charu syllabus

7/25/2019 Charu Syllabus

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Course : M.Sc. Mathematics

Mode : Distance Education

Duration : Two Years

Eligibility : B.Sc. in Mathematics/Statistics/Applied Mathematics

Medium : EnglishCOURSE OF STUDY & SCHEME OF EXAMINATIONS

Subject Code Title TotalMarks

I YEAR

. Algebra !!

." #eal Analysis !!

.$ Di%%erential E&uations and 'umerical Methods !!

.()perations #esearch

!!

.* Mathematical Statistics !!

II YEAR

". +omple, Analysis !!

"." Topology and -unctional Analysis !!

".$ raph Theory !!

".( rogramming in + / + 00 !!

".* Discrete and +ombinatorial Mathematics !!

Total 1000

Paper 1.1: ALGEBRA

UNIT I

roups 1 Subgroups 1 'ormal subgroups 1 2sormorphism theorems 1 ermutation

groups 1 Abelian groups 1 Automorphisms 1 +on3ugate classes 1 Sylow4s theorems 1

Direct products.


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UNIT II

#ings 1 2deals 1 Ma,imal5 prime ideals 1 2ntegral domains 1 Euclidean domains 1

6ni&ue %actorisation domains.

UNIT III

7ector spaces5 linear trans%ormations 1 +anonical %orm5 triangular %orm 1

'ilpotent trans%ormation 1 8ordan %orm 1 9ermitian5 unitary and normal trans%ormations.

UNIT I

-ields5 e,tension %ields5 roots o% polynomials 1 Splitting %ields 1 alois theory5

%inite %ields.

TE!T"##$S AN% RE&ERENCES :

. 9erstein 2 '5 Topics in Algebra5 ed"5 7ias ublications.

". 8ohn B -raliegh5A First Course in Abstract Algebra5 Addision ;esley.

'a(er 1.): REA* ANA*YSIS

UNIT I

)pen balls5 +losed balls in #n1 +losed sets and adherent points 1 The Bolering

theorem 1 +ompactness in #n5 limits and continuity 1 +ontinuous %unctions5 %unctions

continuous on compact sets5 6nion continuity 1 -i,ed point theorem %or contractions.

UNIT II

Deri>ati>es 1 The chain rule5 %unctions with nonati>e5 ?ero deri>ati>es

and local e,trema5 #olle4s theorem5 the Mean=>alue theorem %or deri>ati>es5 intermediate

>alue theorem %or deri>ati>es5 Taylor4s %ormula with remainder5 artial deri>ati>es5

Directional deri>ati>e5 the Total deri>ati>e5 the 2n>erse %unction theorem5 the 2mplicit

%unction theorem.


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UNIT III

The #iemann=Stielt3es 2ntegral 1 De%inition and e,istence o% the integral5

properties o% the integral5 di%%erentiation under integral sign5 interchanging the order o%

integration 1 Se&uence o% %unctions5 uni%orm con>ergence and #iemann=Stielt3es

integration5 uni%orm con>ergence and di%%erentiation.

UNIT I

@ebes&ue measure5 )uter measure5 Measurable sets and @ebesgue measure5

Measurable %unctions5 Egoro%%4s theorem5 @usin4s theorem5 the @ebesgue integral5

Bounded con>ergence theorem5 -atou4s lemma5 Monotone con>ergence theorem5

@ebesgue con>ergence theorem5 +on>ergence in measure.


. Tom M Appostol5Mathematical Analysis5 Addision ;esley5 'arosa.

". ;alter #udin5Principles of Mathematical Analysis5 Mc=raw 9ill.

$. #oyden5Analysis.

'a(er 1.+: %I&&ERENTIA* E,UATI#NS AN% NUMERICA* MET-#%S

UNIT I

)rdinary Di%%erential E&uations 1 2nitial >alue problems %or second order

e&uations5 a %ormula %or the ;ronsian5 The use o% a nown solution to %ind another5@inear e&uations with >ariable coe%%icients5 The method o% undetermined coe%%iciens5 Themethod o% >ariation parameters5 ower series solution 1 The @egendre e&uation5 Bessel

e&uation.

