SYLLABUSM.Sc. (MATHEMATICS) Part-II (Semester III & IV) 2011-12 AND 2012-13

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1

SYLLABUS

M.S
c
. (MATHEMATICS)
Part
-
II (Semester III & IV)

20
11
-
1
2

AND 201
2
-
1
3

Note:

Each paper is of 100 marks. Out of this 20 marks are for continuous internal assessment and 80 marks
are for University Examination at the end of semester.

III
rd
SEMESTER


Five papers from the following list of
optional
:


MM 601

:

Differential equations
-
II



M
M 602

:

Optimization Techniques


MM 603

:

Categor
y Theory
-
I


MM 604

:

Numerical Analysis


MM 605

:

Computer Programming using ‘C’


MM 60
6

:

Algebra
-
II


MM

607

:

Complex Analysis
-
ll


MM 608

:

Topics in Topology

and Analysis.


MM 609

:

Classical Mechanics


MM 610

:


Mathematical Statistics


MM 611

:


Algebraic Topology


MM 612



Solid Mechanics

IV
th
SEMESTER


Five papers from th
e following list of
optional


M
M 701

:

Differentiable Manifolds


MM 702

:

Advanced Numerical Analysis


MM 703

:

Operations Research


MM 704

:

Homology Theory


MM 705

:

Theory of Linear Operators


MM 706

:

Homological Algebra




(Pre
-
requisite: MM

603: Category Theory I)


MM 707

:

Mathem
atical Methods


MM 708

:

Fluid Mechanics




(Pre
-
requisite: MM 609 and MM 612)


MM 709

:

Algebraic Coding Theory


MM 71
0

:

Differential Geometry of Manifolds


MM 711

:

Analytic Number Theory


MM 712

:

Advanced Abstract Algebra


MM 71
3


: Cat
egory Theory
-
II





(Pre
-
requisite: MM

603: Category Theory I)


MM 714


:

Nonlinear Programming




(Pre
-
requisite: MM602: Optimization Techniques)


MM 715

:




Object Oriented Programming using ‘C++’




(
Pre
-
requisite: MM 605: Compute
r Programming using ‘C’
)


MM 716

:

Advanced Optimization


MM717

:

PROGRAMMING LAB
-
II (C++ & Advanced Numerical Analysis)


2

IIIrd SEMESTER


MM 601: DIFFERENTIAL EQUATIONS
-

II


Lectures to be delivered: 60







Maximum Marks: 80

Time Allowed: 3hou
rs









Internal Assessment: 20












Total: 100

INSTRUCTIONS FOR THE PAPER


SETTER

Question paper will consist of five sections A, B, C, D & E.
SECTION
-
A
, B, C & D will have two
questions each from respective section

of syllabus. Section E will consist of 8 to 10 short answer questions
which will cover the entire syllabus uniformly. All questions will carry equal marks.

INSTRUCTIONS FOR THE CANDIDATES

Candidates are required to attempt one question from
SECTION
-
A
, B,

C & D of the question paper and
entire section E.

SECTION
-

A

Existence and uniqueness of solutions of first order differential equations for complex systems. Maximum
and minimum solution. Caratheodory theorem.


SECTION


B

Continuation of solution.
Uni
queness of Solutions and Successive approximations. Variation of Solutions.

SECTION


C

Partial Differential Equations: Occurrence and elementary solution of Laplace equation. Family of
equipotential surfaces. Interior and exterior

Dirichlet boundary valu
e problem for Laplace equation.
Separation of Variables.

SECTION
-

D

Axial symmetry, Kelvin’s inversion theorem. Green’s function for Laplace equation. Dirichlet’s problem
for semi infinite space and for a sphere. Copson’s Theorem (Statement only)

RECOMMEN
DED BOOKS

1.


E. Coddington & N. Levinson, Theory of Ordinary Differential Equations, Tata Mc
-
Graw Hill,
India.

2.

Simmons G. F., Differential Equations with Applications and Historical Notes, Tata McGraw Hill

(1991).

3.

Sneddon I. N., Elements of Partial Di
fferential Equations, Tata McGraw Hill (1957).










3

MM 602: OPTIMIZATION TECHNIQUES

Lectures to be delivered: 60










Maximum Marks: 80

Time Allowed: 3 hours











Internal Assessment: 20

















Total Marks: 100

INSTRUCT
IONS FOR THE PAPER
-
SETTER


The question paper will consist of five sections A, B, C, D and E. Sections A, B, C and D will have
two questions from the respective sections of the syllabus. Section E will consist of 8 to 10 short
-
answer
type questions which w
ill cover the entire syllabus uniformly. All questions will carry equal marks.

INSTRUCTIONS FOR THE CANDIDATES

Candidates are required to attempt one question each from the sections A, B, C and D of the question paper
and the entire section E.

SECTION
-
A

In
troduction, definition of operation research, models in operation research, general methods for solvin
g
O.
R. models
Elementary theory of convex sets, Linear programming problems, examples of LPPs,
mathematical formulation of the
mathematical programming pr
oblems
, Graphical solution

of the problem.
Simplex method, Big M method, Two Phase method
, problem of degeneracy.

SECTION
-
B

Duality in linear programming: Concept of duality, fundamental properties of duality, duality theorems,
complementary slackness theo
rem, duality and simplex method, dual simplex method.

Sensitivity Analysis: Discrete changes in the cost vector, in the

requirement vector and in the co
-
efficient
matrix.

SECTION
-
C

Transportation Problem: Introduction, mathematical formulation of the probl
em, initial basic feasible
solution, optimum solution, degeneracy in transportation problems, transportation algorithm, unbalanced
transportation problems.

Assignment Problems: Introduction, mathematical formulation of an assignment problem, assignment
alg
orithm, unbalanced assignment problems.

SECTION
-
D

Integer Programming: Introduction, Gomory's all
-
IPP method, Gomory's mixed
-
integer method, Branch
and Bound method. Games and Strategies : Introduction,
T
wo person zero sum games, Maximum,
Minimum, Principl
e; Games without saddle points,
M
ixed Strategies,
G
raphical solution,
D
ominance
property,
R
educing the game problem to a LPP.

RECOMMENDED BOOKS

1.

Kanti Swarup
,





:

Operations Research, Sultan Chand

and Sons, New Delhi
,


P. K. Gupta
and Man Mohan


2.

Chander Mohan and Kusum Deep :


Optimization Techniques, New Age International, 2009.


3.

Hadley, G





:

Linear Programming



4

MM
603:

CATEGOR
Y THEORY


I


Lectures to be
delivered:

60










Maximum
Marks:

80

Time
Allowed:

3 hours











Internal
Assessment:

20
















Total
Marks:

100

INSTRUCTIONS FOR THE PAPER
-
SETTER

The question paper will consist of five sections A, B, C, D and E. Sections A, B, C and D will have two
questions from the respective sections of the syll
abus. Section E will consist of 8 to 10 short
-
answer type
questions which will cover the entire syllabus uniformly. All questions will carry equal marks.

INSTRUCTIONS FOR THE CANDIDATES

Candidates are required to attempt one question each from the sections

A, B, C and D of the question paper
and the entire section E.














SECTION
-

A

Categories
:

Introduction with Functions of Sets, Definition and examples of Categories
, Finite Category
,
Additive Categories, The category of modules, The concept of fu
nctor and the category Cat. Isomorhism.
Constructions: Product of two categories, The Dual Category, The Arrow Category, The Slice and Co
-

Slice Category.

Free Categories
: Free Monoids and their Universal Mapping Property, The category Graphs, the categor
y
C (G) generated by a graph, Homomorphism of Graphs and the Universal Mapping Property of C (
G
).

(R.R: Chapter 1)

SECTION

-
B

Abstract Structures
:

Epis and mono, Initial and Terminal objects, Generalized elements, Sections and
Retractions, Product diagram
s and their Universal Mapping Property, Uniqueness up to isomorphism,
Examples of products: Product of Sets, Product in Cat, Poset, Product in Top. Categories with Products,
Hom
-
Sets, Covariant representable functors, Functors preserving binary product.

(R.R 1: Chapter 2 excluding example 6 of section 2.6)

SECTION

-
C

Duality
:
The duality principle
, Coproducts, Examples in Sets, Mon, Top, Coproduct of monoids, of
Abelian Groups and Coproduct in the category of Abelian Groups. Equalizers, Equalizers as a mo
nic,
Coequalizers, Coequalizers as an epic. Coequalizer diagram for a monoid.

Groups and Categories
: Groups in categories, topological group as a group in Top. The category of
groups, Groups as categories, Congruence on a category, quotient category and i
ts univalent mapping
property, finitely presented categories.

