M_Sc Math I to Iv sem syllabai.pdf

8/10/2019 M_Sc Math I to Iv sem syllabai.pdf

1/78

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


!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


2/78

Scheme of Examination for M.Sc. Mathematics

Semester III

Complsor" Papers&

Paper

Code


!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


3/78


!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


4/78

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/


!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


5/78


!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


6/78

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


7/78

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


8/78

Semester-I

MM-02 & 7E$9 $N$9:SIS I


Max. Mar#s & 100

*External !heor" Exam. Mar#s&304 Internal $ssessment Mar#s&20+






Section-I *!wo 6estions+

%e&inition and e#istence o& Rie"ann tielt


9/78

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


10/78

Semester-I

MM-0)& !'P'9';:


Max. Mar#s & 100








%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


11/78

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


!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


12/78

Semester-I

MM-0& C'MP9E< $N$9:SIS-I


Max. Mar#s & 100








*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


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"


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"


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


13/78


14/78

Semester-I

MM-0(& =ifferential E5ations I


Max. Mar#s & 100








*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/


15/78


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)


16/78

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.


17/78

Semester II

MM-0B& $d,anced $stract $lera-II


Max. Mar#s & 100








!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/


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/


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/


18/78


'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


19/78

Semester-II

MM-03 & 7E$9 $N$9:SIS-II


Max. Mar#s & 100

*External !heor" Exam. Mar#s&30








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


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


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


20/78


%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)


21/78

Semester-II

MM-0 & Compter Prorammin *!heor"+










5u"erical constants and variablesL arit."etic e#ressionsL inutBoututL conditional

&loL looing


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


22/78

Semester-II

MM-10 & C'MP9E< $N$9:SIS-II










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


;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


23/78

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,,+


24/78

Semester-II

MM-11& =IE7EN!I$9 E6D$!I'NS-II


Max. Mar#s & 100


N'!E & !he examiner is re5ested to set nine 5estions in allA ta#in two






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/


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


JTe#tbooD o& ;rdinary %i&&erential '(uations> by %eo et al/


!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


JT.eory o& ;rdinary %i&&erential '(uations> by !oddington and 6evinson/


25/78

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


26/78

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.


27/78

SEMES!E7-III

MM-(01 nctional $nal"sis


Max. Mar#s & 100


4 Internal $ssessment Mar#s&20+N'!E & !he examiner is re5ested to set nine 5estions in all ta#in two





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/


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


28/78

;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


29/78

SEMES!E7- III

MM-(02 $nal"tical Mechanics and Calcls of /ariations


Max. Mar#s & 100







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


30/78

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


31/78

SEMES!E7-III

MM-(0) *opt. i+ Elasticit"


Max. Mar#s & 100








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./


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&&/


32/78

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,+$


33/78

SEMES!E7-III

MM-(0) *opt. ii+ =ifference E5ations-I


Max. Mar#s & 100








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


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


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


34/78

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,,)


35/78

SEMES!E7-IIIMM-(0) *opt.iii+ $nal"tic Nmer !heor"


Max. Mar#s & 100








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


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


%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


36/78

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


37/78

SEMES!E7-III

MM-(0) *opt. i,+ Nmer !heor"


Max. Mar#s & 100








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


'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"


'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


*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/


38/78

SEMES!E7- III

MM-(0 *opt. i+ lid Mechanics-I


Max. Mar#s & 100








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


'(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"


39/78

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,+)


40/78

Semester-III

MM & (0 *opt. ii+Mathematical Statistics


Max. Mar#s & 100








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


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


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


'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=AFQjCNGazelLzSCFmSgfs1xBf7CPhdCtDw


41/78

Semester III

MM- (0 *opt. iii+ $leraic Codin !heor"


Max. Mar#s & 100







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/


42/78

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/


43/78

SEMES!E7-III

MM-(0 *opt. i,+ Commtati,e $lera


Max. Mar#s & 100








=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


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


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/


44/78

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/


45/78

SEMES!E7-III

MM-(0( *opt. i+ Interal E5ations


Max. Mar#s & 100








%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/


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/


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/


46/78

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


47/78

Semester-III

MM (0( & *opt. ii+ Mathematical Modelin


Max. Mar#s & 100







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


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


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/


48/78

Semester III

MM-(0( *opt. iii+ 9INE$7 P7';7$MMIN;


Max. Mar#s & 100








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


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"


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


%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


49/78

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,+$


50/78

SEMES!E7-III

MM (0( *opt. i,+ " Sets and $pplications-I


Max. Mar#s & 100








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/


;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/


51/78


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/


u77y RelationsG !ris and &u77y relations ro


52/78

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.


53/78

SEMES!E7-I/

MM-(0B ;eneral Measre and Interation !heor"

Examination %ors & )

%ors









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/


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/


*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/


54/78


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,))


55/78

SEMES!E7- I/

MM-(03 Partial =ifferential E5ations

Examination

%ors & ) %ors

Max. Mar#s & 100







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


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


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

M_Sc Math I to Iv sem syllabai.pdf

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Transcript of M_Sc Math I to Iv sem syllabai.pdf