st. peter’s institute of higher education and research · mean value theorems for riemann -...

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St. Peter’s Institute of Higher Education and Research

(Declared under section 3 of UGC Act 1956)

Avadi, Chennai – 600 054.

M.Sc(MASTER OF SCIENCE)

MATHEMATICS DEGREE PROGRAMME

(I to IV SEMESTERS)

REGULATIONS AND SYLLABI

(CHOICE BASED CREDIT SYSTEM - CBCS)

REGULATIONS – 2016

(Effective from the Academic Year 2016-17)

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M.Sc PROGRAMME OUTCOMES (POs)

PO1: Apply the knowledge of Mathematical concepts in Inter

disciplinary fields.

PO2: Understand the nature of Abstract Mathematics and explore

the concepts in further details.

PO3: Model the real-world problems in to Mathematical Equations

and draw the inferences by finding appropriate solutions.

PO4: Identify challenging problems in Mathematics and find

appropriate solutions.

PO5: Enhances Logical reasoning skills, Arithmetic skills,

Aptitude skills, Self confidence for better employability and

Banking exams.

PO6: Inculcate the curiosity for mathematics in students and to

prepare them for Future Research.

M.SC PROGRAMME SPECIFIC OUTCOMES

PSO1: After completion of the PG course will gain a thorough

Knowledge in preparing themselves for competitive

Examinations.

PSO2: Pursue research in challenging areas of Pure/Applied

Mathematics.

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M.Sc. (MATHEMATICS) DEGREE PROGRAMME

Regulations – 2016

Choice Based Credit System

(Effective from the Academic Year 2016-2017)

1. Eligibility:

Candidates who passed B.Sc. Mathematics in the University or an Examination accepted by the University as

equivalent thereof are eligible for admission to M.Sc. Degree Programme in Mathematics.

2. Duration:

Two years comprising 4 Semesters. Each semester has a minimum of 90 working days with a minimum of 5

hours a day.

3. Medium:

English is the medium of instruction and examinations.

4. Eligibility for the Award of Degree:

A candidate shall be eligible for the award of degree only if he/she has undergone the prescribed course of

study in the University for a period of not less than two academic years (4 semesters), passed the

examinations of all the four semesters prescribed carrying 90 credits and also fulfilled the such candidates as

have been prescribed thereof.

5. Choice Based Credit System:

Choice Based Credit System is followed with one credit equivalent to one hour for theory paper and two

hours for a practical work per week in a cycle of 18 weeks (that is, one credit is equal to 18 hours for each

theory paper and one credit is equal to 36 hours for a practical work in a semester in the Time Table. The total

credit for the M.Sc. Degree Programme in Mathematics (4 semesters) is 90 credits.

6. Weightage for a Continuous and End Assessment:

The weightage for Continuous Assessment (CA) and End Assessment (EA) is 25:75 unless the ratio is

specifically mentioned in the Scheme of Examinations. The question paper is set for a minimum of 100

marks.

7. Course of Study and Scheme of Examinations:

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I SEMESTER

Code No. Course Title Credit Marks

Theory CA EA Total

116PMMT01

Core Sub:

Algebra-I 5 25 75 100

116PMMT02 Real Analysis-I 5 25 75 100

116PMMT03 Ordinary Differential Equations 4 25 75 100

116PMMT04 Graph Theory 4 25 75 100

116PMMT06 Elective – I: (Choose one from Group-A) 4 25 75 100

Total 22 125 350 500

II SEMESTER

Code No. Course Title Credit Marks

Theory CA EA Total

216PMMT01

Core Sub:

Algebra-II 5 25 75 100

216PMMT02 Real Analysis- II 5 25 75 100

216PMMT03 Partial Differential Equations 4 25 75 100

216PMMT04 Probability Theory 4 25 75 100

216PMMT06 Elective – II: (Choose one from Group B)- 4 25 75 100

Total 22 125 350 500

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III SEMESTER

Code No.

Course Title

Credit

Marks

CA EA Total

316PMMT01

Core Sub:

Complex Analysis-I 5 25 75 100

316PMMT02 Topology 5 25 75 100

316PMMT03 Operations Research 5 25 75 100

316PMMT04 Mechanics 5 25 75 100

316PMMT06 Elective –III: (Choose one from Group C) 4 25 75 100

Total 24 125 350 500

IV SEMESTER

Code No.

Course Title

Credit

Marks

CA EA Total

416PMMT01

Core

Sub:

Complex Analysis-II 5 25 75 100

416PMMT02 Differential Geometry 5 25 75 100

416PMMT03 Functional Analysis 4 25 75 100

416PMMT04 Elective – IV: (Choose one from Group D) 4 25 75 100

416PMMT08 Elective – V: (Choose one from Group E) 4 25 75 100

Total 22 125 350 500

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LIST OF ELECTIVES

Elective –I (Group-A) Credits

116PMMT05 Formal Languages and Automata Theory 4

116PMMT06 Discrete Mathematics 4

116PMMT07 Mathematical Economics 4

116PMMT08 Fuzzy sets and Applications. 4

Elective - II (Group B)

216PMMT05 Programming in C ++ and Numerical Methods 4

216PMMT06 Mathematical Programming 4

216PMMT07 Wavelets 4

216PMMT08 Java Programming 4

Elective - III (Group C)

316PMMT05 Algebraic Theory of Numbers 4

316PMMT06 Number Theory and Cryptography 4

316PMMT07 Stochastic Processes 4

316PMMT08 Data Structures and Algorithms 4

Elective - IV (Group D)

416PMMT04 Fluid Dynamics 4

416PMMT05 Combinatorics 4

416PMMT06 Mathematical Statistics 4

416PMMT07 Algebraic Topology 4

Elective - V (Group E)

416PMMT08 Tensor Analysis and Relativity 4

416PMMT09 Mathematical Physics 4

416PMMT10 Financial Mathematics 4

416PMMT11 Calculus of Variations and Integral Equations 4

8. Passing Requirements: The minimum pass mark (raw score) be 50% in End Assessment (EA) and 50% in

Continuous Assessment (CA) and End Assessment (EA) put together. No minimum mark (raw score) in

Continuous Assessment (CA) is prescribed unless it is specifically mentioned in the Scheme of Examinations.

9. Grading System: Grading System on a 10 Point Scale is followed with 1 mark = 0.1 Grade point to

successful candidates as given below.

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CONVERSION TABLE

(1 mark = 0.1 Grade Point on a 10 Point Scale)

Range of Marks Grade Point Letter Grade Classification

90 to 100 9.0 to 10.0 O First Class

80 to 89 8.0 to 8.9 A First Class

70 to 79 7.0 to 7.9 B First Class

60 to 69 6.0 to 6.9 C First Class

50 to 59 5.0 to 5.9 D Second Class

0 to 49 0 to 4.9 F Reappearance

Procedure for Calculation

Cumulative Grade Point Average (CGPA) = Sum of Weighted Grade Points

Total Credits

= ∑ (CA+EA) C

∑C

Where Weighted Grade Points in each Course = Grade Points (CA+EA)

multiplied by Credits

= (CA+EA)C

Weighted Cumulative Percentage of Marks(WCPM) = CGPAx10

C- Credit, CA-Continuous Assessment, EA- End Assessment

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10. Effective Period of Operation for the Arrear Candidates: Two Year grace period is provided for the

candidates to complete the arrear examination, if any.

11. National Academic Depository (NAD): All the academic awards (Grade Sheets, Consolidated Grade sheet, Provisional

Certificate, Degree Certificate (Diploma) and Transfer Certificate are lodged in a digital format in National Academic Depository

organized by Ministry by Ministry of Human Resource Development (MHRD) and University Grants Commission (UGC). NAD is

a 24x7 online mode for making available academic awards and helps in validating its authenticity, safe storage and easy retrieval by

the student, Employing Agencies and Educational institutions.

12. Knowledge levels:

For every course outcome the knowledge level is measure as,

K1 – Remember, K2 – Understand, K3 – Apply, K4 – Analyze, K5 – Evaluate, K6 – Create.

Registrar

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M.Sc. DEGREE COURSE IN MATHEMATICS

SYLLABUS

I Semester

116PMMT01 - ALGEBRA – I

OBJECTIVES: This course is aimed to

(i) Introduce the concepts of Sylow’s theorem.

(ii) Develop the knowledge on Solvable groups and the concepts of inner product space are

also introduced.

(iii) Reduce the matrices using different methods.

UNIT I:

Counting principle - class equation for finite groups and its applications - Sylow's theorems (For theorem 2.12.1,

First proof only).

Chapter 2: Sections 2.11 and 2.12 (Omit Lemma 2.12.5)

UNIT II

Solvable groups - Direct products - Finite abelian groups- Modules

Chapter 5 : Section 5.7 (Lemma 5.7.1, Lemma 5.7.2, Theorem 5.7.1)

Chapter 2: Sections 2.13 and 2.14 (Theorem 2.14.1 only)

Chapter 4: Section 4.5

UNIT III

Linear Transformations - Canonical forms - Triangular form – Nilpotent transformations.

Chapter 6: Sections- 6.4, 6.5

UNIT IV

Jordan form - rational canonical form.

Chapter 6 : Sections 6.6 and 6.7

UNIT V

Trace and transpose - Hermitian, unitary, normal transformations, real quadratic form.

Chapter 6 : Sections 6.8, 6.10 and 6.11 (Omit 6.9)

Recommended Text : 1. I.N. Herstein. Topics in Algebra (II Edition) Wiley, 2002.

Reference Books :

1. M.Artin, Algebra, Prentice Hall of India, 1991.

2. P.B.Bhattacharya, S.K.Jain, and S.R.Nagpaul, Basic Abstract Algebra

(II Edition) Cambridge University Press, 1997. (Indian Edition)

3. I.S.Luther and I.B.S.Passi, Algebra, Vol. I - Groups(1996); Vol. II Rings(1999), Narosa Publishing House

, New Delhi

4. D.S.Dummit and R.M.Foote, Abstract Algebra, 2nd edition, Wiley, 2002.

5. N.Jacobson, Basic Algebra, Vol. I & II W.H.Freeman (1980); also published by Hindustan Publishing

Company, New Delhi.

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Course Outcomes

On the successful completion of the course, students will be able to

CO Number

CO

Statements

Knowledge Level

CO1 Understand the basic concepts of Counting principle, finite groups, applications and Sylow's theorems

K1, K2 ,K3

CO2 Get the knowledge of group theory, modules and uses in various theorems.

K1, K2,K3,K5

CO3 Get the knowledge of Canonical forms and Triangular form K1, K2, K4, K5

C04 Apply the knowledge of Jordan form - rational canonical form K1,K2,K3,K4

CO5 Develop trace and transpose of matrix, various types of transformation

(Hermitian,Unitary,Normal) and to learn Real Quadratic form. K1,K2,K4

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116PMMT02 - REAL ANALYSIS - I

OBJECTIVES : The Aim of the course is to

(i) Work comfortably with functions of bounded variation, Riemann - Stieltjes Integration,

convergence of infinite series, infinite product and uniform convergence and its interplay between various

limiting operations.

UNIT-I : Functions of bounded variation

Introduction - Properties of monotonic functions - Functions of bounded variation - Total variation - Additive

property of total variation - Total variation on [a, x] as a function of x - Functions of bounded variation expressed

as the difference of two increasing functions - Continuous functions of bounded variation.

Chapter – 6 : Sections 6.1 to 6.8

Infinite Series

Absolute and conditional convergence - Dirichlet's test and Abel's test - Rearrangement of series - Riemann's

theorem on conditionally convergent series.

Chapter 8 : Sections 8.8, 8.15, 8.17, 8.18

UNIT-II : The Riemann - Stieltjes Integral - Introduction - Notation - The definition of the Riemann - Stieltjes

integral - Linear Properties - Integration by parts- Change of variable in a Riemann - Stieltjes integral -

Reduction to a Riemann Integral – Euler’s summation formula - Monotonically increasing integrators, Upper and

lower integrals - Additive and linearity properties of upper and lower integrals - Riemann's condition -

Comparison theorems.

Chapter - 7 : Sections 7.1 to 7.14

UNIT-III : The Riemann-Stieltjes Integral - Integrators of bounded variation-Sufficient conditions for the

existence of Riemann-Stieltjes integrals-Necessary conditions for the existence of Riemann-Stieltjes integrals-

Mean value theorems for Riemann - Stieltjes integrals - The integrals as a function of the interval - Second

fundamental theorem of integral calculus-Change of variable in a Riemann integral-Second Mean Value Theorem

for Riemann integral-Riemann-Stieltjes integrals depending on a parameter-Differentiation under the integral

sign-Lebesgue criteriaon for the existence of Riemann integrals.

Chapter - 7 : 7.15 to 7.26

UNIT-IV : Infinite Series and infinite Products - Double sequences - Double series - Rearrangement theorem

for double series - A sufficient condition for equality of iterated series - Multiplication of series - Cesaro

summability - Infinite products.

Chapter - 8 Sec, 8.20, 8.21 to 8.26

Power series - Multiplication of power series - The Taylor's series generated by a function - Bernstein's theorem

- Abel's limit theorem - Tauber's theorem

Chapter 9 : Sections 9.14 9.15, 9.19, 9.20, 9.22, 9.23

UNIT-V: Sequences of Functions - Point wise convergence of sequences of functions - Examples of sequences

of real - valued functions - Definition of uniform convergence - Uniform convergence and continuity - The

Cauchy condition for uniform convergence - Uniform convergence of infinite series of functions - Uniform

convergence and Riemann - Stieltjes integration – Non-uniform Convergence and Term-by-term Integration -

Uniform convergence and differentiation - Sufficient condition for uniform convergence of a series - Mean

convergence.

