department of mathematics arul anandar college …

DEPARTMENT OF MATHEMATICS ARUL ANANDAR COLLEGE (AUTONOMOUS)

M.Sc. MATHEMATICS CBCS & OBE PATTERN (From 2020 – 2021 onwards)

Nature of the paper

Sub.Code Title Hr Cr

FIRST YEAR – FIRST SEMESTER

Core 20PMAC11 Core – 1 Modern Algebra 06 05

20PMAC21 Core – 2 Real Analysis 06 05

20PMAC31 Core – 3 Numerical Analysis 06 05

20PMAC41 Core – 4 Statistics 06 05

Core Elective I

20PMAE11 Core Elective – 1a Graph Theory / 1b Discrete Mathematics

06 04

Total 30 24

FIRST YEAR – SECOND SEMESTER

Core

20PMAC52 Core – 5 Linear Algebra 06 05

20PMAC62 Core – 6 Measure and Integration 06 05

20PMAC72 Core – 7 Differential Equations 06 05

Core Elective II

20PMAE22 Core Elective – 2a Differential Geometry/ 2b Stochastic Process

06 04

20PMAN12 Non-Major Elective -1 Mathematics for Life 04 04

20PLFS12 Life Skills 02+02* 02

Total 30 25

Elective for MCA, M.SC (Physics, Chemistry) students: Mathematics for Life

SECOND YEAR – THIRD SEMESTER

Core

20PMAC83 Core – 8 Topology 06 05

20PMAC93 Core – 9 Classical Mechanics 06 05

20PMAD03 Core – 10 Complex Analysis 06 05

20PMAD13 Core – 11 Operations Research 06 05

Core Elective III

20PMAE33 Core Elective –3a Mathematical Modeling/ 3b Integral Equations

06 04

Total 30 24

SECOND YEAR – FOURTH SEMESTER

Core

20PMAD24 Core – 12 Functional Analysis 06 05

20PMAD34 Core – 13 Fuzzy Sets and Fuzzy Application 06 05

20PMAD44 Core – 14 Project 12 05

Core Elective IV

20PMAE44 Core Elective – 4a Automata Theory / 4b Theory of Numbers

06 04

Total 30 19

Semester Credits I II III IV Total 24 25 24 19 92 Self-Learning Courses 2 The students can undertake any online courses offered by SWAYAM during any of the semesters and can earn extra credit. Credit 2 per course

Maximum 4 credits

ARUL ANANDAR COLLEGE (AUTONOMOUS), KARUMATHUR – 625 514 DEPARTMENT OF MATHEMATICS

Modern Algebra

(For those who joined in June 2020 onwards)

Class : I M.Sc. Mathematics Part : Core-1

Semester : I Hours : 90

Subject Code : 20PMAC11 Credits : 5

Objectives:

This course enables the learners to

1. Gain profound knowledge on the concepts of counting principle.

2. Comprehend the significance of Sylow’s theorem

3. Explain the notion of Euclidean ring and the applications of Fermat theorem

4. Analyze the various forms of polynomial rings.

5. Examine the structure and the properties of field.

Unit 1:Another counting principle –conjugate – conjugate class – normalizer –– center of group-

Cauchy theorem- partition (18 Hours)

Unit 2: Sylow’s theorem – 1st, 2nd, 3rd proofs – p-Sylow subgroup- direct products of finite

abelian groups (18 hours)

Unit 3: Euclidean ring – principle ideal ring – greatest common divisor- prime element –unique

factorization theorem- a particular Euclidean ring– Gaussian integers– Fermat theorem

(18 hours)

Unit 4: Polynomial rings – division algorithm – irreducible polynomials - polynomials over the

rational field– Gauss Lemma- Eisenstein criterion (18 hours)

Unit 5: Fields – extension fields – algebraic extension – roots of polynomials remainder

theorem – splitting field – isomorphism between fields (18 hours)

Book for Study:

HersteinI.N.,Topics in Algebra, John Wiley and Sons Pvt. Ltd., Singapore,2016.

Unit 1: Chapter 2 Section 2.11

Unit 2: Chapter 2 Sections 2.12 -2.14

Unit 3: Chapter 3 Sections 3.7, 3.8

Unit 4: Chapter 3 Sections 3.9, 3.10

Unit 5: Chapter 5 Sections 5.1 -5.3

Books for References:

Vijay Khanna.,A course in Abstract Algebra, V Edition 2018.

Bhattacharya P.B., Jain S.K. and Nagpaul S.R., Basic Abstract Algebra, Cambridge University

Press, II Edition 2005.

LutherI.S. and Passi I.B.S., Algebra, Narosa Publishing House, New Delhi, 2015.

Teaching Learning Methods:

Lecture Method, ICT, Assignment, Quiz, Group Discussion

Course Outcomes (CO):

On completion of this course the students will be able to

Course Outcome No. Course Outcome Knowledge Level

Upto

CO1

Summarize the concepts of counting

principle and explains the

characterization of p groups

K5

CO2 Infer the implications of Sylow’s

theorem to finite group K4

CO3 Justify the theoretical

conceptualization of Euclidean rings K5

CO4 Compare the polynomial rings of

various forms K4

CO5

Make inferences on algebraic

extension of the field, splitting field

and the isomorphism between fields

K5

K1 = Remember, K2 = Understand, K3 = Apply, K4 = Analyze K5 = Evaluate K6 = Create

Mapping Course outcome with

PO 1

PO 2

PO 3

PO 4

PO 5

PO 6

PO 7

PO 8

PSO 1

PSO 2

PSO 3

PSO 4

PSO 5

Sum of COs with

POs & PSOs

CO1 3 2 2 3 2 2 3 3 20

CO2 3 3 2 3 2 3 3 2 21

CO3 3 2 2 3 2 3 3 3 21

CO4 3 3 2 3 2 3 3 2 21

CO5 3 2 2 3 2 2 3 3 20

Grand Total of COs with POs & PSOs 103

Mean Value of COs with POs & PSOs = 2.6

Strong – 3, Medium – 2, Low – 1

Mapping Scale 1 2 3

Relation 0.01-1.0 1.01-2.0 2.1-3

Quality Low Medium Strong

Mean Value of COs with POs & PSOs

2.6

Observation COs of Modern Algebra are strongly correlated with POs & PSOs

ARUL ANANDAR COLLEGE (AUTONOMOUS), KARUMATHUR – 625 514

DEPARTMENT OF MATHEMATICS

Real Analysis





Objectives:

This course enables the students to

1. Obtain insights on the implications of Heine Borel and Weierstrass theorems

2. Acquire intense knowledge of properties of sequences

3. Comprehend various tests of convergence for series

4. Identify the nature of functions and enumerate its infinite limit

5. Illustrate the applications of derivatives and its related theorems

Unit 1: Finite, countable and uncountable sets – metric spaces – compact sets –Heine Borel

theorem-Weierstrass theorem- perfect sets –Cantor set- connected sets

(18 hours)

Unit 2: Convergent sequences – properties-subsequences – Cauchy sequences – upper and

lower limits – some special sequences (18 hours)

Unit 3: Series – comparison test-series of non-negative terms – number e – the root and ratio

tests – power series – Summation by parts – absolute convergence – addition and

multiplication of series –Merten’s theorem- rearrangements –Riemann theorem

(18 hours)

Unit 4: Limits of functions – continuous function – continuity and compactness – continuity

and connectedness – discontinuities – monotonic functions – infinite limits and limits

at infinity (18 hours)

Unit 5: Derivative of a real function – mean value theorem – generalized mean value theorem

- continuity of derivatives – L’Hospital’s Rule – derivatives of higher order – Taylor’s

theorem- differentiation of vector valued functions. (18 hours)

Book for Study:

Walter Rudin,Principles of Mathematical Analysis, McGraw-Hill International Editions, New

Delhi, Third Edition,2017

Unit 1 : Chapter 2 All Sections

Unit 2 : Chapter 3 Sections 3.1-3.20





Bartle R.G.,Introduction to Real Analysis, John Wiley and Sons Inc., New Jersey 2005.

Malik S.C. and Savita Arora., Mathematical Analysis, Wiley Eastern Limited, New Delhi, 2010.


