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DEPARTMENT OF MATHEMATICS


SYLLABUSMASTER’S DEGREE IN MATHEMATICS

2013 – 14 ONWARDS


CHRIST UNIVERSITY


Master’s 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 mathematician’s habit of thought and reasoning and willenlighten students with mathematical ideas relevant for oneself and for thecourse 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 inmathematics such as Algebra, Analysis and Number Theory along with thebasic applied paper ordinary and partial differential equations. In the third andfourth semester focus is on the special papers, elective paper and skill-basedpapers 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 topicFluid Mechanics and Graph Theory is offered in each semester with a project onthese topic in the fourth semester, which will help the students to pursue thehigher studies in these topics. Special importance is given to TeachingTechnology and Research Methodology in Mathematics, MathematicalStatistics 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 willthen appear for subject interview.


Modular Objectives:

MTH 131: NUMBER THEORYThis paper is concerned with the basics of analytical number theory. Topics such asdivisibility, congruence’s, quadratic residues and functions of number theory are covered inthis paper. Some of the applications of the said concepts are also included.

MTH 132: REAL ANALYSISThis paper will help students understand the basics of real analysis. This paper includes suchconcepts as basic topology, Riemann-Stieltjes integral, sequences and series of functions.

MTH 133: ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONSThis helps students understand the beauty of the important branch of mathematics, namely,differential equations. This paper includes a study of second order linear differentialequations, adjoint and self-adjoint equations, Eigen values and Eigen vectors of theequations, power series method for solving differential equations, second order partialdifferential equations like wave equation, heat equation, Laplace equations and their solutionsby Eigen function method.

MTH 134: CONTINUUM MECHANICSThis paper is an introductory course to the basic concepts of continuum mechanics and fluidmechanics. This includes Cartesian tensors, stress–strain tensor, conservation laws andconstitutive relations for linear elastic solid.

MTH 135: ELEMENTARY GRAPH THEORYThis paper is an introductory course to the basic concepts of Graph Theory. This includesdefinition of graphs, vertex degrees, directed graphs, trees, distances, connectivity and paths.

MTH 231: MEASURE THEORY AND INTEGRATIONThis paper deals with various aspects of measure theory and integration by means of theclassical approach. More advanced concepts such as measurable sets, Borel sets, Lebesguemeasure, Lebesgue integration and LP spaces have been included in this paper.

MTH 232: COMPLEX ANALYSISThis paper will help students learn about the essentials of complex analysis. This paperincludes important concepts such as power series, analytic functions, linear transformations,Laurent’s series, Cauchy’s theorem, Cauchy’s integral formula, Cauchy’s residue theorem,argument principle, Schwarz lemma , Rouche’s theorem and Hadamard’s 3-circles theorem.

MTH 233: ADVANCED ALGEBRAThis paper enables students to understand the intricacies of advanced areas in algebra. Thisincludes a study of advanced group theory, polynomial rings, Galois theory and lineartransformation.

MTH 234: FLUID MECHANICSThis 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 andcycles.


MTH 331: GENERAL TOPOLOGYThis paper deals with the essentials of topological spaces and their properties in terms ofcontinuity, connectedness, compactness etc.

MTH 332: NUMERICAL ANALYSISThis paper helps students to have an in-depth knowledge of various advanced methods innumerical analysis. This includes solution of algebraic and transcendental equations, andordinary and partial differential equations.

MTH 333: CLASSICAL MECHANICSThis paper deals with some of the key ideas of classical mechanics. The concepts covered inthe paper include generalized coordinates, Lagrange’s equations, Hamilton’s equations andHamilton - Jacobi theory.

MTH 334: ADVANCED FLUID MECHANICSThis paper helps the students to understand the basic concepts of heat transfer, types ofconvection shear and thermal instability of linear and non-linear problems, dimensionalanalysis. This paper also includes the analysis of Rayleigh –Benard problem with andwithout porous media.

MTH 335: ADVANCED GRAPH THEORYDomination of Graphs, digraph theory, perfect graphs and metroids are dealt with in thedetail in this paper.

MTH 431: DIFFERENTIAL GEOMETRYDifferential geometry is the study of geometric properties of curves, surfaces, and theirhigher dimensional analogues using the methods of calculus. On successful completion ofthis module students will have acquired an active knowledge and understanding of the basicconcepts of the geometry of curves and surfaces in three-dimensional Euclidean space andwill be acquainted with the ways of generalising these concepts to higher dimensions".

MTH 432: ADVANCED NUMERICAL METHODSThis paper includes the advanced and modern methods of solving the linear and non-linear,initial and boundary value differential equations, which includes the methods like homotpyperturbation and differential transforms methods.

MTH 433: FUNCTIONAL ANALYSISThis abstract paper imparts an in-depth analysis of Banach spaces, Hilbert spaces, conjugatespaces, etc. This paper also includes a few important applications of functional analysis toother branches of both pure and applied mathematics.

