M.Sc.Mathematics: Syllabus(CBCS) 76 ....

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M.Sc. Mathematics: Syllabus (CBCS)

VIKRAM UNIVERSITY, UJJAINMASTER OF SCIENCE

M.Sc. MATHEMATICSUNDER CBCS(2016-2017)

The Course of Study and the Scheme of Examinations

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SEMESTERI CIA Uni. Totalr Exam

1 CORE MAT-ClOl 6 6 Advanced Abstract Algebra I 10 40 502 CORE r MAT-Cl02 6 6 Real Analysis I 10 40 503 CORE MAT-Cl03 6 6 Topology I 10 40 504 CORE MAT-C104 6 5 Complex Analysis I 10 40 505 ELECTIVE MAT-E105(A,B,C,D) 6 3+2=5 (to choose lout of 4)

A. Programming in (-I (Theory +Practical) 10 J5+15B. Differential Equation I 10 40 50C. Advanced Discrete Mathematics I 10 40D. Differential Geometry of Manifolds I 10 40

6 tORE lMAT-CI06 6 2 Comprehensive Viva-Voce - 50 50

36 30 50 250 300

CIA Uni. TotalSEMESTERII

Exam7 CORE MAT-C201 6 6 Advanced Abstract Algebra II 10 40 508 CORE MAT-C202 6 6 Lebesgue Measure and Integration 10 40 509 CORE MAT-C203 6 6 Topology II 10 40 5010 CORE MAT-C204 6 5 Complex Analysis II 10 40 50

11 HKTIVE MAT-E205(A,B,C,D) 6 13+2=5(to choose lout of 4)

A. Programming in (-II (Theory +Practical) 10 125+15B. Differential Equation II 10 40 50C. Advanced Discrete Mathemadcs " 10 40D. Differential Geometry of Manifolds II 10 40

12 CORE MAT-C106 6 2 Comprehensive Viva-Voce - 50 5036 30 50 250 300

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I CIA Uni. otalSEMESTER IIIExam

Integration Theory and Functional13 CORE MAT-C301 6 6 Analysis-I 10 40 SO

Fundamentals of ComputerScience(Theory)-1 10 25 35Fundamentals of Computer Science (

14 CORE MAT.C302 6 6 Practical)-I . 15 15MAT-E303

15 ELECTIVEI (A,B,C,D,E) 6 6 (to choose lout of 5) 10 40 SOA. Advanced Functional Analysis-lI B. Partial Differential Equations

I C. Differentiable Structures onmanifolds-I

D. General Theory of Relati vity andrCosmology-I

E. Abstract Harmonic Analysis-Ir

F. Mathematics of Fmance & Insurance~ -I

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(to choose lout of 5)A. Theory of Linear Operator IB. Advanced Numerical Analysis -I

:6 , r I.::TIVE-II MAT-E3046 5 C. Fuzzy Sets and their Applications-I

I (A,B,C,D,E) D. Advanced Graph Theory.!I E. Advanced Special Function-I

I F. Spherical Trigonometry and-. astronomy-I 10 40 SO

, (to choose lout of 5)A. Operations Research -I

'ElECTIVE-III MAT-E30S B. Computational Biology -I17

(A,B,C,D,E) 6 5 C. Fluid Mechanics --ID. Bio- Mechanics -IE. Analytic Number Theory-IF. Integral Transform-I 10 40 SO

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ISEMESTER IV CIA Uni. IrotalExam

19 CORE MAT-C401 6 6 Functional Analysis-II 10 40 SOFundamentals of ComputerScience(Theory)-1I 10 25 35Fundamentals of Computer Science (

20 CORE MAT-C402 6 6 Practical)-II - 15 15MAT-E403

21 ELECTIVE-I (A/B/C/D/E) 6 6 (to choose lout of 6) 10 40 SOA. Advanced Functional Analysis-IIB. MechanicsC. Differentiable Structures on

manifolds-IID; General Theory of Relativity and

Cosmology-IIE. Abstract Harmonic Analysis-IIF.' Mathematics of Finance & Insurance

r -II

(to choose lout of 6).

A. Theory of Linear Operator IIB.~ Advanced Numerical Analysis -II

MAT-E404 , I,..,Fuzzy Sets and their Applications-II22 ELECTIVE-II 6 4

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(A/B/C/D/E) D. Advanced Graph Theory-IIE. Advanced Special Function-IIF. Spherical Trigonometry and

astronomy-II 10 40 SO(to choose lout of 6)A. Operations Research -II

MAT-E405 B. Computational Biology -II23 ELECTIVE-III'

(A/B/C/D/E)6 4 C. Fluid Mechanics -II

D. Bio- Mechanics -IIE. Analytic Number Theory-IIF. Integral Transform-II 10 40 SO

MAT-24 CORE r406 6 2 Comprehensive Viva-Voce - 50 50

MAT-25 CORE P407 2 Job Oriented Project Work - 50 50

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COURSE STRUCTURE FORSchool of Studies in Mathematics

Under CBCS

M.Sc. Mathematics(Regular)

1 & II Semester 2016-2017and

M.Sc. Mathematics

(Regular)

III & IV Semester 2017-2018

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VIKRAM UNIVE~SITY, UJJAIN

MASTER OF SCIENCE


The Course of Study and the Scheme of ExaminationsIns. Cre

S.NO. Study Components :hrs dit Title of the Paper Maximum MarksI

IweekCourse Title ,

I

SEMESTERI CIA Uni. Total"

Examf " 1 CORE MAT-C101 6 6 Advanced Abstract Algebra I 10 40 50~' 2 CORE MAT-Cl02 6 6 Real Analysis I 10 40 50Ic" i

3 CORE MAT-Cl03 6 6 Topology 1 10 40 50,:1'

4 CORE MAT-Cl04 6 5 Complex Analysis I 10 40 505 ELECTIVE MAT-E105(A,B,C,D) 6 /3+2=5 (to choose lout of 4)

,A. Programming in C-I (Theory +Practical) 10 )5+15~:B. Differential Equation I 10 40 50t C. Advanced Discrete Mathematics I 10 40D. Differential Geometry of Manifolds I 10 40

6 CORE MAT-CI06 6 2 Comprehensive Viva-Voce - 50 50

36 30 50 250 300

CIA Uni. TotalSEMESTER"Exam',' 7 CORE MAT-C201 6 6 Advanced Abstract Algebra /I 10 40 50"

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8 CORE MAT-C202 6 6 Lebesgue Measure and Integration 10 40 50,9 CORE MAT-C203 6 6 Topology /I 10 40 50

10 CORE MAT-C204 6 5 Complex Analysis /I 10 40 50

11 ELECTIVE MAT-E205(A,B,C,D) 6 3+2=5 (to choose lout of 4)

A. Programming in C-II (Theory +Praetical) 10 ~5+15B. Differential Equation II 10 40 50C. Advanced Discrete Mathematics II 10 40D. Differential Geometry of Manifolds /I 10 40

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12 CORE jMAT-C106 6 2 Comprehensive Viva-Voce _36 30 50

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Theory Marks: 40C.C.E. Marks: 10

M.Sc. MathematicsUnder CBCS (Only for School of Studies in Mathematics)SEMESTER IPaper I Advanced Abstract Algebra - I

Unit 1 -Automorphisms, Normal and subnormal series of groups, composition series, Jordan-

Holder Theorem.

Unit 2 -

Commutator subgroup, Solvable series and Solvable groups, Cenral series and Nilpotent

groups.

Unit 3 -Extension fields, Roots of polynomials, Algebraic and transcendental extensions, Splitting

fields, Separable ana inseparable extensions.

Unit 4 -

Perfect fields, Finite fields, Algebraically closed fields.

Unit 5-Automorphism of extensions, Galois extensions, Fundamental theorem of Galois theory,

Solution ofpolynomi::ll equations by radicals, Insolvability of the general equation of degree 5

by radicals.

Recommended Books:I. N. Herstein. Topics in algebra, Wiley Eastern Ltd. New Delhi, 1975.Vivek Sahai and Vikas Bist, Algebra, Narosa Publishing House, 1999.

P .B. Bhattacharya, S.K. Jain and S.R ..Nagpaul, Basic Abstract Algebra (2nd Edition),Cambridge University Press, Indian Edition, 1997.

Reference Books:[1] N.Jacobson, Basic Algebra, Vols.1 & II"W.H.freeman, 1980 (also published by

Hindustan Publishing Company).S. Lang, Algebra, Addison- Wesley.I.S. Luther and I.B.S. Passi, Algebra, VoU - Groups, Vol. II - Rings, Narosa PublishingHouse (Vol. I -1996, Vol.lI - 1999 ).

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Theory Marks: 40

c.c.E. Marks: 10

•. M.Sc. MathematicsUnder CBCS (Only' for School of Studies in Mathematics)SEMESTER I

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'1'1; Paper II Real Analysis

)1' Unit 1-

I.

I.!!r; Definition and existence ofRiemann-Stieltjes integral, Properties of integral, integration and.~:~ differentiation, the fundamental theorem of Calculus.

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Unit2 -

.Integration of vector valued functions, Rectifiable curves. Rearrangement of terms of a series,Riemann's theorem. Sequences and series of functions, pointwise and uniform convergence.

Unit 3 -

Cauchy criterian for uniform convergence, Weierstrass M-test, Abel's and Dirichlet's test for uniformconvergence, uniform convergence and cOntinuity, uniform convergence and Riemann -Stielties integration,uniform convergence and differentiation, Weierstrass approximation theorem,

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19nit 4 -

Power series, Uniqueness theorem for power series, Abel's theorem, Functions of several variables,linear transformations, Derivatives in an open subset ofR", chain rule, partial derivatives, interchange of theorder of differentiation, derivatives of higher orders. Taylor's theorem,

UnitS -

Inverse function theorem, Implicit function theorem, Jacobians, Lagrange's multiplier method,Differentiation ofintegrals, partitions of unity, Differential forms, Stoke's theorem.

Recommeilded Books:

[1]. Walter Rudin, Principles of Mathematical Analysis (3rd edition), McGraw-Hill, Kogakusha, 1976,International Student edition.

Referellce Books:

[1] T.M. Apostol, Mathematical Analysis, Narosa Publishing House, New Delhi, 1985.

[2] H.L. Royden, Real Analysis, Macmillan Publishing Co. Inc., 4~1Edition, New York, 1993i'

M.Sc. MathematicsUnder CBCS (Only for School of Studies in Mathematics)SEMESTER I

Countable and Uncountable sets. Infinite sets and the Axiom of Choice. Cardinal numbersand its arithmetic. Schroeder-Bernstein theorem. Cantor's theorem and the continuum hypothesis.Zorn's lemma. Well - ordering theorem.

