Msc Maths Colleges 2013

8/11/2019 Msc Maths Colleges 2013

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M.Sc MATHEMATICS (Revised Syllabus from 2013 Admissions)

LIST OF PAPERS

SEMESTER I

MM"## !inear Al$ebra 9,e4ised S-llabus Attached:MM"#" ,eal Anal-sis ( I 9,e4ised S-llabus Attached:MM"#) Differential Euations 9;ri$inal S-llabus Continues:MM"#0 Topolo$- < I 9,e4ised S-llabus Attached:

SEMESTER ! IIMM""# Al$ebra 9,e4ised S-llabus Attached:MM""" ,eal Anal-sis ( II9,e4ised S-llabus Attached:MM"") Topolo$- < II 9,e4ised S-llabus Attached:MM""0 Computer Pro$rammin$ in C11 9;ri$inal S-llabus Continues:

SEMESTER ! IIIMM")# Comple Anal-sis < I 9;ri$inal S-llabus Continues:MM")" 2unctional Anal-sis < I 9;ri$inal S-llabus Continues:MM")) Electi4e 9;ne amon$ the followin$: Automata Theor- 9;ri$inal S-llabus Continues: Probabilit- 9;ri$inal S-llabus Continues: ;perations ,esearch 9,e4ised S-llabus Attached:MM")0 Electi4e 9;ne amon$ the followin$: =eometr- of numbers 9;ri$inal S-llabus Continues: Differential =eometr- 9;ri$inal S-llabus Continues: =raph Theor- 9>ew S-llabus Attached: Approimation Theor- 9>ew S-llabus Attached:

SEMESTER ! IVMM"0# Comple Anal-sis < II 9;ri$inal S-llabus Continues:MM"0" 2unctional Anal-sis < II 9;ri$inal S-llabus Continues:MM"0) Electi4e 9;ne amon$ the followin$: Mathematical Statistics 9;ri$inal S-llabus Continues: Mechanics 9;ri$inal S-llabus Continues: Theor- of ?a4elets 9;ri$inal S-llabus Continues: Codin$ Theor- 9;ri$inal S-llabus Continues: 2ield Theor- 9,e4ised S-llabus Attached:MM"00 Electi4e 9;ne amon$ the followin$: Commutati4e Al$ebra 9;ri$inal S-llabus Continues: ,epresentation Theor- of 2inite =roups 9;ri$inal S-llabus Continues: Cate$or- Theor- 9;ri$inal S-llabus Continues: Ad4anced =raph Theor- 9;ri$inal S-llabus Continues: Anal-tic >umber Theor- 9;ri$inal S-llabus Continues:


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MM 211 LINEAR ALGEBRA

T"#$% S"'()* A#'"+, Linear Algebra Done Right 2*(E(-$-)*, S+-*/"+.

UNIT 1

5ector spaces7 Definition@ Eamples and properties@ Subspaces@ Sum and Direct sum of subspaces@ Spanand linear independence of 4ectors@ Definition of finite dimensional 4ector spaces@ ases7 Definition andeistence@ Dimension Theorems.BChapters #@" of Tet

UNIT II

!inear maps@ their null spaces and ran$es@ ;perations on linear maps in the set of all linear maps from onespace to another @ ,ank(>ullit- Theorem @ Matri of linear map@ its in4etibilt-.BChapter ) of Tet

UNIT III

In4ariant subspaces@ Definition of ei$en 4alues and 4ectors@ Pol-nomials of operators@ pper trian$ularmatrices of linear operators@ Eui4alent condition for a set of 4ectors to $i4e an upper trian$ular operator@Dia$onal matrices@ In4ariant subspaces on real 4ector spaces.BChapter * of Tet

UNIT IV

Concept of $eneralied ei$en 4ectors@ >ilpotent operators characteristic pol-nomial of an operator@Ca-le-(Familton theorem@ Condition for an operator to ha4e a basis consistin$ of $eneralied ei$en

