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Erode Arts and Science College (Autonomous), Erode-9.

Department of Mathematics (SF)

M.Sc., Mathematics Degree Course

Effective from the Academic yeear 2016--167 and onaards

Scheme of Examination

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PRATICAL/

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ELECTIVE/FC

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SEMESTER II 1 PMA-1 CORE-1 Algebra 6 3 5 25 75

1 PMA-2 CORE-2 Real Analysis 6 3 5 25 75

1 PMA-3 CORE-3 Complex Analysis 6 3 5 25 75

1 PMA-4 CORE-4 Ordinary Differential Equations

6 3 4 25 75

1 ELPM-1 ELECTIVE-1 Numerical Analysis 6 3 4 25 75

SEMESTER IIII 2 PMA-5 CORE-5 Partial Differential

Equations6 3 4 25 75

2 PMA-6 CORE-6 Graph theory 6 3 4 25 75

2 PMA-7 CORE-7 Operations Research 6 3 4 25 75

2 PMA-8 CORE-8 Object Oriented Programming with C++

4 3 3 25 75

2 ELPM-2 ELECTIVE-2 Differential Geometry 6 3 4 25 75

2 PPMA-1 PRACTICAL-1 Programming in C++ 2 3 2 40 60

SEMESTER IIIIII 3 PMA-9 CORE-9 Topology 6 3 5 25 75

3 PMA-10 CORE-10 Mathematical Statistics 6 3 5 25 75

3 PMA-11 CORE-11 Fluid Dynamics 6 3 4 25 75

3 PMA-12 CORE-12 Fuzzy Set Theory 6 3 4 25 75

3 ELPM-3 ELECTIVE-3 Mathematical Methods 6 3 4 25 75

SEMESTER IV

IV 4 PMA-13 CORE-13 Functional Analysis 6 3 5 25 75

4 PMA-14 CORE-14 Number Theory 6 3 5 25 75

4 PMA-15 CORE-15 Mathematical Sofwares 4 3 3 25 75

4 ELPM-4 ELECTIVE-4 Control Theory 6 3 4 25 75

4 PPMA-2 PRACTICAL-2 Mathematical Sofwares 2 3 2 40 60

4 PMA-PV PROJECT Project & Viva-voce 6 3 3 50

Subject & Viva-voce 3 2 50

TOTAL 555 1645

GRAND TOTAL 90 2200

M.Sc., Mathematics (SF)

SEMESTER I PAPER CODE: PMA-1

CORE PAPER I DURATION: 6 HOURS

ALGEBRA

UNIT – I : Group Theory :

Another counting principle, Sylow’s theorem, Direct product

UNIT – II : Ring Theory :

Euclidean ring, principle ideal ring ,a particular Euclidean ring, Fermat’s theorempolynomial rings.

UNIT – III : Field :

Extension field; Definition of extension field and algebra,

UNIT – IV : Linear Transformation :

Algebra of linear transformation- algebra- linear transformation- invertible-Regular transformation- characteristic- roots- Matrices.

UNIT – V :

Nil potent transformation- trace and transpose- Hermitian -Unitary and normaltransformation.

Text Books :

“Topic in Algebra” – I.N. Herstein (II Edition)

UNIT - I - chapter 2 - 2.11 - 2.13

UNIT- II - chapter 3 - 3.7 - 3.9

UNIT- III - chapter 5 - 5.1 -5.2

UNIT- IV - chapter 6 - 6.1 - 6.3

UNIT- V - chapter 6 - 6.5, 6.8, 6.10

ERODE ARTS AND SCIENCE COLLEGE (AUTONOMOUS)

DEPARTMENT OF MATHEMATICS (SF)

QUESTION PATTERN

M.Sc Mathematics

Section A : (10X1=10 marks) Answer all questions

(10 multiple choice question with 4 options, 2 questions from each unit)

Section B: (5X5 =25 marks) Answer all questions

(5 questions Either or type-1 question from each unit)

Section C: (5X8=40 marks) Answer any five questions out of Eight

(At least one question from each unit)


SEMESTER I PAPER CODE:PMA-2

CORE PAPER II DURATION: 6 HOURS

REAL ANALYSIS

UNIT – I :

Monotonically increasing integrations – upper and lower integrals – additive and linear properties of upper and lower integrals – Riemann condition – comparison theorems – integrators of bounded variation – sufficient and necessary conditions for existence of Riemann steiltjes integrals – mean value theorems for r-s integrals.

UNIT – II : Multivariable Differential Calculus

Introduction the directional derivative – Directional derivations and continuity – total derivative – total derivative expressed as partial derivatives - The Jacobian matrix

UNIT – III : Implicit Function And Extremum Problems

Introduction – functions with non-zero Jacobian determinant – the inverse function theorem – Implicit theorem.

UNIT – IV :Lebesque Measure

Outer measure – Measurable sets and lebesque outer measure – Measurable functions.

UNIT – V :

The lebesque integral of a bounded functions over a set of finite measure – Integral of a non-negative function – General lebesque integral.

Text Books: Tom Apostol “Mathematical Analysis” - Addison Wesley(1974)

UNIT- I : chapter 7 - 7.11-7.18

UNIT- II : chapter 12 - 12.1-12.5,12.8

UNIT- III : chapter 13 - 13.1-13.4,13.8

Treatment as in Real Analysis by Royden

UNIT- IV : chapter 3- 3.2-3.4

UNIT- V : chapter 4- 4.2-4.4

Reference: 1.R.G. Bartle, Elements of Real Analysis 2nd Edition, John Wily and sons, Newyork,1976.