MTH 451: PROJECT

MTH 441: CALCULUS OF VARIATIONS AND INTEGRAL EQUATIONSThis paper concerns the analysis and applications of calculus of variations and integralequations. Applications include areas such as classical mechanics and differential equations.

MTH 442: MAGNETOHYDRODYNAMICSThis paper provides the fundamentals of Magnetohydrodynamics, which include theory ofMaxwell’s equations, basic equations, exact solutions and applications of classical MHD.

MTH 443: WAVELET THEORY


This paper aims at studying the fundamentals of wavelet theory. This includes the concept onthe continuous and discrete wavelet transform and wavelet packets like construction andmeasure of wavelet sets and construction of wavelet spaces.

MTH 444: MATHEMATICAL MODELLINGThis paper is concerned with the fundamentals of mathematical modeling. The coverageincludes mathematical modeling through ordinary and partial differential equations.

MTH 445: CRYPTOGRAPHYThis paper introduces basics of number theory and some crypto systems.

MTH 446: ATMOSPHERIC SCIENCEThis paper provides an introduction to the dynamic meteorology, which includes theessentials of fluid dynamics, atmospheric dynamics and atmosphere waves and instabilities.


COURSE STRUCTURE FOR M.SC. (MATHEMATICS)

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 DifferentialEquations

4 100 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


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 andIntegral Equations

MTH 442 Magnetohydrodynamics

MTH 443 Wavelet Theory

MTH 444 Mathematical Modelling

MTH 445 Cryptography

MTH 446 Atmospheric Science

Total 20 500 20


CERTIFICATE COURSES

I Semester

Paper Code Title Total No. ofHours

Credit

MTH 101Teaching Technology andResearch Methodology in

Mathematics

45 2

II Semester


Credit

MTH 201 Statistics45 2

MTH371 : INTERNSHIP 2 credits

III Semester


Credit

MTH 301Introduction to Mathematical

Packages45 2


SYLLABUS (M.SC. MATHEMATICS)

Class : M.Sc (MATHEMATICS) Semester : IPaper : NUMBER THEORY Code : MTH131

Unit I (10 hours)Divisibility: The division algorithm, the Euclidean algorithm, the unique

factorization theorem, Euclid’s theorem, linear Diophantine equations.

Unit II (20 hours)Congruences: Definitions and properties, complete residue system modulo m,

reduced residue system modulo m, Euler’s function, Fermat’s theorem, Euler’sgeneralization of Fermat’s theorem, Wilson’s 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, Gauss’s 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 theoryof 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.


MTH131 : NUMBER THEORY

END SEMESTER EXAMINATION : FORMAT OF THE QUESTION PAPER

PartUnit and No. of subdivisions to be set in the

unit

No. ofsubdivisions 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


Class : M.Sc (MATHEMATICS) Semester : IPaper : 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, monotonicfunctions, 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 valuetheorems 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.

Text Book :

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

Reference Books :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.


MTH132 : REAL ANALYSIS



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 1

Unit IV 1

Total 100


Class : M.Sc (MATHEMATICS) Semester : IPaper : ORDINARY AND PARTIAL Code : MTH133

DIFFERENTIAL EQUATIONS

Unit I (20 hours)Linear Differential Equations:Linear differential equations, fundamental sets of

solutions, Wronskian, Liouville’s theorem, adjoint and self-adjoint equations, Lagrangeidentity, Green’s formula, zeros of solutions, comparison and separation theorems. Eigenvalues 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.

Unit III (10 Hours)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, canonicalforms and general solution.

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

D’Alembert’s solution, domain of influence and domain of dependence, the Cauchy problemfor non-homogeneous wave equation, the method of separation of variables for the one-dimensional wave equation and heat equation. Boundary value problems, Dirichlet andNeumann problems in Cartesian coordinates, solution by the method of separation ofvariables. Solution by the method of eigenfunctions.

Text Books: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).

Reference Books: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, 20038. 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%22


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., NewYork: Pearson Education, 2003.

MTH133 : ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS



unit


be answered



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


Class : M.Sc (MATHEMATICS) Semester : IPaper : CONTINUUM MECHANICS Code : MTH134

Unit I (15 Hours)Cartesian tensors: Co-ordinate transformations, Cartesian tensors, basic properties,

transpose, symmetric and skew symmetric tensors, isotropic tensors, gradient, divergence andcurl in tensor calculus, integral theorems.

Unit II (20 Hours)Stress and Strain: Continuum hypothesis, deformation gradient, strain tensors,

infinitesimal strain, compatibility relations, principal strains, material and local timederivatives, strain-rate tensor, transpose formulas, stream lines, path lines, vorticity andcirculation, stress components and stress tensor, normal and shear stresses, principal stresses.

