St. Peter s Institute of Higher Education and Research

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)

VISION To assist students in acquiring a conceptual understanding of the nature and structure of Mathematics, its processes and applications. MISSION To provide high quality Mathematics graduates who are relevant to industry and Commerce, Mathematics Education and Research in Science and Technology. Program Educational Objectives (PEOs) PEO1 PEO2 PEO3 PEO4 PEO5 Graduates are expected to have the ability to pursue interdepartmental research in Universities in India and abroad. Graduates are expected to have the ability to pursue interdepartmental research in Universities in India and abroad. Graduates are expected to have the caliber to work in foreign Universities. Graduates are expected to shine in higher level of administration like IAS, IPS officers in Nationalized Banks, LIC, and etc,. Graduates are expected to run renowned Educational institutions to serve the society.

Program Outcomes (POs) At the end of the M.Sc., (Mathematics) programme, the graduates will be able to PO1 PO2 PO3 PO4 Identify, formulate, and analyze the complex problems using the principles of Mathematics. Solve critical problems by applying the Mathematical tools. Apply the Mathematical concepts, in all the fields of learning including higher research, and recognize the need and preparefor lifelong learning. Able to crack competitive examinations, lectureship and fellowsip exams approved by UGC like CSIR NET and SET PO5 Apply ethical principles and commit to professional ethics, responsibilities and norms in the society. Program Specific Outcomes(PSOs) PSO1 PSO2 PSO3 To understand the basic concepts of advanced Mathematics. To develops the problem solving skill. To creates Mathematical Models.

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: I SEMESTER Code No. Course Title Credit Marks Theory CA EA Total 116PMMT01 Algebra-I 5 25 75 100

116PMMT02 Real Analysis-I 5 25 75 100 116PMMT03 Core Sub: 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 Algebra-II 5 25 75 100 216PMMT02 Core Sub: 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 III SEMESTER Code No. Course Title Credit Marks CA EA Total 316PMMT01 Complex Analysis-I 5 25 75 100 316PMMT02 Core Sub: 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 Marks Code No. Course Title Credit CA EA Total

416PMMT01 Complex Analysis-II 5 25 75 100 416PMMT02 Core Sub: 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 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 s 4 316PMMT06 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. NVERSION 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 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. levels: For every course outcome the knowledge level is measure as, K1 Remember, K2 Understand, K3 Apply, K4 Analyze, K5 Evaluate and K6 - Create Registrar

M.Sc. DEGREE URSE IN MATHEMATICS SYLLABUS 116PMMT01 - ALGEBRA I OBJECTIVES : To introduce the concepts and to develop working knowledge on class equation, solvability of groups, finite abelian groups, linear transformations, real quadratic forms. 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.

1 2 3 5 Statement Understand the basic concepts of Counting principle, finite groups, applications and Sylow's theorems Learn about groups, abelian groups and Modules Get the knowledge of Canonical forms and Triangular form Apply the knowledge of Jordan form - rational canonical form Learn about Hermitian, and real quadratic form K1, K2,K3 K1, K2,K3,K5 K1, K2,K3.K4,K5 116PMMT02 - REAL ANALYSIS - I OBJECTIVES : To 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 Recommended Text: Tom M.Apostol : Mathematical Analysis, 2 nd Edition, Narosa,1989. Reference Books 1. Bartle, R.G. Real Analysis, John Wiley and Sons Inc., 1976. 2. Rudin,W. Principles of Mathematical Analysis, 3 rd 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. 1 2 3 Statement Understand the basics concept of monotonic functions, continuous functions, properties, Dirichlet's test, Abel's test, and Riemann's theorem Learn about Riemann Integral Euler s summation formula and properties of upper and lower integrals Get the knowledge of Stieltjes integrals, Second Mean Value Theorem and Sufficient condition for uniform convergence and Mean convergence Learn about the infinite Series, infinite Products and Power series K1, K2,K3,K4,K5 K1, K2,K3, K4 K1, K2,K3

