M Scsyllabus CBCS

8/12/2019 M Scsyllabus CBCS

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Regulations and Syllabi

of the

Master of Science in MathematicsBranch I

to be offered from the

academic year 2008 - 09

by theDepartment of Mathematics

Sacred Heart College

(Autonomous)

Tirupattur

Vellore District


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MINUTES OF THE MEETING OF THE

PG BOARD OF STUDIES IN MATHEMATICS

SACRED HEART COLLEGE

TIRUPATTUR

Date : 11 07 2008

Time : 2.00 PM

Venue : Abdul Kalam Research Center

Agenda 1. To approve the Structure of the PG curriculum

2. To approve the Syllabi for the PG Curriculum3. To approve the evaluation pattern4. To approve the syllabi for 2 courses of M.Com

Programme5. To approve the list of Question paper setters and

examiners.

Members Present:

Chairman : Dr.M.Maria Susai Manuel

Reader and Head

Department of Mathematics

Sacred Heart College, Tirupattur

University Nominee : Dr.L.Prathaban

Reader in Mathematics

Muthurangam Govt. Arts College,Vellore

Subject Experts :

1. Dr.E.ThandapaniProfessor in Mathematics

RIAS,University of Madras,Chepauk, Chennai 5

2. Dr.V.Rajkumar DareReader and Head

Department of Mathematics

Madras Christian College

Tambaram, Chennai


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Industrial Representative

Mr.N.Ramakrishnan

Director

NOVA WEIGH India Ltd

Chennai.

Meritorious Alumnus

Mr.C.Thirugnana Sambandam

Senior Manager

Wing Design Group

ARDC, HAL, Bangalore

Members

1. Dr.M.Reni Sagayaraj

2. Dr.K.Ravi

3. Mr.A.George Maria Selvam

4. Mr.R.Murali

5. Mr.S.Joseph

6. Mr.P.Manoharan

Co-opted Member:

1. Mrs.A.Merceline AnithaMr.G.Britto Antony Xavier was absent with prior intimation.Proceedings

The board approved the structure, syllabi for the PG curriculum, syllabi of the two

courses offered for M.Com, Evaluation pattern and the list of question paper setters and

examiners.

The following is the detailed syllabi, question paper pattern, structure, eligibility

conditions approved by the board and recommended to place it before the academic

council.


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M.Sc. Degree Programme in Mathematics

Aims : The M.Sc. Degree in Mathematics curriculum aims to provide

opportunities for students to:

develop flexibility and creativity in applying mathematical ideas and techniquesto unfamiliar problems arising in everyday life, and develop the ability to reflectcritically on the methods they have chosen;

become effective participants in problem-solving teams, learning to express ideas,and to listen and respond to the ideas of others;

develop the characteristics of logical and systematic thinking, and apply these inmathematical and other contexts, including other subjects of the curriculum;

become confident and competent users of information technology in mathematicalcontexts;

develop the skills and confidence to use their own language, and the language ofmathematics, to express mathematical ideas; ensure flexibility of student choice, so far as possible, between degree

programmes and within individual programmes; prepare students for further study in a mathematically related field or for a career

in which clear thinking and problem solving are valued.

Objectives (now intended learning outcomes): At the end of every programme, the

students should:

have a broad knowledge of mathematics and the techniques for solving problemsin several areas, and the ability to apply those techniques with confidence. appreciate logical and precise arguments and the clear writing of

mathematics. be able to use a variety of educational resources such as lectures, books,

tutorial discussion, consulting lecturers, working together and (for nearlyall students) electronic resources such as the Internet.

have knowledge of the applications of mathematics in other subject(s)such as computer science, physics, biology, economics, experience ofmodelling situations in the real world using mathematics.

have an appreciation of the cultural and/or historical context ofmathematics.


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REGULATIONS

1. Nature of the Programme :

This is a post-graduate programme combining both pure and applied oriented

courses in Mathematics with Computer Laboratory courses offered under the Faculty of

Science as the M.Sc., Degree Programme Course in Branch I: Mathematics by the

Department of Mathematics, Sacred Heart College (Autonomous), Tirupattur, Vellore

District.

2. Conditions for Admission:

A candidate who has passed the B.Sc., degree examination in Branch I

Mathematics of Thiruvalluvar University, or an examination of some other University

accepted by the Syndicate of the Thiruvalluvar University as equivalent thereto shall be

permitted to appear and qualify for the Master of Science (M.Sc.,) Degree Examination

after a course of two academic years in Scared Heart College which is affiliated to the

Tiruvalluvar University.

3. Duration of the M.Sc. Degree Programme:

This Programme of study shall be based on semester system with credits underthe Autonomous pattern with internal assessment. This Programme shall consist of four

semesters covering a total of two academic years. For this purpose, each academic year

shall be divided into two semesters; First and Third semester; July to November and

Second and Fourth Semester; December to April.

4. Courses of Study in this M.Sc. Degree Programme:

The Courses of study for the M.Sc. Degree Programme shall be in Branch I

Mathematics Semester System with credits and internal assessment according to a

syllabus to be prescribed from time to time. This Programme consists of Core Courses

and Elective Courses.


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5. Allocation of Credit and Marks:

Credits are assigned to the courses depending on the level and content of the

subject matter in that course.

Total number of credits for the Programme : 90 credits

(inclusive of Theory, and Laboratory components)

Category ofCourses

Credits foreach Course

Number ofCourses

Total number ofCredits in eachCategory of Courses

Total Credits forthe Programme

Core 5 8 40Core 4 7 28 68HumanRights

2 1 2 2

Subject

Elective

4 4 16

Project 4 1 420

90

Total number of Courses : 20Project : 1Total number of marks : 2100 ( 100 marks for each Courses and 100 for the

Project)

6. Examinations:

There shall be four semester examinations, one at the end of each semester in each

academic year. A candidate who does not pass the examination in any course(s) in a

semester will be permitted to appear in such failed courses(s) also, with subsequent

semester examinations.


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7. Scheme of Examinations:

The scheme of examinations for different semesters shall be as follows:

MarksCode Category Paper Title of the Paper

CIA SE

Duration

for SE

Credits

Semester I

Core I Abstract Algebra 25 75 3 Hrs 5Core II Real Analysis 25 75 3 Hrs 5Core III Complex Analysis 25 75 3 Hrs 4Core IV Ordinary Differential

Equations25 75 3 Hrs 4

Elective V Elective-I(Choose one from Group-A)

25 75 3 Hrs 4

Semester II

Core VI Linear Algebra 25 75 3 Hrs 4Core VII Measure Theory and

Integration25 75 3 Hrs 4

Core VIII Classical Mechanics 25 75 3 Hrs 4Core IX Integral and Partial

Differential Equations25 75 3 Hrs 4

Human Rights 25 75 3 Hrs 2Elective X Elective-II

(Choose one from Group-B)

25 75 3 Hrs 4

Semester III

Core XI Mathematical Statistics 25 75 3 Hrs 5Core XII Topology 25 75 3 Hrs 5Core XIII Number Theory and

Cryptography25 75 3 Hrs 5

Core XIV Graph Theory 25 75 3 Hrs 4Elective XV Elective-III

(Choose one from Group-C)

25 75 3 Hrs 4

Semester IV

Core XVI Functional Analysis 25 75 3 Hrs 5Core XVII Operations Research 25 75 3 Hrs 5Core XVIII Finite Element Methods 25 75 3 Hrs 5Elective XVIII Elective-IV

(Choose one from Group-D)

25 75 3 Hrs 4

Project XIX 100 4Total Marks / Credits 2100 90


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Elective - I for Semester I

Any one of the following courses shall be chosen from Group-A as an Elective-I forSemester I.

Group A1. Differential Geometry2. Fuzzy sets and Fuzzy logic3. C++ and Numerical Analysis

Elective-II for Semester II

Any one of the following courses shall be chosen from Group-B as an Elective-II forSemester II.