4ectors@ Minimal pol-nomial. Gordan form of an operator 9=eneral case of Ca-le-(Familton Theoremma- be briefl- sketched from the reference tet:BChapter & of Tet

UNIT V

Chan$e of basis@ trace of an operator@ Showin$ that trace of an operator is eual to the trace if its matri@determinant of an operator@ in4ertibilt- of an operator and its determinant@ relation between characteristicpol-nomial and determinant@ determinant of matrices of an operator w.r.t. two base are the same.Determinant of a matri 9The section 4olumes ma- be omitted:BChapter #% of Tet

R""+"*c"

1. K"**"$ H)* *( R4 K5*6", 7Linear Algebra, P+"*$-c" H'', 1981.

2. I.N H"+$"-*, 7Linear Algebra, W-'"4 E$"+*.

3. S. K5+"*, 7Linear Algebra, P+"*$-c" H', 2000.


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MM 212 REAL ANALYSISI

T"#$% :1; T) M. A)$)',!athemati"al Analysis# Se"ond $dition, N+)

19on(uniforml- con4er$ent series that can be inte$ratedterm b- term@ niform con4er$ence and differentiation@ Sufficient conditions for uniform con4er$ence ofa series.BChapter @ Sections .#(. ecept .+ of tet #. Do more problems to stud- the uniform con4er$ence ofseuences and series

UNIT IV

M5'$-?+-$" C'c5'5%S"5"*c", c)*$-*5-$4 *( '--$.Seuences in @ Sub(seuences and Cauch-seuences@ Compositions of continues functions@ Piecin$ continuous functions on o4erlappin$ subsets@Characteriations of continuit-@ Continuit- and boundedness@ Continuit- and con4eit-@ Continuit- andintermediate 4alue propert- @ niform continuit-@ Implicit function Theorem@ !imits and continuit-.BTet ". Sections ".#@ "." 9ecludin$ Continuit- and monotonicit-@ Continuit-@ ounded 5ariation@ounded i4ariation:@ ".) 9Ecludin$ !imits from a uadrant@ Approachin$ Infinit-:


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UNIT V

P+$-' *( T)$' D-"+"*$-$-)*.Partial deri4ati4e@ Directional deri4ati4es@ Fi$her order partialderi4ati4es@ Fi$her order directional deri4ati4es@ Differentiabilit-@ Ta-lorHs Theorem and Chain rule@2unctions of three 4ariables@ Etensions and analo$ues@ Tan$ent planes normal lines to surfaces.BTet". Chapter ) ecludin$ section ).0 and last subsection of section ).*

R""+"*c".

1. .A D-"5()**", &oundations of !odern Analysis, Ac("-c P+".

2. W. R5(-*,Real and %om'le analysis#T$ McG+ H-''.


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MM 21= TOPOLOGYI

T"#$% S"'()* W.D?-, .o'ology# T$ Mc G+H-'' E(-$-)* 200, T$ MC G+H-''

UNIT 1

M"$+-c c"%Definition and Eamples@ ;pen sets@ Closed sets and their properties@ Con4er$enceof seuences in metric spaces.9Chapter " of the Tet(up to theorem ".)":

UNIT II

Complete metric spaces@ Cantor intersection Theorem@ aire cate$or- Theorem@ Continuit- in metricspaces@ niform continuit-@ anach fied point Theorem.9Chapter " of the Tet < from definition ".)) to corollar- ".*#@ Chapter ):

UNIT III

T))')/-c' c"%Definition and Eamples@ Interior and Closure@ ase for topolo$-@ Subspaces@Continuit- in topolo$ical spaces@ Product topolo$-9Chapter 0 and Chapter * of the Tet:

UNIT IV

S"+$-)* #-)% T%@T#@ T"@ T)and T0spaces@ r-sohnHs !emma@ Tiete etension Theorem.C)c$ c"%Feine(orel theorem@ T-chonoff Theorem.9Chapter ' and Chapter + of the Tet:

UNIT V

Connected spaces@ !ocall- connected spaces@ Pathwise connected spaces@ !ocall- pathwiseconnected spaces.9Chapter and Chapter #% of the Tet:

R""+"*c".

