Curriculum of M. Phil., Mathematics After revision 2017-2018 · [1] Erwin Kreyszig, Introductory...
Transcript of Curriculum of M. Phil., Mathematics After revision 2017-2018 · [1] Erwin Kreyszig, Introductory...
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TIRUCHIRAPPALLI – 620 024
Department of Mathematics
Curriculum Structure
New Syllabus
List of Subjects with codes
Total Credits-
Internal marks-40 External marks-60
* COURSES (CC)
Code Title of the Course Lecture Hours
Tutorial Hours
Practical Hours
Credits Prerequisite (Exposure)
17MP01CC Research Methodology 4 2 0 4 Nil
17MP02CC Algebra and Analysis 4 2 0 4 Nil
17MP03CC Paper on the Topic of Research
(To be framed by the Guide) 4 2 0 4 Nil
17MP04CC Teaching and Learning Skills
(Common Paper) 4 2 0 4 Nil
For each Course other than the Project
Continuous Internal Assessment (CIA) – 40 Marks
End Semester Examination (ESE) – 60 Marks
Total – 100 Marks
ESE Duration – 3 Hours.
For Dissertation
See Annexure (University Regulations)
Question paper pattern and CIA components
10 questions compulsory 10 x 01 = 10 ( 2 from each unit)
5 questions 05 x 04 = 20 (either or, one from each unit)
3 questions from 5 03 x 10 = 30 ( one question from each unit)
Total 60
CIA components
Tests - 30 (2 tests from 3)
Seminar – 5
Assignment - 5
Programme Outcomes
M.Phil. Granduands are well equipped with Research & Development Competences
expressive of their Creative Knowledge, Inventive Skill, Resolute Attitude and Innovative
Pursuits in their chosen fields.
M.Phil. Graduants Collate information from a variety of sources and Enrich a coherent
understanding of the subject concerned pertaining to Novel investigation on the problems
in everyday life.
COURSES
RESEARCH METHODOLOGY
Course Code 17MP01CC
Objectives:
To learn the concept of simply connect spaces and to prove the Riemann mapping
theorem.
To introduce algebraic invariant, homology groups for simplicial complexes.
To demonstrates the power of topological methods in dealing with problems
involving shape and position of continuous mappings.
UNIT – I
Riemann Mapping Theorem: Preservation of angles – Linear
fractional transformations – Normal families - Riemann Mapping Theorem.
UNIT –II
Fundamental Group: Revision on point set topology - Homotopy – Fundamental
group.
UNIT –III
Covering spaces: Covering map - Covering homotopy theorem – Regular covering
spaces.
UNIT –IV
Simplicial complexes: Geometry of Simplicial complexes – Barycentric
subdivisions
UNIT – V
Simplicial complexes: Simplicial approximation theory – Fundamental group
of a Simplicial complex.
UNIT – VI (Advanced topics only for discussion)
Current Contours:
Homology theory
TEXT BOOK(S):
[1] V. Karunakaran, Complex Analysis 2 edn, Narosa, New Delhi, 2005.
[2] I.M. Singer and J.A. Thorpe, Lecture Notes on Elementary Topology and
Geometry, Springer Verlag, 2004.
Unit I - Chapter
Unit II - Chapter 3 (Section: 3.1 to 3.2 )
Unit III - Chapter 3 (Section: 3.3 )
Unit IV - Chapter 4 (Section: 4.1 to 4.2 )
Unit V - Chapter 4 (Section: 4.3 to 4.4 )
REFERENCES
[1] W. Rudin, Real and Complex Analysis, 3rd
edition, McGraw Hill International,
1986.
[2] James R. Munkres, Topology (2nd Edition), Prentice Hall of India, Pvt. Ltd.,
New Delhi, 2004.
Course outcomes : Students will be able to
L T P C
4 2 0 4
Understand the properties of conformal mapping and linear fractional
transformations
Comprehend the important Riemann mapping theorem in Complex analysis
Discuss on the concept of homotopy and homotopy equivalence of topological
spaces.
