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ester -I &III W

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ester -II & IV, Sum

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ATHEM

ATICS

2010V

isit us at ww

w.sgbau.ac.in

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adge Baba

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niversity"

M.A

./M.Sc. Sem

.I to IVProspectus N

o. 2011129

12

Syllabus Prescribed forM

.A./M

.Sc. Part-I & Part-II

Semester I to IV

(Mathem

atics)

M.A

./M.Sc. Part-I (M

athematics)

M.A

./M.Sc. Part-I -Sem

ester I :C

ompulsory Papers

Paper-

1 MTH

1R

eal Analysis

Paper-

1 MTH

2A

dvanced Abstract A

lgebra-IPaper

-1 M

TH3

Com

ples Analysis-I

Paper-

1 MTH

4Topology-I

Optional Papers : C

hoose Any O

ne.Paper

-1 M

TH5

i)D

ifferential Geom

etry-IOR

Paper-

1 MTH

6ii)

Advanced D

iscrete Mathem

atics-IOR

Paper-

1 MTH

7iii)

Differential and Integral Equations-I

M.A

./M.Sc. Part-I -Sem

ester II :C

ompulsory Papers

Paper-

2 MTH

1M

easure and Integration TheoryPaper

-2 M

TH2

Advanced A

bstract Algebra-II

Paper-

2 MTH

3C

omples A

nalysis-IIPaper

-2 M

TH4

Topology-II

Optional Papers : C

hoose Any O

ne.Paper

-2 M

TH5

Rienannian Geom

etryOR

Paper-

2 MTH

6A

dvanced Discrete M

athematics-II O

RPaper

-2 M

TH7

Differential and Integral Equations-II

M.A

./ M.Sc.-I (M

ATHEM

ATICS)

SEMESTER-I

1 MTH

-1 : REA

L AN

ALY

SISUnit-I

:D

efinition and existance of Riem

ann Stieltjes integral,properties of the integral, Integration and differentiation.The fundam

ental theorem of calculas, integral of Vector-

valued function, rectifiable curves.Unit-II

:R

earrangement of term

s of a series, Riem

ann’s theorem.

Power series, U

niqueness theorem for pow

er series, Abel’s

limit theorem

, Tauber’s first theorem.

Unit-III:

Sequences and uniform convergence, C

auchy criterion foruniform

convergence, Weierstrass M

-test, Abel’s and

Dirichlet’s tests for uniform

convergence, uniformconvergence and continuity, uniform

convergence andintegration, uniform

convergence and differentiation,

Weierstrass approxim

ation theorem.

Unit-IV:

Functions of several variables, linear tranformation,

derivatives in an open subset of Rn , C

hain Rule, partial

derivatives, interchange of order of differentiation.D

erivatives of higher order, Taylor’s theorem.

Unit-V

:Inverse function theorem

. Implicite function theorem

,Jacobians, Extrem

um problem

s with constraints, Lagranges

multiplier m

ethod, Differentiation of integrals.

Text Book :(1)

Walter R

udin; Principles of Mathem

atical Analysis, M

c Graw

Hill

Books C

ompany, Third Edition 1976, international student edition.

References :

(1)A

postol T.M., M

athematical A

nalysis, Narosa Publishing H

ouse,N

ew D

elhi, 1985.(2)

Eurl D.R

ainville : Infinite series, The Macm

illan Eompany, N

ewYork.

(3)Friedm

an A., Foundations of M

odern Analysis, H

olt Rinehart and

Winston, Inc, N

ew York, 1970.

(4)H

ewitt E. and Starm

berg, Real and A

bstract Analysis, B

erlin,Springer, 1969.

(5)Jain P.K

. and Gupta V.P., Lebesque M

easure and Integration, New

Age international (P) Ltd., Published, N

ew D

elhi, 1986, (Reprint

2000)(6)

Gabriel K

lambaucer, M

athematical A

nalysis Marcel D

ekkar, Inc.,N

ew York, 1975.

(7)N

atanson I.P., Theory of Function of real variables, Vol.-I, FrederickU

ngar Publishing Co., 1961.

(8)Parthasarathy K

.R., Introduction to Probability and M

easure,M

acmillan Com

pany of India, Delhi, 1977.

(9)Royden H

.L., Real Analysisi, M

acmillian Pub. Co. Inc., 4th Edition,

New

York, 1993.(10)

R.R.Goldberg : Real A

nalysis, Oxford &

I.B.H. Publishing Co., N

ewD

elhi - 1970.(11)

Serge Lang, Analysis I &

II, Addison - W

esley Publishing Company

Inc., 1969.(12)

S.C.Malik : M

athematical A

nalysis, Wiley Fastern Ltd., N

ew D

elhi.(13)

Shani Narayan : A

Course of M

athematical A

nalysis, S.Chand and

Company, N

ew D

elhi.(14)

White A

.J., Real A

nalysis, an introduction.(15)

Karade T.M

. and Salunke J.N., Lectures on A

dvanced Real

Analysis, Sonu N

ilu Publication, 2004.

34

(16)W

alter Rudin, R

eal & C

omplex A

nalysis, Tata McG

raw H

illPublishing C

o. Ltd., New

Delhi.

M.A

./ M.Sc.-I

Semester-I

1MTH

-2 : Advanced A

bstract Algebra-i

Unit-I:

Autom

orphisms, conjugacy and G

-Sets. Norm

al series,solvable groups, N

ilpotent groups.Unit-II

:Sylow

’s theorems, group of order P

2, pq, Canonical form

s,sim

ilarity of linear transformations, invarient subspace,

Reduction to triangular form

s, Nilpotent transform

ations,Index of nilpotency, invarient of a nilpotent transform

ation.Unit-III

:Cyclic m

odules, simple m

odules, Shur’s lemm

a, free module.

Unit-IV:

Noetherian and A

rtiniam M

odule and rings.U

nit-V:

Hilbert basics theorem

, Wedderburn A

rtin theorem, uniform

modules, N

oether Lasker theorem.

Text Book :B

asic Abstract A

lgebra, P.B.B

hattacharya, S.K.Jani, S.R

.Nagpaul.

References :

1)I.N

.Herstein, Topics in A

lgebra, Wiley Eastern Ltd., N

ew D

elhi,1975.

2)M

.Artin, A

lgebra, Pretice-Hall of India, 1991.

3)P.M

.Cohn, Algebra, Vols. I,II &

III, John Wiley &

Sons, 1982, 1989,1991.

4)N

.Jacobson, Basic A

lgebra, Vols. I & II, W

.H. Freem

an, 1980 (alsopublished by H

industan Publishing Com

pany).5)

S.Lang, Algebra, 3rd edition, A

ddison - Wesley, 1993.

6)I.S.Luthar and I.B

.S. Passi, Algebra, Vol. I-G

roups, Vol.II-Rings,

Narosa Publishing H

ouse (Vol.I-1996, Vol.II-1999)7)

D.S.M

alik, J.N.M

ordenson, and M.K

.Sen, Fundamentals of

Abstract A

lgebra, McG

raw-H

ill, International Edition, 1997.8)

K.B.D

atta, Matrix and Linear A

lgebra, Prentice Hall of India Pvt.Ltd.,

New

Delhi, 2000.

9)S.K

.Jain, A. G

unawardena and P.B

.Bhattacharya, B

asic LinearA

lgebra with M

ATLAB, K

ey College Publishing (Springer-Verlag),2001.

10)S.K

umarsena, Linear A

lgebra, A G

eometric A

pproach, Prentice-H

all of India, 2000.11)

Vivek Sahai and V

ikas Bist, A

lgebra, Narosa Publishing H

ouse,1999.

12)I.Stew

art, Galois Theory, 2nd Edition, C

hapman and H

all, 1989.13)

J.P.Escofier, Galois theory, G

TM Vol.204, Springer, 2001.

14)T.Y.Lam

, Lectures on Modules and Rings. G

TM Vol.189, Springer-

Verlag, 1999.15)

D.S.Passm

an, A C

ourse in Ring Theory, W

adsworth and B

rooks/C

ole Advanced B

ooks and Softwares, Pacific G

roves, California,

1991.

M.A

./ M.Sc.-I

SEMESTER-I

1MTH

-3 : CO

MPLEX

AN

ALY

SIS - IUnit-I

:C

omplex Integration : Pow

er Series representation ofanalytic functions, C

auchy’s integral formula, higher order

derivatives, Cauchy’s inequality, Zeros of Analytic function,

Liouvilles theorem, Fundam

ental theorem of algebra.

Unit-II:

Taylor’s theorem, M

aximum

Modulus theorem

, Morera’s

theorem, C

ounting of zeros, Open M

apping theorem,

Cauchy-G

oursal theorem, Schw

arz’s lemna.

Unit-III:

Singularities, Isolated singularities, classification of isolatedsingularities, Laurents series developm

ent, Casorti-

Wierstrass theorem

, Argum

ent principle, Rouches theorem

.Unit-IV

:R

esidue, Cauchy’s residue theorem

, Evaluation of

integration by using residue theorem, B

ranches of many

valued function (Specially arg z, log z, za), B

ilineartransform

ation, Hadam

ard’s three circle theorem.

Unit-V

:Spaces of continuous functions, spaces of analyticfunctions, H

urwitz theorem

, Riem

ann mapping theorem

,W

ierstrass factorization theorem.

Text Book :Functions of one com

plex variable - J.B.C

onway, Springer-

Verlag International Students Edition, Narosa Publishing

House, 1980.

Reference :

1)H

.A.Priestly, Introduction to C

omplex A

nalysis, Clarendon Press,

Oxford, 1990.