Chapter -9 Sec 9.1 to 9.6, 9.8,9.9, 9.10,9.11, 9.13

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Recommended Text:

Tom M.Apostol : Mathematical Analysis, 2nd

Edition, Narosa,1989.

Reference Books

1. Bartle, R.G. Real Analysis, John Wiley and Sons Inc., 1976.

2. Rudin,W. Principles of Mathematical Analysis, 3rd

Edition. McGraw Hill Company, New York, 1976.

3. Malik,S.C. and Savita Arora. Mathematical Anslysis, Wiley Eastern Limited.New Delhi, 1991.

4. Sanjay Arora and Bansi Lal, Introduction to Real Analysis, Satya Prakashan, New Delhi, 1991.

5. Gelbaum, B.R. and J. Olmsted, Counter Examples in Analysis, Holden day, San Francisco, 1964.

6. A.L.Gupta and N.R.Gupta, Principles of Real Analysis, Pearson Education, (Indian print) 2003.

Course Outcomes


CO Number

CO

Statement

Knowledge

Level CO1 Understand the basics concept of monotonic functions, continuous

functions , properties , Dirichlet's test, Abel's test, and Riemann's theorem

K1, K2,K3,K4,K5

CO2 Learn about Riemann Integral – Euler’s summation formula and properties of upper and lower integrals

K1, K2,K3,

K4

CO3 Get the knowledge of Stieltjes integrals, Second Mean Value Theorem and Sufficient condition for uniform convergence and Mean convergence

K1, K2,K3

C04 Learn about the infinite Series, infinite Products and Power series K1,K2,K3

CO5 Apply the knowledge of Sequences of Functions K1,K2,K3,K5

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116PMMT03 - ORDINARY DIFFERENTIAL EQUATIONS

OBJECTIVES: To develop strong background on finding solutions to linear differential equations with constant

and variable coefficients and also with singular points, to study existence and uniqueness of the solutions of first

order differential equations.

UNIT-I Linear equations with constant coefficients

Second order homogeneous equations-Initial value problems-Linear dependence and independence- Wronskian

and a formula for Wronskian-Non-homogeneous equation of order two.

Chapter 2: Sections 1 to 6.

UNIT-II Linear equations with constant coefficients

Homogeneous and non-homogeneous equation of order n –Initial value problems- Annihilator method to solve

non-homogeneous equation.

Chapter 2 : Sections 7 to 11.

UNIT-III Linear equation with variable coefficients

Initial value problems -Existence and uniqueness theorems – Solutions to solve a non- homogeneous equation –

Wronskian and linear dependence – Reduction of the order of a homogeneous equation – Homogeneous equation

with analytic coefficients-The Legendre equation.

Chapter : 3 Sections 1 to 8 (omit section 9).

UNIT-IV Linear equation with regular singular points:

Second order equations with regular singular points –Exceptional cases – Bessel equation .

Chapter 4 : Sections 3, 4 and 6 to 8 (omit sections 5 and 9)

UNIT-V Existence and uniqueness of solutions to first order equations: Equation with variable separated – Exact equation – Method of successive approximations – the Lipschitz

condition – Convergence of the successive approximations and the existence theorem.

Chapter 5 : Sections 1 to 6 ( omit Sections 7 to 9)

Recommended Text

1. E.A.Coddington, An introduction to ordinary differential equations (3rd

Printing) Prentice-Hall of India

Ltd.,New Delhi, 1987.

Reference Books

1. Williams E. Boyce and Richard C. Di Prima, Elementary differential equations and

boundary value problems,John Wiley and sons, New York, 1967.

2. George F Simmons, Differential equations with applications and historical notes,

Tata McGraw Hill, New Delhi, 1974.

3. N.N. Lebedev, Special functions and their applications, Prentice Hall of India, New Delhi, 1965.

4. W.T.Reid. Ordinary Differential Equations, John Wiley and Sons, New York, 1971.

5. M.D.Raisinghania, Advanced Differential Equations, S.Chand & Company Ltd. New Delhi 2001.

6. B.Rai, D.P.Choudhury and H.I. Freedman, A Course in Ordinary Differential

Equations, Narosa Publishing House, New Delhi, 2002.

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Course Outcomes


CO Number

CO STATEMENTS Knowledge Level

CO1 Understand the definition of Linear equations with constant coefficients and the use of Wronskian.

K1,K2,K3

CO2 Classify Ordinary Differential Equations with homogenous and non

homogenous equations. K1,K2,K3

CO3 To have knowledge about Initial value problems ,Existence and uniqueness theorems.

K1,K2,K4

CO4 To learn about the Linear equation with regular singular points. K1,K2,K3,K4

CO5 Get the knowledge of Lipschitz condition Convergence of the successive approximations and the existence theorem.

K1,K2,K3,K4

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116PMMT04 - GRAPH THEORY

OBJECTIVES: To study and develop the concepts of graphs, sub graphs, trees, connectivity, Euler tours,

Hamilton cycles, matching, coloring of graphs, independent sets, cliques, vertex coloring, and planar graphs

UNIT-I : Graphs, sub graphs and Trees :

Graphs and simple graphs – Graph Isomorphism – The Incidence and Adjacency Matrices – Sub graphs – Vertex

Degrees – Paths and Connection – Cycles – Trees – Cut Edges and Bonds – Cut Vertices.

Chapter 1 (Section 1.1 – 1.7)


UNIT-II : Connectivity, Euler tours and Hamilton Cycles :

Connectivity – Blocks – Euler tours – Hamilton Cycles.



UNIT-III: Matching’s, Edge Colorings:

Matchings – Matching’s and Coverings in Bipartite Graphs – Edge Chromatic Number – Vizing’s Theorem.



UNIT-IV: Independent sets and Cliques, Vertex Colorings: Independent sets – Ramsey’s Theorem – Chromatic

Number – Brooks’ Theorem – Chromatic Polynomials.


Chapter 8 (Section 8.1 – 8.2, 8.4)

UNIT-V: Planar graphs:

Plane and planar Graphs – Dual graphs – Euler’s Formula – The Five- Colour Theorem and the Four-Color

Conjecture.

Chapter 9 (Section 9.1 – 9.3, 9.6)

Recommended Book:

1. J.A.Bondy and U.S.R. Murthy, Graph Theory and Applications, Macmillan, London, 1976.

Reference Book

1. J.Clark and D.A.Holton , A First look at Graph Theory, Allied Publishers, New Delhi , 1995.

2. R. Gould. Graph Theory, Benjamin/Cummings, Menlo Park, 1989.

3. A.Gibbons, Algorithmic Graph Theory, Cambridge University Press, Cambridge, 1989.

4. R.J.Wilson and J.J.Watkins, Graphs : An Introductory Approach, John Wiley and Sons, New York, 1989.

5. R.J. Wilson, Introduction to Graph Theory, Pearson Education, 4th

Edition, 2004, Indian Print.

6. 6. S.A.Choudum, A First Course in Graph Theory, MacMillan India Ltd. 1987.

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Course Outcomes


CO Number

CO STATEMENTS Knowledge Level

CO1 Understand and recollect the basic concepts of graphs, sub-graphs, paths ,

cycles and trees. K1,K2,K3,K4

CO2 To learn about the Connectivity, Blocks and Hamilton Cycles. K1,K2,K3,K4

CO3 Apply the basics concepts of Matching, Edge Colorings. K1,K2,K3

CO4 Get the knowledge of Independent sets, Cliques and Vertex colorings and also

the applications of Ramsey’s Theorem and Brooks’ Theorem.

K1,K2,K3,K4

CO5 Analyze the concepts of plane graphs, planar graphs and dual graphs. K1,K2,K3

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ELECTIVE - V

116PMMT06 - DISCRETE MATHEMATICS

OBJECTIVES: This course aims to explore the topics like lattices and their applications in switching circuits, finite fields, polynomials and coding theory.

UNIT-I: Lattices:

Properties of Lattices: Lattice definitions – Modular and distributive lattice; Boolean algebras: Basic properties

– Boolean polynomials, Ideals; Minimal forms of Boolean polynomials.

Chapter 1: § 1 A and B § 2A and B. § 3.

UNIT-II: Applications of Lattices:

Switching Circuits: Basic Definitions - Applications

Chapter 2: § 1 A and B

UNIT-III: Finite Fields

Chapter 3: § 2

UNIT-IV: Polynomials:

Irreducible Polynomials over Finite fields – Factorization of Polynomials

Chapter 3: § 3 and §4.

UNIT-V: Coding Theory:

Linear Codes and Cyclic Codes

Chapter 4 § 1 and 2

Recommended Text:

1. Rudolf Lidl and Gunter Pilz, Applied Abstract Algebra, Spinger-Verlag, New York, 1984.

Reference Books:

1. A.Gill, Applied Algebra for Computer Science, Prentice Hall Inc., New Jersey.

2. J.L.Gersting, Mathematical Structures for Computer Science(3rd

Edn.), Computer Science Press, New

York.

3. S.Wiitala, Discrete Mathematics- A Unified Approach, McGraw Hill Book Co.

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Course Outcomes


CO Number

CO - STATEMENTS Knowledge Level

CO1 Understand the concepts of lattice and Boolean algebra with their applications.

K1,K2,K3,K4

CO2 Understand the basic concepts of switching circuits. K1,K2

CO3 learn about finite field problems K1,K2,K3

CO4 Identify the concept of Polynomials over Finite fields K1,K2,K3,K4,K5

CO5 Analyze and develop linear codes and cyclic codes K1,K2,K3

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II Semester

216PMMT01 - ALGEBRA-II

OBJECTIVES: To study field extension, roots of polynomials, Galois Theory, finite fields, division rings,

solvability by radicals and to develop computational skill in abstract algebra.

UNIT I Extension fields - Transcendence of e.

Chapter 5: Section 5.1 and 5.2

UNIT II Roots or Polynomials.- More about roots

Chapter 5: Sections 5.3 and 5.5

UNIT III Elements of Galois Theory.

Chapter 5: Section 5.6

UNIT IV Finite fields - Wedderburn’s theorem on finite division rings

Chapter 7: Sections 7.1 and 7.2 (Theorem 7.2.1 only)

UNIT V Solvability by radicals – Galois groups over the rationals – A theorem of Frobenius.


Chapter 7: Sections 7.3

Recommended Text: 1. N. Herstein. Topics in Algebra (II Edition) Wiley 2002

Reference Books :

1. M.Artin, Algebra, Prentice Hall of India, 1991.

2. P.B.Bhattacharya, S.K.Jain, and S.R.Nagpaul, Basic Abstract Algebra

(II Edition) Cambridge University Press, 1997. (Indian Edition).

3. I.S.Luther and I.B.S.Passi, Algebra, Vol. I - Groups(1996); Vol. II Rings, (1999) Narosa Publishing

House , New Delhi.

4. D.S.Dummit and R.M.Foote, Abstract Algebra, 2nd edition, Wiley, 2002.

5. N.Jacobson, Basic Algebra, Vol. I & II Hindustan Publishing Company, New Delhi.

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Course Outcomes


CO Number

CO

Statement

Knowledge Level

CO1 Understand the definition of Extension fields K1, K2

CO2 Apply the concepts of Roots or Polynomials K1, K2,K3

CO3 Compute Galois groups of small degree polynomials and describe lattice diagrams of algebraic extensions.

K1, K2,K3.K4

C04 Using the concept of Finite fields - Wedderburn's theorem K1,K2,K5

CO5 Get the knowledge of Galois groups and A theorem of Frobenius K1,K2,K4

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216PMMT02 – REAL ANALYSIS - II

OBJECTIVES: To introduce measure on the real line, Lebesgue measurability and integrability, Fourier series and Integrals, in-depth study in multivariable calculus.

UNIT-I : Measure on the Real line - Lebesgue Outer Measure - Measurable sets - Regularity - Measurable

Functions - Borel and Lebesgue Measurability

Chapter - 2 Sec 2.1 to 2.5 of de Barra

UNIT-II : Integration of Functions of a Real variable - Integration of Non- negative functions - The General

Integral - Riemann and Lebesgue Integrals

Chapter - 3 Sec 3.1,3.2 and 3.4 of de Barra

UNIT-III : Fourier Series and Fourier Integrals - Introduction - Orthogonal system of functions - The

theorem on best approximation - The Fourier series of a function relative to an orthonormal system - Properties of

Fourier Coefficients - The Riesz-Fischer Thorem - The convergence and representation problems in for

trigonometric series - The Riemann - Lebesgue Lemma - The Dirichlet Integrals - An integral representation for

the partial sums of Fourier series - Riemann's localization theorem - Sufficient conditions for convergence of a

Fourier series at a particular point - Cesaro summability of Fourier series- Consequences of Fejes's theorem - The

Weierstrass approximation theorem

Chapter 11 : Sections 11.1 to 11.15 of Apostol

UNIT-IV : Multivariable Differential Calculus - Introduction - The Directional derivative - Directional

derivative and continuity - The total derivative - The total derivative expressed in terms of partial derivatives -

The matrix of linear function - The Jacobian matrix - The chain rule - Matrix form of chain rule - The mean -

value theorem for differentiable functions - A sufficient condition for differentiability - A sufficient condition

for equality of mixed partial derivatives - Taylor's theorem for functions of Rn to R

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Chapter 12 : Section 12.1 to 12.14 of Apostol

UNIT-V : Implicit Functions and Extremum Problems : Functions with non-zero Jacobian determinants – The

inverse function theorem-The Implicit function theorem-Extrema of real valued functions of severable variables-

Extremum problems with side conditions.