2.W.Rudin,Real and Complex analysis,3rdEdition,McGraw- Hill,New york,1986.



QUESTION PATTERN

M.Sc Mathematics









CORE PAPER III DURATION: 6 HOURS

COMPLEX ANALYSIS

Unit I:

Complex Integration: Fundamental Theorems: Line integrals- Rectifiable arcs- Lineintegrals as function of arcs-Cauchy’s theorem for a rectangle- Cauchy’s theorem in a disk- Cauchy’s integral formula-Index of a point w.r.to a closed curve-The integral formula – Higher Derivatives.

UNIT II:

Local properties of Analytic functions: Removable Singularities- Zeros and Poles- Local mapping – The Maximum principle- the Calculus of Residues: The Residue theorem: The Argument Principle

UNIT III:

Evaluation of Definite integrals- Harmonic functions: Definition and Basic properties – the Mean Value property – poisson’s Formula

-Schwarz’s theorem – The reflection principle. .

UNIT IV:

Series and Product Development: Power series Expansions: Weirstrass’s theorem – The Taylor’s series – The Laurent series – Partial fractions – Infinite Products – Canonical products .

UNIT V:

Elliptic Functions: Simply Periodic Functions : Representation by Exponentials - TheFourier developments – Functions of finite order – Doubly Periodic Functions: The period module – Unimodular Transformations – The Canonical Basis – General properties of elliptic functions - The Weirstrass’s theory: The Weirstrass”s P functions – The functions σ(Z) and ς(Z) – The Differential Equation.

Text Book:Content and Treatment as in Complex Analysis by Lars V.Ahlfors, 3rdEdition McGraw Hill.

Unit I - Chapter 4 - 1.1-1.5, 2.1-2.3Unit II - Chapter 4 – 3.1 – 3.4, 5.1 – 5.2Unit III - Chapter 4 - 5.3, 6.1-6.5Unit IV - Chapter5 - 1.1-1.3, 2.1-2.3Unit V - Chapter7 - 1.1-1.3 , 2.1-2.4, 3.1-3.3.



QUESTION PATTERN

M.Sc Mathematics









CORE PAPER IV DURATION: 6 HOURS

ORDINARY DIFFERENTIAL EQUATIONS

UNIT I:

Independent And Dependent Solutions Of Linear Differential Equations:Existenceand uniqueness theorem-Examples-Linearly dependent and independent solutions-Wronskian’s Definition-Theorems-Linear combination of solutions-linear dependence and independence of solutions- Theorems- Wronskian of n functions-Theorems-Problems.

UNIT II:

Exact Differential Equations :Exact differential equation-Definition-Condition of exactness of a linear differential equation of order n -Working rule- Examples-Integrating factor-Examples-Exactness of non-linear equations-Examples.

UNIT III:

Second order linear equations with ordinary points – legendre equation and legendre polynomials – second order equations with regular singular points – Bessel equation .

UNIT IV:

Linear Differential Equations Of Second Order :The general (or standard) form-Complete solution of y”+Py’+Qy=R - Rules-Examples-Removal of the first derivative-Reduction to normal form- working rule-Examples.

UNIT V:

Linear Systems: Uncoupled linear systems- Diagonalization- Exponentials of operators- The fundamental theorem for linear systems Linear systems in R2 - Complex Eigen values.

TEXT BOOKS 1. “Advanced differential equations “, M.D Raisinghania, S.CHAND & Company Ltd. Unit I- Chapter 2 (Sec 2.1 to 2.14). Unit II- Chapter 3 (Sec 3.1 to 3.8). Unit IV- Chapter 4 (Sec 4.1 to 4.8),2. Ordinary Differential Equation and Stability by S.G.Deo and .RaghavendraUnit –III – Chapter 3 – Sec 3.2 to 3.53.“Differential equations and dynamical systems”, Lawrence Peko, Third Edition, Springer. Unit V-(Sec1.1to1.7)

REFERENCE:1.A.Coddington and N.Leinson,Theory of Ordinary Differential Equations,McGrawHill,New york.1955.

2.D.A.Sanchez,Ordinary Differential Equation and Stability Thoery,W.H.Freeman and co.,San Francisco,1968.



QUESTION PATTERN

M.Sc Mathematics







M.Sc. Mathematics (SF)

SEMESTER I PAPER CODE: ELPM-1

ELECTIVE 1 DURATION: 6 HOURS

NUMERICAL ANALYSIS

UNIT I: SOLUTION OF NONLINEAR EQUATIONS:

Interval Halving method-Linear Interpolation method-(Secant method , False Poissonmethod)-Newton’s method-Fixed Pointlteration - Bairstow’s method for quadratic factors-Graeffe’s root-Squaring method- Order of convergence for fixed-point iteration and Newton’smethod.

UNIT II: SOLUTION OF SYSTEM OF LINEAR EQUATIONS:

Direct Methods: Gauss Elimination and Harts-Jordan method-Iteration methods:

Gauss-Jacobi and Gauss-Seidel method-Relaxation method.

UNIT III: SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS:

Solution of ordinary differential equations: Taylor-Series method-Euler and ModifiedEuler Methods-RungeKutta methods-Multistep methods-Milne’s method-Adams-Moultonmethod-Instability of Milne’s Method.

UNIT IV:BOUNDARY VALUE PROBLEMS:

The shooting method (Theory Only)-Solution through a set of equations-Derivativeboundary conditions-Power method-Liebmann’s method.