2)Liang-Shin H

ahn & B

ernard Epstein, Classical C

omplex A

nalysis,Jones &

Barlett Publishers. International London, 1996.

3)L.V.A

hlfors, Complex A

nalysis, McG

raw H

ill, 1979.4)

S.Lang, Com

plex Analysis, A

ddison Wesley, 1977.

5)D

.Sarason, Com

plex Function Theory, Hindustan B

ook, Agency,

Delhi, 1994.

6)M

ark J. Ablow

itz and A.S.Fokar, C

omplex Variables : Introduction

& A

pplications, Cam

bridge University Press, South A

sian Edition,1998.

56

7)E.H

ille, Analytic Function Theory ( 2 Vols), G

onn & co. 1959.

8)W

.H.J.Fuchs, Topics in the Theory of Functions of one com

plexvariable, D

.Van Nostrand C

o., 1967.9)

C.C

aratheodory, Theory of Functions (2 Vols), Chelsea Publishing

Company, 1964.

10)M

.Heins, C

omplex Function Theory, A

cademic Press, 1968.

11)W

alter Rudin, R

eal & C

omplex A

nalysis, McG

raw H

ill Book C

o.,1966.

12)S.Saks &

A.Z

ygmund, A

nalytic Functions, Monografie,

Matem

atyczne, 1952.13)

E.C.Titchmarsh, The Theory of Functions, O

xford University Press,

London.14)

W.A

.Veech, A secord course in Com

plex Analysis, W

.A.B

enjamin,

1967.15)

S.Ponnusamy, Foundation of Com

plex Analysis, N

arosa PublishingH

ouse, 1997.

1MTH

-4 : TOPO

LOG

Y-IUnit-I

:C

ardinal and Ordinal N

umbers : Equipolent sets, cardinal

numbers, order types, ordinal num

bers, Axiom

of choice.Unit-II

:Topological Spaces : D

efinition and examples of topological

spaces. Open sets and Lim

it points. Closed sets and closure.

operators and neighbourhoods. Bases and R

elativeTopologies.

Unit-III:

Connectedness, C

ompactness and C

ontinuity :C

onnected sets and components, com

pact and countablycom

pact spaces. Continuous functions. H

omeom

orphisms.

Arcw

ise connectivity..U

nit-IV:

Seperation and Countability A

xioms :

T0 , T

1 & T

2 spaces. T2 spaces and sequences. First and

Second axiom spaces, separability.

Unit-V

:Seperation and C

ountability Axiom

s (Contd.) :

Regular and norm

al spaces, Urysohn Lem

ma, Tietze

Extension Theorem. C

ompletely regular spaces.

Text Books :(1)Foundations of G

eneral Topology by William

J. Pervin.Publisher : A

cademic Press.

Scope :U

nit-I:

Chapter 2

Unit-II

:C

hapter 3U

nit-III:

Chapter 4

Unit-IV

:C

hapter 5 : From Pg. N

o. 69 to 87.

Unit-V

:C

hapter 5 : From Pg.N

o. 87 to 98.

References B

ooks :(1)

Theory and Problems of Set Theory and R

elated Topics by Semour

Lipshutz, Publisher : SchaumPublishing C

o., New

York.(2)

Topology : A First C

ourse, by J.R.M

unkres, Publishers PrenticeH

all of India.(3)

Introduction to General Topology, B

y K.D

.Joshi, Publisher, Wiley

Eastern Ltd.(4)

A Text B

ook on Topology, By R

.S.Aggarw

al, Publisher : S.Chand

& Com

pany.

1MTH

-5 :(i) DIFFERENTIAL GEO

METRY-I

(OPTIO

NA

L)Unit-I

:Local Intrinsic properties of a surface, D

efinition of surface,curves on a surface, surfaces of R

evolution, Helicoids,

Metric, D

irection Coefficients.

Unit-II:

Families of curves, Isom

etric correspondence, Intrinsicproperties, G

eodesics, Canonical G

eodesic Equation,N

ormal Properties, G

eodesic Existance theorems, G

eodesicparallels.

Unit-III:

Geodesic curvature, G

auss-Bonnet Theorem

, Gaussian

Curvature, Surface of constant curvature, conform

alm

apping, Geodesic m

apping.Unit-IV

:Vector spaces, the dual space, Tensor product of vectorspaces, Transform

ation formulae, contraction special

tensors, Inner product. Associated tensors Exterior A

lgebra.U

nit-V:

Differential m

anifolds, Tangent vectors, Affine Tensors and

Tensorial forms, C

onnexions, covariant differentiation,A

bsolute derivation of Tensorial forms, Tensor connexions.

Text Books :“A

n Introduction to Differential G

eometry”, By T.J.W

ilmore,

Oxford U

niversity Press (1959)

References :

(1)A

course in Differential G

eometry by W

.Klingenberg (Springer)

(2)R

iemannion G

eometry and Tensor C

alculus by Weatherburn, C

.

OR

87

1MTH

5 : (ii) AD

VAN

CED

DISC

RETE M

ATHEM

ATICS -I

(OPTIO

NA

L)Unit-I

:Form

al Logic : Statements, sym

bolic representation andTautologies. Q

uantifiers, Predicates and validity.Propositional logic.

Unit-II:

Semigroups and M

onoids : Definitions and exam

ples ofsem

igroups and monoids (including those pertaining to

concatenation operation). Hom

omorphism

of semigroups

and monoids. C

ongruence relation and Quotient

semigroups. Sub-sem

igroups and submonoids. D

irectproducts. B

asic Hom

omorphism

theorem.

Unit-III:

Lattice Theory : Lattices are partial ordered sets. Theirproperties. Lattices as algebraic system

s. Sublattices. Direct

products and Hom

omorphism

s. Some special lattices, e.g.

complete, com

plemented and distributive lattices.

Unit-IV:

Boolean A

lgebras : Boolean algebra as a lattice. Various

Boolean identities. The sw

itching algebra examples.

Subalgebras. Direct products and H

omom

orphisms. Joint-

irreducible elements.

Unit-V

:B

oolean Algebras (C

ontinue) :A

toms and m

interms. B

oolean forms and their equivalence.

Minterm

Boolean forms. Sum

of products. Canonical forms.

Minim

ization of Boolean functions. Applications of Boolean

algebra of switching theory. (U

sing AN

D, O

R and N

OT

gates). The Karnaugh m

ap method.

References :

(1)J.P.Trem

blay and R.Manohar, D

iscrete Mathem

atical Structure with

Application to C

omputer Science, M

cGraw

Hill B

ook Co. 1997.

(2)Seym

our Lipschutz, Finite Mathem

atics (International Edition 1983).M

cGraw

Hill Book Com

pany.(3)

S.Wiitala, D

iscrete Mathem

atics - A U

nified Approach, M

cGraw

Hill Book Co.

(4)J.L.G

ersting : Mathem

atical Structure for Com

puter Science (3rd

Edition), Com

puter Science Press, New

York.(5)

C.L.Liu, Elements of D

iscrete Mathem

atics, McG

raw H

ill Book Co.

1MTH

5 : (iii)DIFFER

ENTIA

L AN

D IN

TEGR

AL EQ

UATIO

NS-I

(OPTIO

NA

L)Unit-I

:Existence theorem

s, Linear equations of arbitrary order,solutions of linear equations, linear system

with constant

coefficients, operational calculus and solutions of lineardifferential equations, infinite series solutions.

Unit-II:

Solutions of differential equations by definite integrals,B

oundary value problems, G

reen’s functions, expansiontheorem

s, non-linear differential equations.Unit-III

:Fourier Transform

: Definition, properties, evaluation of

Fourier and inverse Fourier transform of functions,

Convolution theorem for Fourier transform

, Sine and CosineFourier transform

s, solving differential and integral equationusing Fourier transform

.Unit-IV

:M

ellin Transform : D

efinition, properties and evaluation oftransform

s, Convolution theorem

for Mellin transform

,application to integral equation.

Unit-V

:H

ankel Transform : D

efinition, properties and evaluation ofH

ankel transform, application to integral equation, Finite

Hankel transform

.

Text Book :(1)Lassy A

ndrews, B

himsen Shivam

osgo, Integral Transformfor Engineers, Prentice H

all of India (2003).

References :

(1)W

.T.Reid, O

rdinary Differential Equation, John W

iley and Sons,N

.Y. (1971)(2)

E.A.C

oddington and N.Levinson, Theory of O

rdinary Differential

Equations, McG

raw H

ill, N.Y. (1955)

(3)I.N

.Sneddon, The use of Integral Transform, Tata M

cGraw

Hill

Publishing Com

pany Ltd.(4)

Zalman R

ubinstein, A C

ourse in Ordinary and Partial D

ifferentialEquations, A

cademic Press, N

.Y. and London.

910

SYLLA

BUS PR

ESCR

IBED FO

R M

.A. / M

.Sc.-IISEM

ESTER-II2M

TH-1 : M

EASU

RE A

ND

INTEG

RATIO

N TH

EORY

Unit-I:

Lebesgue outer measure, m

easurable sets, Regularity,

Measurable functions, B

orel and Lesbesgue measurability.

Unit-II:

Integration of Non-negative function, the general integral,

integration of series, Riem

ann and Lebesgue integrals.Unit-III

:The Four derivatives, continuous non-differentiablefunctions, functions of bounded variation, Lebesguedifferentiation theorem

, differentiation and integration.Unit-IV

:M

easures and outer measures, Extension of a m

easure,uniqueness of Extension, com

pletion of a measure, m

easurespaces, integration w

ith respect to a measure.

Unit-V

:The L

P spaces, convex functions, Jensen’s inequality.H

older and Minkow

ski inequality. Com

pleteness of LP,

convergence in measure. A

lmost U

niform convergence.

