models and computations – Problem of dimensionality in DP

– solution of linear programming problems by DP.

Unit III: (18 hrs)

Decision Theory and Games

Decisions under risk – decision trees – decisions under uncertainty – game theory.

Unit IV: (18 hrs)

Inventory Models

Generalized inventory model – types of inventory models – deterministic models –

probabilistic models.

Unit V: (18 hrs)

Non linear Programming Algorithms

Unconstrained nonlinear algorithms – constrained linear algorithms.

TEXT BOOK:

Hamdy. A. Taha (2000), OPERATIONS RESEARCH – AN INTRODUCTION ,

4thEdition,Pearson Education Publishing Limited, New Delhi.

Unit - I : Chapter 8 (Secs. 8.1 to 8.5)

Unit - II: Chapter 9 (Secs. 9.1 to 9.5)

Unit - III: Chapter 11( Secs. 11.1 to 11.4)

Unit - IV : Chapter 13 (Sec.13.1 to 13.4)

Unit – V : Chapter 19 (Sec.19.1 to 19.2)

REFERENCE BOOKS:

1. Gupta P.K., Hira S. (2005), OPERATION RESEARCH, S Chand &

Co.Limited, New Delhi.

2. MariappanP.(2001), OPERATION RESEARCH METHODS & APPLICATIONS

New Century Book House Private Limited..

3. PanneerSevvam (2003), OPERATION RESEARCH, Prentice Hall of India

Private Limited, New Delhi.

4. Sharma J.K.(2007), OPERATION RESEARCH THEORY & APPLICATIONS,

Macmillan India Limited,Chennai.

5. TahaHamadyA.( 2002), OPERATIONS RESEARCH AN INTRODUCTION,

Pearson Education Publishing Limited,New Delhi.


M.Sc., MATHEMATICS


SEMESTER IV

Major Core 14 : METHODS OF APPLIED MATHEMATICS

Total Hours: 105



Unit I : (21 hrs)

Calculus Of Variations And Applications

Maxima and minima, the simplest case, Illustrative examples, Natural boundary

conditions and transition conditions, the Variational notation, the more general case.

Unit II :(21 hrs) Calculus Of Variations And Applications(continued)

Constraints and Lagrange multipliers, Variable and points, Sturm – Liouville problems,

the Rayleigh – Ritz method, a semi direct method.

Unit III : (21 hrs) Integral Equations

Introduction, relations between differential and integral equations, the green’s function,

fredholm equation with separable kernels, Illustrative examples.

Unit IV: (21 hrs) Integral Equations(continued)

Iterative methods for solving equations of the second kind, the Newmann series, and

special devices.

Unit V: (21 hrs) Fourier Transform

Fourier’s integral theorem – Fourier transforms- cosine transforms – Sine transforms –

Transforms of derivatives – Transforms of some special functions – The Convolution

Theorem – Parseval’s theorem for Cosine and Sine transform.

TEXT BOOKS:

1. Francis B. Hilde brand(1972), METHODS OF APPLIED MATHEMATICS ,

2ndEdition, Prentice –Hall of India Private Limited ,New Delhi.

Unit I :Chapter 2 : Sections 2.1 – 2.6

Unit II : Chapter 2 : Sections 2.7 – 2.9 and 2.19, 2.20

Unit III :Chapter 3 – Sections 3.1 – 3.3, 3.6 and 3.7

Unit IV : Chapter 3 – Sections 3.8 – 3.10 and 3.13

2. Ian N. Sneddon (1974),THE USE OF INTEGRAL TRANSFORMS,

Tata Mc-Graw Hill Publishing CompanyLimited,New Delhi.

Unit V: Chapter 2 – Sections 2.2 – 2.7, 2.9 and 2.10

REFERENCE BOOKS:

1. Doss H.K. (2000), ADVANCED ENGINEERING MATHEMATICS,

S. Chand & Company Ltd. New Delhi.

2. Parashar B.P. (1992),DIFFERENTIAL AND INTEGRAL EQUATIONS,

CBS Publishers,New Delhi.

