Swami Ramanand Teerth Marathwada University, … of Computer Science-II Theory-Paper-XXXVII-Theory of Linear operators-II Theory-Paper-XXXVIII-Project Work (Compulsory for all) 50

Swami Ramanand Teerth Marathwada University,

Nanded. M.A./M. Sc. (Final) Syllabus (Mathematics)

With Effective form June 2009

Semester-III

Paper Periods Per week Max. Marks

Theory Paper-XIII-Functional Analysis 04 50

Theory Paper-XIV-Mechanics 04 50

Three papers to be chosen from XV to XXIV which

are taught in the department.

04 50

Theory-Paper-XV-Difference Equations-I

Theory-Paper-XVI-Fluid Mechanics-I

Theory-Paper-XVII-Analytical Number Theory-I

Theory-Paper-XVIII-Fuzzy Sets and their Applications-I

Theory-Paper-XIX-Wave Prorogation-I

Theory-Paper-XX-Integral Equations

Theory-Paper-XXI-Programming in C (with ANSI features)-I (Theory and Practical)

Theory-Paper-XXII-Advanced Functional Analysis-I

Theory-Paper-XXIII-Fundamentals of Computer Science-I (Theory and Practical)

Theory-Paper-XXIV-Theory of Linear operators-I

Theory-Paper-XXV-Tutorial-III (Compulsory for all) 50 Marks

Semester-IV Paper Periods Per week Max. Marks

Theory Paper-XXVI- Numerical Analysis 04 50

Theory Paper-XXVII- Partial Differential Equations 04 50

Any three Papers to be Chosen from the following

XXVIII to XXXVII which are taught in the

department.

04 50

Theory-Paper-XXVIII-Difference Equations-II

Theory-Paper-XXIX-Fluid Mechanics-II

Theory-Paper-XXX-Analytical Number Theory-II

Theory-Paper-XXXI-Fuzzy Sets and their Applications-II

Theory-Paper-XXXII-Wave Prorogation-II

Theory-Paper-XXXIII-Boundary Value Problems.

Theory-Paper-XXXIV-Programming in C (with ANSI features)-II (Theory and Practical)

Theory-Paper-XXXV-Advanced Functional Analysis-II

Theory-Paper-XXXVI-Fundamentals of Computer Science-II

Theory-Paper-XXXVII-Theory of Linear operators-II

Theory-Paper-XXXVIII-Project Work (Compulsory for all) 50 Marks

Paper-XIII- Functional Analysis Max. Periods: 60

Unit I : Banach Spaces:

The definition an some Examples, continuous linear Transformations,

The Hahn-Banach Theorem, The Natural embedding of N in N**. The

open Mapping Therem, The conjugate of an operator.

Unit II : Hilbert Spaces:

The definition and some simple Properties, Orthogonal complements,

orthonormal sets, The conjugate space H*, The Adjoint of on operator,

self adjoint operators, Normal and Unitary Operators, Projections.

Unit III : Finite Dimensional Spectral Theory:

Introduction, Matrices, Determinants and spectrum of an operator, The

spectral Theorem.

Text Book : Introduction to “Topology and Modern Analysis”

McGraw-Hill Book Company, International student Edition, New York.

Scope : Articles 46 to 62

Reference

Books

: 1. B.V. Limaye, “Functional Analysis”, Wiely Eastern Ltd.

2. G. Bachman and L. Narici “Functional Analysis”

Academic Press 1966.

Paper-XIV-Mechanics Max. Periods: 60

Unit I : Mechanics of System of particles, Generalized co-ordinates, Holonomic and

Noholonomic system, Scleronomic and Rheonomic system, D’Alembert’s

principles and Lagrange’s Equation of Motion, Different forms of Lagrange’s

Equation, Generalized Potential, Conservative fields and its Energy Equation,

Application of Lagrange’s formulation.

Unit II : Functional, Linear Functional, Fundamental lemma of calculus of variations,

Simple variational problems, The variation of functional, The extremum of

functional, Necessary condition for Extreme, Euler Equation. Eulers Equation

of several variables, Invariance of Euler Equation, Motivating Problems of

calculus of variation, Shortest Distance, Minimum surface of Revolution,

Brachistochrone Problem, Isopermetaic Problem, Geodesic.

Unit III : The fixed end point problem for n unknowns function, Variational problems in

Parametric form, Generialization of Euler Equation, Varitional Problems with

subsidary conditions.

Unit IV : Hamilton’s Principle, Hamitton’s canonical Equations, Lagrange’s Equation

from Hamiltons Principle, Extension of Hamiltons Principle to Nonholonomic

systems, Application of Hamiltons formulation, cyclic co-ordinates and

conservation theorems, Routn’s Procedure, Hamilton’s Equations from

variational principle, The principle of least Action.

Text Book : 1. H. Goldstein, Charles Poole, John Sabko, “Classical Mechanics”, Pearson

3rd Edition 2002.

2. I.M. Gelfand and S.V. Fomin “Calculus of Variations” Prentice Hall.

Reference

Books

: 1. N. Rana and B. Joag, “Classical Mechanics”, Tata McGraw Hill 1991.

2. A.S. Ramsey, “Dynamics Part II” The English Language Book Society and

Cambridge University press, 1972.

Paper-XV-Difference Equations-I Max. Periods: 60

Unit I : The Difference calculus:

The Difference operator, Summation, Generating Functions and

Approximate summation.

Unit II : Linear Difference Equations:

First order Equations, General Results for linear Equations, solving linear

equations, Equations with Variable coefficients, Nonlinear Equations, that

can be linearized, The z-Transform.

Unit III : Stability Theory:

Initial value problems for Linear systems, Stability of linear systems,

Stability of Nonlinear systems.

Unit IV : Asymptotic Methods:

Introduction, Asymptotic Analysis of Sums.

Text Book : Walter G. Kelley and Allan C. Peterson, “Difference Equations”,

Academic Press, second Edition.

Scope : 2.1,2.2,2.3,3.1,3.2,3.3,3.5,3.6,3.7,4.1,4.2,4.5,4.6,5.1,5.2.

Reference

Books

: 1. Calvin Ahlbrandt and Allan C. Peterson, “Discrete Hamiltonian

Systems: Difference Equations, Continued Fractions and Riccati

Equations, “Kluwer, Boston, 1996.

