M.Sc in MathematicsBharathiar 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.
Group Theory: Direct products- Group Action on a Set: Isotropy Subgroups- Orbits-
Counting Theorems- p-Groups- The Sylow Theorems
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
Field Theory: Extension Fields-Algebraic and Transcendental Elements-Irreducible
polynomial over F-Simple Extension- Algebraic Extensions: Finite Extensions- Structure
of Finite Fields
Automorphisms of Fields- Conjugation Isomorphisms- Automorphisms and Fixed
Fields- The Frobenius Automorphism- Splitting Fields.
Separable Extensions- Galois Theory: Normal Extensions- The Main Theorem-
Illustrations of Galois Theory: Symmetric Functions
Text book: "A First Course in Abstract Algebra" by J.B.Fraleigh, Fifth Edition, Addition
Wesly Longman, Inc, Reading Massachusetts, 1999.
UNIT-I: Chapter 2, Section: 2.4 (Direct Product only)
Chapter 3, Sections: 3.6, 3.7
UNIT-II: Chapter 4, Section: 4.3, Chapter 5, Sections: 5.5, 5.6
UNIT-III: Chapter 8, Sections: 8.1, 8.3 (Finite Extensions Only), 8.5
UNIT-IV: Chapter 9, Sections: 9.1, 9.3
UNIT-V: Chapter 9, Sections: 9.4, 9.6, 9.7 (Symmetric Functions only)
MATAC02 : REAL ANALYSIS
RIEMANN STILTJES INTEGRAL:
Definition and Existence of the Integral - properties of the integral - Integration and
differentiation - Integration of vector valued function - rectifiable curves
Uniform convergence and continuity - uniform convergence and integration - uniform
convergence and differentiation - equicontinuous families of functions - The Stone
FUNCTIONS OF SEVERAL VARIABLES
Linear transformation - contraction principle - Inverse function theorem - Implicit
function theorem - determinants - derivatives of higher order - differentiation of
Outer measure - Measurable sets and Lebesgue measure - Measurable functions -
The Lebesgue integral of bounded functions over a set of finite measure - integral of a
non - negative function - General Lebesgue Integral - convergence in measure
For Unit-I to III relevant chapters from : Principles of Mathematical Analysis by W.
Rudin, McGraw Hill, New York, 1976
For Unit-IV and V relevant chapters from : Real Analysis by H.L. Roydon, Third Edition,
Macmillan, New York, 1988.MATAC03: ORDINARY DIFFERENTIAL EQUATIONS
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.
Homogeneous and Non - Homogeneous Equations of order n - Initial value problems -
annihilator Method to solve a non - homogeneous equation - Algebra of constant
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.
Linear Equation with regular singular points - Euler Equation - Second order equations
with regular singular points - Exceptional cases - Bessel equation.
Existence and Uniqueness of solutions to first order equations - Equation with variables
separated - Exact Equations - Method of successive approximations - The Lipschitz
condition - convergence of the successive approximations and the existence theorem.
MATAC04: COMPLEX ANALYSIS
Introduction to the concept of analytic function: Limits and continuity - Analytic
functions - Polynomials - Rational functions - Conformality: Arcs and closed curves -
Analytic functions in regions - Conformal Mapping - Length and Area - Linear
Transformations: The Linear group - The Cross ratio - Elementary Riemann Surfaces.
Complex Integration: Line Integrals Rectifiable Arcs - Line Integrals as Functions of
Arcs - Cauchy's theorem for a rectangle - Cauchy's theorem in a disk, Cauchy's Integral
formula: The Index of a point with respect to a closed curve - The Integral formula -
Higher derivatives Removable singularities, Taylor's Theorem - Zeros and Poles - The
Local Mapping - The Maximum principle - chains and cycles.
The Calculus of Residues The Residue theorem - The Argument principle - Evaluation
of definite integrals. Harmonic functions: The Definitions and basic Properties - Mean
value property - Poisson's Formula.
Series and Product Developments: Weierstrass Theorem - The Taylor Series - The
Laurent Series - Partial fractions and Factorization: Partial Fractions - Infinite Products -
The Riemann Mapping Theorem - Statement and Proff - Boundary Behaviour - Use of
the reflection principle - Analytic arcs - Conformal mapping of Polygons: The
Behaviour at an angle - The Schwarz - Christoffel Formula - Mapping on a rectangle.
MATAC05: PARTIAL DIFFERENTIAL EQUATIONS
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.
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.
The Solution of Linear Hyperbolic Equations - Separation of variables - The Method of
Integral Transforms - Nonlinear Equations of the second order.
Laplace's Equation - The occurrence of Laplace's Equation in Physics- Elementary
solution of Laplace's Equation - Families of Equipotential surfaces Boundary value
problems - Separation of variables- Problems with axial symmetry.
The wave equation - The occurrence of wave equation in Physics - Elementary solutions
of the one-dimensional wave equation - Vibrating Membranes: Applications of the
calculus of variations - Three dimensional problems.
The Diffusion Equations: Elementary solutions of the Diffusion Equation - Separation of
variables- The use of Integral transforms.
