m. a./m. sc. in mathematics faculty of science third ...€¦ · m. a./m. sc. in mathematics...

M. A./M. Sc. in MATHEMATICS /Syllabus/ IIIrd Semester/SU

M. A./M. Sc. in MATHEMATICS FACULTY OF SCIENCE

THIRD SEMESTER (ODD SEMESTER)

Eligibility

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MSM 301 CCC Integration Theory and Functional Analysis (I) 6 4 3 00 3 00

MSM 302 CCC Partial Differential Equations & Mechanics (I) 6 4 3 00 3 00

MSM 303 CCC Fluid Dynamics 6 4 3 00 3 00

MSM S02 OSC INTELLECTUAL PROPERTY, HUMAN RIGHTS & ENVIRONMENT:

BASICS 6 4 3 00 3 00

MSM C 01 ECC/CB TRIBAL STUDIES

6 4 3 00 3 00 MSM C 02 ECC/CB Fuzzy Sets and their applications(I)

MSM C 03 ECC/CB Mathematical Modeling

MSM C 04 ECC/CB Operations Research (I)

TOTAL=

30


M. A./M. Sc. (MATHEMATICS) IIIRD SEMESTER

COURSE CODE: MSM 301 COURSE TYPE: CCC

COURSE TITLE: Integration Theory and Functional Analysis (I)

CREDIT:6

HOURS:90

THEORY: 6 PRACTICAL: 0 THEORY: 90 PRACTICAL: 0

MARKS

THEORY: 100 (30+70) PRACTICAL: 00

UN

IT-1

18 h

rs.

Signed measure. Hahn decomposition theorem, mutually singular measures. Radon-Nikodym theorem. Labesgue decomposition. Riesz representation theorem. Extension theorem (Caratheodory).

UN

IT-2

18 h

rs.

Lebesgue-Stieltjes integral, product measures, Fubini’s theorem. Differentiation and Integration. Decomposition into absolutely continuous and singular parts.

UN

IT-3

18 h

rs.

Baire sets. Baire measure, continuous functions with compact support. Regularity of measures on locally compact spaces. Integration of continuous functions with compact support, Riesz- Markoff theorem. Functional Analysis :

UN

IT-4

18 h

rs.

Normed linear spaces. Banach spaces and examples. Quotient space of normed linear spaces and its completeness, equivalent norms. Riesz Lemma, basic properties of finite dimensional normed linear spaces and compactness.

UN

IT-5

18 h

rs.

Weak convergence and bounded linear transformations, normed linear spaces of bounded linear transformations, dual spaces with examples.


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1. P.R. Halmos, Measure Theory, Van Nostrand, Princeton, 1950. 2. B.Choudhary and S.Nanda, Functional Analysis with Applications. Wiley Eastern Ltd. 1989. 3. H.L. Royden, Real Analysis, Macmillan Publishing Co. Inc., New York, 4’h Edition, 1993. 4. S.K. Berberian, Measure and integration, Chelsea Publishing Company, New York, 1965. 5. G. de Barra, Measure Theory and Integration, Wiley Eastern Limited, 1981. 6. P.K. Jain and V.P. Gupta, Lebesgue Measure and Integration, New Age International (P) Limited, New Delhi, 2000. 7. Richard L. Wheeden and Antoni Zygmund, Measure and Integral : An Introduction to Real Analysis, Marcel Dekker Inc. 1977. 8. J.H. Williamson, Lebesgue Integration, Holt Rinehart and Winston, Inc. New York. 1962. 9. T.G. Hawkins, Lebesgue’s Theory of Integration: Its Origins and Development, Chelsea, New York, 1979. 10. K.R. Parthasarathy, Introduction to Probability and Measure, Macmillan Company of India Ltd., Delhi, 1977. 11. R.G. Bartle, The Elements of Integration, John Wiley & Sons, Inc. New York, 1966. 12. Serge Lang, Analysis I & II, Addison-Wesley Publishing Company, Inc. 1967. 13. Inder K. Rana, An Introduction to Measure and Integration, Narosa Publishing House, Delhi, 1997.




COURSE TITLE: Partial Differential Equations and Mechanics (I)

CREDIT:6

HOURS:90


MARKS

THEORY: 100(30+70) PRACTICAL: 00

UN

IT-1

18 h

rs.

