courses of studies - ravenshaw universitycourses of studies for m.a/m.sc part –i examination...
TRANSCRIPT
1
COURSES OF STUDIES
for
M.A/M.Sc Part –I Examination -2014-2015
M.A/M.Sc Part –II Examination -2015-2016
MATHEMATICS
Ravenshaw University
Cuttack
2
M.Sc. (Mathematics)
First Semester
M 1.1.1 Abstract Algebra 50
M 1.1.2 Real Analysis 50
M 1.1.3 Complex Analysis 50
M 1.1.4 Topology 50
M 1.1.5 Linear Algebra 50
Second Semester
M 1.2.6 Discrete Mathematics 50
M 1.2.7 Advanced Analysis 50
M 1.2.8 Numerical Analysis 50
M 1.2.9 Functional Analysis 50
M 1.2.10 Practical 50
Third Semester
M 2.3.11 Optimization Theory 50
M 2.3.12 Differential Equation 50
M 2.3.13 Probability and Stochastic Processes 50
M 2.3.14 Elective - I 50
M 2.3.15 Elective - II 50
Fourth Semester
M 2.4.16 Differential Geometry 50
M 2.4.17 Calculus in vector spaces 50
M 2.4.18 Operator Theory 50
M 2.4.19 Elective - I 50
M 2.4.20 Elective - II 50
Elective – I
A. Computational Finance I and II
B. Theory of Computation I and II
C. Operations Research I and II
D. Fractals I and II
E. Fuzzy sets and their applications I and II
Elective – II
A. Fluid Dynamics I and II
B. Graph Theory I and II
C. Numerical solution of Differential and Integral Equation I and II
D. Design and Analysis of Algorithms I and II
E. Number Theory and Cryptography I and II
3
First Semester Paper - I (M 1.1.1)
Abstract Algebra
FM – 40 + 10 Time : 3 Hrs.
Unit - I 14 Marks
Normal Subgroups: Normal Subgroups & Quotient groups, Isomorphism theorems,
Automorphisms. Normal and Subnormal Series: Normal series, Solvable groups, Nilpotent
groups. Permutation groups: Cyclic decomposition, Alternating group An, Simplicity of An.
Unit - II 13 Marks
Direct products, finitely generated abelian group, Sylow’s theorem, groups of order p,2 pq,
Unique factorization domains Principal Ideal domains, Euclidean domains, Polynomial rings
over UFD.
Unit - III 13 Marks
Field Theory : Algebraic Extensions of fields, algebraically closed fields. Normal and separable
extensions: Splitting fields, Normal extensions, Multiple roots, Finite fields, Galois Theory:
Automorphism groups and fixed fields, fundamental theorem of Galois theory. Applications of
Galois Theory to classical problems: Polynomials solvable by radicals, Ruler and compass
constructions.
Book Prescribed:
Basic Abstract Algebra. P. B. Bhattacharya, S. K. Jain, S.R.Nagpaul 2nd edition (Cambridge
University Press).
Unit - I
Chapter -5 (5.1,5.2, 5.3), 6 (6.1, 6.2, 6.3), 7 (7.1, 7.2, 7.3)
Unit - II
Chapter -8[8.1, 8.2, 8.4 (8.4.1, 8.4.2, 8.4.3, 8.4.4, 8.4.7), 8.5], 11 (11.1 to 11.4)
Unit - III
Chapter - 15 [15.1, 15.2, 15.3, 15.4 (excluding 15.4.2, 15.4.3,15.4.4)], 16 (16.1 to 16.4), 17
(17.1,17.2), 18(18.3, 18.5)
Books for reference:
1. Algebra by M. Artin (PHI)
2. Modern Algebra by Surjeet Singh and Quazzirmuddin
3. Topics in algebra by I. N. Herstein.
4. Basic Algebra Vol-I, Vol-II by M. Jacobson, W.H. Freeman.
5. First Course on Abstract Algebra, John B. Frank. Sh
6. Fundamentals of Abstract Algebra. By D. S. Mallick et. al.
4
Paper - II (M 1.1.2)
Real Analysis
FM – 40 + 10 Time : 3 Hrs.
Unit – I 14 Marks
The Riemann - Stieltjes integral: Definition and Existence of the integral, properties of the
integral, Integration and differentiation.
Sequence and series of Functions, uniform convergence, continuity, integration, Differentiation.
Unit - II 13 Marks
Lebesgue Measure and Integral: Introduction, Outer measure, Measurable sets and Lebesgue
measure, A non-measurable set, measurable function. The Riemann integral, The Lebesgue
integral of a bounded function over a set of finite measure. The integral of a non-negative
function. The general Lebesgue integral.
Unit - III 13 Marks
Differentiation and Integration: The classical Banach spaces
Differentiation of monotone function, Functions of bounded variation.
Differentiation of an integral, Absolute continuity. LP spaces. The Holder and Minkowski’s
inequalities and completeness, Bounded linear functional on the LP spaces.