Chapter 13 : Sections 13.1 to 13.7 of Apostol

Recomwnded Text

1. G. de Barra, Measure Theory and Integration, New Age International, 2003 (for Units I and II)

2. Tom M.Apostol : Mathematical Analysis, 2nd

Edition, Narosa 1989 (for Units III, IV and V)

Reference Books

1. Burkill,J.C. The Lebesgue Integral, Cambridge University Press, 1951.

2. Munroe,M.E. Measure and Integration. Addison-Wesley, Mass.1971.

3. Royden,H.L.Real Analysis, Macmillan Pub. Company, New York, 1988.

4. Rudin, W. Principles of Mathematical Analysis, McGraw Hill Company, New York,1979.

5. Malik,S.C. and Savita Arora. Mathematical Analysis, Wiley Eastern Limited. New Delhi, 1991.

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Course Outcomes


CO Number

CO Statement Knowledge Level

CO1 Understand the definition and Measure on the Real line K1, K2 ,K3,K4, K5

CO2 Apply the concept of Riemann and Lebesgue Integrals K1, K2,K3,K4,K5

CO3 Analyze the Riesz-Fischer Theorem, Riemann's localization theorem and Fejes's theorem

K1, K2,K3.K4,K5

C04 Understand the concept of the Directional derivative, The total derivative and Taylor's theorem

K1,K2,K3

CO5 Have the idea of Implicit Functions and Extremum Problems K1, K2 ,K3,K4, K5

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216PMMT03 - PARTIAL DIFFERENTIAL EQUATIONS

OBJECTIVES: The aim of the course is to introduce to the students the various types of partial differential

equations and how to solve these equations.

UNIT-I : Partial Differential Equations of First Order: Formation and solution of PDE- Integral surfaces –

Cauchy Problem order eqn- Orthogonal surfaces – First order non-linear – Characteristics – Csmpatible system

– Charpit method. Fundamentals: Classification and canonical forms of PDE.

Chapter 0: 0.4 to 0.11 (omit .1,0.2.0.3 and 0.11.1) and Chapter 1: 1.1 to 1.5

UNIT-II : Elliptic Differential Equations: Derivation of Laplace and Poisson equation – BVP – Separation of

Variables – Dirichlet’s Problem and Newmann Problem for a rectangle – Interior and Exterior Dirichlets’s

problems for a circle – Interior Newman problem for a circle – Solution of Laplace equation in Cylindrical and

spherical coordinates – Examples. Chapter 2: 2.1, 2 2 ,2.5 to 2.13 (omit 2.3 and 2.4)

UNIT-III : Parabolic Differential Equations: Formation and solution of Diffusion equation – Dirac-Delta

function – Separation of variables method – Solution of Diffusion Equation in Cylindrical and spherical

coordinates Examples.

Chapter 3: 3.1 to 3.7 and 3.9 (omit 3.8)

UNIT-IV :Hyperbolic Differential equations: Formation and solution of one-dimensional wave equation –

canocical reduction – IVP- d’Alembert’s solution – Vibrating string – Forced Vibration – IVP and BVP for two-

dimensional wave equation – Periodic solution of one-dimensional wave equation in cylindrical and spherical

coordinate systems – vibration of circular membrane – Uniqueness of the solution for the wave equation –

Duhamel’s Principle – Examples

Chapter 4: 4.1 to 4.12(omit 4.13)

UNIT-V: Green’s Function: Green’s function for laplace Equation – methods of Images – Eigen function

Method – Green’s function for the wave and Diffusion equations. Laplace Transform method: Solution of

Diffusion and Wave equation by Laplace Transform. Fourier Transform Method: Finite Fourier sine and

cosine franforms – solutions of Diffusion, Wave and Lpalce equations by Fourier Transform Method.

Chapter 5: 5.1 to 5.6 Chapter 6: 6.13.1 and 6.13.2 only (omit (6.14) Chapter 7: 7.10 to 7.13 (omit 7.14)

Recomended Text

1. S, Sankar Rao, Introduction to Partial Differential Equations, 2nd

Edition, Prentice

Hall of India, New Delhi. 2005

Reference Books

1. R.C.McOwen, Partial Differential Equations, 2nd

Edn. Pearson Eduction, New Delhi, 2005.

2. I.N.Sneddon, Elements of Partial Differential Equations, McGraw Hill, New Delhi, 1983.

3. R. Dennemeyer, Introduction to Partial Differential Equations and Boundary Value Problems, McGraw

Hill, New York, 1968.

4. M.D.Raisinghania, Advanced Differential Equations, S.Chand & Company Ltd., New Delhi, 2001

24

Course Outcomes


CO Number

CO Statement Knowledge

Level

CO1 Understand the rules of Partial Differential Equations of First Order K1, K2 ,K3, K5

CO2 Learn about Dirichlet’s Problem, Solution of Laplace equation in

Cylindrical and spherical coordinates

K1, K2,K3,K4

CO3 An ability to calculate Parabolic Differential Equations K1, K2,K3.K4

C04 Have the idea of D’Alembert’s solution, one-dimensional wave equation

and Duhamel’s Principle with Examples K1,K2,K3

CO5 Apply the concepts of Green’s Function, Laplace Transform and Fourier Transform

K1,K2,K3

25

216PMMT04 - PROBABILITY THEORY

OBJECTIVES: To introduce axiomatic approach to probability theory, to study some statistical characteristics,

discrete and continuous distribution functions and their properties, characteristic function and basic limit

theorems of probability.

UNIT-I : Random Events and Random Variables: Random events – Probability axioms – Combinatorial

formulae – conditional probability – Bayes Theorem – Independent events – Random Variables – Distribution

Function – Joint Distribution – Marginal Distribution – Conditional Distribution – Independent random

variables – Functions of random variables.

Chapter 1: Sections 1.1 to 1.7

Chapter 2 : Sections 2.1 to 2.9

UNIT-II : Parameters of the Distribution : Expectation- Moments – The Chebyshev Inequality – Absolute

moments – Order parameters – Moments of random vectors – Regression of the first and second types.


UNIT-III: Characteristic functions : Properties of characteristic functions – Characteristic functions and

moments – semi invariants – characteristic function of the sum of the independent random variables –

Determination of distribution function by the Characteristic function – Characteristic function of

multidimensional random vectors – Probability generating functions.


UNIT-IV : Some Probability distributions: One point , two point , Binomial – Polya – Hypergeometric –

Poisson (discrete) distributions – Uniform – normal gamma – Beta – Cauchy and Laplace (continuous)

distributions.

Chapter 5 : Section 5.1 to 5.10 (Omit Section 5.11)

UNIT-V: Limit Theorems : Stochastic convergence – Bernaulli law of large numbers – Convergence of

sequence of distribution functions – Levy-Cramer Theorems – de Moivre-Laplace Theorem – Poisson,

Chebyshev, Khintchinep Weak law of large numbers – Lindberg Theorem – Lapunov Theroem – Borel-

Cantelli Lemma - Kolmogorov Inequality and Kolmogorov Strong Law of large numbers.

Chapter 6 : Sections 6.1 to 6.4, 6.6 to 6.9 , 6.11 and 6.12. (Omit Sections 6.5, 6.10,6.13 to 6.15)

Recomended Text:

1. M. Fisz, Probability Theory and Mathematical Statistics, John Wiley and Sons, New York, 1963.

Reference Books:

1. R.B. Ash, Real Analysis and Probability, Academic Press, New York, 1972

2. K.L.Chung, A course in Probability, Academic Press, New York, 1974.

3. R.Durrett, Probability : Theory and Examples, (2nd

Edition) Duxbury Press, New York, 1996.

4. V.K.Rohatgi An Introduction to Probability Theory and Mathematical Statistics, Wiley Eastern Ltd.,

New Delhi, 1988(3rd

Print).

5. S.I.Resnick, A Probability Path, Birhauser, Berlin,1999.

6. B.R.Bhat , Modern Probability Theory (3rd

Edition), New Age International (P)Ltd, New Delhi, 1999.

26

Course Outcomes


CO Number


CO1 Get the knowledge of random events and random variables which includes the basic concepts of probability. K1, K2,K3,K4,K5

CO2 Understand the concepts of moments with the help of

distribution parameters. K1, K2,K3,K4,K5

CO3 Apply the concepts of characteristic functions over the independent

random random variable and distribution functions. K1, K2,K3.K4,K5

C04 Analyze One point, two point, Binomial, Poisson, Uniform, normal,

gamma, Beta and its applications. K1,K2,K3, K4,K5

CO5 Construct various limit theorems such as Levy Cramer theorem,

Chebyshev theorem in order to relate the statistical inference. K1,K2,K3 K4,K5

27

ELECTIVE - V

216PMMT06 - MATHEMATICAL PROGRAMMING

Course Objective: This course is designed to

(i) Understand how to formulate integer programming and solving it.

(ii) Describe important dynamic technology and develop programmingstages.

(iii) Introduces the concepts of multivariable optimization with equality and

inequality constraints.

(iv) Learn advanced topics in Linear and non-linearProgramming

UNIT-I : Integer Linear Programming: Types of Integer Linear Programming Problems – Concept of Cutting

Plane – Gomory’s All Integer Cutting Plane Method – Gomory’s mixed Integer Cutting Plane method – Branch

and Bound Method.

Chapters: 7

UNIT-II :Dynamic Programming: Characteristics of Dynamic Programming Problem – Developing Optimal

Decision Policy – Dynamic Programming Under Certainty – DP approach to solve LPP.

Chapters: 22

UNIT-III :Classical Optimization Methods: Unconstrained Optimization – Constrained Multi-variable

Optimization with Equality Constraints - Constrained Multi-variable Optimization with inequality Constraints

Non-linear Programming Methods: Examples of NLPP – General NLPP – Graphical solution – Quadratic

Programming – Wolfe’s modified Simplex Methods

Chapter 23 and 24 :Sections 24.1 to24.4(Omit Beale’s method)

UNIT-IV :Linear Programming Problem – Simple problems.

Parametric Linear Programming : Variation in the coefficients cj , Variations in the Right hand side, bi .

Chapters 4 :Section 4.1 to 4.3 and Chapter 29

UNIT-V:Goal Programming : Difference between LP and GP approach – Concept of Goal Programming –

Goal Programming Model formulation – Graphical Solution Method of Goal Programming – Modified Simplex

method of Goal Programming.

Chapters 8 :Section 8.1 to 8.5

Recomended Text:

J.K.Sharma, Operations Research , Macmillan (India) New Delhi 2001

References Books:

1. Hamdy A. Taha, Operations Research, (seventh edition) Prentice - Hall of India Private Limited, New

Delhi, 1997.

2. F.S. Hiller & J.Lieberman Introduction to Operation Research (7th Edition) Tata- McGraw Hill

Company, New Delhi, 2001.

3. Beightler. C, D.Phillips, B. Wilde ,Foundations of Optimization (2nd Edition) Prentice Hall Pvt Ltd., New

York, 1979

4. S.S. Rao - Optimization Theory and Applications, Wiley Eastern Ltd. New Delhi. 1990

28

Course Outcomes


CO Number

CO

Statement

Knowledge Level

CO1 Know the formulation of IPP and solving the problems. K1, K2 ,K3, K5

CO2 Understand the concepts of dynamic programming and they can

develop programming stages. K1, K2,K3,K4,K5

CO3 Identify the constrained multi variable optimization with equality and

unequality constraints. K1, K2,K3.K4,K5

C04 Analyze the concepts of Linear Programming problem and also they

know how to apply LPP in Parametric Programming. K1,K2,K3, K4,K5

CO5 Differentiate between linear programming and Goal Programming K1,K2,K3,K4,K5

29

III SEMESTER

316PMMT01 - COMPLEX ANALYSIS – I

OBJECTIVES: To Study Cauchy integral formula, local properties of analytic functions, general form of

Cauchy’s theorem and evaluation of definite integral and harmonic functions

UNIT-I: Cauchy’s Integral Formula: The Index of a point with respect to a closed curve – The Integral formula

– Higher derivatives.

Local Properties of Analytical Functions:

Removable Singularities-Taylors’s Theorem – Zeros and poles – The local Mapping – The Maximum Principle.

Chapter 4: Section 2: 2.1 to 2.3, Section 3: 3.1 to 3.4

UNIT-II: The general form of Cauchy’s Theorem:

Chains and cycles- Simple Connectivity - Homology - The General statement of Cauchy’s Theorem - Proof of

Cauchy’s theorem - Locally exact differentials- Multiply connected regions - Residue theorem - The argument

principle.

Chapter 4: Section 4: 4.1 to 4.7, Section 5: 5.1 and 5.2

UNIT-III: Evaluation of Definite Integrals and Harmonic Functions:

Evaluation of definite integrals - Definition of Harmonic functions and basic properties - Mean value property -

Poisson formula.

Chapter 4: Section 5: 5.3, Section 6: 6.1 to 6.3

UNIT-IV: Harmonic Functions and Power Series Expansions:

Schwarz theorem - The reflection principle - Weierstrass theorem – Taylor Series – Laurent series.