Group B

1. Difference Equations2. Bio Mathematics3. Mathematical Modelling

Elective-III for Semester III

Any one of the following courses shall be chosen from Group-C as Elective III forSemester III.

Group C

1.

Fluid Dynamics2. Formal Languages and Automata3. MATLAB, Practicals

Elective-IV for Semester IV

Any one of the following papers shall be chosen from Group-D as Elective-IV forSemester IV.

Group D

1. Stochastic Processes2. Discrete Mathematics3. Financial Mathematics


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8. TESTING PATTERN

Components of Examination

In each course there are two components, namely, Continuous Internal Assessment

(CA) and Semester Examination. The marks for each component is as follows.For Theory Courses

CA : 25 Marks

Semester Examination : 75 Marks

The Components for CA are as follows

Tests : 15 Marks

Problem Solving : 5 Marks

Seminar : 5 Marks

There is no passing minimum in CA. A student shall be declared to have passed in

a course if he secures 50% and above in the Semester Examination and 50% and above in

the aggregate of CA and Semester Examination.

Question Paper Pattern

The question paper shall contain three parts, Section A, Section B and Section C.

Section A contains 10 questions, 2 questions from each unit. The candidate is

expected to answer all the questions. Each question carries 2 marks.

Section Bcontains 5 questions, one question from each unit and is of the either or

type. The candidate is expected to answer all the questions. Each question carries 5

marks.

Section Ccontains 5 questions; one question from each unit and the candidate is

expected to answer any 3 questions. Each question carries 10 marks.

Computer Laboratory Courses

The Components are as follows

CA : 40 Marks

Semester Examination : 60 Marks

For Computer Laboratory oriented Courses, there shall be two tests in Theory part and

two tests in Laboratory part. Choose one best from Theory part and other best from the


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two Laboratory part. The average of the best two can be treated as the CIA for a

maximum of 40 marks. The duration of each test shall be one / one and a half hour.

Project Evaluation

There shall be two parts

Report : 80 Marks

Viva : 20 marks

There shall be two valuations of the report, one by the guide and another

by an external examiner chosen from a panel of three examiners given by the guide. The

average of the two marks shall be the final mark for the report.

For the Viva Voce examination, the Department will suggest a panel of three

names and one will be chosen by the controller. The respective guide together with the

external member shall conduct the viva and the average of the two marks shall be the

final marks for viva.

There is no improvement for CIA of both theory and laboratory, and, also for

University End Semester Examination.

9. Earning of Credits

A candidate shall earn the credits carried by a paper if he/she passes the paper as

per the passing regulations.

10. Credit requirements for qualifying for the degree:

A candidate shall be declared to have qualified for the M.Sc. degree in

Mathematics (Branch I Semester System under CBCS) if he/she earns 90 credits out of

which 68 credits from the Core courses, 16 credits from the Electives and 4 credits from

the Project.

11. Requirements for Proceeding to subsequent semester.

(i) Candidates shall register their names for the First semester examination after the

admission in the PG courses.

(ii) Candidates shall be permitted to proceed from the First Semester up to the Final

Semester irrespective of their failure in any of the Semester Examination.

(iii) Candidates shall be eligible to proceed to the subsequent semester, only if they earn,

sufficient attendance as prescribed therefore by the University from time to time.


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12. Classification of Successful Candidates:

Candidates who secure not less then 60% of the aggregate marks in the whole

examination shall be declared to have passed the examination in the FIRST CLASS.

Candidates who secure not less than 50% of the aggregate marks in the whole

examination but below 60% shall be declared to have passed the examination in the

SECOND CLASS.

Candidates who obtain 75% of the marks in the aggregate shall be deemed to have passed

the examination in the FIRST CLASS WITH DISTINCTION provided they pass all the

examinations prescribed for the course as per the scheme of examination at the first

appearance.

13. Commencement of this Regulation:

These regulations shall take effect form the academic year 2008-09, i.e., for

students who are admitted to the first year of the course during the academic year 2008-

09 and thereafter.

Mathematics Websites: The following information on the Mathematics Web sites will

be an additional source of information for references and historical development of the

Mathematics. Some biographies of outstanding mathematicians are also available. This is

the common information for both teachers and students of Mathematics.

1. http://scienceworld.wolfram.com/biography/topics/Mathematicians.html2. http://teachers.sduhsd.k12.ca.us/abrown/index2.html3. http://www.maths.tcd.ie/pub/HistMath/People/RBallHist.html

Mathematicians of the 17th and 18th Centuries4. http://www.geometry.net/math.html

A Geometry Site5. http://www-history.mcs.st-andrews.ac.uk/history/Indexes/Full_Alph.html

Site of Biographies of Mathematicians6. http://mathforum.org

This site includes resources in mathematics for school students, teachers, parents.Also contains some research related material on mathematics teaching andlearning. The 'Problems of the Week' contains problems at different levels ofmathematics. It includes selected alternative solutions posted by problem solverswhich is really nice. The `Ask Dr. Math' gives useful explanations of mathconcepts and the discussion groups are about teaching methods.


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7. http://www.cut-the-knot.orgContains interesting puzzles, problems, theorems, proofs, etc. Also has links toother good sites (including all those listed below).

8. http://nrich.maths.orgThe site is run by the University of Cambridge. It contains problems for different

age groups (5 to 18) that one can post solutions to. Selected solutions arepublished at the website. One can also post questions. There is an archive ofquestions posted earlier with answers (in blue coloured font). There are alsoarticles, features, etc.

9. http://archives.math.utk.edu/A fairly comprehensive archive: contains teaching materials, public domainsoftware, shareware, books, articles, etc.

10.http://www-groups.dcs.st-and.ac.uk/~history/The MacTutor history of mathematics archive. The best known website forhistorical information about mathematicians and mathematics.

11.http://www.maa.org/This is the website of the Mathematical Association of America. Contains usefulresources for college mathematics teachers including book reviews.

12.http://e-math.ams.org/Website of the main professional organization in mathematics: AmericanMathematical Society. The journal `Notices of the AMS' is online. PlusInteresting essays.


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Syllabus for M.Sc Mathematics

Semester Course No Code Course Title Credits Category

I 1 M 730 Abstract Algebra 5 CoreInstructional Hours Lecture 6 Lab 0 Total 6Pre requisite UG level Modern AlgebraObjectives of the Course To introduce the concepts and to develop working knowledge on class equation,

solvability of groups, finite abelian groups, Modules, Galois group overrationals.

Course Outline

Unit-I Group Theory (Chapter 2: 2.11 - 2.12 )

Another counting principle - Sylows Theorems .[For Sylows Part I, second proof only]

Unit-II Group Theory (Continued) (Chapter 2: 2.13, 2.14 )

Direct Products - Finite Abelian Groups .

Unit-III Modules and Field Theory(Chapter 4: 4.5, Chapter 5: 5.1 )

Modules - Extension FieldsUnit -IV Field Theory (Continued) (Chapter 5: 5.3 and 5.5) .

Roots of a Polynomial - More about roots.

Unit-V Galois Theory (Chapter 5: 5.6, 5.7, 5.8)

The elements of Galois theory- Solvability by radicals- Galois theory over the rationals.

Recommended Text I. N. Herstein, Topics inAlgebra, 2- e, John Wiley, Reprint 2006.

Reference

B

ooks

1. John B.Fraleigh, A First Course in Abstract Algebra, 4 e, Addition Wesley PublishingCompany London, 1972.

2. I. N. Jacobson, Basic Algebra, Hindustan Publishing Corporation, Delhi, 1984.3. Surjit Singh, Gazi Zameeruddin, Modern Algebra, 2-e, Vikas Publishing House Pvt Ltd .

Delhi, 1975.4. Vasistha - M. L . Kanna, Modern Algebra, 14-e, Jai Prakash Nath & Co . Meerut .

