m.sc mathematics syllabus -2013-14

8/12/2019 M.sc Mathematics Syllabus -2013-14

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DEPARTMENTOF MATHEMATICS

CHRIST UNIVERSITY

DEPARTMENT OF MATHEMATICS

Masters Programme in Mathematics

Course Objective:

The M.Sc. course in Mathematics aims at developing mathematicalability in students with acute and abstract reasoning. The course will enablestudents to cultivate a mathematicians habit of thought and reasoning and will

enlighten students with mathematical ideas relevant for oneself and for the

course itself.

Course Design:

Masters in Mathematics is a two years programme spreading over foursemesters. In the first two semesters focus is on the basic papers in

mathematics such as Algebra, Analysis and Number Theory along with the

basic applied paper ordinary and partial differential equations. In the third and

fourth semester focus is on the special papers, elective paper and skill-based

papers including Topology, Functional Analysis, Advanced Fluid Mechanics,Advanced Graph Theory and Numerical Methods for solving differentialequations. Important feature of the curriculum is that one paper on the topic

Fluid Mechanics and Graph Theory is offered in each semester with a project on

these topic in the fourth semester, which will help the students to pursue the

higher studies in these topics. Special importance is given to TeachingTechnology and Research Methodology in Mathematics, Mathematical

Statistics and Introduction to Mathematical Packages, which are offered ascertificate courses.

Methodology:

We offer this course through Lectures, Seminars, Workshops, GroupDiscussion and talks by experts.

Admission procedure:

Candidates who have secured at least 50% of marks in Mathematics intheir bachelor degree examination are eligible to apply. The candidates will

then appear for subject interview.


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Modular Objectives:

MTH 131: NUMBER THEORY

This paper is concerned with the basics of analytical number theory. Topics such as

divisibility, congruences, quadratic residues and functions of number theory are covered in

this paper. Some of the applications of the said concepts are also included.

MTH 132: REAL ANALYSIS

This paper will help students understand the basics of real analysis. This paper includes such

concepts as basic topology, Riemann-Stieltjes integral, sequences and series of functions.

MTH 133: ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS

This helps students understand the beauty of the important branch of mathematics, namely,

differential equations. This paper includes a study of second order linear differential

equations, adjoint and self-adjoint equations, Eigen values and Eigen vectors of the

equations, power series method for solving differential equations, second order partial

differential equations like wave equation, heat equation, Laplace equations and their solutionsby Eigen function method.

MTH 134: CONTINUUM MECHANICS

This paper is an introductory course to the basic concepts of continuum mechanics and fluid

mechanics. This includes Cartesian tensors, stressstrain tensor, conservation laws and

constitutive relations for linear elastic solid.

MTH 135: ELEMENTARY GRAPH THEORY

This paper is an introductory course to the basic concepts of Graph Theory. This includes

definition of graphs, vertex degrees, directed graphs, trees, distances, connectivity and paths.

MTH 231: MEASURE THEORY AND INTEGRATION

This paper deals with various aspects of measure theory and integration by means of the

classical approach. More advanced concepts such as measurable sets, Borel sets, Lebesgue

measure, Lebesgue integration and LP

spaces have been included in this paper.

MTH 232: COMPLEX ANALYSIS

This paper will help students learn about the essentials of complex analysis. This paper

includes important concepts such as power series, analytic functions, linear transformations,Laurents series, Cauchys theorem, Cauchys integral formula, Cauchys residue theorem,

argument principle, Schwarz lemma , Rouches theorem and Hadamards 3-circles theorem.

MTH 233: ADVANCED ALGEBRA

This paper enables students to understand the intricacies of advanced areas in algebra. This

includes a study of advanced group theory, polynomial rings, Galois theory and linear

transformation.

MTH 234: FLUID MECHANICS

This paper aims at studying the fundamentals of fluid mechanics such as kinematics of fluid,

incompressible flow and boundary layer flows.

MTH 235: ALGORITHMIC GRAPH THEORYThis paper helps the students to understand the colouring of graphs, Planar graphs, edges and

cycles.


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This paper aims at studying the fundamentals of wavelet theory. This includes the concept on

the continuous and discrete wavelet transform and wavelet packets like construction and

measure of wavelet sets and construction of wavelet spaces.

MTH 444:

This paper is concerned with the fundamentals of mathematical modeling. The coverageincludes mathematical modeling through ordinary and partial differential equations.

MTH 445: CRYPTOGRAPHY

This paper introduces basics of number theory and some crypto systems.

MTH 446: ATMOSPHERIC SCIENCE

This paper provides an introduction to the dynamic meteorology, which includes the

essentials of fluid dynamics, atmospheric dynamics and atmosphere waves and instabilities.


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I Semester

Paper Code Title Hrs./week Marks Credit

MTH 131 Number Theory 4 100 4

MTH 132 Real Analysis 4 100 4

MTH 133Ordinary and Partial Differential

Equations4 100 4

MTH 134 Continuum Mechanics 4 100 4

MTH 135 Elementary Graph Theory 4 100 4

Total 20 500 20

II Semester


MTH 231 Measure Theory and Integration 4 100 4

MTH 232 Complex Analysis 4 100 4

MTH 233 Advanced Algebra 4 100 4

MTH 234 Fluid Mechanics 4 100 4

MTH 235 Algorithmic Graph Theory 4 100 4

Total 20 500 20


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III Semester


MTH 331 Topology 4 100 4

MTH 332 Numerical Analysis 4 100 4

MTH 333 Classical Mechanics 4 100 4

MTH 334 Advanced Fluid Mechanics 4 100 4

MTH 335 Advanced Graph Theory 4 100 4

Total 20 500 20

IV Semester


MTH 431 Differential Geometry 4 100 4

MTH 432 Advanced Numerical Methods 4 100 4

MTH 433 Functional Analysis 4 100 4

MTH 451 Project 4 100 4

Elective:

4 100 4

MTH 441Calculus of Variations and

Integral Equations

MTH 442 Magnetohydrodynamics

MTH 443 Wavelet Theory

MTH 444 Mathematical Modelling

MTH 445 Cryptography

MTH 446 Atmospheric Science

Total 20 500 20


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CERTIFICATE COURSES

I Semester

Paper Code Title Total No. of

Hours

Credit

MTH 101

Teaching Technology and

Research Methodology in

Mathematics

45 2

II Semester


Hours

Credit

MTH 201 Statistics45 2

MTH371 : INTERNSHIP 2 credits

III Semester


Hours

Credit

MTH 301Introduction to Mathematical

Packages

45 2


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Class : M.Sc (MATHEMATICS) Semester : I

Paper : NUMBER THEORY Code : MTH131

Unit I (10 hours)

Divisibility: The division algorithm, the Euclidean algorithm, the unique

factorization theorem, Euclids theorem, linear Diophantine equations.

Unit II (20 hours)

Congruences: Definitions and properties, complete residue system modulo m,

reduced residue system modulo m, Eulers function, Fermats theorem, Eulers

generalization of Fermats theorem, Wilsons theorem, solutions of linear congruences, theChinese remainder theorem, solutions of polynomial congruences, prime power moduli,

power residues, number theory from algebraic point of view, groups, rings and fields.

Unit III (18 hours)

Quadratic residues: Legendre symbol, Gausss lemma, quadratic reciprocity, the

Jacobi symbol, binary quadratic forms, equivalence and reduction of binary quadratic forms,

sums of two squares, positive definite binary quadratic forms.

Unit IV (12 hours)

Some functions of number theory: Greatest integer function, arithmetic functions,

the Mobius inversion formula.

Text Book:

Ivan Niven, Herbert S. Zuckerman and Hugh L. Montgomery, An introduction to the theory

of numbers, John Wiley, 2004.

