M.Sc. (Mathematics) Non-Semester

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Oct 31, 2013 (3 years and 9 months ago)

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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.

1.1 Advanced Abstract Algebra




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

Advanced Abstract Alg
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


more about roots.

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


Addition and
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


Steady and
unsteady flows


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


radius of 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


Gradient, Divergence and Curl operators


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.

Statistics: Mangaladas & Others

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.

Operations Research : Mangaladoss.




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


General Quadratue formula for e
quidistant ordinates


Trapezoidal Rule


Simpson’s one third rule


Simpson’s three eight rule


Waddle’s 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)