K.N.GOVERNMENT ARTS COLLEGE FOR WOMEN ( AUTONOMOUS). THANJAVUR-07

M.Sc., MATHEMATICS- CBCS

(Course Structure for students admitted from 2011 onwards)

Semester Code Course Course Title Hrs.

Credit

Int Ext Total

1. 11KP1M01 Core Course–I (CC) Linear Algebra 6 5 25 75 10011KP1M02 Core Course–II (CC) Real Analysis 6 5 25 75 10011KP1M03 Core Course–III (CC) ODE & PDE 6 4 25 75 10011KP1M04 Core Course–IV (CC) Graph Theory 6 5 25 75 10011KP1M05 Core Course–V (CC) Operations

Research 6 4 25 75 100

Total 30 23 125 375 500II 11KP2M06 Core Course–VI (CC) Algebra 6 4 25 75 100

11KP2M07 Core Course–VII (CC)

Complex Analysis 6 5 25 75 100

11KP2M08 Core Course–VIII (CC)

Functional Analysis

6 5 25 75 100

11KP2MELMIP

Elective - I Numerical Analysis with Practical using C

6 5 40 60 100

11KP2MEL01

OECI (For non Maths Students) Numerical Methods & OR

6 5 25 75 100

Total 30 24 140 360 500III 11KP3M09 Core Course–IX (CC) Topology 6 4 25 75 100

11KP3M10 Core Course– X (CC) Integral Equations and Transforms

6 4 25 75 100

11KP3M11 Core Course–XI (CC) Classical Dynamics

7 5 25 75 100

11KP3MELM2

Elective - II Fuzzy Sets and their Applications

6 5 25 75 100

11KP3MEL02

OECII (For non Maths Students) Optimization Techniques

5 5 25 75 100

Total 30 23 125 375 500IV 11KP4M12 Core Course–XII

(CC)Measure and Integration

6 5 25 75 100

11KP4M13 Core Course–XIII (CC)

Discrete Mathematics

6 4 25 75 100

11KP4M14 Core Course–XIV (CC)

Stochastic Processes

6 5 25 75 100

Project Work 12 6 40 60 100Total 30 20 115 285 400

SEMESTER I CORE COURSE-I LINEAR ALGEBRA

UNIT-I

System of linear equations – matrices and elementary row

operations – Row reduced Echelon matrices – Matrix multiplication –

Invertible matrices – Vector spaces – Subspaces – Basis and dimension.

Chapter 1 – Sections 1.2. to 1.6 & Chapter 2 –Section 2.1 to 2.3

UNIT – IILinear transformations – The Algebra of linear transformation –

Isomorphism – Representation of linear transformation by matrices –

Linear functional – The double dual – The transpose of linear

Transformation. Chapter 3

UNIT- III

Polynomials –Algebras – The Algebra of polynomials –

Lagrange Interpolation – Polynomials Ideals – The prime factorization of a

Polynomial. Chapter 4

UNIT – IV

Determinants – Commutative rings – Determinant functions –

Permutations and the uniqueness of determinants –Additional property of

Determinants. Chapter 5 Section 5.1-5.4

UNIT – V

Elementary canonical forms – Introduction – Characteristic

values – Annihilating polynomials - Invariant subspaces – Simultaneous

Inst.Hour 6

Credit 5

Code 11KP1MO1

Triangulation and simultaneous diagonalisation

Chapter6 Section 6.1-6.5

TEXT BOOK:

Kenneth Hoffman and Ray Kunze, Linear Algebra, Second Edition, Prentice

– Hall of India Private Limited New Delhi:1975.

REFFERENCE(S):

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

1975.

2. I.S.Luther and I.B.S.Passi , Algebra, Vol I-Groups , Vol.II- Rings,

Narosa Publishing House (Vol.I-1996,Vol.II-1999)

3. N.Jacobson, Basic Algebra, Vol.I& II.Freeman, 1980 Hindustan

publishing company.

SEMESTER I

CORE COURSE-II REAL ANALYSIS

UNIT-I:

Basic Topology – Metric spaces, Compact sets, Weierstrass theorem perfect sets, the

Cantor sets and connected sets.