Compute the fundamental groups of standard topological spaces.
Learn thoroughly covering homotopy theorem
Understand simplicial complexes and its barycentric subdivisions.
Use simplicial approximations to find the fundamental group of simplicial
complexes.
Appreciate and deduce the important Brouwer’s fixed point theorem.
Algebra and
analysis
Course Code 17MP02CC
Objectives:
To understand the concept of Modules as representation objects for rings.
To learn the notion of Noetherian rings as a important concept in ring theory and
the role it plays in simplifying the ideal structure of rings.
To impart the knowledge of how linear functionals on spaces of continuous function
on a locally compact space are related to measures.
To learn the theory of Fourier transforms in a rigorous setup.
To get a practice in the study of solving the system of linear differential equation
using the concepts from linear algebra.
UNIT I
MODULES: Basic definitions – Group of homomorphisms – Direct products and
sums of modules – Free modules – Vector spaces – The dual space and dual module.
UNIT II
NOETHERIAN RINGS: Basic criteria – Associated primes – Primary
decomposition - Nakayama’s lemma
UNIT III
RIESZ REPRESENTATION THEOREM: Topological preliminaries - Riesz
representation theorem – Regularity properties of Borel measures –Lebesgue measure –
continuity properties of measurable functions
UNIT IV
FOURIER TRANSFORSMS: Formal properties – Inversion theorem – The
Plancherel theorem – Banach Algebra L1
UNIT V
DIFFERENTIAL EQUATIONS: Uncoupled Linear systems – Diagonalization –
Exponentials of operators – Fundamental theorem for Linear systems – Linear Systems in
R2 – Complex Eigen values – Multiple Eigen values – Jordan forms – Stability theory –
Non-homogeneous linear systems.
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4 2 0 4
Unit – VI (Advanced topics only for discussion)
Current Contours:
Modules over PID, Distribution spaces.
TEXT BOOKS
[1] Serge Lang, “Algebra”, Springer - Verlag, Revised Third Edition, 2002.
Unit – I - Chapter III: Sections 1 to 6
Unit – II - Chapter X: Sections 1 to 4.
[2] W. Rudin, Real and Complex Analysis, 3rd
edition, McGraw Hill International,
1986.
Unit III – Chapter 2
Unit IV – Chapter 9
[3] L. Perko, Differential Equations and Dynamical systems, Springer-Verlag, First
Indian Reprint, 2004.
Unit V – Chapter 1: Sections 1.1 – 1.10
REFERENCES
[1] C. Musili, Rings and Modules, 2nd
edition, Narosa, 1994.
[2] P.B. Bhattacharya et al., Basic Abstract Algebra, 2nd
edition, Cambridge University
Press, 1995.
[3] Serge Lang, Complex Analysis, Addison Wesley, 1977.
[4] C.D. Aliprantis and O.Burkinshaw, Priniciples of Real Analysis 2edn, Academic
Press, Inc. New York, 1990.
[5] E.A Coddington and N. Levinson, Theory of Ordinary differential equations , Tata
McGraw Hill, New Delhi, 1972.
Course outcomes :
Students will be able to
Understand the power and elegance of modules structure for instance one
understands the vector spaces as a special type of modules.
Realize Noetherian rings are generalization of skew polynomials.
Understand and discuss the concepts involved in the proof of Nakayama’s lemma
Throughly study Riesz representation theorem and its consequences which imparts
knowledge and skill in abstract measure theory and to apply them in diverse areas.
Comprehend Lucin’s theorem and use it for establishing functional analytic
properties of various function spaces
Rigorously study the Fourier transforms and how it helps to get the analytic tricks
required to solves problems emanating from various applications.
Get well versed in diagonalization and Jordan form in linear algebra.
