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Scheme of Examination for M.Sc. Mathematics
w.e.f 2011-12
Semester I
Paper
Code
Nomenclatre External
!heor"
Exam.
Mar#s
Internal
$ssessme
nt Mar#s
Max.
Mar#s
Examination
%ors
MM-401 Advanced Abstract Algebra I 80 20 100 3 Hours
MM-402 Real Analysis I 80 20 100 3 Hours
MM-403 Toology 80 20 100 3 Hours
MM-404 !o"le# Analysis I 80 20 100 3 Hours
MM-40$ %i&&erential '(uations I 80 20 100 3 HoursMM-40) *ractical-I -- -- 100 4 Hours
Semester II
Paper
Code
Nomenclatre External
!heor"
Exam.
Mar#s
Internal
$ssessme
nt Mar#s
Max.
Mar
#s
Examination
%ors
MM-40+ Advanced Abstract Algebra II 80 20 100 3 Hours
MM-408 Real Analysis II 80 20 100 3 Hours
MM-40, !o"uter *rogra""ing T.eory/ 80 20 100 3 Hours
MM-410 !o"le# Analysis II 80 20 100 3 Hours
MM-411 %i&&erential '(uations II 80 20 100 3 Hours
MM-412 *ractical-II -- -- 100 4 Hours
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Scheme of Examination for M.Sc. Mathematics
Semester III
Complsor" Papers&
Paper
Code
Nomenclatre External
!heor"
Exam.
Mar#s
Internal
$ssessment
Mar#s
Max.
Mar#s
Examination
%ors
MM-$01 unctional Analysis 80 20 100 3 Hours
MM-$02 Analytical Mec.anics and!alculus o& ariations
80 20 100 3 Hours
'ptional Papers&A student can ot one otional aer &ro" MM-$03 ot
i/ to ot iv/ i"ilarly one aer ill be oted eac. &ro" MM-$04 ot i/to ot iv/ and MM-$0$ ot i/ to iv/
Paper Code Nomenclatre External
!heor"
Exam.
Mar#s
Internal
$ssessment
Mar#s
Max.
Mar#s
Examination
%ors
MM-(0)
*'pt. *i++
'lasticity 80 20 100 3 Hours
MM-$03
*'pt. *ii+
%i&&erence '(uations-I 80 20 100 3 Hours
MM-$03
*'pt. *iii+
Analytic 5u"ber T.eory 80 20 100 3 Hours
MM-$03
*'pt. *i,+
5u"ber T.eory 80 20 100 3 Hours
MM-(0
*'pt. *i+
luid Mec.anics I 80 20 100 3 Hours
MM-$04
*'pt. *ii+
Mat.e"atical tatistics 80 20 100 3 Hours
MM-$04*'pt. *iii+ Algebraic !oding T.eory 80 20 100 3 Hours
MM-$04
*'pt. *i,+
!o""utative Algebra 80 20 100 3 Hours
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Paper Code Nomenclatre External
!heor"
Exam.
Mar#s
Internal
$ssessment
Mar#s
Max.
Mar#s
Examination
%ors
MM-(0(*'pt. *i+
Integral '(uations 80 20 100 3 Hours
MM-$0$
*'pt. *ii+
Mat.e"atical Modeling 80 20 100 3 Hours
MM-$0$
*'pt. *iii+
6inear *rogra""ing 80 20 100 3 Hours
MM-$0$
*'pt. *i,+
u77y ets
Alications I
80 20 100 3 Hours
MM-(0 *ractical-III -- -- 100 4 Hours
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Semester I/
Complsor" Papers&
PaperCode
Nomenclatre External!heor"
Exam.
Mar#s
Internal$ssessment
Mar#s
Max.Mar#s
Examination%ors
MM-$0+ 9eneral Measure andIntegration T.eory
80 20 100 3 Hours
MM-$08 *artial %i&&erential '(uations 80 20 100 3 Hours
'ptional Papers&A candidate can ot one otional aer &ro" MM-$0, ot
i/ to ot iv/ i"ilarly one aer ill be oted eac. &ro" MM-$10 ot i/
to ot iv/ and MM-$11 ot i/ to ot iv/
Paper Code Nomenclatre External
!heor"
Exam.
Mar#s
Internal
$ssessment
Mar#s
Max.
Mar#s
Examination
%ors
MM-(0
*'pt. *i+
Mec.anics o& olids 80 20 100 3 Hours
MM-$0,
*'pt. *ii+
%i&&erence '(uations-II 80 20 100 3 Hours
MM-$0,*'pt. *iii+
Algebraic 5u"berT.eory
80 20 100 3 Hours
MM-$0,
*'pt. *i,+
Mat.e"atics &or inance
Insurance
80 20 100 3 Hours
MM-(10
*'pt. *i+
luid Mec.anics-II 80 20 100 3 Hours
MM-$10
*'pt. *ii+
:oundary alue
*roble"s
80 20 100 3 Hours
MM-$10
*'pt. *iii+
5on-!o""utative Rings 80 20 100 3 Hours
MM-$10*'pt. *i,+
Advanced %iscreteMat.e"atics
80 20 100 3 Hours
MM-(11
*'pt. *i+
Mat.e"atical Asects o&
eis"ology
80 20 100 3 Hours
MM-$11
*'pt. *ii+
%yna"ical yste"s 80 20 100 3 Hours
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Paper Code Nomenclatre External
!heor"
Exam.
Mar#s
Internal
$ssessment
Mar#s
Max.
Mar#s
Examination
%ors
MM-$11
*'pt. *iii+
;erational Researc. 80 20 100 3 Hours
MM-$11
*'pt. *i,+
u77y ets
Alications-II
80 20 100 3 Hours
MM-(12 *ractical-I -- -- 100 4 Hours
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Semester I
MM-01& $d,anced $stract $lera-I
Examination %ors & ) %ors
Max. Mar#s & 100*External !heor" Exam. Mar#s&30
4 Internal $ssessment Mar#s&20+
N'!E & !he examiner is re5ested to set nine 5estions in all ta#in two
5estions from each section and one complsor" 5estion. !he complsor" 5estion
will consist of eiht parts and will e distrited o,er the whole s"llas. !he
candidate is re5ired to attempt fi,e 5estions selectin at least one from each
section and the complsor" 5estion.
Section I *!wo 6estions+
Auto"or.is"s and Inner auto"or.is"s o& a grou 9 T.e grous Aut9/ and Inn9/Auto"or.is" grou o& a cyclic grou 5or"ali7er and !entrali7er o& a non-e"ty
subset o& a grou 9 !ons Ist IInd and IIIrdt.eore"s Alication o& ylo t.eory to grous o& s"aller orders
Section II *!wo 6estions+
!.aracteristic o& a ring it. unity *ri"e &ields =B= and C ield e#tensions %egree o&an e#tension Algebraic and transcendental ele"ents i"le &ield e#tensions Mini"al
olyno"ial o& an algebraic ele"ent !on
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Section I/ *!wo 6estions+
olvable grous %erived series o& a grou 9 i"licity o& t.e Alternating grou AnnF$/ 5on-solvability o& t.e sy""etric grou n and t.e Alternating grou An nF$/
Roots o& unity !ycloto"ic olyno"ials and t.eir irreducibility over C Radicals
e#tensions 9alois radical e#tensions !yclic e#tensions olvability o& olyno"ials byradicals over C y""etric &unctions and ele"entary sy""etric &unctions !onstruction
it. ruler and co"ass only
7ecommended 8oo#s&
1 I% Macdonald GT.e t.eory o& 9rous2 *: :.attac.arya
E ?ain R 5agal G :asic Abstract Algebra !a"bridge niversity
*ress 1,,$/
7eference 8oo#s&
1 iveD a.ai and iDas :ist G Algebra 5arosa ublication House/
2 I 6ut.ar and I: *assi G Algebra ol 1 9rous 5arosa ublication
House/3 I5 Herstein G Toics in Algebra iley 'astern 6td/
4 ur
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Semester-I
MM-02 & 7E$9 $N$9:SIS I
Examination %ors & ) %ors
Max. Mar#s & 100
*External !heor" Exam. Mar#s&304 Internal $ssessment Mar#s&20+
N'!E & !he examiner is re5ested to set nine 5estions in all ta#in two
5estions from each section and one complsor" 5estion. !he complsor" 5estion
will consist of eiht parts and will e distrited o,er the whole s"llas. !he
candidate is re5ired to attempt fi,e 5estions selectin at least one from each
section and the complsor" 5estion.
Section-I *!wo 6estions+
%e&inition and e#istence o& Rie"ann tielt
8/10/2019 M_Sc Math I to Iv sem syllabai.pdf
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Section-I/ *!wo 6estions+
*oer eries G ni(ueness t.eore" &or oer series Abel>s and Tauber>s t.eore"
Taylor>s t.eore" '#onential 6ogarit." &unctions Trigono"etric &unctions ourier
series 9a""a &unctioncoe as in !.ater 8 o& J*rinciles o& Mat.e"atical Analysis> by alter Rudin T.ird
'dition/
Integration o& di&&erential &or"sG *artitions o& unity di&&erential &or"s stoDes t.eore"
scoe as in relevant ortions o& !.ater , 10 o& J*rinciles o& Mat.e"atical Analysis>by alter Rudin 3rd 'dition/
7ecommended !ext&
J*rinciles o& Mat.e"atical Analysis> by alter Rudin 3rd 'dition/ Mc9ra-Hill1,+)
7eference 8oo#s &
1 TM Aostol Mat.e"atical Analysis 5arosa *ublis.ing House 5e %el.i 1,8$2 9abriel Ela"bauer Mat.e"atical Analysis Marcel %eDDar Inc 5e KorD 1,+$
3 A? .ite Real AnalysisL an introduction Addison-esley *ublis.ing !o Inc
1,)8
4 ' Heitt and E tro"berg Real and Abstract Analysis :erlin ringer 1,),$erge 6ang Analysis I II Addison-esley *ublis.ing !o"any Inc 1,),
Mc*/Mat.e"atics e"ester-I
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Semester-I
MM-0)& !'P'9';:
Examination %ors & ) %ors
Max. Mar#s & 100
*External !heor" Exam. Mar#s&304 Internal $ssessment Mar#s&20+
N'!E & !he examiner is re5ested to set nine 5estions in all ta#in two
5estions from each section and one complsor" 5estion. !he complsor" 5estion
will consist of eiht parts and will e distrited o,er the whole s"llas. !he
candidate is re5ired to attempt fi,e 5estions selectin at least one from each
section and the complsor" 5estion.
