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

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