INDRODUCTORY CONCEPTS: Mechanical system - Generalized Coordinates -
Constraints - Virtual Work - Energy and Momentum.
LAGRANGE'S EQUATIONS: Derivations of Lagrange's Equations: Derivations of
Lagrange's Equations - Examples - Integrals of Motion.
HAMITON'S EQUATIONS: Hamilton's Principle - Hamilton's Equations.
HAMILTON - JACOBI THEORY: Hamilton's Principle function - Hamilton - Jacobi
Equation - Separability.
CANONICAL TRANSFORMATIONS: Differential forms and Generating Functions -
Lagrange and Poisson Brackets.
MATAC07- TOPOLOGY AND FUNCTIONAL ANALYSIS
Spaces and Maps.
Separability Axioms and Compactness.
Connectedness - Pathwise connectedness - Imbedding and Extension theorems.
Banach spaces - Definition and Examples - continuous linear transformations - Hahn
Banach theorem - Natural Imbedding - open mapping theorem - conjugate of an
Hilbert spaces - Definition and simple properties - Orthogonal Complements -
Orthonormal basis - conjugate space.
MATAC08: 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)
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.
MATAC09: 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
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.
MATAC10 - NON LINEAR DIFFERENTIAL EQUATIONS
First order systems in two variables and linearization: The general phase plane-some
population models - Linear approximation at equilibrium points - Linear systems in
Averaging Methods: An energy balance method for limit cycles - Amplitude and
frequency estimates - slowly varying amplitudes - nearly periodic solutions - periodic
solutions: harmony balance - Equivalent linear equation by harmonic balance - Accuracy
of a period estimate.
Perturbation Methods: Outline of the direct method - Forced Oscillations far from
resonance - Forced Oscillations near resonance with Weak excitation - Amplitude
equation for undamped pendulum - Amplitude Perturbation for the pendulum equation -
Lindstedt's Method - Forced oscillation of a self - excited equation - The Perturbation
Method and Fourier series.
Linear Systems: Time Varying Systems - Constant coefficient System - Periodic
Coefficients - Floquet Theory - Wronskian.
Stability: Poincare stability - solutions, paths and norms - Liapunov stability Stability of
linear systems - Comparison theorem for the zero solutions of nearly - linear systems.MATAC11: CONTROL THEORY
OBSERVABILITY: Linear Systems - Observability Grammian - Constant coefficient
systems - Reconstruction kernel - Nonlinear Systems
CONTROLLABILITY: Linear systems - Controllability Grammian - Adjoint systems -
Constant coefficient systems - steering function - Nonlinear systems
STABILITY: Stability - Uniform Stability - Asymptotic Stability of Linear Systems -
Linear time varying systems - Perturbed linear systems - Nonlinear systems
STABILIZABILITY: Stabilization via linear feedback control - Bass method -
Controllable subspace - Stabilization with restricted feedback
OPTIMAL CONTROL: Linear time varying systems with quadratic performance criteria
- Matrix Riccati equation - Linear time invariant systems - Nonlinear Systems
MATAE01: 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 - Simpson's rules.
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.
SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS: Taylor series method -
Euler and Modified Euler methods - Rungekutta methods - Multistep methods - Milne's
method - Adams Moulton method.
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
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.
MATAE02: COMPUTER PROGRAMMING AND LAB I
Fortran 77 - Representation of Integer and Real constants - Variable names - Arithmetic
operators and modes for Expression - Integer Expressions. Real Expressions - Hierarchy
of Operations in Expressions - Arithmetic statement - Defining Variables - Mixed Mode
Expressions - Special Functions - Input Output statements.
Format description for Read statement - Format description for print statement - multi
record formats - Hollerith field declaration - specifications in a Format - Generalized
input / output statements - Logical constants, variables and Logic Expressions.
Control statements - Relational Operator - Logical IF statement - GO TO Statements -
Nested Logical IF statement - Arithmetic IF statement - computed GOTO statement.
The DO Statement - Rules to be followed in Utilizing Do Loops - RRPEAT WHILE
structure - Subscripted Variables - Subscript Expressions - Dimension statement - DO
loops with subscripts.
Functions and subroutines - Function subprograms - Subroutines - Common declaration
- Implicit declaration - Equivalence declaration.
MATAE03 - COMPUTER PROGRAMMING AND LAB II
Overview of C - Constants. Variables and Data Types - Character set - C tokens -
Keywords & Identifiers - constants - variables - Data types - Declaration of variables -
Assigning values to variables - Defining symbolic constants.
Arithmetic of operators - Relational operators - Logical operators - Assignment
operator - Increment and decrement operators - conditional operator - Bitwise operators
- special operators - Arithmetic Expressions - Evaluation of Expressions - Precedence
of arithmetic operators - Type conversions in Expressions - Operator Precedence and
Associativity - Mathematical Functions.