Section-I *!wo 6estions+
%e&inition and e#a"les o& toological saces 5eig.bour.oods 5eig.bour.ood syste"o& a oint and its roerties Interior oint and interior o& a set interior as an oerator and
its roerties de&inition o& a closed set as co"le"ent o& an oen set li"it ointaccu"ulation oint/ o& a set derived set o& a set de&inition o& closure o& a set as union o&
t.e set and its derived set Ad.erent oint !losure oint/ o& a set closure o& a set as set
o& ad.erent closure/ oints roerties o& closure closure as an oerator and itsroerties boundary o& a set %ense sets A c.aracteri7ation o& dense sets
:ase &or a toology and its c.aracteri7ation :ase &or 5eig.bour.ood syste" ub-base
&or a toology
Relative induced/ Toology and subsace o& a toological sace Alternate "et.ods o&de&ining a toology using Jroerties> o& J5eig.bour.ood syste"> JInterior ;erator>
J!losed sets> EuratosDi closure oerator and Jbase>
irst countable econd countable and searable saces t.eir relations.is and .ereditaryroerty About countability o& a collection o& diss booD given at
r 5o 1/
SEC!I'N-II *!wo 6estions+
%e&inition e#a"les and c.aracterisations o& continuous &unctions co"osition o&continuous &unctions ;en and closed &unctions Ho"eo"or.is" e"bedding
Tyc.ono&& roduct toology in ter"s o& standard de&ining/ subbase ro
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T0 T1 T2Regular and T3 searation a#io"s t.eir c.aracteri7ation and basic
roerties ie .ereditary roerty o& T0 T1 T2 Regular and T3saces and roductive
roerty o& T1and T2saces
Cuotient toology rt a "a !ontinuity o& &unction it. do"ain a sace .aving
(uotient toology About Hausdor&&ness o& (uotient sace scoe as in t.eore"s 1 2 3$ ) 8-11 !.ater 3 and relevant ortion o& c.ater 4 o& Eelley>s booD given at r5o1/
Section-III *!wo 6estions+
!o"letely regular and Tyc.ono&& T 3 1B2/ saces t.eir .ereditary and roductiveroerties '"bedding le""a '"bedding t.eore"
5or"al and T4saces G %e&inition and si"le e#a"les ryso.n>s 6e""a co"lete
regularity o& a regular nor"al sace T4i"lies Tyc.ono&& Tiet7e>s e#tension t.eore"
tate"ent only/ coe as in t.eore"s 1-+ !.ater 4 o& Eelley>s booD given at r 5o
1/%e&inition and e#a"les o& &ilters on a set !ollection o& all &ilters on a set as a o set&iner &ilter "et.ods o& generating &iltersB&iner &ilters ltra &ilter u&/ and its
c.aracteri7ations ltra ilter *rincile */ ie 'very &ilter is contained in an ultra
&ilter I"age o& &ilter under a &unction!onvergence o& &iltersG 6i"it oint !luster oint/ and li"it o& a &ilter and relations.i
beteen t.e" !ontinuity in ter"s o& convergence o& &ilters Hausdor&&ness and &ilter
convergence
Section-I/ *!wo 6estions+
!o"actnessG %e&inition and e#a"les o& co"act saces de&inition o& a co"act subset
as a co"act subsace relation o& oen cover o& a subset o& a toological sace in t.e
sub-sace it. t.at in t.e "ain sace co"actness in ter"s o& &inite intersectionroerty &i/ continuity and co"act sets co"actness and searation roerties
!losedness o& co"act subset closeness o& continuous "a &ro" a co"act sace into a
Hausdor&& sace and its conse(uence Regularity and nor"ality o& a co"act Hausdor&&
sace!o"actness and &ilter convergence !onvergence o& &ilters in a roduct sace
co"actness and roduct sace Tyc.ono&& roduct t.eore" using &ilters Tyc.ono&&
sace as a subsace o& a co"act Hausdor&& sace and its converse co"acti&ication and
Hausdor&& co"acti&ication tone-!ec. co"acti&ication coe as in t.eore"s 1+-1113 14 1$ 22-24 !.ater $ o& Eelley>s booD given at r 5o 1/
8oo#s &
1 Eelley ?6 G 9eneral Toology
2 MunDres ?R G Toology econd 'dition *rentice Hall o& IndiaB *earson
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Semester-I
MM-0& C'MP9E< $N$9:SIS-I
Examination %ors & ) %ors
Max. Mar#s & 100
*External !heor" Exam. Mar#s&304 Internal $ssessment Mar#s&20+
N'!E & !he examiner is re5ested to set nine 5estions in all ta#in two
5estions from each section and one complsor" 5estion. !he complsor" 5estion
will consist of eiht parts and will e distrited o,er the whole s"llas. !he
candidate is re5ired to attempt fi,e 5estions selectin at least one from each
section and the complsor" 5estion.
Section-I *!wo 6estions+
*oer series its convergence radius o& convergence e#a"les su" and roductdi&&erentiability o& su" &unction o& oer series roerty o& a di&&erentiable &unction
it. derivative 7ero e#7 and its roerties log7 oer o& a co"le# nu"ber 7 / t.eirbranc.es it. analyticity
*at. in a region s"oot. at. s"oot. at. contour si"ly connected region
"ultily connected region bounded variation total variation co"le# integration!auc.y-9oursat t.eore" !auc.y t.eore" &or si"ly and "ultily connected do"ains
Section II *!wo 6estions+
Inde# or inding nu"ber o& a closed curve it. si"le roerties !auc.y integral
&or"ula '#tension o& !auc.y integral &or"ula &or "ultile connected do"ain Hig.er
order derivative o& !auc.y integral &or"ula 9auss "ean value t.eore" Morera>st.eore" !auc.y>s ine(uality =eros o& an analytic &unction entire &unction radius o&
convergence o& an entire &unction 6iouville>s t.eore" unda"ental t.eore" o& algebra
Taylor>s t.eore"
Section-III *!wo 6estions+
Ma#i"u" "odulus rincile Mini"u" "odulus rincile c.ar7 6e""aingularity t.eir classi&ication ole o& a &unction and its order 6aurent series !assorati
eiertrass t.eore" Mero"or.ic &unctions *oles and 7eros o& Mero"or.ic
&unctions T.e argu"ent rincile Rouc.e>s t.eore" inverse &unction t.eore"
Section-I/ *!wo 6estions+
Residue G Residue at a singularity residue at a si"le ole residue at in&inity !auc.y
residue t.eore" and its use to calculate certain integrals de&inite integral 0 2@ &cossin/ d -N/d#/ integral o& t.e tye 0/ sin"# d# or 0/ cos"# d# oles ont.e real a#is integral o& "any valued &unctions
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8/10/2019 M_Sc Math I to Iv sem syllabai.pdf
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Semester-I
MM-0(& =ifferential E5ations I
Examination %ors & ) %ors
Max. Mar#s & 100
*External !heor" Exam. Mar#s&304 Internal $ssessment Mar#s&20+
N'!E & !he examiner is re5ested to set nine 5estions in all ta#in two
5estions from each section and one complsor" 5estion. !he complsor" 5estion
will consist of eiht parts and will e distrited o,er the whole s"llas. !he
candidate is re5ired to attempt fi,e 5estions selectin at least one from each
section and the complsor" 5estion.
Section I *!wo 6estions+
*reli"inariesG Initial value roble" and e(uivalent integral e(uation O-aro#i"atesolution e(uicontinuous set o& &unctions
:asic t.eore"sG Ascoli- Ar7ela t.eore" !auc.y *eano e#istence t.eore" and itscorollary 6isc.it7 condition %i&&erential ine(ualities and uni(ueness 9ronall>s
ine(uality uccessive aro#i"ations *icard-6indelP& t.eore" !ontinuation o&
solution Ma#i"al interval o& e#istence '#tension t.eore" Eneser>s t.eore" state"entonly/
Relevant ortions &ro" t.e booD o& JT.eory o& ;rdinary %i&&erential '(uations> by
!oddington and 6evinson/
Section-II *!wo 6estions+
6inear di&&erential syste"sG %e&initions and notations 6inear .o"ogeneous syste"sLunda"ental "atri# Ads &or"ula 6inear e(uation o& order n it.
constant coe&&icients Relevant ortions &ro" t.e booDs o& JT.eory o& ;rdinary
%i&&erential '(uations> by !oddington and 6evinson and t.e booD J%i&&erential
'(uations> by 6 Ross/
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Section I/ *!wo 6estions+
yste" o& di&&erential e(uations t.e n-t. order e(uation %eendence o& solutions oninitial conditions and ara"etersG *reli"inaries continuity and di&&erentiability
Relevant ortions &ro" t.e booD o& JT.eory o& ;rdinary %i&&erential '(uations> by
!oddington and 6evinson/
Ma#i"al and Mini"al solutions %i&&erential ine(ualities A t.eore" o& intner
ni(ueness t.eore"sG Ea"De>s t.eore" 5agu"o>s t.eore" and ;sgood t.eore"Relevant ortions &ro" t.e booD J;rdinary %i&&erential '(uations> by * Hart"an/
7eferneces&
1 'A !oddington and 5 6evinson Theory of Ordinary Differential Equations
Tata Mc9ra-Hill 2000
2 6 RossDifferential Equations ?o.n iley ons3 * Hart"an Ordinary Differential Equations ?o.n iley ons 5K 1,+1
4 9 :irD.o&& and 9! Rota Ordinary Differential Equations ?o.n iley ons1,+8
$ 9 i""onsDifferential Equations Tata Mc9ra-Hill 1,,3
) I9 *etrovsDi Ordinary Differential Equations *rentice-Hall 1,))+ % o"asundara" Ordinary Differential Equations, A first Course 5arosa *ub
2001
8 9 %eo 6aDs."iDant.a" and Rag.avendra Textbook of Ordinary
Differential Equations Tata Mc9ra-Hill 200)
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Semester-I
Paper MM-0 & Practical-I
Examination %ors & horsMax. Mar#s & 100
Part-$ & Prolem Sol,in
In t.is art roble"-solving tec.ni(ues based on aers MM-401 to MM-40$
ill be taug.t
Part-8 & Implementation of the followin prorams in $NSI C.
1 Use of nested i f . . .e lse in f inding the smallest of four numbers.
2 Use series sum to compute sin(x) and cos(x) for given angle x in degrees.Then, check error in veri fy ing sin2x+cos2(x)=1.3 Ver i fy n 3 = n ! 2 , "#here n=$,2 ,. . ,m% & check that p ref i' and pos t fi '
increment operator gives the same result .4 (ompute s imp le in te res t o f a g iven amount fo r the annua l ra te = .$2 i f
amount )=$*,***+ or t ime )=- years = .$- i f amount )=$*,***+ and t ime)=- years and = .$* other#ise.
- . Use ar ray of po in ters for a lphabe t ic so rt ing o f g iven l i st o f /ng l ish #ords .0 . 1rogram for in te rchange o f t#o ro#s or t#o co lumns o f a mat r i'. ead +#r i te
input+output matr i' from+to a f i le. . (a lcu la te the e igenva lues and e igenvec to rs o f a g iven symmetric matr i' o f
order 3.4 . (a lcu la te s tandard dev iat ion fo r a se t o f va lues ' " 5 % 5=l ,2, . . .,n ! hav ing thecorresponding fre6uencies f" 5 % 5= l ,2,. . . ,n!.
7 . 8 ind 9(: o f t#o posi t i ve in teger va lues us ing po in te r to a po in ter .$*. (ompute 9(: of 2 posi t ive integer va lues us ing recursion.$$. (heck a g iven s6uare matr i' for i ts pos i t ive defin i te form.$2. To f ind the inverse of a g iven nonsingular s6uare matr i' .
Note &-E,er" stdent will ha,e to maintain practical record on a file of prolems
sol,ed and the compter prorams done drin practical class-wor#. Examination
will e condcted throh a 5estion paper set >ointl" " the external and internal
examiners. !he 5estion paper will consists of 5estions on prolem sol,in
techni5es?alorithm and compter prorams. $n examinee will e as#ed to write
the soltions in the answer oo#. $n examinee will e as#ed to rn *execte+ one or
more compter prorams on a compter. E,alation will e made on the asis of
the examinee@s performance in written soltions?proramsA exection of compter
prorams and ,i,a-,oce examination.