Text Book :G.de B

arra, Measure Theory and Integration. W

iley EasternLim

ited, 1981.R

eferences :(1)

Bartle R

.G., The Elements of Integration, John W

iley & Sons, Inc.,

New

York, 1966.(2)

Halm

os P.R., M

easure Theory, Van Nostrand Princeton, 1950.

(3)H

awkins T.G., Lebesgue’s Theory of Integration, its origins and

Developm

ent, Chelsea, New

York, 1979.(4)

Inder K. R

ana, An Introduction to M

easure and Integration, Narosa

Publishing House, D

elhi, 1997.(5)

Karade T.M

., Salunke J.N., Lectures on A

dvanced Real A

nalysis,Sonu N

ilu Publication, Nagpur, 2004.

(6)R

oyden H.L., R

eal Analysis, M

acmillan Pub. C

o. Inc., 4th Edition,

New

York, 1993.

SEMESTER-II

2MTH

-2 : AD

VAN

CED

ABSTR

AC

T ALG

EBRA

-II

Unit-I:

Extension fields, Algebraic and trancendutal extensions,

separable and inseparable extensions, normal extensions.

Unit-II:

Perfect Fields, Finite fields, primitive elem

ents, algebraicallyclosed fields, autom

orphism of extensions, G

aloisextensions, Fundam

ental theorem of G

alois theory,Fundam

ental theorem of A

lgebra.

Unit-III:

Roots of U

nity and cyclotomic polynom

ials, cyclicextensions, solution of polynom

ial equations by radicals,Insolvability of the general equations of degree 5 byradicals, R

uler and Com

pass construction.Unit-IV

:Sm

ith Norm

al Form over a PID

and Rank : Preliminaries, row

module, colum

n modul;e and rank, Sm

ith normal form

.U

nit-V:

Fundamental Structure theorem

for finitely generatedm

odules over a PID and its applications to finitely generated

abelian groups.

Text Book :B

asic Abstract A

lgebra, P.B.B

hattacharya, S.K.Jani, S.R

.Nagpaul

Reference Books :

1)I.N

.Herstein, Topics in A

lgebra, Wiley Eastern Ltd., N

ew D

elhi,1975.

2)M

.Artin, A

lgebra, Pretice-Hall of India, 1991.

3)P.M

.Cohn, Algebra, Vols. I,II &

III, John Wiley &

Sons, 1982, 1989,1991.

4)N

.Jacobson, Basic A

lgebra, Vols. I & II, W

.H. Freem

an, 1980 (alsopublished by H

industan Publishing Com

pany).5)

S.Lang, Algebra, 3rd edition, A

ddison - Wesley, 1993.

6)I.S.Luthar and I.B

.S. Passi, Algebra, Vol. I-G

roups, Vol.II-Rings,

Narosa Publishing H

ouse (Vol.I-1996, Vol.II-1999)7)

D.S.M

alik, J.N.M

ordenson, and M.K

.Sen, Fundamentals of

Abstract A

lgebra, McG

raw-H

ill, International Edition, 1997.8)

K.B.D

atta, Matrix and Linear A

lgebra, Prentice Hall of India Pvt.Ltd.,

New

Delhi, 2000.

9)S.K

.Jain, A. G

unawardena and P.B

.Bhattacharya, B

asic LinearA

lgebra with M

ATLAB, K

ey College Publishing (Springer-Verlag),2001.

10)S.K

umarsena, Linear A

lgebra, A G

eometric A

pproach, Prentice-H

all of India, 2000.11)

Vivek Sahai and V

ikas Bist, A

lgebra, Narosa Publishing H

ouse,1999.

12)I.Stew

art, Galois Theory, 2nd Edition, C

hapman and H

all, 1989.13)

J.P.Escofier, Galois theory, G

TM Vol.204, Springer, 2001.

14)T.Y.Lam

, Lectures on Modules and Rings. G

TM Vol.189, Springer-

Verlag, 1999.15)

D.S.Passm

an, A C

ourse in Ring Theory, W

adsworth and B

rooks/C

ole Advanced B

ooks and Softwares, Pacific G

roves, California,

1991.

1112

SEMESTER-II

2MTH

-3 : CO

MPLEX

AN

ALY

SIS-IIUnit-I

:The G

amm

a function and its properties, the Riem

ann ZetaFunction, R

emann’s Functional Equation, Euler’s theorem

,M

ittag-Leffler’s Theorem.

Unit-II:

Analytic C

ontinuation, uniqueness of direct analyticcontinuation, uniqueness of analytic continuation along acurve, pow

er series method of analytic continuation.

Unit-III:

Schewartz R

eflection Principle, monodrom

y theorem and

its consequences, Harm

onic functions on a disk, Harnack’s

inequality, Dirichlet’s problem

, Green’s function.

Unit-IV:

Canonical products, Jensen’s form

ula, Poisson-Jensenform

ula, The genus and order of an entire function, exponentof convergence, H

adamard’s factorization theorem

.U

nit-V:

The range of an analytic function, Bloch theorem

, LittlePicards theorem

, Schottky’s theorem, univalent functions,

Bieberbach’s conjecture (Statem

ent only), Cobe’s “1/4”

theorem

Text Book :Functions of one com

plex variable - J.B.C

onway, Springer-

Verlag International Students Edition, Narosa Publishing

House, 1980.

References :

1)H

.A.Priestly, Introduction to C

omplex A

nalysis, Clarendon Press,

Oxford, 1990.

2)Liang-Shin H

ahn & B

ernard Epstein, Classical C

omplex A

nalysis,Jones &

Barlett Publishers. International London, 1996.

3)L.V.A

hlfors, Complex A

nalysis, McG

raw H

ill, 1979.4)

S.Lang, Com

plex Analysis, A

ddison Wesley, 1977.

5)D

.Sarason, Com

plex Function Theory, Hindustan B

ook, Agency,

Delhi, 1994.

6)M

ark J. Ablow

itz and A.S.Fokar, C

omplex Variables : Introduction

& A

pplications, Cam

bridge University Press, South A

sian Edition,1998.

7)E.H

ille, Analytic Function Theory ( 2 Vols), G

onn & co. 1959.

8)W

.H.J.Fuchs, Topics in the Theory of Functions of one com

plexvariable, D

.Van Nostrand C

o., 1967.9)

C.C

aratheodory, Theory of Functions (2 Vols), Chelsea Publishing

Company, 1964.

10)M

.Heins, C

omplex Function Theory, A

cademic Press, 1968.

11)W

alter Rudin, R

eal & C

omplex A

nalysis, McG

raw H

ill Book C

o.,1966.

12)S.Saks &

A.Z

ygmund, A

nalytic Functions, Monografie,

Matem

atyczne, 1952.13)

E.C.Titchmarsh, The Theory of Functions, O

xford University Press,

London.14)

W.A

.Veech, A secord course in Com

plex Analysis, W

.A.B

enjamin,

1967.15)

S.Ponnusamy, Foundation of Com

plex Analysis, N

arosa PublishingH

ouse, 1997.

SEMESTER-II

2MTH

-4 : TOPO

LOG

Y-IIUnit-I

:M

etric Spaces : Metric Spaces as topological spaces.

Topological properties. Hilbert (e2) space. Frechet space.

Space of continuous functions.Unit-II

:C

omplete M

etric Spaces : Cauchy sequences, com

pletions,Equivalent conditions, B

aire Theorem.

Unit-III:

Product Spaces : Finite Products, product invariantproperties. M

etric Products. Tichonov Topology, TichonovTheorem

.Unit-IV

:Function and Q

uotient Spaces : Topology of pointwise

convergence. Topology of compact convergence. Q

uotienttopology.

Unit-V

:M

etrization and Paracompactness : U

rysohn’s metrization

theorem, paracom

pact spaces, Nagata-Sm

irnov metrization

theorem.

Text Book :Foundation of G

eneral Topology by William

J. Pervin,Publisher : A

cademic Press.

Reference Books :

(1)Topology : A

First Course, by S.R

.Munkres, Publisher : Prentice

Hall of India.

(2)Introduction to G

eneral Topology, by K.D

.Joshi, Publishers : Wiley

Eastern Ltd.(3)

A Text B

ook on Topology, By R

.S.Aggarw

al, Publisher : S.Chand

&Co.

1314

SEMESTER-II

2MTH

-5 : (i) RIEM

AN

NIA

N G

EOM

ETRY

(OPTIO

NA

L)Unit-I

:R

iemannian m

etric, metric tensor, christoffel sym

bol,christoffel sym

bol of first kind, second kind, properties ofchristoffel sym

bols, transformation of christoffel sym

bols,derivatives of tensor, absolute derivative. C

ovariantderivatives, divergence, gradient, laplacian.

Unit-II:

Parallel Vector Fields : Parallel vector field of constantm

agnitude, parallel displacement of covariant vector field,

parallelism of a vector field of variable m

agnitudeG

eodesic : Differential equations of a geodesic, special co-

ordinate system : Local cartesians, Riem

annian co-ordinates,N

ormal co-ordinates, G

eodesic normal co-ordinates.

Unit-III:

Curvature Tensor : C

ovariant curvature tensor of Riem

anntensor, curvature tensor in R

iemannian co-ordinates,

properties of curvature tensors, on a cyclic property, number

of independent companents of R

prmn .

Unit-IV:

Ricci tensor and Einstein tensor, R

icci tensor, curvatureinvariant, Einstein tensor, the B

ianchi identity. Geodesic

deviation : Equations of Geodesic deviation.

Unit-V

:R

iemannian curvature, space of constant curvature, flat

space, cartesian tensor.