Books Prescribed:
1. Principles of Mathematical analysis - Walter Rudin
Chapter - 6 (6.1 to 6.22), 7 (7.1 to 7.18)
2. Real Analysis - H. L. Royden (3rd edition)
Chapter-3 (1 to 5), 4 (1 to 4), 5(1 to 4), 6 (except 4)
5
Paper - III (M.1.1.3)
Complex Analysis
FM – 40 + 10 Time : 3 Hrs.
Unit - I 14 Marks
Power series, analytic function, Analytic functions as mappings, Mobius transformations, power
series representation of Analytic function.
Unit - II 13 Marks
Zeros of analytic function, The index of a closed curve, Cauchy’s theorem and integral formula,
Morera’s theorem, Lioville’s theorem, Fundamental theorem of Algebra, zeros,
Goursat theorem.
Unit - III 13 Marks
Classification of singularities, poles, absolute converagence, Laurents series development,
Casorati Weirstrass theorem, Maximum modulus theorem, Schwartz’s Lemma.
Book Prescribed:
Functions of One complex variable - J. B. Conway
Chapter - III (1-3), IV (2-5,7,8), V (1,3), VI (1,2).
Paper - IV (M 1.1.4)
Topology
FM - 40 + 10 Time : 3 Hrs.
Unit - I 14 Marks
Topological spaces, Basis and order of topology, product and subspace topology, closed set,
limit points, continuous function, Product topology.
Unit - II 13 Marks
Connected spaces, connected sets in real line. Compact spaces. Compact sets in the real Line,
limit point compactness.
Unit - III 13 Marks
The countability axioms. The separation axioms. The Urysohn lemma. The Urysohn metrization
theorem. Tychonof theorem.
Book prescribed:
Topology A First course - J. R. Munkers
Chapter 2 (2.1-2.8), 3(3.1-3.7), 4(4-4.4), 5 (5.1).
6
Paper - V (M 1.1.5)
Linear Algebra
FM – 40 + 10 Time : 3 Hrs.
Unit-I
Dual Space, Finite dimensional vector space, isomorphism of finite vector space, annihilator,
Modules, finitely generated modules, cyclic sub-modules, T-annihilator, Linear Transform(LT),
algebra of LT, Isomorphism of LT, Cauchy Hamilton Theorem, invertible LT, singular LT,
Characteristic Roots, Linear Independence of Characteristic Roots(CR), distinct CR, CR vectors,
Canonical Forms, Triangular Form, Triangular Matrix.
Unit-II
Canonical Forms, Nilpotent Transforms, Index of Nilpotence, A decomposition of V: Jordan
Form, Rational canonical form, Monic.
Unit-III
Inner Product Space: Inner Product, Standard Inner Product, Polarization identities, Euclidean
space, Unitary space, Cauchy-Schwartz inequality, Orthogonal set, Orthonormal set, Orthogonal
complement, Orthogonal projection, Bessel’s inequality, Linear Functional and Adjoints, Linear
Operators, Unitary operator and Normal operators.
Books Prescribed:
1. Topics in Algebra: I.N. Herstein
Chapters: 4(4.3,4.5), 6(6.1,6.2,6.4,6.5,6.6,6.7)
2. Linear Algebra: K. Hoffman & R. Kunz
Chapters: 8(8.1-8.5)
7
Second Semester
Paper -I (M.1.2.6)
Discrete Mathematics
FM – 40 + 10 Time : 3 Hrs.
Unit - I 14 Marks
Graphs : Graphs and Graph models, Graph terminology and special type of graphs, representing
graphs and Graph Isomorphism, connectivity, Euler and Hamilton paths, shortest path problems,
planar graphs
Trees: Introduction to trees, application of trees, Tree traversal, spanning trees, minimum
spanning trees
Unit - II 13 Marks
Boolean Algebra: Boolean functions, representing Boolean functions, Logic gates, minimization
of circuits
Unit - III 13 Marks
Modelling computation: Languages and Grammars and Languages, Finite state machines with
output, finite state machines with no output, language recognition, Turing machines
Book prescribed:
1. Discrete Mathematics and its Applications, Kenneth H. Rosen, Tata Mc-Graw Hill
Education private Limited, Seventh Edition (Indian adaptation by Kamala Krithivasan),
2012
Chapter 8 (8.1-8.7), 9 (9.1-9.5), 10(10.1-10.4), 12(12.1-12.6)
Books for reference:
1. Discrete Mathematical structures with Application to Computer Science - J. P. Tremblay
and R. Monohar
2. Discrete Mathematics for Computer Scientists and Mathematicians by Joe L. Mott,
Abraham Kandel and Theodore P. Baker (Prentice-Halia).
8
Paper - II (M 1.2.7)
Advanced Analysis
FM – 40 + 10 Time : 3 Hrs.
Unit - I 14 Marks
Integration of vector valued functions rectifiable curves. Equi-continuous families of function,
The Stone Weierstrass theorem.
Unit - II 13 Marks
Fourier series, Orthogonal and Orthonormal system of functions, Bersets inequality. Dirichlet –
Kernel, pointwise convergence of Fourier series. Approximation theorems, Parseval’s theorem.