UNIT II

artial Di%%erential E&uations 1 @inear e&uations o% %irst order5 +auchy4s method

o% characteristics5 +harpit4s method5 Solutions satis%ying gi>en conditions5 8acobi4s

method5 Second order e&uations5 E&uation with >ariable coe%%icients5 Separation o%

>ariable5 @aplace4s e&uation5 Boundary >alue problems5 ;a>e e&uation5 Elementary

solution o% one=dimensional wa>e e&uation.

UNIT III

'umerical Analysis 1 System o% e&uations and unconstraint optimi


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UNIT I

'umerical di%%erentiation5 integration5 the solution o% di%%erential e&uations.


. 2an Sneddon5Elements of Partial Differential Equations5 Mcraw=9ill.

". +oddington5An Introduction to Ordinary Differential Equations5 92.

$. Simmons -5Differential Equation ith Applications5 TM9.

(. Elementary !umerical Analysis" An Algorithmic Approach5 Mcraw=9ill.

'a(er 1.: #'ERATI#NS RESEARC-

UNIT I

@inear programming 1 Simple, method 1 Dual simple, method 1 #e>ised simple,method 1 Sensiti>ity or postophmal analysis 1 arametric linear programming 1 2nteger

programming.

UNIT II

Dynamic programming 1 Decisions under ris 1 Decisions under uncertainty 1

ame theory.

UNIT III

ro3ect Scheduling by E#T=+M5 2n>entory models 1 Types o% in>entory models

1 Deterministic models 1 robabilistic models.

UNIT I

ueueing theory 1 ueueing models M/M/2C: D//C5 M/M/2C: D/'/C5

M/M/+C: D//C.


. 9amdyn A Taha5Operations #esearch5 Macmillan.


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'a(er 1./: MAT-EMATICA* STATISTICS

UNIT I

Two dimensional and n=dimensional random >ariable5 Marginal distribution5

Distribution %unctions5 2ndependent random >ariables5 +onditional e,pectation5 rinciple

o% least s&uares5 Discrete distributions5 +ontinuous distributions5 Beta and amma

distributions5 enerating %unctions5 +on>ergence and @imit theorems.

UNIT II

E,act sampling distributions5 t=distribution5 ;ea law o% large numbers and

+entral limit theorem5 E,act distribution o% sample characteristics5 Theory o% estimation5Ma,imum lielihood estimation5 +on%idence inter>als5 @arge sample con%idence

inter>als.

UNIT III

Test o% hypothesis5 +omposite hypothesis5 +omparison o% normal population5

@arge sample tests5 Test o% multinomial distribution.

UNIT I

Statistical &uality control and analysis o% >ariance.

TE!T"##$S AN% RE&ERENCES :. Baisnal A and 8as M5Elements of Probability and $tatistics5 Tata Mcraw=9ill5 'ew Delhi5

$.

". upta S+ and apur 75Fundamentals of Applied $tatistics5 Sultan +hand F Sons.

'a(er ).1: C#M'*E! ANA*YSIS

UNIT I

The geometric representation o% a comple, number 1 The spherical representation and

stereographic pro3ection 1 Analytic %unction 1 +# e&uations 1 9armonic con3ugate 1 To %ind an analytic%unction % i% a harmonic %unction us is gien.

ower series 1 #adius o% con>ergence 1 ower series represents an analytic %unction inside the

circle o% con>ergence 1 Abel4s limit theorem.


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+on%ormal mappings 1 Bilinear trans%ormations 1 -i,ed point o% bilinear trans%ormations 1 +ross

ratio 1 Most general bilinear trans%ormations which trans%orms unit dis onto the unit disH hal% plane

2m


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'a(er ).): T#'#*#Y AN% &UNCTI#NA* ANA*YSIS

UNIT I

Topological spaces 1 De%inition 1 Elementary concepts 1 Bases5 sub=bases5

product spaces 1 +ompactness 1 Tchono%%4s theorem 1 +ompactness %or metric spaces 1@ocally compact spaces.

UNIT II

Separation a,ioms 1 6ryshon4s lemma 1 Tietalent conditions %or

complete orthonormal set 1 +on3ugate space 9L = The ad3oint o% an operator 1 Sel%=

ad3oint operators 1 'ormal unitary operators 1 ro3ections 1 -inite dimensional operator

theory.


. Simmons -5Introduction to Topology and Modern Analysis.