Reference Books :

(1)Lectures on G

eneral Relativity - T.M

.Karade, G.S.K

hadekar andM

aya S.Bendre, Sonu N

ilu Publication.(2)

An Introduction in D

ifferential Geom

etry - T.J.Willm

ore.(3)

Tensor Calculus - Schild, J.L.Synge.

SEMESTER-II

(ii) : AD

VAN

CED

DISC

RETE M

ATHEM

ATICS-II

(OPTIO

NA

L)Unit-I

:G

raph Theory : Definition of (undirected) graphs, paths,

circuits, cycles and subgraphs. Induced subgraphs. Degree

of a vertex. Connectivity planar graphs and their properties.

Trees, Euler formula for connected planar graphs. Com

pleteand com

plete bipartite graphs. Kuratow

ski’s theorem(statem

ent only) and its use.Unit-II

:G

raph Theory (Continue) : Spanning trees, cut sets,

fundamental cut sets, and cycles. M

inimal spanning trees

and Kruskal’s algorithm

. Matrix representations of graphs.

Euler’s theorem on the existence of Eulerian paths and

circuits. Directed graphs. Indegree and outdegree of a

vertex. Weighted undirected graphs. D

ijkstra’s algorithm.

Strong connectivity and Warshall’s algorithm

. Directed

trees. Search trees. Tree traversals.Unit-III

:Introductory C

omputability Theory : Finite state m

achinesand their transition table diagram

s. Equivalence of finitestate m

achines. Reduced m

achines. Hom

omorphism

. Finiteautom

ata acceptors. Non-determ

inistic finite automata and

equivalence of its power to that of determ

inistic finiteautom

ata. Moore and M

ealy machines.

Unit-IV:

Gram

mers and Languages : Phrase structure gram

mars.

Rew

riting rules, Derivations, sentential form

s. Languagegenerated by a gram

mer. R

egular, context free and contextsensitive gram

mers and languages. R

egular sets, regularexpressions and the pum

ping lemm

a. Kleen’s theorem

.U

nit-V:

Turing machine and partial recursive functions.

Notions of syntax analysis, polish notations. C

onversionof infix expressions to polish notations. The reverse polishnotation.

References :

(1)N

.Deo, G

raph Theory with A

pplications to Engineering andC

omputer Sciences, Prentice H

all of India.(2)

J.R.Trem

blay and R.M

anohar, Discrete M

athematical Structure

with A

pplication to Com

puter Science, McG

raw H

ill Book C

o.,1997.

(3)J.E.H

opcroft and J.D.U

llman, Introduction to A

utomata Theory,

Language and Com

putation, Narosa Publishing H

ouse.(4)

C.L.Liu, Elem

ents of Discrete M

athematics, M

cGraw

Hill B

ooksco.

(5)F.H

.Harary - G

raph Theory, Narosa Publishers, N

ew D

elhi (1989)(6)

K.R

.Parthasarthy, Basic G

raph Theory (TMH

)

SEMESTER-II

(iii) : DIFFER

ENTIA

L AN

D IN

TEGR

AL EQ

UATIO

NS-II (O

PTION

AL)

Unit-I:

Fredholm Equations : Som

e Problems w

hich give rise tointegral equations, conversion of ordinary differentialequations into integral equations, integro-differentialequations.

Unit-II:

Degenerated K

ernels, Herm

itian and symm

etric Kernel, the

Hilbert-Schm

idt theorem, H

ermitization and Sym

metrization

of Kernels, Solutions of integral equations w

ith Green’s

function type Kernels.

1516

Unit-III:

Volterra Integral Equation : Types of Volterra equations,R

esolvent Kernel of volterra equations, convolution type

Kernel, som

e miscellaneous type of volterra equations.

Unit-IV:

Non-linear Volterra equations, approxim

ate methods,

application to Volterra equations with com

volution typeK

ernels.U

nit-V:

Existence and uniqueness of solution using fixed pointtheorem

in case of linear and non-linear Volterra andFreedholm

integral equations.

References :

(1)R

.P.Kanw

al, Linear Integral Equation, Theory and Techniques,A

cademic Press, N

.Y. (1971)(2)

S.G.Mikhlin, Linear Integral Equations, H

industan Book A

gency,(1960)

(3)A

.M.Viazw

az, A First course in Integral Equations, W

orld Scientific(1997)

(4)L.I.G. C

hambers, Integral Equation : A

Short Course, International

Text Book Company Ltd. (1976)

(5)Larry A

ndrews, B

himsen Shiram

oggo, Integral Transform for

Engineers, Prentice Hall of India (2003).

SYLLA

BUS PR

ESCR

IBED FO

RM

.A./M

.Sc. Part-II (Mathem

atics)Sem

ester III Com

pulsory Papers3 M

TH-1

:Functional A

nalysis-I3 M

TH-2

:C

lassical Mechanics

Choose A

ny three from the follow

ing optional papers(303)

General R

elativity and Cosm

ology-I3 M

TH-3

:(304)

Fluid Dynam

ics-I3 M

TH-4

:(305)

Operations R

esearch-I3 M

TH-5

:(306)

Difference Equations-I

(307)Fuzzy Sets and A

pplications-I(308)

Wavelet A

nalysis(309)

Banach A

lgebras-I(310)

Non-Com

mulative Rings-I

Semester IV

: Com

pulsory Papers4 M

TH-1

Functional Analysis-II

4 MTH

-2Partial D

ifferential Equations

Choose A

ny three from the follow

ing optional papersPaper

-i)

General R

elativity and Cosm

ology-II4M

TH3

ii)Fluid D

ynamics-II

4MTH

4iii)

Operations R

esearch-II4M

TH5

iv)D

ifference Equations-IIv)

Fuzzy Sets and Applications-II

vi)Lie G

roupsvii)

Banach A

lgebra-IIviii)

Non-Com

mutative Rings-II

SEMESTER-III

3MTH

-1 : FUN

CTIO

NA

L AN

ALY

SIS-IUnit-I

:N

ormal linear spaces, B

anach spaces and examples.

Quotient spaces of norm

ed linear spaces and itscom

pleteness, equivalent norms, R

iesz lemm

a.Unit-II

:B

asic Properties of finite dimensional norm

ed linear spacesand com

pactness. Weak convergence and bounded linear

transformations, norm

ed linear spaces of bounded lineartransform

ations, Dual spaces w

ith example.

Unit-III:

Boundedness theorem

and some of its consequences, open

mapping, H

ahn Banach theorem

for real linear spaces,com

plex linear spaces and normed linear spaces.

Unit-IV:

Reflexive Spaces, W

eak sequential compactness, com

pactoperators, solvability of linear equations in B

anach spaces,the closed graph theorem

.U

nit-V:

Inner product spaces, Hilbert spaces, orthogonal sets,

Bessel’s inequality, com

plete orthogonal sets, Parseval’sidentity, structure of H

ilbert spaces, Projection theorem.

Text Book :E

.Kreyszig, Introductory Functional A

nalysis with

Applications, John W

iley and Sons, New

York, 1978.

References :

1)Serge Lang, A

nalysis I & II, A

ddison-Wesley Publishing Com

pany,Inc. 1967.

2)G.B

achman and L.N

arici, Functional Analysis, A

cademic Press,

1966.3)

N.D

unford and J.T.Schwartz, Linear O

perators, Part-I, Interscience,N

ew York, 1958.

4)R

.E.Edwards, Functional A

nalysis, Holt R

inehart and Winston,

New

York, 1965.5)

C.G

offman and Pedrick, First C

ourse in Functional Analysis,

Prentice Hall of India, N

ew D

elhi, 1987.

1718

6)P.K

.Jain, O.P.A

huja and Khalil A

hmad, Functional A

nalysis, New

Age International (P) Ltd. &

Wiley Eastern Ltd., N

ew D

elhi, 1997.7)

R.B

.Holm

es, Geam

etric Functional Analysis and its A

pplications,Springer-Verlag, 1975.

8)K

.K.Jha, Functional A

nalysis, Students Friends, 1986.9)

L.V.Kantorovich and G.P.A

kilov, Functional Analysis, Pergam

onPress, 1982.

10)B

.K.Lahiri, Elem

ents of functional Analysis, The W

orld Presspvt.Ltd., Calcutta, 1994.

11)B

.Choudhary and Sudarsan N

anda, Functional Analysis w

ithA

pplications, Wiley Eastern Ltd., 1989.

12)B

.V.Limaye, Functional A

nalysis, Wiley Eastern Ltd.

13)L.A

.Lusternik and V.J.Sobolev, Elements of Functional A

nalysis,H

industan Publishing Coorporation, N

ew D

elhi, 1971.14)

G.F.Simm

ons, Introduction to Topology and Modern A

nalysis,M

cGraw

Hill Book Com

pany, New

York, 1963.15)

A.E.Taylor, Introduction to Functional A

nalysis, John Wiley and

Sons, New

York, 1958.16)

K.Yozida, Functional A

nalysis, 3rd Edition, Springer Verlag, New

York, 1971.17)

J.B.C

onway, A

Course in Function A

nalysis, Springer Verlag, New

York, 1990.18)

Walter R

udin, Function Analysis, Tata M

cGraw

Hill, Publishing

Company Ltd. N

ew D

elhi, 1973.19)

A.W

ilansky, Function Analysis, B

laisdell Publishing Com

pany,1964.

20)J.Tinsley O

den & Leszek F., D

emkow

icz, Applied Functional

Analysis, C

RC

Press Inc., 1996.21)

A.H

.Siddiqui, Function Analysis w

ith Applications, Tata M

cGraw

Hill, Publishing Com

pany Ltd. New

Delhi.