2. Saber N. Elaydi “An Introduction to Difference Equations”

Springer, Second Edition.

Paper-XVI- Fluid Mechanics-I Max. Periods:

60

Real Fluids and Ideal Fluids, Velocity of Fluid at a point, Streamlines and Pathlines; Steady

and Unsteady Flows, The Velocity potential, The Vorticity Vector, Local and Particle Rates

of Change, The Equation of Continuity, Worked Examples, Acceleration of Fluid, Conditions

at a Rigid Boundary, General Analysis of Fluid Motion.

Pressure at appoint in a fluid at rest, pressure at a point in a moving fluid, Conditions at a

boundary of two inviscid immisfluids, Euler’s equation of motion, Bernoulli’s equation,

Worked examples, Discussion of the case of steady motion under conserve body forces, some

potential theorems, Some flows involving axial symmetry, Some special two-dimensional

flows, Impulsive motion, Some further aspects of vortex motion.

Meaning of two-dimensional irrigational, incompressible flow, Use of cylindrical polar

coordinates, The stream function, The complex potential for two-dimensional irrigational,

incompressible flow, Complex velocity potentials for standard two-dimensional flows.

(Uniform stream, Line sources and line sinks, Line doublets, Line vortices), some worked

example.

Text Book : 1. “Text book of Fluid Dynamics” by F. Charlton, Reprint 1998 C.B.S.

Publishers and distributors, Delhi-110032.

Reference

Books

: 1. G.K. Batchelor-An Introduction to Fluid Mechanics. (Foundation

Books-New Delhi-1994).

2. W.H. Besaint and A.S. Ramsey-A Treatise on Hydro Mechanics-Part

II, C.B.S. Publishers-1998.

3. S.W. Yuan-Foundations of Fluids Mechanics. Prentice Hall of India

Pvt. Ltd.-New Delhi-1976.

Theory-Paper-XVII-Analytical Number Theory-I

Arithmatical Functions & Dirichlet Multiplication: The Mobius function, The Eular

Totient function, The Managoldt function, Dirichlet Multiplication, Liouville’s function, The

divisor function The Bell series, The Selberg identity.

Averages of Arithmetical functions: The Bigah Notation, Eulers Summation formula,

The average order of divisor functions, The partial sum of Dirichlet product. Chebyshev’s

functions.

Text Book : Introduction to Analytic number theory Tom M. Apostol.

Narosa Publishing house 1980.

Scope : Chapter 2: Articles 2.1 to 2.16, 2.19

Chapter 3: Articles 3.1 to 3.7, 3.9 to 3.12

Chapter 4: Articles 4.1.

Reference

Books

: 1. A course in arithmetic- J.P. Serre. GTM Vol.7, Springer Verlage 1973.

Theory-Paper-XVIII-Fuzzy Sets and their Applications-I

Crisps Sets & Fuzzy Sets: Operation on fuzzy sets

The crisp sets, fuzzy sets, basic concepts of fuzzy sets, fuzzy logic. Operation on fuzzy sets

fuzzy complement, fuzzy union, fuzzy intersection, combination of operation, general

aggregation operations.

Fuzzy Relation : Crisp & fuzzy relation, binary relations, binary relation on a single set,

equivalence & similarity relations. Tolerance relations, ordering morphisms, fuzzy relation

equation.

Text Book : Fuzzy sets, uncertainty & information By George J. Klir & Tina A.

Folger. (Prentice Hall of India Pvt. Ltd.) Sixth Printing 2001.

Scope : Chapter: 1 Complete, Chapter 2 Complete, Chapter 3 Complete.

Reference

Books

: 1. Introduction to Fuzzy control By-D. Drinkov, H. Hellendora & M.

Reinfrank, Narosa Publishing House.

2. Fuzzy Set Theory & Its Applications By-H.J. Zimmermann,

Allied Publishers Ltd. New Delhi-1991.

3. Fuzzy Sets & Fuzzy Logic By-G.J. Klir & B.Yuan.

Prentice Hall of India New Delhi-1995.

Theory-Paper-XIX-Wave Propogation-I

Introduction, SHM, damped harmonic oscillations, viscous damping, damped forced

oscillations, wave equation in one, two & three dimensions, harmonic waves, spherical

waves, super postion of waves & stationary waves, solution of equation of wave motion of

stationary types (1)

Transverses waves on tightly stretched elastic string, derivation of the wave equation,

normal vibration of finite continous sting with fixed ends, Fourier series solution for problem

involving different intial conditions, vibration of a string with damping, expressions for

kinetic & potential energy of a vibrating string, reflection of a waves at discontinuity of

string. (1)

Transverse vibration of thin membrance, normal modes of vibrations of flexible rectangular

drum head with fixed edges, normal vibration of a rectangular flexible drum head with fixed

edges having given initial displacements & released from rest.

Text Book : 1. Gosh P.K., “The mathematics of waves and vibrations,”

Mc Millan Company of India Limited.

2. Ceulson C.A., “Waves. A mathematical account of the common types

of wave motion: Oliver and Boyed.”

3. Ramsay A.S., “A treatise on Hydromechanics part II. (EL.B.S.)

Paper-XX- Integral Equations Max. Periods: 60

Definitions of Integral equations, regularity conditions, Special kinds of kernals,

Eigen values and Eigen functions, convolution integrals. The inner of scalar product

of two functions. Reduction to a system of algebric equations. Examples, An

approximate methods.

Iterative schemes for Fredholm integral equations of second kind. Examples, iterative

scheme for holterra integral equations of second kind, examples, Some results about

resolvent kernel. Classical Fredholm thory. Fredholm Theorem (without proof)

Symmetric kernels. Comlex Hilbert space orthonormal systems of functions,

Fundamental properties of eigen values and eigen functions for symmetric kernels.

Expansion in eigen function and bilinear form. Hilbert Schmdt theorem and some

immediate consequences. Solution of a symmmetric integral equations, examples.

Soultion of the Hilbert type singular integral equation Examples, Integral transforms

method. Fourier transform, Laplace transforms Application to Volterra integral

equations with convolution types of kernels, Examples.

Text Book : 1. R.P. Kanwal, Linera Integral Equantions.

Reference

Books

: 1. S.G. Mikhlin, Linear integral equations (Translated from Russian)

“Hindustan Book Agency 1960.

2. B.L. Moiseiwitsch, Integral Equations, Longman, London & New

York.