Lecture Method ,ICT, Assignment, Quiz, Group Discussion




Upto

CO1 Evaluate the implications of Heine boral

and Weierstrass theorems K5

CO2 Infer on the nature of sequence K4

CO3 Apply the tests of convergence of series

and make inferences K4

CO4 Analyze the nature of functions and find

its limit K4

CO5 Employ the theorems related to

derivatives K3



PO 1

PO 2

PO 3

PO 4

PO 5

PO 6

PO 7

PO 8

PSO 1

PSO 2

PSO 3

PSO 4

PSO 5

Sum of COs with

POs & PSOs

CO1 3 2 2 3 2 2 3 3 20

CO2 3 3 2 3 3 3 3 2 22

CO3 3 3 2 3 3 3 3 2 22

CO4 3 3 2 3 3 3 3 2 22

CO5 3 3 2 3 3 3 2 2 21




Mapping Scale 1 2 3

Relation 0.01-1.0 1.01-2.0 2.1-3



2.7

Observation COs of Real Analysis are strongly correlated with POs & PSOs



Numerical Analysis





Objectives:


1. Attain intense knowledge of different methods of finding approximate solutions.

2. Find solutions to system of linear equations

3. Comprehend the concept of interpolation and different methods of computation

4. Acquire insights on numerical differentiation and integration

5. Solve ordinary differential equations using different methods

Unit 1: Transcendental and polynomial equation – bisection method – secant and Regula –

Falsi method – Newton– Raphson method – iteration methods based on second

degree equation – Muller method – Chebyshev method – multi point iteration

methods – system of non-linear equations – methods for complex roots

(18 hours)

Unit 2: System of linear algebraic equations and eigen value problem – direct methods –

iteration methods – Jacobi-iteration method, Gauss-Seidel iteration method

(18 hours)

Unit 3: Interpolation and approximation – Taylor series – Lagrange and Newton interpolations

– finite difference operators – Hermite interpolation

(18 hours)

Unit 4: Differentiation and integration – numerical differentiation – extrapolation methods –

partial differentiation – numerical integration – methods based on interpolation –

composite integration methods – Romberg integration – double integration

(18 hours)

Unit 5: Ordinary differential equation – initial value problem – difference equation –

numerical methods – Euler method – backward Euler – mid-point method

(18 hours)

Book for Study:

Jain M.K., Iyengar S.R.K. and Jain R.K.,Numerical Methods for Scientific and Engineering

Computations, Fourth Edition, New Age International private Limited, New Delhi, 205.

Unit 1: Chapter 2 Sections 2.1 -2.4, 2.7, 2.8

Unit 2: Chapter 3 Sections 3.1, 3.2, 3.4

Unit 3: Chapter 4 Sections 4.1 - 4.5

Unit 4: Chapter 5 Sections 5.2, 5.4 - 5.7, 5.9 -5.11

Unit 5: Chapter 6 Sections 6.1 - 6.3


Richard L.Barden.J.Bouglas Faires.,Numerical Analysis,IX Edition, Cengage Learning,2011

Radhey.,S.Gupta,Macwillan., Elements of Numerical Analysis, 2009.


Lecture Method, ICT, Assignment, Quiz, Group Discussion.




Up to

CO1 Determine solutions of the system of

equations by applying various methods K3

CO2 Employ the suitable method of computing

solutions to system of linear equations K3

CO3

Apply various methods of interpolation and

interpret on the solutions to the real-life

problems

K4

CO4 Calculate derivatives and integrands using

numerical methods K3

CO5 Solve ordinary differential equations using

different methods K3



PO 1

PO 2

PO 3

PO 4

PO 5

PO 6

PO 7

PO 8

PSO 1

PSO 2

PSO 3

PSO 4

PSO 5

Sum of COs with

POs & PSOs

CO1 3 3 2 2 3 3 2 3 2 2 25

CO2 3 3 2 2 3 3 2 3 2 2 25

CO3 3 3 2 2 3 3 3 3 2 2 26

CO4 3 3 2 2 3 3 2 3 2 2 25

CO5 3 3 2 2 3 3 2 3 2 2 25




Mapping Scale 1 2 3

Relation 0.01-1.0 1.01-2.0 2.1-3



2.5

Observation COs of Numerical Analysis are strongly correlated with POs & PSOs

ARUL ANANDAR COLLEGE (AUTONOMOUS), KARUMATUR – 625 514


Statistics

(For those who have joined in June 2020 onwards)




Objectives

The course enables the students to

1. Acquire knowledge on the underlying concepts of probability

2. Comprehend the association between random variables

3. Get familiarize with various kinds of distributions

4. Attain insights on sampling and its related distributions

5. Explain the aspects of limiting distributions and its applications

Course Outline:

Unit 1: Random variables – probability density function – distribution function – some

probability models – mathematical expectations – Chebyshev’s inequality

(18 hours)

Unit 2: Conditional probability – marginal and conditional distributions – correlation

coefficient – stochastic independence (18 hours)

Unit 3: Binomial, trinomial and multinomial distributions – Poisson distribution –

Gamma and Chi square distributions – Normal distribution – bivariate normal

distribution (18 hours)

Unit 4: Sampling theory – t and F distributions - distributions of order statistics –

moment generating function technique – distributions of X and nS2/2 –

expectations of functions of random variables

(18 hours)

Unit 5: Limiting distributions – stochastic convergence – limiting moment generating

functions – the central limit theorem – theorems on limiting distributions

(18 hours)

Book for Study:

Robert V Hogg, Allen T Craig, Introduction to Mathematical Statistics, Pearson Education, Fifth

Edition, Third Indian Reprint, Singapore, 2020.

Unit :1Chapter 1 Sections :1..5 – 1.10

Unit :2Chapter 2 Sections :2.1 – 2.4

Unit :3Chapter 3 All Sections

Unit :4 Chapter 4 Section4.1, 4.4 – 4.9

Unit :5 Chapter 5 All Sections

Books for Reference:

Sheldon Ross, A first Course in Probability, Collier Macmillan Publication, 2009.

Charles M Grinskad., J.Laurie Snell ., Introduction to Probability, Americal Mathematical Society,

II Revised Edition, 2012.






Upto

CO1 Illustrate the implications of the

concepts of probability K4

CO2 Infer the aspects of stochastic

independence K4

CO3 Compare distributions of various

kinds K4

CO4 Apply the suitable distribution to

compute solutions K3

CO5 Determine solutions to the problems

using central value limit theorem K3



PO 1

PO 2

PO 3

PO 4

PO 5

PO 6

PO 7

PO 8

PSO 1

PSO 2

PSO 3

PSO 4

PSO 5

Sum of COs with

POs & PSOs

CO1 3 3 3 3 3 3 2 3 3 2 28

CO2 3 3 3 3 3 3 2 3 3 2 28

CO3 3 3 3 3 3 3 2 3 3 2 28

CO4 3 3 3 3 3 3 3 3 2 2 28

CO5 3 3 3 3 3 3 3 3 2 2 28




Mapping Scale 1 2 3

Relation 0.01-1.0 1.01-2.0 2.1-3



2.8

Observation COs of Statistics are strongly correlated with POs & PSOs



Graph Theory


Class : I M.Sc. Mathematics Part : Core Elective 1a


Subject ode : 20PMAE11 Credits : 4

Objectives:


1. Acquire the knowledge of spanning trees and its applications.

2. Solve assignment problems using suitable algorithms

3. Gain understanding of the notion of planarity

4. Comprehend the algorithms of vertex coloring and its applications

5. Get familiarize with the concept of edge coloring and directed graphs

Unit 1: Trees and connectivity- Definition and simple properties – Bridges- Spanning

trees – Connector problems- Shortest path problems – Cut vertices and

connectivity.

(18 hours)

Unit 2: Matchings – Matchings and augmenting paths- The marriage problem – The

personnel assignment problem- matching algorithm for bipartite graphs-

Hungarian Algorithm- optimal assignment problem- Kuhn –Munkres algorithm

(18 hours)

Unit3: Planar graphs- Plane and planar graphs- Euler’s formula- The platonic bodies-

Kuratowski’s theorem- Non – Hamiltonian plane graphs- The dual of a plane

graphs. (18 hours)

Unit4: Vertex coloring algorithms: simple sequential coloring algorithm – the largest

first sequential algorithm – the smallest last sequential algorithm.

(18 hours)

Unit5: Edge coloring – Map coloring -Directed graphs- Definitions and more definitions-

In degree and out degree – Tournaments

(18 hours)

Book for Study:

Jhon Clark and Derek Allan Holton, A First Look at Graph Theory, Allied Publishers Limited, New

Delhi, 2005.





Unit 5: Chapter 6 Sections 6.5-6.6

Chapter 7 Sections7.1-7.3


NarasinghDeo, Graph Theory with Applications to Engineering and Computer Science, Prentice-

Hall of India (P) Limited,New Delhi,2019.

Harary, GraphTheory,Narosa Publishing Company, New Delhi, 2008.