SEMESTER-III

3MTH

-2 : CLA

SSICA

L MEC

HA

NIC

SUnit-I

:Variational principle and Lagranges Equations : H

amilton’s

principle, some techniques of the calculus of variations.

Derivation of Lagrange’s Equations from

Ham

ilton’sPrinciple.

Unit-II:

Generalised coordinates, H

alonomic &

Non-holonim

icsystem

s, Scleronomic and Rheonom

ic systems, G

eneralizedpotential, Lagranges Equations of first kind and secondkind, uniqueness of solution, Energy equations forconservative fields.

Unit-III:

Legendre transformations and the H

amilton equations of

motion, cyclic coordinates and conservation theorem

s,Routh’s equations, D

erivation of Ham

ilton’s equations froma variational principle, the principle of least action..

Unit-IV:

Canonical transform

ations : The equations of Canonical

transformation, exam

ples of canonical transformations.

Poisson’s bracket & other canonical invariants (Lagranges

Bracket), Poisson’s identity.

Unit-V

:The H

amilton-Jacobi Equation for H

amilton’s principle

function, The harmonic O

scillator problem as an exam

ple ofthe H

amilton-Jacobi m

ethod. The Ham

ilton-Jacobi Equationfor H

amilton’s characteristic function, Separation of

variables in the Ham

ilton-Jacobi equation.

Text Book :(1)

Classical M

echanics : By H

.Goldstein, Second edition, N

arosaPublishing H

ouse, New

Delhi.

(2)Classical M

echanics : By N.C.Rana &

P.S.Joag, Tata Mc G

raw H

ill,1991.

References :

(1)A

.S.Ramsey D

ynamics Part-II, the English Language Book Society

and Cam

bridge University Press.

SEMESTER-III

3MTH

-3/4/5(i) : GEN

ERA

L RELATIV

ITY AN

D C

OSM

OLO

GY-I

(OPTIO

NA

L)Unit-I

:Principle of Equivalence and M

ach’s, Covariance, Geodesic

Principle, New

tonian approximation of relativistic equations

of motion, E

instein field equation and New

tonianapproxim

ation.Unit-II

:Schw

arzs child exterior solution and its isotopic form,

planetary orbits, General R

elativistic Kepler problem

,A

dvance of Perihelion of a Planet, Bending of Light rays in

a gravitational field, Gravitational Red shift in Spectral lines.

Unit-III:

Schwarzschild interior solution, Linearisation of field

equations, time-independent and spherically sym

metric

field, The Weyl’s solutions to the linearized field equations.

Unit-IV:

Eddington’s form of the Schw

arzschild solution, Einstein’sequations for D

egenerate metrices, the order m

2 equations,Field equations for stationary case.

Unit-V

:The Schw

arzschild and Kerr solutions, other coordinates,

The Kerr solution and R

otation, Distinguished surfaces

and the Rotating B

lack Hole.

1920

References :

(1)Introduction to G

eneral Relativity - R

onald Ader, M

aurice Bazin,

Menahem

Schiffer, 2nd Edtion, M

cGraw

Hill Com

pany.(2)

Lectures of Relativity - T.M

.Karade, et al Einstein Foundtion

International, Nagpur.

(3)G

ravitation and Cosmology : Principles and A

pplications of General

Theory of Relativity - Steven W

einberg, John Wiley Publication.

(4)Relativity, Therm

odynamics and Cosm

ology - R.C.Tolman (O

xfordPress)

(5)G

eneral Relativity and C

osmology - J.V.N

arlikar, Macm

illanCom

pany of India, 1978.(6)

Mathem

atical Theory of Relativity - A

.S.Eddington, Cam

bridgeU

niversity Press, 1965.SEMESTER-III

(ii) : FLUID

DY

NA

MIC

S-I (OPTIO

NA

L)Unit-I

:K

inematics of fluid in M

otion : Real fluids and ideal fluids.

Velocity of a fluid at a point stream lines and pathlines.

Steady and unsteady flows. Velocity potential, vorticity

vector, local and particles rates of change. Equation ofcontinuity, w

orked examples. A

cceleration of a fluid.C

onditions at a rigid boundary, general analysis of fluidm

otion.Unit-II

Pressure of motion of a fluid : Pressure at a point in a fluid

at rest. Pressure at a point in a moving fluid, conditions at a

boundary of two inviscid im

miscible fluids, Euler’s Equation

of motion. B

ernoulli’s equation, worked exam

ples.D

iscussion of the case of steady motion under conservative

body forces, some potential theorem

, some special tw

odim

ensional flow. Some further aspects of vartex m

otion.Unit-III

:Sources, sinks and D

oublets, images in a rigid infinite plane.

Images in a solid spheres. A

sci-symm

etric flow, Stokes

stream function. Som

e two dim

ensional flows, m

eaning oftw

o dimensional flow

, use of cylindrical polar coordinate,the stream

function, the complex potential for tw

odim

ensional, irrotational incompressible flow

. Com

plexvelocity potentials for standard tw

o-dimensional flow

s,uniform

stream, line source and link sinks, link system

.Unit-IV

:The M

ilne-Thomsom

circle theorem, som

e application ofthe circle theorem

, extension of the circle theorem, the

theorem of W

asins, the use of conformal transform

ation.Vortex row

s, single infinite row of line vortices. The K

arnarvortex street.

Unit-V

:Elem

ents of Thermodynam

ics : The equation of state ofsubstance, the first law

of Thermodynam

ics, internel energyof a gas. Specific heat of a gas. Function of state, Entropy,M

axwell’s Therm

odynamics relation. Iso-therm

al Adiabatic

and Isentropic Process.

Text Book :(1)

F.Chorlton, Text B

ook of Fluid Dynam

ics, CB

S Publishers, Delhi,

1985.R

eferences :(1)

Besaint and A

.S.Ram

say, A Treatise on H

ydrodynamics, Part-II,

CBS Publishers, Delhi, 1988.

(2)G.K

.Batchelor, A

n Introduction to Fluid Mechanics, Foundation

Books, New

Delhi, 1994.

(3)H

.Schlichting, Boundary Layer Theory, M

cGraw

Hill B

ookCom

pany, New

York, 1971.(4)

M.D

.Raisinghania, Fluid M

echanics (With H

ydrodynamics),

S.Chand and C

ompany Ltd., N

ew D

elhi.(5)

L.D.Landen and E.M

.Lipschitz, Fluid Mechanics, Pargam

on Press,London, 1985.

(6)R

.K.R

athy, An Introduction to Fluid D

ynamics, O

xford and IBH

Publishing Company, N

ew D

elhi, 1976.(7)

A.D

.Young, B

oundary Layers, A

IAA

Education Series,

Washington, D

C, 1989.

(8)S.W

.Yuan, Foundation of Fluid Mechanics, Prentice H

all of IndiaPrivate Lim

ited, New

Delhi, 1976.

SEMESTER-III

(iii) : OPER

ATION

RESEA

RC

H-I (O

PTION

AL)

Unit-I:

Operation R

esearch & its scope. N

ecessity of operationresearch in industry, linear program

ming. Sim

plex method.

Theory of the simplex m

ethod, Duality and sensitivity

analysis.Unit-II

:O

ther algorithms for linear program

ming-dual sim

plexm

ethod, parametric linear program

ming, upper bound

technique, interior point algorithm, linear goal program

ming.

Unit-III:

Transportation and assignment problem

s, network analysis,

shortest path problem, m

inimum

spanning tree problem,

maxim

um flow

problem.

Unit-IV:

Minim

um cost flow

problem, netw

ork simplex m

ethod,product planning, control w

ith PERT-CPM

.U

nit-V:

Determ

inistic and probabilistic dynamic program

ming.

2122

Text Book :(1)

F.S.Hillier and G.J.Lieberm

ann, Introduction to Operations Research

(6th Ed.) M

c Graw

Hill International Edition, Industrial Engineering

Series, 1995.(2)

Kantisw

aroop, P.K.G

upta and Manm

ohan, Operations R

esearch,Sultan C

hand & Sons, N

ew D

elhi.

Reference Books :

1)G.H

adley, Linear Programm

ing, Narosa publishing H

ouse, 1995.2)

G.Hadle, N

onlinear and Dynam

ic Programm

ing, Addison-W

esley,R

eading Mass.

3)M

okhtar S.Bazaraa, H

ohn J.Jarvis and Hanif D

.Sherali, LinearProgram

ming and N

etwork flow

s, John Wiley &

Sons, New

York,1990.

4)H

.A.Taha, O

perations Research - an Introduction, M

acmillan

Publishing Com

pany, Inc, New

York.5)

S.S.Rao, Optim

ization Theory and Applications, W

iley Eastern Ltd.,N

ew D

elhi.6)

Prem K

umar G

apta and D.S.H

ira, Operations R

esearch - An

Introduction. Chand &

company Ltde, N

ew D

elhi.7)

N.S.K

ambo, M

athematical Program

ming Techniques. A

ffiliatedEast-W

est Press Pvt.Ltd., New

Delhi, M

adras.

SEMESTER-III

(IV) : DIFFERENCE EQUATIO

NS-IUnit-I

:Introduction : D

ifference calculus. The difference operator.G

enerating function and approximate sum

mation.

Unit-II:

Linear Difference Equations : First O

rder Equations, General

results for linear equations. Equations with constant

coefficients. Applications, Equations w

ith variablecoefficients. N

on-linear equations that can be linearized.Unit-III

:The Z-transform

: Properties, initial and final value theorems,

partial sum theorem

, convolution theorem. Inverse Z-

transforms, solution of difference equation w

ith constantcoefficients by Z-transform

s.Unit-IV

:Stability Theory : Initial value problem

s for linear systems.

