PREAMBLE
The post graduate Course is of Three years duration having fifty percent
courses of Mathematics and fifty percent courses of Computer Applications,
one full semester is devoted for Industrial training.
This course is designed as per the guidelines of UGC for creating the
opportunities in the IT field for Mathematics students. As, so far IT
industries used to have Computer Experts and Mathematicians with
separate job profiles, this course eliminates this discrimination and creates
Technologists with both the flavors. It is to be worth mentioning here that
the course took birth due to heavy need of mathematical knowledge and
mathematical aptitude in industries and R & D sectors. This course deals
with heavy mathematics; as such the M. Sc. (Tech.) degree is equivalent to
M. Sc. (Mathematics) course. This degree course is also equivalent to
Master of Computer Applications (MCA) as well as M. Tech. (Computer
Science).
SYLLABUS
M. Sc. (TECH.)
(Industrial Mathematics with Computer Applications)
FIRST YEAR
SEMESTER I (All are compulsory and each course is of 6 credit points)
MIM 101 Advanced Calculus
MIM 102 Topology
MIM 103 Measure and Integration
MIM 104 Algebra
MIM 105 Differential Equations
Total credits: 30 points
SEMESTER II (All are compulsory and each course is of 6 credit points)
MIM 201 Functional Analysis
MIM 202 Complex Analysis
MIM 203 Field Theory
MIM 204 Linear Algebra
MIM 205 Numerical Analysis.
Total credits: 30 points
SECOND YEAR
SEMESTER III
Compulsory Courses: (each course is of 6 credit points)
MIM 301 Theory of Computer Science and Database Fundamentals
MIM 302 Programming in C and C++
MIM 303 Data Structures
Optional Courses: (each course is of 5 credit points)
Any two of the Optional Courses of Second Year (See the list)
Total credits: 28 points
SEMESTER IV
Compulsory Courses : (each course is of 6 credit points)
MIM 401 Java Programming
MIM 402 Operating Systems
MIM 403 Designs and Analysis of Algorithms
Optional Courses : (each course is of 5 credit points)
Any two of the Optional Courses of Second Year (See the list)
Total credits: 28 points
SECOND YEAR OPTIONAL COURSES (each course is of 5 credit points)
MIM 01. Operations Research
MIM 02. Integral Transforms
MIM 03. Number Theory
MIM 04. Coding Theory
MIM 05. Graph Theory
MIM 06. Lattice Theory
MIM 07. Computational Geometry
MIM 08. Cryptography
MIM 09. Financial Mathematics
MIM 10. Modelling and Simulation
MIM 11. Artificial Intelligence
MIM 12. Symmetries
MIM 13. Wavelets
MIM 14. Combinatorics
MIM 15. Partial Differential Equations
MIM 16. Fuzzy Logic
MIM 17. Statistics and Probability
MIM 18. Fluid Dynamics
MIM 19. Banach Algebras
MIM 20. Boundary Value Problems
MIM 21. Baer * Rings
MIM 22.Topics in Matroid Theory
MIM 23.Topics in Sperner Theory
MIM 24.Differential Equation and Dynamical System
MIM 25.Mechanics
THIRD YEAR
SEMESTER V
Compulsory Courses : (each course is of 6 credit points)
MIM 501 Software Engineering
MIM 502 Computer Networks
MIM 503 Computer Graphics and Emerging Technologies
Optional Courses : (each course is of 5 credit points)
Any two of the Optional Courses of Third Year (See the list)
Total credits:28 points
SEMESTER VI
IT PROJECT (In the Industry )
Total credits:26 points
OPTIONAL COURSES: THIRD YEAR
MIM 51. Complex Analysis II
MIM 52. Representation Theory of Groups
MIM 53. Fourier Analysis on Finite Groups
MIM 54. Differential Geometry
MIM 55. Non-Linear Dynamical System
MIM 56. Emerging Technologies
MIM 57. Algebraic Topology
MIM 58. Analysis on Manifolds
MIM 59. Projective Geometry
MIM 60. Algebraic Geometry
MIM 61. Algebraic Number Theory
MIM 62. Algebraic Curves
MIM 63. Commutative Algebra
MIM 64. Advanced Topics in Lattice Theory
MIM 65. Advanced Topics in Graph Theory
MIM 66 Advanced Topics in Matroid Theory
MIM 67. Advanced Topics in Group Theory
MIM 68. Advanced Topics in Ring Theory
MIM 69 Topics in Non Commutative Rings.
SEMESTER I
MIM 101 Advanced Calculus
1. Compact and Connected Subsets of Rn.
2. Differentiation : Derivative, Continuously Differentiable functions, Chain rule,
Inverse function theorem, Implicit function theorem.
3. Integration: integral over a rectangle, Existence of the Integral, evaluation of
the integral, integral over a bounded set and rectifiable sets, improper integrals
4. Change of Variable Theorem (Proof of one variable) and Statement of n-variables
(with Illustrations)
5. Line Integrals with Applications
Textbooks :
1. Analysis on Manifolds - J.R. Munkres : Sections 4 to 15 and Section 17.
2. Calculus (Volume II) - T.M. Apostol, Chapter 10 : Sections 10.1 to 10.9
MIM 102 Topology
1. Cartesian Products, Finite Sets, Countable and Uncountable Sets, Infinite Sets
and Axiom of Choice, Well Ordered Sets.
2. Topological Spaces : Basis for a topology, Order topology, Subspace Topology,
Product topology, closed sets and limit points, Continuous functions, Metric
Topology
3. Connected spaces, Connected Subspaces of Real Line, Components and Local
Connectedness, Compact spaces, Compact Subspaces of the Real Line, Limit point
compactness, Local Compactness.
4. Countablity and Separation Axioms : Countability Axioms, Separation axioms
Normal Spaces, Urysohn's Lemma, Titetz Extension Theorem (Without Proof),
Tychonoff's Theorem , Metrization Theorem (without proof).
Textbook :
J. R Munkres : Topology : A First Course. ( Prentice- Hall)
Chapter 1 : Sections 5,6,7,9,10
Chapter 2 : Sections 12 to 21
Chapter 3 : Sections 23 to 29
Chapter 4 : Sections 30 to 35
Chapter 5 : Section 37
MIM 103 Measure and Integration
1. Preliminaries : Cardinal Numbers and Countability, Properties of Open Sets,
Cantor Like Sets
2. Measure on Real Line : Lebesgue Outer Measure, Measurable Sets, Regularity,
Measurable Functions, Borel and Lebesgue Measurability
3. Integration of Functions on Real Variable : Integration of Non Negative Functions,
General Integral, Integration of Series, Riemann and Lebesgue Integral
4. Differentiation : Functions of Bounded Variation, Lebesgue Differentiation
Theorem, Differentiation Theorem, Differentiation and Integration
5. Abstract Measure Spaces : Measures and Outer Measures, Extension of
Measures, Uniqueness of Extension, Completion of Measure, Measure Spaces,
Integration with respect to Measure
6. Inequaties and Lp spaces : The Lp Spaces, The Convex Functions, Jensen's
Inequalities, Inequalities of Holder and Minkowski, Completion of Lp.
7. Convergence : Convergence in Measure, Almost Uniform Convergence,
Convergence Diagrams, Counter Examples
Textbook :
Measure Theory and Integration : G. de Barra (New Age International Ltd,
Publishers)
Chapter 1 : Sections 1.5 to 1.7.
Chapter 2 : Sections 2.1 to 2.5.
Chapter 3 : Sections 3.1 to 3.4.
Chapter 4 : Sections 4.1 to 4.5.
Chapter 5 : Sections 5.1 to 5.6.
Chapter 6 : Sections 6.1 to 6.5.
Chapter 7 : Sections 7.1 to 7.4.
MIM - 104 Algebra
1. Introduction to Groups, Symmetries of a square, Dihedral Groups, Examples and
properties of Groups, Finite Groups, Subgroups, Cyclic Groups.
2. Permutation Groups and its properties,check digit scheme Based on D5,
Isomorphisms, Cayley's Theorem, Cosets and Lagrange's Theorem,Applications of
Cosets and Permutaion Groups, Rotation Group of a Cube and Soccer.
3. External Direct Products with Applications, Normal subgroups and Factor Groups
and Applications.
4. Group Homomorphisms, First Isomorphism Theorem, Fundamental Theorem of
Finite Abelian Groups.
5. Sylow Theorems, Applications of Sylow Theorems.
6. Introduction to Rings , Integral Domains, Fields, Ideals, Ring Homomorphisms,
Polynomial Rings, Factorisation of Polynomials, Divisibilty in Integral Domains.
Text Book :
Contemporary Abstract Algebra (Fourth Edition)-Joseph Gallian (Narosa Publishing
House).
Chapters: 1 to 18 and 24..
MIM 105 Differential Equations
1. Linear equation of first order.
2. Applications : Orthogonal Trajectories, Population Growth and Decay, Newton's
Law of Cooling, Free Falling Bodies
3. Linear equations with constant coefficients : Second order homogeneous
equations, initial value problems, linear dependence and independence,
nonhomogeneous equations of nth order, Algebra of Constant Coefficients.
4. Applications : Simple Harmonic Motion, Damped Motion, Forced Motion, Other
Applications in Electronics and Pendulum Problem
5. Linear equations with variable coefficients : Initial value problems, solutions of the
homogeneous equation, Wronskian and linear independence, reduction of order
nonhomogeneous equations Legendre equation.
6. Linear Equations with regular singular points : Euler equation, second order
equation with regular singular points, exceptional cases, Bessel equation.
7. Existence and uniqueness of solutions to first order equations : Separation of
variables, exact equations, Method of successive approximations, Lipschitz
condtion, Approximation to and uniqueness of solutions.
8. Existence and uniqueness of solutions to first order equations to systems and 2nd
order equations
9. Laplace transforms: Laplace Transforms, Inverse Laplace Transform, Solving
Initial Value Problems with Laplace Transforms.
Text Book :
1) E. A. Coddington : An Introduction to Ordinary Differential Equatins (Prentice-
Hall).
Chapter 1 : Sections 1 to 7.
Chapter 2 : Sections 1 to 12.
Chapter 3 : Sections 1 to 8.
Chapter 4 : Sections 1 to 4, 7 and 8.
Chapter 5 : Sections 1 to 6.
Chapter 6 : Sections 6,7.
2) Differential Equations with Mathematica - Martha L. Abell and James P. Braselton
(Second Edition) Acadamic Press..
SEMESTER II
MIM 201 Functional Analysis
1. Normed spaces, continuity of linear maps, Hahn - Banach theorems, Banach
spaces.
2. Uniform bounded principle, Application - Divergence of Fourier Series of
Continuous Functions, closed graph theorem, Open mapping theorem, bounded
inverse theorem, spectrum of Bounded Operator.
3. Duals and transposes, duals of LP[a,b] and C[a,b].
4. Inner product spaces, orthonormal sets, approximation and optimization,
projections, Riesz representation theorem.
5. Bounded operators and adjoints on a Hibert space, normal, unitary and self
adjoint operators.
6. Fourier Series and Integrals.
Textbook :
B.V. Limaye : Functional Analysis (Second Edition) - New Age International Limited..
Chapter 2 : Sections 5 to 8.
Chapter 3 : Sections 9 to 12.
Chapter 4 : Sections 13,14.
Chapter 6 : Sections 21 to 24.
Chapter 7 : Sections 25,26.
Chapter 1 : Section 4 (Fourier Series and Integrals).
MIM 202 Complex Analysis
1. Complex number system: Complex plane, polar representation, lines and half
planes, Spherical representation.
2. Analytic functions: Power series, analytic functions, Mobius transformations,
3. Applications of Conformal Mappings : Steady Temperatures, Steady
Temperatures in a wall, Temperatures in a Quadrant with part of one Boundary
Insulated, Electric Potential, Potential in a cylindrical space, Two Dimensional Fluid
Flow, The stream Function, Flow around a Corner, Flow around a Cylinder
4. Integration: Power series representation of analytic functions Zeros of analytic
function, Index of a closed curve, Cauchy's theorem, Homotopy Versions (without
proof), simple Connectivity version (without proof), counting zeroes, Goursats
theorem.
5. Singularities: Classification, residues, argument principle.
6. Maximum Modulus theorem: Maximum principle, Schwarz lemma.
Textbook
1. Complex Analysis - J.B. Conway (Second Edition) Narosa Publishing House..
Chapter 1 : Sections 1 to 6.
Chapter 3 : Sections 1 to 3.
Chapter 4 : Sections 2 to 8.
Chapter 5 : Sections 1 to 3.
Chapter 6 : Sections 1 to 2.
Textbook 2 : Complex Variables and Applications - Ruel V. Churchill (Second
Edition)
Chapter 9..
