CENTRAL UNIVERSITY OF RAJASTHAN - curaj.ac.in · PDF fileHindi, Italian, Russian, Spanish) 2 1 1 0 3 0 20 0 30 50 0 7. ED 102 Senior Secondary Education in India: Status, Challenges

CENTRAL UNIVERSITY OF RAJASTHAN

Int. M. Sc. B.Ed. Mathematics New scheme & Syllabus to be change

Int. M.Sc. B.Ed. Mathematics is a three academic year spread in six semesters. The details of

the courses with code, title and the credits assign are as given below.

First Semester

S.

No

.

Subject

Code

Course Title C

r

e

d

it

Conta

ct

Hours

Exam.

Duration

(Hrs.)

Relative Weights %

L T P The

ory

Pra

ctic

al

C

WS

PR

S

M

TE

ET

E

PR

E

1. MTM

101

Abstract Algebra 4 3 1 0 3 0 20 0 30 50 0

2. MTM

102

Real Analysis 4 3 1 0 3 0 20 0 30 50 0

3. MTM

103

Probability and

Probability

Distribution

4 3 1 0 3 0 20 0 30 50 0

4. MTM

104

Qualitative Theory of

Ordinary Differential

Equations

4 3 1 0 3 0 20 0 30 50 0

5. ED101 Basics of Education 3 3 0 0 3 0 20 0 30 50 0

6. MTM

106

Dynamics of

Communication

(Languages: English,

French, German,

Hindi, Italian,

Russian, Spanish)

2 1 1 0 3 0 20 0 30 50 0

7. ED 102 Senior Secondary Education in India: Status, Challenges and Strategies

3 3 0 3 0 20 0 30 50 0

TOTAL 2

4

1

9

5

Second Semester

S.

No

.

Subject

Code

Course Title Credit Contact

Hours

Exam.

Duration

(Hrs.)

Relative Weights %

L

T P The

ory

Pra

ctic

al

C

WS

PR

S

M

TE

ET

E

PR

E

1. MTM

201

Linear Algebra

4 3 1 0 3 0 20 30 50 50 0

2. MTM

202

Complex Analysis 4 3 1 0 3 0 20 30 50 50 0

3. MTM

203

Topology 4 3 1 0 3 0 20 30 50 50 0

4. MTM

204

Modeling and

Simulation

3 3 0 0 3 0 20 30 50 50 0

5. ED 201 Philosophy of

Mathematics/Physics/C

hemistry

3 3 0 0 3 0 20 30 50 50 0

6. MTM

206

Mathematical

Software Tools

3 2

1 0 3 0 20 30 50 50 0

7 ED 202 Learner and Learning 3 3

0 0 3 0 20 30 50 50 50

TOTAL 24 2

0

4 0

Third Semester

S.

No

.

Subject

Code

Course Title Cr

edi

t

Contact

Hours

Exam.

Duration

(Hrs.)

Relative Weights %

L T P The

ory

Pra

ctic

al

C

WS

PR

S

M

TE

ET

E

PR

E

1. MTM

301

Functional Analysis 4 3 1 0 3 0 20 0 30 50 0

2. MTM

302

Partial Differential

Equations

4 3 1 0 3 0 20 0 30 50 0

3. MTM

303

Numerical Analysis 4 3 1 0 3 0 20 0 30 50 0

4 MTM

304

Differential

Geometry

4 3 1 0 3 0 20 0 30 50 0

5. ED 301 Teaching Approaches

and Strategies

3 3 0 0 2 2 20 0 30 50 0

6. MTM

306

EM-1/EC-1 3 3 0 0 3 0 20 0 30 50 0

7. ED 302 Pedagogy of

Mathematics/Physics

/Chemistry-I

4 3 1 0 3 0 20 30 30 50 50

TOTAL 26 21 5 0

Fourth Semester

S.

No

.

Subject

Code


Hours

Exam.

Duration

(Hrs.)

Relative Weights %

L

T P The

ory

Pra

ctic

al

C

WS

PR

S

M

TE

ET

E

PR

E

1. MTM

401

Mathematical

Programming

4 3 1 0 3 0 20 0 30 50 0

2. MTM

402

Dynamics of Rigid

body

4 3 1 0 3 0 20 0 30 50 0

3. MTM

403

EM -II 3 3 0 0 3 0 20 0 30 50 0

4. MTM

404

EM-III 3 3 0 0 3 0 20 0 30 50 0

5. ED 401 Learning Assessment 3 3 0 0 3 0 20 0 30 50 0

6. MTM

406

Open Elective-II

3 3 0 0 3 0 20 0 30 50 0

7. ED 402 Pedagogy of

Mathematics/Physics/

Chemistry-II

4 3

1 0 3 0 20 0 30 50 0

24 2

1

3 0

Fifth Semester

Sixth Semester

S.

No

.

Subject

Code

Course Title Credi

t

Contact

Hours

EoS Exam.

Duration

(Hrs.)

Relative Weights

%

L

T P Theo

ry

Prac

tical

IA ST

s

EoSE

1. MTM

501

Elective M-IV 4 3 1 0 3 0 20 30 50

2. MTM

502

Open Elective – I 4 3 1 0 3 0 20 30 50

3. ED501 Classroom

Organization and

Management

3 3 0 0 3 0 20 30 50

4. ED502 Internship 12 0 0 12 0

23 9 2 12

S.

No

.

Subject

Code


Hours

EoS Exam.

Duration

(Hrs.)

Relative Weights

%

L

T P Theo

ry

Pra

ctic

al

IA ST

s

EoSE

1. MTM

603

Major project 20 0 500 for project

report and

evalution+100

2. Project/Dissertation

(Review of

Researches in the

subject )

4

24

Note:

1. MTM:302(Mechanics) to be replaced by MTM:401(Dynamics of rigid body)

2. MTM:402 (Mathematical Programing-OLD) to be replaced by MTM:402

(Mathematical Programing-NEW)

3.

MTM 302: Mechanics (Detail Syllabus-OLD) LTP: 3+1+0

UNIT-I: Dynamics of a system of particles: Motion of system of particles, Principal of angular

momentum, motion of a rigid body, angular velocity, rigid body rotation about a fixed axis, rate of

change of a vector in a rotation frame. Moving frame of reference, frames of reference with

translational motion, motion of a particle relative to a rotating frame, uniformly rotating frames,

linear impulse and angular impulse. (15L)

UNIT-II: Kinematics and Dynamics of a rigid body: Moments and products of inertia, moment

of inertia of a body about a line through the origin, Momental ellipsoid, rotation of co-ordinate axes,

principal axes and principal moments. K.E. of rigid body rotating about a fixed points, angular

momentum of a rigid body, Eulerian angle, angular velocity, K.E. and angular momentum in terms

of Eulerian angle. Euler’s equations of motion for a rigid body, rotating about a fixed point, torque

free motion of a symmetrical rigid body (rotational motion of Earth). (15L)

UNIT-III: Lagrangian and Hamiltonian formulation of the dynamics: Classification of

dynamical systems, Generalized co-ordinates systems, geometrical equations, Lagrange’s equation

for a simple system using D’Alembert principle, Deduction of equation of energy, deduction of

Euler’s dynamical equations from Lagrange’s equations, Generalized momentum for a dynamical

system, Hamilton’s canonical equations of motion, Hamiltonian as a sum of K.E. and P.E., Phase

space and Hamiltonian’s variational principle, principle of least action, Canonical transformation,

Hamilton-Jacobi equation, Integral of Hamilton’s equations, Lagrange and passion brackets,

Poisson-Jacobi identity.(15L)

Recommended Readings:

Vectorial Mechanics by E. A. Milne; Methuen & Co. Ltd. London, 1965.

Dynamics (Part II) by A.S. Ramsey; CBS Publishers & Distributors, Delhi, 1985.

A treatise on Analytical Dynamics by L.A. Pars; Heinemann, London, 1968.

Classical mechanics by H. Goldstein; arosa Publishing House, New Delhi, 1990.

Generalized Motion of Rigid Body by N. Kumar; Narosa Publishing House, New Delhi, 2004.

