# M.Sc. (Mathematics) Non-Semester

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M.Sc. (Mathematics)

Non
-
Semester

(To be offered under Distance and Continuing Education)

SCHEME OF EXAMINATIONS

First year

S.No. Paper

Hours

Marks

Passing Min.

3

100

50

1.2 Real Analysis

1.3 Differential
Equations

1.4 Fuzzy Mathematics and Statistics

1.5 Graph Theory and Combinatorics

Second Year

1.1

Programming in C and Numerical Methods

1.2

Measure theory and Complex Analysis

1.3

Topology and Functional Analaysis

1.4

Operations Research

1.5

Mechanics

1.1

ebra

Unit I : Groups

A counting principle

Normal subgroups and quotient groups

homomorphism

isomorphism

Cayley’s theorem

permutation groups.

Unit II : Another counting principle

Sylow’s Theorems

Unit III :

Rings

homomorphism

Ideals and quotient rings

Field of quotients of
an integral domain

Euclidean domain

Polynomial rings.

Unit IV : Vector spaces

Linear transformation and bases

Algebra of linear
transformations

Characteristic roots

trian
gular form.

Unit V: Extension fields

roots of polynomials

Text : I.N. Herstein Topics in Algebra (Second Edition)

Chapter 2, Section 2.1 to 2.12

Chapter 3, section 3. 1 to 3.9

Chapter 4, sections 4.1 , 4.2

Chapter 5, sections 5.1.
, 5.3., 5.5

Chapter 6, sections 6.1 to 6.4

1.2

Real Analysis

Unit I : Basic topology

convergent sequences

subsequences

upper and lower limits

some special sequences

Unit II : Series

Series of non
-
negative terms

The number e

The root and ratio

tests

Power series

suromation by parts

Absolute convergence

multiplication of series

Rearrangements.

Unit III : Continuity

Differentiation.

Unit IV : The Riemann

Steiltjes integral

Sequences and series of functions

Discussio
n of the main problem

Uniform convergence

Uniform convergence and
continuity

Uniform convergence and intergration.

Unit V: Uniform Convergence and differentiation

Equicontinuity

Equicontinuous
family of functions

Stone Weierstrass’ theorem

so
me special functions.

Text : Rudin

Principles of Mathematical Analysis (Tata McGrows Hill) Third Edition,
Chapters 2 to 8.

1.3

Differential Equations

Unit I : Second order linear equations

The general solution of a homogeneous
equation

Use of a known sol
ution to find another

The method of variation of
parameters

Power series solution

Series solution of a first order equation.

Unit II : Second order linear equations

Ordinary points

regular singular points

Legendre polynomials .

Unit III : Besse
l functions and Gamma functions

Linear systems

Homogeneous
linear systems with constant coefficients

The method of successive approximation

Piccard’s theorem.

Unit IV : Partial Differential Equations

Cauchy’s problem for first order equations

L
inear equations of first order

Nonlinear partial differential equations of first order

Cauchy’s method of characteristics

Compatible system of first order equations.

Unit V: Charpit’s method

special types of first order equations

Solutions
satisfying given conditions

Jacobi’s method

Linear Partial Differential Equations
with constant coefficients

Equation with variable coefficients.

Tests: G.F. Simmons, Differential Equations (Torta McGrow Hill )sections 14,15,
16,19 26
-
29, 32
-
35, 37,3
8,55 and 56.

I.N, Senddon, Elements of Partial Differential Equations, (Mc
-
Grow Hill) Chapter 2
sections 1
-
4, 7
-
13, Chapter 3 sections 1,4 and 5.

1.4

Fuzzy Mathematics and Statistics

Unit I : The concept of Fuzziness

Some Algebra of fuzzy sets.

Unit II: Fuz
zy quantities
-

Logical aspects of fuzzy sets.

Unit III: Distribution of random variables.

Unit IV: Conditional Probability and stochastic independence

Some special
distributions

Unit V: Distributions of Functions of random variables

Limiting Distributi
ons.

Texts: H.T. Nguyen and E.A. Walker, A first course in Fuzzy Logic (Second Edition)
CRC Chapters 1 to 4.