Upto

CO1 Determine solutions to shortest path

problems using the notion of spanning tree K3

CO2 Employ various algorithms in finding solution

to assignment problem K3

CO3 Describe and make inferences on planar

graphs K4

CO4 Apply the suitable algorithm of vertex

coloring to solve problems K3

CO5

Summarize the aspects of directed graphs

and edge coloring with illustrations in

scheduling

K5

K1 = Remember, K2 = Understand, K3 = Apply, K4 = Analyze K5 = Evaluate K6= Create


PO 1

PO 2

PO 3

PO 4

PO 5

PO 6

PO 7

PO 8

PSO 1

PSO 2

PSO 3

PSO 4

PSO 5

Sum of COs with

POs & PSOs

CO1 3 3 2 2 3 2 3 3 3 3 2 29

CO2 3 3 2 2 3 2 3 3 3 3 2 29

CO3 3 3 2 2 3 2 3 2 3 3 2 28

CO4 3 3 2 2 3 2 3 3 3 3 2 29

CO5 3 2 2 3 3 2 3 2 2 3 3 28




Mapping Scale 1 2 3

Relation 0.01-1.0 1.01-2.0 2.1-3



2.6

Observation COs of Graph Theory are strongly correlated with POs & PSOs

ARUL ANANDAR COLLEGE (AUTONOMOUS), KARUMATHUR


Discrete Mathematics


Class : I M.Sc. Mathematics Part : Core Elective 1b


Subject Code : 20PMAE11 Credits : 4

Course Objectives :


1. Obtain intense knowledge of principles & applications of inclusion and exclusion

2. Comprehend the applications of Polya’s theory of counting

3. Gain understanding of Boolean algebra

4. Acquire the conceptualization of Boolean functions

5. Get acquainted with coding theory

Unit 1: The principle of inclusion and exclusion – general formula – derangements –

permutations with restrictions on relative positions

(18 hours)

Unit 2: Polya’s theory of counting – equivalence classes under a permutation group –

equivalence classes of functions – weights and inventories of functions – Polya’s

fundamental theorem and its generalization (18 hours)

Unit 3 : Boolean Algebra: Definitions and examples- Sub algebra, Direct Product, and

Homomorphism-Boolean functions-Boolean forms and free Boolean Algebras-

Values of Boolean expressions and Boolean functions

(18 hours)

Unit 4: Representations of Boolean functions – Minimization of Boolean functions –

Design examples using Boolean algebra – Finite state machines: Introductory

sequential circuits – Equivalence of finite –State Machines

(18 hours)

Unit 5: Coding Theory: Preliminaries and basic Definitions – Error Detection and Error

Correction – Linear Codes – Generator matrix for Linear Code Encoding –

Decoding using Cosets and Syndromes – Some special codes.

(18 hours)

Book for Study:

1.Liu C.L., “Introduction to Combinatorial Mathematics”, Mcgraw-Hill Book Company, New

York, 1968.

Unit 1: Chapter 4 sections 4.1 – 4.5

Unit 2 :Chapter 5 sections 5.1, 5.3 – 5.7

2. Tremblay.J.B and Manohar., ”Discrete mathematical structures with applications to

computer science Tata McGraw- Hill,

Unit 3: Chapter 4 sections 4.2, 4.3, 4.4

Unit 4: Chapter 4 sections 4.5, 4.6

3.N.Chandrasekaran and M.Umaparvathi,” Discrete Mathematics”,PHI Learning Private Limited,

New Delhi, 2010

Unit 5: Chapter 13 sections 13.3.1-13.3.6

Books for Reference :

1. Kenneth.H.Rosen, Discrete Mathematics and its Applications, Mc Graw -Hill, Fourth

Edition, 2010.

2. Sharma.J.K., Discrete mathematics, Trinity, Fourth Edition, 2011.




Upto

CO1 Summarize the concepts of inclusion

and exclusion with illustrations K5

CO2 Employ the notion of Polya’s theory

of counting in other domains K3

CO3 Analyze the characteristics of Boolean

algebra K4

CO4 Infer the aspects of Boolean functions K4

CO5 Write fundamental codes in finding

solutions to the problems K3



PO 1

PO 2

PO 3

PO 4

PO 5

PO 6

PO 7

PO 8

PSO 1

PSO 2

PSO 3

PSO 4

PSO 5

Sum of COs with

POs & PSOs

CO1 3 2 3 2 2 3 3 18

CO2 3 3 3 3 3 2 3 20

CO3 3 3 3 2 3 3 2 19

CO4 3 3 3 3 3 3 2 3 3 2 28

CO5 3 3 3 3 3 3 3 3 2 3 29




Mapping Scale 1 2 3

Relation 0.01-1.0 1.01-2.0 2.1-3



2.8

Observation COs of Discrete Mathematics are strongly correlated with POs & PSOs



Linear Algebra



Semester : II Hours : 90


Objectives:


1. Acquire in-depth knowledge of the concept of dual space.

2. Comprehend the process of orthogonalization

3. Get acquainted with the notion of linear transformation

4. Obtain understanding of various forms of matrix

5. Gain insights on the concepts of matrix and determinants

Unit 1: Dual spaces – vector space homomorphism – dimension – dual spaces – linear

functional – dual basis and annihilator

(18 hours)

Unit 2: Inner product spaces – definition – norm – Schwarz-inequality – orthogonal

compliment – orthonormal set and Gram–Schmidt orthogonalization process

(18 hours)

Unit 3: Linear transformation – algebra of linear transformations – invertible-minimal

polynomial- range and rank - characteristic roots and matrices

(18 hours)

Unit 4: Canonical Forms – triangular form – definition – similar - invariant– nilpotent

transformations – Jordan form – rational canonical form – characteristic

polynomial (18 hours)

Unit 5: Trace of the matrix – definition -transpose of the matrix- definition – adjoint-

determinants- properties of determinants-Crammer’s rule – secular equation

(18 hours)

Book for Study:

Herstein I.N., Topics in Algebra, Second Edition, John Wiley and Sons Pvt. Ltd., Singapore, 2016

Unit 1 : Chapter 4 Section 4.3

Unit 2 : Chapter 4 Section 4.4

Unit 3 : Chapter 6 Sections 6.1 - 6.3


Unit 5 : Chapter 6 Sections 6.8, 6.9

Books for Reference:

Joseph A.Galiien .,Contemporary Abstract Algebra, Narosa Publications, Reprinted,2009.

Nathan Jacobson, Lectures in Abstract Algebra, Affiliated East-West Press Pvt. Ltd.,2003.

Vijay Khanna.,A course in Abstract Algebra, V Edition 2018.






Upto

CO1 Explore the concepts of dual space

with illustrations K4

CO2

Test the independence of the vectors

by applying Gram–Schmidt

orthogonalization process

K5

CO3

Characterize the nature of the roots

and determine the linear

transformation of the functions

K4

CO4

Infer on the nature of characteristic

polynomial and compares various

forms of matrix representation

K4

CO5 Conceptualize and apply the notion of

matrix and secular equation K4



PO 1

PO 2

PO 3

PO 4

PO 5

PO 6

PO 7

PO 8

PSO 1

PSO 2

PSO 3

PSO 4

PSO 5

Sum of COs with

POs & PSOs

CO1 3 3 2 3 2 3 3 2 21

CO2 3 2 2 3 2 2 3 2 19

CO3 3 3 2 3 2 3 3 2 21

CO4 3 3 2 2 2 3 2 3 3 3 26

CO5 3 3 2 2 2 3 2 3 3 3 26




Mapping Scale 1 2 3

Relation 0.01-1.0 1.01-2.0 2.1-3



2.6

Observation COs of Linear Algebra are strongly correlated with POs & PSOs



Measure and Integration





Objectives :


1. Obtain profound knowledge of differentiation – integration of vector-valued functions

2. Gain insights on continuity and convergence of sequences and series of functions

3. Comprehend the concept of Lebesgue measure

4. Acquire acquaintance of integration of functions

5. Familiarize with the notion of measure space

Unit 1: Riemann -Stieltjes Integral: definition and existence of the integral-properties –

integration and differentiation – integration of vector-valued functions –

rectifiable curves (18 hours)

Unit 2: Sequences and series of functions: uniform convergence – uniform convergence

and continuity – uniform convergence and integration – uniform convergence

and differentiation - Equicontinuous families of functions – Stone-Weierstrass

theorem (18 hours)

Unit 3: Measure on the real line – Lebesgue outer measure – σ – algebra- Borel set -

measurable sets and functions- Lebesgue measurable function – Borel function

(18 hours)

Unit 4: Integration of functions of a real variable: integration of non-negative functions –

general integral – integration of series – Riemann and Lebesgue integrals

(18 hours)

Unit 5: Measure spaces – integration with respect to a measure – Lp spaces – convex

functions – Jensen’s inequality (18 hours)

Books for Study:

Walter Rudin, Principles of Mathematical Analysis, Third Edition, Tata McGraw-Hill International

Editions, New Delhi, 2016.

Unit 1: Chapter 6

Unit 2: Chapter 7

De Barra,G., Measure Theory and Integration, First reprint, Wiley Eastern Limited, New Delhi,

2019.

Unit 3: chapter2 Sections 2.1, 2.2, 2.4

Unit 4: Chapter3 Sections 3.1 -3.4

Unit 5: Chapter5 Sections 5.5, 5.6

Chapter6 Sections 6.1- 6.3


Gupt.S.L., Nisha Rani., Principles of Real Analysis,S.D.R.Printers, 2004.

Royden.H.L., Real Analysis, Prentice Hall of India, III Edition, 2007.

Munroe,M.E. Measure and Integration. Addison-Wesley Mass. Stockholm, 2009.