Classical Mechanics by K. Sankara Rao; PHI learning Private Ltd., New Delhi, 2009

MTM 402: Dynamics of Rigid Body (Detail Syllabus- NEW) LTP: 3+1+0

UNIT-I: Moments and products of inertia, moment of inertia of a body about a line through the origin,

Momental ellipsoid, rotation of co-ordinate axes, principal axes and principal moments. K.E. of rigid body

rotating about a fixed points, angular momentum of a rigid body, Eulerian angle, angular velocity, K.E. and

angular momentum in terms of Eulerian angle. Euler’s equations of motion for a rigid body, rotating about a

fixed point, torque free motion of a symmetrical rigid body (rotational motion of Earth). (15L)

UNIT-II: Classification of dynamical systems, Generalized co-ordinates systems, geometrical equations,

Lagrange’s equation for a simple system using D’Alembert principle, Deduction of equation of energy,

deduction of Euler’s dynamical equations from Lagrange’s equations, Hamilton’s equations, Ignorable co-

ordinates, Routhian Function. (15L)

UNIT-III: Hamiltonian’s principle for a conservative system, principle of least action, Hamilton-Jacobi

equation, Phase space and Liouville’s Theorem, Canonical transformation and its properties, Lagrange and

passion brackets, Poisson-Jacobi identity. (15L)


1. Vectorial Mechanics by E. A. Milne; Methuen & Co. Ltd. London, 1965.

2. Dynamics (Part II) by A.S. Ramsey; CBS Publishers & Distributors, Delhi, 1985.

3. A treatise on Analytical Dynamics by L.A. Pars; Heinemann, London, 1968.

4. Classical mechanics by H. Goldstein; arosa Publishing House, New Delhi, 1990.

5. Generalized Motion of Rigid Body by N. Kumar; Narosa Publishing House, New Delhi, 2004.

6. Classical Mechanics by K. Sankara Rao; PHI learning Private Ltd., New Delhi, 2009

MTM 402: Mathematical Programming (Detail syllabus- OLD) LTP: 3+0+0

Unit I. Linear Programming, optimal solutions of linear programming problems. Duality in linear

programming. The dual simplex method. Integer programming; Importance of integer programming

problems. (15 Lectures)

Unit II. Dynamic programming, application of dynamic programming, Bellman’s principle of

optimality, solution of problems with a finite number of stages, Inventory control and solving linear

programming problems. Network Analysis: Shortest path problem. Minimum spanning tree

problem. Maximum flow problems. Network Simplex method. Project planning and control with

PERT-CPM.

(15 Lectures)

Unit III. Nonlinear programming problems: the general linear programming problems of

constrained maxima and minima. Quadratic programming: General quadratic programming problem,

Kuhn Tucker conditions of quadratic programming problems, examples based on Wolfe’s method

and Beale’s method. (15 Lectures)

Recommended Reading:

1. S. D. Sharma: Operations Research, Kedar Nath Ram Nath and Co.

2. Kanti Swarup, P. K. Gupta and Manmohan: Operations Research, S. Chand & Co.

3. Hamady Taha: Operations Research, Mac Millan Co.

4. S. D. Sharma: Nonlinear and Dynamic Programming, Kedar Nath Ram Nath & Co.

5. G. Hadley: Linear Programming, Oxford and IBH Publishing Co.

6. S. I. Gass: Linear Programming, Mc Graw Hill Book Co.

MTM 402 : Mathematical Programming (Detail Syllabus- NEW) LTP: 3+1+0

Unit I. Linear Programming, Theoretical foundation of Simplex Method, Revised Simplex Method,

Post Optimality Analysis, Computational Complexity of Simplex Algorithm, Karmarkar's

Algorithm for Linear programming, Linear Fractional programming problem.

(15 Lectures)

Unit II. Multi-Objective Optimization Theory, Goal Programming, Network Models- Minimum

Spanning Tree Problem, Maximum flow problem, Project planning and Control with PERT-CPM,

Deterministic Inventory Control.

(15 Lectures)

Unit III. Convex Programs and Duality, Quadratic programming and Complementarity Problem-

Wolfe's method, Beale's method, Fletcher's method. Nonlinear Programming Methods, Frank-Wolfe

Method, Gradient Projection Method, Penalty and Barrier Function Method.

(15 Lectures)


1. Operations Research: An Introduction, Hamady A. Taha, Prentice Hall of India, 8th ed.,

2006.

2. Numerical Optimization with Applications, S. Chandra, Jayadeva and A. Mehra, Narosa

Publishing House, 2009.

3. G. Hadley: Linear programming, Addison-Wesley Pub. Co., 1962.

4. Introduction to Operations Research, F.S. Hillier and G.J. Lieberman, 2001.

Detailed Syllabus for

M. Sc. Tech. Mathematics

Semester-I

MTM 101: Abstract Algebra LTP: 3+1+0

Unit I: Groups, subgroups, Cosets, Lagrange’s theorem, normal subgroups, quotient groups,

homomorphism, isomorphism theorems, Conjugacy, Class equation, simple groups. Sylow

theorems, Normal and subnormal series, composition series, Jordan holder theorem. Solvable

groups, simplicity of An ( n > 5). (15 Lectures)

Unit II: Rings, homomorphisms, ideals,Quotient rings, prime ideals, maximal ideals, field of

quotients of an integral domain,Euclidean rings, unique factorization domains, principal ideal

domain. Polynomial rings. Eisenstenin’s criterion of irreducibility.Chain conditions, on rings.

Noetherian and Artinian rings. Modules, Submodules. Quotient modules. (15 Lectures)

Unit III: Homomorphism and Isomorphism theorems. Extension fields, algebraic element and

transcendental elements, Simple extension, Algebraic extension, Roots of a polynomial and splitting

of a polynomial over fields. (15 Lecture)

Recommended Reading :

1. Topics in algebra by I. N. Herstein. Wiley Eastern Limited.

2. A first course in Abstract Algebra by John Fraleigh (3rd Edition), Narossa Publishing House.

3. Basic Abstract Algebra by Bhattacharya, Jain and Nagpal, 2nd Edition.

4. Algebra by S.Mclane and G.Birkhoff, 2nd Edition,

5. Basic Algebra by N.Jacbson, Hind.Pub.Corp.1984.

MTM 102: Real Analysis LTP: 3+1+0

Unit-I: Euclidian space ℝn: Open ball and open sets, closed sets, adherent points, accumulation

points, closure of sets, derived sets, Bolzano Weierstrass theorem. Cantor intersection theorem.

Lindeloff covering theorem, Heine Borel theorem, Compactness in ℝn. Metric spaces: open sets,

closed sets, compact subsets of a metric space. (15L)

Unit-II: Functions of bounded variations: Monotonic functions and its properties, types of

discontinuity functions of bounded variations and its properties, total variations. Functions of

several variables: continuity, partial derivatives, differentiability, derivatives of functions in an open

set of ℝn into ℝn as a linear transformations, chain rule, Taylor’s theorem, inverse function theorem,

implicit function theorem and explicit function theorem, Jacobians. (15L)

Unit-III: Riemann-Stieltjes integral: Definition and existence of R-S integration, conditions of R-

S integrability, properties of R-S integrals, integration and differentiation. Sequence and series of

functions: Pointwise and uniform convergence, Uniform convergence and continuity, Uniform

convergence and integration, Uniform convergence and differentiation, Uniform convergence and

R-S integration. (15L)


1. W. Rudin , Principles of Mathematical Analysis (3rd Ed.)McGraw Hill International Edition,

1976

2. Mathematical Analysis by T. M. Apostol( 2nd Ed.), Narosa Publishing House , 1985

3. Theory of Functions of a Real Variable, Volume 1 by I. P. Natanson, Frederick Pub. Co.,

1964

4. H.L. Royden, Real Analysis, McMillan Publication Co. Inc. New York

5. Malik, S.C. Mathematical Analysis, Wiley Eastern, New Delhi, 1984.

MTM 103: Probability and Probability Distributions LTP: 3+1+0

1. Exploratory data analysis: summary statistics, box and whisker plots, histogram, P-P and Q-Q

plots

2. Random Experiment and its sample space, probability as a set function on a collection of

events, stating basic axioms, random variables, c.d.f., p.d.f., p.m.f., absolutely continuous and

discrete distributions, Some common distributions (Negative Binomial, Pareto, lognormal,

beta, etc). Transformations, moments, m.g.f., p.g.f., quantiles and symmetry. Random vectors,

Joint distributions, copula, joint m.g.f. mixed moments, variance covariance matrix.

3. Independence, sums of independent random variables, conditional expectation and variances,

compound distributions, prior and posterior distribution, best predictors.

4. Sampling distributions of statistics from univariate normal random samples, chi-square, t and

F distributions

5. Order statistics and the distribution of rth order statistic, joint distribution of rth and sth order

statistics.

6. Statement and application of central limit theorem for a sequence of independent and

identically distributed random variables.