(R.R 1: Chapter 3 and 4)


5











SECTION
-
D

Limits
and Co
-
limits

Subojects, Pullbacks, Properties of Pullbacks, Pullback as a functor, Limits, Cone
to a diagram, limit for a diagram, Co
-
cones and Colimits. Preservation of limits, contra variant functors.
Direct limit of groups. Functors Creating limits and co
-
limits.

(R.R. Chapter 5 Excluding Example 5.33 and Proposition 5.35)

RECOMMENDED BOOK


Steven Awodey
:
Category Theory,
(Oxford Lo
gic Guides, 49, Oxford University Press.)






6

MM 604: NUMERICAL ANALYSIS


Lectures to be
delivered:

60










Maximum
Marks:

80

Time
Allowed:

3 hours











Internal
Assessment:

20

















Total:

100


INSTRUCTIONS FOR THE PAPER
-
S
ETTER


The question paper will consist of five sections A, B, C, D and E. Sections A, B, C and D will have two
questions from the respective sections of the syllabus. Section E will consist of 8 to 10 short
-
answer type
questions which will cover the entire

syllabus uniformly. All questions will carry equal marks.

INSTRUCTIONS FOR THE CANDIDATES

Candidates are required to attempt one question each from the sections A, B, C and D of the question paper
and the entire section E.

Use of Scientific (non
-
programma
ble) calculator is allowed.

SECTION
-
A

Number System, Error in evaluating a function, Absolute, Relative, Truncation and round off errors,
Floating Point Arithmetic,
Bounds on error,
Arithmetic accuracy in computers, Loss of significan
ce and
error propagati
on
, Condition and
I
nstability, Principle of equal effects.


SECTION
-
B

Bisection

method
, Regula
-
falsi

method
, Secant

method
, Fixed
-
point iteration and Newton
-
Raphson
method and their convergence,
Methods for
multiple roots
: Newton Raphson Method, Muller's m
ethod
,
Solution of Non
-
linear simultaneous equations: Fixed point iteration method, Seidel Method and Newton
Raphson Method.


SECTION
-
C

Gauss
-
elimination method (with pivoting), Triangularisation method, Error analysis: ill conditioning, norm
of matrix, co
ndition number. Matrix inversion using Triangularisation and partitioning methods.


Jacobi method, Gauss
-
Seidel method
,
SOR method,
Power method for finding largest eigenvalue, Jacobi
and Householder method for finding eigenvalues of symmetric matrices. Ma
trix inversion using Iterative
method.

SECTION
-
D

Interpolation:
Finite Differences, Newton Gregory Forward and Backward formula,
Lagrange's formulae
with error, Divided Difference
s,

Newton's formulae, Central Differences,
Hermite

interpolation.

Spline and
its application to integration,
Integration using Newton Cote's formula with errors, Gauss Quadrature
formulae.



7

RECOMMENDED BOOKS


1.

C. F. Gerald & P. O. Wheatl
e
y,
Applied Numerical Analysis
,
6th edition, Addison Wesley Publishing
Company, New York.

2.

S
. D. Conte & C. D. Boor,
Elementary Numerical Analysis
,
(An Algorithmic Approach), 3rd edition,
Mc
-
Graw Hill International Company, New York.

3.

M. K. Jain, S. R. K. Iyengar and R. K. Jain,
Numerical Methods for Scientific and Engineering
Computation
, New
Age Publishers, New Delhi.

4.

F.B.
Hildebrand,
Introduction to Numerical Analysis,

McGraw

Hill, New York.

5.

R.S. Gupta, Elements of Numerical Analysis, Macmillan India Ltd., 2009.


8

MM
605:

COMPUTER PROGRAMMING USING 'C'



Lectures to be delivered: 60





University Exam:

30

Time Allowed: 3 hours






Internal Assessment:

20

Min.Pass Marks: 35%






Max. Marks:

50


INSTRUCTIONS FOR THE PAPER SETTER


The question paper will consist of five sections A, B, C, D and E. Sections A, B, C and D will have two
ques
tions from the respective sections of the syllabus. Section E will consist of 8 to 10 short
-
answer type
questions which will cover the entire syllabus uniformly. All questions will carry equal marks.

INSTRUCTIONS FOR THE CANDIDATES

Candidates are required
to attempt one question each from the sections A, B, C and D of the question paper
and the entire section E.





SECTION
-
A


I
ntroduction to computer and its components, bits and word ,computer memory and its types, data
representation and storag
es, binary codes, binary system and its relationship with Boolean algebra,
different number systems and arithmetic operations.


Programming process: Problem definition, program design, coding, compilation debugging and
documentation. Problem Solving with C
omputer: Algorithm, Pseudocodes and Flowcharts



SECTION
-
B

Operators:

arithmetic, unary, logical and relational. Identifiers, keywords, Data types, constants, variables,
logical expressions, data input and output, assignment statements, conditional stateme
nts,
iteration,

case
control statement, break and continue statements.


SECTION
-
C

Storage classes, Arrays, String processing, User defined data types, function: call, definition, prototype,
scope, Recursion, Parameter Passing by reference & by value, libra
ry functions.


SECTION
-
D

Bitwise Logical operators: AND, OR, complement, shift operators, Precedence and associativity.

Structures: Nested structures, Array of structures, Unions, Pointers: character pointers. Pointers to arrays,
Array of pointers, pointer
s to structures. Files: Reading, Writing text and binary files.

RECOMMENDED BOOKS

1.

E. Balagurusamy ,"Programming in ANSI C ", Tata McGraw Hill.

2.

Byron Gottfried ," Programming with C",

Tata McGraw Hill

3.

Richie and Kerningham: "C Programming",PHI.


9

4.

D
Dromey: How to solve it by Computer ( Prentice
-
Hall 1985).

5.

Kanetkar:"Let us C", BPB Publications.

6.

William H. Press
,
Brian P. Flannery
,
Saul A. Teukolsky

and
William T. Vetterling
,
Numerical Recipes
in C
& C++ Source Code CD
-
ROM with Windows, DOS, or Mac Single Screen License (CD
-
ROM),
Cambridge University Press.




PROGRAMMING LAB
-
I (C & NUMERICAL
ANALYSIS)



No. of Lab Units:
20






University Exams
:
20

Time Allowed: 3 hours






Internal Assessment:
30

Min.Pass Marks: 35%







Max. Marks:
50



INSTRUCTIONS FOR CONDIDATES

This laboratory course will mainly comprise of exerci
ses on what is learnt under the papers:

(1)

Computer Programming using ‘C’.

(2)

Numerical Analysis.



10

MM 60
6
: ALGEBRA
-

II


Lectures to be delivered: 60










Maximum
Marks:

80

Time
Allowed:

3 hours











Internal
Assessment:

20

Total:

1
00

INSTRUCTIONS FOR THE PAPER
-
SETTER

The question paper will consist of five sections A, B, C, D and E. Sections A, B, C and D will have two
questions from the respective sections of the syllabus. Section E will consist of 8 to 10 short
-
answer type
questio
ns which will cover the entire syllabus uniformly. All questions will carry equal marks.

INSTRUCTIONS FOR THE CANDIDATES

Candidates are required to attempt one question each from the sections A, B, C and D of the question paper
and the entire section E.

SE
CTION
-
A

Fields, examples,
Algebraic and transcendental elements, Irreducible polynomials. Gauss Lemma,
Eisenstein's criterion, Adjunction of roots, Kronecker's theorem, algebraic extensions, algebraically closed
fields.

SECTION
-

B

Splitting fields, Norm
al extensions, multiple roots, finite fields, Separable extensions, perfect fields,
primitive elements, Lagrange's theorem on primitive elements.


SECTION
-

C

Automorphism

groups and fixed fields, Galois extensions, Fundamental theorem of Galois theory,
F
undamental theorem of algebra, Roots of unity and cyclotomic polynomials.


SECTION
-

D

Cyclic extension, Polynomials solvable by radicals, Symmetric functions, cyclotomic extension, quintic
equation and solvability by radicals.


RECOMMENDED BOOKS

1.

Bhattacharya & Jain

:


Basic abstract algebra (Chapters 15
-
17, Chapter


and Nagpaul




18 : excluding section 5)

2.

M. Artin


:


Algebra








11

MM
607:

COMPLEX ANALYSIS

II


Lectures to be
delivered:

60










Maximum
Marks:

80

Time
Allowe
d:

3 hours











Internal
Assessment:

20

Total:

100

INSTRUCTIONS FOR THE PAPER
-
SETTER

The question paper will consist of five sections A, B, C, D and E. Sections A, B, C and D will have two
questions from the respective sections of the syllabus
. Section E will consist of 8 to 10 short
-
answer type
questions which will cover the entire syllabus uniformly. All questions will carry equal marks.