Course Outcome No. Course Outcome Knowledge Level Upto

CO1 Interpret on the implications of Riemann - Stieltjes Integral

K4

CO2 Analyze equicontinuous families of functions

K4

CO3 Examine the concept of Lebesgue measure with illustrations

K4

CO4 Distinguish the implications of Riemann and Lebesgue integrals

K5

CO5 Make inferences on the conceptualization of measure space

K4



PO 1

PO 2

PO 3

PO 4

PO 5

PO 6

PO 7

PO 8

PSO 1

PSO 2

PSO 3

PSO 4

PSO 5

Sum of COs with

POs & PSOs

CO1 3 3 2 3 2 3 3 2 21

CO2 3 3 2 3 2 3 3 2 21

CO3 3 3 2 3 2 3 3 2 21

CO4 3 2 2 3 2 3 3 2 20

CO5 3 3 2 2 3 2 3 3 2 23

Grand Total of Cos with POs & PSOs 106

Mean Value of Cos with POs & PSOs = 2.6


Mapping Scale 1 2 3

Relation 0.01-1.0 1.01-2.0 2.1-3



2.6

Observation COs of Measure & Integration are strongly correlated with POs & PSOs



Differential Equations




Subject Code : 20PCHC72 Credits : 5

Objectives:


1. Understand the concepts of homogeneous equations and its characteristics.

2. Solve homogeneous equations using various methods

3. Acquire knowledge of Bessel’s equations and different methods of solving.

4. Gain insights on finding approximate solutions to differential equations

5. Apply different methods to solve linear and non-linear partial differential equations.

Unit 1: Initial value problems for the homogeneous equation – solutions of the

homogeneous equation – Wronskian and linear independence – reduction of the

order of a homogeneous equation – non-homogeneous equation

(18 hours)

Unit 2: Homogeneous equations with analytic coefficients – the Legendre equation – the

Euler equation – second order equation with regular singular points

(18 hours)

Unit 3: The Bessel’s equation – regular singular points at infinity – equations with

variables separated – exact equations – the method of successive

approximations – Lipschitz condition (18 hours)

Unit 4: Convergence of the successive approximations – non-local existence of solutions

– approximations to solutions – equations with complex valued functions –

partial differential equations – Cauchy’s problem for first order equations –

linear equations of first order (18 hours)

Unit 5: Integral surfaces passing through a given curve – surfaces orthogonal to a given

system of surfaces – non-linear partial differential equations of the first order –

Cauchy’s method of characteristics – compatible systems of first order equations

– Charpit’s method – special types of first order equations.

(18 hours)

Books for Study:

Earl A Coddington, An Introduction to Ordinary Differential Equations, Prentice-Hall India

Private Limited, New Delhi, 2014

Unit 1 : Chapter 3 Sections 3.1 – 3.6


Chapter 4 Sections 4.1 – 4.4


Chapter 5 Sections 5.1 - 5.5

Unit 4 : Chapter 5 Sections 5.6 to 5.9

Ian Sneddon, Elements of Partial Differential Equations, McGraw-Hill International Editions,

Singapore, 2006




Deo S.G. and Raghavendra.V, Text Book of Ordinary Differential equations and Stability Theory,

Tata McGraw Hill Publishing Company Ltd., New Delhi,2007.

SankarRao.S, Introduction to Partial Differential Equations, Prentice Hall of India, New Delhi.

2016. ]





Course Outcome No. Course Outcome Knowledge Level Upto

CO1 Solve homogeneous equations by using suitable methods

K3

CO2 Apply Legendre and Euler equations and make inferences on the solutions to the differential equations

K4

CO3 Calculate solutions to Bessel’s equation K3

CO4 Find solutions to Cauchy problem and linear equations

K3

CO5 Determine solutions to linear and non-linear partial differential equations

K3



PO 1

PO 2

PO 3

PO 4

PO 5

PO 6

PO 7

PO 8

PSO 1

PSO 2

PSO 3

PSO 4

PSO 5

Sum of COs with

POs & PSOs

CO1 3 3 2 2 3 3 3 3 2 3 27

CO2 3 3 2 2 3 3 2 3 3 3 27

CO3 3 3 2 2 3 3 3 3 2 3 27

CO4 3 3 2 2 3 3 3 3 2 3 27

CO5 3 3 2 2 3 3 3 3 2 3 27




Mapping Scale 1 2 3

Relation 0.01-1.0 1.01-2.0 2.1-3



2.7

Observation COs of Differential Equations are strongly correlated with POs & PSOs



Differential Geometry


Class : I M.Sc. Mathematics Part : Core Elective 2a


Subject Code : 20PMAE22 Credits : 4

Objectives:


1. Gain comprehension on the concepts of spaces, surfaces.

2. Acquire acquaintance of the notion of tangent surfaces, involutes and evolutes.

3. Get familiarize with intrinsic properties of surfaces

4. Comprehend the knowledge of Geodesics and their properties.

5. Obtain in-depth conceptualization of developable curves and minimal surfaces.

Unit 1: Space curves-Definition of a space curve - Arc length - tangent - normal and

binormal - curvature and torsion - contact between curves and surfaces

(18 hours)

Unit 2: Tangent surface - Involutes and evolutes - Intrinsic equations - Fundamental

Existence Theorem for space curves - Helices.

(18 hours)

Unit 3: Local intrinsic properties of a surface - Curves on a surface - Surface of revolution

- Helicoids - Metric - Direction coefficients - Families of curves - Isometric

correspondence - Intrinsic properties. (18 hours)

Unit 4: Geodesics - Canonical geodesic equations - Normal property of geodesics -

Geodesics curvature - Gauss - Bonnet Theorem - Gaussian curvature.

(18 hours)

Unit 5: Local non-intrinsic properties of a surface-Principal curvature - Lines of

curvature - Developable - Developable associated with space curves and with

curves on surface - Minimal surfaces - Ruled surfaces.

(18 hours)

Book for Study:

WillmoreT.J., An Introduction to Differential Geometry, Oxford University Press, New Delhi

2014.

Unit 1 : Chapter 1 Sections 1 -4

Unit 2 : Chapter 2 Sections 5 - 9





Somasundaram.D., Differential Geometry A First Course, Narosa Publications, 2018.

Kobayashi.S. and Nomizu. K., Foundations of Differential Geometry, Interscience Publishers,

Gneva, 2007.






Upto

CO1 Illustrate the concepts of spaces,

surfaces K4

CO2

Characterize the nature of the surfaces

and find the involute and evolute of

various surfaces

K3

CO3 Infer on the intrinsic properties of

surfaces and Helicoids K4

CO4 Determine Geodesics of different

surfaces K3

CO5 Compare the properties of developable

curves and minimal surfaces K5



PO 1

PO 2

PO 3

PO 4

PO 5

PO 6

PO 7

PO 8

PSO 1

PSO 2

PSO 3

PSO 4

PSO 5

Sum of COs with

POs & PSOs

CO1 3 3 2 3 2 3 3 2 21

CO2 3 3 2 3 3 3 2 2 21

CO3 3 3 2 3 2 3 3 2 21

CO4 3 3 2 3 3 3 2 2 21

CO5 3 2 2 3 2 3 3 2 20




Mapping Scale 1 2 3

Relation 0.01-1.0 1.01-2.0 2.1-3



2.6

Observation COs of Differential Geometry are strongly correlated with POs & PSOs



Stochastic Processes


Class : I M.Sc. Mathematics Part : Core Elective 2b Semester : II Hours : 90 Subject Code : 20PMAE22 Credits : 4

Objectives:


1. Acquire intense knowledge on the underlying concepts of Stochastic processes

2. Familiarize with Markov chain and system

3. Obtain in-depth understanding of birth and death process

4. Develop the acquaintance with applications of Markov process

5. Comprehend the concept of renewal process

Unit 1: Stochastic Processes -Specification of stochastic processes – stationary processes

– MarkovChains: Definitions and Examples – Higher transition probabilities –

Generalization of independent Bernoulli trials.

(18 hours)

Unit 2: Markov Chains-Stability of Markov system – Graph theoretic approach – Markov

chain with denumerable number of state – Reducible chains – Statistical

inference for Markov chains. specification of stochastic processes – stationary

processes – MarkovChains: Definitions and Examples – Higher transition

probabilities – Generalization of independent Bernoulli trials.

(18 hours)

Unit 3: Poisson process: Poisson process and related distributions – Generalizations of

Poisson process – Birth and death process – Markov process with discrete state

space (Continuous time Markov chain).

(18 hours)

Unit 4: Markov Process with continuous state space -Brownian motion – Weiner process

– Differential equations for Weiner Process – Kolmogorov equations – First

passage time distribution for Weiner process. (18 hours)

Unit 5: Renewal process and renewal equation – Stopping time – Wald’s equation –

Renewal theorem – Delayed and equilibrium renewal process. (18 hours)

Book for Study:

Medhi.J.,Stochastic Processes, New Age International, Cochin, 2nd edition 2017.


Chapter 3 Sections 3.1 - 3.5




Unit 5: Chapter 6 Sections6.1 to 6.6


Leo Breiman., Probability and Stochastic Processes, Houghton Mifflin,2008

Athanasios Papoulis., Probability Random Variable & Stochastic Process, Mc Graw

Hill, International ,II Edition,2004.