Reference Books:

1. Kenneth Ireland and Michael Rosen, A classical introduction to modern number theory,

Springer, 2010.

2. Neal Koblitz,A course in number theory and cryptography, Springer, 2010 (Reprint).

3. Gareth A. Jones and J. Mary Jones,Elementary number theory, Springer, 1998.

4. Joseph H. Silverman, A friendly introduction to number theory, Pearson Prentice Hall,

2006.


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MTH131 : NUMBER THEORY

END SEMESTER EXAMINATION : FORMAT OF THE QUESTION PAPER

PartUnit and No. of subdivisions to be set in the

unitNo. of

subdivisions to

be answered

Marks for eachsubdivision

Max. marks forthe part

A

Unit I 1

5 2 10Unit II 2

Unit III 1

Unit IV 1

B

Unit I 2

10 5 50Unit II 4

Unit III 4

Unit IV 2

C

Unit I 1

4 10 40Unit II 1

Unit III 1

Unit IV 1

Total 100


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Paper : REAL ANALYSIS Code : MTH132

Unit I (18 hours)

Basic Topology and sequences and series: Finite, countable and uncountable sets,metric spaces, compact sets, perfect sets, connected sets, convergent sequences,

subsequences, Cauchy sequences, upper and lower limits, some special sequences, series,

series of nonnegative terms, absolute convergence.

Unit II (12 hours)Continuity and Differentiability: Limits of functions, continuous functions,

continuity and compactness, continuity and connectedness, discontinuities, monotonic

functions, derivative of a real function, mean value theorems, continuity of derivatives.

Unit III (15 hours)

The Riemann-Stieltjes Integral: Definition, existence and linearity properties, the

integral as the limit of sums, integration and differentiation, integration by parts, mean value

theorems on Riemann-Stieltjes integrals, change of variable.

Unit IV (15 hours)

Sequences and Series of Functions: Pointwise and uniform convergence, Cauchy

criterion for uniform convergence, Weierstrass M-test, uniform convergence and continuity,

uniform convergence and Riemann-Stieltjes integration, uniform convergence anddifferentiation.

Walter Rudin, Principles of Mathematical Analysis, 3rd ed., New York: McGraw-Hill, 1976.

:

1. T.M. Apostol,Mathematical Analysis, New Delhi: Narosa, 2004.

2. E.D. Bloch, The Real Numbers and Real Analysis, New York: Springer, 2011.3. J.M. Howie,Real Analysis, London: Springer, 2005.

4. J. Lewin,Mathematical Analysis, Cambridge: Cambridge University Press, 2003.

5. F. Morgan,Real Analysis, New York: American Mathematical Society, 2005.


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MTH132 : REAL ANALYSIS



unitNo. of

subdivisions to

be answered



A

Unit I 2

5 2 10Unit II 1

Unit III 1

Unit IV 1

B

Unit I 3

10 5 50Unit II 3

Unit III 3

Unit IV 3

C

Unit I 1

4 10 40Unit II 1

Unit III 1

Unit IV 1

Total 100


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Paper : ORDINARY AND PARTIAL Code : MTH133

DIFFERENTIAL EQUATIONS

Unit I (20 hours)

Linear Differential Equations:Linear differential equations, fundamental sets ofsolutions, Wronskian, Liouvilles theorem, adjoint and self-adjoint equations, Lagrange

identity, Greens formula, zeros of solutions, comparison and separation theorems. Eigen

values and Eigen functions, related examples.

Unit II (10 hours)

Power series solutions: Solution near an ordinary point and a regular singular point

by Frobenius method, hypergeometric differential equation and its polynomial solutions,

standard properties.

Partial Differential Equations: Basic concepts and definitions, mathematical models

representing stretched string, vibrating membrane, heat conduction in solids and thegravitational potentials, second-order equations in two independent variables, canonical

forms and general solution.

Unit IV (20 Hours)Solutions of PDE: The Cauchy problem for homogeneous wave equation,

DAlemberts solution, domain of influence and domain of dependence, the Cauchy problem

for non-homogeneous wave equation, the method of separation of variables for the one-

dimensional wave equation and heat equation. Boundary value problems, Dirichlet and

Neumann problems in Cartesian coordinates, solution by the method of separation of

variables. Solution by the method of eigenfunctions.

:

1. E. A. Coddington, Introduction to ordinary differential equations, McGraw Hill, 2006

(Reprint) (Unit I and II).

2. G. F. Simmons, Differential equations with applications and historical notes, Tata

McGraw Hill, 2003. (Unit I and II).

3. Tyn Myint-U and L. Debnath, Linear Partial Differential Equations, Boston: Birkhauser,

2007. (Unit III and IV).

4. Christian Constanda, Solution Techniques for Elementary Partial Differential Equations,

New York: Chapman & Hall, 2010. (Unit III and IV).

:

1. M.S.P. Eastham, Theory of ordinary differential equations, London:Van Nostrand, 1970.

2. E. D. Rainville and P. E. Bedient,Elementary differential equations, New York: McGraw-

Hill, 1969.

3. W. E. Boyce and R. C. DiPrima, Elementary differential equations and boundary value

problems, Fourth Edition, New York: Wiley, 1986.

4. K. F. Riley, M. P. Hobson and S. J. Bence, Mathematical Methods for Physics and

Engineering, Cambridge, 2005.

5. Edwards Penney, Differential Equations and Boundary Value Problems, Pearson

Education, 2005.

6. J. David Logan, Partial Differential Equations, 2nd ed., New York: Springer, 2002.7. Alan Jeffrey,Applied Partial Differential Equations: An Introduction, California:

Academic Press, 2003

8. M. Renardy and R.C. Rogers,An Introduction to Partial Differential Equations, 2nd ed.,

New York: Springer, 2004.
http://www.google.co.in/search?tbo=p&tbm=bks&q=inauthor:%22Alan+Jeffrey%22http://www.google.co.in/search?tbo=p&tbm=bks&q=inauthor:%22Alan+Jeffrey%22


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9. L.C. Evans, Partial Differential Equations, 2nd ed., American Mathematical Society,

2010.

10. K. Sankara Rao,Introduction to Partial Differential Equations, 2nd ed., New Delhi:

Prentice-Hall of India, 2006.

11. R.C. McOwen, Partial Differential Equations: Methods and Applications, 2nd ed., New

York: Pearson Education, 2003.

MTH133 : ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS



unit

No. of

subdivisions to

be answered

Marks for each

subdivision

Max. marks for

the part

A

Unit I 2

5 2 10Unit II 1

Unit III 1

Unit IV 1

B

Unit I 4

10 5 50Unit II 2

Unit III 4

Unit IV 2

C

Unit I 1

4 10 40Unit II 1

Unit III 1

Unit IV 1

Total 100


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Paper : ELEMENTARY GRAPH THEORY Code : MTH135

Unit I (15 hours)

Introduction to Graphs: Definition and introductory concepts, Graphs as Models,Matrices and Isomorphism, Decomposition and Special Graphs, Connection in Graphs, Bipartite

Graphs, Eulerian Circuits.

Unit II (15 hours)

Vertex degrees and directed Graphs: Counting and Bijections, Extremal Problems,

Graphic Sequences, Directed Graphs, Vertex Degrees, Eulerian Digraphs, Orientations and

Tournaments

Unit III (15 hours)Trees and Distance: Properties of Trees, Distance in Trees and Graphs, Enumeration of

Trees, Spanning Trees in Graphs, Decomposition and Graceful Labellings, Minimum Spanning

Tree, Shortest Paths

Unit IV (15 hours)Connectivity and Paths: Connectivity, Edge-Connectivity, Blocks, 2-connected Graphs,

Connectivity in Digraphs, k-connected and k-edge-connected Graphs, Maximum Network Flow,

Integral Flows

:

1. Kenneth H. Rosen,Discrete mathematics and its applications, McGraw-Hill, 2008.

2. R.P. Grimaldi,Discrete and combinatorial mathematics: An applied introduction, Pearson

Education Inc., 2008.

