m.sc. mathematics 2014-15 · mth-112real analysis-i max. marks : 80 time:3 hours unit -i (2...

Scheme of Examination

M.Sc. (Mathematics) (w.e.f.Session 2014-15)

SEMESTERI

CourseNo. CourseTitle Theory+Int.

MTH-111 Advanced Abstract Algebra-I

80+20

MTH-112 RealAnalysis-I 80+20 MTH-113 Topology-I 80+20

MTH-114 IntegralEquations and Calculus of Variations 80+20

MTH-115 Mathematical Statistics 80+20

Total Marks 500

SEMESTER II

MTH- 211 AdvancedAbstract Algebra-II 80+20

MTH- 212 Real Analysis-II 80+20 MTH- 213 Topology-II 80+20

MTH- 214 Ordinary Differential Equations 80+20

MTH- 215 Operation Research Techniques 80+20

Total Marks 500

SEMESTER III

MTH- 311 Functional Analysis-I 80+20

MTH- 312 Partial Differential Equations and Mechanics 80+20

MTH- 313 Complex Analysis-I 80+20

MTH- 314 One paper out of either Group A1 or Group B1

A11)Advanced Functional Analysis-I

A12)AdvancedDiscreteMathematics–I

A13) Algebraic Coding Theory-I

A14) Wavelets-I

A15) Soboley Spaces-I

B11) Mechanics of Solids-I

B12) Continuum Mechanics-I

B13) Computational Fluid Dynamics-I

B14) Difference Equations-I

B15) Information Theory-I

80+20

MTH-315 One paper out of either Group C1 or Group D1

C11) Theory of Linear Operations-I C12) Analytical Number Theory-I C13) Fuzzy Sets and their Applications-I C14) Bases in Banach Spaces-I C15) Algebraic Topology-I D11) Fluid Dynamics-I D12) Biomechanics-I D13) Integral Equations and Boundary Value Problems-I D14) Mathematics for Finance and Insurance-I D15) Space Dynamics-I

80+20

Total 500

SEMESTERIV

MTH- 411 Functional Analysis-II 80+20

MTH-412 Classical Mechanics 80+20

MTH-413 Complex Analysis-II 80+20

MTH- 414 One paper out of either Group A2 or Group B2

A21)AdvancedFunctional Analysis-II

A22)AdvancedDiscreteMathematics–II A23) Algebraic Coding Theory-II A24) Wavelets-II A25) Soboley Spaces-II A26) Frames and Riesz Bases-II B21) Mechanics of Solids-II B22) Continuum Mechanics-II B23) Computational Fluid Dynamics-II B24) Difference Equations-II B25) Information Theory-II

80+20

MTH-415 One paper out of either Group C2 or Group D2

C21) Theory of Linear Operations-II C22) Analytical Number Theory-II C23) Fuzzy Sets and their Applications-II C24) Bases in Banach Spaces-II C25) Algebraic Topology-II D21) Fluid Dynamics-II D22) Biomechanics-II D23) Integral Equations and Boundary Value Problems-II D24) Mathematics for Finance and Insurance-II D25) Space Dynamics-II

80+20

Total Marks 500

GrandTotal 2000

NOTE: Eitherofthe paperMTH-115-A orMTH-115-B to beselected. Note 1:The marks ofinternalassessmentofeach papershallbesplitedas under:

A) One classtestof10 marks. The classtestwillbe heldin the middle ofthe semester. B) Assignment&Presentation : 5 marks

C) Attendance : 5 marks

65%butupto 75% : 1 marks

Morethan 75%butupto 85% : 2 marks

Morethan 85%butupto 90% : 3 marks Morethan 90%butupto 95% : 4 marks

Above 95% : 5 marks Note2:Thesyllabusofeachpaperwillbedividedintofourunitsoftwoquestionseach.The

questionpaperofeachpaperwillconsistoffiveunits.Eachofthefirstfourunitswill contain two

questions and the students shall be asked to attempt one question from

eachunit.Unitfiveofeachquestionpapershallcontaineighttotenshortanswertype

questionswithoutanyinternalchoiceanditshallbecoveringtheentiresyllabus.As such

unitfive shallbe compulsory.

Note3:AsperUGCrecommendations,theteachingprogramshallbesupplementedbytutorials

andproblemsolvingsessionsforeachtheorypaper. Forthispurpose,tutorialclasses

shallbeheld foreachtheorypaper in groups of8 studentsforhalf-hourperweek.

Note 4: In orderto passaparticularpaper,thestudenthasto passin theorypaperseparately.

2

M.Sc. (Mathematics) SEMESTER-I

MTH-111Advanced Abstract Algebra-I Max. Marks : 80

Time:3 hours

Unit -I (2 Questions) Groups : Zassenhauslemma, Normal and subnormal series, Composition series,

Jordan-Holdertheorem, Solvableseries,Derived series,Solvable groups, SolvabilityofSn – the

symmetricgroupof degreen≥ 2.

Unit -II (2 Questions)

Nilpotentgroup:Centralseries,Nilpotentgroupsandtheirproperties,Equivalentconditionsforafinit

egrouptobenilpotent,Upperandlowercentralseries,Sylow-psub

groups,Sylowtheoremswithsimpleapplications.Descriptionof groupoforderp2

andpq,

wherepand q aredistinct primes(Ingeneral surveyofgroups upto order 15).

Unit -III (2 Questions)

Field theory, Extension of fields, algebraic and transcendental extensions.

Splittingfields,Separableandinseparableextensions,Algebraicallyclosedfields,Perfect fields.

Unit -IV (2 Questions)

Finite fields, Automorphismof extensions, Fixed fields, Galois extensions,

Normal extensions and their properties, Fundamental theorem of Galois theory,

Insolvabilityof thegeneral polynomial of degreen ≥ 5 byradicals.

Note : The question paper will consist of five units. Each of the first four units will

containtwoquestionsfromunitI,II,III,IVrespectivelyandthestudentsshall

beaskedtoattemptonequestionfromeachunit.Unitfivewillcontaineightto ten short

answer type questions without anyinternal choice covering the entire syllabus and

shallbecompulsory.

Books Recommended:

1. I.N.Herstein, Topics in Algebra, WileyEasternLtd., New Delhi, 1975.

2. P.B. Bhattacharya, S.K. Jain and S.R. Nagpaul, Basic Abstract Algebra (2nd

Edition), CambridgeUniversityPress,Indian Edition, 1997.

3. P.M. Cohn, Algebra,Vols.I,II& III, John Wiley&Sons, 1982, 1989, 1991.

4. N.Jacobson, BasicAlgebra,Vol.I&II,W.HFreeman,1980(alsopublishedby

Hindustan PublishingCompany).

5. S.Lang, Algebra, 3rd editioin, Addison-Wesley, 1993.

6. I.S. Luther and I.B.S.Passi, Algebra, Vol. I-Groups, Vol. II-Rings, Narosa

PublishingHouse (Vol.I– 1996, Vol.II–1990).

7. D.S.Malik,J.N.Mordenson,andM.K.Sen,FundamentalsofAbstract Algebra,

McGrawHill,International Edition, 1997.

8. VivekSahai and VikasBist, Algebra, NarosaPublishingHouse, 1999.

3


MTH-112Real Analysis-I Max. Marks : 80

Time:3 hours

Unit -I (2 Questions)

Riemann-Stieltjes integral, its existence and properties, Integration and

differentiation, The fundamental theorem of calculus, Integration of vector-valued

functions, Rectifiable curves.


Set functions, Intuitive idea of measure, Elementary properties of measure,

Measurable sets and their fundamental properties. Lebesgue measure of a set of real

numbers, Algebraofmeasurablesets, Borelset, Equivalentformulationofmeasurable sets in

terms of open,Closed, Fand Gsets, Nonmeasurable sets.


Measurable functions and their equivalent formulations. Properties

ofmeasurablefunctions. Approximationofameasurablefunctionbyasequenceofsimple

functions, Measurable functions as nearlycontinuous functions, Egoroff’stheorem,

Lusin’stheorem, Convergence in measure and F. Riesztheorem. Almost uniform convergence.


ShortcomingsofRiemannIntegral,LebesgueIntegralofaboundedfunctionovera set of finite

measure and its properties. Lebesgue integral as a generalization of Riemann integral,

Bounded convergencetheorem, Lebesguetheoremregarding points of discontinuities of

Riemann integrablefunctions, Integral of non-negative functions,

Fatou’sLemma,Monotoneconvergencetheorem, GeneralLebesgueIntegral, Lebesgue

convergencetheorem.


containtwoquestionsfromunitI,II,III,IVrespectivelyandthestudentsshallbeaskedtoatte

mptonequestionfromeachunit.Unitfivewillcontaineightto ten short answer type

questions without anyinternal choice covering the entire syllabus and

shallbecompulsory.

Books Recommended:

1. Walter Rudin, Principles of Mathematical Analysis (3rd edition) McGraw-Hill,

Kogakusha, 1976, International Student Edition.

2. H.L. Royden, Real Analysis, Macmillan Pub. Co., Inc. 4th Edition, New York,1993.

3. P. K. Jain and V. P. Gupta, Lebesgue Measure and Integration, New Age

International (P)LimitedPublished,New Delhi, 1986.

4. G.DeBarra, MeasureTheoryandIntegration, WileyEasternLtd., 1981.

5. R.R. Goldberg, MethodsofReal Analysis, Oxford &IBH Pub. Co. Pvt.Ltd.

6. R. G. Bartle, TheElements of Real Analysis, Wiley International Edition.


4

MTH-113 Topology-I

Max. Marks : 80

Time:3 hours


Statementsonlyof(Axiomofchoice,Zorn’slemma,WellorderingtheoremandContinnumhypothes

is).

Definition and examples of topological spaces, Neighbourhoods, Interior point and

interiorof aset , Closed set asa complementof an open set ,Adherentpointand limit

pointofaset,Closureofaset,Derivedset,PropertiesofClosureoperator,Boundary ofaset ,

Dense subsets,Interior,Exterior and boundaryoperators.

Base and subbase for a topology, Neighbourhoodsystem of a point and its

properties,BaseforNeighbourhoodsystem.

Relative(Induced) topology, Alternative methods of defining a topology in terms of

neighbourhoodsystem and Kuratowskiclosureoperator.

Comparison of topologies on aset,Intersection and union of topologies onaset.


Continuous functions, Open and closedfunctions , Homeomorphism.

Tychonoffproduct topology, Projection maps, Characterization of Product topology as

smallest topology, Continuityof a function fromaspaceinto aproduct ofspaces.

Connectedness and its characterization, Connected subsets and their properties,

Continuityandconnectedness,Connectednessandproductspaces,Components, Locally

connected spaces,Locallyconnectedand productspaces.


Firstcountable,secondcountableandseparablespaces,hereditary andtopological

property, Countabilityof a collection of disjoint open sets in separable and secondcountable

spaces, Product space as first axiom space, Lindeloftheorem. T0, T1, T2(Hausdorff)separation

axioms,their characterization and basic properties.


Compactspacesandsubsets,Compactnessintermsoffiniteintersectionproperty,

Continuity and compact sets, Basic properties of compactness, Closednessof compact

subsetandacontinuousmapfromacompactspaceintoaHausdorffanditsconsequence.

Sequentiallyandcountablycompactsets,Localcompactness,Compactnessandproduct space,

Tychonoffproduct theorem and one point compactification. Quotient topology,

Continuityoffunctionwithdomain-aspacehavingquotienttopology, Hausdorffnessof

5

quotient space.





shallbecompulsory.

Books Recommended:

1. George F. Simmons, Introduction to Topology and Modern Analysis, McGraw-

HillBook Company, 1963.

2. K.D. Joshi, Introduction to General Topology, WileyEasternLtd.

3. J. L. Kelly, General Topology, Affiliated East West Press Pvt.Ltd., NewDelhi.

4. J. R. Munkres, Toplogy,Pearson Education Asia, 2002.

5. W.J.Pervin,FoundationsofGeneralTopology,AcademicPressInc.NewYork,1964.

6


MTH-114 IntegralEquations andCalculusofVariations

Max. Marks : 80

Time:3 hours


Linearintegralequations,Somebasicidentities,Initialvalueproblemsreducedto

Volterraintegralequations, Methodsofsuccessivesubstitution and successive

approximationtosolveVolterraintegralequationsofsecondkind, Iteratedkernelsand

Neumann series for Volterraequations. Resolventkernel as a series in, Laplace

transformmethodforadifferencekernel,SolutionofaVolterraintegralequationofthe first kind.


Boundary value problems reduced to Fredholmintegral equations, Methods of

successive approximation and successive substitution to solve Fredholmequations of

second kind, Iterated kernels and Neumann series for Fredholmequations. Resolvent

kernelasasumofseries. Fredholmresolventkernelasaratiooftwoseries.Fredholmequations

with separable kernels, Approximation of a kernel by a separable kernel,

FredholmAlternative, Non homogenousFredholm equations with degenerate kernels.


Green’s function, Use of method of variation of parameters to construct the

Green’s function for a nonhomogeneous linear second order boundary value problem,

BasicfourpropertiesoftheGreen’sfunction,OrthogonalseriesrepresentationofGreen’s

function,AlternateprocedureforconstructionoftheGreen’sfunctionbyusingitsbasic

fourproperties. ReductionofaboundaryvalueproblemtoaFredholmintegralequation with

kernel as Green’s function.Hilbert-Schmidt theoryforsymmetric kernels.


Motivating problems of calculus of variations, Shortest distance, Minimum

surfaceof revolution, Branchistochroneproblem,Isoperimetric problem,

Geodesic. Fundamental lemma of calculus of variations, Euler’s equation for one

dependantfunctionanditsgeneralizationto‘n’dependantfunctionsandtohigherorder

derivatives, Conditional extremum under geometric constraints and under integral

constraints.



7



shallbecompulsory.

Books Recommended:

1. Jerri, A.J., Introduction to Integral Equations with Applications, A Wiley-

IntersciencePub.

2. Kanwal, R.P., Linear Integral Equations, Theory and Techniques, AcademicPress,

New York.

3. Gelfand,J.M.andFomin,S.V.,CalculusofVariations,PrenticeHall,NewJersy,1963.

4. Weinstock , Calculus of Variations, McGraw Hall.

5. Abdul-Majid wazwaz, A first courseinIntegral Equations, World ScientificPub.

6. David, P. and David, S.G. Stirling, Integral Equations, Cambridge UniversityPress.

7. Tricomi, F.G.,Integral Equations, Dover Pub., New York.

8


Max. Marks : 60

Time:3 hours

MTH-115-A: Programming inC (ANSIFeatures)


An overview of Programming, Programming Language, Classification. Basic

structureof a C Program, Clanguagepreliminaries.

Operators and Expressions, Two’s compliment notation, Bit - Manipulation Operators,

BitwiseAssignment Operators, MemoryOperators.


Arrays and Pointers, Encryption and Decryption.Pointer Arithmetic,Passing

PointersasFunctionArguments, AccessingArrayElementsthroughPointers, Passing

ArraysasFunctionArguments. MultidimensionalArrays. ArraysofPointers, Pointersto

Pointers.


Storage Classes –Fixed vs. Automatic Duration. Scope. Global Variables.

Definitions and Allusions. The register Specifier. ANSI rules for the Syntax and

Semantics ofthe Storage-Class Keywords. DynamicMemoryAllocation.

Structures and Unions. enumdeclarations. Passing Arguments to a Function,

Declarations and Calls, Automatic Argument Conversions, Prototyping.Pointers to

Functions.


The C Preprocessors, Macro Substitution. Include Facility. Conditional

Compilation. Line Control.

Input andOutput -Streams.Buffering.ErrorHandling. OpeningandClosinga File.

ReadingandWritingData. SelectinganI/OMethod. UnbufferedI/O. Random Access.

TheStandardLibraryforI/O.



mptonequestionfromeachunit.Unitfivewillcontaineighttoten short answer type


shallbecompulsory.

Books Recommended:

9

1. PeterA.DarnellandPhilipE.Margolis, C:ASoftwareEngineeringApproach,

NarosaPublishingHouse(SpringerInternational Student Edition) 1993.

2. Samuel P. Harkisonand GlyL. Steele Jr., C : A Reference Manual, SecondEdition,

PrenticeHall, 1984.

