# M Sc Mathematics with Computer Applications & Compulsory Diploma in Computational Software

Bharathiar University
In Coimbatore

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MASTER OF PHILOSOPHY IN MATHEMATICS
INTERNATIONAL INSTITUTE OF MANAGEMENT & ENGINEERING

MPhil
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## Important information

Typology | Master |

Location | Coimbatore |

- Master
- Coimbatore

Where and when

Starts | Location |
---|---|

On request |
CoimbatoreBharathiar University, Coimbatore, 641046., Tamil Nadu, India See map |

Starts | On request |

Location |
CoimbatoreBharathiar University, Coimbatore, 641046., Tamil Nadu, India See map |

## Frequent Asked Questions

**· Requirements**

A candidate who has passed the Degree Examination in B.Sc. (Mathematics) or B. Sc. (Mathematics with Computer Applications) of this University or an examination of some other University accepted by the syndicate as equivalent thereto shall be eligible for admission to the Master Degree of this University.

## Course programme

MATBC1 - ALGEBRA

UNIT-I: Group Theory: Direct products- Group Action on a Set: Isotropy Subgroups- Orbits-

Counting Theorems- p-Groups- The Sylow Theorems.

UNIT-II: Applications of the Sylow Theory: Applications to p-Groups and the Class Equation-

Further Applications.

Ring Theory: Ring of Polynomials: Polynomials in an Inderterminate- The Equation

Homomorphisms- Factorization of Polynomials over a Field.

UNIT-III: Field Theory: Extension Fields - Algebraic and Transcendental Elements-

Irreducible polynomial over F-Simple Extension- Algebraic Extensions: Finite Extensions-

Structure of Finite Fields.

UNIT-IV: Automorphisms of Fields - Conjugation Isomorphisms- Automorphisms and

Fixed Fields- The Frobenius Automorphism- Splitting Fields.

UNIT-V: Separable Extensions- Galois Theory: Normal Extensions- The Main Theorem-

Illustrations of Galois Theory: Symmetric Functions.

MATBC2: ANALYSIS

RIEMANN STILTJES INTEGRAL:

Unit-I. Definition and Existence of the Integral - properties of the integral - Integration

and differentiation - Integration of vector valued function - rectifiable curves

Unit-II. Uniform convergence and continuity - uniform convergence and integration - uniform

convergence and differentiation - equicontinuous families of functions - The Stone Weirstrass

theorem

FUNCTIONS OF SEVERAL VARIABLES

Unit-III. Linear transformation - contraction principle - Inverse function theorem -

Implicit function theorem - determinants - derivatives of higher order - differentiation of

integrals

LEBESGUE MEASURE:

Unit-IV. Outer measure - Measurable sets and Lebesgue measure - Measurable functions -

Littlewood's Theorem.

Unit-V. The Lebesgue integral of bounded functions over a set of finite measure - integral of a

non - negative function - General Lebesgue Integral - convergence in measure.

MATBC3: DIFFERENTIAL EQUATIONS

Unit-I. Linear Equations with constant coefficients - Second order Homogeneous equations

- Initial value problems - Linear dependence and independence Wronskian and a formula

for Wronskian - Non Homogeneous equation of order two.

Unit-II. Homogeneous and Non - Homogeneous Equations of order n - Initial value problems -

Annihilator Method to solve a non - homogeneous equation - Algebra of constant coefficients

operators.

Unit-III. Linear Equations with variable coefficients - Initial value problems - Existence

and Uniqueness Theorems - Solutions to a non - homogeneous equation - Wronskian and

Linear dependence - reduction of the order of a homogeneous equation - Homogeneous

equation with analytic coefficients - The Legendre equation.

Unit IV: Nonlinear Partial Differential Equations of the first order - Cauchy's method of

characteristics - Compatible systems of first order equations - Charpit's method- Special types

of First order equations - Jacobi's method.

Unit V: Partial Differential Equations of Second order - The origin of Second-order Equations -

Linear Partial Differential Equations with constant coefficients - Equations with variable

coefficients - Characteristics curves of second - order equations- Characteristics of equations in

three variables.

MATBE1: NUMERICAL METHODS

Unit I

Solution of Nonlinear Equations: Newton's method - Convergence of Newton's method -

Bairstow"s Method for quadratic factors. Numerical Differentiation and Integration:

Derivatives from Differences tables - Higher order derivatives - Divided difference, Central-

Difference formulas - Composite formula of Trapezoidal rule - Romberg integration -

Simpson's rules.

Unit II

Solution of System of Equations: The Elimination method - Gauss and Gauss Jordan

methods - LU Decomposition method - Matrix inversion by Gauss-Jordan method - Methods

of Iteration - Jacobi and Gauss Seidal Iteration - Relaxation method - Systems of Nonlinear

equations.

Unit III

Solution of Ordinary Differential Equations: Taylor series method - Euler and Modified

Euler methods - Rungekutta methods - Multistep methods - Milne's method - Adams

Moulton method.