:

1. F. Harary, Graph theory, Addison Wesley, 1969.2. J.P. Tremblay and R.P. Manohar, Discrete mathematical structures with applications to

computer science, McGraw-Hill, 1975.

3. C. L. Liu,Elements of discrete mathematics, Tata McGraw-Hill, 2000.

4. V.K. Balakrishnan, Combinatorics, Schaums ouline series, 2001.

5. D.B. West,Introduction to graph theory, 2nd Ed., Pearson Education Asia, 2002.

6. Alan Tucker,Applied combinatorics, John Wiley and Sons, 2005.

7. D.S. Chandrasekharaiah, Graph theory and combinatorics, Prism Books, 2005.


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MTH135: ELEMENTARY GRAPH THEORY



unit

No. of

subdivisions to

be answered

Marks for each

subdivision

Max. marks for

the part

A

Unit I 2

5 2 10Unit II 1

Unit III 1

Unit IV 1

B

Unit I 3

10 5 50Unit II 3

Unit III 3

Unit IV 3

C

Unit I 1

4 10 40Unit II 1

Unit III 1

Unit IV 1

Total 100


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Class : M.Sc (MATHEMATICS) Semester : II

Paper : MEASURE THEORY AND INTEGRATION Code : MTH231

Unit ILebesgue Measure (20 hours)

The axiom of choice, extended real numbers, algebras of sets, Borel sets, outer measure,

measurable sets, Lebesgue measure, a non-measurable set, measurable functions,Littlewoods principles.

Unit IIThe Lebesgue Integral (15 hours)

The Riemann integral, the Lebesgue integral of a bounded function over a set of finite

measure, the integral of a nonnegative function, the general Lebesgue integral, convergence

in measure.

Unit IIIDifferentiation and Integration (15 hours)

Differentiation of monotone functions, functions of bounded variation, differentiation of

an integral, absolute continuity.

Unit IV

The Classical Banach Spaces (10 hours)The LP spaces, the Minkowski and Hlder inequalities, convergence and completeness,

bounded linear functionals on the LP spaces.

Text Book

H.L. Royden,Real analysis, Third Edition, Macmillan, 1988.

Reference Books

1. Paul R. Halmos,Measure theory, Van Nostrand, 1950.

2. M.E. Munroe,Introduction to measure and integration, Addison Wesley, 1959.

3. G. de Barra,Measure theory and integration, New Age, 1981.

4. P.K. Jain and V.P. Gupta,Lebesgue measure and integration, New Age, 1986.

5. Frank Morgan, Geometric measure theory A beginners guide, Academic Press, 1988.6. Frank Burk,Lebesgue measure and integration: An introduction, Wiley, 1997.

7. D.H. Fremlin,Measure theory, Torres Fremlin, 2000.

8. M.M. Rao,Measure theory and integration, Second Edition, Marcel Dekker, 2004.

MTH231: MEASURE THEORY AND INTEGRATION



unit

No. of

subdivisions tobe answered



A

Unit I 2

5 2 10Unit II 1

Unit III 1

Unit IV 1

B

Unit I 4

10 5 50Unit II 3

Unit III 3

Unit IV 2

C

Unit I 1

4 10 40

Unit II 1

Unit III 1

Unit IV 1

Total 100


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Paper : COMPLEX ANALYSIS Code : MTH232

Unit I (18 hours)Power Series: Power series, radius and circle of convergence, power series and analytic

functions, Line and contour integration, Cauchys theorem, Cauchy integral formula, Cauchyintegral formula for derivatives, Cauchy integral formula for multiply connected domains,

Moreras theorem, Gauss mean value theorem, Cauchy inequality for derivatives, Liouvillestheorem, fundamental theorem of algebra, maximum and minimum modulus principles.

Unit II (15 hours)Singularities: Taylors series, Laurents series, zeros of analytical functions,

singularities, classification of singularities, characterization of removable singularities and poles.

Unit III (15 hours)Mappings: Rational functions, behavior of functions in the neighborhood of an essential

singularity, Cauchys residue theorem, contour integration problems, mobius transformations,

conformal mappings.Unit IV (12 hours)

Meromorphic functions: Meromorphic functions and argument principle, Schwarz

lemma, Rouches theorem, convex functions and their properties, Hadamard 3-circles theorem.

:

1. M.J. Ablowitz and A.S. Fokas, Complex Variables: Introduction and Applications, Cambridge:

Cambridge University Press, 2003.

2. J.B. Conwey, Functions of One Complex Variable, 2nd ed., New York: Springer, 2000.

1. J.H. Mathews and R.W. Howell, Complex Analysis for Mathematics and Engineering,6th ed., London: Jones and Bartlett Learning, 2011.

2. J.W. BROWN AND R.V. CHURCHILL, COMPLEXVARIABLES ANDAPPLICATIONS, 7TH ED., NEW

YORK: MCGRAW-HILL, 2003.

3. L.S. Hahn and B. Epstein, Classical Complex Analysis, London: Jones and Bartlett

Learning, 2011.

4. A. David Wunsch, Complex Variables with Applications, 3rd ed., New York: Pearson

Education, 2009.

5. D.G. Zill and P.D. Shanahan,A First Course in Complex Analysis with Applications, 2nd

ed., Boston: Jones and Bartlett Learning, 2010.

6. E.M. Stein and Rami Sharchi, Complex Analysis, New Jersey: Princeton University Press,

2003.

7. T.W.Gamblin, Complex Analysis, 1st

ed., Springer, 2001.


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MTH232: COMPLEX ANALYSIS



unit

No. of

subdivisions to

be answered

Marks for each

subdivision

Max. marks for

the part

A

Unit I 2

5 2 10Unit II 1

Unit III 1

Unit IV 1

B

Unit I 4

10 5 50Unit II 2

Unit III 3

Unit IV 3

C

Unit I 1

4 10 40

Unit II 1

Unit III 1

Unit IV 1

Total 100


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Paper : ADVANCED ALGEBRA Code : MTH233

Unit I (15 hours)Advanced Group Theory: Automorphisms, Cayleys theorem, Cauchys theorem,

permutation groups, symmetric groups, alternating groups, simple groups, conjugate elements

and class equations of finite groups, Sylow theorems, direct products, finite abelian groups,solvable groups.

Unit II (15 hours)Polynomial Rings and Fields: Polynomial rings, polynomials rings over the rational

field, polynomial rings over commutative rings, extension fields, roots of polynomials,

construction with straightedge and compass, more about roots.

Unit III (15 hours)Galois theory: The elements of Galois theory, solvability by radicals, Galois group over

the rationals, finite fields.

Unit IV (15 hours)Linear transformation: Algebra of linear transformations, characteristic roots, canonical

forms - triangular, nilpotent and Jordan forms, Hermitian, unitary and normal transformations,

real quadratic forms.

Text Book :I. N. Herstein, Topics in algebra, Second Edition, John Wiley and Sons, 2007.

Reference Books :1. S. Lang,Algebra, 3rd revised ed., Springer, 2002.

2. S. Warner,Modern Algebra, Reprint, Courier Dover Publications, 1990.

3. G. Birkhoff and S.M. Lane,Algebra, 3rd ed., AMS, 1999.

4. J. R. Durbin,Modern algebra: An introduction, 6th ed., Wiley, 2008.

5. N. Jacobson,Basic algebraI, 2nd ed., Dover Publications, 2009.6. S. Singh and Q. Zameeruddin,Modern algebra, revised ed., Vikas Publishing House, 1994.