3. BrianW.Kernighan&DennisM.Ritchie,TheCProgrammeLanguage,SecondEdition

(ANSIfeatures) ,PrenticeHall1989.

4. BalagurusamyE : Programming in ANSI C, Third Edition, Tata McGraw-

HillPublishingCo.Ltd.

5. Byron, S. Gottfried : Theory and Problems of Programming with C, SecondEdition

(Schaum’s OutlineSeries), Tata McGraw-Hill PublishingCo.Ltd.

6. VenugopalK. R. and Prasad S. R.: Programming with C , Tata McGraw-

HillPublishingCo.Ltd.

10


PRACTICALS: Based on

MTH-115-A:Programming inC (ANSIFeatures)

Max. Marks : 40

Time4 Hours

Notes :

a) Thequestionpapershallconsistoffourquestionsandthecandidateshallbe required

to attempt anytwo questions.

b) Thecandidatewillfirstwriteprograms in C ofthequestions in theanswer-

bookandthenrunthesameonthecomputer,andthenaddtheprint-outsin

theanswer-book. Thisworkwillconsistof20marks,10marksforeach question.

c) The practical file of each student will be checked and viva-voce

examinationbaseduponthepracticalfileandthetheorywillbeconductedby

external and internal examiners jointly. This part of the practical

examination shallbeof 20marks.

11


MTH-115 :Mathematical Statistics


Max. Marks : 80

Time:3 hours

Probability: Definition of probability-classical, relative frequency, statistical and

axiomatic approach, Addition theorem, Boole’s inequality, Conditional probability and

multiplicationtheorem,Independentevents,Mutualandpairwiseindependenceofevents,

Bayes’ theorem and its applications.


Random Variable and Probability Functions: Definition and properties of random

variables, discrete and continuous random variables, probability mass and density

functions, distribution function. Concepts of bivariate random variable: joint, marginal

and conditional distributions. Mathematical Expectation: Definition and its properties.

Variance, Covariance, Moment generating function- Definitions and their properties.

Chebychev’s inequality.


Discrete distributions: Uniform, Bernoulli, binomial, Poisson and geometric

distributions with theirproperties.

Continuous distributions: Uniform, Exponential and Normal distributions with their

properties. CentralLimitTheorem (Statement only).


Statisticalestimation:Parameterandstatistic,samplingdistributionandstandarderrorof estimate.

Pointand interval estimation, Unbiasedness, Efficiency.

Testing of Hypothesis: Null and alternative hypotheses, Simple and composite

hypotheses, Critical region, Level of significance, Onetailed and two tailed tests, Two

types oferrors.

Tests of significance: Large sample tests for single mean, single proportion, difference

between two means and two proportions;





shallbecompulsory.

Books Recommended: 1. Mood, A.M., Graybill, F.A. and Boes, D.C., Mc GrawHill Book Company. 2. Freund,J.E., Mathematical Statistics, PrenticeHallofIndia.

3. Gupta S.C.andKapoorV.K.,Fundamentalsof Mathematical Statistics, S.ChandPub.,

New Delhi.

4. Speigel, M., Probabilityand Statistics, Schaum Outline Series.

12

NOTE: Eitherofthe paperMTH-215-A (Pre-requisite PaperMTH-115-A) or

MTH-215-B (Pre-requisitePaper AMT-115-B) to beselected.

Note1 :Themarks of internal assessment of eachpaper shallbesplitedasunder :

A) One class test of 10 marks. The class test will be held in the middle of thesemester.

B) Assignment&Presentation : 5 marks


65%but upto 75% : 1 marks

Morethan 75%but upto85% : 2 marks



Above95% : 5 marks

Note2:Thesyllabusofeachpaperwillbedividedintofourunitsoftwoquestionseach.

Thequestionpaperofeachpaperwillconsistoffiveunits.Eachofthefirst four

unitswillcontaintwoquestionsandthestudentsshallbeaskedtoattemptone

questionfromeachunit.Unitfiveofeachquestionpapershallcontaineightto ten short

answer type questions without any internal choice and it shall be coveringthe

entiresyllabus. As such unitfive shall becompulsory.

Note3:AsperUGCrecommendations,theteachingprogramshallbesupplementedby

tutorialsandproblemsolvingsessionsforeachtheory paper. Forthispurpose,

tutorialclassesshallbeheldforeachtheorypaperingroupsof8studentsfor half-

hourper week.

Note4 :In order to passaparticular paper,the student has to pass in theorypaper

separately.

13

M.Sc. (Mathematics) SEMESTER-II

MTH-211: Advanced Abstract Algebra-II

Max. Marks : 80

Time:3 hours

Unit -I (2 Questions) Cyclicmodules,Simpleandsemi-simplemodules,Schur’slemma,Freemodules,

Fundamentalstructuretheoremoffinitelygeneratedmodulesoverprincipalidealdomain and its

applications to finitelygeneratedabeliangroups.


NeotherianandArtinianmodulesandringswithsimplepropertiesandexamples,

Nil and Nilpotent ideals in Neotherianand Artinian rings, Hilbert Basistheorem.


HomR(R,R), Opposite rings, Wedderburn– Artintheorem, Maschk’stheorem,

Equivalent statement for left Artinian rings having non-zero nilpotent ideals, Uniform

modules, Primarymodules andNeother-Laskertheorem.


Canonical forms : Similarity of linear transformations, Invariant subspaces,

Reductiontotriangularform,Nilpotenttransformations,Indexofnilpotency,Invariantsof nilpotent

transformations, The primary decomposition theorem, Rationalcanonical forms, Jordan blocks

and Jordan forms.


containtwoquestionsfromunitI,II,III,IVrespectivelyandthestudentsshallbeaskedtoattemptoneq

uestionfromeachunit.Unitfivecontain eighttoten short answer type questions without any

internal choice covering the entire syllabus and shallbe compulsory.

Books Recommended:

1. I.N.Herstein, Topics in Algebra, WileyEasternLtd., New Delhi, 1975.

2. P.B. Bhattacharya, S.K. Jain and S.R. Nagpaul, Basic Abstract Algebra

(2nd

Edition), CambridgeUniversityPress,Indian Edition, 1997.

3. M. Artin, Algebra, Prentice-HallofIndia, 1991.

4. P.M. Cohn, Algebra,Vols.I,II& III, John Wiley&Sons, 1982, 1989, 1991.

5. I.S. Luther and I.B.S.Passi, Algebra, Vol. I-Groups, Vol. II-Rings,

NarosaPublishingHouse (Vol.I– 1996, Vol.II–1990).

6. D.S.Malik,J.N.Mordenson,andM.K.Sen,FundamentalsofAbstract Algebra,

McGrawHill,International Edition, 1997.

7. K.B. Datta,Matrix and Linear Algebra, PrenticeHallof India Pvt.,New Dlehi,2000.

8. VivekSahai and VikasBist, Algebra, NarosaPublishingHouse, 1999.

9.T.YLam,Lectures on Modules and Rings, GTMVol. 189, Springer-Verlag, 1999.


MTH-212: Real Analysis-II

Max. Marks : 80

Time:3 hours

14

Unit -I (2 Questions) Rearrangementsoftermsofaseries,Riemann’stheorem.Sequenceandseriesoffunctions,Pointwise

and uniform convergence, Cauchycriterion foruniform

convergence,Weirstrass’sMtest,Abel’sandDirichlet’stestsforuniformconvergence, Uniform

convergence and continuity, Uniformconvergenceand differentiation,

Weierstrass approximation theorem.


Powerseries,itsuniformconvergenceanduniquenesstheorem,Abel’stheorem,Tauber’s theorem.

Functionsofseveralvariables,LinearTransformations,EuclideanspaceRn,Open balls and open

sets in Rn, Derivatives in an open subset of R

n, Chain Rule, Partial

derivatives, ContinuouslyDifferentiable Mapping, Young’s andSchwarz’s theorems.


Taylor’s theorem. Higher order differentials, Explicit and implicit functions. Implicitfunction

theorem, Inverse function theorem. Change of variables, Extreme values of explicit

functions, Stationary values of implicit Functions.Lagrange’smultipliers method. Jacobian

and its properties, Differential forms, Stoke’s Theorem.


Vitali’scovering lemma, Differentiation of monotonic functions, Function of

boundedvariationanditsrepresentationasdifferenceofmonotonicfunctions, Differentiationof

indefinite integral, Fundamental theorem of calculus, Absolutely continuous functions and

theirproperties.

Lp

spaces,Convexfunctions,Jensen’sinequalities, Measurespace,GeneralizedFatou’slemma,

Measureand outermeasure, Extensionof ameasure, Caratheodoryextension theorem.



mptonequestionfromeachunit.Unitfivewillcontaineighttoten short answer type


shallbecompulsory.

Books Recommended:

1. S.C. Malik and Savita Arora, Mathematical Analysis, New Age

InternationalLimited, NewDelhi.

2. T. M. Apostol, Mathematical Analysis, NarosaPublishingHouse, NewDelhi.

3. H.L. Royden, Real Analysis, Macmillan Pub. Co., Inc. 4th Edition, New

York,1993.

4. G. DeBarra, MeasureTheoryandIntegration, WileyEasternLimited, 1981.

5. R.R. Goldberg, MethodsofReal Analysis, Oxford &IBH Pub. Co. Pvt.Ltd.

6. R. G. Bartle, TheElements of Real Analysis, Wiley International Edition.

15


MTH-213: Topology-II Max. Marks : 80

Time:3 hours

Unit -I (2 Questions) Regular,Normal,T3 andT4

separationaxioms,theircharacterizationandbasicproperties,Urysohn’slemma and

Tietzeextension theorem, Regularity and normality of a compactHausdorffspace, Complete

regularity, Complete normality, T1and T5 spaces, theircharacterization and basicproperties.


Nets :Netsintopologicalspaces,Convergenceofnets,Hausdorffnessandnets,Subnetand cluster

points, Compactness and

nets,Filters:Definitionandexamples,Collectionofallfiltersonasetasaposet,Finerfilter,

Methodsofgeneratingfiltersandfinerfilters,ultrafilteranditscharacterizations,Ultrafilter

principle, Image of filter under a function, Limit point and limit of a filter, Continuityin

termsofconvergenceoffilters,Hausdorffnessandfilters,Convergenceoffilterinaproduct

space,Compactnessandfilterconvergence,Canonicalwayofconvertingnetstofiltersand viceversa,

Stone-Cechcompactification.

Unit -III (2

Questions) Covering of a space, Local finiteness, Paracompact spaces, Michaelltheorem on

characterizationofparacompactnessinregularspaces,Paracompactnessasnormalspace,A.H.

Stonetheorem, Nagata-SmirnovMetrization theorem.

Unit -IV (2

Questions)

Embedding and metrization: Embedding lemma and Tychonoffembedding

theorem,Metrizable spaces, Urysohn’smetrization theorem.

Homotopyand Equivalence of paths, Fundamental groups, Simply connected spaces,

Coveringspaces,Fundamental groupof circle and fundamental theorem of algebra.

Note:Thequestionpaper willconsistoffiveunits.Eachofthefirstfourunitswillcontain

twoquestionsfromunitI,II,III,IVrespectivelyandthestudentsshallbeaskedto

attemptonequestionfromeachunit.Unitfivewillcontaineighttotenshortanswer

typequestionswithoutanyinternal choicecoveringtheentiresyllabusandshallbe compulsory. Books Recommended:

1. George F. Simmons, Introduction to Topologyand Modern Analysis, McGraw-

HillBook Company, 1963.

2. K.D. Joshi, Introduction to General Topology, WileyEasternLtd.

3. J. L. Kelly, General Topology,Springer Verlag,New York, 1991.

4. J. R. Munkres, Toplogy,Pearson Education Asia, 2002.

5. W.J. Pervin, Foundations of General Topology, Academic Press Inc. New

York,1964.

M.Sc. (Mathematics)

16

SEMESTER-II

MTH-214:OrdinaryDifferential Equations

Max. Marks : 80

Time:3 hours


Preliminaries :Initial valueproblem and equivalent integralequation.

-approximate solution, Cauchy-Euler construction of an -approximate solution,

Equicontinuousfamily of functions, Ascoli-Arzelalemma, CauchyPeanoexistence

theorem.

Uniqueness of solutions, Lipschitz condition, Picard-Lindelofexistence and

uniqueness theorem for dy = f(t,y), Dependence of solutions on initial conditions

dt

and parameters, Solution ofinitial-value problems byPicard method.


Sturm-LiouvilleBVPs, Sturmsseparation and comparison theorems, Lagrange’s identity and

Green’s formula for second order differential equations, Properties of eigenvalues and

eigenfunctions, Pruffertransformation, Adjointsystems, Self-adjointequations of second

order.

Linear systems, Matrix method for homogeneous first order system of linear

differentialequations,Fundamentalsetandfundamentalmatrix,Wronskianofasystem, Method

of variation of constants for a nonhomogeneous system with constant coefficients,

nth order differentialequation equivalent to a first order system.


Nonlinear differential system, Plane autonomous systems and critical points,

Classification of critical points – rotation points, foci, nodes, saddle points. Stability,

Asymptotical stabilityand unstabilityof critical points,


Almost linear systems, Liapunovfunction and Liapunov’smethod to determine stability for

nonlinear systems, Periodic solutions and Floquettheory for periodic systems, Limit

cycles, Bendixsonnon-existence theorem, Poincare-Bendixsontheorem (Statement

only),Indexof acritical point.





shallbecompulsory.

17

Books Recommended:

1. Coddington,E.A.andLevinson,N.,,TheoryofOrdinaryDifferentialEquations, Tata

McGraw Hill, 2000.

2. Ross, S.L., Differential Equations, JohnWileyandSonsInc., New York, 1984.

3. Deo, S.G., Lakshmikantham, V. and Raghavendra, V., Textbook of

OrdinaryDifferential Equations, Tata McGrawHill, 2006.

4. Boyce,W.E.andDiprima,R.C.,ElementaryDifferentialEquationsandBoundaryValue

Problems, John Wileyand Sons,Inc., NewYork, 1986,4th edition.

5. Goldberg, J. and Potter, M.C., Differential Equations – A System Approach,

PrenticeHall, 1998

6. Simmons, G.F., Differential Equations, Tata McGraw Hill, New Delhi, 1993.

7. Hartman, P., OrdinaryDifferential Equations, John Wiley&Sons, 1978.

8. Somsundram, D., Ordinary Differential Equations, A First Course, NarosaPub.Co.,

2001.

18


MTH-215-A: Object OrientedProgramming with C++

Max. Marks : 60 Time :3 hours


Basic concepts of Object-Oriented Programming (OOP).Advantages and

applications of OOP. Object-oriented languages. Introduction to C++.Structure of a

C++program. Creatingthe sourcefiles. Compilingand linking.C++ programming

basics: Input/Output, Data types, Operators, Expressions, Control

structures,Libraryfunctions.


Functions in C++ : Passing arguments to and returning values from functions,

Inlinefunctions, Defaultarguments, Function overloading.

Classesandobjects:Specifyingandusingclassandobject,Arrayswithinaclass,Arraysofobjects,

Objectas a function arguments, Friendlyfunctions, Pointers tomembers.


Constructors and destructors. Operator overloadingand type conversions.

Inheritance : Derived class and their constructs, Overriding member functions, Class

hierarchies, Public and privateinheritancelevels.

Polymorphism,Pointerstoobjects,thispointer, Pointerstoderivedclasses,virtual functions.


Streams, stream classes, Unformatted I/O operations, Formatted console I/Ooperations,

Managingoutputwith manipulators.

Classesforfilestreamoperations,OpeningandClosingafile. Filepointersand

theirmanipulations,Randomaccess. Errorhandlingduringfileoperations,Command- line

arguments. Exceptional handling.


containtwoquestionsfromunitI,II,III,IVrespectivelyandthestudentsshall

beaskedtoattemptonequestionfromeachunit.Unitfivewillcontaineightto ten short

answer type questions without anyinternal choice covering the entire syllabus and

shallbecompulsory.

Books Recommended:

1. I.S. Robert Lafore, Object Oriented Programming using C++, Waite’s Group

GalgotiaPub.

2. E.Balagrusamy,ObjectOrientedProgrammingwithC++,2ndEdition,TataMcGrawHill

Pub. Co.