MIM 203 Field Theory
1. Fields - Example, Algebraic and Transcendental Elements, Degree of Field
Extension, Construction with Ruler and Compass, Symbolic Adjunction of Roots,
Finite Fields
2. The Main Theorem of Galois Theory - Cubic Equations, Summetric Functions,
Primitive Elements, Proof of Main Theorem, Kummer Extension, Cyclotomic
Extension, Quintic Extension.
Textbook :
Algebra - Michael Artin (Prentice Hall of India Private Limited).
Chapter 13 : Sections : 1 to 6.
Chapter 14 : Sections : 1 to 9.
MIM 204 Linear Algebra
1. Modules : Definition and Examples, Further notions and Results
2. Free Modules : Linear Independence, Bases of Free Modules, Matrices and
Homomorphisms
3. Vector Spaces : Definition and Examples, Subspaces, Bases and Dimensions,
Linear Transformations, Quotient Spaces, Direct Sum, The matrix of Linear
Transformation, Duality
4. Canonical Forms : Eigenvalues and Eigenvectors, The minimal Polynomial,
Diagonalisability, Triangularisable Operators, Jordan Forms, The Rational Forms
5. Inner Product Spaces : Inner Product Spaces, Orthogonality, The Adjoint of Linear
Transformation, Unitary operators, Self Adjoints and Normal Operators, Polar and
Singular Value Decomposition
6. Bilinear Forms : Definition and Examples, The matrix of a Bilinear Form,
Orthogonality, Classification of Bilinear Forms
Textbook 1 : Modules - Luther and Passi (Narosa Publishing House). Chapters 1, 2.
Textbook 2 : Linear Algebra - Bisht and Sahai, Chapters 2 to 5.
MIM 205 Numerical Analysis
1. 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.
2. Polynomial interpolation: interpolation polynomial, divided difference
interpolation, Aitken's formula, finite difference formulas, Hermite's interpolation,
double interpolation.
3. Linear systems of Equations: Gauss Elimination, Gauss-Jordan method, LU
decomposition, iterative methods, and Gauss- Seidel iteration.
4. 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.
5. 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.
6. Eigenvalue Problem : Power method, Jacobi method, Householder method.
7. Mupad Computer practicals.
Textbooks .
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..
SEMESTER III (Compulsory Courses)
MIM 301 Theory of Computer Science
and Database Fundamentals
1. Lattices and Boolean Algebra : Introduction, Hasse Diagram, Boolean
Expressions, Boolean Identities, d.n.f., c.n.f., Logic Gates.
2. Finite State Machines - Introduction to Graphs, State Diagram, Transition Table,
Finite State Automata, Deterministic and non-deterministic
3. Formal Language : Alphabets, Language, Grammar, Kleene Closure, Regular
Expression, Finite Automata with output, Non deterministic automata, Moore
Machines, Mealy Machines
4. Turing Machine - Tape Head Program, Deterministic, Non Deterministic, Accept,
Reject, Loop
5. Grammar and Languages : Phase Structure Grammar, Types, Language
Recognition, Regular Sets, Kleen's Theorem, Regular Grammar, Pumping Lemma
Chomsky Hierarchy, Context Sensitive Grammar, Polish notation, Postfix, prefix,
Grammars and Cohen Languages
6. Introduction to DBMS : File System Vs DBMS, Storing Data, Levels of
abstraction, data independence, Structure of DBMS, Advantages of DBMS
7. Conceptual Design (ER Model) Additional constraints, weal entities,
Aggregation, generalisation, conceptual design using ER, Entituy Vs relationship,
binary Vs Ternary, Contraints beyond ER, Conceptual design for small to large
enterprises, Case Studies,
8. Relational data model : Conversion of ER to Relational model, integrity
constraints
9. Relational algebra : Preliminaries, Relational algebra
10. Relational calculus : Tuple calculus, Calculus Versus Relational Algebra
11. SQL : DDL, forms of a basic SQL query, union/ intersection/ except, nested
queries, Aggregate Operators, Aggregate functions, Null Values, impact on SQL
commands, outer joins, disallowing NULL, examles on SQL, Creating functions in
PLSQL, cursors, triggers.
12. Functional dependency : Introductory to schema refinement, use of
decomposition, problems relation to decomposition, functional dependencies,
lossless-join decomposition, Normalisation and its forms.
Reference Books :
1. Applied Abstract Algebra - Lid and Pilz 2nd Edition
2. Introduction to Computer Theory by Colen
MIM 302 Programming in C and C++
1. Introduction – Variables and Constants, Assignment Statements
2. Data Types – Integers, Floating Point Numbers, Void Data Type
3. Operators and Expressions – Operators, Assignment Operators, Precedence and
Associativity
4. Control Statements – If Statements, Looping, Switch, Break, Continue
Statements, Infinite Loops
5. Arrays and Pointers – Arrays, Initializing Arrays, Pointer Arithmetic, String
Manipulations, Multidimensional Arrays, Pointers to Pointers, Dynamic Memory
Allocation,
6. Structures and Unions – Structures and Functions, Array of Structures, Unions.
7. Files in C - File Operations, Command Line Arguments
8. Graphics in C - Concepts, Simple Programs
9. Preprocessor – Macro Substitution, Include Facility, Line Control
10. Types of Files – Logical File, Memory Hierarchy
12. File Organization – Fields and Record Organization, Overview of Indexes,
Types.
13. Tree Structured Indexing – Sequential and Binary Searching, ISAM, B+ Tree
Indexing, Static Hashing, Dynamic Hashing, Linear and Extendible Hashing
14. Database System – Data Independence, Relational Systems
15. Database System Architecture – Three Levels, Mapping, DBA, Database
Management Systems.
16. Object Oriented Programming Concepts - Data Types, New Operators, Classes
and access Specifiers, Array of Objects, Managing Console I/O stream class,
Formatted and Unformatted Console I/O – Usage manipulator
17. Functions in C++ : Call by reference, Function Overloading and default
argument, static lass member, friend function.
18. Constructors and Destructors – Types, Usage of destruct
19. Operator Overloading – Overloading Unary and Binary Operators, Overloading
using friend function, Usage of this pointer, Overloading insertion and extraction.
20. Inheritance – Types, Virtual Base class and abstract class, Virtual function and
pure virtual function.
21. Working with Files – File Operations, File Pointers and manipulations, File
Updation with random access
22. Templates : Class Template, function Templates, Overloading of Template
Functions, Basics of Exception Handling.
Textbooks :
1. Let us C - Kanitkar
2. Object Oriented Programming with C++ - E. Balagurswamy.
MIM 303 DATA STRUCTURES
Basics of digital Electronics: Bits, Bytes, Binary Addition ,Subtraction, Gates, Boolean
Algebra, Half Adder, Half-subtractor, Full-adder, 2's Compliment method of subtraction,
De-Morgan's Theorems, Flip-Flops, Registers
1. Uni-Processors: Computer Organization, History of microprocessors, Microprocessor
Architecture, RAM ,ROM, Instructions set, Machine and Assembly Language, Instruction
Decoding, Microprocessor Performance and trends, Progression of Intel Microprocessor,
Architecture of 8086,80286,80386,80486,80586.
Storage Devices : Magnetic Storage, Optical CR - R/ CD - RW
2. Input Output Devices: Keyboard, Mouse, Introduction to Computer Monitors: The
Basics, Display Technology: Background, VGA, DVI, Viewable Area, Maximum
Resolution and Dot Pitch, Power Consumption, Monitor Trends: Flat Panels, Introduction
to Printers: Inkjet Printers ,Working, Impact vs. Non-impact, Paper and Ink, Introduction
to Laser Printers: The Basics of Static Electricity, Drum, Fuser, Toner.
3. Introduction to Parallel computing: Concepts and Terminology, von Neumann
Computer Architecture, Flynn's Taxonomy ,Some General Parallel Terminology, Parallel
Computer Memory Architectures: Shared Memory ,Distributed Memory, Parallel
Programming Models: Overview Shared Memory Model ,Threads Model .Message
Passing Model ,Data Parallel Model Designing Parallel Programs :Automatic vs. Manual
Parallelization ,Understand the Problem and the Program ,Partitioning Communications,
Synchronization ,Data Dependencies ,Load Balancing Parallel Examples : Sistolic and
Wave front Array Processor ,PI Calculation Simple Heat Equation ,l-D Wave Equation
4. Introduction – Data Types, Data Structures, Arrays as Abstract Data Type
1-d,2-d, Multidimensions
5. Stacks – Push and Pop Operations, Stack Implementations using C, Stack
Applications, Recursions,
6. Queues – Concepts, Queue Applications, Priority Queues
7. Linked Lists – Static implementation using arrays, Doubly Linked List, Circular List,
Linked List Applications, Polynomial Representation, Stacks and Queues as Linked Lists
8. Trees – Terminology and Concepts, BinaryTree Representation, Linked
Representation , Binary Search Tree, Operations, Tree Traversals, Representing
General Trees as binary Tree
9. Sorting – Need and concepts, Techniques like Bubble, Quick, Selection, Insertion,
Tree, Merge, Radix
10. Graphs – Concepts, Graph Representation, Adjacency Matrix, Multilist, Traversals.
Text Books :
1. K Hwang and FA Briggs : Computer Architecture
2. Data Structures using C - Tenenbaum, Langsam, Augenstein.
3.Art of Computer Programming, D.Knuth (Prentice hall publishers).
SEMESTER IV (Compulsory Courses)
MIM 401 - JAVA PROGRAMMING
1. Introduction to Java Programming - Overview, Java Tools, Java Byte Code
2. Elementary Programming Concepts - Variables & Identifiers, Java keywords,
Data Types, Operators, Expression, Constants, Statements, Arrays
3. Classes and Packages - Defining classes, Static Members, Using packages,
Access Specifiers, Constructors, Finalisers referencing objects
4. Inheritance, nested and inner class - Extending classes, Abstract Class
Interface, Super Keyword, Final classes, Constructors and Inheritance, Dynamic
Binding, Overriding methods
5. Exception and Input and Output - Byte streams, Character streams, File i/o
basics, Introduction to exception, Try and catch block and finally block, Inbuilt
Exception.
6. String Handling and Exploring Java.lang - String Operations, Character
Extractions, Data Conversions, Modifying strings.
7. Applet and Event Handling and Controls
8. Input and Output package - Object serialization, reader and writer
9. Swings - Layout Manager Layout Manager swing Controls Components
Organizers, Jlish, Jtree, Jtables, Dialogue, File chooser, color chooser.
10. JDBC - The design of JDBC, JDBC programming concepts making the
connection, statement and result set class, Executing SQL commands, Executing
Queries.
11. Multithreading - Running multiple threads, The runnable interface Threads
priorities Daemon, Thread States, thread groups Synchronization and Interthread
Communication Deadlocks.
Textbook :
A Complete Reference Java 2 - Herbert Schildt.
MIM 402 - OPERATING SYSTEMS
1. Introduction to Operating Systems - Batch System, Time sharing system,
personal computer system, Parallel system, Distributed System, Real Time
System
2. File System - File Systems, File Concepts, Allocation Methods, Access
Methods, Directory Structure
3. Threads - Overview, Multithreading models, Threading Issues.
4. CPU Scheduling - Basics, Scheduling criteria, Scheduling Algorithms, Multiple
Processor Scheduling.
5. Disk and Drum Scheduling
6. Memory Management - Background, Swapping, Contiguous allocation,
Paging, Segmentation, Segmentation and paging- combined system, virtual
memory concept, Demand paging, Paging replacement algorithms.
7. Concurrent Processing and programming - Review of process concepts,
Hierarchy of processes, Problem and solution algorithm, Semaphores, Overview
of Concurrent programming, Modularization, Synchronization in Windows 2000,
Concurrent languages.
8. Deadlocks - System model, Deadlock characterization, methods of handling
Deadlocks, Deadlock Prevention, Deadlock Avoidence, Deadlock Detection,
Recovery from Deadlock.
Textbook :
Operating System concepts by Silberschatz 6th edition
MIM 403 DESIGN AND ANALYSIS OF ALGORITHMS
1. Mathematical Foundation, Growth Functions, Summations, Recurrences
Substitutions, Iterations, Master Methods, Counting and probability
2. Sorting, Heap Sort, Quick Sort, Merge Sort, Sorting in linear Time, Medians
and Order Statistics
3. Dynamic Programming - Matrix chain Multiplication, longest common
subsequence, optimal polygon triangularisation.
4. Greedy Algorithm.
5. Graphs - Traversals, Topological sort, Minimum spanning trees, single source
shortest path, All pair shortest path, Maximum flow problems.