Unit III (15 Hours)Fundamental Law: Law of conservation of mass, principles of linear and angular

momentum, balance of energy.

Unit IV (10 Hours)Linear Elastic Solids: Constitutive relations for a linear elastic solid, generalized

Hooke’s law, governing equations, Navier’s equation, stress formulation, Beltrami-Michellequation.

Text Book:

D. S. Chandrasekharaiah and L. Debnath, Continuum mechanics, Academic Press, 1994(Reprint).

Reference Books:

1. W. Michel Lai, David Rubin and Erhard Krempl, Introduction to Continuum Mehcanics,Fourth Edition, Elsevier, 2009.

2. Oscar Gonzalez and Andrew M Stuart, A first course in continuum Mechanics, Cambridge,2010.

3. Morton E. Gurtin, An Introduction to continuum Mechanics, Academic Press, 2008.4. J. N. Reddy, An Introduction to continuum Mechanics, Cambridge, 2008.5. P. Chadwick, Continuum mechanics, Allen and Unwin, 1976.6. A. J. M. Spencer, Continuum mechanics, Longman, 1980.


MTH134: CONTINUUM MECHANICSEND SEMESTER EXAMINATION : FORMAT OF THE QUESTION PAPER


unit


be answered



A

Unit I 2

5 2 10Unit II 1

Unit III 1

Unit IV 1

B

Unit I 5

10 5 50Unit II 3

Unit III 1

Unit IV 3

C

Unit I 1

4 10 40Unit II 1

Unit III 1

Unit IV 1

Total 100


Class : M.Sc (MATHEMATICS) Semester : IPaper : 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, BipartiteGraphs, 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 andTournaments

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 SpanningTree, 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

Text Books:

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.

Reference Books:

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, Schaum’s 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.


MTH135: ELEMENTARY GRAPH THEORY



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 1

Unit IV 1

Total 100


Class : M.Sc (MATHEMATICS) Semester : IIPaper : MEASURE THEORY AND INTEGRATION Code : MTH231

Unit I – Lebesgue 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,Littlewood’s principles.

Unit II – The 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, convergencein measure.

Unit III – Differentiation 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) (10 hours)The LP spaces, the Minkowski and Hölder inequalities, convergence and completeness,

bounded linear functionals on the LP spaces.

Text BookH.L. Royden, Real analysis, Third Edition, Macmillan, 1988.

Reference Books1. 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 beginner’s 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


be 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 40Unit II 1

Unit III 1

Unit IV 1

Total 100


Class : M.Sc (MATHEMATICS) Semester : IIPaper : 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, Cauchy’s theorem, Cauchy integral formula, Cauchyintegral formula for derivatives, Cauchy integral formula for multiply connected domains,Morera’s theorem, Gauss mean value theorem, Cauchy inequality for derivatives, Liouville’stheorem, fundamental theorem of algebra, maximum and minimum modulus principles.

Unit II (15 hours)Singularities: Taylor’s series, Laurent’s 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, Cauchy’s residue theorem, contour integration problems, mobius transformations,conformal mappings.

Unit IV (12 hours)Meromorphic functions: Meromorphic functions and argument principle, Schwarz

lemma, Rouche’s theorem, convex functions and their properties, Hadamard 3-circles theorem.

Text Books: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.

Reference Books: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, COMPLEX VARIABLES AND APPLICATIONS, 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.


MTH232: COMPLEX ANALYSISEND SEMESTER EXAMINATION : FORMAT OF THE QUESTION PAPER


unit


be answered



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 40Unit II 1

Unit III 1

Unit IV 1

Total 100


Class : M.Sc (MATHEMATICS) Semester : IIPaper : ADVANCED ALGEBRA Code : MTH233

Unit I (15 hours)Advanced Group Theory: Automorphisms, Cayley’s theorem, Cauchy’s theorem,

permutation groups, symmetric groups, alternating groups, simple groups, conjugate elementsand 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 algebra – I, 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 ALGEBRAEND SEMESTER EXAMINATION : FORMAT OF THE QUESTION PAPER


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 4

Unit III 2

Unit IV 3

C

Unit I 1

4 10 40Unit II 1

Unit III 1

Unit IV 1

Total 100


Class : M.Sc (MATHEMATICS) Semester : IIPaper : 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 mechanicsof fluids, ideal fluids, viscous incompressible fluids, non-Newtonian fluids. Kinematics offluids: Methods of describing fluid motion - Lagrangian and Eulerian methods, translation,rotation and rate of deformation, stream lines, path lines and streak lines, material derivativeand acceleration, vorticity, vorticity in polar coordinates and orthogononal curvilinearcoordinates. 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-Stokesequations), 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, Euler’s equation, the Bernoulli equation,applications of Bernoulli equation, basic equations and concepts of flow, equation ofcontinuity, Eulerian equation of motion, circulation theorems, circulation concept, Stoke’stheorem, Kelvin’s theorem, constancy of circulation, velocity potential, irrotational flow,integration of the equations of motion, Bernoulli’s equation, steady motion, irrotational flow,the momentum theorem, the moment of momentum theorem, Laplace equations, streamfunctions 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 orsink.