INSTRUCTIONS FOR THE CANDIDATES

Candidates are required to attempt one question each from the sections A,
B, C and D of the question paper
and the entire section E.

SECTION
-
A

Normal families of analytic functions. Montel's theorem, Hurwitz's theorem, Riemann Mapping theorem,
Univalent functions. Distoration and growth theorems for the class S of normalized uni
valent functions.
Koebe 1/4 theorem. Bieberbach Conjecture (statement only) Littlewood's inequality for the class S.
Coefficient inequalities for functions in S in case of real coefficients only.

SECTION
-
B

Principle of analytic continuat
ion, The general de
finition of
an analytic function. Analyti
c continuation by
power series
method. Natural boundary. Schwarz reflection principle, Monodromy theorem.

Mittag
-
Leffler's theorem (only in the case when the set of isolated singularities admits the point at infinit
y
alone as an accumulation point). Cauchy's method of expansion of meromorphic functions. Partial fraction
decomposition of cosec Z, Representation of an integral function as an infinite

product. Infinite product for
sin z.

SECTION
-
C

The factorization of i
ntegral functions. Weierstrass
theorem regarding construction of an integral function
with prescribed zeros.

The minimum modules of an integral function. Hadamard's three circle theorem. The order of an integral
function. Integral functions of finite orde
r with no zeros. Jensen's inequality. Exponent of convergence.
Borel's theorem on canonical products. Hadmard's factorization theorem.

SECTION
-
D

Basic properties of harmonic functions, maximum and minimum prin
ciples, Harmonic functions on a
disc.
Harnack's

inequality and theorem. Subharmonic and superharmonic functions. Dirichlet prob
lem.
Green's
function.


12

RECOMMENDED BOOKS

1.

Zeev Nihari

:

Conformal Mapping, Chap.III (section 5), Chap.lV, Chap.V (pages 173
-
178, 209
-
220)

2.

G. Sansone and

:

Lectures on the
theory of functions of


J. Gerretsen


a complex variable, sections 4.11.1 and 4.11.2 only.

3.

J. B. Conway

:

Functions of one complex variable. Springer
-
vertag
-
International student
edition, Narosa Publishing House, 1980 (Chap.X only)

4.

E. T. Copson

:

Th
eory of Functions of a Complex Variable (Oxford University Press),
Chapter IV (4.60, 4.61, 4.62) Chap. VII (excl. Section 7.7) Chap.VIII
(Section 8.4).











13

MM
608:

TOPICS IN TOPOLOGY

AND ANALYSIS


Lectures to be
delivered:

60










Max
imum
Marks:

80

Time
Allowed:

3 hours











Internal
Assessment:

20

Total:

100

INSTRUCTIONS FOR THE PAPER
-
SETTER

The question paper will consist of five sections A, B, C, D and E. Sections A, B, C and D will have two
questions from the respectiv
e sections of the syllabus. Section E will consist of 8 to 10 short
-
answer type
questions which will cover the entire syllabus uniformly. All questions will carry equal marks.

INSTRUCTIONS FOR THE CANDIDATES

Candidates are required to attempt one question
each from the sections A, B, C and D of the question paper
and the entire section E.

SECTION
-

A

Uniform
Spaces:

Diagonal Uniformities, surroundings, examples
;
usual, metric, discrete and trivial
uniformities, uniform topology, uniform isomorphism, unifor
m covers, base and sub
-
base for a covering
uniformities, metrizability of a unifo
rmity, fine uniformity and para
compactness.

(R.R 1: Sections 35 and 36 (Statement only of Theorem 36.11)


SECTION
-

B

Function
Spaces
:

The compact
-

ope
n
topology, on
Y
X
, Ha
u
sdorf
f
ness and regularity of
Y
X
, Continuity of
composition; the Evaluation Map. C
artesion

Products, Application to Identification Topologies
;

Theorem
of Whitehead, Basis of
Z
Y

and Compact Subsets of
Z
Y
. Equicontinuity and Arzela
-

Ascoli Theorem.

Sequentia
l convergence in the c
-
topology, Comparison of topologies on Z
Y
.

(R.R 2 : Chapter XII Section 1to 6,(Excluding Theorems 4.4, 5.2 and 5.3). Section 7: Theorem 7.2 only.
Statements only of theorems 8.2 and 8.3 and Section 10)












SECTION
-

C

Spaces C
(Y)

:
Continuity of the Algebr
aic operations,
Ĉ
(Y;C) as a locally convex linear topological,
Space, Algebras in
Ĉ

(Y
;C), Unitary algebra in C
(Y;C), the Stone
-

Weierstr
ass theorem, the metric space
C
(
Y), The ring Ĉ(Y) and the Gelfan
d
-

Kolmogoroff result.

(R.R 2: Chapter XIII : Sections 1 to 4 and S
ection 6 excluding Theorem 6.5)


SECTION
-

D

H
-

Spaces
:

Homotopy and Function Spaces, Path Spaces, H
-
structures, H
-
homomorphism, H
-
Spaces,
Units, inversion and associativity, path spaces on H
-
spaces.

(R.R 2 : Chapter XV Section3 and Chapter XIX)


14

RECOMMEN
DED BOOKS

1.

Stephen Willard

:

General Topology

2.

James Dugundji

:

Topology


15

MM 60
9
: CLASSICAL MECHANICS

Lectures to be
delivered:

60










Maximum
Marks:

80

Time
Allowed:

3 hours











Internal
Assessment:

20

Total:

100

INSTRU
CTIONS FOR THE PAPER
-
SETTER

The question paper will consist of five sections A, B, C, D and E. Sections A, B, C and D will have two
questions from the respective sections of the syllabus. Section E will consist of 8 to 10 short
-
answer type
questions which
will cover the entire syllabus uniformly. All questions will carry equal marks.

INSTRUCTIONS FOR THE CANDIDATES

Candidates are required to attempt one question each from the sections A, B, C and D of the question paper
and the entire section E.

SECTION
-
A


Basic Principles
: Mechanics of a Particle and a System of Particles, Constraints , Generalized
Coordinates, Holonomic and Non
-
Holonomic Constraints. D’Alemberts Priciple and Lagrange’s
Equations, Velocity Dependent Potentials and the Dissipation Function,
Simple Applications of the
Lagrangian formulation.

Variational Principles and Lagrange’s Equations
: Hamilton’s Principle, Derivation of Lagrange’s
Equations from Hamilton’s Principle, Extension of Hamilton’s Principle to Non
-
Holonomic Systems.


SECTION
-
B

Conservation Theorems and Symmetry Properties
: Cyclic Coordinates, Canonical Momentum and its
Conservation, The Generalized Force, and Angular Momentum Conservation Theorem.

The Two
-
Body Central Force Problem
: Reduction to the Equivalent One
-
Body Problem,
The Equation
of Motion, The Equivalent One Dimensional Problem and the Classification of Orbits, The Virial
Theorem, Conditions for Closed Orbits, Bertrand’s Theorem.


SECTION
-
C

The Kepler Problem
: Inverse Square Law of Force, The Motion in Time in the Ke
pler Problem, Kepler’s
Laws, Kepler’s Equation, The Laplace
-
Runge
-
Lenz Vector.

Scattering in a Central Force Field
: Cross Section of Scattering, Rutherford Scattering Cross Section,
Total Scattering Cross Section, Transformation of the Scattering Problem t
o Laboratory Coordinates.


SECTION
-
D

The Kinematics of Rigid Body Motion
: The Independent Coordinates of Rigid Body, The
Transformation Matrix, The Euler Angles, The Cayley
-
Klein Parameters and Related Quantities, Euler’s
Theorem on the Motion of Rigid Bo
dies, Finite Rotations, Infinitesimal Rotations, The Coriolis Force.

RECOMMENDED BOOKS

1.

Herbert Goldstein:

Classical Mechanics,

Chapter
s

1 to 4.


16

M
610:

MATHEMATICAL
S
TATISTICS


Lectures to be delivered
: 60












Maximum Marks: 80

Time A
llowed: 3 hours











Internal Assessment
: 20



Total
: 100

INSTRUCTIONS FOR THE PAPER
-
SETTER

T
he question paper will consist of five sections A, B, C, D and E. Sections A, B, C and D will have two
questions from the respective sections of the syllabus. Section E will consist of 8 to 10 short
-
answer type
questions which will cover the entire syllabu
s uniformly. All questions will carry equal marks.

INSTRUCTIONS FOR THE CANDIDATES

Candidates are required to attempt one question each from the sections A, B, C and D of the question paper
and the entire section E.

