department of mathematicsdiscrete mathematics, real analysis and solve problems efficiently....

DEPARTMENT OF

MATHEMATICS

Programme Specific Outcomes

1 Help the students to enhance their knowledge in

soft skills and Computing skills.

2 Enable the students to equip knowledge in various

concepts involved in algebra, differential equations

and graph theory.

3 Enable the students to acquire knowledge in C

programming.

4 Students are trained in an effective manner to

attend the competitive exams in order to brighten

their future.

5 Facilitate students to acquire a flair knowledge in

discrete mathematics, real analysis and solve

problems efficiently.

ALAGAPPA UNIVERSITY, KARAIKUDI

NEW SYLLABUS UNDER CBCS PATTERN (w.e.f. 2014-15)

M.Sc., MATHEMATICS – PROGRAMME STRUCTURE

Sem Course Cr. Hrs./

Week

Marks Total

Subject

code

Name Int. Ext.

I

4MMA1C1 Core – I – Algebra – I 5 6 25 75 100

4MMA1C2 Core – II – Analysis – I 5 6 25 75 100

4MMA1C3 Core – III – Differential Geometry 5 6 25 75 100

4MMA1C4 Core – IV – Differential Equations 5 6 25 75 100

Elective – I 4 6 25 75 100

Total 24 30 -- -- 500

II

4MMA2C1 Core – V – Algebra – II 5 6 25 75 100

4MMA2C2 Core – VI – Analysis – II 5 6 25 75 100

4MMA2C3 Core – VII – Probability and Statistics 5 6 25 75 100

Elective – II 4 6 25 75 100

Elective – III 4 6 25 75 100

Total 23 30 -- -- 500

III

4MMA3C1 Core – VIII – Complex Analysis 5 6 25 75 100

4MMA3C2 Core – IX – Topology – I 5 6 25 75 100

4MMA3C3 Core – X – Operations Research 5 6 25 75 100

4MMA3C4 Core – XI – Number Theory 5 6 25 75 100

Elective – IV 4 6 25 75 100

Total 24 30 -- -- 500

IV

4MMA4C1 Core – XII – Functional Analysis 5 8 25 75 100

4MMA4C2 Core – XIII – Topology – II 5 8 25 75 100

4MMA4C3 Core – XIV – Numerical Analysis 5 7 25 75 100

Elective – V 4 7 25 75 100

Total 19 30 -- -- 400

Grand Total 90 120 -- -- 1900

Semester –I Name of the subject (4MMA1C1) Algebra I

Course Description:

This course will cover the topics contained in the five syllabus. Several additional topics will also be covered and

the presentation will not necessarily follow the text. Attendance is required and the exams will be over the lectures

and homework. The course topics include:

Group theory, subgroups, homomorphism, permutation group.

Another counting principle.

Ring theory.

Ideals and Quotient rings.

Enclidean rings, Polynomials over the rational field.

Course Objectives:

The objectives of this subject is to expose student to understand several important concepts in algebra.

Including group, ring, field, homomorphism, automorphism, quotient structure and to apply some of these

concepts to real world problems.

Use results from binary operations, and its properties, definition of a group, subgroups, homomorphism,

cayley’s theorem, permutation groups, to solve contempotary problems.

Explain from elementary principle, sylow’s theorem why certain algebraic facts are true.

Ring theory ,definition, examples of rings.

Ideals and Quotient ring, more ideals and quotient rings, integral domain.

Enclidean rings, prime ideal, maximal ideal prime avoidance theorem, polynomial rings over commutative

rings.

Course outcomes:

Students will able to:

Use sylow’s theorem to describe the structure of certain finite groups.

Explain the notation of extention field.

Give the definition of concepts related to group, homomorphism, ring, ideals and field.

Understand the structure of fields do computations in specific examples of finite fields.

Understand the congrurence modulo concept, fermat’s theorem.

Use the concept of decomposition of composite number as a product of primes uniquely and euler

functions.

Text Book:

Topics in Algebra by Herstein I.N

NAME OF THE SUBJECT: (4MMA1C2) ANALYSIS-I

COURSE DISCEIPTION:

This course covers Basic Topology, Numerical Sequences and Series, Power Series, Continuity,

Differentiation.

COURSE OBJECTIVE:

To apply Basic Topology on Metric Space, Compact Sets, Perfect Sets,

connected Set.

Acquired Knowledge on Convergent Sequences, Sub Sequences, Cauchy

Sequences, Upper and Lower Limits, Special Sequences.

Distinguish of Continuity and Compactness.

Acquired Knowledge Continuity, Discontinuity, Monotonic Function.

COURSE OUTCOMES: On complete of this course

To know the Basic of Analysis.

To learn the different types of Numerical Sequences and Series.

Understand the Bounded, Convergences, Divergence Definetion.

Understand and perform simple derivative of a Real Function.

Prove a theorem Mean Value Theorems, Taylor’s Theorems,

Differentiation of Vector-Valued Function.

TEXT BOOK:

Principle of Mathematical Analysis – Walter Rudin, III Edition (Relevant portions of chapters II,III,IV &

V), McGRAW – Hill Book Company.

Name of the subject (4MMA1C3) Differential geometry

Course Description:




Arc length, serret - frenet formulae

Osculating circle.

Surfaces and class of surfaces.

Geodesics and geodesic curvature.

Course Objectives:

The objectives of this subject is to expose student to understand several important concepts in Differential

Geometry.

Arc length and tangent,normal Binormal to apply some of these concepts to real world problems.

Use results from geodesics curvature, and its properties, definition of a osculating sphere, intrinsic

equations.

Explain from serret_frenet formulae isometric correspondents.

Canonical geodesics equations ,definition, examples .

Direction coefficients and family of curves.

Fundamental form of principal curvature,line curvature and developables.

Course outcomes:


Use gauss bonnet theorem to describe the second fundamental form and developables.

Explain the geodesics parallels.

Give the definition of concepts related to normal propertyof geodesics.

Understand the structure of arc length binormal to the curvature.

Understand the developable associated with the space curves.

Gaussian curvature to use developable associated with the space curves.

TEXT BOOk:

Topics in Differential geometry by T.G. Willmore .

Name of the subject ﴾4MMA1C4)Differential Equation

Course Description:

Student will be able to

Linear Equation with variable coefficients – the wronskian and linear

dependence reduction of the order of a homogeneouse Equation.

Linear Equation with regular singular points- the Euler Equation.

Partial Differentiual Equations of the first order –Linear Equation of the first

order.

Second order equation –linear partial equations with constant coefficiant

equations with variable co-efficients.

Laplace’s Equation-The wave equation.

Course objective: This course intended to expose you to the basic ideas of Differential equation. In

particular to learn about

Difference methods for solving ordinary and partial differential equations.

Ordinary Differential Equations: Methods for Initial Value problems,

Stability and convergence analysis, Linear and nonlinear boundary value

problems.

Partial Differential Equations: Partial differential Equations of the first

order-linear equation of the first order

Linear partial equation with constand coefficient equation with variable

coefficients.

Elemetary solutions of laplaces equation-The wave equation.

Course outcome:After completion of course,the students will be able to

Solve Homogeneous equations, find solutions of exact equations, use the

method of reduction of order to find a second linearly independent solution

of a second order, linear homogeneous equation when one solution is given.

Use the method of undetermined coefficients to solve second order, linear

homogeneous equations with constant coefficients, Use the method of

variation of parameters to find particular solutions of second order, linear

homogeneouse equations.

Solve first order linear differential equations, find the solution of

homogeneous linear systems of equations.

Form a partial differential equations by eliminating the arbitrary constants

and functions, find different types of solutions like complete integral,

singular integral and general integral. Solve Lagrange’s equation.

Solve partial differential equations using Charpit’s method, classify partial

differential equations to five special types.

Text book:

1. An introduction to Ordinary Differential Equation by Earl A.Coddington

Prentice Hall of india﴾1987).

2. Elements of Partial Differential Equations by Ian SneddenMcGRAW-Hill

book company

NAME OF THE SUBJECT: (7MMA2C4) MECHANIS COURSE DISCEIPTION:

Understand the basic concepts of Velocity, Acceleration and Motion of Particles in all planes. The

concept of forces and impact of the particles. The concept of forces and impact of the particles. Topics include:

The Mechanical System.

Hamilton Principle.

Differential forms and Generation Functions.

COURSE OBJECTIVE:

Acquired Knowledege on Virtual Work, Energy and Momentum.

Distingaish Hamilton’s Principle and Hamilton’s Equations.

To provide students with basic skills and knowledge of Mechanics.

COURSE OUTCOMES:

On complete of this course,

The learner will be able to Understand the Generalized coordinates, constraints.

Derivative of Langrange’s Equation and slove the examples.

Understand the Special Transformation, Lagrange and Poisson Brackets.

TEXT BOOK:

D.Greenwood, Classical Dynamics. Prentice Hall of India.New Delhi, 1985.

REFERENCE BOOK:

1. H.Goldstein, Classical Mechanics. 2th edition , Narosa Publishing House, New

Delhi.

2. N.C.Rane and P.S.C.Joag,Classical Mechanics, Tata McGraw Hill, New

Delhi,1991.

3. J.L.Synge and B.A.Griffth, Principles of Mechanics, McGraw Hill Book Co.,

New York, 1970.

SEMSTER:-II Name of the subject (4MMA2C1) Algebra II

Course Description:

The course will be covers the topics:

Vectorspace , elementary basic concepts.

Dual spaces, inner product spaces.

Fields,roots of polynomials.

The elements of Galois theory.

Linear transformations, hermitian, unitary,normal transformations.

Course objectives:

The objective of this subject is to expose student to understand the importance of algebra to improve ability to think

logically, analytically and abstractily.

Vector space: elementary basic concepts, linear independence and basis, dual spaces.

Learn about and work with subspaces, linear transformation, inner products and their uses.

Learn to solve systems of linear equations and application problems requiring them.

Work with matrices and determine if a given squre matrix is invertible.

Course outcome:

Upon completion of the course the student will be able to:

Use the definition of vector space to determine of a given set of vectors is a vector space.