Unit IV (15 Hours)The Laminar Flow of Viscous Incompressible Fluids and the Laminar Boundary

Layer: Similarity of flows, the Reynolds number, viscosity from the point of view of thekinetic theory, flow between parallel flat plates, Couette flow, plane Poiseuille flow, steadyflow in pipes, flow through a pipe, the Hagen-Poiseuille flow, flow between two concentricrotating cylinders, properties of Navier-Stokes equations, boundary layer concept, theboundary layer equations in two-dimensional flow, the boundary layer along a flat plate, theBlasius solution.

Text Book: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.Reference Books: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.


MTH234 : FLUID mechanicsEND SEMESTER EXAMINATION : FORMAT OF THE QUESTION PAPER


unit


be answered



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


Class : M.Sc (MATHEMATICS) Semester : IIPaper : 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 Turan’sTheorem, Colour-Critical Graphs, Counting Proper Colourings

Unit II (15 hours)Matchings and Factors: Maximum Matchings, Hall’s Matching Condition, Min-Max

Theorem, Independent Sets and Covers, Maximum Bipartite Matching, Weighted BipartiteMatching, Tutte’s 1-factor Theorem.

Unit III (15 hours)Planar Graphs: Drawings in the Plane, Dual Graphs, Euler’s Formula, Preparation for

Kuratowski’s 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 inDirected Graphs, Tait’s Theorem, Grinberg’s 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,

20064. 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, 20086. 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, Schaum’s outlines, New Delhi:Tata Mcgrahill, 2004.


MTH235: ALGORITHMIC GRAPH THEORY



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 1

Unit IV 1

Total 100


Class : M.Sc (MATHEMATICS) Semester : IIIPaper : GENERAL TOPOLOGY Code : MTH331

Unit I (15 hours)Topological Spaces: Elements of topological spaces, basis for a topology, the ordertopology, 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 extensiontheorem.

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 TOPOLOGYEND SEMESTER EXAMINATION : FORMAT OF THE QUESTION PAPER


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 1

Unit IV 1

Total 100


Class : M.Sc (MATHEMATICS) Semester : IIIPaper : NUMERICAL ANALYSIS Code : MTH332

Unit I (20 Hours)Solution of algebraic and transcendental equations: Fixed point iterative method,

convergence criterion, Aitken’s 2 -process, Sturm sequence method to identify the numberof real roots, Newton-Raphson methods (includes the convergence criterion for simple roots),Bairstow’s method, Graeffe’s root squaring method, Birge-Vieta method, Muller’s method.Solution of Linear System of Algebraic Equations: LU-decomposition methods (Crout’s,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-Moultonmethod, 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 Poisson’sequations, iterative methods of solution by Jacobi and Gauss-Siedel, solution techniques forrectangular 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.

Text Books:

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

Reference Books:

1. R.L. Burden and J. Douglas Faires, Numerical Analysis, 9th ed., Boston: CengageLearning, 2011.

2. S.C. Chopra and P.C. Raymond, Numerical Methods for Engineers, New Delhi: TataMcGraw-Hill, 2010.

3. C.F. Gerald and P.O. Wheatley, Applied Numerical Methods, 7th ed., New York: PearsonEducation, 2009.


MTH332 : NUMERICAL ANALYSISEND SEMESTER EXAMINATION : FORMAT OF THE QUESTION PAPER


unit


be 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 2

Unit IV 3

C

Unit I 1

4 10 40Unit II 1

Unit III 1

Unit IV 1

Total 100


Class : M.Sc (MATHEMATICS) Semester : IIIPaper : 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 - Squire’s theorem forviscous and inviscid theory – Rayleigh stability equation – Derivation of Orr-Sommerfeldequation assuming that the basic flow is strictly parallel.

Unit II: (20 Hours)Dimensional Analysis and Similarity: Non-dimensional parameters determined fromdifferential equations – Buckingham’s Pi Theorem – Nondimensionalization of the BasicEquations - 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 ofstabilities – first variation principle – Different boundary conditions on velocity andtemperature.

Unit III (10 Hours)Porous Media: Introduction to porous medium, porosity, Darcy’s Law, Extension of DarcyLaw – accelerations and inertial effects, Brinkman’s equation, effects of porosity variations,Bidisperse porous media.

Unit IV (15 Hours)Non – Newtonian Fluids: Constitutive equations of Maxwell, Oldroyd, Ostwald , Ostwaldde waele, Reiner – Rivlin and Micropolar fluid. Weissenberg effect and Tom’s 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.


