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CENTRAL UNIVERSITY OF RAJASTHAN Scheme of Integrated M. Sc. Mathematics (5 years) programme for semester 7th, 8th, 9th and
10th (to be start from session 2016-17)
Int. M.Sc. Mathematics is a five academic year spread in ten semesters. The details of the
courses of 7th, 8th, 9th, and 10th with code, title and the credits assign are as given below.
7th 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. MT 701 Abstract Algebra-I 4 3 1 0 3 0 20 0 30 50 0
2. MT 702 Probability and
Probability
Distribution
4 3 1 0 3 0 20 0 30 50 0
3. MT 703 Programming in C 4 3 1 0 3 0 20 0 30 50 0
4. MT 704 Qualitative Theory of
Ordinary Differential
Equations
4 3 1 0 3 0 20 0 30 50 0
5. MT 705 Complex Analysis- I
3 2 1 0 3 0 20 0 30 50 0
6. MT 706 EM01 3 3 0 0 3 0 20 0 30 50 0
7. MT 707 Practical related to
MT 703
2 0 0 4 0 4 50 0 0 0 50
TOTAL 2
4
1
7
5 4
8th Semester
S.
No
.
Subject
Code
Course Title Credits 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. MT 801 Linear Algebra -I
4 3 1 0 3 0 20 30 50 50 0
2. MT 802 Dynamics of Rigid
Body
4 3 1 0 3 0 20 30 50 50 0
3. MT 803 Topology 4 3 1 0 3 0 20 30 50 50 0
4. MT 804 Modelling and
Simulations
3 3 0 0 3 0 20 30 50 50 0
5. MT 805 EM02 3 3 0 0 3 0 20 30 50 50 0
6. MT 806 EM03 3 3 0 0 3 0 20 30 50 50 0
7. MT 807 Mathematical Tools
and Software
3 2 1 0 2 0 20 30 50 50 0
TOTAL 24 2
0
4 0
9th 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. MT 901 Functional Analysis 4 3 1 0 3 0 20 0 30 50 0
2. MT 902 Partial Differential
Equations
4 3 1 0 3 0 20 0 30 50 0
3. MT 903 Differential
Geometry
4 3 1 0 3 0 20 0 30 50 0
4 MT 904 EM04 4 3 1 0 3 0 20 0 30 50 0
5. MT 905 EM05 4 3 1 0 3 0 20 0 30 50 0
6. MT 906 EM06 4 3 1 0 3 0 20 0 30 50 0
TOTAL 24 18 6 0
10th 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. MT
1001
EM07 4 3 1 0 3 0 20 0 30 50 0
2. MT
1002
EM08 4 3 1 0 3 0 20 0 30 50 0
3. MT
1003
Major Project 16 0 0 16 0 16 50 0 0 50 50
Total 24 6 2 16 6 16
Detailed Syllabus for
Int. M. Sc. Mathematics (5 years)
Semester-VII
MT 701: Abstract Algebra –I LTP: 3+1+0
Unit I: Automorphism and inner automorphism, Class equation, Cauchy’s theorem, Sylow’s theorems,
Direct product of groups, Structure theorem for finitely generated abelian groups, Normal and subnormal
series, Composition series, Solvable groups, Jordan-Holder theorem. (15 Lectures)
Unit II: Euclidean domain, Unique factorization domain, Principal ideal domain. Chain conditions on rings.
Noetherian and Artinian rings, Fields, Finite fields, Extension fields, Algebraic and transcendental
extensions. (15 Lectures)
Unit III: Splitting fields, Simple and normal extensions, Perfect fields, Primitive elements, Algebraically
closed fields, Automorphisms of extensions, Galois group, Fundamental theorem of Galois theory, Galois
group over the rationals. (15 Lectures)
Recommended Readings:
1. Contemporary Abstract Algebra by Joseph A. Gallian, Narosa Publishing House.
2. Topics in Algebra 2nd Edition by I. N. Herstein. Wiley Eastern Limited.
3. A first course in Abstract Algebra by John B. Fraleigh (3rd Edition), Narossa Publishing House.
4. Basic Abstract Algebra 2nd Edition by Bhattacharya, Jain and Nagpaul, Cambridge University Press.
5. Abstract Algebra by Dummit and Foote, Wiley.
6. Algebra by Michael Artin, PHI.
MT 702: Probability and Probability Distribution LTP: 3+1+0
MT 703: Programming in C LTP: 3+1+0
Unit I: Basic concepts of programming languages: Programming domains, language evaluation
criterion and language categories, Describing Syntax and Semantics, formal methods of describing
syntax, recursive descent parsing, attribute grammars, Dynamic semantics (operational semantics,
denotational semantics, axiomatic semantics) (10 Lectures)
Unit II: Names, Variables, Binding, Type checking, Scope and lifetime data types, array types,
record types, union types, set types and pointer types, arithmetic expressions, type conversions,
relational and Boolean expressions, assignment statements, mixed mode assignment. Statement
level control structures, compound statements, selection statement, iterative statements,
unconditional branching. (10 Lectures)
Unit III: Programming in C: Character set, variables and constants, keywords, Instructions,
assignment statements, arithmetic expression, comment statements, simple input and output,
Boolean expressions, Relational operators, logical operators, control structures, decision control
structure, loop control structure, case control structure, functions, subroutines, scope and lifetime of
identifiers, parameter passing mechanism, arrays and strings, Pointers, Pointers to Function,
Function returning Pointers. (25 Lectures)
Recommended Reading:
Basic Reading:
1. Concepts of Programming Language by Robert W. Sebesta, Addison Wesley, pearson
Education Asia, 1999.