Solve system of differential equations as an application of algebraic concepts.
GUIDE PAPER
ADVANCED FUNCTIONAL ANALYSIS
Course Code: 17MP03CC:01
Objectives:
To demonstrate the Banach fixed point theorem is important as a source of existence
and uniqueness theorem in difference branches of analysis.
To realize the concept of compact operator, weak topology, and adjoint operator.
To learn the relation between distribution derivative and the distribution generated
by classical derivative.
UNIT – I,
Banach Fixed point theorem - Applications of Banach’s Theorem to Linear
Equations - Applications of Banach Theorem to Differential Equations Applications
- Applications of Banach Theorem to Integral Equations.
UNIT II,
Compact Linear Operators on Normed Spaces – Further properties of Compact
Linear Operators – Spectral Properties of Compact Linear Operator on Normed
Linear Spaces -
UNIT III
Operator Equations Involving Compact Linear Operators – Theorem of Fredholm
Type – Fredholm Alternative – Weak and Weak* Topology – Banach Alaoglu
Theorem.
UNIT IV
Distribution: Introduction - Test Function and Distributions - Some operation with
Distributions - Supports and Singular Support of Distributions - Convolution of
Function and Distributions.
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4 2 0 4
UNIT V
Convolution of Distributions – Fundamental solutions – The Fourier Transform –
Schwartz spaces – The Fourier Inversion Formula – Tempered Distributions.
Unit – VI (Advanced topics only for discussion)
Current Contours: Locally convex spaces. Topological Vector spaces.
TEXT BOOK
[1] Erwin Kreyszig, Introductory Functional Analysis with Applications, University of
Winndsor, Wiley Classics library Edition published 1989.
[2] Bella Bollobas, Linear Analysis , Cambridge University press,Second Edition
2004.
[3] S. Kesavan, Topics in Functional Analysis and Its Applications, TIFR Bangalore,
India, Wiley Eastern Limited.
UNIT I Chapter 8 [1]
UNIT II Chapter 8 [1]
UNIT III Chapter 8 [1],Chapter 8 [2]
UNIT IV Chapter 1 (Section 1.1 – 1.6) [3]
UNIT V Chapter 1 (Section 1.7 – 1.11) [3].
Course Outcomes:
Students will be able to
Apple the existence and uniqueness of solution of the problems in differential
equations, linear algebraic equations and integral equations by using the Banach
fixed point theorem.
To understand the theory of compact operators and utilize its importance
Comprehend spectral theorem on compact operators.
To prove the Banach Alaoglu theorem by using the concept of weak* topology.
Realize the importance of Schwartz theory of distributions.
Get the working knowledge on various distributions such as dirac, tempered etc..
Workout on fundamental solutions of plenty of distributions.
Comprehend Fourier inversion formula and work on tempered distributions.
ADVANCED NUMERICAL ANALYSIS
Course Code 17MP03CC:02
Objectives:
To learn numerical analysis is concerned with obtaining approximate solutions
while maintaining reasonable bounds on errors.
To numerically obtain the roots of an equation using several methods these Muller
method and Regula-Falsi method.
UNIT – I
Transcendental and polynomial equations: Iteration methods based on second
degree equation - Rate of convergence - iterative methods – Methods for finding
complex roots – iterative methods : Birge-Vieta method, Bairstow’s method,
Graeffe’s root squaring method
UNIT – II
System of Algebraic linear equations: Direct methods - Gauss Jordan Elimination
Method – Triangularization method – Cholesky method – partition method. Error
Analysis – Iteration methods : Jacobi iteration method – Gauss - Seidal iteration
method – SOR method. Jacobi method for symmetric matrices – Power method –
Inverse power method.
UNIT – III
Interpolation and Approximation : Hermite Interpolations – Piecewise and Spline
Interpolation – Appproximation – Least Square Approximation..