MTH333 : ADVANCED FLUID MECHANICSEND SEMESTER EXAMINATION : FORMAT OF THE QUESTION PAPER


unit


be answered



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 1

Unit III 1

Unit IV 1

Total 100


Class : M.Sc (MATHEMATICS) Semester : IIIPaper : ADVANCED GRAPH THEORY Code : MTH334

Unit I (15 hours)Domination in Graphs: Domination in Graphs, Bounds in terms of Order, Bounds in termsof 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 φ-DecomposableDigraphs, Planar Digraphs.

Unit III (15 hours)Perfect Graphs: The Perfect Graph Theorem, Chordal Graphs Revisited, Other Classes ofPerfect 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, MatroidUnion

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,

20064. 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, 20086. 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, Schaum’s outlines, New Delhi:Tata Mcgrahill, 2004.


MTH334:ADVANCED GRAPH THEORY



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 1

Unit IV 1

Total 100


Class : M.Sc (MATHEMATICS) Semester : IIIPaper : 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 - Smalloscillations. Special Applications of Lagrange’s Equations: Rayleigh’s dissipation function -impulsive motion - velocity dependent potentials.

UNIT III (13 Hours)Hamilton's equations: Hamilton's principle - Hamilton’s equations - Other variationalprinciples - phase space.

UNIT IV (15 Hours)Hamilton - Jacobi Theory: Hamilton's Principal Function – The 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


be answered



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 40Unit II 1

Unit III 1

Unit IV 1

Total 100


Class : M.Sc (MATHEMATICS) Semester : IVPaper : DIFFERENTIAL GEOMETRY Code : MTH431

Unit I (15 hours)Calculus on Euclidean Geometry: Euclidean Space – Tangent Vectors – Directional

derivatives – Curves in E3 – 1-Forms – Differential Forms – Mappings.

Unit II (15 hours)Frame Fields and Euclidean Geometry: Dot product – Curves – vector field - The

Frenet Formulas – Arbitrary speed curves – cylindrical helix – Covariant Derivatives – Framefields – Connection Forms - The Structural equations.

Unit III (15 hours)Euclidean Geometry and Calculus on Surfaces: Isometries of E3 – The derivative

map of an Isometry - Surfaces in E3 – patch computations – Differential functions andTangent vectors – Differential forms on a surface – Mappings of Surfaces.

Unit IV (15 hours)Shape Operators: The Shape operator of M E3 – Normal Curvature – Gaussian

Curvature - Computational Techniques – Special curves in a surface – Surfaces of revolution.

Text Book

B.O’Neill, 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 GEOMETRYEND SEMESTER EXAMINATION : FORMAT OF THE QUESTION PAPER


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 1

Unit IV 1

Total 100


Class : M.Sc (MATHEMATICS) Semester : IVPaper : ADVANCED NUMERICAL METHODS Code : MTH432

Unit I: (20 Hours)Numerical Solutions of Nonlinear system of Equations: Fixed points for functions ofseveral variables, Newton’s Method, Quasi-Newton methods, Homotopy and ContinuationMethods.

Unit II: (15 Hours)

Initial value problems for Ordinary Differential Equations: Solution by Runge-KuttaFehlberg method, Runge-Kutta- Gill method. Solutions of higher order differential equationsand system of differential equations by Runge-Kutta methods.

Unit III: (15 Hours)Boundary value problems for Ordinary Differential Equations: Linear Shooting method,Shooting method for nonlinear Problems, Finite –difference methods for non-linear problems,Rayleigh-Ritz method.

Unit IV: (10 Hours)Modern methods for linear and non-linear differential equations: Homotopy Perturbationmethod and Differential Transforms methods.

Text books:1. Richard L. Burden and J. Douglas Faires Numerical Analysis, Fourth Edition, P.W.S. Kent

Publishing Company, 2007.

2. S. J. Liao, Beyond Perturbations, CRC Press, 2003.

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

7. Vedat Suat Ertürk, Differential Transformation Method For Solving Differential Equationsof Lane - Emden Type, Mathematical and computer applications, vol. 12(3), 135-139,2007.


MTH432 :ADVANCED NUMERICAL METHODS



unit


be 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 2

Unit IV 3

C

Unit I 1

4 10 40Unit II 1

Unit III 1

Unit IV 1

Total 100


Class : M.Sc (MATHEMATICS) Semester : IVPaper : FUNCTIONAL ANALYSIS Code : MTH433

Unit I (15 hours)Banach spaces: Normed linear spaces, Banach spaces, continuous linear

transformations, isometric isomorphisms, functionals and the Hahn-Banach theorem, thenatural embedding of a normed linear space in its second dual.

Unit II (12 hours)Mapping theorems: The open mapping theorem and the closed graph theorem, the

uniform boundedness theorem, the conjugate of an operator.