Partial Differential Equations: Examples of PDE. Classification. Transport Equation-Initial value Problem. Non-homogeneous Equation. Laplace’s Equation-Fundamental Solution, Mean Value Formulas, Properties of Harmonic Functions, Green’s Function, Energy Methods.

UN

IT-2

18 h

rs.

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

UN

IT-3

18 h

rs.

Analytical Dynamics: Generalized coordinates. Holonomic and Non-holonomic systems. Scleronomic and Rheonomic sytems. Generalized potential. Lagrange’s equations of first kind. Lagrange’s equations of second kind. Uniqueness of solution. Energy equation for conservative fields. Hamilton’s variables. Donkin’s theorem. Hamilton canonical equations. Cyclic coordinates. Routh’s equations.

UN

IT-4

18 h

rs.

Analytical Dynamics: Poisson’s Bracket. Poisson’s Identity. Jacobi-Poisson Theorem. Motivating problems of calculus of variations, Shortest distance. Minimum surface of revolution. Brachistochrone problem. Isoperimetric problem. Geodesic. Fundamental lemma of calculus of variations. Euler’s equation for one dependent function and its generalization to (1) ‘n’ dependent functions, (ii) higher order derivatives. Conditional extremum under geometric constraints and under integral constraints.

UN

IT-5

18 h

rs.

Gravitation: Attraction and potential of rod, disc, spherical shells and sphere. Surface integral of normal attraction (application & Gauss’ theorem). Laplace and Poisson equations. Work done by selfattracting systems. Distributions for a given potential. Equipotential surfaces. Surface and solid harmonics. Surface density in terms of surface harmonics


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1. L.C. Evans, Partial Differential Equations, Graduate Studies in Mathematics,Volume 19, AMS, 1998. 2. F. Gantmacher, Lectures in Analytic Mechanics, MIR Publishers, Moscow, 1975. 3. R. C. Mondal, Classical Mechanics, Prentice Hall of India 4. S. L. Loney, An Elementary Treatise on Statics, Kalyani Publishers, New Delhi, 1979. 5. Books on Partial differential equation by 1.N. Sneddon, F. John, P. Prasad and R. Ravindran, Amarnath etc. 6. A.S. Ramsey, Dynamics Part II, The English Language Book Society and Cambridge University Press, 1972. 7. H. Goldstein, Classical Mechanics (2nd edition), Narosa Publishing House, New Delhi. 8. I. M. Gelfand and S.V. Fomin, Calculus of Variations, Prentice Hall. 9. Narayan Chandra Rana & Pramod Sharad Chandra Joag, Classical Mechanics, Tata McGraw Hill, 1991. 10. Louis N. Hand and Janet D. Finch, Analytical Mechanics, Cambridge University Press, 1998. 11. A.S. Ramsey, Newtonian Gravitation, The English Language Book Society and the Cambridge University Press.




COURSE TITLE: FLUID DYNAMICS

CREDIT: 6

HOURS:90


MARKS

THEORY:100 (30+70) PRACTICAL : 00

UN

IT-1

18 h

rs.

Kinematics of Fluids in motion. Real fluids and Ideal fluids - Velocity of a fluid at a

point, Stream lines , path lines , steady and unsteady flows- Velocity potential - The vorticity vector- Local and particle rates of changes - Equations of continuity -

Worked examples - Acceleration of a fluid - Conditions at a rigid boundary.

Chapter 2. Sections 2.1 to 2.10.

UN

IT-2

18 h

rs.

EQUATIONS OF MOTION OF A FLUID

Pressure at a point in a fluid at rest. - Pressure at a point in a moving fluid - Conditions at a boundary of two inviscid immiscible fluids- Euler’s equation of

motion - Discussion of the case of steady motion under conservative body forces.

Chapter 3. Sections 3.1 to 3.7

UN

IT-3

18 h

rs. Some three dimensional flows. Introduction- Sources, sinks and doublets - Images in

a rigid infinite plane - Axis symmetric flows - stokes stream function Chapter 4 Sections 4.1, 4.2, 4.3, 4.5.

UN

IT-4

18 h

rs.

