Govt. (Autonomous) Holkar Science College, Indore O4I.P.)(Syllabus)

(Approved by Board of Studies of College)(2018- 1e)

Class :

Subject:

Paper:

Title:

&ot 8-tS

M.Sc. III Sem.

Mathematics

I (Compulsory)

Functional Analysis(Theory + CCE) 75+25 =100

Normed linear spaces, Pioperties ofNormal linear spaces, Branch Space & Example,T]NIT-I Convex Sets inNormal linear spaces. [TB-1: Art 3.r,3.2,2.2,2.3 &TB-2:

VIFinite Dimensional Normed linear spaces & Subspaces. Basic Properties of Finite

LINIT-II Dimensional Normed linear spaces. Equivalent norms, Riesz lernma andco -1: Art. 2.4,2.5 8.TB-2Quotient Space of normed linear spaoes and its completeness, Liner operato-isardtheir elemen ro l: Art.2.5 &TB -2Normed Linear operators & Bor:nded Linear operators & continuous operatots.[TB-1: Art. 2.7,2.10-l ,2.10-2 & TB -2 Chapter VIIILinearfirnctiona1,boundedLinearfunctiona1,DualspaceSwitti@Art. 2.8, 2.10-3 to 2.L0-7 & TB -2 chapter vIII up to tur.3.l

TB : TEXT BOOK

Text Books:

1. E- Kreyszig, Introductory Functional Analysis with applications, John Wiley & SonsNew York

.

2 . B.K. Lahiri, The world press private Limited calcutta ( w.B. )

Reference Books:

1. B. Cho$dhary and Sudarshan Nanda. Functional Analysis with applications, Wiley Eastem Ltd

2. G.F. Simmons, Introduction to To

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GoW. (Autonomous) Holkar Science College, Indore (M.p.)

[semester wise svllabus (cBCS) f- M. s" Apprfrl"liul;?oard of studies of conege for session 20 1 8- 1 9r

Class :

Subject:

Paper:

Tiile:Max Marks:

lrI.Sc. Sem. IIIMathematics

1I ( Compulsory)Group: Advanced Special Functions(Theory + CCE) 75+25:10G

UN-'IT-I ;;, ffi,iJ"n',,1T,llTT.eflcn.lr".oto Jtr*li^^il^- f^_^^--r ?

ffi;ffi,l;;;;,;ffi;F'.rrler Tnf.-..rr"ai f^- F- D^r-^ -a '-r--)tI,esgn{e's duplication fornauia , Gauss murtiplication theorern.

N7-12, Art 15-20.

Sir,ple integral formffii-;'lIIT-II

The contiguous function relations. The hyper geometric diffbrentiaiSirnpie'iransformai:ons, Relation

Eleraentary series marripulations.functioa z and l-2.

{a,b,c, i ),eQualion.bew,eelr

Art 30-34, Arr 37-39.

saaischiitz''iheolem.A$ 44-46, Arr 49-53.

lLri 57 -64.Jontl

W: The flrncttonwruppte's t hesre:a, Dixon's Ttreorem..

LI\{IT.IV

iivIT-'ri

Bessei fi:nctionrtrecurrence relatigF' A pure rocutrt,nc, reiarion. A generaring function" E3esso--i,sintegral Index haif an odd inteser.

Basic properties of 1Fl, fr***r,, flrsi fid o*ho

nrsi ibruluia. Kuramer's second forrcr-ria ,

crthogunaiity, An eq r-ii trr*i etru: i,

poiynoraiais, Expansion ct' j

orthogonal poiynomiais; simple se:s *i loly=ro*iuir,condidon ibr ,_ortsrgonalif,v, zeros *f' oitilogonai

noiynoroials. ?te Three-tenn reo-'r,=ence relarion.Ari 68-7S, Art Z8-83.

BcoS<s Resor$mended

i. R-ainvilie, E-D. ; speciar furiction, The rda;,ailia='Co.. New york 19?i "

-,?eferencSBooks

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Govt.(Autonomous) Holkar Science Collegeo Indore O,I.P.)(Syliabus)

(Approved by Board of Studies of €ollege)(201 8- 1e)

20tg-ts

M.Sc. III Sem.

Mathematics

III (Optional)Advanc el F uzzy Mathem ati c s

(Theory + CCE) 7 5+25 :100

Class:

Subject:

Paper:Title:Max Marks:

LNIT-V

Fuzzy Versus Crisp Number System, Interval Sets, Representation of A Set Type of i

Sets, Subsets, Universal Set. Operations On Sets, Some irnportant Results on Venn ,

Diagrams Fuzzy Sets Fvzzy Set Definition Different types of Fuzzy Sets General i

Definitions and Properties of Fuzzy Sets. ;

Other Important Operations General Properties : Fuzzy Vs Crisp Some ImportantTheorems Extension Principal For Fuzzy Sets Fuzzy Compliments Further OperationsonFuzzv Sets.