3. M. Krasnov, A Kiselev, G.Makaregko, Problems and Exercises in

integral equations (Translated from Russian) by George Yankovsky)

MIR Publishers Moscow, 1971.

Paper XXI-Programming in C (with ANSI features)-I Theory & Practical.

An overview of programming, Programming language classification, C-Essentials-Program

development functions, Anatomy of a C function, Variables and constant, Expressions,

Assignment statements, Formatting source field, Continuation character, The preprocessor.

Scalar data types-Declarations, Different types of integers, Different kinds of integer

constants, Floating point types, Initialization, Mixing types Enumeration types, The void data

type, Typedefs, Find the address of an object, Pointers.

Operators and expressions-precedence and associativity, Unary plus and minus operators,

Binary arithmetic operators, Arithmetic assignment operators, Increment operations, And

decrement operators, Comma operator, Relational operators, Logical operators, Bit-

Manipulation operators, Bitwise assignment operators, Cost operator size of operator,

Conditional operator, Memory operators, Control flow-conditional branching, The switch

statement, looping, nested loops, The “break” and “continue” statements, the go to statement,

Infinite loops. Arrays and Pointers: Declaring an array, Arrays and memory, initializing

array, Multidimensional arrays.

Text Book : “Programming in ANSI C,”-E Balaguruswamy, Second Edition, Tata-

McGraw Hill Publications.

Scope : Chapter 1: Complete

Chapter 2: 2.4 to 2.9

Chapter 3: 3.1 to 3.9

Chapter 5: 5.1 to 5.4, 5.7, 5.9

Chapter 6: Complete

Chapter 7: Complete

Reference

Books

: 1. Peter A. Darnell and Pholip E. Margolis “C:A Software Engineering

Approach” Narosa Publishing House 1993.

2. Brain W.Kernighan and Dennis M. Ritchie. “The C Programme

Language, 2nd Edition (ANSI features).

3. Samuel P. Harkison and Gly L. Steele Sr. “C: A Reference Manual”,

2nd Edition. Pub-Prentice Hall 1984.

Paper XXII-Advanced Functional Analysis-I

Defination & example of topological vector speces. Convex, balanced and absorbing sets and

their properties. Minkowsiski’s functional, subspace, product space and quotient space of a

topological vector space.

Locally convex topological vector spaces. Normable and Metrizable topological vector

spaces. Complete topological vector spaces & Frechet spaces. Linear transformations and

Linear functional and their continuity.

Reference

Books

: 1. John Horvath, Topological Vector Spaces & Distribution.

Addison-Wesely Publishing Company 1966.

2. F. Treves, Topological Vector spaces, Distribution, Kernel,

Academic Press, Inc., New York, 1967.

3. G. Kothe, Topological Vector spaces, Vol.1, Springer, New York, 1969.

4. R. Larsen, Functional Analysis, Marcel Dekker, Inc., New York, 1973.

5. Walter Rudein, Functional Analysis, TMH edition, 1974.

Paper XXIII-Fundamentals of Computer Science-I (Theory & Practical)

Object oriented programming-class and scope nested classes pointer class members, Class

initialization, assignment and destruction. Overloaded function and operators, Templates

including class templates, Class Inheritance and subtyping, Multiple and virtual inhabitance.

Data structures: Analysis of algorithms, Lists, Stacks & queues Introduction to Relational

Algebra and Relational Calculus.

Text Book : 1. “Object-oriented programming with C++”-E, Balaguruswamy,

Tata-McGraw Hill Publishing Co., Ltd, New Delhi (11th reprint, 1998)

Reference

Books

: 1. “Theory & Problems of Data Structures” – Seymour Lipschutz,

International Edition 1986, Schaum’s outline series.

2. “An Introduction to database Systems”-Bipin C. Desai, 1998,

Galgotia Publication Pvt. Ltd.

3. “Database System Concepts” – Silberschatz, Korth, Sudavshan, 3rd

Edition McGraw Hill International Editions.

Paper XXIV-Theory of Linear Operators

Spectral theory in normed linear spaces, resolvent set and spectrum, spectral properties of

bounded linear operators. Properties of resolvent and spectrum. Spectral maping theorem for

polynomials. Spectral radius of a bounded linear operator on a complex banach space.

Elementary theory banach algebra.

General properties of compact linear operators. Spectral properties of compact linear

operators on nomed spaces. Behaviors of compact linear operators with respect to solvability

of operator equations. Fredholm type theorems. Fredholm alternative theorm. Fredholm

alternative for integral equations. Spectral properties of bounded self-adjoint linear operators

on a complex Hibert space. Positive operators. Monotone Sequences theorem for bounded

self-adjoint operators on a complex Hilbert space, Square roots of a positive operator.

Reference

Books

: 1. E. Kreyszig, Introductory functional analysis with applications,

Johan-Wiley & Sons, New York, 1978.

2. P.R. Halmos, Introduction to Hilbert space and the theory of spectral

multiplicity, 2nd Edn. Chelsea Pub., Co., N.Y. 1957.

3. N. Dunford and J.T. Schwartz, Linear operators-3 parts,

Interscience Wiley, New York, 1958-71.

4. G. Bachman & Narici, Functional analysis, Aca-demic Press, New

York, 1966.

5. Akniezer, N.I. and I.M. Glazman, Theory of linear operators in

Hilbert space, Frederick Ungar Pub. Co. NY, Vol. 1 (1961), Vol.

2(1963).

6. P.R. Halmos, A Hilbert space problem book, D.Van Nostrand Co. Inc,

1967.

Paper No. XXV-Tutorial-III Paper Marks

Functional Analysis-XIII 10

Partial Differential equation-XIX 10

Elective paper 10

Elective paper 10

Elective paper 10

Total 50

The format for scheme of marking for tutorial of 10 marks in each paper.

Tutorial ____________________ Name of the paper & No. ______________________

Name of the teacher ____________________________

Submission Attendance Contents Viva Total

Marks

Sr.

No.

Name of the

Student

Seat

No.

Marks 2 Marks 3 Marks 3 Marks 2 10

Head of the Department Teacher

The Format, in which the marks obtained by students in tutorial _________ out of 50 marks

to be submitted by HOD through the Principal, to the department of Examination

S.R.T.M.U., Nanded.

Tutorial Paper No.

Paper No.

Paper No.

Paper No.

Paper No.