Stability of linear systems. Stability of non-linear system

.C

haotic behaviour.U

nit-V:

Asym

ptotic Methods : Introduction, A

symptotic analysis

of sums, linear equations, non-linear equations.

Text Book :W

alter G. Kelley and A

llan C. Peterson, Difference Equations

: An Introduction w

ith Applications, A

cademic Press, Inc.

Harcourt B

race Joranovich Publishers, 1991.

Reference Books :

(1)C

alvin Ahlbrandt and A

llan C-Peterson, D

iscrete Ham

iltoniansystem

s. Difference Equations. C

ontinued Fractions and Riccati

Equations, Kluw

er, Boston, 1996.

(2)Saber Elaydi, A

n Introduction to Difference Equations, Springer,

1999.(3)

Pundir S.K. and Pundir R., D

ifference Equations, Pragati PrakashanM

eerut, 2006.SEM

ESTER-III(V

) : FUZZY SETS A

ND

APPLIC

ATION

S-I (OPTIO

NA

L)

Unit-I:

Fuzzy sets - basic definitions, a-level sets. Convex fuzzy

sets, basic operations on fuzzy sets, cartesian products,A

lgebraic products, bounded sum and difference t-norm

sand t-conorm

s ( [1] Cha.1).

Unit-II:

The Extension Principle - The Zade’s extension principle,im

age and inverse image of fuzzy sets, Fuzzy num

bers andelem

ents of fuzzy arithmetic ( [1] C

ha.2).Unit-III

:Fuzzy relations and fuzzy graphs - Fuzzy relations fuzzysets, com

position of fuzzy relations. Min-M

ax composition

and its properties, fuzzy equivalence relations, fuzzycom

putability relations, fuzzy relations equations, fuzzygraphs, sim

ilarity relations ( [1] Cha.3).

Unit-IV:

Possibility theory - fuzzy measures, evidence theory,

possibility theory and fuzzy sets ( [1] Cha.4).

Unit-V

:Fuzzy Logic - A

n Overview

of classical logic, multivalued

logics, fuzzy propositions, fuzzy quantifiers ( [2] Cha.8,

8.1-8.4).

Text Books :(1)

H.J.Zim

merm

ann, Fuzzy Set Theory and its applications, Allied

Publ. Ltd., New

Delhi, 1991.

(2)T.Terano, Fuzzy system

and its applications, Academ

ic Press, 2001.

Reference Books :

(1)G.J.K

lir and B.Yuan, Fuzzy Sets and Fuzzy Logic, Prentice H

all ofIndia, N

ew D

elhi, 1995.SEMESTER-III

(VI) : W

AVELET A

NA

LYSIS (O

PTION

AL)

Unit-I:

Preliminaries - Linear algebra, H

ilbert spaces, Fourier series,Fourier integral and signal processing. [C

ha.1 (1.1-1.4)]Unit-II

:W

indowed Fourier transform

- Motivation and definition,

time frequency localization, the reconstruction form

ula.[Cha.2 (2.1-2.3)]

2324

Unit-III:

Continuous Wavelet Transform

s - Motivation and definition

of Wavelet Transform

s, the constructions formula,

frequency localization. [Cha.3 (3.1-3.3)]Unit-IV

:G

eneralized Frames - From

resolution of unity to frames,

reconstruction formula and consistancy condition,

Recursive construction. [C

ha.4 (4.1, 4.2, 4.4)]U

nit-V:

Discrete Tim

e - Frequency analysis, Shannon sampling

theorem, sam

pling in the time frequency dom

ain, time

sampling verses frequency sam

pling. [Cha.5 (5.1-5.3)]

Text Book :(1)

Gerald K

aiser : A Friendly G

uide to Wavelets, B

irkhauser, 1994.

Reference Books :

(1)Eugenio H

ernandez & G

uido Weiss, A

First Course on W

avelets,CRC Press, N

ew York, 1996.

(2)C

hui C.K

., An Introduction to W

avelets, Academ

ic Press, 1992.(3)

M.W

.Wang : W

avelet Transforms &

Localization Operators,

Berkhauser BVerleg.

SEMESTER-III

(VII) : BA

NA

CH

ALG

EBRA

S-I (OPTIO

NA

L)

Unit-I:

Definition of B

anach Algebra and Exam

ples. Singular andnon-singular elem

ents. The abstract index. The spectrumof an elem

ent.Unit-II

:The Spectral radius. G

elfund formula. M

ultiplicative linearfunctionals and the m

aximal ideal space. G

leason Kahane

Zelazko theorem.

Unit-III:

The Gelfand Transform

s, the spectral mapping theorem

.Isom

etric Gelfand transform

. Maxim

al ideal spaces for discalgebra and the algebra 1

1 (Z).Unit-IV

:C* - algebras : D

efinition and examples, self-adjoint, unitary,

normal, positive and projection elem

ents in C* - algebras.

Unit-V

:C

omm

utative C* algebras. C

* - homom

orphisms.

Representation of com

mutative C

*-algebras.

Text Book :M.A

.Naim

ark, Norm

ed Algebras, G

roningen, Netherlands, 1972.

Reference Books :

(1)C

.E.Rickart, G

eneral Theory of Banach A

lgebras, Von Nostrand,

1960.(2)

T.W.Palm

er, Banach A

lgebras Vol.-I, Cam

bridge University Press,

1994.

SEMESTER-III

(VIII) : N

ON

CO

MM

UTATIV

E RIN

GS-I (O

PTION

AL)

Unit-I:

Basic Terminology and exam

ples. Semi sim

plicity (x1, x2 of[1]).

Unit-II:

Structure of Semi sim

ple rings. (x3 of [1] ).Unit-III

:The Jacobson R

adical (x4 of [1]).Unit-IV

:The prim

e radical; prime and sem

i prime rings. Structure of

primitive rings; the D

ensity Theorem (x10, x11 of [1]).

Unit-V

:Sub-direct products and com

mutativity theorem

s. (x12 of[1]).

Text Book :(1)

A First C

ourse in Non-com

mutative R

ings by T.Y.Lam, Springer-

Verlag, 1991.

Reference Books :

(1)I.N

.Herstein, N

on comm

utative Rings, Carus Monographs of A

MS,

1968.(2)

N.Jacobson, B

asic Algebra II, W

H Freem

an, 1989.(3)

D.Passm

an, A C

ourse in Ring Theory, W

adsworth and B

rooks /Cole Pacific G

rove Colif, 1991.(4)

Louis H. R

owen, R

ing Theory, (Student Edition), Academ

ic Press,1991.

SYLLA

BUS PR

ESCR

IBED FO

RM

.A./M

.Sc.IISEM

ESTER-IV4M

TH-1 : FU

NC

TION

AL A

NA

LYSIS-II

Unit-I:

Riesz R

epresentation theorem, adjoint of an operator on a

Hilbert space, R

eflexivity of Hilbert spaces, self adjoint

operators, positive, projection, normal and unitary

operators.Unit-II

:Spectral properties of bounded linear operators, basicconcepts, further properties of resolvent and spectrum

, useof com

plex analysis in spectral theory.Unit-III

:C

ompact linear operators on norm

ed spaces, furtherproperties of com

pact linear operators, spectral propertiesof com

pact linear operators on normed spaces.

Unit-IV:

Spectral properties of bounded self-adjoint linear operators,further spectral properties of bounded self-adjoint linearoperators.

Unit-V

:Positive operator, square root of positive operator,projection operators, spectral fam

ily.

2526

Text Book :(1)

E.Kreyszig : Introductory functional analysis w

ith applications,John W

iley & Sons, N

ew York, 1978.

References :

1)Serge Lang, A

nalysis I & II, A

ddison-Wesley Publishing Com

pany,Inc. 1967.

2)G.B

achman and L.N

arici, Functional Analysis, A

cademic Press,

1966.3)

N.D

unford and J.T.Schwartz, Linear O

perators, Part-I, Interscience,N

ew York, 1958.

4)R

.E.Edwards, Functional A

nalysis, Holt R

inehart and Winston,

New

York, 1965.5)

C.G

offman and Pedrick, First C

ourse in Functional Analysis,

Prentice Hall of India, N

ew D

elhi, 1987.6)

P.K.Jain, O

.P.Ahuja and K

halil Ahm

ad, Functional Analysis, N

ewA

ge International (P) Ltd. & W

iley Eastern Ltd., New

Delhi, 1997.

7)R

.B.H

olmes, G

eametric Functional A

nalysis and its Applications,

Springer-Verlag, 1975.8)

K.K

.Jha, Functional Analysis, Students Friends, 1986.

9)L.V.K

antorovich and G.P.Akilov, Functional A

nalysis, Pergamon

Press, 1982.10)

B.K

.Lahiri, Elements of functional A

nalysis, The World Press

pvt.Ltd., Calcutta, 1994.11)

B.C

houdhary and Sudarsan Nanda, Functional A

nalysis with

Applications, W

iley Eastern Ltd., 1989.12)

B.V.Lim

aye, Functional Analysis, W

iley Eastern Ltd.13)

L.A.Lusternik and V.J.Sobolev, Elem

ents of Functional Analysis,

Hindustan Publishing C

oorporation, New

Delhi, 1971.

14)G.F.Sim

mons, Introduction to Topology and M

odern Analysis,

McG

raw H

ill Book Company, N

ew York, 1963.

15)A

.E.Taylor, Introduction to Functional Analysis, John W

iley andSons, N

ew York, 1958.

16)K

.Yozida, Functional Analysis, 3rd Edition, Springer Verlag, N

ewYork, 1971.