7. Simulation techniques such as Monte Carlo, Resampling techniques.


1. Ross, Sheldon M. (2003) Introductory Statistics

2. Hogg, R. V. and Craig, T. T. (1978) Introduction to Mathematical Statistics (Fourth Edition)

(Collier-McMillan)

3. Rohatgi, V. K. (1988) Introduction to Probability Theory and Mathematical Statistics (Wiley

Eastern)

4. C. R. Rao (1995) Linear Statistical Inference and Its Applications (Wiley Eastern) Second

Edition

5. H, Cramer (1946) Mathematical Methods of Statistics,( Prinecton).

6. J. D. Gibbons & S. Chakraborti (1992) Nonparametric statistical Inference (Third Edition)

Marcel Dekker, New York

MTM 104: Qualitative Theory of Ordinary Differential Equations LTP: 3+1+0

UNIT-I: Existence and uniqueness theorems-solution to non –homogeneous equations, Wronskian

and linear dependence, Reduction of the order of a homogeneous equation, Cauchy-Euler equation,

Pfaffian Differential equation, separation and comparison theorems, system of equations existence

theorems, Homogeneous linear systems, Non homogeneous Linear systems, Linear systems with

constant coefficients. (15 Lectures)

UNIT-II: Two-point boundary-value problem, Green's functions, Construction of Green's

functions, Non homogeneous boundary conditions, Orthogonal sets of function and Strum Liouville

problem, Eigen values and Eigen functions, Eigen function expansions convergence in the

Mean. (15 Lectures)

UNIT-III: Stability of autonomous system of differential equations, Stability for Linear systems with

constant coefficients, linear plane autonomous systems, perturbed systems, Method of Lyapunov for

nonlinear systems. Limit cycles of Poincare Bendixson Theorem. (15 Lectures)


1. Simmons: Ordinary Differential Equations.

2. Lakshmikantham, Deo and Raghavendra, Ordinary Differential Equations.

ED 101: Basics of Education LTP: 3+1+0

MTM 106: Dynamics of Communication (Languages) LTP: 1+1+0

ED 102: Senior Seco. Edu. in India: Status, Challenges & Strategies LTP: 3+0+0

Semester-II

MTM 201: Linear Algebra LTP: 3+1+0

Unit I: Review of Vector Spaces, The algebra of linear transformations, Isomorphism, Linear

functional, dual and double dual, Eigen values and eigen vectors, Annihilating polynomials,

diagonalization, Tringularization. (15 L)

Unit II: Determinants and its geometric properties, Laplace expansion, rational and Jacobian

canonical form, primary decomposition theorem, nilpotent matrices, canonical form for nilpotent

matrix, computation of invariant factors, non-negatives matrices, generalized inverse of a matrix,

diagonalization of symmetric bilinear forms. (15 L)

Unit III: The Adjoint of Linear Transformation, Unitary operators, Self Adjoints and Normal

Operators, spectral theorem of normal operators, Polar and Singular Value, Decomposition, spectral

theory of normal operators on finite dimensional vector space. ( 15 L)


1. Algebra by S. Mclane and G. Birkhoff

2. Linear algebra by S. Lang, Springer

3. Linear Algebra by Bisht and Sahai 4. Linear Algebra by Hoffman and Kunze, P.H.I 5. Matrix Analysis and Applied linear Algebra, by Carl D. Meyer 6. Linear Algebra by S. Lang. 7. Linear Algebra by P. Lax. 8. Linear Algebra: a geometric approach by S. Kumaresan.

MTM 202: Complex Analysis LTP: 3+1+0

Unit-I: Functions of a complex Variable, Differentiability and analyticity, Cauchy Riemann

Equations, Power series as an analytic function, properties of line integrals, Goursat Theorem,

Cauchy theorem, consequence of simply connectivity, index of a closed curve (15 Lectures)

Unit-II: Cauchy’s integral formula, Morera’s theorem, Liouville’s theorem, Fundamental theorem

of Algebra, Harmonic functions, Existence of Harmonic conjugate, Taylor’s theorem,

Zeros of Analytic functions, Laurent series, singularities, classification of singularities (15

Lectures)

Unit-III: Maximum modulus theorem, Minimum modulus theorem, Hadamard three circle

theorem, Schwarz’s Lemma, Rouche’s theorem, Calculation of residues, Residue theorem,

Evaluation of integrals of the form , , Conformal mappings.

(15 Lectures)


1. Complex Analysis ( Third edition) by L. V. Ahlfors, McGraw Hill Book Company, 1979

2. Complex Analysis by J. B. Conway, Narosa Publishing House,

3. Complex Analysis by Serg Lang, Addison Wesley

4. Foundations of Complex analysis ( Second Edition), S. Ponnusamy, Narosa Publishing

House.

5. Complex variables and Applications by Ruel V. Churchill,

MTM 203: Topology LTP: 3+1+0

Unit-I: Topological spaces. Open sets, closed sets. Interior points, Closure points. Limit points,

Boundary points, exterior points of a set, Closure of a set, Derived set, Dense subsets. Basis, sub

base, relative topology. (15 Lectures)

Unit-II: Continuous functions, open & closed functions, homeomorphism, Lindelof‘s, Separable

spaces, Connected Spaces, locally connectedness, Connectedness on the real line, Components,

Compact Spaces, one point compactification, compact sets, properties of Compactness and

Connectedness under a continuous functions, Compactness and finite intersection property,

Equivalence of Compactness. (15 Lectures)

Unit-III: Separation Axioms: T0 , T1, and T2 spaces, examples and basic properties, First and

Second Countable Spaces, Regular, normal, T3 & T4 spaces, Tychnoff spaces, Urysohn’s Lemma,

Tietze Extension Theorem, finite product topological spaces and some properties. (15 Lectures)


1. G.F.Simmons: Topology and Modern Analysis, McGraw Hill (1963)

2. W. J. Pervin, Foundations of General Topology

3. Willard, Topology, Academic press

4. Vicker , Topology via logic (School of Computing, Imperial College, London)

5. Topology, A First Course By: J. R. Munkers Prentice Hall of India Pvt. Ltd.

MTM 204 – Modeling and Simulation LTP: 3+0+0

Unit I: Introduction to modelling and simulation. Definition of System, classification of systems,

classification and limitations of mathematical models and its relation to simulation, Methodology of

model building Modelling through differential equation: linear growth and decay models, non linear

growth and decay models, Compartment models. (15 Lectures)

Unit II: Checking model validity, verification of models, Stability analysis, Basic model relevant

to population dynamics, Ecology, Environment Biology through ordinary differential equation,

Partial differential equation and Differential equations (15 Lectures)

Unit III: Basic concepts of simulation languages, overview of numerical methods used for

continuous simulation, Stochastic models, Monte Carlo methods. (15 Lectures)


1. D. N. P. Murthy, N. W. Page and E. Y. Rodin, Mathematical Modeling, Pergamon Press.

2. J. N. Kapoor, Mathematical Modeling, Wiley Estern Ltd.

3. P. Fishwick: Simulation Model Design and Execution, PHI, 1995, ISBN 0-13-098609-7

4. A. M. Law, W. D. Kelton: Simulation Modeling and Analysis, McGraw-Hill, 1991, ISBN

0-07-100803-9

5. J. A. Payne, Introduction to Simulation, Programming Techniques and Methods of

Analysis, Tata McGraw Hill Publishing Co. Ltd.

6. F. Charlton, Ordinary Differential and Differential equation, Van Nostarnd.

ED 202: Philosophy of Maths./Phy./Chem. LTP: 3+0+0

MTM 206 : Mathematical Tools and Software LTP: 3+0+0

Unit 1: MATLAB: Basic Introduction: Simple arithmetic calculations, Creating and working with

arrays, numbers and matrices, Creating and printing simple plots, Function files, Applications to

Ordinary differential equations: A first order ODE, A second order ODE, ode23, ode45, Basic 2-D

plots and 3-D plots. (15 Lectures)

Unit 2: Mathematica: Basic introduction: Arithmetic operations, functions, Graphics: 2-D plots, 3-

D plots, Plotting the graphs of different functions, Matrix operations, Finding roots of an equation,

Finding roots of a system of equations, Solving differential equations.

(15 Lectures)

Unit 3: LaTeX: Basic Introduction: Mathematical symbols and commands, Arrays, Formulas, and

Equations, Spacing, Borders and Colors, Using date and time option in LaTeX, To create

applications and Letters, PPT in LaTeX, Writing an article, Pictures and Graphics in LaTeX.

(15Lectures)

Reference books:

1. R. Pratap: Getting started with MATLAB, Oxford University Press, 2010.

2. S. Lynch, Dynamical Systems with Applications using MATLAB, Birkhäuser, 2014.

3. M. L. Abell, J.P. Braselton, Differential Equations with Mathematica, Elsevier Academic

Press, 2004.

4. I. P. Stavroulakis, S.A. Tersian, An Introduction with Mathematica and MAPLE, World

Scientific, 2004.

5. L.W. Lamport, LaTeX: A document Preparation Systems, Addison-Wesley Publishing

Company, 1994.