6. Sorting Networks - Comparision, bitonic sort and merge sort networks
7. Parallel Algorithms - CRCW, EREW algorithms efficiency sorting linear system
problem, Matrix Operations, Strassens Algorithm and matrix inversion.
7. FFT - Polynomials DFT, FFT
8. Number Theoretic Algorithm - Rabin - Karp, KMP, Bower - Moore algorithms
9. Geometric Algorithms - Finding convex hall, closes pair of points, linear
programming problem
10. NP Completeness - P and NP classes, NP completeness and reducibility
11. Approximation Algorithms - Vertex cover problem, traveling salesman
problem, set covering and subset sum problems
Textbook :
1. Introduction to Algorithms : T. H Coreman, Leiserson, Rivest.
OPTIONAL SUBJECTS FOR SECOND YEAR
MIM 01 - OPERATIONS RESEARCH
Unit I - Kuhn – Tucker conditions of Optimality – Quadratic Programming
(Sections 19.2.2B, 20.2.2)
Unit II - Inventory Models
(Sections 14.1 to 14.3)
Unit III - Queueing Models
(Section 15.1, 15.2, 15.4, 15.5)
Unit IV - Project Scheduling By PERT – CPM
(Sections 13.1 to 13.4)
Unit V - Simulation Modeling with SIMNET – II
(Sections 17.1 to 17.10)
Textbook : Hamdy A.Taha, Operations Research, Fifth Edition, Prentice
Hall of India.
MIM 02- INTEGRAL TRANSFORMS
A) Classification of Liner Integral Equations : Fredholm, Volterra, Integro-
Differential Equations, Singular Integral Equations, Converting Volterra
Equation to ODE, Conversion of IVP to Volterra equation Conversion of
BVP to Fredholm equation
B) Fredholm Intergral Equations - Decomposion method, Direct
Computation method, successive approximation method, method of
successive substitutions, Homogeneous Fredholm Equations, Comarison
between alternative methods.
C) Volterra Interal Equation - Adomian Decomposion method, Series
solution method, converting Volterra equation to VIP, Successive
Approximation method, successive substitution method, comparison
between alternative methods.
D) Integro-Differential Equations - Introduction, Direct Computation
method, Adomian Decompostion Method. Conversion to Fredholm
integral Equation.Volterra Intego-Differential equations Series Solution,
Decomposition Method, Conversion to IVP.
E) Singular Integral Equations - Abel problem , Genralized Abel Integral
Equation, Weakly-singular Volterra Equations.
F) Non Liner Integral Equation`s - Non liner Fredholm Integral
equations,Direct Computation, decomposition method, Non liner Volterra
Integral Equation, Series solution, Decomposition method
G) Existence and uniqueness of solutions using fixed-point theorems in
cash of Liner and nonliner Volterra and Fredholm integral equations.
H) Fourier Transforms : [FT] Definition Properties evaluation of Fourier
and inverse Fourier transforms of functions, Convolution theorem for FT.
Sine and Cosine Fourier transforms. Solving differential equations and
interal equations using FT.
I) Laplace Transform :Definition Properties, evaluation of Laplace and
Inverse Laplace transforms of functions. Convolution theorem for Laplace
Transforms.Solving initial value problem using Laplace Transforms.Solving
integral equation using Laplace Transforms.
J) Mellin Transforms : Definition, properties and evaluation of
transforms, Convolution theorem for Mellin transforms, Applications to
integral equations.
Reference Books:
1) A First course in integral equations –A.M. Wazwaz (1997) (world
Scientific)
2) Introduction to Integral Equation with Applications –A.J. Jerri
(1999)Second edition Wiley Interscience.
\
MIM 03 - NUMBER THEORY
1. Congruences : Solutions of congruencs. Chinese Remainder
Theorem,Techniques of numerical calculation Public-Key Crytography.
2. Prime power moduli. Prime modulus. Primitive roots and power residues,
Congruences of degree two.
3. Quadratic Residues, Quadratic Reciproctiy.
4. Greatest integer function, Arithmetic functions, Multiplicative functions,Dirichlet
multiplication. Mobius Inversion Formula.
5. Diophantine equations. The equation ax + by = c, Pythagorean triangles,
Assorted examples. Rational points on curves.
Books: 1.Niven and Zuckerman, An introduction to the Theory of Numbers,Wiley
Publishers.
2.David Burton,Elementary Number Theory.
MIM 04 - CODING THEORY
1. Error detection: correction and decoding: Communication channels,Maximum
likelihood decoding,Hamming distance,Nearest neighbour / minimum distance
decoding, Distance of a code.
2. Linear codes: Vector spaces over finite fields,Linear codes, Hamming weight,
Bases of linear codes, Generator matrix and parity check matrix, Equivalence of
linear codes, Encoding with a linear code,Decoding of linear codes, Cosets,
Nearest neighbour decoding for linear codes, Syndrom decoding.
3. Cyclic codes: Definitions, Generator polynomials, Generator and parity check
matrices, Decoding of cyclic codes, Burst-error-correcting codes.
4. Some special cyclic codes: BCH codes, Defintions, Parameters of BCH codes,
Decoding of BCH codes.
Reference: 1. San Ling and Chaoing xing, Coding Theory- A First Course
2. Applied Abstract Algebra - Lid and Pilz 2nd Edition
MIM 05 - GRAPH THEORY
1. Fundamental concepts : Definitions and examples, graphs as models,
matrices and isomorphism, paths, connected graphs, bipartite graphs,
extremality vertex degree, the Pigeonhole principal, Turan`s theorem, degree
sequences, graphic sequences, degree and digraphs.
2. Tree and Distances : Properties of tree, distance in graphs, stronger results,
disloint spanning trees, shortest paths, tress in computer science, Eulerian
circuits.
3. Matching and Factors : Matching in bipartite graphs, maximum matchings,
Hall`s matching conditons, Min-Matching in bipartite graphs, sets, applications
and algorithms, maximum bipartite matching, weighted bipartite matching, in
general graphs, Tutte`s 1-factor theorem,f-factors of graphs.
4. Connectivity and Paths : Cuts, connectivity, edge-connectivity, blocks, 2-
connected graphs, connectivity of digraphs, k connected and k-edge
connected graphs, applications of Meger`s theorem, Network flow problems
maximum network flow, integral flows.
5. Edges and cycles : Line graph and edge-colouring, Hamiltonian cycles:
necessary conditions, Sufficient conditions.
Recommended Book:
Douglas B. West, Introduction to Graph Theory Prentice- Hall, New Delhi (1999)
Reference Books:
1. John Clarke and D.A. Holton, A First Look at Graph Theory, Allied Publisher
(1991)
2. Nora Harsfield and Gerhard Ringel , Pearls Theory, Academic Press (1990)
3. Harary, Graph Theory, Narosa Publishers, New Delhi (1989)
MIM 06 - LATTICE THEORY
Two definitions of lattices, Hasse diagrams, homorphism, isotone maps,
ideals, congruence relations, congruence lattices, the homorphism theorem,
product of lattices, complete lattice, ideal lattice, distributive –modular
inqualities and identifies, complemets, pseudocomplements, Boolean lattice
of pseudocomplements, join and meet-irreducible elements.
Characterizatin theorems and representation theorems-Dedekind`s modularity
criterion Bikhoff`s distributivity criterion, hereditary subsets, rings of sets,
Stone theorems,Nachbin theorem, staements of Hashimoto`s theorem.
Modular lattices, isomorphism theorem, Upper and lower covering conditions,
Kurosone theorem, independent sets (Drops results involving projectivity and
sublattice generated by sets / elements )
Semimodular lattices Jordan-Holder chain condition, Modular pair, Msysmmetric
lattices.
Book- Genral Lattice Theory
Auther- G. Gratzer (Academic Press 1978)
Chap. 1 Section 1,2,3,4,6, Cha. 2 Section-1, Chap.3. Section –1,2.
MIM 07 - COMPUTATIONAL GEOMETRY
The course text will be We will cover most of the book, adding some additional material.
1. Geometric primitives [Chap. 1]
2.Line intersection [Chaps. 2] plus randomized incremental
3. Triangulation and visibility and [Chaps. 3,15]
4. Linear programming in two and three dimensions [Chap. 4]
5. Orthogonal range searching [Chaps. 5,10]
6. Point location and Binary Space Partitions [Chaps. 6,12]
7. Voronoi diagrams and Delaunay triangulation [Chaps. 7,9]
8. Convex hulls [Chap. 11]
9. Non-orthogonal range searching [Chap. 16]
10. Curved Elements (Bezier, B-Splines)
11. Curve Reconstruction (reconstruction a curve(surface) from sample
points)
12. 3-Dimensional Geometry
Text Book :
Computational Geometry Algorithms and Applications, 2nd ed., by de
Berg, van Kreveld, Overmars, and Schwarzkopf (Springer-Verlag, 2000).
MIM 08 - CRYPTOGRAPHY
1. Introduction : Overview of course, Classical cryptography [parts of
Chapter 1].
2. Secret Key Encryption : Perfect Secrecy - One time pads [Chapter 2.1],
Stream ciphers and the Data Encryption Standard (DES) [Chapter 3
(excluding 3.6)], The Advanced Encryption Standard (AES) - adopted
September 2000.
3. Public Key Encryption : Factoring and the RSA encryption [Chapter 4.1 -
4.4], Discrete log. Diffie-Hellman Key Exchange [Chapter 8.4 (only pages
270-273)]. ElGamal encryption [Chapter 5 (only pages 162-164)] , Digital
Signatures [Chapter 6 (excluding 6.5 - 6.6)], One-time signatures, Rabin
and ElGamal signatures schemes, Digital Signature Standard (DSS).
4. Hashing : Motivation and applications. Cryptographically Secure
Hashing. [Chapter 7.1-7.3,7.6], Message Authentication Codes (MAC).
HMAC, Network Security , Secure Socket Layer (SSL), I Psec.,Secret
Sharing, Definition. Shamir's threshold scheme [Chapter 11.1], Visual
secret sharing schemes.
Reerence Book :
D. R. Stinson. CRYPTOGRAPHY: Theory and Practice. CRC Press.
1995.
MIM 09 - FINANCIAL MATHEMATICS
1. Introduction to options and markets: types of options, interest rates and
present values.
2. Black Sholes model : arbitrage, optionvalues, 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.
3. Binomial methods : option valuation, dividend paying stock, general
formulation and implementation.
4. Monte Carlo simulation : valuation by simulation
5. 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.
6. Lab component: implementation of the option pricing algorithms and
evaluations for Indian companies.
References:
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.
MIM 10 - MODELLING AND SIMULATION
1. Introduction to modelling and simulation. System analysis, clasification
of systems. System theory basics, its relation to simulation.
2. Model classification: conceptual, abstract, and simulation models.
Heterogeneous models. Methodology of model building
3. Simulation systems and languages, means for model and experiment
description. Principles of simulation system design
4. Parallel process modelling. Using Petri nets and finite automata in
simulation
Models o queuing systems. Discrete simulation models. Model time,
simulation experiment control.
5. Continuous systems modelling. Overview of numerical methods used for
continuous simulation.
6. Combined simulation. The role of simulation in digital systems design
7. Special model classes, models of heterogeneous systems.
8. Checking model validity, verification of models. Analysis of simulation
results
9. Simulation results visualization. Interactive simulation
10. Design and control of simulation experiments. Model optimization
11. Generating, transformation, and testing of pseudorandom numbers.
Stochastic models, Monte Carlo method
12. Overview of commonly used simulation systems.\
References :
1. Fishwick P.: Simulation Model Design and Execution, PrenticeHall,
1995, ISBN 0-13-098609-7
2. Law A., Kelton D.: Simulation Modelling and Analysis, McGraw-Hill,
1991, ISBN 0-07-100803-9
3. Rábová Z. a kol: Modelování a simulace, VUT Brno, 1992, ISBN 80-
214-0480-9
4. Ross, S.: Simulation, Academic Press, 2002, ISBN 0-12-598053-1.
MIM 11 - ARTIFICIAL INTELLIGENCE
1. Overview of history and goals of AI : Tentative definitions. Turing's test.
Knowledge vs. Symbolic Level. Relations with other disciplines, from
Philosophy, to Linguistics, to Engineering. Review of AI successes and
failures.
2. State Spaces, Production Systems, and Search : State Space
representation of problems. Problem solving as search. Constraints.
Definition and examples of Production Systems. Heuristic search
techniques. Two person games.
3. Knowledge Representation Issues: Procedural Knowledge
Representation vs. Declarative Knowledge + Reasoning. Facts, General
Assertions, Metaknowledge. The Frame Problem.