R.V. Hagg and A.T. Craig, Introduction to Mathematical Statistics (fourth Edition)
Macmillan, Chapters 1 to 3, Chapter 4 (except section 4.6) and C
hapter 5.

1.5

Graph Theory and Combinatorics

Unit I : Graphs and Subgraphs

Trees

Connectivity.

Unit II : Euler Tours and Hamilton Cycles

Matchings

Edge Colorings.

Unit III : Independent sets amd cliques

Verter Colorings.

Unit IV: Generating Function
s

Recurrence relations.

Unit V: The principle of Inclusion and Exclusion

Polya’s theory of counting.

Texts: J.A. Bondy and U.S. R. Murty, Graph theory with Applications (Macmillan),
Chapter 1, (sections 1 to 7), chapters 2,3,4,5 and 6. (Exclusing secti
ons dealing with
Applications), Chapters 7 (Sections 1 to 3) , Chapter 8. C.L. Liu, introduction to
combinatorics (Mc. Grow Hill) chapters 2 to 5.

1.1

Programming in C and Numerical Methods

Unit I :

Overview of C

Constants, Variables and data types

Operators and
expressions

managing input and output operations

Decision making branching
and looping.

Unit II: Arrays

Handling of character strings

User defined functions

Structures and Union
s.

Unit III: Pointers

File management in C.

Unit IV : Interpolation

Lagrange’s interpolation formula

Numerical solution
of ordinary differential equations

Taylor series method

Piccard’s method

Euler’s method.

Unit V: Runge

Kuttar Fourth order

method

Predictor

Corrector methods

Milne’s method.

Texts: E. Balagurusamy, Programming in Ansi C (Tata Mc. Graw Hill) Chapters
1 to 12. M.K. Jain, S.R.K. Iyengar and R.K. Jain, Numerical methods for
scientific and engineering computation. (Second Ed
ition) Wiley Eastern, Sections
4.1, 4.2, 6.1, 6.2 and 6.3.

1.2

Measure Theory and complex Analysis

Unit I : Lebesgue Measure

Lebesgue Integral

Unit II: Measure and Integration

Measure and Outer Measure.

Unit III: Complex numbers

Analytic functions

Elem
entary Theory of Power
series .

Unit IV: Cauchy’s theorem

Cauchy’s integral formula

singularities.

Unit V: Taylor’s theorem

Maximum principle

The calculus of Residues .

Texts: Royden

Real Analysis Third Edition(PHI) Chapters 3,4,11 and 12.
Ahlfo
rs

Complex analysis (Tata

McGraw Hill) Second Edition, Chapter 1,
Chapter 2, sections 1 and 2. Chapter 4 sections 1,2,3 and 5.

1.3

Topology and Functional analysis

Unit I : Topological Spaces

Compactness.

Unit II: Separation

connectedness

Unit III: Ba
nach Spaces

Unit IV: Hilbert spaces

Finite dimensional spectral Theory

Unit V: General Preliminaries on Banach Algebras

The structure of
commutative Banach Algebra.

Text: G.F. Simmons, introduction to Topology and Modern Analysis (Mc. Graw
Hill)

Chapte
rs 3,4,5,6,9, 10, 11, 12 and 13.

1.4

Operations Research

Unit I : Linear Programming

Simplex method

Transportation and its variation

Unit II: Network Models

CPM

PERT

Unit III : Integer Programming

Unit IV: Inventory models

Dicision Analysis and
Games

Unit V: Queueing Models.

Text: Taha

Operations Research

An Introduction (sixth Edition) PHI,
Chapters 2,3,5,6,9,11,14 and 17.

1.5

Mechanics

Unit I: Kinematics

Kinetic Energy and Angular Momentum

Methods of
Dynamics in space.

Unit II: The simple
pendulum

The spherical pendulum

Motion of a rigid body.

Unit III: The equations of Lagrange and Hamilton

Hamiltonian methods.

Unit IV: Real fluids and ideal fluids

Velocity

Stream lines

velocity potential.

Unit V: Vo
rticity vector

Equation of continuity

Euler’s equation of motion

Bernoulli’s equation

some three dimensional flows.