3. Byron,S.Gottfried,ObjectOrientedProgrammingusingC++,Schaum’sOutlineSeries,

Tata McGrawHill Pub. Co.

4. J.N.Barakaki,ObjectOrientedProgrammingusingC++,PrenticeHallofIndia,1996.

5. Deitel and Deitel, C++: How to program, PrenticeHallofIndia

M.Sc. (Mathematics)

19

SEMESTER-II

PRACTICALS:Based onMTH215-A:ObjectOrientedProgramming with C++

Max. Marks : 40

Time4 Hours Notes :

a) Thequestionpapershallconsistoffourquestionsandthecandidateshallbe required

to attempt anytwo questions.

b) The candidate will first write programs in C++ of the questions in the

answer-book and then run the same on the computer, and then add the

print-outs in the answer-book. This work will consist of 20 marks, 10

marks foreach question.

c) The practical file of each student will be checked and viva-voce

examinationbaseduponthepracticalfileandthetheorywillbeconductedby

external and internal examiners jointly. This part of the

practicalexamination shallbeof 20marks.

20

20


MTH-215-B: OperationsResearch Techniques

Max. Marks : 80 Time:3 hours


Operations Research: Origin, definition, methodologyand scope.

Linear Programming: Formulation and solution of linear programming problems by

graphicalandsimplexmethods,Big-Mandtwophasemethods,Degeneracy,Dualityin linear

programming.


Transportation Problems: Basic feasible solutions, optimum solution by stepping stone and

modified distribution methods, unbalanced and degenerate problems, transhipment problem.

Assignment problems: Solution by Hungarian method, unbalanced problem,

caseof maximization, travellingsalesmanand crew assignment problems.


Queuingmodels:Basiccomponentsofaqueuingsystem,Generalbirth-deathequations, steady-

state solution of Markovian queuing models with single and multiple servers

(M/M/1. M/M/C, M/M/1/k, M/MC/k )

Inventorycontrol models: Economic order quantity(EOQ) model with uniform demand

andwithdifferentratesofdemandsindifferentcycles,EOQwhenshortagesareallowed, EOQwith

uniform replenishment,Inventorycontrol with pricebreaks.


Game Theory : Two person zero sum game, Game with saddle points, the rule of

dominance; Algebraic, graphical and linear programming methods for solving mixed

strategygames. Sequencingproblems:Processingofnjobsthrough2machines,njobs through 3

machines, 2 jobs through m machines, n jobs through m machines.

Note : The question paper will consist of five units. Each of the first four units will contain

two questions from unit I , II , III , IV respectivelyand the students shall be askedtoattempt

onequestionfromeachunit.Unitfivewillcontain eighttoten short

answertypequestionswithoutanyinternalchoicecoveringtheentiresyllabusandshall

becompulsory.

Books recommended:

1. Taha, H.A.,Operation Research-An introducton, PrinticeHall ofIndia.

2. Gupta, P.K. and Hira,D.S., Operations Research,S. Chand &Co.

3. Sharma, S.D., Operation Research,KedarNathRam NathPublications.

4. Sharma, J.K., Mathematical Model in Operation Research, Tata McGraw Hill.

21

21

M.Sc. (Mathematics)

SEMESTER-III

GroupA1

A11 AdvancedFunctional Analysis-I

A12 AdvancedDiscrete Mathematics-I

A13 Algebraic CodingTheory-I

A14 Wavelets-I

A15 Sobolev Spaces-I

A16 Framesand RieszBases-I

Group B1

B11 Mechanics ofSolids-I

B12 Continuum Mechanics-I

B13 Computational Fluid Dynamics-I

B14 DifferenceEquations-I

B15 Information Theory-I

GroupC1(Pre-requisite:GroupA1)

C11 TheoryofLinear Operators-I

C12 Analytical NumberTheory-I

C13 FuzzySetsand Applications-I

C14 Bases inBanachSpaces-I

C15 AlgebraicTopology-I

GroupD1(Pre-requisite:GroupB1)

D11 Fluid Dynamics-I

D12 Bio-Mechanics-I

D13 Integral Equations andBoundaryValueProblems-I

D14 Mathematics of FinanceandInsurance-I

D15 SpaceDynamics-I

Note1: Themarks of internal assessment of eachpaper shallbesplitas under :

A) Oneclasstestof10marks. Theclasstestwillbeheldinthemiddleofthe

22

22

semester.






Morethan 90%butupto95% : 4 marks

Above95% : 5 marks

Note2:Thesyllabusofeachpaperwillbedividedintofourunitsoftwoquestions each.

The question paper will consist of five units. Each of the first four

unitswillcontaintwoquestionsandthestudentsshallbeaskedtoattempt

onequestionfromeachunit.Unitfiveofeachquestionpapershallcontain

eighttotenshortanswertypequestionswithoutanyinternalchoiceandit shall be

coveringtheentiresyllabus. As such unitfiveshall becompulsory.

Note 3: As per UGC Recommendations, the teaching program shall be

supplemented by tutorials and problem solving sessions for each theory

paper. Forthispurpose,tutorialclassesshallbeheldforeachtheory paper

in groups of8 students for half-hourper week.

Note 4: Optional papers can be offered subject to availability of requisite

resources/ faculty.

23

23

M.Sc. (Mathematics) SEMESTER-III

MTH-311: Functional Analysis-I

Max. Marks : 80

Time:3 Hours


Normed linear spaces, Metric on normed linear spaces, Completion of a normed space,

Banachspaces,subspace of a Banachspace, Holder‟sand Minkowski‟sinequality,

Completeness of quotientspaces of normed linear spaces.

Completeness of lp,Lp, R

n, C

n and C[a,b]. Incompletenormed spaces.


Finite dimensional normed linear spaces and Subspaces, Bounded linear transformation,

Equivalent formulation of continuity, Spaces of bounded linear transformations, Continuous

linear functional, Conjugate spaces, Hahn-Banachextension theorem (Realand Complexform).


RieszRepresentation theorem for bounded linear functionalson Lp

and

C[a,b].Secondconjugatespaces,Reflexivespace,Uniformboundednessprinciple and its

consequences, Open mapping theorem and its application projections, Closed Graph theorem.


Equivalent norms, Weak and Strong convergence, their equivalence in finite dimensional

spaces. Weaksequential compactness, Solvability of linear

equations in Banach spaces.

Compactoperatoranditsrelationwithcontinuousoperator. Compactnessof linear transformation on

a finite dimensional space, properties of compactoperators, compactness of the limit of the

sequence of compact operators, the closed rangetheorem.

Note:Thequestion paper will consistoffive units. Each of thefirst fourunits will containtwo

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24

questions from unitI , II , III , IVrespectivelyand the students shall

beaskedtoattemptonequestion fromeachunit. Unit fivewillcontain eight to ten short answer

type questions without any internal choice coveringthe entiresyllabus and shallbecompulsory.

BooksRecommended

1. H.L. Royden,RealAnalysis,MacMillanPublishingCo., Inc., New

York,4thEdition, 1993.

2. E. Kreyszig, Introductory Functional Analysis with Applications, JohnWiley.

3. George F. Simmons, Introduction to Topology and Modern Analysis,

McGraw-HillBook Company, 1963.

4.

A.H.Siddiqi,KhalilAhmadandP.Manchanda,IntroductiontoFunctionalAnal

ysiswith Applications.

25

25


MTH-312: Partial Differential Equations and Mechanics

Max. Marks : 80

Time:3 Hours

Unit – I(2 Questions)

Method of separation of variables to solve B.V.P. associated with one dimensional heat

equation. Solution of two dimensional heat equation and two dimensional

Laplaceequation.Steadystatetemperatureinarectangular plate,in the circular disc, in a semi-

infinite plate. The head equation in semi-infinite and infinite regions. Temperature

distribution in square plate and infinite cylinder. Solution of three dimensional Laplace

equation in Cartesian, cylindrical and

sphericalcoordinates.Dirichletsproblemforasolidsphere.(Relevanttopicsfromthe books

byO‟Neil)

Unit -II(2 Questions) Method of separation of variables to solve B.V.P. associated with motion of a vibrating string.

Solution of wave equation for Semi-infinite and infinite strings. Solutionof

waveequationintwodimensions.Solutionofthreedimensionalwave equation in Cartesian,

cylindrical and spherical coordinates. Laplace transform

solutionofB.V.P..Fouriertransformsolutionof B.V.P.(Relevanttopicsfromthe books byO‟Neil)

Unit-III(2 Questions)

Kinematicsofarigidbodyrotatingaboutafixedpoint,Euler‟stheorem, general rigid body motion

as a screw motion, moving coordinate system - rectilinear moving frame, rotating frame

of reference, rotating earth. Two- dimensional rigid bodydynamics – problems

illustratingthe laws of motion and impulsivemotion.(Relevant topics from the bookof

Chorlton).

Unit -IV(2 Questions)

Moments and products of inertia, angular momentum of a rigid body,

principalaxesandprincipalmomentofinertiaofarigidbody,kineticenergyofa rigid body rotating

about a fixed point, momentalellipsoid and equimomentalsystems, coplanar mass

distributions, general motion of a rigid body. (Relevant topics from thebook ofChorlton).

26

26



beaskedtoattemptonequestion fromeach unit. Unit fivewillcontain eight to ten short answer

type questions without any internal choicecoveringthe entiresyllabus and shallbecompulsory.

BooksRecommended

1. Sneddon,I.N. ElementsofPartialDifferentialEquations,

McGrawHill, New York.

2. O‟Neil, PeterV. Advanced Engineering Mathematics,ITP.

3. F. Chorlton Textbook of Dynamics, CBS Publishers,

NewDelhi.

4. H.F. Weinberger A First Course in Partial Differential

Equations, JohnWiley&Sons, 1965.

5. M.D. Raisinghania AdvancedDifferentialequations,S.Chand

&Co.

27

27


MTH-313: ComplexAnalysis-I

Max. Marks: 80

Time:3 hours

Unit -I(2 Questions)

Function of a complex variable, continuity, differentiability. Analytic

functionsandtheirproperties,Cauchy-Riemann equationsinCartesianandpolar

coordinates.Powerseries,Radiusofconvergence,Differentiabilityofsumfunctionofapowerseries.

Branchesofmany valuedfunctionswithspecialreferencetoargz, logzand za.

Unit -II(2 Questions)

Path in aregion,Contour, Simplyand multiplyconnectedregions, Complex integration.Cauchy

theorem. Cauchy‟sintegral formula. Poisson‟sintegral formula. Higher order

derivatives. Complex integral as a function of its upper limit, Morera‟stheorem

Cauchy‟sinequality. Liouville‟stheorem. The fundamental theorem ofalgebra. Taylor‟s

theorem.

Unit -III(2 Questions)

Zeros of an analytic function, Laurent‟sseries. Isolated singularities. Cassorati-

Weierstrasstheorem,Limitpointof zeros and poles.

Maximum modulus principle, Minimum modulus principle. Schwarzlemma.Meromorphicfunctions.

The argument principle. Rouche‟stheorem,Inversefunction theorem.


Calculus of residues. Cauchy‟s residue theorem. Evaluation of integrals.Bilinear

transformations, their properties and classifications. Definitions and examples of Conformal

mappings.

Space of analytic functions and their completeness, Hurwitz‟stheorem. Montel‟s theorem.

Riemann mappingtheorem.

Note:Thequestion paper will consistof five units. Each of thefirst fourunits will containtwo




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28

BooksRecommended

1. H.A. Priestly, Introduction to Complex Analysis, Clarendon Press, Oxford,1990.

2. J.B. Conway, Functions of one Complex variable, Springer-Verlag,

Internationalstudent-Edition, NarosaPublishing House, 1980.

3. Liang-shin Hann &BernandEpstein, Classical Complex Analysis, Jones

andBartlett PublishersInternational,London, 1996.

4. E.T. Copson, An Introduction to the Theory of Functions of a ComplexVariable,

OxfordUniversityPress,London.

5. E.C. Titchmarsh, TheTheoryofFunctions, Oxford UniversityPress,London.

6. L.V.Ahlfors, ComplexAnalysis, McGrawHill, 1979.

7. S.Lang, ComplexAnalysis, Addison Wesley, 1977.

8. Mark J. Ablowitzand A.S. Fokas, Complex Variables : Introduction

andApplications, CambridgeUniversityPress, South Asian Edition, 1998.

8. S. Ponnusamy,Foundations of ComplexAnalysis, NarosaPublishing House,1997.

9. RuelV. Churchill and James Ward Brown, Complex Variables

andApplications, McGraw-Hill PublishingCompany.

29

29


MTH-314: (OptionA12) AdvancedDiscreteMathematics–I

Max. Marks : 80

Time:3 Hours

Unit -I(2 Questions)

GraphTheory–Definitionsandbasicconcepts,specialgraphs,Subgraphs, isomorphism of graphs,

Walks, Paths and Circuits, Eulerian Paths and Circuits,

HamiltonianCircuits,matrixrepresentationofgraphs,Planargraphs,Colouringof Graph.


Directed Graphs, Trees, Isomorphism of Trees, Representation of Algebraic Expressions

byBinaryTrees, SpanningTree of a Graph, Shortest PathProblem, Minimal spanningTrees,Cut

Sets, TreeSearching..


Introductory Computability Theory - Finite state machines and their

transitiontablediagrams,equivalenceoffinitestatemachines,reducedmachines, homomorphism,

finite automata acceptors, non-deterministic finite automata and

equivalenceofitspowertothatofdeterministicfiniteautomataMooreandMealy machines.


Grammars and Languages – Phrase-structure grammar rewriting rules,

derivations,sententialforms,Languagegeneratedbyagrammar,regular,context- free and context

sensitive grammars and languages, regular sets, regularexpressions and pumping lemma,

Kleene‟s theorem.



beaskedtoattemptonequestion fromeach unit. Unitfivewillcontain eight to ten short answer


30

30

BooksRecommended

1. J.P. Tremblay &R. Manohar, Discrete Mathematical Structures

withApplications to Computer Science, McGraw-HillBook Co., 1997.

2. J.L.Gersting,MathematicalStructuresforComputerScience,(3rd

edition),

Computer SciencePress,New York.

3. Seymour Lipschutz, Finite Mathematics (International edition 1983),

McGraw-HillBook Company,New York.

4. C.L.Liu, Elements of Discrete Mathematics, McGraw-HilllBook Co.

5. Babu Ram, Discrete Mathematics, VinayakPublishers and Distributors,

Delhi, 2004.

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31


MTH-314: (OptionA13) :Algebraic Coding Theory-I

Max. Marks: 80

Time:3 hours


The communication channel. TheCodingProblem. Types of Codes. Block Codes. Error-

Detecting and Error-CorrectingCodes. Linear Codes. Hamming Metric. Description

ofLinearBlock Codes byMatrices. Dual Codes.


Hamming Codes, GolayCodes, perfect and quasi-perfect codes. ModularRepresentation.

Error-Correction Capabilities of Linear Codes. Tree Codes. . Description ofLinear Tree


Bounds onMinimum DistanceforBlockCodes. PlotkinBound. HammingSphere Packing

Bound.Varshamov-Gilbert – Sacks Bound. Bounds for Burst-Error

Detectingand Correcting Codes.


Convolutional Codes and Convolutional Codes by Matrices. Standard Array.Bounds on

minimum distance for Convolutional Codes. V.G.S. bound. BoundsforBurst-

ErrorDetectingandCorrectingConvolutionalCodes.TheLee metric, packingbound for

Hammingcodew.r.t.Leemetric.




type questions without any internal choice coveringthe entiresyllabus and shallbe

compulsory.

32

32

BooksRecommended

1. Ryamond Hill, A First Coursein CodingTheory,Oxford

UniversityPress,1986.

2. ManYoungRhee, ErrorCorrectingCodingTheory,McGrawHillInc.,

1989.

3. W.W.Petersonand E.J. Weldon, Jr., Error-CorrectingCodes. M.I.T.

Press, CambridgeMassachuetts, 1972.

4. E.R. Berlekamp, Algebraic CodingTheory, McGraw HillInc., 1968.

5. F.J. Macwilliamsand N.J.A. Sloane, Theoryof ErrorCorrectingCodes,

North-HolandPublishing Company.