Chapter -2

UNIT-II:

The Riemann-Stieltjes Integral- Definition and existence of the integral, Properties of the

Integral, Change of variables- Integration and Differentiation, The fundamental theorem of

calculus – Integration by parts – Integration of vectors – Valued function and Rectifiable

curves.

Chapter -6

UNIT-III:

Sequences and series of functions –Discussion of main problem, Uniform convergence

,uniform convergence and continuity, uniform convergence and integration, uniform

convergence and differentiation – Equicontinuous families of function – the Stone

Weierstrass theorem.

Chapter -7

UNIT-IV:

Multivariable Differential Calculus – Directional derivative – Directional derivatives and

continuity Total derivative – the total derivative expressed in terms of partial derivatives –

an application to complex valued functions, Matrix of linear function, the Jacobian matrix

the chain rule- the matrix form of the chain rule.

Chapter -12 section 12.1 to 12.10 ( Book 2)

UNIT-V:

Mean-value theorem for differential functions, A Sufficient condition for differentiability –

a sufficient conditions for equality of mixed partial derivatives, Taylor’s formula for

functions.

Chapter -12 section 12.11 to 12.14

Inst.Hour 6

Credit 5

Code 11KP1MO2

TEXT BOOKS:

1. W.Rudin, Principles of mathematical Analysis, IIIEd.,1976, McGrawHillBookCo.

(Chapter 2(2.15 to 2.47), Chapter 6,7(complete))

2. Tom.M.Apostal, Mathematical Analysis – IIEd Narosa Publishing House-1974.

(Chapter 12(complete))

REFERENCE BOOKS:

1. A.J. White, Real Analysis: An Introduction, Addison Wesley Publishing Co., Inc 1968.

SEMESTER I

CORE COURSE-III

ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS

UNIT-I:

The general solution of the homogeneous equation – the use of known solution to find

another – the method of variation of parameter – power series solution.

Chapter 3: sections 15, 16, 19, and Chapter 5: Sections 26 ( Book1)

UNIT-II:

Regular singular points – Gauss’s hypergeometric equation – the point at infinity –

Legendre polynomial – Bessel functions – Properties of Legendre polynomials – Bessel

Functions.

Chapter 5: sections 29 to 32 and Chapter8 : Sections 44 to 47 ( Book1)

UNIT-III:

First order partial differential equations-Paffian differential equations – Compactability

Systems – Charpit’s Method.

Chapter 1: section 1.1 to 1.7( Book2)

UNIT-IV:

Jacobi’s Method Integral Surface through a given curve – Quasilinear equaions –

Nonlinear equations.

Chapter 1 : section 1.8 to 1.11 ( Book2)

UNIT-V:

Second order Partial Differential Equation – General Solutions of second order partial

differential equation – Classification – One dimensional wave equation – Laplace equation

– Heat Conduction problem – Duhamel’s principle

Chapter 2 : section 2.1 to 2.3 .3 , 2.3.5 , 2.4 to 2.4.1 to 2.4.11 , 2.5 , 2.6 ( Book2)

TEXT BOOKS:

1. G.F.Simmons, Equation with application and historical notes, TMH,

New Delhi,1984

2. T. Amarnath, An elementary course in partial differential equation,

Narosa,1999

Inst.Hour 6

Credit 4

Code 11KP1MO3

SEMESTER I

CORE COURSE IV

GRAPH THEORY

UNIT-I:

Basic results and Directed graphs.

UNIT-II:

Connectivity

UNIT-III:

Trees & Independent sets and Matchings.

UNIT-IV:

Eulerian and Hamiltonian graphs.

UNIT-V:

Graph colorings and Planarity.

TEXT BOOK:

R.Balakrishnan and K.Renganathan, A text book of Graph theory, Springer – Verlag, New

York (2000)

Chapter I to VII and sections 8.0 to 8.4 of chapter VIII,

Omitting sections 5.4,5.5,6.3,6.4,7.4,7.5,7.6 and 7.7.