3. Raisinghania M.D. (2007),INTEGRAL EQUATIONS AND BOUNDARY

VALUE PROBLEMS, S. Chand & Company Ltd. New Delhi.

4. Veerarajan T. (2003), ENGINEERING MATHEMATICS, Tata Mc- Graw

Hill Publishers, New Delhi.

5. Dr.Venkataraman M.K. (2001), HIGHER MATHEMATICS FOR

ENGINEERING ANDSCIENCE,The National Publishers,Cheenai.


M.Sc., MATHEMATICS


SEMESTER IV

MAJOR ELECTIVE 2: CODING THEORY

Total Hours: 90



GENERAL OBJECTIVE:

To give an introduction to basic concepts and techniques of coding theory such as,

Double Error-Correcting B.C.H. code , cyclic codes, The Group of a code, Quadratic

residue codes and Bose-Chaudhuri- Hocquenghem codes.

UNIT I - INTRODUCTORY CONCEPTS (18HRS)

Introduction - Basic Definitions - Weight,Minimum Weight and Maximum-Likelihood

Decoding - Syndrome Decoding - Perfect Codes, Hamming Codes, Sphere-

Packing Bound -General Facts- Self-Dual Codes, the GolayCodes.

UNIT II - DOUBLE ERROR-CORRECTING B.C.H. CODE AND

FINITE FIELDS POLYNOMIALS (18 HRS)

A Finite Field of 16 Elements - The Double-Error-Correcting Bose-Chaudhuri-

Hocquenhem (B.C.H.) Code Problems - Groups - The Structure of a Finite Field -

Minimal Polynomials - Factoring xn−1 .

UNIT III-CYCLICCODES (18 HRS)

The Origin and Definition of Cyclic Codes - How to Find Cyclic Codes:

The Generator Polynomial - The Generator Polynomial of the Dual Code - Idempotents

and Minimal Ideals for Binary Cyclic CodesProblems.

UNIT IV - THE GROUP OF A CODE AND QUADRATIC RESIDUE

(Q.R.) CODES (18 HRS)

Some Cyclic Codes WeKnow - Permutation Groups - The Group of a Code -

Definition of Quadratic Residue (Q.R.) Codes - Extended Q. R. Codes, Square

Root Bound and Groups of Q.R. Codes - PermutationDecoding.

UNIT V - BOSE-CHAUDHURI-HOCQUENGHEM (B.C.H.) CODES

(18 HRS)

Cyclic Codes Given in Termsof Roots - Vandermonde Determinants - Definition and

Properties of B.C.H. Codes - Preliminary Concepts and a Theorem on Weightsin

Homogeneous Codes - The MacWilliams Equations - PlessPower Moments - Gleason

Polynomials.

TEXT BOOK

Vera Pless, Introduction to the Theory of Error-Correcting Codes, John Wiley &

Sons, New York, 1982.

Unit I: Chapters 1 and 2

Unit II: Chapters 3 and 4

Unit III: Chapter 5

Unit IV: Chapter 6

Unit V: Chapter 7 and 8

REFERENCE BOOKS :

1) I.F. Blake and R.C. Mullin, Introduction to Algebraic and Combinatorial

Coding Theory, Academic Press, INC, New York,1977.

2) F.J. MacWilliams and N.J.A. Sloane, The Theory of Error-Correcting

Codes, Vols. I and II, North-Holland, Amsterdam,1977.

3) Ling, S. and Xing, C.: "Coding Theory: A First Course", Cambridge University Press

4) Roth, R. M.: “Introduction to Coding Theory”, Cambridge University Press


M.Sc., MATHEMATICS


SEMESTER IV

Major Elective 2: THEORY OF FUZZY SETS

Total Hours: 90



Objective:

To introduce the concept of fuzzy sets and import the features of fuzzy in various

representations like Relations, Numbers & Decision Making.