Unit IV

Boundary Value Problems and Characteristic Value Problems: The shooting method -

solution through a set of equations - Derivative boundary conditions - Characteristic value

problems - Eigen values of a matrix by Iteration - The power method.

Unit V

Numerical Solution of Partial Differential Equations: (Solutions of Elliptic, Parabolic and

Hyperbolic partial differential equations) Representation as a difference equation - Laplace's

equation on a rectangular region - Iterative methods for Laplace equation - The Poisson

equation - Derivative boundary conditions - Solving the equation for time-dependent heat

flow (i) The Explicit method (ii) The Crank Nicolson method - solving the wave equation

by Finite Differences.

MATBC4: DISCRETE MATHEMATICS

Unit I: Logic

Propositions - Logical Connectives - Compound statements - Conditional and

Biconditional Propositions - Truth tables - Tautologies and Contradictions - Logical

equivalence and implications - Demorgan's Law - Normal forms - PDNF and PCNF - Predicate

Calculus - Free and bound variables - Quantifiers - Universe of discourse - Theory of inference

- Rules of universal specification and generalization - Arguments - Validity of Arguments.

Unit II: Set Theory

Basic concepts - Notations - Algebra of sets - The power sets - Ordered pairs

and Cartesian products - Relation and its types - Properties - Relational Matrix and the

graph of relation - Partitions -Equivalence elations - Poset - Hasse diagram - Lattices

and their properties - Sublattice -Boolean Algebra - Homomorphism.

Unit III: Functions

Definitions of functions and its Classification - Types - Examples - Composition

of functions - Inverse functions - Binary and n-ary operations - Characteristic function of a

set - Hashing functions - Recursive functions - Permutation functions.

Unit IV: Graph Theory

Graph Theory: Introduction - Basic terminology - Representation of graphs -

connectivity - Eulerian and Hamiltonian graphs - Planar graphs- Directed graphs-Application of

Graphs. Trees: Binary tree - traversals of a binary tree - Expansion trees.

Unit V: Grammars and Languages

Definitions - Types of Grammars - Productions - Regular Grammar and Languages

- Finite state Automata (FSA) - Deterministic and Non-Deterministic FSA - Conversion

of NDFSA to DFSA.

MATBC5: FLUID DYNAMICS

Unit-I:

Introductory Notions- Velocity- Stream Lines and Path Lines- Stream Tubes and

Filaments- Fluid Body- Density- Pressure. Differentiation following the Fluid- Equation of

continuity- Boundary conditions- Kinematical and physical- Rate of change of linear

momentum- Equation of motion of an inviscid fluid.

Unit-II:

Euler's momentum Theorem- Conservative forces- Bernoulli's theorem in steady motionenergy

equation for inviscid fluid- circulation- Kelvin's theorem- vortex motion-

Helmholtz equation.

Unit-III:

Two Dimensional Motion- Two Dimensional Functions- Complex Potential Basic

singularities- source- sink- Vortex- doublet- Circle theorem. Flow past a Circular cylinder with

circulation- Blasius Theorem- Lift force (Magnus effect). Irrotational Motion in Three

Dimensions- Spherical Harmonics- Axially Symmetric Field- Stokes's Stream Function- Motion

of a Sphere.

Unit-IV:

Viscous flows- Navier - stokes equations- some exact solutions of Navier Stokes

equations- Flow between parallel flat plates- Couette flow- Plane Poiseuille flow- Steady flow

in pipes: Flow through a pipe- The Hagen Poiseuille flow.

Unit-V:

Laminar Boundary Layer in incompressible flow: Boundary Layer concept- Boundary

Layer equations- Boundary Layer along a flat plate- The Blasius solution- Shearing stress and

boundary layer thickness- Displacement thickness, momentum thickness- Momentum integral

theorem for the boundary layer- The Von Karman Integral relation, The Von Karman

Integral relation by momentum law.

MATBC6: PROGRAMMING LANGUAGES

Unit I:

Principles of object-Oriented Programming: Software crisis - Software evolution -

A look at procedure-oriented Programming - Object-oriented Programming Paradigm -

Basic Concept of Object-Oriented Programming - Benefits of OOP - Object-Oriented

languages - Applications of OOP.

Unit II:

Tokens, Expressions and Control structure: Introduction - Tokens - Keywords -

Identifiers and constants - basic data types - User defined data types - Derived data

types - Symbolic constants - Type compactability - Declaration of variables - Dynamic

insulation of variables - Reference variables - operations in C++ - Scope resolution

operator - member Dereferencing operators - memory management operators - Manipulators

- type cast operator - expressions and their types - Special assignment expressions - implicit

conversions - operator over loading - operator precedence - Control structures.

Unit III:

Functions in C++: Introduction - The main function - Function prototyping - call

by reference - return by reference inline functions - default arguments - constant

arguments - function over loading - friend and virtual functions - Math library functions

Managing Console I/O operations: Introduction - C++ streams - C++ stream classes -

Unformatted I/O operations - Formatted I/O operations - Managing output with manipulators.