UNIT.IIIAggregation Operations Extension Principle For Fvzzy Sets Surnmary.

IINIT.IV Extension for sets - The zadeh's extension principle.Fvzzy Numbers Algebraic Operations With Fuzzy Numbers Binary Operations WithFvzzy Numbers Binary Operations of Two Fuzry Numbers Some Special ExtendedOperations Extended Operations For L-R Representation of Fuzzy Sets Fuzzy

i fuithmetic Arithmetic operations on Fuzzy Numbers in the Form of a- Cut Sets.

Text Book:1. Fwzy Set Theory and its Application by F{.J. Zimrnermarul, Ailied Publication ltd. }Jew

Delhi, 199i2" Fuzzy Sets and FuzzyLogic by G. J. Kiir and B. Yuan Prentice- Hail of India, Nevr Deihi,

l tes

Book Recommended:Fuzzy Sets and uncertainty and Information by G.J. Kalia tina A. \olj.r- Prentice-Ha'' oflndia'

rr ) o"rt---:€i.,w Ldo\- f Jr.'-1$k" hK ^ ,$4 W,t,-V

',,4.r, e eL>-\ \ A CI- ,^.- , nr/ ^ \N\\ tJ A.,-o (

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t-norms and t-conoilns Definition Of Intersectiog and Union by Hamacher Yager'sUnion and Intersection of Two Fuzzy Sets as Definbd by Defined by Dubois and Prade

Govt. (Model, Autonomous) Holkar science, college rndore (M.P.)

(Department of lligher Educationo Govt' of M'P')CfiolCE BASED CREDIT SYSTEM (CBCS)

Class: M.Sc. (Mathematics)Title: Theory of Linear OPerators -I

Syllabus(Session 2018-19)

Semester - III Paper IVMax Marks: 100 (Theory + CCE = 75 + 25)

T]1\[IT - ISpectral theory in finite dimensional normed space. Basic concept: Resolvent set and Spectrum'

Spectral properties of bounded linear operators. Properties of resolvent and spectrum'

( Art. No. 7.1 -7.4 from 1)

I]NIT - IIUse of complex analysis in spectral theory: Banach algebras. Properties of Banach algebras'

( Art. No. 7.5 - 7 .7 from 1)

UNIT - IIICompact linear operators on normed linear spaces and their p$perties.

( Art. No. 8.1 - 8.2frbm 1)

UNIT - rySpectral properties of compact linear operators on normed spaces. Operator equations' Theorem

ofFredholems tyPe.( tut. No. 8.3 - 8.5 from l)

UNIT - VTheorems of Fredholms type. Fredholm alternative and fredholm alternative theorem.

( Art. No. 8.6 - 8.7 from 1)

Text Book:1. E. Kreyszing: Introductory Fr:nction Analysis with applications, John-Wiley & Sons,

Ne)fs York, t978.\Reference Books:

Z. B. K. Lahiri : Elements of Fgnctional analysis, The word press Pvt. Ltd. Calcutta, 1998.

3. B. Choudhary and S. Nanda: Functional analysis with applications, \Viley Eastem, Ltd.,

4. N. Dunford & J.T. Schwartz:Linear operators - 3 part,Interqcience Wiley, New York,

1958 -71.wv'^'"'^'''

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Govt (Autonomous) Ilolkar Science College, Indore (M'P')(Syllabus)

[Semesier wise syllabus (CBCS) for M.Sc Approved by Board of Studies of College for session 2018-i9]

Class :

Subject:

Paper:Title:

Max Marks:

M.Sc. iII Sem.

Mattrematics

V (Optional)Analytic Number Theory

(Theory = CCE) 75+25:100

UNIT.I

lNIT.II

chlet Multipiication: Introduction , Definitions of the

Mobius function p(n), Eluer totient function i(n).A retration connecting !"r and i. The

product formal for 0 (n), The Dirichlet product of arithmetical functions. Dirichiet

lrrn.6.t and Mobins inversion formula.Tire Mangoidt fi:nction. Multiplicative functions.

Multiplicative functions and Dirichlet Multiplicatioqis. The big 'oh' notation, Euler's

Summation formul4 some elementary asymp'totic forrnulas.