7. M. Artin,Algebra, 1st ed., Pearson, 1991.

8. J. B. Fraleigh,A first course in abstract algebra, 7th ed., Addison-Wesley Longman, 2002.

9. D.M. Dummit and R.M.Foote,Abstract Algebra, 3rd ed., John Wiley and Sons, 2003.

MTH233 : ADVANCED ALGEBRA



unit

No. ofsubdivisions to

be answered



A

Unit I 2

5 2 10Unit II 1

Unit III 1

Unit IV 1

B

Unit I 3

10 5 50Unit II 4

Unit III 2

Unit IV 3

C

Unit I 1

4 10 40Unit II 1

Unit III 1Unit IV 1

Total 100


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Paper : FLUID MECHANICS Code : MTH234

Unit I (15 Hours)

Introduction: General description of fluid mechanics, continuum mechanics. Fluid

properties: Pressure, density, specific weight, specific volume, specific gravity, viscosity,temperature, thermal conductivity, specific heat, surface tension. Regimes in the mechanics

of fluids, ideal fluids, viscous incompressible fluids, non-Newtonian fluids. Kinematics of

fluids: Methods of describing fluid motion - Lagrangian and Eulerian methods, translation,

rotation and rate of deformation, stream lines, path lines and streak lines, material derivative

and acceleration, vorticity, vorticity in polar coordinates and orthogononal curvilinear

coordinates. Stress and rate of strain: Nature of stressess, transfomation of stress components,

nature of strain, transformation of the rate of strain, relation between stress and rate of strain.

Unit II (10 Hours)

Fundamental Equations of the Flow of Compressible and Incompressible Fluids:

The equation of continuity, conservation of mass, equation of motion (Navier-Stokes

equations), conservation of momentum, the energy equation, conservation of energy.

Unit III (20 Hours)

One, Two and Three Dimensional, Inviscid Incompressible Flow: Equation of

continuity, stream tube flow, equation of motion, Eulers equation, the Bernoulli equation,

applications of Bernoulli equation, basic equations and concepts of flow, equation of

continuity, Eulerian equation of motion, circulation theorems, circulation concept, Stokes

theorem, Kelvins theorem, constancy of circulation, velocity potential, irrotational flow,

integration of the equations of motion, Bernoullis equation, steady motion, irrotational flow,

the momentum theorem, the moment of momentum theorem, Laplace equations, stream

functions in two and three dimensional motion. Two dimensional flow: Rectilinear flow,

source and sink, radial flow, the Milne-Thomson circle theorem and applications, the theoremof Blasius. Three dimensional axially symmetric flow: Uniform flow, radial flow, source or

sink.

Unit IV (15 Hours)

The Laminar Flow of Viscous Incompressible Fluids and the Laminar BoundaryLayer: Similarity of flows, the Reynolds number, viscosity from the point of view of the

kinetic theory, flow between parallel flat plates, Couette flow, plane Poiseuille flow, steady

flow in pipes, flow through a pipe, the Hagen-Poiseuille flow, flow between two concentric

rotating cylinders, properties of Navier-Stokes equations, boundary layer concept, the

boundary layer equations in two-dimensional flow, the boundary layer along a flat plate, the

Blasius solution.

:

1. S. W. Yuan, Foundations of fluid mechanics, Prentice Hall of India, 2001.

2. M. D. Raisinghania, Fluid Dynamics, S. Chand and Company Ltd., 2010.

:

1. R.K. Rathy, An introduction to fluid dynamics, New Delhi: Oxford and IBH Publishing

Company, 1976.

2. G.K. Batchelor,An introduction to fluid mechanics, New Delhi: Foundation Books, 1984.

3. F. Chorlton, Text book of fluid dynamics, New Delhi: CBS Publishers & Distributors,

1985.

4. J.F. Wendt, J.D. Anderson, G. Degrez and E. Dick, Computational fluid dynamics: An

introduction, Springer-Verlag, 1996.

5. Pijush Kundu and Cohen, Fluid Mechanics, Elsevier, 2010.

6. Frank M White, Fluid Mechanics, Tata Mcgraw Hill. 2010.


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MTH234 : FLUID mechanics



unit

No. of


Marks for each

subdivision

Max. marks for

the part

A

Unit I 1

5 2 10Unit II 1

Unit III 1

Unit IV 2

B

Unit I 2

10 5 50Unit II 2

Unit III 4

Unit IV 4

C

Unit I 1

4 10 40Unit II 1

Unit III 1

Unit IV 1

Total 100


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Paper : ALGORITHMIC GRAPH THEORY Code : MTH235

Unit I (15 hours)

Colouring of Graphs: Definition and Examples of Graph Colouring, Upper Bounds,

Brooks Theorem, Graph with Large Chromatic Number, Extremal Problems and TuransTheorem, Colour-Critical Graphs, Counting Proper Colourings

Unit II (15 hours)

Matchings and Factors: Maximum Matchings, Halls Matching Condition, Min-Max

Theorem, Independent Sets and Covers, Maximum Bipartite Matching, Weighted Bipartite

Matching, Tuttes 1-factor Theorem.

Unit III (15 hours)

Planar Graphs: Drawings in the Plane, Dual Graphs, Eulers Formula, Preparation forKuratowskis Theorem, Convex Embeddings, Coloring of Planar Graphs, Crossing Number

Unit IV (15 hours)

Edges and Cycles Edge: Colourings, Characterisation of Line Graphs, Necessary

Conditions of Hamiltonian Cycles, Sufficient Conditions of Hamiltonian Cycles, Cycles in

Directed Graphs, Taits Theorem, Grinbergs Theorem, Flows and Cycle Covers

Textbook

D.B. West, Introduction to Graph Theory, New Delhi: Prentice-Hall of India, 2011.

Reference Books

1. B. Bollabas,Modern Graph Theory, Springer, New Delhi, 2005.

2. F. Harary, Graph Theory, New Delhi: Narosa, 2001.

3. G. Chartrand and P.Chang,Introduction to Graph Theory, New Delhi: Tata McGraw-Hill,

2006

4. G. Chatrand and L. Lesniak, Graphs and Digraphs, Fourth Edition, Boca Raton: CRC Press,

2004.5. J. A. Bondy and U.S.R. Murty, Graph Theory, Springer, 2008

6. J. Clark and D.A. Holton,A First Look At Graph Theory, Singapore: World Scientific, 1995.

7. R. Balakrishnan and K Ranganathan,A Text Book of Graph Theory, New Delhi: Springer, 2008.

8. R. Diestel, Graph Theory, New Delhi: Springer, 2006.

9. R. J. Wilson,Introduction To Graph Theory, Edinburgh: Oliver and Boyd, 1979.

10. V. K. Balakrishnan Graph Theory, Schaums outlines, New Delhi:Tata Mcgrahill, 2004.


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Class : M.Sc (MATHEMATICS) Semester : III

Paper : GENERAL TOPOLOGY Code : MTH331

Unit I (15 hours)

Topological Spaces: Elements of topological spaces, basis for a topology, the order

topology, the product topology on X x Y, the subspace topology, Closed sets and limit points.

Unit II (15 hours)

Continuous Functions: Continuous functions, the product topology, metric topology.

Unit III (15 hours)

Connectedness and Compactness: Connected spaces, connected subspaces of the Real Line,

components and local connectedness, compact spaces, Compact Subspaces of the Real Line,

limit point compactness, local compactness.

Unit IV (15 hours)

Countability and Separation Axioms: The countability axioms, the separation axioms,normal spaces, the Urysohn lemma, the Urysohn metrization theorem, Tietze extension

theorem.