4. Using First-Order Logic for Knowledge Representation : Propositional
Logic: Semantics and Deduction. First Order Logic: Semantics and
Deduction. Unification. Resolution-based theorem proving. Using theorem
proving to answer questions about the truth of sentences or to identify
individuals that satisfy complex constraints. Logic Programming.
5. Common Sense Reasoning : Nonmonotonic reasoning and modal logics
for nonmonotonic reasoning. How to deal with Agents and their Beliefs.
6. Weak Slot-and-Filler Structures: Semantic Nets and Frames. Scripts for
representing prototypical combinations of events and actions.
7. Rule-Based Systems: Pattern-matching algorithms. The problem of
Control in Rule Based Systems. The Rete Algorithm.
8. Planning: Representing plans. Partial order planning. Planning
applications.
9. Statistical Reasoning: Use of Certainty Factors in Rule-Based Systems.
Associating probabilities to assertions in first-order logic. Bayesian
Networks. Fuzzy Logic.
10. Learning: Learning to classify concepts using features of their
instances. Learning a concept [Induction] from examples. Explanation-
Based Learning. Version Spaces. Neural Nets with back propagation.
Text Book :
Artificial Intelligence: A Modern Approach : Prentice-Hall, 1995
MIM 12 - SYMMETRIES
1. Symmetry of plane figures of motions of the plan, finite groups of
motions, discrete groups of motion, symmetry, cosets, counting formula,
permutation representations, finite subgroups of the generators and
relations
2. Operation of a group on itself, class equation of the isocahedral groups
operations on subsets groups of order12, free group generators and
relations.
3. Bilinear forms, symmetric forms, orthogonality, geometry associated to a
positive from, Hermitian forms, spectral theorem, conics and quadrics,
normal operators, skew symmetric forms.
Text: Artin: Algebra (Prentice-Hall)
Chapter 5, 6 (sections 1, 2, 3 and7)
MIM 13 - WAVELET ANALYSIS
1. Fourier Transform : Fourier transform on L1(R) and L2(R) and basic
properties and examples
2. Windowed Fourier Transform : Motivation and definition of Windowed
Fourier Transform and examples, Time frequency localization, the
reconstruction formula.
3. Continuous Wavelet Transform : Motivation and Definition of the
wavelet transform and examples, Basic properties, The reconstruction
formula, Frequency localization, Orthonormal Wavelets.
4. Multiresolution Analysis : Definition of MRA and examples, Properties of
scaling functions and orthonormal wavelets bases, Construction of
orthonormal wavelets.
Textbook :
1. Bachman G, L. Narici & E. Beckensterin: Fourier and Wavelet
Analysis, Springer-Verlage (2000)
2. Chui C. K. : An Introduction to Wavelets, Academic Press (1992)
.
MIM 14 - COMBINATORICS
1. Counting Methods for selections arrangements : Basic counting principles,
simple arrangements and selections, arrangements and selection with
repetition , distributions, binomial, generating permutations and combinations
and programming projects.
2. Generating function : Generating function models, calculating of generating
functions, partitions exponential generating functions, a summation method.
3. Recurrence Relations : Recurrence relation model, divide and conquer
relations, solution of inhomogeneous recurrence relation, solution with
generating functions.
4. Inclusion-exclusion : Counting with Venn diagrams inclusion formula,
restricted positions and rook polynomials.
5. Ramsey Theory : Ramsey theorem, applications to geometrical problems.
Recommended Books:
1. Alan Tucker, Applied Combinatorics (third edition), John Wiley & sons , New
York (1995)
2. V. Krishnamurthy, Combinatorial, Theory and Applications, East West Press,
New Delhi (1989) Scientific, (1996)
MIM 15 - PARTIAL DIFFERENTIAL EQUATIONS
1. First order PDE, Linear Equations of first order, Charpit’s method, Jacobi’s
method, Quasi-linear equations, Non-linear first order PDE.
2. Second ordered PDE: Genesis, Classification, One dimensional Wave
equation, Laplace equation, Boundary Value Problems, Maximum and Minimum
Principles, Cauchy Problem,
3. Heat Conduction Problem, Duhamel’s Principle
Text Book :-
An Elementary Course in Partial Differential Equations by T. Amaranath
(Narosa) Chapters 1-2.
MIM 16 - FUZZY LOGIC
1. Fuzzy Sets and Operations on Them
2. Fuzzy Relations
3. Fuzzy Rules
4. Approximate Reasoning
5. Fuzzy Logic
6. Fuzzy Systems (e.g., Fuzzy Logic Control)
7. Fuzzy Logic in Pattern Recognition
8. Fuzzy Decision Making
9. Fuzzy Logic Applications
TextBook :
George J. Klir, Bo Yuan, Fuzzy Sets and Fuzzy Logic: Theory and
Applications, Prentice Hall PTR, 1995.
Hao Ying, Fuzzy Control and Modeling: Analytical Foundations and
Applications, IEEE Press, 2000. (Do not need to buy it. A handout will be
distributed).
MIM 17 - STATISTICS AND PROBABILITY
1. Introduction to Discrete Probability : Intuitive concepts: probability of an
event as a measure between 0 and 1; random variable; probability
distribution; frequency interpretation of probability; random numbers; coins,
dice, and other games; simulations; odds; historical development of
probability; random walks.
2. Formal concepts: sample space, outcomes, and events; random
variable; discrete distribution functions and axioms of probability; unions,
intersections, and complements; properties of probabilities, principle of
inclusion and exclusion; tree diagrams; uniform distributions over finite
sets, symmetry; infinite sample spaces with discrete probabilities.
3. Introduction to Continuous Probability : The intuitive problems with
probabilities over space (line, plane, Rn in general). Monte Carlo
simulations, Buffon's needle. Formal concepts: density function for a
continuous random variable; integration; cumulative distribution functions;
derivatives; exponential density function;
4. Conditional Probability : Intutive concept of conditional probability;
formal definition of conditional probability; Bayes' formula for inverting
conditional probabilities; independent events; joint distribution functions;
independent random variables; independent trials. Conditional density
functions for continuous distributions; the beta distribution
5. Distributions and Densities : Uniform continuous distributions; geometric
distribution; Poisson distribution; exponential and gamma distributions;
introduction to queueing theory; normal (Gaussian) distribution; Chisquared
distribution
6. Expected Value and Variance : Expected value for discrete random
variables, expectation; linearity of expectation; expectation of independent
random variables; conditional expectation; variance and standard
deviation; variance of various distributions. Expectation and variance for
continuous random variables.
7. Sums of Random Variables : Analysis of sums of independent random
variables with identical distributions, that is, independent trials.
8. Law of Large Numbers : Chebychev inequality, law of averages, law of
large numbers.
9. The Central Limit Theorem : The central limit theorem for Bernoulli trials,
binomial distributions again, the normal distribution, the general central
limit theorem.
TextBook :
Charles M. Grinstead and J. Laurie Snell's textbook Introduction to
Probability, published by the American Mathematical Society, 1997
.
MIM 18 - FLUID DYNAMICS
1. Physical Properties of fluids. Concept of fluids, Continuum Hypothesis,
density, specific weight, specific volume.
2. 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.
3. 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.
4. Conservation Laws : Equation of conservation of mass, equation of
conservation of momentum, Navier Stokes equation, equation of moments of
momentum, Equation of energy, Basic equations in different co-ordinate
systems, boundary conditions.
5. 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.
TextBooks :
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.
MIM 19 - BANACH ALGEBRA
1. Relatively compact sets, compactly continuous operators, finite
dimensional operators, transformation that is bounded but not completely
continuous, a type of transformation that is always completely continuous,
further properties of completely continuous transformations.
2. Spectra and the resolvent set, Approximate proper values, Banach
Algebra With identity, compactness of the spectrum, the resolvent
operator, Spectral radius and spectral mapping theorem for polynomials,
the Gelfand Theory.
3. Sesquilinear functions : Spectral results for normal and completely
continuous operators, numerical range
4. The Fredholm alternative theory, the spectral theorem for bounded,
normal Finite dimensional operators.
5. Commutative Banach Algebras, ideals and homomorphisms.
TextBook :
1. Walter Rudin: Functional Analysis, Tata MeGrow Hill Publishing co.
New Delhi.
\
MIM 20 - BOUNDARY VALUE PROBLEMS
1. Definition of boundary Value Problems, the heat equation, wave
equation, Laplac`s equation, the Fourier method, Liner Operators,
Principal of Superpositin, series solutions, uniform convergence
(weierstrass M-test), separation of variables, non homogeneous
conditions, Sturm-Liouville problems, formal solutions, the vibrating
string.
2. Orthogonal sets of functions, Generalized Fourier series, Best
approximation in the mean, Convergence in the mean, the orthonormal
trigonometric functions, other types of orthogonality.
3. sturm-Liouville Problem and applications, orthogonality and
uniqueness of eigenfunctions, method of solutions, surface heat
transfer other boundary value problems.
4. Bessel function Jn, recurrence relation, the zero of Jo (X) and related
functions, Fourier-Bessel series, Temperatures in a long cylinder.
5. Legendre polynomials, orthogonality of Legendre polynomicals,
Legendre series, Dirichlet Problem in spherical regions.
Prescribed Text Book
R.V. Churchill and J. Brown.: Fourier Series and Boundary Value Problems
(4th edtion)(Publisher: McGraw-Hill Book Company)
MIM 21 - BAER* RINGS
1. Rings with involution
2. Poset of projections
3. Proper involutions and C*-algebras
4. Rickart *-rings and Bear *-ring
5. Weakly Rickart *-rings and unitification
6. Central cover
7. Additivity of projections
8. Comparability axioms and parallelogram law
9. Finite and abelian projections
10. Structure Theorem
Prescribed Book: Bear *-rings, S.K. Berberian, Springer
MIM 22 - MATROID THEORY
1. Basic definitions and examples
Independent sets and circuits, bases, rank, closure, geometric
representations of matroids of small rank, transversal matroids, the lattice
of flats, the greedy algorithm.
2. Duality
The definition and basic properties, duals of representable matroids, duals
of graphic matroids, duals of traversal matroids.
3. Minors
Contraction, Minors of certain matroids, flats and the sum theorem
4. Connectivity
Connectivity, for graphs and matroids, properties of matroid connectivity,
more properties of connectivity.
Text Book:
James G. Oxley, Matroid Theory Science Publications, Oxford
(1992)(Chapter 1 to 4)
MIM 23 - SPERNER THEORY
1. Introduction and sperner`s Theorem: A Simple intersection result,
Sperner`s theorem, Theorem of Bollobaas.
2. Normalized Matching and rank numbers: Sperner`s proof,system
of distinct representatives, L Y M inequalities, and normalized
matching property. Rank numbers, some examples.
3. Systemmebric Chain: Sysmmertric chain decompositions,
Dilworth`s theorem, symmetric chains of sets, Application to
Nested chains, posets with sysmmetric chain decompositions.
4. Rank numbes of multisets. Unimodality and log connectivity, the
normalized matching property. The largest size of a rank number.
Prescribed Text Book
Ian Aderson : Combinatorics of Finite Sets. (Oxferd Science
Publications)
Reference Book.
Konrad Engel: Sperner Theory (Cambride University Press)
MIM 24 - DIFFERENTIAL EQUATIONS AND DYNAMICAL
SYSTEMS
1) Liner Systems: Uncoupled Liner Systems, Diagonalization,
Exponential of operators Fundamental theorem for liner systems, liner
systems in R, Complex eigenvalues, multiple eigenvalues, jorden
Canonical Forms, stability theory Nohomogeneous Liner systems.
2) Nonlinear Systems: Local Theory, Fundamental existence theorem
dependence on initial conditions and parameters, the maximal interval
of existence, Flow defined by a differential equation. Linearization,
stable manifold theorem, Hartman-Grobman theorem, Stability and
Liapunov functions, Saddles, Nodes, Foci and centers, Nomhyperbolic
critical points in Rn, Gradient and Hamiltonian system.
Text (1) L. Perko- Differential Equations and Dynamical systems (1991)
Springer-verlag
(2) Hirsch and Smale – Differential Equations, Dynamical Systems,
and Liner Algebra - Academic Press, New York, (1974)
MIM 25 - MECHANICS
1. Sec 1.1-1.6 Survey of Elementary Particles
2. Sec. 2.1-2.7 Variational Principles & Lagranges` Equation
3. Sec.3.1-3.7 Central Force problem
4. Sec. 4.1-4.10 Kinematics of rigid body motion
5. Sec. 8.1-8.2 Hamilton Equations of motion
6. Sec.9.1-9.9 Canonical Transformations
Text Book: Classical Mechanics by Godlstein, Poole and Safko (Third
Edition) 2002, Person Education Inc. Supplementary Reading (1) Rana &
Joag Classical Mechanics (Tata McGraw Hill)
The post graduate Course is of Three years duration having fifty percent
courses of Mathematics and fifty percent courses of Computer Applications,
one full semester is devoted for Industrial training.