UNIT V: SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS:

Heat equation and wave equation-The explicit method-Solving the Vibrating String Problem-Crank-Nicolson method-Convergence of explicit method.

TEXT BOOK: Curtis F.Gerald and Patrick O.Wheatley, APPLIED NUMERICAL ANALYSIS, Sixth Edition,Pearson Education Asia,2002

UNIT I - Chapter 1 - 1.2-1.4, 1.6-1.8, 1.11UNIT II - Chapter 2 - 2.3-2.4, 2.10, 2.11UNIT III - Chapter 6 - 6.2-6.7, 6.11UNIT IV - Chapter 7 - 7.2-7.5, 7.7, 7.8UNIT V - Chapter 8 - 8.2, 8.3, 8.4, 8.7

REFERENCES: 1. Burden, R.L. and Faries, T.D., Numerical Analysis , Seventh Edition, Thomson Asia Pvt. Ltd. Singapore 2002.

2. Venkatraman M.K., Numerical Methods, National Pub.Co.Chennai,1991.



QUESTION PATTERN

M.Sc Mathematics








SEMESTER II PAPER CODE: PMA-5

CORE PAPER V DURATION: 6 HOURS

PARTIAL DIFFERENTIAL EQUATIONS

UNIT I:

Compatibility of first order Partial differential equation-Solution of non-linear partial differential equation of first order: Charpits method, Jacobi method, special types of first order equation.

UNIT II:

Second order partial differential equation: Origin of second order P.D.E., linear P.D.E,with constant Coefficient ,Method of solving linear P.D.E, classification of second order P.D.E., canonical forms.

UNIT III:

Elliptic differential equation: Occurrence of Laplace and Poisson equation-Derivation of Laplace equation, Derivation of Poisson equation, Boundary value problem-.Separation of variables method-Laplace equation in cylindrical coordinates, Laplace equation in spherical coordinates.

UNIT IV:

Parabolic differential equations: Occurrence and derivation of the diffusion equation, Boundary conditions, separation of variation method, diffusion equation in cylindrical coordinates- diffusion equation in spherical coordinates.

UNIT V:

Hyperbolic differential equation: Occurrence of the wave equation, Derivation of one

dimensional wave equation, Reduction of one dimensional wave equation of canonical form and its solution, D’Alembert solution of one dimensional wave equation, separation of variable methods.Text Book: Partial differential equations for Engineers and Scientists by J.N. Sharma,Keharsingh

(Narosa Publishing House)

Unit I Chapter 1 1.7-1.9, ( 1.9.1-1.9.3)Unit II Chapter 2 2.2-2.4Unit III Chapter 3 3.1-3.5Unit IV Chapter 4 4.1-4.5Unit V Chapter 5 5.2-5.5

References:1. I.N.Sneddon,Elements of Partial Differential Equations,McGraw Hill,Londan.19572. L.c.Evans,Partial Differential Equations ,AMS,Providence,R.I,2003.



QUESTION PATTERN

M.Sc Mathematics








SEMESTER II PAPER CODE:PMA-6

CORE PAPER VI DURATION: 6HOURS

GRAPH THEORY

UNIT I

Introduction-Paths and Circuits

UNIT II

Trees and Fundamental Circuits- Cut sets and Cut vertices

UNIT III

Planar and Dual Graphs – Vector Spaces of a Graphs

UNIT IV

Matrix Representation of Graphics-Coloring

UNIT V

Directed Graphs

Treatment as in:

Narasingh Deo-Graph Theory with Application to Engineering and Computer Science-Prentice Hall 1974.

UNIT I - Chapter 1&2 Sec: 1.1-1.5 , 2.1-2.10UNIT II - Chapter 3&4 Sec : 3.1-3.8,4.1-4.6UNIT III - Chapter 5&6 Sec : 5.1 – 5.7, 6.1,6.2,6.4-6.6 UNIT IV - Chapter7&8 Sec:7.1- 7.4,7.6, 8.1-8.3UNIT V - Chapter 9 Sec : 9.1-9.6Reference:J.A.Bondy and U.S.R.Murty,Graph Theory with Applications,AmericanElseier Publishing Company,Inc,New york,1976.



QUESTION PATTERN

M.Sc Mathematics









CORE PAPER VII DURATION: 6 HOURS

OPERATIONS RESEARCH

UNIT I:

Linear programming problems: Simplex method-Artificial variable techniques-Method of penalties-Two phase simplex method.

UNIT II:

Replacement problem: Replacement of equipment that deteriorates gradually- Replacementof equipment that fails suddenly- Individual replacement policy- Group replacement policy.

UNIT III:

Two- person zero-sum games-The Maximin- Minimax principle-Games without saddlepoints-mixed strategies-solution of 2 ´2 rectangular games- Graphical method- Dominanceproperty.

UNIT IV:

Dynamic Programming: Dynamic programming algorithm-Optimal subdivision problem-Solution of diacrete D.P.P – Solution of L.P.P by D.P

UNIT V:

Non-linear programming techniques: Kuhn-Tucker conditions-Non-negative constraints-Solution of NLPP using Lagrangian multiplier method.

