M Sc Mathematics with Computer Applications & Compulsory Diploma in Computational SoftwareBharathiar University
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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.
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-
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.
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
FUNCTIONS OF SEVERAL VARIABLES
Unit-III. Linear transformation - contraction principle - Inverse function theorem -
Implicit function theorem - determinants - derivatives of higher order - differentiation of
Unit-IV. Outer measure - Measurable sets and Lebesgue measure - Measurable functions -
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
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
MATBE1: NUMERICAL METHODS
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 -
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
Solution of Ordinary Differential Equations: Taylor series method - Euler and Modified
Euler methods - Rungekutta methods - Multistep methods - Milne's method - Adams
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.
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
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.
Euler's momentum Theorem- Conservative forces- Bernoulli's theorem in steady motionenergy
equation for inviscid fluid- circulation- Kelvin's theorem- vortex motion-
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.
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.
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
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.
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.
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.
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.
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
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.
Advanced Linear Programming - Generalized simplex Tableau in matrix form -
Decomposition algorithm - Matrix definition of dual problem - optimal dual solution.
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.
Classical optimization Theory - unconstraint problems - Necessary and sufficient
conditions - The Newton-Raphson method - constrained problems - Equality constraints (Jacobi
method and Lagrangian method).
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)
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.
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
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
Introduction - Delivery process - System processes - Process architecture.
Database Schema - Partitioning strategy - Aggregations - Data Marting.
Metadata - System and Data warehouse process managers - Hardware architecture - Physical
Security - Backup recovery - Service level agreement - Operating the data warehouse.
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 -
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)
Running mathematica - Numerical calculations - Building Up calculations - Using the
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.
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.
Graphics: Basic plotting - options - Redrawing and combining plots - manipulating
options - Three-dimensional surface plots - converting between types of Graphics.
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