Unit - III 13 Marks
Harmonic functions, Basic properties of Harmonic functions, Harmonic functions on a disk,
Dirichlet problem, Green’s function. Entire functions, Jenson’s formula, Genus & order of an
entire function Hardamard factorization theorem.
Books Prescribed:
1. Principles of Mathematical Analysis – W. Rudin (3rd
edition)
Chapter 6(23,24,25,26,27), 7(19 to 33), 8(8-9 to 8-16)
2. Function of one complex variable – J.B. Conway
Chapter X (1,2,4,5), XI(1,2,3)
9
Paper - III (M 1.2.8)
Numerical Analysis
FM – 40 +10 Time : 3 Hrs.
Unit - I 14 Marks
Approximation of functions: Weierstrass theorem and Taylor’s theorem, Minimax approximation
problem, least square approximation problem, orthogonal polynomials,
least square approximation problem(Continued).
Unit - II 13 Marks
Numerical solution of systems of linear equations: Gaussian Elimination, pivoting and scaling in
Gaussian Elimination, variants of Gaussian Elimination, Error analysis, Residual correction
method, iteration methods, Error prediction and acceleration.
Unit - III 13 Marks
Eigen value location, error and stability results; Hermite interpolation, Piecewise polynomial
interpolation(Cubic spline interpolation, B-spline curves).
Book Prescribed:
An introduction to Numerical Analysis (2nd Edition) – Kendall E Atkinson (Wiley)
Chapter 3(3.6, 3.7), 4(4.1-4.5), 8(8.1-8.7), 9(9.1).
10
Paper - IV (M 1.2.9)
Functional Analysis
FM – 40 + 10 Time : 3 Hrs.
Unit-I
Metric Space, Definition and examples, Open Set, Close Set, Neighborhood, Convergence,
Cauchy Sequence, Completeness Definition of a Continuous mapping space, Banach Space,
Properties of normed Space, Finite dimensional normed Spaces and subspaces. Compactness and
finite dimensional Linear operator, Bounded and Continuous linear operators, Linear functional,
linear functional and operator on finite dimensional spaces, Normed spaces of operators, Dual
space.
Unit-II
Inner product space and its properties, Hilbert space, Orthonormal Sets and sequences, Total
orthonormal sets and sequences, Representation of functional on Hilbert Spaces, Hilbert adjoint
operators self adjoint, Unitary and normal operator.
Unit-III
Fundamental Theorems for normed and Banach Spaces. Zern`s Lemma, Hahn Banach theorems,
Hahn Normed space, Application to bounded linear functional on C[a,b], Adjoint operator,
Reflexive spaces, Bair’s Category theorem, Uniform Boundedness theorem , Strong and weak
convergence open mapping Theorem, Closed linear operator, closed graph theorem.
Book Prescribed:
Introductory Functional Analysis with Applications-Erwin Kreyszig
Chapters: 1(1.1-1.5), 2(2.2-2.10), 3(3.1-3.4,3.6,3.8-3.10), 4(4.1-4.8,4.12, 4.13)
11
Paper - V (M 1.2.10)
Practical-Programming in C
(Exp-30-Viva - 10 - Record – 10)
FM - 50 Time : 3 Hrs
List of Experiments :
1. To calculate mean & Standard deviation.
2. To calculate Pearson’s coefficient of correlation
3. To find area under a curve.
4. Lagrange’s interpolation.
5. Gauss elimination.
6. Inverse of a matrix.
7. To find Eigen value & Eigen vectors.
8. Runge Kutta method
9. Finding minimax approximation to e by Chebyshev’s polynomials.
10. Approximating definite integral by Newton cotes, Gauss quadrature rule.
12
Third Semester Paper -I (M 2.3.11)
Optimization Theory
FM – 40 + 10 Time : 3 Hrs.
Unit - I 13 Marks
One dimensional Optimization: Introduction, function comparison methods, polynomial
interpolation, iterative methods
UNIT-II 13 Marks
Gradient based optimization methods (I) : Calculus of Rn, method of steepest descent, conjugate
gradient method, The generalized gradient method, gradient projection method
Gradient based optimization methods (II) : Newton type methods (Newton’s method,
Marquardt’s method), Quasi- Newton methods.
UNIT-III 14 Marks
Linear programming: Convex analysis, simplex method, two phase simplex method, Duality,
Dual simplex method
Constrained optimization methods: Lagrange multipliers, Kuhn-Tucker conditions, convex
Optimization, Penalty function techniques, methods of multiplier, linear constrained problems-
cutting plane method
Text recommended
1. M.C. Joshi and K. Moudgalya, Optimization: Theory and Practice, Narosa Publishing
House, New Delhi, 2004
2. J.A. Snyman, Practical Mathematical Optimization, Springer Sciences, 2005
13
Paper -II(M 2.3.12)
Differential Equation FM – 40 + 10 Time : 3 Hrs.
UNIT-I 14 Marks
Differential Equations:
Existence and Uniqueness of solutions, Lipschitz condition, Gronwall inequality, Successive
approximations, Picard’s theorem, Continuation and dependence on initial conditions, Existence
of solutions in the large, Existence and uniqueness of solution of systems, Fixed point method,
Sytems of linear differential equations, nth order equation as a first order systems, system of first
order equations, Existence and uniqueness theorem, fundamental matrix, non-homogeneous
linear systems, linear systems with constant coefficients.