Websites and

e-learning

Source

http://en.wikipedia.org/wiki/abstract_algebrahttp://mathworld.wolfram.com


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I 2 M 731 Real Analysis 5 CoreInstructional Hours Lecture 6 Lab 0 Total 6Pre requisite UG level Real Analysis

Objectives of the Course To study infinite series and infinite products, sequence of functions,multivariable differential calculus, implicit functions and extremum problemsand multiple Riemann integral and to develop computational skills in RealAnalysis

Course Outline

Unit I Infinite series and infinite products (Chapter 8: 8.17 8.23)Rearrangement of series - Riemanns theorem on conditionally convergent series - Sub series, doublesequences, double series - Rearrangement theorem for double series - A Sufficient condition for equality ofiterated series

Unit II Sequence of Functions (Chapter 9: 9.13 9.15, 9.19, 9.20, 9.23).Mean convergence , power series - Multiplication of power series - The Taylors series generated by a

function - Bernsteins Theorem - Taubers TheoremUnit III Multivariable differential Calculus (Chapter 12: 12.1 12.11)

The directional derivative and continuity, The total derivative expressed in terms of partial derivative, Anapplication to complex valued functions, The matrix of a linear function, The Jacobian matrix ,the chain rule,Matrix form of the chain rule, MVT for differential functions

Unit IV Implicit Functions and Extremum Problems (Chapter 13:13.1 13.6)Introduction, Functions with non zero Jacobian determinant, The inverse function theorem, The implicitfunction theorem, Extrema of real valued function of one and several variable

Unit V Multiple Riemann Integral (Chapter 14: 14.1 14.3, 14.5, 14.6, 14.10).Introduction, The measure of a bounded interval in Rn , The Riemann integral of a bounded function defined on

a compact interval in Rn , Evaluation of multiple integral by iterated integration, Jordan measurable sets in Rn,

MVT for multiple integrals

Recommended Text Tom .M. Apostal, Mathematical Analysis, 4-e,Addison Wesley PublishingCompany, 1979.

Reference

Books

1.Robert G.Bartle and Donald R.Sherbert, Introduction to Real Analysis by 2-e,John Wiley and Sons,1994.

2.H.L. Royden, Real Analysis, 3-e Prentice Hall of India Pvt Ltd., 1997, New Delhi.3.Serge Lang, Real Analysis, 2- e Addision Wesley Publishing company, 1997.4. Richard R.Goldberg, Method of Real Analysis ,Indian edition 1970,Waltham, Mas, U.S.A.5.Walter Rudin ,Real and Complex Analysis, 2-e, Tata McGraw Hill Publishing Company, 1979.

Websites and

e-learning

Source

www.expocentral.com/directory/Science/Math/Analysis/Real_Variable/www.dmoz.org/Science/Math/Analysis/Real_Variable/


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I 3 M 732 Complex Analysis 4 CoreInstructional Hours Lecture 6 Lab 0 Total 6Pre requisite UG level Complex AnalysisObjectives of the Course To study general form of Cauchys theorem, evaluation of definite integral,

harmonic functions, normal families, Riemann mapping theorem, conformalmapping of polygons, elliptic functions and Weiestrass theory and analyticcontinuation

Course Outline

Unit-I Complex Integration (Chapter 4: 4.4, 4. 5, 4. 6)The General Form of Cauchys Theorem, The Calculus of Residues , Harmonic Functions.Unit-II Power Series(Chapter 5 : 5.1, 5. 2, 5. 3)Power Series Expansions, Partial Fractions and Factorization, Entire FunctionsUnit-III Normal Families and Conformal Mapping (Chapter 5 : 5.5, Chapter 6 : 6.1, 6 .2)Normal Families, The Riemann Mapping Theorem, Conformal Mapping of PolygonsUnit-IV Elliptic Functions (Chapter 7 : 7.1, 7. 2, 7. 3,)Simply Periodic Functions, Doubly Periodic Functions, The Weierstrass Theory

Unit-V Global Analytic Functions (Chapter 8: 8.1, 8. 2, 8. 3)Analytic Continuation, Algebraic Functions, Picards Theorem

Recommended Text Lars V.Ahlfors, Complex Analysis, 3 e, McGraw Hill International BookCompany, 1979.

Reference

Books

1. B.Choudhary, The Elements of Complex Analysis, 2-e, Wiley Eastern Limited.2. Boston , Complex variables, Silverman Houghton Mifflin Company .3. Serge Lang, Complex Analysis, 2 e , Springer Verlag, New York4. John B. Conway, Functions of One Complex Variable, 2-e, Springer International student

Edition.Websites and

e-learning

Source

http://en.wikipedia.org/wiki/complex_analysiswww.usfca.edu/vca/websites.html


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I 4 M 733 Ordinary Differential Equations 4 CoreInstructional Hours Lecture 6 Lab 0 Total 6Pre requisite UG level Calculus , Differential EquationsObjectives of the Course To study the Differential equation of higher order, to find the power series

solutions of special type of Differential equations, to solve the system of linearDifferential equations, existence and uniqueness of the solutions of first orderDifferential equations and to appreciate the concept of oscillation in secondorder equations.

Course Outline

Unit:I: Linear Differential Equation of Higher order (Chapter 2: 2.1 to 2.5)Introduction . Linear dependence and Wronskian, Basic theory for linear equations, Method of variation of

parameters, Two useful formulae.Unit-II Solution in Power Series (Chapter 3: 3.1 - 3.5)

Introduction, Second order Linear equations with ordinary points, Legendre equation and Legendrepolynomials, Second order equation with Regular singular points, Bessel Equation.Unit-III Systems of Linear Differential Equations (Chapter 4 :4.2 - 4.6)

Systems of first order equations, Existence and uniqueness theorem, Fundamental matrix, Non-homogeneous

linear systems, Linear systems with constant coefficients.Unit -IV Existence and Uniqueness of solutions(Chapter 5:5.2 - 5.6)Preliminaries Gronwall inequality, Successive Approximations, Picards theorem, Non uniqueness of

solutions, Continuation and Dependence on initial conditions.Unit - V Oscillations of second order equations (Chapter: 6 : 6.1 - 6.5)

Fundamental results, Sturms comparison theorem, Elementary Linear oscillations, Comparison theorem of

Hille-Wintner, Oscillations of ( ) 0x a t x + = .

Recommended Text S.Deo and V.Raghavendra,Ordinary Differential Equations and Stability

Theory,Tata McGraw Hill Publishing Company, New Delhi.

Reference

Books

1. Ean A.Coddington, An Introduction to ODE, Prentice Hall of India Pvt Ltd., New Delhi,1992.2. M.D.Rasingania, Advanced Differential Equations, 4-e, Tata McGraw Hill Publishing

Company, New Delhi.3. G.F.Simmons,Differential Equations, S.Chand and Company Ltd, New Delhi,1974.4. M.Rana Mohana Rao, Ordinary Differential Equations Theory and Applications,Affiliated

East-WestPress private Ltd,Chennai.5. D.Somasundaram, Ordinary Differential Equations,Narosa Publishing House,Chennai,2002.6 D.Ravi,D,.P.Choudary and H.I.Freedman,A Course in Ordinary Differential Equations Narosa

Publishing House,Chennai,2004.Websites and

e-learning

Source

http://en.wikipedia.org/wiki/ordinary_differential_equation


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II 6 M 830 Linear Algebra 4 Core

Instructional Hours Lecture 6 Lab 0 Total 6Pre requisite UG level AlgebraObjectives of the Course To study linear transformation, linear functionals inner product spaces, unitary

operators, normal operators and canonical forms and Jordan canonical formsand to develop computational skill in Linear Algebra

Course Outline

Unit- I (Chapter 3: 3.1 3.3)Linear Transformation , Algebra of linear transformation , Isomorphism

Unit -II(Chapter 3:3.4 , 3.5 , 3.6 ).Representation of transformation by matrices , Linear functionals, Double Dual

Unit -III (Chapter 3: 3.7, Chapter 8: 8.1, 8.2, 8.3)Transpose of a linear transformation, Inner products , Inner product spaces, Linear functionals and

adjoints .Unit -IV (Chapter 8: 8.4 , 8.5 ,Chapter 9: 9.2) .Unitary operators, Normal operators, Forms on inner product spaces .Unit-V Canonical forms

Triangular form , Invariance, Invariant direct sum Decompositions Primary Decomposition -NilpotentOperators - Jordan Canonical forms,Problems 10.1-10.20 [Chapter 10 of Linear Algebra by Seymour Lipschutz -Schaums outline series]

Recommended Text Kenneth Hoffman and Ray Hunze, Seymour Lipschutz, Linear AlgebraSchaums Outline Series .