Use linear transformations to prove that vector spaces are isomorphic.

Solve the eigenvalues and corresponding eigenvectors for a linear transformation.

Understand the concepts on skew symmetric, orthogonal matrix, eigenvalues, eigenvectors and workout

problems related to it.

Solve problems on root of polynomials equations transformation of equations.

Text Book:


NAME OF THE SUBJECT: (4MMA2C2) ANALYSIS-II

COURSE DISCEIPTION:

This course covers Riemann-Stieltjes Integral, Sequences and Series of Functions, Some Special

Functions, Lebesgue Measure, Lebesgue Integral.

COURSE OBJECTIVE:

Acquired Knowledge on Riemann-Stieltjes Integration and

Differentiation.

To apply Integration of Vector Valued Functions, Rectifiable Curves.

Discussion of main problem Sequences and Series of Function.

Uniform Convergence, Continuity Integration and Differentiation.

Power Series, Logarithmic and Trigonometric Functions.

Prove Convergence theorems on Measurable Functions.



To learn the different types of Sequences and Series of Functions,

Equicontinuous Families of Functions.


Prove a theorem the Stone Weierstrass theorem, Convergence theorem

on Measurable Functions.

Define Algebra of Sets, Measurable Space, Lebesgue Measure.

Distinguish Lebesgue Measurable Sets and non Measurable Sets.

TEXT BOOK:

1. Principles of Mathematical Analysis by Walter Rudin (3nd Edition) McGRAW Hill

1976 (For Analysis part chapters VI,VII and VIII)

2. Real Analysis by Royden (For Measure Theory chapter III & I

Name of the subject ﴾4MMA2C3): Probability and statistics. Course discription:


Probability and distribution-Chebyshev’sInequadity.

Multivariate distributions-distribution of two random variables.

Some special distribution-the gamma and chi square distributions.

Distribution of functions of Random variables-expectations of

functions of random variables.

Limited distribution-some theorems on limiting distribution.

Course objective:

The objective of this subject is to expose student to understand the

importance of probability and statistical methods.

Axiomatic definition, properties conditional probability, Independent

events, Baye’s theorem, Density function, Distribution function,

Expectation, Moments, Moment generating function, Characteristic

function, Chebyshev’s inequality, Law of large numbers.

Special distributions: Binomial, Negative Binomial, Geometric,

Poisson, Uniform, Normal, Gamma,Exponential, Joint distributions,

Marginal and conditional density functions.

Sampling theory for small and large samples, Sampling distributions,

Estimation theory and interval estimation for population parameters

using normal, t, F and Chi square distributions.

Testing of hypothesis and test of significance.

Course outcome:

Understand the concepts of Mechanics and its mathematical

application by Newton’s laws of motion, resultant of two forces on a

particle, Equilibrium of a particle- Limiting Equilibrium of a particle

on an inclined plane.

Understand the different types of forces, parallel forces and

equivalent systems and rigid body.

Learn the Resultant of several coplanar forces- equation of the line of

action of the resultant, Equilibrium of a rigid body under three

coplanar forces – Reduction of coplanar forces into a force

Understand the basic concepts of centre of gravity and its

applications.

Learn the operations of Virtual work, hanging strings, equilibrium of

a uniform homogeneous string and suspension bridge.

Text book:

1. Introduction to Mathematical Statistics,by Robert V.Hogg and

AllenT.Craig Pears

Name of the subject (4MMA2E1) Applied Algebra Course Description:

The course provides and introduction to topics involving. Finite state machine , equivalent states and minimization

procedure.

Programming language, identifies, assignment statements, compiling arithmetic expressions.

Boolean algebra, direct products, morphism.

Optimization and computer design, NAND gates and NOR gates.

Binary group codes, hamming codes.

Course objectives:

To provide the standard methods for solving applied algebra as well as method based on the use of NAND gates and

NOR gates.

Finite state machines, binary devices and states, Turing machines.

Programminglanguages , FOR statements, The ALGOL grammer.

Boolean algebras, disjunctive normal form.

Optimization and computer design, logic design, sequential mechine design.

Binary group codes, encoding and decoding, group codes.

Course outcome:


Knowledge of various techniques of applied algebra and ability of solving certain type of practical

problems.

Use the applied algebra techniques that are best suited to solve certain problems.

Acquiring knowledge of the various techniques of applied algebra and training in solving practical

problems using this techniques.

Text Books:

Modern applied algebra by Garret Birkhoff and Thomas C. Bartee, McGRAW Hill International Student Edition.

Name of the subject (4MMA2E2) GRAPH THEORY

Course Description: The course covers basic theory and applications of combinatory and graph theory. Combinatorics is a study

of different enumeration techniques of finite but large sets. Topics that will be studied include principle of inclusion

and exclusion, generating functions and methods to solve difference equations. Graph theory is a study of graphs,

trees and networks. Topics that will be discussed include Euler formula, Hamilton paths, planar graphs and coloring

problem; the use of trees in sorting and prefix codes; useful algorithms on networks such as shortest path algorithm,

minimal spanning tree algorithm and min-flow max-cut algorithm. Course Objectives: The successful student will

know the definitions of relevant vocabulary from graph theory and combinatory, and know the statements and

proofs of many of the important theorems in the subject, and be able to perform related calculations.

Course Objectives: The successful student will know the definitions of relevant vocabulary from graph theory and

combinatorics, and know the statements and proofs of many of the important theorems in the subject, and be able to

perform related calculations.

Goals of the Course: • gain an understanding of the fundamental concepts of graph theory,

• gain an understanding of when a graph is a useful mathematical tool to solve problems in mathematics, the

sciences and the environment,

• develop the ability to write a logical and coherent proof, including proof by contradiction and induction,

• introduce topics suitable for a senior thesis (and see ones that have been),

• develop a desire for further study in related areas, including combinatorics and computer science.

Text Books:

Graph Theory with application by A.Bondy and U.S.R Murty,Macmillan Press Ltd.

Name of the subject ﴾4MMA2E3﴿Latex:

Course Description:

Student will be express to:

TEX and its offspring what’s different in LATEX 2c, Basic of a

LATEX file.

Commends and environments commend names and

arguments,environment,declarationlengths,specialcharacter’s,fragile

commends, exercises.

Documend Layout and organization-document class,pagestyle,part of

the document,

Displayed Changing font,centering and indenting,Theorem-like

declaretions,tabulatorstops,Boxes.

Tables,Printing literal text,footnotes,drawing pictures with LATEX.

Mathematical formulas-mathemeticalenvironments,Fine-tuning

mathematics.

Course objective:

This course intended to expose you to the basic ideas of Modern Analysis. In


Understand the purpose and nature of LATEX user interface of

LATEX.

Understand how LATEX difference from a word processor.

Format text in various way and learn how to use LATEX to format

mathematical equations.

Course outcome:

After completion of coures, the students will be able to

Explain and use TEX and LATEX.

Describs the deelopement process of TEX and LATEX.

Explaine the difference between TEX and LATEX.

Tells the advantages of LATEX over other more traditional

softwares.

Wribe documents containing mathematical formulas.

Type equations and formulas.

Type mathematical symbols in prographs.

Labels and refer the equations.

Text book:

A Guide to LATEX by H.Kopka and P.W.Daly,Third edition,addison-

wesley,London,1995 Name of the subject (4MMA2E4) Discrete Mathematics

Course Description:

Discrete mathematics has as its goal the algebraic systems, binary operations, semi groups and monoids.

Logic : TF- statements , tautology, replacement process, normal forms.

Theory of inference, open statements.

Lattices: modular and distributive lattices.

Boolean algebra: Boolean polynomials.

Course objectives:

The objective of this subject is to expose student to understand the important discrete mathematical structure.

Algebraic systems homomorphism and isomorphism of semi - groups and monoids, properties of

homomorphisms.

Logic, connectives, well -formed formulae, truth table of a formulae, replacement process, theory of

inference, open statements, statements involving more than one quantifier.

Lattices, new lattices, modular and distributive lattices.

Boolean algebra, uniqueness of finite Boolean algebra, Boolean expression, Boolean polynomials,

karnaugh map, switching circuits.

Course outcome:

Upon completing requirements for this course the student will be able to:

Solve the Boolean algebra for the two element Boolean algebra disjunctive normal form, conjunctive

normal form of a Boolean expression.

Use logical gates, construct and recognize truth tables for these gates and simple combinations of them with

up to four inputs.

Understanding the concept of TF statements, logic.

Understand the definition of homomorphism, semi-groups, monoid, lattices, Boolean algebra.

Text Books:

Discrete Mathematics by Dr. M. K. Venkataraman, Dr. N. Sridharan and Dr.N.Chandrasekaran, The National

Publishing Company.

Name of the subject (4MMA2E5) Programming in Java- Theory and Practical

Course Description:

This course introduces computer programming using the java programming language.

Java and internet, simple programs.

Object orientation in java, exception handling.

Threads, synchronization.

Input/Output, Applets

Java networking, concepts of advanced java programming.

Course objectives:

Implementing programs for user interface and application development using care java programming principles.

Focus on object oriented concepts and jav programs structure and its installation, fundamentals user defined

exceptions.

Comprehension of java programming constructs control structure of java.

Thread model, creating a thread, multiple threads, suspending, resuming, and stopping threads.

Implementing object oriented constructs such as various class string handling, AWT classes, Event

handling.

Java networking, basics, TCP/IP sever sockets, URL, JAVA SCRIPTS, JDBC, EJB, JSP.

Course outcome:


Create software application uing the java programming language.

Dubug a software application written in the java programming.

Test a software application written in the java programming language.

Understanding of thread concepts an Input / Output in java.

Understanding of various components of java AWT and swing and writtingcode snippets using them.

Text Book:

Partrickaughton, Herbert Schildt, “JAVA 2- The complete reference” Tata McGraw Hill Fifth Edition, New Delhi

2002.

Reference Books:

Deitel H M and Deiltel P J “JAVA – How to program” Pearson Education, New Delhi 2003.

Chitra A “Internet and Java Programming” ISTE 2002.