REFERENCE BOOKS:

1. S.A.Choudum, A First Course in Graph Theory, Mac millan India

Limited, 1987.

2.R.J.Wilson & J.J.Watkins, Graphs: An Inroductory Approach, John Wiley & Sons, 1989.

Inst.Hour 6

Credit 5

Code 11KP1MO4

SEMESTER I

CORE COURSE V

OPERATIONS RESEARCH

UNIT-I:

Methods of Integer Programming, Cutting plane Algorithms, Branch and Bound Method.

(Chapter 8 – Integer Programming, Sections 8.2 to 8.4)

UNIT - II:

Dynamic (Multistage) Programming – Elements of the DP model – The Capital Budgeting

Example, More on the Definition of the State, Examples of DP models and Computations,

(Chapter 9 – Sections 9.1 to 9.3)

UNIT-III:

Decision theory and Games – Decisions under Risk – Decision Trees – Decisions under

uncertainity – Game Theory.

(Chapter 11 : Sections 11.1 to 11.4)

UNIT-IV:

Inventory models – A Generalized Inventory model – Types of Inventory Models –

Deterministic Models.

(Chapter 13 : Sections 13.1 to 13.3)

UNIT-V:

Non-linear Programming Algorithm – Unconstrained Non-linear Algorithms – Constrained

Non-linear Algorithms.

(Chapter 19 : Sections 19.1 and 19.2.4)

TEXT BOOK : Operations research by Hamdy A.Taha (Third Edition)

REFERENCE :

1. Prem Kumar Gupta & D.S.Hira, Operations Research : An Introduction, S.Chand and

Co., Ltd., New Delhi.

2. S.S.Rao, Optimization Theory and Applications, Wiley Eastern Limited, New Delhi.

Inst.Hour 6

Credit 4

Code 11KP1MO5

SEMESTER II

CORE COURSE – VI ALGEBRA

UNIT-I-GROUP THEORY

A Counting Principle – Normal subgroups and Quotient groups – Homomorphism –

Cayley’s theorem – Permutation groups – Another counting principle – Sylow’s theorem.

Chapter 2: section 2.5, 2.6, 2.7, 2.9, 2.10, 2.11, 2.12

UNIT-II-RING THEORY

Homomorphism of rings – More ideals and Quotient rings – Polynomial rings –

Polynomials over the rational field – Polynomials over commutative rings

Chapter 3: section 3.3, 3.4, 3.5, 3.9, 3.10, 3.11

UNIT-III-MODULES

Inner Product Spaces – Orthogonal complement – Orthogonal Basis – Left module over a

ring - Sub module – Quotient Module – Cyclic Module – Structure theorem for finitely

generated Modules over Euclidean Rings.

Chapter 4 section 4.4, 4.5

UNIT-IV-FIELDS

Extension fields – Roots of Polynomials – More about roots – The elements of Galois

Theory. Finite fields.

Chapter 5 : section 5.1, 5.3, 5.5, 5.6 & Chapter 7 7.1

UNIT-V-TRANSFORMATIONS

Triangular Form – Hermitian, Unitary and Normal Transformations.

Chapter 6 : 6.4 & 6.10

TEXT BOOK : “Topics in Algebra” by I.N.Herstein – Second Edition – Wiley Eastern

Limited.

REFERENCE

1. Modern Algebra – Surjeet Singh Qasi Zameeruddin VIKAS Publishing House Pvt.Ltd.,

2. A First Course in Abstract Algebra – John.B.Fraleign - Addison – Wesley Publishing

Company.

Inst.Hour 6

Credit 4

Code 11KP2MO6

SEMESTER II

CORE COURSE - VII COMPLEX ANALYSIS

UNIT-I

Arcs & closed curves – Analytic functions in regions – Conformal mapping – Length and

area - Line integrals – Rectifiable arcs – Line integrals as functions of arcs – Cauchy’s

Theorem for a Rectangle – Cauchy’s Theorem in a disk.