5 Apply the knowledge of Sequences of Functions,K4, K5 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 (3 rd 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. 1 STATEMENTS Understand the definition of Linear equations with constant coefficients,k4,k5 2 To study the function of Homogeneous, non-homogeneous functions and calculate the Initial value problems,k4 3 To have knowledge about Initial value problems,existence and uniqueness,k4 theorems 4 To learn about the Linear equation with regular singular points,k4 5 Get the knowledge of Lipschitz condition Convergence of the successive approximations and the existence theorem,k4 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 ana Bonds Cut Vertices. Chapter 1 (Section 1.1 1.7) Chapter 2 (Section 2.1 2.3) UNIT-II : Connectivity, Euler tours and Hamilton Cycles : Connectivity Blocks Euler tours Hamilton Cycles. Chapter 3 (Section 3.1 3.2) Chapter 4 (Section 4.1 4.2) UNIT-III: Matching s, Edge Colorings: Matchings Matching s and Coverings in Bipartite Graphs Edge Chromatic Vizing s Theorem. Chapter 5 (Section 5.1 5.2) Chapter 6 (Section 6.1 6.2) UNIT-IV: Independent sets and Cliques, Vertex Colorings: Independent sets Ramsey s Theorem Chromatic Brooks Theorem Chromatic Polynomials. Chapter 7 (Section 7.1 7.2)

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, 4 th Edition, 2004, Indian Print. 6. 6. S.A.Choudum, A First Course in Graph Theory, MacMillan India Ltd. 1987. 1 2 STATEMENTS Understand the various types of Graphs, sub graphs,trees and paths To learn about the Connectivity, Blocks and Hamilton Cycles,K4,K4 3 Apply the basics concepts of Matching, Edge Colorings. 4 To understand about Ramsey s Theorem and Brooks Theorem,K4 5 Knows the fundamental of Planar graphs.

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(3 rd Edn.), Computer Science Press, New York. 3. S.Wiitala, Discrete Mathematics- A Unified Approach, McGraw Hill Book Co. STATEMENTS 1 Understand the definition of Lattice and Boolean algebra.,k4 2 Using Bayes problems to solve conditional probability problem and applications. K1,K2 3 To learn about finite field problems

4 Identified the concept of Polynomials over Finite fields,k4,k5 5 Understand the concept of Linear Codes and Cyclic Codes 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 5: Sections 5.7 and 5.8 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.

Statement 1 Understand the definition of Extension fields K1, K2,K3, K5 2 Apply the concepts of Roots or Polynomials K1, K2,K3, 3 5 To understand about Elements of Galois theory Using the concept of Finite fields - Wedderburn's theorem Get the knowledge of Galois groups and A theorem of Frobenius K4,K5 K1, K2,K3.K4,K5 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 R n to R 1 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, 2 nd 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. Statement 1 Understand the definition Measure on the Real line K1, K2,K3,K4, K5 2 Apply the concept of Riemann and Lebesgue Integrals K1, K2,K3, 3 5 Analyze the Riesz-Fischer Theorem,, The Riesz-Fischer Theorem, Riemann's localization theorem and Fejes's theorem Understand the concept of the Directional derivative, The total derivative and Taylor's theorem Have the idea of Implicit Functions and Extremum Problems K4,K5 K1, K2,K3.K4,K5 K1, K2,K3,K4, K5 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, 2 nd Edition, Prentice Hall of India, New Delhi. 2005 Reference Books 1. R.C.McOwen, Partial Differential Equations, 2 nd 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 Statement 1 Understand the rules of Partial Differential Equations of First Order K1, K2,K3, K5 2 Learn about Dirichlet s Problem, Solution of Laplace equation in K1, K2,K3,K4 Cylindrical and spherical coordinates 3 An ability to calculate Parabolic Differential Equations K1, K2,K3.K4

5 Have the idea of D Alembert s solution, one-dimensional wave equation and Duhamel s Principle with Examples Apply the concepts of Green s Function, Laplace Transform and Fourier Transform 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. Chapter 3 : Sections 3.1 to 3.8 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. Chapter 4 : Sections 4.1 to 4.7 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, (2 nd Edition) Duxbury Press, New York, 1996. 4. V.K.Rohatgi An Introduction to Probability Theory and Mathematical Statistics, Wiley Eastern Ltd., New Delhi, 1988(3 rd Print). 5. S.I.Resnick, A Probability Path, Birhauser, Berlin,1999. 6. B.R.Bhat, Modern Probability Theory (3 rd Edition), New Age International (P)Ltd, New Delhi, 1999.