Name of the subject ﴾4MMA2E6)FLUID DYNAMICS

Course Description: The course will be covered the topic contained in the five

units of syllabus several additional topic will also be covered and resentation will

not necessarily follow the text attendance is required and the exams will be over

the lectures and home work. The course topic include


Kinetic of fluids in motion-real fluids and ideal fluids, velocity of a

fluid at a point.

Equation of the motion of a fluid-conservation body forces, some

flows involving axial symmetry.

Some three dimensional flows-complex velocity potentials for

standard two.

Viscous fluid-the co-efficient of a viscous fluid.

Some two dimention flows-Thomson circle theorem.

Course objective:The objective of this subject is to expose student to understand

the importance of fluid dynamics in Science & Engineering.

Kinematicsof fluids in motion: Real fluids and ideal fluids, Velocity

of a fluid at a point , Stream linesand path lines, Steady and unsteady

flows, The velocity potential, The velocity vector, Local andparticle

rates of change, Equation of continuity, Acceleration of fluid ,

Conditions at a rigid boundary.

Equations of motion of fluid: Euler’s equations of motion,

Bernoulli’s equation,Some flowsinvolving axial symmetry, Some

special two-dimensional flows.

Some three dimensional flows: Introduction, Sources, sinks and

doublets, Axisymmetric flows,Stokes’ stream function. The Milne-

Thomson circle theorem, Theorem of Blasius.

Viscous flows: Stress analysis in fluid motion, Relations between

stress and rate of strain, Coefficientof viscocity and laminar flow,

Navier-Stokes’ equations of motion of viscous fluid, Steady

motionbetween parallel planes and through tube of uniform cross

section.Flow between concentric rotatingcylinders.

Steady viscous flow in tube having uniform elliptic cross section,

Tube having equilateral triangularcross section, Steady flow past a

fixed sphere.

Course outcome:After completion of course the students will be able to

Understand the basic concepts of velocity, Acceleration and motion

of particlesin all planes.

Acquired adequate knowledge on Work, Energy and Simple

Harmonic Mean.They can able to solve problems on that.

Understood the concept of forces and impact of the particles.

Acquired knowledge on conical pendulum, central orbits and general

orbits.

Acquired knowledge on Moment of Inertia of various shapes.

Text book:

Fluid dynamics,FrankChorlton,CBS publishers &Distributors,2009

Semester –III

NAME OF THE SUBJECT: (4MMA3C1) COMPLEX ANALYSIS COURSE DISCEIPTION:

This course covers the Fundamental Principles of Analytic Functions. Topics include

Conformability, Linear Transformations.

Cauchy Integral Formula, Residues.

Local Properties of Analytic Functions.

Canonical Products, Jensen’s Formula.

COURSE OBJECTIVE:

To introduce Elementary Theory of Power Series.

To equip with necessary knowledge and skills to enable them handle

Mathematical Operations, Analysis and problems involving Complex

Number.

Know and Understand Analytic Function and Local properties of Analytic

Function.

COURSE OUTCOMES:


The learner will be able to comprehend the local and global properties of

Analytic Functions.

Improve and outline the complex integration.

Prove a Cauchy Integral Formula.

Understand local properties of Analytic Functions.

The learner will be able to solve the problems, Power series expansions,

Jensen’s Methods.

TEXT BOOK:

Complex Analysis by Lars V.Ahlfors, McGRAW Hill Book co.

Name of the subject (4MMA3C2) TOPOLOGY - I

Course Description:




Topological spaces.

Continuous functions.

Connected spaces and real line.

The countability axioms and the separation axioms.

Course Objectives:

The objectives of this subject is to expose student to understand several important concepts in

Topology -I.

To study the student will also being topological spaces and the order topology.

Closed sets and the product topology of components.

Explain compact spaces and the compact sets in the real line.

Components and path components and linit point compactness.

The contability axioms and the separation axioms urtsohn’s lemma.

Course outcomes:

Students will be able to:

To understand Topological spaces and basic of a topology and the product topology.

Identify the concept of connected sets in the real line components and path components.

Understand the countability axioms and the separation axioms.

Understand Urysohn’s lemma and Urysohn’s metrization theorm.

Text Book:

Topology a first course by James R. Munkres

Name of the subject (4MMA3C3)Operation Research

Course Description:

This course will covers the topics include:

Network models, scope and definition of network models.

Route problem, examples of the shortest route application.

CPM/PERT, network representation linear programming formulation of CPM/PERT calculations.

Deterministic inventory models, static economic, classic EOQ models.

Queuing system, Elements of a queuing model, pure birth, death model.

Generalized poissonqueuing model specialized poisson queues, steady state measures of performance,

(M/M/R):(GD/K/K), R>K- (M/G/1):(GD/∞/∞).

Course objectives:

The objectives of this subject is to expose student to understand the importance optimization techniques and the

theory behind it.

To introduce students to the techniques of operation research in mining operations.

To provide students with basic skills and knowledge of Operation Researchand its application in mineral

industry.

Network models, enumeration of cuts, maximal flow algorithm, linear programming formulation of

maximal flow mode, CPM computations.

Deterministic inventory models, EOQ models, no step model, setup model.


Generalized poisson queuing model specialized poisson queues, single server model, multi - server model,

mechine servicing model, pollaczek, khintchine(P-K) formula, other queuing models.

Course outcome:

The students will able to :

Explain the meaning of operation research know the various techniques of operations research.

Use operations research to solve transportation problems during the allocation of trucks to the formulate

operation research models to solve real life problem.

Solve linear programming problem by using algebraic graphical method.

Understand the concept of minimal spanning tree algorithm and maximal flow algorithm.

Understand Queuing theory basic concepts and solve queuing theory problems.

Compute CPM and PERT through project network diagram.

Understand test the optimality for degeneracy by using machine servicing model (M/M/R): (GD/K/K).

Text Books:

Operation Research VIII Edition by HamdyA.Taha ,Prentice –Hall of India Pvt.Ltd.

NAME OF THE SUBJECT: (4MMA3C4) NUMBER THEORY

COURSE DISCEIPTION:

Number Theory was studied for it’s long and rich history, its wealth of easily accessible and fascinating

questions, and its intellectual appeal. But , in recent years Number Theory has been studied for both for the

traditional reasons and for the compelling reason that Number Theory has become essential for cryptology. Topics

include:

The Fundamental Theorem of Arithmetic.

Arithmetical Functions and Dirichlet Multiplication.

Averages of Arithmetical Functions.

Quadratic Residuces and the Quadratic Reciprocity Law.

COURSE OBJECTIVE:

To expose students to this beautiful theory.

To understand what inspired this quote from Gauss.

To allow students to experience mathematics as a creative, empirical science.

Acquired Knowledage on Divisibility, Greatest Common Divisor, Prime

Numbers.

Distinguish Dirichlet Inverse and the Mobiles Inverse.

Derive the Mangoldt Inversion.

Derivatives of Arithmetical Functions the Selberg Identity.

Euler’s Summation Formula Some Elementary Asymptotic Formulas.

Definition and Basic properties of Congruences Residue Classes and complete

Residue Systems Linear Congruences.

COURSE OUTCOMES:


The Series of reciprocals of the primes the Euclidean Algorithm – The

Greatest Common Divisors of more than two numbers.

Definition of Mobius Function 𝜇(n),Euler Function (n),Mangoldt Function

(n),Liouville’s Function λ(n) and Division Function 𝜎𝛼(n).

Prove a theorem Mobiles Inverse.

Distinguish The Average Order of d(n) - The Average Order of the Division

Function 𝜎𝛼(n).

Applications to 𝜇(n) and ˄ (n).

Definition and Basic properties of congruences Residue classes and complete

residue systems linear congruences – reduced residue systems.

Prove a theorem the Eular – Fermat Theorem, Polynomial congruences

modulo Lagrange’s Theorem.

Evaluation of (−1

𝑝 ) and (

2

𝑝 ).

Prove a Theorem Gauss’s Lemma and The Quadratic Reciprocity law.

TEXT BOOK:

Introduction to Analytic Number theory by Tom M.Apostal, Springer Verlag.

Name of the subject (4MMA3E1) combinatorial Mathematics

Course Description:

The course will cover the topics contained in the five syllabus. The course topics include:

Generating function

Recurrence relation

The principle of inclusion and exclusion.

Polya theory of counting.

Block designs.

Course objectives:

The students should gain an understanding of fundamental concepts of combinatorial.

Combinatorial deals with the existence of certain configurations in in structure and when it exists it counts

the number of such configurations.

In this course we deal with the generating function,The principle of inclusion and exclusion.

Polya theory of counting and Block designs.

Course outcome:

After completing this course, the student will be able to:

Understand the rules of sum and product of permutations and combinations.

Identify solutions by the technique of generating functions and recurrence relations with two indices.

Understand the concepts of permutations with restrictions on relative positions and rook polynomials.

Explain the principle of inclusion and exclusion.

Enumerate configuration using poly’s theory.

Understand the concept of Block designs.

Text Books:

Introduction to combinatorial Mathematics by CL. Liu, Mc GRAW Hill.

Name of the subject ﴾4MMA3E2): Stochastic Processes. Course Discription: The course will be covered the topic contained in the five





Stochastic processes-Basic concepts,markovchains.

Define –Transition Matrix,order of markov chain.

Stability of a markov chain,limiting behaviourmarkov process.

Generalization of poisson processes-birth death process.

Renewal processes –Renewal equation,renewal theorems.

Course objective:

This course is intended to expose you to the basic ideas of

stochastic process.In particular to learn about.

Stochastic processes and their applications.

Course outcome:

The course learning outcomes, specific knowledge, skills and

competences of an appropriate level, which the students will acquire

with the successful completion of the course are described. Consult

Appendix A.

Description of the level of learning outcomes for each qualifications

cycle, according to the Qualifications Framework of the European

Higher Education Area.

Descriptors for Levels 6, 7 & 8 of the European Qualifications

Framework for Lifelong Learning and Appendix B.

Guidelines for writing Learning Outcomes.