SOME TWO DIMENSIONAL FLOWS

Meaning of two dimensional flow - Use of Cylindrical polar coordinate - The stream

function - The complex potential for two dimensional, irrotational incompressible flow - Complex velocity potentials for standard two dimensional flows - Some

worked examples - Two dimensional Image systems - The Milne Thompson circle

Theorem. Chapter 5. Sections 5.1 to 5.8

UN

IT-5

18 h

rs.

VISCOUS FLOWS Stress components in a real fluid. - Relations between Cartesian components of

stress- Translational motion of fluid elements - The rate of strain quadric and

principal stresses - Some further properties of the rate of strain quadric - Stress analysis in fluid motion - Relation between stress and rate of strain - The coefficient

of viscosity and Laminar flow - The Navier - Stokes equations of motion of a

Viscous fluid. Chapter 8. Sections 8.1 to 8.9

SU

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Books Recommended : 1. R.W.Fox and A.T.McDonald. Introduction to Fluid Mechanics, Wiley, 1985.

References

1. E.Krause, Fluid Mechanics with Problems and Solutions, Springer, 2005. 2. B.S.Massey, J.W.Smith and A.J.W.Smith, Mechanics of Fluids, Taylor and

Francis, New York, 2005

3. P.Orlandi, Fluid Flow Phenomena, Kluwer, New Yor, 2002. 4. T.Petrila, Basics of Fluid Mechanics and Introduction to Computational Fluid

Dynamics, Springer, berlin, 2004.



COURSE CODE : MSMS02 COURSE TYPE : OSC

COURSE TITLE: INTELLECTUAL PROPERTY RIGHTS, HUMAN RIGHTS &

ENVIRONMENT: BASICS

CREDIT: 06

THEORY: 06

HOURS : 90

THEORY: 90

MARKS : 100

THEORY: 70 CCA : 30

OBJECTIVE:

- Understands the concept and place of research in concerned subject

- Gets acquainted with various resources for research

- Becomes familiar with various tools of research

- Gets conversant with sampling techniques, methods of research and techniques of analysis of data.

UN

IT -

1

12 H

rs

Patents :- Introduction & concepts, Historical Overview.

Subject matter of patent.

Kinds of Patents.

Development of Law of Patents through international treaties and conventions including TRIPS

Agreement.

Procedure for grant of patents & term of Patent.

Surrender, revocation and restoration of patent.

Rights and obligations of Patentee

Grant of compulsory licenses

Infringement of Patent and legal remedies

Offences and penalties

Discussion on leading cases.

UN

IT -

2

24 H

rs

Meaning of Copyright, Historical Evolution,

Subject matter of copyright.

Literary works

Dramatic Works & Musical Works

Computer Programme

Cinematographic films

Registration of Copyrights

Term of Copyright and Ownership of Copyrights

Neighboring Rights

Rights of Performers & Broadcasters

Assignment of Copyright.

Author’s Special Rights (Moral Rights)

Infringement of Copyrights and defenses

Remedies against infringement (Jurisdiction of Courts and penalties)

International Conventions including TRIPS Agreement WIPO, UCC, Paris Union, Berne

Convention, UNESCO.

Discussion on leading cases.


UN

IT -

3

10 H

rs

Rights: Meaning

Human Rights- Meaning & Essentials

Human Rights Kinds

Rights related to Life, Liberty, Equals & Disable

UN

IT -

4

2

4 H

rs

National Human Rights Commission

State Human Rights Commission

High Court

Regional Court

Procedure & Functions of High & Regional Court.

UN

IT -

5

20 H

rs

Right to Environment as Human Right

International Humanitarian Law and Environment

Environment and Conflict Management

Nature and Origin of International Environmental Organisations (IEOs)

Introduction to Sustainable Development and Environment

Sustainable Development and Environmental Governance

SU

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1. G.B.Reddy, Intellectual Property Rights and Law, Gogia Law Agency, Hyderabad.

2. S.R.Myneni, Intellectual Property Law, Eastern Law House, Calcutta

3. P Narayanan Intellectual Property Rights and Law (1999), Eastern Law House, Calcutta, India

4. Vikas Vashistha, Law and Practice of Intellectual Property,(1999) Bharat Law House, New

Delhi.