1 2 3 5 Statement Apply the knowledge of Random Events and Random Variables Get the knowledge of Parameters of the Distribution Describe fundamental properties,characteristic functions and Properties Apply the concepts of One point, two point, Binomial, Poisson, Uniform, normal, gamma, Beta and its applications. Construct the rigorous methods Stochastic convergence, Levy- Cramer Theorems, de Moivre-Laplace, Chebyshev Theorem K1, K2,K3,K4,K5 K1, K2,K3,K4,K5 K1, K2,K3.K4,K5, K4,K5 K4,K5 ELECTIVE - V 216PMMT06 - MATHEMATICAL PROGRAMMING OBJECTIVES: This course introduces advanced topics in Linear and non-linear Programming 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. Zero-One Integer Programming. Dynamic Programming: Characteristics of Dynamic Programming Problem Developing Optimal Decision Policy Dynamic Programming under Certainty DP approach to solve LPP. Chapters: 7 and 21. UNIT-II : 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 Beale s Method. Chapters: 22 and 23 UNIT-III: Theory of Simplex method: Canonical and Standard form of LP Slack and Surplus Variables Reduction of any feasible solution to a Basic Feasible solution Alternative Optimal solution Unbounded solution Optimality conditions Some complications and their resolutions Degeneracy and its resolution. Chapter 24 UNIT-IV : Revised Simplex Method: Standard forms for Revised simplex Method Computational procedure for Standard form I comparison of simplex method and Revised simplex Method. Bounded Variables LP problem: The simplex algorithm Chapters 25 and 27

UNIT-V: Parametric Linear Programming: Variation in the coefficients c j, Variations in the Right hand side, b i. 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 28 and 29. Recomended Text: 1. 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 (7 th Edition) Tata- McGraw Hill Company, New Delhi, 2001. 3. Beightler. C, D.Phillips, B. Wilde,Foundations of Optimization (2 nd Edition) Prentice Hall Pvt Ltd., New York, 1979 4. S.S. Rao - Optimization Theory and Applications, Wiley Eastern Ltd. New Delhi. 1990 1 2 3 5 Statement Understand the definition of Integer Linear Programming and Dynamic K1, K2,K3, K5 Programming Use the Classical Optimization Methods and Non-linear Programming Methods K1, K2,K3,K4,K5 Understand the Theory of Simplex method and Some complications K1, K2,K3.K4,K5 Have the idea of Revised Simplex Method Calculate the Parametric Linear Programming and Goal Programming, K4,K5,K4,K5

III SEMESTER 316PMMT01 - MPLEX 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. Chapter 4: Sections 6.4 and 6.5 Chapter 5: Sections 1.1 to 1.3 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, (3 rd 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 1 2 3 Statement To study the Cauchy s Integral Formula and Local Properties of Analytical Functions Understand the general form of Cauchy s Theorem Learn the basic concept of integrals, Harmonic functions, Poisson formula and basic properties. K1, K2,K3, K4 K5 K1, K2,K3, K4,K5 K1, K2,K3.K4,K5 Define and recognize the harmonic functions and Power Series Expansions 5 Get the knowledge of Partial Fractions and Entire Functions 316PMMT02 - TOPOLOGY OBJECTIVES: To study topological spaces, continuous functions, connectedness, compactness, countability and separation axioms. 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: T 1 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 (2 nd 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 Statement 1 Learn the basic concept of metric Spaces and Topological Spaces. K1, K2,K3, K5 2 Get the knowledge of Compact spaces and its uses. K1, K2,K3, 3 5 Apply the concepts of Tychonoff s theorem and Ascoli s theorem An understanding of Urysohn s lemma and the Tietze extension theorem Have the knowledge of the Weierstrass approximation Theorem and its uses. K4,K5 K1, K2,K3,K5