Unit IV:

Classes and Objects: Introduction - C Structures Revisited - Specifying a class -

Defining Member Functions - A C++ Program with class - Making an outside Function Inline -

Nesting of Member Functions - Private Member Functions - Arrays within a class - Memory

Allocation for Objects - Static Data Members - Static Member Functions - Arrays of Objects -

Objects as Function Arguments - Friendly functions - Returning Objects - Constant Member

Functions. Constructors and Destructors: Introduction - Constructors - Parameterized

Constructors - Multiple Constructors in a class - Constructors with Default Arguments -

Dynamic Initializations of Objects - Copy Constructor - Constructing Two dimensional arrays

- Constant Objects - Destructors.

Unit V:

Operators Overloading and Type Conversions: Introduction - Defining Operator

Overloading - Overloading Unary Operators - Overloading Binary Operators -

Overloading Binary Operators Using Friends - manipulating of strings Using Operators -

Rules of Overloading Operators.

Inheritance: Extending Classes: Introduction - Defining Derived Classes - Single

inheritance - Making a Private Member Inheritable - Multilevel Inheritance - Multiple

Inheritance - Hierarchical Inheritance - Hybrid Inheritance - Virtual Base Classes - Abstract

Classes - Constructors in Derived Classes - Member Classes: Nesting of Classes.

MATBC7: OPERATIONS RESEARCH

Unit I:

Introduction to L.P. - Graphical L.P. Solution - Sensitivity analysis - Simplex Method -

L.P. solution space in equation form - Transition from graphical to algebra solution -

The simplex method - artificial starting solution - Special cases in simplex method applications.

Duality - Primal and Dual - relationships - additional simplex algorithm for L.P.

Unit II:

Advanced Linear Programming - Generalized simplex Tableau in matrix form -

Decomposition algorithm - Matrix definition of dual problem - optimal dual solution.

Unit III:

Integer L.P. and Dynamic Programming - Integer Programming - Integer Programming

algorithm - Gomory cutting plane algorithm - Branch and Bound algorithm - Solution of the

Traveling salesperson problem - Deterministic Dynamic programming - Recursive nature

of computation in D.P. - Forward and Backward recursion.

Unit IV:

Classical optimization Theory - unconstraint problems - Necessary and sufficient

conditions - The Newton-Raphson method - constrained problems - Equality constraints (Jacobi

method and Lagrangian method).

Unit V:

Non-linear programming - unconstrained algorithms Direct search method - Gradient

method - constraint algorithms - Separable programming - Quadartic programming.

MATBC8: DATA STRUCTURES

Unit I: INTRODUCTION & ARRAYS

Introduction: Overview - SPARKS - How to Create Programs - How to Analyze

Programs. Arrays: Axiomatization - Ordered Lists - Sparse Matrices - Representation of Arrays.

Unit II: STACKS, QUEUES & LINKED LISTS

Stacks and Queues: Fundamentals - A Mazing Problem - Evaluation of Expressions -

Multiple Stacks and Queues. Linked Lists : Singly Linked Lists - Linked Stacks and Queues -

Polynomial Addition - Doubly Linked Lists and Dynamic Storage Management.

Unit III: TREES

Basic Terminology - Binary Trees - Binary Tree Representations - Binary Tree Traversal

- More on Binary Trees - Threaded Binary Trees - Representation of Binary Trees -

Applications of Trees - Counting Binary Trees.

Unit IV: SORTING

Internal Sorting : Searching - Insertion sort - Quick sort - 2-way Merge sort - Heap sort.

External Sorting : Storage Devices - Sorting with Disks.

Unit V: GRAPHS & FILES

Graphs : Terminology and Representations - Traversals, Connected Components and

Spanning Trees - Shortest paths and Transitive Closure - Activity Networks, Topological Sort

and Critical Paths. Files : Files, Queries and Sequential Organizations - Hashed Indexes - File

Organizations: Sequential Organizations - Random organizations.

MATBE3: JAVA PROGRAMMING (Theory and Practical)

Unit-I

Basic concepts of object oriented programming - benefits & applications of oop. JAVA

evolution: java features - java and c - java and C++ - java and internet Overview of JAVA

language: introduction- implementation of java program - creating, compiling, running the

program , JVM .

Unit - II

Data Types - operators and Expressions - Branching: Decision making with if statement,

if...else statement, nesting if...else statements, the else if ladder, switch statement. Looping: The

while statement, do statement, for statement- additional features of for loop: nesting of for

loops; jumps in loops - jumping out of a loop; skipping a part of loop; labeled loops

Unit - III

Classes and Objects: Introduction; adding variables, creating and adding methods,

constructors, overloading; Inheritance - defining a subclass, multilevel inheritance,

hierarchical inheritance, overriding methods, visibility control, rules of thumb.

Unit - IV

Packages - Multithreaded Programming: creating threads, extending the thread classimplementing

the run() method, starting new thread, stopping and blocking a thread- life

cycle of a thread - new born state, running state, blocked state, dead state.

Unit -V

Applet: Basics - Architecture - Passing parameters to Applets - Skeleton - simple

Applet - AWT.