UNIT-V: Partial Fractions and Entire Functions:

Partial fractions - Infinite products – Canonical products – Gamma Function- Jensen’s formula – Hadamard’s

Theorem

Chapter 5: Sections 2.1 to 2.4, Sections 3.1 and 3.2

Recommended Text

Lars V. Ahlfors, Complex Analysis, (3rd

edition) McGraw Hill Co., New York, 1979

Reference Book

1. H.A. Priestly, Introduction to Complex Analysis, Clarendon Press, Oxford, 2003.

2. J.B.Conway, Functions of one complex variable Springer International Edition, 2003

3. T.W Gamelin, Complex Analysis, Springer International Edition, 2004.

4. D.Sarason, Notes on complex function Theory, Hindustan Book Agency, 1998

30

Course Outcomes


CO Number CO Statement Knowledge Level

CO1 To study the Cauchy’s Integral Formula and Local Properties of Analytical Functions.

K1, K2 ,K3, K4 K5

CO2 Understand the general form of Cauchy’s Theorem. K1, K2,K3,K4,K5

CO3 Learn the basic concept of integrals, Harmonic functions, Poisson

formula and basic properties. K1, K2,K3.K4,K5

C04 Define and recognize the harmonic functions and Power Series Expansions

K1,K2,K3

CO5 Get the knowledge of Partial Fractions and Entire Functions K1,K2,K3

31

316PMMT02 - TOPOLOGY

OBJECTIVES:

The aim of the course is to :

(i) Define the convergent sequence and demonstrate equivalence of metric space. (ii) explainTopological space with the examples. (iii) Introduce the concept of open and closed basis.

(iv) Develop the Urysohn’s lemma, the Tietze extension theorem and Weierstrass approximation Theorem.

UNIT-I : Metric Spaces:

Convergence, completeness and Baire’s Theorem; Continuous mappings; Spaces of continuous functions;

Euclidean and Unitary spaces. Topological Spaces: Definition and Examples; Elementary concepts.

Chapter Two (Sec 12 - 15) Chapter Three (Sec 16 & 17)

UNIT-II: Topological Spaces (contd…)

Open bases and sub bases; Weak topologies; the function algebras C(X, R) and C(X, C): Compact spaces.

Chapter Three (Sec 18 - 20) Chapter Four (Sec 21)

UNIT-III:

Tychonoff’s theorem and locally compact spaces; Compactness for metric spaces; Ascoli’s theorem.

Chapter Four (Sec 23 - 25)

UNIT-IV:

T1 – spaces and Hausdorff spaces; completely regular spaces and normal spaces; Urysohn’s lemma and the

Tietze extension theorem; The Urysohn imbedding theorem.

Chapter Five (Sec 26 - 29)

UNIT-V:

The Stone – Cech compactification; Connected spaces; The components of a space; Totally disconnected spaces;

Locally connected spaces; The Weierstrass approximation Theorem.

Chapter Five (Sec 30) Chapter Six (Sec 31 - 34)

Chapter Seven (Sec 35)

Recommended Text

George F.Simmons, Introduction to Topology and Modern Analysis, Tata-McGraw Hill. New Delhi, 2004

Reference Books

1. J.R. Munkres, Topology (2nd

Edition) Pearson Education Pvt. Ltd., Delhi-2002 (Third Indian Reprint)

2. J. Dugundji , Topology , Prentice Hall of India, New Delhi, 1975.

3. J.L. Kelly, General Topology, Springer

4. S.Willard, General Topology, Addison - Wesley, Mass., 1970

32

Course Outcomes


CO Number CO Statement Knowledge

Level

CO1 Define convergence of sequences in a metric space and determine whether a given sequence in metric space converges.

K1, K2 ,K3, K5

CO2 Work with sets and functions in topology. K1, K2

CO3 Know how the topology on a space is determined by the collection of open sets, closed sets, or by a basis of neighbourhood at each point.

K1, K2,K3,

K4,K5

C04 Cheracterise metrizable spaces using Urysohn lamma and the Tietze extension theorem.

K1, K2,K3,K5

CO5 Have the knowledge of the Weierstrass approximation Theorem and its uses.

K1,K2,K3

33

316PMMT03 - OPERATION RESEARCH

OBJECTIVES: This course is designed to

(i) Identify the steps and analyze alternatives in a decision making process.

(ii) Develop linear programming problem for shortest path, maximum flow, minimal

spanning tree, critical path, minimum cost flow.

(iii) To determine the optimum level of inventories that should be maintained in a production

process, ordering, deciding on quantity of goods.

(iv) Identify by the presence of a group of customers who arrive randomely to receive

some service and also to Find out the optimum service rate and the number of servers.

(iv) Direct the organization towards profit maximization or cost minimization

.

UNIT-I : Decision Theory :

Steps in Decision theory Approach – Types of Decision-Making Environments – Decision Making Under

Uncertainty – Decision Making under Risk – Posterior Probabilities and Bayesian Analysis – Decision Tree

Analysis – Decision Making with Utilities.

Chapter 10: Sec. 10.1 to 10.8

UNIT-II: Network Models:

Scope of Network Applications – Network Definition – Minimal spanning true Algorithm – Shortest Route

problem – Maximum flow model – Minimum cost capacitated flow problem - Network representation – Linear

Programming formulation – Capacitated Network simplex Algorithm.


UNIT-III: Deterministic Inventory Control Models:

Meaning of Inventory Control – Functional Classification – Advantage of Carrying Inventory – Features of

Inventory System – Inventory Model building - Deterministic Inventory Models with no shortage – Deterministic

Inventory with Shortages .

Probabilistic Inventory Control Models:

Single Period Probabilistic Models without Setup cost – Single Period Probabilities Model with Setup cost.



UNIT-IV : Queueing Theory :

Essential Features of Queueing System – Operating Characteristic of Queueing System – Probabilistic

Distribution in Queueing Systems – Classification of Queueing Models – Solution of Queueing Models –

Probability Distribution of Arrivals and Departures – Erlangian Service times Distribution with k-Phases.

Chapter 15 : Sec. 15.1 to 15.8

UNIT-V : Replacement and Maintenance Models:

Failure Mechanism of items – Replacement of Items that deteriorate with Time – Replacement of items that

fail completely – other Replacement Problems.


Recommended Text

1. For Unit 2 : H.A. Taha, Operations Research, 6th

edition, Prentice Hall of India

2. For all other Units: J.K.Sharma, Operations Research , MacMillan India, New Delhi, 2001.

34

Reference Books

1. F.S. Hiller and J.Lieberman -,Introduction to Operations Research (7th

Edition), Tata McGraw Hill Publishing

Company, New Delhui, 2001.

2. Beightler. C, D.Phillips, B. Wilde ,Foundations of Optimization (2nd

Edition) Prentice Hall Pvt Ltd., New

York, 1979

3. Bazaraa, M.S; J.J.Jarvis, H.D.Sharall ,Linear Programming and Network flow, John Wiley and sons, New

York 1990.

4. Gross, D and C.M.Harris, Fundamentals of Queueing Theory, (3rd

Edition), Wiley and Sons, New York, 1998.

Course Outcomes


CO Number


CO1 Apply decision theory to a variety of practices of complex decision making in business firms and other organizations in everyday situations and during crisis

K1,K2,K3,K5

CO2 Investigate network analysis on elements of scheduling by CPM and PERT K1, K2,K3,K5

CO3 Identify various types of inventory models and they can demonstrate their applications

K1, K2,K3, K5

C04 Expose the basic characteristic features of a queuing system and acquire

skills in analyzing queuing models K1, K2,K3, K5

CO5 Get the knowledge of Replacement and Maintenance Models in which time is machine replaced or repaired.

K1, K2,K3, K5

35

316PMMT04 - MECHANICS

OBJECTIVES: To study mechanical systems under generalized coordinate systems, virtual work, energy and momentum, to study mechanics developed by Newton, Langrange, Hamilton Jacobi and Theory of Relativity due to Einstein.

UNIT-I: Mechanical Systems:

The Mechanical system- Generalized coordinates – Constraints - Virtual work - Energy and Momentum


UNIT-II : Lagrange's Equations:

Derivation of Lagrange's equations- Examples- Integrals of motion.

Chapter 2 : Sections 2.1 to 2.3 (Omit Section 2.4)

UNIT-III : Hamilton's Equations :

Hamilton's Principle - Hamilton's Equation - Other variational principles.

Chapter 4 : Sections 4.1 to 4.3 (Omit section 4.4)

UNIT – IV : Hamilton-Jacobi Theory :

Hamilton Principle function – Hamilton-Jacobi Equation - Separability


UNIT-V : Canonical Transformation :

Differential forms and generating functions – Special Transformations– Lagrange and Poisson brackets.

Chapter 6 : Sections 6.1, 6.2 and 6.3 (omit sections 6.4, 6.5 and 6.6)

Recommended Text

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

Reference Books

1. H. Goldstein, Classical Mechanics, (2nd

Edition) Narosa Publishing House, New Delhi.

2. N.C.Rane and P.S.C.Joag, Classical Mechanics, Tata McGraw Hill, 1991.

3. J.L.Synge and B.A.Griffth, Principles of Mechanics (3rd

Edition) McGraw Hill Book Co., New York, 1970.

36

Course Outcomes


CO Number

CO

Statement

Knowledge

Level

CO1 Understand the basic of Mechanical Systems. K1, K2 ,K3

CO2 Understand the effect of co-ordinate transformations and visualize the tensor as a linear transformation.

K1, K2,K3,K5

CO3 Understand the conventions like summation convention and comma notations. Also, students shall learn the concepts of tensor calculus.

K1,K2,K3.K4

C04 To learn about Hamilton-Jacobi Theory and lemma. K1,K2,K3

CO5 To get the knowledge of Canonical Transformation and able to apply the knowledge in solving real world problems related to continuum

mechanics.

K1,K2,K3

37

ELECTIVE - V

316PMMT06 - NUMBER THEORY AND CRYPTOGRAPHY

OBJECTIVES: This course aims to give elementary ideas from number theory which will have applications in cryptology.

UNIT-I :

Elementary Number Theory: Time Estimates for doing arithmetic – divisibility and Euclidean algorithm –

Congruence’s – Application to factoring. (Chapter 1)

UNIT-II:

Introduction to Classical Crypto systems – Some simple crypto systems – Enciphering matrices DES

(Chapter 3)

UNIT-III:

Finite Fields, Quadratic Residues and Reciprocity (Chapter 2)

UNIT-IV:

Public Key Cryptography (Chapter 4)

UNIT-V:

Primality, Factoring, Elliptic curves and Elliptic curve crypto systems (Chapter 5, sections 1,2,3 &5 (omit

section 4), Chapter 6, sections 1& 2 only)

Reference Books

Neal Koblitz, A Course in Number Theory and Cryptography, Springer-Verlag, New York,1987

Rec ommended Text

1.I. Niven and H.S.Zuckermann, An Introduction to Theory of Numbers (Edn. 3), Wiley Eastern Ltd., New

Delhi,1976

2. David M.Burton, Elementary Number Theory, Brown Publishers, Iowa,1989

3. K.Ireland and M.Rosen, A Classical Introduction to Modern Number Theory, Springer Verlag, 1972

4. N.Koblitz, Algebraic Aspects of Cryptography, Springer 1998

38

Course Outcomes


CO Number


Level CO1 Understand the basic concept of Euclidean algorithm and its applications. K1, K2 ,K3

CO2 Introduction to Classical Crypto systems and Enciphering matrices. K1, K2,K3

CO3 To get the knowledge of Finite Field. K1, K2, K3.

C04 To learn about the Public Key Cryptography. K1,K2,K3

CO5 Get the knowledge of Elliptic curve. K1,K2,K3

39

IV SEMESTER

416PMMT01 - COMPLEX ANALYSIS - II

OBJECTIVES: To study Riemann Theta Function and normal families, Riemann mapping theorem, Conformal mapping of polygons, harmonic functions, elliptic functions and Weierstrass Theory of analytic continuation.

UNIT-I: Riemann Zeta Function and Normal Families:

Product development – Extension of (s) to the whole plane – The zeros of zeta function – Equicontinuity –

Normality and compactness – Arzela’s theorem – Families of analytic functions – The Classical Definition

Chapter 5: Sections 4.1 to 4.4, Sections 5.1 to 5.5

UNIT-II: Riemann mapping Theorem:

Statement and Proof – Boundary Behaviour – Use of the Reflection Principle. Conformal mappings of

polygons: Behaviour at an angle Schwarz-Christoffel formula – Mapping of a rectangle.Harmonic Functions:

Functions with mean value property – Harnack’s principle.

Chapter 6: Sections 1.1 to 1.3 (Omit Section 1.4)

Sections 2.1 to 2.3 (Omit section 2.4), Section 3.1 and 3.2

UNIT-III: Elliptic functions:

Simply periodic functions – Doubly periodic functions

Chapter 7: Sections 1.1 to 1.3, Sections 2.1 to 2.4

UNIT-IV: Weierstrass Theory:

The Weierstrass -function – The functions (s) and (s) – The differential equation – The modular equation

() – The Conformal mapping by ().


UNIT-V: Analytic Continuation:

The Weierstrass Theory – Germs and Sheaves – Sections and Riemann surfaces – Analytic continuation along

Arcs – Homotopic curves – The Monodromy Theorem – Branch points.


Recommended Text

Lars V. Ahlfors, Complex Analysis, (3rd

Edition) McGraw Hill Book Company, New York, 1979.