Text Book:

OPERATIONS RESEARCH by KantiSwarp, P.K. Gupta, ManMohan.(Sultan Chand & Sons)

Reference:

1. Problem in Operations Research (Methods & Solutions)_

2. Operations Research V.Sundaresan, K.S. GanapathySubramanaian ,K.Ganesan.



QUESTION PATTERN

M.Sc Mathematics









CORE PAPER VIII DURATION: 4 HOURS

OBJECT ORIENTED PROGRAMMING WITH C++

UNIT I: Software Evolution –Procedure oriented programming-object oriented programming paradigm –

basic concepts of object oriented programming - Benefits of oops-Object oriented Languages-Applications of OOP-Beginnings with C++ - what is C++ - Application of C++ - A simple C++ Program- More C++ statements -An Example with class-Structure of C++ Program.UNIT II:

Token, Expressions and control Structures: Tokens-Keywords-Identifiers and constants-Basic Data types-User defined data types-Derived data types-Symbolic constants in C++-Scope Resolution Operator-Manipulators-Type Cost operator-Expressions and their types- assignment expressions-Implicit Conversions-Operators Overloading-Operator precedence-Control Structure.

UNIT III: Function in C++: Main Function- Function Prototyping Call by reference-Return by reference-

Inline functions-default arguments-Const arguments-Function overloading-Friend and Virtual functions-Math library function. Class and Objects: Specifying a class-Defining member functions-A C++ program with class-Making an outside function online-Nesting of member functions-Private Member functions Arrays within a class-Memory Allocation for objects-Static Data Members- Static Member functions-Array of the object-Object as Function Arguments-Friendly Function –Returning Objects-ConstMember functionsUNIT IV:

Constructors and destructors: Constructors-Parameterized Constructors in a Constructors-Multiple constructors in a class – Constructors with default arguments-Dynamic Initialization of objects-Copy constructors-Dynamic Constructors-Constructing Two-Dimensional Arrays-Const Objects-Destructors.

Operator overloading and type Conversions: Defining operator overloading-Overloading-Overloading Unary operators-Overloading Binary operators -Overloading Binary operators using Friends-Manipulation of string using operators-Rules for overloading operators- Type Conversions.UNIT V:

Files: Introduction-Class for File Stream operations-Opening and Closing a file-Detecting End of file-More about open () File modes-File pointer and their Manipulations-Sequential Input and Output Operations. Exception Handling: Introduction-Basic of Exception Handling-Exception Handling Mechanism-Throwing Mechanism-Catching Mechanism-Rethrowing an Exception.Text Book: “Object Oriented Programming with C++”(Second Edition) E.Balagurusamy. Tata Mcgrew Hill

Reference books:

1. Robert lafore-“The waite group’s object oriented programming in turbo c++” – Galgotia publication pvt ltd-1998.

2. Allan neibaver – Office 2000



QUESTION PATTERN

M.Sc Mathematics








SEMESTER II PAPER CODE:ELPM-2

ELECTIVE 2 DURATION: 6 HOURS

DIFFERENTIAL GEOMETRYUnit I:

Curves: Analytic representation - Arc Length - Tangent - Osculation plane - Curvature torsion - Formulas of Frenet

Unit II:

Contact - Natural equations - Helices - General Solutions of Natural equations - Evaluates and Involutes.

Unit III:

Elementary theory of surface: Analytic representation - First fundamental form - Normal, Tangent plane - Developable surfaces.

Unit IV:

Second fundamental from –Meusnier's theorem - Euler's Theorem - Dupin'sindicatrixSome surfaces - The fundamental equations - The equations of Gauss-- Weingarten.

Unit V:

The theorem of Guass and the equations of Codazzi - Some applications of the Gauss and Codazziequations.The fundamental theorem of surface theory - Geodesic curavature - Geodesics.

Treatment as in:

D. Struik, Lectures on Classical Differential Geometry, Addison Wesley Publishing Company, 1961.



QUESTION PATTERN

M.Sc Mathematics








SEMESTER II PAPER CODE:PPMA-1

PRACTICAL -1 DURATION: 2 HOURS

PRACTICAL -PROGRAMMING IN C++

SAMPLE LIST OF PRACTICALS

1. DISTANCE CONVERSION PROBLEM: Create two classes DM and DB which store thevalue of distances. DM stores the value of distances. DM stores distances in meters andcentimeters in DB in feet and inches. Write a program that can create the values of the classobjects and add one object DM with another object DB. Use a friend function to carry outaddition operation. The object that stores the result may be DM object or DB object dependingon the units in which results are required. The display should be in the order of meter andcentimeter and feet or inches depending on the order of display.

2. OVERLOADING OBJECTS: Create a class FLOAT that contains one float data memberoverload all the four arithmetic operators so that operate on the objects of FLOAT.

3. OVERLOADING CONVERSIONS: Design a class polar which describes a point in a planeusing polar Co-ordinates radius and angle. A point in polar Co-ordinates is as shown below. Usethe over loader + operator to add two objects of polar. Note that we cannot add polar values oftwo points directly. This requires first the conversion Points into rectangular co-ordinates andfinally converting the result into polar coordinates. You need to use following trigonometricformulas.X= r * cos (a); Y= r * sin (a); a = tan-1(Y/X) ; r = √( X 2 +Y 2 );

4. POLAR CONVERSION: Define two classes polar and rectangular coordinates to representpoints in the polar and rectangular systems. Use conversion routines to convert from one systemto another.

5. OVRELOADING MATRIX: Create a class MAT of size M*N. Define all possible matrixoperations for MAT type objects. Verify the identity.(A-B)^2 = A^2+B^2 – 2*A*B

6. AREA COMPUTATION USING DERIVED CLASS :Area of rectangle = X*Y

Area of triangle = ½ * X * Y

7. VECTOR PROBLEM: Define a class for vector containing scalar values. Apply overloadingconcepts for vector addition, Multiplication of a vector by a scalar quantity, replace the values ina position vector.

