VIRUDHUNAGAR HINDU NADARS’ SENTHIKUMARA NADAR COLLEGE
(An Autonomous Institution Affiliated to Madurai Kamaraj University)
[Re-accredited with ‘A’ Grade by NAAC]
Course Name : M.Phil
Discipline
: Maths
COURSE SCHEME:
Semester Part
Core 1
Core 2
Elective
I
Subject
Research
Methodology
Stochastic Processes
Generalized Inverses:
Theory and its
Application/
Advanced Algebra/
Advanced Analysis/
Functional Analysis
Library
Semester Subject
Hour
II
30
Dissertation & Viva Voce
Hour Int+Ext=Total
5
40+60=100
Subject Code
M1MAC11
5
5
40+60=100
40+60=100
M2MAC12
M1MAE11/
M1MAE12/
M1MAE13/
M2MAE14
15
-
-
Viva+Dissertation Subject Code
=Total
50+150=200
M1MA2PV
TITLE OF THE PAPER: RESEARCH METHODOLOGY
Subject Code: M1MAC11
Unit I:
Research in Mathematics
15-Hrs
Research in Mathematics – Proof Techniques – Proof by induction – Proof by
contradiction – Proof by construction – Mathematical Journals –
AMS Subject Classification Impact factor – Search engines – Thesis and dissertation.
Unit II: Bounds on Domination
15-Hrs
(Omit theorems, lemmas and propositions that are stated without proof)
Bounds in terms of order – Bounds in terms of order, degree and packing – Bounds in
terms of order and size – Bounds in terms of degree diameter and girth.(Omit Theorem 2.6).
Unit III: Planarity
15-Hrs
Introduction – Planar and non planar graphs – Euler formula and its conseguences –
dual of a plane graph – The four colour theorem and the Heawood five colour theorem –
Kuratowski’s theorem.
Unit IV: Triangulated graphs
15-Hrs
Introduction – perfect graphs – triangulated graphs – interval graphs – bipartite graph of a
graph – circular arc graphs.
VIRUDHUNAGAR HINDU NADARS’ SENTHIKUMARA NADAR COLLEGE
(An Autonomous Institution Affiliated to Madurai Kamaraj University)
[Re-accredited with ‘A’ Grade by NAAC]
Unit V:
Irregular graphs
15-Hrs
Introduction – Different types of irregular graphs – Definitions and examples – Some basic
results – main results.
Text Book:
1.Fundamentals of domination in graphs, Terasa W.Haynes, Stephen T.Hedetniemi and Peter
J.Slater, Marcel Dekker, Inc. (1998) New York.
Chapter 2 – Sections 2.1 -2.4 (omit Theorem 2.6)
2.A Text book of Graph Theory, R.Balakrishnan and K.Ranganathan, Springer International
Edition (2000)
Chapters VIII – Section 8.0 – 8.6
Chapters IX – Section 9.0 – 9.5
3. Irregular graphs – Selvam Avadayappan and Bhuvaneshwari (2012).
Reference Book:
1.Highly irregular graphs – Yousef Alavi and others, Journal of Graph theory, Vol.11 No.2
(235 -249) 1987.
2.Highly irregular bipartite graphs – Selvam .A, Indian Journal of Pure and Applied
Mathematics, 27(6) June 1996 (527 -536).
3.Neighbourly irregular graphs, S. Gnaana Bhragsam and S.K.Ayyasamy, Indian Journal of
Pure and Applied Mathematics, 35(3) March 2004 (389 -399).
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Stochastic Processes
Contact hours per semester:5
Subject Code: M2MAC12
Contact hours per week:5
Credit:5
Unit I:
Stochastic process: Introduction – Specification of stochastic processes, stationary processes,
Martingales.
Markov chains:
Definitions, examples, Higher transition probabilities,
classification of states and chains.
15-Hrs
Unit II:
Stability – Markov chains with denumerable number of states, Poisson process.
Unit III:
Poisson process and related distributions. Markov chain with discrete state
space.Generalisations of Poisson Process, Birth and death Process.
Unit IV:
15-Hrs
15-Hrs
Queueing concepts: Queueing systems, M/M/1 steady state behaviour, transient
behaviour of M/M/1 model.