SECTION
-
A

Concept of random variables an
d probability distributions: Two

dimensional random variables, joint,
marginal and conditional distributions, independence of random variables, expectation, conditional
expectation, moments, product moments, probability generating functions, moment generat
ing function
and its properties. Moment inequalities, Techebyshey's, inequalities, characteristic function and its
elementary properties.

SECTION
-
B

Study of various discrete and continuous d
istributions

:

hyper

geometric, binomial, poison, negative
binomial, geometric, rectangular, normal, exponential, beta and gamma distributions.

SECTION
-
C

Concept of sampling distribution and its standard error, Derivation of sampling distributions of

C
hi
-
square, t and F (null case

only) distribution of sample mean and sample variance and their in random
sampling from a normal distribution.

SECTION
-

D

Elementary concepts in testing of statistical hypotheses. Tests of significance : tests based on normal

distribution, Chi
-
square, t and F statistic and transformation of correlation coefficient, tests for regression
coefficients and partial and multiple correlation coefficients.

Analysis of variance : One way classification, two way classification with one

observation per cell:

RECOMMENDED BOOKS

1.

Goon, Gupta and

Dasgupta


:

An Outline of Statistical Theory.

2.

Gupta and Kapoor




:

Fundamental of Mathematical

Statistics

3.

Goon, Gupta and Das Gupta


:

Fundamentals of Statistics Vol
.
-
II





17

M
M
611:

ALGEBRAI
C TOPOLOGY

Lectures to be
delivered:

60








Maximum Marks: 80

Time Allowed: 3 hours










Internal Assessment: 20













Total: 100


INSTRUCTIONS FOR THE PAPER
-
SETTER

The question paper will consist of
four sections: A, B, C, D and E Sections A B, C and D will have two
questions each from the respective sections of the syllabus. All questions will carry equal marks.


INSTRUCTIONS FOR THE CANDIDATES

Candidates are required to attempt one question each fro
m the sections A, B, C, and D of the question
paper.


SECTION
-
A


The Fundamental group: Homotopy of paths, Homotopy classes, The Fundamental group, change of base
point, Topological invariance, covering spaces, local homeomoerphisms.


SECTION
-
B


The Fund
amental group of the circle. Retractions and fixed points, No Retraction Theorem, The Bro
uwer
Fixed Point Theorem for the disc, The Fundamental theorem of Algebra.


SECTION
-
C


The Borsuk
-
Ulam theorem, The Bisection theorem, Deformation Retracts and Homo
topy type,
Homotopy invariance. Direct sums of Abelian Groups, Free products of groups and uniqueness, least
normal subgroup.


SECTION
-
D


Free groups, generators and relations, The Seifert
-
Van Kampen theorem, also classical version, The

Fundamental grou
p of a wedge of circles.


RECOMMENDED BOOKS


1.
James R. Munkres:

Topology, Pearson Prentice Hall, Chapter


9 (51
-
58), Chapter

11 (67
-
71),








18

MM 611: SOLID MECHANICS

Lectures to be delivered: 60







Maximum Marks: 80

Time Allowed: 3

hours










Internal Assessment: 20












Total: 100

INSTRUCTIONS FOR THE PAPER
-
SETTER

The question paper will consist of four sections: A, B, C, D and E Sections A B, C and D will have two
questions each from the r
espective sections of the syllabus. All questions will carry equal marks.


INSTRUCTIONS FOR THE CANDIDATES

Candidates are required to attempt one question each from the sections A, B, C, and D of the question
paper.

SECTION
-
A



Tensor Algebra: Coordinate
-
transformation, cartesian Tensor of different order. Properties of Tensors,
Isotropic tensors of different orders and relation between them, symmetric and skew
-
symmetric tensors.
Tensor Invariants, Deviatoric tensors, eigenvalues and eigen
-
vectors of a ten
sor.

Tensor analysis: scalar, vector, tensor functions, Comma notation, gradient, divergence and curl of a
vector/ tensor field.

(Relevant portions of
C
hapters 2 and 3 of book by D.S.
C
handrasekharaiah and l
Debnath)

SECTION
-
B


Analysis of strain: Affine
transformation, Infinitesimal affine deformation, Geometrical Interpretation of
the compo
nents of strain. Strain quadric
of Cauchy. Principal strains and invariance, General infinitesimal
deformation, Saint
-
Venants equations of compatibility, Finite deform
ations

Analysis of Stress: Stress tensor, Equations of equilibrium, Transformation of coordinates, Stress quadric
of Cauchy, Principal stress and invariants, Maximum normal and shear stresses.
(Relevant

portion of
chapter
1

&
2

of book by I.S. Sokolnikoff)
.


SECTION
-
C


Equations of Elasticity: Generalized Hooks Law, Anisotropic medium, Homogeneous isotropic media,
Elasticity, moduli for Isotropic media. Equilibrium and dynamic equations, for and isotropica elastic solid,
Strain energy function and its conn
ection with Hooke's Law, Uniqueness of solution. Beltrami
-
Michell
compatibilty equations, Saint
-
Venant's principle.

(Relevant portion of Chapter
3

of book by I.S.Sokonikoff).

SECTION
-
D


Two dimensional problems : Plane stress, Generalized plane stress, Ai
ry stress function. General solution
of biharmonic equation. Stresses and displacements in terms of complex potentials. The structure of
functions of


(z) and (z). First and second boundary
-
valuc problems in plane elasticity.
Existence

and
uniqueness

of

the solutions, (Section 65
-
74 of

I.S.

Sokolnikoff).


19


RECOMMENDED BOOKS


1.

I.S. Sokolnikoff, Mathematical Theory of Elasticity, Tata
-
McGraw Hill Publishing company Ltd.

Ne
w
Delhi, 1977.

2.

A.E.H. Love, A

Treatise on the Mathematical theory of Elasticity,
D
over

Publications, New York.

3.

Y.C. Fung. Foundations of Solid Mechanics, Prentice Hall, New delhi, 1965.

4.


D.S. Chandrasekharai and L. Debnath, Continuum Mechanics, Academic Press, 1994.

5.

Shanti Narayan, Text Book of Cartesian Tensor, S. Chand & Co., 1950.

6.

S. Tim
eshenk
o

and N. Goodier. Theory of Elasticity, McGraw Hill, New York, 1970.

7.

I.H. Shames, Introduction to Solid Mechanics, Prentice Hall, New Delhi, 1975.


20


IVth SEMESTER


MM
701:

DIFFERENTIABLE MANIFOLD


Lectures to be
delivered:

60










Max
imum
Marks:

80

Time
Allowed:

3 hours











Internal
Assessment:

20

Total:

100

INSTRUCTIONS FOR THE PAPER
-
SETTER

The question paper will consist of five sections A, B, C, D and E. Sections A, B, C and D will have two
questions from the respectiv
e sections of the syllabus. Section E will consist of 8 to 10 short
-
answer type
questions which will cover the entire syllabus uniformly. All questions will carry equal marks.

INSTRUCTIONS FOR THE CANDIDATES

Candidates are required to attempt one question
each from the sections A, B, C and D of the question paper
and the entire section E.

SECTION
-
A

Differentiable Manifolds, examples of differentiable manifolds, Differentiable maps on manifolds, tangent
vectors and tangent space,
cotangent

space, Vector Fiel
ds, Lie
-
bracket of vector fields. Jacobian of a map.
Curves and integral curves, Immersions and embeddings.

SECTION
-
B

Tensors and forms. Exterior product and Grassman algebra, connections. Difference tensor, existence of
parallelism and geodesics, covarian
t derivative, exterior derivative contraction, Lie
-
derivative.

SECTION
-
C

Torsion tensor and curvature tensor of a connection, properties of torsion and curvature tensor, Bianchi's
identities, Cartan's approach and structure equations of cartan. Riemannian
manifolds, Fundamental
theorem of Riemannian geometry, Riemannian connection.

SECTION
-
D

Riemannian curvature tensor and its properties. Bianchi's identities, Sectional curvature, Theorem of
Schur, Sub
-
manifolds and hyper
-
surfaces, Normals, induced connecti
on, Gauss and Weingartan formulae.

RECOMMENDED BOOKS

1.

Hicks, N. J.

:

Notes on Differential Geometry (Relevant Portion).

2.

H. B. Sinha

:

An
I
ntroduction to modern Differential Geometry, Kalyani Pub. N. Delhi
(Relevant Portion).

3.
Y. Matsushima

:

Diff
erentiable Manifolds.



21

MM 702: ADVANCED NUMERICAL ANALYSIS


Lectures to be delivered: 60










Maximum Marks: 80

Time Allowed: 3 hours











Internal Assessment: 20

Total: 100

INSTRUCTIONS FOR THE PAPER
-
SETTER

The question paper

will consist of five sections A, B, C, D and E. Sections A, B, C and D will have two
questions from the respective sections of the syllabus. Section E will consist of 8 to 10 short
-
answer type
questions which will cover the entire syllabus uniformly. All
questions will carry equal marks.