6. J.H. Van Lint, Introduction to Coding Theory, Graduate Texts

inMathematics, 86, Springer, 1998.

7. L.R. Vermani, ElementsofAlgebraicCodingTheory, Chapman and Hall,1996.

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33


MTH-314: (OptionA14):Wavelets –I

Max. Marks : 80

Time:3 hours


Definition and Examples ofLinear Spaces,Basesand Frames, Normed Spaces,TheLp-Spaces,

Definitionand Examples ofInner Product Spaces, Hilbert Spaces,Orthogonal and Orthonormal Systems.


TrigonometricSystems,Trigonometric Fourier Series, ConvergenceofFourierSeries, Generalized

Fourier Series.

Fourier Transforms in L1(R) and L

2(R), Basic Properties of FourierTransforms, Convolution,

PlancherelFormula,PoissionSummation Formula, SamplingTheorem andGibbs Phenomenon.

Unit -III (2 Questions) Definition and Examples of Gabor Transforms,Basic Properties ofGabor Transforms.

Definition and Examples of Zak Transforms,BasicProperties ofZak Transforms,Balian-

LowTheorem.

Unit-IV (2 Questions)

Wavelet Transform, Continuous Wavelet Transforms, Basic Properties of Wavelet

Transforms, Discrete Wavelet Transforms, Partial Discrete Wavelet Transforms, Maximal

Overlap Discrete Wavelet Transforms.



beaskedtoattemptonequestion fromeach unit. Unit fivewillcontain eight to ten short answer type

questions without any internal choice coveringthe entiresyllabus and shallbecompulsory.

BooksRecommended

1. K. Ahmad andF. A. Shah,Introduction to Wavelet AnalysiswithApplications, Anamaya

Publishers, 2008.

2. Eugenio Hernandez and Guido Weiss, A first Course on Wavelets, CRC Press, New York,

1996.

3. C.K. Chui, AnIntroduction to Wavelets, AcademicPress, 1992.

4. I.Daubechies, TenLecturesonWavelets,CBS-NSFRegionalConferences in Applied

Mathematics,61, SIAM, 1992.

5. Y. Meyer, Wavelets, Algorithms and Applications (translated by R.D. Rayan, SIAM, 1993).

34

34


MTH-314: (OptionA15):Sobolev Spaces-I

Max. Marks : 80

Time:3 hours

Unit-I (2 Questions)

Distributions– Test function spaces anddistributions, convergence

distributional derivatives.

Unit-II (2 Questions)

Fourier Transform–L1-Fourier transform. Fourier transform ofa

Gaussian,L2-Fourier transform,Inversion formula. L

p-Fourier transform,

Convolutions.

Unit-III (2 Questions) SobolevSpaces - Thespaces Wl,p( ) and Wl,p( ). Their simple characteristic properties,densityresults. Min and Maxof Wl,p– functions. The spaceH1( ) and its properties, densityresults.


ImbeddingTheorems- Continuous and compact imbeddings of Sobolev

spaces intoLebesguespaces. SobolevImbedding Theorem, Rellich–Kondrasov

Theorem.

Note:Thequestion paper will consistoffive units. Each of thefirst fourunits will

containtwo questions from unitI , II , III , IVrespectivelyand the students

shall beaskedtoattemptonequestion fromeach unit. Unit fivewillcontain

eight to ten short answer type questions without any internal choice

coveringthe entiresyllabus and shallbecompulsory.

35

35

BooksRecommended

1. R.A. Adams, Sobolev Spaces, AcademicPress,Inc. 1975.

2. S. Kesavan, Topics in Functional Analysis and Applications,

WileyEasternLimited, 1989.

3.

A.Kufner,O.JohnandS.Fucik,FunctionSpaces,NoordhoffInternationalPubli

shing,Leyden, 1977.

4. A. Kufner, Weighted Sobolev Spaces, John Wiley&SonsLtd., 1985.

5. E.H.Lieband M.Loss,Analysis, NarosaPublishingHouse, 1997.

6. R.S. Pathak, A Course in Distribution Theory and Applications,

NarosaPublishingHouse, 2001.

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36


MTH-314: (Option B11): Mechanics of Solids-I

Max. Marks: 80

Time:3 hours


Cartesian tensors of different order. Properties of tensors. Symmetric and skew-

symmetrictensors.Isotropictensorsofdifferentordersandrelationbetween

them.Tensorinvariants.Eigen-valuesandeigenvectorsofasecondordertensor.

Scalar,vector,tensorfunctions.Commanotation.Gradiant,divergenceandcurlofatensor field.

Unit-II(2 Questions)

Analysis of Stress : Stress vector, stress components. Cauchy equations of equilibrium.

Stresstensor. Symmetryofstresstensor. StressquadricofCauchy. Principal stress and invariants.

Maximum normal and shear stresses. Mohr‟s diagram.Examples of stress.

Unit-III (2 Questions)

Analysis of Strain : Affine transformations. Infinitesimal affine deformation.

Geometricalinterpretationofthecomponentsofstrain. StrainquadricofCauchy. Principal strains

andinvariants. General infinitesimal deformation. Saint-Venant‟sequationsofCompatibility.

Finitedeformations. Examplesofuniformdilatation, simple extension and shearingstrain.


EquationsofElasticity: GeneralizedHooke‟slaw. Hooke‟slawinmediawith

oneplaneofsymmetry,orthotroicandtransverselyisotropicmedia,Homogeneous isotropic media.

Elastic moduli for isotropic media. Equilibrium and dynamic

equationsforanisotropicelasticsolid. Beltrami-Michellcompatibilityequations. Strain

energyfunction. Clapeyron‟s theorem.Saint-Venant‟sPrinciple.



37

37


type questions without any internal choicecoveringthe entiresyllabus and shallbecompulsory.

1. I.S. Sokolnikoff, Mathematical Theory of Elasticity, Tata McGraw HillPublishingCompany Ltd., New Delhi, 1977.

2. TeodarM. Atanackovicand ArdeshivGuran, Theory of Elasticity forScientists

and EngineersBirkhausev, Boston, 2000.

3. Y.C.Fung, FoundationsofSolidMechanics, PrenticeHall,New,Delhi,1965.

4. Jeffreys, H. , Cartesian tensor.

5. ShantiNarayan, Text Book ofTensors, S. chand&co.

6. Saada, A.S., Elasticity-Theoryandapplications, PergamonPress, NewYork.

7. A.E.H. Love, A Treatise on a Mathematical Theoryof Elasticity,

DoverPub.,New York.

8. D.S. Chandersekhariahand L. Debnath, ContinumMechanics, AcademicPress,

1994.

38

38


MTH-314: (Option B12) : ContinuumMechanics-I

Max. Marks : 80

Time:3 hours


Cartesian tensors andtheir elementaryproperties.

Analysis of Stress : Stress vector, stress components, Cauchy equations of

equilibrium, stress tensor,symmetryof stress tensor, principal stressand invariants,

maximumnormal and shear stresses.


Analysis of Strain : Affine transformation, infinitesimal affine deformation,

geometric interpretation of the components of strain, principal strains and

invariants, general infinitesimal deformation, equations of compatibility,

finitedeformations.

Unit-III(2 Questions)

EquationsofElasticity: GeneralizedHooke‟slaw,Hooke‟slawinmediawith one

plane of symmetry, orthotropic andtransversely isotropic

media,

Homogeneousisotropicmedia,elasticmoduliforisotropicmedia,equilibriumand

dynamicalequationsforanisotropicelasticsolid,Beltrami–Michellcompatibility

equations.


Two-DimensionalElasticity: Planestrain,planestress,generalizedplanestress,

Airystressfunction,problemofhalf-planeloadedbyuniformlydistributedload, problem

of thick wall tube under the action of internal and external pressures, Rotatingshaft.

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39






BooksRecommended

1. S. Valliappan, Continuum Mechanics, Fundamentals, Oxford &IBH

PublishingCompany,1981.

2. G.T. Maseand G.E. Mase, Continuum Mechanics for Engineers, CRC Press,

1999.

3. Atanackovic, T.M. A. Guran, Theory of Elasticity for scientists

andEngineers,Birkhausev,2000.

4. D.S. Chandrasekharaiah, Continuum Mechanics, Academic Press,

PrismBooks Pvt.Ltd.,Bangalore.

5. L.S. Srinath, Advanced Mechanics of Fields, Tata McGraw-Hill, NewDelhi.

40

40


MTH-314: (Option B13):Computational FluidDynamics-I

Max. Marks : 80

Time:3 hours

Unit-I (2

Questions)

BasicequationsofFluid

dynamics.Analyticaspectsofpartialdifferential equations- classification,

boundary conditions, maximum principles, boundary layer theory.

Unit-II (2

Questions)

Finite difference and Finite volume discretizations. Vertex-

centred discretization. Cell-centreddiscretization. Upwind discretization.

Nonuniformgrids in one dimension.

Unit-III (2

Questions)

Finitevolumediscretizationofthestationaryconvection-

diffusionequationinonedimension. Schemesofpositivetypes. Defectcorrection. Non-

stationary convection-diffusion equation. Stability definitions.The discrete

maximum principle.

Unit-IV (2

Questions)

Incompressible Navier-Stokes equations. Boundaryconditions. Spatial

discretization on collocated and on staggered grids. Temporal discretization on

staggeredgridand on collocatedgrid.

Note:Thequestion paper will consistoffive units. Each of thefirst fourunits

will containtwo questions from unitI , II , III , IVrespectivelyand

the students shall beaskedtoattemptonequestion fromeach unit.

Unit fivewillcontain eight to ten short answer type questions

without any internal choice coveringthe entiresyllabus and

41

41

shallbecompulsory.

BooksRecommended

1. P. Wesseling: Principles of Computational Fluid Dynamics, SpringerVerlag,

2000.

2. J.F. Wendt,J.D.Anderson, G. Degrezand E.Dick : Computational Fluid Dynamics :

AnIntroduction, Springer-Verlag, 1996.

3. J.D. Anderson, Computational Fluid Dynamics: Thebasics with applications,

McGraw- Hill, 1995.

4. K. Muralidherand T.Sundarajan : Computational Fluid Flow andHeatTransfer,

NarosaPub. House.

5. T.J. Chung:Computational Fluid Dynamics, CambridgeUni. Press.

6. J.N. Reddy:An introduction to the Finite Element

Methods,McGrawHillInternationalEdition, 1985.

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42


MTH-314: (Option B14):DifferenceEquations-I

Max. Marks : 80

Time:3 hours

Unit-I (2

Questions)

Introduction, Difference Calculus – The difference operator,

Summation, Generating functions andapproximate summation.

LinearDifferenceEquations - Firstorderequations. Generalresultsforlinear

equations.

Unit-II (2

Questions)

Equations with constant coefficients.Applications Equations with variable

coefficients.

StabilityTheory- Initialvalueproblemsforlinearsystems. Stabilityoflinear

systems.

Unit-III (2

Questions)

Stabilityof nonlinear systems. Chaoticbehaviour.

Asymptotic methods - Introduction, Asymptotic analysis of sums.

Linear equations. Nonlinearequations.

Unit-IV (2

Questions)

Self-adjointsecond order linear equations –Introduction. SturmianTheory.

Green‟sfunctions. Disconjugacy. TheRiccatiEquations. Oscillation.

Note:Thequestion paper will consistoffive units. Each of thefirst fourunits willcontaintwo questions from unitI , II , III , IVrespectivelyand the students shall beaskedtoattemptonequestion fromeach unit. Unit fivewillcontain eight to ten short answer type questions without any internal choice coveringthe entiresyllabus and shallbecompulsory.

43

43

BooksRecommended

1. Walter G. Kelleyand Allan C. Peterson-DifferenceEquations. An

Introduction with Applications, AcademicPressInc., HarcourtBrace

Joranovich Publishers, 1991.

2. Calvin Ahlbrandt and Allan C. Peterson. DiscreteHamiltonian Systems,

DifferenceEquations, Continued Fractions and Riccatti Equations. Kluwer,

Boston, 1996.

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44


MTH-314: (Option B15):InformationTheory-I

Max. Marks: 80

Time:3 hours


Measure of Information – Axioms for a measure of uncertainty. The Shannon

entropyanditsproperties. Jointandconditionalentropies. Transformationandits

properties.AxiomaticcharacterizationoftheShannonentropyduetoShannonand

Fadeev.


Noiseless coding-Ingredients of noiseless coding problem. Uniquely

decipherable codes. Necessary and sufficient condition for the existence of

instantaneous codes. Construction ofoptimal codes.


Discrete Memoryless Channel - Classification of channels. Information

processedbyachannel.Calculationofchannelcapacity. Decodingschemes. The

idealobserver.ThefundamentaltheoremofInformationTheoryanditsstrongand weak

converses.


Continuous Channels - The time-discrete Gaussian channel. Uncertainty of an

absolutelycontinuous random variable. The converseto the codingtheorem for

time-discrete Gaussian channel. The time-continuous Gaussian channel. Band-

limited channels.





45

45


BooksRecommended

1. R. Ash,Information Theory,IntersciencePublishers, NewYork, 1965.

2. F.M. Reza, An Introduction to Information Theory, MacGraw-Hill

BookCompany Inc., 1961.

3. J. AczeladnZ. Daroczy, On Measures of Information and

theirCharacterizations, AcademicPress, New York.

46

46


MTH-315: (OptionC11): Theory ofLinearOperators-I

Max. Marks : 80

Time:3 hours


Spectral theory in normed linear spaces, resolventset and spectrum, spectral

properties of bounded linear operators, Properties of resolventand spectrum,

Spectral mapping theorem for polynomials, Spectral radius of a bounded linear

operator ona complexBanach space.


Elementary theory of Banachalgebras. Properties of Banachalgebras.

General properties of compact linear operators. Spectral properties of

compact linear operators on normed spaces.


Behaviourof compact linear operators with respect to solvability of

operator equations. Fredholmtype theorems. Fredholmalternative theorem.

Fredholmalternative for integral equations. Spectral properties of bounded self-

adjointlinear operators on a complexHilbert space.


Positiveoperators,Monotonesequencetheoremforboundedself-adjointoperatorson a

complex Hilbert space. Square roots of a positive operator. Projection operators,

Spectral family of a bounded self-adjointlinear operator and its properties.




eight to ten short answer type questions without any internal

choicecoveringthe entiresyllabus and shallbecompulsory.

47

47

BooksRecommended

1. E. Kreyszig, Introductory Functional Analysis with Applications, John-

Wiley&Sons, New York, 1978.

2. P.R. Halmos, Introduction to Hilbert Space and the Theory of

SpectralMultiplicity, Second-Edition, ChelseaPublishingCo., New York, 1957.

3. N. Dunford and J.T. Schwartz, Linear Operators -3 Parts,

Interscience/Wiley, NewYork, 1958-71.

4. G. Bachman and L. Narici, Functional Analysis, Academic Press, York,1966.

48

48


MTH-315: (Option C12): Analytical NumberTheory-I

Max. Marks : 80

Time:3 hours


Distributionofprimes. Fermat‟sandMersennenumbers, Fareyseriesandsome

resultsconcerningFareyseries. Approximationofirrationalnumbersbyrations,

Hurwitz‟stheorem. Irrationality of e and .(Relevant portions from the Books

Recommendedat Sr. No.1 and 4)


Diophantine equations ax + by = c, x2+y2 = z2 and x4+y4 = z4.

The representationofnumberbytwoorfoursquares. Warig‟sproblem,Foursquare

theorem, thenumbersg(k)& G(k). Lower bounds forg(k)&G(k). Simultaneous

linearandnon-linearcongruencesChineseRemainderTheoremanditsextension.

(Relevant portions from theBooksRecommendedat Sr. No. 1 and 4)

Unit-III (2 Questions) Quadraticresiduesandnon-residues. Legender‟sSymbol. GaussLemma

anditsapplications. Quadratic Law of ReciprocityJacobi‟sSymbol. The arithmetic

inZn. ThegroupUn .Congruenceswithprimepowermodulus,primitiverootsand their

existence. (Scope asin Book at Sr. No. 5)


ThegroupUpn (p-odd)andU2

n. ThegroupofquadraticresiduesQn , quadratic

residuesforprimepowermoduliandarbitrarymoduli. ThealgebraicstructureofUn

and Qn . (Scope as in Book at Sr. No. 5)


containtwo questions from unitI , II ,III , IV respectivelyand the students


49

49



BooksRecommended

1. Hardy, G.H.and Wright,E.M., AnIntroduction to theTheoryofNumbers

2. Burton, D.M., ElementaryNumber Theory.

3. McCoy, N.H.,TheTheoryofNumber byMcMillan.

4. Niven, I. And Zuckermann, H.S., An Introduction to the Theory

ofNumbers.