8. To Find the roots of a Quadratic Equation and nCr.


SEMESTER III PAPER CODE: PMA-9

CORE PAPER IX DURATION: 6 HOURS

TOPOLOGY

UNIT I

Metric spaces - definition and some examples - open sets-closed sets-convergencecompleteness and baire’s theorems-continuous mappings - spaces of continuous functionsEuclidean and unitary spaces.

UNIT II

Topological spaces-definition and some examples-Elementary concepts-open bases andopen sub bases-weak topologies-the function algebras c(x,r) and c(x,c).

UNIT III

Compactness-compact spaces-product spaces-Tychonoff,’s theorem and locallycompact spaces-compactness for metric spaces-Ascoli’s theorem.

UNIT IV

Separation –Ti spaces-and Hausdroff spaces completely regular spaces and normalspaces-Uryshon’s lemma and Tietez the stone cech compactification.

UNIT V

Connectedness- connected spaces- the components of a space- totally disconnectedspaces-locally connected spaces.

TEXT BOOK: Introduction to Topology and Modern Analysis- by G.F. Simmons

Unit I - Chapter 2

Unit I - Chapter 3

Unit III - Chapter 4

Unit iv - Chapter 5

Unit V - Chapter 6

References:1. A first course by James R. Munkres, Prentice Hall of India Private Limited , New Delhi.2. J.L.Kelley,General Topology, Van Nostrand, Reinhold Co.. New York,1995.3. Sze- Tsen Hu, elements of general Topology , Holden – Day , Inc , 1965.



QUESTION PATTERN

M.Sc Mathematics









CORE PAPER X DURATION: 6 HOURS

MATHEMATICAL STATISTICS

UNIT I:

Preliminary remarks-random events and operation performed on them-the system ofaxiom of the theory of probability-conditional probability-bayes theorem- independent events.The concepts of random variable -The distribution function-random variables-conditionaldistributions- Independent random variables- Functions of multidimensional randomvariables.Expected values-moments of random vectors-Regressions.

UNIT II:

Characteristic function-properties of characteristic functions-characteristic function andmoments-invariants-the characteristic functions of sum of independent random variables-determination of distribution function of the characteristic function. Probability generatingfunction.One point and two point distributions - The bernoualli scheme-the binomialdistributions

UNIT III:

The poisson distribution –the uniform distribution the normal distribution-te gammadistribution-beta distribution- the Cauchy and laplace distribution-compound distributions.Numbers-the levy cramer theorem-the demovirelaplace theorem-the lindberg levy theorem.

UNIT IV:

Sample moments and their functions-The notion of sample-the notion of a statistic – thedistributed random variables-x2 distribution-the distribution of the statistic(x,s)- Students”t”distribution fisher’s z distribution.

UNIT V:

Theory of estimates :consistant estimates-unbiased estimates-the sufficiency of anestimate-the efficiency of an estimate - Asymptotically most efficient estimates.The powerfunction and dc function-Most powerful tests - Uniformly most powerful tests-Unbiasedestimates.

Trearment as in :

“Probability and mathematical Statistics” Bymarekfisz-john wiely-III Edition.



QUESTION PATTERN

M.Sc Mathematics







M.Sc. Mathematics(SF)


CORE PAPER XI DURATION: 6 HOURS

FLUID DYNAMICS

UNIT I: Introductory Notions: Velocity-Stream lines and Path lines – Stream tubes and filaments- Fluid

body-Density-Pressure. Differentiation following the Fluid- Equation of Continuity-Boundary conditions-kinematical Boundary conditions- Physical Boundary conditions-Rate of change of Linear Momentum-Euler’s equation of motion of an in viscid fluid. UNIT II:

Euler’s Momentum Theorem-Conservative forces-Bernoulli”s Theorem in Steady Motion-The Energy equation for In viscid fluid-Circulation, Kelvin’s Theorem-Vortex Motion-Permanence of velocity(Helmholtz equation)UNIT III :

Two Dimensional Motion: Two dimensional functions-Velocity Potential - Complex Potential-Basic Singularities-Source-Sink-[Complex potential of the flow due to a source of strength m at the origin] Doublet:[Complex Potential of the due to doublet of strength in two dimensional motion-Complex Potential of the flow due to a rectilinear vortex filament about a point a two dimensional flow]-Vortex-Mixed Flow.UNIT IV:

Method of images[Image of system-Image of Doublet w.r.t a Plane]- Circle theorem-Flow past a circular cylinder with circulation- Conformal Transformation- Transformation of flow field- Transformation of Source- Transformation of Doublet-The Aerofoil[Introduction]-[The Joukowski Transformation]-Thin Aerofoil- Joukowski's hypotheses- Blasius theorem-Lift force (Magnus effect)[Kutta and Joukowski theoremUNIT V :

Dynamics of real fluid : Viscusity and Reynolds Number-Viscous flows-The Equations of motion for viscous flow: The Stress Tensor in a Viscous fluid –Navier – Stokes equations(Equations of continuity,momentumequations,Equation of energy conservation)-Vorticity and circulation in a viscous fluid [Helmholtz’s equation for rate of change of vorticity,Rate of change of circulation with viscousity].Some exact solutions of Navier – Stokes equations:Steady flow through an arbitrary cylinder under pressure-Poiseuille’s law-steady Coutte flow between cylinders in relative motion- steady flow between parallel flat planes [plane Coutte flow-plane Poiseuille flow ]-Steady flow in pipes: Flow through a pipe –The Hagen Poiseuille flow.