Total Marks 50

Sr. No.

Name of the Student

Seat No.

Marks 10

Marks 10

Marks 10

Marks 10

Marks 10

Head of the Department Principal

Paper –XXVI- Numerical Analysis Max. Periods: 60

Transceridental and Polynomial equations: Introduction, Bisection method, Iteration

methods based on first degree equations and second degree equations, Rate of

convergence, Polynomial Equations, Model problems.

System of Linear algebraic equations and Eigen value problems: Introduction, direct

methods, Iteration methods, Eigenvalue and Eigen vectors, Model problems.

Interpolations and approximations: Introduction, Lagrange’s, Newtonian

Interpolation, finite difference operators, Interpolating polynomials using finite

differences, Approximations, Least Square approximations.

Text Book : 1. M.K. Jain, SRK Iyengar, R.K. Jain, “Numerical methods for Scientific

and Engineering computations.” New Age International Limited Pub.

Scope : Chap2: Art. 2.1 to 2.5, 2.8, 2.9

Chap3: Art 3.1, 3.2 3.4 3.5 3.6

Chap4 Art 4.1 to 4.4, 4.8, 4.9

Reference

Books

: 1. S.S. Satry, “Introductory methods of Numerical Analysis” Prentice-

Hall of India Private Ltd. (Second Edition) 1997.

2. E.V. Krishnamurthi & Sen. “Numerical Algorithm,”

Affiliate East. West press. Private Limited 1986.

Paper-XXVII- Partial Differential Equations Max. Periods: 60

Partial Differential Equations of the second order:

The origin of second order Equations, Second-order Equation in Physics, Linear

Partial Differential equation with constant, coefficients, Equation with variable

coefficient Characteristic curves of second-order Equations, Separation of variables,

Non linear equations of second order.

Laplace Equation:

Occurance of Laplace’s Equation in Physics, Elementry solution of Laplace’s

Equation, Families of Equipotential surfaces, Boundary value problems, Separation of

variable, the theory of Green’s function for Laplace’s Equation, Two dimensional

Laplace Equation, Green’s function for the two-dimensional equation.

The wave equation:

The occurrence of Wave equation in Physics, Elementry solution for the one

dimensional wave equations, General solution of the wave equations, Green’s

function for the wave equation.

Text Book : IAN. Sneddon- Elements of Partial Differential Equation,

McGraw –Hill Book company.

Scope : Chap3: Art 1,2,4,5,6,9,11

Chap4: Art 1,2,3,5,8,11,13

Chap5: Art 1,2,6,7

Reference

Books

: 1. L.C. Evance, “Partial differential Equation Graduate studes in

Mathematics” Vol: no. 11-AMS 1998.

2. T. Amarnath, “Partial Differential equation”, Narosa Publication,

second Edition 2003

3. P. Prasad, R. Ravindran, “partial Differential equations.”

Paper-XXVIII- Difference Equations-II Max. Periods: 60

Unit I : The self-Adjoint second order Linear Equation:

Inroduction, Sturmian Theory, Green’s Functions, Disconjugacy, The

Riccati Equations.

Unit II : The sturm - Liouville Problem:

Introduction, Finite fourier Analysis, Nonhomogeneous Problem.

Unit III : Discrete calculus of variation:

Intrduction, Necessary conditions, sufficient conditions and Disconjugacy.

Unit IV : Boundary value Problems for Nonlinear Equations:

Introduction, The Lipschitz case, Existence of solutions, Boundary value

Problems for differential Equations.

Text Book : 1. Walter G. Kelley and Allon C. Peterson, “Difference Equations,”

Academic press, Second Edition.

Scope 6.1, 6.2, 6.3, 6.4, 6.5, 7.1, 7.2, 7.3, 8.1, 8.2, 8.3, 9.1,9.2,9.3,9.4

Reference

Books

1. Calvin Ahlbrandt and Allan C. Peterson, “Discrete Hamittonian systems,

Difference Equations, Continued Fractions and Riccat Equations,”

Kluwer, Boston, 1996.

2. Saber N. Elyadi, “An Introduction to difference Equations,”

Springer, second Edition.

Paper-XXIX-Fluid Mechanics-II Two-dimensional image systems, The Milne-Thomson circle theorem (Some applications of

the circle theorem, Extension of the circle theorem), Thye theorem of Blasius.

Compressibility effects in real fluids, The elements of wave motion(The one-dimensional

wave equation, Wave equations in two and in three-dimensions, Spherical waves, Progressive

and stationary waves). The speed of sound in a gas, Equations of motion of a gas, Subsonic,

Sonic and Supersonic flows, Lsentropic gas flow, Reservoir discharge through a channel of

varying section (Investigation of maximum mass flow through a nozzle, Nozzle with

different mass flows), Shockwaves (Formation of shock waves, Elementary analysis of

normal shock waves, Elementary analysis of oblique shock waves, use of shock charts).

Stress components in a real fluid, Relations between Cartesian components of stress,

Translational motion of fluid element, The rate of strain quadric and principle stresses, Some

further properties of the rate of strain quadric, Stress analysis in fluid motion, Relations

between stress and rate of strain, The coefficient of viscosity and laminar flow. The navier-

stokes equations of motion of a viscous fluid, some solvable problems in viscous flow (steady

motion between parallel planes, steady flow through tube of uniform circular cross-section,

Steady flow between concentric rotating cylinders). Steady viscous flow between concentric

rotating cylinders (steady viscous flow in tubes of uniform cross-section, Tube having

uniform elliptic cross-section, Tube having equilateral triangular cross-section, Use of

harmonic functions). Diffusion of vorticity, Energy dissipation due to viscosity, Steady flow

past a fixed sphere,

Dimensional analysis; Reynolds number, Prandtl’s Boundary Layer (Karman’s integral

equation)

Text Book : “Text book of Fluid Dynamics” by F. Charlton, Reprint 1998, C.B.S.

Publishers and distributors, Delhi-110032.

Reference

Books

1. G.K. Batchelor-An Introduction to Fluid Mechanics.

Foundation Books-New Delhi-1994).

2. W.H. Besaint and A.S. Ramsey-A Treatise on Hydro Mechanics-Part II,

C.B.S. Publishers 1988.

3. S.W. Yuan-Foundations of Fluids Mechanics,

Prentice Hall of India Pvt. Ltd.-New Delhi-1976.