6. H. Kopka, P.W. Daly, Guide to LATEX, Fourth Edition, Addison Wesley, 2004

ED 202: Learner and Learning LTP: 3+0+0

Draft syllabus of third and fourth semester

III-SEMESTER

MTM 301: Functional Analysis LTP: 3+1+0

Unit-I. Inner product spaces, Normed linear spaces, Banach spaces, Quotient norm spaces,

continuous linear transformations, equivalent norms, the Hahn-Banach theorem and its

consequences. Conjugate space and separability, second conjugate space, Weak *topology on the

conjugate space (15 Lectures)

Unit-II. The natural embedding of the normed linear space in its second conjugate space, The open

mapping Theorem, The closed graph theorem, The conjugate of an operator, The uniform

boundedness principle, Definition and examples of a Hilbert space and simple properties,

orthogonal sets and complements (15 Lectures)

Unit -III. The projection theorem, separable Hilbert spaces. Bessel's inequality, the conjugate

space, Riesz's theorem, The adjoint of an operator, self adjoint operators, Normal and unitary

operators, Projections, Eigen values and eigenvectors of on operator on a Hilbert space, The

spectral theorem on a finite dimensional Hilbert space (15 Lectures)


1. G.F.Simmons: Topology and Modern Analysis, McGraw Hill (1963)

2. G.Bachman and Narici : Functional Analysis, Academic Press 1964

3. A.E.Taylor : Introduction to Functional analysis, John Wiley and sons (1958)

4. A.L.Brown and Page : Elements of Functional Analysis, Van-Nastrand Reinehold Com

5.B.V. Limaye: Functional Analysis, New age international.

6.Erwin Kreyszig, Introductory functional analysis with application, Willey.

MTM 302: Partial Differential Equation LTP: 3+1+0

UNIT-I: Formation of PDE, First order PDE in two independent variables, Derivation of PDE by

elimination method of arbitrary constants and arbitrary functions, Lagrange’s LPDE and Non Linear

PDE of first order. Charpit’s method, Monge’s method Jacobi’s method and Cauchy’s method.

(15 Lecture)

UNIT-II: PDE of second order with variable coefficients, Classification of second order PDEs, Cauchy

problem, Method of separation of variables, Canonical form, Elliptic, Qualitative behavior of solution to

Parabolic and Hyperbolic PDE, Eigen values and Eigen functions of BVP, Strum-Liouville boundary

value problem, Orthogonality of Eigen

function. (15 Lecture)

UNIT-III: Initial value problem and characteristics, Green,s function for IVP’s and BVP’s, Solution of

partial transform by Laplace, Solution of BVP in spherical and cylindrical coordinates, Variational

formulation of boundary value problem. (15 Lecture)


1. K, Sankara, Rao, Introduction to Partial Differential Equations, Phi Learning.

2. Ian N. Sneddon, Elements of Partial Differential Equations, Dover Publications.

3. Garrett Birkhoff and Gian-Carlo Rota, Ordinary Differential Equations.

MTM 303. Numerical Analysis LTP: 3+1+0

Unit I. Iterative solutions of nonlinear equation: bisection method. Fixed-point interation, Newton's

method, secant method, acceleration of convergence, Newton's method for two non linear equations,

polynomial equation methods, Polynomial interpolation: interpolation polynomial, divided

difference, interpolation, Aitken's formula, finite difference formulas, Hermite's interpolation,

double interpolation. (15 Lecture)

Unit II. Linear systems of Equations: Gauss Elimination, Gauss-Jordan method, LU decomposition,

iterative methods, and Gauss- Seidel iteration, Numerical Calculus : Numerical differentiation,

Errors in numerical differentiation, Numerical Integration, Trapezoidal rule, Simpson's 1/3 - rule,

Simpson's 3/8 rule, error estimates for Trapezoidal rule and Simpson's rule. (15 Lecture)

Unit III. Numerical Solution of Ordinary differential Equations : Solution by Taylor series, Picard

Method of successive approximations, Euler's Method, Modified Eular Method, Runge- Kutta

Methods, Predicator-Corrector Methods, Eigenvalue Problem : Power method, Jacobi method,

Householder method, Mupad Computer practicals. (15 Lecture)

Recommended Reading: 1. K.E. Atkinson: An Introduction to Numerical Analysis.

2. J. I. Buchaman and P. R. Turner: Numerical Methods and Analysis..

3. S. S. Sastry, Introduction Methods of Numerical Analysis (4th Edition) Prentice-Hall.

MTM 304. Differential Geometry LTP: 3+1+0

Unit-I : Differential Calculus, Tangent space. Vector fields, Cotangent space and differentials on

Charts and atlases. Differential manifolds, Induced topology on manifolds, functions and maps,

some special functions of class Para compact manifolds and partition of unity. Pullback

functions, local coordinates systems and partial derivatives. (15 Lecturers)

Unit-II : Tangent vectors and Tangent space, differential of a map, the tangent bundle, pullback

vector fields, Lie bracket, the cotangent space, the cotangent bundle, the dual of the differential

map. One parameter group and vector fields. (15 Lectures)

Unit-III : Lie derivatives, tensors, tensor fields, connections, parallel translation, covariant

differentiation of tensor fields, torsion tensor, curvature tensor, Bianchi and Ricci identities ,

Geodesics, Riemannian manifolds. (15 Lecturers)

Recommended books:

1. K. S. Amur, D. J. Shetty, C. S. Bagewadi, An introduction to differential geometry, Narosa

Publishing house, 2010.

2. B. O’Neill, Elementary differential geometry, Academic Press , New York, 1966

3. J. A. Thorpe, Elementary topics in differential geometry, Undergraduate text in Mathematics,

Springer Verlag , 1979.

4. T. J. Willmore, An introduction to differential geometry, Oxford University Press, 1965.

5. U. C. De, A. A. Shaikh, Differential geometry of Manifolds, Narosa Pub. House, 2009

6. D. Somasundaram, Differential Geometry: a first course, Narosa Pub. House, 2010.

ED-301: Teaching approaches and strategies LTP: 3+0+0

MTM-306 : EM-I LTP: 3+0+0

ED 302: Pedagogy of Maths./Phy./Chem.-I LTP: 3+1+0

IV SEMESTER

MTM 402 : Mathematical Programming LTP: 3+1+0

Unit I. Linear Programming, Theoretical foundation of Simplex Method, Revised Simplex Method,

Post Optimality Analysis, Computational Complexity of Simplex Algorithm, Karmarkar's

Algorithm for Linear programming, Linear Fractional programming problem. (15 Lectures)

Unit II. Multi-Objective Optimization Theory, Goal Programming, Network Models- Minimum

Spanning Tree Problem, Maximum flow problem, Project planning and Control with PERT-CPM,

Deterministic Inventory Control. (15 Lectures)

Unit III. Convex Programs and Duality, Quadratic programming and Complementarity Problem-

Wolfe's method, Beale's method, Fletcher's method. Nonlinear Programming Methods, Frank-Wolfe

Method, Gradient Projection Method, Penalty and Barrier Function Method. (15 Lectures)



2006.





MTM 402: Dynamics of Rigid Body LTP: 3+1+0

UNIT-I: Moments and products of inertia, moment of inertia of a body about a line through the origin,

Momental ellipsoid, rotation of co-ordinate axes, principal axes and principal moments. K.E. of rigid body

rotating about a fixed points, angular momentum of a rigid body, Eulerian angle, angular velocity, K.E. and

angular momentum in terms of Eulerian angle. Euler’s equations of motion for a rigid body, rotating about a

fixed point, torque free motion of a symmetrical rigid body (rotational motion of Earth). (15L)

UNIT-II: Classification of dynamical systems, Generalized co-ordinates systems, geometrical equations,

Lagrange’s equation for a simple system using D’Alembert principle, Deduction of equation of energy,

deduction of Euler’s dynamical equations from Lagrange’s equations, Hamilton’s equations, Ignorable co-

ordinates, Routhian Function. (15L)

UNIT-III: Hamiltonian’s principle for a conservative system, principle of least action, Hamilton-Jacobi

equation, Phase space and Liouville’s Theorem, Canonical transformation and its properties, Lagrange and

passion brackets, Poisson-Jacobi identity. (15L)


1. Vectorial Mechanics by E. A. Milne; Methuen & Co. Ltd. London, 1965.

2. Dynamics (Part II) by A.S. Ramsey; CBS Publishers & Distributors, Delhi, 1985.

3. A treatise on Analytical Dynamics by L.A. Pars; Heinemann, London, 1968.

4. Classical mechanics by H. Goldstein; arosa Publishing House, New Delhi, 1990.

5. Generalized Motion of Rigid Body by N. Kumar; Narosa Publishing House, New Delhi, 2004.

6. Classical Mechanics by K. Sankara Rao; PHI learning Private Ltd., New Delhi, 2009

MTM 403: EM-II LTP: 3+0+0

MTM 404: EM-III LTP: 3+0+0

ED 401: Learning Assessment LTP: 3+0+0

MTM 406: OE-II LTP: 3+0+0

ED 402: Pedagogy of Maths./Phy./Chem.-II LTP: 3+1+0

Semester-V

MTM 501: EM-IV LTP: 3+1+0

MTM 502: OE-I LTP: 3+1+0

ED 501: Class room organization and management LTP: 3+0+0

ED 502: Internship LTP: 12+0+0

Semester-VI

MTM 603: Major Project LTP: 20+0+0

ED: Project/Dissertation LTP: 4+0+0

List of Elective/Open Elective Papers

Sr.