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Semester II
MM-0B& $d,anced $stract $lera-II
Examination %ors & ) %ors
Max. Mar#s & 100
*External !heor" Exam. Mar#s&304 Internal $ssessment Mar#s&20+
N'!E & !he examiner is re5ested to set nine 5estions in all ta#in two
5estions from each section and one complsor" 5estion. !he complsor" 5estion
will consist of eiht parts and will e distrited o,er the whole s"llas. !he
candidate is re5ired to attempt fi,e 5estions selectin at least one from each
section and the complsor" 5estion.
Section-I *!wo 6estions+
!o""utators and .ig.er co""utators !o""utators identities !o""utator subgrous%erived grou T.ree subgrous 6e""a o& *Hall !entral series o& a grou 9 5ilotent
grous !entre o& a nilotent grou ubgrous and &actor subgrous o& nilotent grousinite nilotent grous er and loer central series o& a grou 9 and t.eir roerties
ubgrous o& &initely generated nilotent grous ylo-subgrous o& nilotent grous
coe o& t.e course as given in t.e booD at r 5o 2/
Section-II *!wo 6estions+
i"ilar linear trans&or"ations Invariant subsaces o& vector saces Reduction o& a lineartrans&or"ation to triangular &or" 5ilotent trans&or"ations Inde# o& nilotency o& a
nilotent trans&or"ation !yclic subsace it. resect to a nilotent trans&or"ation
ni(ueness o& t.e invariants o& a nilotent trans&or"ation
*ri"ary deco"osition t.eore" ?ordan blocDs and ?ordan canonical &or"s !yclic
"odule relative to a linear trans&or"ation !o"anion "atri# o& a olyno"ial /Rational !anonicals &or" o& a linear trans&or"ation and its ele"entary divisior
ni(ueness o& t.e ele"entary divisior ections )4 to )+ o& t.e booD Toics in Algebra
by I5 Herstein/
Section-III *!wo 6estions+
Modules sub"odules and (uotient "odules Module generated by a non-e"ty subset o&an R-"odule initely generated "odules and cyclic "odules Ide"otents
Ho"o"or.is" o& R-"odules unda"ental t.eore" o& .o"o"or.is" o& R-"odules
%irect su" o& "odules 'ndo"or.is" rings 'nd=M/ and 'ndRM/ o& a le&t R-"oduleM i"le "odules and co"letely reducible "odules se"i-si"le "odules/ initely
generated &ree "odules RanD o& a &initely generated &ree "odule ub"odules o& &ree
"odules o& &inite ranD over a *I% ections 141 to 14$ o& t.e booD :asic Abstract
Algebra by *: :.attac.arya E ?ain and R 5agal/
8/10/2019 M_Sc Math I to Iv sem syllabai.pdf
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Section-I/ *!wo 6estions+
'ndo"or.is" ring o& a &inite direct su" o& "odules initely generated "odules
Ascending and descending c.ains o& sub "odules o& an R-"odule Ascending and
%escending c.ange conditions A!! and %!!/ 5oet.erian "odules and 5oet.erianrings initely co-generated "odules Artinian "odules and Artinian rings 5il and
nilotent ideals Hilbert :asis T.eore" tructure t.eore" o& &inite :oolean rings
edeerburn-Artin t.eore" and its conse(uences sections 1,1 to 1,3 o& t.e booD :asicAbstract Algebra by *: :.attac.arya E ?ain and R 5agal/
7ecommended 8oo#s&
1 :asic Abstract Algebra G *: :.attac.arya R ?ain and R 5agal
2 T.eory o& 9rous G I% Macdonald
3 Toics in Algebra G I5 Herstein
4 9rou T.eory G R cott
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Semester-II
MM-03 & 7E$9 $N$9:SIS-II
Examination %ors & ) %ors
Max. Mar#s & 100
*External !heor" Exam. Mar#s&30
4 Internal $ssessment Mar#s&20+
N'!E & !he examiner is re5ested to set nine 5estions in all ta#in two
5estions from each section and one complsor" 5estion. !he complsor" 5estion
will consist of eiht parts and will e distrited o,er the whole s"llas. !he
candidate is re5ired to attempt fi,e 5estions selectin at least one from each
section and the complsor" 5estion.
Section-I *!wo 6estions+
6ebesgue outer "easure ele"entary roerties o& outer "easure Measurable sets and
t.eir roerties 6ebesgue "easure o& sets o& real nu"bers algebra o& "easurable sets:orel sets and t.eir "easurability c.aracteri7ation o& "easurable sets in ter"s o& oen
closed and 9 sets e#istence o& a non-"easurable set 6ebesgue "easurable &unctions and t.eir roerties c.aracteristic &unctions si"le
&unctions aro#i"ation o& "easurable &unctions by se(uences o& si"le &unctions
"easurable &unctions as nearly continuous &unctions :orel "easurability o& a &unction
Section-II *!wo 6estions+
Al"ost uni&or" convergence 'goro&&>s t.eore" 6usin>s t.eore" convergence in"easure Ries7 t.eore" t.at every se(uence .ic. is convergent in "easure .as an
al"ost every.ere convergent subse(uence
T.e 6ebesgue Integral G
.ortco"ings o& Rie"ann integral 6ebesgue integral o& a bounded &unction over a set o&
&inite "easure and its roerties 6ebsegue integral as a generali7ation o& t.e Rie"annintegral :ounded convergence t.eore" 6ebesgue t.eore" regarding oints o&
discontinuities o& Rie"ann integrable &unctions
Section-III *!wo 6estions+
Integral o& a non negative &unction atou>s le""a Monotone convergence t.eore"
integration o& series t.e general 6ebesgue integral 6ebesgue convergence t.eore"%i&&erentiation and Integration G
%i&&erentiation o& "onotone &unctions itali>s covering le""a t.e &our %ini
derivatives 6ebesgue di&&erentiation t.eore" &unctions o& bounded variation and t.eirreresentation as di&&erence o& "onotone &unctions
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Section-I/ *!wo 6estions+
%i&&erentiation o& an integral absolutely continuous &unctions conve# &unctions ?ensen>sine(uality
T.e 6saces
T.e 6saces MinDosDi and Holder ine(ualities co"leteness o& 6 saces :ounded
linear &unctionals on t.e 6 saces Ries7 reresentation t.eore"
7ecommeded !ext &
JReal Analysis> by H6Royden 3rd'dition/ *rentice Hall o& India 1,,,
7eference 8oo#s &
1 9de :arra Measure t.eory and integration illey 'astern 6td1,81
2 *RHal"os Measure T.eory an 5ostrans *rinceton 1,$0
3 I*5atanson T.eory o& &unctions o& a real variable ol I redericD ngar
*ublis.ing !o 1,)14 R9:artle T.e ele"ents o& integration ?o.n iley ons Inc5e KorD
1,))
$ ER*art.sart.y Introduction to *robability and "easure Mac"illan!o"any o& India 6td%el.i 1,++
*E?ain and *9uta 6ebesgue "easure and integration 5e age
International */ 6td *ublis.ers 5e %el.i 1,8)
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Semester-II
MM-0 & Compter Prorammin *!heor"+
Examination %ors & ) %ors
Max. Mar#s & 100*External !heor" Exam. Mar#s&30
4 Internal $ssessment Mar#s&20+
N'!E & !he examiner is re5ested to set nine 5estions in all ta#in two
5estions from each section and one complsor" 5estion. !he complsor" 5estion
will consist of eiht parts and will e distrited o,er the whole s"llas. !he
candidate is re5ired to attempt fi,e 5estions selectin at least one from each
section and the complsor" 5estion.
Section-I *!wo 6estions+
5u"erical constants and variablesL arit."etic e#ressionsL inutBoututL conditional
&loL looing
Section-II *!wo 6estions+
6ogical e#ressions and control &loL &unctionsL subroutinesL arrays
Section- III *!wo 6estions+
or"at seci&icationsL stringsL array argu"ents derived data tyes
Section- I/ *!wo 6estions+
*rocessing &ilesL ointersL "odulesL ;RTRA5 ,0 &eaturesL ;RTRA5 ,$ &eatures
7ecommended !ext&
Ra
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Semester-II
MM-10 & C'MP9E< $N$9:SIS-II
Examination %ors & ) %ors
Max. Mar#s & 100*External !heor" Exam. Mar#s&30
4 Internal $ssessment Mar#s&20+
N'!E & !he examiner is re5ested to set nine 5estions in all ta#in two
5estions from each section and one complsor" 5estion. !he complsor" 5estion
will consist of eiht parts and will e distrited o,er the whole s"llas. !he
candidate is re5ired to attempt fi,e 5estions selectin at least one from each
section and the complsor" 5estion.
Section-I *!wo 6estions+
aces o& analytic &unctions and t.eir co"leteness Hurit7>s t.eore" Montel>s
t.eore" Rie"ann "aing t.eore" in&inite roducts eierstrass &actori7ation t.eore"actori7ation o& sine &unction 9a""a &unction and its roerties &unctional e(uation &or
ga""a &unction Integral version o& ga""a &unction
Section- II *!wo 6estions+
Rei"ann-7eta &unction Rie"ann>s &unctional e(uation Runge>s t.eore" Mittag-
6e&&ler>s t.eore"Analytic continuation uni(ueness o& direct analytic continuation uni(ueness o& analytic
continuation along a curve *oer series "et.od o& analytic continuation c.ar7
re&lection rincile
Section III *!wo 6estions+
Monodro"y t.eore" and its conse(uences Har"onic &unction as a disD *oisson>s
Eernel HarnacD>s ine(uality HarnacD>s t.eore" !anonical roduct ?ensen>s &or"ula
*oisson-?ensen &or"ula Hada"ard>s t.ree circle t.eore" %iric.let roble" &or a unit
disD %iric.let roble" &or a region 9reen>s &unction
Section I/ *!wo 6estions+
;rder o& an entire &unction '#onent o& convergence :orel t.eore" Hada"ard>s
&actori7ation t.eore" T.e range o& an analytic &unction :loc.>s t.eore" 6ittle-*icard
t.eore" c.ottDy>s t.eore" Montel-!arat.edory t.eore" 9reat *icard t.eore"nivalent &unctions :ieberbac.>s con
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8oo#s recommended &
1 A.l&ors 6 !o"le# Analysis Mc9ra-Hill :ooD !o"any 1,+,2 !.urc.ill R and :ron ? !o"le# ariables and Alications Mc9ra
Hill *ublis.ing !o"any 1,,0
3 !onay ?: unctions o& ;ne co"le# variables 5arosa *ublis.ing 2000
7eference 8oo#s &
1 *riestly HA Introduction to !o"le# Analysis !laredon *ress ;r&ord 1,,0
2 6iang-s.in Hann :ernard 'stein !lassical !o"le# Analysis ?ones and
:artlett *ublis.ers International 6ondon 1,,)3 %arason !o"le# unction T.eory Hindustan :ooD Agency %el.i 1,,4
4 MarD ?Ableit7 and AoDas !o"le# ariables G Introduction
Alications !a"bridge niversity *ress out. Asian 'dition 1,,8
$ '!Titc."arsn T.e T.eory o& unctions ;#&ord niversity *ress 6ondon) *onnusa"y oundations o& !o"le# Analysis 5arosa *ublis.ing House
1,,+
8/10/2019 M_Sc Math I to Iv sem syllabai.pdf
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Semester-II
MM-11& =IE7EN!I$9 E6D$!I'NS-II
Examination %ors & ) %ors
Max. Mar#s & 100
*External !heor" Exam. Mar#s&304 Internal $ssessment Mar#s&20+
N'!E & !he examiner is re5ested to set nine 5estions in allA ta#in two
5estions from each section and one complsor" 5estion. !he complsor" 5estion
will consist of eiht parts and will e distrited o,er the whole s"llas. !he
candidate is re5ired to attempt fi,e 5estions selectin at least one from each
section and the complsor" 5estion.