Text book:

1. Stochastic Processes by J.Medhi,Wileyeastern,June 1987.

Name of the subject ﴾4MMA3E3): Fuzzy mathematics

Course description:

Studentwill be exposed to mathematics mapping on fuzzy sets crisp sets

and fuzzy sets. Operation on fuzzy sets, fuzzy relations, fuzzy measures,

decision making in fuzzy environments. Course objective:

The objectivesof this course are to: Provide an understanding of the basic mathematical

elements of the theory of fuzzy sets.

Provide an emphasis on the differences and similarities

between fuzzy sets and classical sets theories.

Cover fuzzy logic inference with emphasis on their use in

the design of intelligent or humanistic systems.

Provide a brief introduction to fuzzy arithmetic concepts.

Provide an insight into fuzzy inference applications in the

area of control and robotics.

Course outcome:

Be able to distinguish between the crisp set and fuzzy

set concepts through the learned differences between

the crisp set characteristic function and the fuzzy set

membership function.

Be able to draw a parallelism between crisp set

operations and fuzzy set operations through the use of

characteristic and membership functions respectively.

Be able to define fuzzy sets using linguistic words and

represent these sets by membership functions.

Know how to perform mapping of fuzzy sets by a

function and also use the α-level sets in such instances.

Know fuzzy-set-related notions; such as α-level sets,

convexity, normality, support, etc.

Know the concept of a fuzzy number and how it is

defined.

Become familiar with the extension principle, its

compatibility with the α-level sets and the usefulness of

the principle in performing fuzzy number arithmetic

operations (Additions, multiplications, etc.).

Become familiar with fuzzy relations and the properties

of these relations.

Become capable of drawing a distinction between

binary logic and fuzzy logic at the conceptual level.

Become capable of representing a simple classical

propositionusing crisp set characteristic function and

likewise representing a fuzzy proposition using fuzzy

set membership function.

Become knowledgeable of conditional fuzzy

propositions and fuzzy inference systems.

Become aware of the use of fuzzy inference systems in


Become aware of the application of fuzzy inference in

the area of control.

Have acquired the ability of thinking differently and

have become capable, when necessary, to apply a new

thinking methodology to real life problems including

enginering ones.

Text book: 1. Fuzzy sets-un certainity and information by George J.Klir and

tina A.Folger Prentice-Hall of India Pvt﴾2006).

2. Fuzzy set theory and its applications-H.J.Zimmermann-

Springer-Fourth Edition ﴾2001).

SEMESTER:-IV

NAME OF THE SUBJECT:(4MMA4C1)FUNCTIONAL ANALYSIS

COURSE DESCRIPTION:

Topics include:

Duals and Transposes.

Duals of LP([a,b,]) and c([a,b]).

Reisz Representation Theorems.

COURSE OBJECTIVES:

The objectives of the course are the study of the main properties of

bounded operators between Banach and Hilbert Spaces.

The basic result associated to different type of convergences in

normed spaces and the spectral theorem and some of its applications.

COURSE OUTCOMES:

On completion of this course the learner will

Be able to Normed Space and Continuity of Linear Maps.

Uniform Boundedness Principle.

Closed Graph and Open Mapping Theorems.


Orthonormal Sets, Projection and Reisz Representation Theorems.

TEXT BOOK:

Functional Analysis by B.V.Limaye, Second Edition, New Age International Pvt. Ltd., Publishers.

Name of the subject (4MMA4C2) Topology - II

Course Description:


the presentation will not necessarily follow the text. Attendance is required and the exams will be over the lectures .

The course topics include:

Connectedness and compactness: the Tychonoff Theorm .

Completely regular spaces, The Stone – cech compactification.

Metrization theorems.

Complete Metric spaces and function spaces

The compact open topology ascoli’s theorem .

Course Objectives:


Topology - II.

Connectedness and compactness: the Tychonoff Theorm.

Use results from completely regular spaces , definition of a local finiteness.

Explain from Nagata- smirnov metrization theorem.

The compact open topology Ascoli’s theorem .

Complete metric spaces and functions of a space filling curve and nowhere differential functions.

Course outcomes:


Use local compactness and the tychonoff theorem describe the completely regular spaces.

Explain the Nagata semirnov theorem for necessary and sufficient .

Give the definition of concepts related to complete metric space.

Understand the Metrization theorems and parcompactness.

Understand the open topology and Baire spaces.

To use A nowhere differential functions.

TEXT BOOk: Topology a first course by James R. Munkres.

NAME OF THE SUBJECT: (4MMA4C3) NUMERICAL ANALYSIS

COURSE DESCRIPTION:

Numerical Analysis was studied for its rich problems its wealth of easily accessible and fascinating questions and its

intellectual appeal.

This course covers optimization and steepest descent Newton’s Method.

Fixed Point Iteration.

Approximation, Differentiation and Integration.

The solutions of Differential Equation.

COURSE OBJECTIVES:

To introduce students to the technique of Numerical Analysis in

systems of equations and unconstrained optimization.

To provide students with basic skills and knowledge of Numerical

Analysis and its applications in mineral industry.

To introduce students topractical application of Numerical Analysis

in big mining projects.

To solving the problem Relaxation Method Numerical Integration,

Shooting Methods. COURSE OUTCOMES:

On complete of this course

The learner will be able to optimization and steepest

descentNewton’s Method.

Solved the problems of Numerical Integration, Gaussian Rules,

Numerical Integration by Taylor’s Series.

Understand the Multistep formula, Corrector Methods, Finite

Difference Methods, Shooting Methods.

TEXT BOOK

Elementary Numerical Analysis –An algorithimic Approach by S.C. Conte and Carl De

Boor, Mcgraw Hill (1981) Third Edition Name of the subject (4MMA4E1) Advanced Statistics

Course Description:

A course dealing with statistical concepts

Introduction statistical inference, point estimation.

Sufficient statistics, measures of quality of estimators.

More about estimation, Bayesian estimation.

Theory of statistical tests, certain best tests.

Inferences about normal model, non- central chi-squareand non - central F.

Course objectives:

The course is designed to introduce more advanced statistical methods that are used in data analysis. both statistical

theories and inference technique will be covered.

Introduction to statistical inference, test of statistical hypothesis.

Sufficient statistics, measures of quality of estimators, functions of a parameter.

More about estimation, fisher information and the rao.

Theory of statistical tests, likelihood ratio tests.

Inferences about normal models, a test of independence.

Course outcome:

On successful completion of this module, a student will be expected to be able to:

Be competent on a variety of well- known distributions and the calculation involved.

Understand the theories of statistical inferences and apply the approximate models in different setting to

solve real life problems.

Perform test of hypothesis as well as calculate confidence interval for single sample and two sample cases.

Understand the concepts non-central chi-square and non-central F.

Understand both the meaning and applicability of a dummy variable and the assumptions which underline a

regression model.

Understand the empty advanced statistical methods such as the analysis of variances , ratio test, the

sequential probability ratio test.

Text Books:

Introduction to Mathematical Statistics (Fifth Edition) by Robert V.Hogg and Allen T.Craig, Pearson Education

Asia.

Name of the subject ﴾4MMA4E3):Automata theory.

Course discription::The course will be covered the topic contained in the

five units of syllabus several additional topic will also be covered and resentation

will not necessarily follow the text attendance is required and the exams will be

over the lectures and home work. The course topic include


Definition of automata –The equivalence of DFA and NDFA.

Formal languages-chomsky classification of languages.

Recular expressions-finite atomata and regular expansions.

Recursive and recursively Enumerable sets-operation on languages.

Context-free languages

Course objective: The fundamental topics to be covered in this course include regular

expressions, finite automata, (non)regular languages, context-free

grammars, regular grammars, Chomsky normal forms, pushdown

automata, (non-)context-free languages, parsing and Turing machines.

These fundamentals are essential prerequisite for those who may

pursue more advanced topics and applications of Computer Science.

Since the ultimate goal of automata theory is the construction of

efficient program languages, no study of automata is complete without

some experience designing grammars.

For this purpose, a medium-scale program language design project will

be assigned as a class project.

The design project is an essential part of the successful course

completion. The grading will be based on the following criteria.

Course outcome:Aftercomplition of the course, the students will be able to

Understand the concept of formal languages through such mechanism

as regular expression, recursive definitions, finite automata, transition

graph, Mealy machine and Moore machine.

Apply Kleene’s theorem and pumping lemma for the design and

management of regular and non-regular languages.

Construct context free, regular, Chomsky normal form grammars to

design computer languages

Design and construct a pushdown automata and a Turing machine for

a computer language.

Text book:

1. Theory of computer science ﴾Auto mata,language and computation)III

Edition by KLP Mishra And N.Chadrasekaran,Prentice Hall of

India﴾2007).