Chap-art -2.1 -2.1 0, Chap -uk-Z .Z,Z .3,3 .4

Abei's identity for arithmetical fi:nctions"Construction of subgroups. The charactercharacters"

Theorem 4.2, Chap6-6.4-6.7

Dirichtet characters. Sums involving Diriciriet charasters, The non-vanishing

UNIT-III for reatr non principal X. Dirichlet theorem on primes, primes of fomr 4n-1 and

I Art 6.8-6.10, Chap 7-art7.t,

The plan of the proof of Diri theorem on primes. Distribution of primes in arithmetic

trN-IT,ry progression.Art - 7.3- 7.9"

Finite abeliangroup. The

groups & their characters;-orthogonaltty reiations for

oi i-{.1, X)4n+1.

I

i!t.:

-. UNIT.Yi:

i

Diiicirtet Series and Euler Products:- Defiaition and different types of Dirichiet series.

The half -piane of absolute convergence of a Dirictrlet series. Ttie function defineC bl a :

Dirichiet series, Uniqueness tJreorem. muitipiicatioa of Dirictriet series, Euler prociuet.

Chapli -art ii.1 - 11.5.

Book Recomme4ded:i.T"M. Apostol, Introduction to Analytic Number Theory, Narosa pub. Ho.use,1989.

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Govt. (Model, Autonomous) Holkar Scienceo College Indore (M.P.)(Department of Higher Education, Govt. of M.P.)

Semester wise Syllabus for post graduatesCHOICE BASED CREDIT SYSTEM (CRCS)

(With effect from the Academic year 201&"11I)

SyllabusClass - M.Sc. A{A (Semester-Il!Mathematics Modeling

Max Marks: 100 (Theory + CCE = 75+25)

Syllabus :

UNIT I:Simple situation requiring mathematicdl modeiling. Techniques, classification and characteristicsof mathematical models. Limitations of mathematics modellins.

;

UNIT II: qSefiing up first order differential equffions, qualitative solution, stabilewweity of solutions,growth and decay population models,.exponential and logistic population models.

UNIT lil:Comparfment models through system ef ODE. Mathematics models in medicine, arms race,battles and intemational trade in terms of system of ODE.

e"TINIT IV: i

Non-linear difference equations models for population growth probability generating function,fourth method of obtaining partial differentiai equation models. PDE model for a stochasticepidemic process with no removal. \4odel for traffic on a highway.-s..

USIIT V: 3General Communication networks, general weighted digraphs, 5nap-colorxing problems.

Reference Books:i. Kapoor JN: Mathematics models in biology and medicine EWP (1985)2. Mdtin Brann C.S Coleman , DA Drew (Eds) differential equation models.,?. C.L.Liu, Elements of discrete mathematics.4. A.M.Law and W.D Ketton, Simulation rnodeling and analysis, McGraw Hill Intl. Ed(Second

edition) 1991 $ >_+1D \M O"-h-'r (

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Subjecfi MathematicsPaper- Interdisciplinary Theory Paper

-,3:iffi1'6"",181ffi:,i!;!TillJ,3'";i;iilii"';", ro* - ts

As recommended by central Board of studies and approved by the

Governor of M'P' Ma* n4"115731pro-tm 316 : 50

fffft Class

++'g{ Semester

frqq qW or qfrfir Title of SubjecV Group

gST Wi @. PaPer No.

srffi/ ffio- Compulsory/ OPtional Optional Gr-IV (i)

n and DeveloPment of

Operations Research, Characteristics of Operations Research'

u.* of Op.rations Researcho Uses anrtr

Limitations of operation Research, Linear programming Problerus''

Mathematical Formulation, Gnaptrical Solution Method'

: SirnPlex Method excePtional

I

I ."r.., artificial variable techniques ; Big M method, two phase Metho6I

i and Cyctic Problems, problem of degeneracy.

dualitY and theorem of dualitY'

Recommended Books :-1_ Kanti swarup, p.K. Gupta and Manmohan, operations Research,

Sultan Chand & Sons, New Delhi'Reference Books:-1- S.D, Sharma, OPeration Researcho

2- F.So l{iiler and b.J. Lieberrnan, Industrial Engineering serieso 1995

ffhis,book comes with a cD containing software)

i: - (

C. Hadley , Linear Prograrnrning, Narosa Publishing Elouse. 1995'

4- G. I{adley, Linear uod Dynamic programming, Addison \ileslerv

Reading Mass5- H.A. Tahao operations Research An introductiono Macmillan

Publishing co. Inc. New York6- prem Kumar Gupta and D.s. Hirao operation Reasearch, an

Introdution, S. Chand & Company Ltd. New Delhi'

1- I{.S. Kambo, Mathematiial Programming Techniques' Affilated East

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*-.. ^vxd {>v r1\'- h o r-\ nn/ \r n t"$t' \^l'y'.r\ \N\ nL{" \-N\'ffi^, & M q

M. Sc./iM.A Mathematics)

Operations Research-I

TUTTI]TV N

ation, GraPhical Solution Method'