Text Book:

J.R. Munkres, Topology, Second Edition, Prentice Hall of India, 2007.

Reference Books:

1. Simmons,G.F.Introduction to topology and modern analysis, Tata McGraw Hill, 1963.

2. Dugundji,J. Topology, Prentice Hall of India, 1966.3. Willard, General topology, Addison-Wesley, 1970.

4. Crump, W. Baker,Introduction to topology, Krieger Publishing Company, 1997.

MTH331 : GENERAL TOPOLOGY



unit


be answered



A

Unit I 2

5 2 10Unit II 1

Unit III 1

Unit IV 1

B

Unit I 3

10 5 50Unit II 3

Unit III 3

Unit IV 3

C

Unit I 1

4 10 40Unit II 1

Unit III 1Unit IV 1

Total 100


28/55



Paper : NUMERICAL ANALYSIS Code : MTH332

Unit I (20 Hours)

Solution of algebraic and transcendental equations: Fixed point iterative method,convergence criterion, Aitkens 2 -process, Sturm sequence method to identify the numberof real roots, Newton-Raphson methods (includes the convergence criterion for simple roots),

Bairstows method, Graeffes root squaring method, Birge-Vieta method, Mullers method.

Solution of Linear System of Algebraic Equations: LU-decomposition methods (Crouts,

Choleky and Delittle methods), consistency and ill-conditioned system of equations, Tri-

diagonal system of equations, Thomas algorithm.

Unit II (15 Hours)

Numerical solution of ordinary differential equations: Initial value problems,

Runge-Kutta methods of second and fourth order, multistep method, Adams-Moulton

method, stability (convergence and truncation error for the above methods), boundary valueproblems, second order finite difference method, linear shooting method.

Unit III (10 Hours)

Numerical solution of elliptic partial differential equations: Difference methods

for elliptic partial differential equations, difference schemes for Laplace and Poissons

equations, iterative methods of solution by Jacobi and Gauss-Siedel, solution techniques for

rectangular and quadrilateral regions.

Unit IV (15 Hours)

Numerical solution of parabolic and hyperbolic partial differential equations:

Difference methods for parabolic equations in one-dimension, methods of Schmidt,

Laasonen, Crank-Nicolson and Dufort-Frankel, stability and convergence analysis forSchmidt and Crank-Nicolson methods, ADI method for two-dimensional parabolic equation,

explicit finite difference schemes for hyperbolic equations, wave equation in one dimension.

:

1. M.K. Jain, S.R.K. Iyengar and R.K. Jain,Numerical Methods for Scientific and

Engineering Computation, 5th ed., New Delhi: New Age International, 2007.

2. S.S. Sastry,Introductory Methods of Numerical Analysis, 4th ed., New Delhi: Prentice-

Hall of India, 2006.

:

1. R.L. Burden and J. Douglas Faires,Numerical Analysis, 9th ed., Boston: Cengage

Learning, 2011.

2. S.C. Chopra and P.C. Raymond,Numerical Methods for Engineers, New Delhi: Tata

McGraw-Hill, 2010.

3. C.F. Gerald and P.O. Wheatley,Applied Numerical Methods, 7th ed., New York: Pearson

Education, 2009.


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MTH332 : NUMERICAL ANALYSIS



unit

No. of

subdivisions to

be answered

Marks for each

subdivision

Max. marks for

the part

A

Unit I 2

5 2 10Unit II 1

Unit III 1

Unit IV 1

B

Unit I 4

10 5 50Unit II 3

Unit III 2

Unit IV 3

C

Unit I 1

4 10 40Unit II 1Unit III 1

Unit IV 1

Total 100


30/55



Paper : ADVANCE FLUID MECHANICS Code : MTH333

Unit I: (15 Hours)

Heat Transfer: Introduction to heat transfer, different modes of heat transfer- conduction,

convection and radiation, steady and unsteady heat transfer, free and forced convection.Shear Instability: Stability of flow between parallel shear flows - Squires theorem for

viscous and inviscid theory Rayleigh stability equation Derivation of Orr-Sommerfeld

equation assuming that the basic flow is strictly parallel.

Unit II: (20 Hours)

Dimensional Analysis and Similarity: Non-dimensional parameters determined from

differential equations Buckinghams Pi Theorem Nondimensionalization of the Basic

Equations - Non-demensional parameters and dynamic similarity.

Thermal Instability: Basic concepts of stability theory Linear and Non-linear theories

Rayleigh Benard Problem Analysis into normal modes Principle of Exchange of

stabilities first variation principle Different boundary conditions on velocity and

temperature.

Unit III (10 Hours)

Porous Media: Introduction to porous medium, porosity, Darcys Law, Extension of Darcy

Law accelerations and inertial effects, Brinkmans equation, effects of porosity variations,

Bidisperse porous media.

Unit IV (15 Hours)

NonNewtonian Fluids:Constitutive equations of Maxwell, Oldroyd, Ostwald , Ostwaldde waele, Reiner Rivlin and Micropolar fluid. Weissenberg effect and Toms effect.

Equation of continuity, Conservation of momentum for non-Newtonian fluids.

Text Books:

1. Drazin and Reid,Hydrodynamic instability, Cambridge University Press, 2006.

2. S. Chardrasekhar, Hydrodynamic and hydrodmagnetic stability, Oxford University

Press, 2007 (RePrint).

References :

1. D. J. Tritton, Physical fluid Dynamics, Van Nostrand Reinhold Company, 1979.

2. Drazin .Introduction to Hydrodynamic Stability, Cambridge University Press, 2006.

3. Pijush Kundu and Cohen, Fluid Mechanics, Elsevier, 2010.

4. Frank M White, Fluid Mechanics, Tata Mcgraw Hill. 2010.

5. Donald A. Nield and Adrian Bejan, Convection in Porous Media, Third edition,

Springer, 2006.


31/55


MTH333 : ADVANCED FLUID MECHANICS



unit

No. of

subdivisions to

be answered

Marks for each

subdivision

Max. marks for

the part

A

Unit I 1

5 2 10Unit II 1

Unit III 1

Unit IV 2

B

Unit I 3

10 5 50Unit II 3

Unit III 3

Unit IV 3

C

Unit I 1

4 10 40Unit II 1Unit III 1

Unit IV 1

Total 100


32/55



Paper : ADVANCED GRAPH THEORY Code : MTH334

Unit I (15 hours)Domination in Graphs: Domination in Graphs, Bounds in terms of Order, Bounds in terms

of Order, Degree and Packing, Bounds in terms of Order and Size, Bounds in terms ofDegree, Diameter and Girth, Bounds in terms of Independence and Covering

Unit II (15 hours)Advaced Digraph theory: Acyclic Digraphs, Multipartite Digraphs and Extended Digraphs,

Transitive Digraphs, Line Digraphs, Series-Parallel Digraphs, Quasi-Transitive Digraphs,

Path-Mergeable Digrpahs, Locally Semicomplete Digrraphs, Totally -Decomposable

Digraphs, Planar Digraphs.

Unit III (15 hours)

Perfect Graphs: The Perfect Graph Theorem, Chordal Graphs Revisited, Other Classes of

Perfect Graphs, Imperfect Graphs, The Strong Perfect Graph Conjecture

Unit IV (15 hours)Matroids: Hereditary Systems and Examples, Properties of Matroids, The Span Function,

The Dual of a Matroid, Matroid Minors and Planar Graphs, Matroid Intersection, Matroid

Union

Textbooks

1. D.B. West, Introduction to Graph Theory, New Delhi: Prentice-Hall of India, 2011.

2. T.W. Haynes, S. T. Hedetniemi and P. J. Slater, Fundamentals of Domination in Graphs.New York: Marcel Dekker, Inc., 1998.