MATBC10: MATHEMATICAL METHODS

Unit-I: FOURIER TRANSFORMS: Fourier Transforms - Defn. Inversion theorem -

Fourier cosine transforms - Fourier sine transforms - Fourier transforms of derivatives -

Fourier transforms of some simple functions - Fourier transforms of rational functions - The

convolution integral - convolution theorem - Parseval's relation for Fourier transforms -

solution of PDE by

Fourier transform.

Laplace's Equation in Half plane

Laplace's Equation in an infinite strip

The Linear diffusion equation on a semi-infinite line

The two-dimensional diffusion equation.

Unit-II: HANKEL TRANSFORMS: Definition - Elementary properties of Hankel Transforms -

Hankel Transforms of Derivatives of functions - Hankel Transforms of some elementary

functions - The Parseval relation for Hankel transforms - Relation between Fourier and Hankel

transforms - Application to PDE.

Axisymmetric Dirichlet problem for a half - space.

Axisymmetric Dirichlet problem for a thick plate

Unit-III: INTEGRAL EQUATIONS: Types of Integral equations - Equation with

separable kernel - Fredholm Alternative Approximate method - Volterra integral equations

- Classical Fredholm theory - Fredholm's First, Second, Third theorems.

Unit-IV: Application of Integral equation to ordinary differential equation - initial value

problems - Boundary value problems - singular integral equations - Abel Integral equation

Unit-V: CALCULUS OF VARIATIONS: Variation and its properties - Euler's equation

- Functionals of the integral forms Functional dependent on higher order derivatives -

functionals dependent on the functions of several independent variables - variational problems in

parametric form.

MATBE4-DATA WAREHOUSING

Unit I

Introduction - Delivery process - System processes - Process architecture.

Unit II

Database Schema - Partitioning strategy - Aggregations - Data Marting.

Unit III

Metadata - System and Data warehouse process managers - Hardware architecture - Physical

layout.

Unit IV

Security - Backup recovery - Service level agreement - Operating the data warehouse.

Unit V

Capacity planning - Tuning the Data warehouse - Testing the Data warehouse - Data warehouse futures.

MATLAB (Theory-60 marks and Practical-40 marks)

Unit - I

Introduction - Basics of MATLAB, Input - Output, File types - Platform dependence -

General commands.

Unit - II

Interactive Computation: Matrices and Vectors - Matrix and Array operations - Creating

and Using Inline functions - Using Built-in Functions and On-line Help - Saving and loading

data - Plotting simple graphs.

Unit - III

Programming in MATLAB: Scripts and Functions - Script files - Functions files-

Language specific features - Advanced Data objects.

Unit - IV

Applications - Linear Algebra - Solving a linear system - Finding Eigen values and Eigen

vectors - Matrix Factorizations.

Unit - V

Applications - Data Analysis and Statistics - Numerical Integration - ordinary

differential equations - Nonlinear Algebraic Equations.

MATHEMATICA (Theory-60 marks and Practical-40 marks)

Unit -I

Running mathematica - Numerical calculations - Building Up calculations - Using the

Mathematica system.

Unit -III

Algebraic calculations - systematic computation - values of symbols - transforming

Algebraic expressions - simplifying Algebraic expressions - putting expression into different

forms - simplifying with assumption - symbolic mathematics - sums and products - relation and

logical operators - solving equations - Inequalities - differential equations - power series -

limits - Integral transforms - recurrence equations - mathematical notation and note books.

Unit -III

Numerical Mathematics: Basic operations - Numerical sums, product and integrals -

Numerical equation solving - Numerical differential equations - Numerical optimization -

Manipulating numerical data - Statistics.

Functions and Programs: Defining functions - functions as procedures - Repetitive

operations - Transformation rules for functions - Lists - Collecting objects together - Making

tables of values - Vectors and matrices - Getting pieces of lists - Testing and searching list

elements - adding, removing and modifying list elements - combining lists - rearranging lists -

ordering in lists.

Unit -IV

Graphics: Basic plotting - options - Redrawing and combining plots - manipulating

options - Three-dimensional surface plots - converting between types of Graphics.

Unit -V

Input and output in Notebooks: Entering Greek letters - Two dimensional inputs - editing

and evaluating two - dimensional expressions - entering formulas - entering tables and matrices

- subscripts, bars and other modifiers - Non-English characters and key boards - other

mathematical Notation - Forms of input and output - mixing text and formulation - displaying and printing mathematica notebooks

VISUAL BASIC (Theory - 60marks and Practical- 40 marks)

Unit - I

Visual basic fundamentals - Data types, constants and variables - Conditional statements -

Loops - Arrays.

Unit - II

Strings and Typecasting - Intrinsic controls - Visual basic menus - Forms and dialog boxes.

Unit - III

Handling keyboard and mouse input - Time and timers - Subroutines and functions.

Unit - IV

Visual basic Database programming - Data environment - DAO,RDO,ADO,ADO.

Unit - V

Building interfaces from the database - Data reports - Making reports.