Chapter – III : Sec 2.1 to 2.4 Chapter - IV : Sec 1.1 to 1.5

UNIT-II

Cauchy’s Integral Formula – The Index of a point with respect to a Closed Curve - The

integral formula – Higher Derivatives – Morera’s theorem – Liouville’s theorem -

Cauchy’s estimates – Fundamental theorem of algebra.

Chapter-IV : Sections 2.1 to 2.3

UNIT-III

Local properties of analytical functions – Removable singularities – Taylor’s theorem

Zeros and poles – Meromorphic functions – Essential singularities – The Local Mapping

Theorem – The Maximum Principles.

Chapter –IV : Sec 3.1 – 3.4

UNIT-IV

The General form of Cauchy’s theorem – Chains and Cycles – Simply connected sets –

Homology – The general statement of Cauchy;s theorem and it’s proof – Locally exact

differentials – Multiply connected Evaluation of Definite Integrals.

Chapter 4 : Sec 4.1 to 4.7 & 5.1 to 5.3

UNIT-V

Harmonic functions - Basic properties – Polar form mean value property – Poisson’s

formula – Schwartz’s Theorem – Reflection Principle – Weierstrass Theorem – The

Taylor’s series – The Laurent series.

Chapter 4 : Sec 6.1 to 6.5 & Chapter 5 : Sec 1.1 to 1.3

TEXT BOOK

L.V.Ahlfors – Complex Analysis – Third Edition Mc Graw Hill International 1979.

REFERENCES :

1. SergeLang, Complex Analysis, Addison Wesley, 1977.

2. S.Ponnusamy, Foundations of Complex Analysis, Narosa Publishing House, 1977.

Inst.Hour 6

Credit 5

Code 11KP2MO7

SEMESTER II

CORE COURSE – VIII FUNCTIONAL ANALYSIS

UNIT I

Algebraic Systems: Groups – Rings – The structure of rings -Linearspaces – The

dimension of a linear space – Linear Transformations – Algebras – Banach Spaces: The

definition and some examples

Chapter 8 & Chapter 9: Section 46

UNIT II

Continuous linear transformations – The Hahn-Banach theorem – The natural imbedding of

N in N** - The open mapping theorem – The conjugate of an operator.

Chapters 9 section : 47 to 51

UNIT III

Hilbert Spaces: The definition and some simple properties – Orthogonal complements –

Orthonormal sets – The conjugate space H* - The adjoint of an operator – Self-adjoint

operators – Normal and unitary operators – Projections.

Chapter 10

UNIT IV

Finite – Dimensional Spectral Theory: Matrices – Determinants and the spectrum of an

operator –The spectral theorem – A survey of the situation.

Chapter 11

UNIT V

General Preliminaries on Banach Algebras: The definition and some examples – Regular

and singular elements – Topological divisors of zero – The spectrum – The formula for the

spectral radius – The radical and semi – simplicity.

Chapter 12

Inst.Hour 6

Credit 5

Code 11KP2MO8

TEXT BOOK(S):

Introduction to Topology and Modern Analysis, G.F.Simmons, McGraw-Hill International

Ed.2004.

REFERENCE(S)

[1] Walter Rudin, Functional Analysis, TMH Edition,1974.

[2] B.V. Limaye, Functional Analysis , Wiley Eastern Limited, Bombay, Second

print,1985.

[3] K.Yosida, Functional Analysis, Springer – Verlag, 1974.

[4] Laurent Schwarz, Functional Analysis, Courant Institute of Mathematical Sciences,

New York University,1964.

SEMESTER II

ELECTIVE COURSE-I

NUMERICAL ANALYSIS WITH PRACTICALS USING C

1. False position method

2. Fixed point iteration

3. Newton-Raphson method

4. Lagrange Interpolation

5. Newton’s Forward and Backward Difference Formula

6. Gauss Elimination Method

7. Gauss Jordan Method

8. Jacobi’s method

9. Gauss Seidal Method

10.Trapezoidal Rule

11.Simpson’s 1/3 Rule

12. Euler’s Method

13.Runge-Kutta Method of order second and fourth

14.Predictor-Corrector Method

Inst.Hour 6

Credit 5

Code 11KP2MELMIP

15.Payroll problem

16.Electricity Bill

17.Marks Statement

18.Standard deviation

19.Correlation Coefficient

20.Method of least squares(straight line)

Reference(s):

1. Numerical method for scientific and Engineering computation by

M.K.Jain S.R.K.Iyengar and R.K.Jain, New age international

publishers.