Unit 2 -Defintion and examples of topological spaces. Closed sets, Closure. Dense subsets.

Neighbourhoods. Interior, exterior and boundary. Accumulation points and derived sets.

Unit3-r

Bases and sub bases. Subspaces and relative topology, Product Topology, MetricTopology, Continuous fuqctions and homomorphism.

Unit 4 -

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Unit 1 -

Topology - I

RegularTheory Marks: 40C.C.E. Marks: 10

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,First and SeconpCountable ~paces. Covering and Lindelofs spaces, Separable spaces,

second countability and Separability.

Unit 5-Connected spaces, connectedness on real line, components, Path connectedness, locally

connected spaces.

Recommended Books:

[1] James R. Munkres, Topology: A First Course, Prentice Hall of India Pvt. Ltd. NewDelhi, 2000.

Reference Books:

K.D. Joshi, Introduction to General Topology,Willey Eastern Limited, 1983.

George F. Simmons, Introduction to Topology and Modern Analysis, McGraw-Hill BookCompany, 1963.

J. Dugundji, Topology, Allyn and Bacon, 1966 (Reprinted in India by Prentice-Hall ofIndia Pvt. Ltd.) 1111444555566

N. Bourbaki, General Topology part-I (Trans!.) Addison Wesley Reading 1966.

B. Mendelson, Introduction to Topology, Allyn & Becon, Inc., Boston, 1962.

E.H. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966.

J.L. Kelley, General Topology, Van Nostrand, Reinhold Co., New York, 1995.

M.J. Mansfield, Introduction to Topology, D.Van Nostrand Co. Inc., Princeton, N.J.1963.

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RegularTheory Marks: 40c.c.E. Marks: 10

r...M.Sc. MathematicsUnder CBCS (Only for School of Studies in Mathematics)SEMESTER I

Paper IV. Complex Analysis-1

Unit 1 -

Complex integration. Cauchy-Gorsat Theorem. Cauchy's integral Formula. Higher Orderderivatives.

Unit 2 -

Morera's Theorem. Cauchy's inequality and Liouville's theorem. The fundamental theoremof Algebra. Taylor's theorem.

Unit 3 -

Maximum modulus principle. Schwarz lemma. Laurent's series. Isolated singularities.Meromorphic functions. The argument principle. Rouche's theorem inverse function theorem.

Unit 4 -

Mobius Transformations. Fixed Points, Cross Ratio, Bilinear transformations, theirproperties and classifications. Definitio'ns and Examples of Conformal mappings.

Unit 5 -

Residues. Cauchy's residue theorem. Evaluation of integrals. Branches of many valuedfunctions with special reference to arg z, log z and Z3.

Recommended Books:

[1] J.B. Conway, Functions of one Complex variable, Springer-Verlag, International StudentEdition, Narosa Publishing House, 1980.

[2] Brijendra Singh, Yarsha Karanjgaokar and R. S. Chandel, Complex Analysis, Gaura PustakSadan, Agra - 7.

Reference Books:

[1] S. Ponnusamy, Foundations of Complex Analysis, Narosa Publishing House, 1997.[2] L.Y. Ahlfors, Complex Analysis, McGraw-Hi!!, 1979.

[3] B. Singh, Varsha Karanjgoakar and R.S.Chandel, Complex analysis,Golden YalleyPublications.

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C.C.E. Marks: 10 Practical Marks:15

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M.Sc. Mathematics

Under CBCS (Only for School of Studies in Mathematics)SEMESTER I

Optional Paper V (A) Programming in C (Theory and Practical) I

Unit-1

An overview of programming languagesUnit-2

Classification. C Essentials - Programs development, FunctionsUnit-3

Anatomy of a Function. Variables and Constants Expressions. Assignment Statements.Formatting Source files Continuation Character. the Preprocessor.

Unit-4,

Scalar Data types - Declarations, Different Types of integers. Different kinds ofIntegerConstants Floating - point type Initialization

Unit-5

mixing types Explicit conversions - casts. Enwneration Types. the void data type, Typedefs.Pointers.

Reference Books:

(1) Sanmel P. Harkison and Oly L Steele Jr. C; A Reference manual, 2ao Edition Prentice hall1984.

(2) Brain W Kemigham & Dennis M Ritchie the C Programmed Language 2nd Edition (ANSI.features), Prentice Hall 1989.

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Basic Theorems- Ascoli- Arzela Theorem. A theorem on convergence of solutions ofafamily of initial value problems.


Picard-Lindeloftheorem-Peano's existence theorem and corollary. Maxi,mal intervals ofexistence. Extension theorem and corollaries. Kamke's convergence theorem. Kneser's theorem(Statement only)

Unit 4 -

rUnit 3-


Optional Paper V (B) Differential Equations - I

Unit 1 -

Initial value Problem and the equivalent integral equation, mth order equation in d -dimension as a first order system, concepts of local existence, existence in the large anduniqueness of solutions with examples.

Unit 2 -

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Differential inequalities and Uniqueness - Gronwall's inequality. Maximal and MinimalSolutions. Differential inequalities. A theorem of Winter. Uniqueness Theorems. Nagumo's andOsgoods's criteria.

Unit 5-

Egres points and Lyapunov functions. Successive approximations.

Linear Differential Equations - Linear systems, Variation of Constants, reduction to smallersystems. Basic inequalities, Constant coefficients. Floquet theory. Adjoint systems, Higher Orderequations.

Recommended Books:

[1] P. Hartman, Ordinary Differential Equations, John Wiley (1964).

Reference Books:

[1] W.T. Reid, Ordinary Differential Equations, John Wiley & Sons, NY (1971).

[2] E.A. Coddington and N. Levinson, Theory of Ordinary Differential Equaions, Mc GrawHill, NY (1955).

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Optional Paper V (C) Advanced Discrete Mathematics - I

Unit I -

Semigroups & Monoids - Definitions and examples of Semi groups and Monoids (includingthose pertaining to concatenation operation). Homomorphism of semigroups and Monoids.Congruence relation andQuotient Semigroups. Subsemigroup and submonoids. Direct products.Basic Homomorphism Tl ~orem.Unit 2 -

Lattices - Lattices as partially ordered sets. Their properties. Lattices as Algebraics~'st~ms. Sublatti<.:cs, Direct products,and Homomorphisms. Some Special Lattices e.g .. ,Complete, Complemented and Distributive Lattices.Unit 3-

Boolean Algebras Boolean Algebras as Lattices. Various Boolean Id~ntitil'<;.The SwitchingAlgebra example. Suhalgebras, Direct products and Homon.urphisms.join- irreducible elements,Atoms and Minterms. booleean forms and Teir Equivalence. Minterm Boolean forms, Sum ofproducts Canonical forms. Minimizaticn of Boolean Fuctions. Applications of boolean Alegebrato Switching Theory. ( using AND, (lR & NOT gates). the Karnaugh Map method.

Unit4-

Graph Theury- .).:finiition of (undirected) Graphs, Paths, Circuits C) c' 'S & <-:ubgraphs.Induced Subgraphs. Jl"gree ofa ver~.;x. Connectivity. Planar Graphs and their properties.Trees.

Unit 5 -

Eulers Formula for connected Planar Graphs. Complete & Complete Bipartite Graphs.Kuratowskis Theorem ( statement only) and its use. Spanning trees, cut-sets. FundamentalCut- Sets, and Cycles. minimal Spanning trees and Kruskals Algorithm. Matrix Representationsof Graphs.Recommended Books:ll] J.P.Trembly & R.Manohar, Discrete mathematical Structures with Applications to

Computer Science, McGraw Hill Book Co. 1997.

12] N. Deo, Graph Theory with applications to Engineering and Computer Sciences, PrenticeHall of India.

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Reference Books:III J.L.Gersting, Mathematical Structures for Computer Science, (3rd cdittion), Computer

Science Press, New york.12] Seymour Lepschutz, Finite Mathematics (International edition 1983) McGraw- Hill Book

Company, Newyork.13J S. Wiitala,Discrete Mathematics - A Unified Approach, MC graw- Ilill Book Co.[4] J.E.Hopcroft and J,D. UIIman, Introduction to Automata Theory Languages & Compulation

Narosa Publishing House.[5] B. Singh, R.S.Chandel and Akhilesh Jain, Advanced Discrete Mathematics,Golden Valley

Publications.

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RegularTheory Marks: 40r--C.C.K Marks: 10


Optional Paper V (D) Differential Geometry of Manifolds - IUnit I -

Definition and examples of differentiable manifolds. Tangent spaces. Jacobian map. Oneparameter goup of transformations.

Unit 11-

Lie derivatives. Immersions and Embeddings. Distributions. Exterior algebra. Exteriorderivative.

Unit 111-

Topological Groups. Lie groups and lie algebras. Product of two Liegroups.

Unit IV -

"One parameter subgroup and exponential maps. Examples of Lie groups.

UnitV -

Homomorphism and Isomorphism. Lie transformation groups. General Linear groups.Principal fibre bundle. Linear frame bundle.

Recommellded Books:

[1] B.B. Sinha, An Introduction to Modern differential Geometry, Kalyani Publishers, NewDelhi, 1982.

[2] K. Yallo and M. Kon, Structure of Manifolds, World Scientific Publishing Co. Pvt. Ltd.,1984.

Referellces Books:

[1] R.S. Mishra, A Course in tensors with applications to Riemannian Geometry, Pothishaia(Pvt.) Ltd., 1965.

[2] R.S. Mishra, Structures on a differentiable manifold and their applications, ChandramaPrakashan, Allahabad, 1984.


VIKRAM UNIVE~SlTV, UJJAIN

MASTER OF SCIENCE


The Course of study and the Scheme of Examinations~ - .

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II ~ " 'ltudv Component~ hrs dit Title of the Paper I Maximum Marks

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SEMESTERI Exam

1 CORE MAT-ClOl • 6 6 Advanced AbstractAgebra I 10 40 50

2 CORE MAT-Cl02 6 6 Real Analysis I / 10 40 50

3, CORE MAT-Cl03 6 6 ITopology I -/ 10 40 50

4 CORE MAT-Cl04 6 5 Comple~alysis I 10 40 50

5 ELECTIVE MAT-El05(A,B,C,D) 6 3+2=5 zse lout of 4)

V. Programming in C-I (Theory +Praetical) I 10 125+15

B. Differential Equation I : :0 40 50

IC. Advanced Discrete Mathematics I ] 10 40

/ D. Differential Geometry of Manifolds I _ 10 40.