(1) G.G S-)*, .o'ology and !odern Analysis, Mc G+H-'' I*c, N" Y)+, 193

(2) S$""* W-''+(, eneral .o'ology, A((-)*W"'"4, R"(-*/, 19c,Lynn Arthur Steen# %ounter $am'les in .o'ology, D)?"+

P5>'-c$-)*, 199


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MM 222 REAL ANALYSISII

T"#$% :1; G.(".B++,!easure .heory and 4ntegration#N" A/" I*$"+*$-)*'P5>'-"+, N" D"'-, 1981.

UNIT I

!ebes$ue ;uter Measure@ Measurable sets@ ,e$ularit-@ Measurable functions@ orel and !ebesueMeasurabilit-9Chapter "@ ".#(".* of Tet:

UNIT II

Inte$ration of >on(ne$ati4e functions@ The =eneral Inte$ral@ Inte$ration of Series@ ,iemann and!ebes$ue Inte$rals@ The 2our Deri4ati4es@ !ebes$ueHs Differentiation Theorem@ Differentiations andInte$ration.9Chapter )@ ).# to ).0@ Chapter 0@ 0.#@ 0.0@0.* of the Tet:

UNIT III

A>$+c$ M"5+" Sc"%Measures and ;uter Measures@ Etension of a measure@ niueness ofthe Etension@ Completion of the Measure@ Measure spaces@ Inte$ration with respect to a Measure9Chapter *@ *.#(*.' of Tet:

UNIT IV

The !p Spaces@ Con4e 2unctions@ GensenHs Ineualit-@ The Ineualities of Folder and Minkowski@Completeness of !p9K:.9Chapter '@ '.#('.* of Tet:

UNIT V

Con4er$ence in Measure@ Si$ned Measures and the Fahn Decomposition@ The GordanDecomposition@ The ,adon(>ikod-m Theorem@ Some Applications of the ,adon(>ikod-mTheorem.9Chapter +@+.# Chapter &@&.#(&.0 of Tet:

R""+"*c"%

:1; H.L.R)4()*,Real Analysis# .hird $dition#McM-''*

:2; W.R5(-*,-rin"i'les of !athemati"al Analysis#T-+( E(-$-)*

:3;P.R H'),!easure .heory#S+-*/"+.


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MM223 TOPOLOGYII

T"#$% :1; S"'()* W D?-, .o'ology#T$ Mc G+H-'' E(-$-)* 200,

T$ Mc G+ H-'' :2; I.M S-*/"+, .A T)+",Le"ture ,otes on $lementary .o'ology and

eometry#S+-*/"+ I*$"+*$-)*' E(-$-)*, S+-*/"+ : I*(-; P+-?$"

L--$"(, N" D"'-, 200=

UNIT 1

L)c''4 C)c$ Sc"% Definition and Eamples@ Aleandroff one(point compactification@ aire(Cate$or- Theorem.9Chapter & of Tet #:

UNIT II

5)$-"*$ *( +)(5c$% Luotient Topolo$-@ Product Topolo$-@ Embeddin$@ Embeddin$ !emma.9Chapter #* of Tet #:

UNIT III

C)*?"+/"*c"%>et@ !imit point and Cluster Point of the >et@ 2ilter@ ltrafilter@ ,elationship betweennet and filter.9Chapter #' of Tet #:

UNIT IV

F5*("*$' G+)5 *( C)?"+-*/ c"%Fomotop-@ 2undamental =roup@ Co4erin$ spaces.9Chapters ) of Tet ":( The proof of Theorem 0 is omitted:

UNIT V

S-'-c-' C)'"#"%=eometr- of Simplicial Complees@ ar-centric Subdi4isions@ Simplicialapproimations Theorem@ 2undamental $roup of a simplicial comple.9Chapter 0 of Tet ":

R""+"*c"%

:1; S$""* W-''+(, eneral .o'ology#A((-)*W"'"4, R"(-*/, 19


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M22= COMPUTER PROGRAMMING IN C