3. J. Bang-Jensen and G. Gutin,Digraphs. London: Springer, 2009.

Reference Books

1. B. Bollabas,Modern Graph Theory, Springer, New Delhi, 2005.

2. F. Harary, Graph Theory, New Delhi: Narosa, 2001.

3. G. Chartrand and P.Chang,Introduction to Graph Theory, New Delhi: Tata McGraw-Hill,

2006

4. G. Chatrand and L. Lesniak, Graphs and Digraphs, Fourth Edition, Boca Raton: CRC

Press, 2004.

5. J. A. Bondy and U.S.R. Murty, Graph Theory, Springer, 2008

6. J. Clark and D.A. Holton,A First Look At Graph Theory, Singapore: World Scientific,

1995.

7. R. Balakrishnan and K Ranganathan,A Text Book of Graph Theory, New Delhi: Springer,

2008.

8. R. Diestel, Graph Theory, New Delhi: Springer, 2006.

9. R. J. Wilson,Introduction To Graph Theory, Edinburgh: Oliver and Boyd, 1979.

10. V. K. Balakrishnan Graph Theory, Schaums outlines, New Delhi:Tata Mcgrahill, 2004.


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MTH334:ADVANCED GRAPH THEORY



unit

No. of

subdivisions to

be answered

Marks for each

subdivision

Max. marks for

the part

A

Unit I 2

5 2 10Unit II 1

Unit III 1

Unit IV 1

B

Unit I 3

10 5 50Unit II 3

Unit III 3

Unit IV 3

C

Unit I 1

4 10 40Unit II 1

Unit III 1

Unit IV 1

Total 100


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Paper : CLASSICAL MECHANICS Code : MTH335

UNIT I (12 Hours)

Introductory concepts: The mechanical system - Generalised Coordinates - constraints -

virtual work - Energy and momentum.UNIT II (20 Hours)Lagrange's equation: Derivation and examples - Integrals of the Motion - Small

oscillations. Special Applications of Lagranges Equations: Rayleighs dissipation function -

impulsive motion - velocity dependent potentials.

UNIT III (13 Hours)

Hamilton's equations: Hamilton's principle - Hamiltons equations - Other variational

principles - phase space.

UNIT IV (15 Hours)

Hamilton - Jacobi Theory: Hamilton's Principal FunctionThe Hamilton - Jacobi equation

- Separability.

Text Book:

Donald T. Greenwood, Classical Dynamics, Reprint, USA: Dover Publications, 1997.

Reference Books:

1. H. Goldstein, Classical Mechanics, Second edition, New Delhi : Narosa Publishing House,

2001.

2. N.C. Rana and P.S. Joag, Classical Mechanics, 29th

Reprint, New Delhi: Tata McGraw-

Hill, 2010.

3. J.E. Marsden, R. Abraham, Foundations of Mechanics, 2nd ed., American MathematicalSociety, 2008.

MTH335 : CLASSICAL MECHANICS



unit

No. of

subdivisions to

be answered

Marks for each

subdivision

Max. marks for

the part

A

Unit I 1

5 2 10Unit II 2

Unit III 1

Unit IV 1

B

Unit I 2

10 5 50Unit II 4

Unit III 3

Unit IV 3

C

Unit I 1

4 10 40

Unit II 1

Unit III 1

Unit IV 1

Total 100


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Class : M.Sc (MATHEMATICS) Semester : IV

Paper : DIFFERENTIAL GEOMETRY Code : MTH431

Unit I (15 hours)

Calculus on Euclidean Geometry: Euclidean SpaceTangent VectorsDirectional

derivativesCurves in E31-FormsDifferential FormsMappings.Unit II (15 hours)

Frame Fields and Euclidean Geometry: Dot productCurvesvector field - The

Frenet FormulasArbitrary speed curvescylindrical helixCovariant DerivativesFrame

fieldsConnection Forms - The Structural equations.

Unit III (15 hours)

Euclidean Geometry and Calculus on Surfaces: Isometries of E3The derivative

map of an Isometry - Surfaces in E3 patch computations Differential functions and

Tangent vectorsDifferential forms on a surfaceMappings of Surfaces.

Unit IV (15 hours)

Shape Operators: The Shape operator of M E3 Normal Curvature Gaussian

Curvature - Computational TechniquesSpecial curves in a surfaceSurfaces of revolution.

Text Book

B.ONeill, Elementary Differential geometry, 2nd

revised ed., New York: Academic Press,

2006.

Reference Books

1. J.A. Thorpe,Elementary topics in differential geometry, 2nd

ed., Springer, 2004.

2. A. Pressley,Elementary differential geometry, 2nd ed., Springer, 2010.

3. Mittal and Agarwal,Differential geometry, 36th

ed., Meerut: Krishna Prakashan Media (P)

Ltd., 2010.

MTH431 : DIFFERENTIAL GEOMETRY



unit

No. of

subdivisions to

be answered

Marks for each

subdivision

Max. marks for

the part

A

Unit I 2

5 2 10Unit II 1

Unit III 1

Unit IV 1

B

Unit I 3

10 5 50Unit II 3

Unit III 3

Unit IV 3

C

Unit I 1

4 10 40Unit II 1

Unit III 1

Unit IV 1

Total 100


36/55



Paper : ADVANCED NUMERICAL METHODS Code : MTH432

Fourth Edition, P.W.S. KentPublishing Company, 2007.

Reference Books:1. Kandasamy, P., Thilagavathy, K. and Gunavathy, K.,Numerical Methods, New Delhi:

S. Chand Co. Ltd., 2003.

2. R.L. Burden and J. Douglas Faires, Numerical Analysis, Fourth Edition, P.W.S. Kent

Publishing Company, 2007.

3. S.C. Chopra and P.C. Raymond, Numerical methods for engineers, Tata McGraw-Hill,

2000.4. C.F. Gerald and P.O. Wheatley,Applied numerical methods, Pearson Education, 2002.

5. L. C. Andrews, and R. L. Philips, Mathematical Techniques for Engineers and Scientists,

Prentice Hall of India, 2006.

6. Ji Huan He,Homotopy perturbation technique, Computer Methods in Applied Mechanics

and Engineering Vol. 178, Issues 3-4, August 1999, Pages 257-262.

7. Vedat Suat Ertrk,Differential Transformation Method For Solving Differential Equations

of Lane - Emden Type, Mathematical and computer applications, vol. 12(3), 135-139,

2007.


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40/55



Paper : CALCULUS OF VARIATIONS Code : MTH441

AND INTEGRAL EQUATIONS

Euler equations and variational notations: Maxima and minima, method of

Lagrange multipliers, the simplest case, Euler equation, extremals, stationary function,

geodesics, Brachistochrone problem, natural boundary conditions and transition conditions,

variational notation, the more general case.

Advanced variational problems: Constraints and Lagrange multipliers, variable end

points, Sturm-Liouville problems, Hamiltons principle, Lagranges equation, the Rayleigh-Ritz method.

Linear integral equations: Definitions, integral equation, Fredholm and Volterra

equations, kernel of the integral equation, integral equations of different kinds, relations

between differential and integral equations, symmetric kernels, the Greens function.

Methods for solutions of linear integral equations: Fredholm equations with

separable kernels, homogeneous integral equations, characteristic values and characteristic

functions of integral equations, Hilbert-Schmidt theory, iterative methods for solving integralequations of the second kind, the Neumann series.

F.B. Hildebrand,Methods of Applied Mathematics, New York: Dover, 1992.

1. B. Dacorogna,Introduction to the Calculus of Variations, London: Imperial College Press,

2004.