Upto

CO1 Correlate the concepts of stochastic

processes with illustrations K4

CO2 Illustrate Markov chain and its

applications K4

CO3 Compare the conceptualization of

pure birth and death process K4

CO4 Apply Markov process in solving

problems K3

CO5 Summarize the concepts of renewal

process and its applications K5



PO 1

PO 2

PO 3

PO 4

PO 5

PO 6

PO 7

PO 8

PSO 1

PSO 2

PSO 3

PSO 4

PSO 5

Sum of COs with

POs & PSOs

CO1 3 3 2 2 2 3 2 3 3 2 25

CO2 3 3 2 2 2 3 2 3 3 2 25

CO3 3 3 2 2 2 3 2 3 3 2 25

CO4 3 3 2 2 2 3 2 3 2 2 24

CO5 3 2 2 2 2 3 2 3 3 2 24




Mapping Scale 1 2 3

Relation 0.01-1.0 1.01-2.0 2.1-3



2.5

Observation COs of Stochastic Processes are strongly correlated with POs & PSOs



Mathematics for Life


Class : I M.Sc. (Other Major) Part : NME-1


Subject Code : 20PMAN12 Credits : 4

Objectives:


1. Acquire knowledge of applications of differentiation in modeling solution

2. Comprehend the implications of integration in finding solutions to financial problems

3. Gain insights on the significance of correlation

4. Use various techniques of regression

5. Enrich their computational skills

Unit 1: Differentiation – laws of differentiation – second order derivatives – applications

– economic analysis – rates of growth – price elasticity – maxima and minima

(12 hours)

Unit 2: Integration – techniques of integration – applications – cost-revenue – profit

function – investment and capital formation rate of sales

(12 hours)

Unit 3: Correlation analysis: correlation and their calculation (positive, negative and

simple) – Karl Pearson’s correlation coefficient and rank correlation – merits and

demerits – (12 hours)

Unit 4: Regression analysis: simple, linear and their properties and their computations –

applications to life and social science problems. (12 hours)

Unit 5: Numerical ability:simplification – problems on ages – time and work- time and

distance-simple interest and compound interest – percentage – ratio and

proportion- calendar. (12 hours)

Books for Study:

HarbansLal, Business Mathematics, Sultan Chand Education Publishers, New Delhi, 2013

Unit 1: Chapter 9 Sections9.1-9.

Unit 2: Chapter 11 Sections11.1-11.

Gupta S.P., Statistical Methods, Sultan Chand Educational Publishers, New Delhi, 2005.

Unit 3: Volume – I Chapter:10 (pages 378-394;399-412)

Volume – I Chapter:11 (pages 436 – 451)

Unit 4: Volume – II Chapters: 4, 5

Aggarwal.R.S,QuantitativeAptitude,Narosa Publishing House, New Delhi,2020

Books for Reference

Gupta, B.N., Statistics (Theory and Practice), SahityaBhavan, Agra, 4th Edition, 2004.

Sancheti, D.C. and Kapoor, V.K., Statistics (Theory, Methods and Applications), Sultan Chand and

Sons, New Delhi, 7th Edition, 2017.






Upto

CO1 Determine solutions to the economic

and growth rate problems K3

CO2 Solve financial problems K3

CO3

Apply the techniques of correlation to

find solutions to social science

problems

K3

CO4

Calculate solutions and make

inferences for decision making using

regression

K4

CO5 Find solutions to computational

problems. K3



PO 1

PO 2

PO 3

PO 4

PO 5

PO 6

PO 7

PO 8

PSO 1

PSO 2

PSO 3

PSO 4

PSO 5

Sum of COs with

POs & PSOs

CO1 3 3 2 3 3 3 2 19

CO2 3 3 2 3 3 3 2 19

CO3 3 3 2 3 3 3 3 2 22

CO4 3 3 3 2 3 3 3 2 22

CO5 3 3 2 3 3 3 2 19


Mean Value of COs with POs & PSOs =

2.72


Mapping Scale 1 2 3

Relation 0.01-1.0 1.01-2.0 2.1-3



2.72

Observation COs of Mathematics for Life are strongly correlated with POs & PSOs



TOPOLOGY


Class : II M.Sc. Mathematics Part : Core - 8 Semester : III Hours : 90 Course Code : 20PMAC83 Credits : 5

Objectives:

To enable the learners comprehend the notion of topology and its kinds

To create conducive environment for the students to learn about continuity

To make the pupil understand the concept of connected spaces

To bestow the knowledge of compact spaces to the students

To assist the learners in being acquainted with the applications of countability

and separation axioms

Unit 1 Topological spaces – definition of topological space – basis for a topology – order

topology – product topology – projections – subspace topology

(18 hours)

Unit 2: Closed set and limit point – closure and interior of a set – Hausdorff space

continuous functions – homeomorphism- the pasting lemma – metric topology –

sequence lemma – uniform limit theorem (18 hours)

Unit 3: Connected spaces – definition – connected subsets in the real line –

intermediate value theorem – path connected – components and path

components – local connectedness (18 hours)

Unit 4: Compact spaces – definition – tube lemma – finite intersection property –

compact subspaces of the real line – Lebesgue number lemma – uniform

continuity theorem – limit point compactness – local compactness

(18 hours)

Unit 5: The countability axioms – Lindelof space – regular and normal space – the

separation axioms – Urysohn’s lemma – completely regular – Urysohn’s

metrization theorem – imbedding theorem – the Tietze extension theorem

(18 hours)

Book for Study:

James, R. Munkres, Topology, II Edition, Pearson India Education Services Pvt.Ltd,2015.

Unit 1 : Chapter 2 sections 12 - 16



Unit 4: Chapter 3 sections 26 - 29

Unit 5: Chapter 4 sections 30 - 35

References:

1. George F. Simmons, Introduction to Topology and Modern Analysis, Tata McGraw-Hill,

16th Reprint, 2011.

2. Chandrasekhara Rao, K., Topology, Narosa Publishing House, 2nd Reprint, 2015.






Upto

CO1 Explain various kinds of topologies

with illustrations K3

CO2 Deduce the implications of lemmas

related to continuous functions K4

CO3 Interpret on the theorems associated

with connectedness K4

CO4 Infer on the concept of compactness

and its related theorems K4

CO5

Examine the interventions of

separation axioms in proving Urysohn

lemma and Urysohn Metrization

theorem

K4

K1 = Remember, K2 = Understand, K3 = Apply, K4 = Analyze, K5 = Evaluate, K6 = Create

Mapping Course Outcomes with

PO 1

PO 2

PO 3

PO 4

PO 5

PO 6

PO 7

PO 8

PSO 1

PSO 2

PSO 3

PSO 4

PSO 5

Sum of COs with

POs & PSOs

CO1 3 2 3 2 3 3 3 19

CO2 3 2 3 2 3 2 2 17

CO3 3 2 3 2 3 2 2 17

CO4 3 2 3 2 2 3 2 17

CO5 3 2 3 2 3 3 2 18




Mapping Scale 1 2 3

Relation 0.01-1.0 1.01-2.0 2.1-3



2.5

Observation COs of Topology are strongly correlated with POs & PSOs



CLASSICAL MECHANICS


Class : II M.Sc. Mathematics Part : Core - 9

Semester : III Hours : 90

Course Code : 20PMAC93 Credits : 5

Objectives :

To introduce the concept of D’ Alembert’s Principle and applications of Lagrangian

Formulation to the learners

To enlighten the student’s comprehension on the concept Lagrangian Formulation for

Non Holonomic System

To help the students understand central orbits and related theorems

To expose the students, the applications of Kepler’s law

To familiarize the applications of Legendre Transformation

Unit 1: D’Alembert’s principle and Lagrange’s equations – velocity dependent potentials

– dissipative function – applications of Lagrangian formulation.

(18 hours)

Unit 2: Hamilton’s principle – some techniques of the calculus of variations – derivation

of Lagrange’s equations forms – Hamilton’s principle – Hamilton’s principle to

non holonomic systems.

(18 hours)

Unit 3: The two-body central force problem – classification of orbits – Virial theorem –

differential equation for the orbit and integrable power – law potentials.

(18 hours)

Unit 4: Betrand’s theorem – Kepler’s problem inverse square law force – Kepler’s

equation of motion & first integrals – Laplace – Runge – Lenz Vector.

(18 hours)

Unit 5: Legendre Transformations – the Hamilton equation of motion – Routh Procedure

– derivation of Hamilton equation from variation principle – the principle of least

action. (18 hours)

Book for Study:

Herbert Goldstein, “Classical Mechanics, 2nd Edition”, Twentieth Reprint, Narosa Publishing

House, New Delhi, 2007.




Unit 4: Chapter 3 Section 3.6 – 3.9

Unit 5: Chapter 8 Sections 8.1 – 8.3, 8.5, 8.6

References:

1. D. T. Greenwood, “Classical Dynamics”, Prentice Hall of India, New Delhi, 1985.

2. N.C.Rane and P.S.C.Joag, “Classical Mechanics”, Tata McGraw Hill, 1991.






Upto

CO1 Apply Lagrangian’s equation to

various dynamical systems K3

CO2 Employ Hamilton’s principle to non-

holonomic system K3

CO3

Determine the differentiation of

central orbits and apply the

respective theorems to the problems

K3

CO4 Analyse the implications of Kepler’s

law K4

CO5 Summarize the applications of

Legendre’s transformation K5



PO 1

PO 2

PO 3

PO 4

PO 5

PO 6

PO 7

PO 8

PSO 1

PSO 2

PSO 3

PSO 4

PSO 5

Sum of COs with

POs & PSOs

CO1 2 2 3 3 2 3 2 2 19

CO2 2 2 3 2 3 2 3 2 2 21

CO3 2 2 3 3 2 3 2 2 19

CO4 3 2 3 3 2 3 2 2 20

CO5 2 2 2 2 2 3 2 3 2 2 22




Mapping Scale 1 2 3

Relation 0.01-1.0 1.01-2.0 2.1-3



2.3

Observation COs of Classical Mechanics are strongly correlated with POs & PSOs



Complex Analysis




Course Code : 20PMAD03 Credits : 5

Objectives :

To introduce the idea of analytic functions and related concepts to the learners.