ORACLE (Theory - 60marks and Practical- 40 marks)

Unit - I

ORACLE 8.0 - Data types - Basic parts of SQL, DDL, DML, and TCL.

Unit - II

String functions - Group value functions - Single value functions - Data functions and views.

Unit - III

Indexes - Sequences - Sub queries - Reports in SQL PLUS.

.

Unit - IV

Introduction to PL/SQL - PL/SQL Block - Exception handling.

Unit - V

Triggers - Procedures - Cursors

UNIT-I: Group Theory: Direct products- Group Action on a Set: Isotropy Subgroups- Orbits-

Counting Theorems- p-Groups- The Sylow Theorems.

UNIT-II: Applications of the Sylow Theory: Applications to p-Groups and the Class Equation-

Further Applications.

Ring Theory: Ring of Polynomials: Polynomials in an Inderterminate- The Equation

Homomorphisms- Factorization of Polynomials over a Field.

UNIT-III: Field Theory: Extension Fields - Algebraic and Transcendental Elements-

Irreducible polynomial over F-Simple Extension- Algebraic Extensions: Finite Extensions-

Structure of Finite Fields.

UNIT-IV: Automorphisms of Fields - Conjugation Isomorphisms- Automorphisms and

Fixed Fields- The Frobenius Automorphism- Splitting Fields.

UNIT-V: Separable Extensions- Galois Theory: Normal Extensions- The Main Theorem-

Illustrations of Galois Theory: Symmetric Functions.

MATBC2: ANALYSIS

RIEMANN STILTJES INTEGRAL:

Unit-I. Definition and Existence of the Integral - properties of the integral - Integration

and differentiation - Integration of vector valued function - rectifiable curves

Unit-II. Uniform convergence and continuity - uniform convergence and integration - uniform

convergence and differentiation - equicontinuous families of functions - The Stone Weirstrass

theorem

FUNCTIONS OF SEVERAL VARIABLES

Unit-III. Linear transformation - contraction principle - Inverse function theorem -

Implicit function theorem - determinants - derivatives of higher order - differentiation of

integrals

LEBESGUE MEASURE:

Unit-IV. Outer measure - Measurable sets and Lebesgue measure - Measurable functions -

Littlewood's Theorem.

Unit-V. The Lebesgue integral of bounded functions over a set of finite measure - integral of a

non - negative function - General Lebesgue Integral - convergence in measure.

MATBC3: DIFFERENTIAL EQUATIONS

Unit-I. Linear Equations with constant coefficients - Second order Homogeneous equations

- Initial value problems - Linear dependence and independence Wronskian and a formula

for Wronskian - Non Homogeneous equation of order two.

Unit-II. Homogeneous and Non - Homogeneous Equations of order n - Initial value problems -

Annihilator Method to solve a non - homogeneous equation - Algebra of constant coefficients

operators.

Unit-III. Linear Equations with variable coefficients - Initial value problems - Existence

and Uniqueness Theorems - Solutions to a non - homogeneous equation - Wronskian and

Linear dependence - reduction of the order of a homogeneous equation - Homogeneous

equation with analytic coefficients - The Legendre equation.

Unit IV: Nonlinear Partial Differential Equations of the first order - Cauchy's method of

characteristics - Compatible systems of first order equations - Charpit's method- Special types

of First order equations - Jacobi's method.

Unit V: Partial Differential Equations of Second order - The origin of Second-order Equations -

Linear Partial Differential Equations with constant coefficients - Equations with variable

coefficients - Characteristics curves of second - order equations- Characteristics of equations in

three variables.

MATBE1: NUMERICAL METHODS

Unit I

Solution of Nonlinear Equations: Newton's method - Convergence of Newton's method -

Bairstow"s Method for quadratic factors. Numerical Differentiation and Integration:

Derivatives from Differences tables - Higher order derivatives - Divided difference, Central-

Difference formulas - Composite formula of Trapezoidal rule - Romberg integration -

Simpson's rules.

Unit II

Solution of System of Equations: The Elimination method - Gauss and Gauss Jordan

methods - LU Decomposition method - Matrix inversion by Gauss-Jordan method - Methods

of Iteration - Jacobi and Gauss Seidal Iteration - Relaxation method - Systems of Nonlinear

equations.

Unit III

Solution of Ordinary Differential Equations: Taylor series method - Euler and Modified

Euler methods - Rungekutta methods - Multistep methods - Milne's method - Adams

Moulton method.

Unit IV

Boundary Value Problems and Characteristic Value Problems: The shooting method -

solution through a set of equations - Derivative boundary conditions - Characteristic value

problems - Eigen values of a matrix by Iteration - The power method.

Unit V

Numerical Solution of Partial Differential Equations: (Solutions of Elliptic, Parabolic and

Hyperbolic partial differential equations) Representation as a difference equation - Laplace's

equation on a rectangular region - Iterative methods for Laplace equation - The Poisson

equation - Derivative boundary conditions - Solving the equation for time-dependent heat

flow (i) The Explicit method (ii) The Crank Nicolson method - solving the wave equation

by Finite Differences.