5. Gareth,A.JonesandJ.MaryJones,ElementaryNumberTheory,SpringerEd. 1998.

50

50


MTH-315: (OptionC13): Fuzzy Sets andtheirApplications -I

Max. Marks : 80

Time:3 hours


DefinitionofFuzzySet,ExpandingConceptsofFuzzySet,StandardOperationsofFuzzySet,

FuzzyComplement,FuzzyUnion,Fuzzy Intersection,OtherOperationsin FuzzySet, T-

norms and T-conorms. (Chapter1of [1])


Product Set, Definition of Relation, Characteristics of Relation, Representation

Methods of Relations, Operations on Relations, Path and Connectivityin Graph,

Fundamental Properties, Equivalence Relation, CompatibilityRelation, Pre-order

Relation, Order Relation, Definition and Examples of Fuzzy Relation, Fuzzy

Matrix,OperationsonFuzzyRelation,CompositionofFuzzyRelation, -cutof

Fuzzy Relation, Projection and Cylindrical Extension, Extension by Relation,

ExtensionPrinciple,ExtensionbyFuzzyRelation,FuzzydistancebetweenFuzzy

Sets.(Chapter2,3 of[1])


GraphandFuzzyGraph,FuzzyGraphandFuzzyRelation, -cutofFuzzyGraph,

Fuzzy Network, Reflexive Relation, Symmetric Relation, Transitive Relation,

Transitive Closure, Fuzzy Equivalence Relation, Fuzzy Compatibility Relation,

Fuzzy Pre-order Relation, Fuzzy Order Relation, Fuzzy Ordinal Relation,

DissimilitudeRelation,FuzzyMorphism,ExamplesofFuzzyMorphism.(Chapter

4 of [1])


Interval,FuzzyNumber,Operation ofInterval, Operation of - cutInterval,

Examples of FuzzyNumber Operation,, Definition ofTriangularFuzzyNumber,

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51

Operation ofTriangularFuzzyNumber, Operation of General FuzzyNumbers,

Approximation ofTriangular FuzzyNumber, Operations of Trapezoidal Fuzzy

Number, Bell Shape FuzzyNumber.

Function with Fuzzy Constraint, Propagation of Fuzziness by Crisp

Function,FuzzifyingFunctionofCrispVariable,MaximizingandMinimizingSet,

Maximum Value of Crisp Function, Integration and Differentiation of Fuzzy

Function. (Chapter5,6 of [1])


containtwo questions from unitI , II , III , IVrespectivelyand the studentsshall

beaskedtoattemptonequestion fromeach unit. Unit fivewillcontaineight to ten

short answer type questions without any internal choice coveringthe

entiresyllabus and shallbecompulsory.

BooksRecommended

1.

KwangH.Lee,FirstCourseonFuzzyTheoryandApplications,SpringerInterna

tional Edition, 2005.

2. H.J.Zimmerman,FuzzySetTheoryanditsApplications,AlliedPublishersLtd.,

New Delhi, 1991.

3. John Yen, Reza Langari, Fuzzy Logic - Intelligence, Control

andInformation, Pearson Education.

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52


MTH-315: (OptionC14):Bases inBanachSpaces–I

Max. Marks : 80

Time:3 hours

Unit-I (2Questions)

Hamelbases. Thecoefficientfunctionalsassociatedtoabasis. Schauder bases.

Bounded bases and normalized bases. Examples of bases in concrete

Banach spaces.

Unit-II (2Questions)

Biorthogonalsystems. Associatedsequencesofpartialsumoperators -E-

complete,regularandirregularbiorthogonalsystems. Characterizationsofregular

biorthogonalsystems. Basicsequences. Banachspace(separableornot)andbasic

sequence.

Unit-III (2Questions)

Some types of linear independence of sequences - Linearly independent(finitely) W-

linearly independent and minimal sequences of elements in Banach spaces. Their

relationship together with examples and counter-examples.

Problemofuniquenessofbasis - Equivalentbases,Stabilitytheoremsof Paley-

Winertype. Blockbasicsequenceswithrespecttoasequence(basis)and their existence.

Bessaga-Pelczynski theorem.


Properties of strong duality. Weak bases and weak Schauderbases in a

Banachspace. Weakbasistheorem. Weak*basesinconjugatespacesandtheir

properties.

Shrinking bases and boundedlycomplete bases together with their

relationship.


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53





BooksRecommended

1. Jurgt.Marti, IntroductiontoTheoryofBases,

SpringerTractsinNaturalPhilosophy18, 1969.

2. IvanSinger,BasesinBanachSpaces I,Springer-Verlag, Berlin,Vol.1541970.

3. Ivan Singer,Bases inBanachSpacesII, Springer-Verlag,Berlin, 1981.

4. J.LinderstraussandI.Tzafriri,ClassicalbanachSpaces(Sequencespaces),

SpringerVerlag, Berlin, 1977.

5. IvanSinger,BestApproximationinNormedLinearSpacesbyElementsofLinear

Spaces, Springer-Verlag,Berlin, 1970.

54

54


MTH-315: (OptionC15): Algebraic Topology-I

Max. Marks : 80

Time:3 hours


Fundamental group function, homotopyof maps between topological

spaces, homotopyequivalence, contractible and simple connected spaces,

fundamentalgroups ofS1, and S

1xS

1 etc.

Unit-II (2

Questions)

Calculation of fundamental group of Sn, n >1 using Van

Kampen‟stheorem, fundamental groups of a topological group. Brouwer‟sfixed

point theorem,fundamentaltheoremofalgebra,vectorfieldsonplanersets. Frobenius

theorem for3 x3 matrices.

Unit-III (2

Questions)

Covering spaces, uniquepath lifting theorem,covering

homotopytheorems, group of covering transformations, criterianof lifting of

maps in terms of fundamental groups, universal covering, its existence, special

cases of manifolds and topological groups.

Unit-IV (2

Questions)

Singular homology, reduced homology, EilenbergSteenrodaxioms of

homology (no proof for homotopyinvariance axiom, excision axiom and exact

sequence axiom) and their application, relation between fundamental group and

first homology.

55

55

CalculationofhomologyofSn,Brouwer‟sfixedpointtheoremforf:E

n

En,applicationspheres,vectorfields,Mayer-Vietorissequence(withoutproof)andits

applications.






BooksRecommended

1. JamesR.Munkres,Topology–AFirstCourse,PrenticeHallofIndiaPvt.Ltd., New

Delhi, 1978.

2. MarwinJ. Greenberg and J.R. Harper, Algebraic Topology – A FirstCourse,

Addison-WesleyPublishingCo., 1981.

3. W.S.Massey,AlgebraicTopology–

AnIntroduction,Harcourt,BraceandWorldInc. 1967, SV, 1977.

56

56


MTH-315: (Option D11):FluidDynamics-I

Max. Marks : 80

Time:3 hours


Kinematics - Velocity at a point of a fluid. Eulerian and Lagrangian

methods. Streamlines,pathlinesandstreaklines. Velocitypotential. Irrotationaland

rotational motions. Vorticity and circulation. Equation of continuity.

Boundarysurfaces. Acceleration atapoint ofafluid. Components of acceleration in

cylindrical and spherical polar co-ordiantes.


Pressureatapointofamovingfluid. Euler‟sandLagrange‟sequationsofmotion.Equations

of motion in cylindricaland spherical polar co-ordinates.

Bernoulli‟sequation. Impulsive motion. Kelvin‟scirculation theorem. Vorticity

equation. Energyequation forincompressible flow.


Acyclic and cyclic irrotationalmotions. Kinetic energy of irrotationalflow.

Kelvin‟sminimumenergytheorem.Meanpotentialoverasphericalsurface.K.E.ofinfinitefl

uid.Uniquenesstheorems.Axiallysymmetricflows. Liquidstreaming part a fixed sphere.

Motion of a sphere through a liquid at rest at infinity.

Equationofmotionofasphere.K.E.generatedbyimpulsivemotion. Motionoftwo

concentricspheres.


Three-dimensional sources, sinks and doublets. Images of sources, sinks and

doubletsinrigidimpermeableinfiniteplaneandinimpermeablesphericalsurface.

Twodimensionalmotion,Kineticenergyofacyclicandcyclicirrotationalmotion. Useof

cylindricalpolarco-ordinates.Streamfunction.Axisymmetricflow.Stoke‟s stream

function. Stoke‟sstream function ofbasic flows.


57

57





BooksRecommended

1.W.H.BesaintandA.S.Ramasey,ATreatiseonHydromechanics,PartII,CBS Publishers,

Delhi, 1988.

2. F. Chorlton, Text Book of Fluid Dynamics, C.B.S. Publishers, Delhi, 1985

3.O‟Neill,M.E.andChorlton,F., IdealandIncompressibleFluid Dynamics,

EllisHorwoodLimited, 1986.

4. S.W. Yuan, Foundations of Fluid Mechanics, Prentice Hall of India

PrivateLimited, NewDelhi, 1976.

5. R.K.Rathy,AnIntroductiontoFluid Dynamics,OxfordandIBHPublishingCompany,

New Delhi, 1976.

6. G.K.Batchelor,AnIntroductiontoFluidMechanics,FoundationBooks,NewDelhi,

1994.

58

58


MTH-315: (Option D12): Biomechanics-I

Max. Marks : 80

Time:3 hours


Newton‟sequations of motion. Mathematicalmodeling. Continuum

approach. Segmental movement and vibrations. Lagrange‟sequations. Normal

modes ofvibration. Decouplingof equations of motion.


Flow around an airfoil. Flow around bluff bodies. Steady state aeroelastic

problems. Transient fluid dynamicsforces duetounsteadymotion. Flutter. Kutta-

Joukowskitheorem. Circulation and vorticity in the wake. Vortex system

associated with a finite wingin nonsteadymotion. Thin wingin steadyflow.


Blood flow in heart, lungs, arteries, and veins. Field equations and

boundaryconditions. PulsatileflowinArteries.

Progressivewavessuperposedonasteadyflow.

Reflectionandtransmissionofwavesatjunctions. Velocityprofileofasteadyflowinatube.

Steadylaminarflowinanelastictube. Velocity profileofPulsatileflow.

TheReynoldsnumber,Stokesnumber,andWomersleynumber. Systematicblood pressure.

Flow in collapsibletubes.


Micro-and macrocirculationRheological properties of blood. Pulmonary

capillaryblood flow. Respiratorygas flow. Intractionbetween convection and

diffusion. Dynamics of theventilation system.

Note:Thequestion paper will consistoffive units. Each of thefirstfourunits will





BooksRecommended

1. Y.C.Fung,Biomechanics:Motion,Flow,StressandGrowth,Springer-Verlag, New

59

59

YorkInc., 1990.


MTH-315: (Option D13): Integral Equations andBoundary ValueProblems-I

Max. Marks: 80

Time:3 hours


Definition of integral equations and their classification. Eigenvalues and

eigenfunctions. Convolutionintegral. Fredholmintegralequationsofthesecond

kindwithseparablekernelsandtheirreductiontoasystemofalgebraicequations.

Fredholmalternative. Fredholmtheorem, Fredholmalternative theorem. An

approximate method.


Method of successive approximations. Iterative scheme for Fredholmintegral

equationsofthesecondkind. Neumannseries,iteratedkernels,resolventkernel,

IterativeschemeforVolterraintegralequationsofthesecondkind. Conditionsof

uniform convergenceanduniqueness of series solution.


Classical Fredholmtheory. Fredholm‟sfirst, second and third theorems.

Applications integral equations to ordinary differential equations. Initial value

problems transformed to volterraintegral equations. Boundary value problems

equivalent to Fredholmintegral equations. Diracdelta function.


Construction of Green‟sfunction for a BVP associated with a nonhomogeneous

ordinary differential equation of second order with homogeneous boundary

conditionsbyusingthemethodofvariationofparameters,Basicfourpropertiesof

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60

theGreen‟sfunction.

AlternativeprocedureforconstructionofaGreen‟sfunctionbyusingitsbasicfourproperti

es. Green‟sfunctionapproachforIVPforsecondorder equations. Green‟sfunction

for higher order differential equations. Modified Green‟s function.






BooksRecommended

1. Kanwal, R.P., Linear Integral Equations – Theory and Technique, AcademicPress,

1971.

2. Kress, R., LinearIntegral Equations, Springer-Verlag,New York, 1989.

3. Jain,D.L.andKanwal,R.P.,MixedBoundaryValueProblemsinMathematicalPhysics.

4. Smirnov,V.I., IntegralEquationsandPartialDifferentialEquations,Addison- Wesley,

1964.

5. Jerri, A.J., Introduction to Integral Equations with Applications,

SecondEdition,John-Wiley&Sons, 1999.

6. Kanwal, R.P., LinearIntegration Equations, (2ndEd.)Birkhauser,Boston, 1997.

61

61


MTH-315: (Option D14):Mathematics forFinanceandInsurance-I Max.

Marks : 80

Time:3 hours


Financial Management – AN overview. Nature and Scope of Financial

Management. Goals of Financial Management and main decisions of financial

management. Differencebetween risk, speculation and gambling.

TimevalueofMoney- Interestrateanddiscountrate. Presentvalueand

futurevalue-discretecaseaswellascontinuouscompoundingcase. Annuitiesand

its kinds.


Meaningofreturn. ReturnasInternalRateofReturn (IRR). NumericalMethods like

Newton Raphson Method to calculate IRR. Measurement of returns under

uncertainty situations. Meaningofrisk. Differencebetweenriskanduncertainty.

Typesofrisks. Measurementsofrisk. CalculationofsecurityandPortfolioRisk

andReturn-MarkowitzModel. Sharpe‟sSingleIndexModel-SystematicRiskand

UnsystematicRisk.


Taylor series and Bond Valuation. Calculation of Duration and Convexity of

bonds.InsuranceFundamentals–Insurancedefined. Meaningofloss. Chancesofloss,

peril,hazard,andproximatecauseininsurance. Costsandbenefitsofinsuranceto

thesocietyandbranchesofinsurance-lifeinsuranceandvarioustypesofgeneral insurance.

Insurable lossexposures-featureofalossthat is ideal forinsurance.


Life Insurance Mathematics – Construction of MoralityTables. Computation

ofPremium ofLifeInsurance fora fixed duration and forthe whole life.Determination

of claims for General Insurance – Using

PoissonDistribution and NegativeBinomial Distribution–the

PolyaCase.DeterminationoftheamountofClaimsofGeneralInsurance–

CompoundAggregateclaimmodelanditsproperties,andclaimsofreinsurance.

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62

Calculationofa compound claimdensityfunction F,recursiveand approximate formulae

forF.






BooksRecommended

1. AswathDamodaran,CorporateFinance-

TheoryandPractice,JohnWiley&Sons,Inc.

2. John C. Hull, Options, Futures, and Other Derivatives, Prentice-Hall ofIndian

PrivateLimited.

3. Sheldon M. Ross, An Introduction to Mathematical Finance,

CambridgeUniversityPress.

4. Mark S. Dorfman, Introduction to Risk Management and Insurance,

PrenticeHall, EnglwoodCliffs, New Jersey.

5. C.D. Daykin, T. Pentikainenand M. Pesonen, Practical Risk Theory

forActuaries, Chapman&Hall.

6. SalihN. Neftci, An Introduction to the Mathematics of

FinancialDerivatives, AcademicPress,Inc.

7. RobertJ.ElliottandP.EkkehardKopp,MathematicsofFinancialMarkets,

Sprigner-Verlag, New YorkInc.

8. Robert C. Merton, Continuous– Time Finance, BasilBlackwellInc.

9. Tomasz Rolski, HanspterSchmidli, Volker Schmidt and JozefTeugels,

Stochastic Processes for Insurance and Finance, John Wiley &Sons

Limited.