Non-linear differential equations, Existence theorems, External solutions, Upper and lower
solutions, Monotone iterative method, and method of quasi-linearization. Stability of liner and
non-linear systems, Critical points, systems of equations with constant coefficients, linear
equations with constant coefficients, Lyapunov stability.
UNIT-II 13 Marks
Boundary value problems for ordinary differential equations, Sturm-Liouville problem, Eigen
value and Eigen functions, Expansion in eigen functions, Green’s function, Picard’s theorem for
boundary value problem, series solution of Legendre and Bessel equations.
UNIT-III 13 Marks
Laplace’ equation: Boundary value problem for Laplace’ equation, Fundamental solution,
integral representation and mean value formula a for harmonic function, Green’s function for
Laplace’s equation, solution of Dirichlet’s problem for a ball, solution by separation of variables,
solution of Laplace’s equation for a disc, the wave equation and its solution by the method of
separation of variables, D’ Alembert’s solution, of the wave equation,
Books recommended
1. S.D. Deo, V, Lakhmikanthan and V. Raghavendra Textbook of rdinary Diffrential
equation, 2nd
edition, TMH, Chapter: 4 (4.1-4.7), 5, 6 (6.1-6.5), 7 (7.5), 9 (9.1-9.5)
2. J. Sinharoy and S. Padhy: A course of ordinary and partial differential equation. Kalyani
Publishers, Chapters: 10, 15, 16 and 17.
14
Paper -III(M 2.3.13)
Probability and Stochastic Processes FM – 40 + 10 Time : 3 Hrs.
UNIT-I 14 Marks
Random Variables : Introduction, Function of Random variables, moments and generating
functions.
Multiple Random Variables: Independent random variables, functions of several random
variables, Covariance, Correlation and moments, conditional Expectation
Some Special distributions : The Bivariate and Multivariate Normal Distributions. The
exponential Family of distributions.
Unit - II 13 Marks
Limit Theorems : Modes of convergence, the weak law of large numbers, Strong Law of Large
numbers, Limiting Moment generating functions, central limit theorems
Sample moments and their distributions : Random sampling sample characteristics and their
distribution, chi-square, t and F distributions: Exact sampling distribution
UNIT-III 13 Marks
Stochastic processes: Definition with examples, Markov chains, Chapman Kolmogorov
equations, Classification of states, Limiting probabilities, some applications: The gambler’s Ruin
problem
Continuous-time Markov chains. Birth-and-death processes, transition probability function
Limiting Probbailities
Book Prescribed:
1. An introduction to Probability and Statistics by V. K. Rohatgi and A.K. Md. Ehasanes
Saleh, Second edition, John Wiley and Sons.
Chapter 3, 4 (4.1-4.6), 5, 6, 7 (7.1-7.4)
2. An introduction to Probability Models by Sheldon M. Ross, Academic Press Harcourt
India Private Limited) Chapter 4 (4.1-4.5.1), 6 (6.1-6.5)
15
Paper - IV (M 2.3.14)
A- Computational Finance-I
(Elective - I)
FM - 40 + 10 Time : 3 Hrs.
UNIT-I
Basic concepts of Financial – Stock options, Forward and futures, Speculation, Hedging ,
Put-call parity, Principle of non-arbitrage pricing, Computation of volatility
Derivation of Black-Scholes differential equation and Black –Scholes Option Pricing
formula, Greeks and Hedging strategies.
UNIT-II
Finite difference methods for partial differential methods- finite difference approximation to
derivatives, Explicit and Implicit methods for parabolic equations, Iterative methods for
solution of a system of liner algebraic equations, two dimensional parabolic equations,
alternating-direct implicit method, convergence, stability and consistency of finite difference
schemes.
UNIT-III
Binomial pricing models, Explicit and implicit finite difference methods for European and
American options, Monte Carlo simulation
Books recommended:
1. J Bax and G Chacko- Financial derivatives: Pricing, applications and Mathematics-
Cambridge University Press, 2004.
2. G. D. Smith: Numerical Solution of Partial Differential Equations, Oxford University
Press.
3. P. Wilmott: Qualitative Finance- John Wiley, 2000.
16
Paper - IV (M 2.3.14)
B- Theory of Computation – I
(Elective -I)
FM – 40 + 10 Time : 3 Hrs.
Unit - I 14 Marks
Introduction of finite automata, Central concepts of automata theory, Informal picture of finite
automata, Deterministic finite automata, Non - deterministic finite automata, Application.
Unit - II 13 Marks
Regular expressions, Finitet automata and regular expressions, Application of regular
expressions, Algebraic law of regular expressions, Pumping lemma and its application for
regular language, Closure and Decision properties of regular languages.
Unit - III 13 Marks
Context - free Grammars, Parse trees, ambiguity in grammar & Languages, Pushdown
automation, The language of PDA, Equivalence of PDAs and CFS’S, Deterministic pushdown
automata, Change key normal form, The pumping lemma for context free languages, Decision
properties of CFL’s.