15-Hrs
VIRUDHUNAGAR HINDU NADARS’ SENTHIKUMARA NADAR COLLEGE
(An Autonomous Institution Affiliated to Madurai Kamaraj University)
[Re-accredited with ‘A’ Grade by NAAC]
Unit V: Birth death processes in queueing theory: Multi-channel models – non Birth-Death
queueing processes.
15-Hrs
Text Book:
“Stochastic Processes” – By Prof. J. Medhi
UnitI: Chapter 2: 2.1 to 2.4, Chapter 3: 3.1, 3.2, 3.3, 3.4
Unit II: Chapter 3: 3.6, 3.7, 3.8, Chapter 4: 4.1 (Pages 157 – 169)
Unit III: Chapter 4: 4.2, 4.3, 4.4, 4.5
Unit IV: Chapter 10: 10.1, 10.2, 10.3
Unit V: Chapter 10: 10.4 – 10.5.1 (Pages 431 – 446)
Reference Books:
1. “Introduction to Stochastic Processes” – By Prof. N. P. Basu
2. “First Course in Stochastic Processes” – By Samuel Karlin and Taylor, Wiley Eastern
Limited, 2000
3. “Stochastic Processes” – By Srinivasan and Mehta, Tata McGrawHill Publishing
Company, New Delhi 1999.
4. “Elements of applied Stochastic Process” – By U. Bhat, G. Miller, Wiley, New york,
1984.
Generalized Inverses: Theory and Applications (Elective)
Contact hours per week-5
Contact hours per semester-75
CREDIT-5
Subject Code: M1MAE11
Objectives:
To enable the students to
1. know the concept of generalized inversed
2. prove the existence of{1} and {1,2} inversed
3. solve linear equations
4. know the bolt-duffin inverses
5. know about the applications
UNIT-1:
15-Hrs
Existence and construction of generalized inverses; The Penrose existence and
construction of {1}-inverses-properties of {1}-inverses –Bases for the range and null space of a
matrix-existence and construction {1, 2}-inverses.
VIRUDHUNAGAR HINDU NADARS’ SENTHIKUMARA NADAR COLLEGE
(An Autonomous Institution Affiliated to Madurai Kamaraj University)
[Re-accredited with ‘A’ Grade by NAAC]
UNIT -2:
15-Hrs
Existence and construction of {1, 2 ,3} –{1, 2 ,4}-{1, 2, 3, 4}-inverses –full rank
factorization Explicit formula for AT Construction of {2} – inverses of prescribed rank –An
Application of {2} inverses in iterative methods for solving non linear equations.
UNIT-3:
15-Hrs
Solution of linear systems-characterization of A {1,3},A{1,4} –Characterization of A
{2},A{1,2} And other subsets of A {2}-idempotent matrices and projectors-Generalized inverses
with prescribed range and null Space.
UNIT-4:
15-Hrs
Orthogonal projections and orthogonal projectors –efficient characterization of classes of
generalized inverses-restricted generalized inverses-the bolt-duffin inverse.
UNIT-5:
15-Hrs
An application of {1} –inverses in interval linear programming –A {1,2}-Inverses for the
Integral Solution of linear equations-An application of the Bolt-Duffin inverse to electrical
Networks-least square solution of inconsistent linear systems-solutions of minimum norm.
TEXT BOOK:
Generalized Inverses: Theory and applications By di Ben-Israel, Thomas N.E.Greville.