Texts: J.L. Synge and B.A Griffith, Principles of Mechanics (Mc. Graw Hill)
Chapters 11,12 and 13 *sections 2 and 3), Chapters 14,15
, and 16 (Section 1) F.
Chorlton, Text book of Fluid Dynamics (CBs Publishers) Chapter 2, sections 2.1.
to 2.7, Chapter 3, sections 3.1 and chapter 4.

1.6

Operations Research

Unit I: Linear Programming

Simplex method

Transporation model and its
variation

Unit II: Networks Models

CPM

PERT

Unit III : Integer Programming

Unit IV: Inventory models

Decision Analysis and Games

B. Sc Mathematics Main (Non
-

Semester)

(to be offered under DD& CE)

Scheme of Examination

I Year ( 3 papers)

Paper I Calculus

Paper II Classical Algebra

Paper III Analytical Geometry 3D and Vector Calculus

II Year (3 papers)

Paper IV Modern algebra

Paper V Statistics

Paper VI operations Research

III Year (5 papers)

Paper VII Analysis

Paper VIII Mechanics

Paper
IX Astronomy

Paper X Numerical Analysis

Paper XI Discrete Mathematics

I Year

Paper I

Calculus

Unit I : Curvature

Cartesian and polar

centre of curvature

Involute and evolute

Asymptotes in Cartesian co
-
ordinates

Multipl
e points

double
points.

Unit II:

Evaluation of double and triple integrals

jocobians, change of variables.

Unit III:

First order differential: equations of higher degree

solvable for p,x and y

Clairaut’s
form/ linear differential equations of seco
nd order

particular integrals for functions of
the form, Xn, eax, eax (f(x). Second order differential equations with variable
coefficients.

Unit IV: Laplace transform
-
Inverse transform

Properties

Solving differential
equations. Simultaneous equatio
ns of first order using Laplace transform.

Unit V: Partial differential equations of first order

formation

different kinds of
solution

four standard forms

Lagranges method.

Books:

1.

Calculus 1,2 & 3, T.K. Manickavachagom pillai & others.

2.

Calculus 1
&2, S. Arumugam and Isaac.

I year

Paper II

Classical Algebra

Unit I :

Theory of Equations: Every equation f(x) =0 of nth degree has ‘n’ roots. Symmetric
functions of the roots in terms of the coefficients

sum of the rth powers of the roots

Newt
on’s
theorem

Descartes rule of sign

Rolle’s theorem.

Unit II

Reciprocal Equations

Transformation of equations

solution of cubic and biquadratic equation

Cardon’s land Ferrari’s methods

Approximate solution of numerical equations
-

Netwon’s
and

Horner’s methods.

Unit III

Sequences and series: Sequences

limits, bounded, monotonic, convergent, oscillatory and
divergent sequence

algebra of limits

subsequences

Cauchy sequences in R and Cauchy’s
general principle of convergence.

Unit IV

Seri
es

convergence, divergence

geometric, harmonic, exponential, binomial and logratithmic
series

Cauchy’s general principle of convergence

comparison test

tests of convergence of
positive termed series

Kummer’s test, ratio test, Raabe’s test, Cauc
hy’s root test, Cauchy’s
condensation test.

Summation of series using exponential, binomial and logarithmic series.

Books for reference:

1.

Sequences and series, S. Arumugam & Others

2.

Algebra

Vol. I, T.K. Manickavachagom pillai & Others

3.

Real Analysis

Vol.
I, K. Chandrasekara Rao & K.S. Narayanan

4.

Infinite series, Bromwich.

I Year

Paper III

Analytical Geometry 3D and Vector Calculus

Unit I : Rectangular Cartesian Coordinates in space

Distance formula

Direction ratio and
cosines

Angle between lines

simple problems.

Plane

different forms of equation

angle between two planes

perpendicular distance from a
point on a plane

projection of a line or a point on a plane.

Unit II: Lines

symmetrical form

plane and a straight line

The perpendicu
lar from a point on
a line

Coplanar lines

shortest distance between two skew lines and its equation.

Sphere

Different forms of equations
-

plane section

the circle and its radius and centre

tangent plane

condition for tangency

touching spheres

common tanget plane

point of
orthogonality of intersection of two spheres.