This course is designed as per the guidelines of UGC for creating the
opportunities in the IT field for Mathematics students. As, so far IT
industries used to have Computer Experts and Mathematicians with
separate job profiles, this course eliminates this discrimination and creates
Technologists with both the flavors. It is to be worth mentioning here that
the course took birth due to heavy need of mathematical knowledge and
mathematical aptitude in industries and R & D sectors. This course deals
with heavy mathematics; as such the M. Sc. (Tech.) degree is equivalent to
M. Sc. (Mathematics) course. This degree course is also equivalent to
Master of Computer Applications (MCA) as well as M. Tech. (Computer
Science).
SYLLABUS
M. Sc. (TECH.)
(Industrial Mathematics with Computer Applications)
FIRST YEAR
SEMESTER I (All are compulsory and each course is of 6 credit points)
MIM 101 Advanced Calculus
MIM 102 Topology
MIM 103 Measure and Integration
MIM 104 Algebra
MIM 105 Differential Equations
Total credits: 30 points
SEMESTER II (All are compulsory and each course is of 6 credit points)
MIM 201 Functional Analysis
MIM 202 Complex Analysis
MIM 203 Field Theory
MIM 204 Linear Algebra
MIM 205 Numerical Analysis.
Total credits: 30 points
SECOND YEAR
SEMESTER III
Compulsory Courses: (each course is of 6 credit points)
MIM 301 Theory of Computer Science and Database Fundamentals
MIM 302 Programming in C and C++
MIM 303 Data Structures
Optional Courses: (each course is of 5 credit points)
Any two of the Optional Courses of Second Year (See the list)
Total credits: 28 points
SEMESTER IV
Compulsory Courses : (each course is of 6 credit points)
MIM 401 Java Programming
MIM 402 Operating Systems
MIM 403 Designs and Analysis of Algorithms
Optional Courses : (each course is of 5 credit points)
Any two of the Optional Courses of Second Year (See the list)
Total credits: 28 points
SECOND YEAR OPTIONAL COURSES (each course is of 5 credit points)
MIM 01. Operations Research
MIM 02. Integral Transforms
MIM 03. Number Theory
MIM 04. Coding Theory
MIM 05. Graph Theory
MIM 06. Lattice Theory
MIM 07. Computational Geometry
MIM 08. Cryptography
MIM 09. Financial Mathematics
MIM 10. Modelling and Simulation
MIM 11. Artificial Intelligence
MIM 12. Symmetries
MIM 13. Wavelets
MIM 14. Combinatorics
MIM 15. Partial Differential Equations
MIM 16. Fuzzy Logic
MIM 17. Statistics and Probability
MIM 18. Fluid Dynamics
MIM 19. Banach Algebras
MIM 20. Boundary Value Problems
MIM 21. Baer * Rings
MIM 22.Topics in Matroid Theory
MIM 23.Topics in Sperner Theory
MIM 24.Differential Equation and Dynamical System
MIM 25.Mechanics
THIRD YEAR
SEMESTER V
Compulsory Courses : (each course is of 6 credit points)
MIM 501 Software Engineering
MIM 502 Computer Networks
MIM 503 Computer Graphics and Emerging Technologies
Optional Courses : (each course is of 5 credit points)
Any two of the Optional Courses of Third Year (See the list)
Total credits:28 points
SEMESTER VI
IT PROJECT (In the Industry )
Total credits:26 points
OPTIONAL COURSES: THIRD YEAR
MIM 51. Complex Analysis II
MIM 52. Representation Theory of Groups
MIM 53. Fourier Analysis on Finite Groups
MIM 54. Differential Geometry
MIM 55. Non-Linear Dynamical System
MIM 56. Emerging Technologies
MIM 57. Algebraic Topology
MIM 58. Analysis on Manifolds
MIM 59. Projective Geometry
MIM 60. Algebraic Geometry
MIM 61. Algebraic Number Theory
MIM 62. Algebraic Curves
MIM 63. Commutative Algebra
MIM 64. Advanced Topics in Lattice Theory
MIM 65. Advanced Topics in Graph Theory
MIM 66 Advanced Topics in Matroid Theory
MIM 67. Advanced Topics in Group Theory
MIM 68. Advanced Topics in Ring Theory
MIM 69 Topics in Non Commutative Rings.
SEMESTER I
MIM 101 Advanced Calculus
1. Compact and Connected Subsets of Rn.
2. Differentiation : Derivative, Continuously Differentiable functions, Chain rule,
Inverse function theorem, Implicit function theorem.
3. Integration: integral over a rectangle, Existence of the Integral, evaluation of
the integral, integral over a bounded set and rectifiable sets, improper integrals
4. Change of Variable Theorem (Proof of one variable) and Statement of n-variables
(with Illustrations)
5. Line Integrals with Applications
Textbooks :
1. Analysis on Manifolds - J.R. Munkres : Sections 4 to 15 and Section 17.
2. Calculus (Volume II) - T.M. Apostol, Chapter 10 : Sections 10.1 to 10.9
MIM 102 Topology
1. Cartesian Products, Finite Sets, Countable and Uncountable Sets, Infinite Sets
and Axiom of Choice, Well Ordered Sets.
2. Topological Spaces : Basis for a topology, Order topology, Subspace Topology,
Product topology, closed sets and limit points, Continuous functions, Metric
Topology
3. Connected spaces, Connected Subspaces of Real Line, Components and Local
Connectedness, Compact spaces, Compact Subspaces of the Real Line, Limit point
compactness, Local Compactness.
4. Countablity and Separation Axioms : Countability Axioms, Separation axioms
Normal Spaces, Urysohn's Lemma, Titetz Extension Theorem (Without Proof),
Tychonoff's Theorem , Metrization Theorem (without proof).
Textbook :
J. R Munkres : Topology : A First Course. ( Prentice- Hall)
Chapter 1 : Sections 5,6,7,9,10
Chapter 2 : Sections 12 to 21
Chapter 3 : Sections 23 to 29
Chapter 4 : Sections 30 to 35
Chapter 5 : Section 37
MIM 103 Measure and Integration
1. Preliminaries : Cardinal Numbers and Countability, Properties of Open Sets,
Cantor Like Sets
2. Measure on Real Line : Lebesgue Outer Measure, Measurable Sets, Regularity,
Measurable Functions, Borel and Lebesgue Measurability
3. Integration of Functions on Real Variable : Integration of Non Negative Functions,
General Integral, Integration of Series, Riemann and Lebesgue Integral
4. Differentiation : Functions of Bounded Variation, Lebesgue Differentiation
Theorem, Differentiation Theorem, Differentiation and Integration
5. Abstract Measure Spaces : Measures and Outer Measures, Extension of
Measures, Uniqueness of Extension, Completion of Measure, Measure Spaces,
Integration with respect to Measure
6. Inequaties and Lp spaces : The Lp Spaces, The Convex Functions, Jensen's
Inequalities, Inequalities of Holder and Minkowski, Completion of Lp.
7. Convergence : Convergence in Measure, Almost Uniform Convergence,
Convergence Diagrams, Counter Examples
Textbook :
Measure Theory and Integration : G. de Barra (New Age International Ltd,
Publishers)
Chapter 1 : Sections 1.5 to 1.7.
Chapter 2 : Sections 2.1 to 2.5.
Chapter 3 : Sections 3.1 to 3.4.
Chapter 4 : Sections 4.1 to 4.5.
Chapter 5 : Sections 5.1 to 5.6.
Chapter 6 : Sections 6.1 to 6.5.
Chapter 7 : Sections 7.1 to 7.4.
MIM - 104 Algebra
1. Introduction to Groups, Symmetries of a square, Dihedral Groups, Examples and
properties of Groups, Finite Groups, Subgroups, Cyclic Groups.
2. Permutation Groups and its properties,check digit scheme Based on D5,
Isomorphisms, Cayley's Theorem, Cosets and Lagrange's Theorem,Applications of
Cosets and Permutaion Groups, Rotation Group of a Cube and Soccer.
3. External Direct Products with Applications, Normal subgroups and Factor Groups
and Applications.
4. Group Homomorphisms, First Isomorphism Theorem, Fundamental Theorem of
Finite Abelian Groups.
5. Sylow Theorems, Applications of Sylow Theorems.
6. Introduction to Rings , Integral Domains, Fields, Ideals, Ring Homomorphisms,
Polynomial Rings, Factorisation of Polynomials, Divisibilty in Integral Domains.
Text Book :
Contemporary Abstract Algebra (Fourth Edition)-Joseph Gallian (Narosa Publishing
House).
Chapters: 1 to 18 and 24..
MIM 105 Differential Equations
1. Linear equation of first order.
2. Applications : Orthogonal Trajectories, Population Growth and Decay, Newton's
Law of Cooling, Free Falling Bodies
3. Linear equations with constant coefficients : Second order homogeneous
equations, initial value problems, linear dependence and independence,
nonhomogeneous equations of nth order, Algebra of Constant Coefficients.
4. Applications : Simple Harmonic Motion, Damped Motion, Forced Motion, Other
Applications in Electronics and Pendulum Problem
5. Linear equations with variable coefficients : Initial value problems, solutions of the
homogeneous equation, Wronskian and linear independence, reduction of order
nonhomogeneous equations Legendre equation.
6. Linear Equations with regular singular points : Euler equation, second order
equation with regular singular points, exceptional cases, Bessel equation.
7. Existence and uniqueness of solutions to first order equations : Separation of
variables, exact equations, Method of successive approximations, Lipschitz
condtion, Approximation to and uniqueness of solutions.
8. Existence and uniqueness of solutions to first order equations to systems and 2nd
order equations
9. Laplace transforms: Laplace Transforms, Inverse Laplace Transform, Solving
Initial Value Problems with Laplace Transforms.
Text Book :
1) E. A. Coddington : An Introduction to Ordinary Differential Equatins (Prentice-
Hall).
Chapter 1 : Sections 1 to 7.
Chapter 2 : Sections 1 to 12.
Chapter 3 : Sections 1 to 8.
Chapter 4 : Sections 1 to 4, 7 and 8.
Chapter 5 : Sections 1 to 6.
Chapter 6 : Sections 6,7.
2) Differential Equations with Mathematica - Martha L. Abell and James P. Braselton
(Second Edition) Acadamic Press..
SEMESTER II
MIM 201 Functional Analysis
1. Normed spaces, continuity of linear maps, Hahn - Banach theorems, Banach
spaces.
2. Uniform bounded principle, Application - Divergence of Fourier Series of
Continuous Functions, closed graph theorem, Open mapping theorem, bounded
inverse theorem, spectrum of Bounded Operator.
3. Duals and transposes, duals of LP[a,b] and C[a,b].
4. Inner product spaces, orthonormal sets, approximation and optimization,
projections, Riesz representation theorem.
5. Bounded operators and adjoints on a Hibert space, normal, unitary and self
adjoint operators.
6. Fourier Series and Integrals.
Textbook :
B.V. Limaye : Functional Analysis (Second Edition) - New Age International Limited..
Chapter 2 : Sections 5 to 8.
Chapter 3 : Sections 9 to 12.
Chapter 4 : Sections 13,14.
Chapter 6 : Sections 21 to 24.
Chapter 7 : Sections 25,26.
Chapter 1 : Section 4 (Fourier Series and Integrals).
MIM 202 Complex Analysis
1. Complex number system: Complex plane, polar representation, lines and half
planes, Spherical representation.
2. Analytic functions: Power series, analytic functions, Mobius transformations,
3. Applications of Conformal Mappings : Steady Temperatures, Steady
Temperatures in a wall, Temperatures in a Quadrant with part of one Boundary
Insulated, Electric Potential, Potential in a cylindrical space, Two Dimensional Fluid
Flow, The stream Function, Flow around a Corner, Flow around a Cylinder
4. Integration: Power series representation of analytic functions Zeros of analytic
function, Index of a closed curve, Cauchy's theorem, Homotopy Versions (without
proof), simple Connectivity version (without proof), counting zeroes, Goursats
theorem.
5. Singularities: Classification, residues, argument principle.
6. Maximum Modulus theorem: Maximum principle, Schwarz lemma.
Textbook
1. Complex Analysis - J.B. Conway (Second Edition) Narosa Publishing House..
Chapter 1 : Sections 1 to 6.
Chapter 3 : Sections 1 to 3.
Chapter 4 : Sections 2 to 8.
Chapter 5 : Sections 1 to 3.
Chapter 6 : Sections 1 to 2.
Textbook 2 : Complex Variables and Applications - Ruel V. Churchill (Second
Edition)
Chapter 9..