No.

Title of Course Credits

1 Advanced Numerical Method 4

2 Advanced Real Analysis 4

3 Celestial Mechanics 4

4 Computational ODE 4

5 Computational PDE 4

6 Complex Dynamics 4

7 Dynamical Systems 4

8 Fluid Dynamics 4

9 Integral Equations and calculus of variations 4

10 Linear and nonlinear programing 4

11 Measure Theory and Integration 4

12 Advance Complex Analysis 3

13 Automata Theory and Formal Languages 3

14 Bio-Mathematics 3

15 Financial Mathematics 3

16 Fractional Calculus and Geometric Function Theory 3

17 Fuzzy Logic and its Applications 3

18 Game Theory 3

19 Graph Theory 3

20 Number Theory-I 3

21 Number Theory-II 3

22 Nonlinear Dynamics and its application to Information Technology 3

23 Operation Research 3

24 Special Functions 3

25 Module Theory 3

1. Advanced Numerical Method LTP: 3+1+0

UNIT-I: Numerical solution of algebraic and transcendental equations: Introduction- iteration

method, Newton-Raphson method, Graeffe’s root square method, acceleration of convergence.

Numerical Solution of systems of nonlinear equations: iteration method, Newton-Raphson method.

Linear Systems of equations: Introduction- Gauss elimination method, LU decomposition, Solution

of tridiagonal system, Ill-conditioned linear systems and method for Ill-conditioned matrix. Eigen

Value problem: Power method, Jacobi Method, Householder method.

(15-Lectures)

UNIT-II: Polynomial Interpolation: introduction- finite difference formulas, divided difference

interpolation, Aitken’s formula, Hermite’s interpolation, double interpolation, Spline interpolation

(linear, quadratic and cubic spline), Error in cubic Spline. Numerical differentiation, Errors in

numerical differentiation, cubic spline method; Numerical Integration: introduction to trapezoidal,

Simpson’s rules and error estimates, use of cubic splines, numerical double integration.

(15-Lectures)

UNIT-III: Boundary value problem: Introduction, BVP governed by second order ordinary

differential equations, Finite difference method, shooting method, cubic splines method. IVP and

BVP in partial differential equations: classification of linear second order partial differential

equations, Finite difference methods for Laplace and Poisson equations - Jacobi method, Gauss-

Seidel method and ADI (alternating direction implicit) method , Finite difference method for heat

conduction equation - Bender- Schmidt recurrence relation, Crank-Nicolson formula, and Jacobi

Iteration formula, Finite difference method for wave equation.

(15-Lectures)


1. K. E. Atkinson: An Introduction to Numerical Analysis.

2. J. I. Buchaman and P. R. Turner: Numerical Methods and Analysis.

3. S. S. Sastry: Introductory Methods of Numerical Analysis.

4. S. R. K. Iyengar and P. K. Jain: Numerical Methods.

2. Advanced Real Analysis LTP: 3+1+0

Unit I: Metric spaces revisited; Baire Category theorem, completion of Metric spaces, Banach

contraction principle and some of its applications. Compactness, Total boundedness,

characterization of compactness for arbitrary Metric spaces; Arzella-Ascoli theorem, Stone

Weierstrass theorem.

Unit II: Integrations : Lebesgue’s criterion of Riemann integrability over a bounded closed interval

[a, b] and its consequence, length of a rectifiable curve in a plane, Riemann-Stieltjes integral over

[a, b] and its properties, Integrators of bounded variation, Integration by parts, Stieltjes integral as a

Riemann integral, Step function as integrator, Riesz theorem.

Unit III: Cesaro’s Method of Summability and Fourier Series: Cesaro’s method of summability of

order 1 and order 2, Some specific examples, Regularity of Cesaro’s method, Definition of Fourier

series and some examples, Dirichlet’s Kernel, Fejer’s Kernel, Fejer’s theorem, Dini’s and Jordan’s

tests for point wise convergence of Fourier series.

Recommended reading:

[1] A. M. Bruckner, J. Bruckner & B. Thomson : Real Analysis, Prentice-Hall, N.Y. 1997.

[2] R. R. Goldberg : Methods of Real Analysis, Oxford-IBH, New Delhi, 1970.

[3] I. P. Natanson : Theory of Functions of a Real Variable, Vol-I, F.Ungar, N.Y. 1955.

[4] E. Hewitt and K. Stromberg : Real and Abstract Analysis, John-Willey, N.Y. 1965.

[5] J. F. Randolph : Basic Real and Abstract Analysis. Academic Press, N.Y. 1968.

[6] P. K. Jain and K. Ahmad : Metric Spaces, Narosa Publishing House.

[7] G. Tolstov : Fourier Series, Dover Publication, N.Y. 1962.

3. Celestial Mechanics LTP: 3+1+0

UNIT-I: Introduction, Kepler’s Laws of Planetary Motion, Newton’s law of gravitation, Central

force motion, Integral of energy, Differential equation of orbit, Inverse square force, Geometry of

orbits, Two body problem, Motion of center of mass, Relative motion, Earth bound satellite circular

orbit, Classical orbital elements, Position in elliptic orbit, Position in parabolic orbit and Position in

hyperbolic orbit. (15 Lectures)

UNIT-II: N-body problem, Mathematical formulation of N-body problem, Integrals of motion,

The Virial theorem, Equation of relative motion, Three body problem, Stationary solution of three

body problem, Restricted three body problem- formulation and its solution, Restricted three body

problem, Stability of motion near Lagrangian points. (15 Lectures)

UNIT-III: Theory of perturbations, Variation of parameter, Properties of Lagrange’s brackets,

Evaluation of Lagrange’s brackets, Solution of the perturbation equations, Perturbation function,

Earth-Moon system, Potential due to an oblate spheroid, Perturbations due to oblate planet,

Perturbation due atmospheric drag, Perturbation due to solar radiation. (15 Lectures)


1. Introduction to Celestial Mechanics by S. W. McCuskey, Addison-Wesley Publishing

Company, 1963.

2. Solar System Dynamics by C. D. Murray and S. F. Dermott, Cambridge University

Press, 2000.

3. An Introduction to Celestial Mechanics by F. R. Moulton, the MacMillan Company,

1914.

4. Theory of orbits. The Restricted problem of three bodies by V. Szebehely, New York

Academic Press, 1967.

5. Classical Mechanics by K. Sankara Rao, PHI Learning Pvt. Ltd., 2009

4. Computational ODE LTP: 3+1+0

Unit-I

Numerical solutions of system of simultaneous first order differential equations and second order initial

value problems (IVP) by Euler and Runge-Kutta (IV order) explicit methods, Numerical solutions of second

order boundary value problems (BVP) of first, second and third types by shooting method.

Unit-II

Types Finite difference schemes of second order BVP based on difference operators (solutions of tri-

diagonal system of equations), Solutions of such BVP by Newton-Cotes and Gaussian integration rules,

Convergence and stability of finite difference schemes.

Unit-III

Variational principle, approximate solutions of second order BVP of first kind by Reyleigh-Ritz,

Galerkin, Collocation and finite difference methods, Finite Element methods for BVP-line segment,

triangular and rectangular elements, Ritz and Galerkin approximation over an element, assembly of

element equations and imposition of boundary conditions.

References:

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

Computations, New Age Publications, 2003.

2. M. K. Jain, Numerical Solution of Differential Equations, 2nd

edition, Wiley-Eastern.

3. S. S. Sastry, Introductory Methods of Numerical Analysis,

4. D.V. Griffiths and I. M. Smith, Numerical Methods for Engineers, Oxford University Press, 1993.

5. C. F. Gerald and P. O. Wheatley, Applied Numerical Analysis, Addison- Wesley, 1998.

6. A. S. Gupta, Text Book on Calculas of Variation, Prentice-Hall of India, 2002.

7. Naveen Kumar, An Elementary Course on Variational Problems in Calculus, Narosa, 2004.

5. Computational PDE LTP: 3+1+0

Unit-I

Numerical solutions of parabolic equations of second order in one space variable with constant

coefficients:- two and three levels explicit and implicit difference schemes, truncation errors and

stability, Difference schemes for diffusion convection equation, Numerical solution of parabolic

equations of second order in two space variable with constant coefficients-improved explicit schemes,

Implicit methods, alternating direction implicit (ADI) methods.

Unit-II

Numerical solution of hyperbolic equations of second order in one and two space variables with

constant and variable coefficients-explicit and implicit methods, alternating direction implicit (ADI)

methods.

