MM/AM/MCS -101                      SEMESTER – I                         Paper-I : Abstract Algebra

Course Objectives: 

The course aims to:

1. Introduce students to the fundamental concepts of abstract algebra, including groups, rings, and fields.

2. Develop a deep understanding of structural properties of algebraic systems such as solvable groups, nilpotent groups, and finite abelian groups.

3. Familiarize students with advanced topics like ideals, homomorphisms, unique factorization domains, principal ideal domains and Euclidean domains.

4. To understand the concepts and properties of unique factorization, principal ideal, and Euclidean domains for applications in advanced algebra and related fields.

Course Outcomes: 

After successful completion of this course, students will be able to:

1. Understand and apply the concepts of automorphisms, conjugacy, solvable and nilpotent groups.

2. Analyze group structures using direct products, Sylow’s theorems, and classification of finite abelian groups.

3. Apply the concepts of unique factorization, principal ideal, and Euclidean domains to solve problems in ring theory.

4. Apply abstract algebraic methods to solve problems in advanced mathematics, and theoretical computer science.

Unit – I

Automorphisms – Conjugacy and G-sets – Normal series and Solvable groups – Nilpotent groups. (Page Nos. 104 – 128)

Unit – II

Structure theorems of groups: Direct products – Finitely generated abelian groups – Invariants of a finite abelian group – Sylow’s theorems – Groups of orders p2 , pq. (Page Nos. 138 – 155)

Unit – III

Ideals and Homomorphisms: Sum and direct sum of ideals – Maximal and Prime ideals – Nilpotent and nil ideals – Zorn’s lemma. (Page Nos. 179 – 211)

Unit – IV

Unique factorization domains (UFD) – Principal ideal domains – Euclidean domains – Polynomial rings over UFD – Rings of fractions. (Page Nos. 212 – 228)

Text Book:

Basic Abstract Algebra by P. B. Bhattacharya, S. K. Jain and S. R. Nagpaul, Cambridge University Press, Second Edition.

Reference Books:

1. Topics in Algebra by I.N. Herstein, Wiley India Pvt. Ltd, New York.

2. Elements of Modern Algebra by Gilbert and Gilbert, PWS Publishing Company Boston.

3. Abstract Algebra by Joffrey Bergen, Academic Press, Boston.

4. Basic Abstract Algebra by Robert B. Ash

MM/AM/MCS 102   SEMESTER-I Paper - II: Mathematical Analysis

Course Objectives:

The course aims to:

1. Understand the fundamental concepts of metric spaces, including compactness, connectedness, and perfect sets.

2. Develop a conceptual understanding of continuity, limits, and properties of continuous and monotonic functions.

3. Apply the theory of the Riemann-Stieltjes integral to real and vector-valued functions.

4. Analyze the convergence of sequences and series of functions and explore their impact on integration and differentiation.

Course Outcomes:

After successful completion of this course, students will be able to

1. Describe and classify metric space structures.

2. Explain and examine the role of continuity and discontinuity in function behavior across different topological structures.

3. Compute the Riemann-Stieltjes integral and apply it to practical examples involving rectifiable curves and vector functions.

4. Evaluate the uniform convergence of sequence of functions and apply the Stone-Weierstrass theorem and contraction principle in problem-solving.

Unit-I

Basic Topology: Metric spaces - Compact sets - Perfect sets - Connected sets. (Page Nos. 24- 46)

Unit-II

Continuity: Limits of functions - Continuous functions - Continuity and compactness, Continuity and connectedness - Discontinuities - Monotonic functions. (Page Nos. 83-102)

Unit-III

The Riemann - Steiltjes integral: Definition and Existence of the integral - Properties of the integral –Integration of vector valued functions - Rectifiable curves.

(Page Nos. 120-133 & 135-142)

Unit-IV

Sequences and series of functions: Discussion of Main Problem, Uniform convergence, Uniform convergence and Continuity, Uniform convergence and integration - Uniform convergence and differentiation – The Stone-Weierstrass theorem, The Contraction Principle. (Page Nos. 143-154, 159-161, 165-166 & 220-221)

Text Book:

Principles of Mathematical Analysis (3rd Edition) By Walter Rudin, McGraw-Hill International Edition.