Section I *!wo 6estions+
6inear second order e(uationsG *reli"inaries sel& ads e(uation *rQ&&er trans&or"ation 7ero o& a
solution ;scillatory and non-oscillatory e(uations Abel>s &or"ula !o""on 7eros o&solutions and t.eir linear deendence
Relevant ortions &ro" t.e booD J%i&&erential '(uations> by 6 Ross and t.e booD
JTe#tbooD o& ;rdinary %i&&erential '(uations> by %eo et al/
Section II *!wo 6estions+
tur" t.eoryG tur" searation t.eore" tur" &unda"ental co"arison t.eore" and itscorollaries 'le"entary linear oscillations
Autono"ous syste"sG t.e .ase lane at.s and critical oints Tyes o& critical ointsL
5ode !enter addle oint iral oint tability o& critical oints !ritical oints andat.s o& linear syste"sG basic t.eore"s and t.eir alications
Relevant ortions &ro" t.e booD J%i&&erential '(uations> by 6 Ross and t.e booD
JTe#tbooD o& ;rdinary %i&&erential '(uations> by %eo et al/
Section-III *!wo 6estions+
!ritical oints and at.s o& non-linear syste"sG basic t.eore"s and t.eir alications6iaunov &unction 6iaunov>s direct "et.od &or stability o& critical oints o& non-linear
syste"s
6i"it cycles and eriodic solutionsG 6i"it cycle e#istence and non-e#istence o& li"itcycles :enedi#son>s non-e#istence criterion Hal&-at. or e"iorbit 6i"it set *oincare-
:enedi#son t.eore" Inde# o& a critical oint
Relevant ortions &ro" t.e booD J%i&&erential '(uations> by 6 Ross and t.e booD
JT.eory o& ;rdinary %i&&erential '(uations> by !oddington and 6evinson/
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Section- I/ *!wo 6estions+
econd order boundary value roble"s:*/G 6inear roble"sL eriodic boundaryconditions regular linear :* singular linear :*L non-linear :* tur"-6iouville
:*G de&initions eigen value and eigen &unction ;rt.ogonality o& &unctions
ort.ogonality o& eigen &unctions corresonding to distinct eigen values 9reen>s &unctionAlications o& boundary value roble"s se o& I"licit &unction t.eore" and i#ed
oint t.eore"s &or eriodic solutions o& linear and non-linear e(uations
Relevant ortions &ro" t.e booD JTe#tbooD o& ;rdinary %i&&erential '(uations> by %eoet al/
7eferneces&
1 'A !oddington and 5 6evinson Theory of Ordinary Differential Equations
Tata Mc9ra-Hill 2000
2 6 RossDifferential Equations ?o.n iley ons
3 9 %eo 6aDs."iDant.a" and Rag.avendra Textbook of OrdinaryDifferential Equations Tata Mc9ra-Hill 200)
4 * Hart"an Ordinary Differential Equations ?o.n iley ons 5K 1,+1$ 9 :irD.o&& and 9! Rota Ordinary Differential Equations ?o.n iley ons
1,+8
) 9 i""onsDifferential Equations Tata Mc9ra-Hill 1,,3+ I9 *etrovsDi Ordinary Differential Equations *rentice-Hall 1,))
8 % o"asundara" Ordinary Differential Equations, A first Course 5arosa *ub
2001
8/10/2019 M_Sc Math I to Iv sem syllabai.pdf
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Semester-II
Paper MM-12 & Practical-II
Examination %ors & hors
Max. Mar#s & 100
Part-$ & Prolem Sol,in
In t.is art roble" solving tec.ni(ues based on aers MM-40+ to MM-411
ill be taug.t
Part-8 & Implementation of the followin prorams in '7!7$N-0
$. (alculate the area of a triangle #ith given lengths of its sides.2. 9iven the centre and a point on the boundary of a circle, find its perimeter and area.3. To check an e6uation a'2; by2;2c';2dy;e=* in "', y% plane #ith given coefficients for
representing parabola+ hyperbola+ ellipse+ circle or else.ointl" " the external and internal
examiners. !he 5estion paper will consists of 5estions on prolem sol,in
techni5es?alorithm and compter prorams. $n examinee will e as#ed to write
the soltions in the answer oo#. $n examinee will e as#ed to rn *execte+ one or
more compter prorams on a compter. E,alation will e made on the asis of
the examinee@s performance in written soltions?proramsA exection of compter
prorams and ,i,a-,oce examination.
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SEMES!E7-III
MM-(01 nctional $nal"sis
Examination %ors & ) %ors
Max. Mar#s & 100
*External !heor" Exam. Mar#s&30
4 Internal $ssessment Mar#s&20+N'!E & !he examiner is re5ested to set nine 5estions in all ta#in two
5estions from each section and one complsor" 5estion. !he complsor" 5estion
will consist of eiht parts and will e distrited o,er the whole s"llas. !he
candidate is re5ired to attempt fi,e 5estions selectin at least one from each
section and the complsor" 5estion.
SEC!I'N-I *!wo 6estions+
5or"ed linear saces :anac. saces and e#a"les subsace o& a :anac. saceco"letion o& a nor"ed sace (uotient sace o& a nor"ed linear sace and its
co"leteness roduct o& nor"ed saces &inite di"ensional nor"ed saces and
subsaces e(uivalent nor"s co"actness and &inite di"ension Ries7>s le""a
:ounded and continuous linear oerators di&&erentiation oerator integral oeratorbounded linear e#tension linear &unctionals bounded linear &unctionals continuity and
boundedness de&inite integral canonical "aing linear oerators and &unctionals on&inite di"ensional saces nor"ed saces o& oerators dual saces it. e#a"les coe
o& t.is section is as in relevant arts o& !.ater 2 o& JIntroductory unctional Analysis
it. Alications> by 'Ereys7ig/
SEC!I'N-II *!wo 6estions+
Ha.n-:anac. t.eore" &or real linear saces co"le# linear saces and nor"ed linear
saces alication to bounded linear &unctionals on !abS Ries7-reresentation t.eore"
&or bounded linear &unctionals on !abS ad by 'Ereys7ig/
Inner roduct saces Hilbert saces and t.eir e#a"les yt.agorean t.eore"Aolloniu>s identity c.ar7 ine(uality continuity o& innerroduct co"letion o& an
inner roduct sace subsace o& a Hilbert sace ort.ogonal co"le"ents and direct
su"s ro
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;rt.onor"al sets and se(uences :essel>s ine(uality series related to ort.onor"al
se(uences and sets totalco"lete/ ort.onor"al sets and se(uences *arseval>s identity
searable Hilbert sacesReresentation o& &unctionals on Hilbert saces Ries7reresentation t.eore" &or bounded linear &unctionals on a Hilbert sace ses(uilinear
&or" Ries7 reresentation t.eore" &or bounded ses(uilinear &or"s on a Hilbert sace
Hilbert ad
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SEMES!E7- III
MM-(02 $nal"tical Mechanics and Calcls of /ariations
Examination %ors & ) %ors
Max. Mar#s & 100
*External !heor" Exam. Mar#s&304 Internal $ssessment Mar#s&20+
N'!E & !he examiner is re5ested to set nine 5estions in all ta#in two
5estions from each section and one complsor" 5estion. !he complsor" 5estion
will consist of eiht parts and will e distrited o,er the whole s"llas. !he
candidate is re5ired to attempt fi,e 5estions selectin at least one from each
section and the complsor" 5estion.
SEC!I'N-I *!wo 6estions+Motivating roble"s o& calculus o& variationsG s.ortest distance Mini"u" sur&ace o& revolution
:rac.istoc.rone roble" Isoeri"etric roble" 9eodesic unda"ental 6e""a o& calculus o&
variation 'uler>s e(uation &or one deendent &unction o& one and several indeendent variables
and its generali7ation to i/ unctional deending on Jn> deendent &unctions ii/ unctional
deending on .ig.er order derivatives ariational derivative invariance o& 'uler>s e(uationsnatural boundary conditions and transition conditions !onditional e#tre"u" under geo"etric
constraints and under integral constraints ariable end oints
SEC!I'N-II *!wo 6estions+ree and constrained syste"s constraints and t.eir classi&ication 9enerali7ed coordinates
Holono"ic and 5on-Holono"ic syste"s clerono"ic and R.eono"ic syste"s 9enerali7ed
*otential *ossible and virtual dislace"entsideal constraints 6agrange>s e(uations o& &irst
Dind *rincile o& virtual dislace"ents %>Ale"bert>s rincile Holono"icyste"s indeendent
coordinates generali7ed &orces 6agrange>s e(uations o& second Dind ni(ueness o& solution
T.eore" on variation o& total 'nergy *otential 9yroscoic and dissiative &orces 6agrange>s
e(uations &or otential &orces e(uation &or conservative &ields
SEC!I'N-III *!wo 6estions+Ha"ilton>s variables %on Din>s t.eore" Ha"ilton canonical e(uations Rout.>s e(uations
!yclic coordinates *oisson>s :racDet *oisson>s Identity ?acobi-*oisson t.eore" Ha"ilton>s
*rincile second &or" o& Ha"ilton>s rincile *oincare-!arton integral invariant .ittaDer>s
e(uations ?acobi>s e(uations *rincile o& least action
SEC!I'N-I/ *!wo 6estions+
!anonical trans&or"ations &ree canonical trans&or"ations Ha"ilton-?acobi e(uation ?acobi
t.eore" Met.od o& searation o& variables &or solving Ha"ilton-?acobi e(uation Testing t.e
!anonical c.aracter o& a trans&or"ation 6agrange bracDets !ondition o& canonical c.aracter o& a
trans&or"ation in ter"s o& 6agrange bracDets and *oisson bracDets i"licial nature o& t.e
?acobian "atri# o& a canonical trans&or"ations Invariance o& 6agrange bracDets and *oisson
bracDets under canonical trans&or"ations
8oo#s&
1 9ant"ac.er 6ectures in Analytic Mec.anics E.osla *ublis.ing House 5e
%el.i
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2. H 9oldstein !lassical Mec.anics 2nd edition/ 5arosa *ublis.ing House 5e
%el.i
3 IM 9el&and and o"in !alculus o& ariations *rentice Hall
4 rancis : Hilderbrand Met.ods o& alied "at.e"atics *rentice Hall
$ 5arayan !.andra Rana *ra"od .arad !.andra ?oag !lassical Mec.anics Tata
Mc9ra Hill 1,,1
) 6ouis 5 Hand and ?anet % inc. Analytical Mec.anics !a"bridge niversity
*ress 1,,8
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SEMES!E7-III
MM-(0) *opt. i+ Elasticit"
Examination %ors & ) %ors
Max. Mar#s & 100
*External !heor" Exam. Mar#s&304 Internal $ssessment Mar#s&20+
N'!E & !he examiner is re5ested to set nine 5estions in all ta#in two
5estions from each section and one complsor" 5estion. !he complsor" 5estion
will consist of eiht parts and will e distrited o,er the whole s"llas. !he
candidate is re5ired to attempt fi,e 5estions selectin at least one from each
section and the complsor" 5estion.