Paper-XXX-Analytical Number Theory-II

Some elementary theorems on the distribution of prime numbers: equivalent forms of prime

number theorem, Shapiors Tauberian theoem, brief sketch of an elementary proof of prime

number theory, Selberg’s asymptotic formula.

Riemann Zeta function: Hurwitz Zeta function, Analytic continuation, Hurwitz formula,

functions, equations, Bemoulli numbers & polynomials,

Analytic proof of prime number theorem: The plan of the proof, A countour integral

representation, non vanishing of Z(S) on the line S=1, completion of the proof of prime

number theorem.

Text Book : Introduction to analytic number theorem, By- Tom M. Apostol

Narosa publishing House 1980.

Scope Chapter 4: Articles 4.6, 4.7, 4.11.

Chapter 12: Articles 12.1, 12.3, 12.5 to 12.13.

Chapter 13 : Articles 13.1 to 13.7.

Reference

Books

1. A course in Arithmetic-J.P. Serre, GTM Vol.7, Springer Verlage, 1973.

Paper XXXI-Fuzzy Sets and their applications-II

Fuzzy Measures Belief & plausibility measures, probability measures, possibility & necessity

measures, relationship among classes of fuzzy measures.

Uncertainty & Information, types of uncertainty, measures of fuzziness, classical measure of

uncertanity, measures of dissonance, measure of non specificity, uncertinity & information

information and complexity, Principles of uncertainty and information, Applications, General

discussion, Natural, life & Social, Sciences, Engineering, Medicine, Management & decision

making.

Text Book : Fuzzy sets, Uncertinity & Information, By George J.Klir & Tina A. Folger.

(Prentice Hall of India Pvt. Ltd.) Sixth printing 2001.

Scope Chapter 4: Complete, Chapter 5: Complete, & Chapter 6:6.1 to 6.5.

Reference

Books

1. Introduction to Fuzzy Control, By-D. Drinkov, H. Hellendora & M.

Reinfrank Narosa Publishing House.

2. Fuzzy Set Theory & its Applications, By-H.J. Zimmerman Allied

Publishers Ltd. New Delhi-1991.

3. Fuzzy Sets & Fuzzy Logic, By- G.J. Klir & B. Yuan. Prentice Hall of

India, New Delhi-1995.

Paper XXXII-Wave Propogation-II

Expressions for kinetic & potential energy of vibration membrance. Vibration of a stretched

circular drum head fastened at the circumference & releases initially from rest. Propogation

of waves in elastic solid media waves of dilatation & waves of distortion in isotropic elastic

media plane waves, Rayleight waves, Love waves (1).

Types of liquid waves, Tidal waves, oscillator waves, surface waves, capillary waves, group

waves, introduction, general form of wave equation, wave equation for plane waves,some

examples of normal vibrations of air in a pipe, spherical waves, energy of sound waves,

illustrative examples (1).

Solution for a source free empty space, solution for a homogeneous isotropic medium in

which there are free charges but no conduction current and the field vectors are independent

of time, Uniform plane waves propagation in free space (vacuo), wave in a conducting

medium, Electromagnetic waes in a rectangular wave guide. (1) (2)

Reference

Books

: 1. Gosh P.K. “The mathematics of waves and Vibrations,”

Mc Millan Company of India Limited.

2. Ceulson C.A., “Waves. A mathematical account of the common types of

wave motion”, Oliver and Boyed.

3. Ramsay A.S., “A treatise on Hydromechanics”, part II, (E.L.B.S.)

Paper-XXXIII-Boundary value problems (Max Periods 60)

Preliminaries, Definition of a boundary value problem for an ordinary differential

equation of the second order and its reduction to Fredholm integral equations of the

second kind, Examples, Dirac Delta function, Green’s function approach to reduce

boundary value problems of self-adjoint differential equation with homogeneous

boundary conditionts to integral equations form Exampes Auxiliary problem satified

by Green’s function

Integral equation formulation for the boundary value problem with more general and

in homogeneous boundary conditions, Examples, the Strum-Liouville problem,

Example, Modified Green’s Function, Examples.

Integral representation formulas for the solutions of the Laplace and Poisson

equations. The Newtonian single-layer and double-layer potentials. Interior and

exterior Dirichlet and Numann boundary value problems for Laplace’s equation,

examples.

Green’s function for Laplace’s equation in a frees pace as well as a space bouded by a

ground vessel. Integral equation formulation of boundary value problems for Laplce’s

equation. Poission’s integral formula mixed boundary value problem, Tow-part and

three-part boundary value problem.

Text Book : R.P. Kanwal, “Linear integral Equation: Theory and Technique”, A.P. 1971

Reference

Book

: 1. S.G. Mikhlin, Linear Integral Equation (translated from Russion).

Hindustan Books Agency 1960.

2. I.N. Sneddon, “Mixed Boundary Value Problem in Potential Theory “

Nostn Holland 1966.

3. I. Stakgold, “Boundary Value Problems of mathematical Physics” Vol- I

& II MacMillan 1961.

Paper-XXXIV-Programming in C (With ANSI Feature)-II

(Theory and Practical) Max. No.

Periods: 60

Theory part 35 marks +Practical 15 marks = total 50 marks

Encryption and Decryption, Pointer arithmetic, Passing Pointers as function arguments,

Accessing array elements through pointers, Passing arrays as function arguments; Sorting

algorithms, Strings, Multidimensional array, Arrays of pointers, Pointers to pointers, Storage

classes-Fixed VS. Automatic duration, Scope Global Variable, The register variables, ANSI

rule for syntax and semantics of the storage class key-words. Dynamic memory allocation.

Structures and union-Structures linked lists, Unions enum declarations, Functions-Passing

arguments, Declarations and class, Pointers to functions, Recursion, The Main () function

complex declarations.

The C preprocessor-Macro Substitution, Conditional Compilation, Include facility, Line

control. Input and output streams buffering <stdio.h> header file, Error handling, Opening

and choosing a file, Reading and writing data, Selecting and I/O method, Unbuffered I/O

random access, The standard library for Input/Output.

Text Books : 1. “Programming in ANSI C” –E Balaguruswamy, Second Edition, Tata McGraw

Hill Publications.