To make the students comprehend on conformal mappings and transformations.

To enable the pupil understand and evaluate partial derivatives and integrals of multi

variable functions.

To assist the learners gain knowledge on the implications of Taylors theorem and residue

theorem.

To facilitate pupil in knowing about power series expansions and Weierstrass theorem

Unit 1: Analytic functions – polynomials – rational functions – sequences – series –

uniform convergence – power series – Abel’s limit theorem. (18 Hours)

Unit 2: Analytic functions in regions – conformal mapping – length and area – linear

transformations – cross ratio – elementary conformal mappings – elementary

Riemann surfaces. (18 Hours)

Unit 3: Complex integration – Cauchy’s theorems – Cauchy’s integral formula – higher

derivatives. (18 hours)

Unit 4: Removable singularities – Taylor’s theorem – zeros and poles – general form of

Cauchy’s theorem – calculus of residues – residue theorem – evaluation of

definite integrals. (18 hours)

Unit 5: Harmonic functions – Poisson’s formula – Schwarz’s theorem – power series

expansions – Weierstrass theorem – Taylor’s series – Laurent series

(18 hours)

Book for Study:

Lars V Ahlfors, “Complex Analysis”, Tata McGraw-Hill International Edition, Singapore, Third

Edition, 1979.

Unit 1: Chapter 2 Sections 1.2 – 1.4, 2.1 – 2.5

Unit 2: Chapter 3 Sections 2.2 – 2.4, 3.1 – 3.5, 4.1 – 4.3

Unit 3: Chapter 4 Sections 1.1, 1.3 – 1.5, 2.1 – 2.3

Unit 4: Chapter 4 Sections 3.1 – 3.4, 4.1, 4.4 – 4.6, 5.1 – 5.3

Unit 5: Chapter 4 Sections 6.1 – 6.4 &

Chapter5 Sections1.1 – 1.3

References:

1. J.B. Conway, “Functions of one complex variables”, Springer - Verlag, International student

Edition, Narosa Publishing Co., 1978.

2. E. Hille, “Analytic function Theory”,(2 vols.), Gonm & Co, 1959.






Upto

CO1

Analyze the nature of the analytical

functions and find the radius of

convergence of power series

K4

CO2 Test the conformality of mappings and

compare various transformations K4

CO3

Evaluate complex contour integrals by

applying Cauchy’s theorem and Integral

formula

K3

CO4

Classify singularities and poles, find residues

and evaluate complex integrals using the

residue theorem. K4

CO5

Illustrate and express the functions of a

complex variable as Taylor’s series and

Laurent’s series

K3



PO 1

PO 2

PO 3

PO 4

PO 5

PO 6

PO 7

PO 8

PSO 1

PSO 2

PSO 3

PSO 4

PSO 5

Sum of COs with

POs & PSOs

CO1 3 2 3 2 3 2 2 17

CO2 3 3 2 3 2 3 2 2 20

CO3 3 2 2 3 2 3 2 2 19

CO4 3 3 2 3 2 3 2 2 20

CO5 3 2 3 2 3 2 2 17




Mapping Scale 1 2 3

Relation 0.01-1.0 1.01-2.0 2.1-3



2.4

Observation COs of Complex Analysis are strongly correlated with POs & PSOs


OPERATIONS RESEARCH (For those who joined in June 2020 onwards)

Class : II M.Sc. Mathematics Part : Core - 11 Semester : III Hours : 90 Course Code : 20PMAD13 Credits : 5

Objectives:

To enable the learners understand the characteristics of network models, and comprehend the concept of PERT-CPM.

To familiarize the concept of probabilistic inventory model to pupil

To disseminate the knowledge of queuing models to the pupil.

To facilitate the learners understand the notion of classical optimization theory.

To assist the pupil in applying the non-linear programming algorithms. Unit 1: Network models - definitions – minimal spanning tree problem – shortest route

algorithms - problems – maximal flow problem – capacitated network model – linear programming formulation- PERT-CPM- critical path calculations – construction of the time chart and resource leveling –Linear programming formulation of CPM- PERT calculation. (18 hours)

Unit 2: Probabilistic inventory models- continuous review models- probabilistic EOQ model – Single period models- No-setup model (Newsvendor model)- Setup policy (s-S policy)- Multiperiod model. (18 hours)

Unit 3: Queuing models - definitions – Poisson and Exponential distributions – pure birth model and pure death model – generalized Poisson model – steady-state measures – Single server models – Multiple server models. (18 hours)

Unit 4: Classical optimization theory - unconstrained extremal problems – Newton – Raphson method – constrained extremal problems – equality constraints. (18 hours)

Unit 5: Nonlinear programming algorithms - unconstrained nonlinear algorithms – direct search method – gradient method – constrained nonlinear algorithms – separable programming – quadratic programming (18 hours)

Book for Study: Taha H.A., “Operations Research – An Introduction”, IX Edition, Pearson Education Inc,2011 Unit :1 Chapter 6 Sections : 6.1 – 6.5.5 Unit :2 Chapter 14 Sections : 14.1-14.3 Unit :3 Chapter 15 Sections : 16.1 – 16.6 Unit :4 Chapter 18 Sections: 18.1, 18.2 Unit :5 Chapter 19 Sections : 19.1, 19.2.1,19.2.2

References: 1. Kantiswaroop, P.K.Gupta and Manmohan, “Operations Research”, Sultan Chand & Sons, New Delhi,15th edition, reprinted 2011. 2. Sharma., “Operations Research”, 2nd Edition, Vikas Publishing House Private Limited, New Delhi, 2002.






Upto

CO1 Find optimal solutions to network problems using algorithmic approaches

K3

CO2 Calculate optimal order quantity of probabilistic inventory models

K3

CO3 Analyze the applications of different types of queues in real life situations.

K4

CO4 Solve the problems using classical optimization theory

K3

CO5 Apply various methods to solve for non linear programming problems.

K3



PO 1

PO 2

PO 3

PO 4

PO 5

PO 6

PO 7

PO 8

PSO 1

PSO 2

PSO 3

PSO 4

PSO 5

Sum of COs with

POs & PSOs

CO1 3 2 2 2 2 3 3 3 2 2 24

CO2 3 2 2 2 2 3 3 3 2 2 24

CO3 3 2 2 2 2 3 3 3 2 2 24

CO4 3 2 2 2 2 3 3 3 2 2 24

CO5 3 2 2 2 2 3 3 3 2 2 24




Mapping Scale 1 2 3

Relation 0.01-1.0 1.01-2.0 2.1-3



2.4

Observation COs of Operations Research are strongly correlated with POs & PSOs



MATHEMATICAL MODELLING

Class : II M.Sc. Mathematics Part : Core Elective – 3a

Semester : IV Hours : 90

Course Code : 20PMAE33 Credits : 4

Objectives :

Comprehend the basic modelling concepts in day to day life.

Familiarize the modeling techniques using statistical methods.

Acquire profound knowledge of probabilistic modelling

Obtain the comprehension of modelling using graphs of functions

Formulate models using differential equations

Unit 1 : Modelling change with difference equations- approximating change with

difference equations – Solutions to Dynamical systems- Systems of difference

equations. (18 hours)

Unit 2 : Mathematical Models – Modeling using proportionality – Modeling using

Geometric similarity – Automobile Gasoline Mileage- Body weight & height,

Strength & Agility – fitting models to data graphically – Analytic Methods of

model fitting – applying the least squares Criterion – choosing a best model.

(18 hours)

Unit 3 : Probabilistic modelling with discrete systems- Modelling component and system

Reliability- Linear Regression. (18 hours)

Unit 4 : An Arms race- Modelling an arms race in stages – Managing non- renewable

Resources- Effects of Taxation on the Energy crisis- A Gasoline shortage and

Taxation. (18 hours)

Unit 5 : Population growth- prescribing Drug Dosage- Braking Distance Revisited-

Graphical solutions of Autonomous Differential equations- Numerical

Approximation Methods. (18 hours)

Book for study: Giardano, Weir & William, A First Course in Mathematical Modeling, Thompson

Asia & China press (First published by Brookes & Cole, Thompson’s Learning).

Unit 1: Chapter 1: Section 1.1 to 1.4

Unit 2: Chapter 2: Section 2.1 to 2.3 & Chapter 3 – Section 3.1 to 3.4





1. Principles of Mathematical Modeling (Ideas, Methods, Examples) A.A. Samarskii, A.P.

Mikhailov © 2002 by Taylor & Francis Group, LLC

2. Mathematical modelling- Applications with GeoGebra., Jonas Hall and Thomas

Lingefjärd @ 2017 by John Wiley & Sons.