Books Recommended:
1. J E. Hoperoft, R MOtwani J. D. Uliman- Introduction to Automata Theory Languages and
compulaton (2nd Edition)Pearson Education 2001
2. M. Sipson Introduction to Theory of compulation Thomson Leamings.
3. R. Greenlan H. J. Hooer - Fundamentals of the Theory of computation, principles and practice
- Harcourt India Pvt.
4. Peter linz - An introduction to Forml Languages and Automata Narosa Publishing Hosue
1998.
17
Paper - IV (M 2.3.14)
C- Operation Research – I
(Elective -1)
FM – 40 + 10 Time : 3 Hrs.
Unit - I 14 Marks
Integer Programming and Dynamic Programming
Unit – II 13 Marks
Sequencing problems and games and strategies.
Unit - III 13 Marks
Queuing Theory- Introduction to Poisson Queuing Systems.
Book Prescribed:
Operation Research (Ninth Edn. 2001) Kanti Swarup, P.K. Gupta and Manmohan (S.C)
Chapter 7, 12, 13,17 (17.1-17.9), 20(20.1-20.8)
18
Paper - IV (M 2.3.14)
D- Fractals – I
(Elective -I)
FM – 40 + 10 Time : 3 Hrs.
Unit - I 14 Marks
Metric Spaces, Equivalent spaces, classification of subsets and the space of fractals,
Transformation on Metric Spaces, contraction mappings and the construction of Fractals.
Unit - II 13 Marks
Chaotic Dynamics on Fractals
Unit - III 13 Marks
Fractal Dynamics
Book Prescribed:
Fractal Everywhere - Michacl F. Bamsley
Chapter - II, III, IV, V
19
Paper - IV (M 2.3.14)
E- Fuzzy Sets and their Application-I
(Elective - I)
FM – 40 + 10 Time : 3 Hrs.
Unit - I 14 Marks
Fuzzy sets - Basic definition a -level sets. Convex fuzzy sets. Baic operations Fuzzy sets. Type
of Fuzzy sets. Cartesian products. Algebraic products. Bounded sum and difference t-norms and
t-conorms.
The extension Principle- The Zadeh’s extension principle image and inverse image of Fuzzy
arithmetic.
Unit - II 13 Marks
Fuzzy Relation and Fuzzy Graphs-Fuzzy equivalence equations. Fuzzy graphs, Similarity
relation.
Unit - III 13 Marks
Possibility theory-Fuzzy measures, Evidence theory necessity measure, Possibility theory versus
probability theory.
Books Prescribed:
1. Fuzzy set theory and its application allied publisher rd New Delhi - 1991 - U. Z. Zimmermann
2. Fuzzy set and fuzzy logic prentice Hall of Indi New Delhi 1995- G J Klir & Bo Yuan
20
Paper - V (M 2.3.15)
A - Fluid Dynamics - I
(Elective - II)
FM – 40 + 10 Time : 3 Hrs.
Unit - I 14 Marks
Basic Concepts
Unit - II 13 Marks
Fundamental Equations of the flow of viscous fluids. Dynamical similarity, inspection and
Dimensional Analysis.
Unit - III 13 Marks
Exact solutions of Navier - Stokes Equations (Restricted)
Book Prescribed:
Viscous Fluid Dynamics - J.L. Bansal, Oxford & IBH Publishing. Co.
Chapter (2.1-2.6), 3 (3.1-3.4, 3.8, 3.9), (4.1-4.6).
21
Paper - V (M 2.3.15)
B - Graph Theory- I
(Elective - II)
FM – 40 + 10 Time : 3 Hrs.
UNIT-I
Fundamental Concepts : Basic Definitions. Graphs, Vertex degrees, Walks, Paths, Trails,
Cycles, Circuits, Subgraphs, Induced subgraph, Cliques, Components, Adjacency Matrices,
Incidence Matrices, Isomorphisms. Graphs with special properties : Complete Graphs.
Bipartite Graphs. Connected Graphs, k-connected Graphs, Edge-connectivity, Cut-vertices, Cut-
edges. Eulerian Trails, Eulerian Circuits, Eulerian Graphs : characterization, Hamiltonian
(Spanning) Cycles, Hamiltonian Graphs : Necessary condition, Sufficient conditions (Dirac, Ore,
Chvatal, Chvatal-Erdos), Hamiltonian Closure, Traveling Salesman Problem.
UNIT-II
Trees : Basic properties, distance, diameter. Rooted trees, Binary trees, Binary Search Trees.
Cayley’s Formula for counting number of trees. Spanning trees of a connected graph, Depth first
search (DFS) and Breadth first search (BFS) Algorithms, Minimal spanning tree, Shortest path
problem, Kruskal’s Algorithm, Prim’s Algorithm, dijkstra’s Algorithm. Chinese Postman
Problem.
UNIT-III Coloring of Graphs : Vertex coloring : proper coloring, k-colorable graphs, chromatic number,
upper bounds, Cartesian product of graphs, Structure of k-chromatic graphs, Mycielski’s
Construction, Color-critical graphs, Chromatic Polynomial, Clique number, Independent (Stable)
set of vertices, Independence number, Clique covering, Clique covering number. Perfect graphs :
Chordal graphs, Interval graphs, Transitive Orientation, Comparability graphs. Edge-coloring,
Edge-chromatic number, Line Graphs.