Unit-III

Numerical solutions of elliptic equations, Solutions of Dirichlet, Neumann and mixed type problems

with Laplace and Poisson equations in rectangular, circular and triangular regions, Finite element

methods for Laplace, Poisson, heat flow and wave equations.

References:

1. M. K. Jain, S. R. K. Iyenger and R. K. Jain, Computational Methods for Partial Differential

Equations, Wiley Eastern, 1994.

2. M. K. Jain, Numerical Solution of Differential Equations, 2nd

edition, Wiley Eastern.

3. S. S. Sastry, Introductory Methods of Numerical Analysis, , Prentice-Hall of India, 2002.

4. D. V. Griffiths and I. M. Smith, Numerical Methods of Engineers, Oxford University Press, 1993.

5. C. F. General and P. O. Wheatley, Applied Numerical Analysis, Addison- Wesley, 1998.

6. K. J. Bathe and E. L. Wilson, Numerical Methods in Finite Element Analysis, Prentice-HalI,

1987.

7. A. S. Gupta, Text Book on Calculas of Variation, Prentice-Hall of India, 2002.

8. Naveen Kumar, An Elementary Course on Variational Problems in Calculus, Narosa, 2004.

6. Complex Dynamics LTP: 3+1+0

UNIT –I: Iteration of a Mobius transformation, attracting, repelling and indifferent fixed points.

Iterations of R(z) = z2, z2+c, z + . The extended complex plane, chordal metric, spherical metric,

rational maps, Lipschitz condition, conjugacy classes of rational maps, valency of a function, fixed

points, Critical points, Riemann Hurwitz relation. (15 Lectures)

UNIT –II: Equicontinuous functions, normality sets , Fatou sets and Julia sets, completely invariant

sets, Normal families and equicontinuity, Properties of Julia sets, exceptional points Backward

orbit, minimal property of Julia sets. (15 lectures)

UNIT -III : Julia sets of commuting rational functions, structure of Fatou set, Topology of the

Sphere, Completely invariant components of the Fatou set , The Euler characteristic, Riemann

Hurwitz formula for covering maps, maps between components of the Fatou sets, the number of

components of Fatou sets, components of Julia sets. (15 lectures)

Recommended Books:

1. A. F. Beardon, Iteration of rational functions, Springer Verlag , New York, 1991.

2. L. Carleson and T . W. Gamelin, Complex dynamics, Springer Verlag, 1993.

3. S. Morosawa, Y. Nishimura, M. Taniguchi, T. Ueda, Holomorphic dynamics,

Cambridge University Press, 2000.

4. X. H. Hua, C. C. Yang, Dynamics of transcendental functions, Gordan and Breach Science

Pub. 1998.

.

7. Dynamical System LTP: 3+1+0

Unit-I Linear Systems: Exponentials of operators, Linear systems in R2, Complex eigenvalues,

Multiple eigenvalues, Jordon forms, Stability theory, generalized eigenvectors and invariant

subspaces, Non-homogeneous linear systems.

(15 lectures)

Unit-II: Non-linear Systems: local analysis: the fundamental existence-uniqueness theorem, The

flow defined by a differential equation, Linearization, The stable manifold theorem, The Hartman-

Grobman theorem, Stability and Liapunov functions, Saddles, Nodes, Foci, and Centers.

(15 lectures)

Unit-III: Non-linear Systems: global analysis: Dynamical systems and global existence theorem,

Limit sets and Attractors, Periodic orbits, Limit Cycles, and Seperatrix cycles, the Poincare map,

the stable manifold theorem for periodic orbits, the Poincare-Bendixon theory in R2, Lineard

Systems, Bendixon’s Criteria.

(15 lectures)

Recommended reading:

1. Differential Equations and Dynamical Systems by Lawrence Perko, Springer-Verlag, 2006.

2. Differential Equations, Dynamical Systems and an Introduction to Chaos by Morris W.

Hirsch, Stephen Smale and Robert L. Devaney, Academic Press, 2013

3. Dynamical Systems and Numerical Analysis by A.M. Stuart and A.R. Humphries,

Cambridge University Press, 1998.

4. Dynamical Systems with Applications using MATLAB by S. Lynch, Birkhause press, 2004.

5. Nonlinear Dynamics and Chaos with Applications to Physics, Biology, Chemistry and

Engineering, by Steven H. Strogatz, Westview Press.

8. Fluid Dynamics LTP: 3+1+0

Unit I. Physical Properties of fluids. Concept of fluids, Continuum Hypothesis, density, specific

weight, specific volume, Kinematics of Fluids : Eulerian and Lagrangian methods of description of

fluids, Equivalence of Eulerian and Lagrangian method, General motion of fluid element,

integrability and compatibility conditions, strain rate tensor, stream line, path line, streak lines,

stream function¸ vortex lines, circulation,

Unit II. Stresses in Fluids : Stress tensor, symmetry of stress tensor, transformation of stress

components from one co-ordinate system to another, principle axes and principle values of stress

tensor Conservation Laws : Equation of conservation of mass, equation ofconservation of

momentum, Navier Stokes equation, equation of moments of momentum, Equation of energy, Basic

equations in different co-ordinate systems, boundary conditions.

Unit III. Irrotational and Rotational Flows : Bernoulli’s equation, Bernoulli’s equation for

irrotational flows, Two dimensional irrotational incompressible flows, Blasius theorm, Circle

theorem, sources and sinks, sources sinks and doublets in two dimensional flows, methods of

images.


1. An introduction to fluid dynamics, R.K. Rathy, Oxford and IBH Publishing Co. 1976.

2. Theoretical Hydrodynamics, L. N. Milne Thomson, Macmillan and Co. Ltd.

3. Textbook of fluid dynamics, F. Chorlton, CBS Publishers, Delhi.

4. Fluid Mechanics, L. D. Landau and E.N. Lipschitz, Pergamon Press, London, 1985.

9. Integral equation and Calculus of variation LTP: 3+1+0

Unit-I: The variation of a functional and its properties, Euler’s equations and application,

Geodesics, Variational problems for functional involving several dependent variables, Hamilton

principle, Variational problems with moving (or free) boundaries, Approximate solution of

boundary value problem by Rayleigh-Ritz method.

(15 Lectures)

Unit-II: Linear integral equation and classification of conditions, Volterra integral equation,

Relationship between linear differential equation and Volterra integral equation, Resolvent kernel of

Volterra integral equation, solution of integral equation by Resolvent kernel, The method of

successive approximations, Convolution type equation. (15

Lectures)

Unit-III Fredholm integral equation, Fredholm equation of the second kind, Fundamentals-iterated

kernels, constructing the resolvent kernel with the aid of iterated kernels, Integral equation with

degenerated kernels, solutions of homogeneous integral equation with degenerated kernel.

(15 Lectures)


1. Applied Mathematics for Engineers and Physicists by L. A. Pipe (McGraw Hill)

2. Introduction to Mathematical Physics by Charlie Harper, P.H.I. , New Delhi

3. Higher Engineering Mathematics by B.S. Grewal, Khanna Publications, Delhi

4. Mathematical Methods for Physicists by George Arfken (Academic Press)

5. Mathematical Methods by Potter and Goldberg (Prentice Hall of India)

6. Calculus of Variations by IM Gelfand, SV Fomin, and Richard A Silverman

7. Introduction to the Calculus of Variations by Bernard Dacorogna, World Scientific

8. Calculus of Variations with Applications to Physics and Engineering by Robert Weinstock,

Dover Publications.

9. Linear Integral equations By R. P. Kanwal, Academic Press, New- York.

10. Linear and Nonlinear programing LTP:3+1+0

Unit I. Linear Programming, Theoretical foundation of Simplex Method: Proof of Theorems,

Revised Simplex Method, Duality Theorems and Post Optimality Analysis, Computational

Complexity of Simplex Algorithm, Karmarkar's Algorithm for Linear programming.

(15 Lectures)

Unit II. Degeneracy in Transportation Problem, Unbalanced Transportation Problem, Duality in

Assignment problems, Integer Linear programming: Gomory's Cutting Plane Method, Multi-

Objective Optimization Theory, Goal Programming, Computer Programming for Simplex Method,

Dual simplex Method, Branch & Bound Method and Hungarian Method.

(15 Lectures)

Unit III. Convex Optimization Problems and Duality, Quadratic programming and

Complementarity Problem-Wolfe's method, Beale's method, Fletcher's method. Nonlinear

Programming Methods, Frank-Wolfe Method, Gradient Projection Method, Penalty and Barrier

Function Method.

(15 Lectures)



2006.