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63


MTH-315: (Option D15): SpaceDynamics-I

Max. Marks : 80

Time:3 hours


BasicFormulaeofaspherical triangle -Thetwo-bodyProblem :TheMotion of the

CenterofMass. Therelativemotion. Kepler‟sequation. SolutionbyHamilton Jacobi

theory.


TheDetermination ofOrbits–Laplace‟sGaussMethods. TheThree-Bodyproblem–GeneralThreeBodyProblem. RestrictedThreeBodyProblem.


Jacobi integral. CurvesofZero velocity. Stationarysolutions and theirstability.

The n-Body Problem – The motion of the centreof Mass. Classical

integrals.


Perturbation – Osculating orbit, Perturbing forces, Secular &

Periodic perturbations.

Lagrange‟sPlanetoryEquationsintermsofpertainingforcesandin terms of aperturbed

Hamiltonian.

Note:Thequestion paper will consistof five units. Each of thefirst fourunits will





BooksRecommended

64

64

1. J.M. A. Danby, Fundamentals of Celestial Mechanics. The

MacMillanCompany, 1962.

2. E. Finlay, Freundlich, Celestial Mechanics. The MacMillan Company,

1958.

3.

TheodoreE.Sterne,AnIntroductionofCelestialMechanics,IntersciencesPubli

shers. INC., 1960.

4. ArigeloMiele, Flight Mechanics – Vol . 1 - Theory of Flight Paths,

Addison-WesleyPublishingCompany Inc., 1962.

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65

M.Sc. (Mathematics) SEMESTER-IV

GroupA2 (Pre-requisitepaperA1i)

A21 AdvancedFunctional Analysis-II

A22 AdvancedDiscreteMathematics-II

A23 Algebraic CodingTheory-II

A24 Wavelets-II

A25 Sobolev Spaces-II

A26 Framesand RieszBases-II

Group B2 (Pre-requisitepaperB1i)

B21 Mechanics ofSolids-II

B22 Continuum Mechanics-II

B23 Computational Fluid Dynamics-II

B24 DifferenceEquations-II

B25 Information Theory-II

GroupC2 (Pre-requisite:GroupA2andpaperC1i)

C21 TheoryofLinear Operators-II

C22 Analytical NumberTheory-II

C23 FuzzySetsand Applications-II

C24 Bases inBanachSpaces-II

C25 Algebraic Topology-II

GroupD2 (Pre-requisite:GroupB2and paperD1i)

D21 Fluid Dynamics-II

D22 Bio-Mechanics-II

D23 Integral Equations andBoundaryValueProblems-II

D24 Mathematics of FinanceandInsurance-II

D25 SpaceDynamics-II

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66

Note1: Themarks of internal assessment of eachpaper shallbesplitas under :

A) Oneclasstestof10marks. Theclasstestwillbeheldinthemiddleofthesemester.







Above95% : 5 marks

Note2:Thesyllabusofeachpaperwillbedividedintofourunitsoftwoquestionseach. The question paper will consist of five units. Each of the first four unitswillcontaintwoquestionsandthestudentsshallbeaskedtoattempt onequestionfromeachunit.Unitfiveofeachquestionpapershallcontain eighttotenshortanswertypequestionswithoutanyinternalchoiceanditshall be coveringtheentiresyllabus. As such unitfiveshall becompulsory. Note 3: As per UGC Recommendations, the teaching program shall be

supplemented by tutorials and problem solving sessions for each theory paper. Forthispurpose,tutorialclassesshallbeheldforeachtheory paperin groups of8 students for half-hourper week.

Note 4: Optional papers can be offered subject to availability of requisite

resources/ faculty.

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67


MTH-411:Functional Analysis–II

Max. Marks : 80

Time:3 Hours


Signed measure, Hahn decomposition theorem, Jordan decomposition theorem,

Mutually signed measure, Radon – Nikodyntheorem Lebesgue decomposition,

Lebesgue -Stieltjes integral, Product measures, Fubini‟s theorem.


Bairesets, Bairemeasure, continuous functions with compact support,

Regularityofmeasures on locallycompact spaces, Riesz-Markofftheorem.

Hilbert Spaces:Inner product spaces, Hilbert spaces, Schwarz‟s

inequality,Hilbert spaceas normed linear space.


Convex sets in Hilbert spaces, Projection theorem. Orthonormal sets, Bessel‟s

inequality, Parseval‟sidentity, conjugate of a Hilbert space, Rieszrepresentation

theorem in Hilbert spaces.


AdjointofanoperatoronaHilbertspace,ReflexivityofHilbertspace,Self-adjoint

operators, Positive and projection operators, Normal and unitary operators,

Projections on Hilbert space, Spectral theorem onfinite dimensional space.






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68

BooksRecommended

1. H.L. Royden,RealAnalysis,MacMillanPublishingCo., Inc., New

York,4th

Edition, 1993.

2. E. Kreyszig, Introductory Functional Analysis with Applications, John

dispatchWiley.

3. S.K. Berberian, Measure and Integration, Chelsea Publishing Company, New

York, 1965.

4. G. Bachman andL.Narici, Functional Analysis,AcademicPress, 1966.

5. George F. Simmons, Introduction to Topology and Modern Analysis,

McGraw-HillBook Company,1963.

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69


MTH-412: ClassicalMechanics

Max. Marks: 80

Time: 3hours

Unit –I(2 Question)

Free&constrainedsystems,constraintsandtheirclassification,holonomic and

non-holonomic systems, degree of freedom and generalisedcoordinates, virtual

displacement and virtual work, statement of principle of virtual work (PVW),

possiblevelocityand possible acceleration, D‟Alembert‟s principle,

LagrangianFormulation:Idealconstraints,generalequationofdynamics

forideal constraints,Lagrange‟sequations of thefirst kind.

Unit –II(2 Question)

Independent coordinates and generalized forces, Lagrange‟sequations

ofthesecondkind,generalizedvelocitiesandaccelerations. Uniquenessofsolution,

variation oftotal energyfor conservative fields.

Lagrange‟svariableandLagrangianfunctionL(t,qi,q i),Lagrange‟sequationsforpotential

forces, generalized momenta pi, Hamiltonian variable and Hamiltonian function H(t,

qi, pi), Donkin‟s theorem, ignorablecoordinates.

Unit -III(2 Question)

Hamilton canonical equations, Routh variables and Routh function R,

Routh‟sequations,PoissonBracketsandtheirsimpleproperties,Poisson‟sidentity,

Jacobi– Poisson theorem.

Hamilton action and Hamilton‟sprinciple, Poincare – Carton integral

invariant, Whittaker‟sequations, Jacobi‟sequations, Lagrangianaction and the

principle ofleast action.

Unit -IV(2 Question)

Canonical transformation, necessary and sufficient condition for a

canonical transformation, univalent Canonical transformation, free canonical

transformation,Hamilton-

Jacobiequation,Jacobitheorem,methodofseparationofvariablesinHJequation,Lagrang

70

70

ebrackets,necessaryandsufficientconditionsof canonical character of a

transformation in terms of Lagrange brackets, Jacobian

matrixofacanonicaltransformation,conditionsofcanonicityofatransformation

in terms of Poison brackets, invariance of Poisson Brackets under canonical

transformation.

Note:Thequestion paper will consistof five units. Each of thefirst fourunits will





BooksRecommended

1. F. Gantmacher Lectures in Analytic Mechanics, MIR

Publishers, Moscow, 1975.

2. P.V. Panat Classical Mechanics, NarosaPublishing

House, NewDelhi, 2005.

3. N.C. Rana and P.S. Joag Classical Mechanics, Tata McGraw- Hill,

New Delhi, 1991.

4. LouisN.Handand

JanetD. Finch Analytical Mechanics, CUP, 1998.

5. K. SankraRao Classical Mechanics, Prentice Hall of

India, 2005.

6. M.R. Speigal Theoretical Mechanics, SchaumOutline

Series

71

71


MTH-413:ComplexAnalysis-II

Max. Marks: 80

Time:3 hours

Unit -I(2 Question) IntegralFunctions. Factorization of an integral function.

Weierstrass‟factorisation theorem. Factorization of sine function. Gamma function and its properties. Stirlingformula. Integral version of gamma function. Riemann Zeta function. Riemann‟sfunctional equation. . Runge‟s theorem. Mittag-Leffler‟s theorem.

Unit -II(2 Question)

Analytic Continuation. Natural Boundary. Uniqueness of direct analytic

continuation. Uniqueness of analytic continuation along a curve. Power

series method of analytic continuation. Schwarz Reflection principle. Germ of an

analytic function.

Monodromytheoremanditsconsequences. Harmonicfunctionsonadisk. Poisson

kernel. TheDirichlet problem foraunitdisc.

Unit -III(2 Question)

Harnack‟sinequality. Harnack‟stheorem. Dirichlet‟sregion. Green‟sfunction.

Canonical product. Jensen‟sformula. Poisson-Jensen formula.

Hadamard‟sthree circles theorem. Growth and order of an entire function. An

estimate of number of zeros. Exponent of Convergence. Borel‟stheorem.

Hadamard‟sfactorization theorem.

Unit -IV(2 Question)

Therangeofananalyticfunction. Bloch‟stheorem. Schottky‟stheorem. Little

Picard theorem. Montel Caratheodorytheorem. Great Picard theorem. Univalent

72

72

functions. Bieberbach‟sconjecture(Statement only) and the “1/4 theorem” .






BooksRecommended

1. H.A. Priestly, Introduction to Complex Analysis, Clarendon Press,

Oxford, 1990.

2. J.B. Conway, Functions of one Complex variable, Springer-Verlag,

International student-Edition, NarosaPublishing House, 1980.

3. Liang-shin Hann &BernandEpstein, Classical Complex Analysis, Jones and

Bartlett PublishersInternational,London, 1996.

4. E.T. Copson, An Introduction to the Theory of Functions of a

ComplexVariable, Oxford UniversityPress,London.

5. E.C. Titchmarsh, The Theory of Functions, Oxford University Press, London.

6. L.V.Ahlfors, ComplexAnalysis, McGrawHill, 1979.

7. S.Lang, ComplexAnalysis, Addison Wesley, 1977.

8. Mark J. Ablowitzand A.S. Fokas, Complex Variables : Introduction

andApplications, CambridgeUniversityPress, South Asian Edition, 1998.

9. S. Ponnusamy,Foundations of ComplexAnalysis, NarosaPublishing

House, 1997.


MTH-414:(OptionA22) AdvancedDiscreteMathematics II

Max. Marks : 80

Time:3 Hours


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73

Formal Logic – Statements. Symbolic Representation and Tautologies.

Quantifier, Predicates and Validty. Propositional Logic.

Unit -II (2 Questions) Semigroups &Monoids-Definitions and Examples of Semigroups

andMonoids(includingthosepertainingtoconcatenationoperation). Homomorphismof

semigroups and monoids. Congruence relation and Quotient Semigroups.

Subsemigroupandsubmonoids. Directproducts. BasicHomomorphismTheorem.

Pigeonhole principle, principleof inclusion and exclusion, derangements.


Lattices- Lattices as partially ordered sets. Their properties. Lattices as

Algebraic systems. Sublattices, Direct products, and Homomorphisms. Some

Special Lattices e.g., Complete. Complemented and Distributive Lattices. Join-

irreducibleelements. Atoms and Minterms.


Boolean Algebras – Boolean Algebras as Lattices. Various Boolean

Identities. The switching Algebra example. Subalgebras, Direct Products and

Homomorphisms. Boolean Forms and Their Equivalence. MintermBoolean

Forms, Sum of Products Canonical Forms. Minimization of Boolean Functions.

Applications of Boolean Algebra to SwitchingTheory(usingAND, OR &NOT

gates). TheKarnaughMap method.






BooksRecommended

1. J.P. Tremblay &R. Manohar, Discrete Mathematical Structures

withApplications to ComputerScience, McGraw-HillBook Co., 1997.

2. J.L.Gersting,MathematicalStructuresforComputerScience,(3rd

edition),

Computer SciencePress,New York.

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74

3. Seymour Lipschutz, Finite Mathematics (International edition 1983),

McGraw-HillBook Company,New York.

4. C.L.Liu, Elements of Discrete Mathematics, McGraw-HilllBook Co.

5. Babu Ram, Discrete Mathematics, VinayakPublishers and Distributors,

Delhi, 2004.

75

75


MTH-414: (Option A23):Algebraic Coding Theory-II

Max. Marks : 80

Time:3 hours


Cyclic Codes. CyclicCodes as ideals. MatrixDescription of CyclicCodes.

HammingandGolayCodesasCyclicCodes. ErrorDetectionwithCyclicCodes. Error-

Correction procedure for Short Cyclic Codes. Short-ended Cyclic Codes. Pseudo

CyclicCodes.


Quadraticresiduecodesofprimelength,HadamardMatricesandnon-linearCodes derived

from them. Product codes. Concatenated codes.


Code Symmetry. Invariance of Codes under transitive

group of permutations. Bose-Chaudhary-Hoquenghem(BCH)Codes.

BCHbounds. Reed- Solomon (RS) Codes. Majority-Logic Decodable

Codes. Majority- Logic Decoding. Singleton bound. TheGriesmer

bound.


Maximum – Distance Separable (MDS) Codes. Generator and Parity-check

matrices of MDS Codes. Weight Distribution of MDS code. Necessary and

SufficientconditionsforalinearcodetobeanMDSCode. MDSCodesfromRS codes.

AbramsonCodes. Closed-loopburst-errorcorrectingcodes(Firecodes).

ErrorLocatingCodes.


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76





BooksRecommended

1. Ryamond Hill, A First Coursein CodingTheory,Oxford

UniversityPress,1986.

2. ManYoungRhee,ErrorCorrectingCodingTheory,McGrawHillInc.,

1989.

3. W.W.Petersonand E.J. Weldon, Jr., Error-CorrectingCodes. M.I.T.

Press, CambridgeMassachuetts, 1972.

4. E.R. Berlekamp, Algebraic CodingTheory, McGraw HillInc., 1968.

5. F.J. Macwilliamsand N.J.A. Sloane, Theoryof ErrorCorrectingCodes,

North-HolandPublishing Company.

6. J.H. Van Lint, Introduction to Coding Theory, Graduate Texts

inMathematics, 86, Springer, 1998.

7. L.R. Vermani, ElementsofAlgebraicCodingTheory, Chapman and Hall,1996.

77

77


MTH-414: (OptionA24):Wavelets –II

Max. Marks : 80

Time:3 hours


Definition and Examples of Multiresolution Analysis, Properties ofScaling

Functions and Orthonormal Wavelet Bases,TheHaar MRA,Band-Limited MRA,

TheMeyer MRA.


Haar Wavelet and its Transform, DiscreteHaarTransforms, Shannon Wavelet andits

Transform.

Haar Wavelets, Spline Wavelets, Franklin Wavelets, Battle-Lemarie

Wavelets, DaubechiesWavelets and Algorithms.


Biorthogonal Wavelets,Newlands Harmonic Wavelets, Wavelets in Higher

Dimensions, Generalized Multiresolution Analysis, Frame Multiresolution

Analysis, AB- Multiresolution Analysis, WaveletsPackets, Multiwaveletand

MultiwaveletPackets, Wavelets Frames.


Applications of Wavelets inImageProcessing,Integral Opertors, Turbulence,

Financial Mathematics : Stock Exchange, Statistics, Neural Networks,Biomedical

Sciences.

Note:Thequestion paper will consist of five units. Each of thefirst fourunits will





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78

BooksRecommended

1. K. Ahmad andF. A. Shah,Introduction to Wavelet Analysiswith

Applications, Anamaya Publishers, 2008.

2. EugenioHernandezandGuido Weiss, A first Courseon Wavelets,CRC

Press, New York, 1996.

3. C.K. Chui, AnIntroduction to Wavelets, AcademicPress, 1992.

4. I.Daubechies, TenLecturesonWavelets,CBS-NSFRegionalConferencesin

Applied Mathematics,61, SIAM, 1992.

5. Y. Meyer, Wavelets, Algorithms and Applications (translated by R.D.Rayan,

SIAM, 1993).

79

79


MTH-414:(OptionA25):Sobolev Spaces-II

Max. Marks : 80

Time:3 hours


Other SobolevSpaces- Dual Spaces,FractionalOrder Sobolev spaces,

Tracespaces andtracetheory.