References:

The Real Numbers by John Stillwell. Real Analysis by Barry Simon

Mathematical Analysis Vol - I by D J H Garling.

Measure and Integral by Richard L. Wheeden and Antoni Zygmund.

  MM/AM 103                               SEMESTER-I                   Paper-III: Ordinary Differential Equations

Course Objectives:

The course aims to:

1. To understand the fundamental theory of existence and uniqueness of solutions for ordinary differential equations (ODEs).

2. To learn methods for solving higher-order linear differential equations, including those with constant and variable coefficients.

3. To familiarize students with power series solutions and special functions like Legendre polynomials and Bessel functions.

4. To study systems of first-order ordinary differential equations and boundary value problems

Course Outcomes: 

After successful completion of this course, students will be able to

1. Apply Picard’s theorem and successive approximations to prove the existence and uniqueness of solutions to ODEs.

2. solve homogeneous linear equations with constant coefficients and use the method of variation of parameters for variable coefficients.

3. find power series solutions for second-order linear equations and understand the properties of Legendre and Bessel functions.

4. analyze linear systems with constant or periodic coefficients and solve Sturm-Liouville boundary value problems.

Unit I:

Existence and Uniqueness of Solutions: Preliminaries , Successive approximations , Picard’s theorem , Some examples , Continuation and dependence on initial conditions , Existence of solutions in the large , Fixed point technique, Classical Peano existence theorem.

Unit II:

Linear Differential Equations of Higher Order: Introduction , Higher order linear differential equations, Linear dependence and Wronskian, Homogeneous linear equations with constant coefficients, Equations with variable coefficients, Method of variation of parameters, Laplace transforms.

Unit III:

Solutions in Power Series: Introduction, Second order linear equations with ordinary points, Legendre equation and Legendre Polynomials - Second order equations with regular singular points, Bessel functions - Oscillations of Second Order Equations: Introduction, Sturm’s comparison theorem - Sturm’s separation theorem, Elementary linear oscillations.

Unit IV:

Systems of ordinary differential equations: Introduction, Systems of first-order equations, Existence and uniqueness theorem, Fundamental matrix, Non homogeneous linear systems, Linear systems with constant coefficients - Linear systems with periodic coefficients - Boundary value problems: Sturm – Liouville problems.

Text Book:

Textbook of Ordinary Differential Equations by S.G. Deo, V. Raghavendra, Rasmita Kar and V. Lakshmikantham, Third Edition, McGraw-Hill Education (India) Private Limited, New Delhi,2015.

References: 

1. Differential Equations with Applications with Historical Notes by George F. Simmons, Second Edition,1991.

2.An Introduction to Ordinary Differential Equations by Earl A Coddington, Dover Publications 1989

MM/AM 104 SEMESTER-I Paper – IV: Numerical Analysis

Course Objectives: 

This course aims to:

1. Introduce numerical techniques for solving algebraic, transcendental, and differential equations.

2. Develop proficiency in iterative and direct methods for solving systems of linear equations.

3. Explore interpolation and approximation methods for data analysis and function estimation.

4. Equip students with numerical integration techniques and methods for solving ordinary differential equations.

Course Outcomes

Upon successful completion of this course, students will be able to:

1. Apply root-finding algorithms such as Bisection, Newton-Raphson, and Chebyshev methods to solve nonlinear equations.

2. Solve systems of linear equations using both direct (e.g., Gauss Elimination) and iterative (e.g., Jacobi, Gauss-Seidel) methods with convergence analysis.

3. Construct interpolating polynomials using techniques like Lagrange, Newton, and spline interpolation, and perform least squares approximation.

4. Solve initial value problems for ordinary differential equations using Euler, Runge-Kutta, and multistep methods such as Adams-Bashforth and Milne-Simpson.