Unit III

Vector differentiation

solenoidal and irrotational
fields
-

formulas involving the laplace operator.

Unit IV: Double and tri
ple integrals

Jacobian

change of variables

Vector integration

single scalar variables

line, surface and volume integrals.

Unit V: Gauss’s Stoke’s and Green’s theorems

statements and verification only.

Books for reference:

1.

Analytical Geometry of

3D
-
Part II, Manickavachagom Pillai

2.

Analytical Geometry of 3D & Vector Calculus

P. Duraipandian & Others

3.

Analytical Geometry of 3D & Vector Calculus

S. Arumugam & Others

4.

Vector Analysis, K. Viswanathan.

II year

PaperIV

Modern Algebra

Unit I :

Sets

functions

relations

partitions

composition of functions

groups

subgroups

cyclic
groups.

Unit II:

Normal subgroups

cosets

lagrage’s theorem

Quotient groups

Homomorphism

Kernel

Cayley’s theorem

Fundamental theorem of homomorphism
.

Unit III:

Rings

types

subring

ordered integral domain

ideals

Quotient rings

P.I.D.

Homomorphism of rings

fundamental theorem of homomorphism

Euclidean rings.

Unit IV: Definition and example of vector speaces

subspaces

sum and direct

sum of
subspaces

linear span, linear dependence, independence and their basic properties

Basis

finite dimensional vector spaces

dimension of sums of subspaces

Quotient space and its
dimension.

Unit V:

Linear transformation and their representati
on as matrices

Algebra of linear transformations

dual spaces

Eigen values & eigen vectors of a linear transformation

inner product spaces

Schwartz inequality

orthogonal sets and basis

Gram Schmidt orthogonalization process.

Reference books:

1
.Modern Algebra, S. Arumugam and Issac

2. Modern Algebra, Vasistha

3. Topics in Algebra, I.N. Herstein, Vikas Publishers

II year

Paper V

Statistics

Unit I :

Correlation

Karl Pearson’s coefficient of Correlation, Lines of Regression

Regression
coeff
icients

Rank correlation.

Unit II:

Probability

Definition

application of addition and multiplication, theorems

conditional,
Probability

Mathematical Expectations

Moment generating function

special distributions,
(Binomial distribution, Poisso
n distribution, Normal distribution

properties).

Unit

III:

Association of attributes

Coefficient of association

consistency

time series

Definition

Components of a time series

Seasonal and cyclic variations.

Unit

IV

Sampling

definition

large samples. Small samples

Population with one sample and
population with two samples

students

t
-
test
-
applications

chi

square test and goodness of
fit

applications.

Unit

V

Index numbers

Types of Index Numbers

Tests

Unit test, Commodit
y reversal test, time
reversal test, factor reaversal test

Chain index numbers

cost of living index
-

Interpolation

finite differences operators
--------------

-

Newton’s forward, backward interpolation formulae,
Lagrange’s formula.

Books:

1.

Statistics:

S. Arumugam & others

2.

Statistics: D.C. Saucheti & Kapoor

3.

4.

Statistics: T. Sankaranarayana & Others.

II Year

Paper VI
-

Operations Research

Unit: 1

Linear programming problem

Mathematical formulation

Graphical method of so
lution
-

simplex method

The big M method (Charnes method of penalties)

Two phase simplex
method

Duality

Dual simplex method

integer programming.

Unit

II

Transportation problem

mathematical formulation

North

west corner rule

Vogel’s
appr
oximation method (unit penalty method )

method of matrix minima

optimality test

maximization

Assignment problem

mathematical formulation

method of solution

maximization of the effective matrix.

Unit III:

Sequencing problem

introduction

n
jobs and two machines

n jobs and three machines

two
jobs and n machines

graphical method

inventory models: types of inventory modals:
Deterministic: 1) Uniform rate of demand, infinite rate of production and no shortage

2)
Uniform rate of demand,

finite rate of replenishment and no shortage

3) Uniform rate of
demand, instantaneous production with shortages

4) Uniform rate of demand, instantaneous
production with shortage and fixed time.

Unit IV:

Probabilistic Models: Newspaper boy problem

discrete and continuous type cases
-

Inventory
models with one price break.