17)J.B

.Conw

ay, A C

ourse in Function Analysis, Springer Verlag, N

ewYork, 1990.

18)W

alter Rudin, Function A

nalysis, Tata McG

raw H

ill, PublishingCom

pany Ltd. New

Delhi, 1973.

19)A

.Wilansky, Function A

nalysis, Blaisdell Publishing C

ompany,

1964.20)

J.Tinsley Oden &

Leszek F., Dem

kowicz, A

pplied FunctionalA

nalysis, CR

C Press Inc., 1996.

21)A

.H.Siddiqui, Function A

nalysis with A

pplications, Tata McG

rawH

ill, Publishing Company Ltd. N

ew D

elhi.

SEMESTER-IV

4MTH

-2 : PARTIA

L DIFFER

ENTIA

L EQU

ATION

S

Unit-I:

Four important linear partial differential equations, transport

equation - initial value problem, N

on homogeneous problem

,Laplace equation, Fundam

ental solution, Mean Value

formulas, Properties of harm

onic functions, Green’s

function, Energy methods.

Unit-II:

Heat Equation : Fundam

ental solution, Mean Value form

ula,Properties of solutions, Energy m

ethods.Unit-III

:W

ave Equation : Solution by spherical m

eans,N

onhomogeneous problem

, Energy methods.

Unit-IV:

Nonlinear first order PD

E : Complete integral, new

solutionsfrom

envelopes, derivation of characteristic OD

E, Examples,

Boundary conditions, Local solution, A

pplications.U

nit-V:

Calculus of variations, Ham

ilton’s OD

E, Legendre transform,

Hopf-Lex form

ula, Weak solutions, uniqueness.

Text Book :Law

rence C. Evans : Partial Differential Equations, G

raduatestudies in M

athematics, Vol.19. A

merican M

athematical

Society, Providence Rhode Island.

References :

(1)I.M

.Sreddon : Elements of Partial D

ifferential Equation, Mc G

rawH

ill, International Editon.(2)

Phoolan Prasad : Partial Differential equations, N

ew A

ge andR

enuka Ravindram

, International Publishers.

SEMESTER-IV

4MTH

-3/4/5 (i) : GEN

ERA

L RELATIV

ITY AN

D C

OSM

OLO

GY-II

(OPTIO

NA

L)Unit-I

:Einstein Field Equations w

ith Cosm

ological term, static

cosmological m

odels of Einstein and De-sitter, their

derivations, properties and comparision w

ith the actualU

niverse.Unit-II

:C

osmological principle, H

ubble’s law, Weyls Postulate,

Steady State Cosmological m

odels, Derivation of Roberson-

Walker M

etric, Further Properties.Unit-III

:H

ubble and deceleration parameters, R

edshift, matter

dominated era of the U

niverse steady state cosmology,

Friedmann m

odel. Fundamental Equation of dynam

ical

2728

Cosmology, Critical density, Closed and open universe, A

geof the U

niverse.Unit-IV

:R

elativistic Stellar Structure, A Sim

ple Stellar model - The

interior Schwartzchild solution, stellar m

odels and stability.U

nit-V:

The field of a charged mass point, W

eyl’s generalizaton ofR

iemannian geom

etry, Weyl’s theory of electrom

agnetism.

References :

(1)Lectures on R

elativity : T.M.K

arade, et al Einstein FoundationInternational, N

agpur.(2)

Introduction to General R

elativity - Ronald A

der, Maurice B

azin,M

enahem, Schiffer.

(3)M

athematical Theory of R

elativity : A.S.Eddington, C

ambridge

University Press, 1965.

(4)R

elativity : The General Theory - J.L.Synge, N

orth Holland

Publishing Company, 1976.

(5)The C

lassical Theory of Fields - I.D.Landau and E.M

.Lifshitz,Pergam

on Press, 1980.(6)

An Introduction to R

iemannian G

eometry and the Tensor C

alculus- C

.E. Weatherburn, C

ambridge U

niversity Press, 1950.

SEMESTER-IV

4MTH

-3/4/5 (ii) : FLUID

DY

NA

MIC

S-II (OPTIO

NA

L)

Unit-I:

Gas D

ynamics : C

ompressibility effects in real fluids, the

elements of w

ave motion, one dim

ensional wave equation,

wave equation in tw

o and in three dimensions, spherical

waves, progressive and stationary w

aves, the speed ofsound in gas equation of m

otion of a gas, subsonic, sonicand supersonic flow

s, isentropic gas flow, R

eservoirdischarge through a channel of varying section.Investigation of m

aximum

mass flow

through a nozzle.Shockw

aves, formation of shockw

aves, elementary analysis

of normal shock w

aves.Unit-II

:V

iscous Flow : Stress com

ponents in a real fluid, relationbetw

een cartesian components of stress, translation m

otionof fluid elem

ent, the rate of strain quadric and principalstresses. Som

e further properties of the rate of strain quadricand principal stresses, stress analysis in fluid m

otion,relation betw

een stress and rate of strain, the coefficient ofviscosity and Lam

inar flow.

Unit-III:

The Navier stokes equations of m

otion of a viscous fluid,som

e solvable problem in viscous flow

, steady motion

between parallel planes, steady flow

through tube of

uniform circular cross section, steady flow

between

cocentric rotating cylinders, diffusion of vorticity energydissipation due to viscosity steady flow

past a fixed sphere.Unit-IV

:M

agnetohydrodynamics

: N

ature of

Magnetohydrodynam

ics, Maxw

ell’s electromagnetic field

equation, medium

at rest, medium

in motion, the equation

of motion of a conducting fluid rate of flow

of charge,sim

plification of the electromagnetic field equations, the

magnetic R

enolds number, A

lfvens theorem, the m

agneticbody force, Ferraro’s law

s of isorotation.U

nit-V:

Dynam

ical similarity, B

uckingham p-theorem

, Reynold

number, Prandt’s boundary layer, B

oundary layer equationsin tw

o dimensions, B

lasing solutions, boundary layerthickness, displacem

ent thickness, Karm

ar integralconditions, seperation of boundary layer flow.

Text Book :(1)

Chorlton, Text B

ook of Fluid Dynam

ics, CB

S Publishers, Delhi,

1985.

References :

(1)W

.H.B

esaint and A.S.R

amsay, A

Treatise on Hydrodynam

ics, Part-II, CBS Publishers, D

elhi, 1988.(2)

G.K.B

atchelor, An Introduction to Fluid M

echanics, FoundationBooks, N

ew D

elhi, 1994,(3)

H.Schlichting, B

oundary Layer Theory, McG

raw H

ill Book

Company, N

ew York, 1971.

(4)M

.D.R

aisinghania, Fluid Mechanics (W

ith Hydrodynam

ics)S.C

hand and Com

pany Ltd., New

Delhi.

(5)L.D

.Landen and E.M.Lipschite, Fluid M

echanics, Pargamon Press,

London, 1985.(6)

R.K

.Rathy, A

n Introduction to Fluid Dynam

ics, Oxford and IB

HPublishing Com

pany, New

Delhi, 1976.

(7)A

.D.Y

oung, Boundary L

ayers, AIA

A E

ducation Series,W

ashington, DC

, 1989.(8)

S.W.Yuan, Foundation of Fluid M

echanics, Prentice Hall of India,

Private Limited, N

ew D

elhi, 1976.

SEMESTER-IV

4MTH

-3/4/5 (iii) : OPER

ATION

RESEA

RC

H-II (O

PTION

AL)

Unit-I:

Gam

e Theory - Two Person, Zero sum

games, gam

es with

mixed strategies, G

raphical solution, solution by linearprogram

ming.

Unit-II:

Integer Programm

ing : Branch and B

ound technique,Q

ueueing theory and sequencing, applications to industrialproblem

s, optimal product m

ix and activity levels, petroleumrefinary operations, blending problem

s.Unit-III

:Econom

ic Interpretation of dual linear programm

ingproblem

s, input - output analysis, Leontief system,

Indecomposable and decom

posable economics.

Unit-IV:

Non-Linear Program

ming : O

ne and multi-variable

unconstrained optimization. K

uhn-Tucker conditions forconstrained optim

ization.U

nit-V:

Quadratic Program

ming, separable program

ming, convex

programm

ing, non-convex programm

ing.

Text Book :(1)

F.S.Hillier and G.J.Lieberm

ann, Introduction to Operations research

(6th Edition), M

cGraw

Hill international edition, Industrial

Engineering series, 1995.(2)

Kantisw

arup, P.K.G

upta and Man M

ohan, Operations R

esearch,Sultan C

hand & Sons, N

ew D

elhi.

Reference Books :

1)G.H

adley, Linear Programm

ing, Narosa publishing H

ouse, 1995.2)

G.Hadle, N

onlinear and Dynam

ic Programm

ing, Addison-W

esley,R

eading Mass.

3)M

okhtar S.Bazaraa, H

ohn J.Jarvis and Hanif D

.Sherali, LinearProgram

ming and N

etwork flow

s, John Wiley &

Sons, New

York,1990.

4)H

.A.Taha, O

perations Research - an Introduction, M

acmillan

Publishing Com

pany, Inc, New

York.5)

S.S.Rao, Optim

ization Theory and Applications, W

iley Eastern Ltd.,N

ew D

elhi.6)

Prem K

umar G

apta and D.S.H

ira, Operations R

esearch - An

Introduction. Chand &

company Ltde, N

ew D

elhi.7)

N.S.K

ambo, M

athematical Program

ming Techniques. A

ffiliatedEast-W

est Press Pvt.Ltd., New

Delhi, M

adras

SEMESTER-IV

(IV) : DIFFERENCE EQUATIO

NS-IIUnit-I

:The Self-adjoint Second O

rder Linear Equations :Introduction, Sturm

ian theory, Green’s functions.