MATBC4: DISCRETE MATHEMATICS

Unit I: Logic

Propositions - Logical Connectives - Compound statements - Conditional and

Biconditional Propositions - Truth tables - Tautologies and Contradictions - Logical

equivalence and implications - Demorgan's Law - Normal forms - PDNF and PCNF - Predicate

Calculus - Free and bound variables - Quantifiers - Universe of discourse - Theory of inference

- Rules of universal specification and generalization - Arguments - Validity of Arguments.

Unit II: Set Theory

Basic concepts - Notations - Algebra of sets - The power sets - Ordered pairs

and Cartesian products - Relation and its types - Properties - Relational Matrix and the

graph of relation - Partitions -Equivalence elations - Poset - Hasse diagram - Lattices

and their properties - Sublattice -Boolean Algebra - Homomorphism.

Unit III: Functions

Definitions of functions and its Classification - Types - Examples - Composition

of functions - Inverse functions - Binary and n-ary operations - Characteristic function of a

set - Hashing functions - Recursive functions - Permutation functions.

Unit IV: Graph Theory

Graph Theory: Introduction - Basic terminology - Representation of graphs -

connectivity - Eulerian and Hamiltonian graphs - Planar graphs- Directed graphs-Application of

Graphs. Trees: Binary tree - traversals of a binary tree - Expansion trees.

Unit V: Grammars and Languages

Definitions - Types of Grammars - Productions - Regular Grammar and Languages

- Finite state Automata (FSA) - Deterministic and Non-Deterministic FSA - Conversion

of NDFSA to DFSA.

MATBC5: FLUID DYNAMICS

Unit-I:

Introductory Notions- Velocity- Stream Lines and Path Lines- Stream Tubes and

Filaments- Fluid Body- Density- Pressure. Differentiation following the Fluid- Equation of

continuity- Boundary conditions- Kinematical and physical- Rate of change of linear

momentum- Equation of motion of an inviscid fluid.

Unit-II:

Euler's momentum Theorem- Conservative forces- Bernoulli's theorem in steady motionenergy

equation for inviscid fluid- circulation- Kelvin's theorem- vortex motion-

Helmholtz equation.

Unit-III:

Two Dimensional Motion- Two Dimensional Functions- Complex Potential Basic

singularities- source- sink- Vortex- doublet- Circle theorem. Flow past a Circular cylinder with

circulation- Blasius Theorem- Lift force (Magnus effect). Irrotational Motion in Three

Dimensions- Spherical Harmonics- Axially Symmetric Field- Stokes's Stream Function- Motion

of a Sphere.

Unit-IV:

Viscous flows- Navier - stokes equations- some exact solutions of Navier Stokes

equations- Flow between parallel flat plates- Couette flow- Plane Poiseuille flow- Steady flow

in pipes: Flow through a pipe- The Hagen Poiseuille flow.

Unit-V:

Laminar Boundary Layer in incompressible flow: Boundary Layer concept- Boundary

Layer equations- Boundary Layer along a flat plate- The Blasius solution- Shearing stress and

boundary layer thickness- Displacement thickness, momentum thickness- Momentum integral

theorem for the boundary layer- The Von Karman Integral relation, The Von Karman

Integral relation by momentum law.

MATBC6: PROGRAMMING LANGUAGES

Unit I:

Principles of object-Oriented Programming: Software crisis - Software evolution -

A look at procedure-oriented Programming - Object-oriented Programming Paradigm -

Basic Concept of Object-Oriented Programming - Benefits of OOP - Object-Oriented

languages - Applications of OOP.

Unit II:

Tokens, Expressions and Control structure: Introduction - Tokens - Keywords -

Identifiers and constants - basic data types - User defined data types - Derived data

types - Symbolic constants - Type compactability - Declaration of variables - Dynamic

insulation of variables - Reference variables - operations in C++ - Scope resolution

operator - member Dereferencing operators - memory management operators - Manipulators

- type cast operator - expressions and their types - Special assignment expressions - implicit

conversions - operator over loading - operator precedence - Control structures.

Unit III:

Functions in C++: Introduction - The main function - Function prototyping - call

by reference - return by reference inline functions - default arguments - constant

arguments - function over loading - friend and virtual functions - Math library functions

Managing Console I/O operations: Introduction - C++ streams - C++ stream classes -

Unformatted I/O operations - Formatted I/O operations - Managing output with manipulators.

Unit IV:

Classes and Objects: Introduction - C Structures Revisited - Specifying a class -

Defining Member Functions - A C++ Program with class - Making an outside Function Inline -

Nesting of Member Functions - Private Member Functions - Arrays within a class - Memory

Allocation for Objects - Static Data Members - Static Member Functions - Arrays of Objects -

Objects as Function Arguments - Friendly functions - Returning Objects - Constant Member

Functions. Constructors and Destructors: Introduction - Constructors - Parameterized

Constructors - Multiple Constructors in a class - Constructors with Default Arguments -

Dynamic Initializations of Objects - Copy Constructor - Constructing Two dimensional arrays

- Constant Objects - Destructors.