Text Books: [1] Theoretical Hydrodynamics by L.M.MilneThomsor- Macmillan company,V Edition(1968).UNIT I - Chapter 1&3 1.1(1.11,1.12,1.13),3.10,3.20,3.30,3.31,3.40,3.41.UNIT II - Chapter 3 - 3.42,3.43,3.45,3.50,3.51,3.52,3.53.[2] Modern Fluid Dynamics-vol-1 by N.Curle and H.J.Davies,D Van Nostrand Company Ltd.,London(1968).UNIT III - Chapter 3 - 3.2.1-3.2.3,3.3,3.3.1-3.3.4UNIT IV - Chapter 3 - 3.5,3.5.1,3.5.2,3.6,3.6.1-3.6.3,3.7,3.7.2-3.7.5UNIT V - Chapter 5 - 5.1.1,5.2.1,5.



QUESTION PATTERN

M.Sc Mathematics







M.Sc .Mathematics(SF)

SEMESTER III PAPER CODE:PMA-12

CORE PAPER XII DURATION: 6 HOURS

FUZZY SET THEORY

UNIT I:

Crisp sets and fuzzy sets: The notation of fuzzy sets-Basic concepts of fuzzy sets. Operations on fuzzy sets: Fuzzy complement-Fuzzy union-Fuzzy intersection-Combinations of operations.

UNIT II:

Fuzzy relations: Crisp and Fuzzy relations-Relations-Binary relations on single set-Equivalence and Similarity relations-Orderings-Fuzzy relation equations.

UNIT III:

Fuzzy measures: Belief and plausibility-Probability measure-Possibility and necessary measures.

UNIT IV:

Applications: Natural, life and social sciences-Engineering-Medicine-computer science

UNIT V:

Fuzzy Arithmetic: Fuzzy Numbers-linguistic variables-Arithmetic Operations on intervals-Arithmetic operations on fuzzy numbers-Lattice of fuzzy numbers-Fuzzy Equations.

Treatment an in

1.GeorgeJ.Klir and Tina A.Folger,Fuzzy Sets, Uncertainty and information, PHI Learning Pvt.Ltd., 2009 (Unit I to Unit IV)

Unit I - Chapter 1&2-Section 1.3 -1.4,2.2 -2.5

Unit I - Chapter 3-Section 3.2-3.4,3.6-3.7

Unit III - Chapter 4-Section 4.2-4.4

Unit IV - Chapter 6-Section 6.2-6.4,6.6

GerogeJ.Klir and Boyvan, Fuzzy sets and fuzzy logic, PHZL earning Pvt Ltd, 2009. (Unit V)

Unit V - Chapter4-Section 4.1-4.6

References:

1. GerogeJ.Klir and Bo Yuan , Fuzzy Sets and Fuzzy logic, PHI Learning Pvt,ltd.,2009

2. H.J.Zimmermann, Fuzzy Set Theory and its applications, Allied Publishers Chennai 1996.

3. V.Novak, Fuzzy Set and their applications, AdamHieqer,Bristol 1989.



QUESTION PATTERN

M.Sc Mathematics








SEMESTER III PAPER CODE: ELPM-3

ELECTIVE-3 DURATION: 6 HOURS

MATHEMATICAL METHODS

Unit-I: Fourier Transforms: Fourier sine and cosine transforms – Fourier transforms of

derivatives - Fourier transforms of simple functions- The convolution integral –Parseval’stherom– solution of PDE by Fourier transform.- Laplace’s equation in half plane -Laplace’s equation in an infinite strip - The linear diffusion equation on a semi-infinite line - Thetwo-dimensional diffusion equation.Unit-II:

HankelTransforms : properties of Hankel Transforms - Hankel Transforms ofderivatives of functions - Hankel Inversion Theorem (Statement only)- The Parseval’s relation -relation between Fourier and Hankel transforms – Asymmetric Dirichlet problem for a thickplate.Unit-III:

Integral Equations: Types of integral equations –Integral Fredholm Alternative –Approximate Method – Equation with separable kernels – Volterra integral equations -Fredholm’s theory- Fredholm’s first, second, third theorems.Unit-IV:

Applications Of Integral Equations: Application to ordinary differential equation –Initial value problems, boundary value problems –Singular integral equations – Abel integralequation.Unit-V:

Calculus Of Variations: Variation and its properties - Euler’s equation, Functions of theintegral forms – Functional dependent on higher order derivatives – functional dependent on the functions of several independent variables – variables variational problem in parametric form – applications.Text Books:“The Use of Integral Transforms” by I.N.Sneddon, Tata McGraw-Hill Book Company, New Delhi,1974.

Unit – I - Chapter 2 - Sec 2.4 - 2.7, 2.9-2.10 ,2.16-2. (a,b,c)Unit- II - Chapter 5 - Sec 5.2, 5.4 5.6- 5.7, 5.10-5.12.

” Linear Integral Equations Theory and Technique” by R.P.Kanwal, Academic press,New York 1971.

Unit – III - Chapter 2&3 - sec 2.3-2.5,3.3-3.4 Unit- IV - Chapter 5&8 - sec 5.1,-5.2, 8.1-8.2 ,

“Calculus of Variations” by L.Elsgolts , Mir publishers , Moscow , 1970 Unit- V - Chapter 6 – sec 6.1-6.7



QUESTION PATTERN

M.Sc Mathematics








SEMESTER IV PAPER CODE: PMA-13

CORE PAPER XIII DURATION: 6 HOURS

FUNCTIONAL ANALYSIS

UNIT I :BANACH SPACES:

The definition and some Examples-Continuous Linear Transformation-The Hanh-Banach Theorem-The Natural Imbedding of N in N

UNIT II:

The Open Mapping theorem-The Closed Graph theorem the Conjugate of an Operator.