11. MEASURE THEORY AND INTEGRATIONS LTP:3+1+0

UNIT-I: Countable and uncountable sets, cardinality and cardinal arithmetic, Schr der–Bernstein

theorem, the Canter’s ternary set, semi-algebras, algebras, monotone class,

algebras, measure and outer measures, Carathe dory extension process of extending a measure

on a semi-algebra to generated algebras, Borel sets (10 Lectures)

Unit-II: Lebesgue outer measure and Lebesgue measure on R, translation invariance of Lebesgue

measure, existence of a non-measurable set, characterizations of Lebesgue measurable sets, the

Cantor-Lebesgue function, measurable functions on a measure space and their properties, Borel and

Lebesgue measurable functions, Simple functions and their integrals, Littlewood’s three principle

(statement only) (10 Lectures)

UNIT-III: Lebesgue integral on R and its properties, bounded convergence theorem, Fatou’s

lemma, Lebesgue monotone convergence theorem, Lebesgue dominated convergence theorem,

L_p-spaces, Holder-Minkowski inequalities, parseval’s identity, Riesz Fisher’s theorem. (10

Lectures)

Books Recommended:

1. H. L. Royden amd P. M. Fitzpatrick, Real Analysis (Fourth edition), PHI 2010.

2. P. R. Halmos, Measure Theory, Springer, 1994.

3. E. Hewit and K. Stromberg, Real and Abstract Analysis, Springer, 1975.

4. K. R. Parthasarathy, Introduction to Probability and Measure, Hindustan Book Agency,

2005.

5. I. K. Rana, An Introduction to Measure and Integration (2nd Edition) Narosa Publishing

House, 2005.

12. Advanced Complex Analysis LTP: 3+0+0

Unit-I. Analytic Continuation, Analytic Continuation along Paths via Power Series, Monodromy Theorem,

Picard theorem, Poisson integral, Mean value theorem, Schwarz reflection principle, Analytic continuation

via reflexion. (15 Lectures)

Unit-II. Infinite sums and infinite product of complex numbers, Infinite product of analytic functions,

Factorization of entire functions, The Gamma functions, The Zeta functions.

(15 Lectures)

Unit-III. The Riemann mapping theorem (Statement only), Area Theorem, Biberbach Theorem and

conjecture, Distortion theorem, Koebe ¼ theorem, Starlike and convex functions. Coefficient estimates and

distortion theorem. (15 Lectures)

Recommended Books:

1. S. Ponnusamy, Foundation of Complex Analysis, 2nd edition, Narosa Publishing House.

2. L. R. Ahlofrs, Complex Analysis, McGraw Hill

13. Automata Theory and Formal Languages LTP:3+0+0

Unit-I: Theory of Computation: Finite automata, Deterministic and non-deterministic finite

automata, equivalence of deterministic and non-deterministic automata, Moore and Mealy

machines, Regular expressions, Grammars and Languages, Derivations, Language generated by a

grammar. (15L)

Unit-II: Regular Language and regular grammar, Regular and Context free grammar, Context

sensitive grammars and Languages, Pumping Lemma, Kleene’s theorem. (15L)

Unit-III: Turing Machines: Basic definitions, Turing machines as language acceptors, Universal

Turing machines, decidability, undecidability, Turing Machine halting problem. (15L)


1) D. Kelly, Automata and Formal Languages: An Introduction, Prentice-Hall, 1995.

2) J. E. Hopcroft, R. Motwani, and J. D. Ullman, Introduction to Automata, Languages, and

Computation

(2nd edition), Pearson Edition, 2001.

3) P. Linz, An Introduction to Formal Languages and Automata, 3rd Edition.

14. Bio- Mathematics LTP: 3+0+0

Unit 1: Introduction: Goals and Challenges of mathematical modeling in biology and ecology.

Idealization and general principle of model building, deterministic and stochastic models, different

types of mathematical models and differential and difference equations as relevant mathematical

techniques, complex network dynamics, biological and ecological examples.

(15 Lectures)

Unit 2: Continuous growth models for single species: The linear model, Logistic population model,

Stability of equilibrium states and bifurcation analysis, Constant Harvesting and bifurcations, Delay

models, Linear analysis of delay models: periodic solutions.

(15 Lectures)

Unit 3: Discrete population models for single species: Simple models, Discrete logistic-type model:

Chaos, stability, periodic solutions and bifurcations, discrete delay models. Models with interacting

populations: predator-prey interactions, Analysis of a predator-prey model with limit cycle periodic

behaviour, Harvesting in two species models.

(15 Lectures)

Reference books:

1. J.D. Murray, Mathematical biology: An introduction, Springer, 2007.

2. F. Brauer, C.C-Chavez, Mathematical Models in Population Biology and Epidemiology,

Springer, 2000.

3. N.F. Britton, Essential mathematical biology, Springer, 2004.

4. M. Kot, Elements of mathematical ecology, Cambridge University Press, 2001.

5. A. Okubo, S.A. Levin, Diffusion and ecological problems, Springer, 2002.

6. S.V. Petrovskii, B.L. Li, Exactly solvable models of biological invasions, CRC Press/

Chapman and Hall, 2005.

7. R.W. Sterner, J.J. Elser, Ecological stoichiometry: the biology of elements from molecules

to the biosphere, Princeton University Press, 2002.

8. H. Smith, An Introduction to Delay Differential Equations with Applications to Life

Sciences, Springer, 2010

15. Financial Mathematics LTP: 3+0+0

Unit I. Introduction to options and markets: types of options, interest rates and present values,

Black Sholes model : arbitrage, option values, pay offs and strategies, putcall parity, Black Scholes

equation, similarity solution and exact formulae for European options, American option, call and

put options, free boundary problem. (15L)

Unit II. Binomial methods: option valuation, dividend paying stock, general formulation and

implementation, Monte Carlo simulation : valuation by simulation, Lab component: implementation

of the option pricing algorithms and evaluations for Indian companies. (15L)

Unit III. Finite difference methods: explicit and implicit methods with stability and conversions

analysis methods for American options- constrained matrix problem, projected SOR, time stepping

algorithms with convergence and numerical examples. (15L)


1. D. G. Luenberger, Investment Science, Oxford University Press, 1998. 2. J. C. Hull , Options, Futures and Other Derivatives, 4th ed., Prentice- Hall ,New York, 2000.

3. J. C. Cox and M. Rubinstein, Option Market, Englewood Cliffs, N. J.: Prentice-Hall, 1985.

4. C.P. Jones. Investments, Analysis and Measurement, 5th ed., John Wiley and Sons, 1996.

16. Fractional Calculus and Geometric Function Theory LTP: 3+0+0

Unit I: Fractional derivatives and Integrals, application of fractional calculus, Laplace transforms of

fractional integrals and fractional derivatives, fractional ordinary differential equations, fractional

integral equations, Initial value problem of fractional differential equations. (15 Lectures)

Unit II: Univalence in Complex plane. Area theorem. Growth, covering and distortion results.

Starlike and Convex functions. Starlike and Convex functions of order α. Alpha convexity. Close

to convexity, spirallikeness and Φ-likeness in unit disk. (15 Lectures)

Unit III: Subordination. Application of subordination principle. First and second order differential

subordination. Briot-Bouquet differential subordinations. Briot-Bouquet application in Univalent

function theory. (15 Lectures)


1. Loknath Debnath and Dambaru Bhatta, Intgral Transforms and Special Functions, CRC

press, 2010.

2. I. Graham and G. Kohr, Geometric function theory in one and higher dimensions, Marcel

Dekker, 2003.

3. S. S. Miller and P. T. Mocanu, Differential Subordinations theory and Applications, Marcel

Dekker, 2000.

17. Fuzzy Logic and its Applications LTP:3+0+0

18. GAME THEORY LTP: 3+0+0

UNIT-I

A General Introduction to Game Theory-its Origin, Representation of Games, Types of Game,

Static Games with Complete and Incomplete Information, Strategic Form Game with Illustrations,

Solution Concept- Pure and Mixed Strategies, Dominance and Best Response, Pareto Optimality,

Maxmin and Minmax Strategies, Pure and Mixed Strategies Nash Equilibrium, Correlated

Equilibrium, Bayesian Games, Market Equilibrium and Pricing: Cournot and Bertrand Game.

(15 Lectures)

UNIT-II

Existence and Properties of Nash Equilibrium, Two-person Zero-Sum Games-its Solution; Dynamic

Games of Perfect Information, Extensive Form Game, Nash Equilibrium, Sub-game Perfection,

Backward Induction (looking forward), Stackelberg Model of Duopoly.

(15 Lectures)

UNIT-III

Bargaining Problem, Dynamic Games with Imperfect Information, Finitely and Infinitely Repeated

Games, The Folk Theorem, Illustrations, Stochastic Games, Coalition Games, Core and Shapley

Value, Illustrations.

(15 Lectures)

Text Books:

1. M.J. Osborne, An Introduction in Game Theory, Indian Ed.

2. M. J. Osborne and A. Rubinstein, A course in Game Theory, MIT Press, 1994

3. D. Fudenberg and J. Tirole, Game Theory, MIT Press, 1991.

Reference Books:

1. J. Von Neumann and O. Morgenstern, Theory of Games and Economic Behaviour, New

York: John Wiley and Sons., 1944.