SEC!I'N-I *!wo 6estions+
Tensor AlgebraG !oordinate-trans&or"ation !artesian Tensor o& di&&erent order
*roerties o& tensors Isotroic tensors o& di&&erent orders and relation beteen t.e"
y""etric and sDe sy""etric tensors Tensor invariants %eviatoric tensors 'igen-
values and eigen-vectors o& a tensorTensor AnalysisG calar vector tensor &unctions !o""a notation 9radient divergence
and curl o& a vector B tensor &ield Relevant ortions o& !.aters 2 and 3 o& booD by %!.andraseD.araia. and 6 %ebnat./
SEC!I'N-II *!wo 6estions+
Analysis o& train G A&&ine trans&or"ation In&initesi"al a&&ine de&or"ation 9eo"etricalInterretation o& t.e co"onents o& strain train (uadric o& !auc.y *rincial strains and
invariance 9eneral in&initesi"al de&or"ation aint-enantUs e(uations o& co"atibility
inite de&or"ations
Analysis o& tress G tress ecotr tress tensor '(uations o& e(uilibriu"Trans&or"ation o& coordinates
Relevant ortion o& !.ater I II o& booD by I oDolniDo&&/
SEC!I'N-III *!wo 6estions+
tress (uadric o& !auc.y *rincial stress and invariants Ma#i"u" nor"al and s.ear
stresses Mo.r>s circles e#a"les o& stress '(uations o& 'lasticity G 9eneralised HooDs
6a Anisotroic sy""etries Ho"ogeneous isotroic "ediu"Relevant ortion o& !.ater II III o& booD by I oDolniDo&&/
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SEC!I'N- I/ *!wo 6estions+
'lasticity "oduli &or Isotroic "edia '(uilibriu" and dyna"ic e(uations &or an isotroicelastic solid train energy &unction and its connection it. HooDe>s 6a ni(ueness o&
solution :eltra"i-Mic.ell co"atibility e(uations !laeyro">s t.eore" aint-enantUs
rincileRelevant ortion o& !.ater III o& booD by IoDolniDo&&/
8oo#s&
1 I oDolniDo&& Mat.e"atical T.eory o& 'lasticity Tata-Mc9ra Hill *ublis.ing
!o"any 6td 5e %el.i 1,++
2 A'H 6ove A Treatise on t.e Mat.e"atical T.eory o& 'lasticity %over*ublications 5e KorD
3 K! ung oundations o& olid Mec.anics *rentice Hall 5e %el.i 1,)$
4 % !.andraseD.araia. and 6 %ebnat. !ontinuu" Mec.anics Acade"ic *ress
1,,4$ .anti 5arayan Te#t :ooD o& !artesian Tensor !.and !o 1,$0
) Ti"es.enDi and 5 9oodier T.eory o& 'lasticity Mc9ra Hill 5e KorD1,+0
+ IH .a"es Introduction to olid Mec.anics *rentice Hall 5e %el.i 1,+$
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SEMES!E7-III
MM-(0) *opt. ii+ =ifference E5ations-I
Examination %ors & ) %ors
Max. Mar#s & 100
*External !heor" Exam. Mar#s&30
4 Internal $ssessment Mar#s&20+N'!E & !he examiner is re5ested to set nine 5estions in all ta#in two
5estions from each section and one complsor" 5estion. !he complsor" 5estion
will consist of eiht parts and will e distrited o,er the whole s"llas. !he
candidate is re5ired to attempt fi,e 5estions selectin at least one from each
section and the complsor" 5estion.
SEC!I'N-I *!wo 6estions+
Introductiont.e di&&erence calculusG T.e di&&erence oerator&alling &actorial oer rt
bino"ial coe&&icient
r
t su""ation de&inition roerties and e#a"les Abel>s
su""ation &or"ula 9enerating &unctions 'uler>s su""ation &or"ula :ernoulli
olyno"ials and e#a"les aro#i"ate su""ation
SEC!I'N-II *!wo 6estions+
6inear %i&&erence '(uationG irst order linear e(uations general results &or linear
e(uations solution o& linear di&&erence e(uation it. constant coe&&icients and it.
variable coe&&icients 5on-6inear '(uations t.at can be lineari7ed alications
SEC!I'N-III *!wo 6estions+
tability T.eory G Initial value *roble"s &or 6inear syste"s eigen values eigen vectors
and sectral radius !aylay-Ha"ilton T.eore" *ut7er algorit." olution o&non.o"ogeneous syste" it. initial conditions tability o& linear syste"s stable
subsace t.eore" and e#a"le tability o& non-linear syste" !.aotic be.aviour
SEC!I'N-I/ *!wo 6estions+T.e =-Trans&or" de&inition *roerties initial and &inal value T.eore" !onvolationT.eore" olving t.e initial value roble"s olterra su""ation e(uation and red.ol"
su""ation e(uation by use o& =-Trans&or"
Asy"totic Met.ods G Introduction Asy"totic Analysis o& u"s and e#a"lesAsy"totic be.aviour o& solutions o& .o"ogeneous linear e(uations *oincare>s
T.eore" *erron T.eore" tate"ent only/ non-linear e(uations
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7ecommended !ext&
9 Eelley and A! *etersonG %i&&erence '(uationsL An introduction it.
Alications Acade"ic *ress Harcourt 1,,1 Relevant ortions o& c.aters 1-$/
7eference 8oo#&
!alvin A.lbrandt Allan ! *eterson %iscreet Ha"iltonian syste"s %i&&erence
'(uations !ontinued ractions Ricati '(uation Eluer :otson 1,,)
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SEMES!E7-IIIMM-(0) *opt.iii+ $nal"tic Nmer !heor"
Examination %ors & ) %ors
Max. Mar#s & 100
*External !heor" Exam. Mar#s&30
4 Internal $ssessment Mar#s&20+N'!E & !he examiner is re5ested to set nine 5estions in all ta#in two
5estions from each section and one complsor" 5estion. !he complsor" 5estion
will consist of eiht parts and will e distrited o,er the whole s"llas. !he
candidate is re5ired to attempt fi,e 5estions selectin at least one from each
section and the complsor" 5estion.
SEC!I'N-I *!wo 6estions+
Arit."etical &unctions Mobius &unction 'uler totient &unction relation connecting
Mobius &unction and 'uler totient &unction *roduct &or"ula &or 'uler totient &unction
%iric.let roduct o& arit."etical &unctions %iric.let inverses and Mobius inversion
&or"ula Mangoldt &unction "ultilicative &unctions Multilicative &unctions and%iric.let "ultilication Inverse o& co"letely "ultilicative &unction 6iouville>s
&unction divisor &unction generali7ed convolutions or"al oer-series :ell series o&an arit."etical &unction :ell series and %iric.let "ultilication %erivatives o&
arit."etical &unctions elberg identity Asy"totic e(uality o& &unctions 'uler>s
su""ation &or"ula so"e ele"entary asy"totic &or"ulas average order o& divisor&unctions average order o& 'uler totient &unction
SEC!I'N-II *!wo 6estions+
Alication to t.e distribution o& lattice oints visible &ro" t.e origin average order o&
Mobius &unction and Mangoldt &unction *artial su"s o& a %iric.let *roduct alicationsto Mobius &unction and Mangoldt &unction 6egendre>s identity anot.er identity &or t.e
artial su"s o& a %iric.let roduct !.ebys.ev>s &unctions Abel>s identity so"e
e(uivalent &or"s o& t.e ri"e nu"ber t.eore" Ine(ualities &or n/ and * n SEC!I'N-III *!wo 6estions+
.airo>s Tauberian t.eore" Alications o& .airo>s t.eore" An asy"totic
&or"ula &or t.e artial su"s
xp p
1 *artial su"s o& t.e Mobius &unction :rie& sDetc.
o& an ele"entary roo& o& t.e ri"e nu"ber t.eore"L elberg>s asy"totic &or"ula
'le"entary roerties o& grous construction o& subgrous c.aracters o& &inite abelian
grous t.e c.aracter grou ort.ogonality relations &or c.aracters %iric.let c.aracters
u"s-involving %iric.let c.aracters 5onvanis.ing o& 61/ &or real nonrincial
SEC!I'N-I/ *!wo 6estions+
%iric.let>s t.eore" &or ri"es o& t.e &or" 4n-1 and 4nV1 %iric.let>s t.eore"unctions eriodic "odulo E '#istence o& &inite ourier series &or eriodic arit."etical
&unctions Ra"anus su" and generali7ations "ultilicative roerties o& t.e su"s
k n/ 9auss su"s associated it. %iric.let c.aracters %iric.let c.aracters it.
nonvanis.ing 9auss su"s Induced "oduli and ri"itive c.aracters roerties o&
induced "oduli conductor o& a c.aracter *ri"itive c.aracters and searable 9auss su"s
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inite &ourier series o& t.e %iric.let c.aracters *olya>s ine(uality &or t.e artial su"s o&
ri"itive c.aracters
7ecommended 8oo#&
To" M Aostol Introduction to Analytic 5u"ber T.eory
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SEMES!E7-III
MM-(0) *opt. i,+ Nmer !heor"
Examination %ors & ) %ors
Max. Mar#s & 100
*External !heor" Exam. Mar#s&30
4 Internal $ssessment Mar#s&20+N'!E & !he examiner is re5ested to set nine 5estions in all ta#in two
5estions from each section and one complsor" 5estion. !he complsor" 5estion
will consist of eiht parts and will e distrited o,er the whole s"llas. !he
candidate is re5ired to attempt fi,e 5estions selectin at least one from each
section and the complsor" 5estion.
SEC!I'N-I *!wo 6estions+
T.e e(uation a#Vby W c si"ultaneous linear e(uations *yt.agorean triangles assortede#a"les ternary (uadratic &or"s rational oints on curves
SEC!I'N-II *!wo 6estions+
'llitic curves actori7ation using ellitic curves curves o& genus greater t.an 1 areyse(uences rational aro#i"ations Hurit7 t.eore" irrational nu"bers 9eo"etry o&
5u"bers :lic.&eldt>s rincile MinDosDi>s !onve# body t.eore" 6agrange>s &ours(uare t.eore"
SEC!I'N-III *!wo 6estions+
'uclidean algorit." in&inite continued &ractions irrational nu"bers aro#i"ations to
irrational nu"bers :est ossible aro#i"ations *eriodic continued &ractions *ell>s
e(uation
SEC!I'N-I/ *!wo 6estions+
*artitions errers 9ra.s or"al oer series generating &unctions and 'uler>s
identity 'uler>s &or"ula bounds on *n/ ?acobi>s &or"ula a divisibility roerty7ecommended !ext&An Introduction to t.e T.eory o& 5u"bers Ivan 5iven
Herbert =ucDer"anHug. 6G Montgo"ery
?o.n iley onsAsia/*te6td
i&t. 'dition/
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SEMES!E7- III
MM-(0 *opt. i+ lid Mechanics-I
Examination %ors & ) %ors
Max. Mar#s & 100
*External !heor" Exam. Mar#s&30
4 Internal $ssessment Mar#s&20+N'!E & !he examiner is re5ested to set nine 5estions in all ta#in two
5estions from each section and one complsor" 5estion. !he complsor" 5estion
will consist of eiht parts and will e distrited o,er the whole s"llas. !he
candidate is re5ired to attempt fi,e 5estions selectin at least one from each
section and the complsor" 5estion.