Reference Books

1. H.A. Priestly, Introduction to Complex Analysis, Clarendon Press,Oxford, 2003.

2. J.B.Conway, Functions of one complex variable, Springer International Edition, 2003

3. T.W Gamelin, Complex Analysis, Springer International Edition, 2004.

4. D.Sarason, Notes on Complex function Theory, Hindustan Book Agency, 1998

40

Course Outcomes


CO Number

CO

Statement

Knowledge Level

CO1 Understand the zeros of zeta function and Arzela’s theorem. K1, K2 ,K3,

CO2 Learn about the Riemann mapping Theorem. K1, K2,K3,K4

CO3 Get the knowledge of Elliptic functions. K1, K2,K3

C04 To understand about Weierstrass -function – The functions (s). K1,K2,K3

CO5 Get the knowledge of Analytic Continuation. K1,K2,K3

41

416PMMT02 - DIFFERENTIAL GEOMETRY

OBJECTIVES: This course introduces space curves and their intrinsic properties of a surface and geodesics. Further the non-intrinsic properties of surfaces are explored.

UNIT-I: Space curves:

Definition of a space curve – Arc length – tangent – normal and binormal – curvature and torsion – contact

between curves and surfaces- tangent surface- involutes and evolutes- Intrinsic equations – Fundamental

Existence Theorem for space curves- Helices.

Chapter I: Sections 1 to 9.

UNIT-II: Intrinsic properties of a surface:

Definition of a surface – curves on a surface – Surface of revolution – Helicoids – Metric- Direction coefficients

– families of curves- Isometric correspondence- Intrinsic properties.

Chapter II: Sections 1 to 9.

UNIT-III: Geodesics:

Geodesics – Canonical geodesic equations – Normal property of geodesics- Existence Theorems – Geodesic

parallels – Geodesics curvature- Gauss- Bonnet Theorem – Gaussian curvature- surface of constant curvature.

Chapter II: Sections 10 to 18.

UNIT-IV: Non-intrinsic properties of a surface:

The second fundamental form- Principal curvature – Lines of curvature – Developable - Developable associated

with space curves and with curves on surfaces - Minimal surfaces – Ruled surfaces.

Chapter III: Sections 1 to 8.

UNIT-V: Differential Geometry of Surfaces:

Compact surfaces whose points are umblics- Hilbert’s lemma – Compact surface of constant curvature –

Complete surfaces and their characterization – Hilbert’s Theorem – Conjugate points on geodesics.

Chapter IV: Sections 1 to 8

Recommended Text

1.T.J.Willmore, An Introduction to Differential Geometry, Oxford University Press,(17th

Impression) New Delhi

2002. (Indian Print)

Reference Books

1. Struik, D.T. Lectures on Classical Differential Geometry, Addison – Wesley, Mass. 1950.

2. A.Pressley, Elementary Differential Geometry, Springer International Edition, 2004

3. Wilhelm Klingenberg: A course in Differential Geometry, Graduate Texts in Mathematics, Springer-Verlag

1978.

4. J.A. Thorpe Elementary Topics in Differential Geometry, Springer International Edition, 2004.

42

Course Outcomes


CO Number


Level

CO1 Understand the basic concept of definition of a space curve curvature, torsion, involutes and evolutes.

K1, K2 ,K3.S

CO2 To get the knowledge of intrinsic properties of a surface K1, K2,K3,K4

CO3 To get the knowledge of Geodesics and its properties. K1,

K2,K3.K4,K5

C04 To understand about the second fundamental form K1,K2,K3

CO5 To have knowledge about Differential Geometry of Surfaces and

characterization of Complete surfaces K1,K2,K3

43

416PMMT03 - FUNCTIONAL ANALYSIS

OBJECTIVES: To study the details of Banach and Hilbert Spaces and to introduce Banach algebras.

UNIT-I: Normed Spaces:

Definition – Some examples – Continuous Linear Transformations – The Hahn-Banach Theorem – The

natural embedding of N in N**

Chapter 9 : Sections 46 to 49

UNIT-II: Banach spaces and Hilbert spaces:

Open mapping theorem – conjugate of an operator – Definition and properties – Orthogonal complements

– Orthonormal sets

Chapter 9: Sections 50 and 51

Chapter 10: Sections 52, 53 and 54.

UNIT-III: Hilbert Space:

Conjugate space H* - Adjoint of an operator – Self-adjoint operator – Normal and Unitary Operators –

Projections


UNIT-IV : Preliminaries on Banach Algebras :

Definition and some examples – Regular and singular elements – Topological divisors of zero – spectrum

– the formula for the spectral radius – the radical and semi-simplicity.


UNIT-V: Structure of commutative Banach Algebras: Gelfand mapping – Spectral radius formula

Involutions in Banach Algebras – Gelfand-Neumark

Theorem.


Recommended Text

1. G.F.Simmons , Introduction toTopology and Modern Analysis, McGraw Hill International Book

Company, New York, 1963.

Reference Books

1. W.Rudin Functional Analysis, Tata McGraw-Hill Publishing Company, New Delhi , 1973

2. G. Bachman & L.Narici, Functional Analysis Academic Press, New York ,1966.

3.C. Goffman and G.Pedrick, First course in Functional Analysis, Prentice Hall of India, New Delhi, 1987

4. E. Kreyszig Introductory Functional Analysis with Applications, John wiley & Sons, New York.,1978.

5. M.Thamban Nair, Functional Analysis. A First Course, Prentice Hall of India, New Delhi, 2002.

44

Course Outcomes


CO Number


Level CO1 Understand the basic concept of Normed Spaces with example. K1, K2 ,K3,

CO2 Learn about open mapping theorem and properties. K1, K2,K3,

K4,K5

CO3 To learn the concept of Conjugate space , Normal and Unitary Operators K1, K2,K3.K4

C04 An understanding the concept of Preliminaries on Banach Algebras K1,K2,K3

CO5 To get the knowledge about Gelfand-Neumark Theorem and its uses. K1,K2,K3

45

ELECTIVE (1)

416PMMT04 - FLUID DYNAMICS

OBJECTIVES: This course aims to discuss kinematics of fluids in motion, Equations of motion of a fluid, three

dimensional flows, two dimensional flows and viscous flows.

UNIT-I : Kinematics of Fluids in motion. Real fluids and Ideal fluids- Velocity of a fluid at a point, Stream

lines , path lines , steady and unsteady flows- Velocity potential - The vortices vector- Local and particle rates of

changes - Equations of continuity - Worked examples - Acceleration of a fluid - Conditions at a rigid body.

Chapter 2. Sec 2.1 to 2.10.

UNIT-II: Equations of motion of a fluid: Pressure at a point in a fluid at rest. - Pressure at a point in a moving

fluid - Conditions at a boundary of two in viscid immiscible fluids- Euler’s equation of motion - Discussion of the

case of steady motion under conservative body forces.

Chapter 3. Sec 3.1 to 3.7

UNIT-III : Some three dimensional flows. Introduction- Sources sinks and doublets - Images in a rigid infinite

plane - Axis symmetric flows - Stokes stream function

Chapter 4 Sec 4.1, 4.2, 4.3, 4.5.

UNIT-IV : Some two dimensional flows : Meaning of two dimensional flow - Use of Cylindrical polar

coordinates - The stream function - The complex potential for two dimensional , irrotational incompressible flow

- Complex velocity potentials for standard two dimensional flows - Some worked examples - Two dimensional

Image systems - The Milne Thompson circle Theorem.


UNIT-V: Viscous flows: Stress components in a real fluid. - Relations between Cartesian components of stress-

Translational motion of fluid elements - The rate of strain quadric and principle stresses - Some further

properties of the rate of strain quadric - Stress analysis in fluid motion - Relation between stress and rate of strain-

The coefficient of viscosity and Laminar flow - The Navier – Stokes equations of motion of a Viscous fluid.


Reference Books F. Chorlton, Text Book of Fluid Dynamics, CBS Publications. Delhi, 1985.

Recommended Text 1. R.W.Fox and A.T.McDonald. Introduction to Fluid Mechanics, Wiley, 1985.

2. E.Krause, Fluid Mechanics with Problems and Solutions, Springer, 2005.

3. B.S.Massey, J.W.Smith and A.J.W.Smith, Mechanics of Fluids,

Taylor and Francis, New York, 2005

4. P.Orlandi, Fluid Flow Phenomena, Kluwer, New Yor, 2002.

5. T.Petrila, Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics, Springer, berlin,

2004.

46

Course Outcome


CO Number


Level CO1 Understand the basic concept of Kinematics of Fluids in motion. K1, K2

CO2 Learn the concept of Pressure at a point in a fluid at rest and Euler’s equation of motion of a Fluid.

K1, K2,K3,K4

CO3 Get the knowledge of rigid body & steady motion under conservative body forces and axis symmetric flows and Stokes stream function.

K1, K2,K3.K4

C04 Learn the applications of two & three dimensional flows. K1,K2,K4,K5

CO5 Get the knowledge of relations between stream & stress and rate of strain. K1,K2,K4,K5

47

ELECTIVE (2)

416PMMT08 - TENSOR ANALYSIS AND RELATIVITY

OBJECTIVES: The course aims to introduce vector algebra and vector calculus and special relativity and relativistic kinematics, dynamics and accelerated systems.

UNIT-I : Tensor Algebra : Systems of Different orders – Summation Convention – Kronecker Symbols -

Transformation of coordinates in Sn - Invariants – Covariant and Contravariant vectors - Tensors of Second

Order – Mixed Tensors – Zero Tensor – Tensor Field – Algebra of Tensors – Equality of Tensors – Symmetric

and Skew-symmetric tensors - Outer multiplication, Contraction and Inner Multiplication – Quotient Law of

Tensors – Reciprocal Tensor – Relative Tensor – Cross Product of Vectors.

Chapter I: I.1 – I.3, I.7 and I.8 and Chapter II: II.1 – II.19

UNIT-II: Tensor Calculus: Riemannian Space – Christoffel Symbols and their properties.

Chapter III: III.1 and III.2

UNIT-III: Tensor Calculus (contd): Covariant Differentiation of Tensors – Riemann–Christoffel Curvature

Tensor – Intrinsic Differentiation

Chapter III: III.3 – III.5

UNIT-IV: Special Theory of Relativity: Galilean Transformations – Maxwell’s equations – The ether Theory –

The Principle of Relativity.

Relativistic Kinematics : Lorentz Transformation equations – Events and simultaneity – Example – Einstein

Train – Time dilation – Longitudinal Contraction - Invariant Interval - Proper time and Proper distance - World

line - Example – twin paradox – addition of velocities – Relativistic Doppler effect.


UNIT-V : Relativistic Dynamics : Momentum – Energy – Momentum – energy four vector – Force -

Conservation of Energy – Mass and energy – Example – inelastic collision – Principle of equivalence –

Lagrangian and Hamiltonian formulations.

Accelerated Systems: Rocket with constant acceleration – example – Rocket with constant thrust.


Reference Books

U.C. De, Absos Ali Shaikh and Joydeep Sengupta, Tensor Calculus, Narosa Publishing House, New Delhi, 2004.

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

Rec ommended Text

1. J.L.Synge and A.Schild, Tensor Calculus, Toronto, 1949.

2. A.S.Eddington. The Mathematical Theory of Relativitity, Cambridge University Press, 1930.

3. P.G.Bergman, An Introduction to Theory of Relativity, Newyor, 1942.

4. C.E.Weatherburn, Riemannian Geometry and the Tensor Calculus, Cambridge, 1938.

48

Course Outcomes


CO Number


Level CO1 Understand the basic concept of invariants ,covariant and contra variant

vector K1, K2 ,K3, K5

CO2 To get the knowledge of Riemannian Space , Christoffel Symbols and their properties

K1, K2,K3,

K4,K5

CO3 Calculate the value of Tensor and their properties. K1,

K2,K3.K4,K5

C04 To learn about Maxwell’s equations and the Principle of Relativity.

K1,K2,K3

CO5 Get the knowledge of Momentum ,Energy Momentum , Lagrangian and

Hamiltonian K1,K2,K3

49

ELECTIVE PAPERS

ELECTIVE - I (Group – A)

116PMMT05 - FORMAL LANGUAGES AND AUTOMATA THEORY

COURSE OBJECTIVES:

1. To define mathematical methods of computing devices, called abstract machines, namely Finite

Automata, Pushdown Automata.

2. To classify machines by their power to recognize languages.

3. Employ finite state machines to solve problems in computing

4. Explain deterministic and nondeterministic machines.

5. Identify different formal language classes and their relationships

6. Design grammars and recognizers for different formal languages

UNIT-I:

Finite automata, regular expressions and regular grammars: Finite state systems – Basic definitions – Nondeterministic finite automata – Finite automata with move – Regular expressions – Regular grammars.

Chapter 2. Sections 2.1 to2.5 and Chapter 9 Section 9.1

UNIT-II:

Properties of regular sets.

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. Chapter 3: Sections 3.1 to 3.4

UNIT-III: Context-free grammars

Motivation and introduction – Context-free grammars – Derivation trees- Simplification of

context-free grammars – Chomsky normal form – Greibach normal form. Chapter 4 : Section 4.1 to 4.6

UNIT-IV: Pushdown automata Informal description- Definitions-Pushdown automata and context-free languages – Normal

forms for deterministic pushdown automat. Chapter 5 : Sections 5.1 to 5.3

UNIT-V: Properties of context-free languages

The pumping lemma for CFL’s – Closure properties for CFL’s – Decision algorithms for CFL’s.


Recomended Test: 1. John E.Hopcraft and Jeffrey D.Ullman, Introduction to Automata Theory, Languages

and Computation, Narosa Publishing House, New Delhi, 1987.

Reference Books:

1. A. Salomaa, Formal Languages, Academic Press, New York, 1973. 2. John C. Martin, Introduction to Languages and theory of Computations (2

nd Edition)

Tata-McGraw Hill Company Ltd., New Delhi, 1997.