INSTRUCTIONS FOR THE CANDIDATES

Candidates are required to attempt one question each from the sections A, B, C and D of the question paper
and the entire section E.

Use of Scientific (non
-
programmable) calculator is allowe
d.

SECTION
-
A

Solution of Differential Equations
:
Tayler's series, Euler's

method
,
Improved Euler method, M
odified

Euler method
, and Runge
-
Kutta methods (upto fourth order), Predictor Corrector methods. Stability and
convergence of Runge
-
Kutta and
P
redictor

Corrector Methods.

SECTION
-
B

Parabolic Equation:
Explicit and
I
mplicit schemes for solution of one dimensional equations, Crank
-
Nicolson, Du fort and Frankel schemes for one dimension equations. Discussion of their compatibility,
stability and convergence
. Peaceman
-
Rachford A.D.I. scheme for two dimensional equations.

SECTION
-
C

Elliptic Equation:
Finite difference replacement and reduction to block tridiagonal form and its solution;
D
I
richlet and Neumann boundary conditions. T
reatment of curved boundaries;

S
olution by

A.D.I. method.

SECTION
-
D

Hyperbolic equations:

Solution by finite difference methods on rectangular and characteristics grids and
their stability.

Approximate methods:

Methods of weighted residual, collocation, Least
-
squares and Galerkin' s
me
thods. Variational formulation of a given boundary value problem, Ritz method. Simple examples from
ODE and PDE.

RECOMMENDED BOOKS

1.

Smith, G D,
Numerical solution of partial differential equations, Oxford Univ. Press (1982).

2.


R.S. Gupta, Elements of N
umerical Analysis, Macmillan India Ltd., 2009.

3
.

Mitchell, A. R.,
Computational methods in partial differential equations, John Wiley (1975).

4
.

Froberg, C. E
.,
Introduction to Numerical Analysis, Addision
-
Wesley, Reading, Mass (1969).

5
.

Gerald, C. F.
,
A
pplied Numerical Analysis Addision Wesley, Reading, Mass (1970).

6
.

Jain, M. K.
,
Numerical solutions of Differential equations, John Wiley (1984).

7
.

Collatz, L.
,
Numerical Treatment of Differential Equations, Springer
-

Verlag, Berlin (1966)


22


MM 703: O
PERATION
S

RESEARCH


Lectures to be
delivered:

60










Maximum
Marks:

80

Time
Allowed:

3 hours











Internal Assessment : 20

Total:

100

INSTRUCTIONS FOR THE PAPER
-
SETTER


The question paper will consist of five sections A, B, C,

D and E. Sections A, B, C and D will have two
questions from the respective sections of the syllabus. Section E will consist of 8 to 10 short
-
answer type
questions which will cover the entire syllabus uniformly. All questions will carry equal marks.

INSTR
UCTIONS FOR THE CANDIDATES


Candidates are required to attempt one question each from the sections A, B, C and D of the question paper
and the entire section E.

SECTION
-
A

Queueing problems:
C
haracteristics of queueing system. Distributions in queueing syst
ems, poisson
arrivals and exponential service times, the M/M/I, M/M/S queueing systems, steady state solutions and
their measures of effectiveness.

SECTION
-
B

Inventory problems, definition, the nature and structure of inventory system, deterministic models

and their
solution, multi item inventory problems, stochastic inventory models.

SECTION
-
C

Replacement and maintenance problems: replacement of capital equipment, discounting cost, replacement
in anticipation of failure, preventive maintenance, the general

renewal process.

SECTION
-
D



Network Analysis:
Introduction to Networks, Minimal Spanning Tree Problem,
S
h
ortest Path problem
:

Dijkstra's Algorithm
,
Floyd's Algorithm
,
M
aximum
F
low
P
roblem,
P
roject
Management: Critical Path
method, Critical Path Computati
ons, Optimal Scheduling by CPM, Project Evaluation and Review
Techniques (PERT).

RECOMMENDED BOOKS


1
.

Sharma, S.D.

:


Operation research, Kedar Nath and Co., Meerut
.

2
.

Kanto Swaroop, P.K.

:
Operations Researc
h, Sultan Chand and Sons.


Gupta and Man M
ohan


23

3
.

Hamdy A. Taha

:


Operations Research; An Introduction, PHI, New Delhi.

4.

Kasana and Kumar

:
Introductory Operation Research, Springer
.

5.

Chander Mohan and Kusum Deep
:
Optimization Techniques, New Age International, 2009.








24

MM
704:

HOM
OLOGY THEORY

Lectures to be
delivered:

60










Maximum
Marks:

80

Time
Allowed:

3 hours











Internal
Assessment:

20

















Total:

100

INSTRUCTIONS FOR THE PAPER
-
SETTER

The question paper will consist of five sections A,

B, C, D and E. Sections A, B, C and D will have two
questions from the respective sections of the syllabus. Section E will consist of 8 to 10 short
-
answer type
questions which will cover the entire syllabus uniformly. All questions will carry equal marks.

INSTRUCTIONS FOR THE CANDIDATES

Candidates are required to attempt one question each from the sections A, B, C and D of the question paper
and the entire section E.

SECTION
-

A

Singular Homology
Theory
:

Euclidean Simplexes
,

Linear Maps,

Singular p
-
simplex
, the group
Cp (E;
G
),
induced homomorphism on chains. The Boundary Operator d, the boundary of a singular simplex, the
boundary of a p
-
chain. Cycles & Homology; the group Zp(E;

G
), the homology groups Hp(E;
G),
Hp(E,
F;
G
). induced homomorphism on relative ho
mology groups, the dimension theorem, the Exactness
theorem; exact sequence, the boundary homomorphism & the exactness of the singular homology
sequences.

(R.R. : Sec 1
-
1 to 1
-
9)

SECTION
-

B

Singular and Simplicial
Homology
:

Homotopic maps of pairs, the p
ri
sm operator

P
. The homotopy
theorem.
The Ex
cision Theorem; the barycentric subdivision operator B.

The Axiomatic Approach;
Simplicial Complexes, traingulable space,

triangulation, The Direct Sum Theorem. The Direct Sum
Theorem for complexes.

(R.R. : Sec

1
-
10 to 2
-
4)

SECTION
-

C

Simplicial Homology

:
Homology groups of cells and spheres, Orientation, Homology groups of a
Simplicial pair, Formal description of Simplicial Homology; the oriented chain group, the oriented
boundary operator, the oriented simpl
icial homology group, simplicial map, cell complexes, canonical
Basis, the Betti group Bp and the Torsion Group Tp.

(R.R. : Sec 2
-
5 to 2
-
10)

SECTION
-

D

Chain Complexes

:
Singular chain complex, oriented simplicial chain complex, the group K
p

of
p
-
chains
o
f a

c
hai
n Complex, the group K
p

of Co
-
chains,
Co
-
boundary operator
, the co
-
chain complex & the p
th

Co
-
homology group H
p
(K), Chain homomorphism, induced homomorphism on homology and co
-
homology
groups. Chain homotopy

and the Algebraic homotopy theorem
.

(
R.R. : Sec 3
-
1 to 3
-
6)


25

RECOMMENDED BOOKS



1.
A.H. Wallace:


Algebraic
Topology:

Homology and Cohomology


26

MM
705:

THEORY OF LINEAR OPERATORS


Lectures to be
delivered:

60










Maximum
Marks:

80

Time
Allowed:

3 hours











Intern
al
Assessment:

20

Total:

100

INSTRUCTIONS FOR THE PAPER
-
SETTER


The question paper will consist of five sections A, B, C, D and E. Sections A, B, C and D will have
two questions from the respective sections of the syllabus. Section E will consist of 8 to 1
0 short
-
answer
type questions which will cover the entire syllabus uniformly. All questions will carry equal marks.

INSTRUCTIONS FOR THE CANDIDATES

Candidates are required to attempt one question each from the sections A, B, C and D of the question paper
a
nd the entire section E.

SECTION
-
A

Spectral theory in normed linear spaces, resolvant set and spectrum. Spectral properties of bounded linear
operator. Properties of resolvant and spectrum. Spectral mapping theorem for polynomials, spectral radius
of bound
ed linear operator on a complex Banach space.

SECTION
-
B

Elementary theory of Banach algebras. Resolvant set and spectrum. Invertible elements, Resolvant
equation. General properties of compact linear operators.

SECTION
-
C

Spectral properties of compact line
ar operators on normed space. Behaviour of compact linear operators
with respect to solvability of operator equations. Fredholm type theorems. Fredholm alternative theorems.