6 120RE lMAT-CI06 1/6 2 Comprehensive Viva-Voce - 50 50

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7, CIA Uni. Total

SEMESTER11 Exam

7 CORE MAT-GOl 6 6 Advanced Abstract Algebra 11 10 40 50

8 CORE MAT-G02 6 6 Lebesgue Measure and Integration 10 40 50

9 CORE MAT-C203 6 6 Topology 11 10 40 50

10 CORE MAT-C204 6 5 Complex Analysis 11 10 40 50

11 ELECTIVE MAT-E205(A,B,C,D) 6 3+2==5(to choose lout of 4)

A. Programming in C-II (Theory +Practieal) 10 125+15

B. Differential Equation II 10 40 50

C. Advanced Discrete Mathematics II 10 40

D. Differential Geometry of Manifolds II 10 40

12 tORE MAT-C106 6 2 Comprehensive Viva-Voce - 50 50

36 30 50 250 300-

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r.M.Sc. MathematicsUnder CBCS (Only for School of Studies in Mathematics)SEMESTER IIPaper I Advanced Abstract Algebra - II

Unit 1 -

Introduction to Modules, Examples, Sub-modules and direct sums, Examples of sub-modules, Quotient Modules, R-Homomorphism and Examples ofR-Homomorphism,

Unit 2-

Fi'nitely generated modules. Cyclic modules, Simple modules, Schur's Lemma, Freemodules.

Unit 3 t

Noetherian and A.rtinian modules and rings, I-lilbert basis theorem.r

Unit 4 ,

Uniform modules, primary modules and Noether-Lasker theorem.

Unit 5 -

Algebra oflinear transformations, Characteristic roots, Similarity oflinear transformations,Invariant subspaces, Reduction to triangular forms, Nilpotent transformations, Index ofnilpotency, Invariants of a nilpotent transformation, The primary decomposition theorem.

Recommended Books:..

[I] I. N. Herstein. Topics in algebra, Wiley Eastern Ltd. New Delhi, 1975.[2] Vevek Sahai and Vikas Bist, Algebra, Narosa Publishing House, 1999.

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Reference Books:

[1] P.B. Bhattacharya, S.K. Jain and S.R ..Nagpaul, Basic Abstract Algebra (2nd Edition),Cambridge University Press, Indian Edition, 1997.

[2] S. Kumaresan, Linear Algebra - A geometric approach, Prentice Hall of India, Ltd.

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Lebesgue Measure and Integration

~.~ <,ThcOl1' Marks: 40C.C.E. Marks: 10

M.Sc. Mathematics

Under CBCS (Only for School of Studies in Mathematics)SEMESTER II

Paper IIUnit 1-

Lebesgue outer measure. Measurable sets. Regularity. Measurable functions. Borel and Lebesguemeasurability. Non-measurable sets.Unit2 -

Integration of Non-negative functions. The General integral. Integration of Series. Reimann andLebesgue integrals.

Unit3 -

The Four derivatives. Functions of bounded variation. Lebesgue Differentiation Theorem.Differentiation and IntegratR:m.Unit4 -

The e spaces, Con~ex functions, Jensen's inequality, Holder and Minkowski inequalities,Completeness of e.Unit 5-

Dual of space, Convergence inMeasure, Unifonn convergence and Almost tmif()rmconvergence.

Recommended Books:

[1] G.de Barra, Measure Theory and Integration, Wiley Eastern Limited, 1981.

Reference Books:

[I]. Walter Rudin, Principles of Mathematical Analysis (3rdedition), McGraw-Hill, Kogakusha, 1976,International Student edition.

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

Inder K. Rana, An Introduction to Measure and Integration, Narosa Publishing House, 1997.

P.R. Halmos, Measure Theory, Van Nostrand, Princeton, 1950.

P.K. Jain and V.P. Gupta, Lebesgue Measure and Integration, New Age International (P) LimitedPublished New Delhi, 1986 (Reprint 2000).

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M.Sc. MathematicsUnder CBCS (Only for School of Studies in Mathematics)SEMESTER II

Paper III Topology - II

Unit 1 -

Separation axioms To' T I' T2' T31/2 , T4 their characterization and basic properties.Urysohn's lemma. Tietze extension theorem.

Unit 2-

Compactness. Continuous functions and compact sets. Basic properties of compactness.Compactness and finite intersection property. Sequentially and countably compact sets.Local Compactness and one point compactification. Stone-Cech compactification.

Unit 3 -

Tychonoffproduct, Projection maps. Separation axioms and product spaces.Connectedness and product spaces. Compactness and product spaces (TychonoffTheorem). Embedding lemma and Tych~noff embedding.

Unit 4 -

Nets and Filters. Topology and Convergence of nets. Hausdorffness and nets.Compactness and nets. Filters and their convergence. Canonical way of converting netsto filters and vice versa. Ultrafilters and compactness.

Unit 5 -r.li~i~ The fundamental group and covering spaces-Homotopy of paths. The fundamental group.i;I.'( Covering spaces. The fundamental group of the circle and the fundamental theorem ofI!; algebra. .

Recommended Books:

[I] James R. Munkres, Topology: A First Course, Prentice Hall of India Pvt. Ltd. NewDelhi,2000.

[2] K.D. Joshi, Introduction to General Topology, Willey Eastern Limited, 1983.Reference Books:

[I] George F. Simmons, Introduction to Topology and Modern Analysis, McGraw-Hill BookCompany, 1963.

[2] J. Dugundji, Topology, Allyn and Bacon, 1966 (Reprinted in India by Prentice-Hall ofIndia Pvt. Ltd.)

[3] N. Bourbaki, General Topology part-I (Trans!.) Addison Wesley Reading 1966.

[4] B. Mendelson, Introduction to Topology, Allyn & Becon, Inc., Boston, 1962.

[5] E.H. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966.

[6] J.L. Kelley, General Topology, Van Nostrand, Reinhold Co., New York, 1995.

[7] M.J. Mansfield, Introduction to Topology, D.Van Nostrand Co. Inc., Princeton, N.J.1963.

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Unit3 -Schwarz Reflection principle. Monodromy theorem and its consequences. Harmonic

functions on a disk.

M.Sc. MathematicsUnder CBCS (Only for School of Studies in Mathematics)SEMESTER IIPaper IV Complex Analysis - II

Unit 1-


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Weierstrass' factorisation theorem. Gamma function and its properties. Riemann Zeta function.Riemann's functional equation.Unit2- .

Runge's theorem. Mittag-Leffler's theorem. Analytic Continuation. Uniqueness of directanalytic continuation. Uniqueness of analytic continuation along a curve. Power series methodof analytic continuation.

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Unit 4-Harnack's inequality and theorem. Dirichlet problem. Green's function. Canonical products.

Jensen's formula. Poisson - Jensen formula. Hadamard's three circles theorem. Order ofan entirefunction. Exponent of Convergence. Borel's theorem. Hadamard's factorization theorem.

Dnit 5-The range of an analytic function. Bloch's theorem. The little Picard theorem. Schottky's

theorem. Montel Caratheodary and great Picard theorem. Univalent function. Bieberbachconjecture and the 1/4 theorem.'

Recommended Books:[1] J.B. Conway, Functions of one Complex variable, Springer-Verlag, International Student

Edition, Narosa Publishing House, 1980.

Reference Books :[1] S. Ponnusamy, Foundations of Complex Analysis, Narosa Publishing House, 1997.

[2] H.A. Priestly, Introduction to complex analysis, Clarendon Press, Oxford, 1990.

[3] D, Sarason, Complex Function Theory, Hindustan Book Agency, Delhi, 1994.

[4] E.C. Titchmarsh, The Theory of Functions, Oxford University Press, London.

[5] L.V. Ahlfors, Complex Analysis, McGraw-Hili, 1979.

[6] Walter Rudin, Real and Complex Analysis, McGraw-Hili Book Co., 1966.

[7] S. Saks and Zygmund, Analytic Functions, Monografie matematyczne, 1952 .

[8] B. Singh, Varsha Karanjgoakar and R.S.Chandel, Complex analysis,Golden Valley

Publications.

...._ ..._._..._...._._, .. .__ .._.• _---.----'--~_.-. ~:C--. -. .,.....

Operators and Expressions - Precedence and associatively. Unary plus and Minus operators.Binary Arithmetic operators arithmetic assignment operators. Lr1crementand decre~entoperators. Comma Operator Relational (lperators logical operators bit- Manipulation operatorsBitwise assignment operators. Cast operators size of Operators, Conditional Operators,memory operator.

Unit-4 ...


Optionall>aper V (A) Programming in C (Theory & Practical)-II

Unit-lControl Flow - Conditional Branching, the Switch Statement. looping. nested loops

Unit-2

The Break and Continue statement. the goto statement infinite loops.Unit-3

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Arrays and multidimensional Arrays. Storage Classes - fixed VS. Automatic Duration Scope,global variable

Unit-5

The Register Specifier Structures and Unions.

Recommended Books:

(1) Peter A Darnell and Philip E. Margolis, C; A Software Engineering Approched narosa PublishingHouse (Springer International Student Edition) 1993.

Reference Books:

(1) Samuel P. Harkison and Gly L Steele Jr. C; A Reference manual, 2an Edition Prentice hall1984.

(2) Brain W Kernigham & Dennis M Ritchie the C Programmed Language 2nd Edition (ANSIfeatures), Prentice Hall 1989.

~lI',



Optional Paper V (B) Differential Equations - II

Unit 1 -

Dependence on initial conditions and parameters, preliminaries. Continuity. Differentiability.Higher Order Differentiability.

Unit 2 -

Poincare-Bendixson Theory - Autonomous Systems. Umlanfsatz. Index ofa stationarypoint.

Poincare-Bendixson Theorem. Stability of periodic solutions, rotation points, foci, nodesand saddle points.

if Unit3-'j

Linear second order equations-preliminaries. Basic facts. Theorems of Sturm. Sturm-Liouvelle Boundary Value Problems. Number of Zeros. Nonoscillatory equations and principai.solutions. Nonoscillation theorems.

Unit 4-

Use ofImplicit function and fixed point theorems- periodic solutions. Linear equations.Nonlinear problems.

Unit 5 ->

Second Order Boundary Value Problems- Linear Problems. Nonlinear problems. Aproribounds.

Recommended Books:

[il [1] P. Hartman, Ordinary Differential Equations, John Wiley (1964).

Reference Books:

[1] W.T. Reid, Ordinary Differential Equations, John Wiley & Sons, NY (1971).