50

Course Outcomes


CO Number


CO1 Students would be able to explain basic concepts in formal language

theory, grammars, automata theory, computability theory, and complexity

theory

K1, K2 ,K3, K5

CO2 The student will be able to demonstrate abstract models of computing,

including deterministic (DFA), non-deterministic (NFA), Push Down

Automata(PDA) and their power to recognize the languages.

K1, K2,K3,K4,K5

CO3 The student will be able to explain the application of machine models and

descriptors to compiler theory and parsing. K1, K2,K3,K4

CO4 . Students will be able to relate practical problems to languages, automata,

computability, and complexity. K1, K2,K3.K4

C05 Students will demonstrate an increased level of mathematical

sophistication. K1,K2,K3

CO6 Students will be able to explain the relationship among language classes

and grammars with the help of Chomsky Hierarchy. K1,K2,K3

51

116PMMT07 - MATHEMATICAL ECONOMICS

Objective: To initiate the study on consumer behavior, Theory of firms, Market equilibrium and Monopolistic Composition

UNIT-I: The Theory of Consumer Behaviour: Utility function – Indifference Curves – Rate of Commodity Substitution – Existence of Utility Function – Maximizatin of Utility – Choice of a utility Index – Demand function – Income and Leisure – Substitution and Income Effects – Generalization to n variables – Theory of Revealed Preference – Problem of Choice in Risk.

Chapter 2: 2.1 to 2.10

UNIT-II : The Theory of Firm: Production Function – Productivity Curves – Isoquents –

Optimization behavior – Input Demand Functions – Cost Functions (short-run and long-

run) – Homogeneous Production functions and their properties – CES Production Function

and their Properties – Joint Products – Generalization to m variables.

UNIT-III : Market Equilibrium: Assumptions of Perfect Competition – Demand

Functions – Supply Functions – Commodity Equilibrium – Applications of the Analysis –

Factor Market Equilibrium – Existence of Uniqueness of Equilibrium – Stability of

Equilibrium

– Dynamic Equilibrium with lagged adjustment.

UNIT-IV: Imperfect Competition: Monopoly and its Applications – Duopoly and

Oligopoly – Monopolistic Composition – Monopsory, Duopoly and Oligopoly –

Bilateral Monopoly


UNIT-V: Welfare Economics: Parato Optimality and the efficiency of Perfect competition

– The efficiency of Imperfect competition – External Effects in comsumption and Production – Taxes, Subsidies and Compensation – Social Welfare functions – The theory of

Second Best.


Recommended Text:

1. J.M.Henderson and R.E.Quandt, Micro Economic Theory- A Mathematical Approach,

(2nd

Edn) McGraw Hill, New York, 1971.

Reference Books

1. William J. Baumol. Economic Theory and Operations Analysis, Prentice Hall of

India, New Delhi, 1978

2. A.C.Chiang, Fundamental Methods of Mathematical Economics, McGraw Hill,

New York, 1984

3. Michael D. Intriligator, Mathematical Optimization and Economic Theory,

Prentice Hall, New York, 1971. 4. A. Kautsoyiannis, Modern Microeconomics (2

nd edn) MacMillan, New York, 1979

52

Course Outcomes


CO Number


CO1 Understand the concept of Utility function, Substitution and Income Effects and

Theory of Revealed Preference K1, K2 ,K3, K5

CO2 To get the knowledge of production Function, Homogeneous Production functions

and their properties. K1, K2,K3, K4,K5

CO3 To learn about Applications of the Analysis , Factor Market Equilibrium and

Existence of Uniqueness of Equilibrium K1, K2,K3.K4,K5

C04 An ability to apply knowledge of Monopolistic Competition,

Duopoly and Oligopoly and Bilateral Monopoly.

K1,K2,K3,K4

CO5 Get the knowledge of the efficiency of Imperfect competition and Social Welfare functions.

K1,K2,K3

53

116PMMT08 - FUZZY SETS AND APPLICATIONS

OBJECTIVES: The course aims to introduce Fundamental Notions, Fuzzy Graph, Relations, Logic

and Fuzzy Logic.

UNIT-I: Fundamental Notions:

Chapter I: Sec. 1 to 8

UNIT-II: Fuzzy Graphs:

Chapter II: Sec. 10 to 18

UNIT-III: Fuzzy Relations:

Chapter II: Sec. 19 to

UNIT-IV: Fuzzy Logic:

Chapter III: Sec.31 to 40 (omit Sec. 37, 38, 41)

UNIT-V: Fuzzy Logic:

Chapter IV: Sec.43 to 49

Recommended Text:

1. A.Kaufman, Introduction to the theory of Fuzzy subsets, Vol.I, Academic Press, New York,

1975.

Reference Books:

1. H.J.Zimmermann, Fuzzy Set Theory and its Applications, Allied Publishers, Chennai, 1996

2. George J.Klir and Bo Yuan, Fuzzy sets and Fuzzy Logic-Theory and

Applications, Prentice Hall India, New Delhi, 2001.

Course Outcomes


CO Number

CO

Statement

Knowledge Level

CO1 Acquire basic understanding of the various algorithms involved in Neural

Networks & Fuzzy Systems Fundamental Notions K1, K2 ,K3

CO2 Acquire basic understanding of the various learning methods K1, K2,K3

CO3 Acquire basic understanding of the back propagation & how to use it to

analyze a network K1, K2,K3.K4,K5

C04 Appreciate the importance of fuzzy data K1,K2,K3,K4

CO5 Analyze how to apply the concept of fuzzy & neural in biomedical

applications K1,K2,K3,K4

54

ELECTIVE - II (GROUP - B)

216PMMT05 - PROGRAMMING IN C++ AND NUMERICAL METHODS

Course Objectives:

1. Make the students familiar with basic concepts and techniques of object oriented programming in

C++.

2. Develop an ability to write programs in C++ for problem solving.

3. Develop types of inheritance like single, multiple, multilevel hierarchical and hybrid inheritance.

4. Develop operator overloading with unary and binary operator overloading.

5. Develop program to use of constructor and destructor with its types.

6. To provide a basic understanding of the derivation, analysis, and use of numerical methods.

UNIT-I : Tokens, Expressions and Control Structures – Functions in C++

Chapters: 3 and 4 (Balagurusamy)

UNIT-II : Classes and Objects – Constructors and Destructors – Operator Overloading and Type conversions.

Chapters: 5, 6 and 7(Balagurusamy)

UNIT-III: Inheritance – Pointers – Virtual Functions and Polymorphism.

Chapters 8 and 9(Balagurusamy)

UNIT-IV: The solution of Nonlinear Equations f(x)=0

Chapter2: Sec. 2.1 to 2.7(John H.Mathews)

Interpolation and Polynomial Approximation

Chapter 4: 4.1 to 4.4 (omit Sec. 4.5 & 4.6)(John H.Mathews)

UNIT-V : Curve Fitting.

Chapter 5: Sec. 5.1 to 5.3 (omit Sec. 5.4)( John H.Mathews).

Solution of Differential Equations.

Chapter 9: Sec. 9.1 to 9.6 (omit 9.7 to 9.9) (John H.Mathews)

Recomended Text:

1. E. Balagurusamy, Object Oriented Programming with C++, Tata McGraw Hill,

New Delhi, 1999.

2. John H.Mathews, Numerical Methods for Mathematics, Science and Engineering

(2nd

Edn.), Prentice Hall, New Delhi, 2000

55

Reference Books: 1. 1.D. Ravichandran, Programming with C++, Tata McGraw Hill, New Delhi, 1996

2. Conte and de Boor, Numerical Analysis, McGraw Hill, New York, 1990

Computer Laboratory Practice Exercises :

Section I : Computer Language Exercises for Programming in C++ :

1. Write a class to represent a vector (a series of float values). Include member

functions to perform the following tasks: To create the vector, To modify the value

of a given element, To multiply by a scalar value, To display the vector in the form

(10, 20, 30,…). Write a program to test your class.

2. Create a class FLOAT that contains one float data member. Overload all the

four arithmetic operators so that they operate on the objects of FLOAT.

3. Write a class called employee that contains a name and an employee number.

Include a member function to get data from the user for insertion into object, and

another function to display the data. Write a main() program to create an array of

employee information and accept information from the user and finally print the

information.

4. Write a program which shows the days from the start of year to date specified. Hold

the number of days for each month in an array. Allow the user to enter the month

and the day of the year. Then the program should display the total days till the day.

5. Write a program to use a common friend function to exchange the private values of

two classes.

6. Write a program to include all possible binary operator overloading using friend

function.

7. Write a program to read an array of integer numbers and sort it in descending order.

Use readdata, putdata, and arraymax as member functions in a class.

8. Write a program to read two character strings and use the overloaded ‘+’ operator to

append the second string to the first.

9. Write a function that takes two Distance values as arguments and returns the larger

one. Include a main() program that accept two Distance values from the user,

compare them and displays the larger.

10. Write a program to implement the concept of object as function argument and

returning objects.

11. Develop a program Railway Reservation System using Hybrid Inheritance and

Virtual Function.

56

12. Using overloaded constructor in a class write a program to add two complex

numbers.

13. Create a class MAT of size(m,n). Define all possible matrix operations for MAT

type objects.

14. Write a program that determines whether a given number is a prime number or not

and then prints the result using polymorphism.

Sections II : Numerical Methods Exercises for Programming in C++:

1. Non-Linear Equations

Bisection Method Regula-falsi Method Newton-Raphson Method Secant Method Fixed Point Iteration

2. Interpolation

Lagrange’s Interpolation Formula

Newton Interpolation

Formula 2.3

3. Curve Fitting

Least-Square line

Least-Square polynomial

Non linear curve fitting

4. Numerical Solution to Differential Equations

Euler’s Method

Taylor’s Method of order 4

Runge-Kutta Method of order 4

Milne-Simpson Method

57

Course Outcomes


CO Number


Level CO1 Describe the principles of object oriented programming. K1, K2 ,K3, K5

CO2 Apply the concepts of data encapsulation, inheritance in C++.

K1, K2, K3, K4,

K5

CO3 Understand basic program constructs in C++.

K1, K2, K3. K4,

K5

C04 Apply the concepts of classes, methods and inheritance to write C++

programs. K1,K2,K3

CO5 Using appropriate numerical methods, determine the solutions to given

non-linear equations K1, K2, K3. K4,

K5

CO6 Using appropriate numerical methods, determine approximate solutions to

systems of polynomial approximation. K1,K2,K3,K4

CO7 Using appropriate numerical methods, determine approximate solutions to

ordinary differential equations K1,K2,K3,K4

CO8 Demonstrate the use of interpolation methods to find intermediate values

in given graphical and/or tabulated data. K1,K2,K3,K4

216PMMT07 – WAVELETS

Objective: To initiate the study on the Discrete Fourier Transforms, Wavelets on Znm ., Wavelets on Z,

Wavelets and Differential Equations

UNIT-I: The Discrete Fourier Transforms Chapter 2: 2.1 to 2.3

UNIT-II: Wavelets on Znm . Chapter 3: 3.1 to 3.3

UNIT-III: Wavelets on Z. Chapter 4: 4.1 to 4.7

UNIT-IV: Wavelets on R. Chapter 5: 5.1 to 5.5

UNIT-V: Wavelets and Differential Equations. Chapter 6: 6.1 to 6.3 Recomended Text:

1. Michael W.Frazier, An Introduction to Wavelets through Linear Algebra, Springer

Verlag, Berlin, 1999

58

Reference Books

1. C.K.Chui, An Introduction to Wavelets, Academic Press, 1992 2. E.Hernandez and G.Weiss, A First Course in Wavelets, CRC Press, New York,1996 3. D.F.Walnut, Introduction to Wavelet Analysis, Birhauser, 2004.

Course Outcomes


CO Number

CO

Statement

Knowledge

Level

CO1 Understand the knowledge of the Discrete Fourier Transforms K1, K2 ,K3,K4

CO2 To learn about Wavelets on Znm K1, K2,K3

CO3 To understand the concept of Wavelets on Z and its applications. K1, K2, K3, K4

C04 To learn about the Wavelets on R K1,K2,K3,K4

CO5 To get the knowledge of Wavelets and Differential Equations K1,K2,K3

59

216PMMT08 - JAVA PROGRAMMING

Objective: To initiate the study of Java statements, Decision making and Branching, Arrays – Strings –

Vectors and Applet Programming.

UNIT-I: Java Tokens – Java statements – Constants – Variables – Data types. Chapters 3 and 4

UNIT-II: Operators – Expressions – Decision making and Branching. Chapters 5,6 and 7

UNIT-III: Classes – Objects – Methods – Arrays – Strings – Vectors – Multiple Inheritance. Chapters 8, 9 and 10

UNIT-IV : Multithreaded Programming – Managing errors and Exceptions. Chapters 12 and 13

UNIT-V : Applet Programming. Chapter 14

Recommended Text:

1. E. Balagurusamy, Programming with Java – A primer , Tata McGraw Hill

Publishing Company Limited, New Delhi, 1998.

Reference Books 1. Mitchell Waite and Robert Lafore, Data Structures and 2. Algorithms in Java, Techmedia (Indian Edition), New Delhi, 1999 3. Adam Drozdek, Data Structures and Algorithms in Java, (Brown/Cole),

Vikas Publishing House, New Delhi, 2001.

Computer Laboratory Exercises Section 1. CLASSES,

OBJECTS, INHERITANCE, INTERFACE 1. Design a class to represent a bank Account. Include the following members:

Data Members: Methods: (1) Name of the Depositor (1) To Assign initial values. (2) Account Number (2) To deposit an amount. (3) Type of account (3) To withdraw an amount after checking

tbe balance.