ALAGAPPA UNIVERSITY, KARAIKUDI

NEW SYLLABUS UNDER CBCS PATTERN (w.e.f.2017-18)

M.Sc., MATHEMATICS – PROGRAMME STRUCTURE

Sem. Course Code Name of the Course Cr. Hrs./

Week

Max. Marks

Int. Ext. Total

I

7MMA1C1 Core–I– Algebra – I 5 6 25 75 100

7MMA1C2 Core–II– Analysis – I 5 6 25 75 100

7MMA1C3 Core – III–Differential Geometry 5 6 25 75 100

7MMA1C4 Core –IV–Ordinary Differential Equations 5 6 25 75 100

7MMA1E1

7MMA1E2

7MMA1E3

Elective–I: (Choose One out of Three)

A) Number Theory (or)

B) Calculus of Variations and

Special Functions (or)

C) Data Structures and Algorithms

Theory and Practical

4 6 25 75 100

Total 24 30 -- -- 500

II

7MMA2C1 Core –V–Algebra – II 5 6 25 75 100

7MMA2C2 Core –VI–Analysis – II 5 6 25 75 100

7MMA2C3 Core–VII–Partial Differential Equations 5 6 25 75 100

7MMA2C4 Core –VIII–Mechanics 5 6 25 75 100

7MMA2E1

7MMA2E2

7MMA2E3

Elective–II:(Choose One out of Three)

A) Graph Theory

B) Applied Algebra

C) Difference Equations

4 6 25 75 100

Total 24 30 -- -- 500

III

7MMA3C1 Core–IX–Complex Analysis 5 6 25 75 100

7MMA3C2 Core–X–Topology – I 5 6 25 75 100

7MMA3C3 Core–XI–Probability and Statistics 5 6 25 75 100

7MMA3E1

7MMA3E2

7MMA3E3

Elective–III:(Choose One out of Three)

A) Discrete Mathematics (or)

B) Fluid Dynamics (or)

C) Automata Theory

4 6 25 75 100

7MMA3E4

7MMA3E5

7MMA3E6

Elective–IV:(Choose One out of Three)

A) Fuzzy Mathematics (or)

B) Stochastic Processes (or)

C) Combinatorial Mathematics

4 6 25 75 100

Total 23 30 -- -- 500

IV

7MMA4C1 Core – XII–Functional Analysis 5 8 25 75 100

7MMA4C2 Core – XIII–Operations Research 5 8 25 75 100

7MMA4C3 Core – XIV–Topology II 5 7 25 75 100

7MMA4E1

7MMA4E2

7MMA4E3

Elective–V:(Choose One out of Three)

A) Advanced Statistics

B) Stochastic Differential Equations

C) Numerical Methods

4 7 25 75 100

Total 19 30 -- -- 400

Grand Total 90 120 -- -- 1900

Semester –I Name of the subject (7MMA1C1) Algebra I

Course Description:




Group theory, subgroups, homomorphism, permutation group.

Another counting principle.

Ring theory.

Ideals and Quotient rings.

Enclidean rings, Polynomials over the rational field.

Course Objectives:

The objectives of this subject is to expose student to understand several important concepts in algebra.

Including group, ring, field, homomorphism, automorphism, quotient structure and to apply some of these

concepts to real world problems.

Use results from binary operations, and its properties, definition of a group, subgroups, homomorphism,

cayley’s theorem, permutation groups, to solve contempotary problems.

Explain from elementary principle, sylow’s theorem why certain algebraic facts are true.

Ring theory ,definition, examples of rings.

Ideals and Quotient ring, more ideals and quotient rings, integral domain.

Enclidean rings, prime ideal, maximal ideal prime avoidance theorem, polynomial rings over commutative

rings.

Course outcomes:


Use sylow’s theorem to describe the structure of certain finite groups.

Explain the notation of extention field.

Give the definition of concepts related to group, homomorphism, ring, ideals and field.

Understand the structure of fields do computations in specific examples of finite fields.

Understand the congrurence modulo concept, fermat’s theorem.

Use the concept of decomposition of composite number as a product of primes uniquely and euler

functions.

Text Book:


NAME OF THE SUBJECT: (7MMA1C2) ANALYSIS-I

COURSE DISCEIPTION:

This course covers Basic Topology, Numerical Sequences and Series, Power

Series, Continuity, Differentiation.

COURSE OBJECTIVE:

To apply Basic Topology on Metric Space, Compact Sets, Perfect Sets,

connected Set.

Acquired Knowledge on Convergent Sequences, Sub Sequences, Cauchy

Sequences, Upper and Lower Limits, Special Sequences.

Distinguish of Continuity and Compactness.

Acquired Knowledge Continuity, Discontinuity, Monotonic Function.

COURSE OUTCOMES:



To learn the different types of Numerical Sequences and Series.


Understand and perform simple derivative of a Real Function.

Prove a theorem Mean Value Theorems, Taylor’s Theorems,

Differentiation of Vector-Valued Function.

TEXT BOOK:

Principle of Mathematical Analysis – Walter Rudin, III Edition (Relevant portions of

chapters II,III,IV & V), McGRAW – Hill Book Company. Name of the subject (7MMA1C3) Differential geometry

Course Description:




Arc length, serret - frenet formulae

Osculating circle.

Surfaces and class of surfaces.

Geodesics and geodesic curvature.

Course Objectives:

The objectives of this subject is to expose student to understand several important concepts in Differential

Geometry.

Arc length and tangent,normal Binormal to apply some of these concepts to real world problems.

Use results from geodesics curvature, and its properties, definition of a osculating sphere, intrinsic

equations.

Explain from serret_frenet formulae isometric correspondents.

Canonical geodesics equations ,definition, examples .

Direction coefficients and family of curves.

Fundamental form of principal curvature,line curvature and developables.

Course outcomes:


Use gauss bonnet theorem to describe the second fundamental form and developables.

Explain the geodesics parallels.

Give the definition of concepts related to normal propertyof geodesics.

Understand the structure of arc length binormal to the curvature.

Understand the developable associated with the space curves.

Gaussian curvature to use developable associated with the space curves.

TEXT BOOk:

Topics in Differential geometry by T.G. Willmore

Name of the subject ﴾7MMA1C4): Ordinary

Differential Equations. Course discription:

Student will be exposed linear equations with variable

coefficients. This is a one term introduction to ordinary

differential equation with applications.

Homogeneous equation – Homogeneous equation of order

n-initial value problems for 𝑛𝑡ℎ order equation.

Linear equations with reqular singular points-Euler

equation and Besel equation.

Existence and uniqueness of solution to first order

equations-Equation with variable seperation.

Nonlocal existence of solution-exstence and uniqueness of

solution to system.

Coures objective:

Objective of this subject is to expose student to understand the

importence of

The general purpose of this course is to introduce basis

concepts of theory of ordinary differential equations, to

give several methods including the series method for

solving linear and nonlinear differential equations, to

learn about existence and uniqueness of the solution to

thenonlinear initial value problems.

First order differential equations, Linear differential

equations of higher order, Linear dependence and

Wronskian, Basic theory for linear equations, Method of

variation of parameters, Linear equations with variable

coefficients.

Systems of differential equations, existence and

uniqueness theorems, Fundamental matrix, Non-

homogeneous linear systems, Linear systems with

constant coefficients and periodic coefficients, Existence

and uniqueness of solutions, Gronwall inequality,

Successive approximation, Picard’s theorem,

Nonuniqueness of solutions,Continuous.

Boundary Value Problems: Green's function, Sturm-

Liouville problem, eigenvalue problems.Stability of

linear and nonlinear systems: Lyapunov stability,

Sturm’s Comparison theorem.

Corse outcome:After completion of this course, students will be able

to:

The study of Differential focuses on the existence and

uniqueness of solutions and also emphansizes the

rigorous justification of methods for approximating

solutions in pure and applied mathematics.

It plays an important role in modelling virtually every

physically technical orbiological process from celestial

motion to bridge design to interactions between neurons.

Theory of differential equations is widely used in

formulating many fundamentallaws of physics and

chemistry.

Theory of differential equation is used in economics and

biology to model thebehaviour of complex systems.

Differential equations have a remarkable ability to

predicts the world around us.They can describe

exponential growth and decay population growth of

species orchange in investment return over time.

Text book:

EarlA.Coddington,an introduction to Ordinary Differential Equation-

Prentice Hall of india,1987.

Refference book:

1. D.Somasundaram,ordinery Differential equations,Narosa

Publishing house,chennai,2002.

2. M.D.Raisinghania,Advance Differential Equations.S.Chand

and company Ltd,New delhi.2001.

NAME OF THE SUBJECT: (7MMA1E1) NUMBER THEORY

COURSE DISCEIPTION:

Number Theory was studied for it’s long and rich history, its wealth of easily

accessible and fascinating questions, and its intellectual appeal. But , in recent years Number

Theory has been studied for both for the traditional reasons and for the compelling reason that

Number Theory has become essential for cryptology. Topics include:

The Fundamental Theorem of Arithmetic.

Arithmetical Functions and Dirichlet Multiplication.

Averages of Arithmetical Functions.

Quadratic Residuces and the Quadratic Reciprocity Law.

COURSE OBJECTIVE:

To expose students to this beautiful theory.

To understand what inspired this quote from Gauss.

To allow students to experience mathematics as a creative, empirical science.

Acquired Knowledage on Divisibility, Greatest Common Divisor, Prime

Numbers.

Distinguish Dirichlet Inverse and the Mobiles Inverse.

Derive the Mangoldt Inversion.

Derivatives of Arithmetical Functions the Selberg Identity.

Euler’s Summation Formula Some Elementary Asymptotic Formulas.

Definition and Basic properties of Congruences Residue Classes and complete

Residue Systems Linear Congruences.

COURSE OUTCOMES:


The Series of reciprocals of the primes the Euclidean Algorithm – The

Greatest Common Divisors of more than two numbers.

Definition of Mobius Function 𝜇(n),Euler Function (n),Mangoldt Function

(n),Liouville’s Function λ(n) and Division Function 𝜎𝛼(n).

Prove a theorem Mobiles Inverse.

Distinguish The Average Order of d(n) - The Average Order of the Division

Function 𝜎𝛼(n).

Applications to 𝜇(n) and ˄ (n).

Definition and Basic properties of congruences Residue classes and complete

residue systems linear congruences – reduced residue systems.

Prove a theorem the Eular – Fermat Theorem, Polynomial congruences

modulo Lagrange’s Theorem.

Evaluation of (−1

𝑝 ) and (

2

𝑝 ).

Prove a Theorem Gauss’s Lemma and The Quadratic Reciprocity law.

TEXT BOOK:

Introduction to Analytic Number theory by Tom M.Apostal, Springer Verlag.

Semester –II Name of the subject (7MMA2C1) Algebra II

Course Description:

The course will be covers the topics:

Vectorspace , elementary basic concepts.

Dual spaces, inner product spaces.

Fields,roots of polynomials.

The elements of Galois theory.

Linear transformations, hermitian, unitary,normal transformations.

Course objectives:

The objective of this subject is to expose student to understand the importance of algebra to improve ability to think

logically, analytically and abstractily.

Vector space: elementary basic concepts, linear independence and basis, dual spaces.

Learn about and work with subspaces, linear transformation, inner products and their uses.

Learn to solve systems of linear equations and application problems requiring them.

Work with matrices and determine if a given squre matrix is invertible.