2. Introductory methods of Numerical Analysis by S.S.Sastry-

Prentice hall of India Pvt.Ltd.,

SEMESTER II

OEC1 FOR NON-MATHS STUDENTS

NUMERICAL METHODS AND OPERATIONS

RESEARCH

UNIT-I:

Solution of Algebraic and Transcedental Equations – Bisection Method – The Iteration

Method. Method of False position – Newton Raphson Method.

(Chapter 2: Sections:2.1,2.2,2.3,2.4,2.5.,)

UNIT-II:

Interpolation – finite Differences – Forward Differences – Backward Differences-

Central Differences

(Chapter 3: Sections : 3.3.1,3.3.2,3.3.3)

UNIT-III: Transportation

Transportation Model – Mathematical formulation –Northwest Corner Rule – Least Cost

Method

UNIT-IV: Assignment

Assignment algorithm.

UNIT-V: Networking scheduling by PERT/CPM

Network and basic components – numbering the events – time calculations in networks -

Critical path Network Calaculations – PERT Network-

***(In all the units application of Concept only.No Bookwork)

Book for References:

1. An introductory methods for Numerical Analysis by S.S.Sastry

2. Operations research by Kanti Swarup , Gupta . P.K & Manmohan ( 8th edition

1997)

3. Problems in operation Research by Gupta P.K. & Manmohan

4. Resource Management Techniques by Prof. V.Sundaresan , K.S.Ganapathy

Subramaniyam , K.Ganesan

Inst.Hour 6

Credit 5

Code 11KP2MELO1

SEMESTER III

CORE COURSE – IX TOPOLOGY

UNIT-I

Topological spaces – Bases for a Topology – The order topology – Product topology of X x

Y – The subspace Topology – Limit points – closed sets – continuous – Continuous

Functions – Homeomorphism – Properties of continuous functions.

Chapter II : Sections 2.1 to 2.7

UNIT-II

Connected spaces – connected sets on the real line – components of a space – locally

connected spaces – compact spaces – compactness in the real line – limit point

compactness – Lebesgue numbers – uniform continuity.

Chapter 3 : Sections 3.1, 3.2, 3.3, 3.5, 3.6, 3.7

UNIT-III

The Countability axioms – Lindelo’f spaces – Seperable spaces – The seperation axioms –

Regular and normal spaces.

Chapter 4 : Sections 4.1, 4.2

UNIT-IV

The Urysohn’s lemma – Tietze Extension Theorem – The Urysohn’s metrization theorem –

Embedding theorem.

Chapter 4 : Section 4.3, 4.4

UNIT-V

The Tychonoff theorem – completely regular spaces – The Stonecech compactification –

Complete metric spaces - Compactness in metric spaces – Ascoli’s theorem ( Classical

version) Chapter 5 : Sections 5.1, 5.2, 5.3 & Chapter 7 : Sections 7.1, 7.3

TEXT BOOK : A First course in Topology : James R.Munkres.

Prentice Hall of India(p)Ltd., New Delhi, 1988.

REFERENCE:

1. George.F.Simmons, Introduction to Topology and Modern Analysis, Mc.Graw Hill Co.,

1963.

2. J.L.Kelly, General Topology, Van Nostrand, Rein Hold Co., Newyork.

Inst.Hour 6

Credit 4

Code 11KP3MO9

SEMESTER III

CORE COURSE X

INTEGRAL EQUATIONS AND TRANSFORMS

UNIT-I

Linear Integral Equations – Definitions Regularity conditions – special kind of kernels –

Eigen values and Eigen functions – convolution Integral – The Inner and scalar product of

two functions – Notations – Reduction to a system of algebraic equations – examples –

Fredholm alternative – Examples – An approximate method.