Managing Input and Output operators - Reading a character - Writing a character -
formatted input - formatted output - Decision making IF statement - IF - ELSE -
statement - Nesting of IF ELSE statements - The Switch statement - The GO TO
The WHILE statement DO statement - FOR statement -Jumps in Loops - Onedimensional
Array - Two dimensional Arrays - Initializing two dimensional arrays -
Need for User defined functions - A multi-function program - the form of C Functions -
Return Values and their Types calling a function - Category of functions - Arguments
but no return values - Arguments with return values In file management in C - Defining
and with return values - In file management in C - Defining and opening a file - closing
a file- Input / Output operations on files.
COMPUTER PROGRAMMING AND LAB II
PAPER - I
Text formatting, TEX and its offspring, What's different in LATEX 2є, Distinguishing
LATEX 2є , Basics of a LATEX file.
Commands and Environments-Command names and arguments, Environments,
Declarations, Lengths, Special Characters, Fragile Commands, Exercises.
Document Layout and Organization - Document class, Page style, Parts of the document,
Table of contents, Fine - Tuning text, Word division.
Displayed Text - Changing font, Centering and indenting, Lists, Generalized lists,
Theorem-like declarations, Tabulator stops, Boxes.
Tables, Printing literal text, Footnotes and marginal notes. Drawing pictures with
Mathematical Formulas - Mathematical environments, Main elements of math mode,
Mathematical symbols, Additional elements, Fine-tuning mathematics.
PAPER - II
Unit - I
Introduction - Basics of MATLAB, Input - Output, File trypes - 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 - Curve fitting and Interpolation - Data analysis and
Statistics - Numerical Integration - Ordinary differential equations - Nonlinear
Unit - V
Graphics: Basic 2-D Plots - Using subplot to Layout multiple graphs - 3 - D Plots -
Handle Graphics - Saving and printing Graphs - Errors.
PAPER - III
Unit - I: Introduction to Mathematica
Running Mathematica - Numerical Calculations - Building Up calculations - Using the
Mathematica system - Algebraic calculations - Symbolic Mathematics - Numerical
Unit - II
Functions and Programs - Lists - Graphics - Input and Output in Notebooks - The
structure of Graphics.
Unit - III: Advanced Mathematics in Mathematica
Mathematical Functions - Algebraic Manipulation - Manipulating Equations - Calculus.
Unit - IV
Series, Limits and Residues - Linear Algebra - Constructing matrices - Getting pieces of
matrices - Scalars, Vectors and Matrices - Operations on scalars, vectors and matrices -
Multiplying Vectors and matrices - Matrix inversion - Basic matrix operations - Solving
linear systems - Eigen values and Eigen vectors.
Unit - V
Numerical operations on data - Curve fitting - Approximate functions and Interpolation
- Fourier Transforms.
Numerical operations on functions - Numerical Integration - Numerical evaluation of
sums and products - Numerical Solution of Polynomial equations - Numerical root
finding - Numerical solution of Differential equations -
Implementing the Algorithms of any one of the software in Papers I to III above.
SUPPORTIVE: APPLIED MATHEMATICS - I
UNIT I: ORDINARY DIFFERENTIAL EQUATIONS
Second and higher order linear ODE - Homogeneous linear equations with constant and
variable coefficients - Nonhomogeneous equations - Solutions by variation of
UNIT II: FUNCTIONS OF SEVERAL VARIABLES
Partial derivatives - Total differential - Taylor's expansions - Maxima and Minima of
functions - Differentiation under integral sign.
UNIT III: PARTIAL DIFFERENTIAL EQUATIONS
Formation of PDE by elimination of arbitrary constants and functions - Solutions -
General and singular solution- Lagrange's Linear equation - Linear PDE of second and
higher order with constant coefficients.
UNIT IV: FOURIER SERIES
Dirichlet's conditions - General Fourier series - Half range Sine and Cosine series -
Parseval's identity - Harmonic Analysis.
UNIT V: BOUNDARY VALUE PROBLEMS
Classifications of PDE - Solutions by separation of variables - One dimensional heat and wave equation.
SUPPORTIVE: APPLIED MATHEMATICS - II
UNIT I: LAPLACE TRANSFORM
Transform of elementary functions - Transforms of derivatives and integrals - Initial and
final value theorems - Inverse Laplace transform - Convolution theorem - Solutions of
linear ODE with constant coefficients.
UNIT II: FOURIER TRANSFORMS
Fourier integral theorem - Fourier transform pairs- Fourier Sine and Cosine transforms -
Properties - Transforms of simple functions - Convolution theorem - Parseval's identity.
UNIT III: MULTIPLE INTEGRALS
Double integration - Cartesian and polar co-ordinates - Change of order of integration -
Area as a double integral - Triple integration - Volume as a triple integral.
UNIT IV: VECTOR CALCULUS
Gradient, Divergence and Curl - Directional derivative - Irrotational and solenoidal
vector fields - Vector integration - Green's theorem, Gauss divergence theorem and
UNIT-V: NUMERICAL SOLUTIONS OF ODEs
Solution by Taylor's series Method - Euler's Method - Modified Euler Method, Runge-
Kutta Method - Solving simultaneous equations.