Scope : Chapter 8: Articles-8.1 to 8.4, 8.6 to 8.8

Chapter 9: Complete

Chapter 10: Articles 10.1 to 10.4, 10.6,10.7,10.9 & 10.10

Chapter 11: Complete, Chapter 12 Complete, Chapter 13: 13.1, 13.2

Chapter 14: 14.1 to 14.4

Reference

Books

: 1. Peter A. Darnell and Pholip E. Margolis “C: A Software Engineering

Approach” Narosa Publishing House 1993.

2. Brain W. Kernighan and Dennis M. Ritchie. “The C Programme Language,

2nd Edition (ANSI features). Pub-Prentice Hall 1989.

3. Samuel P. Harkison and Gly L. Steele Sr. “C:A Reference Manual”, 2nd

Edition. Pub. Prentice Hall 1984.

Paper-XXXIV-Advanced Functional Analysis-II Max. Periods-60

Finite dimensional topological vector spaces. Linear varities and hyperplanes, geometric form

of Hahn-banach theorem. Uniform boundedness principle. Open Maping theorem and closed

graph theorem for Frechet Spaces, Banach- Alaoglu theorem. Extrem points and extremal

sets. Krein-Milman’s theorem Duality polar, Bipolar theorem, baralled and bornological

spaces, Mackey spaces, Semireflexive and reflexive topological vector spaces, Montel spaces

and Schwarz Spaces, Quasicompletenesss, Inverse limit and inductive limit of locally convex

spaces, distribution.

Reference

Book

: 1. John Horvath, “Topological Vector Spaces & Distribution,”

Addison-Wesely Publishing Company, 1966.

2. F. Treves, “Toplogical Vector Spaces, Distribution, Kernel,”

Academic Press, Inc., New York, 1967.

3. G. Kothe, “Topological Vector Spaces, Vol.1,”

Springer, New York, 1969.

4. R. Larsen, “Functional Analysis,” Marcel Dekker, Inc., New York, 1973.

Walter Rudein, “Functional Analysis,” TMH edition, 1974.

Paper –XXXVI-Fundamentals of Computer Science-II

(Theory and Practical) Theory-35 Marks and Practical -15 marks Max. Periods-60

Sequential and linked representations Trees: Binary tree-search tree implementations, B-tree

(Concept only), Hashing-Open and closed sorting: Insertion sort, quit sort, heap-sort, and

their analysis.

Database system: Role of database systems. Architecture operating system: user interface,

processor management, I/O Management, Memory management, Concurrency and security,

Network and distributed systems.

Text Book : “Object-oriented programming with C+” –E Balaguruswamy, Tata-

McGraw Hill Publishing Company, Ltd, New Delhi (Eleventh reprint,

1998.)

Reference

Books

: 1. “Theory & Problems of Data Structures” –Seymour Lipschutz,

International Edition 1986, Schaum’s outline series.

2. “An Introduction to Database Systems” –Bipin C. Desai, 1998,

Galgota Publication Pvt. Ltd.

3. “Database System Concepts”-Silberschatz, Korth, Sudavshan, third

Edition McGraw Hill International Editions.

4. “Operating systems”-Stuart E. Madnick, John. J.Donovan.

International Edition, 1974 McGraw Hill.

5. “Operating Systems: Concepts and Design”-Milan Milen Kovic,

Second Edition, Tata McGraw Hill.

Paper –XXXVII-Theory of Linear Operators -II Max. Periods-60

Projection operators. Spectral family of a bounded self-adjoint linear operator and its

properties. Spectral representation of bunded self-adjoint linear operators. Spectral theorems.

Spectral measures. Spectral integrals. Regular spectral measures. Real and complex spectral

measures. Complex spectral integrals. Description of the spectral subspaces. Characterization

of the spectral subspaces. The spectral theorem for bounded normal operators.

Unbounded linear operators in Hilbert space. Hellinger-Toeplitz theorem. Hilbert adjoint

operators. Symetric and self-adjoint linear operators.Closed linear operators and closures.

Spectrum of an unbounded self-adjoint linear operators. Spectral theorem for unitary and

self-adjoint linear operators. Multiplication operator and differentiation operator.

Reference

Book

: 1. E. Kreyszing, “Introductory Functional Analysis with applications,” John-

Wiley & Sons. New York. 1978.

2. P.R. Halmos, “Introduction to Hilbert Space and the theory of Spectral

multiplicity,” 2nd Edition, Chelsea Publishing Co, N.Y. 1957.

3. N. Dunford and J.T. Schwartz, “Linear operators-3 parts,” Interscience Wiley,

New York, 1958-71.

4. G. Bachman & L. Narici, “Functional Analysis,” Academic Press, New York,

1966

5. Akniezer, N.I. and I.M. Glazman, “Theory of Linear Operators in Hilbert

Space,” Frederick Ungar Pb. Co. NY.Vol. 1 (1961) Vol.2 (1963)

6. P.R. Halmos , “A Hilbert Space Problem Book,” D.Van Nostrand Co. Inc.

1967.

Paper –XXXVIII-Project Work Note:- To be assessed by External Examiners

Scheme of Marking

Project Work Submission – 40 Marks

Viva -10 Marks

Total – 50 Marks

Swami Ramanand Teerth Marathwada

University,

Nanded.

M.A./M.Sc. (Final)

Syllabus (Mathematics)

Effective from June -2009

Swami Ramanand Teerth Marathwada

University,

Nanded.

B.A./B.Sc. Second Year

Syllabus (Mathematics)

Effective from June -2009

B.A./B.Sc. S.Y. (Mathematics

Theory Paper-IV- Advanced Calculus

No. of Periods 120 Max. Marks: 100 Unit I : Definition of a Sequence, Theorems on Limits of Sequences, Bounded and

Monotonic Sequences, Non convergent sequences, Cauchy’s general

principle of convergence. Series of nonnegative terms, Comparison tests,

D’Alembert’s ratio test, Raabe’s Test, Logarithmic test, Alternating Series,

Leibnitz’s test, Absolute and Conditional Convergence.

Unit II : Continuity, Sequential Continuity, Properties of Continuous Functions,

Uniform Continuity.

Unit III : Differentiation of Function of single variable, Darboux’s theorem, Rolle’s

theorem and it’s interpretation, Lagrange’s mean value theorem and it’s

geometrical interpretation, Cauchy’s mean value theorem and it’s

geometrical interpretation, Taylor’s theorem for function of single variable

with various forms of remainders.