Weight Functions - Definiton, motivation, examples ofpractical

importance. Special weights of power type. General Weights.

Weighted Spaces- WeightedLebesguespaceP( , ) , and their

properties.


Domains - Methods of local coordinates, the classes Co, C

o,k,

Holder‟scondition, Partition ofunity, the class K(x0) includingConeproperty.


Inequalities – Hardy inequality, Jensen‟sinequality, Young‟sinequality,Hardy-

Littlewood- Sobolevinequality, Sobolevinequality and its various

versions.

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80






BooksRecommended

1. R.A. Adams, Sobolev Spaces, AcademicPress,Inc. 1975.

2. S. Kesavan, Topics in Functional Analysis and Applications,

WileyEasternLimited, 1989.

3.

A.Kufner,O.JohnandS.Fucik,FunctionSpaces,NoordhoffInternationalPubli

shing,Leyden, 1977.

4. A. Kufner, Weighted Sobolev Spaces, John Wiley&SonsLtd., 1985.

5. E.H.Lieband M.Loss,Analysis, NarosaPublishingHouse, 1997.

6. R.S. Pathak, A Course in Distribution Theory and Applications,

NarosaPublishingHouse, 2001.

81

81


MTH-414:(Option B21): Mechanics of Solids-II

Max. Marks : 80

Time:3 hours

Note:-Thequestionpaperwillconsistoffiveunits. Eachofthefirstfourunits

willcontaintwoquestionsfromunitI,II,III,IVrespectively

andthestudentsshallbeaskedtoattemptonequestionfromeachunit.

Unitfivewillcontaineightto tendshort answer type questions without anyinternal

choice covering the entire syllabus and shallbe compulsory.

Unit-I (2 Questions) Two-dimensionalProblems: PlanestrainandPlanestress.Generalized planestress. Airy stress function for plane strain problems. General solutions of a BiharmonicequationusingFouriertransformaswellasin termsoftwoanalytic functions. Stresses and displacements in terms of complex potentials. Thick walled tube underexternal and internal pressures. Rotatingshaft.

Unit-II (2 Questions) TorsionofBeams: Torsionofcylindricalbars. Torsionalrigidity. Torsionandstress functions. Lines of shearingstress. Simple problems related tocircle, ellipse and equilateral triangle cross-section. Circulargroovein a circular shaft.

ExtensionofBeams:Extensionofbeamsbylongitudinalforces. Beamstretchedbyits own

weight.

Unit-III (2 Questions) Bending ofBeams:BendingofBeams byterminal Couples,Bendingof abeam bytransverseload at thecentroid of the end section alongaprincipal axis. VariationalMethods: Variationalproblems and Euler‟sequations. The Ritz

method-one dimensional case, the Ritz method-Two dimensional case, The

Galerkinmethod, Applicationstotorsionofbeams,ThemethodofKantrovitch.

(Relevant topics from the Sokolnikoof‟s book)

Unit-IV(2 Questions) Waves: Simple harmonic progressive waves, scalar wave equation, progressivetype solutions, plane waves and spherical waves, stationary type solutions inCartesian and Cylindricalcoordinates. ElasticWaves:Propagationofwavesinanunboundedisotropicelasticsolid. P. SV and

SH waves. Wavepropagation in two-dimensions.

BooksRecommended: 1. I.S. Sokolnikof, Mathematical theory of Elasticity. Tata McGraw Hill

publishingCompany Ltd. New Delhi, 1977.

2. TeodarM. Atanackovicand ArdeshivGuran, Theory of Elasticity forScientists

and Engineers, Birkhausev, Boston, 2000.

3. A.K. Mal &S.J. Singh, Deformation of Elastic Solids, Prentice Hall, NewJersey,

82

82

1991

4. C.A. Coluson, Waves

5. A.S. Saada, Elasticity-Theory and Applications, PergamonPress, New

York,1973.

6. D.S. Chandersekhariahand L. Debnath, Continuum Mechanics, AcademicPress.

7. S. Valliappan, Continuum Mechanics-Fundamentals,Oxford &IBH

PublishingCompany, New Delhi-1981

83

83


MTH-414: (Option B22) : ContinuumMechanics-II

Max. Marks : 80

Time:3 hours

Unit -I (2

Questions)

Thermoelasticity: Basic concepts of thermoelasticity, Stress-strain relation for

thermo-elasticity,Navierequationsforthermoelasticity,thermalstressesinalong

circularcylinder and inasphere.

Unit -II (2

Questions)

Viscoealsticity: Viscoelastic models – Maxwell model, Kelvin model and

Standardlinearsolidmodel. Creepcompliance andrelaxationmodulus, Hereditary

integrals, visco-elastic stress-strain relations, correspondence principle and its

application to thedeformation ofaviscoelastic thick-walled tube in plane strain.

Unit -III (2

Questions)

FluidDynamics : Viscousstresstensor,StokesianandNewtonian fluid,Basic

equationofviscousflow,Navierstokesequations,specializedfluid,steadyflow,

irrotationalflow,potentialflow,Bernolliesequation,Kilvin‟stheorem(AsChapter7

of theBook byMaseand Mase).

Unit -IV (2

Questions)

Plasticity : Basic concepts, yield criteria, yield surface, equivalent stress and

equivalentstrain,elastic–plasticstress-strainrelation,plasticstress-

strainrelatin,plasticflowofanisotropicmaterial,specialcasesofplanestress,planestraina

nd axis-symmetry.



84

84




BooksRecommended

1. S. Valliappan, Continuum Mechanics, Fundamentals, Oxford &IBH

PublishingCompany, 1981.

2. G.T. Maseand G.E. Mase, Continuum Mechanics for Engineers, CRC Press,

1999.

3. Atanackovic, T.M. A. Guran, Theory of Elasticity for scientists

andEngineers,Birkhausev,2000.

4. D.S. Chandrasekharaiah, Continuum Mechanics, Academic Press,

PrismBooks Pvt.Ltd.,Bangalore.

5. L.S. Srinath, Advanced Mechanics of Fields, Tata McGraw-Hill, NewDelhi.

85

85

M.Sc. (Mathematics) SEMESTER-IV MTH-414:(Option B23):Computational FluidDynamics-II

Max. Marks : 80

Time:3 hours


Iterative methods. Stationary methods. Krylovsubspace methods.

Multigrademethods. Fast Poisson solvers.


Iterativemethods for incompressible Navier-Stokes equations.

Shallow-water equations – One and two dimensional cases. Godunov

„sorderbarriertheorem.


Linearschemes.Scalar conservationlaws.Eulerequationinonespacedimension– analytic

aspects. Approximate Riemann solverof Roe.


Osherscheme. Fluxsplittingscheme. Numericalstability.Jameson–Schmidt–Turkel

scheme. Higher order schemes.


containtwo questions fromunitI , II , III , IVrespectivelyand the students




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86

Books Recommended

1. P. Wesseling: Principles of Computational Fluid Dynamics, SpringerVerlag,

2000.

2. J.F. Wendt,J.D. Anderson, G. Degrezand E.Dick, Computational Fluid Dynamics :

AnIntroduction, Springer-Verlag, 1996.

3. J.D. Anderson, Computational Fluid Dynamics: Thebasics with applications,

McGraw- Hill, 1995.

4. K. Muralidher, Computational Fluid Flow andHeat Transfer,NarosaPub. House.

5. T.J. Chung, Computational Fluid Dynamics, CambridgeUni. Press.

6. J.N. Reddy, An introduction to the Finite Element

Methods,McGrawHillInternationalEdition, 1985.

87

87


MTH-414: (Option B24):DifferenceEquations-II

Max. Marks : 80

Time:3 hours

Unit -I (2

Questions)

Sturm-Liouville problems - Introduction, Finite Fourier Analysis. A

non-homogeneous problem.

Discrete Calculus of Variations - Introduction. Necessary conditions.

Sufficient Conditions and Disconjugacy.


Nonlinear equations that can belinearized. Thez-transform.

Boundary value problems for Nonlinear equations. Introduction.

TheLipschitzcase. Existenceof solutions.


Boundaryvalue problems fordifferential equations. Partial Differential Equations.


Discretization of Partial Differential Equations. Solution of

PartialDifferential Equations.






BooksRecommended

1. Walter G. Kelleyand Allan C. Peterson-DifferenceEquations. An

Introduction with Applications, AcademicPressInc.,

HarcourtBrace Joranovich Publishers, 1991.

2. Calvin Ahlbrandt and Allan C. Peterson. DiscreteHamiltonian

Systems, DifferenceEquations, Continued Fractions and Riccatti

Equations. Kluwer,Boston, 1996.

88

88


MTH-414:(Option B25):InformationTheory-II

Max. Marks : 80

Time:3 hours


Some imuitiveproperties of a measure of entropy – Symmetry,

normalization, expansibility, boundedness, recursivity, maximality, stability,

additivity, subadditivity, nonnegativity, continuity, branching, etc.

and interconnectionsamongthem.

AxiomaticcharacterizationoftheShannonentropy dueto Shannon and Fadeev.


Information functions, the fundamental equation of information, information

functions continuous at the origin, nonnegative bounded information functions,

measurableinformationfunctionsandentropy. Axiomaticcharacterizationsofthe

Shannon entropyduetoTverbergandLeo.


Thegeneralsolutionofthefundamentalequationofinformation. Derivationsand

their role in the study of information functions. The branching property. Some

characterisationsof the Shannon entropy based upon the branching property.

Entropies with the sum property.


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89

The Shannon inequality. Subadditivie, additive entropies. The Renjientropies.

Entropiesandmeanvalues. Averageentropiesandtheirequality,optimalcoding and the

Renjientropies. Characterisationof some measures of average code

length.






BooksRecommended

1. R. Ash,Information Theory,IntersciencePublishers, NewYork, 1965.

2. F.M. Reza, An Introduction to Information Theory, MacGraw-Hill

BookCompany Inc., 1961.

3. J. AczeladnZ. Daroczy, On Measures of Information and

theirCharacterizations, AcademicPress, New York.

90

90

M.Sc. (Mathematics) SEMESTER-IV MTH-415:(Option C21): Theory ofLinearOperators II

Max. Marks : 80

Time:3 hours


Spectral representation of bounded self adjointlinear operators. Spectral

theorem. Properties of thespectral familyof a bounded self-adjointlinear operator.

Unbounded linear operators in Hilbert Space. Hellinger - Toeplitz theorem.

Hilbert adjointoperators.


Symmetricandself-adjointlinearoperators,Closedlinearoperatorsandclosures.

Spectrum of an unbounded self-adjointlinear operator. Spectral theorem for

unitary and self-adjoint linear operators. Multiplication operator

and differentiation operator.


Spectral measures. Spectral integrals. Regular spectral measures. Real and

complex spectral measures. Complex spectral integrals. Description of the

spectralsubspaces. Characterizationofspectralsubspaces. Thespectraltheorem

forbounded normal operators.


TheProblemofUnitaryEquivalence,MultiplicityFunctionsinFinite-dimensional

Spaces, Measures, Boolean Operations on Measures, MultiplicityFunctions, The

CanonicalExampleofaSpectralMeasure,FinitedimensionalSpectralMeasures, Simple

finite dimensional Spectral Measures.




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91



BooksRecommended

1. E. Kreyszig, Introductory Functional Analysis with Applications, John-

Wiley&Sons, New York, 1978.

2. P.R. Halmos, Introduction to Hilbert Space and the Theory of

SpectralMultiplicity, Second-Edition, ChelseaPublishingCo., New York, 1957.

3. N. Dunford and J.T. Schwartz, Linear Operators -3 Parts,

Interscience/Wiley, NewYork, 1958-71.

4. G. Bachman and L. Narici, Functional Analysis, Academic Press, York,1966.

92

92


MTH-415: (Option C22): Analytical NumberTheory-II

Max. Marks : 80

Time:3 hours


Riemann Zeta Function (s) and its convergence. Application to prime

numbers. (s)asEuler‟sproduct. Evaluationof (2)and (2k). Dirichletseries with

simple properties. Eulersproducts and Dirichletproducts, Introduction to modular

forms. (Scope as in Book at Sr. No.5).

Unit -II (2 Questions) AlgebraicNumberandIntegers :Gaussianintegersanditsproperties. Primes

and fundamental theorem in the ring of Gaussian integers. Integers

and fundamental theorem in Q( ) where 3

= 1. Algebraic fields. Primitive

polynomials. ThegeneralquadraticfieldQ( m),UnitsofQ( 2). Fieldsinwhich

fundamental theorem is false. Real and complex Euclidean fields.

Fermat'‟theorem ithe ring of Gaussian integers. Primes of Q( 2) and Q( 5)

Series of Fibonacci and Lucas. Luca'‟test for the primality of the

mersenneprimes. (Relevant sections of RecommendedBookat Sr.No. 1).


Arithmeticfunctions (n), (n), (n)and k(n),u(n),N(n),I(n). Definition and

examples and simple properties. Perfect numbers the Mobius inversion

formula.TheMobiusfunction n,Theorderandaverageorderofthefunction (n),(n) and

(n). (Scope as in books at Sr. No. 1 and4).


Thefunctions (n), (n)and

(n)BertrandPostulate,Merten‟stheorem,Selberg‟stheorem and

Primenumber Theorem. (Scopeas in Books at Sr. No. 1 and4).

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93



shall beaskedtoattemptonequestion fromeach unit. Unitfivewillcontain eight

to ten short answer type questions without any internal choice coveringthe


BooksRecommended

1. Hardy, G.H.and Wright,E.M., AnIntroduction to theTheoryofNumbers

2. Burton, D.M., ElementaryNumber Theory.

3. McCoy, N.H.,TheTheoryofNumber byMcMillan.

4. Niven, I. And Zuckermann, H.S., An Introduction to the Theory

ofNumbers.

5. Gareth,A.JonesandJ.MaryJones,ElementaryNumberTheory,SpringerEd. 1998.

94

94


MTH-415: (OptionC23): Fuzzy Sets andtheirApplications-II

Max. Marks : 80

Time:3 hours


Probability Theory, Probability Distribution, Comparison of Probability and

Possibility, Fuzzy event, Crisp Probabilityof FuzzyEvent, Fuzzy Probabilityof

FuzzyEvent,UncertaintyLevelofElement,FuzzinessofFuzzySet,Measureof

Fuzziness,MeasureusingEntropy,MeasureusingMetricDistance.(Chapter7of book at

serial no.1 )


PropositionLogic,LogicFunction,Tautology andInferenceRule,PredicateLogic,

Quantifier,FuzzyExpression,OperatorsinFuzzyExpression,SomeExamplesof Fuzzy

Logic Operations, Linguistic Variable, Fuzzy Predicate, Fuzzy Modifier, Fuzzy

Truth Values, Examples of Fuzzy Truth Quantifier, Inference and

KnowledgeRepresentation,RepresentationofFuzzyPredicatebyFuzzyRelation,

Representation ofFuzzyRule.

ExtensionPrincipleandComposition,CompositionofFuzzySets,Compositionof Fuzzy

Relation, Example of Fuzzy Composition, Fuzzy if-then Rules, Fuzzy

Implications,ExamplesofFuzzyImplications,DecompositionofRuleBase,Two-

Input/Single-OutputRuleBase,CompositionalRuleofInference,FuzzyInference

withRuleBase,InferenceMethods,MamdaniMethod,LarsenMethod,Tsukamoto

Method, TSK Method. (Chapter 8,9of book at serial no. 1)

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95


Advantage of Fuzzy Logic Controller, Configuration of Fuzzy Logic Controller,

Choice of State Variables and Control Variables, FuzzificationInterface

Component, Data Base, Rule Base, Decision Making Logic, MamdaniMethod,

Larsen Method, Tsukamoto Method, TSK Method, Mean of Maximum Method,

CenterofAreaMethod(COA),BisectorofArea,LookupTable,DesignProcedureof

Fuzzy Logic Controller, Application Example of FLC Design, Fuzzy

ExpertSystems. (Chapter 10 ofbook at serial no. 1 )


Applications of Fuzzy Set Theory in Natural, Life and Social Sciences,

Engineering, Medicine, Management and Decision Making, Computer Science,

System Sciences.(Chapter 6 ofbook at serial no.2 )



shall beaskedtoattemptonequestion fromeach unit.Unit fivewillcontain eight

to ten short answer type questions without any internal choice coveringthe


BooksRecommended

1.