ReferenceBooks 1. I . N . Herstein,Topics in Algebra,2-e,Vikas Publishing House Pvt Ltd ,Chennai 6.

2. Serge Lang, Linear Algebra ,6-e,Addition Wesley Publishing Company , London .3. K.P .Gupta, Linear Algebra,2-e, Pragati Prakashan ,Meerut , India .4. V.Krishnamurthy, V.Manira, J.L.Arora, An Introduction to Linear Algebra, Affiliated

East-West Press pvt Ltd,New Delhi,Chennai, 1976.5. P.P.Gupta,S.K.Sharma, Linear Algebra, S.Chand and Company Ltd, New Delhi,1982.

6. L.Mirsky, An Introductionto Linear Algebra, Oxford at the Clarendon Press,1955.Websites and

e-learning

Source

http://en.wikipedia.org/wiki/Linear_algebra


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II 7 M 831 Measure Theory and Integration 4 Core

Instructional Hours Lecture 6 Lab 0 Total 6Pre requisite Real AnalysisObjectives of the Course To introduce measure on the real line ,measure on space ,Lebesgue

measurability and integrbility ,inner and outer measure, signed measures.Course Outline

Unit I Lebesgue Measure (Chapter 3: 3.1 3.5)Introduction, Outer measure, Measurable sets and Lebesgue measure, A Non Measurable set, Measurable

functionsUnit II The Lebesgue Integral (Chapter 4:4.1 4.4)

The Riemann Integral, The Lebesgue integral of a bounded function over a set of finite measure, The integralof a non-negative function, The general Lebesgue integralUnit III Differentiation and Integration (Chapter 5: 5.1 5.5)

Differentiation of Monotone functions, Functions of Bounded Variation, Differentiation of an integral,Absolute continuity, Convex functionsUnit -IV Measure and Integration (Chapter 11:11.1 11.6)

Measure space, Measurable functions, Integration, General convergence theorem, Signed measures, The

Radon Nikodym theoremUnit -V Measure and Outer Measure ( Chapter 12:12.1 12.4)

Outer measure and measurability of sets, The extension theorem, The Lebesgue Stieltjes integral, Productmeasures

Recommended Text H.L.Royden, Real Analysis , 3-e, Macmillan Publishing Company, New Delhi, 2005.

Reference

Books

1. Paul.R.Halmos, Measure Theory ,Springer International Student Edition.2. G. De. Barra, Measure Theory and Integration, Ellis Harwood Limited Publisher.3. Inder K.Rana,An Introduction to Measure and Integration,2-e,Narosa Publishing House.4. P.K.Jain and V.P.Gupta,Lebesgue Measure and Integration,Wiley Eastern Ltd.5. P.P.Gupta,G.S.Malik and S.K.Mittal, Measure Theory , Kedar Nath Publication, Meerut.

Websites and

e-learning

Source

http://en.wikipedia.org/wiki/measurehttp://www.uni_math.gwdg.dewww.math.uconn.edu/~bass/lecture.html


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II 8 M 832 Classical Mechanics 4 Core

Instructional Hours Lecture 6 Lab 0 Total 6Pre requisite UG level Calculus, Differential Equations, DynamicsObjectives of the Course To study mechanical systems under generalized coordinate systems, virtual

work, energy and momentum, mechanics developed by Newton, Lagrange,Hamiltion, Jacobi

Course Outline

Unit -I Mechanical Systems( Chapter 1: 1.1 to 1.5)The Mechanical System, Generalized Co-ordinates, Configuration Space, Constraints, Virtual work, Principle

of virtual work, DAlemberts principle, Generalized force, Energy and MomentumUnit:II Lagranges Equations( Chapter:2: 2.1 to 2.3)

Derivation of Lagranges equations, Integrals of the motion, Ignorable co-ordinates, The Routhian function,Conservative systems, Natural systemsUnit:III Hamiltons equation (Chapter:4: 4.1 to 4.3)

Hamiltons principle, Derivation of Hamiltons equations, The Legendre transformation, Modified Hamiltonsprinciple, Principle of least actionUnit:IV Hamilton Jacobi Theory (Chapter:5: 5.1 to 5.3)

Hamiltons principal function, Pfaffian differential forms, The Hamilton Jacobi equation, Jacobis theorem,Seperability, Stackels theoremUnit:V Introduction to Theory of Relativity (Chapter:7: 7.1 to 7.3)

Galilean transformation, The principle of relativity, The Lorentz transformation equations, Events andsimultaneity, Time dilation, Longitudinal contraction, Proper time and proper distance, Addition of velocities,The relativistic Doppler effect, Relativistic dynamics

Recommended Text Donald T.Greenwood, Classical Dynamics, Prentice Hall of India Pvt. Ltd., NewDelhi 1985.

Reference

Books

1. H.Goldstein: Classical Mechanics, second edition, Narosa Publishing House , New Delhi.2. J.L.Synge and B.A.Griffth,Principles of Mechanics,3e, McGraw Hill Book Company, New york .3. J.L.Synge and P.S.C.Joag, Classical Mechanics , Tata Mc Graw Hill, New Delhi, 1991.4. P.G.Bergmann, Introductionto Theory of Relativity, Prentice Hall of India, Eddington, New

Delhi, 1969.Websites ande-learning

Source

http://en.wikipedia.org/wiki/classical_mechanics


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II 9 M 833 Integral and Partial DifferentialEquations

4 Core

Instructional Hours Lecture 6 Lab 0 Total 6Pre requisite Knowledge in Differential equations

Objectives of the Course To develop skills in solving integral and partial differential equations.Course Outline

Unit I Elliptic Differential Equations (Chapter 2:2.1 2.5 and 2.13)Occurrence of the Laplace and Poisson equations, Boundary value problems, Some important mathematical

tools, Properties of harmonic functions, Separation of variables, Miscellaneous examples 2.10 to 2.20Unit II Parabolic Differential Equations (Chapter 3:3.1 3.5)

Occurrence of the diffusion equation, Boundary conditions, Elementary solutions of the diffusion equations,Dirac delta function, Separation of variables method with examples up to 20Unit IIIHyperbolic Differential Equations (Chapter 4:4.1 4.6)

Occurrence of the wave equation, Derivation of one-dimensional wave equation, Solution of one dimensionalwave equation by canonical equation, The initial value problem, DAlemberts solution, Vibration string-variable separable solution, Forced vibration-solution of non homogeneous equation with examples upto 4.7.Unit -IV Classification of Integral equations and connection with differential equations

(Chapter 1 and 2:1.1 1.6, 2.1)Historical introduction, Linear integral equations, Special type of kernel, Square integrable functions andkernels, Singular integral equations, Non linear equations, Linear differential equations.Unit -V Integral equations of the convolution type and the method of Successive approximations

( Chapter 3:3.1 3.6)Integral transforms, Fredholm equation of the second kind, Volterra equation of the second kind, Fredholmequation of the first kind, Stieltjes integral equation, Volterra equation of the first kind, Abels integral equation,Foxs integral equation

Recommended Text For Unit 1, 2, 3Sankara Rao.K, Introduction to Partial Differential Equations,Prentice Hall of India,1995.For Unit 3, 4 B.L.Moiseiwitsch,Integral Equation, Longman Group Limited,

London.