T"#$% 1. R)>"+$ L)+", /b5e"t /riented -rogramming in %66 (.hird $dition)

G'/)$- P5>'-c$-)* P?$. L$( :2003;

2% K.V V"*5/)' *( S5*("+ A+)+,-rogramming *ith %#T$ Mc G+ H-''

O5$'-*" S"+-"

UNIT 1

Characteristics of ;b6ect ;riented !an$ua$es( C11 Pro$rammin$ asics@ asic Pro$ramConstruction( Comments@ 5ariables @ Constants@ Epressions@ Statements@ cin and cout@Manipulators@ T-pe con4ersion@ Arithmetic operators@ !ibrar- functions@ !oops and decisions@,elational operators@ !o$ical operators@ ;ther control statements9Chapters #@ "@ and ) of Tet #:

UNIT 2

Structures( Declarin$ structures@ Definin$ structure 4ariables@ Accessin$ structure members@ ;therstructure features@ Structure within structures@ Enumerated data t-pes. 2unctions( Simple functions@Passin$ a $arments to functions@ ,eturnin$ 4alued from functions@ ,eference ar$uments@ ;4erloadedfunctions@ Inline functions@ Default ar$uments@ 5ariables and stora$e classes@ ,eturnin$ b-reference.9Chapters 0 and * of the Tet#:

UNIT 3

;b6ect Classes( Simple class@ Specif-in$ the class@ C11 ob6ects as ph-sical ob6ects@ C11 ob6ects asdata t-pes@ Constructors@ Destructors@ ;b6ects as function ar$uments@ ,eturnin$ ob6ects fromfunctions@ Structures and classes@ ;b6ects and memor-@ Static class data@ Arra-s(Arra-

fundamentals@ Multidimensional arra-s@ Passin$ arra-s to functions@ 2unction declaration with arra-ar$uments@ Arra-s of structures@ Arra-s as class members data@ Arra-s of ob6ects@ C(strin$s@ Arra-sof strin$s@ Strin$s as class members@ a user defined strin$ t-pe.9Chapters ' and + of Tet #:

UNIT =

;perator o4erloadin$( ;4erloadin$ unar- operators@ ;4erloadin$ binar- operators@ Arithmeticoperators@ Multiple o4erloadin$ @ Data con4ersion@ Inheritance( Deri4ed class and basic class@Deri4ed class constructors@ ;4er ridin$ member function@ Class Fierarchies@ Public and pri4ateinheritance@ !e4els of inheritance@ Multiple inheritance@ Class within class.9Chapters & and of Tet #:

UNIT

Pointers(Addresses and pointers@ Pointers and arra-s@ Pointers and functions@ Passin$ simple4ariable@ Passin$ arra-s@ Pointers and C t-pe strin$s@ !ibrar- strin$ functions@ Memor- mana$ement7>ew and delete pointers to ob6ects@ Pointers to pointers9Chapter #% of Tet #:


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MM 231 COMPLE ANALYSIS ! I

T"#$% )*. B. C)*4, &un"tions of %om'le 7ariables, S+-*/"+V"+'/, N" Y)+, 19


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MM 232 FUNCTIONAL ANALYSIS ! 1

T"#$% B.V L-4",&un"tional Analysis (2nd$dition)

UNIT I

>ormed Spaces and Continuit- of !inear maps. 9Section * and ' of the Tet@;mittin$ '.+ and '.&:

UNIT II

Fahn(anach Theorem and anach Spaces. 9Section + and & of the Tet@;mittin$ Subsection anach limits:

UNIT III

niform ounded Principle < Closed and ;pen Mappin$ Theorems@ ounded in4erse Theorems9Section .#@."@.)@#% and ##.# onl-:

UNIT IV

Spectrum of a ounded ;perator < Dual and Transposes 9Sections #"@ #).#@ #)."@ #).)@ #).0and #).* onl-:

UNIT V

,eflei4it- < Compact !inear Maps@ Spectrum of a Compact ;perator 9Sections #'.#@ #+.#@ #+."@ #+.)@#&.#@ #&." and #&.):

R""+"*c"%

1. D5*)+( M *( .T Sc+6,Linear /'erators -art 2, W-'"4.

2. T4')+ A.E,4ntrodu"tion to &un"tional Analysis#W-'"4.

3. G.F. S-)*,.o'ology and !odern Analysis#Mc G+ H-''.


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MM 2=1 COMPLE ANALYSIS ! II

T"#$% )*. B. C)*4, &un"tions of %om'le 7ariables#S+-*/"+V"+'/, N" Y)+, 19


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MM 233 AUTOMATA THEORY :E'"c$-?";

T"#$% .E. H))+)$ *( .D. A'*,4ntrodu"tion to Automata .heory Languages and

%om'utation#N+), 1999.

UNIT I

Strin$s@ Alphabets and !an$ua$es 9Section #.# of the Tet:2inite Automata 9Chapters "@ Sections ".# to ".0:

UNIT II

,e$ular epressions and Properties of ,e$ular sets.9Sections ".* to ".& and ).# to ).0:

UNIT III

Contet 2ree $rammars 9Section 0.# to 0.*:

UNIT IV

Pushdown Automata properties of Contet free lan$ua$esTheorem *.)@ *.0 9without proof:@ 9Section is *.# to *.) and '.# to '.):

UNIT V

Turnin$ Machine and Choamski hierarch-@ 9Sections +.# to +.) and ." to .0:

R""+"*c"

1. G.E R"?"6 ,4ntrodu"tion to &ormal Languages

2. P.L-*6 ,4ntrodu"tion to &ormal Languages and Automata#N+) 2000

3. G.L''"*$,Semigrou's and A''li"ations


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19

R""+"*c"%

:1;. A R.B asi" -robability .heory, )* -'"4, N" Y)+ :19'-"+ M)c) :199;

:=;. L5cc.E%hara"teristi" &un"tions, H*"+, N" Y)+ :S"c)*( E(-$-)*, 19


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MM 233 OPERATIONS RESEARCH :E'"c$-?";

T"#$% 1; R?-*(+*, P-'-, S)'>"+/# /'erations Resear"h# -rin"i'les and -ra"ti"e, S"c)*(

E(-$-)*, )* W-'"4 & S)*.

2; K. V. M-$', C. M)*. /'timi;ation !ethods in /'erations Resear"h and Systems

Analysis, T-+( E(-$-)*, N" A/" I*$"+*$-)*' P5>'-"+, N" D"'-.

UNIT I

L-*"+ P+)/+-*/7 2ormulation of !inear Pro$rammin$ Models@ =raphical solution of !inearPro$rams in two 4ariables@ !inear pro$rams in standard form@ basic 4ariable@ basic solution@ basicfeasible solution@ Solution of !inear Pro$rammin$ problem usin$ simple method@ i$ ( Msimplemethod@ The two phase simple method.BChapter " of tet # @ sections ".# to "@

UNIT II

T+*)+$$-)* P+)>'"7 !inear pro$rammin$ formulation@ Initial basic feasible solution@ de$enerac-in basic feasible solution@ Modified distribution method@ ;ptimalit- test. A-/*"*$ P+)>'"7Standard assi$nment problems@ Fun$arian method for sol4in$ an assi$nment problem.BChapter ) of tet #@ sections ).# to ).)

UNIT III

P+)@"c$ */""*$8 Pro$ramme E4aluation and ,e4iew Techniue 9PE,T:@ Critical Path Method9CPM:BChapter ) of tet #@ section ).+

UNIT IV

K5* ! T5c"+ T")+4 *( N)*'-*"+ P+)/+-*/7 !a$ran$ian function@ saddle point@ Nuhn '-c$-)*.


