Page 81 - CatalogNEP-PS

UNIT II: (12 Hours)

Definition and Existence of Riemann-Stieltjes Integral. Properties of The Integral. Integration and
Differentiation. The Fundamental Theorem of Calculus. Change of Variable. Rectifiable Curves.
UNIT III: (18 Hours)
Problem of Interchange of Limit Processes for Sequences of Functions, Pointwise and Uniform
Convergence, Cauchy Criterion for Uniform Convergence. Weierstrass M-Test. Abel‘s and
Dirichlet‘s Tests for Uniform Convergence. Uniform Convergence and Continuity. Uniform
Convergence and Riemann – Stieltjes Integration. Uniform Convergence and Differentiation. The
Weierstrass Approximation Theorem.

*TUTORIAL (15 Hours (1 Hour per week))

SUGGESTED READINGS:

1. Walter Rudin, Principles of Mathematical Analysis (3rd Edition), McGraw-Hill,
Kogakusha, 1976, International Student Edition.
2. T.M. Apostol, Mathematical Analysis, Narosa publishing House, New Delhi, 1985.
3. S. Lang, Analysis-I, Addison – Wesley Publishing Company, Inc. 1969
4. Robert G. Bartle, Donald R. Sherbest, Introduction to Real Analysis (Fourth Edition-
(2015), John Wiley & Sons, Inc.
5. S.C. Malik, Savita Arora, Mathematical Analysis (Third edition-2008), New Age
International (P) Ltd., New Delhi.

Math.412 Advanced Algebra 3+1*

LEARNING OBJECTIVES:

The primary objective of this course is to introduce:
 The concept of group actions on sets, normal series, solvable group, nilpotent group, and
their application.
 Finitely generated Abelian groups which are decomposable as a finite direct sum of cyclic
groups.
 The ring theory and special classes of rings such as Quotient rings, Euclidean rings, ring of
Gaussian integers and Polynomial rings over the Rational fields; Commutative rings.

LEARNING OUTCOMES:

On completion of the course, students shall be able to
 Develop the understanding about the importance of group actions on sets, describe the
normal series, solvable groups, nilpotent groups, and their applications to characterize
some classes of groups.
 Attain the knowledge about finitely generated Abelian groups which are decomposable as
a finite direct sum of cyclic groups which enables the students to find the number of non-
isomorphic Abelian groups of given order.
 Provide the comprehensive understanding of ring theory and some special classes of rings
such as Quotient rings, Euclidean rings, ring of Gaussian integers and Polynomial rings
over the Rational fields; Commutative rings.

THEORY (45 Hours)

UNIT I (15 Hours)

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