Unit V:

Operators Overloading and Type Conversions: Introduction - Defining Operator

Overloading - Overloading Unary Operators - Overloading Binary Operators -

Overloading Binary Operators Using Friends - manipulating of strings Using Operators -

Rules of Overloading Operators.

Inheritance: Extending Classes: Introduction - Defining Derived Classes - Single

inheritance - Making a Private Member Inheritable - Multilevel Inheritance - Multiple

Inheritance - Hierarchical Inheritance - Hybrid Inheritance - Virtual Base Classes - Abstract

Classes - Constructors in Derived Classes - Member Classes: Nesting of Classes.

MATBC7: OPERATIONS RESEARCH

Unit I:

Introduction to L.P. - Graphical L.P. Solution - Sensitivity analysis - Simplex Method -

L.P. solution space in equation form - Transition from graphical to algebra solution -

The simplex method - artificial starting solution - Special cases in simplex method applications.

Duality - Primal and Dual - relationships - additional simplex algorithm for L.P.

Unit II:

Advanced Linear Programming - Generalized simplex Tableau in matrix form -

Decomposition algorithm - Matrix definition of dual problem - optimal dual solution.

Unit III:

Integer L.P. and Dynamic Programming - Integer Programming - Integer Programming

algorithm - Gomory cutting plane algorithm - Branch and Bound algorithm - Solution of the

Traveling salesperson problem - Deterministic Dynamic programming - Recursive nature

of computation in D.P. - Forward and Backward recursion.

Unit IV:

Classical optimization Theory - unconstraint problems - Necessary and sufficient

conditions - The Newton-Raphson method - constrained problems - Equality constraints (Jacobi

method and Lagrangian method).

Unit V:

Non-linear programming - unconstrained algorithms Direct search method - Gradient

method - constraint algorithms - Separable programming - Quadartic programming.

MATBC8: DATA STRUCTURES

Unit I: INTRODUCTION & ARRAYS

Introduction: Overview - SPARKS - How to Create Programs - How to Analyze

Programs. Arrays: Axiomatization - Ordered Lists - Sparse Matrices - Representation of Arrays.

Unit II: STACKS, QUEUES & LINKED LISTS

Stacks and Queues: Fundamentals - A Mazing Problem - Evaluation of Expressions -

Multiple Stacks and Queues. Linked Lists : Singly Linked Lists - Linked Stacks and Queues -

Polynomial Addition - Doubly Linked Lists and Dynamic Storage Management.

Unit III: TREES

Basic Terminology - Binary Trees - Binary Tree Representations - Binary Tree Traversal

- More on Binary Trees - Threaded Binary Trees - Representation of Binary Trees -

Applications of Trees - Counting Binary Trees.

Unit IV: SORTING

Internal Sorting : Searching - Insertion sort - Quick sort - 2-way Merge sort - Heap sort.

External Sorting : Storage Devices - Sorting with Disks.

Unit V: GRAPHS & FILES

Graphs : Terminology and Representations - Traversals, Connected Components and

Spanning Trees - Shortest paths and Transitive Closure - Activity Networks, Topological Sort

and Critical Paths. Files : Files, Queries and Sequential Organizations - Hashed Indexes - File

Organizations: Sequential Organizations - Random organizations.

MATBE3: JAVA PROGRAMMING (Theory and Practical)

Unit-I

Basic concepts of object oriented programming - benefits & applications of oop. JAVA

evolution: java features - java and c - java and C++ - java and internet Overview of JAVA

language: introduction- implementation of java program - creating, compiling, running the

program , JVM .

Unit - II

Data Types - operators and Expressions - Branching: Decision making with if statement,

if...else statement, nesting if...else statements, the else if ladder, switch statement. Looping: The

while statement, do statement, for statement- additional features of for loop: nesting of for

loops; jumps in loops - jumping out of a loop; skipping a part of loop; labeled loops

Unit - III

Classes and Objects: Introduction; adding variables, creating and adding methods,

constructors, overloading; Inheritance - defining a subclass, multilevel inheritance,

hierarchical inheritance, overriding methods, visibility control, rules of thumb.

Unit - IV

Packages - Multithreaded Programming: creating threads, extending the thread classimplementing

the run() method, starting new thread, stopping and blocking a thread- life

cycle of a thread - new born state, running state, blocked state, dead state.

Unit -V

Applet: Basics - Architecture - Passing parameters to Applets - Skeleton - simple

Applet - AWT.

MATBC10: MATHEMATICAL METHODS

Unit-I: FOURIER TRANSFORMS: Fourier Transforms - Defn. Inversion theorem -

Fourier cosine transforms - Fourier sine transforms - Fourier transforms of derivatives -

Fourier transforms of some simple functions - Fourier transforms of rational functions - The

convolution integral - convolution theorem - Parseval's relation for Fourier transforms -

solution of PDE by

Fourier transform.