[2] E.A. Coddington and N. Levinson, Theory of Ordinary Differential Equaion~, Mc GrawHill, NY (1955).

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Optional Paper V (C) Advanced Discrete Mathematics - II

Unit 1 -

Directed Graphs. Indegree and Outdegree of a Vetex. Weighted Undirected Graphs,Dijkstra's Algorithms. Strong connectivity and Warshall's Algorithms. Directed Tress. SearchTrees. Tree Traversals.Unit 2 -

Introductry Computability Theory- Finite State Machines and their Transition TableDiagrams. Equivalence ofFintie State Machines. Reduced Machines. Homomorphism. FiniteAutomata. Acceptors.Unit 3 -

Non- deterministic finite Automata and equivalence of its power to that of Deter minis ticsFinte- Automata Moore and Mealy Machines.Unit 4 -

Turing Machine and Partial Recursive Functions.

Grammars and Languages - Phrase- Structure Grammars. Rewriting Rules. Derivations.Unit 5-

Sentential forms. Language generated by a Grammar. Regular, Context -Free, andContext Sensitive Grammers and Languages Regular Sets, Regular Expressions and the PumpingLemma Kleenes Theorem.

Notions of Syntax Analysis. Polish Notaticns Conversion ofInfix ExperessionS' to PolishNatations. The Reverse Polish Notation.

Recommended Books:

[1] J.P.Trembly & R.Manohar, Discrete mathematical Structures with Applications toComputer Science, McGraw Hill Book Co. 1997.

[2] N. Deo, Graph Theory with applications to Engineering and Computer Sciences, PrenticeHall of India.

Reference Books:

[1] J.L.Gersting, Mathematical Structures for Computer Science, (3rd edittion), ComputerScience Press, New york.

[2] Seymour Lepschutz, Finite Mathematics (International edition 1983) McGraw- Hill BookCompany, Newyork.

[3] S.Wiitala,Discrcte Mathematics - A Unified Approach, MC graw- Hill Book Co.[4] J.E.Hopcroft and J,D. Ullman, Introduction to Automata Theory Languages & Compulation

Narosa Publishing House.

[5] B. Singh, R.S.Chandel and Akhilesh Jain, Advanced Discrete Mathematics,Golden ValleyPublications.

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Optional Paper V (D) Differential Geometry of Manifolds - II

Unit 1-Associated fibre bundle. Vector bundle. Induced undle. Bundle homomorphisms.

Unit II -

Riemannian manifolds. Riemannian connection. Curvature tensors. Sectional Curvature.Schur's theorem.

Unit 111-Geodesics in a Riemannian manifold. Projective curvature tensor. Conformal curvature

tensor.

Unit IV-

Submanifolds & Hypersurfaces. Normals. Gauss' formulae. Weingarten equations. Linesof Curvature. Generalised Gauss and Mainardi-Codazzi equations.

Unit V-Almost Complex manifolds. Nijenhuis tensor. Contravariant and covariant almost analytic

vector fields. F-co:mection.

Recommended Books:[1] B.B. Sinha, An Introduction to Modern differential Geometry, Kalyani Publishers, New

Delhi, 1982.[2] K. Yano and M. Kon, Structure of Manifolds, World Scientific Publishing Co. Pvt. Ltd.,

1984.References Books:[1] R.S. Mishra, A Course in tensors with applications to Riemannian Geometry, Pothishala

(Pvt.) Ltd., 1965.[2] R.S. Mishra, Structures on a differentiable manifold and their applications, Chandrania

. Prakashan, Allahabad, 1984.

•

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, ."I

I CIA Uni. IrotalSEMESTER III

Exam" Integration Theory and Functional.

13 CORE MAT-C301 6 6 Analysis-I 10 40 50Fundamentals of ComputerScience(Theory)-1 10 25 35Fundamentals of Computer Science (

14 CORE MAT-C302 6 6 Practical)-I - 15 15MAT.E303

15 ELECTIVE-I (A,B,C, 0, E) 6 6 (to choose lout of 5) 10 40 50A. Advanced Functional Analysis-IB. Partial Differential Equations .-C. Differentiable Structures on

manifolds-I.D . General Theory of Relativity andr

Cosmology-IT E. Abstract Harmonic Analysis-I

F. Mathematics ofrinance & Insurance,

y -I- ,(to choose lout of 5)A. Theory of Linear Operator IB. Advanced Numeric<ll Analysis-I

16 ELECTIVE-IIMAT-E304

6 5 C. Fuzzy Sets and their Applications-I(A,B,C,D,E) D. Advanced Graph Theory-I

E. Advanced Special Function-IF. Spherical Trigonometry and

astronomy-I 10 40 50(to choose lout of 5)A. Operations Research -I

MAT-E30S B. Computational Biology -I17 ELECTIVE-III 6 5 t" Fluid Mechanics -1(A,B,C,D,E) --

D. Bio- Mechanics -IE. Analytic Number Theory-IF. Integral Transform-I 10 40 50

MAT-18 tORE K:306 6 :2 Comprehensive Viva-Voce - 50 50

36 30 50 250 300" , ;'/ ' . .'. ' , " I'-0- ., •... A ••••

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Unit-1Signed measure. Hahn decomposition theorem, mutually singular measures. Radon-Nikodim

theorem. Lebesgue decomposition. Riesz representation theorem.Unit-2

Outer measure, Extension theorem Caratheodory theorem, Lebesgue -Stieltjes integral, productmeasures, Fubini's theorem.Unit-3

Nomled linear spaces. Banach spaces, Further prope11iesofNOlmed Spaces, Finite dimensionalNormed Spaces and Subspaces and Quotientnormed linear space.Unit-4

Compactness and finite dimension, Linear Operators, B0u.nded and Continuous Linear Opera-tors.Unit-5 ,

Linear Functional~, Lmear Operators and functional on fini'tedimensional Spaces, NormedSpaces of Operators and Dual Space.

Recommended Books:[1] B.L. Royden, Real Analysis, Prentic Hall of India Publishing Co. Inc.[2] E. Kreyszig, Introductory Functional Analysis with applications, Jolm Wiley & Sons Ne\v York.[3] G.F. Simmons, Introduction to Topology & Modern Analysis Me Graw Hill, New York.

Reference:[1] B. Choudhary and Sudarshan Nanda. Functional AnalysiJ with applications, Wiiey Eastern Ltd.

M.Sc. MathematicsUnder CBCS (Only for School of Studies in Mathematics)SEMESTER III

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;,

Paper II Fundamentals of Computer Science-IUnit 1 -

Object Oriented Programming Paradigm, Basic Concepts, Benefits and Applications of ObjectOriented Programming.

Unit 2-C++ - Introduction, Tokens, Keywords, Identifiers and Constants, Basic Data Types, User-

Defined Data Types, Derived Data Types, Vari~bles, Operators in C++, Expressions, ImplicitConversions. j

Unit3 - ,Operator Overloading, Operator Precedence, Control Structure - The if Statement, The switch

Statement, The do ...while Statement, The while Statement, The for statement.

Unit 4-Functions in C++, The main Function, Function Prototyping, Call by Reference, Inline Fllliction,

Function Overloading, Friend and Virtual Functions.

Unit 5-Classes and Objects: Specifying a Class, Defining Member Function, Nesting of Member

Function, Private Member Functions, Arrays within a Class, Static Data Members, Static MemberFunctions, Pointers to Members.

Reference Books:[J] E. Balagurusamy, Object Oriented Programming with C++, III Edition, Tata McGraw-HilI

Publishing Company Limited, New Delhi.[2] 13.Stroustrup, The C++ programming Language, Addition Wesley.

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RegularTheory Marks : 40C.C.E. Marks: 10

M.Sc. MathematicsUnder CBCS (Only for School of Studies in Mathematics)SEMESTER II I

Optional Paper I (A) Advanced Functional Analysis-I

Unit-lDefinition and examples of topological vector Spaces Convex, Balanced and absorbing

sets and their properties.

Unit-2Minkowski's functionals, Subspace product space and quotient space of a topological

Vector space. Chapte.r I of R. Larsen.

Unit-3Locally convex topological Vector Spaces. Normable a,"1dmetrizable topological vector

spaces.Unit-4

Complete topological vector spaces and Frechet space.Chapter 2 and 3 ofR. Larsen.

Unit-5Linear transformations and linear functionals and their continuity. Chapter 2 and 3 ofR.

Larsen.

Recommended Books:[1] R.Larsen, Functional Analysis, Marcel Dekker, Inc. New york, 1973.[2] Walter Rudin, Functional Analysis, TMH Edition,1974.[3] L.V.Kantorovich and G.P. Akilov, Functional Analysis, Pergamon Press 1982.

Reference Books:1. Laurent Schwartz, Functional Analysis Courant Institute of Mathematical Sciences, New

York University, 1964.

•


Optional Paper I (B) Partial Differential EquationsUnit-l

Transport Equation-Intial value Problem Non-homogeneous Equation.Laplace's Equations-Fundamental Solution, Mean Value Formula, Properties of Harmonic functions, Green'sFunctions, Energy M~thods.

Unit-2

Heat Equation - Fundamental Solution, Mean Value Formula, Properties of Solutions, EnergyMethods. Wave Equation- Solution by Spherical Means, Non-homogeneous Equations,..Energy Methods.

Unit-3

Nonlinear First Order PDE. Complete integrals, Envelopes, Characteristics, Hami}ton-Jacobi Equations (Calculus of Variations, Hamiltons ODE, Legepdre Transform,Hopf-Lax formulae)

Unit-4

Conservation Laws (Shocks, Entropy Condition Lax-Oleinic formula, RiemannsProblem, Long Time behaviour).Represenation of Solutions- Separation of Variables , Similarity Solutions (Plane andTrayelling Waves, Soluitions, Similarity under Scalling)

Unit-5

I::

Max. Marks 50

RegularTheory Marks: 40C.C.K Marks: 10

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Fourier and Laplace Transform, Hopf- Cole Transform, Hodographand, lcgen-dreTransforms, Potential Functions, Power Series (non - charcateristic surface, RealAnalytic Functions, Cauchy - Kovalevskaya Theorem).

Recommended Books:- (1) L.c. Evans, Partial Differential Equations, 1998.

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RegularTheory MarKs: 40C.C.E. Marks: 10

M.'Sc. MathematicsUnder CBCS (Only for School of Studies in Mathematics)SEMESTER III

Optional Paper I (C) Differentiable Structures on manifolds-I

Unit-lSubmanifolds & Hyper surfaces. Normals. Gauss's formulae, Weingarten equations .