SEC!I'N-I *!wo 6estions+
Eine"atics o& &luid in "otionGelocity at a oint o& a &luid 6agrangian and 'ulerian"et.ods trea" lines at. lines and streaD liens vorticity and circulation orte# lines
Acceleration and Material derivative '(uation o& continuity vector or !artesian &or"/
Reynolds transort T.eore" 9eneral analysis o& &luid "otion *roerties o& &luids- static
and dyna"ic ressure :oundary sur&aces and boundary sur&ace conditions Inotationaland rotational "otions elocity otential
SEC!I'N-II *!wo 6estions+
'(uation o& Motion G 6agrangeUs and 'ulerUs e(uations o& Motion vector or in !artesian
&or"/ :ernculliUs t.eore" Alications o& t.e :ernoulli '(uation in one di"ensional
&lo roble"s Eelvins circulation t.eore" vorticity e(uation 'nergy e(uation &orinco"ressible &lo Einetic energy o& irrotational &lo Eelvins "ini"u" energy
t.eore" "ean otential over a s.erical sur&ace Einetic energy o& in&inite li(uid
ni(ueness t.eore"s
SEC!I'N III *!wo 6estions+
tress co"onents in a real &luid Relations beteen rectangular co"onents o& stress
!onnection beteen stresses and gradients o& velocity5avier- toDe>s e(uations o&
"otion teady &los beteen to arallel lates *lane *oiseuille and !ouette &los
SEC!I'N I/ *!wo 6estions+
Reduction o& 5avier-tocD e(uations in &los .aving a#is o& sy""etry steady &lo in
circular ieG t.e Hagen-*oiseuille &lo steady &lo beteen to coa#ial cylinders &lobeteen to concentric rotating cylinders teady&los t.roug. tubes o& uni&or" cross-
section in t.e &or" i/ 'llise ii/ e(uilateral triangle iii/ rectangle under constant
ressure gradient uni(ueness t.eore"
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8oo#s &
1 H :esant and A Ra"sey A Treatise on Hydro"ec.anics *art-II !:
*ublis.ers %el.i 1,88
2 !.orlton Te#t-booD o& luid %yna"ics !: *ublis.ers %el.i 1,8$
3 Mic.ael '; 5eill and !.orlton Ideal and Inco"ressible luid %yna"ics?o.n iley ons 1,8)
4 9E :atc.elor An Introduciton to luid Mec.anics oundation :ooDs 5e
%el.i 1,,4$ A? !.orin and A Marsden A Mat.e"atical Introduction to luid %yna"ics
ringer-erlag 5e KorD 1,,3
) 6% 6andau and 'M 6isc.it7 luid Mec.anics *erga"on *ress 6ondon 1,8$+ H c.lic.ting :oundary 6ayer T.eory Mc9ra Hill :ooD !o"any 5e
KorD 1,+,
8 RE Rat.y An Introduction to luid %yna"ics ;#&ord and I:H *ublis.ing
!o"any 5e %el.i 1,+),, A% Koung :oundary 6ayers AIAA 'ducation eries as.ington %! 1,8,
10 Kuan oundations o& luid Mec.anics *rentice Hall o& India 6td 5e
%el.i 1,+)
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Semester-III
MM & (0 *opt. ii+Mathematical Statistics
Examination %ors & ) %ors
Max. Mar#s & 100
*External !heor" Exam. Mar#s&30
4 Internal $ssessment Mar#s&20+N'!E & !he examiner is re5ested to set nine 5estions in all ta#in two
5estions from each section and one complsor" 5estion. !he complsor" 5estion
will consist of eiht parts and will e distrited o,er the whole s"llas. !he
candidate is re5ired to attempt fi,e 5estions selectin at least one from each
section and the complsor" 5estion.
Section-I *!wo 6estions+
Rando" distributionG reli"inaries *robability density &unction *robability "odels
Mat.e"atical '#ectation !.ebys.ev>s Ine(ualityL !onditional robability Marginal
and conditional distributions !orrelation coe&&icient toc.astic indeendence
Section-II *!wo 6estions+
re(uency distributionsG :ino"ial *oissson 9a""a !.i-s(uare 5or"al :ivariatenor"al distributions
%istributions o& &unctionsG a"ling Trans&or"ations o& variablesG discrete andcontinuousL t distributionsL !.ange o& variable tec.ni(ueL %istribution o& orderL
Mo"ent-generating &unction tec.ni(ueL ot.er distributions and e#ectations
Section-III *!wo 6estions+
6i"iting distributionsG toc.astic convergence Mo"ent generating &unction Related
t.eore"s
IntervalsG Rando" intervals !on&idence intervals &or "ean di&&erences o& "eans andvarianceL :ayesian esti"ation
Section-I/ *!wo 6estions+
'sti"ation su&&iciencyG *oint esti"ation su&&icient statistics Rao-:lacDell T.eore"!o"leteness ni(ueness '#onential *% unctions o& ara"etersL toc.asticindeendence
8oo#s&
1 R Hogg AT !raigG Introduction to Mat.e"atical tatistics A"erind *ub!o *vt 6td 5e %el.i 1,+2 !.aters 1 to +/
2 ! 9uta E EaoorGunda"entals o& Mathematical Statistics ultan !.and ons 200+/
http://www.google.co.in/url?q=http://www.flipkart.com/fundamentals-mathematical-statistics-gupta-kapoor-book-8180540049&sa=U&ei=qqGiTfzoFYzPrQfnkszrAg&ved=0CBQQFjAC&usg=AFQjCNGazelLzSCFmSgfs1xBf7CPhdCtDwhttp://www.google.co.in/url?q=http://www.flipkart.com/fundamentals-mathematical-statistics-gupta-kapoor-book-8180540049&sa=U&ei=qqGiTfzoFYzPrQfnkszrAg&ved=0CBQQFjAC&usg=AFQjCNGazelLzSCFmSgfs1xBf7CPhdCtDwhttp://www.google.co.in/url?q=http://www.flipkart.com/fundamentals-mathematical-statistics-gupta-kapoor-book-8180540049&sa=U&ei=qqGiTfzoFYzPrQfnkszrAg&ved=0CBQQFjAC&usg=AFQjCNGazelLzSCFmSgfs1xBf7CPhdCtDwhttp://www.google.co.in/url?q=http://www.flipkart.com/fundamentals-mathematical-statistics-gupta-kapoor-book-8180540049&sa=U&ei=qqGiTfzoFYzPrQfnkszrAg&ved=0CBQQFjAC&usg=AFQjCNGazelLzSCFmSgfs1xBf7CPhdCtDwhttp://www.google.co.in/url?q=http://www.flipkart.com/fundamentals-mathematical-statistics-gupta-kapoor-book-8180540049&sa=U&ei=qqGiTfzoFYzPrQfnkszrAg&ved=0CBQQFjAC&usg=AFQjCNGazelLzSCFmSgfs1xBf7CPhdCtDw8/10/2019 M_Sc Math I to Iv sem syllabai.pdf
41/78
Semester III
MM- (0 *opt. iii+ $leraic Codin !heor"
Examination %ors & ) %ors
Max. Mar#s & 100
*External !heor" Exam. Mar#s&30
4 Internal $ssessment Mar#s&20+N'!E & !he examiner is re5ested to set nine 5estions in all ta#in two
5estions from each section and one complsor" 5estion. !he complsor" 5estion
will consist of eiht parts and will e distrited o,er the whole s"llas. !he
candidate is re5ired to attempt fi,e 5estions selectin at least one from each
section and the complsor" 5estion.
SEC!I'N I *!wo 6estions+
:locD !odes Mini"u" distance o& a code %ecoding rincile o& "a#i"u" liDeli.ood
:inary error detecting and error correcting codes 9rou codes Mini"u" distance o& a
grou code " "V1/ arity c.ecD code %ouble and trile reition codes Matri# codes
9enerator and arity c.ecD "atrices %ual codes *olyno"ial codes '#onent o& aolyno"ial over t.e binary &ield :inary reresentation o& a nu"ber Ha""ing codes
Mini"u" distance o& a Ha""ing code !.ater 1 2 3 o& t.e booD given at r 5o 1/
SEC!I'N II *!wo 6estions+
inite &ields !onstruction o& &inite &ields *ri"itive ele"ent o& a &inite &ield
Irreducibility o& olyno"ials over &inite &ields Irreducible olyno"ials over &inite &ields*ri"itive olyno"ials over &inite &ields Auto"or.is" grou o& 9(n/ 5or"al basis o&
9(n/ T.e nu"ber o& irreducible olyno"ials over a &inite &ield T.e order o& an
irreducible olyno"ial 9enerator olyno"ial o& a :ose-!.aud.uri-Hoc(.eng.e" codes
:!H codes/ construction o& :!H codes over &inite &ields !.ater 4 o& t.e booD givenat r 5o 1 and ection +1 to +3 o& t.e booD given at r 5o 2/
SEC!I'N III *!wo 6estions+
6inear codes 9enerator "atrices o& linear codes '(uivalent codes and er"utation"atrices Relation beteen generator and arity-c.ecD "atri# o& a linear codes over a
&inite &ield %ual code o& a linear code el& dual codes eig.t distribution o& a linear
code eig.t enu"erator o& a linear code Hada"ard trans&or" Macillia"s identity &orbinary linear codes
Ma#i"u" distance searable codes M% codes/ '#a"les o& M% codes
!.aracteri7ation o& M% codes in ter"s o& generator and arity c.ecD "atrices %ual
code o& a M% code Trivial M% codes eig.t distribution o& a M% code 5u"ber o&code ords o& "ini"u" distance d in a M% code Reed solo"on codes !.ater $ ,
o& t.e booD at r 5o 1/
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SEC!I'N I/ *!wo 6estions+
Hada"ard "atrices '#istence o& a Hada"ard "atri# o& order n Hada"ard codes &ro"
Hada"ard "atrices !yclic codes 9enerator olyno"ial o& a cyclic code !.ecDolyno"ial o& a cyclic code '(uivalent code and dual code o& a cyclic code Ide"otent
generator o& a cyclic code Ha""ing and :!H codes as cyclic codes *er&ect codes T.e
9ilbert-vars.a-"ove and *lotDin bounds el& dual binary cyclic codes !.ater ) 11o& t.e booD given at r 5o 1/
7ecommended !ext &
1 6R er"ani G 'le"ents o& Algebraic !oding T.eory !.a"an and Hall
Mat.e"atics/
2 teven Ro"an G !oding and In&or"ation T.eory ringer erlag/
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SEMES!E7-III
MM-(0 *opt. i,+ Commtati,e $lera
Examination %ors & ) %ors
Max. Mar#s & 100
*External !heor" Exam. Mar#s&30
4 Internal $ssessment Mar#s&20+N'!E & !he examiner is re5ested to set nine 5estions in all ta#in two
5estions from each section and one complsor" 5estion. !he complsor" 5estion
will consist of eiht parts and will e distrited o,er the whole s"llas. !he
candidate is re5ired to attempt fi,e 5estions selectin at least one from each
section and the complsor" 5estion.