'a(er ).+: RA'- T-E#RY

UNIT I

raphs 1 ;al5 path5 cycle 1 Bipartite graphs 1 Trees 1 +utest 1 -undamental

circuits 1 Spanning trees 1 +ayley4s %ormula 1 rusal4s algorithm.


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UNIT II

+onnecti>ity 1 Blocs 1 Euler tours 1 9amiltonian cycles 1 +losure o% a graph 1

+ha>atal theorem %or 'on=9amiltonian simple graphs.

UNIT III

2ndependent sets 1 +li&ues 1 #amsey4s numbers 1 7erte, colouring 1 Broo4s

theorem 1 9a3o4s con3ecture 1 +hromatic polynomials.

UNIT I

lanar graphs 1 Dual graphs 1 Euler4s %ormula 1 The %i>e colour theorem 1 'on=

9amiltonian planar graphs 1 Directed graphs 1 'etwors o% %lows 1 Ma,=%low Min=cut

theorem.

TE!T"##$S AN% RE&ERENCES :. Bondy and Murty5 &raph Theory and Its Applications.

". Balarishnan #5 &raph Theory.

$. Arumugam S5In'itation to &raph Theory.

'a(er ).: 'R#RAMMIN IN CC22

UNIT I2ntroduction: A computer program 1 rogramming languages 1 +ompilers and

interpreters 1 ;hy +/+00 = -unction libraries 1 )b3ect oriented programming 1 Steps inprogram de>elopment 1 Synta, o% language and logic programming.

+/+00 Basics: Structure o% a + program 1 #eturn C %unction 1 +omments in + and +00

= include command 1 +haracters5 integers5 decimal numbers 1 eywords 1 +onstants and

>ariables and their declaration 1 Data types and %unctions 1 @iterals.

UNIT II)utput and 2nput in +/+00: utsC and putcharC %unctions 1 +ontrol codes 1 rint%C

%unction 1 -ormatted output 1 )utput in +00 = getsC and getcharC %unctions 1 Scan%C %unction1 2nput in +00 = 6se%ul input %unctions.


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Arithmetic )perations and -unctions: Arithmetic operators 1 2nteger di>ision 1

)perators and data types 1 E,tensions 1 )rder o% precedence 1 +ounters5 increment and

assignment operators 1 6sing %unctions 1 @ocal and global >ariables.

UNIT III+ontrol Structures: i% and i% N. else statements 1 'ested i% statements 1 #elational

operators 1 @ogic operators 1 Switch command 1 %or5 do N. while5 while loops 1 'ested doloops 1 +ombining loop types 1 6sing %lags and brea statement.

Arrays and Strings: Arrays 1 De%inition5 declaration5 entering >ariables in manipulating

arrays 1 E,amining and passing an array 1 Strings 1 +omparing two strings 1 Determining string

length 1 Assigning and combining strings 1 String arrays.

UNIT IStructures and ointers: Structures 1 De%inition 1 Assigning structure >ariable 1

Assigning initial >alues 1 6sing a structure 1 Structure ways 1 Structure and %unctions 1

6nderstanding pointers 1 ointers and %unctions.

-ile operations: 6nderstanding %iles 1 Declaring a %ile 1 )pening a %ile 1 +losing a %ile 12nput and output %unctions 1 -ormatted input and output 1 ;oring with structures 1 Adding data

to a %ile 1 #eading and printing a dis %ile.

TE!T"##$S AN% RE&ERENCES :. Allan #. 'eibauer5 (our First C)C** Program.

". aul M. +hirian5Programming in C**.


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'a(er )./: %ISCRETE AN% C#M"INAT#RIA* MAT-EMATICS

UNIT I

enerating %unctions %or combinations 1 Enumerators %or permutations 1

Distributions o% distinct ob3ects into non=distinct cells 1 artitions o% integers 1 The-errers graph 1 #ecurrence relations 1 @inear recurrence relations with constant

coe%%icients 1 'on=linear di%%erence e&uations 1 #ecurrence relations with two indices.

UNIT II

The principle o% inclusion and e,clusion 1 Derangements 1 ermutations with

restrictions on relati>e positions 1 ermutation with %or%idden positions.

UNIT III

olya4s theory o% counting 1 E&ui>alence classes under a permutation group 1

olya4s %undamental theorem 1 enerali

charu syllabus

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