Unit-2Lines of Curvature. Generalized Gauss and Mainrdi - Codazzi equations .

Unit-3almost complex manifolds, Njcnhuis tensors. Contravariant and covariant almost analytic vectorfields.

Unit-4F-connection, almost Hermit manifolds.

Unit-5almost analytic vector fields. cUrvature tensor, Linera connection.

Recommelided Books.1. B.B. Sinha, An Introduction to modem Differential Geometry, Kalyani Publishers, new Delhi,

19822. K. Yano and M. Kon structure of Manifolds. world scientific Publishing C. Pvt. Ltd. 19843. A. Behaneu, Geomatry ofCR- sub manifolds, D. Reidel Publishing company, Dordrccht,

1986

Reference Books:(i) R.S. Mishra, A course in tensor with application to Riemannian geometry

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Optional Paper I (D) General Theory of Relativity and Cosmology-I

Unit-1 Transformation of coordinates. Tensors. Algebra of Tensors. Symmetric and skew symmetricTensors.

Unit-2 Contraction of tensors and quotient law. Riemannian metric. Christoffel SymbolsUnit-3 Covariant derivatives. Gradient, Divergence and Curl.Unit-4 Intrinsic derivatives and geodesics, Riemann Christoffel cwvature tensor and its symmetry

properties.Unit-5 Intrinsic derivatives and geodesics, Riemann Christoffel cwvature tensor and itc;symmetry

properties.

RecoInmended Books:[1] S..R.Roy and Raj Bali: Theory of Relativity Jaipur Publishing House,Jaipur, 1987.[2] S: K. Shrivastva: General Relativity and Cosmology, PHI, New Delhi.[3] J.V. Narlikar, General Relativity and Cosmology: The Macmillan Company of India Limited, 1978.

References:[1] C.E. Weatherbmn, An Introduction to Riemannian Geometry and the tensor Calculus, Cambridge

University, Press 1950.[2] H. Stephani, General Relativity: An Introduction to the theory of the gravitational field,

Cambridge University Press 1982.[2] A.S. Eddington, The Mathematical Theory of Relativity. Cambridge University Press, 1965.[4] R. Adler, M. Bazin, M. Schiffer, Introduction to general relativity, McGraw Hill Inc.,1975.

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M.Sc. Mathematics

Under CBCS (Only for School of Studies in Mathematics)SEMESTER III

Optional Paper I (E) Abstract Harmonic Analysis I

Unit-l

Topological groups, Examples of topological groups and its basic Properties. Subgroups andquotient groups.

Unit-2

Product groups and projective limits. (See G. Bachman[l]) Continuity, homeomorphism.left translate, right translate, inversion mapping, inner automorphism,

Unit-3

Homogenous topological group. Properties of topological groups involving connectedness.Invariant pseudo-ipetrics and separation axioms.

Unit-4

Symmetric neighbomhood of identity, compact sets, Structure theory for compact and locallycompact Abelian groups. (See Hewitt and Ross [3j), Locally compact topological groups

Unit-5

Compact support subgroups and quotient groups of topological groups, topology for quotientgroup, open sets, Open mapping, Hausdorff quotient group compact quotient group.

Reconunended Books.

1- George Baclunan Elements of Abstract Hannonic Analysis Acadmic Press, New Yom. 1964 2-Taqdir Hussain Introduction to Topological Group W.D. Saudss Company 1966 to ok W.O. 3-Walter Rudin, Fourier Analysis On Group Intersceince publisher, John wiley, New York,1967

Reference Books.1- Edwin Hewit and Kenneth A. Ross. Abstract Hanllonic Analysis -1, Springer - Verlag, Berlin,

1963.

2- lynn H. Loomis: An Introduction to Abstract Ha.l1nonicAnalysis, D, Van Nostrand Co. Princet

Max. Marks 50

Regularj'.! Theory Marks : 40

C.C.K Marks: 10M.Sc. Mathematics


Optional Paper I (F) Mathematics of Finance and Insurance- I

T .

Unit-l Elements of Theory of InterestUnit-2 Flow Valuation Annuities

Unit-3 Amortization and Sinking Funds, brief review of probability theory.

Unit-4 Survival Distributions, Life Tables, Valuing Contingent Payment Life insurance,Unit-5 life annuities, Net Premiums Insurance Models including Expenses.

TextBooks:

1 Options, Futures and other Drivativesby Jhon C. Hull Prentice -Hall of India Pvt. Ltd.2 An,introduct~on to Mathematic Finance by Che1donM. Ross, Cambridge University Press.

Reference Books:

1 An Introduction to Mathematics of Financial Derivatives by Salih N.Neftci, Academic Press, Inc.2 Mathematics of Financial markets by Ribert J. Elliot & P.E. Kopp Springer Verlag, New Yorlk

Inc


Max. Marks 50

M.Sc. Mathematics


Optional Paper II (A) Theory of Linear Operators IUnit-l

Spectral theory in normed linear spaces, resolvent set and spectrum, Spectral properties ofbounded linear operators. Properites of resolvent and spectrum, Spectral maping theoremfor polynomials.

Unit-2

Spectral radius of a bounded linear operator on a complex Banach space. Elenientarytheory of Banach alegbras. General properties of compact linear operators. Spectral prop-erties of compact linear operators on normed spaces. Chapter 7,8 (E. Kreyszig).

Unit-3

Behaviours ofCompa~t linear operators with respect to solvability of operators equations.Fredholm type theorems. Fredholm alternative theorem. Fredholm alternative for integral

T .

equationsUnit-4

Spectral properties of bounded self -adjoint linear operators on a complex J-Iilbert space.Positive operators. Monotone sequence theorem for bounded self -adjoint operators on acomplex Hilbert space.

Unit-5

Square roots of a positive operator. projection operators. Spectral family of a boundedself -adjoint linear operator and its properties.

Recommended Books:(1) E. Krcyszig Introductory functional analysis with applications, 1hon wiley & Sons, Nwe York,

1978.

Referance Books:(1) P. R. Halmos Introduction to Hilbert space and the theory of Spectral Multiplicity, Second

edition, Chelsea publishing co. N.Y. 1957.(2) N. Dundford and J.T. Schwartz, linear operator -3 part, Interscience / Wiley, New York 1958-

71.

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Optional Paper II (B) Advanced Numerical Analysis I

Unit-I

Piece wise and spline interpolationUnit-2

Bivariate inter polation Approximation,Unit-3

Least squares approximationUnit-4

UnifOlm approximation Rational approximAtion, choice of methodUnit-5

Numerical differentiation optimum choke of step length

Text book-

(1) Numerical Methods for scientific and Engineering computation by M.K. Jain, S.R.K.lyenger,R.K. Jain south Edition (2003) New Age .

l-t..

Max. Marks 50

RegularTheory Marks : 40c.c.E. Marks: 10

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Optional Paper 1J (C) Fuzzy Sets and Their Applications I

Unit-lFuzzy Sets-Basic definitions, A-level sets, convex fuzzy sets,

Unit-2Basic operations on fuzzy sets Types offuzzy sets, Cartesian, Product, Algebraic products,

Unit-3Bow1dedsum and difference, t-norms and T - co norms.

Unit-4The Extension Principle - The Zadeh's extension principle,

Unit-5Image and inverse image offuzzy sets, fuzT.jnumbers, EJentents offuzzy arithmetic.

TextBooks:(l) Fuzzy set theory and its Applications by H.J. Zimmennarm, Allied Publishers Ltd., New Delhi,

1991.

(2) Fuzzy sets and Fuzzy logic byGJ. Klir and B. Yuan Prentice - Hall of India, New Delhi, 1995

Reference Books:-(1) Fuzzy sets and Uncertainty and Information by q.J. Kalia Tina A. Foljer - Prentice - Hall of

India.

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Optional Paper II (D) Advanced Graph Theory- I

Unit-l Revision of graph theoretic preliminaries, Operations on graphs. Graph IsomorphismDisconnected graph and their Components. Traveling salesman problem, round tableproblem,

Unit-2 Eulerian and Hanliltonian Paths and circuits.

Unit-3 Properties of trees, Distance centre, radius, diameter eccentricity and related theorems, Graphas Metric space Rooted and binary trees,

Unit-4 Labelled graph and trees spanning tree, weighted spanning tree, Shortest path,Unit-5 fundamental cutsets. Rank and nullity, cutsets and cut vertices, fundamental cut;ets,

Text Book:-

1- Graph Theory with Application to Engineering and Computer Science byNarsingh Deo.

Reference Book:-1- Graph Theory by Harary.

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Optional Paper II (E) Advanced Special Function I

Unit-lGamma and Beta Functions: The Euler or Mascheroni Constant? , Ganuna Flillction, A series

for' (z) / r (z), Difference equation ?(z+ 1) = z?(z),

Unit-2Beta function, value of?(z) '1(1-z),Factorial Function, Legendre's duplication formula, Gauss

multiplicationtheorem.Unit-3 Hypergoemetric and Generalized Hypergeometric functions: FlUlction 2Fl (a,b;c;z) A simple

integral form evaluation of2Fl (a,b;c;z)Unit-4 Contiguous function relations, Hyper geometrical differential equation and its solutions, F

(a,b;c;z) as function of its parameters,Unit-5 Eiementary series manipulations, Simple transfOml<ltion,Relations between functions ofz and 1

-z

:BooksReconunended;1- Rainville, E.D, ; Special Functbns, The Macmill';ln co., New york 1971,2- Srivastava, H.M. Gupta, K.c. and Goyal, S.P.; The H-functions ofbne and Two Variables with

applications, South Asian Publication, New Delhi.3- Saran, N., Sharma S.D. and Trivedi, - Special Functions with application, Pragati prakashan,

1986.

Reference Books.1- Lebdev, N.N, Special Functions and Their Applications, Prentice Hall, Englewood Cliffs, New

jersey, USA 1995.2- Whittaker, E.T. and Watson, G.N., A Course of Modern Anal

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Optional Paper II (F) Spherical Trigonometry an Astronomy- I

Unit-1 Fundamentalof Spherical TrigonometryUnit-2 solution of right angled triangleUnit-3 Properties of Right angle triangle

Unit-4 Relation between Sides & angles of a Spherical triangleUnit-5 Application of Spherical triangle & Examples.

Text Books:-

1- A text book of spherical trigonometry : Gorakh Prasad.2-(\ text book of spherical Astronomy: Gorakh Prasad.Reference Book.