Chapter 1 (1.1-1.7), Chapter 2 (2.1-2.4)

UNIT-II

Method of successive approximations – Iterative scheme – examples – Volterra Integral

Equation – Examples – some results about the resolvent kernal. Classical Fredholm theory

– The method of solution of Fredholm – Fredholm’s first theorem – second theorem.

Chapter 3 (3.1-3.5), Chapter 4 (4.1-4.4)

UNIT-III

Applications to Ordinary differential Equations – Initial value problems – Boundary value

problems – Examples – Singular Integral equations – The Abel Integral equations –

Examples.

Chapter 5 (5.1-5.3), Chapter 8 (8.1&8.2)

TEXT BOOK : Ram. P.Kanwal – Linear Integral Equations Theory and Practice.

Academic Press 1971 – Chapters 1, 2, 3, 4 and 5.1, 5.3 and 8.1, 8.2.

UNIT-IV

Fourier Transforms – Dirichlets Conditions – Fourier series – Fourier Integral formula –

Fourier Transform – Fourier sine Transform – Inversion Formula for Fourier Sine

Inst.Hour 6

Credit 4

Code 11KP3M1O

Transforms – Fourier cosine transform – Inversion formula for Fourier cosine Transform –

Linearity property – Change of scale property – Shifting property – Modulation theorem –

Examples.

Chapter 6 section 6.1-6.5

UNIT-V

Finite Fourier Transform – Sine Transform – Inversion formula for sine Transform – Finite

Fourier Cosine Transform – Inversion formula for cosine transform – Examples – Multiple

finite Fourier Transforms – Operational properties of Sine & Cosine Transforms –

Combined properties of Finite Fourier Sine and Cosine Transform – Convolution –

Examples.

Chapter 7 Section7.1-7.9

TEXT BOOK : Integral Transforms, A.R.Vasistha & R.K.Gupta, Krishnapragasam

Publications.

REFERENCES:

1.S.J.Mikhlin, Linear Integral Equations, Hindustan Book Agency, 1960.

2. I.N.Sneddon, Mixed Boundary Value Problem & Potential Theory, North Holland, 1966

SEMESTER III

CORE COURSE XI CLASSICAL DYNAMICS

UNIT-I

Introduction concepts - The mechanical system – Generalised coordinates – constraints-

virtual work –energy and momentum.

Chapter I : Sec 1.1 to 1.5

UNIT-II

Lagrange’s equation – Derivations of Lagrange’s equation – examples – Integrals of

motion-small oscillations.

Chapter 2 : Sec 2.1 to 2.4

UNIT-III

Special applications of Lagrange’s equation’s : Equation – Rayleighs disspation functions –

Impulsive motion – Gyroscopic sysytems – velocity – dependent potentials.

Chapter 3 : Sec 3.1 to 3.4

UNIT-IV

Hamilton’s equation – Hamilton’s principle

Chapter 4 : 4.1 to 4.2

UNIT-V

Other variational principles – Phase space

Chaper 4 : 4.3, 4.4

TEXT BOOK :

Classical Dynamics, Donald T.Greenwood, PHI Pvt.Ltd., New Delhi.

REFERENCE BOOK:

Classical Mechanics, Goldstein Poole & Safco, Pearson Education.

Inst.Hour 7

Credit 5

Code 11KP3M11

SEMESTER III

ELECTIVE COURSE II

FUZZY SETS AND THEIR APPLICATIONS

UNIT-I

Fuzzy sets – Definitions – Different types of Fuzzy sets – General Definitions and

Properties of Fuzzy sets – Other important Operations.

Chapter1 : Sec 1.16-1.20

UNIT-II

Operations on Fuzzy sets – Introduction – Some important theorems - Extension Principle

for Fuzzy sets – Fuzzy Compliments – Further operations on Fuzzy Sets – T-norms and T-

Conorms.

Chapter 2 : Sec 2.1-2.6

UNIT-III

Fuzzy Numbers and Arithmetic – Introduction – Fuzzy Numbers – Algebraic Operations

with Fuzzy Numbers – Binary operations of two Fuzzy numbers – Fuzzy Arithmetic.