Laplace's Equation in Half plane

Laplace's Equation in an infinite strip

The Linear diffusion equation on a semi-infinite line

The two-dimensional diffusion equation.

Unit-II: HANKEL TRANSFORMS: Definition - Elementary properties of Hankel Transforms -

Hankel Transforms of Derivatives of functions - Hankel Transforms of some elementary

functions - The Parseval relation for Hankel transforms - Relation between Fourier and Hankel

transforms - Application to PDE.

Axisymmetric Dirichlet problem for a half - space.

Axisymmetric Dirichlet problem for a thick plate

Unit-III: INTEGRAL EQUATIONS: Types of Integral equations - Equation with

separable kernel - Fredholm Alternative Approximate method - Volterra integral equations

- Classical Fredholm theory - Fredholm's First, Second, Third theorems.

Unit-IV: Application of Integral equation to ordinary differential equation - initial value

problems - Boundary value problems - singular integral equations - Abel Integral equation

Unit-V: CALCULUS OF VARIATIONS: Variation and its properties - Euler's equation

- Functionals of the integral forms Functional dependent on higher order derivatives -

functionals dependent on the functions of several independent variables - variational problems in

parametric form.

MATBE4-DATA WAREHOUSING

Unit I

Introduction - Delivery process - System processes - Process architecture.

Unit II

Database Schema - Partitioning strategy - Aggregations - Data Marting.

Unit III

Metadata - System and Data warehouse process managers - Hardware architecture - Physical

layout.

Unit IV

Security - Backup recovery - Service level agreement - Operating the data warehouse.

Unit V

Capacity planning - Tuning the Data warehouse - Testing the Data warehouse - Data warehouse futures.

MATLAB (Theory-60 marks and Practical-40 marks)

Unit - I

Introduction - Basics of MATLAB, Input - Output, File types - Platform dependence -

General commands.

Unit - II

Interactive Computation: Matrices and Vectors - Matrix and Array operations - Creating

and Using Inline functions - Using Built-in Functions and On-line Help - Saving and loading

data - Plotting simple graphs.

Unit - III

Programming in MATLAB: Scripts and Functions - Script files - Functions files-

Language specific features - Advanced Data objects.

Unit - IV

Applications - Linear Algebra - Solving a linear system - Finding Eigen values and Eigen

vectors - Matrix Factorizations.

Unit - V

Applications - Data Analysis and Statistics - Numerical Integration - ordinary

differential equations - Nonlinear Algebraic Equations.

MATHEMATICA (Theory-60 marks and Practical-40 marks)

Unit -I

Running mathematica - Numerical calculations - Building Up calculations - Using the

Mathematica system.

Unit -III

Algebraic calculations - systematic computation - values of symbols - transforming

Algebraic expressions - simplifying Algebraic expressions - putting expression into different

forms - simplifying with assumption - symbolic mathematics - sums and products - relation and

logical operators - solving equations - Inequalities - differential equations - power series -

limits - Integral transforms - recurrence equations - mathematical notation and note books.

Unit -III

Numerical Mathematics: Basic operations - Numerical sums, product and integrals -

Numerical equation solving - Numerical differential equations - Numerical optimization -

Manipulating numerical data - Statistics.

Functions and Programs: Defining functions - functions as procedures - Repetitive

operations - Transformation rules for functions - Lists - Collecting objects together - Making

tables of values - Vectors and matrices - Getting pieces of lists - Testing and searching list

elements - adding, removing and modifying list elements - combining lists - rearranging lists -

ordering in lists.

Unit -IV

Graphics: Basic plotting - options - Redrawing and combining plots - manipulating

options - Three-dimensional surface plots - converting between types of Graphics.

Unit -V

Input and output in Notebooks: Entering Greek letters - Two dimensional inputs - editing

and evaluating two - dimensional expressions - entering formulas - entering tables and matrices

- subscripts, bars and other modifiers - Non-English characters and key boards - other

mathematical Notation - Forms of input and output - mixing text and formulation - displaying and printing mathematica notebooks

VISUAL BASIC (Theory - 60marks and Practical- 40 marks)

Unit - I

Visual basic fundamentals - Data types, constants and variables - Conditional statements -

Loops - Arrays.

Unit - II

Strings and Typecasting - Intrinsic controls - Visual basic menus - Forms and dialog boxes.

Unit - III

Handling keyboard and mouse input - Time and timers - Subroutines and functions.

Unit - IV

Visual basic Database programming - Data environment - DAO,RDO,ADO,ADO.

Unit - V

Building interfaces from the database - Data reports - Making reports.

ORACLE (Theory - 60marks and Practical- 40 marks)

Unit - I

ORACLE 8.0 - Data types - Basic parts of SQL, DDL, DML, and TCL.

Unit - II

String functions - Group value functions - Single value functions - Data functions and views.

Unit - III

Indexes - Sequences - Sub queries - Reports in SQL PLUS.

.

Unit - IV

Introduction to PL/SQL - PL/SQL Block - Exception handling.

Unit - V

Triggers - Procedures - Cursors