Reference

Books

1. Snedon.I.N, Elements of Partial Differential Equations, Tata McGraw Hill, New Delhi, 1991.

2. M.D.Raisingania, Advanced Differential Equations, 4-e, Tata McGraw Hill Publishing Company,New Delhi, 2001.

3. Amarnath.T, An Elementary Course in Partial Differential Equations, Narosa PublishingHouse, 1997.

4. .Shanti Swaraj,linear Integral Equations, Krishna Prakashan Mandir, Meerut.5. Smithiees.F,Integral Equations, Cambridge University Press, London.

Websites and

e-learning

Source

http://en.wikipedia.org/wiki/partial_differential_equation


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III 11 M 933 Mathematical Statistics 5 Core

Instructional Hours Lecture 6 Lab 0 Total 6Pre requisite Probability TheoryObjectives of the Course To study sampling theory, significance tests, estimation, testing of hypothesis

and linear models.Course Outline

Unit I Sampling and Sampling Distribution (Chapter 4: 4.2 4. 5)Sampling, Sample Mean, Sampling from the Normal Distributions, Order Statistics

Unit -II Parametric Point Estimation (Chapter 7: 7.2-7. 5)Methods of Finding Estimators, Properties of Point Estimators, Sufficiency, Unbiased Estimation

Unit-III Parametric Point and Interval Estimators (Chapter7:7.7, Chapter 8: 8. 2-8.6)Bayes Estimators, Confidence Intervals, Sampling from the Normal distribution, Methods of Finding

Confidence Intervals, Large Sample Confidence Intervals, Bayesian Interval estimateUnit IV Tests of Hypotheses (Chapter 9: 9.4 9.6)

Test of Hypotheses - Sampling from the Normal distribution, Chi square testsTest of Hypotheses and Confidence Intervals

Unit -V Linear models (Chapter 10:10.2 10.7)Definitions and Examples of linear models, Point Estimation Case A, Confidence Intervals Case A , Tests

of Hypotheses Case A, Point Estimation Case B

Recommended Text Alexander M.Mood , Franklin, A. Graybill andDuane C. Bose , Introduction

to the Theory of Statistics, 3 e, Tata McGraw Hill, 1974.

Reference

Books

1. Paul G.Hoel, Introduction to Mathematical Statistics2. S.S.Wilks, Mathematical Statistics3. V.K.Rohatgi,An Introduction to Probability theory and Mathematical Statistics.4. Marker Fisz, Probability and Mathematical Statistics.5. Simmons and Schuster, Probability Statistics and Random Process.6. Ebevier, Probability and Statistics.

Websites and

e-learning

Source

http://en.wikipedia.org/wiki/Mathematical_statisticshttp://www.uconn.edu/~ericskey/361F98.htmlhttp://en.wikipedia.org/wiki/Recurrence_relation


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III 12 M 934 Topology 5 Core

Instructional Hours Lecture 6 Lab 0 Total 6Pre requisite Real AnalysisObjectives of the Course To study topological spaces, Haudroff spaces, continuous functions,

connectedness, compactness, regular spaces, normal spaces, countability andseparation axiom.

Course Outline

Unit 1 Topological Spaces (Chapter 2: Sections 12 - 17)

Topological spaces , Basis for a topology, The order topology, The product topology on X Y, The subspacetopology , Closed sets and limit points.Unit -II Continuous Functions(Chapter 2 : Sections 18 - 21)

Continuous functions, The Product topology, The metric topology.Unit -III Connectedness (Chapter 3 : Sections 23 - 25)

Connected spaces , Connected subspaces of the Real line, Components and local connectedness.Unit -IV Compactness (Chapter 3 : Sections 26 - 29)

Compact spaces, Compact sets in the Real line, Limit point Compactness, Local CompactnessUnit-V Countability And Separation Axiom(Chapter 4: Sections 30 - 35)

The Countability Axioms, The separation Axioms, Normal spaces, The Urysohn Lemma, The Urysohnmetrization, The Tietz extension theorem.

Recommended Text James R.Munkres, Topology,2 e, Prentice Hall of India, Private Limited,New Delhi, 2003.

ReferenceBooks

1. J.Dugundji, Topology, Prentice Hall of India, New Delhi, 1975,.2. George F. Sinmons, Introduction to Topology and Modern Analysis,

McGraw Hill Book Co., 19633. J.L. Kelly, General Topology, Van Nostrand, Reinhold Co., New York4. L.Steen and J.Seeback, Counter Examples in Topology, Hoit, Rinehart and Winston,

New York, 1970.5. S.Willard, General Topology6. Sze- Tsen Hu, Introduction to Topology TMH Edition , TMH Publishing Company Ltd.

Websites ande-learning

Source

http://archives.math.utk.eduhttp://mathworld.wolfram.com


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III 13 M 935 Number Theory and Cryptography 5 Core

Instructional Hours Lecture 6 Lab 0 Total 6Pre requisite Basic operation on numbers, basic calculus, basic knowledge on finite fieldsObjectives of the Course This course aims to give elementary ideas from number theory which will have

applications in cryptography, enciphering, deciphering methods shift, affinetransformation, enciphering matrices, using public key, secrete key and givingdiscrete log

Course Outline

Unit: I Some Topics In Elementary Number Theory (Chapter 1 )Time estimate for doing arithmetic , Divisibility and Euclidian Algorithm, Congruences , Someapplications to factorizing .Unit - II Finite Fields Quadratic Residues (Chapter 2)Finite fields , Quadratic residues and reciprocity .Unit - III Cryptography ( Chapter 3)Some simple Cryptosystem, Enciphering Matrices.Unit -IV Public Key (Chapter 4: 4.1 , 4.2 )

The idea of Public key cryptography, R S A .Unit -V Public Key ( Continued ) (Ch 5: 5.1, 5.2)

Discrete log, Knapsack, Zero Knowledge protocols and oblivious transfer.

Recommended Text Neal Koblitz, Number theory and cryptography 2 - e by SpringerVerlag, New Delhi, 1994.

Reference

Books

1. Graham.R ,L . Knuth , D. E. Patachink . O,Concrete Mathematics2 e, Pearson education Asia , 2002

2. Brensoud , D . Wagon S. A course in Computational Number Theory Key

Collage Publishing , 2000.

Websites and

e-learning

Source

http://en.wikipedia.org/wiki/Number_theoryhttp://mathworld.wolfram.com


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III 14 M 936 Graph Theory 4 Core

Instructional Hours Lecture 6 Lab 0 Total 6Pre requisite Basic level mathematics at UG levelObjectives of the Course To develop the concepts of graphs, sub graphs, trees, connectivity, Euler tour,

Hamiltion cycles, matching, colouring of graphs, independent sets, cliques,vertex colouring, and planar graphs.

Course Outline

Unit I Graphs and Subgraphs ( Chapter 1: 1.1 1.8)Graphs and Simple graphs, Graphs isomorphism, The incident and adjacent matrices, Sub graphs, Vertex

degrees, Paths and connection, Cycles, The Shortest path problems.Unit II Trees and Connectivity (Chapter 2: 2.1-2.5,Chapter 3:3.1,3.2)

Trees, Cut edges and bonds, Cut vertices, Cayleys formula, The connector problems, Connectivity, BlocksUnit III Euler Tours and Hamilton Cycles(Chapter 4 4.1-4.4)

Eulers Tour, Hamilton Cycles, The Chinese postman problem, The traveling salesman problemUnit IV Matching,Independent Sets and Cliques(Chapter5:5.1-5.5,Chapter7:7.1)

Matchings, Matchings and Coverings in bipartite graphs, Perfect matchings, The Personal assignment

Problems, The Optimal assignment problems, Independent setsUnit V Vertex Colouring and Planar graph (Chapter 8:8.1, 8.2, 8.4, Chapter9 : 9.1-9.3, 9.6)

Chromatic Number, Brooks theorem, Chromatic polynomials, Plane and planar graphs, Dual graphs, EulersFormula, The five colour theorem and four colour conjecture.