(4) Balance (4)To display the name and

balance. Write a Java program for handling 10 customers.

2. Java lacks a complex datatype. Write a complex class that represents a single

Complex number and includes methods for all the usual operations, ie:

addition, subtraction, multiplication, division.

Section 2 : EXCEPTION HANDLING, MULTITHREADING AND PACKAGES 3. Write a Java program to handle different types of exceptions using try, catch

and finally statements

4. Write a Java program to implement the behavior of threads. (a) To create and run threads. (b) To suspend and stop threads. (c) To move a thread from one state to another. (d) By assigning a prioity for each thread.

5. Create three classes Protection, Derived and SamePackage all in same package.

Class Protection is a base class for the class Derived and SamePackage is a seperate

60

class. Class Protection has three variables each of type private,protected and

public. Write a program that shows the legal protection modes of all the different

variables.

Section 3: APPLET PROGRAMMING 6. Write an applet to draw the following shapes : (a) Cone (b) Cylinder (c) Cube

(d) Square inside a circle (e) Circle inside a square. 7. Creating a Java applet which finds palindromes in sentences. Your applet will have

two input controls; One input will be a text field for entering sentences, the other

input will be a text field or scroll bar for selecting the minimum length a

palindrome to be shown. Your applet will output the first 10 palindromes it finds in

the sentence.

8. Write a program which displays a text message coming down the screen by

moving left to right and modify the above program instead of text moving from

left to right it moves top to bottom.

Section 4 : AWT FORMS DESIGN USING FRAMES 9. Create a frame that contains 3 text fields and four buttons for basic arithmetic

operations. You have to enter two numbers in first two text fields. On clicking

the respective button that answer should be displayed in the last text filed.

10. Create a frame with check box group containing Rectangle, Circle, Triangle,

Square. If the particular value is true then the corresponding shape should be

displayed.

Course Outcomes


CO Number

CO

Statement

Knowledge

Level

CO1 Understand the concept of Java Tokens, Java statements, Constants,

Variables and Data types. K1, K2 ,K3, K5

CO2 To learn about Operators, Expressions, Decision making and Branching. K1, K2,K3,

K4,K5

CO3 To learn the concept of c Objects – Methods – Arrays – Strings – Vectors –

Multiple Inheritance. K1,

K2,K3.K4,K5

C04 Explore the use of various operating systems commands on different platforms. K1,K2,K3

CO5 Students will be able to a better understanding of essential problem solving and

programming concepts K1,K2,K3

61

Elective - III (Group C)

316PMMT05 - ALGEBRAIC THEORY OF NUMBERS

Objective: To initiate the study of Algebraic background, Algebraic Numbers, Quadratic and

Cyclotomic Fields, Euclidean Quadratic fields , Consequences of unique factorization ,The Ramanujan

,Nagell Theorem and Prime Factorization of Ideals.

UNIT-I : Algebraic background: Rings and Fields- Factorization of Polynomials –

Field Extensions – Symmetric Polynomials – Modules – Free Abelian Groups.


UNIT-II : Algebraic Numbers: Algebraic numbers –Conjugates and Discriminants –

Algebraic Integers – Integral Bases – Norms and Traces – Rings of Integers.

Chapters 2: Sec. 2.1 to 2.6

UNIT-III : Quadratic and Cyclotomic Fields :Quadratic fields and cyclotomic fields

Factorization into Irreducibles: Trivial factorization – Factorization into irreducibles –

Examples of non-unique factorization into irreducibles.

Chapter 3: Sec. 3.1 and 3.2 ; Chapter 4: Sec. 4.1 to 4.4 UNIT-IV : Prime Factorization – Euclidean Domains – Euclidean Quadratic

fields - Consequences of unique factorization – The Ramanujan –Nagell Theorem. Chapter 4: Sec. 4.5 to 4.9

UNIT-V : Ideals: Prime Factorization of Ideals – The norms of an Ideal – Non-unique

Factorization in Cyclotomic Fields..

Chapter 5 : Sec. 5.1 to 5.4

Reference Books

1. IStewart and D.Tall. Algebraic Number Theory and Fermat’s Last Theorem (3rd

Edition) A.K.Peters Ltd., Natrick, Mass. 2002.

Recommended Text 1. Z.I.Borevic and I.R.Safarevic, Number Theory, Academic Press, New York, 1966. 2. J.W.S.Cassels and A.Frohlich, Algebraic Number Theory, Academic Press, New

York, 1967. 3. P.Ribenboim, Algebraic Numbers, Wiley, New York, 1972. 4. P. Samuel, Algebraic Theory of Numbers, Houghton Mifflin Company, Boston, 1970.

62

Course Outcomes


CO Number

CO

Statement Knowledge

Level

CO1 Understand the knowledge of Algebraic background K1, K2 ,K3, K5

CO2 To learn about the algebraic numbers, Conjugates,

discriminates, Algebraic Integers and Rings of Integers.

K1, K2,K3,

K4,K5

CO3 To get the knowledge of Quadratic and Cyclostomes Fields. K1, K2,K3

C04 To learn about the Prime Factorization and The Ramanujan ,Nagell

Theorem K1,K2,K3

CO5 To get the knowledge of Prime Factorization of Ideals. K1,K2,K3

63

316PMMT07 - STOCHASTIC PROCESSES

Objective: To initiate the study on Markov Chains, Limit Theorems of Markov Chains, Continuous

Time Markov Chains, Renewal Processes and Brownian Motion.

UNIT-I : Markov Chains :Classification of General Stochastic Processes – Markov Chain

– Examples – Transition Probability Matrix – Classifications of States – Recurrence –

Examples of recurrent Markov Chains.

Chapter 1 : Section 3 only. Chapter 2 : Sections 1 to 6 (Omit section 7)

UNIT-II: Limit Theorems of Markov Chains: Discrete renewal equation and its proof –

Absorption probabilities – criteria for recurrence – Queueing models – Random walk.

Chapter 3 : Sections 1 to 7

UNIT-III : Continuous Time Markov Chains: Poisson Process – Pure Birth Process

– Birth and Death Process – Birth and Death process with absorbing states – Finite

State Continuous time Markov Chains.

Chapter 1 : Section 2 (Poisson Process) Chapter 4 : Sections 1,2 and 4 to 7 (Omit sections 3 and 8)

UNIT-IV : Renewal Processes: Definition and related concepts – Some special Renewal

processes – Renewal equation and Elementary Renewal Theorem and its applications.


UNIT-V : Brownian Motion : Definition – Joint probabilities for Brownian Motion –

Continuity of paths and the maximum variables – Variations and extensions – Computing

some functional of Brownian Motion by Martingale methods. Chapter 1 : Section 2 (Brownian Motion) Chapter 6 : Sections 1 to 5 and 7A only (Omit Sections 6, and 7B,C)

Reference Books

S.Karlin and H.M.Taylor. A First Course in Stochastic Processes( 2nd

edition),

Academic Press, New York, 1975.

Rec ommended Text 1. Cinler E., Introduction to Stochastic Processes, Prentice Hall Inc., New Jersey, 1975 2. Cox D.R. & H.D.Miller, Theory of Stochastic Processes (3

rd Edn.), Chapman and Hall,

London, 1983 3. Kannan D., An Introduction to Stochastic Processes, North Holland, New York 1979 4. Ross S.M., Stochastic Processes, John Wiley and Sons, New ork,1983 5. H.W.Taylor and S.Karlin, An Introduction to Stochastic Modeling (3

rd Edition),

Academic Press, New York, 1998

64

Course Outcomes


CO Number

CO

Statement Knowledge

Level

CO1 Understand the knowledge of Classification of General Stochastic

Processes and Classifications of States. K1, K2 ,K3, K5

CO2 To learn about Limit Theorems of Markov Chains K1, K2, K3, K4,

K5

CO3 To get the knowledge of Birth and Death Process and

Finite State Continuous time Markov Chains. K1, K2,K3,K4

C04 To learn about Renewal equation and Elementary Renewal

Theorem and its applications.

K1,K2,K3,K4

CO5 To get the knowledge of some functional of Brownian Motion by

Martingale methods K1,K2,K3,K4

65

316PMMT08 - DATA STRUCTURES AND ALGORITHMS

COURSE OBJECTIVES:

1. To impart the basic concepts of data structures and algorithms

2. To understand concepts about searching and sorting techniques

3. To Understand basic concepts about stacks,queues,lists,trees and graphs

4. To understanding about writing algorithms and step by step approach in solving problems with the

help of fundamental data structures

UNIT-I : Algorithms and Elementary Data Structures :

Algorithms – Structures programs – Analysis of algorithms – Stacks and Queues – Trees

– Heaps and Heapsort – Sets and disjoint set union – Graphs – Hashing. Chapter 1 : Sections 1.1 to 1.4 Chapter 2 : Sections 2.1 to 2.6

UNIT-II : The Divide and Conquer Method : The general method – Binary search –

Finding the maximum and minimum – Mergesort – Quicksort – Selection sort –

Strassen’s matrix multiplication.


UNIT-III : The Greedy Method : The General method – Optimal storage on tapes –

Knapsack problem – Job Scheduling with deadlines – Optimal merge pattern –

Minimum spanning trees – Single source shortest paths. Chapter 4 : Sections 4.1 to 4.7

UNIT-IV : Backtracking : The general methods – The 8-queens problem - sum of

subsets – Graph colouring – Hamiltonian Cycles – Knapsack problem. Chapter 7 : Section 7.1 to 7.6

UNIT-V : Branch-and-Bound and NP-Hard and NP-Complete problems : Branch and

Bound Method – 0/1 knapsack problem – Traveling salesperson – Efficiency

Considerations

– Basic concepts of NP-Hard problems – Cook’s theorem - NP-Hard graph problems -

NP- Hard Scheduling Problems. Chapter 8 : Sections 8.1 to 8.4

Chapter 11: Sections 11.1 to 11.4 (omit 11.5 and 11.6)

Reference Books orowitz and S.Sahni. Fundamentals of Computer Algorithm, Galgotia lications New Delhi, 1984

Recommended Text 1. D.E.Knuth, The Art of Computer Programming, Sorting and Searching.

Vol.3. Addism tresher mass.1973.

66

2. A.Nijenhuis and H.S.Wilf, Combinatorial Algorithms, Academic Press. New York 1975. 3. M.Garey and D.Johnson, Computers and Intractability: A Guide to the theory

of NP.Completeners. Johnson, Freeman and San Francisco, 1979.

4. A.V.Aho, J.E.Hoperoft, JD Ullman, The Design and Analysis of Computer

Algorithms. Addison – Wesley, Reading’, MASS. 1974.

Course Outcomes


CO Number

CO

Statement

Knowledge

Level

CO1 Ability to analyze algorithms and algorithm correctness. K1, K2 ,K3

CO2 Ability to summarize searching and sorting techniques

K1, K2,K3

CO3 Ability to describe stack, queue and linked list operation. K1, K2, K3.

K4, K5

C04 Ability to have knowledge of tree and graphs concepts.

K1,K2,K3,K4

67

ELECTIVES - IV (GROUP D)

416PMMT05 - COMBINATORICS

Objective: To initiate the study Classical Techniques, Polya Theory, Schur Functions,

Character Theory of Sn and Inversion Techniques.

UNIT-I : Classical Techniques : Basic combinatorial numbers - Generator functions and

Recurrence Relations – Symmetric functions – Multinomials – Inclusion and Exclusion

Principle.

Chapter 1 : Sections 1 to 5 only (avoid 6 ).

UNIT-II : Polya Theory : Necklace problem and Burnside’s lemma – cycle Index of

Permutation group – Polya’s Theorems and their applications – Binary operations on

permutation Groups.

Chapter 2 : Sections 1 to 4 only

UNIT-III: Schur Functions: Robinson–Schensted–Knuth correspondence-Combinatorics

of the Schur Functions. More on Schur functions: Little wood – Richardson Rule –

Plethysm and Polya process – The Hook formula.

Chapter 3 : Sections 1 & 2 only .

Chapter 5 : Sections 1 to 3 only

UNIT-IV : Character Theory of Sn : Character Theory of finite groups .

Matching Theory : Partially ordered set – Basic Existence Theory

Chapter 6 : Sections 1 & 2 only

UNIT-V : Inversion Techniques : Classical Inversion Formulae Inversion via Mobius

Function.

Designs : Existence and construction . Chapter 7 : Sections 1 & 2 only. Chapter 8 : Section 1 only Ramsey Theory : Ramsey Theorem. Chapter 9 : Section 1 only

Reference Books

V. Krishnamurthy, Combinatorics – Theory and Applications, Affiliated East – West Press

Pvt Ltd, New Delhi . 1985.