Disconjugacy, the R

iccati equations. Oscillation.

Unit-II:

The Sturm-Liouville Problem

: Introduction, Finite Fourieranalysis, A

non-homogeneous problem

.:Unit-III

:D

iscrete Calculation of Variation : Introduction. N

ecessary

conditions. Sufficient conditions and disconjugacy.Unit-IV

:B

oundary Value Problems for N

on Linear Equations :Introduction, the Lipschitz case. Existence of solutions.B

oundary value problems for differential equations.

Unit-V

:Partial D

ifferential Equations. Discretization of partial

differential equations. Solution of partial differentialequations.

Text Book :W

alter G. Kelley and A

llan C.Peterson, Difference Equations

: An Introduction w

ith Applications, A

cademic Press, Inc.,

Harcourt B

race Joranovich Publishers, 1991.R

eferences :(1)

Calvin A

hlbrandt and Allan C

. Peterson, Discrete H

amiltonian

Systems. D

ifference Equations, continued Fractions and Riccati

Equations : Kluw

er, Boston, 1996.

(2)Pundir S.K

. and Pundir R., Difference Equations, Pragati Prakashan,

Meerut, 2006.

SEMESTER-IV

(V) : FU

ZZY SETS AN

D A

PPLICATIO

NS-II (O

PTION

AL)

Unit-I:

Possibility Theory : Fuzzy sets and Possibility Distributions.

Possibility and necessity measures. Possibility vs

Probability.Unit-II

:Linguistic Variables and hedges. Inference from

conditionalfuzzy propositions. The com

positional rule for inference.Unit-III

:A

pproximate reasoning - A

n overview of fuzzy expert

system. Fuzzy im

plications and their selection. Multi

conditional approximate reasoning. The role of fuzzy

relation equations.Unit-IV

:A

n Introduction to fuzzy control - fuzzy controllers. Fuzzyrule base. Fuzzy inference engine fuzzification.D

efuzzification and the various defuzzification methods (the

centre of area, the centre of maxim

a, and the mean of m

axima

methods)

Unit-V

:D

ecision making in Fuzzy Environm

ent - Individual decisionm

aking. Multiperson decision m

aking. Multicriteria decision

making. M

ultistage decision making. Fuzzy ranking m

ethod.Fuzzy linear program

ming.

Reference Books :

(1)H

.J.Zimm

ermann, Fuzzy set Theory and Its A

pplications, 2nd revised

edition, Allied Publishers Ltd., N

ew D

elhi, 1996.(2)

G.J.Klir and Bo Yuan, Fuzzy Sets and Fuzzy Logic, Prentice-H

all ofIndia Pvt. Ltd., N

ew D

elhi, 1995,

2930

SEMESTER-IV

(VI): LIE G

RO

UPS

Lie G

roups : Topics for Review

Only : (N

o question to be set on thistopic) C

harts and coordinates, analytic structures. Real

functions on a manifold. Tangent vectors. The dual vector

space. Differentials. Infinitesim

al. Transformations and

differential forms. M

appings of manifolds. Subm

anifolds.Product of m

anifolds.Unit-I

:Topological G

roups. The family of nuclei of a topological

group. Subgroups and homom

orphic images. C

onnectedtopological groups.

Unit-II:

Local Groups : Lie groups. Local lie groups. A

nalyticsubgroups of a lie group. O

ne dimensional lie groups.

Unit-III:

The Com

mutator of tw

o infinitesimal transform

ations. Thealgebra of infinitesim

al right translations. Lie groups oftransform

ations.Unit-IV

:The lie algebra of sub-group. O

ne parameter subgroup.

Taylor’s theorem for Lie groups. The Exponential m

apping.U

nit-V:

The Exterior algebra of a vector space. The algebra ofdifferential form

s. Exterior differentiation. Maurer-C

hartanform

s. The Maurer C

artan relations. Statement of the lie

fundamental theorem

s. The converses of Lie’s first andsecond theorem

s.

Text Books :(1)

Lie Groups by P.M

.Cohn, C

ambridge U

niversity Press, 1961.(2)

Introduction to Lie Groups and Lie A

lgebras by A.S.Sagle and

R.E.Walde, A

cademic Press, 1973.

Reference Books :

(1)Lie G

roups and Com

pact Groups by John F.Price (C

ambridge

University Press)

(2)Theory of Lie G

roups by Claude C

herallay (Princeton University

Press)SEM

ESTER-IV (V

II) : BAN

AC

H A

LGEBR

AS-II (O

PTION

AL)

Unit-I

:Sub algebras of C

* - algebra and the spectrum. The spectral

theorem. The continuous functional calculus. Positive linear

functionals and states in C*-algebras. T

he GN

Sconstruction.

Unit-II:

Strong and weak operator topologies. Von N

ewm

annA

lgebras. Monotone Sequence of O

perators. Range

Projections.

Unit-III:

The Com

mutant. The double com

mutant theorem

. TheK

aplansky Density theorem

. L¥ as Von N

ewm

ann Algebra,

Maxim

al Abelian A

lgebras.Unit-IV

:A

belian Von New

man A

lgebras. Cycling and Seperating

vectors. Representation of Abelian Von N

ewm

ann Algebras,

the L¥ functional calculus. C

onnectedness of the Unitary

group.U

nit-V:

The Projection lattice. Kaplansky’s form

ula. The centre of aVon N

ewm

ann Algebra. Various types of projections.

Centrally orthogonal projections, type decom

position.Text Book :

M.A

.Naim

ark, Norm

ed Algebras, N

oordhoff, Groningen,

Netherlands, 1972..

Reference Books:

(1)C

.E.Ricart, G

eneral Theory of Banach A

lgebras, Von-Nostrand,

1960.(2)

T.W.Palm

er, Banach A

lgebras, Vol.-I, Cam

bridge University Press,

1994.

SEMESTER-IV

4MTH

-3/4/5 (viii) : NO

N-C

OM

MU

TATIVE R

ING

S-II(O

PTION

AL)

Unit-I

:D

ivision rings, tensor products and maxim

al subfields [x13,x15 of [1] ].

Unit-II:

Polynomials over division rings. [x16 of [1] ].

Unit-III:

Local rings, Semi local rings [x19, x20 of [1] ].

Unit-IV:

The theory of idempotents. C

entral idempotents and block

decompositions. [x21, x22 of [1] ].

Unit-V

:Perfect and sem

iperfect rings. [x23 of [1] ].

Text Book :T.Y.Lam

, A First Course in Non Com

mutative Rings, Springer-Verlag,

1991.

Reference Books :

(1)I.N

.Herstein, N

on Com

mutative R

ings, Carns M

onographs ofA

MS, 1968.

(2)N

.Jacobson, Basic A

lgebra-II, W H

Freemann, 1989.

(3)D

.Passman, A

Course in R

ing Theory, Wardsw

orth and Brooks /

Cole Pacific Grove Colif, 1991.

(4)Louis H

. Row

en, Ring Theory (Student Edition), A

cademic Press,

1991.*****

3132

IND

EXM

.A./M

.Sc.Part-I & Part-II (Sem

ester I to IV)

Examinations in M

athematics

(Prospectus No.2011129)

Sr.No.

PaperPage N

os.

1.Special N

ote1

2.O

rdinance No. 4 of 2008

3D

irection No. 26 &

27 of 2010

M.Sc. Part-I Sem

ester I : Com

pulsory Papers3.

1MTH

1R

eal Analysis

14.

1MTH

2A

dvanced Abstract A

lgebra-I3

5.1M

TH3

Com

ples Analysis-I

46.

1MTH

4Topology-I

5O

ptional Papers : Choose A

ny One.

7.1M

TH5

Differential G

eometry-I

61M

TH5

Advanced D

iscrete Mathem

atics-I7

1MTH

5D

ifferential and Integral Equations-I8

M.Sc. Part-I

Semester-II : C

ompulsory Papers

8.2M

TH1

Measure and Integration Theory

99.

2MTH

2A

dvanced Abstract A

lgebra-II9

10.2M

TH3

Com

ples Analysis-II

1111.

2MTH

4Topology-II

12O

ptional Papers : Choose A

ny One.

12.2M

TH5

Rienannian G

eometry O

R13

2MTH

5A

dvanced Discrete M

athematics-II O

R13

2MTH

5D

ifferential and Integral Equations-II OR

14

M.Sc. Part-II

Semester-III : C

ompulsory Papers

13.3M

TH1

Functional Analysis-I

1614.

3MTH

2C

lassical Mechanics

17C

hoose Any three from

the following optional papers

15.3M

TH3

General R

elativity and Cosm

ology-I18

3MTH

4Fluid D

ynamics-I

193M

TH5

Operations R

esearch-I20

Difference Equations-I

21Fuzzy Sets and A

pplications-I22

Wavelet A

nalysis22

Banach A

lgebras-I23

Non-Com

mulative Rings-I

24M

.Sc. Part-II Semester-IV

: Com

pulsory Papers16.

4 MTH

1Functional A

nalysis-II24

17.4 M

TH2

Partial Differential Equations

26C

hoose Any three from

the following optional papers

18.4 M

TH3

i)G

eneral Relativity and C

osmology-II

264 M

TH4

ii)Fluid D

ynamics-II

274 M

TH5

iii)O

perations Research-II

28iv)D

ifference Equations-II29

v)Fuzzy Sets and A

pplications-II30

vi)Lie Groups

31vii)B

anach Algebra-II

31viii)N

on-Comm

utative Rings-II32