References :
1. Introduction to Graph Theory, Douglas B. West, Prentice-Hall of India Pvt. Ltd.,
New Delhi 2003.
2. Graph Theory, F. Harary, Addison-Wesley, 1969.
3. Basic Graph Theory, K.R. Parthasarathi, Tata McGraw-Hill Publ. Co. Ltd., New
Delhi, 1994.
4. Graph Theory Applications, L.R. Foulds, Narosa Publishing House, New
Delhi,1993.
5. Graph Theory with Applications, J.A. Bondy and U.S.R. Murty, Elsevier science,
1976.
10. Graph Theory with Applications to Engineering and Computer Science, Narsingh
Deo, Prentice-Hall of India Pvt.Ltd., New Delhi, 1997
22
Paper - V (M 2.3.15)
C: Numerical Solution of
Differential and Integral Equation - I
(Elective-II)
FM – 40 + 10 Time : 3 Hrs.
Unit - I 14 Marks
Numerical methods for ordinary differential equation: Existence, uniqueness and stability
theory, Euler’s method, multistep methods, midpoint methods, Trapezoidal method, low-order
predictor-corrector algorithm, derivation of higher order multistep methods, convergence and
stability for multistep methods.
Unit - II 13 Marks
Numerical Integration: Corrected Trapezoidal rule and its error formula, Peano kernel error
formulas, Newton- Cotes integration formulas, Gaussian quadrature, asymptotic error formulas
and their applications.
Unit - III
Integral equation: Volterra integral equation, Fredholm integral equation, singular integral
equation, non-linear integral equation, convolution integral, differentiation of a function under an
integral sign, relation between differential and integral equation.
Books Prescribed:
1.An introduction to Numerical Analysis (2nd Edition) –Kendall E Atkinson (Wiley)
Chapter 6(6.1-6.8), 5(5.1-5.4).
2. Integral Equations - Shanti Swarup (Krishna Prakashan)
Chapter 1(1.1-1.3).
23
Paper – V (M 2.3.15)
D - Design and Analysis of Algorithms – I
(Elective -II)
FM – 40 + 10 Time : 3 Hrs.
Unit - I 14 Marks
Mathematical Foundations - Growth functions, summations and recurrences - substitution,
iteration and master methods, counting and probability, amortized analysis.
Sorting - Heap sort, quick, merge sort, sorting in liner time, median and order statistics.
Unit - II 13 Marks
Advanced Data Structures -B- trees, red-black trees, hashing, dynamic order statistics, binomial
and fibonacci heap, disjoint sets.
Unit - III 13 Marks
Dynamic Programming - Matrix chain multiplication, longest common subsequence, optimal
polygon triangulation. Greedy Algorithms - Huffman coding and task scheduling
Problems Graphs - Traversal, topological sort, minimum spanning trees, single source shortest
paths Dijkstra’s and Bellman Ford algorithms, all-pairs shortest path, maximum flow problems.
Book Prescribed:
Introduction to Algorithms, Prentice Hall of India – TH Core man, C.E. Leiserson, R.L. Rivest.
Books Recommended: 1. Computer Algorithms, Introduction to Design and Analysis Addision Wesley - S. Basse, AV.
Gelger
2. Algorithms, Addison Wesley - S. Sadgeerick
3. Designing Efficment Algorithms for Parallel computers - M T Quinn.
24
Paper – V (M 2.3.15)
E - Number Theory and Cryptography – I
(Elective -II)
FM – 40 + 10 Time : 3 Hrs.
Unit - I 14 Marks
Some Topics in Elementary Number Theory Divisibility and the Euclidean algorithm
congruence’s, some applications to factoring.
Unit - II 13 Marks
Finite Fields and Quadratic Residues Finite fields, Quadratic residues and reciprocity.
Unit - III 13 Marks
Cryptography Same simple cryptosystem Enchiphering matrices
Book Prescribed:
A Course in Number Theory and cryptography (Second Edition) Springer - Neal Koblitz
Chapter -I (2,3,4), II, III
25
Fourth Semester Paper - I (M 2.4.16)
Differential Geometry
FM – 40 + 10 Time : 3 Hrs.
Unit - I 14 Marks
Topological manifold, cutting and pasting abstract manifolds, The space of tangent vectors at a
point of Rn,
Inverse function theorem. Another definition of Ta (Rm
), vector fields on open
subsets of Rn.
Unit - II 13 Marks
Differentiation on differential manifolds, The tangent space at a point of a manifold, vector
fields, Tangent co-vectors, Bilinear forms, Riemann manifold as metric space.
Unit - III 13 Marks
Tensor field, multiplication of Tensors, Exterior differentiation, Differentiation of vector fields
on sub manifolds of Rn, Differentiation on Riemann manifolds.
Book Prescribed :
An introduction to Differentiable manifolds and Riemannian Geometry by William Boothby,
Academic Press, New York.