Unit III (15 hours)Inner products: Inner products, Hilbert spaces, Schwarz inequality, parallelogram

law, orthogonal complements, orthonormal sets, Bessel’s inequality, complete orthonormalsets.

Unit IV (18 hours)Conjugate space: The conjugate space, the adjoint of an operator, self-adjoint,

normal and unitary operators, projections, finite dimensional spectral theory.

Text Book:G.F. Simmons, Introduction to topology and modern Analysis, Reprint, Tata McGraw-Hill,2004.

Reference Books :1. K. Yoshida, Functional analysis, 6th ed., Springer, 1996.2. E. Kreyszig, Introductory functional analysis with applications, 1st ed., John Wiley, 1978.3. B.V. Limaye, Functional analysis, 2nd ed., New Age International, 1996.4. W. Rudin, Functional analysis, 2nd ed., McGraw Hill, 2010.5. S. Karen, Beginning functional analysis, Reprint, Springer, 2002.

MTH433 : FUNCTIONAL ANALYSISEND SEMESTER EXAMINATION : FORMAT OF THE QUESTION PAPER


unit


be answered



A

Unit I 1

5 2 10Unit II 1

Unit III 1

Unit IV 2

B

Unit I 3

10 5 50Unit II 2

Unit III 3

Unit IV 4

C

Unit I 1

4 10 40Unit II 1

Unit III 1

Unit IV 1

Total 100


Class : M.Sc (MATHEMATICS) Semester : IVPaper : PROJECT Code : MTH451

The objective of this paper is to develop positive attitude, knowledge and competencefor the research in Mathematics. Through this project students will develop analytical andcomputational skills. Students are exposed to the mathematical softwares like Mathematica,Scilab and Matlab. Students are given with their choice of topic either on Fluid Mechanics orGraph theory or can select any other topic from other fields with the approval of HOD /Coordinator. Each candidate will work under the supervision of the faculty. Coordinator willallot the supervisor for each candidate in consultation with the HOD at the end of thirdsemester.

Project need not be based on original research work. Project could be based on thereview of advanced text book or advanced research papers.

Each candidate has to submit a dissertation on the project topic followed by viva voceexamination. The viva voce will be conducted by the committee constituted by the head ofthe department which will have an external and an internal examiner. The student must secure50% of the marks to pass the project examination. The candidates who fail must redo theproject as per the university regulation.


ELECTIVE PAPERS

Class : M.Sc (MATHEMATICS) Semester : IVPaper : CALCULUS OF VARIATIONS Code : MTH441

AND INTEGRAL EQUATIONS

Unit I (18 hours)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.

Unit II (16 hours)Advanced variational problems: Constraints and Lagrange multipliers, variable end

points, Sturm-Liouville problems, Hamilton’s principle, Lagrange’s equation, the Rayleigh-Ritz method.

Unit III (12 hours)Linear integral equations: Definitions, integral equation, Fredholm and Volterra

equations, kernel of the integral equation, integral equations of different kinds, relationsbetween differential and integral equations, symmetric kernels, the Green’s function.

Unit IV (14 hours)Methods for solutions of linear integral equations: Fredholm equations with

separable kernels, homogeneous integral equations, characteristic values and characteristicfunctions of integral equations, Hilbert-Schmidt theory, iterative methods for solving integralequations of the second kind, the Neumann series.

Text book :

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

Reference Books :

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: Birkhäuser,

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.


MTH441: CALCULUS OF VARIATIONS AND INTEGRAL EQUATIONS



unit


be 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 2

Unit IV 3

C

Unit I 1

4 10 40Unit II 1

Unit III 1

Unit IV 1

Total 100


Class : II - M.Sc Semester : IVPaper : MAGNETOHYDRODYNAMICS Code : MTH442

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

Gauss law, Faraday’s law, Ampere’s 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 andmagnetic 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, Hartmanncouette flow, magnetostatic-force free magnetic field, abnormality parameter, Chandrashekartheorem, 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, earth’s magnetic field,Alfven wave equation in an incompressible conducting fluid in the presence of an verticalmagnetic field, solution of Alfven wave equation, Alfven wave equation in a compressibleconducting non-viscous fluid, Helmholtz vorticity equation, Kelvin’s circulation theorem,Bernoulli’s equation.

Text Books:

1. V.C.A. Ferraro and Plumpton, An introduction to magnetofluid mechanics, ClarendonPress, 1966.

2. P.H. Roberts, An introduction to magnetohydrodynamics, Longman, 1967.3. Allen Jeffrey, Magnetohydrodynamics, Oliver Boyds, 1970.

Reference Books:

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.