Recommended Text J.A.Bondy & U.S.R.Murty, Graph Theory with Application, Macmillian Press,1976

Reference

Books

1. P.Harray, Graph Theory, Narosa Publishing House, New Delhi 1998.2. K.R.Parthasarthy, Basic Graph Theory, Tata McGraw Hill,1994.3. S.Arumugam & S.Ramachandran, Invitation to Graph Theory4. V.K.Balakrishnan, Graph Theory, Tata McGraw Hill Publishing Company, New Delhi.5. Reinhard Diestel, Graph Theory, 2 e, Springer International Edition. 2000.

Websites and

e-learning

Source

http://en.wikipedia.org/wiki/Graph_theoryhttp://www.math.fau.edu/locko/graphite.htm


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IV 16 M 1032 Functional Analysis 5 Core

Instructional Hours Lecture 6 Lab 0 Total 6Pre requisite Linear Algebra, TopologyObjectives of the Course The aim of this course is to give students basic knowledge in Banach and

Hilbert Spaces.Course Outline

Unit- I Banach Spaces (Chapter 9, Sections 46 49)Definition and some examples, Continuous Linear Transformations, The Hahn-Banach Theorem, The natural

embedding of N in N**.Unit - II Banach Spaces and Hilbert Spaces (Chapter 9: Sections 50, 51 Chapter 10 : Sections 52. 53, 54)

Open Mapping theorem, Conjugate of an Operator, Definition and some simple examples, OrthogonalComplements, Orthogonal Sets.Unit -III Hilbert Spaces (Chapter 10 :Sections 55 59)

Conjugate Space H*, Adjoint of an operator, Self-adjoint operators, Normal and Unitary operators, ProjectionsUnit - IV Preliminaries on Banach Algebras (Chapter 12: Sections 64 69)

Definition and examples, Regular and singular elements, Topological divisors of zero, Spectrum, The formula

for the spectral radius, The radical and semi-simplicitys.Unit - V Struture of Commutative Banach Algebras (Chapter 13, Sections 70 73)

Gelfand Mapping, Applications of the formula r (x)=|xn|1/ n, Involution in Banach Algebras, Gelfand-NeumarkTheorem

Recommended Text Simmons G.F., Introduction to Topology and Modern Analysis, McGraw Hill BookCompany, Singapore, 1963.

R

eferenceBooks 1. Kreyszig E, Introductory Functional Analysis with Applications, John Wiley & Sons.

2. Ponnusamy S., Foundations of Functional Analysis, Narosa Publishing House.3. Rudin W., 2 e, Functional Analysis, Tata McGraw Hill Book Company, New Delhi,

1991.4. B.V.Limaye, Functional Analysis, 2- e, New Age International Ltd, Publishers 1996.5. Chandrasekara RaoK., Functional Analysis, Narosa Publishing House, 2006.6. Somasundaram D., A First Course in Functional Analysis, Narosa Publishing House, 2006.

Websites and

e-learning

Source

http://en.wikipedia.org


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IV 18 M 1034 Finite Element Method 5 Elective

Instructional Hours Lecture 6 Lab 0 Total 6Pre requisite Knowledge of Differential equationsObjectives of the Course The aim of this course is to give students a basic expertise in the use of FEM

Course OutlineUnit-I Introduction to Finite Element Method (Chapter 1: 1.3 1.6 )

General applicability of the method, Engineering Applications of the Finite Element Method, General

Description of the Finite Element Method, Comparison of Finite Element Method with other methods of

Analysis.

Unit-II Solution of finite Element Equations (Chapter 2 : 2.1 2.3 )

Introduction, Solution of equilibrium Problems, Solution of eigen value problems.

Unit-III General procedure of Finite Element Method (Chapter 3: 3.1, 3.2)

Discretization of the domains, Interpolation Polynomials

Unit -IV General procedure of Finite Element Method (contd)(Chapter 3: 3.3, 3.4)

Formulation of element characteristic matrices and vectors, Assembly of element matrices and vectors and

derivation of system equations.

Unit-V Applications (Chapter 4, 7, 8, 11; 4.0, 7.0, 8.0, 11.0)

Solutions of equations axially loaded (1 D ), Solutions for plane steady flow problems, Solutions of the

conduction equations over a rectangular area, Solutions of a forced vibration of a beam.

Recommended Text For Units I IV

Rao S.S, The Finite element Method in Engineering, 2-e, Pergamon Press, 1989.

For Unit V

Smith I.M & Griffith D.V, Programming the Finite Element Method, John Wiley

and Sons, West Sussex, UK, 1998.

Reference

Books

1. O.C.Zienkiewicz, R.L.Taylor, The Finite Element Method, Vol.I & II, Butterworth, Heinemann,Oxford, 2000.

2. George R.Buchanan, Finite Element Analysis, Schaums Outlines , McGraw Hill InternationalEditions,

Websites and

e-learning

Source

http://documents.wolfram.com/applications/structuralhttp://en.wikipedia.org/wiki/finite_elements_analysis


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II 10 M 834 (B1) Difference Equations 4 Elective

Instructional Hours Lecture 4 Lab 0 Total 4Pre requisite UG level Modern Algebra and CalculusObjectives of the Course To introduce the process of discretization, discrete version of Differential

Equations, Discrete oscillation and the asymptotic behavior of solutions ofcertain class of difference equations for linear cases only. Solution of differenceequations using Z-transforms is stressed

Course Outline

Unit I Linear Difference Equation of Higher Order (Chapter 2: 2.1-2.4)Difference Calculus - General Theory of linear Difference Equations - Linear Homogeneous Equationswith constant coefficients - Linear non-homogeneous equations - Method of Undetermined coefficients,the method of variation of constants

Unit II System of Difference Equations (Chapter 3: 3.1-3.3)Autonomous System - The basic theory - The Jordan form

Unit III The Z- Transform Method (Chapter 5: 5.1, 5.2)

Definition, Examples and properties of Z-Transform - The inverse Z- Transform and solution of -Difference Equations - Power series method, partial fraction method, the inverse integral method.

Unit IV Asymptotic Behaviour of Difference Equation (Chapter 8: 8.1 - 8.3)Tools and Approximations - Poincares Theorem - Second order Difference Equations

Unit V Oscillation Theory (Chapter 7: 7.1-7.3)Three term difference equations - Nonlinear Difference Equations - Self adjoint second order equations

Recommended Text Saber N.Elaydi, An Introduction to Difference Equations, Springer Verlag, 1996.

ReferenceBooks 1. Peterson, Difference Equations an introduction with applications, Academic Press, 1991.

2. V.LakshmiKantham and Trigiante, Theory of Difference Equations, Academic Press, 19883. S.Goldberg, Introduction to Difference Equations, Dover Publications, 1986.4. Charles Jordan, Calculus of Finite Differences, 2 e , Chelsea Publishing Company, New

York N.Y, 1950.

