∫…∆i… M……b˜M…‰ §……§…… +®…Æ˙…¥…i…“ ¥…t
……“`ˆ
SAN
T GA
DG
E B
AB
A A
MR
AVATI U
NIV
ER
SITY
¥…Y……x… ¥…t…∂……J……
(FAC
ULTY
OF SC
IENC
E)
+¶™……∫…GÚ ®…EÚ…
¥…Y……x… ……Æ∆M…i… … Æ˙I…… (M… h…i…)∫…j… - 1 i…‰ ∫…j… - 4
PRO
SPEC
TU
SOF
MASTER O
F SCIENCE EXAMINATIO
NSem
ester -I &III W
inter 2010,Sem
ester -II & IV, Sum
mer 2011
INM
ATHEM
ATICS
2010V
isit us at ww
w.sgbau.ac.in
Price Rs. /-
PUBLISH
ED BY
Dineshkum
ar JoshiR
egistrarSant G
adge Baba
Am
ravati University
Am
ravati-444602
©"™…… +
¶™……∫…GÚ ®…E‰Úi…“±… (Prospectus) EÚ…‰h…i……Ω˛“ ¶……M… ∫…∆i… M……b˜M…‰ §……§…… +
®…Æ˙…¥…i…“
¥…t……“`ˆ…S™……
…⁄¥……«x…÷®…i…“ ∂…¥……™…
EÚ…‰h……∫…Ω˛“ …÷x…®…÷« p˘i…
ÀEÚ¥…… |…EÚ… ∂…i…
EÚÆ˙i…… ™…‰h……Æ˙
x……Ω˛“.'
©"N
o part of this prospectus can be reprinted or published without
specific permission of Sant G
adge Baba A
mravati U
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