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4 2 0 4
UNIT – IV
Differentiation and Integration : Numerical Differentiation – Numerical
Integration – Methods based on Interpolation.
UNIT - V
Ordinary Differential Equations :- Multi – step method – Predictor – Corrector
method – Boundary value problem – Initial value methods – Shooting method –
Finite Difference method..
Unit – VI (Advanced topics only for discussion)
Current Contours: Numerical methods for partial differential equations -
Eigen value problems – Finite Element Methods.
TEXT BOOK(S):
[1] M.K. Jain, S.R.K. Iyengar and R.K. Jain, Numerical Methods for Scientific
and Engineering Computation, III Edn. Wiley Eastern Ltd., 1993.
Unit I – Chapter 2 – 2.4 to 2.8
Unit II - Chapter 3- 3.2– 3.5
Unit III – Chapter 4 4.4 – 4.6, 4.8 to 4.9
Unit IV – Chapter 5 – 5.2, 5.6, 5.7
Unit V – Chapter 6 – 6.4, 6.5, 6.8, 6.9, 6.10
REFERENCES
[1] Kendall E. Atkinson, An Introduction to Numerical Analysis, II Edn., John
Wiley & Sons, 1988.
[2] M.K. Jain, Numerical Solution of Differential Equations, II Edn., New Age
International Pvt Ltd., 1983.
[3] Samuel. D. Conte, Carl. De Boor, Elementary Numerical Analysis, Mc
Course outcomes:
Students will be able to
Understand the theoretical and practical aspects on the use of numerical methods.
Analyze the errors obtained in the numerical solutions of problems.
Demonstrate the use of interpolation methods to find intermediate values in the
given graphical and/or tabulated data.
Numerically solve lot of practical problems where exact solution is unknown.
Determine approximate solutions of system of linear algebraic equations using the
method of matrix decomposition.
Understand the idea of interpolation namely the deviation of the given function
from the approximating polynomial.
Evaluate the derivative of the given function by approximating polynomial.
Study in detail the numerical methods involved in shooting method, finite difference
method.
ALGEBRAIC CODING THEORY
Course Code 17MP03CC:03
Objectives:
To learn how codes in mathematics are used for error correction and data
transmission.
To comprehend the algebraic structure of linear codes viewed as a vector space over
a finite field.
UNIT – I
Linear Code: Introduction - Linear codes, Encoding, Decoding - Check Matrices
and Dual Code
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4 2 0 4
UNIT II
Linear Code continued: Classification by Isometry – Semilinear Isometry Classes
of Linear codes – The Weight Enumerator
UNIT III
Bounds and Modifications: Combinatorial Bounds for the Parameters - New
Codes from Old and Minimum distance – Further Modifications and Constructions.
UNIT IV
Reed Muller Codes – MDS Codes
UNIT V
Cyclic Codes: Cyclic Codes as Group Algebra Codes – Polynomila Representaion of
Cyclic Codes – BCH Codes and Reed-Solomon Codes
Unit – VI (Advanced topics only for discussion)
Current Contours: Graph Codes – Identifying codes.
TEXT BOOK
[1] Anton Betten, Michael Braun, Harald Fripertinger, Adalbert Kerber, Axel Kohnert
and Alfred Wassermann, Error-Correcting Linear Codes, Classification by Isometry
and Applications, Springer-Verlag berlin Heidelberg 2006.
UNIT I Chapter 1 (Section 1.1 – 1.3)
UNIT II Chapter 1 (Section 1.4 – 1.6)
UNIT III Chapter 2 (Section 2.1 – 2.3)
UNIT IV Chapter 2 (Section 2.4 & 2.5)
UNIT V Chapter 4 (Section 4.1 – 4.3)
REFERENCE(S)
.
1. F. J Mac Williams and N. J. A. Sloane, The Theory of Error-Correcting Codes,
North Holland Publishing Company 1977.