Upto

CO1 Apply difference equations in model

formulation K3

CO2 Use statistical methods in framing

mathematical models K3

CO3 Infer the applications of probabilistic

modeling to discrete systems K4

CO4

Illustrate the applications of mathematical

models using graphs of functions in

resource management

K4

CO5 Interpret the intervention of differential

equations in mathematical modeling. K4



PO 1

PO 2

PO 3

PO 4

PO 5

PO 6

PO 7

PO 8

PSO 1

PSO 2

PSO 3

PSO 4

PSO 5

Sum of COs with

POs & PSOs

CO1 3 3 2 2 2 3 3 3 2 3 26

CO2 3 3 2 2 2 3 3 3 2 3 26

CO3 3 3 2 2 2 3 3 3 2 3 26

CO4 3 3 2 2 2 3 3 3 2 3 26

CO5 3 3 2 2 2 3 3 3 2 3 26




Mapping Scale 1 2 3

Relation 0.01-1.0 1.01-2.0 2.1-3



2.6

Observation COs of Mathematical Modelling are strongly correlated with POs & PSOs

ARUL ANANDAR COLLEGE (AUTONOMOUS), KARUMATHUR


Integral Equations


Class : II M.Sc. Mathematics Part : Core Elective – 3b



Objectives:

To facilitate the learners understand the different concepts and methods in convolution

integrals.

To give idea on Fredholm integrals in solving problems to the pupil.

To equip students with the concept of symmetric kernel, Eigen values and functions in

complex Hilbert spaces.

To assist students in finding solution of Hilbert type singular integral equations.

To help out students understand Integral transforms method.

Unit 1: Definitions of integral equations- regularity conditions- special kinds of kernels-

eigenvalues and eigen functions- convolution integrals- reduction to a system of

algebraic equations-examples- an approximate method. (18 hours)

Unit 2: Iterative schemes for Fredholm integral equations of second kind- examples –

iterative scheme for Volterra integral equations of second kind, examples- Some

results about resolvent kernel-classical Fredholm theory-Fredholm theorem .

(18 hours)

Unit 3: Symmetric kernels- complex Hilbert space-ortho normal systems of functions-

fundamental properties of eigen values and eigen functions for symmetric

kernels-expansion in eigen function and bilinear form. (18 hours)

Unit 4: Hilbert Schmidt theorem and some immediate consequences- solution of a

symmmetric integral equations, examples-solution of the Hilbert type singular

integral equation, examples. (18 hours)

Unit 5: Integral transforms method- Fourier transform- Laplace transforms application

to Volterra integral equations with convolution types of kernels, examples.

(18 hours)

Book for study:

R.P. Kanwal,” Linear Integral Equations-Theory and technique.” Springer Science and Business

Media, 2 ndedition, 2012.

Unit 1 : Chapter 1, 2 Sections 1.1 – 1.5, 2.1 – 2.5

Unit 2 : Chapter 3, 4 Sections 3.1 – 3.5, 4.1 - 4.5


Unit 4 : Chapter 7, 8 Sections 7.4 – 7.6, 8.1, 8.2

Unit 5 : Chapter 9 Sections 9.1 – 9. 5

References:

1. S.G. Mikhlin, “Linear integral equations” (Translated from Russian) Hindustan Book Agency,

2005.

2. B.L. Moiseiwitsch, “Integral Equations” Longman, London & New York, 2003.






Upto

CO1 Apply the concept and classification of

integral equations and kernels K3

CO2 Solve the Volterra and Fredholm integral

equations by different techniques K3

CO3

Analyze the concept of eigen values and

eigen functions for symmetric kernels and

their properties

K4

CO4 Calculate solutions to Hilbert type singular

integral equation K3

CO5 Use various transformations to solve

integral equations K3



PO 1

PO 2

PO 3

PO 4

PO 5

PO 6

PO 7

PO 8

PSO 1

PSO 2

PSO 3

PSO 4

PSO 5

Sum of COs with

POs & PSOs

CO1 3 2 2 3 2 3 2 2 19

CO2 3 2 2 3 2 3 2 2 19

CO3 3 2 2 3 2 3 2 2 19

CO4 3 2 2 3 2 3 2 2 19

CO5 3 2 3 3 2 3 2 2 20




Mapping Scale 1 2 3

Relation 0.01-1.0 1.01-2.0 2.1-3



2.4

Observation COs of Integral Equations are strongly correlated with POs & PSOs

RUL ANANDAR COLLEGE (AUTONOMOUS), KARUMATHUR – 625 514


Functional Analysis





Objectives:

To strengthen the student’s knowledge on linear spaces, algebras and linear

transformations

To introduce Banach spaces and to make them understand Hahn-Banach and open

mapping theorems and their applications

To explicate the notion of Hilbert spaces

To institute in the learners the concepts of operators and projections

To expound spectral theory and related concepts to the students

Unit 1: Algebraic systems – Linear spaces – dimension of a linear space – linear

transformations – algebras (18 hours)

Unit 2: Banach spaces – continuous linear transformations – Hahn – Banach theorem –

natural imbedding of N in N** – open mapping theorem – conjugate of an

operator (18 hours)

Unit 3: Hilbert spaces – orthogonal complements – orthogonal sets – Bessel’s inequality

– conjugate space H* (18 hours)

Unit 4: Adjoint of an operator – self adjoint operators – normal and unitary operators –

projections (18 hours)

Unit 5: Finite dimensional spectral theory – matrices – determinants – spectrum of an

operator – spectral theorem (18 hours)

Book for Study:

Simmons, G. F., Introduction to Topology and Modern Analysis, Tata McGraw-Hill Publishing Co.

Ltd., New Delhi, 2006.






References:

1. Walter Rudin, Functional Analysis, Tata McGraw-Hill publishing Co. Ltd., New Delhi, 2006.

2. Casper Goffman and George Pedrick, First Course in Functional Analysis, Prentice Hall of India

Private Ltd., 1987.






Upto

CO1 Apply their knowledge on linear spaces and linear transformations

K3

CO2 Examine the theoretical justifications in Hahn-Banach and Open Mapping theorems and deduce a few applications

K4

CO3 Infer the geometrical properties of orthogonality in Hilbert Spaces

K3

CO4 Classify various operators on Hilbert Spaces K3

CO5 Illustrate the concepts and implications of finite dimensional spectral theory

K4



PO 1

PO 2

PO 3

PO 4

PO 5

PO 6

PO 7

PO 8

PSO 1

PSO 2

PSO 3

PSO 4

PSO 5

Sum of COs with

POs & PSOs

CO1 3 3 2 2 2 3 2 3 20

CO2 3 3 2 3 3 3 2 3 22

CO3 3 3 2 3 3 3 2 3 22

CO4 3 3 2 3 3 3 2 3 22

CO5 3 3 2 2 3 3 2 3 21




Mapping Scale 1 2 3

Relation 0.01-1.0 1.01-2.0 2.1-3



2.7

Observation COs of Functional Analysis are strongly correlated with POs & PSOs



Fuzzy Sets and Applications





Objectives :

To enable learners compare fuzzy and crisp sets.

To familiarize the students with operations on fuzzy sets.

To make the pupil acquainted with fuzzy number and fuzzy relations.

To equip students with the knowledge of fuzzy controller.

To enable the learners apply fuzzy sets in decision making

Unit 1: Fuzzy sets- types-terminology- properties of α-cuts - representation of fuzzy sets–

decomposition theorems of fuzzy sets-extension principle for fuzzy sets.

(18 hours)

Unit 2: Operation on fuzzy sets – types of operations – fuzzy compliments –First, Second

Characterization theorem of fuzzy complements - fuzzy intersections: t-norms – fuzzy

union: t-co norms - combination of operations. (18 hours)

Unit 3: Fuzzy arithmetic – fuzzy number – linguistic variables – arithmetic operation on

intervals – arithmetic operations on fuzzy numbers – binary fuzzy relation- fuzzy

equivalence relation-fuzzy compatibility relation-fuzzy ordering relation

(18 hours)

Unit 4: Overview of fuzzy controller and examples– fuzzy systems and neural networks -

fuzzy neural networks- fuzzy automata- fuzzy dynamic systems (18 hours)

Unit 5: Individual Decision Making - Multi person Decision Making - Multi Criteria Decision

Making – Multi Stage Decision Making – Fuzzy Ranking Methods. (18 hours)

Book for Study :

01. George J.Klir and Bo Yuan, “Fuzzy Sets and Fuzzy Logic Theory and Applications”, PHI

Learning Private Limited, New Delhi, 2009.








References :

01. Zimmermann, “Fuzzy set theory and its applications” Affiliated East West Press Pvt Ltd, 2nd

Edition, 1996.

02. George J.Klir and Tina A.Folger, “Fuzzy sets, Uncertainty and information” PHI Learning Pvt

limited, New Delhi, 2009.