Course outcome:


Use the definition of vector space to determine of a given set of vectors is a vector space.

Use linear transformations to prove that vector spaces are isomorphic.

Solve the eigenvalues and corresponding eigenvectors for a linear transformation.

Understand the concepts on skew symmetric, orthogonal matrix, eigenvalues, eigenvectors and workout

problems related to it.

Solve problems on root of polynomials equations transformation of equations.

Text Book:


NAME OF THE SUBJECT: (7MMA2C2) ANALYSIS-II

COURSE DISCEIPTION:

This course covers Riemann-Stieltjes Integral, Sequences and Series of Functions,

Some Special Functions, Lebesgue Measure, Lebesgue Integral.

COURSE OBJECTIVE:

Acquired Knowledge on Riemann-Stieltjes Integration and

Differentiation.

To apply Integration of Vector Valued Functions, Rectifiable Curves.

Discussion of main problem Sequences and Series of Function.

Uniform Convergence, Continuity Integration and Differentiation.

Power Series, Logarithmic and Trigonometric Functions.

Prove Convergence theorems on Measurable Functions.



To learn the different types of Sequences and Series of Functions,

Equicontinuous Families of Functions.


Prove a theorem the Stone Weierstrass theorem, Convergence theorem

on Measurable Functions.

Define Algebra of Sets, Measurable Space, Lebesgue Measure.

Distinguish Lebesgue Measurable Sets and non Measurable Sets.

TEXT BOOK:

3. Principles of Mathematical Analysis by Walter Rudin (3nd Edition) McGRAW Hill

1976 (For Analysis part chapters VI,VII and VIII)

4. Real Analysis by Royden (For Measure Theory chapter III & IV)

Name of the subject ﴾7MMA2C3) Partial differential

equation

Course discription:Student will be able to

Ordinary differential equations in more than two

variable:method of solution of 𝑑𝑥

𝑃=

𝑑𝑦

𝑄=

𝑑𝑧

𝑅 orthogonal

trajectories of a system of curves on a surface.

Partial differential equation of the first order: cauchy’s

problem for first order equations.

Compatiblesystemof first order equations-charpits method

Partial differential equation of the second order-method of integral

transforms.

Laplace’s equation –elemetary solution of Laplace’s equation.

Course objective:

This course intended to expose you to the basic ideas of Differential equation. In



importance of partialdifferential equations.

Formulation, Linear and quasi-linear first order partial differential

equations, Paffianequation,Condition for integrability, Lagrange’s

method for linear equations. First order non-linear

equations,method of Charpit - method of characteristics.

Equations of higher order: Method of solution for the case of

constant coefficients.

Equations of second order reduction to canonical forms,

Characteristic curves and the Cauchyproblem, Riemann’s method

for the solution of linear hyperbolic equations, Monge’s method for

thesolution of non-linear second order equations,Method of solution

by separation of variables.

Laplace’s equations: Elementary solutions, Boundary value

problems, Green’s functions for

Laplace’s equation, Solution using orthogonal functions.

Wave equations: One dimensional equation and its solution in

trigonometric series,

Riemann-Volterra solution, vibrating membrane.

Diffusion equations: Elementary solution, Solution in terms of

orthogonal functions.

Course outcome:Upon completion of this course, students should be able to:

Identify, analyze and subsequently solve physical situations whose

behaviour can be described byordinary differential equations

Understand the order, degree and various standard forms of

differential equations

Determine solutions to first order separable differential equations

Determine solutions to first order linear differential equations

Explain an integrating factor, which may reduce the given

differential equation into an exact one andeventually provide its

solutions Familiarize the orthogonal trajectory of the system of

curves on a given surface

Determine solutions to first order exact differential equations

Determine solutions to second order linear homogeneous

differential equations with constantcoefficients

Understand the basic knowledge of complimentary function and

particular integral Determine solutions to second order linear non-

homogeneous differential equations with constantcoefficients

Evaluate and apply linear differential equations of second order

(and higher)

Obtain power series solutions of differential equations

Develop the ability to apply differential equations to significant

applied and/or theoretical problems

Investigate the qualitative behavior of solutions of systems of

differential equations Identify and obtain the solution of Clairaut’s

equation

Be familiar with the modeling assumptions and derivations that

lead to PDEs Describe the origin of partial differential equation and

distinguish the integrals of first order linear

Partial differential equation into complete, general and singular

integrals

Familiarize with the various techniques of finding the solution of

the differential equation𝑑𝑥

𝑃=

𝑑𝑦

𝑅=

𝑑𝑧

𝑄. Acquire the idea of

Lagrange’s method for solving the first order linear partial

differential equations

Recognize the major classification of PDEs and the qualitative

differences between the classes ofequations

Be competent in solving linear PDEs using classical solution

methods.

Text book:

1. I.N.Sneddon Element of Partial differential equation,McGraw Hill

Book.

Reference book:

1. M.D.Raisinghania,Advance Differential Equation,S.Chand&

Company Ltd.,New Delhi,2001.

2. K.SankaraRao,Introduction to partial differential

equation,secondEdition,Prentice-Hall of India,new delhi,2006.

NAME OF THE SUBJECT: (7MMA2C4) MECHANIS

COURSE DISCEIPTION:

Mechanics was studied for the basic concepts of Velocity, Acceleration and Motion

of particles in all planes. Acquired adequate knowledge on work ,Energy and Simple Harmonic

mean. They can be able to solve problems on that. Topics include:

Application of Lagrange Formula.

Classification of Oribits.

COURSE OBJECTIVE:

To introduce D’ Alembert’s Principle and Lagranges Equations.

Understand the Kinematics of a Fluid through Equation of Motion of the Fluid.

To equip with necessary knowledge and skills to enable them handle potentials,

symmetry properties, classification of oribits.

To solving the Kepler Problem and its applications.

COURSE OUTCOMES:


The learner will be able to comprehend the local and global.

Application of Lagrange Formulation and properties of Symmetry.

Prove a Hamilton’s Principle and Conservation Theorems.

Prove a Virtual Theorem, Bertrand’s Theorem.

Sloving a problems Kepler problem, The Laplace – Runge - Lunz vector –

stress.

TEXT BOOK:

Classical Mechanics by H.Goldstein , Second Edition, Addison Wesley.

Name of the subject (7MMA2E1) GRAPH THEORY

Course Description: The course covers basic theory and applications of combinatory and graph theory.

Combinatorics is a study of different enumeration techniques of finite but large sets. Topics that

will be studied include principle of inclusion and exclusion, generating functions and methods to

solve difference equations. Graph theory is a study of graphs, trees and networks. Topics that

will be discussed include Euler formula, Hamilton paths, planar graphs and coloring problem;

the use of trees in sorting and prefix codes; useful algorithms on networks such as shortest path

algorithm, minimal spanning tree algorithm and min-flow max-cut algorithm. Course Objectives:

The successful student will know the definitions of relevant vocabulary from graph theory and

combinatory, and know the statements and proofs of many of the important theorems in the

subject, and be able to perform related calculations.

Course Objectives: The successful student will know the definitions of relevant vocabulary from graph

theory and combinatorics, and know the statements and proofs of many of the important

theorems in the subject, and be able to perform related calculations.

Goals of the Course: • gain an understanding of the fundamental concepts of graph theory,

• gain an understanding of when a graph is a useful mathematical tool to solve problems in

mathematics, the sciences and the environment,

• develop the ability to write a logical and coherent proof, including proof by contradiction and

induction,

• introduce topics suitable for a senior thesis (and see ones that have been),

• develop a desire for further study in related areas, including combinatorics and computer

science. Text Books: Graph Theory with application by A.Bondy and U.S.R Murty,Macmillan Press Ltd

Name of the subject (7MMA2E2) Applied Algebra

Course Description:

The course provides and introduction to topics involving. Finite state machine , equivalent states and minimization

procedure.

Programming language, identifies, assignment statements, compiling arithmetic expressions.

Boolean algebra, direct products, morphism.

Optimization and computer design, NAND gates and NOR gates.

Binary group codes, hamming codes.

Course objectives:

To provide the standard methods for solving applied algebra as well as method based on the use of NAND gates and

NOR gates.

Finite state machines, binary devices and states, Turing machines.

Programminglanguages , FOR statements, The ALGOL grammer.

Boolean algebras, disjunctive normal form.

Optimization and computer design, logic design, sequential mechine design.

Binary group codes, encoding and decoding, group codes.

Course outcome:


Knowledge of various techniques of applied algebra and ability of solving certain type of practical

problems.

Use the applied algebra techniques that are best suited to solve certain problems.

Acquiring knowledge of the various techniques of applied algebra and training in solving practical

problems using this techniques.

Text Books:

Modern applied algebra by Garret Birkhoff and Thomas C. Bartee, McGRAW Hill International Student Edition.

Name of the subject ﴾7MMA2E3)Differential Equation

Course Description:


Linear Equation with variable coefficients – the wronskian and linear

dependence reduction of the order of a homogeneouse Equation.

Linear Equation with regular singular points- the Euler Equation.

Partial Differentiual Equations of the first order –Linear Equation of the first

order.

Second order equation –linear partial equations with constant coefficiant

equations with variable co-efficients.

Laplace’s Equation-The wave equation.

Course objective: This course intended to expose you to the basic ideas of Differential equation. In


Difference methods for solving ordinary and partial differential equations.

Ordinary Differential Equations: Methods for Initial Value problems,

Stability and convergence analysis, Linear and nonlinear boundary value

problems.

Partial Differential Equations: Partial differential Equations of the first

order-linear equation of the first order

Linear partial equation with constand coefficient equation with variable

coefficients.

Elemetary solutions of laplaces equation-The wave equation.

Course outcome:After completion of course,the students will be able to

Solve Homogeneous equations, find solutions of exact equations, use the

method of reduction of order to find a second linearly independent solution

of a second order, linear homogeneous equation when one solution is given.

Use the method of undetermined coefficients to solve second order, linear

homogeneous equations with constant coefficients, Use the method of

variation of parameters to find particular solutions of second order, linear

homogeneouse equations.