UNIT I: (18 hrs)

Fuzzy sets and its operations

Fuzzy sets - Basic concepts – addition properties of α cuts – Extension principle for

fuzzy sets – types of operations – fuzzy complements .

UNIT II: (18 hrs)

Fuzzy Arithmetic

Fuzzy numbers – Linguistic variable – Arithmetic operations on intervals – Arithmetic

operation on fuzzy numbers .

UNIT III: (18 hrs)

Possibility theory

Fuzzy measures - Evidence Theory – Possibility Theory – Fuzzy Sets and Possibility

Theory – Possibility Theory vs Probability Theory.

UNIT IV: (18 hrs)

Fuzzy Decision Making

Individual Decision Making -Multi person Decision Making - Multi Criteria Decision

Making – Multi Stage Decision Making – Fuzzy Ranking Methods.

UNIT V : (18 hrs)

Fuzzy LPP and Transportation Problem

Fuzzy Linear Programming Problem – Fuzzy Transportation Problem.

TEXTBOOKS:

For Units 1 to 5:

1. George J.Klir / Bo Yaun (1995), FUZZY SETS AND FUZZY LOGIC ,THEORY AND

APPLICATIONS , Prentice Hall of India Pvt. Ltd, New Delhi.

Unit I: Chapter 1 (Sec 1.4), Chapter 2 (Sec 2.1 ,2.3 ) & Chapter 3 (Sec 3.1, 3.2)

Unit II: Chapter 4 (Sec 4.1 – 4.4 )

Unit III : Chapter 7 ( Sec 7.1 - 7.5 )

Unit IV : Chapter 15 ( Sec 15.1 – 15.6 )

Unit V:Chapter15(Sec.15.7)

For Fuzzy Transportation problem

2. Zimmermann. H.J..(2006), FUZZY SET THEORY AND ITS APPLICATIONS,

Springer International Edition, New York.

Chapter 15: Sec 15.3.2

REFERENCE BOOKS:

1. Kaufmann.A (1994), INTRODUCTION TO THE THEORY OF

FUZZY SETS , Academic press ,Newyork .

2. Klir&Bouyal(2003), FUZZY SET ,UNCERTAINITY AND INFORMATION,

Prentice Hall Of India .

3. Klir J. & Bo Yuan(2000),UNCERTAINITY AND FUZZY LOGIC ,

Wiley western publishers

4. Zimmermann. H.J..(2006), FUZZY SET THEORY AND ITS APPLICATIONS,

Springer International Edition, New York.


M.Sc., MATHEMATICS


SEMESTER IV

Major Elective 3 : STOCHASTIC PROCESSES

Total Hours: 90



Objectives:

1. To introduce the concept of discrete and continuous time Marko chains and their properties to study the renewal process and related results and their applications.

2. To learn more about several queuing models and their performance measures.

Unit I :Stochastic Processes (18 hrs)

Some notions - Specifications of stochastic processes- Stationary processes- Markov Chains –

Definition and examples-Higher transition probabilities-Generalization of

Independent Bernoulli trials

Unit II :Markov chain (18 hrs)

Sequences of chain –Dependent trails - Classification of states and chains - determination of

higher transition probabilities- Stability of Markov system.

Unit III Markov processes with discrete state space (18 hrs)

Poisson process and their extensions - Poisson process and related distributions- Generalization

of Poisson process- Birth and death process- - Markov processes with discrete state [

continuous time Markov chains].

Unit IV: Renewal Processes and theory (18 hrs)

Renewal Processes - Renewal Processes with continuous time – Renewal equation –Wald’s

equation: stopping time- Renewal theorems.