Unit IV : Line integral, plane curve definitions, properties of line integral, Double

integral, Integral over a rectangle, Partition of rectangle, integral as limit of

sum, prosperities, change of variables.

Beta, Gamma Functions and their properties.

Text Book : 1. Mathematical Analysis, By- S.C. Malik, Savita Arora.

(Second reprint 2002), New age international.

Scope Chapter 3: 1,1.1,2,2.1,2.2,2.3,2.4,3,3.1, 3.3,4,4.1,5,6,7,7.1,8,10,10.1

Chapter 4: 1,1.1,1.2,1.3,1.4,2.1,2.2,3,3.1,3.2,3.3,5,6,7,10,10.1,10.2.

Chapter 5: 1,1.1,1.2,1.3,2,2.1,2.2,2.3,2.4,3,3.1,3.2,3.3,3.4,3.5,4,4.1

Chapter 6: 1,1.1,1.2,2.1,2.2,3,3.1,3.2,4,5,5.1,6,6.1,6.2,7,8,8.1.

Chapter 17: 1,1.1,1.2,1.4,2,2.2,2.4,2.6,2.5, Appendix-I Page No. 872 to 878.

Reference

Books

: 1. Text Book on Integral Calculus-By: Gorakh Prasad. 14th edition 2000,

Pothishala Pvt. Ltd. Allahabad.

B.A./B.Sc. S.Y. (Mathematics)

Theory Paper-V

DIFFERENTIAL EQUATIONS

No. of Periods 120 Max. Marks: 100 Unit I : Series solutions of differential equations:

Special Functions, power Series Solution, Validity of Power Series. Bessel’s equation solution, Bessel’s function, Recurrence formulae, Orthogonality of functions, Generating function. Legendre’s equation, Legendre’s polynomial, Legendre’s function of second kind. General solution of Legendre’s equation, Rodrigus’s formula, Generating function of Legendre’s polynomial, Orthogonality, Recurrence formulae, Orthogonality of Eigen function.

Unit II : Laplace Transformations Laplace transformation: Formulae, Properies. Laplace transformation of the derivative of f(t). Laplace transformation of derivative of order n. Laplace transform of integrals of f(t), transform of t.f(t),1/t f(t). Unit step function. Shifting theorem, Convolution theorem. Evaluation of integrals. Inverse Laplace transform: Formulae, Multiplication by s, division by s, shifting properities. Inverse Laplace transform of derivatives and integrals, partial fraction method. Solution of differential equations and solution of simultaneous differential equations.

Unit III : Partial differential equation of the first order: Definition, Derivation by elimination of constant, derivation by elimination of arbitrary function. Integrals of the non-linear equation: Complete and particular integral, singular integral, general integral. Intergral of linear equation, Equation equivalent to linear equation, Lagrange’s solution, Verification of Lagrange’s solution, Linear Equation involving more than two independent variables. Geometrical meaning of linear partial differential equation. Special methods of solution: Standard I,II,III,IV, general method of solution

Unit IV : Partial Differential equation of second order: General method of solving, General linear partial differential equation of an order higher than the first. Homogeneous equations with constant coefficients: Complementary functions, Complementary function for repeated, imaginary roots. The particular integral. Non-homogeneous equations with contant coefficient: The complementary function. The particular integral. Transformation of equation. Calculus of varations: Functionals, Definition Euler’s equation, Extremal, Isopermetric problems.

Text Book : 1. Advanced Engineering Mathematics By: H.K. Dass, 9th Revised Edition 2001 (S. Chand and Co.)

Scope : Chapter 8: 8.1, to 8.4, 8.6 to 8.9, 8.11, 8.12, 8.17.1, 8.18 to 8.25, 8.29to 8.30. Chapter 13: 13.1 to 13.12,13.15,13.17 to 13.28,13.30,13.31 Chapter 17: 17.1 to 17.7,

2. Introductory course in differential equations. By: D.A. Murray. Scope : Art.-107 to 133 References : 1. Advanced Engineering Mathematics-Ervin Kreyszig. Johan Wiley

and sons Inc. New York. 2. Calculus of Variations with Applications-A.S. Gupta Prentice Hall of

India. 3. Operational Mathematics-R.V. Churchill. 4. Elements of Partial differential equations-I.N. Sneddon. McGraw-

Hill Book Company. 5. Differential equations-Jane Cronin. (Publisher-Marcel Dekkar,

Co.1994.) 6. Theory and Problems of Differential equations.-Richard Bronson.

McGraw-Hill, inc.1973.

B.Sc. S.Y. (Mathematics)

Theory Paper-VI

MACHANICS

No. of Periods 120 (Only for B.Sc. Students) Max. Marks: 100 Unit I : (Statics)

Forces acting on a rigid body: Introduction, Moment of a force, couples, Equivalent Couples, conditions of equilibrium of forces acting on rigid body, Equilibrium of forces that are coplanar and acting on a rigid body. Centroid of weighted points, centre of inertia, centre of gravity, C.G. of uniform bodies, (Uniform rod, triangular lamina, Uniform parallelogram)

Unit II : (Dynamics) Kinematics and Dynamics of a particle (Two Dimension) Introduction, Definition of velocity, acceleration, curvature and principle normal, Tangential and normal components of velocity and acceleration, Angular speed, angular velocity, angular acceleration, components of velocity and accelerations along radial and transverse directions. Areal speed, Areal velocity.

Unit III : Kinetics of a particle: Introduction, Newton’s Laws of motion and deductions. Linear momentum, angular momentum, Impulsive force and its impulse, conservation of linear momentum, Impact of two bodies, work, work done by a variable force, power, Energy, Scalar point function, vector point function, Field of force, conservative field of force, Potential function.

Unit IV : Projectile motion and central orbits. Rectilinear motion, Motion under gravity, Projectile, its motion and equation of trajectory, Parabola of safety. Definition of central orbit, properties, Differential equations of central orbit, Apses, Finding law of force in different cases. Kepler’s Laws of planetary motion and deductions.

Text Book : Mechanics and Differential Geometry-V. Tulsani, Wanrhekar, N.N. Saste.

Pub., S. Chand and Company (Pvt.) Ltd. Ram Nagar, New Delhi, (Edn. 1987).