Solve first order linear differential equations, find the solution of

homogeneous linear systems of equations.

Form a partial differential equations by eliminating the arbitrary constants

and functions, find different types of solutions like complete integral,

singular integral and general integral. Solve Lagrange’s equation.

Solve partial differential equations using Charpit’s method, classify partial

differential equations to five special types.

Text book:

3. An introduction to Ordinary Differential Equation by Earl A.Coddington

Prentice Hall of india﴾1987).

4. Elements of Partial Differential Equations by Ian SneddenMcGRAW-Hill

book company﴾1986).

SEMSTER:-III

NAME OF THE SUBJECT: (7MMA3C1) COMPLEX ANALYSIS

COURSE DISCEIPTION:

This course covers the Fundamental Principles of Analytic Functions. Topics include

Conformability, Linear Transformations.

Cauchy Integral Formula, Residues.

Local Properties of Analytic Functions.

Canonical Products, Jensen’s Formula.

COURSE OBJECTIVE:

To introduce Elementary Theory of Power Series.

To equip with necessary knowledge and skills to enable them handle

Mathematical Operations, Analysis and problems involving Complex

Number.

Know and Understand Analytic Function and Local properties of Analytic

Function.

COURSE OUTCOMES:


The learner will be able to comprehend the local and global properties of

Analytic Functions.

Improve and outline the complex integration.

Prove a Cauchy Integral Formula.

Understand local properties of Analytic Functions.

The learner will be able to solve the problems, Power series expansions,

Jensen’s Methods.

TEXT BOOK:

Complex Analysis by Lars V.Ahlfors, McGRAW Hill Book co.

Name of the subject (7MMA3C2) TOPOLOGY - I

Course Description:




Topological spaces.

Continuous functions.

Connected spaces and real line.

The countability axioms and the separation axioms.

Course Objectives:


Topology -I.

To study the student will also being topological spaces and the order topology.

Closed sets and the product topology of components.

Explain compact spaces and the compact sets in the real line.

Components and path components and linit point compactness.

The contability axioms and the separation axioms urtsohn’s lemma.

Course outcomes:


To understand Topological spaces and basic of a topology and the product topology.

Identify the concept of connected sets in the real line components and path components.

Understand the countability axioms and the separation axioms.

Understand Urysohn’s lemma and Urysohn’s metrization theorm.

Text Book:

Topology a first course by James R. Munkres

Name of the subject ﴾7MMA3C3): Probability and statistics. Course discription:


Probability and distribution-Chebyshev’sInequadity.

Multivariate distributions-distribution of two random variables.

Some special distribution-the gamma and chi square distributions.

Distribution of functions of Random variables-expectations of

functions of random variables.

Limited distribution-some theorems on limiting distribution.

Course objective:


importance of probability and statistical methods.

Axiomatic definition, properties conditional probability, Independent

events, Baye’s theorem, Density function, Distribution function,

Expectation, Moments, Moment generating function, Characteristic

function, Chebyshev’s inequality, Law of large numbers.

Special distributions: Binomial, Negative Binomial, Geometric,

Poisson, Uniform, Normal, Gamma,Exponential, Joint distributions,

Marginal and conditional density functions.

Sampling theory for small and large samples, Sampling distributions,

Estimation theory and interval estimation for population parameters

using normal, t, F and Chi square distributions.

Testing of hypothesis and test of significance.

Course outcome:

Understand the concepts of Mechanics and its mathematical

application by Newton’s laws of motion, resultant of two forces on a

particle, Equilibrium of a particle- Limiting Equilibrium of a particle

on an inclined plane.

Understand the different types of forces, parallel forces and

equivalent systems and rigid body.

Learn the Resultant of several coplanar forces- equation of the line of

action of the resultant, Equilibrium of a rigid body under three

coplanar forces – Reduction of coplanar forces into a force

Understand the basic concepts of centre of gravity and its

applications.

Learn the operations of Virtual work, hanging strings, equilibrium of

a uniform homogeneous string and suspension bridge.

Text book:

2. Introduction to Mathematical Statistics,by Robert V.Hogg and

AllenT.Craig Pearson Educations.

Name of the subject (7MMA3E1) Discrete Mathematics

Course Description:

Discrete mathematics has as its goal the algebraic systems, binary operations, semi groups and monoids.

Logic : TF- statements , tautology, replacement process, normal forms.

Theory of inference, open statements.

Lattices: modular and distributive lattices.

Boolean algebra: Boolean polynomials.

Course objectives:

The objective of this subject is to expose student to understand the important discrete mathematical structure.

Algebraic systems homomorphism and isomorphism of semi - groups and monoids, properties of

homomorphisms.

Logic, connectives, well -formed formulae, truth table of a formulae, replacement process, theory of

inference, open statements, statements involving more than one quantifier.

Lattices, new lattices, modular and distributive lattices.

Boolean algebra, uniqueness of finite Boolean algebra, Boolean expression, Boolean polynomials,

karnaugh map, switching circuits.

Course outcome:


Solve the Boolean algebra for the two element Boolean algebra disjunctive normal form, conjunctive

normal form of a Boolean expression.

Use logical gates, construct and recognize truth tables for these gates and simple combinations of them with

up to four inputs.

Understanding the concept of TF statements, logic.

Understand the definition of homomorphism, semi-groups, monoid, lattices, Boolean algebra.

Text Books:

Discrete Mathematics by Dr. M. K. Venkataraman, Dr. N. Sridharan and Dr.N.Chandrasekaran, The National

Publishing Company.

Name of the subject ﴾7MMA3E2)FLUID DYNAMICS

Course Description: The course will be covered the topic contained in the five





Kinetic of fluids in motion-real fluids and ideal fluids, velocity of a

fluid at a point.

Equation of the motion of a fluid-conservation body forces, some

flows involving axial symmetry.

Some three dimensional flows-complex velocity potentials for

standard two.

Viscous fluid-the co-efficient of a viscous fluid.

Some two dimention flows-Thomson circle theorem.

Course objective:The objective of this subject is to expose student to understand

the importance of fluid dynamics in Science & Engineering.

Kinematicsof fluids in motion: Real fluids and ideal fluids, Velocity

of a fluid at a point , Stream linesand path lines, Steady and unsteady

flows, The velocity potential, The velocity vector, Local andparticle

rates of change, Equation of continuity, Acceleration of fluid ,

Conditions at a rigid boundary.

Equations of motion of fluid: Euler’s equations of motion,

Bernoulli’s equation,Some flowsinvolving axial symmetry, Some

special two-dimensional flows.

Some three dimensional flows: Introduction, Sources, sinks and

doublets, Axisymmetric flows,Stokes’ stream function. The Milne-

Thomson circle theorem, Theorem of Blasius.

Viscous flows: Stress analysis in fluid motion, Relations between

stress and rate of strain, Coefficientof viscocity and laminar flow,

Navier-Stokes’ equations of motion of viscous fluid, Steady

motionbetween parallel planes and through tube of uniform cross

section.Flow between concentric rotatingcylinders.

Steady viscous flow in tube having uniform elliptic cross section,

Tube having equilateral triangularcross section, Steady flow past a

fixed sphere.

Course outcome:After completion of course the students will be able to

Understand the basic concepts of velocity, Acceleration and motion

of particlesin all planes.

Acquired adequate knowledge on Work, Energy and Simple

Harmonic Mean.They can able to solve problems on that.

Understood the concept of forces and impact of the particles.

Acquired knowledge on conical pendulum, central orbits and general

orbits.

Acquired knowledge on Moment of Inertia of various shapes.

Text book:

Fluid dynamics,FrankChorlton,CBS publishers &Distributors,2004

Name of the subject ﴾7MMA3E3) :Automata theory.

Course discription::The course will be covered the topic contained in the

five units of syllabus several additional topic will also be covered and resentation

will not necessarily follow the text attendance is required and the exams will be

over the lectures and home work. The course topic include


Definition of automata –The equivalence of DFA and NDFA.

Formal languages-chomsky classification of languages.

Recular expressions-finite atomata and regular expansions.

Recursive and recursively Enumerable sets-operation on languages.

Context-free languages

Course objective: The fundamental topics to be covered in this course include regular

expressions, finite automata, (non)regular languages, context-free

grammars, regular grammars, Chomsky normal forms, pushdown

automata, (non-)context-free languages, parsing and Turing machines.

These fundamentals are essential prerequisite for those who may

pursue more advanced topics and applications of Computer Science.

Since the ultimate goal of automata theory is the construction of

efficient program languages, no study of automata is complete without

some experience designing grammars.

For this purpose, a medium-scale program language design project will

be assigned as a class project.

The design project is an essential part of the successful course

completion. The grading will be based on the following criteria.

Course outcome:Aftercomplition of the course, the students will be able to

Understand the concept of formal languages through such mechanism

as regular expression, recursive definitions, finite automata, transition

graph, Mealy machine and Moore machine.

Apply Kleene’s theorem and pumping lemma for the design and

management of regular and non-regular languages.

Construct context free, regular, Chomsky normal form grammars to

design computer languages

Design and construct a pushdown automata and a Turing machine for

a computer language.

Text book:

2. Theory of computer science ﴾Auto mata,language and computation)III

Edition by KLP Mishra And N.Chadrasekaran,Prentice Hall of

India﴾2007).

Name of the subject ﴾4MMA3E3): Fuzzy mathematics Course description:

Studentwill be exposed to mathematics mapping on fuzzy sets crisp sets

and fuzzy sets. Operation on fuzzy sets, fuzzy relations, fuzzy measures,

decision making in fuzzy environments. Course objective:

The objectivesof this course are to: Provide an understanding of the basic mathematical

elements of the theory of fuzzy sets.

Provide an emphasis on the differences and similarities

between fuzzy sets and classical sets theories.

Cover fuzzy logic inference with emphasis on their use in


Provide a brief introduction to fuzzy arithmetic concepts.

Provide an insight into fuzzy inference applications in the

area of control and robotics.

Course outcome:

Be able to distinguish between the crisp set and fuzzy

set concepts through the learned differences between

the crisp set characteristic function and the fuzzy set

membership function.