1- Spherical Astronomy - Smarat2 - spherical Astronomy - Bell

:, Max. Marks 50


M.Sc. MathematicsURder CBCS (Only for School of Studies in Mathematics)SEMESTER III

Optional Paper III (A) Operations Research I

Unit-I. Operations Research and its scope ,Nature and Meaning of OR, Origin and Development ofOR, Necessity of OR in Industry, Case studies of OR. Model in OR, Main Face of OR ,Uses and

limitation of OR, Scope of OR, and role of OR in decision making.Unit-2. Linear Programming Problem, MaL,ematical Formulation, Graphical Solution Method. Graphi-cal Solution in some exceptional cases. Geometrical propertit?sofL.P.P, General Formulation ofL.P.P, Slack and Surplus Variables, Standard form ofL.P.P. Asswnptions in L.P.P, Limitation of

L.P.P.Unit-3. Linear Progranuning Problem -Simplex Method with exceptional ca~es, Computational proce--dureof simplex method, artificial variable techniques; Big M method, two phase Method, Problem of

degeneracy.Unit-4. Duality: Fundamental properties of Duality and Theorem of Duality.Unit-So Transportation Problems, Initial Feasible Solution to T.P., North- West comer rule, Rowminima, Colwn Minima, Matrix Minima, VAM; Optimality test for the initial Feasible solution, Degen-

eracy in TP., Assignment Problems ,Hungarian Method fo~assignment Problem and unbalanced

assignment Problem.Recommended Books:-1- Kanti Swarup, P.K. Gupta and Manmohan, Operations Research, Sultan Chand & Sons, New

Delhi. I

Reference Books:-1- S.D, Sharma, Operation Research,2- F.S, I-Iillerand GJ. Lieberman, Industrial Engineering Series, 1995 (This book comes with a CD

containing software)3- G. Hadley, Linear Programming, Narosa Publishing House. 1995.4- G. Hadley, Linear and Dynamic programming, Addison - Wesley Reading Mass.S _ H.A. Taha, Operations Research - An introduction, Macmillan Publishing co. Inc. New york.6- Prem Kumar Gupta and D.S. Hira, Operation Reasearch, an Introdution, S. Chand & Company

Ltd. New Delhi.7- N .S. Kambo, Mathematical Progranll11ingTeclmiques, Affilated East - West Pvt. Lt

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Max. Marks 50

./ RegularTheory Marks: 40

C.C.E. Marks: 10M.Sc. MathematicsUader CBCS (Only for School of Studies in Mathematics)SEMESTER III

Optional Paper III (B) Computational Biology- I

Unit-l Basic concepts of Molecular biologyUnit-2 DNA and Proteins, The Central Dogma, Gene and Genome Sequences.Unit -3 Restriction Maps - Graphs, Interval graphs. Measuring Fragment sizes,Unit-4 Algorithms for double digest proolem (DDP) - Algorithms and complexity,Unit-5 Approaches to DDP.

r

Text B09ks:-! 1- Introduction to Computational Biology by M.S, Waterman Chapman & Hall, 1995.

2- Bio informatics - A practical Guide to the analysis of Genes and Proteins by A. Baxevanisand B. Ouelette, Wileylnterscience (1998).

Reference Books:-1- Introduction to Bio informatics by Attwood.2- Bioinformatics-Sequence and Genome analysis by David W.Mount.

"j/).);r'" III

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M.Sc. MathematicsUDder CBCS (Only for School of Studies in Mathematics)SEMESTER ITT

Optional Paper III (C) Fluid Mechanics- I

Unit-1 Lagrangian and Eulerian MethodsUnit-2 equation of continuity, types offlow lines, velocity potential, C

Unit-3 stream function irrigational and rotational motions, vortex lines.Unit-4 Lagrange's and Euler's equation of motion, bumoulli's theorem,

Unit-5 irrotational motion in two dimensions,j

TextBooks.1- A text book offluid Mechanics in SI units by R.K, Rajput.2- An introduction to Fluid Dynamics by R.K. Rathy,:Oxford and IBH Published Co.

Reference Books:1- Fluid Mechanics (Springer) By Joseph H. Spurk.2- Fluid Mechanics by Irfan A Khan (H.R. W.)3- An Introduction to Fluid Mechanics by O.K. Batchelor, FouJldation Books, New Delhi,

1994.

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Max. Marks 50


M.Sc. MathematicsU n d ere B C s (Only for School of Studies in Mathematics)SEMESTER III

Optional Paper III (D) Bio-Mechanics- I

Unit-1 Bio-physics of Human Cardio - vascular system: Types of Blood Vessels, Properties of BloodUnit-2 Flow in Tubes, Poiseuibles law, Erythrocyte Sedimentation Rate, Stroke's law, Palatial flow in

elastic vessels.Unit-3 Bio - physics of Human Thermo- Regulation Head Flow in Human Dermal and Subdermal

partst

Unit-4 Derivation of Governing partial differential equations IncorporatingUnit-5 Microcirculation and perspiration.

Text books: ~.

I - Introduction to Mathematical Biology by S.l. Rubinow, J. Wiley & Sons.2- Biomechanics by Y.C, Fung, Springer - Verlag.

3- Introduction to Biomathematics by V.P. Saxena, Vishwa Prakashan (Wiley eastern)Reference Book :-

1- Biofluid Dynamics byMazumdar.

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/ RegularTheory Marks: 40C.C.E. Marks: 10


Optional Paper III (E) Analytic Number Theory- I

Unit-1 Characters offInite abelian groupsUnit-2 The Character Group, DiricWet charactersUnit-3 Sums involving Dirichilet characters.Unit-4 Dirichlet Theorem on primes in arithmetic progressions.Unit-5 DiricWet series and Euler products,

t

Book Rocommneded :1- T.M. Apostol, Introduction to Analytic Number Tilleory,Narosa Pub, House, 1989.

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M.Sc. MathematicsUader CBCS (Only for School of Studies in Mathematics)SEMESTER III

Optional Paper III (F) Integral Transform I

Unit-lApplication of Laplace Transfonns

Unit-2Laplace's equations,

Unit-3Laplace's wave equation

Unit-4Application of Laplace Transforms

Unit-5Heat conduction equation.

Books reconunendcd:-[1] Integral Trans,forms by Goyal & Gupta.[2] !ntegral TransfOlIDSby Sneddon

.---- -ISEMESTER IV

CIA Uni. TotalExam

19 CORE MAT-C401 6 6 Functional Analysis-II 10 40 SOFundamentals of ComputerScience(Theory)-1I 10 25 35Fundamentals of Computer Science (

20 CORE MI\T-CI102 6 6 Practical)-II - 15 15MAT-E403

21 ELECfIVE-1 (A,B,C,D,E) 6 6 (to choose lout of 6) 10 40 SOA. Advanced Functional Analysis-IIB. MechanicsC. Differentiable Structures on

I manifolds-III D. General Theory of Relativity and

Cosmology-II

IE. Abstract Harmonic Analysis-IIF. Mathematics of Finance & Insurance. -II

I• -

(to choose lout of 6)A. Theory of Linear Operator IIB. Advanced Numerical Analysis-II ,

')') MAT-E404 6 4C. Fuzzy Sets and their Applications-II ,

I... ELECTIVE-II (A,B/C,D,E) D. Advanced Graph Theory-II IE. Advanced Special Function-!!

I

IF. Spht:rical Tri~onom~try andastronomy-II .0 40 SO

(to choose lout of 6) I

I A. Operations Research -II

I MAT-E405B. Computational Biology -II

23 LECTIVE-1I1 6 4 C. Fluid Mechanics -II(A,B,C,D,E) D. Bio- Mechanics -II

E. Analytic Number Theory-IIF. Integral Transform-II 10 40 SO

\1AT-24 -'ORE C406 6 2 Comprehensive Viva-Voce - SO SO

- MAT-25 ""ORE P407 2 Job Oriented Project Work - 50 SO

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C.C.K Marks: 10M.Sc. MathematicsUnder CBCS (Only for School of Studies in Mathematics)SEMESTER IVPaper I Functional Analysis-II

Unit-1

Hahn-Banach theorem, Hahn-Banach theorem for complex vector space and normed spaces,

Reflexive spaces, Catigory Theorem and Uniform boundedness theorem.

Unit-2

Strong and Weak convergence, Open Mapping Theorem, Closed Linear Operators and closed

Graph Theorem., Closed Range Theorem.

Unit-3

Inner product spaces, Hilbert spaces, further properties of inner product spaces, Orthogonal

complements and Direct Sums (Projection Operator)

Unit-4

Complete Orthonormal sets and Bessel's Inequality, Convergence Theorems and fourier

coeifficients ,total orthonormal sets and sequences Parsval's Relation Riesz representation theorem."

Unit-5

Representation ofFunctionals on Hilbert space (Riesz theorem, Riesz representation). Hilbert

adjoint operator, Self-adjoint operators, Unitary operators and N01TI1aloperators

Recommended Books:

[1] E. Kreyszig, Introductory Functional Analysis with applications, John Wiley & Sons.

[2] G.F. Simmons, Introduction to Topology & Modem Analysis Mc Graw Hill, New

Reference:

[1] B. Choudhary and Sudarshan Nanda. Functional Analysis with applications, Wiley

Eastern Ltd ..

Max. Marks 50

.

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M.Sc. MathematicsUnder CBCS (Only for School of Studies in Mathematics)

SEMESTER IVPaper II Fundamentals of Computer Science - II (Theory and Practical)

SQL -Basic Features including views, Integrity Constraints, Key, Functional Depedency,Multivalued Functional Dependency, Database Design- Normalization up to BCNF.

Unit 5-Operating Systems _User Interface, Processor Mangement, Memorj management, Network

and Distributed Systems.

Reference Books:[1] E. Balagurusam y, 0 hject Oriented Pro gramming with C++, III Edi ti0n, Tata McGraw- Hill

Publishing Company Limited, New Delhi.[1] S.B.Lipman,] Lajoi; C++ Primer Addison Wesley.[2] C.J. Date; Introduction to Database systems, Addition.Wesley.[3] C. Ritchie; Operating Systems, Incorporating UNIX and Windows, BPB publications.

Unit 1-Inheritance, Single Inheritance, Multilevel Inheritance, Multiple Inheritance, Hierarchical

Inherita.'1ce,Hybrid Inheritance, Templates including Class Templates.

Unit 2-C++ Streams, C++ Stream Classes, putO and getO Functions, getlineO and write() Functions.

Unit 3-Database Systems _Role of Database Systems, Dat~base Systems Architecture.