Scope : Scope : (Statics) Chapter 3: Art. 3.1 to 3.11, Chapter 4: Art. 4.1 to 4.6

Dynamics (Unit-II,III,IV): Chapter 1: Art 1.01 to 1.14, Chapter 2 : Art. 2.01 to 2.25, Chapter 3 : Art. 3.01 to 3.15, Chapter 4: Art. 4.01 to 4.08 and 4.18.

References : 1. Mechanics (R.P. Unified) –B.R. Thakur and G.P Shrivastav, Pub. Ramprasad and son’s.

2. Mechanics, By- Shanti Narayan, S. Chand and Co.

B.Sc. Second Year

Practical-Paper No. –VII. (Only for B.Sc. Students)

Note:

1. For a batch of 20 students, two periods per week, will be the work load.

2. A record book, consisting of at least 50% of the practical given below given

below be maintained by each student.

3. The theory part required for the practical be explained to the students, by the

teacher concerned, from the reference books.

PARCTICALS:

1. Show that the sequence n

n

11 is convergent and its limit lies between 2 and 3.

2. Test the convergence of the series

........)2)(1()2)(1(

11

3. Find the limits given below

i) x

exOx

Lim x

1

)1( ii)

222 sin11

xxOxLim

iii) xx

OxLim x

1

)(tan

iv)

xxexe

OxLim xx

sin.cos2

4. Prove that J-n(x) = (-1)n Jn(x) were n is positive integer.

5. Show that i) P2n (0) = (-1)n

nn2.......6.4.2

)12.......(5.3.1

ii) P2n+1 (0) =0

6. Evaluate

0

t. e-3t sint dt

7. Find the inverse Laplace transform of )4)(1(

42

sss

s

8. Evaluate y dx dy over the part of the plane bounded by the lines y = x and the

parabola y = 4x-x2

9. Trace the curve “witch of Agnesi” : xy2 =4a2(2a-x).

10. Trace the curve Cissoid : y2(2a-x) = x3.

11. Trace the Cycloid : x =a(θ + sin θ), y = a(1-cos θ)

12. Trace the Parabola: x = at2,y =2at.

13. Trace the curve r = asing2θ

14. Trace the curve r2 = asing2θ

15. Trace the Cardiod r = a(1+cosθ)

16. Evaluate , ][ dxdyyxR over the rectangle R= [0,1;0,2].

17. Trace the curve x = a cos3θ, y = a sin3θ, and find total length of the curve.

18. Trace the cardioid r = a (1-cosθ) and find its perimeter.

19. Trace the cycloid, x =a (θ-cosθ), and find the area included.

20. Find the area common to the two Parabolas y2 = 4x and x2 = 4y.

21. Find the area of the loop of the curve ay2 = x2 (a-x)

22. Find the area of one loop of the curve r = a cos4θ.

23. A function f is defined on the rectangle R = [0,1;0,1] as follows:

rational isywhen

irrationalisy Whenxyxf ,

21

,),(

Show that the double integral R f(x,y) dx dy does not exist.

24. The area bounded by the parabola y2=4x and the line y = 2x is revolved about x-axis;

find the volume and the area of the surface so generated.

25. Find R (x+2y) dx dy, when R = [1,2; 3, 5]

26. Find the area of the surface formed by the revolution of x2+2y2=16 about its major

axis.

27. Find the volume of the spindle shaped solid generated by revolving the asteroid about

the x-axis,

28. Find the volume of the solid generated by revoving the ellipse 12

2

2

2

by

ax about y-

axis.

29. Find the surface area generated by revolving the circle x = a cosθ, y = a sin about x-

axis

30. if n is a positive integer, prove that the ratio of the areas enclosed by the curves

x2n + y2 = 1, x2n + y2n = 1 is n21/n/(n+1).

Pattern of the theory Question Paper

B.Sc. (Second Year) Mathematics (Yearly Pattern) Maximum Marks: 100 Duration: 3:00 Hrs.

N.B.:- i) All questions are compulsory. ii) Figures to the right indicate full marks. Q. 1 Ten Multiple choice Questions based on all units of 2 marks each Marks (20) Q. 2 A) Attempt any one of the following

a) Theory Marks (10) b) Theory Marks (10) B) Attempt any one of the following a) Theory Marks (05) b) Problem Marks (05) c) Problem Marks (05)

Q. 3 A) Attempt any one of the following






*****

Swami Ramanand Teerth Marathwada University, Nanded. M.A./M. Sc. (Previous) Syllabus (Mathematics)

Semester-I


Theory Paper-I- Advance Abstract Algebra-I 04 50

Theory Paper-II-Real Analysis 04 50

Theory Paper-III- Topology-I 04 50

Theory Paper-IV- Complex Analysis-I 04 50

Any one of the following of VA to VD. 04 50

Theory-Paper-V(A)-Difference Equations-I

Theory-Paper-V(B) -s-Advanced Discrete Mathematrs-I

Theory-Paper-V (C) –Differential Geometry of Manifolds- I

Theory-Paper-V (D)- Dynamics and Continuum Mechanics-I

Theory-Paper-VI-Tutorial-I-(Compulsory for all) 50

Semester-II


Theory Paper-VII- Advance Abstract Algebra-II 04 50

Theory Paper-VIII-Integration Theory 04 50

Theory Paper-IX- Topology-II 04 50

Theory Paper-X- Complex Analysis-II 04 50

Any one of the following of XIA to XI D. 04 50

Theory-Paper-XI (A)-Differential Equations-II

Theory-Paper-XI (B)-Advanced Discrete Mathematics-II

Theory-Paper-XI (C) –Differential Geometry of Manifolds- II

Theory-Paper-XI (D)- Dynamics and Continuum Mechanics-II

Theory-Paper-XII-Tutorial-II-(Compulsory for all) 50

Pattern of theory question paper for all semesters

M.A./M. Sc. (First Year & Second Year) Mathematics

Maximum Marks: 50 Duration: 3:00 Hrs.

N.B.:-

i) All questions are compulsory.

ii) Figures to the eight indicate full marks.

Q. 1 Attempt any one of the following

a) Theory Marks (08)

b) Theory Marks (08)

Q. 2 Attempt any one of the following


b) Theory Marks (08)

Q. 3 Attempt any two of the following


b) Theory/Problem Marks (06)

c) Problem Marks (06)









*****

Documents

Swami Ramanand Teerth Marathwada University, … of Computer Science-II Theory-Paper-XXXVII-Theory of Linear operators-II Theory-Paper-XXXVIII-Project Work (Compulsory for all) 50