2. D.G. Hoffman et al, Coding Theory: The Essentials, Marcel Dekker, Inc, 1991
3. S. Ling and C. Xing, Coding Theory : A first course, Cambridge University press,
2004.
Course Outcomes:
Coding theory demonstrates how mathematics is used in real life applications.
Students get thorough idea about channel, noise, encoding, decoding etc and also
error detection and error correction which are involved in a data communication
One can have a clear cut idea about parameters of codes and the aims of
constructing codes with minimum length and maximum number of codewords and
maximum distance.
To learn how finite fields, vector space, number theory and probability theory are
applied in Coding Theory.
To acquire the knowledge of efficient decoding schemes of various linear codes.
To understand various bounds involved in coding theory and how it is used for
proving non-existence of codes under certain situation.
To practice how to construct new codes from the existing codes and learn more
about the parameters of the newly obtained codes.
To get expertise in some important codes like Hamming code, Golay code and
Reed-Muller code and thier applications to information theory.
FIXED POINT THEORY
Course Code 17MP03CC:04
Objectives:
To understand the concept of fixed point theorem for various spaces.
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4 2 0 4
To learn the concept of uniformly convex banach spaces and Tarsiki’s fixed point
theorem.
UNIT I
Banach’s contraction principle – Further extensions- Caristi – Ekeland principle -
Equivalance of Caristi- principles.
UNIT II
Tarsiki’s Fixed point theorem - Hyperconvex spaces – Properties – fixed point
theorems – intersection of hyper convex spaces – Isbell’s convex hull.
UNIT III
Uniformly convex Banach spaces – Fixed point theorem of Browder, Gohde and
Kirk. Reflexive Banach spaces –Normal structure- Fixed point theorems.
UNIT IV
Generalized Banach Fixed-point theorem- Upper and lower semi continuity of
multivalued maps –Generalized Schauder Fixed point theorem – Variational
Inequalities and the Browder Fixed-Point theorem – Extremal Principle –
Applications to Game Theory – Michael’s selection theorem
UNIT V
Fixed point theorem for continuous functions- Brouwer's theorem -Schauder's
theorem - applications - Hairy ball theorem - pancake problems- Kyfan's best
approximation theorem.
Unit – VI (Advanced topics only for discussion)
Current Contours: Generalizations of Brouwer’s fixed point theorem.
Best proximity pairs.
TEXT BOOK(S)
1. E. Zeidler, Nonlinear Functional Analysis and its applications, Vol. I Springer –
Verlag New york (1986)
2. M. A. Khamsi & W. A. Kirk, An introduction of Metric spaces and Fixed point
theory, John Wiley & sons (2001).
UNIT – I - Chapter 3 (3.1 - 3.4) from [2]
UNIT – II - Chapter 4 from [2]
UNIT – III - Chapter 10 (10.1 -10.3) from[1] and chapter 5 (5.1 -5.4) from [2]
UNIT – IV - Chapter 9 from [1]
UNIT – V - Chapter 2 from [1]
REFERENCES
1. D.R. Smart, Fixed point theory, Cambridge University Press, (1974).
2. V.I. Istratescu, Fixed point theory, D. Reidel Publsihing Company, Boston (1979).
Course Outcomes:
Students will be able to
Appreciate how the study of fixed point theory helps to solve problems which are
theoretical as well as practical
Realize contraction, contractive maps have elegant results on the existence and
uniqueness of fixed points.
Study the theory of non-expansive fixed point theorems and understand the
geometry of the spaces involved.
Understand the depth of mathematical concepts required to give the proof Brouwer’s
fixed point theorem due to Milnor.
Appreciate the generalizations of Brouwer’s fixed point theorem, viz., Schauder and
the use of it in analysis and differential equations.
Comprehend the fixed point theory on multivalued maps and see the interconnection
with single valued cases.
Thoroughly understand the idea behind Michael’s selection theorem.
Understand Kyfan’s best approximation theorem and its consequences.