Unit4,..f

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Optional Paper I (A) Advanced }"'unctionalAnalysis-IIUnit-l

Finite - dimensional topological vector spaces. Linear Varieties and Hyperplanes. Geometricform of Hahn -Banch theorem .. Chapter 2(2.2),5(5.1,5.2) 6(6.2), 7 and 9 (9.4) ofR.Larsen.

Unit-2Uniform - Boundedness principle. Open Mapping theorem and closed graph theorem for Frehetspaces, Banach - Alaouglu theorem. Chapter 6(6.2), 7 and 9 (9.4) ofR. Larsen.

Unit-3 t

Extreme points and Extremal sets. Krein- Milman's theorem. Duality polar. Bipolar theorem.r

Baralled and Bornological spaces.

Unit-4 ,

.. .ty1acekeySpaces. Sami-reflexive and Reflexive topological vector spaces. Montel Spaces andSchwarz spaces. Quasi-completeness. Chapter II (11. I, 11.2) ofR. Larsen

Unit-5

Inverse Limit and inductive limit oflocally convex spaces. Distributions. [Walter Rudin andL.V. Kantorovich and G.P. Akilov].

Recommended Books:[I] R.Larsen, Functional Analysis, Marcel Dekker, Inc. New york, 1973.[2] Walter Rudin, Functional Analysis, TMI--IEdition,1974.[3] L.V.Kantorovich and G.P. Akilov, Functional Analysis, Pergamon Press 1982.

Reference Books:

I. Laurent Schwartz, Functional Analysis Courant Institute of Mathematical Sciences, NewYork University, 1964.

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Optional Paper I (B) MechanicsUnit-l

Generalized cordinates. Holonomic and Non-holonornic systems. Scleronornic and RheonornicSystem Generalized Potential. Lagrange's equations of first kind. Lagrange's equations of secondKind. Uniqueness of solution. Energy equation for conservative fields.

Unit-2Hamilton's variables, Hamilton's canonical equations, Donkin's theorem, Matovating probelms ofcalculus of van ations, Shortest distance. Minimwn surface of revolution. Brachistochrone problem.Fundamental lemma of cal~ulus of variations. Euler's equation for one dependent function and itsgeneralization to (i) n dependent functions. (ii)higher order dei)vatives.

Unit-3 r

Hamilton's Principle. Principle of least action, Hamilton-Jacobi equation (time-dependent andtime-independent), Whittaker's equations, Statement of Lee HWA Chung's theorem, PoincareCarten Integal invariant.

Unit 4-Poisson's Bracket. Poisson's Identity. Jacobi-Poisson theorem, Lagrange Brackets. Condition ofcanonical character of a transformation in terms of Lagrange brackets and Poisson brackets,Invariance of Lagrange brackets and Poisson brackets under canonical transformations.

Unit-5Hamilton-Jacobi Theory: Solution of Hamilton-Jacobi eqmtion, Jacobi theorem. Method of saperationof variables.

Attraction and Potential of rod, disc, Spherical shells and sphere.

Reference Books:(1) Narayanan Chandra Rana & Pramod Sharad Chandra loag, Classical Mechanics, Tata

Mcgraw Hill 1991.(2) F. Gantmacher, Lectures in Analytic Mechanics MIR Publishers.(3) M. Ray, Attraction and Potential, Student's Friends and Company, Agra.(4) H. Goldstein Cla<;sicalMachanies (2nd Edition), Narosa Publishing House, .

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Max. Marks 50


Reference Books.

K. Yano and M. Kon, Structure of Manifolds, World Scientific Publishing co-Pvt. Ltd. 1984.

A. Bejaneu, Geometry ofCr- Submanifolds, D. Reidel Publishing Company" 1986

Optional Paper I (C) Differentiable Structures on manifolds-IIUnit-l

Kahler manifolds. Affine conncetion

Unit-2

Holomorphic sectional curvature. Curvature tensor. Almost analytic vector fields.

Unit-3

Nearly Kahler manifolds, Curvature identities. Constant Holomorphic sectional curvature

Unit-4 ;-

Almost analytic vector fields Almost Kahler Manifold Anilities vector fields, Almo~t Contact

manifolds: Lie derivative normal contact structure r

Unit-5

Affinely almost almost cosymplectic manifOld, Almost Grayn manifolds: D-conformal transforma-

ti9-n,Perticular affined connection K- Contact Rumanian manifolds.


I- B.B, Sinha, An Introduction to Modem Differential Geometry, Kalyani Publishers, New Delhi.1982.

2-3-

Reference Books:

I- R.S, Mishra, A course in tensons with application to Riemannian geometry pothishala Pvt. Ltd.1965.

2- R.S. Misru'a, Structures on Differentiable manifold and theire applications, Chandrema Prakashan, 1984.

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....M.Sc. MathematicsUnder CBCS (Only for School of Studies in Mathematics)SEMESTER IV

Optional Paper I (D) General Theory of Relativity and Cosmology-II

Unit-lReview of the special theory of relativity and the Newtonian Theory of gravitati~n. Principle

of equivalence and general covariance, geodesic principle.

Unit-2Newtonian approximation of relativistic equations of motion. Einstein's field equations and

its Newtonian ap~)foximation.

Unit-3Schwarzschild external solution and its isotropic form. Planetary orbits and anologues of

Kepler's Laws in general relativity. Advance of perihelion of a planet

Unit-4Bending oflight rays in a gravitational field. Gravitational redshift of spectral lines. Radar echo delay.

Unit-5Energy-momentum tensor of a pelfect fluid. Schwarzschild internal solution. Boundary conditions.

Recommended Books:[1] S.R.Roy and Raj: Theory of Relativity Jaipur Publishing House,Jaipur, 1987.

[2] S. K. Shrivastva: General Relativity and Cosmology, PHI, .

[3] J.V. Narlikar, General Relativity and Cosmology: Tbe Macmillan Company of Limited, 1978.


I.

Max. Marks 50


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Optional Paper I (E) Abstract Harmonic Analysis IIUnit-l

The Haar covering function Existence and properties ofHaar covering function DefInitionand properties of the function Ig (f).Existence and Uniqueness of the Haar integral,

Unit-2

Translation in Lp(G), uniform continuity of translation character, Characters, Characters group,properties of characters

Unit-3

Character group or dual group Locally compact abelian group non - trivial complexhomomorphism.

Unit-4

The Fourier transform, Convolution, convolution of function set A (r )of all Fourier transfOlmsinvariance, of A (r ),

Unit-5

Fourier Stieltjes transform set B( r )of all Fomier Stieltjes transform, invariance ofB (r ),Duality Theorem.

Recommended Books.

1- George Bachman Elements of Abstract Harmonic Analysis Acadmic Press, New Your. 19642- Taqdir Hussain Introduction to Topological Group W.D. Saudss Company 1966 to ok W.O.3- Walter Rudin, Fourier Analysis On Group lntersceince publisher, John wiley, New York, 1967

Reference Books.

I - Edwin Hewit and Kenneth A. Ross. Abstract Harmonic Analysis -1, Springer - Verlag, Berlin,1963.

2- lynn H. Loomis: An Introduction to Abstract Harmonic Analysis, D, Van Nostrand Co.Princeton.


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Optional Paper I (F) Mathematics of Finance and Insurance- IIUnit-l

A Briefintroduction to financial Markets,

Unit-2

basics of Securities, Stocks Bonds and fInancial derivatives,

Unit-3

Viz forwards, Futures, Options and Swaps.

Unit-4

An Introduction to stochastic Calculus stochastic process, geometric Brownian motion stochastic

integration and! ito's lemma

Unit-5

Option Pricing models- Binamial Models and Black Scholes Option Pricing Model for European

Options, Black Scholes formula and computation of greeks.

TextBooks:

Options, Futures and other Drivatives by nlOn c. Hull Prentice -Hall of India Pvt. Ltd. An introduction

to Mathematic Finance by Cheldon M. Ross, Cambridge University Press.

Reference Books:

An Introduction to Mathematics of Financial Derivatives by Salih N .Neftci, Academic Press, Inc.

mathematics of Financial markets by Ribert]. Elliot & P.E. Kopp Springer Verlag, New Yorlk Inc.

, •.

Max. Marks 50


Optional Paper II (A) Theory of Linear Operators IIUnit-l

Spectral representation of bounded self- adjoint linear operators. Spectral theorem. Spectralmeasures. Spectral Integral.

Unit-2

Regular Spectral Measure. Real and Complex Spectral Measure. Complex Spectral IntegralDescription of the Spectral Subspaces. Characterization of the Spectral Subspaces.

Unit-3

The Spectral theorem for bounded Normal Operators. Unbounded linear operators in Hilbertspece. Hellinger- Toeplitz theorem. Hilbert adjoint operators.

Unit-4 .

Symmetric and self -adjoint linear operators. Closed linear operators and closures. Spectrumof an unbounded self-adjoint linear operators.

Unit-5

Spectral theorem for unitary and self- adjoint linear operators. Multiplication operator andDifferentiation Operator. Chapter 10, E. Kreyszig.


(

Recommended Books: ,

(1) E. Kreyszig Introductory functional analysis with applications, ]hon .wiley& Sons, Nwe York,1978.

G.Bachman and L. Narci, FWlctuional analysis, Academic press, 1966

Referance Books:

P. R. Halmos Introduction to Hilbert space and the theory ofSpec):ral Multiplicity, Secondedition, Chelsea publishing co. N.Y. 1957.

N. Dundford and J.T. Schwartz, linear operator -3 part, Interscience / Wiley, New York 1958-71.

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Optional Paper II (B) Advanced Numerical Analysis II

Unit-lExtrapolation methods ordinary differential equations. multi step methods Predicator and corrector

methodUnit-2

stability analysis of multistep methods. Ordinary differential equation

Unit-3boundary value problems shooting method.

Unit-4Finte difference methods

Upit-5finite element method

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Textbook -(1) Numerical Mmethod for scientific and Engineering computation by M.K. Jain, S.R.K. Iyenger,

R.K. Jain south Edition (2003) New p.ge.

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Max. Marks 50

RegularTheory Marks..: 40c.c.E. Marks: 10

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Optional Paper II (C) Fuzzy Sets and Their Applications IIUnit-I

Fuzzy Relation and fuzzy graphs - Fuzzy relation on Fuzzy sets, Composition of Fuzzy relation,

Unit-2

Min-Max .composition and its properties, Fuzzy equivalence relation Fuzzy compatibility relation

Fuzzy relation equation Fuzzy graphs, Similarity relation.

Unit-3

Possibility Theory-Fuzzy measures, Evidence theory, Necessity Measure, possibility measure,

Unit-4

possibilitY distribution, possibility theory and fuzzy sets possibility theory versus probability theory.