4. R.P.Agarwal, Difference equations and Inequalities,Marcel Dekker, 1999.

Websites and

e-learning

Source

http://en.wikipedia.org/wiki/Difference_equations


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II 10 M 835 (B2) Bio Mathematics 4 Elective

Instructional Hours Lecture 4 Lab 0 Total 4Pre requisite Basic knowledge in Differential and Difference equations

Objectives of the Course To train students in applying Differential and Difference equations in Biologicalstudies.Course Outline

Unit - I Discrete Population Growth Models (Chapter 2: 2.2 2.5)Arithmetic Growth Model - Geometric Growth Model Generalizations - Age Structured Populations

Unit - II Continuous Growth Models (Chapter 3: 3.2 3.5)The Linear Model - The Exponential Model - Model for the Distribution of drugs in the body - CoalitionModels

Unit III Continuous Growth Models (contd) (Chapter 3: 3.8 3.11 )Environmental Resistance - A Model for the Spread of Technological Innovations - The Gomertz Model -Bertalanffy Growth Model

Unit - IV Qualitative behaviour of Populations (Chapter 5: 5.2 5.7 )Autonomous Equations - Steady and Equilibrium States - Stability of Equilibrium States - Logistic Model

with Harvesting - Fixed Points and their stability - The Logistic MapUnit - V Mathematical Models in Epidemiology (Chapter 7: 7.2 7.5)Plant Epidemics - Some features of Human Epidemics - A Simple Deterministic Epidemic Model - AMore General Epidemic: SIR Disease

Recommended Text C.R.Ranganathan , A First Course in Mathematical Models of Population

Growth (With MATLAB Programs), Associated Publishing Company, New Delhi,2006.

Reference

Books

1. Pundir, Pundir, Bio Mathematics, A Pragati Edition, 2006.2. J.N.Kapur, Mathematical Models in Biology and Medicine, Affiliated East West Press

PVT LTD, New Delhi, 1985.3. Nicholas F.Britton, Essential Mathematical Biology, Springer International Edition, First

Indian Reprint, 2004.

4. Murray, Mathematical Biology, Springer International Edition, First Indian Reprint, 2004.Websites and

e-learning

Source

http://en.wikipedia.org/wiki/bio_mathematics


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III 15 M 937 (C1) Fluid Dynamics 4 ElectiveInstructional Hours Lecture 6 Lab 0 Total 6Pre requisite Differential Equations, Vector Calculus, Complex Analysis at UG level

Objectives of the Course This course aims to Kinematics of fluids in motion, equations of motion of afluid, three dimensional flows, two dimensional flows and viscous flows

Course Outline

Unit I Kinematics of Fluids in Motion (Chapter 2: 2.1 - 2.8)Real Fluids and Ideal fluids, Velocity of a Fluid at a point, Stream lines and path lines , steady and unsteady

flows, The Velocity potential , The Vorticity vector, Local and particle rates of change, The equations ofcontinuity, Worked ExamplesUnit II Equation of Motion of a Fluid (Chapter 3: 3.1 3.6)

Pressure at a point in a Fluid at rest, Pressure at a point in a moving Fluid, Conditions at a boundary of twoinviscid immiscible Fluids, Eulers Equation of Motion, Benoullis equation, Worked examplesUnit III Some Three Dimensional Flows(Chapter 4: 4.1,4.2, 4.5)

Introduction, Sources, sinks and doublets, Axis Symmetric flows, Stokes Stream function

Unit IV Some two Dimensional Flows (Chapter 5:5.1 5.6 )Meaning of two Dimensional Flows, Use of Cylindrical polar coordinates, The Stream function, The complex

potential for two dimensional, irrotational, incompressible flows, Complex velocity potentials for standard twodimensional flows.

Unit V Viscous Flows ( Chapter 8:8.1-8.7,8.9)Stress components in real fluids, Relation between Cartesian components of stress, Translation motion of a

fluid element, The rate of strain quadric and principle stresses, Some Further properties of the rate of strainquadric Stress analysis in fluid motion, Relation between stress and rate of strain, The Coefficient of Viscosityand Laminar flow, The Navier Strokes equations of motion of a viscous fluid

Recommended Text F.Chorlton, Text Book of Fluid Dynamics,CBS Publication , New Delhi,1985.

ReferenceB

ooks

1. G.K.Batchaelor,An Introduction of Fluid Mechanics , Foundation Books , New Delhi,19932. S.W.Yuan, Foundation of Fluid Mechanics, Prentice Hall Private Ltd, New Delhi, 1976.3.

R.K.Rathy, An Introduction to Fluid Dynamics, IBH Publishing Company, New Delhi1976.

4. A.R.Paterson, A First Course in Fluid Dynamics, Cambridge University Press, New York,1987.

5. R.Von Mises,K,O.Friedrichs,Fluid Dynamics, Springer International Student Edition,Narosa Publishing House, New Delhi.

Websites and

e-learning

Source

http://en.wikipedia.org/wiki/Fluid_dynamics


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III 15 M 938 (C2) Formal Languages and Automata 4 Elective

Instructional Hours Lecture 6 Lab 0 Total 6Pre requisite UG level Algebra and CalculusObjectives of the Course This course aims to introduce finite automata, regular expressions and regular

grammars, properties of regular sets, context-free grammar, push downautomata and properties of context free languages

Course Outline

Unit I Finite Automata and Regular Expressions ( Chapter 2:2.1 2.5)

Finite state systems, Basic definitions , Non deterministic finite automata,Finite automata with e moves , Regular expressionsUnit II Properties of Regular Sets (Chapter 3:3.1 3.4 )

The pumping lemma for regular sets, Closure properties of regular sets, Decision algorithms for regular

sets, The Myhill-Nerode theorem and minimization of finite automata

Unit III Context Free Grammars (Chapter 4: 4.1 4.6 )

Motivation and introduction, Context free grammars, Derivation trees, Simplification of Context freegrammars, Chmosky normal form, Greibach normal formUnit IV Pushdown Automata (Chapter 5:5.1 5.3 )

Informal descripition, Definitions- Pushdown automata and context-free languagesUnit V Properties of Context free Languages (Chapter 6:6.1 6.3 )

The pumping lemma for CFLs, Closure properties for CFLs , Decision algorithm of CFLs

Recommended Text John E.Hopcraft and Jeffrey D.Ullman, Introduction to Automata Theory,Languages and Computation , Narosa Publishing House, New Delhi,1987.

Refe

renceBooks

1. A.Salomaa,Formal Languages , Academic Press, New York,1973.2. John C.Martin, Introduction to Languages and Theory of Computation 2-e, Tata McGraw

Hill Company Ltd, NewDelhi, 1997.3. Ranisionmoney, Formal Languages and Automata, The Christian Literature Society, 1984.4. Abib.M.A, Theories of Abstract Automata, Prentice Hall Ltd, Engle wood Cliffs, 1970.

5. Ginsburg.S, Algebraic and Automata, the arithmetic Properties of Formal Languages, NorthHolland, Amsterdam, 1975.

Websites and

e-learning

Source

http://en.wikipedia.org/wiki/formal_language_automata


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List of Practical with MATLAB

1. Solving a linear system of equations

2. Finding Eigen values and eigen vectors

3. Linear Curve fitting

4. Least square curve fitting

5. Interpolation formula

6. Numerical evaluation of the integral

7. Evaluation of double integral

8. Solving an ODE using ode 23 and ode 45

9. Solving a Delay Differential equation

10. Plotting 2 D curves

11. 3-D plots and plotting of surfaces

12. Solving the Riccati difference equation associated with the Kalman filter

13. Bifurcation analysis for the logistic difference equation

14. Solving discrete logistic equation and cobweb analysis

15. Finding geodesics on a right cylinder with a circular cross section, a right cone witha circular base and a sphere.

16. Finding Mean curvature and Gaussian curvature of a surface


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IV 19 M 1037 (D3) Financial Mathematics 4 Elective

Instructional Hours Lecture 6 Lab 0 Total 6Pre requisite Basic Mathematics at UG level.Objectives of the Course To introduce topics in Financial mathematics - single period models, binomial

trees and discrete parameter martingales, Brownian Motion, StochasticCalculus, Black Scholes model.