Upto

CO1 Compare the properties of crisp and fuzzy

sets K4

CO2 Examine the various operations of fuzzy

sets and characterization theorems K4

CO3 Differentiate the concepts of fuzzy number

with illustrations and distinguish fuzzy

relations from crisp

K4

CO4 Analyze the implications of crisp and fuzzy

sets in various perspectives K4

CO5 Construct fuzzy decision-making models K3



PO 1

PO 2

PO 3

PO 4

PO 5

PO 6

PO 7

PO 8

PSO 1

PSO 2

PSO 3

PSO 4

PSO 5

Sum of COs with

POs & PSOs

CO1 3 3 3 3 3 2 2 19

CO2 3 3 3 3 3 2 2 19

CO3 3 3 3 3 3 2 2 19

CO4 3 3 3 3 3 3 3 3 2 3 29

CO5 3 2 3 3 3 3 3 3 2 3 28




Mapping Scale 1 2 3

Relation 0.01-1.0 1.01-2.0 2.1-3



2.8

Observation COs of Fuzzy sets & Applications are strongly correlated with POs & PSOs



Project



Semester : IV Credits : 5

Course Code : 20PMAD44

To make the learners acquire intense knowledge on the nuances of research

To facilitate the pupil apply the mathematical concepts to design solutions to

social problems

To explore the intervention of mathematical ideas in various fields

To enrich the skill of documentation

To enable the students appreciate the manifestation of mathematics in realm

Outline:

The students undertake the project during the IV semester after the preliminary steps of

student allotment to staff and topic selection in the III semester.

The students must attend atleast one conference/seminar at international/national/state level

and it is made mandatory for internal assessment.

The student’s progress is periodically assessed by the project guide through tests and

presentation.

The significant research work is encouraged for presentations and publications in conferences

and journals

Evaluation:

Internal – 50 Marks

Certificate of Participation / Presentation in conferences / seminars at international / national /

state level – 10 Marks

Internal Viva-Voce- 15 Marks

Dissertation – 25 Marks

External Viva-Voce – 50 Marks

Total – 100 Marks




Upto

CO1 Gain new insights and apply in the

respective field of study K3

CO2 Illustrate the concept of lab to land in the

project K3

CO3

Develop and apply the nuances of

documentation of the works based on

mathematical conceptualizations and

implications

K3

CO4 Appraise and appreciate mathematical

interventions in real life scenario K4

CO5

Design innovative projects with the

application of mathematical concepts

towards scientific and societal

development

K6



PO 1

PO 2

PO 3

PO 4

PO 5

PO 6

PO 7

PO 8

PSO 1

PSO 2

PSO 3

PSO 4

PSO 5

Sum of COs with

POs & PSOs

CO1 3 2 2 2 3 2 3 3 2 3 3 28

CO2 3 2 3 3 3 2 3 3 3 3 3 31

CO3 2 2 2 2 3 2 13

CO4 3 2 3 3 3 2 3 3 3 3 3 31

CO5 3 2 3 2 3 2 3 3 3 3 3 30




Mapping Scale 1 2 3

Relation 0.01-1.0 1.01-2.0 2.1-3



2.7

Observation COs of Project are strongly correlated with POs & PSOs



Automata Theory


Class : II M.Sc. Mathematics Part : Core Elective - 4a



Objectives

Acquire profound knowledge on the conceptualization of finite automata

Comprehend various kinds of grammar

Familiarize with the notion of context free languages

Obtain the conceptualization of lexical analyzer

Attain the comprehension of parsing techniques

Unit 1 : Finite Automata and Regular expressions- Definitions and examples -

Deterministic and Nondeterministic finite Automata- Finite Automata with

moves. (18 hours)

Unit 2 : Context free grammar- Regular expressions and their relationship with

automation - Grammar - Ambiguous and unambiguous grammars - Derivation

trees – Chomsky Normal form. (18 hours)

Unit 3 : Pushdown Automaton- - Definition and examples - Relation with Context free

languages. (18 hours)

Unit 4 : Finite Automata and lexical analysis- Role of a lexical analyzer - Minimizing the

number of states of a DFA -Implementation of a lexical analyzer.

(18 hours)

Unit 5 : Basic parsing techniques- Parsers - Bottom up Parsers - Shift reduce - operator

precedence - Top down Parsers - Recursive descent - Predictive parsers.

(18 hours)

Books for Study:

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

Computations, Narosa Publishing House, Chennai, 2000.


Unit 2 : Chapters 2, 4 Sections 2.5, 4.1 – 4.3, 4.5


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

Chennai, 2002.



References:

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.






Upto

CO1 Illustrate finite automata and its kinds K3

CO2 Compare various kinds of grammars and

its implications K4

CO3 Establish automation in relation with

context free languages K3

CO4 Analyze the role of lexical analyzer K4

CO5 Apply the parsing techniques in generating

strings K3



PO 1

PO 2

PO 3

PO 4

PO 5

PO 6

PO 7

PO 8

PSO 1

PSO 2

PSO 3

PSO 4

PSO 5

Sum of COs with

POs & PSOs

CO1 3 2 3 2 3 2 2 17

CO2 3 3 3 2 3 2 2 18

CO3 3 2 3 2 3 2 2 17

CO4 3 2 3 2 3 2 2 17

CO5 3 3 3 2 3 2 2 18




Mapping Scale 1 2 3

Relation 0.01-1.0 1.01-2.0 2.1-3



2.5

Observation COs of Automata Theory are strongly correlated with POs & PSOs



Theory of Numbers


Class : II M.Sc. Mathematics Part : Core Elective - 4b



Objectives

To make the learners acquire knowledge of Euclidean algorithm and its related concepts

To equip them with the conceptualization Dirichlet product and convolutions

To assist the students in solving congruences using various methods.

To enable the learners determine the solution of quadratic residues and solve

certain types of Diophandine equations.

To facilitate pupil understand the concept of partitions and representations

Unit 1: Fundamental theorem of arithmetic – divisibility – Euclidean algorithm – gcd of

more than two numbers – Mobius function µ(n) – Euler totient function φ(n) –

product function for ψ(n) (18 Hours)

Unit 2: Dirichlet product – Mangoldt function Λ(n) – multiplicative functions – inverse of

a completely multiplicative function – Liouville’s function λ(n) – generalized

convolutions – Bell series and Dirichlet multiplication – Seltag identity

(18 Hours)

Unit 3: Congruences – residue classes and complete residue systems – linear

congruences – Euler – Fermat theorem – simultaneous linear congruences –

polynomial congruences with prime power moduli (18 Hours)

Unit 4: Quadratic residues – Legendre’s symbol and its properties – Gauss’s lemma –

quadratic reciprocity law and its applications – Jacobi’s symbol – applications to

Diophandine equations (18 Hours)

Unit 5: Partitions – geometric representation – generating functions for partitions –

Euler’s pentagonal number theorem – combinatorial proof – Euler’s recursion

formula for p(n) (18 Hours)

Book for Study:

1. Tom M Apostol, “Introduction to Analytical Number Theory”, Narosa Publishing House, New

Delhi, 8th Reprint, 1998.

Unit 1: Chapter 1 Sections 1.1-1.8 &


Unit 2: Chapter 2 Sections 2.6 – 2.19




References:

1. Z.I.Bosevic and I.R.Safarevic, “Number Theory”, Academic Press, New York, 1996.

2. A.Weil. “Basic Number Theory”, Springer, New York, 2008.






Upto

CO1 Illustrate the implications of

properties of divisibility K3

CO2

Analyse the conceptualization

Dirichlet product and convolutions

functions

K4

CO3 Examine the properties of

congruences K4

CO4 Infer the Law of Quadratic Reciprocity

& Quadratic Residues K4

CO5 Examine the concept of Partitions K4



PO 1

PO 2

PO 3

PO 4

PO 5

PO 6

PO 7

PO 8

PSO 1

PSO 2

PSO 3

PSO 4

PSO 5

Sum of COs with

POs & PSOs

CO1 3 2 3 2 2 2 2 16

CO2 3 2 3 2 2 2 2 16

CO3 3 2 3 2 2 2 2 16

CO4 3 2 3 2 2 2 2 16

CO5 3 2 3 2 2 2 2 16




Mapping Scale 1 2 3

Relation 0.01-1.0 1.01-2.0 2.1-3



2.3

Observation COs of Theory of Numbers are strongly correlated with POs & PSOs


M.Sc.

Internal Test Question Paper Pattern for Core, Core Elective and Non-Major Elective

SECTION – A (4x 3 = 12)

FIVE questions without choice; each question carries 3 marks.

SECTION – B (4x 7= 28)

FIVE questions with internal choice have to be answered; each question carries 7 marks.

Semester Examination Question Paper Pattern for Core, Core Elective and Non-Major Elective (For those who joined in June 2020 onwards)

SECTION – A (10 x 1 = 10)

TWO question from each unit.

SECTION – B (5 x 6 = 30) FIVE questions have to be answered with internal choice; each question carries 8 marks; one question from each unit.

SECTION – C (5 x 12 = 60) FIVE questions have to be answered with internal choice; each question carries 10 marks; one question from each unit.


M.Sc.

SCHEME OF EVALUATION


Continuous Internal Assessment

Marks Test – 1 20 Test – 2 20 Seminar 10

------- Total 50

-------

Semester Examinations 50 Marks

A candidate must score a minimum of 23 marks out of 50 in the semester examination and an overall aggregate minimum of 50 percent for a pass.

department of mathematics arul anandar college …

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