SEC!I'N-I *!wo 6estions+
=ero divisors nilotent ele"ents and units *ri"e ideals and "a#i"al ideals 5il radicaland ?acobson radical !o"a#i"al ideals !.ineese re"ainder t.eore" Ideal (uotients
and anni.ilator ideals '#tension and contraction o& ideals '#act se(uences Tensor
roduct o& "odule Restriction and e#tension o& scalars '#actness roerty o& t.e tensor
roduct Tensor roducts o& algebrasSEC!I'N-II *!wo 6estions+Rings and "odules o& sections 6ocali7ation at t.e ri"e ideal * *roerties o& t.elocali7ation '#tended and contracted ideals in rings o& &ractions
*ri"ary ideals *ri"ary deco"osition o& an ideal Isolated ri"e ideals Multilicatively
closed subsets
SEC!I'N-III *!wo 6estions+
Integral ele"ents Integral closure and integrally closed do"ains 9oing-u t.eore" and
t.e 9oing-don t.eore" valuation rings and local rings 5oet.er>s nor"ali7ation le""a
and eaD &or" o& nullstellensat7 !.ain condition 5oet.erian and Artinian "odulesco"osition series and c.ain conditions
SEC!I'N-I/ *!wo 6estions+
5oet.erian rings and ri"ary deco"osition in 5oet.erian rings radical o& an ideal 5ilradical o& an Artinian ring tructure T.eore" &or Artinian rings %iscrete valuation
rings %edeDind do"ains ractional ideals
coe o& t.e course is as given in !.ater 1 to , o& t.e reco""ended te#t/
8/10/2019 M_Sc Math I to Iv sem syllabai.pdf
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7ecommended !ext&
MAtiya. R and I9Macdonald Introduction to !o""utative Algebra
Addison-esley *ublis.ing!o"any/
7eference 8oo#s&
1 59oal Eris.nan ;#onian *ress *vt 6td !o""utative Algebra2 =arisDi an 5ostrand *rinceton1,$8/ !o""utative Algebraol I/
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SEMES!E7-III
MM-(0( *opt. i+ Interal E5ations
Examination %ors & ) %ors
Max. Mar#s & 100
*External !heor" Exam. Mar#s&30
4 Internal $ssessment Mar#s&20+N'!E & !he examiner is re5ested to set nine 5estions in all ta#in two
5estions from each section and one complsor" 5estion. !he complsor" 5estion
will consist of eiht parts and will e distrited o,er the whole s"llas. !he
candidate is re5ired to attempt fi,e 5estions selectin at least one from each
section and the complsor" 5estion.
SEC!I'N-I *!wo 6estions+
%e&inition o& Integral '(uations and t.eir classi&ications 'igen values and 'igen&unctions ecial Dinds o& Eernel !onvolution Integral T.e inner or scalar roduct o&
to &unctions Reduction to a syste" o& algebraic e(uations red.ol" alternative
red.ol" t.eore" red.ol" alternative t.eore" An aro#i"ate "et.od
Relevant ortions &ro" t.e !.aters 1 2 o& t.e booD X6inear Integral '(uationsT.eory Tec.ni(ues by R*EanalY/
SEC!I'N-II *!wo 6estions+
Met.od o& successive aro#i"ations Iterative sc.e"e &or red.ol" and olterrra
Integral e(uations o& t.e second Dind !onditions o& uni&or" convergence and
uni(ueness o& series solution o"e results about t.e resolvent Eernel Alication o&iterative sc.e"e to olterra integral e(uations o& t.e second Dind
!lassical red.ol">s t.eory t.e "et.od o& solution o& red.ol" e(uation red.ol">s
irst t.eore" red.ol">s second t.eore" red.o">s t.ird t.eore"
Relevant ortions &ro" t.e !.ater 3 4 o& t.e booD X6inear Integral '(uation T.eoryand Tec.ni(ues by R*EanalY/
SEC!I'N-III *!wo 6estions+
y""etric Eernels Introduction !o"le# Hilbert sace An ort.onor"al syste" o&&unctions Ries7-is.er t.eore" A co"lete to-%i"ensional ort.onor"al set over t.e
rectangle a A dtcbs unda"ental roerties o& 'igenvalues and 'igen&unctions&or sy""etric Eernels '#ansion in eigen &unctions and :ilinear &or" Hilbert-c."idtt.eore" and so"e i""ediate conse(uences
%e&inite Eernels and Mercer>s t.eore" olution o& a sy""etric Integral '(uation
Aro#i"ation o& a general 2 -Eernel5ot necessarily sy""etric/ by a searable
Eernel T.e oerator "et.od in t.e t.eory o& integral e(uations Rayleig.-Rit7 "et.od
&or &inding t.e &irst eigenvalue
Relevant ortions &ro" t.e !.ater + o& t.e booD X6inear Integral '(uation T.eory and
Tec.ni(ues by R*EanalY/SEC!I'N-I/ *!wo 6estions+
T.e Abel Intergral '(uation Inversion &or"ula &or singular integral e(uation it.
Eernel o& t.e tye .s/-.t/ 0ZZ1 !auc.y>s rincial value &or integrals solution o& t.e!auc.y-tye singular integral e(uation closed contour unclosed contours and t.e
Rie"ann-Hilbert roble" T.e Hilbert-Eernel solution o& t.e Hilbert-Tye singularIntergal e(uation
Relevant ortions &ro" t.e !.ater 8 o& t.e booD X6inear Integral '(uation T.eory and
Tec.ni(ues by R*EanalY/
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7eferences&
1 R*Eanal 6inear Integral '(uations T.eory and Tec.ni(ues Acade"ic*ress 5e KorD
2 9MiD.lin 6inear Integral '(uations translated &ro" Russian/ Hindustan
:ooD Agency 1,)03 I5neddon Mi#ed :oundary alue *roble"s in otential t.eory 5ort.
Holland 1,))
4 I taDgold :oundary alue *roble"s o& Mat.e"atical *.ysics olI IIMacMillan 1,),
$ *undir and *undir Integral '(uations and :oundary value roble"s *ragati
*raDas.an Meerut
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Semester-III
MM (0( & *opt. ii+ Mathematical Modelin
Examination %ors & ) %ors
Max. Mar#s & 100
*External !heor" Exam. Mar#s&304 Internal $ssessment Mar#s&20+
N'!E & !he examiner is re5ested to set nine 5estions in all ta#in two
5estions from each section and one complsor" 5estion. !he complsor" 5estion
will consist of eiht parts and will e distrited o,er the whole s"llas. !he
candidate is re5ired to attempt fi,e 5estions selectin at least one from each
section and the complsor" 5estion.
Section-I (Two Questions)
T.e rocess o& Alied Mat.e"aticsL "at.e"atical "odelingG need tec.ni(ues classi&ication and
illustrativeL "at.e"atical "odeling t.roug. ordinary di&&erential e(uation o& &irst orderL (ualitative
solutions t.roug. sDetc.ing2
Section-II *!wo 6estions+
Mat.e"atical "odeling in oulation dyna"ics eide"ic sreading and co"art"ent "odelsL
"at.e"atical "ode1ing t.roug. syste"s o& ordinary di&&erential e(uationsL "at.e"atical
"ode1ing in econo"ics "edicine ar"-race battle2
Section-III *!wo 6estions+
Mat.e"atical "odeling t.roug. ordinary di&&erential e(uations o& second order2 Hig.er order
linear/ "odels2 Mat.e"atical "odeling t.roug. di&&erence e(uationsG 5eed basic t.eoryL
"at.e"atical "odeling in robability t.eory econo"ics &inance oulation dyna"ics and
genetics2
Section-I/*!wo 6estions+
Mat.e"atical "odeling t.roug. artial di&&erential e(uationsG si"le "odels "ass-balance
e(uations variational rinciles robability generating &unction tra&&ic &lo4 roble"s initial 8
boundary conditions2
8oo# recommended &
?252 EaurG Mat.e"atical Modeling Iiley 'astern 6i"ited 1,,0 Relevant ortions "ainly
&ro" !.aters 1 to )2/
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Semester III
MM-(0( *opt. iii+ 9INE$7 P7';7$MMIN;
Examination %ors & ) %ors
Max. Mar#s & 100
*External !heor" Exam. Mar#s&30
4 Internal $ssessment Mar#s&20+N'!E & !he examiner is re5ested to set nine 5estions in all ta#in two
5estions from each section and one complsor" 5estion. !he complsor" 5estion
will consist of eiht parts and will e distrited o,er the whole s"llas. !he
candidate is re5ired to attempt fi,e 5estions selectin at least one from each
section and the complsor" 5estion.
Section I *!wo 6estions+
i"ultaneous linear e(uations :asic solutions 6inear trans&or"ations *oint sets 6inesand .yerlanes !onve# sets !onve# sets and .yerlanes !onve# cones Restate"ent
o& t.e 6inear *rogra""ing roble" lacD and surlus variables *reli"inary re"arDs on
t.e t.eory o& t.e si"le# "et.od Reduction o& any &easible solution to a basic &easible
solution %e&initions and notations regarding linear rogra""ing roble"s I"roving abasic &easible solution nbounded solutions ;ti"ality conditions Alternative oti"a
'#tre"e oints and basic &easible solutions
Section-II *!wo 6estions+
T.e si"le# "et.od election o& t.e vector to enter t.e basis %egeneracy and breaDing
ties urt.er develo"ent o& t.e trans&or"ation &or"ulas T.e initial basic &easiblesolution-----arti&icial variables Inconsistency and redundancy Tableau &or"at &or
si"le# co"utations se o& t.e tableau &or"at !onversion o& a "ini"i7ation roble"
to a "a#i"i7ation roble" Revie o& t.e si"le# "et.od
T.e to-.ase "et.od &or arti&icial variables *.ase I *.ase II 5u"erical e#a"les o&t.e to-.ase "et.od Re(uire"ents sace olutions sace %eter"ination o& all
oti"al solutions nrestricted variables !.arnes> erturbation "et.od regarding t.e
resolution o& t.e degeneracy roble"
Section-III *!wo 6estions+
election o& t.e vector to be re"oved %e&inition o& b[/ ;rder o& vectors in b[/ se o&
erturbation tec.ni(ue it. si"le# tableau &or"at 9eo"etrical interretation o& t.eerturbation "et.od T.e generali7ed linear rogra""ing roble" T.e generali7ed
si"le# "et.od '#a"les ertaining to degeneracy An e#a"le o& cycling
Revised si"le# "et.odG tandard or" I !o"utational rocedure &or tandard or"
I Revised si"le# "et.odG tandard or" II !o"utational rocedure &or tandardor" II Initial identity "atri# &or *.ase I !o"arison o& t.e si"le# and revised
si"le# "et.ods T.e roduct &or" o& t.e inverse o& a non-singular "atri# Alternative
&or"ulations o& linear rogra""ing roble"s
Section-I/ *!wo 6estions+
%ual linear rogra""ing roble"s unda"ental roerties o& dual roble"s ;t.er
&or"ulations o& dual roble"s!o"le"entary slacDness nbounded solution in t.eri"al %ual si"le# algorit." Alternative derivation o& t.e dual si"le# algorit."