316PMMT03 - OPERATION RESEARCH OBJECTIVES: This course aims to introduce decision theory, PERT, CPM, deterministic and probabilistic inventory systems, queues, replacement and maintenance problems. 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. Chapter 6: Sections 6.1 to 6.6 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. Chapter 13: Sec. 13.1 to 13.8 Chapter 14: Sec. 14.1 to 14.3 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. Chapter 16: Sec. 16.1 to 16.5 Recommended Text 1. For Unit 2 : H.A. Taha, Operations Research, 6 th edition, Prentice Hall of India 2. For all other Units: J.K.Sharma, Operations Research, MacMillan India, New Delhi, 2001. Reference Books 1. F.S. Hiller and J.Lieberman -,Introduction to Operations Research (7 th Edition), Tata McGraw Hill Publishing Company, New Delhui, 2001. 2. Beightler. C, D.Phillips, B. Wilde,Foundations of Optimization (2 nd 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, (3 rd Edition), Wiley and Sons, New York, 1998. 1 2 3 Statement Explain the basics concepts of Decision Theory and solve Bayesian problems Get the knowledge and reproduce of Network Model and Linear Programming problem. Demonstrate the basic concepts of Deterministic Inventory Control Models problem and their application. Apply the concepts of Queueing theory in day to day life, K5 K1, K2,K3, K5 K1, K2,K3, K5 K1, K2,K3, K5 5 Get the knowledge of Replacement and Maintenance Models in which time is machine replaced or repaired. K1, K2,K3, K5 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 Chapter 1 : Sections 1.1 to 1.5 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 Chapter 5 : Sections 5.1 to 5.3 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, (2 nd 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 (3 rd Edition) McGraw Hill Book Co., New York, 1970. Statement 1 Understand the basic of Mechanical Systems. K1, K2,K3 2 Learn about the Lagrange's Equations K1, K2,K3,K5 3 5 To understand the Hamilton's Equation and uses. To learn about Hamilton-Jacobi Theory and lemma. To get the knowledge of Canonical Transformation.K4 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 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 Theory and Cryptography, Springer-Verlag, New York,1987 Rec ommended Text 1.I. Niven and H.S.Zuckermann, An Introduction to Theory of s (Edn. 3), Wiley Eastern Ltd., New Delhi,1976 2. David M.Burton, Elementary Theory, Brown Publishers, Iowa,1989 3. K.Ireland and M.Rosen, A Classical Introduction to Modern Theory, Springer Verlag, 1972 4. N.Koblitz, Algebraic Aspects of Cryptography, Springer 1998 Statement 1 Understand the basic concept of Euclidean algorithm and its applications. K1, K2,K3 2 Introduction to Classical Crypto systems and Enciphering matrices. K1, K2,K3 3 5 To get the knowledge of Finite Field. To learn about the Public Key Cryptography. Get the knowledge of Elliptic curve. K1, K2, K3. IV SEMESTER 416PMMT01 - MPLEX 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 ( ). Chapter 7: Sections 3.1 to 3.5 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. Chapter 8: Sections 1.1 to 1.7 Recommended Text Lars V. Ahlfors, Complex Analysis, (3 rd 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 Statement 1 Understand the zeros of zeta function and Arzela s theorem. K1, K2,K3, 2 Learn about the Riemann mapping Theorem. K1, K2,K3, 3 5 Get the knowledge of Elliptic functions. To understand about Weierstrass -function The functions (s). Get the knowledge of Analytic Continuation. K4 K1, K2,K3

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: Nonintrinsic 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,(17 th 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.

1 2 3 5 Statement Understand the basic concept of definition of a space curve curvature, torsion, involutes and evolutes. To get the knowledge of intrinsic properties of a surface To get the knowledge of Geodesics and its properties. To understand about the second fundamental form To have knowledge about Differential Geometry of Surfaces and characterization of Complete surfaces K1, K2,K3.S K1, K2,K3,K4 K1, K2,K3.K4,K5 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 Chapter 10 : Sections 55 to 59. 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. Chapter 12: Sections 64 to 69. UNIT-V: Structure of commutative Banach Algebras: Gelfand mapping Spectral radius formula Involutions in Banach Algebras Gelfand-Neumark Theorem. Chapter 13: Sections 70 to 73. Recommended Text 1. G.F.Simmons, Introduction totopology and Modern Analysis, McGraw Hill International Book Company, New York, 1963.