Unit V: Stochastic processes in queuing (18 hrs)

Queueing process systems: General concepts- The queueing model M/M/I: Steady state

behavior – Transient behavior of M/M/I model

TEXT BOOK

Medhi J. (1994), STOCHASTIC PROCESSES, second edition, Wiley eastern ltd

New Delhi

UnitI : Chapter 2: 2.1 to 2.3

Chapter 3: 3.1 to 3.2

UnitII: Chapter 3: 3.3 to 3.6 & 3.8,

Unit III: Chapter 4: 4.1 to 4.5

Unit IV: Chapter 6: 6.1 to 6.5

Unit V: Chapter 10: 10.1 to 10.3.

REFERENCE BOOKS:

1. Basu A. K(2003),INTRODUCTION TO STOCHASTIC PROCESS

Narosa Publishing House, New Delhi.

2. Richard Bron Son, GovindasamiNaadimuthu(2004),

OPERATION RESEARCH Second edition , Tata Mc. Graw Hill

Publishing Company Ltd.,New Delhi.

3. Samuel Karlin& Howard M.Taylor(1981.), A FIRST COURSE IN STOCHASTIC

PROCESSES, Academic Press.

4. Samuel Karlin& Howard M.Taylor(1981), A SECOND COURSE IN STOCHASTIC

PROCESSES, Academic Press..

5. Sheldon Ross M.(1996), STOCHASTIC PROCESS, second edition,

John WilkeysSon INC.

6. Srinivasan SK .&Medhi J.(1978), STOCHASTIC PROCESS, Second edition

Tata McGraw -Hill Publishing Company ltd.


M.Sc., MATHEMATICS


SEMESTER IV

MAJOR ELECTIVE 3 :DIFFERENTIAL GEOMETRY

Total Hours: 90



OBJECTIVE:

1. To introduce the notion of surfaces and theirproperties. 2. ToexplainthevariousintrinsicconceptsofDifferentialGeometry.

3. To understand the theory of DifferentialGeometry on surfaces

UNITI: SPACECURVES (18 hrs)

Definition of a space curve - Arc length - tangent - normal and binormal - curvature and

torsion - contact between curves and surfaces- tangent surface- involutes and evolutes-

Intrinsic equations - Fundamental Existence Theorem for space curves- Helics.

UNITII: INTRINSIC PROPERTIES OF ASURFACE (18 hrs)

Definition of a surface - curves on a surface - Surface of revolution - Helicoids - Metric-

Direction coefficients - families of curves- Isometric correspondence- Intrinsic properties.

UNITIII: GEODESICS (18 hrs)

Geodesics - Canonical geodesic equations - Normal property of geodesics- Existence

Theorems - Geodesic parallels - Geodesics curvature- Gauss- Bonnet Theorem - Gaussian

curvature- surface of constant curvature.

UNITIV: NON INTRINSIC PROPERTIES OF ASURFACE (18 hrs)

The second fundamental form- Principal curvature - Lines of curvature - Developable –

Developable associated with space curves and with curves on surface - Minimal surfaces

- Ruled surfaces.

UNITV: DIFFERENTIAL GEOMETRY OFSURFACES (18 hrs)

Compact surfaces whose points are umbilics- Hilbert's lemma - Compact surface of

constant curvature - Complete surface and their characterization - Hilbert's Theorem -

Conjugate points on geodesics.

TEXT BOOK

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

Impression) New Delhi 2002.

UNIT–I : Chapter I : Sections 1 to 9.

UNIT–II : Chapter II: Sections 1 to 9.

UNIT–III : Chapter II: Sections 10 to 18.

UNIT–IV : Chapter III: Sections 1 to 8.

UNIT–V : Chapter IV : Sections 1 to8.

REFERENCES

1. Struik, D.T. Lectures on Classical Differential Geometry, Addison - Wesley, Mass.

1950.

2. WihelmKlingenberg: A course in Differential Geometry, Graduate Texts in

Mathematics, Springer Verlag,1978.

3. J.A. Thorpe Elementary topics in Differential Geometry, Under - graduate Texts in

Mathematics, Springer - Verlag1979.

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