Be able to draw a parallelism between crisp set

operations and fuzzy set operations through the use of

characteristic and membership functions respectively.

Be able to define fuzzy sets using linguistic words and

represent these sets by membership functions.

Know how to perform mapping of fuzzy sets by a

function and also use the α-level sets in such instances.

Know fuzzy-set-related notions; such as α-level sets,

convexity, normality, support, etc.

Know the concept of a fuzzy number and how it is

defined.

Become familiar with the extension principle, its

compatibility with the α-level sets and the usefulness of

the principle in performing fuzzy number arithmetic

operations (Additions, multiplications, etc.).

Become familiar with fuzzy relations and the properties

of these relations.

Become capable of drawing a distinction between

binary logic and fuzzy logic at the conceptual level.

Become capable of representing a simple classical

propositionusing crisp set characteristic function and

likewise representing a fuzzy proposition using fuzzy

set membership function.

Become knowledgeable of conditional fuzzy

propositions and fuzzy inference systems.

Become aware of the use of fuzzy inference systems in


Become aware of the application of fuzzy inference in

the area of control.

Have acquired the ability of thinking differently and

have become capable, when necessary, to apply a new

thinking methodology to real life problems including

enginering ones.

Text book: 3. Fuzzy sets-un certainity and information by George J.Klir and

tina A.Folger Prentice-Hall of India Pvt﴾2006).

4. Fuzzy set theory and its applications-H.J.Zimmermann-

Springer-Fourth Edition ﴾2001).

Name of the subject ﴾7MMA3E5):Stochastic Processes. Course discription: The course will be covered the topic contained in the five





Stochastic processes-Basic concepts, markovchains.

Define –Transition Matrix, order of markov chain.

Stability of a markov chain, limiting behaviourmarkov process.

Generalization of poisson processes-birth death process.

Renewal processes –Renewal equation,renewal theorems.

Course objective:

This course is intended to expose you to the basic ideas of

stochastic process.In particular to learn about.

Stochastic processes and their applications.

Course outcome:

The course learning outcomes, specific knowledge, skills and

competences of an appropriate level, which the students will acquire

with the successful completion of the course are described. Consult

Appendix A.

Description of the level of learning outcomes for each qualifications

cycle, according to the Qualifications Framework of the European

Higher Education Area.

Descriptors for Levels 6, 7 & 8 of the European Qualifications

Framework for Lifelong Learning and Appendix B.

Guidelines for writing Learning Outcomes.

Text book:

2. Stochastic Processes by J.Medhi,Wileyeastern,June 1987.

Name of the subject (7MMA3E6) combinatorial Mathematics

Course Description:

The course will cover the topics contained in the five syllabus. The course topics include:

Generating function

Recurrence relation

The principle of inclusion and exclusion.

Polya theory of counting.

Block designs.

Course objectives:

The students should gain an understanding of fundamental concepts of combinatorial.

Combinatorial deals with the existence of certain configurations in in structure and when it exists it counts

the number of such configurations.

In this course we deal with the generating function,The principle of inclusion and exclusion.

Polya theory of counting and Block designs.

Course outcome:

After completing this course, the student will be able to:

Understand the rules of sum and product of permutations and combinations.

Identify solutions by the technique of generating functions and recurrence relations with two indices.

Understand the concepts of permutations with restrictions on relative positions and rook polynomials.

Explain the principle of inclusion and exclusion.

Enumerate configuration using poly’s theory.

Understand the concept of Block designs.

Text Books:

Introduction to combinatorial Mathematics by CL. Liu, Mc GRAW Hill.

SEMESTER:-IV

NAME OF THE SUBJECT:(7MMA4C1) FUNCTIONAL ANALYSIS

COURSE DESCRIPTION:

Topics include:

Duals and Transposes.


Reisz Representation Theorems.

COURSE OBJECTIVES:

The objectives of the course are the study of the main properties of

bounded operators between Banach and Hilbert Spaces.

The basic result associated to different type of convergences in

normed spaces and the spectral theorem and some of its applications.

COURSE OUTCOMES:

On completion of this course the learner will

Be able to Normed Space and Continuity of Linear Maps.

Uniform Boundedness Principle.

Closed Graph and Open Mapping Theorems.


Orthonormal Sets, Projection and Reisz Representation Theorems.

TEXT BOOK:

Functional Analysis by B.V.Limaye, Second Edition, New Age International Pvt. Ltd.,

Publishers.

Name of the subject (7MMA4C2 )Operation Research

Course Description:

This course will covers the topics include:

Network models, scope and definition of network models.

Route problem, examples of the shortest route application.

CPM/PERT, network representation linear programming formulation of CPM/PERT calculations.

Deterministic inventory models, static economic, classic EOQ models.


Generalized poissonqueuing model specialized poisson queues, steady state measures of performance,

(M/M/R):(GD/K/K), R>K- (M/G/1):(GD/∞/∞).

Course objectives:

The objectives of this subject is to expose student to understand the importance optimization techniques and the

theory behind it.

To introduce students to the techniques of operation research in mining operations.

To provide students with basic skills and knowledge of Operation Researchand its application in mineral

industry.

Network models, enumeration of cuts, maximal flow algorithm, linear programming formulation of

maximal flow mode, CPM computations.

Deterministic inventory models, EOQ models, no step model, setup model.


Generalized poisson queuing model specialized poisson queues, single server model, multi - server model,

mechine servicing model, pollaczek, khintchine(P-K) formula, other queuing models.

Course outcome:

The students will able to :

Explain the meaning of operation research know the various techniques of operations research.

Use operations research to solve transportation problems during the allocation of trucks to the formulate

operation research models to solve real life problem.

Solve linear programming problem by using algebraic graphical method.

Understand the concept of minimal spanning tree algorithm and maximal flow algorithm.

Understand Queuing theory basic concepts and solve queuing theory problems.

Compute CPM and PERT through project network diagram.

Understand test the optimality for degeneracy by using machine servicing model (M/M/R): (GD/K/K).

Text Books:

Operation Research VIII Edition by HamdyA.Taha ,Prentice –Hall of India Pvt.Ltd.

Name of the subject (7MMA4C3) Topology - II

Course Description:


the presentation will not necessarily follow the text. Attendance is required and the exams will be over the lectures .

The course topics include:

Connectedness and compactness: the Tychonoff Theorm .

Completely regular spaces, The Stone – cech compactification.

Metrization theorems.

Complete Metric spaces and function spaces

The compact open topology ascoli’s theorem .

Course Objectives:


Topology - II.

Connectedness and compactness: the Tychonoff Theorm.

Use results from completely regular spaces , definition of a local finiteness.

Explain from Nagata- smirnov metrization theorem.

The compact open topology Ascoli’s theorem .

Complete metric spaces and functions of a space filling curve and nowhere differential functions.

Course outcomes:


Use local compactness and the tychonoff theorem describe the completely regular spaces.

Explain the Nagata semirnov theorem for necessary and sufficient .

Give the definition of concepts related to complete metric space.

Understand the Metrization theorems and parcompactness.

Understand the open topology and Baire spaces.

To use A nowhere differential functions.

TEXT BOOk: Topology a first course by James R. Munkres

Name of the subject (7MMA4E1) Advanced Statistics

Course Description:

A course dealing with statistical concepts

Introduction statistical inference, point estimation.

Sufficient statistics, measures of quality of estimators.

More about estimation, Bayesian estimation.

Theory of statistical tests, certain best tests.

Inferences about normal model, non- central chi-squareand non - central F.

Course objectives:

The course is designed to introduce more advanced statistical methods that are used in data analysis. both statistical

theories and inference technique will be covered.

Introduction to statistical inference, test of statistical hypothesis.

Sufficient statistics, measures of quality of estimators, functions of a parameter.

More about estimation, fisher information and the rao.

Theory of statistical tests, likelihood ratio tests.

Inferences about normal models, a test of independence.

Course outcome:

On successful completion of this module, a student will be expected to be able to:

Be competent on a variety of well- known distributions and the calculation involved.

Understand the theories of statistical inferences and apply the approximate models in different setting to

solve real life problems.

Perform test of hypothesis as well as calculate confidence interval for single sample and two sample cases.

Understand the concepts non-central chi-square and non-central F.

Understand both the meaning and applicability of a dummy variable and the assumptions which underline a

regression model.

Understand the empty advanced statistical methods such as the analysis of variances , ratio test, the

sequential probability ratio test.

Text Books:

Introduction to Mathematical Statistics (Fifth Edition) by Robert V.Hogg and Allen T.Craig, Pearson Education

Asia.

NAME OF THE SUBJECT: (7MMA4E3) NUMERICAL METHODS

COURSE DESCRIPTION:

Numerical Analysis was studied for its long and rich problems its wealth of easily accessible and

fascinating questions and its intellectual appeal. Topics include

Transcendental and Polynomial Equation.

System of Linear Algebraic Equations and Eigen Value Problems.

Differentiation and Integration, Ordinary Differential Equations.

COURSE OBJECTIVES:

To introduce students to the technique of Numerical Methods in Vita

Method, Bait Stow’s Method, Graeffe’s Root Squaring Method.

To provide students with basic skills and knowledge of Numerical Methods

and its applications.

To solving the problem Error Analysis of Direct and Iteration Methods,

Finding Eigen Values and Vectors, Jacobi.

COURSE OUTCOMES:


The learner will be able to Rate of Convergence of Iterative

Methods,Graeffe’s Root Squaring Method.

Improve the Hermit Interpolations, Bivariate Interpolation.

Understand the problems Extrapolation Methods, Gauss Method.

Local Truncation Error-Euler, Backward Euler Midpoint.

Runge Kutta Method, Stability Analysis

TEXT BOOK

M.K.Jain, S.R.K.Iyengar and R.K.Jain, Numerical Methods for Scientific and

Engineering Computation. III Edn Wiley Eastern Ltd.,993.

REFERENCE BOOK:

1. Kendall E.Atkinson, An Introduction to Numerical Analysis, IIEdn., John

Wiley & Sons,1983.

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