2. R.D. Luce and H. Raiffa, Games and Decisions, New York: John Wiley and Sons.,1957.

3. G. Owen, Game Theory, (Second Edition), New York: Academic Press, 1982.

19. Graph Theory LTP: 3+0+0

Unit I: Graphs and simple graphs, Vertex Degrees, Subgraphs, Graph Isomorphism, The

Incidence and Adjacency Matrices, Paths and Circuits, Trees, Cut Edges and Cut Vertices, Euler

and Hemilton circuits, Bipartite and Complete graphs. Spanning trees, Minimal spanning trees,

Kruskal’s Algorithm, Directed graphs, Weighted undirected graphs, Dijkstra’s algorithm,

Warshal’s Algorithm.

Unit II: Connectivity: Connectivity of graphs, Cut-sets, Edge Connectivity and Vertex

Connectivity, Planarity: Planar Graphs, Testing of Planarity, Euler’s formula for connected planar

graphs, Kuratowski Theorem for Planar graphs, Random Graphs.

Unit III: Coloring of graphs: Chromatic number and chromatic polynomial of graphs, Brook’s

Theorem, Five Color Theorem and Four Color Theorem.


1) F. Harary, Graph Theory, Narosa Publ.

2) R. Diestel, Graph Theory, Springer, 2000.

3) Narsingh Deo, Graph Theory with Applications to Engineering and Computer Science, Prentice-

Hall of India.

4) W. T. Tutte, Graph Theory, Cambridge University Press, 2001.

5) Kenneth H. Rosen, Discrete Mathematics and its Applications, 6th ed., Tata McGraw-Hill.

20. Module Theory LTP: 3+0+0

Unit I: Modules over a ring, submodules, Quotient Modules, module homomorphism and

isomorphism theorems for modules, cyclic modules, simple modules and semisimple modules and

rings, Schur’s lemma.

Unit II: Exact sequences, Products, Coproducts and their universal property, External and internal

direct sums, Free modules, Left exactness of Hom sequences and counter-examples for non-right

exactness.

Unit III: Noetherian and Artinian modules and rings. Hilbert basis theorem, Projective and

injective modules, Divisible groups, Example of injective modules.


1. I. N. Herstein, Topics in Algebra, Wiley Eastern, 1975.

2. P. B. Bhattacharya, S. K. Jain and S. R. Nagpaul, Basic Abstract Algebra (2nd Edition),

Cambridge

University Press, 1997.

3. D. S. Malik, J. N. Mordeson, and M. K. Sen, Fundamentals of Abstract Algebra, McGraw-Hill

International Edition, 1997.

4. D.S. Dummit and R.M. Foote, Abstract Algebra, John Wiley, 2003.

5. J.S. Golan, Modules & the Structures of Rings, Marcl Dekkar. Inc.

21. Number Theory-I LTP: 3+0+0

Unit I: Primes, Divisibility, Greatest common divisor, Euclidean algorithm, Fundamental theorem

of arithmetic, Perfect numbers, Mersenne primes and Fermat numbers, Farey sequences.

Unit II: Congruence and modular arithmetic, Residue classes and reduced residue classes, Chinese

remainder theorem, Fermat's little theorem, Wilson's theorem, Euler's theorem and its application to

cryptography, Arithmetic functions , Möbius inversion formula, Greatest

integer function.

Unit III: Primitive roots and indices, quadratic residues, Legendre symbol, Euler's criterion,

Gauss's lemma, Quadratic reciprocity law, Jacobi symbol, Representation of an integer as a sum of

two and four squares, Diophantine equations ax+by=c, x2+y2=z2, x4+y4=z4. Binary quadratic forms

and Equivalence of quadratic forms.


1) David M. Burton, Elementary Number Theory, Wm. C. Brown Publishers, Dubuque,

Iowa 1989.

2) G.A. Jones and J.M. Jones, Elementary Number Theory, Springer-Verlag, 1998.

3) W. Sierpinski, Elementary Theory of Numbers, North-Holland, Ireland, 1988.

4) Niven, S.H. Zuckerman and L.H. Montgomery, An Introduction to the Theory of

Numbers, John Wiley, 1991.

5) Joseph H. Silverman, A Friendly Introduction to Number Theory, 4th ed., Pearson.

6) Thomas Koshy, Elementary Number Theory with Applications, 2nd ed., Academic Press.

22. Number Theory – II LTP: 3+0+0

Unit-I: Continued fractions, Approximation of real numbers by rational numbers, Pell's equations,

Partitions, Ferrers graphs, Jacobi's triple product identity. (15L)

Unit-II: Congruence properties of p(n), Rogers-Ramanujan identities, Minkowski's theorem in

geometry of numbers and its applications to Diophantine inequalities, Order of magnitude and

average order of arithmetic functions. (15L)

Unit-III: Euler's summation formula, Abel's identity, Elementary results on distribution of primes,

Characters of finite Abelian groups, Dirichlet's theorem on primes in arithmetical progression.

(15L)


1. Thomas Koshy, Elementary Number Theory with Applications, 2nd ed., Academic Press.

2. Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976.

23. Non linear Dynamics and its applications to Information Technology LTP: 3+0+0

Unit I: Introduction to nonlinear Dynamics. Non linear Maps. Bass Model for technology Diffusion

(in both Differential and Difference form). (15 Lectures)

Unit II: Existence of chaos in Logistic and Base Models in Discrete forms. Effects of variable

externals and internal influence in Bass Model (in both continuous and discrete forms, conditions

for chaos). (15 Lectures)

Unit III: Modelling of two or more competing Technologies and their coexistence, Application to

markets. Models for Virus in Communication and Computer Networks. Network Crimes and

Control. (15 Lectures)


1. P. Glendenning, Stability, Instability and Chaos, Cambridge University Press (1994).

2. M. Laxshmanan and S. Rajsekher, Nonlinear Dynamics, Springer-Verlag, Heidelberg (2003)

24. Operations Research LTP: 3+0+0

Unit I: Nonlinear Programming: Unconstrained algorithms; direct search method, gradient method .

Constrained methods; Separable programming, quadratic programming.

General Inventory models, role of demand in the development of inventory; Static Economic-

Order-Quantity (EOQ) models; Dynamic EOQ models. Continuous review model, single period

models, multi period models. (15 Lecture)

Unit-II: Elements of Queuing models, role of exponential, pure birth and death models.

Generalized Poisson Queuing models, Specialized Poisson Queues: Steady state measures of

performance, single server model multi server models, machine servicing models-(M/M/R):

(GD/K/K), R< K.

Replacement and maintenance models; gradual failure, sudden failure, replacement due to

efficiency deteriorate with time, staffing problems, equipment renewal problems. (15 Lecture)

Unit-III: Project Scheduling by PERT-CPM

Simulation modelling: Monte Carle Simulation, Types of simulations, Elements of discrete-events

simulation, generation of random numbers. Mechanics of discrete simulation, Methods of gathering

statistical observations: subinterval method, replication method, regeneration method.

Sequencing Problems: notions, terminology, and assumptions, processing n jobs through m

machines. (15 Lecture)

Recommended Books:

1. Operations Research an Introduction –Hamady A. Taha, Prentice Hall.

2. Operations Research Theory and Applications-J.K.Sharma, Macmillan Publishers.

3. Non linear Programming –S.D. Sharma, Kedar Nath Ram Nath & Co.

4. Mathematical Programming Theory and Methods-S.M.Sinha.

5. Operations Research, - Kanti Swarup, P. K. Gupta and Man Mohan, Sultan Chand &

Sons

25. SPECIAL FUNCTIONS LTP:3+0+0

UNIT-I: Beta and Gamma Functions, Euler Reflection Formula, Stirling’s Asymptotic Formula,

Gauss’s Multiplication Formula, Ratio of two gamma functions, Integral Representations for

Logarithm of Gamma function and Beta functions.

(10 Lectures)

UNIT-II. Hypergeometric Differential Equations, Gauss Hypergeometric Function, Elementary

Properties, Conditions of convergence, Integral Representation, Gauss Theorem, Vandermonde’s

theorem, Kummer’s theorem, Linear transformation, Generalized Hypergeometric Functions,

Elementary Properties, Integral Representation.

(10 Lectures)

UNIT-III: Legendre polynomials and functions, Solution of Legendre’s differential equations,

Generating Functions, Rodrigue’s Formula, Orthogonality of Legendre polynomials, Recurrence

relations.

Bessel functions, Bessel differential equation and it’s solution, Recurrence relation, Generating

functions, Integral representation. (10 Lectures)

Recommended Books:

1. G. E. Andrews, R. Askey, Ranjan Roy, Special Functions, Encyclopedia of Mathematics

and its Applications, Cambridge University Press, 1999.

2. E. D. Rainville, Special Functions, Macmillan, New York, 1960.

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