MIM 203 Field Theory
1. Fields - Example, Algebraic and Transcendental Elements, Degree of Field
Extension, Construction with Ruler and Compass, Symbolic Adjunction of Roots,
Finite Fields
2. The Main Theorem of Galois Theory - Cubic Equations, Summetric Functions,
Primitive Elements, Proof of Main Theorem, Kummer Extension, Cyclotomic
Extension, Quintic Extension.
Textbook :
Algebra - Michael Artin (Prentice Hall of India Private Limited).
Chapter 13 : Sections : 1 to 6.
Chapter 14 : Sections : 1 to 9.
MIM 204 Linear Algebra
1. Modules : Definition and Examples, Further notions and Results
2. Free Modules : Linear Independence, Bases of Free Modules, Matrices and
Homomorphisms
3. Vector Spaces : Definition and Examples, Subspaces, Bases and Dimensions,
Linear Transformations, Quotient Spaces, Direct Sum, The matrix of Linear
Transformation, Duality
4. Canonical Forms : Eigenvalues and Eigenvectors, The minimal Polynomial,
Diagonalisability, Triangularisable Operators, Jordan Forms, The Rational Forms
5. Inner Product Spaces : Inner Product Spaces, Orthogonality, The Adjoint of Linear
Transformation, Unitary operators, Self Adjoints and Normal Operators, Polar and
Singular Value Decomposition
6. Bilinear Forms : Definition and Examples, The matrix of a Bilinear Form,
Orthogonality, Classification of Bilinear Forms
Textbook 1 : Modules - Luther and Passi (Narosa Publishing House). Chapters 1, 2.
Textbook 2 : Linear Algebra - Bisht and Sahai, Chapters 2 to 5.
MIM 205 Numerical Analysis
1. 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.
2. Polynomial interpolation: interpolation polynomial, divided difference
interpolation, Aitken's formula, finite difference formulas, Hermite's interpolation,
double interpolation.
3. Linear systems of Equations: Gauss Elimination, Gauss-Jordan method, LU
decomposition, iterative methods, and Gauss- Seidel iteration.
4. 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.
5. 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.
6. Eigenvalue Problem : Power method, Jacobi method, Householder method.
7. Mupad Computer practicals.
Textbooks .
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..
SEMESTER III (Compulsory Courses)
MIM 301 Theory of Computer Science
and Database Fundamentals
1. Lattices and Boolean Algebra : Introduction, Hasse Diagram, Boolean
Expressions, Boolean Identities, d.n.f., c.n.f., Logic Gates.
2. Finite State Machines - Introduction to Graphs, State Diagram, Transition Table,
Finite State Automata, Deterministic and non-deterministic
3. Formal Language : Alphabets, Language, Grammar, Kleene Closure, Regular
Expression, Finite Automata with output, Non deterministic automata, Moore
Machines, Mealy Machines
4. Turing Machine - Tape Head Program, Deterministic, Non Deterministic, Accept,
Reject, Loop
5. Grammar and Languages : Phase Structure Grammar, Types, Language
Recognition, Regular Sets, Kleen's Theorem, Regular Grammar, Pumping Lemma
Chomsky Hierarchy, Context Sensitive Grammar, Polish notation, Postfix, prefix,
Grammars and Cohen Languages
6. Introduction to DBMS : File System Vs DBMS, Storing Data, Levels of
abstraction, data independence, Structure of DBMS, Advantages of DBMS
7. Conceptual Design (ER Model) Additional constraints, weal entities,
Aggregation, generalisation, conceptual design using ER, Entituy Vs relationship,
binary Vs Ternary, Contraints beyond ER, Conceptual design for small to large
enterprises, Case Studies,
8. Relational data model : Conversion of ER to Relational model, integrity
constraints
9. Relational algebra : Preliminaries, Relational algebra
10. Relational calculus : Tuple calculus, Calculus Versus Relational Algebra
11. SQL : DDL, forms of a basic SQL query, union/ intersection/ except, nested
queries, Aggregate Operators, Aggregate functions, Null Values, impact on SQL
commands, outer joins, disallowing NULL, examles on SQL, Creating functions in
PLSQL, cursors, triggers.
12. Functional dependency : Introductory to schema refinement, use of
decomposition, problems relation to decomposition, functional dependencies,
lossless-join decomposition, Normalisation and its forms.
Reference Books :
1. Applied Abstract Algebra - Lid and Pilz 2nd Edition
2. Introduction to Computer Theory by Colen
MIM 302 Programming in C and C++
1. Introduction – Variables and Constants, Assignment Statements
2. Data Types – Integers, Floating Point Numbers, Void Data Type
3. Operators and Expressions – Operators, Assignment Operators, Precedence and
Associativity
4. Control Statements – If Statements, Looping, Switch, Break, Continue
Statements, Infinite Loops
5. Arrays and Pointers – Arrays, Initializing Arrays, Pointer Arithmetic, String
Manipulations, Multidimensional Arrays, Pointers to Pointers, Dynamic Memory
Allocation,
6. Structures and Unions – Structures and Functions, Array of Structures, Unions.
7. Files in C - File Operations, Command Line Arguments
8. Graphics in C - Concepts, Simple Programs
9. Preprocessor – Macro Substitution, Include Facility, Line Control
10. Types of Files – Logical File, Memory Hierarchy
12. File Organization – Fields and Record Organization, Overview of Indexes,
Types.
13. Tree Structured Indexing – Sequential and Binary Searching, ISAM, B+ Tree
Indexing, Static Hashing, Dynamic Hashing, Linear and Extendible Hashing
14. Database System – Data Independence, Relational Systems
15. Database System Architecture – Three Levels, Mapping, DBA, Database
Management Systems.
16. Object Oriented Programming Concepts - Data Types, New Operators, Classes
and access Specifiers, Array of Objects, Managing Console I/O stream class,
Formatted and Unformatted Console I/O – Usage manipulator
17. Functions in C++ : Call by reference, Function Overloading and default
argument, static lass member, friend function.
18. Constructors and Destructors – Types, Usage of destruct
19. Operator Overloading – Overloading Unary and Binary Operators, Overloading
using friend function, Usage of this pointer, Overloading insertion and extraction.
20. Inheritance – Types, Virtual Base class and abstract class, Virtual function and
pure virtual function.
21. Working with Files – File Operations, File Pointers and manipulations, File
Updation with random access
22. Templates : Class Template, function Templates, Overloading of Template
Functions, Basics of Exception Handling.
Textbooks :
1. Let us C - Kanitkar
2. Object Oriented Programming with C++ - E. Balagurswamy.
MIM 303 DATA STRUCTURES
Basics of digital Electronics: Bits, Bytes, Binary Addition ,Subtraction, Gates, Boolean
Algebra, Half Adder, Half-subtractor, Full-adder, 2's Compliment method of subtraction,
De-Morgan's Theorems, Flip-Flops, Registers
1. Uni-Processors: Computer Organization, History of microprocessors, Microprocessor
Architecture, RAM ,ROM, Instructions set, Machine and Assembly Language, Instruction
Decoding, Microprocessor Performance and trends, Progression of Intel Microprocessor,
Architecture of 8086,80286,80386,80486,80586.
Storage Devices : Magnetic Storage, Optical CR - R/ CD - RW
2. Input Output Devices: Keyboard, Mouse, Introduction to Computer Monitors: The
Basics, Display Technology: Background, VGA, DVI, Viewable Area, Maximum
Resolution and Dot Pitch, Power Consumption, Monitor Trends: Flat Panels, Introduction
to Printers: Inkjet Printers ,Working, Impact vs. Non-impact, Paper and Ink, Introduction
to Laser Printers: The Basics of Static Electricity, Drum, Fuser, Toner.
3. Introduction to Parallel computing: Concepts and Terminology, von Neumann
Computer Architecture, Flynn's Taxonomy ,Some General Parallel Terminology, Parallel
Computer Memory Architectures: Shared Memory ,Distributed Memory, Parallel
Programming Models: Overview Shared Memory Model ,Threads Model .Message
Passing Model ,Data Parallel Model Designing Parallel Programs :Automatic vs. Manual
Parallelization ,Understand the Problem and the Program ,Partitioning Communications,
Synchronization ,Data Dependencies ,Load Balancing Parallel Examples : Sistolic and
Wave front Array Processor ,PI Calculation Simple Heat Equation ,l-D Wave Equation
4. Introduction – Data Types, Data Structures, Arrays as Abstract Data Type
1-d,2-d, Multidimensions
5. Stacks – Push and Pop Operations, Stack Implementations using C, Stack
Applications, Recursions,
6. Queues – Concepts, Queue Applications, Priority Queues
7. Linked Lists – Static implementation using arrays, Doubly Linked List, Circular List,
Linked List Applications, Polynomial Representation, Stacks and Queues as Linked Lists
8. Trees – Terminology and Concepts, BinaryTree Representation, Linked
Representation , Binary Search Tree, Operations, Tree Traversals, Representing
General Trees as binary Tree
9. Sorting – Need and concepts, Techniques like Bubble, Quick, Selection, Insertion,
Tree, Merge, Radix
10. Graphs – Concepts, Graph Representation, Adjacency Matrix, Multilist, Traversals.
Text Books :
1. K Hwang and FA Briggs : Computer Architecture
2. Data Structures using C - Tenenbaum, Langsam, Augenstein.
3.Art of Computer Programming, D.Knuth (Prentice hall publishers).
SEMESTER IV (Compulsory Courses)
MIM 401 - JAVA PROGRAMMING
1. Introduction to Java Programming - Overview, Java Tools, Java Byte Code
2. Elementary Programming Concepts - Variables & Identifiers, Java keywords,
Data Types, Operators, Expression, Constants, Statements, Arrays
3. Classes and Packages - Defining classes, Static Members, Using packages,
Access Specifiers, Constructors, Finalisers referencing objects
4. Inheritance, nested and inner class - Extending classes, Abstract Class
Interface, Super Keyword, Final classes, Constructors and Inheritance, Dynamic
Binding, Overriding methods
5. Exception and Input and Output - Byte streams, Character streams, File i/o
basics, Introduction to exception, Try and catch block and finally block, Inbuilt
Exception.
6. String Handling and Exploring Java.lang - String Operations, Character
Extractions, Data Conversions, Modifying strings.
7. Applet and Event Handling and Controls
8. Input and Output package - Object serialization, reader and writer
9. Swings - Layout Manager Layout Manager swing Controls Components
Organizers, Jlish, Jtree, Jtables, Dialogue, File chooser, color chooser.
10. JDBC - The design of JDBC, JDBC programming concepts making the
connection, statement and result set class, Executing SQL commands, Executing
Queries.
11. Multithreading - Running multiple threads, The runnable interface Threads
priorities Daemon, Thread States, thread groups Synchronization and Interthread
Communication Deadlocks.
Textbook :
A Complete Reference Java 2 - Herbert Schildt.
MIM 402 - OPERATING SYSTEMS
1. Introduction to Operating Systems - Batch System, Time sharing system,
personal computer system, Parallel system, Distributed System, Real Time
System
2. File System - File Systems, File Concepts, Allocation Methods, Access
Methods, Directory Structure
3. Threads - Overview, Multithreading models, Threading Issues.
4. CPU Scheduling - Basics, Scheduling criteria, Scheduling Algorithms, Multiple
Processor Scheduling.
5. Disk and Drum Scheduling
6. Memory Management - Background, Swapping, Contiguous allocation,
Paging, Segmentation, Segmentation and paging- combined system, virtual
memory concept, Demand paging, Paging replacement algorithms.
7. Concurrent Processing and programming - Review of process concepts,
Hierarchy of processes, Problem and solution algorithm, Semaphores, Overview
of Concurrent programming, Modularization, Synchronization in Windows 2000,
Concurrent languages.
8. Deadlocks - System model, Deadlock characterization, methods of handling
Deadlocks, Deadlock Prevention, Deadlock Avoidence, Deadlock Detection,
Recovery from Deadlock.
Textbook :
Operating System concepts by Silberschatz 6th edition
MIM 403 DESIGN AND ANALYSIS OF ALGORITHMS
1. Mathematical Foundation, Growth Functions, Summations, Recurrences
Substitutions, Iterations, Master Methods, Counting and probability
2. Sorting, Heap Sort, Quick Sort, Merge Sort, Sorting in linear Time, Medians
and Order Statistics
3. Dynamic Programming - Matrix chain Multiplication, longest common
subsequence, optimal polygon triangularisation.
4. Greedy Algorithm.
5. Graphs - Traversals, Topological sort, Minimum spanning trees, single source
shortest path, All pair shortest path, Maximum flow problems.