KwangH.Lee,FirstCourseonFuzzyTheoryandApplications,SpringerInterna

tional Edition, 2005.

2. George J. Klirand Tina A. Folger, Fuzzy Sets, Uncertainty

andInformation, Prentice Hall of India Private Limited, New Delhi-110 001,

2005. 3. H.J.Zimmerman,FuzzySetTheoryanditsApplications,AlliedPublishersLtd.,

New Delhi, 1991.

4. John Yen, Reza Langari, Fuzzy Logic - Intelligence, Control

andInformation, Pearson Education.

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96


MTH-415: (Option C24):Bases inBanachSpaces–II

Max. Marks : 80

Time:3 hours

Unit -I (2Questions)

Different types of convergence in a Banachspace. Unconditional bases.

Unconditional basissequences. SymmetricBases.

Bases and structure of the space Bases and completeness. Bases and

reflexivity.


Generalized bases. Generalized basic sequences.

Boundedlycomplete generalizedbases andshrinkinggeneralizedbases. M-Bases

(Markusevicbases) and M-Basic sequences.

Bases of subspaces (Decomposition-Existence of a decomposition in

Banachspaces. The sequences of co-ordinate projections associated to a

decomposition.

Unit -III (2Questions)

Schauderdecompositions. Characterization of a Schauderdecomposition.

ExamplethatadecompositionisnotalwaysSchauder. Variouspossibilitiesfora

Banachspace to possess a Schauderdecomposition. Shrinking decompositions

boundedlycompletedecompositionsandunconditionaldecompositions. Reflexivity of

Banachspaces having a Schauderdecomposition). Bases and decompositions in the

spaceC[0,1].


Best approximation in Banachspaces. Existence and uniqueness of

element of best approximation by a subspace. Proximinalsubspaces. Semi

Cebysev subspacesandCebysev subspaces.

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97

Monotone and strictly monotone bases, co-monotone and strictly co-

monotonebases. Examples and counter-examples.

T-norm, K-norm and KT-norm on Banachspaces having bases. Their

characterizationintermsofmonotoneandcomonotonebases. Variousequivalent

norms on aBanach spacein terms oftheirbases.






BooksRecommended

1. Jurgt.Marti, IntroductiontoTheoryofBases,

SpringerTractsinNaturalPhilosophy18, 1969.

2. IvanSinger,BasesinBanachSpaces I,Springer-Verlag, Berlin,Vol.1541970.

3. Ivan Singer,Bases inBanachSpacesII, Springer-Verlag,Berlin, 1981.

4. J.LinderstraussandI.Tzafriri,ClassicalbanachSpaces(Sequencespaces),

SpringerVerlag, Berlin, 1977.

5. IvanSinger,BestApproximationinNormedLinearSpacesbyElementsofLinear

Spaces, Springer-Verlag,Berlin, 1970.

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98


MTH-415:(Option C25): Algebraic Topology-II

Max. Marks : 80

Time:3 hours


MayerVietorissequenceanditsapplicationtocalculationofhomologyof

graphs,torusandcompactsurfaceofgenusg,collared

pairs,constructionofspacesbyattaching ofcells,sphericalcomplexeswithexamplesofSn,

r-leavedrose,torus,RPn, CP

netc.


Computation of homology of RPn, CP

n, torus, suspension space, XVY,

compact surface of genus g and non-orientable surface of genus h using Mayer

Vietorissequence,BettinumbersandEulercharacteristicsandtheircalculationforSn, r-

leaved rose,RPn, CP

n, S

2 x S

2, X + Y etc.

Unit -III(2 Questions)

Singular cohomology modules, Kronecker product,

connecting homomorphism, contra-functorialityofsingular

cohomologymodules, naturalityof connecting homomorphism, exact

cohomologysequence of pair, homotopy

invariance,excisionproperties,cohomologyofapoint. MayerVietorissequenceand its

application in computation ofcohomology ofSn, RP

n,

CPn,torus,compactsurfaceofgenus g and non-orientablecompact surface.


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99

Compact connected 2-manifolds, their orientabilityand non-orientability,

examples,connectsum,constructionofprojectivespaceandKlein‟sbottlefroma square,

Klein‟s bottle asunityof two Mobius strips, canonical form ofsphere, torusand

projective palnes. Kelin‟sbottle. Mobiousstrip, triangulation of compact surfaces.

Classification theorem for compact surfaces, connected sum of torus and

projective plan as the connected sum of three projective planes. Euler

characteristic as a topological invariationof compact surfaces. Connected sum

formula,2-

manifoldswithboundaryandtheirclassificationEulercharacteristicofabordered

surface, models of compact bordered surfaces inR3

.






BooksRecommended

1. JamesR.Munkres,Topology–AFirstCourse,PrenticeHallofIndiaPvt.Ltd., New

Delhi, 1978.

2. MarwinJ. Greenberg and J.R. Harper, Algebraic Topology – A FirstCourse,

Addison-WesleyPublishingCo., 1981.

3. W.S.Massey,AlgebraicTopology–

AnIntroduction,Harcourt,BraceandWorldInc. 1967, SV, 1977.

100

100


MTH-415:(Option D21):FluidDynamics-II

Max. Marks : 80

Time:3 hours


Irrotationalmotionintwo-dimensions. Complexvelocity potential. Milne-

Thomson circle theorem. Two-dimensional sources, sinks, doublets and their

images. Blasius theorem. Two-dimensional irrotation motion produced bymotionof

circular, co-axial and elliptical cylinders in an infinite mass of liquid.


Vortex motion. Kelvin‟sproof of permanence. Motions due to circular and

rectilinearvortices.Spiralvortex. Vortexdoublet. Imageofavortex.Centroidof

vortices. Single and double infinite rows of vortices. Karman vortex street.

Applications of conformal mappingto fluid dynamics.


Stress components in a real fluid.Relations between rectangular

components of stress. Gradients of velocity. Connection between stresses and

gradientsofvelocity. Navier-Stoke‟sequationsofmotion. Equationsofmotionin

cylindrical and sphericalpolar co-ordinates.

PlanePoiseuilleandCouetteflowsbetweentwoparallelplates. Theoryof

lubrication. Flow through tubes of uniform cross-section in form of circle,

annulus,ellipseandequilateraltriangleunderconstantpressuregradient. Unsteady

flowover a flat plate.

Unit -IV (2 Questions) Dynamicalsimilarity.Inspectionanalysis. Reynoldsnumber. Dimensional analysis.

Buckingham -theorem. Prandtl‟sboundary layer. Boundary layer

equation in two-dimensions. Blasius solution. Boundary layer thickness,

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101

displacement thickness, momentum thickness. Karman integral conditions.

Karman-Pohlmansenmethod. Separation ofboundarylayer flow. Note:Thequestion paper willconsist offive units. Each of thefirst fourunits will





BooksRecommended

1.W.H.BesaintandA.S.Ramasey,ATreatiseonHydromechanics,PartII,CBS Publishers,

Delhi, 1988.

2. F. Chorlton, Text Book of Fluid Dynamics,C.B.S. Publishers, Delhi, 1985

3.O‟Neill,M.E.andChorlton,F., IdealandIncompressibleFluid Dynamics,

EllisHorwoodLimited, 1986.

4. O‟Neill, M.E. and Chorlton, F. , Viscous and Compressible Fluid Dynamics,Ellis

HorwoodLimited,1989.

5. S.W. Yuan, Foundations of Fluid Mechanics, Prentice Hall of India

PrivateLimited, NewDelhi, 1976.

6. H. Schlichting, Boundary-Layer Theory, McGraw Hill Book Company, NewYork,

1979.

7. R.K.Rathy,AnIntroductiontoFluid Dynamics,OxfordandIBHPublishingCompany,

New Delhi, 1976.

8. G.K.Batchelor,AnIntroductiontoFluidMechanics,FoundationBooks,NewDelhi,

1994.

102

102


MTH-415: (Option D22): Biomechanics-II

Max. Marks : 80

Time:3 hours

Unit -I (2 Questions) Lawsofthermodynamics. GibbsandGibbs–Duhemequations. Chemical potential. Entropyinasystemwithheatandmasstransfer. Diffusion,filtration, andfluidmovementininterstitialspaceinthermodynamicview. Diffusionfrom molecularpointof view.


Masstransportincapillaries,tissues,interstitialspace,lymphatics,indicator

dilution method, and peristalsis. Tracer motion in a model of pulmonary

microcirculation.


Descrptionofinternaldeformationandforces. EquationsofmotioninLagrangian

description. Work andstrain energy. Calculation of stresses from strain energy

function.


Stress,strainandstabilityoforgans. Stressandstrainsinbloodvessels. Strength,

Trauma,andtolerance. Shockloadingandstructuralresponse. Vibrationandthe

amplification spectrum of dynamicstructural response. Impact andelasticwaves.






BooksRecommended

1. Y.C.Fung,Biomechanics:Motion,Flow,StressandGrowth,Springer-Verlag, New

YorkInc., 1990.

103

103


MTH-415:(OptionD23):Integral Equations andBoundary ValueProblems-II

Max. Marks : 80

Time:3 hours

Unit -I (2

Questions)

Application to Partial Differential Equations. Integral representation

formulasforthesolutionsoftheLaplaceandPoissonequations. Newtoniansingle layer

and double layer potentials. Interior and exterior Dirichletand Neumann

problems for Laplace equation. Green‟sfunction for Laplace equation in a free

space as well as in a space bounded by a grounded vessel. Integralequation

formulation ofBVPs forLapalceequation. TheHelmholftzequation.

(Relevant topics from thechapters 5and 6 ofthe book byR.P. Kanwal).


Symmetrickernels. ComplexHilbertspace. Orthonormalsystemoffunctions. Riesz-

Fischer theorem (statement only). Fundamental properties of eigenvalues and

eigenfunctionsfor symmetric kernels. Expansion in

eigenfunctionsandbilinearform.

AnecessaryandsufficientconditionforasymmetricL2-kerneltobe separable.Hilbert-

Schmidttheorem. Definiteand indefinitekernels.

Mercer‟s

theorem(statementonly).Solutionofintegralequationswithsymmetrickernelsbyusin

gHilbert-Schmidttheorem.


Singular integral equations. The Abel integral equation. Inversion formula for

singular integral equation with kernel of the type [h(s) – h(t)] with0< <1.

Cauchyprincipalvalueforintegrals. SolutionoftheCauchytype singular integral

equations. The Hilbert kernel. Solution of the Hilbert-type singular integral

equations. Integral transform methods. Fourier transform. Laplace transform.

Applications to Volterraintegral equations with convolution typekernels. Hilbert

transforms and their usetosolve integral equations.


Applications to mixed BVP‟s. Two-part BVP‟s, Three-part BVP‟sGeneralized

104

104

two-partBVP‟s.Perturbationmethod. ItsapplicationstoStokesandOseenflows,

and to Navier-Cauchy equations of elasticity for elastostaticand elastodynamic

problems.

(Relevant topics from thechapters 9 to 11 ofthe book byR.P. Kanwal).






BooksRecommended

1.Kanwal,R.P., LinearIntegralEquations–TheoryandTechnique,AcademicPress,

1971.

2. Kress, R., LinearIntegralEquations, Springer-Verlag,New York, 1989. 3.Jain,D.L.andKanwal,R.P.,MixedBoundaryValueProblemsinMathematicalPhysics

.

4.Smirnov,V.I., IntegralEquationsandPartialDifferentialEquations,Addison-

Wesley, 1964.

5. Jerri, A.J., Introduction toIntegral Equations with Applications, Second Edition,

John-Wiley&Sons, 1999.

6. Kanwal, R.P., LinearIntegration Equations, (2nd

Ed.)Birkhauser,Boston, 1997.

105

105


MTH-415: (Option D24): Mathematics forFinanceandInsurance-II

Max. Marks : 80

Time:3 hours

Unit -I (2

Questions)

Financial Derivatives – Futures. Forward. Swaps and Options. Call and

Put Option. Call and Put Parity Theorem. Pricing of contingent claims through

Arbitrageand ArbitrageTheorem.

FinancialDerivatives–AnIntroduction;TypesofFinancialDerivatives-Forwards and

Futures; Options and its kinds; and SWAPS.


TheArbitrageTheoremandIntroductiontoPortfolioSelectionandCapitalMarket

Theory: Static and Continuous– Time Model.

PricingArbitrage - ASingle-PeriodoptionPricingModel;Multi-Period

PricingModel- Cox–Ross- RubinsteinModel;BoundsonOptionPrices.The

lto‟sLemmaand theIto‟s integral.


The Dynamics of Derivative Prices - Stochastic Differential

Equations (SDEs)-

MajorModelsofSDEs:LinearConstantCoefficientSDEs;Geometric

SDEs;SquareRootProcess;Mean RevertingProcessand Omstein-

UhlenbeckProcess.

MartingaleMeasuresandRisk-

NeutralProbabilities:PricingofBinomialOptions with equivalent

martingalemeasures.


TheBlack-ScholesOptionPricingModel- usingnoarbitrageapproach,limiting

caseofBinomial Option Pricingand Risk-Neutralprobabilities.

TheAmericanOptionPricing- ExtendedTradingStrategies;Analysisof

American Put Options; early exercise premium and relation to free boundary

problems.

106

106


containtwo questions from unitI, II , III , IV respectivelyand the students




BooksRecommended

1. AswathDamodaran,CorporateFinance-

TheoryandPractice,JohnWiley&Sons,Inc.

2. John C. Hull, Options, Futures, and Other Derivatives, Prentice-Hall

ofIndian PrivateLimited.

3. Sheldon M. Ross, An Introduction to Mathematical Finance,

CambridgeUniversityPress.

4. Mark S. Dorfman, Introduction to Risk Management and Insurance,

PrenticeHall, EnglwoodCliffs, New Jersey.

5. C.D. Daykin, T. Pentikainenand M. Pesonen, Practical Risk Theory

forActuaries, Chapman&Hall.

6. SalihN. Neftci, An Introduction to the Mathematics of

FinancialDerivatives, AcademicPress,Inc.

7. RobertJ.ElliottandP.EkkehardKopp,MathematicsofFinancialMarkets,

Sprigner-Verlag, New YorkInc.

8. Robert C. Merton, Continuous– Time Finance, BasilBlackwellInc.

9. Tomasz Rolski, HanspterSchmidli, Volker Schmidt and JozefTeugels,

Stochastic Processes for Insurance and Finance, John Wiley &Sons

Limited.

107

107


MTH-415: (Option D25): SpaceDynamics-II

Max. Marks : 80

Time:3 hours

Unit -I (2

Questions)

Motionofthemoon–Theperturbingforces. PerturbationsofKeplerianelementsofthe

Moon bytheSun.


Flight Mechanics – Rocket Performance in a Vacuum. Vertically

ascending paths. Gravity Twin trajectories. Multi stage rocket in a Vacuum.

Definitonspertinentto single stage rocket.


Performance limitations of single stage rockets, Definitions pertinent to multi

stage rockets. Analysis of multi stage rockets neglecting gravity. Analysis

of multistage rockets including gravity.


Rocket Performance with Aerodynamic forces. Short range non-lifting missiles.

Ascentofasoundingrocket. Someapproximateperformanceofrocket-poweredair-craft.


containtwo questions from unitI , II , III , IVrespectivelyand the

studentsshall beaskedtoattemptonequestion fromeach unit. Unit

fivewillcontain eight to ten short answer type questions without any internal

choice coveringthe entiresyllabus and shallbecompulsory.

108

108

BooksRecommended

1. J.M. A. Danby, Fundamentals of Celestial Mechanics. The

MacMillanCompany, 1962.

2. E. Finlay, Freundlich, Celestial Mechanics. The MacMillan

Company,1958.

3.

TheodoreE.Sterne,AnIntroductionofCelestialMechanics,IntersciencesPu

blishers. INC., 1960.

4. ArigeloMiele, Flight Mechanics – Vol . 1 - Theory of Flight Paths,

Addison-WesleyPublishingCompany Inc., 1962.

m.sc. mathematics 2014-15 · mth-112real analysis-i max. marks : 80 time:3 hours unit -i (2...

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