2. How to Program C by Deitel and Deitel, Addison Wesley, Pearson Education Asia.
Reference Books:
1. Introduction to Computer Science by Ramon A. Mata-Toledo and Pauline K. Cushman, Mc
Graw Hill International Edition.
2. Programming Languages by D. Appleby and JJ Vande Kopple, Tata McGraw Hill, India.
3. C Programming a Modern Approach by KN King, WW Norton & Co.
4. Programming in C by Yashwant Kanetkar, B.P.B Publications.
MT 704: 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)
Recommended Reading: 1. Simmons: Ordinary Differential Equations.
2. Lakshmikantham, Deo and Raghavendra, Ordinary Differential Equations.
MT 705: Complex Analysis -I LTP: 2+1+0
UNIT-I. Mappings of Elementary Functions and Conformal Mappings. Linear Functions, the function
1/z, Bilinear transformations, their properties and classification, the function z2, the function z1/2, the
transformation w=exp z, the transformation w= sin z, Conformal mappings, basic properties and examples.
(15 Lectures)
UNIT-II. Zeros and Poles. The number of zeros and poles, The argument principle, Rouche's throrem,
Zeros of polynomials, Meromorphic functions, Essential Singularities and Picard’s Theorem.
Univalent functions and mapping of unit disk. Univalent functions and it’s properties, Schwarz theorem,
Schwarz Pick’ Lemma, Open mapping theorem and Hurwitz’ theorem. (15 Lectures)
UNIT-III. Applications of conformal mapping. Harmonic conjugate, Transformation of Harmonic
Functions, Transformation of Boundary Conditions, Steady temperatures, Steady temperature in a half plane
and related problems, Electrostatic potential, Potential in cylindrical Space, Two dimensional fluid flow.
(15 Lectures)
Recommended Reading:
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,
MT 706: EM1 LTP: 3+0+0
MT 707: Lab LTP: 0+0+2
Lab work based on paper MTM 105
Semester-VIII
MT 801: Linear Algebra-I LTP: 3+1+0
Unit I: Finite dimensional vector spaces over arbitrary fields, Linear Transformation, Linear
Isomorphism, Linear Functionals, Dual and Double dual, Canonical Isomorphism, Polynomial and
its prime factorization, Characteristic and Minimal Polynomial, Diagonalization of Linear
operators, (15 Lectures)
Unit II: Annihilating polynomials, Invariant Subspaces, Direct Sum Decomposition, Invariant
Direct Sums, The Primary Decomposition Theorem, Cyclic Decomposition, Nilpotent Operator,
Jordan Form, Computation of Invariant Factors, Bilinear Forms, Symmetric and Skew-symmetric
Bilinear Forms. (15 Lectures)
Unit III: The Adjoint of Linear Transformation, Unitary Operators, Self Adjoints and Normal
Operators, Polar and Singular Value Decomposition. Semi-simple Operators. Spectral Theory of
Normal Operators on Finite Dimensional Vector Spaces. (15 Lectures)
1. Recommended Books:
2. Linear Algebra, Hoffman and Kunze, Prentice-Hall, Inc., 2009.
3. Linear Algebra, Bist and Sahai.
4. Linear Algebra, S.H. Friedberg, A.J. Insel and L. E. Spence, Pearson, 4th Ed., 2015.
5. Topics in Algebra, I.N. Herstein, 2nd ed., Wiley Eastern LTD., 1988.
6. Linear Algebra, Surjeet Singh, Vikas Publishing House Ltd.
MT 802: 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)
Recommended Readings:
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
MT 803: 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)
Recommended Reading:
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.
MT 804: 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, nonlinear
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)
Recommended Reading: 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.
MT 805: EM2 LTP: 3+0+0
MT 806: EM3 LTP: 3+0+0
MT 807: 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
SEMESTER-IX
MT 901: 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)
Recommended Reading:
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.
MT 902: 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)
Recommended Reading: 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.
MT 903: 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.
MT-904: EM04 LTP:3+1+0
MT-905: EM05 LTP:3+1+0
MT-906: EM06 LTP:3+1+0
Semester-X
MT-1001: EM07 LTP:3+1+0
MT-1002: EM08 LTP:3+1+0
MT-1003: Major Project Credits:16
Elective Mathematics Papers of 3 & 4 credits
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
26 Topology
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)
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: 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)
Recommended Reading:
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.
Recommended Reading:
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)
Recommended Reading :
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)
Recommended Reading:
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.
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)
Recommended Reading:
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)
Recommended Reading:
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)
Recommended Reading: 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.
Recommended Reading:
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.
Recommended Reading:
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.
Recommended Reading:
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)
Recommended Reading:
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)
Recommended Reading: 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.