Queueing Theory: General concept and definitions

classification of queues

Poisson process,
properties of poisson process

models: 1) (M/M/1) : (∞/FCFS), 2)(M/M/1): (N/FCFS),
3)(µ/M/S): (∞/FCFS).

Unit V: Network Analysis: Drawing network diagram

Critical path method

labelling method

concept of slack and floats on network

PERT

Algorithm for PERT

Differences in PERT
and CPM.

Resource Analysis in Network Scheduling :
Project cost

Crashing cost

Time
-
cost
optimization algorithm

Resource allocation and scheduling.

Books for Reference:

1.

Operations Research : Kantiswarup, P.K. Gupta and Man Mohan.

2.

Operations Research : P.K. Gupta, D.S. Hira.

3.

Operations Research : V.K.

Kapoor

4.

Operations Research : S.D. Sharma

5.

III Year

Paper VII

Analysis

Unit I:

Metric spaces

open sets

Interior of a set

closed sets

closure

completeness

Cantor’s
intersections theorem

Baire

Category

Theorem.

Unit II:

Continuity of functions

Continuity of compositions of functions

Equivalent conditions for
continuity

Algebra of continuous functions

hemeomorphism

uniform continuity

discontinuities connectednon

connected subsets of R

Con
nectedness and continuity

continuous image of a connected set is connected

intermediate value theorem.

Unit III:

Compactness

open cover

compact metric spaces

Herni Borel theorem. Compactness and
continuity

continuous image of compact metric spa
ce is compact

Continuous function on a
compact metric space in uniformly continuous

Equivalent forms of compactness

Every
compact metric space is totally bounded

Bolano

Weierstrass property

sequentially compact
metric space.

Unit IV:

Algebra o
f complex numbers

circles and straight lines

regions in the complex plane

Analytic functions Cauchy

Rienann equations

Harmonic functions

Bilinear transformation
translation, rotation, inversion

Cross

ratio
-

Fixed points

Special bilinear t
ransformations.

Unit V: Complex Integration

Cauchy’s integral theorem

Its extension

Cauchy’s integral
formula

Morera’s theorem

Liouville’s theorem

fundamental theorem of algebra

Taylor’s
series

Laurent’s series

Singularities. Residues

R
esidue Theorem

Evaluation of definite
integrals of the following types.

0

F (Cos x, sin x) dx

2 ∫
-

dx

Books for reference:

1.

Modern Analysis

Arumugam and Issac.

2.

Real Analysis

Vol. III

K. Chandrasekhara Rao and K.S. Narayanan, S. V
iswanathan
Publisher.

3.

Complex Analysis

Narayanan & Manicavachagam Pillai

4.

Complex Analysis

S. Arumugam & Issac.

5.

Complex Analysis

P. Durai Pandian

6.

Complex Analysis

Karunakaran, Narosa Publishers.

III Year

Paper VIII

Mechanics

Unit I:

Forces acting at a point

parallelogram of forces

triangle of forces

Lami’s theorem, Parallel
forces and moments

Couples

Equilibrium of three forces acting on a rigid body

Coplanar
forces

Reduction of any number of Coplanar forces theorems. Ge
neral conditions of quilibrium
of a system of Coplanar forces.

Unit II:

Friction

Laws of friction

Equilibrium of a particle (i) on a rough inclined plane. (ii) under a
force parallel to the plane (iii) under any force

Equilibrium of strings

Equatio
n of the
common catenary

Tension at any point

Geometrical properties of common catenary

uniform chain under the action of gravity

Suspension bridge.

Unit III:

Dynamics

Projectiles

Equation of path, Range etc

Range on an inclined plane

Motio
n on
an inclined plane. Impulsive forces

Collision of elastic bodies

Laws of impact

direct and
oblique impact

Impact on a fixed plane.

Unit IV:

Simple harmonic motion in a straight line

Geometrical representation

Composition of SHM’s
of the sam
e period in the same line and along two perpendicular directions

Particles suspended
by spring

S.H.M. on a curve

Simple pendulum

Simple Equivalent pendulum

The seconds
pendulum.

Unit V:

Motion under the action of Central forces

velocity and a
cceleration in polar coordinates

Differential equation of central orbit

Pedal equation of central orbit

Apses

Apsidal
distances

Inverse square law.

