M.A. /M. Sc. (II) Syllabus (Mathematics) Semester-III

Paper Periods Per week Max. Marks Theory Paper-XIII - Algebra II (Field Theory and Galois Theory )

04 50

Theory Paper-XIV--Functional Analysis 04 50 Three papers to be chosen from XV to XXII which will be taught in the department.

04 50

Theory-Paper- XV- Graph Theory Theory-Paper- XVI- Operations Research Theory-Paper- XVII- Advanced Number Theory Theory-Paper- XVIII- Integral Equations Theory-Paper- XIX -Programming in C (with ANSI features) (Theory and Practical) Theory-Paper- XX - Wave Propagation Theory-Paper- XXI-Theory of Linear operators Theory-Paper- XXII-Commutative Algebra Theory-Paper- XXIII-Tutorial-II (Compulsory for all) 125 Marks

Semester-IV

Paper Periods Per week Max. Marks

Theory Paper-XXIV- Boundary Value Problems 04 50

Theory Paper-XXV- Mechanics 04 50

Any three Papers to be Chosen from the following three

groups which will be taught in the department.

Theory Paper-XXVI To XXXVIII

04 50

A) Computational Mathematics

B) Applied Mathematics

C) Pure Mathematics

Theory- Paper-XXXIX- Project Work (Compulsory for all) 125 Marks.

Distribution of marks Semester III:

Name of Paper External

Exam(Theory

conducted

by University)

Internal Exam

(Tests+Assignments

+ Class Performance)

(By Dept.)

Project or Practical

(External Exam )

Algebra II

(Field Theory and Galois

Theory)

50 Marks 50 Marks 25 Marks

Functional Analysis 50 Marks 50 Marks 25 Marks

Optional I 50 Marks 50 Marks 25 Marks

Optional II 50 Marks 50 Marks 25 Marks

Optional III 50 Marks 50 Marks 25 Marks

Total Marks for one semester 625

Paper-XIII Algebra II (MTU-301)

(Field Theory and Galois Theory)

(Maximum Number of Periods: 60)

UNIT 1:

Concepts of Rings and Modules as a Pre-requisites, Field, Examples of Fields, Algebraic and Transcendental Elements, The Degree of a Field Extension, Construc- tion with Ruler and Compass, Symbolic Adjunction of Roots, Finite Fields, Tran- scendental Extensions. UNIT 2:

The Fundamental Theorem of Galois Theory, Composite Extensions and Simple Extensions, Cyclotomic Extensions and Abelian Extensions, Cubic Equations, Symmetric Functions, Primitive Elements, Proof of the Main Theorem, Quartic Equa- tions, Cyclotomic Extensions. Text Book:

M. Artin, Algebra, Prentice-Hall of India Pvt. Ltd.

Scope: chapter 13 and 14. Reference Books: 1. D. S. Dummit and R. M. Foote, Abstract Algebra, 2nd Ed., John Wiley, 2002. 2. T. A. Hungerford, Algebra, Graduate Texts in Mathematics, Vol. 73, Springer- Verlag, 1980 (Indian Reprint 2004). 3. S. Lang, Algebra, 3rd Edition, Addison-Wesley, 1999 4. I. S. Luthar, I. B. S. Passi, Algebra, Vol. 1, Groups, Narosa Publishing House. 5. P. B. Bhattacharyya, S. K. Jain and S. R. Nagpaul, Basic Abstract Algebra (2e), Camb.Univ. Press, Indian Edition, 1997.

Paper- XIV Functional Analysis (MTU 302)

(Maximum Number of Periods: 60)

Unit I : Banach Spaces:

Definition and some Examples, continuous linear transformations, The

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

Mapping Theorem, 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 an 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” By G.F. Simmons

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

Scope : Articles 46 to 62

Reference books (1) B.V. Limaye, “Functional Analysis”, Wiley Eastern Ltd.

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

(3) Kreyszig , Introductory Functional Analysis with Applications, John Wiley & Sons, New York, 1978Academic Press 1966.

(4) J. B. Conway, A course in functional analysis, Springer-Verlag, New York 1990.

(5) Foundations Of Functional Analysis , by S.Ponnusamy, Narosa Publishing House.

Paper XV –Graph Theory (MTU-303)

(Maximum Number of Periods: 60)

Unit I: Graphs, Subgraphs and Components, Degrees and Distances, Operations on Graphs,

Trees and Cycles

Unit II: Connectivity and Planarity, Eulerian and Hamiltonian Graphs, Perfect graph and

Ramsey Theory

Unit III: Matrices and vector Spaces, Groups and polynomials.

Unit IV: Digraphs, Networks.

Text Book:

Basic Graph Theory By K. R. Parthasarthy, Tata Mc Graw- Hill Pub Comp Limited

Delhi.