The uniform boundedness theorem Hilbert spaces the definition and simple properties

UNIT III

Orthogonal Complements-Orthonormal Sets- Bessel’s Inequality Gram-Schmidt Orthogonal Process- The Conjugate Spaces H*

UNIT IV

The adjoint of a operator –self adjoint operators-normal operators-unitary operators-projections.

UNIT V: FINITE DIMENSIONAL SPECTRAL THEORY

Matrices- Determinations and the spectrum of an operators-the spectral theorem.

Text Book : Treatment to topology and modern analysis by G.F.Simmons(mc raw hill-1963)

References: Functional Analysis by Dr.Somasundaram (S.V pub-1999)



QUESTION PATTERN

M.Sc Mathematics








SEMESTER IV PAPER CODE:PMA-14

CORE PAPER XIV DURATION: 6 HOURS

NUMBER THEORY

UNIT I

DIVISIBLITY : Introduction- divisibility-primes

CONGRUENCES : Definition-solution of congruences- congruences of degree one.

UNIT II

The function q(n) congruences of higher degree- prime power moduli-prime modules-congrunces of degree two, prime modules-power –power residue.

UNIT III

Number theory from an algebraic view point: multiplicative groups, rings and fields. Quadratic reciprocity: quadratic residues-quadratic reciprocity –The Jacobi symbol.

UNIT IV

Some functions of number theory: Greatest integer function-Arithmatic functions- Themoebios inversion formula-The multiplication of Arithmatic functions- Recurrence functions.

UNIT V

Some Diophantine equation: Definition of Diophantine equations The equation ax+by=cpositive solutions –other linear equations The equation x2+y2=n2- The x4 +y4 =z2-Sums of fourand five squares- Waring’s problem.

Text Book:Introduction to number theory byNiven and Zuckerman.UNIT I- Chr. I (1.1-1.3)

UNIT II- Chr. II (2.1-2.3)UNIT III-Chr. III (2.10-2.11)UNIT IV-Chr. IV (4.1-4.5)UNIT V-Chr. V (5.1-5.10)

REFERENCES:

1.Kennath and Rosan,Elementary Number Theory and its Applications,Addison Wesley Publishing Company,1968.

2.GeorgeE.Andrews,NumberTheory,HindustanPublishing,New Delhi,1989.



QUESTION PATTERN

M.Sc Mathematics







M.Sc.Mathematics(SF)

SEMESTER IV PAPER CODE:PMA-15

CORE PAPER XV DURATION: 4 HOURS

Mathematical Softwares

Unit IIntroduction : Basics of Matlab, Matlab windows – input – output – file types – platform

dependents –General commands.Unit II

Interactive computation: Matrices and vectors [Input ,indexing (or subscripting),matrix manipulation, creating vectors]-matrix and array operations [arithmetic operations, relational operations, logical operations, elementary math functions-matrix functions]-command-line functions[inline functions,anonymous functions]-using built in functions and online help-saving and loading data [saving into and loading from the binary mat-files, importing from the binary mat – files, importing data file]-plotting simple graphs. Unit IIIProgramming in MATLAB: Scripts and functions : Script files –function files [executing a function [more on functions M – Lint code analyzer, sub functions ,nested functions ,compiled (parsed ) functions:The p-code, the profiler]-language – specific features][use of comments to create on-line help-continuation, global variables, loops, branches and control flow-Interactive input – recursion, input/output]-Applications: Solving linear systems: ordinary differential equations.Unit IVDimensional Graphs : Basic 2-D Plots[style options, labels, title, legend, and other text objects, Axis control, zoom in, and zoom out, Modify plots with the plot editor, overlay plots, specialized 2-D plots]-using subplot fi=or multiple graphs.

Three –Dimensional Graphs : 3-D Plots [view, rotate view, mesh and surface plots ,vector field and volumetric plot, interpolated surface plots]-handle graphics[the object hierarchy ,object handles, object properties, modifying an existing plot, complete n control over the graphics layout]saving and printing graphs[saving graphs to reusable files]Unit V

Mathematica commands(General form, Examples, Uses only): Differential calculus: Limit [f(x), x →a]-D[f[x] ,x ]- D[ f[x] ,{x,n}].Integral Calculus :integrate [f[x], x] – Integrate [ f[x],{x,a,b}]-N Integrate [f[x] ,{x,a,b}].Partial Derivatives : D [f,x]-D[f,{x,n}]-Total Differential : Dt[f[x,y]] – Dt [f[x,y],x].

Multiple Integrals : integrate [f[x,y],{x,a,b},{y,c,d}].Differential Equations:Dsolve [eqn,y[x],x]-NDSolve [eqn,y,{x, xmin, xmax}].Algebra : Expand [ ] – Factor [ ] – Roots [ ] –Solve [eqn,var].Matrices : Determinant [ mx ] – Inverse [ mx ] –Eigen values [ mx ] –Eigenvectors [ mx ] – graphics : Plot [ f[x] , {x,a,b} ] – Plot [ f[x], [g[x], {x,a,b} – ParametricPlot [ {x[t], y[t] ,{t,a,b} ] – Plot3D[ { f[x,y], {x,a,b}, {y,c,d}].Treatment as in:

1) Getting started with Matlab – a quick introduction for scientist and engineers by Rudrapratap – Oxford University press – 2003

2) Eugene Don,Ph.D.Mathematica (schaum’s outlines), Mc.Graw.Hill.