Unit –I: Chapter I (3,4,5), II (3,4,5,6)
Unit –II: Chapter III (1), IV (1,2), V (1,2,3)
Unit –III: Chapter V (5,6,8), VII (2,3)
26
Paper - II (M 2.4.17)
Calculus in Vector Spaces
FM – 40 + 10 Time : 3 Hrs.
Unit-I
Functions on n-space:
Partial derivatives, Differentiability and the Chain Rule, potential function, curve integrals,
Taylor series, maxima and derivatives.
Unit-II
Derivatives in Vector Space:
The space of Continuous linear maps, The derivative as a lines map, Properties of the derivative,
Mean value Theorem, the second derivative, Higher derivative and Taylor`s formula, Partial
derivatives, Differentiating under the integral sign.
Unit-III
Inverse Mapping Theorem and Ordinary Differential Equation:
The Shrinking Lemma, Inverse mappings, linear case, The inverse mapping theorem, Implicit
function theorem and charts, Local existence and uniqueness, Approximate solutions, linear
differential equations, Dependence on initial Conditions.
Book of Prescribed :
Undergraduate Analysis-S. Lang (Springer &Verlag)
Chapters: 15(15.1-15.6), 16(16.1-16.8), 17(17.1-17.5), 18(18.1-18.4)
27
Paper - III (M.2.4.18)
Operator Theory
FM – 40 + 10 Time : 3 Hrs.
Unit - I 14 Marks
Spectral Theory in dimensional normed spaces: Basic concepts, Spectral properties of Bounded
linear operators, Further properties of resolvent and spectrum, Banach algebra, Further properties
of Banach Algebra.
Unit - II 13 Marks
Compact linear operator on normed spaces, Further properties of compact linear operators,
Spectral properties of compact linear operators, Spectral properties of Bounded
Self Adjoint linear operators, Further spectral properties of Bounded Self Adjoint linear
operators.
Unit - III 13 Marks
Positive operators, Projection operators, Unbalanced linear operators and their Hilbert Adjoint
operators, Hilbert Adjoint operators, Symmetric and Self-Adjoint linear operators, Closed linear
operators and closures, Spectral properties of Self-Adjoint linear operators.
Book prescribed:
Introductory Functional Analysis with Applications-Erwin Kreyszig
Chapter 7(7.1-7.7), 8(8.1-8.3), 9(9.1-9.3, 9.5), 10(10.1-10.4).
28
Paper - IV (M.2.4.19)
A- Computational Finance-II
(Elective-I)
FM – 40 + 10 Time : 3 Hrs.
UNIT-I 13marks
Exotic and Path dependent options, Introduction, Barrier options, Asian options, Look back
options, computational Schemes, option on stock indices, Currencies and futures.
UNIT-II 14marks
Extensions of Black-Scholes model- Limitations of Black-Scholes model, Discrete Hedging,
Transaction costs, volatility smiles, stochastic volatility, Jump diffusion, dividend modeling,
pricing models for multi-asset options.
UNIT-III 13marks
Interest rates and their derivatives: Fixed income products and analysis (yield , duration and
convexity), swaps, one factor and multi-factor interest rate models, interest rate derivatives,
Heath-Jarrow-Merton model
Risk measurement and management, Portfolio management, value at risk, credit risk, credit
derivatives, risk metrics and credit metrics.
Books recommended:
1. J Bax and G Chacko- Financial derivatives: Pricing, applications and Mathematics-
Cambridge University Press, 2004.
2. Y.K. Kwok- Mathematical Models for financial derivatives-Springer Verlang
3. J.C. Hull-Options, Futures and other derivatives- Prentice Hall of India, 2003
29
Paper - IV (M 2.4.19)
B- Theory of Computation - II
(Elective - I)
FM – 40 + 10 Time : 3 Hrs.
Unit - I 14 Marks
The Turing machine, Programming techniques for Turing machines, Extensions to the basis
Turing machine, Restricted Turing machines, Turing machines and computers.
Unit - II 13 Marks
Non - Recursively emmerable languages, undecidable problem that is recursively enumerable
undecidable problems about Turing machines. Post’s correspondence problems, other
undecidable problems.
Unit - III 13 Marks
Mapping Reducibility, Measuring complexity, The Class P & the class NP.
Books Recommended:
1. J. E. Hopcroft, R. Motwani, J. D. Ullman - Introduction to Automata theory, Languages and
Computation, 2nd Edition, Pearson Education, 2001
2. M. Sipser - Introduction to Theory of Computation Thomson Leamings.
3. R. Greenlan, H. J. Hooer - Fundamentals of the Theory of Computation, Principles and
Practice - Harcourt India Pvtt.
4. Peter Linz - An Introduction to Formal Languages & Automata, Narsora Publising House,
1998.
30
Paper – IV (M 2.4.19)
C - Operation Research - II
( Elective - I )
FM – 40 + 10 Time : 3 Hrs.
Unit - I 14 Marks
Non - linear programming and methods
Unit - II 13 Marks
Non linear programming methods (contd.), Quadratic programming (Wolfe’s and Beale’s
Method), Separable convex programming, Separable programming algorithm.