SECTION
-
D

Spectral properties of bounded se
lf
-
adjoint linear operators on
a comple
x Hilbert space. Positive operators.
Monotone sequence theorem for bounded self
-
adjoint operators on a complex Hilbert space. Square roots
of positive operators. Spectral family of a bounded self
-
adjoint linear operator and its properties, Spectral
theorem
.

RECOMMENDED BOOKS

1.

E. Kreyszic

:

Introductory Functional Analysis with Applications.

2.
Bachman and Narici

:

Functional Analysis




27

MM
706:

HOMOLOGICAL ALGEBRA

Lectures to be
delivered:

60










Maximum
Marks:

80

Time
Allowed:

3 hour
s











Internal Assessment : 20

Total:

100

INSTRUCTIONS FOR THE PAPER
-
SETTER

The question paper will consist of five sections A, B, C, D and E. Sections A, B, C and D will have two
questions from the respective sections of the syllabus. Sectio
n E will consist of 8 to 10 short
-
answer type
questions which will cover the entire syllabus uniformly. All questions will carry equal marks.

INSTRUCTIONS FOR THE CANDIDATES

Candidates are required to attempt one question each from the sections A, B, C and

D of the question paper
and the entire section E.

SECTION
-
A

Homology functors

: Diagrams over a ring, Translations of diagrams, Translation category, split exact
sequence, images and kernel as functors, Homology functors, The connecting homomorphism, Comp
lexes,
boundary homomorphism, differentiation homomorphism, homology modules, right and left complexes,
exact homology sequence and Homotopic translations.

(R. R. : Chapter 4 of Northcott)

SECTION
-
B

Projective and injective modules

: Projective modules, in
jective modules, An existence theorem for
injective modules, Complexes over a modules, right and left complexes over a module, augmentation
translation and augmentation homomorphism, acyclic right and acyclic left complexes over a module,
Projective and in
jective resolutions of a module, Properties of resolutions of a module.

(R.R ; Sections 5.1 to 5.5)

SECTION
-
C

Derived Functors
:
Projective and injective resolutions of an exact sequence, Propert
ies of resolutions of
sequences, Functors of complexes, Assoc
ited translations, Functors of two complexes, Right
-
derived
functors, the defining systems and the connecting homomorphisms, the functor R
0
T
, Left
-
derived functors,
the functor L
0
T.

(R.R. Sections 5.6 to 6.4)











SECTION
-
D

Torsion and Extension Funct
ors
:
Connected sequences of functors, Connected right

and
left

sequences
of covariant and contravariant functors, homomorphism
and isomorphism as a natural equivalence
between
connected sequences of functors
. Torsion functors Tor
n

, Basic properties of T
orsion functors, Extension
functors Ext
n

and Basic properties of extension functors.

(R.R. Sections 6.5 to 7.4)

RECOMMENDED BOOKS

1.

D. G. Northcott : An introduction to Homological Algebra.


28

MM
-
707: MATHEMATICAL METHODS


Lectures to be delivered: 60








Maximum Marks: 80

Time Allowed: 3

hours












Internal Assessment: 20











Total: 100


INSTRUCTIONS FOR THE PAPER


SETTER

Question paper will consist of five sections A, B, C, D & E.
SECTION
-
A
,

B, C & D will have two
questions each from respective section of syllabus. Section E will consist of 8 to 10 short answer questions
which will cover the entire syllabus uniformly. All questions will carry equal marks.

SECTION

A

Linear integral equations
of first and second kind, Abel’s problem, Relation between linear differential
equation and Volterra’s equation, Non linear and Singular equations, Solution by successive substitutions,
Volterra’s equation, iterated and reciprocal functions, Volterra’s sol
ution of Fredholm’s equation.


SECTION

B

Fredholm’s equation as limit of finite system of linear equations, Hadamard’s theorem, convergence proof,
Fredholm’s two fundamental relations, Fredholm’s solution of integral equation when D(

)

0,Fredholm’s
soluti
on of Dirichlet’s problem and Neumann’s problem, Lemmas on iterations of symmetric kernel,
Schwarz’s inequality and its applications

SECTION

C

Simple variational problems, Necessary condition for an extremum, Euler’s equation, End point problem,
Variation
al derivative, Invariance of Euler’s equation, Fixed end point problem for n
-
unknown functions,
Variational problem in parametric form, Functionals depending on higher order derivatives.


SECTION

D

Euler Lagrange equation, First integral of Euler
-
Lagrange

equation, Geodesics, The brachistochrone,
Minimum surface of revolution, Brachistochrone from a given curve to a fixed point, Snell’s law, Fermat’s
principle and calculus of variations.


RECOMMENDED BOOKS

1.

F.B. Hildebrand, Method of Applied Mathematics. Pr
entice Hall, India.

2.

I.M. Gelfand & S.V. Fomin, Calculus of Variations, Prentice Hall, India.

3.

W.W. Lovitt, Linear Integral Equations, Tata
-
McGraw Hill, India.

4.

Robert Weinstock, Calculus of Variations, McGraw Hill, London.

5.

L.B. Chambers, Integral Equations,
International Text Book Co.




29

M
M
708:

FLUID MECHANICS

Lectures to be
delivered:

60










Maximum
Marks:

80

Time
Allowed:

3 hours











Internal
Assessment:

20

Total:

100

INSTRUCTIONS FOR THE PAPER
-
SETTER


The question paper will
consist of five sections A, B, C, D and E. Sections A, B, C and D will have
two questions from the respective sections of the syllabus. Section E will consist of 8 to 10 short
-
answer
type questions which will cover the entire syllabus uniformly. All questi
ons will carry equal marks.

INSTRUCTIONS FOR THE CANDIDATES

Candidates are required to attempt one question each from the sections A, B, C and D of the question paper
and the entire section E.

SECTION
-
A

Equations of Fluid Mechanics : Real and continuous fl
uids, differentiation following the motion, equation
of continuity, Stream function, Stream lines, Pressure, Euler's equation of motion. Bernoulli's theorem
Steady irrotational non
-
viscous compressible flow.

SECTION
-
B

Vorticity, circulation, Kelvin's theor
em on constancy of circulation, Kinetic energy. Three dimensional
problems : Laplace's equation. Three dimensional sources and dipoles. Spherical obstacle in a uniform
steam Moving sphere, images.

SECTION
-
C

Application of complex variable method : Conjugat
e functions in plane, complex potential, incompressible
flow in two dimensions, uniform stream, Source and sink, Vortex, Two dimensional dipole, Superposition,
Joukowski's transformation. Milne Thomson circle theorem, Blasius theorem, Drag and lift.

SECTIO
N
-
D

Source and vortex filaments, vortex pair, rows of vortices, Karman cortex street. Viscous flow : Navier
Stokes equations, Dissipation of energy. Diffusion of vorticity in an incompressible fluid, condition of no
slip, Steady flow between two parallel i
nfinite flat plates, steady flow through a straight circular pipe
(Poiseuille Flow).

RECOMMENDED BOOKS

1.

D. E. Rutherford

:

Fluid Dynamics, Ch.
. (excluding
Sec. 0) Ch.II (excluding Sec.22 to 29)
C
h.III (excluding Sec.35 to 40)
Ch.V (excluding Sec.61 to 63
).

2.

F. Chorlton

:

Fluid Dynamics, (Relevant portion).




30

MM
709:

ALGEBRAIC CODING THEORY


Lectures to be
delivered:

60










Maximum
Marks:

80

Time
Allowed:

3 hours











Internal
Assessment:

20

Total:

100

INSTRUCTIONS FOR THE
PAPER
-
SETTER

The question paper will consist of five sections A, B, C, D and E. Sections A, B, C and D will have two
questions from the respective sections of the syllabus. Section E will consist of 8 to 10 short
-
answer type
questions which will cover the
entire syllabus uniformly. All questions will carry equal marks.

INSTRUCTIONS FOR THE CANDIDATES

Candidates are required to attempt one question each from the sections A, B, C and D of the question paper
and the entire section E.

SECTION
-
A

Introduction to
error
-
correcting codes, The main coding theory problem, An introduction to finite fields,
Introduction
-
to Linear codes, Encoding & Decoding with a linear code.

SECTION
-
B

The dual code, the parity
-
check matrix and syndrome decoding, incomplete decoding.

SEC
TION
-
C

Hamming codes, extended binary Hamming codes, Q
-
ary Hamming codes, Perfect codes, Golay codes,
sphere packing bound.

SECTION
-
D

Cyclic codes, Hamming codes as cyclic codes, BCH codes, Quadratic residue codes.

RECOMMENDED BOOKS


1.