Books for Reference:

1.

Statics and Dynamics: S. Narayanan

2.

Statics and Dynamics : M.K. Cenkataraman

3.

S
tatics: Manickavachagom pillai

4.

Dynamics: Duraipandian.

III Year

Paper IX

Astronomy

Unit I:

Spherical Trigonometry (only formulae) celestial sphere

four systems of coordinates

Diurnal
motion

Zones of the earth

Perpetual day and night

Terrestrial longitude and latitude

International date line.

Unit II:

Dip of horizon

effects

Twili
ght

shortest twilight.

Unit III:

Refraction

Tangent formula

Cassini’s formula

Effects

Horizontal refraction

Geocentric
parallax.

Unit IV:

Kepler’s laws

verification

Newton’s deductions

Anomalies

Planets

Inferior and
superior planet

Bode’s law

Elongation

Sidereal period

Synodic period

Phase of the
planet

Stationary positions of a planet.

Unit V :

Moon

Phase

sidereal and synodic period

elongation

Metonic cycle

golden number

Eclipses

Lunar and solar eclipses

co
nditions

Synodic period of the nodes

Ecliptic limits

Maximum and minimum number of eclipses near a node and in a year

Saros

Lunar and solar
eclipses compared.

Books:

1.

Astronomy : S. Kumaravelu & Susheela Kumaravelu.

2.

Astronomy: G.V. Ramachandran

3.

As
tronomy: K. Subramanian and L.V. Subramanian

III Year

Paper X

Numerical Analysis

Unit I :

Finite differences

difference table

operators E,∆ and
-

Relations between these operatous

Factorial notation

Expressing a given polynomial in

factorial notation

Difference equation

Linear difference equations

Homogeneans linear difference equation with constant
coefficients.

Unit II

Interpolation using finite differences

Newton

Gregory formula for forward interpolation

Divided diffe
rences

Properties

Newton’s formula for unequal intervals
-

Lagrange’s formula

Relation between ordinary differences and divided differences

inverse interpolation.

Unit III

Numerical differentiation and integration

quidistant ordinates

Trapezoidal Rule

Simpson’s one third rule

Simpson’s three eight rule

Cote’s method.

Unit IV:

Numerical solution of ordinary differential equations of first and second orders

Piccards
method. Eulers method and

modified Euleis method

Taylor’s series method

Milne’s method

Runge

Kutta method of order 2 and 4

Solution of algebraic and transcendent equations.
Finding the initial approximate value of the root

Iteration method

Newton Raphson’s
method.

Un
it V:

Simultaneous linear algebraic equations

Different methods of obtaining the solution

The
elimination method by Gauss

Jordan method

Grouts’ method

Method of factorization .

Books:

Calculus of finite differences and Numerical Analysis, P.P. Gu
pta & G.S. Malik, Krishna
Prakasham Mardin, Mecrutt.

III Year

Paper XI

Discrete Mathematics

Unit I:

Definition and examples of graphs

degrees

subgraphs

ismorphims

Ramsey numbers

independent sets and coverings

intersection graphs and
line graphs

matrices

operations in
graphs

degree sequences, graphic sequences.

Unit II:

Walks

trails and paths

connectedness and components

blocks

connectivity

Eulerian
graphs

Hamiltonian graphs

trees

characterization of trees

centr
e of a tree.

Unit III:

Planas graph and their properties

characterization of planas graphs

thickness

crossing and
outerplanarity

Chromatic number

chromatic index

five colour theorm

four colour
problem

chromatic polynomials

Directed
graphs and basic properties

paths and connections
in digraphs

digraphs and matrices

tournaments.

Unit IV:

Permutations

ordered selections

unordered selections

further remarks on binomial theorem

Pairings within a set

pairings between sets,
-

an optimal assignment problem.

Unit V:

Recurrence relations

Fibonacci type relations

Using generating functions

miscellaneous
methods

The inclusion exclusion principle and rook polynomials.

Text Books:

1.

Invitation to graph theory, S. Arumugam and
S. Ramachandran, Scitech Publications.

2.

A first course in combinational mathematics, Ian Anderson (Oxford applied Math. Series)