6. Sorting Networks - Comparision, bitonic sort and merge sort networks
7. Parallel Algorithms - CRCW, EREW algorithms efficiency sorting linear system
problem, Matrix Operations, Strassens Algorithm and matrix inversion.
7. FFT - Polynomials DFT, FFT
8. Number Theoretic Algorithm - Rabin - Karp, KMP, Bower - Moore algorithms
9. Geometric Algorithms - Finding convex hall, closes pair of points, linear
programming problem
10. NP Completeness - P and NP classes, NP completeness and reducibility
11. Approximation Algorithms - Vertex cover problem, traveling salesman
problem, set covering and subset sum problems
Textbook :
1. Introduction to Algorithms : T. H Coreman, Leiserson, Rivest.
OPTIONAL SUBJECTS FOR SECOND YEAR
MIM 01 - OPERATIONS RESEARCH
Unit I - Kuhn – Tucker conditions of Optimality – Quadratic Programming
(Sections 19.2.2B, 20.2.2)
Unit II - Inventory Models
(Sections 14.1 to 14.3)
Unit III - Queueing Models
(Section 15.1, 15.2, 15.4, 15.5)
Unit IV - Project Scheduling By PERT – CPM
(Sections 13.1 to 13.4)
Unit V - Simulation Modeling with SIMNET – II
(Sections 17.1 to 17.10)
Textbook : Hamdy A.Taha, Operations Research, Fifth Edition, Prentice
Hall of India.
MIM 02- INTEGRAL TRANSFORMS
A) Classification of Liner Integral Equations : Fredholm, Volterra, Integro-
Differential Equations, Singular Integral Equations, Converting Volterra
Equation to ODE, Conversion of IVP to Volterra equation Conversion of
BVP to Fredholm equation
B) Fredholm Intergral Equations - Decomposion method, Direct
Computation method, successive approximation method, method of
successive substitutions, Homogeneous Fredholm Equations, Comarison
between alternative methods.
C) Volterra Interal Equation - Adomian Decomposion method, Series
solution method, converting Volterra equation to VIP, Successive
Approximation method, successive substitution method, comparison
between alternative methods.
D) Integro-Differential Equations - Introduction, Direct Computation
method, Adomian Decompostion Method. Conversion to Fredholm
integral Equation.Volterra Intego-Differential equations Series Solution,
Decomposition Method, Conversion to IVP.
E) Singular Integral Equations - Abel problem , Genralized Abel Integral
Equation, Weakly-singular Volterra Equations.
F) Non Liner Integral Equation`s - Non liner Fredholm Integral
equations,Direct Computation, decomposition method, Non liner Volterra
Integral Equation, Series solution, Decomposition method
G) Existence and uniqueness of solutions using fixed-point theorems in
cash of Liner and nonliner Volterra and Fredholm integral equations.
H) Fourier Transforms : [FT] Definition Properties evaluation of Fourier
and inverse Fourier transforms of functions, Convolution theorem for FT.
Sine and Cosine Fourier transforms. Solving differential equations and
interal equations using FT.
I) Laplace Transform :Definition Properties, evaluation of Laplace and
Inverse Laplace transforms of functions. Convolution theorem for Laplace
Transforms.Solving initial value problem using Laplace Transforms.Solving
integral equation using Laplace Transforms.
J) Mellin Transforms : Definition, properties and evaluation of
transforms, Convolution theorem for Mellin transforms, Applications to
integral equations.
Reference Books:
1) A First course in integral equations –A.M. Wazwaz (1997) (world
Scientific)
2) Introduction to Integral Equation with Applications –A.J. Jerri
(1999)Second edition Wiley Interscience.
\
MIM 03 - NUMBER THEORY
1. Congruences : Solutions of congruencs. Chinese Remainder
Theorem,Techniques of numerical calculation Public-Key Crytography.
2. Prime power moduli. Prime modulus. Primitive roots and power residues,
Congruences of degree two.
3. Quadratic Residues, Quadratic Reciproctiy.
4. Greatest integer function, Arithmetic functions, Multiplicative functions,Dirichlet
multiplication. Mobius Inversion Formula.
5. Diophantine equations. The equation ax + by = c, Pythagorean triangles,
Assorted examples. Rational points on curves.
Books: 1.Niven and Zuckerman, An introduction to the Theory of Numbers,Wiley
Publishers.
2.David Burton,Elementary Number Theory.
MIM 04 - CODING THEORY
1. Error detection: correction and decoding: Communication channels,Maximum
likelihood decoding,Hamming distance,Nearest neighbour / minimum distance
decoding, Distance of a code.
2. Linear codes: Vector spaces over finite fields,Linear codes, Hamming weight,
Bases of linear codes, Generator matrix and parity check matrix, Equivalence of
linear codes, Encoding with a linear code,Decoding of linear codes, Cosets,
Nearest neighbour decoding for linear codes, Syndrom decoding.
3. Cyclic codes: Definitions, Generator polynomials, Generator and parity check
matrices, Decoding of cyclic codes, Burst-error-correcting codes.
4. Some special cyclic codes: BCH codes, Defintions, Parameters of BCH codes,
Decoding of BCH codes.
Reference: 1. San Ling and Chaoing xing, Coding Theory- A First Course
2. Applied Abstract Algebra - Lid and Pilz 2nd Edition
MIM 05 - GRAPH THEORY
1. Fundamental concepts : Definitions and examples, graphs as models,
matrices and isomorphism, paths, connected graphs, bipartite graphs,
extremality vertex degree, the Pigeonhole principal, Turan`s theorem, degree
sequences, graphic sequences, degree and digraphs.
2. Tree and Distances : Properties of tree, distance in graphs, stronger results,
disloint spanning trees, shortest paths, tress in computer science, Eulerian
circuits.
3. Matching and Factors : Matching in bipartite graphs, maximum matchings,
Hall`s matching conditons, Min-Matching in bipartite graphs, sets, applications
and algorithms, maximum bipartite matching, weighted bipartite matching, in
general graphs, Tutte`s 1-factor theorem,f-factors of graphs.
4. Connectivity and Paths : Cuts, connectivity, edge-connectivity, blocks, 2-
connected graphs, connectivity of digraphs, k connected and k-edge
connected graphs, applications of Meger`s theorem, Network flow problems
maximum network flow, integral flows.
5. Edges and cycles : Line graph and edge-colouring, Hamiltonian cycles:
necessary conditions, Sufficient conditions.
Recommended Book:
Douglas B. West, Introduction to Graph Theory Prentice- Hall, New Delhi (1999)
Reference Books:
1. John Clarke and D.A. Holton, A First Look at Graph Theory, Allied Publisher
(1991)
2. Nora Harsfield and Gerhard Ringel , Pearls Theory, Academic Press (1990)
3. Harary, Graph Theory, Narosa Publishers, New Delhi (1989)
MIM 06 - LATTICE THEORY
Two definitions of lattices, Hasse diagrams, homorphism, isotone maps,
ideals, congruence relations, congruence lattices, the homorphism theorem,
product of lattices, complete lattice, ideal lattice, distributive –modular
inqualities and identifies, complemets, pseudocomplements, Boolean lattice
of pseudocomplements, join and meet-irreducible elements.
Characterizatin theorems and representation theorems-Dedekind`s modularity
criterion Bikhoff`s distributivity criterion, hereditary subsets, rings of sets,
Stone theorems,Nachbin theorem, staements of Hashimoto`s theorem.
Modular lattices, isomorphism theorem, Upper and lower covering conditions,
Kurosone theorem, independent sets (Drops results involving projectivity and
sublattice generated by sets / elements )
Semimodular lattices Jordan-Holder chain condition, Modular pair, Msysmmetric
lattices.
Book- Genral Lattice Theory
Auther- G. Gratzer (Academic Press 1978)
Chap. 1 Section 1,2,3,4,6, Cha. 2 Section-1, Chap.3. Section –1,2.
MIM 07 - COMPUTATIONAL GEOMETRY
The course text will be We will cover most of the book, adding some additional material.
1. Geometric primitives [Chap. 1]
2.Line intersection [Chaps. 2] plus randomized incremental
3. Triangulation and visibility and [Chaps. 3,15]
4. Linear programming in two and three dimensions [Chap. 4]
5. Orthogonal range searching [Chaps. 5,10]
6. Point location and Binary Space Partitions [Chaps. 6,12]
7. Voronoi diagrams and Delaunay triangulation [Chaps. 7,9]
8. Convex hulls [Chap. 11]
9. Non-orthogonal range searching [Chap. 16]
10. Curved Elements (Bezier, B-Splines)
11. Curve Reconstruction (reconstruction a curve(surface) from sample
points)
12. 3-Dimensional Geometry
Text Book :
Computational Geometry Algorithms and Applications, 2nd ed., by de
Berg, van Kreveld, Overmars, and Schwarzkopf (Springer-Verlag, 2000).
MIM 08 - CRYPTOGRAPHY
1. Introduction : Overview of course, Classical cryptography [parts of
Chapter 1].
2. Secret Key Encryption : Perfect Secrecy - One time pads [Chapter 2.1],
Stream ciphers and the Data Encryption Standard (DES) [Chapter 3
(excluding 3.6)], The Advanced Encryption Standard (AES) - adopted
September 2000.
3. Public Key Encryption : Factoring and the RSA encryption [Chapter 4.1 -
4.4], Discrete log. Diffie-Hellman Key Exchange [Chapter 8.4 (only pages
270-273)]. ElGamal encryption [Chapter 5 (only pages 162-164)] , Digital
Signatures [Chapter 6 (excluding 6.5 - 6.6)], One-time signatures, Rabin
and ElGamal signatures schemes, Digital Signature Standard (DSS).
4. Hashing : Motivation and applications. Cryptographically Secure
Hashing. [Chapter 7.1-7.3,7.6], Message Authentication Codes (MAC).
HMAC, Network Security , Secure Socket Layer (SSL), I Psec.,Secret
Sharing, Definition. Shamir's threshold scheme [Chapter 11.1], Visual
secret sharing schemes.
Reerence Book :
D. R. Stinson. CRYPTOGRAPHY: Theory and Practice. CRC Press.
1995.
MIM 09 - FINANCIAL MATHEMATICS
1. Introduction to options and markets: types of options, interest rates and
present values.
2. Black Sholes model : arbitrage, optionvalues, 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.
3. Binomial methods : option valuation, dividend paying stock, general
formulation and implementation.
4. Monte Carlo simulation : valuation by simulation
5. 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.
6. Lab component: implementation of the option pricing algorithms and
evaluations for Indian companies.
References:
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.
MIM 10 - MODELLING AND SIMULATION
1. Introduction to modelling and simulation. System analysis, clasification
of systems. System theory basics, its relation to simulation.
2. Model classification: conceptual, abstract, and simulation models.
Heterogeneous models. Methodology of model building
3. Simulation systems and languages, means for model and experiment
description. Principles of simulation system design
4. Parallel process modelling. Using Petri nets and finite automata in
simulation
Models o queuing systems. Discrete simulation models. Model time,
simulation experiment control.
5. Continuous systems modelling. Overview of numerical methods used for
continuous simulation.
6. Combined simulation. The role of simulation in digital systems design
7. Special model classes, models of heterogeneous systems.
8. Checking model validity, verification of models. Analysis of simulation
results
9. Simulation results visualization. Interactive simulation
10. Design and control of simulation experiments. Model optimization
11. Generating, transformation, and testing of pseudorandom numbers.
Stochastic models, Monte Carlo method
12. Overview of commonly used simulation systems.\
References :
1. Fishwick P.: Simulation Model Design and Execution, PrenticeHall,
1995, ISBN 0-13-098609-7
2. Law A., Kelton D.: Simulation Modelling and Analysis, McGraw-Hill,
1991, ISBN 0-07-100803-9
3. Rábová Z. a kol: Modelování a simulace, VUT Brno, 1992, ISBN 80-
214-0480-9
4. Ross, S.: Simulation, Academic Press, 2002, ISBN 0-12-598053-1.
MIM 11 - ARTIFICIAL INTELLIGENCE
1. Overview of history and goals of AI : Tentative definitions. Turing's test.
Knowledge vs. Symbolic Level. Relations with other disciplines, from
Philosophy, to Linguistics, to Engineering. Review of AI successes and
failures.
2. State Spaces, Production Systems, and Search : State Space
representation of problems. Problem solving as search. Constraints.
Definition and examples of Production Systems. Heuristic search
techniques. Two person games.
3. Knowledge Representation Issues: Procedural Knowledge
Representation vs. Declarative Knowledge + Reasoning. Facts, General
Assertions, Metaknowledge. The Frame Problem.