UNIT-1: Sections 1.1 to 1.5
UNIT-2: Sections 1.6 to 1.0
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ADVANCED ALGEBRA
Subject Code: M1MAE12
UNIT 1:
(15 hours)
Associative algebra-Group algebra-Endimorphism algebras-Matrixalgebras-Finite
Dimensional algebras-Quternion algebras-Isomorphism of Quaternion algebras
UNIT 2:
(15hours)
Modules-Change of scalars –Lattice submodules-simple modules-semisimplemodulesStructure of semisimplemodules-Chain conditons-The Raddical of a ring –Tensor product of
modules
UNIT 3:
(15hours)
Structure of semisimple algebras-Semisimple-Minimal right ideals-simple algebrasMatrices of homomorphisms-The density theorem-Wedderburn structure theorem-Mascheke’s
theorem
UNIT 4:
(15hours)
The Raddical-Raddical of an algebra-Nakayam’s lemma-The Jacobson Raddical-The
Radical of an artinin algebra-Nilpotent algebras-The raddical of Group algebra-ideals in artinian
rings-Direct decompositions-Local algebras-Fitting’s lemma
UNIT 5:
(15hours)
Simple algebras-centers of simple algebras-The density theorem-The Jacobson-Bourbaki
theorem-Central simple algebras-The Brauer Group-The Noehter-Skolem Theorem-The Double
centralizer theorem
VIRUDHUNAGAR HINDU NADARS’ SENTHIKUMARA NADAR COLLEGE
(An Autonomous Institution Affiliated to Madurai Kamaraj University)
[Re-accredited with ‘A’ Grade by NAAC]
TEXT BOOK:
“The Associative algebras”-R.S.Pierce-GIM 88,Springer-Verlag,1982
Unit -1: Chapteer 1(pp.1-20)
Unit-2: Chapter 2&9(section9.1)(pp.21-39 & 157-163)
Unit -3: Chapter 3(pp.40-54)
Unit-4: Chapter 4(pp.55-71) & Chapter 5(pp.72-76)
Unit-5: Chapter 12(pp.218-233)
ADVANCED ANALYSIS
Subject Code: M1MAE13
SYLLABUS
Objectives
To enable the students to
(i)
Know the concepts of measurability.
(ii)
Know the concepts of Borel measures.
(iii) Acquire the knowledge of 𝐿 𝑃 spaces.
(iv)
Concept of Elementary Hilbert space theory.
(v)
Understand the Banach space Techniques.
UNIT – I ABSTRACT INTEGRATION
Chapter I Pages 6 to 30
UNIT – II POSITIVE BOREL MEASURE
Chapter II Pages 33 to 56
UNIT – III 𝐿𝑃 SPACES
Chapter III Pages 61 to 70
UNIT –IV ELEMENTARY HILBERT SPACE THEORY
Chapter III Pages 76 to 91
UNIT – V BANACH SPACE TECHNIQUES
Chapter III Pages 95 to 111
TEXT BOOK:
Real and Complex Analysis, 3rd Edition by Walter Rudin.
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Functional Analysis
Contact hours per semester:5
Subject Code: M2MAE14
Contact hours per week:5
Credit:5
Unit I: Topological vector spaces: Introduction – Separation properties – Linear Mappings –
Finite dimensional spaces – Metrization
15-Hrs
Unit II: Boundedness and continuity – Semi-norms and local convexity – Quotient spaces –
Examples 1.44, 1.45, 1.46.
15-Hrs
VIRUDHUNAGAR HINDU NADARS’ SENTHIKUMARA NADAR COLLEGE
(An Autonomous Institution Affiliated to Madurai Kamaraj University)
[Re-accredited with ‘A’ Grade by NAAC]
Unit III: Completeness: Baire Category theorem – The Banach-Steinhaus Theorem – The open
mapping theorem – The closed graph theorem – Bilinear mappings.
15-Hrs
Unit IV: Convexity: The Hahn-Banach Theorems – Weak topologies – The weak topology on a
topological vector space – The weak* topology of a dual space – Compact convex sets. 15-Hrs
Unit V: Vector valued integration – Holomorphic functions – A continuity theorem – Kakutani’s
Fixed point theorem – fixed point theorem due to Markov and Kakutani - The SchauderTychonoff fixed point theorem.
15-Hrs
Text Book:
“Functional Analysis” Second Edition – By Walter Rudin, Tata McGraw-Hill Edition, 2006,
New Delhi.
Chapters: 1, 2 and 3; Chapter 5: Theorems 5.1, 5.11, 5.23, 5.28 (only)
Reference Books:
1. “Topological vector spaces, Second Edition” –By Lawrence Narici and Edward
Beckenstein, Chapman and Hall/CRC, (July 2010)
2. “Topological Vector spaces, Second Edition” – By H.H. Schaefer and M.P. Wolff,
Publisher: Springer -Graduate Texts in Mathematics 3, (1999).
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