MTH442 : MagnetohydrodynamicsEND SEMESTER EXAMINATION : FORMAT OF THE QUESTION PAPER


unit


be answered



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


Class : M.Sc (MATHEMATICS) Semester : IVPaper : WAVELET THEORY Code : MTH443

Unit I (15 hours)Introduction: Complex numbers and basic operation, the space L2(R), inner products, basesand projections, Euler’s formula and complex exponential function, Fourier series, Fouriertransforms, 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, generalHaar space, Haar wavelet space, general Haar wavelet space, discrete Haar wavelettransforms and applications.

Unit IV (15 hours)Wavelet packets: The construction of wavelet sets, the measure of the closure of a waveletset, constructing wavelet packet spaces, wavelet packet spaces.

Text books1. 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 books1. 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.


MTH443: WAVELET THEORYEND SEMESTER EXAMINATION : FORMAT OF THE QUESTION PAPER


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 1

Unit IV 1

Total 100


Class : M.Sc (MATHEMATICS) Semester : IVPaper : MATHEMATICAL MODELLING Code : MTH444

Unit I (15 Hours)Concept of mathematical modeling: Definition, Classification, Characteristics AndLimitations , Mathematical Modelling Through Ordinary Differential Equations Of FirstOrder: Linear And Nonlinear Growth and Decay Models Compartment Models, DynamicsProblems, Geometrical Problems

Unit II (12 hours)Mathematical modelling through systems of ordinary differential equations of firstorder: Population Dynamics, Epidemics, Compartment Models, Economics, Medicine, ArmsRace, 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 ModellingThrough Linear Differential Equations Of Second Order, Miscellaneous MathematicalModels

Unit IV (20 Hours)Mathematical Modelling leading to linear and nonlinear partial differential equations:Simple models, conservation law – Traffic flow on highway – Flood waves in rivers – glacierflow, roll waves and stability, shallow water waves – Convection diffusion –processesBurger’s equation, Convection – reaction processes – Fisher’s equation.Telegraphersequation 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, Dover5. Walter J. Meyer , Concepts of Mathematical Modeling


MTH444: MATHEMATICAL MODELLINGEND SEMESTER EXAMINATION : FORMAT OF THE QUESTION PAPER


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 1

Unit IV 1

Total 100


Class : M.Sc (MATHEMATICS) Semester : IVPaper : 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: Finitefields, 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.

Text Book:

N. Koblitz, A course in number theory and cryptography, Graduate Texts in Mathematics,No.114, Springer-Verlag, 1987.

Reference Books:

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

3. D.R. Stinson, Cryptography: Theory and Practice, CRC Press, 19954. 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.


MTH445: CryptographyEND SEMESTER EXAMINATION : FORMAT OF THE QUESTION PAPER


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 1

Unit IV 1

Total 100


Class : M.Sc (MATHEMATICS) Semester : IVPaper : ATMOSPHERIC SCIENCE 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 ofthe atmosphere, adiabatic process, potential temperature, the Classius-Clapyeron equation, laws ofblack body radiation, solar and terrestrial radiation, solar constant, Albedo, greenhouse effect, heatbalance 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 boundarylayer.

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 momentumtransport, 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 ofsound and gravity waves, linear barotropic and baroclinic instability.

Text Books:

1. Joseph Pedlosky, Geophysical fluid dynamics, Springer-Verlag, 1979.2. J.R. Holton, An introduction to dynamic meteorology, 3rd Ed., Academic Press, 1992.

Reference Books:

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.


MTH446: ATMOSPHERIC SCIENCEEND SEMESTER EXAMINATION : FORMAT OF THE QUESTION PAPER


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 1

Unit IV 1

Total 100


CERTIFICATE COURSE

I Semester

MTH 101: Teaching Technology and Research Methodology in Mathematics(45 Hours)

Unit I:-Teaching Technology (15 Hours)Development of concept of teaching, Teaching skills, Chalk board skills, Teaching practices,Effective teaching, Models of teaching, Teaching aids(Audio-Visual), Teachingaids(projected and non-projected), Communication skills, Feed back in teaching, Teacher’srole and responsibilities, Information technology for teaching.

Unit II:-Research Methodology (15 Hrs)Introduction to research and research methodology, Scientific methods, Choice of researchproblem, Literature survey and statement of research problem, Reporting of results, Rolesand responsibilities of research student and guide.

Unit III:-Mathematical research methodology (15 Hrs)Theorems and proofs, Reading and understanding Mathematical Proofs, Proofing Methods,Introducing mathematics Journals, Reading a Journal article, Mathematics writing skillsStandard Notations and Symbols, Using Symbols and Words, Organizing a paper, Definingvariables, Symbols and notations, Different Citation Styles, IEEE Referencing Style in detailPackage for Mathematics Typing, MS Word, Math Type, Open Office Math Editor, Tex, yEdGraph Editor, Tex in detail, Installation and Set up, Text, Formula, Pictures and Graphs,Producing various types of documents using Tex.