4. Using First-Order Logic for Knowledge Representation : Propositional
Logic: Semantics and Deduction. First Order Logic: Semantics and
Deduction. Unification. Resolution-based theorem proving. Using theorem
proving to answer questions about the truth of sentences or to identify
individuals that satisfy complex constraints. Logic Programming.
5. Common Sense Reasoning : Nonmonotonic reasoning and modal logics
for nonmonotonic reasoning. How to deal with Agents and their Beliefs.
6. Weak Slot-and-Filler Structures: Semantic Nets and Frames. Scripts for
representing prototypical combinations of events and actions.
7. Rule-Based Systems: Pattern-matching algorithms. The problem of
Control in Rule Based Systems. The Rete Algorithm.
8. Planning: Representing plans. Partial order planning. Planning
applications.
9. Statistical Reasoning: Use of Certainty Factors in Rule-Based Systems.
Associating probabilities to assertions in first-order logic. Bayesian
Networks. Fuzzy Logic.
10. Learning: Learning to classify concepts using features of their
instances. Learning a concept [Induction] from examples. Explanation-
Based Learning. Version Spaces. Neural Nets with back propagation.
Text Book :
Artificial Intelligence: A Modern Approach : Prentice-Hall, 1995
MIM 12 - SYMMETRIES
1. Symmetry of plane figures of motions of the plan, finite groups of
motions, discrete groups of motion, symmetry, cosets, counting formula,
permutation representations, finite subgroups of the generators and
relations
2. Operation of a group on itself, class equation of the isocahedral groups
operations on subsets groups of order12, free group generators and
relations.
3. Bilinear forms, symmetric forms, orthogonality, geometry associated to a
positive from, Hermitian forms, spectral theorem, conics and quadrics,
normal operators, skew symmetric forms.
Text: Artin: Algebra (Prentice-Hall)
Chapter 5, 6 (sections 1, 2, 3 and7)
MIM 13 - WAVELET ANALYSIS
1. Fourier Transform : Fourier transform on L1(R) and L2(R) and basic
properties and examples
2. Windowed Fourier Transform : Motivation and definition of Windowed
Fourier Transform and examples, Time frequency localization, the
reconstruction formula.
3. Continuous Wavelet Transform : Motivation and Definition of the
wavelet transform and examples, Basic properties, The reconstruction
formula, Frequency localization, Orthonormal Wavelets.
4. Multiresolution Analysis : Definition of MRA and examples, Properties of
scaling functions and orthonormal wavelets bases, Construction of
orthonormal wavelets.
Textbook :
1. Bachman G, L. Narici & E. Beckensterin: Fourier and Wavelet
Analysis, Springer-Verlage (2000)
2. Chui C. K. : An Introduction to Wavelets, Academic Press (1992)
.
MIM 14 - COMBINATORICS
1. Counting Methods for selections arrangements : Basic counting principles,
simple arrangements and selections, arrangements and selection with
repetition , distributions, binomial, generating permutations and combinations
and programming projects.
2. Generating function : Generating function models, calculating of generating
functions, partitions exponential generating functions, a summation method.
3. Recurrence Relations : Recurrence relation model, divide and conquer
relations, solution of inhomogeneous recurrence relation, solution with
generating functions.
4. Inclusion-exclusion : Counting with Venn diagrams inclusion formula,
restricted positions and rook polynomials.
5. Ramsey Theory : Ramsey theorem, applications to geometrical problems.
Recommended Books:
1. Alan Tucker, Applied Combinatorics (third edition), John Wiley & sons , New
York (1995)
2. V. Krishnamurthy, Combinatorial, Theory and Applications, East West Press,
New Delhi (1989) Scientific, (1996)
MIM 15 - PARTIAL DIFFERENTIAL EQUATIONS
1. First order PDE, Linear Equations of first order, Charpit’s method, Jacobi’s
method, Quasi-linear equations, Non-linear first order PDE.
2. Second ordered PDE: Genesis, Classification, One dimensional Wave
equation, Laplace equation, Boundary Value Problems, Maximum and Minimum
Principles, Cauchy Problem,
3. Heat Conduction Problem, Duhamel’s Principle
Text Book :-
An Elementary Course in Partial Differential Equations by T. Amaranath
(Narosa) Chapters 1-2.
MIM 16 - FUZZY LOGIC
1. Fuzzy Sets and Operations on Them
2. Fuzzy Relations
3. Fuzzy Rules
4. Approximate Reasoning
5. Fuzzy Logic
6. Fuzzy Systems (e.g., Fuzzy Logic Control)
7. Fuzzy Logic in Pattern Recognition
8. Fuzzy Decision Making
9. Fuzzy Logic Applications
TextBook :
George J. Klir, Bo Yuan, Fuzzy Sets and Fuzzy Logic: Theory and
Applications, Prentice Hall PTR, 1995.
Hao Ying, Fuzzy Control and Modeling: Analytical Foundations and
Applications, IEEE Press, 2000. (Do not need to buy it. A handout will be
distributed).
MIM 17 - STATISTICS AND PROBABILITY
1. Introduction to Discrete Probability : Intuitive concepts: probability of an
event as a measure between 0 and 1; random variable; probability
distribution; frequency interpretation of probability; random numbers; coins,
dice, and other games; simulations; odds; historical development of
probability; random walks.
2. Formal concepts: sample space, outcomes, and events; random
variable; discrete distribution functions and axioms of probability; unions,
intersections, and complements; properties of probabilities, principle of
inclusion and exclusion; tree diagrams; uniform distributions over finite
sets, symmetry; infinite sample spaces with discrete probabilities.
3. Introduction to Continuous Probability : The intuitive problems with
probabilities over space (line, plane, Rn in general). Monte Carlo
simulations, Buffon's needle. Formal concepts: density function for a
continuous random variable; integration; cumulative distribution functions;
derivatives; exponential density function;
4. Conditional Probability : Intutive concept of conditional probability;
formal definition of conditional probability; Bayes' formula for inverting
conditional probabilities; independent events; joint distribution functions;
independent random variables; independent trials. Conditional density
functions for continuous distributions; the beta distribution
5. Distributions and Densities : Uniform continuous distributions; geometric
distribution; Poisson distribution; exponential and gamma distributions;
introduction to queueing theory; normal (Gaussian) distribution; Chisquared
distribution
6. Expected Value and Variance : Expected value for discrete random
variables, expectation; linearity of expectation; expectation of independent
random variables; conditional expectation; variance and standard
deviation; variance of various distributions. Expectation and variance for
continuous random variables.
7. Sums of Random Variables : Analysis of sums of independent random
variables with identical distributions, that is, independent trials.
8. Law of Large Numbers : Chebychev inequality, law of averages, law of
large numbers.
9. The Central Limit Theorem : The central limit theorem for Bernoulli trials,
binomial distributions again, the normal distribution, the general central
limit theorem.
TextBook :
Charles M. Grinstead and J. Laurie Snell's textbook Introduction to
Probability, published by the American Mathematical Society, 1997
.
MIM 18 - FLUID DYNAMICS
1. Physical Properties of fluids. Concept of fluids, Continuum Hypothesis,
density, specific weight, specific volume.
2. 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.
3. 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.
4. Conservation Laws : Equation of conservation of mass, equation of
conservation of momentum, Navier Stokes equation, equation of moments of
momentum, Equation of energy, Basic equations in different co-ordinate
systems, boundary conditions.
5. 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.
TextBooks :
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.
MIM 19 - BANACH ALGEBRA
1. Relatively compact sets, compactly continuous operators, finite
dimensional operators, transformation that is bounded but not completely
continuous, a type of transformation that is always completely continuous,
further properties of completely continuous transformations.
2. Spectra and the resolvent set, Approximate proper values, Banach
Algebra With identity, compactness of the spectrum, the resolvent
operator, Spectral radius and spectral mapping theorem for polynomials,
the Gelfand Theory.
3. Sesquilinear functions : Spectral results for normal and completely
continuous operators, numerical range
4. The Fredholm alternative theory, the spectral theorem for bounded,
normal Finite dimensional operators.
5. Commutative Banach Algebras, ideals and homomorphisms.
TextBook :
1. Walter Rudin: Functional Analysis, Tata MeGrow Hill Publishing co.
New Delhi.
\
MIM 20 - BOUNDARY VALUE PROBLEMS
1. Definition of boundary Value Problems, the heat equation, wave
equation, Laplac`s equation, the Fourier method, Liner Operators,
Principal of Superpositin, series solutions, uniform convergence
(weierstrass M-test), separation of variables, non homogeneous
conditions, Sturm-Liouville problems, formal solutions, the vibrating
string.
2. Orthogonal sets of functions, Generalized Fourier series, Best
approximation in the mean, Convergence in the mean, the orthonormal
trigonometric functions, other types of orthogonality.
3. sturm-Liouville Problem and applications, orthogonality and
uniqueness of eigenfunctions, method of solutions, surface heat
transfer other boundary value problems.
4. Bessel function Jn, recurrence relation, the zero of Jo (X) and related
functions, Fourier-Bessel series, Temperatures in a long cylinder.
5. Legendre polynomials, orthogonality of Legendre polynomicals,
Legendre series, Dirichlet Problem in spherical regions.
Prescribed Text Book
R.V. Churchill and J. Brown.: Fourier Series and Boundary Value Problems
(4th edtion)(Publisher: McGraw-Hill Book Company)
MIM 21 - BAER* RINGS
1. Rings with involution
2. Poset of projections
3. Proper involutions and C*-algebras
4. Rickart *-rings and Bear *-ring
5. Weakly Rickart *-rings and unitification
6. Central cover
7. Additivity of projections
8. Comparability axioms and parallelogram law
9. Finite and abelian projections
10. Structure Theorem
Prescribed Book: Bear *-rings, S.K. Berberian, Springer
MIM 22 - MATROID THEORY
1. Basic definitions and examples
Independent sets and circuits, bases, rank, closure, geometric
representations of matroids of small rank, transversal matroids, the lattice
of flats, the greedy algorithm.
2. Duality
The definition and basic properties, duals of representable matroids, duals
of graphic matroids, duals of traversal matroids.
3. Minors
Contraction, Minors of certain matroids, flats and the sum theorem
4. Connectivity
Connectivity, for graphs and matroids, properties of matroid connectivity,
more properties of connectivity.
Text Book:
James G. Oxley, Matroid Theory Science Publications, Oxford
(1992)(Chapter 1 to 4)
MIM 23 - SPERNER THEORY
1. Introduction and sperner`s Theorem: A Simple intersection result,
Sperner`s theorem, Theorem of Bollobaas.
2. Normalized Matching and rank numbers: Sperner`s proof,system
of distinct representatives, L Y M inequalities, and normalized
matching property. Rank numbers, some examples.
3. Systemmebric Chain: Sysmmertric chain decompositions,
Dilworth`s theorem, symmetric chains of sets, Application to
Nested chains, posets with sysmmetric chain decompositions.
4. Rank numbes of multisets. Unimodality and log connectivity, the
normalized matching property. The largest size of a rank number.
Prescribed Text Book
Ian Aderson : Combinatorics of Finite Sets. (Oxferd Science
Publications)
Reference Book.
Konrad Engel: Sperner Theory (Cambride University Press)
MIM 24 - DIFFERENTIAL EQUATIONS AND DYNAMICAL
SYSTEMS
1) Liner Systems: Uncoupled Liner Systems, Diagonalization,
Exponential of operators Fundamental theorem for liner systems, liner
systems in R, Complex eigenvalues, multiple eigenvalues, jorden
Canonical Forms, stability theory Nohomogeneous Liner systems.
2) Nonlinear Systems: Local Theory, Fundamental existence theorem
dependence on initial conditions and parameters, the maximal interval
of existence, Flow defined by a differential equation. Linearization,
stable manifold theorem, Hartman-Grobman theorem, Stability and
Liapunov functions, Saddles, Nodes, Foci and centers, Nomhyperbolic
critical points in Rn, Gradient and Hamiltonian system.
Text (1) L. Perko- Differential Equations and Dynamical systems (1991)
Springer-verlag
(2) Hirsch and Smale – Differential Equations, Dynamical Systems,
and Liner Algebra - Academic Press, New York, (1974)
MIM 25 - MECHANICS
1. Sec 1.1-1.6 Survey of Elementary Particles
2. Sec. 2.1-2.7 Variational Principles & Lagranges` Equation
3. Sec.3.1-3.7 Central Force problem
4. Sec. 4.1-4.10 Kinematics of rigid body motion
5. Sec. 8.1-8.2 Hamilton Equations of motion
6. Sec.9.1-9.9 Canonical Transformations
Text Book: Classical Mechanics by Godlstein, Poole and Safko (Third
Edition) 2002, Person Education Inc. Supplementary Reading (1) Rana &
Joag Classical Mechanics (Tata McGraw Hill)
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