Scope : Chapters 1,2,3,4,5,6, 7, 11, 12,13,14 15 . Reference Books: (1) Graph Theory With Applications to Engineering and Computer Science, By Narsing Deo , Prentice Hall of India. (2) Graph Theory, By F. Harare, Addison Wesley. (3) Douglas B. West, Introduction to Graph Theory Prentice- Hall, New Delhi (1999) (4) John Clarke and D.A. Holton, A First Look at Graph Theory, Allied Publisher (1991) (5) Nora Harsfield and Gerhard Ringel , Pearls Theory, Academic Press (1990)

Paper-XVI- Operations Research (MTU304) (Maximum Number of Periods: 60)

Unit 1 (Prerequisites):

Operations research & its scope, Necessity of operations research in industry Introductions to Linear programming problems, General linear programming problems , Mathematical Formulation of L.P.P., Basic solution, Important theorems, solution of linear programming problem, Graphical method for solution , convex set , some important theorems, Revised simplex method, dual simplex method. Unit 2: Theory of Simplex methods:

Introduction, slack and surplus variables, some definitions and notations , Fundamental theorems of linear programming, BSF from F.S., Improved B.S.F. Unbounded solution, optimality of solutions, computational procedure of simplex method for the solution of a maximization L.P.P., artificial variable technique., duality and sensitivity analysis. Unit 3: Transportation and assignment Problems. Unit 4: Game Theory: Introduction, competitive game, finite and infinite game, two person zero sum game, rectangular game , solution of game, saddle point, solution of a rectangular game with saddle point. Unit 5;

PERT-CPM, product planning control with PERT-CPM. Text Books:

1. Dr. R. K. Gupta “Linear Programming”,Krishna Prakashan Mandir. 2. F.S.Hillier and G.J.Liebermann, Introduction to Operations Research

(6th Ed.) Mc Graw Hill International Edition, Industrial Engineering Series, 1995.

3. Kantiswaroop, P.K.Gupta and Manmohan, Operations Research, Sultan Chand & Sons, New Delhi.

Reference Books: 1) G.Hadley, Linear Programming, Narosa publishing House, 1995.

2) G.Hadley, Nonlinear and Dynamic Programming, Addison-Wesley, Reading Mass. 3) H.A.Taha, Operations Research - An Introduction, Macmillan Publishing Company, Inc, New York. 4) S.S.Rao, Optimization Theory and Applications, Wiley Eastern Ltd., New Delhi. 5) Prem Kumar Gupta and D.S.Hira, Operations Research - An Introduction. Chand & company Ltd, New Delhi. 6) N.S.Kambo, Mathematical Programming Techniques. Affiliated East-West Press Pvt.Ltd, New Delhi, Madras.

Paper-XVII - Advanced Number Theory (MTU305) (Maximum Number of Periods: 60)

Unit I:

Elementary theorems, Prime numbers, Distribution of prime numbers, The function (x), its properties, The Prime Number Theorem.

Unit II:

Riemann Zeta Function:-zeta function, Riemann zeta function, Riemann Hypothesis, connection between (s) and primes, Dirichlet L-functions, Dedekind zeta function, prime number theorem and RH ,Extended Riemann Hypothesis(ERH),Generalized Riemann Hypothesis(GRH),zeros of the Riemann zeta function on the critical line.

Unit III:

Study on RH:-Equivalent forms of RH and ERH, study on claims to prove RH, product formula for ,proof of Riemann’s main formula, derivative of the Riemann zeta function, the theorem of Hadamard and De le Vallee Poussin and its consequences, Lindelof Hypothesis, RH and Finite field theory. Unit IV:

Applications:-Applications of RH and ERH in different fields, consequences of RH, RH for the Hilbert Spaces of entire functions. Text Book: Riemann zeta function, H.M.Edward, (Academic Press) Scope: Reference Books: 1) The distribution of prime numbers, G.H.Hardy, E.Cunningham, (Cambridge University Press) 2) Theory of Riemann zeta function, E.C.Titchmarsh,(Oxford Press) 3) Little book of bigger primes, Paulo Ribenboim,(Springer International Edition)

Paper-XVIII - Integral Equations (MTU 306)

(Maximum Number of Periods: 60)

Unit I:

Definitions of Integral equations, regularity conditions, Special kinds of kernels, Eigen

values and Eigen functions, convolution integrals, Reduction to a system of algebraic

equations. Examples, An approximate method.

Unit II:

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

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

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

Unit III:

Symmetric kernels. Complex 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 Schmidt theorem and some

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

Unit IV:

Solution 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, Linear 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 XIX -Programming in C (with ANSI features) (MTU307)

Theory & Practical.

(Maximum Number of Periods: 60)

Unit I:

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.

Unit II:

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.

Unit III:

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.

Theory-Paper-XX-Wave Propagation (MTU308)

(Maximum Number of Periods: 60)

Unit 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)

Unit II:

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)

Unit III:

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 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. (EL.B.S.)

Paper XXI -Theory of Linear Operators (MTU 309)

(Maximum Number of Periods: 60)

Unit I:

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.

Unit II:

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.

Unit III:

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 XXII –Commutative Algebra (MTU 310)

(Maximum Number of Periods: 60)

Unit I:

Introduction to Module Theory, Basic Definitions and Examples,Quotient Modules and

Module Homomophisms, Generators of a Module, Direct sums, and free Modules. Tensor

Product of Modules, Exact sequence – Projective, Injective and Flat Modules.

Unit II:

Commutative Rings, Noetherian Rings and Affine Algebraic Sets, Radicals and Affine Varieties, Integral Extensions and Hibert`s Nulstellensatz thm, Localization, The Prime Spectrum of a Ring.

Unit III:

Artinian Rings, Discrete Valuation Rings and Dedekind Domains,

Text Book:

1) Abstract Algebra by D.S. Dummit and R.M. Foote, Second, John Wiley & Sons.

Scope: Chapter 10 to Chapter 16.

Reference Books:

1) Commutative Algebra – N. S. Gopalkrishnan

2) Undergraduate Commutative Algebra – Miles Reid

3) Introduction to Commutative Algebra and Algebraic Geometry – Ernst Kunze

4) Introduction to Commutative Algebra M- Atiyah and I. Macdonald.