Course Outline

UNIT-I : Single Period Models (Chapter 1)Definitions from Finance , Pricing a forward , One-step Binary Model, a ternary Model , Characterization of no

arbitrage , Risk-Neutral probability measure.UNIT-II : Binomial Trees and Discrete parameter martingales (Chapter 2)

Multi-period binary model , American options , Discrete parameter Martingales and Markov processes ,Martingale theorems , Binomial representation theorem , Overture to continuous models.UNIT-III : Brownian Motion (Chapter 3)

Definition of the process , Levys construction of Brownian motion , The reflection principle and scaling ,Martingales in continuous time.

UNIT-IV : Stochastic Calculus (Chapter 4)Stock prices are not differentiable , Stochastic integration , Itos formula , Integration by parts and Stochastic

Fubini theorem , Girsanov Theorem., Brownian Martingale representation theorem , Geometric Brownianmotion, The Feynman - Kac representation.UNIT-V : Block-Scholes Model (Chapter 5)

Basic Block-Scholes model , Block-Scholes price and hedge for European options , Foreign Exchange Dividends, Bonds, Market price of risk.

Recommended Text Alison Etheridge, A Course in Financial Calculus, Cambridge University Press,Cambridge, 2002.

ReferenceB

ooks

1.Martin Boxter and Andrew Rennie, Financial Calculus: An Introduction to Derivatives Pricing,Cambridge University Press, Cambridge, 1996.

2. Damien Lamberton and Bernard Lapeyre, (Translated by Nicolas Rabeau and FrancoisMantion ), Introduction to Stochastic Calculus Applied to Finance, Chapman and Hall, 1996.3. Marek Musiela and Marek Rutkowski, Martingale Methods in Financial Modeling,Springer

Verlag, New York, 1988.4. Robert J.Elliott and P.Ekkehard Kopp, Mathematics of Financial Markets, Springer Verlag, New

York, 2001 (3rdPrinting).Websites and

e-learning

Source

http://mathforum.org, http://ocw.mit.edu/ocwweb/Mathematics,http://www.opensource.org, http://en.wikiepedia.org


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I CM102 ADVANCED BUSINESS STATISTICSFor M.Com

4 Core

Instructional Hours Lecture 6 Lab 0 Total 6Pre requisite Basic MathematicsObjectives of the Course To apply statistical techniques for interpreting and drawing conclusion for

business problems.

Course Outline

UNIT I Correlation (18 Hrs)

Partial correlation - Partial correlation coefficient - Partial correlation in case of four variables Multiple

correlation Multiple regression.

UNIT II Theory of probability (18 Hrs)

Theory of probability probability rules Bayes theorem probability distribution Characteristics and

application of Binomial, Poisson and Normal Distributions.

UNIT III Sampling (18 Hrs)

Sampling Sampling methods Sampling error and standard error relationship between sample size and

standard error. Testing of hypothesis testing of means and proportions large and small samples z test and t

test.

UNIT IV Chi square Distribution (18 Hrs)

Chi square distribution Characteristics and application test of goodness of fit and test of independence test

of homogeneity.

UNIT V F- Distribution (18 Hrs)

F distribution testing equality of population variances Analysis of variance one way and two way

classifications.

Note:The proportion between theory and problems shall be 20:80

Recommended Text S.P.Gupta, Statistical Methods, Sultan Chand & Sons, New Delhi, 2000.

ReferenceBooks 1. D.C Sancheti and V.K.Kapoor, Business Statistics,2-e, Sultan Chand & Sons, New Delhi,1979.

2. Richard I Levin and David S. Rubit, Statistics for management, Seventh edition, Pearson

Education, New Delhi, 2001.

3. S.C.Gupta and V.K.Kapoor, Fundamentals of Mathematical Statistics, 11-e, Sultan Chand &

Sons, New Delhi, 2004.

Websites ande-learning Source http://mathworld.wolfram.com


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II CM202

QUANTITATIVE TECHNIQUES FOR

BUSINESS DECISIONS

For M.Com4

Core

Instructional Hours Lecture 6 Lab 0 Total 6

Pre requisite Basic MathematicsObjectives of the Course To apply OR techniques for interpreting and drawing conclusion for business

problems.

Course Outline

UNIT ILinear programming and Network Analysis (18 Hrs)

Linear programming and network analysis- PERT and CPM- Simplex Method- Application of simplex technique.

UNIT II Inventory Models (18 Hrs)

Inventory models - General concepts and definitions - Various cost concepts- the technique of inventory control -

EOQ models.

UNIT III Transportation model (18 Hrs)

Transportation model Definitions - Formulation and solution of transportation models - North West Corner- Matrix

minimum method, Vogels Approximation method, Optimality test, MODI method.

UNIT IV Assignment Model (18 Hrs)

Assignment model- Definitions - Formulation and solution of Assignment models - simplex and Hungarian method.

UNIT V Queuing theory (18 Hrs)

Queuing theory - meaning - objectives- limitations - elements of queuing system - queuing models, M/M/1:/FIFO,

M/M/1: /SIRO, M/M1:M/FIFO, Birth death process.

Note:The proportion between theory and problems shall be 20:80.

Recommended Text P.K.Gupta , Kanthi Swarup and Manmohan, Operations research, Sultan Chand &

Sons, New Delhi, 1992.

ReferenceBooks 1. P.K.Gupta and D.S.Hira, Operations Research, S.Chand & Company, New Delhi, 2000.

2. J.K.Sharma, Operations Research Theory and Applications, 2-e, Macmillian Business Books, 2003.

3. Hamdy A.Taha, Operations Research, Pearson Education, New Delhi, 2002.

4. S.K.Mittal and B.S.Goel, Operations Research, 13-e, Pragrati Pragasan, Meerut,1995.

5. P.K.Gupta, Operations Research, 8-e, Krishna Prakasan Mandir, Meerut, 1993.

Websites and

e-learning Sourcehttp://mathworld.wolfram.com


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List of Question Paper Setters and Examiners (PG)

1. Dr.E.Thandapani,Professor in Mathematics,RIAS, University of Madras,

Chepauk, Chennai-5.

2. Dr.V.Rajkumar Dare,Reader and Head,Department of Mathematics,Madras Christian College,Tambaram, Chennai.

3. Dr.L.Prathaban,Reader,Department of Mathematics,

Muthurangam Govt. ArtsCollege, Vellore.

4. Dr.Ebinezar,Principal,Govt.College of Education,Vellore.

5. Dr.S.Pandian,Reader in Mathematics,Presidency College,

Chennai-5.

6. Dr.Selvaraj,Reader in Mathematics,Presidency College,Chennai-5.

7. Dr.U.Rizwan,Reader in Mathematics,Islamiah College,Vaniyambadi,Vellore Dist.

8. Dr.P.Chandra Sekharan ,Reader in Statistics,Loyola College,Chennai-600034.

9. Dr.V.N.Saradhamani,HOD & Controller of Exam,Queen Marys College,Chennai-600005.

10.Dr.R.Arul,Reader in Mathematics,Kandasami Kandar College,Paramathy Velur,Namakkal Dist.

11.Dr.P.Ramachandran,Reader in Mathematics,Govt. Arts College for Men,Salem.

12.Dr.R.Murthy,Reader in Mathematics,Presidency College (Autonomus)Chennai-600 005.

13.Dr.S.J.Venkatesan,Reader,Dept. of Mathematics,Arignar Anna Govt. ArtsColllege, Cheyyar.

14. Dr.S.Rajalakshmi,HOD of Mathematics,Govt Arts College for Women,Chennai-6000001.

15. Dr.K.Srinivasan,Reader, Presidency College,Chennai-600005.

16. Dr.N.Rahima,Reader in Mathematics,Govt Arts College for Women,Wallaja, Vellore.

17. Dr.Allah Pitchai,Reader in Mathematics,Abdul Hakeem College

Documents

M Scsyllabus CBCS