k.n.government arts college for women ( … · 11kp1m02 core course–ii (cc) real analysis 6 5 25...
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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).