Unit-5

FuzzY Logic-An overview of classical logic, multivalued logics, Fuzzy proposition Fuzzy quantifiers

Linguistic variables and hedges, Inference from conditional fuzzy proposition, the compositional rule of

inference.

TextBooks:

(l) Fuzzy set theory and its Applications by H.J. Zimmermann, Allied Publishers Ltd., , 199 j.

(2) Fuzzy sets and Fuzzy logic by G.J. KJir and B. Yuan Prentice - Hall ofIndia, ,1995

Reference Books:-

(I) Fuzzy sets and Uncertainty and Information by G.J. Kalia Tina A. Foljer - Prentice - Hall of

India.

II

Max. Marks 50


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Optional Paper II (D) Advanced Graph Theory- II

Unit-1Connectivity and separability in graphs Abstract graphs geometric graphs planar graphsKurtowski rNO graphs embedding and regions of a planar graphs Detection of planarity

Unit-2Geometric dual and combinationa dual.

Unit-3Coloring and covering of grapps, Chromatic, Polynomial chromatic partitioning Dimmerproblem Domination sets independent sets, Four colour conjecture.

Unit-4Digraph and types of digraphs, Digrapl) and binary relation Equiyalence relation in a l:,1faphDirected path walk circu~t and connectedness Eulerian digraph, arborscence matlices A, B and C

of digraphs.Unit-5Adjacency metric of a digraph, Alogorithms, Kruskal algorithm, Prism algoritlun, Dijkastra Algo-

rithm.

TextBook :-1- Graph Theory with Application to Engineering and Computer Science byNarsingh Deo.

Reference Book:-1- Graph Theory by Harary,

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Optional Paper II (E) Advanced Special Function IIUnit-1

Bessel function and Legendre polynomials: Definition ofJn (z), Bess.el's differential equation,Generating function, Bessel's integral wit~ index half and an odd integer,

Unit-2

Generating function for Legendre polynomials Rodrigues formula, Bateman's generatingfunction, Additional generating funtions, Hypergeometric forms ofPn (X) , Special propertiesofPnX), Some more generating functions, Lapla~e's first integral form, Othergonality.

Unit-3

Special properties ofPnX), Some more generating functions, Laplace's first integral form,Othergonality.Unit-4

Definition of Hermite polynomials Hn(x), Pure recurrence relations, Differential recurrence relations,Rodrigue's formula, Other generating functions, Othogonality, Expansion of polynomials, more generat-ing functions.Unit-5

Laguerre Polynomials: The Laguerre Polynomials Ln(X), Generating functions, Pure recurrencerelation~:Differential recurrence relation, Rodrigo's formula, Orthogonal, Expansion of polynomials,Special properties, Other generating functions.

Books Recommended;1- Rainville, E.D, ; Special Functions, Tbe Macmillan co., New york 1971,2- Srivastava, H.M. Gupta, K.C. and Goyal, S.P.; The H-functions of One and Two Variables with

applications, South Asian Publication, New Delhi.

3- Saran, N., Sharma S.D. and Trivedi, - Special Functions with application, Pragati prakashan,1986.

Reference Books.

1- Lebdev, N.N, Special Functions and l11eirApplications, Prentice Hall, Englewood Cliffs, Newjersey, USA 1995.

2- Whittaker, E.T. and Watson, G.N., A Course of Modem Analysis Can1bridge University Press,London, 1963

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Optional Paper II (F) Spherical Trigonometry an Astronomy- n

Unit-lSpherical Astronomy - Various system of references and related topics.

Unit-2Celestial sphere,

Unit-3Transit instrument. Atmospheric Retraction. Time planetary phenomcrna.

Unit-4Atmospheric Retraction.

Unit-5Time planetary phenomena.

Text Books:-1- A text book of spherical trigonometTY : Gorakh Prasad.2- A text book of spherical Astronomy: Gorakh Prasad.

Reference Book.1- Spherical Astronomy - Smarat2- spherical Astronomy - Bell

Max. Marks 50

RegularTheory Marks: 40C.C.E. Marks: 10M.Sc. Mathematics

Under CBCS (Only for School of Studies in Mathematics)SEMESTER IV

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Optional Paper III (A) Operatiolls Research IIUnit-l

Network analysis, constraints in Network, Construction of network, Critical Path Method(CPM) PERT, PERT Calculation, Resource Levelling by Network Techniques andadvances of network (PERT /CPM).

Unit-2

Dynamic Programming - recursive equation approach, Characteristic of Dyna!TIic Programriling, Computational procedure, Integer programming Gomory's all I.P.P. m~thod,Branch and Bound Technique.

Unit-3

Game theory - Two person Zero-sum games, Maximix-Minimax principle, games without saddle points - Mixed strategies, Graphical solution of2Xn and Mx2 Games,Solution by Linear Programming,

Unit-4

Non-linear programming: Mathematical Formulation, General Non-linear ProgrammingProblems, Problems of Constrained Maxima and Minima (Kuhn-Tucker Condition),Non-negative Constraints,

Unit-5Quadratic programming: Wolfe's Modified Simplex method, Beale's Method, Separableprogramming, Convex programming, Separable programming algorithms.

Recommended Books :-1- Kanti Swamp, P.K. Gupta and Manmohan, Operations Research, Sultan Chand & Sons, New

Delhi.Reference Books:-1- S.D, Sharma, Operation Research,2- F.S, Hiller and GJ. Lieberman, Industrial Engineering Series, 1995 (This book comes with a CD

containing software)

1- G. Hadley, Linear Programming, NarosaPublishing House. 1995.4- G. Hadley, Linear and Dynamic programming, Addison - Wesley5- H.A. Taha, Operations Research - An introduction, Macmillan Publishing co. Inc..6- Prem Kumar Gupta and D.S. Hira, Operation Reasearch, an Introdution, S. Chand & Company

Ltd.

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RegularTheory Marks : ~OC.C.E. Marks: 10


Optional Paper III (B) Computational Biology- II

Unit-l

Integer programming, Partition Problems, Traveling Salesman Problem (TSP) simulated annealing,

Sequence.

Unit-2

Assembly - Sequencing strategies,

Unit-3

Traveling Salesman Problem (TSP) simulated annealing Sequence.

Unit-4

Fragment alignment, Sequence accuracy, , .

Unit-5

sequence comparisons Methods - Local and global alignment, Dynamic prohTfamming method.

Text Books:-

I-Introduction to Computational Bi()logy by M.S, Waterman Chapman & Hall, 1995.

2-Bio informatics - A practical Guide to the analysis of Genes and Proteins by A. Baxevanis and B.

Ouelette, WileyInterscience (1998).

Reference Books:-

1- Introduction to Bio informatics by Attwood.

2 - Bioinfom1atics-Sequence and Genome analysis by David W.Mow1t.

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Max. Marks 50

M.Sc. Mathematics


Optional Paper III (C) Fluid Mechanics- II

Unit-1

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Motion of a sphere through agapsquid at rest as infinity. equation of motion of a sphere, stresscomponents in a real fluid.

Unit-2

Relations between rectangular components of stress convection between streses and gradients ofvelocity,

Unit-3

plane Poiseuille and coquette flows between two parallel plate, flow through tubes of uniform,cross - section in the fonner of circle, annulus under constant pressure gradient.

..Unit-4

Dynamical similarity, Reynolds nwnber, Prandt's boundary layer, boundary layer equations in twodimension, blasius solution

Unit-5

boundary layer thickness, displacement thickness, Karman itegral conditions, separation ofboundary layer flow.

TextBooks.

1- A text book of Fluid Mechanics in SI units by R.K, Rajput.2- An introduction to Fluid Dynamics by R.K. Rathy, Oxford and IBH Published Co.

Reference Books:1- Fluid Mechanics (Springer) By Joseph H. Spurk.2- Fluid Mechanics by Irfan A Khan (I-LR.W.)

3- An Introduction to Fluid Mechanics by G.K. Batchelor, Foundation Books" 1994.


C.C.E. Marks: 10""

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Optional Paper III (D) Bio-Mechanics-II

Unit-l

Solution of steady state and Unsteady - state flow problems in one dimesion, application of finiteelement method and exact solutions.

Unit-2

Diffusion processes in biology; diffusion in Tissue Fick's principle,tUnit-3

One, two and three Dimensional diffusion problems and their solution, Water Transport, Diffusionthrough membranes.

Unit-4

Respiratory Gas Flows, flow in Airways, Interaction Between convection and diffusion Exchangebetween Alvoclar Gas and Erythrocytes,

Unit-5

Pulmonary function Test, Dynamics of Ventilation system.

Textbooks: '

1- Introduction to Mathematical Biology by S.L Rubinow, J. Wiley & Sons.2- Biomechanics by Y.C, Fung, Springer- Verlag.

3- Introduction to Biomathematics by V.P. Saxena, VishwaPrakashan (Wiley eastern)

Reference Book :-

I- Biofluid Dymunics by Mazumdar.

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Optional Paper III (E) Analytic Number Theory- II

Unit-lDiricWet series and Euler products,

Unit-2the fimction defined by DiricWet series, the hal:tplane of convergence of a DiricWet series.

Unit-3Integral formwa for the coefficients of Dirichlet series

Unit-4 r

Analytic properties ofDirichilet series, Mean value formula for Dirichilet series.Unit-5 r •

Pr9perties of the gamma fimction, Integral representations ofHwwitz zeta fimctions, Analyticcontinuation ofHwWitz zeta function. '

Book Rocommneded :1- T.M. Apostol, Introduction to Analytic Nwnber Theory, Narosa Pub, House, 1989.

m.,1'~II~,. Max. Marks 50

RegularTheory Marks: 40C.CK Marks: 10

Option~1 Paper III (F) Integral Transform II

M.Sc. Mathematics


Unit-l

Application of Laplace Transform to Boundary Value Problems.Unit-2

Electric Circuits. Application to Beams.Unit-3

rThe complex Fourier Transform, Inversion Formula, Fourier cosine and sine transform,Unit-4

properties offourier Transoforms, Convolution & Parseval's identity. -~ ,

Unit-5 1

Fourier Transform of the derivatives, Finite Fourier Sine & Cosine Transform, Inversion Operationaland combined properties Fourier transform.

}

Books recommended :-

(1] Integral Transforms by Goyal & Gupta.(2] Integral Transforms by Sneddon

M.Sc.Mathematics: Syllabus(CBCS) 76 ....

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Transcript of M.Sc.Mathematics: Syllabus(CBCS) 76 ....