2. F. Wan,Introduction to the Calculus of Variations and Its Applications, New York:

Chapman/Hall, 1995.

3. J. Jost and X. Li-Jost, Calculus of Variations, Cambridge: Cambridge University Press,

1998.

4. R.P. Kanwal,Linear Integral Equations: Theory and Techniques, New York: Birkhuser,

2013.

5. C. Corduneanu,Integral Equations and Applications, Cambridge: Cambridge University

Press, 2008.

6. A.J. Jerry,Introduction to Integral Equations with Applications, 2nd ed., New York: John

Wiley & Sons, 1999.


41/55


MTH441: CALCULUS OF VARIATIONS AND INTEGRAL EQUATIONS



unit


be answered

Marks for each

subdivision

Max. marks for

the part

A

Unit I 2

5 2 10Unit II 1

Unit III 1

Unit IV 1

B

Unit I 4

10 5 50Unit II 3

Unit III 2Unit IV 3

C

Unit I 1

4 10 40Unit II 1

Unit III 1

Unit IV 1

Total 100


42/55


Class : II - M.Sc Semester : IV

Paper : MAGNETOHYDRODYNAMICS Code : MTH442

Unit I (12 Hours)Electrodynamics: Outline of electromagnetic units and electrostatics, derivation of

Gauss law, Faradays law, Amperes law and solenoidal property, dielectric material,

conservation of charges, electromagnetic boundary conditions.

Unit II (13 Hours)

Basic Equations: Outline of basic equations of MHD, magnetic induction equation,

Lorentz force, MHD approximations, non-dimensional numbers, velocity, temperature and

magnetic field boundary conditions.

Unit III (20 Hours)

Exact Solutions: Hartmann flow, generalized Hartmann flow, velocity distribution,

expression for induced current and magnetic field, temperature discribution, Hartmann

couette flow, magnetostatic-force free magnetic field, abnormality parameter, Chandrashekar

theorem, application of magnetostatic-Bennett pinch.

Unit IV (15 Hours)

Applications: Classical MHD and Alfven waves, Alfven theorem, Frozen-in-

phenomena, Application of Alfven waves, heating of solar corana, earths magnetic field,

Alfven wave equation in an incompressible conducting fluid in the presence of an vertical

magnetic field, solution of Alfven wave equation, Alfven wave equation in a compressible

conducting non-viscous fluid, Helmholtz vorticity equation, Kelvins circulation theorem,Bernoullis equation.

:

1. V.C.A. Ferraro and Plumpton, An introduction to magnetofluid mechanics, Clarendon

Press, 1966.

2. P.H. Roberts,An introduction to magnetohydrodynamics, Longman, 1967.

3. Allen Jeffrey,Magnetohydrodynamics, Oliver Boyds, 1970.

:

1. Sutton and Sherman,Engineering magnetohydrodynamics, McGraw-Hill, 1965.

2. H.K. Moffat,Magnetic generation in electrically conducting fluids, Cambridge University

Press, 1978.

3. David J. Griffiths,Introduction to electrodynamics, Prentice Hall of India, 1997.


43/55


MTH442 : Magnetohydrodynamics



unit

No. of


Marks for each

subdivision

Max. marks for

the part

A

Unit I 1

5 2 10Unit II 1

Unit III 2

Unit IV 1

B

Unit I 3

10 5 50Unit II 2

Unit III 4

Unit IV 3

C

Unit I 1

4 10 40Unit II 1

Unit III 1

Unit IV 1

Total 100


44/55



Paper : WAVELET THEORY Code : MTH443

Unit I (15 hours)

Introduction: Complex numbers and basic operation, the space L2(R), inner products, basesand projections, Eulers formula and complex exponential function, Fourier series, Fourier

transforms, Convolutions and B-Splines, the wavelet, requirements for wavelet.

Unit II (15 hours)

The Continuous wavelet transform: The wavelet transform, the inverse wavelet transform,

wavelet transform in terms of Fourier transform, Complex wavelets: the Morlet wavelet.

Unit III (15 hours)

The discrete wavelet transform: Frames and orthogonal wavelet bases, Haar space, general

Haar space, Haar wavelet space, general Haar wavelet space, discrete Haar wavelet

transforms and applications.

Unit IV (15 hours)Wavelet packets: The construction of wavelet sets, the measure of the closure of a wavelet

set, constructing wavelet packet spaces, wavelet packet spaces.

Text books

1. David K .Ruch and Patrick J. Van Fleet, Wavelet Theory: An elementary approach with

Applications, Wiley, 2009.

2. Paul S. Addison, The Illustrated Wavelet Transform Handbook, IOP, 2002.

Reference books

1. Rao R.M. &Bopardikar A.S., Wavelet Transforms-Introduction to Theory and

Applications, Pearson Education Asia, 1999.

2. Sidney Burrus, Gopinath R.A. &HaitaoGuo,Introduction to Wavelets and Wavelet

Transforms, Prentice Hall International, 1998.

3. Chan Y.T., Wavelet Basics, Kluwer Academic Publishers, 1995.

4. Goswami J.C. & Chan A.K., Fundamentals of Wavelets - Theory Algorithms and

Applications, New York: John Wiley, 1999.


45/55


MTH443:


Part Unit and No. of subdivisions to be set in theunit

No. of


Marks for eachsubdivision Max. marks forthe part

A

Unit I 2

5 2 10Unit II 1

Unit III 1

Unit IV 1

B

Unit I 3

10 5 50Unit II 3

Unit III 3

Unit IV 3

C

Unit I 1

4 10 40Unit II 1

Unit III 1

Unit IV 1

Total 100


46/55



Paper : MATHEMATICAL MODELLING Code : MTH444

Unit I (15 Hours)Concept of mathematical modeling: Definition, Classification, Characteristics And

Limitations , Mathematical Modelling Through Ordinary Differential Equations Of First

Order: Linear And Nonlinear Growth and Decay Models Compartment Models, Dynamics

Problems, Geometrical Problems

Unit II (12 hours)

Mathematical modelling through systems of ordinary differential equations of firstorder: Population Dynamics, Epidemics, Compartment Models, Economics, Medicine, Arms

Race, Battles and International Trade and Dynamics

Unit III (13 Hours)Mathematical modelling through ordinary differential equations of second order:

Modeling Of Planetary Motions Circular Motion Of Satellites, Mathematical Modelling

Through Linear Differential Equations Of Second Order, Miscellaneous Mathematical

Models

Unit IV (20 Hours)

Mathematical Modelling leading to linear and nonlinear partial differential equations:

Simple models, conservation lawTraffic flow on highwayFlood waves in riversglacier

flow, roll waves and stability, shallow water waves Convection diffusion processesBurgers equation, Convection reaction processes Fishers equation.Telegraphers

equation heat transfer in a layered solid. Chromatographic models sediment Transport inrivers reaction-diffusion systems, travelling waves, pattern formation, tumour growth.

Text Books:

1. M. Braun, C.S. Coleman and D. A. Drew,Differential equation Models, 1994

2. J.N.Kapur,Mathematical Modelling, Springer, 2005

3. J.N.Kapur,Mathematical Models in Biology and Medicine, East-West Press, New Delhi,

1981

Reference Books:

1. W. F. Lucas, F S Roberts and R.M. Thrall,Discrete and system models, Springer, 1983.

2. H.M. Roberts & Thompson,Life science models, Springer, 1983.

3. A.C. Fowler,Mathematical Models in Applied Sciences, Cambridge University Press,

1997.