5. Comish W.R Intellectual Property,3rd

ed, (1996), Sweet and Maxwell

6. P.S. Sangal and Kishor Singh, Indian Patent System and Paris Convention,

7. Comish W.R Intellectual Property, Patents, Copyrights and Allied Rights, (2005)

8. Bibeck Debroy, Intellectual Property Rights, (1998), Rajiv Gandhi Foundation.


MASTER OF SCIENCE

(THIRD SEMESTER)

COURSE CODE: MSMC01 COURSE TYPE : ECC/CB

COURSE TITLE: TRIBAL STUDIES

CREDIT: 06

THEORY: 06

HOURS : 90

THEORY: 90

MARKS : 100

THEORY: 70 CCA : 30

OBJECTIVE:

- Understands the concept and place of research in concerned subject

- Gets acquainted with various resources for research

- Becomes familiar with various tools of research

- Gets conversant with sampling techniques, methods of research and techniques of analysis of data

- Achieves skills in various research writings

- Gets acquainted with computer Fundamentals and Office Software Package .

UN

IT -

1

12 H

rs

Tribal Studies : Meaning, Nature, Scope, Need & importance of tribal studies. Meaning,

Definition & characteristics of Tribe, Caste & Race.

UN

IT -

2

24 H

rs

Scheduled Tribe in India : Population Composition of tribal, classification of Indian Tribe –

Racial, Lingual, Geographical, Cultural.

Some Major Tribes in India : Santhal, Khasi, Munda, Bhils.

Some Major Tribes in Central India : Gond, Baiga, Bharia, Korkus.

UN

IT -

3

10 H

rs

Iliteracy :Poverty, Indebness, Unemployment, migration & Exploitation Environmental &

Degradation.

Problem of Health and sanitation :

Prostitution, Culture Decay due to assimilation. Replacement & Rehabilitation of Tribal

population.

UN

IT -

4

2

4 H

rs

Welfare-Concept, Characteristics: Tribal Welfare in post independence period. Constitutional

provision & safe guard after independence, Legislation & Reservation Policy.


UN

IT -

5

20 H

rs

Tribal Development Programs for Scheduled Tribes : Medical, Education, Economy,

Employment & Agriculture Evaluation of Programs

Tribal Welfare & Advisory Agencies in India : Role of NGO’s in tribal development, Role of

Christian missionaries in tribal welfare & development. Tribal Welfare Administration.

SU

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1. Tribal Development In India (Orissa) by Dr. Taradutt

2. Books on Tribal studies by PK Bhowmik

3. Books on 'Tribal Studies' by W.G. Archer



COURSE CODE: MSM C02 COURSE TYPE: ECC/CB

COURSE TITLE: FUZZY SETS AND ITS APPLICATIONS (I)

CREDIT: 6 HOURS:90

THEORY: 6 PRACTICAL:0 THEORY: 90 PRACTICAL: 0

MARKS

THEORY:100(30+70) PRACTICAL: 00

UN

IT-1

18 h

rs.

Fuzzy sets-Basic definitions, -level sets. Convex fuzzy sets. Basic Operations on fuzzy sets. Types of fuzzy sets. Cartesian products, Algebraic products. Bounded sum and difference, t-norms and tconorms.

UN

IT-2

18 h

rs. The Extension Principle- The Zadeh’s extension principle. Image

and inverse image of fuzzy sets. Fuzzy numbers. Elements of fuzzy arithmetic.

UN

IT-3

18 h

rs. Fuzzy Relations on Fuzzy sets, Composition of Fuzzy relations. Min-

Max composition and its properties.

UN

IT-4

18 h

rs. Fuzzy equivalence relations. Fuzzy compatibility relations. Fuzzy

relation equations. Fuzzy graphs, Similarity relation.

UN

IT-5

18 h

rs. Possibility Theory-Fuzzy measures. Evidence theory. Necessity

measure. Possibility measure. Possibility distribution. Possibility theory and fuzzy sets. Possibility theory versus probability theory.

SU

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S 1. H.J. Zimmermann, Fuzzy set theory and its Applications, Allied

Publishers Ltd. New Delhi, 1991. 2. G.J. Klir and B. Yuan- Fuzzy sets and fuzzy logic, Prentice-Hall ol India, New Delhi, 1995.