Raymond Hill

:

Intr
oduction to Error

Correcting Codes (Ch 1
-
0 & 12)

2.

F. J. Macwilliams

:

T
heory of Error Correcting Codes


& NJA Sloane









31

MM
710:

DIFFERENTIAL GEOMETRY OF MANIFOLDS


Lectures to be
delivered:

60










Maximum
Marks:

80

Time
Allowed:

3
hours











Internal
Assessment:

20

Total:

100

INSTRUCTIONS FOR THE PAPER
-
SETTER

The question paper will consist of five sections A, B, C, D and E. Sections A, B, C and D will have two
questions from the respective sections of the syllabus. Sec
tion E will consist of 8 to 10 short
-
answer type
questions which will cover the entire syllabus uniformly. All questions will carry equal marks.

INSTRUCTIONS FOR THE CANDIDATES

Candidates are required to attempt one question each from the sections A, B, C
and D of the question paper
and the entire section E.

SECTION
-
A

Topological groups, Lie groups and lie algebras. Product of two Liegroups, One parameter subgroups and
exponential maps. Examples of Lie groups, Homomorphism and Isomorphism, Lie transformatio
n groups,
General Linear groups.

SECTION
-
B

Principal fibre bundle, Linear frame bundle, Associated fibre bundle, Vector bundle, Tangent bundle,
Induced bundle, Bundle
homomorphism
.

SECTION
-
C

Sub
-
manifolds, induced connection and second fundamental form. No
rmals, Gauss formulae, Weingarten
equations, Lines of curvature, Generalized Gauss and Mainardi

Codazzi equations.

SECTION
-
D

Almost Complex manifolds, Nijenhuis tensor, Contravariant and

covariant almost analytic vector fields, F
-
connection.

RECOMMENDED BO
OKS

1.

B. B. Sinha

:

An Introduction to Modem Differential Geometry, Kalyani Publishers, New
Delhi, 1982 (Rel.Portion ).

2.

K. Yano and M. Kon

:

Structure of Manifolds, World. Scientific Publishing Co. Pvt. Ltd., 1984
(Rel. Portion).

3.

Y. Matsushima

:

Dif
ferentiable Manifolds













32

MM 71
1
:
ANALYTIC NUMBER

THEORY


Lectures to be
delivered:

60










Maximum
Marks:

80

Time
Allowed:

3 hours











Internal
Assessment:

20

Total:

100


INSTRUCTIONS FOR THE PAPER
-
SETTER

The question
paper will consist of five sections A, B, C, D and E. Sections A, B, C and D will have two
questions from the respective sections of the syllabus. Section E will consist of 8 to 10 short
-
answer type
questions which will cover the entire syllabus uniformly.

All questions will carry equal marks.

INSTRUCTIONS FOR THE CANDIDATES

Candidates are required to attempt one question each from the sections A, B, C and D of the question paper
and the entire section E.

SECTION
-
A

Arithmetical functions: Mobius function,
Euler’s totient function,

Mangoldt function, Liouville’s
function, The divisor functions, Relation connecting


and

, product formula for

(n), Dirichlet product
of arithmetical functions, Dirichlet inverses and Mobius inversion formula, Multiplicative f
unctions,
Dirichlet multiplication, The inverse of a completely multiplicative function, Generalized convolutions.
(Ch.2, Sec.2.1
-
2.14)

SECTION
-
B

Averages of arithmetical functions: The big oh notation, Asymptotic equality of functions, Euler’s
summation
formula, Elementary asymptotic formulas, Average order of d(n),

(n),


(n),

(n),

n), The
Partial sums of a Dirichlet product, applications to

(n) and

(n), Legendre’s identity. (Ch.3, Sec.3.1
-
3.7,
3.9
-
3.11)

SECTION
-
C

Some elementary theorems on the dis
tribution of prime numbers: Chebyshev’s functions

(x) &

(x),
Relation connecting

(x) and

(x), Abel’s identity, equivalent forms of Prime number theorem,
inequalities for

(n) and P
n,
Shapiro’s Tauberian theorem, applications of Shapiro’s theorem, Asymp
totic
formula for the partial sums

p

x
(1/p). (Ch.4, Sec.4.1
-
4.8)

SECTION
-
D

Elementary properties of groups , Characters of finite abelian groups, The character group, Orthogonality
relations for characters, Dirichlet characters, Dirichlet’s theorem for p
rimes of the form 4n
-
1 and 4n+1,
Dirichlet’s theorem in primes on Arithmetical progression, Distribution of primes in Arithmetical
progression. (Ch.6, Sec.6.1
-
6.8), (Ch.7)

RECOMMENDED BOOKS


1. T.M. Apostol : Introduction to Analytic Number Theory




33

MM
7
12:

ADVANCED ABSTRACT ALGEBRA


Lectures to be
delivered:

60










Maximum
Marks:

80

Time
Allowed:

3 hours











Internal
Assessment:

20

Total:

100

INSTRUCTIONS FOR THE PAPER
-
SETTER


The question paper will consist of five section
s A, B, C, D and E. Sections A, B, C and D will have
two questions from the respective sections of the syllabus. Section E will consist of 8 to 10 short
-
answer
type questions which will cover the entire syllabus uniformly. All questions will carry equal ma
rks.

INSTRUCTIONS FOR THE CANDIDATES

Candidates are required to attempt one question each from the sections A, B, C and D of the question paper
and the entire section E.

SECTION
-
A

Modules, sub modules, Direct sums, Free modules, vector spaces, Difference
between modules and vector
spaces, R homomorphisms, Fundamental theorem of R homomorphisms, Completely reducible modules
and Schur's Lemma, Representation of linear mappings, Rank of a linear mapping.

SECTION
-
B

The isomorphism theorem for the ring of endo
morphisms of a direct sum of modules, Noetherian and
Artinian modules, Hilbert basis theorem, Wedderburn
-
Artin theorem.

SECTION
-
C

Uniform modules, Primary modules
.

Free modules over PID, row module, column module and rank, Smith
normal form.

SECTION
-
D

De
composition theorem for finitely generated modules over PID, application to finitely generated abelian
groups, Rational canonical form, Generalized Jordan form over any field.


RECOMMENDED BOOKS


1.

Bhattacharya, Jain and Nagpaul :
Basic Abstract Algebra

(
Chap. 14, 19
-
21).

2.

C.Musili :
Introduction to Rings and Modules
(Section 5.5)

3.

S. Lang : Algebra


34

MM 713
: CATEGORY

THEORY
-
II





(Pre
-
requisite: MM603: Category Theory I)


Lectures to be
delivered:

60










Maximum
Marks:

80

Time
Allowed:

3 hours











Internal
Assessment:

20

Total:

100

INSTRUCTIONS FOR THE PAPER
-
SETTER

The question paper will consist of five sections A, B, C, D and E. Sections A, B, C and D will have two
questions from the respective sections of the sy
llabus. Section E will consist of 8 to 10 short
-
answer type
questions which will cover the entire syllabus uniformly. All questions will carry equal marks.

INSTRUCTIONS FOR THE CANDIDATES

Candidates are required to attempt one question each from the sectio
ns A, B, C and D of the question paper
and the entire section E.

SECTION A

Naturality

:
Exponential in a category
, Cartesian Closed categories, Category of Categories,
Representable Structure, Stone Duality; ultrafillers in Boolean Algebra, Naturality, Exa
mples
of natural transformations, the functor category Fun (C D) and natural isomorphism. (R.R:
Sections 6.1, 6.2 and 7.1 to 7.5)

SECTION
B

Equivalence

Exponentials of Categories, The

Bifunctor Lemma,

Cat is cartesian closed,

Functor Categories,


Equival
ence of Categories, Examples of Equivalence :
Sets
fin

and Ord
fin.
,
Pointed

Set

and partial maps, slice categories and indexed families, stone duality.(R.R 7.6 to
7.9)

SECTION
C

Categories of Diagrams

Set
-
valued functor categories, The Yoneda embedding,
The Yoneda Lemma,
Applications of the Yoneda lemma, Limits, Colimits and Exponentials in Categories of diagrams. Hom (X,
G
P
) and Hom (X x P , Q) .

(R.R.: Sections 8.1 to 8.7)

SECTION
D

Adjoints
:

Adjunction between categories, left and right adjoints, Hom
-
Set definition of adjoints.
Examples of Adjoints, Uniqueness up to isomorphism. Order Adjoints and interior operation in Topology
as an order adjoint. Preservation of Limits (Co limits) by Right (Left) Adjoints. UMP of the Yoneda
Embedding and Kan Extensio
ns. Statement only of the Adjoint Functor Theorem.

(R.R. Relevant part from Chapter 9)


RECOMMENDED BOOK

Steven Awodey
:
Category Theory,
(Oxford Logic Guides, 49, Oxford University Press.)