Initial solution &or dual si"le# algorit." T.e dual si"le# algorit."L an e#a"le
geo"etric interretations o& t.e dual linear rogra""ing roble" and t.e dual si"le#
algorit." A ri"al dual algorit." '#a"les o& t.e ri"al-dual algorit."Transortation roble" its &or"ulation and si"le e#a"les
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8oo#s &
1 9Hadley G 6inear *rogra""ing 5arosa ublis.ing House 1,,$/
2 I 9auss G 6inear *rogra""ing G Met.ods and Alications 4t.'dition/ Mc9ra Hill 5e KorD 1,+$
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SEMES!E7-III
MM (0( *opt. i,+ " Sets and $pplications-I
Examination %ors & ) %ors
Max. Mar#s & 100
*External !heor" Exam. Mar#s&30
4 Internal $ssessment Mar#s&20+N'!E & !he examiner is re5ested to set nine 5estions in all ta#in two
5estions from each section and one complsor" 5estion. !he complsor" 5estion
will consist of eiht parts and will e distrited o,er the whole s"llas. !he
candidate is re5ired to attempt fi,e 5estions selectin at least one from each
section and the complsor" 5estion.
SEC!I'N-I *!wo 6estions+
u77y etsG :asic de&initions -cuts strong -cuts level set o& a &u77y set suort o& a&u77y set t.e core and .eig.t o& a &u77y set nor"al and subnor"al &u77y sets conve#&u77y sets cutort.y roerty strong cutort.y roerty standard &u77y set oerations
standard co"le"ent e(uilibriu" oints standard intersection standard union &u77y setinclusion scalar cardinality o& a &u77y set t.e degree o& subset.ood coe as in relevantarts o& sections 13-14 o& !.ater 1 o& t.e booD given at r5o1/
Additional roerties o& -cuts involving t.e standard &u77y set oerators and t.estandard &u77y set inclusion Reresentation o& &u77y sets t.ree basic deco"osition
t.eore"s o& &u77y sets '#tension rincile &or &u77y setsG t.e =eda.>s e#tension
rincile I"ages and inverse i"ages o& &u77y sets roo& o& t.e &act t.at t.e e#tensionrincile is strong cutort.y but not cutort.y coe as in relevant arts o& !.ater 2
o& t.e booD "entioned at t.e end/
SEC!I'N-II *!wo 6estions+
;erators on &u77y setsG tyes o& oerations &u77y co"le"ents e(uilibriu" o& a &u77y
co"le"ent e(uilibriu" o& a continuous &u77y co"le"ent &irst and secondc.aracteri7ation t.eore"s o& &u77y co"le"ents &u77y intersections t-nor"s/ standard
&u77y intersection as t.e only ide"otent t-nor" standard intersection algebraic roductbounded di&&erence and drastic intersection as e#a"les o& t-nor"s decreasing generator
t.e *seudo-inverse o& a decreasing generator increasing generators and t.eir *seudo-
inverses convertion o& decreasing generators and increasing generators to eac. ot.erc.aracteri7ation t.eore" o& t-nor"sstate"ent only/ u77y unionst-conor"s/ standard
union algebraic su" bounded su" and drastic union as e#a"les o& t-conor"s
c.aracteri7ation t.eore" o& t-conor"s tate"ent only/ coe as in relevant arts o&
sections 31 to 34 o& !.ater 3 o& t.e booD "entioned at t.e end/
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SEC!I'N-III *!wo 6estions+
u77y nu"bers relation beteen &u77y nu"ber and a conve# &u77y set c.aracteri7ation
o& &u77y nu"bers in ter"s o& its "e"bers.i &unctions as ieceise de&ined &unctions&u77y cardinality o& a &u77y set using &u77y nu"bers arit."etic oerators on &u77y
nu"bers e#tension o& standard arit."etic oerations on real nu"bers to &u77y nu"bers
lattice o& &u77y nu"bers R MI5 MA\/ as a distributive lattice &u77y e(uationse(uation AV\ W : e(uation A\ W : coe as in relevant arts o& sections !.ater 4 o&
booD "entioned at t.e end/
SEC!I'N-I/ *!wo 6estions+
u77y RelationsG !ris and &u77y relations ro
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Semester-III
Paper MM- (0 & Practical-III
Examination %ors &
hors
Max. Mar#s & 100
Part-$ & Prolem Sol,inIn t.is art roble" solving tec.ni(ues based on aers MM-$01 to MM-$0$
ill be taug.t
Part-8 & Implementation of the followin prorams in '7!7$N-0?(
$. Use a function program for simple interest to display year#ise compound interest andamount, for given deposit, rate and time.
2. Use logical operators in computing the compound interest on a given amount for rate ofinterest varying #ith amount as #ell as time of deposit.
3. Hrite a subroutine program to check "logical output% #hether the three given points in a planeare collinear.
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SEMES!E7-I/
MM-(0B ;eneral Measre and Interation !heor"
Examination %ors & )
%ors
Max. Mar#s & 100*External !heor" Exam. Mar#s&30
4 Internal $ssessment Mar#s&20+
N'!E & !he examiner is re5ested to set nine 5estions in all ta#in two
5estions from each section and one complsor" 5estion. !he complsor" 5estion
will consist of eiht parts and will e distrited o,er the whole s"llas. !he
candidate is re5ired to attempt fi,e 5estions selectin at least one from each
section and the complsor" 5estion.
SEC!I'N-I *!wo 6estions+
Measures so"e roerties o& "easures outer "easures e#tension o& "easures
uni(ueness o& e#tension co"letion o& a "easure t.e 6: o& an increasingly directed
&a"ily o& "easurescoe as in t.e ections 3-) ,-10 o& !.ater 1 o& t.e booD JMeasureand Integration> by E:erberian/
Measurable &unctions co"binations o& "easurable &unctions li"its o& "easurable&unctions locali7ation o& "easurability si"le &unctions coe as in !.ater 2 o& t.e
booD JMeasure and Integration> by E:erberian/
SEC!I'N-II *!wo 6estions+
Measure saces al"ost every.ere convergence &unda"ental al"ost every.ere
convergence in "easure &unda"ental in "easure al"ost uni&or" convergence
'goro&&>s t.eore" Ries7-eyl t.eore" coe as in !.ater 3 o& t.e booD JMeasure and
Integration> by E:erberian/Integration it. resect to a "easureG Integrable si"le &unctions non-negative
integrable &unctions integrable &unctions inde&inite integrals t.e "onotone convergence
t.eore" "ean convergence coe as in !.ater 4 o& t.e booD JMeasure andIntegration> by E:erberian/
SEC!I'N-III *!wo 6estions+
*roduct MeasuresG Rectangles !artesian roduct o& to "easurable saces "easurablerectangle sections t.e roduct o& to &inite "easure saces t.e roduct o& any to
"easure saces roduct o& to - &inite "easure sacesL iterated integrals ubini>st.eore" a artial converse to t.e ubini>s t.eore" coe as in !.ater ) e#cet
section 42/ o& t.e booD JMeasure and Integration> by E:erberian/
igned MeasuresG Absolute continuity &inite singed "easure contractions o& a &inite
signed "easure urely ositive and urely negative sets co"arison o& &inite "easures6ebesgue deco"osition t.eore" a reli"inary Radon-5iDody" t.eore" Ha.n
deco"osition ?ordan deco"osition uer variation loer variation total variation
do"ination o& &inite signed "easures t.e Radyon-5iDody" t.eore" &or a &inite "easure
sace t.e Radon-5iDody" t.eore" &or a - &inite "easure sace coe as in !.ater +e#cet ection $3/ o& t.e booD JMeasure and Integration> by E:erberian/
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SEC!I'N-I/ *!wo 6estions+
Integration over locally co"act sacesG continuous &unctions it. co"act suort 9 Js and >s :aire sets :aire &unction :aire-sandic. t.eore" :aire "easure :orelsets Regularity o& :aire "easures Regular :orel "easures Integration o& continuous
&unctions it. co"act suort Ries7-MarDo&&>s t.eore" coe as in relevant arts o&t.e sections $4-$+)0)2)) and ), o& !.ater 8 o& t.e booD JMeasure and Integration> byE:erberian/
7ecommended !ext&
E:erberianG Measure and Integration !.elsea *ublis.ing !o"any 5e KorD 1,)$
7eferences&
1 H6RoydenG Real Analysis *rentice Hall o& India 3rd'dition 1,88
2 9de :arraG Measure T.eory and Integration iley 'astern 6td1,81
3 *RHal"osG Measure T.eory an 5ostrand *rinceton 1,$04 IERanaG An Introduction to Measure and Integration 5arosa *ublis.ing
House %el.i 1,,+
$ R9:artleG T.e 'le"ents o& Integration ?o.n iley and ons Inc 5eKorD 1,))
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SEMES!E7- I/
MM-(03 Partial =ifferential E5ations
Examination
%ors & ) %ors
Max. Mar#s & 100
*External !heor" Exam. Mar#s&304 Internal $ssessment Mar#s&20+
N'!E & !he examiner is re5ested to set nine 5estions in all ta#in two
5estions from each section and one complsor" 5estion. !he complsor" 5estion
will consist of eiht parts and will e distrited o,er the whole s"llas. !he
candidate is re5ired to attempt fi,e 5estions selectin at least one from each
section and the complsor" 5estion.
SEC!I'N-I *!wo 6estions+*%' o& Dt. orderG %e&inition e#a"les and classi&ications Initial value roble"s Transort
e(uations .o"ogeneous and non-.o"ogeneous Radial solution o& 6alace>s '(uationG
unda"ental solutions .ar"onic &unctions and t.eir roerties Mean value or"ulas *oissons
e(uation and its solution strong "a#i"u" rincile uni(ueness local esti"ates &or .ar"onic
&unctions 6iouvilles t.eore" HarnacD>s ine(uality
SEC!I'N-II *!wo 6estions+
9reen>s &unction and its derivation reresentation &or"ula using 9reen>s &unction sy""etry o&
9reen>s &unction 9reen>s &unction &or a .al& sace and &or a ball 'nergy "et.odsG uni(ueness
%ric.let>s rincile Heat '(uationsG *.ysical interretation &unda"ental solution Integral o&
&unda"ental solution solution o& initial value roble" %u.a"el>s rincile non-.o"ogeneous
.eat e(uation Mean value &or"ula &or .eat e(uation strong "a#i"u" rincile and uni(ueness
'nergy "et.ods
SEC!I'N-III *!wo 6estions+
ave e(uation- *.ysical interretation solution &or one di"entional ave e(uation d>Ale"berts&or"ula and its alications re&lection "et.od olution by s.erical "eans 'uler-
*oisson]%arbou# e(uation Eirc..o&&>s and *oisson>s &or"ulas &or nW2 3 only/ olution o& non
.o"ogeneous ave e(uation &or nW13 'nergy "et.od ni(ueness o& solution &inite
roagation seed o& ave e(uation 5on-linear &irst order *%'- co"lete integrals enveloes
!.aracteristics o& i/ linear ii/ (uasi
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