Text and Reference books:1. R.Varma, Modern trends in teaching technology, Anmol publications Pvt.Ltd., New Delhi2003.2. Usha Rao, Educational teaching, Himalaya Publishing house, New Delhi 2001.3. J. Mohanthy, Educational teaching, Deep & Deep Publications Pvt.Ltd., New Delhi 20014. K.J.Sree and D. B. Rao, Methods of teaching sciences, Discovery publishing house, Delhi,1981.5. E.B.Wilson J., An introduction to scientific research, Dover publications, inc., New York1990.6. Ram Ahuja, Research Methods, Rawat Publications, New Delhi 2001.7. Gopal Lal Jain, Research Methdology, Mangal Deep Publictions, Jaipur 2003.8. B. C. Nakra and K. K. Chaudhry, Instrumentation, measurement and analysis, TMHpublishing Co.Ltd., New Delhi 1985.9. S. L. Mayers, Data analysis for Scientists, John Wiley & Sons, 1976.


CERTIFICATE COURSE

II Semester

MTH 201: STATISTICS(45 Hours)

Unit I (15 Hours)Random Variables and Expectation: Discrete and continuous random variables,distribution functions, probability mass and density functions, bivariate distributions,marginal and conditional distributions, expected value of a random variable, independence ofrandom variables, conditional expectations, covariance matrix, correlation coefficients andregression, Chebyshev’s inequality, moments, moment generating functions, characteristicfunctions.

Unit II (15 Hours)Probability: Sample spaces, events, probability of an event, theorems on probability,conditional probability, independent events, Bayes theorem. Boole’s inequality.Discrete Probability Distribution: Introduction, uniform, Bernoulli, Binomial, negativeBinomial, geometric, Hypergeometric and Poisson distribution. Continuous ProbabilityDistributions: Introduction, uniform, gamma, exponential, beta and normal distributions.

Unit III (15 Hours)Sampling distributions: t, F and chi-square distributions, standard errors and large sampledistributions.

Text Books:1. E. Freund John, Mathematical Statistics, 5th Ed., Prentice Hall of India, 2000.2. Gupta S.C. and Kapoor V.K., Fundamentals of mathematical Statistics, Sultan Chand and

Sons, New Delhi, 2001.

Reference Books:1. Paul G. Hoel, Introduction to mathematical Statistics, Wiley, 1984.2. M. Spiegel, Probability and statistics, Schaum’s Outline Series, 2000.3. Neil Weiss, Introductory Statistics, Addison-Wesley, 2002.4.Sheldon M. Ross, A first course in probability, Pearson Prentice Hall, 2005.5. Ronald E. Walpole, Raymond H. Myers and Sharon L. Myers, Probability and Statistics

for Engineers and Scientists, Pearson Prentice Hall, 2006.6. Dennis Wackerly, William Mendenhall and Richard L. Scheaffer, Mathematical Statistics

with Applications, Duxburry Press, 2007.


CERTIFICATE COURSE

III Semester

MTH 301: INTRODUCTION TO MATHEMATICAL PACKAGES(45 Hours)

Unit I (15 Hours)Algebraic Computation: Simplification of algebraic expression, simplification ofexpressions involving special functions, built-in functions for transformations ontrigonometric expressions, definite and indefinite symbolic integration, symbolic sums andproducts, symbolic solution of ordinary and partial differential equations, symbolic linearalgebra, matrix operations.

Unit II (15 Hours)Mathematical Functions: Special functions, inverse error function, gamma and betafunction, hypergeometric function, elliptic function, Mathieu function. NumericalComputation: Numerical solution of differential equations, numerical solution of initial andboundary value problems, numerical integration, numerical differentiation, matrixmanipulations, optimization techniques.

Unit III (15 Hours)Graphics: Two- and three-dimensional plots, parametric plots, typesetting capabilities forlabels and text in plots, direct control of final graphics size, resolution etc. Packages: Linearalgebra, 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.


Internship in PG Mathematics courseSemester: IIICode: 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 45working days at any of the following: reputed research centres, recognized educationalinstitutions, summer research fellowships, programmes like M.T.T.S. or any other approvedby the P.G. coordinator and H.O.D.

The internship is to be undertaken at the end of second semester (during first yearvacation). The report submission and the presentation on the report will be held during thethird 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, natureof internship, period on internship, relevant permission letters, if available, name of thementor in the organization, email, telephone and mobile numbers of the person in theorganization 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 secondsemester.

The coordinator of the programme in consultation with the HOD will assign facultymembers from the department as guides at least two weeks before the end of secondsemester.

The students will have to be in touch with the guides during the internship period eitherthrough 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 ofthe University for the third semester.

Within 20 days from the day of reopening, the department must hold a presentation by thestudents. During the presentation the guide or a nominee of the guide should be present andbe 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 thepresenter. The maximum limit is left to the discretion of the evaluation committee. Studentswill get 2 credits on successful completion of internship.

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