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MM 23= GEOMETRY OF NUMBERS :E'"c$-?";

T"#$ B))% D.D O'(,A**"'- L# *( G5-'-* P. D?-(), .he eometry of ,umbers#

T" M$"$-c' A)c-$-)* ) A"+-c 2000

UNIT 1

!attice points and strai$ht lines@ Countin$ of lattice points 9Chapters # and ":

UNIT 2

!attice points and area of pol-$ons@ !attice points in circles 9Chapter ) and 0:

UNIT 3

Minkowski fundamental Theorem and Applications 9Chapters * and ':

UNIT =

!inear transformation and inte$ral lattices@ =eometric interpretations of Luadratic forms9Chapters + and &:

UNIT

lichfieldts and applications@ Tcheb-che4Hs and conseuences 9Chapter and #%:

R""+"*c"

1. .W.S C"'',4ntrodu"tion to eometry of ,umbers#S+-*/"+ V"+'/ 199on Parametric Methods Chi suare Test of $oodness of fit@ Empirical distribution function@ 2n9: as anestimator of population distribution function 29:@ its eact and as-mptotic distributions for fied @Noimo$ro4e test@ Si$n test@ ?ilcoon < Mann(?hitne- Test

R""+"*c"%

1) R)$/- V.K , An 4ntrodu"tion to -robability .heory and !athemati"al Statisti"s# W-'"4

E$"+* N" D"'- : 198;

2) R), C.L , Liner Statisti"al 4nferen"e and 4ts A''li"ations7, W-'"4 E$"+* ,

N" D"'- : 19


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MM 2=3 FIELD THEORY :E'"c$-?";

T"#$% )" R)$*, alois .heory#S"c)*( E(-$-)*, S+-*/"+, 1998.

UNIT 1

S)'?>'" /+)5 :A"*(-# B ) $" $"#$;%Isomorphism Theorems@ Correspondence Theorem@ S-lowp(sub$roup@ commutator sub$roups and Fi$her sub$roups@ S*is not sol4able.9The followin$ results are included7 =*@ =' @ =+@ =&@ =@ =#0@ =#*@ =#'@ =#+@ =#&@ =# @="%@ ="# @ =""@=") @ =)#@ =)0@ =)'@ =)+@ =)&@=):

UNIT 2

P)'4*)-' R-*/ )?"+ F-"'(% Principal ideal@ =reatest common di4isor@ !CM@ ,emainder Theorem@Prime and maimal ideals@ Splittin$@ prime fields@ Characteristic@ Irreducible and primiti4e pol-nomials@Content@ Eisenstein Criterion@ C-clotomic pol-nomial.9The followin$ results are included7 Theorem #) to Theorem ""@ Theorem "0 to Theorem ))@ Theorem )*

to Corollar- 0":

UNIT 3

S'-$$-*/ F-"'(% De$ree of an etension@ Simple etension@ Al$ebraic etension and transcendental e(tension@ Splittin$ field@ Seperable etension@ =alois field@ =alois $roup.9The followin$ results are included7 !emma 00 to Corollar- *)@ !emma *0 to Theorem *&:

UNIT =

R))$ ) U*-$4 *( S)'?>-'-$4 >4 R(-c'% C-clic $roup of nthroots of unit-@ Primiti4e element@2obenius automorphism@ ,adical etension@ Sol4abilit- b- radicals@ nsol4able uintic.

9The followin$ results are included7 Theorem '" to Corollar- +"@ !emma +) to Theorem +*:

UNIT

F5*("*$' T")+" ) G')- T")+4% =alois etensions@ 2undamental Theorem@ 2undamentalTheorem of al$ebra@ =aloisTheorem on sol4abilit-.9The followin$ results are included7 Theorem to Corollar- )@ !emma 0 to Theorem &:

R""+"*c"%

1. H+)'( M. E(+(, alois .heory#S+-*/"+, 198=.

2. )" . A. G''-*, %ontem'orary Abstra"t Algebra#


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MM 2=3 MECHANICS :E'"c$-?";