Chapter 3 : Sec 3.1-3.5

UNIT-IV

Fuzzy Relations And Fuzzy Graphs – Introduction – Projection and Cylindrical Fuzzy

relations – Composition – Properties of Min.max Composition – Binary Relations on a

Single set - Compatibility Relations – Fuzzy ordering Relation – Fuzzy Morphisms –

Fuzzy Relation Equations.

Chapter 4 : Sec 4.1-4.9

UNIT-V

Decision Making in Fuzzy Environment – Introduction - Individual Decision Making –

Multiperson Decision Making – Multicriteria Decision Making – Fuzzy Ranking Method –

Fuzzy Linear Programming. Chapter 9 : Sec 9.1-9.6

Inst.Hour 6

Credit 5

Code 11KP3MELM2

TEXT BOOK :

Fuzzy sets and their applications By Pundir & Pundir, First Edition 2006.

REFERENCE :

1. H.J.Zimmermann, Fuzzy set theory and its applications, Allied Publishers Ltd.,

NewDelhi, 1991.

2. A.Kaufman, Introduction to the theory of Fuzzy Subsets, Vol.I, Academic Press,

Newyork 1975.

3. George J.Klir and Boyuan, Fuzzy sets and Fuzzy Logic, Prentice Hall of India,

NewDelhi, 2004.

III SEMESTER

OEC2

OPTIMIZATION TECHNIQUES

(OPEN Elective for P.G offered by MATHS Department)

For Non Maths Students

UNIT-I

Formulation of LPP – Graphical Solution

(Chapter II)

UNIT-II

General LPP – Simplex Method

(Chapter III – Section 3.1)

UNIT-III

Simplex Method –Big M Method or The Method of Penalties

(Chapter III – Section 3.2.1)

UNIT-IV

Inventory Models Deterministic Models Purchasing Model with no Shortages –

Manufacturing Model with no shortages.

UNIT – V

Deterministic Model – Purchasing Model with Shortages – Manufacturing Model with

Shortages.

Chapter 12 – Section 12.1-12.7

**(In all units application of Concepts Only No Bookwork)

TEXT BOOK:

1. Content and Treatment as in Resource Management and Techniques (OR) by

Professor V.Sundaresan, K.S.Ganapathy Subramanian, K.Ganesan – (A.R.Publications

Arpakkam- 609 111) Second Edition.

REFERENCE BOOK:

1. Operations Research by Hamdy A.Taha (Third Edition)

Inst.Hour 5

Credit 5

Code 11KP3MELO2

SEMESTER IV

CORE COURSE – XII

MEASURE AND INTEGRATION

UNIT-I

Measures on Real line – Lebesque outer measure – measurable sets – Regularity –

Measurable function Borel and Lebesgue measurability.

Chapter – 2: Sections 2.1 to 2.5

UNIT-II

Abstract measure spaces – Measures and outer measures – Extension of a Measure –

Uniqueness of the Extension – Completion of a measure – measure spaces - Integration

with respect to a measure.

Chapter – 5 : Sections 5.1 to 5.6

UNIT-III

LP spaces – Convex functions – Jensen’s inequality – Inequalities of Holder & Minkowski

– Completeness of LP (u).

Chapter 6 : Section 6.1 – 6.5

UNIT-IV

Signed measures – Hahn decomposition, The Jordan Decomposition – The Radon –

Nyeodym Theorem

Chapter VIII : Sections 8.1 to 8.3

UNIT-V

Some applications – Measurability in a product space – Fubini’s Theorem.

Chapter VIII : Section 8.4

Chapter X : Section 10.1, 10.2

TEXT BOOK: Measure Theory and Integration – G.De Barra

REFERENCE :

Measure and Integration second editon by M.E.Manroe Addison – Wesley Publishing

Company 1971.

Inst.Hour 6

Credit 5

Code 11KP4M12

SEMESTER IV

CORE COURSE XIII DISCRETE MATHEMATICS

UNIT-I

Connectives – Negation – conjunction –Disjunction – Statement formulas and truth tables –

logical capabilities of programming languages – Conditional and Bi-conditional – well

formed formulas – tautologies – Equivalence of formulas – Duality law – tautological

implications- formulas with distinct truth tables – functionally complete set of connectives

– Other connectives.