Unit - III 13 Marks
Geometric Programming - Goal Programming
Book Prescribed :
Operation Research (Ninth Edn. 2001) - Kanti Swarup, P. K. Gupta, Manmohan (S. Chand)
Chapter 24, 25(25.8), 26(26.1-26.6)
31
Paper - IV (M 2.4.19)
D- Fractals - II
( Elective -I)
FM – 40 + 10 Time : 3 Hrs.
Unit - I 14 Marks
Fracta Interpolation.
Unit - II 13 Marks
Julia sets.
Unit - III 13 Marks
Parameter spaces and Mandelbrot sets.
Book Recommended:
Fractals Everywhere - Michael F. Bamslury
32
Paper - IV (M 2.4.19)
E- Fuzzy sets and their applications-II
( Elective -I)
UNIT-I
Fuzzy Logic and Approximate reasoning: Linguistic variables, Fuzzy logic, Truth tables and
Linguistic approximation, approximate reasoning, fuzzy languages
UNIT-II
Decision making in fuzzy environments: Fuzzy decisions, fuzzy linear programming, symmetric
fuzzy LP, Fuzzy dynamic programming
UNIT-III
Fuzzy set models in Operations Research: Introduction, fuzzy set models in Logistics, fuzzy
approach to transportation problems, fuzzy linear programming in logistics, fuzzy set decision
model as optimization criterion, and other associated models
Books Prescribed:
1. Fuzzy set theory and its application allied publisher rd New Delhi - 1991 - U. Z. Zimmermann
2. Fuzzy set and fuzzy logic prentice Hall of Indi New Delhi 1995- G J Klir & Bo Yuan
33
Paper - V (M 2.4.20)
A- Fluid Dynamics - II
(Elective - II)
FM – 40 + 10 Time : 3 Hrs.
Unit - I 14 Marks
Exact solutions of Navier – Stokes Equations .
Unit - II 13 Marks
Theory of Laminar Boundary Layers
Unit - III 13 Marks
Integral methods for the Approximate solution of laminar Boundary Layer equations, thermal
Boundary Layers in Two – Dimensional Flow.
Book Prescribed:
Viscous Fluid Dynamics – J. L .Bansal, Oxford 7 IBH Publishing Co.
Chapter 4(4.8-4.13), 6(6.1-6.4), 7(7.1-7.4), 8(8.1,8.2).
------------------------------------------------------------------------------------------------------------
34
Paper - V (M 2.4.20)
B- Graph Theory II
(Elective - II)
FM – 40 + 10 Time : 3 Hrs
UNIT-I 14marks
Planarity of Graphs : Drawing graphs in a plane, Planar Graphs, Planar embeddings, Dual
Graphs, Euler’s Formula, Maximal Planar Graphs. Subdivisions, Kuratowski’s Theorem, Convex
Embedding. Planarity Testing Algorithm. Coloring of planar graphs, Edge-contraction, Five
color theorem, Kempe’s chain, Four Color Theorem (Statement only). Crossing Number.
UNIT-II 13marks
Directed Graphs : Definitions and examples, Vertex degrees, Eulerian Digraphs, Orientations
and Tournaments, Network and Flow problem, Max Flow – Min Cut Theorem, Algorithm for
finding maximum flow.
UNIT-III 13marks
Matching : Maximum Matching Problem, Hall’s Marriage Theorem, Minimum covering
problems : Vertex Cover, Konig-Egervary Theorem, Edge Cover and its characterization in
terms of independence number.
Books for References :
1. Introduction to Graph Theory, Douglas B. West, Prentice-Hall of India Pvt. Ltd.,
New Delhi 2003.
2. Graph Theory, F. Harary, Addison-Wesley, 1969.
35
Paper - V (M 2.4.20)
C: Numerical Solution of Differential and Integral Equation – II
(Elective-II)
FM – 40 + 10 Time : 3 Hrs.
Unit - I 14 Marks
Numerical solution of ordinary differential equations by boundary value problems.
Numerical solution of partial differential equations by finite difference methods.
Unit - II 13 Marks
Finite Element method.
Unit - III 13 Marks
Solution of integral equations.
Books Prescribed :
1. Introductory methods of Numerical Analysis S. S. Sastry (PHI)
Chapter 7(7.10), 8, 10.
2. Integral Equations - Shanti Swarup (Krishna Prakashan)
Chapter 2(2.1-2.9)
36
Paper - IV (M 2.4.20)
E - Number Theory and Cryptography - II
( Elective - II)
FM – 40 + 10 Time : 3 Hrs.
Unit - I 14 Marks
Public Key, the idea of public key cryptography, RSA, Discrete log, knapsack.
Unit - II 13 Marks
Primality and factoning, pseudo primes, the rho method, Format factorization and factor bases,
continued fraction method.
Unit - III 13 Marks
Elliptic curves, Basic facts, Elliptic curve cryptosystem, Elliptic curve primality test
Books Prescribed :
A course in Number Theory and Cryptography - Neal Koblitz (Springer)
Chapter IV (1,2,3,4), V (1,2,3,4), VI (1,2,3)