QUESTION PATTERN

M.Sc Mathematics







M.Sc ., Mathematics(SF)

SEMESTER IV PAPER CODE:ELPM-4

ELECTIVE-4 DURATION:6 HOURS

CONTROL THEORY

Unit I:

Introduction: Motivation-Basic results of Differential Equation-Fixed point methods-Exercises.

Unit II:

Observability: Linear system-Non Linear Systems- Definition-Examples-Exercises.

Unit III:

Controllability: Linear system-Nonlinear Systems- Definition-Examples-Exercises.

Unit IV:

Stability: Linear system-Perturbed linear systems-Nonlinear system-Definition examples-Exercises.

Unit V:

Stabilizability: Stabilization via linear feedback control-The Controllable subspace-Stabilization with Restricted Feedback-Definition Examples.

Text Book: Elements of Control Theory-K.Balachandran, J.P.Dauer.,

Narosa Publishing House, 1999.Unit I - Chapter 1 - Sec 1.1-1.4

Unit II - Chapter 2 - Sec 2.1-2.3

Unit III - Chapter 3 - Sec 3.1-3.5

Unit IV - Chapter 4 - Sec 4.1-4.5

Unit V - Chapter 5 - Sec 5.1-5.4

Reference:

1. Linear Differential Equations and control by R. Conti, Academic Press, London , 1976.

2. Functional Analysis and Modern Applied Mathematics by R.F. Curtain and A.J. Pritchard, Academic Press , New York,1977



QUESTION PATTERN

M.Sc Mathematics







M.Sc .Mathematics(SF)

SEMESTER IV PAPER CODE:PPMA-2

PRACTICAL 2 DURATION: 2 HOURS

PRACTICAL’S IN MATHEMATICAL SOFTWARESPROGRAMS IN MATLAB

Program 1 Student’s t-distribution Test

Write a Matlab program to perform the t-statistic and degrees of freedom for Student's distribution, in Matlab code.

The calculations are based on anyone of the three hypotheses given below:

(1) Assume that on of the population mean is equal to a given value.

(2) Compare two samples. In both tests the means of the two populations and the standard deviations are equal respectively.

(3) In both tests the means of the two populations are equal, but the standard deviations are different.

Program 2 Matlab Program on ‘Graph Theory’

Write matlab program to(1) Find all shortest paths in a graph (2) Find strongly or weakly connected components in a graph(3) Test for cyclesindirected graph (4) Find isomorphism between two graphs (5) Determine whether the given tree is a spanning tree (6) Calculate maximum flow in a directed graph (7) Find minimal spanning tree in a graph (8) Convert predecessor indices to paths (9) Solve shortest path problem in a graph (10) Perform topological sort of directed acyclic graph

Program 3 Heat Transfer Visualization

Write a Matlab program to illustrate the use of theSimulink 3D Animation with the interface to read matrix-type data, transferred between MATLAB and a virtual reality world.

Use pre-calculated data of time-based temperature distributions in an L-shaped metal block and send that data to the virtual world.

Program 4 Perform various 2D plots

Write a Matlab program using the following features in an output: (1) Plot

(2) Subplot (3) Legend, Title (4) Style, Depth (5) Color (6) Axis labels (7) Linkaxes(8) Plotyy(9) Vector field plots (plots the slope lines ODE defined in function)

Program 5 Perform various 3D plots

Write a matlab program to input a set of data and plot the graph using (1) Mesh Plot,

(2) Surface Plot, (3) Surface Plot (with Shading),(4) Contour Plot, (5) Quiver Plot (6) Slices through 3-D Volumes, Fix the camera angle, Edit the properties of a graph

Program 6 Visualizing Four-Dimensional Data

Write a matlab program to (1) Visualize 4-D Data with One Discrete Variable

(2) Visualize 4-D Data with Multiple Plots (3) Visualize Function of Three Variables (4) Visualize Data in a Volume (5) Plot the Function of a Complex Variable

Program 7 Program on Image Processing

Write a Matlabprog m to

(1) Retrieve agray scale image. (2) Denoise the image using mean and median, high-pass, low-pass/Gaussian,

Butterworth) filters (3) Enhance the image using Histogram equalization(4) Detect the edges of an image (5) Save the output with specific resolution and size in (a) .BMP, (b) .tiff, (c) .jpg, (d)

MAT file formats using various storage commands.

Program 8 Building Systems with Fuzzy Logic Toolbox Software

Use the Fuzzy Logic Toolbox GUI tools to build a Fuzzy Inference System (FIS) for the tipping example described in The 'Basic Tipping Problem' .

Write a Mathematica program from numerical solution of the advection partial differential equation, finite differences, and fixed step method.

Program 9 Programs in mathematicaPrograms on partial differential equations

Write the mathematica program for numerical solution of the advection partial differential equations,finite differences and fixed step method.

Program 10 Numerical Integration

Write a Mathematica program for Numerical integration(Newton-Cotes)method to calculate

an approximate value of a definite integral I=∫a

b

f ( x ) dx. Compare various Newton-Cotes

methods to approximate the integrals of several different functions over the interval [a ,b] .Show that, the error decreases as the order of the method is increased. Likewise, show thatmore segments lead to a more accurate approximation of the integral.

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