Recommended Text 1. Aigner, M. Combinatorial Theory, Springer Verlag, Berlin 1979. 2. Liu, C.L. Introduction to combinatorial Mathematics . MC 3. Grimaldi,R.P. Discrete and Combinatorial Mathematics : An applied Introduction (

4th

Edition ).Pearson, (8th

Indian Print)

68

Course Outcomes


CO Number

CO

Statement

Knowledge

Level

CO1 Understand the Basic combinatorial numbers, Symmetric functions and

Multinomial’s. K1, K2 ,K3

CO2 To learn the Necklace problem and Burnside’s lemma, cycle Index of

Permutation group,Polya’s Theorems and their applications K1, K2,K3,

K4,K5

CO3 An ability to use the techniques in Schur functions and Richardson Rule

K1, K2, K3,

K4,K5

C04 An understanding the Character Theory of finite groups

K1,K2,K3,K4

CO5 Knowledge of Classical Inversion Formulae Inversion via

Mobius Function. K1,K2,K3,K4,

K5

69

416PMMT06 - MATHEMATICAL STATISTICS

Objective: To initiate the study on Sample Moments and their Functions, Significance Test,

Estimation, Analysis of Variance and Sequential Analysis

UNIT-I : Sample Moments and their Functions:

Notion of a sample and a statistic – Distribution functions of X, S2 and ( , S

2 ) -

2

distribution – Student t-distribution – Fisher’s Z-distribution – Snedecor’s F- distribution –

Distribution of sample mean from non-normal populations


UNIT-II : Significance Test : Concept of a statistical test – Parametric tests for small

samples and large samples - 2 test – Kolmogorov Theorem – Smirnov Theorem – Tests

of Kolmogorov and Smirnov type – The Wald-Wolfovitz and Wilcoxon-Mann-Whitney

tests – Independence Tests by contingency tables. Chapter 10 : Sections 10.11 Chapter 11 : 12.1 to 12.7.

UNIT-III : Estimation : Preliminary notion – Consistency estimation – Unbiased

estimates – Sufficiency – Efficiency – Asymptotically most efficient estimates – methods

of finding estimates – confidence Interval.

Chapter 13 : Sections 13.1 to 13.8 (Omit Section 13.9)

UNIT-IV : Analysis of Variance : One way classification and two-way classification.

Hypotheses Testing: Poser functions – OC function- Most Powerful test – Uniformly

most powerful test – unbiased test.

Chapter 15 : Sections 15.1 and 15.2 (Omit Section 15.3) Chapter 16 : Sections 16.1 to 16.5 (Omit Section 16.6 and 16.7)

UNIT-V : Sequential Analysis : SPRT – Auxiliary Theorem – Wald’s fundamental identity

– OC function and SPRT – E(n) and Determination of A and B – Testing a

hypothesis concerning p on 0-1 distribution and m in Normal distribution.

Chapter 17 : Sections 17.1 to 17.9 ( Omit Section 17.10)

Reference Books

M. Fisz , Probability Theory and Mathematical Statistics, John Wiley and sons, New

Your, 1963.

Recommended Text 1. E.J.Dudewicz and S.N.Mishra , Modern Mathematical Statistics, John Wiley

and Sons, New York, 1988.

2. V.K.Rohatgi An Introduction to Probability Theory and Mathematical Statistics,

Wiley Eastern New Delhi, 1988(3rd

Edn ) 3. G.G.Roussas, A First Course in Mathematical Statistics,

Addison Wesley Publishing Company, 1973 4. B.L.Van der Waerden, Mathematical Statistics,

G.Allen & Unwin Ltd., London, 1968.

70

Course Outcomes


CO Number

CO

Statement

Knowledge

Level

CO1 Understand the basic concept of Distribution functions of X, S2, Student t-

distribution and Fisher’s Z-distribution. K1, K2 ,K3, K5

CO2 To learn the Concept of a statistical test, Kolmogorov Theorem, Tests of

Kolmogorov and Smirnov type. K1, K2,K3,

K4,K5

CO3 Ability to Asymptotically most efficient estimates and its applications.

K1,

K2,K3.K4,K5

C04 An understanding of Poser functions and Uniformly most

powerful test , unbiased test. K1,K2,K3

CO5 A knowledge of Auxiliary Theorem and Wald’s fundamental identity.

K1,K2,K3

71

416PMMT07 - ALGEBRAIC TOPOLOGY

Objective: To initiate the study on Homotopy of paths, Deformation Retracts and Homotopy, The

Fundamental Group of a wedge of circles and constructing compact surfaces

UNIT-I : Homotopy of paths - Fundamental Group – Covering space -The Fundamental Group of the circle

– Retractions and Fixed points.

Chapter 9: Sections 51 – 55.

UNIT-II : The Fundamental Theorem of Algebra – Borsuk–Ulam Theorem – Deformation

Retracts and Homotopy Type – The Fundamental Group of S n - Fundamental Groups of

some surfaces.

Chapter 9 : Sections 56 – 60

UNIT-III : Direct sums of Abelian Groups – Free products of Groups – Free Groups – The Seifert–

van Kampen Theorem – The Fundamental Group of a wedge of circles.

Chapter 11 : Sections 67 -71.

UNIT-IV : Fundamental groups of surfaces – Homology of surfaces – cutting and pasting

– The classification theorem – constructing compact surfaces. Chapter 12 : Sections 74 – 78

UNIT-V : Equivalence of covering spaces – The Universal covering space –

covering transformations – Existence of covering spaces

Chapter 13 : Sections 79 – 82

Reference Books J.R.Munkres, Toplogy, Pearson Education Asia , Second Edition 2002.

Rec ommended Text 1. M.K.Agoston, Algebraic topology – A First Course, Marcel Dekker, 1962. 2. Satya Deo, Algebraic Topology , Hindustan Book Agency, New Delhi, 2003. 3. M.Greenberg and Harper, Algebraic Topology – A First course,

Benjamin/Cummings, 1981. 4. C.F. Maunder, Algebraic topology, Van Nostrand, New York, 1970. 5. A.Hatcher, Algebraic Topology, Cambridge University Press, South Asian Edition 2002. 6. W.S.Massey, Algebrai Topology : An Introduction, Springer 1990

72

Course Outcomes


CO Number

CO

Statement

Knowledge

Level

CO1 Understand the concept of Homotopy of paths ,Fundamental Group – and

Covering space K1, K2 ,K3

CO2 To learn about the Fundamental Theorem of Algebra – Borsuk–Ulam

Theorem – Deformation Retracts and Homotopy Type K1, K2,K3,

K4,K5

CO3 Knowledge the concept of Fundamental Group of a wedge of circles. K1,

K2,K3.K4,K5

C04 To learn the fundamental groups of surfaces and Homology of surfaces K1,K2,K3,K4

CO5 A recognition of the need for Equivalence of covering spaces ,the Universal

covering space and covering transformations K1,K2,K3

73

ELECTIVE - V (GROUP E)

416PMMT09 - MATHEMATICAL PHYSICS

Objective: To initiate the study on Integral Equations, Sturm, Methods of Non linear Dynamics and Non linear Differential Equations and their solutions.

UNIT-I : Integral Equations, Sturm–Liouville Theorem and Green’s Functions Chapter 4: Sections 4.1 – 4.4 only.

UNIT-II : Methods of Non linear Dynamics – I : Phase Portraits .

Chapter 6 : Sections 6.1 – 6.4 only.

UNIT-III : Methods of Non linear Dynamics - II : Stability and Bifurcation. Chapter 7 : Sections 7.1 -7.4 only .

UNIT-IV : Non linear Differential Equations and their solutions. Chapter 8 : Sections 8.1 - 8.3 only

UNIT-V : Non linear Integral Equations and their solutions. Chapter 9 : Sections 9.1 – 9.7 only.

Reference Books R.S.Kaushal and D.Parashar, Advanced Methods of Mathematical Physics.

Recommended Text

1. Arfken,G (1966) Mathematical Methods for Physicists, A.P.NY. 2. Butkor,E. (1968) Mathematical Physics, Addison –Wesley. 3. Strogatz,S.H.(1994) Non linear Dynamics and Chaos : With Applications to

Physics, Biology, Chemistry and Engineering Addison – Wesley

4. Tabor, M (1989) Chaos and integrability in Non linear systems: An Introduction,

John Wiley & Sons,NY. 5. Lakshmanan, M (1988). Solitons : Introduction and Applications, Springer Verlag, Berlin 6. Debnath, Lokenath ( 1997). Introduction to Non linear PDE for

Scientists and Engineers , Birkhauser , Boston

Course Outcomes


CO Number

CO

Statement

Knowledge

Level

CO1 Understand the concept of Integral Equations, Sturm–Liouville Theorem and

Green’s Functions K1, K2 ,K3

CO2 To learn the Methods of Non linear Dynamics – I and Phase Portraits. K1, K2,K3,K4

CO3 An understanding the Methods of Non linear Dynamics – I, Stability and

Bifurcation. K1, K2,K3.K4

C04 To get the knowledge of Non linear Differential Equations and their solutions K1,K2,K3

CO5 A knowledge of Non linear Integral Equations and their solutions K1,K2,K3

74

416PMMT10 - FINANCIAL MATHEMATICS

Objective: To initiate the study on Binary Model, American Options, Discrete parameter martingales and

Markov processes, Levy’s Construction of Brownian Motion and Foreign Exchange Dividends

UNIT-I : Binary Model – a ternary Model – Characterization of no arbitrage – Risk-Neutral

Probability Measure

Chapter 1

UNIT-II : Binomial Trees and Discrete parameter martingales

Multi-period Binary model – American Options – Discrete parameter martingales and

Markov processes – Martingale Theorems – Binomial Representation Theorem – Overture

to Continuous models

Chapter 2

UNIT-III : Brownian Motion : Definition of the process – Levy’sConstruction

of Brownian Motion – The Reflection Principle and Scaling – Martingales in Continuous

time.

Chapter 3

UNIT-IV : Stochastic Calculus : Stock Prices are not differentiable – Stochastic

Integration – Ito’s formula – Integration by parts and Stochastic Fubini Theorem–Girsanov

Theorem – Brownian Martingale Representation Theorem – Geometric Brownian Motion –

The Feynman-Kac Representation

Chapter 4

UNIT-V : Block-Scholes Model : Basic Block-Scholes Model – Block-Scholes price and

hedge for European Options – Foreign Exchange – Dividends – Bonds – Market price of

risk. Chapter 5

Reference Books

Alison Etheridge, A Course in Financial Calculus, Cambridge University Press,

Cambridge, 2002.

Recommended Text

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

Farancois Mantion ), 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 (3rd

Printing)

75

Course Outcomes


CO Number

CO

Statement

Knowledge

Level

CO1 To understand the binary Model, Characterization of no

arbitrage, Risk-Neutral Probability Measure.

K1, K2 ,K3, K5

CO2 An ability to function on Multi-period Binary model – American

Options, Binomial Representation Theorem K1, K2,K3,K4.

CO3 To get the definition of the process and The Reflection Principle and Scaling. K1, K2, K3, K4,

K5

C04 An understanding Integration by parts and Stochastic Fubini Theorem

and Brownian Martingale Representation Theorem. K1,K2,K3,K4

CO5 To get the knowledge of Basic Block-Scholes Model, Block-Scholes price and hedge for European Options and s Foreign Exchange

K1,K2,K3,K4

76

416PMMT11 - CALCULUS OF VARIATIONS AND INTEGRAL EQUATIONS

Objective: To initiate the study on The Method of Variations in Problems with Fixed Boundaries, Variational Problems with Moving Boundaries and and Integral Equations with Separable

UNIT-I : The Method of Variations in Problems with Fixed Boundaries

Chapter 6 : Sections 1 to 7 (Elsgolts)

UNIT-II : Variational Problems with Moving Boundaries and certain other problems

and Sufficient conditions for an Extremum

Chapter 7 : Sections 1 to 4 (Elsgolts)

Chapter 8 : Sections 1to 3(Elsgolts)

UNIT-III : Variational Problems Involving a conditional Extremum

Chapter 9 : Sections 1 to 3. (Elsgolts)

UNIT-IV : Integral Equations with Separable Kernels and Method of successive

approximations.

Chapter 1 : Sections 1.1 to 1.7 (Kanwal) Chapter 2 : Sections 2.1 to 2.5 (Kanwal)

Chapter 3 : Sections 3.1 to 3.5 (Kanwal)

NIT-V: Classical Fredholm Theory , Symmetric Kernels and Singular Integral Equations

Chapter 4 : Sections 4.1 to 4.5 (Kanwal) Chapter 7 : Sections 7.1 to 7.6 (Kanwal)

Chapter 8 : Sections 8.1 to 8.5 (Kanwal)

Recommended Text

For Units I,II and III : L. Elsgolts , Differential Equations and the Calculus of

Variations, Mir Publishers, Moscow, 1973 (2nd Edition)

For Units IV and V :Ram P.Kanwal,Linear Integral Equations, Academic Press, New

York, 1971.

Reference Books

1. I.M.Gelfand and S.V.Fomin, Calculus of Variations, Prentice-Hall Inc. New Jersey, 1963. 2. A.S.Gupta, Calculus of Variations with Applications, Prentice-Hall of India, New

Delhi, 1997. 3. M.Krasnov, A.Kiselev and G.Makarenko, Problems and Exercises in Integral Equations,

Mir Publishers, Moscow, 1979.

77

4. S.G.Mikhlin, Linear Integral Equations, Hindustan Publishing Corp. Delhi,1960. 5. L.A.Pars, An Introduction to the Calculus of Variations, Heinemann, London, 1965. 6. R.Weinstock, Calculus of Variations with

Applications to Physics and Engineering, McGraw-

Hill Book Company Inc. New York, 1952.

Course Outcomes


CO Number

CO

Statement

Knowledge

Level

CO1 Understand the concept of The Method of Variations in Problems with Fixed Boundaries.

K1, K2 ,K3

CO2 To learn the concept of Variational Problems with Moving Boundaries and

certain other problems and Sufficient conditions for an Extremum. K1, K2,K3,

K4,K5

CO3 An understanding of Variational Problems Involving a conditional Extremum

problems .

K1,

K2,K3.K4,K5

C04 To learn the concept of Integral Equations with Separable Kernels and Method

of successive approximations. K1,K2,K3,K4

CO5 To get the knowledge of Classical Fredholm Theory , Symmetric Kernels and

Singular Integral Equations K1,K2,K3,K4

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