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Algebra

Unit I: Group of permutations, Examples, Alternating Groups, Simple groups, simplicity of An (n > 4), Applications, Subnormal and Normal Series, Jordan-Holder Theorem, The Center and the Ascending Central Series, Isomorphism Theorems. 


Unit II: The Zassenhaus (Butterfly) Lemma, Schreier Theorem, Group action on a set, fixed sets and isotropy subgroups, Orbits, Applications of G-Sets to Counting, Burnside theorem, p-groups, The Sylow Theorems. 


Unit III: Applications of Sylow theorems to p-Groups and the Class equation, Further Applications, Polynomial in an Indeterminate, Polynomial rings, The evaluation Homomorphisms, Factorization of Polynomials over Fields, The Division Algorithm in F[x], Irreducible Polynomials, Eisenstein criteria, Ideal Structure in F[x], Uniqueness of Factorization in F[x]. 


Unit IV: Principal Ideal Domain (PID), Uniqueness of Factorization Domain(UFD), Gauss lemma, Introduction and Definition of Euclidean Domain, Arithmetic in Euclidean Domains. Definitions and Examples of Modules, Direct Sums, Free Modules, sub-modules, Quotient Modules, Homomorphism, Simple Modules. 

Recommended Book(s):
1. John B. Fraleigh , A first course in Abstract Algebra (Third Edition), Narosa publishing house, New Delhi.
2. C. Musili, Introduction to Rings and Modules (Second Revised Edition), Narosa Publishing house, New Delhi.


Reference Books:
1. Joseph A. Gallian, Contemporary Abstract Algebra (Fourth Edition), Narosa Publishing
house, New Delhi.
2. Bhattacharya, Jain and Nagpal, Basic Abstract Algebra, 2nd edition, Narosa Publishing
House, New Delhi.
3. I. N. Herstein, Topics in Algebra, Vikas Publishing House.
4. N. Jacobson, Basic Algebra, Hind Publishing Corporation, 1984.

Skill Level: Beginner

Research Methodology

Unit I: Research Methodology: An Introduction : Meaning of Research , Objectives of
Research, Motivation in Research ,Types of Research , Research Approaches , Significance of
Research , Research Methods versus Methodology, Research and Scientific Method , Importance
of Knowing How Research is Done , Research Process , Criteria of Good Research , Problems
Encountered by Researchers in India. 15 lectures
 
Unit II: Defining the Research Problem: What is a Research Problem, Selecting the Problem ,
Necessity of Defining the Problem, Technique Involved in Defining a Problem, An Illustration ,
Conclusion. Research Design, Meaning of Research Design, Need for Research Design ,
Features of a Good Design, Important Concepts Relating to Research Design, Different
Research Designs, Basic Principles of Experimental Designs, Conclusion, Developing a
Research Plan. 15 lectures
 
Unit III: Meaning of Interpretation, Why Interpretation?, Technique of Interpretation:
Precaution in Interpretation, Significance of Report Writing, Different Steps in Writing Report,
Layout of the Research Report, Types of Reports, Oral Presentation, Mechanics of Writing a
Research Report, Precautions for Writing Research Reports. The Computer: Its Role in
Research, The Computer and Computer Technology, The Computer System, Important
Characteristics, The Binary Number System, Computer Applications, Computers and
Researcher. 15 lectures
 
Unit IV: What is LATEX? Simple typesetting, Fonts, Type size , Tables, Basic Typesetting of
Mathematics: Superscripts and subscripts, Roots, Mathematical symbols , Custom Commands,
Single equations , Groups of equations , Numbered equations, Matrices, Typesetting Theorems.
15 lectures
 
Recommended Book:
1. C. R. Kothari, Research Methodology – Methods and Techniques, (Second Revised
Edition), New Age International Publications,
2. LATEX Tutorials A Primer, Indian TEX Users Group, Trivandrum, India,
2003 September.
References:
1. Higham Nicholas J., Handbook of writing for the mathematical sciences, SIAM, 1961.
2. Michael Alley, The Craft of Scientific Writing (3rd Edition), Springer, New York, 1996
3. Philip Reubens (General editor), Science and Technical Writing – A Manual of Style (2nd
Edition), Routledge, New York, 2001
Skill Level: Beginner

Combinatorics

Unit I: The sum rule and product rule, permutations and combinations, the Pigeonhole principle,
Ramsay numbers, Catalan numbers, sterling numbers. 15 Lectures
 
Unit II: Further basic tools, generalized permutations and combinations sequences and
selections, the inclusion and exclusion principle, systems of distinct representatives, solved
problems derangements and other constrain derangements. 15 Lectures
 
Unit III: Combinatorial number theory, the permanent of a matrix, Rook polynomials and Hit
polynomials, SDR and coverings, (Sperners theorem and Symmetric chain decomposition, posets
and Dilworth's theorem) statements. 15 Lectures
 
Unit IV: Generating functions and recurrence relations, ordinary and exponential generating
functions, partitions of a positive integer, recurrence relations, algebraic solutions of linear
recurrence relations with constant coefficients and solutions of recurrence relations using
generating functions. 15 Lectures
 
Seminars, Tutorials, Problem solving session and group discussions on above four units
Recommended Books:
1. V. K. Balkrishnan: Combiactorics, Shaums Outlines Series, Mc Grow Hill Inc.
Reference Books:
1. Richard Brualdi – Introductory Combinatosics North Holland.
2. V. Krishnamurthy: Combinatorics, E. W. Press
3. A. Tucker: Combinatorics, John Wiley & Sons, Inc
4. C. Vasudev, Theory and Problems of Combinatorics, New Age International
Skill Level: Beginner

Numerical Analysis-I

Unit I: Transcendental & polynomial equations: Bisection method, Iteration methods based on
First degree equation (Secant method, Regula-Falsi method and Newton-Raphson method). Rate
of Convergence, Iterative methods (Birge-Vieta method and Bairstow method). 15 Lectures
 
Unit II: System of linear algebraic equations and eigen value problems: Matrix factorization
methods (Doolittle’s method, Crout’s method), Iteration methods (Jacobi iteration method,
Gauss-Seidel iteration method), convergence analysis of iterative methods, Eigen values and
eigenvectors, Gerschgorin theorem, Brauer theorem, Jacobi method for symmetric matrices,
Power method. 15 Lectures
 
Seminars, Tutorials, Problem solving session and group discussions on above four units
Recommended Books:
1. M. K. Jain, S. R. K. Iyengar, R. K. Jain, Numerical methods for scientific and Engineering
 
Computation (Fifth Edition), New Age International Publishers 2007.
Reference Books:
1. S. S. Sastry, Introductory methods of Numerical Analysis (Fifth Edition), PHI learning Private
 
Limited, New Delhi 2012.
2. D. Kincaid, W. Cheney, Numerical Analysis Mathematics of Scientific Computing (Third
 
Edition), American Mathematical Society.
3. J.C. Butcher, Numerical methods for ordinary differential equations (Second Edition), John
 
Wiley & Sons Ltd, 2008.
4. Kendall E. Atkinson, An Introduction to Numerical Analysis (Second Edition), John Wiley &
 
Sons 1988.
Skill Level: Beginner