35

MM 714
:

Nonlinear Programming

(Pre
-
requisite: MM602: Optimization
Techniques)


Lectures to be delivered: 60










Maximum Marks: 80

Time Allowed: 3 hours











Internal Assessment: 20

Total: 100


INSTRUCTIONS FOR THE PAPER
-
SETTER

The question paper will consist of five sections A, B, C, D and E
. Sections A, B, C and D will have two
questions from the respective sections of the syllabus. Section E will consist of 8 to 10 short
-
answer type
questions which will cover the entire syllabus uniformly. All questions will carry equal marks.

INSTRUCTIONS
FOR THE CANDIDATES

Candidates are required to attempt one question each from the sections A, B, C and D of the question paper
and the entire section E.


SECTION
-
A


Problem statement and basic definition, Fritz John and Kuhn Tucker optimality conditions fo
r
unconstrained optimization problems and problems with equality and inequality constrained, Lagrange's
duality and saddle point conditions.


SECTION

-
B



The concept of computational algorithm, computational algorithm for unconstrained optimization
prob
lems, Penalty and
B
arrier function methods and methods of feasible directions for constrained
optimization problem.

SECTION
-
C


Computational Techniques for L
inear
C
omplimentarily

P
roblem, Q
uadratic

Programming, Linear

F
ractional
Programming P
roblems.

SE
CTION

-
D


Optimization of Nonlinear programming problems by dynamic programming approach.


RECOMMENDED BOOKS


1.

Bazara
a
, M.S., Sherali, Hanif D and Shetty, C.M., Nonlinear programming
:
Theory and
Algorithm, John Wiley, Second Edition, 1993.

2.

Simm
ons
, D.
M., Non
-
Linear Programming for Operations research, Prentice
-

Hall, 1975.

3.
Avriel, M. Non
-
linear programming, Analysis & methods, Englewo
od Cliffs, Prentice Hall, 1976.

4.
Chander Mohan and Kusum Deep, Optimization Techniques, New Age Inter
national, 2009.






36

MM 715
:

Object Oriented Programming using ‘C++’

(Pre
-
requisite: MM

605: Computer Programming using ‘C’)


Lectures to be delivered: 60










Maximum Marks: 80

Time Allowed: 3 hours











Internal Assessment: 2
0

Total: 100


INSTRUCTIONS FOR THE PAPER
-
SETTER

The question paper will consist of five sections A, B, C, D and E. Sections A, B, C and D will have two
questions from the respective sections of the syllabus. Section E will consist of 8 to 10 short
-
answer t
ype
questions which will cover the entire syllabus uniformly. All questions will carry equal marks.

INSTRUCTIONS FOR THE CANDIDATES

Candidates are required to attempt one question each from the sections A, B, C and D of the question paper
and the entire se
ction E.

SECTION
-
A


Evolution of OOP, OOP Paradigm, Advantages of OOP, Comparison between Functional Programming
and OOP Approach, Characteristics of Object Oriented Language
-
objects, Classes, Inheritance,
Reusability, User defined Data Types,
Polymorphis
m,

Overloading.

Introduction to C++ , Identifier and keywords, Constants, C++ Operators, type conversion, Variable
declaration, statements, expressions, features of iostream.h and iomanip.h, input and output, conditional
expression loop statements, breaki
ng control statements.

SECTION
-

B

Defining a function, types of functions, storage class specifiers, recursion, pre
-
processor, header files and
standard functions, Arrays, pointer arithmetic's, structures, pointers and structures, unions, bit fields typed,

enumerations.


SECTION
-

C

Classes, member functions, objects, arrays of class objects, pointers and classes, nested classes,
constructors, destructors, inline member functions, static class member, friend functions, dynamic memory
allocation.

Inheritance,

single inheritance, types of base classes, types of derivation, multiple inheritance, container
classes, member access control.


SECTION
-
D

Function overloading, operator overloading, polymorphism, early binding, polymorphism with pointers,
virtual functi
ons, late binding, pure virtual functions, opening and closing of files, stream state member
functions, binary file operations, structures and file operations, classes and file operations, random access
file processing.


37

RECOMMENDED BOOKS


1.

Herbert Scheldt
,”The Complete Reference C++”, Tata McGraw Hill.

2.

Robert Lafore, "Object Oriented Programming in Turbo C++", Galgotia Publications,
1994.

3.

E. Balagurusamy,”Object
-
oriented programming with C++”, Tata McGraw Hill.

4.

D. Ravichandran, "Programming with C++", TMH
,

1996
.

5.

Timothy Budd, “An Introduction to Object Oriented Programming", 2nd edition, Addison
-

Wesley
-

1997.

6.

William H. Press
,
Brian P. Flannery
,
Saul A. Teukolsky

and
William T. Vetterling
,
Numerical
Recipes in C & C++ Source Code CD
-
ROM with Windows, DOS, or Mac Single Screen License
(CD
-
ROM),
Cambridge University Press.


38

MM 716
:
Advanced Optimi
zation


Lectures to be delivered: 60










Maximum Marks: 80

Time Allowed: 3 hours











Internal Assessment: 20

Total: 100


INSTRUCTIONS FOR THE PAPER
-
SETTER

The question paper will consist of five sections A, B, C, D and E. Sec
tions A, B, C and D will have two
questions from the respective sections of the syllabus. Section E will consist of 8 to 10 short
-
answer type
questions which will cover the entire syllabus uniformly. All questions will carry equal marks.

INSTRUCTIONS FOR T
HE CANDIDATES

Candidates are required to attempt one question each from the sections A, B, C and D of the question paper
and the entire section E.


SECTION
-
A


One
-
Dimensional Minimization Methods: Elimination Methods and Interpolation Methods, Exhaustive
Search, Dichotomous Search, Interval Halving Method, Fibonacci Method, Golden Section Method,
Newton Method, Quasi
-
Newton Method, Secant Method


SECTION
-
B


Unconstrained Optimization Techniques: Direct Search Methods and Descent Search Methods, Random
Sea
rch Methods, Hooke and Jeeves Method, Powells Method, Steepest Descent (Cauchy) Method,
Conjugate Gradient (Fletcher
-
Reeves) Method, Newton's Method, Marquardt Method, Quasi
-
Newton
Methods Constrained Optimization Techniques: Direct and Indirect Methods, R
andom Search Methods,
Rosens Gradient Projection Method, Penalty Function Methods, Augmented Lagrange Multiplier Method


SECTION
-
C


No free lunch theorem
,
Genetic Algorithm (GA): Theory and working procedure of binary and real valued
Genetic Algorithm, St
udy of different Crossover and Mutation operators, Particle Swarm Optimization
(PSO): Theory and Algorithms of real and binary PSO, Local and global versions of PSO, Basic Variations
of PSO, Mathematics behind PSO.


SECTION
-
D


Ant Colony Optimization (ACO
): Theory and Algorithm for real valued ACO, Simulated Annealing (SA):
Theory and Algorithm

Differential Evolution (DE: Theory and Algorithm DE)

Hybrid Algorithms: Importance of hybrid algorithms, Hybrid algorithms of GA, PSO, ACO, SA and DE.






39

RECOMMEND
ED BOOKS


1.

Bazaraa, M.S. and Sherali, H.D. and Shetty, CM, "Nonlinear programming: theory and
algorithms", John Wiley and Sons, 2006

2.

Rao S.S., "Engineering optimization: theory and practice", Wiley
-
IEEE, 1996

3.

Sivanandam. S. N. , Deep S. N., "Introduction to

genetic algorithms", Springer Verleg, 2007

4.

Engelbrecht Andries, "Fundamentals of computational swarm intelligence", Wiley, 2005

5.

Engelbrecht Andries, "Computational intelligence: an introduction", John Wiley and Sons, 2007
item Marco Dorigo, Thomas Sttzle
"Ant colony optimization", MIT Press, 2004

6.

Deb Kalyanmoy, "Optimization for Engineering Design: Algorithms and Examples", Prentice
-
Hall of India Pvt. Ltd, 2004.

7.

Price, K.V. and Storn, R.M. and Lampinen, J.A., "Differential evolution", Springer 2005.



40


MM7
17:
PROGRAMMING LAB
-
II (C++ &

Advanced Numerical Analysis
)



No. of Lab Units: 20








University Exam:


20

Time Allowed:
2
h
ours







Internal Assessment:

30

Min.Pa
s
s Marks: 35%







Max. Marks:

50




INSTRUCTIONS FOR CONDIDATES

This laboratory course will mainly comprise of exercises on what is learnt under the papers:


(1) Object
Oriented Programming

using ‘C++’.

(2)

Advanced Numerical analysis.