4. Stanely J. Farlow, Partial Differential Equations for Scientists and Engineers, Dover

5. Walter J. Meyer , Concepts of Mathematical Modeling


47/55


MTH444: MATHEMATICAL MODELLING



unit


be answered

Marks for each

subdivision

Max. marks for

the part

A

Unit I 2

5 2 10Unit II 1

Unit III 1

Unit IV 1

B

Unit I 3

10 5 50Unit II 3

Unit III 3

Unit IV 3

C

Unit I 1

4 10 40Unit II 1

Unit III 1

Unit IV 1

Total 100


48/55



Paper : CRYPTOGRAPHY Code : MTH445

Unit I (15 hours)Some Topics in Elementary Number Theory: Elementary concepts of number

theory, time estimates for doing arithmetic, divisibility and the Euclidian algorithm,

congruences, some applications to factoring. Finite fields and quadratic residues: Finite

fields, quadratic residues and reciprocity.

Unit II (15 hours)

Cryptography: Some simple cryptosystems, enciphering matrices.

Unit III (15 hours)

Public Key: The idea of public key cryptography, RSA, discrete log., knapsack,

zero-knowledge protocols and oblivious transfer.

Unit IV (15 hours)

Elliptic Curves: Basic facts, elliptic curve cryptosystems, elliptic curve primality

test, elliptic curve factorization.

:

N. Koblitz, A course in number theory and cryptography, Graduate Texts in Mathematics,

No.114, Springer-Verlag, 1987.

:

1. A. Baker, A concise introduction to the theory of numbers, Cambridge University Press,

1990.

2. A.N. Parshin and I.R. Shafarevich (Eds.), Number theory, encyclopedia of mathematics

sciences, Vol. 49, Springer-Verlag, 1995.

3. D.R. Stinson, Cryptography: Theory and Practice, CRC Press, 1995

4. H.C.A. van Tilborg,An introduction to cryptography, Kluwer Academic Publishers, 1998.5. Wade Trappe and Lawrence C. Washington, Introduction to Cryptography with Coding

Theory, Prentice hall, 2005.


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50/55



Paper : Code : MTH446

Unit I (15 Hours)

Essential Fluid Dynamics: Thermal wind, geostrophic motion, hydrostatic approximation,consequences, Taylor-Proudman theorem, Geostrophic degeneracy, dimensional analysis and non-

dimensional numbers. Physical Meteorology: Atmospheric composition, laws of thermodynamics of

the atmosphere, adiabatic process, potential temperature, the Classius-Clapyeron equation, laws of

black body radiation, solar and terrestrial radiation, solar constant, Albedo, greenhouse effect, heat

balance of earth-atmosphere system.

Unit II (15 Hours)Atmosphere Dynamics: Geostrophic approximation, pressure as a vertical coordinate,

modified continuity equation, balance of forces, non-dimensional numbers (Rossby, Richardson,

Froude, Ekman etc.), scale analysis for tropics and extra-tropics, vorticity and divergence equations,

conservation of potential vorticity, atmospheric turbulence and equations for planetary boundary

layer.

Unit III (15 Hours)General Circulation of the Atmosphere: Definition of general circulation, various

components of general circulation, zonal and eddy angular momentum balance of the atmosphere,

meridional circulation, Hadley-Ferrel and polar cells in summer and winter, North-South and East-

West (Walker) monsoon circulation, forces meridional circulation due to heating and momentum

transport, available potential energy, zonal and eddy energy equations.

Unit IV (15 hours)Atmospheric Waves and Instability: Wave motion in general, concept of wave packet,

phase velocity and group velocity, momentum and energy transports by waves in the horizontal andvertical, equatorial, Kelvin and mixed Rossby gravity waves, stationary planetary waves, filtering of

sound and gravity waves, linear barotropic and baroclinic instability.

:

1. Joseph Pedlosky, Geophysical fluid dynamics, Springer-Verlag, 1979.

2. J.R. Holton,An introduction to dynamic meteorology, 3rd Ed., Academic Press, 1992.

1. F.F. Grossard and W.H. Hooke, Waves in the atmosphere, Elsevier, 1975.

2. Ghil and Chidress, Topics in geophysical fluid dynamics, Applied Mathematical Science,

Springer Verlag, 1987.

3. S. Friedlander, Geophysical fluid dynamics, Lecture Notes, Springer, 1998.


51/55


MTH446:



unit

No. of

subdivisions to

be answered

Marks for each

subdivision

Max. marks for

the part

A

Unit I 2

5 2 10Unit II 1

Unit III 1

Unit IV 1

B

Unit I 3

10 5 50Unit II 3

Unit III 3

Unit IV 3

C

Unit I 1

4 10 40Unit II 1

Unit III 1

Unit IV 1

Total 100


52/55


53/55


54/55


(45 Hours)

Unit I (15 Hours)

Algebraic Computation: Simplification of algebraic expression, simplification of

expressions involving special functions, built-in functions for transformations on

trigonometric expressions, definite and indefinite symbolic integration, symbolic sums and

products, symbolic solution of ordinary and partial differential equations, symbolic linear

algebra, matrix operations.

Unit II (15 Hours)

Mathematical Functions: Special functions, inverse error function, gamma and betafunction, hypergeometric function, elliptic function, Mathieu function. Numerical

Computation: Numerical solution of differential equations, numerical solution of initial and

boundary value problems, numerical integration, numerical differentiation, matrix

manipulations, optimization techniques.

Unit III (15 Hours)

Graphics: Two- and three-dimensional plots, parametric plots, typesetting capabilities for

labels and text in plots, direct control of final graphics size, resolution etc. Packages: Linear

algebra, calculus, discrete math, geometry, graphics, number theory, vector analysis,

statistics.

Text Book:

1. Stephen Wolfram, The mathematica book, Wolfram Research Inc., 2008.

Reference Books:

1. Michael Trott, The Mathematica guide book for programming, Springer, 2004.

2. P. Wellin, R. Gaylord and S. Kamin,An introduction to programming with Mathematica,

Cambridge, 2005.


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Internship in PG Mathematics course

Semester: III

Code: MTH371

Objectives:

To expose students to the field of their professional interest To give an opportunity to get practical experience of the field of their interest

To strengthen the curriculum based on internship feedback where relevant

To help student choose their career through practical experience

Level of Knowledge: Working knowledge of Mathematics

M.Sc. Mathematics students have the option to undertake an internship of not less than 45

working days at any of the following: reputed research centres, recognized educational

institutions, summer research fellowships, programmes like M.T.T.S. or any other approved

by the P.G. coordinator and H.O.D.

The internship is to be undertaken at the end of second semester (during first year

vacation). The report submission and the presentation on the report will be held during the

third semester and the credits will appear in the mark sheet of third semester.

The students will have to give an internship proposal with the following details:

Organization where the student proposes to do the internship, reasons for the choice, nature

of internship, period on internship, relevant permission letters, if available, name of thementor in the organization, email, telephone and mobile numbers of the person in the

organization with whom Christ University could communicate matters related to internship.

Typed proposals will have to be given at least one month before the end of the second

semester.

The coordinator of the programme in consultation with the HOD will assign faculty

members from the department as guides at least two weeks before the end of second

semester.

The students will have to be in touch with the guides during the internship period either

through personal meetings, over the phone or through email.

At the place of internship, students are advised to be in constant touch with their mentors.

At the end of the required period of internship, the candidates will submit a report in not

less than 750 words. The report should be submitted within first 10 days of the reopening of

the University for the third semester.

Within 20 days from the day of reopening, the department must hold a presentation by the

students. During the presentation the guide or a nominee of the guide should be present and

be one of the evaluators. Students should preferably be encouraged to make a power pointpresentation of their report. A minimum of 10 minutes should be given for each of the

presenter. The maximum limit is left to the discretion of the evaluation committee. Students

will get 2 credits on successful completion of internship.

m.sc mathematics syllabus -2013-14

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