Unit- I

Transcendental and Polynomial Equations: Introduction, Bisection Method - Iteration Methods Based on First Degree Equation: Secant Method, Regular Falsi Method, Newton- Raphson Method - Iteration Methods Based on Second Degree Equation: Muller’s Method, Chebyshev Method, Multipoint Iteration Methods, Rate of convergence - Iteration Methods. (Sec. 2.1-2.5)

Unit- I I

System of Linear Algebraic Equations: Introduction - Direct Methods: Gauss Elimination Method, Gauss Jordan Method, Triangularization Method, Cholesky Method, Partition Method - Iteration Methods:  Jacobi Iteration Method, Gauss Seidel Iteration Method, SOR Method, Convergence Analysis for iterative Methods. (Sec.3.1-3.2, 3.4)

Unit- I I I

Interpolation and Approximation: Interpolation: Introduction - Lagrange and Newton Interpolations, Finite Difference Operators - Interpolating Polynomials using Finite Differences - Hermite Interpolations, Piecewise and Spline Interpolations. Approximation: Least Squares Approximation.( Sec. 4.1-4.6, 4.8-4.9)

Unit- I V

Numerical Integration: Methods Based on Interpolation: Newton- Cotes Methods - Methods Based on Undetermined Coefficients: Gauss- Legendre Integration Methods - Composite Integration Methods. (Sec. 5.6, 5.7,5.8, 5.9)

Numerical Solution of O D E s : Introduction - Numerical Methods:  Euler Methods- Midp oint Method Single Step Methods: Taylor series method, Runge-Kutta Method (2nd and  Simpson Method ( Sec. 6.3, 6.4, 6.6)

Text Book:

Numerical Methods for Scientific and Engineering computation  by M.K.  Jain,

S.R.K. Iyengar, R. K. Jain, 7th Edition, New Age International Publishers,2019.

Reference book:

Introductory Methods of Numerical Analysis by S. S. Sastry, Fifth Edition 2012 by PHI Learning Private Limited, New Delhi.

Applied Numerical Analysis, by Curtis F. Gerald, Patrick O. Wheatley, Seventh Edition, 2003, Pearson Publishers.

MM -105                                             SEMESTER – I                        Paper-V:Elementary Number Theory

Course Objectives:

The course aims to:

1. To introduce the notions of divisibility and the Euclidean algorithm

2. To study relation between various arithmetical functions like mobius function, Euler totient function.

3. To understand linear congruences and their applications

4. To apply Fermat theorem, Wilson theorem and quadratic residues to determine the primality of numbers.

Course Outcomes: 

After completion of this course, students will be able to

1. understand the basic concepts of number theory and prove the fundamental theorem of arithmetic

2. To apply the Dirichlet product of arithmetical functions and its applications

3. Study congruences and establish some remarkable theorems such as Euler theorem, Fermat theorem, Wilson theorem and Chinese remainder theorem and their applications

4. Explore connections between quadratic residues and number theoretic-functions such as Legendre symbol and Jacobi symbol and their applications.

Unit – I

The Fundamental Theorem of Arithmetic: Divisibility, Greatest Common Divisor - Prime Numbers- Fundamental Theorem of Arithmetic- the series of reciprocal of the Primes- The Euclidean Algorithm. 

Unit – II

Arithmetic function and Dirichlet Multiplication: The Mobius function µ(n) - The Euler Totient

function φ(n) and a relation connecting them -A product formula for φ(n) - The Dirichlet product of

arithmetical Functions – Dirichlet Inverses - The Mobius inversion formula- The Mangoldt function

Λ(n) - Multiplicative functions - Multiplicative functions and Dirichlet multiplication- The Inverse

of a completely multiplicative function- The Liouville's function λ(n) - the divisor function σα (n) . 

Unit – III

Congruences: Definition and basic properties of congruences- Residue classes and complete residue systems - Linear Congruences - Reduced residue system and Euler Format theorem- Polynomial congruence modulo p- Lagrange's theorem- Applications of Lagrange's theorem - Simultaneous Linear Congruences -The Chinese remainder theorem – Applications of the Chinese Remainder Theorem - polynomial congruences with prime power moduli.

Unit – IV

Quadratic residue and quadratic reciprocity law- Quadratic residues- Legendre's symbol and its properties- evaluation of (-1/p) and (2/p) - Gauss Lemma The quadratic reciprocity law and its applications.

Text Book:

1. Introduction to Analytic Number Theory by Tom M. Apostol, Springer International, Student Edition (Ch. 1, 2,5,9).

References:

1. Number Theory by Joseph H. Silverman.

2. Theory of Numbers by K. Ramachandra.

3. Elementary Number Theory by James K. Strayer.

4. Elementary Number Theory by James Tattersall. regards