COURSE TITLE: MATHEMATICAL MODELLING

CREDIT:6 HOURS:90


MARKS


UN

IT-1

18 h

rs.

Mathematical Modelling through Systems of Ordinary differential Equations of the First Order (Chapter 3: 3.1, 3.2, 3.5, and 3.6)

Mathematical modeling in population dynamics, Mathematical modeling of epidemics

through systems of ordinary differential equations of first order - Mathematical Models in Medicine, Arms Race, Battles and international Trade in terms of Systems

of ordinary differential equations - Mathematical modeling in dynamics through

systems of ordinary differential equations of first order.

UN

IT-2

18 h

rs.

Mathematical Modelling through difference equations (Chapter 5: 5.1 to 5.3)

The need for Mathematical modelling through difference equations - some simple models - Basic theory of linear difference equations with constant coefficients -

Mathematical modelling through difference equations in economics and finance

UN

IT-3

18 h

rs.

Mathematical Modelling through difference equations (contd.) (Chapter 5: 5.4 to 5.6)

Mathematical modelling through difference equations in population dynamics and

genetics. Mathematical Modelling through difference equations in probability theory.

Miscellaneous examples of Mathematical modelling through difference equations

UN

IT-4

18 h

rs.

Mathematical modelling through Graphs (Chapter 7: 7.1 to 7.4)

Situations that can be modeled through graphs - Mathematical models in terms of

directed graphs - Mathematical models in terms of signed graphs - Mathematical

models in terms of weighted graphs

UN

IT-5

18 h

rs.

Mathematical Modelling through calculus of Variations and Dynamic Programming

(Chapter 9: 9.1 to 9.3) Optimization principles and techniques - Mathematical modelling through calculus of variations - Mathematical Modelling through dynamic

programming.


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1. D. J. G. James and J. J. Macdonald, Case studies in Mathematical Modelling, Stanly Thames, Cheltonham.

2. J.N. Kapur, Mathematical entropy Models.

3. M. Crossand A. O. Moscrcadini, The art of Mathematical Modelling, Ellis Harwood and John Wiley.

4. C. Dyson, Elvery, Principles of Mathematical Modelling, Academic Press, New York.

5. D. N. Burghes, Modelling with Difference Equations, Ellis Harwood and John Wiley.




COURSE TITLE: OPERATION RESEARCH I

CREDIT: 6 HOURS:90


MARKS


UN

IT-1

23 h

rs. Operations Research and its Scope. Necessity of Operations

Research in Industry. Linear Programming-Simplex Method. Theory of the Simplex Method. Duality and Sensitivity Analysis.

UN

IT-2

20 h

rs. Other Algorithms for Linear Programmrng-Dual Simplex Method.

UN

IT-3

17 h

rs. Parametric Linear Programming. Upper Bound Technique. Interior

Point Algorithm. Linear Goal Programming.

UN

IT-4

12 h

rs. Transportation and Assignment Problems.

UN

IT-5

18 h

rs.

Network Analysis-Shortest Path Problem. Minimum Spanning Tree Probiem. Maximum Flow I Problem. Minimum Cost Flow Problem. Network Simplex Method. Project Planning and Control I with PERT-CPM.

SU

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1. F.S. Hillier and G.J. Ueberman. Introduction to Operations ResBareft (SixthEdition), McGraw Hill International Edition, Industrial Engineering Series, 1995. 2. G. Hadley, Linear Programming, Narosa Publishing House, 1995. 3. G. Hadly, Nonlinear and Dynamic Programming, Addison-Wesley, Reading Mass. 4. H.A. Taha, Operations Research -An introduction, Macmillan Publishing Co., Inc., New Yark. 5. Kanti Swarup, P.K. Gupta and Man Mohan, Operations Research, Sultan Chand & Sons, New Delhi 6. S.S. Rao, Optimization Theory and Applications, Wiley Eastern Ltd., New Delhi. 7. Prem Kumar Gupla and D.S. Hira, Operations Research-An Introduction. S. Cliand & Company Ltd., New Delhi.

m. a./m. sc. in mathematics faculty of science third ...€¦ · m. a./m. sc. in mathematics...

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