T"#$% H"+>"+$ G-)'($"-*, %lassi"al !e"hani"s#A((-)* W"'"4

UNIT I

Mechanics of a particle@ Mechanics of a s-stem of particles@ Constraints@ DHAlembertHs principles and!a$an$eHs Euations@ 5elocit- dependent potentials and dissipation functions@ Simple applications of!a$ran$ian formulation.9Chapter # of Tet:

UNIT II

FamiltonHs principle@ Deri4ation of !a$ran$eHs euation@ Some techniues of Calculus of 5ariation@Etension of Familton principle@ Conser4ation Theorems.9Secitons ".#@ "."@ ".)@ ".0 and ".' of Tet:

UNIT III

The two bod- Central force problem@ ,eduction to eui4alent one bod- problem euation of notation@The eui4alent one dimensional problem@ The 5irial Theorems@ the differential euations for the orbits @The Neplar problem.9sections).# to ).' of Tet:

UNIT IV

The Ninematics of ri$id bod- motion@ the independent coordinates of a ri$id bod- ortho$onaltransformations@ The Eulerian an$les@ The Ca-le-(Nlein parameters@ EulerHs Theorem on the motion of ari$id bod-@ The Coriolis force9Sections 0.#@ 0."@ 0.0@ 0.*@ 0.'@ 0. of Tet:

UNIT V

The ri$id bod- euations of motion@ An$ular momentum@ Tensor and d-namics@ The inertia tensor@ Theei$en 4alues of the inertia tensor@ Methods of sol4in$ ri$id bod- problem and Euler euations of motion.9Sections *.# to *.' of Tet:

R""+"*c"%

S4*/" .L *( G+--$ B.A,-rin"i'les of !e"hani"s#MC G+H-''


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MM 2=3 THEORY OF WAVELETS :E'"c$-?";

T"#$ B))%

M-c"' F+6-"+,An 4ntrodu"tion to =avelets through Linear Algebra#S+-*/"+

Prereuisites7 !inear Al$ebra@ Discrete 2ourier Transforms@ elementar- Filbert Space Theorems9>o uestions from the pre(reuisites:

UNIT I

Construction of ?a4elets on J> the first sta$e. 9Section ).#:

UNIT II

Construction of ?a4elets on Jn the iteration sets@ Eamples ( Shamon@ Daubiehie and Faar9Sections7 )." and ).):

UNIT III

Q"9J:@ Complete ;rthonormal sets@ !"B(R@R and 2ourier Series.9Sections7 0.#@0." and 0.):

UNIT IV

2ourier Transforms and con4olution on Q"9J:@ 2irst sta$e wa4elets on J.

9Section7 0.0 and 0.*:

UNIT V

The iteration step for wa4elets on J@ Eamples@ Shamon Faar and Daubiehie

R""+"*c"%

M4)+ :1993;, =avelets and /'erators#C>+-(/" U*-?"+-$4 P+"

C5-. C: 1992;, An 4ntriodu"tion to =avelets#Ac("-c P+", B)$)*


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MM 2=3 CODING THEORY :E'"c$-?";

T"#$% D. H)* "$'., %oding .heory .he $ssentials#P5>'-"( >4 M+c"' D""+ I*c 1991

UNIT I

Detectin$ and correctin$ error patterns@ Information rate@ The effects of error detection and correction@2indin$ the most likel- code word transmitted@ ?ei$ht and distance@ M!D@ Error detectin$ andcorrectin$ codes.9Chapter # of the Tet:

UNIT II

!inear codes@ bases for C S and CU@ $eneratin$ and parit- check matrices@ Eui4alent codes@ Distanceof a linear code@ M!D for a linear code@ ,eliabilit- of IM!D for linear codes.9Chapter " of the Tet:

UNIT III

Perfect codes@ Fammin$ code@ Etended codes@ =ola- code and etended =ola- code@ ,ed Fulles Codes.9Chapter ) sections7 # to &of the Tet:

UNIT IV

C-clic linear codes@ Pol-nomial encodin$ and decodin$@ Dual c-clic codes.9Chapter 0 and Appendi A of the Tet:

UNIT V

CF Codes@ C-clic Fammin$ Code@ Decodin$ " error correctin$ CF codes9Chapter * of tet:

R""+"*c"

1. E.R B"+'", Algebria" %oding .heory#Mc G+H-'', 198

2. P. C"+)* *( .H V* L-*$, ra'hs#C)("( *( D"-/* CUP

3. H. H-'',A &irst %ourse in %oding .heory#OUP 198.


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MM 2== ANALYTIC NUMBER THEORY :E'"c$-?";

T"#$% T).M. A)$)'4ntrodu"tion to Analyti"al ,umber .heory#S+-*/"+V"+'/

UNIT I

The 2undamethal Theorem of Arithmetic 9chapter # of Tet:

UNIT II

Arithmetical function and Dirichlet multiplication9Section ".# to ".#+ of Tet:

UNIT III

Con$ruences@ Chinese ,emainder Theorem9Sections *.# to *.#% of Tet:

UNIT IV

Luadratic residues@ ,eciprocit- law@ Gacobi s-mbol9Sections .# to .& of Tet:

UNIT V

Primiti4e roots@ Eistence and number of primiti4e roots.9Sections #%.# to #%. of tet:

R""+"*c"

1 E( G+)'(, .o'i"s from the .heory of ,umbers#B-+5"

2 G.H H+(4 *( E.M W+-/$ ,4ntrodu"tion to the .heory of ,umbers#O#)+(


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MM 2== COMMUTATIVE ALGEBRA :E'"c$-?";

T"#$% N.S G)'+-*,%ommutative Algebra#O#)*-* P+"

UNIT I

Modules@ 2ree pro6ecti4e@ Tenser product of modules@ 2lat modules9Chapter # of Tet:

UNIT II

Ideals@ !ocal rin$s@ !ocaliation and applications9Chapter " of Tet:

UNIT III

>oetherian rin$s@ modules@ Primar- decomposition@ Artinian modules9Chapter ) of Tet:

UNIT IV

Inte$ral domains@ Inte$ral etensions@ Inte$rall- closed domain@ 2initeness of inte$ral closure9Chapter 0 of Tet:

UNIT V

5aluation rin$s@ Dedikind domain

9Chapter * of Tet@ Theorems 0 and * omitted:R""+"*c"%

1 M.F A$-4 *( I.G Mc D)*'(,4ntrodu"tion to %ommuni"ation Algebra#A((-)* W"'"4

2 T.W H5*/"+)+(,Algebra#S+-*/"+V"+'/


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MM 2== REPRESENATION THEORY OF FINITE GROUPS

:E'"c$-?";

T"#$% W'$"+ L"("+**,4ntrodu"tion to rou' %hara"ters#C>+-(/" U*-?"+-$4 P+"

UNIT I

=(module@ Characters@ ,educibilit-@ Permutation representations@ Complete reducibilit-@ SchurHs !emma9Sections #.# to #.+ of Tet:

UNIT II

The commutant al$ebra@ ;rtho$onalit- relations@ The $roups al$ebra9Section #.&@ ".#@ "." of Tet:

UNIT III

Character table@ Character of finite abelian $roups@ The liftin$ process@ !inear characters.9Section ".)@ ".0@ ".*@ ".' of Tet:

UNIT IV

Induced representations@ ,eciproca- law@ A* @ >ormal sub$roups@ Transiti4e $roups @ Induced charactersof Sn9Sections ).#@ )."@ ).)@ ).0@ 0.#@ 0."@ 0.) of Tet:

UNIT V

=roup theoretical applications@ runsideHs 9p@: Theorem@ 2robenius $roups.9Chapter * of Tet:

R""+"*c"% S.L*/, Algebra#A((-)* W"'"4


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MM 2== CATEGORY THEORY :E'"c$-?";

T"#$ B))% S. Mc'*", %ategories for the *or:ing !athemati"ian#S+-*/"+, 19+-(/" U*-?"+-$4 P+", 1991


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Msc Maths Colleges 2013