Chapter 1 : Section 1.2.1 - 1.2.14

UNIT – II

Normal forms – Disjunctive normal forms-Conjunctive normal forms – Principal

disjunctive normal forms principal conjunctive normal forms – Ordering and uniqueness of

normal forms – Completely parenthesized Infix notation polish notation – the theory of

inference for the statement calculus- validity using truth tables – rules of inference –

consistency of premises and indirect method of proof.

Chapter 1 : Section 1.3.1- 1.3.6 & 1.4.1-1.4.3

UNIT-III

The predicate calculus – predicates – The statement function, variables and Quantifiers –

Predicate Formulas- free and bund variables – The universe of discourse- inference theory

of predicate calculus – valid formulas and equivalence – some valid formulas over finite

universe –Special valid Formulas involving Quantifiers-Theory of inference for predicate

calculus – Formulas involving more then one Quantifier.

Chapter – I Section 1.5.1-1.5.5 & 1.6.1 – 1.6.5

UNIT –IV

Boolean Algebra – Definition and examples – Sub Algebra, direct product and

homomorphism – Boolean forms and Free Boolean Algebras – Values of Boolean

expression and Boolean Functions.

Chapter – 4 Sections 4.2.1 -4.2.2 & 4.3.1 -4.3.2

Inst.Hour 6

Credit 4

Code 11KP4M13

UNIT – V

Group codes – The communication model and basic notion of error correction- generation

of course by using parity checks – error recovery in group codes

Chapter 3 : Section 3.7.1 to 3.7.3

TEXT BOOK:

1. P. Trembly and R. Manohar : Discrete Mathematical Structures with Applications to

Computer Science Mc.Graw Hill International Edition.

SEMESTER IV

CORE COURSE XIVSTOCHASTIC PROCESSES

UNIT- I

Stochastic processes: Some notions – specifications of Stochastic

processes- stationary processes – Markov chains – Definitions and examples

– Higher Transition probabilities.

Chapter-II: sec2.1 to2.3.Chapter-III: sec3.1 to3.2.

UNIT –II

Markov Chains: classification of states and chains –

determination of Higher transitions probabilities – stability of a Markov

system. Chapter –III: 3.4,3.5,3.6.

UNIT – III

Markov processes with Discrete state space: Poisson processes

and their extensions – Poisson process and related distribution –

Generalisation of poisson process – Birth and Death process

Chapter – IV: sec 4.1 to 4.4

UNIT –IV

Renewal processes and theory; Renewal process – Renewal

processes in continuous time – Renewal equation – stopping time – Wald’sequation – Elementary Renewal theorem. Chapter- VI: sec 6.1 to 6.5.1

UNIT – V

Some basic Mathematical Results: Difference Equations – Homogeneous

Inst.Hour 6

Credit 5

Code 11KP4M14

Difference equation with constant coefficients- Difference Equations in

Probability theory – Differential Difference Equations- Spectral

Representation of a Matrix. Stochastic processes in Queuing and Reliability:

Queuing system – general concepts – the Queuing model M/M/1 – Steady

state behavior – Transient behavior of M/M/1 Model – Difference Equation

Techniques – method of Generating Functions – Busy period – Zero

avoiding state probability.

Appendix – A: A.2.1, A.2.2, A.2.4,A.3,A.4.4.

Chapter - X: sec 10.1 to 10.3,(omit sec . 10.2.3&10.2.3.1)

TEXT BOOK: J. Medhi, Stochastic Processes _ Wiley Eastern Ltd., second

edition.

REFFERENCE BOOKS:

1. A first course in Stochastic Processes – Samuel Korlin, Howard M. Taylor

- Second edition.

2. Elements of applied Stochastic processes – Narayan Bhat.

3. Stochastic processes – Srinivasan and Mehta.

4. Stochastic process –N.V. Prabhu – Macmillan(New York).