ABSTRACT ALGEBRA-M101
Paper I
Course Outcomes
1. To combine polynomial by additions or subtractions.
2. To solve problems of simple inequalities.
3. Interpret basic absolute value expression.
4. To simplify algebraic expression,using the commutative,Associative and Distributive properties.
Unit-I
Automorphisms. , Conjugacy and G-sets. Normal series Solvable groups. Nilpotent groups.
Unit-II
Structure theorems of groups. Direct product. Finitely generated abelian group. Invariants of a finite abelian group. Sylow's theorems-Groups of orders p2, pq.
Unit-III
Ideals and homomorphisms. Sum and direct sum of ideals. Maximal and prime ideals. Nilpotent and nil ideals.
Zorn'slemma.
Unit-IV
Unique factorization domains. Principal ideal domains. Euclidean domains. Polynomial rings over UFD.
Rings of Fractions.
TextBook:
Basic Abstract Algebra byP.B.Bhattacharya,S.K.Jainand S.R.Nagpaul.
Reference:
[1] Topics in Algebra by I.N.Herstein.
[2] Elements of Modern Algebra by Gibert & Gilbert.
[3] Abstract Algebra byJe reyBergen.
[4] Basic Abstract Algebra by RobertBAsh.
Mathematical Analysis-M102
Paper-II
Course Outcomes
1.Define and recognize the basic properties of the field of real numbers and Series of real numbers and convergence
2.The basic topological properties
3.The real functions and its limits, the continuity of real functions
4.The differentiability of real functions and its related theorems
Unit-I
Metric spaces.
Compact sets.
Perfect sets.
Connected sets.
Unit-II
Limits of functions.
Continuous functions.
Continuity and compactness, Continuity and connectedness.
Discontinuities.
Monotone functions.
Unit-III
Riemann - Steiltjes integral.
Definition and Existence of the Integral.
Properties of the integral.
Integration of vector valued functions.
Rectifiable curves.
Unit-IV
Sequences and series of functions.
Uniform convergence.
Uniform convergence and continuity.
Uniform convergence and integration.
Uniform convergence and differentiation.
Approximation of a continuous function by a sequence of polynomials.
Text Book:
Principles of Mathematical Analysis (3rd Edition(Chapters 2, 4, 6 ) By Walter Rudin, Mc Graw - Hill Internation Edition.
References:
[1] The Real Numbers by John Stillwel.
[2] Real Analysis by Barry Simon
[3] Mathematical Analysis Vol - I by D J H Garling.
[4] Measure and Integral by Richard L.Wheeden and Antoni Zygmund.
Ordinary and Partial Differential Equations-M103
Paper - III
Course Outcomes
1.Apply the fundamental concepts of Ordinary Differential Equations and Partial Differential Equations and the basic numerical methods for their resolution.
2.Solve the problems choosing the most suitable method.
3.Understand the difficulty of solving problems analytically and the need to use numerical approximations for their resolution.
4.Use computational tools to solve problems and applications of Ordinary Differential Equations and Partial Differential Equations.
Unit-I
Existence and Uniqueness of solution of dy dx = f(x, y) and problems there on.
The method of successive approximations - Picard’s theorem.
Non - Linear PDE of order one.
Charpit’s method.
Cauchy’s method of Characteristics for solving non - linear partial differential equations.
Linear Partial Differential Equations with constant coefficients.
Unit-II
Partial Differential Equations of order two with variable coefficients.
Canonical form.
Classification of second order Partial Differential Equations.
separation of variables method of solving the one - dimensional Heat equation, Wave equation and Laplace equation.
Sturm - Liouville’s boundary value problem.
Unit-III
Power Series solution of O.D.E. – Ordinary and Singular points.
Series solution about an ordinary point.
Series solution about Singular point - Frobenius Method.
Lagendre Polynomials: Lengendre’s equation and its solution.
Lengendre Polynomial and its properties - Generating function - Orthogonal properties - Recurrance relations.
Laplace’s definite integrals for Pn(x).
Rodrigue’s formula.
Unit-IV
Bessels Functions: Bessel’s equation and its solution.
Bessel function of the first kind and its properties - Recurrence Relations - Generating function - Orthogonality properties.
Hermite Polynomials: Hermite’s equation and its solution.
Hermite polynomial and its properties - Generating function - Alternative expressions (Rodrigue’s formula) - Orthogonality properties.
Recurrence Relations.
Text Books:
[1] Ordinary and Partial Differential Equations, By M.D. Raisingania, S. Chand Company Ltd., New Delhi.
[2] Text book of Ordinary Differential Equation, By S.G.Deo, V. Lakshmi Kantham, V. Raghavendra, Tata Mc.Graw Hill Pub. Company Ltd.
[3] Elements of Partial Differential Equations, By Ian Sneddon, Mc.Graw - Hill International Edition.
Reference:
[1] Worldwide Differential equations by Robert McOwen .
[2] Differential Equations with Linear Algebra by Boelkins, Goldberg, Potter.
[3] Differential Equations By Paul Dawkins.
Elementary Number Theory-M104
Paper - IV
Course Outcomes
1.Prove results involving divisibility and greatest common divisors.
2.Solve systems of linear congruences
3.Apply Euler-Fermat’s Theorem to prove relations involving prime numbers.
4.Apply the Wilson’s theorem.
Unit-I
The Fundamental Theorem of arithmetic: Divisibility, GCD, Prime Numbers,
Fundamental theorem of Arithmetic,
The series of reciprocal of the Primes,
The Euclidean Algorithm.
Unit-II
Arithmetic function and Dirichlet Multiplication.
The functions φ(n), µ(n) and a relation connecting them.
Product formulae for φ(n), Dirichlet Product, Dirichlet inverse and Mobius inversion formula.
Mangoldt function Λ(n), multiplication function.
multiplication function and Dirichlet multiplication.
Inverse of a completely multiplication function.
Liouville’s function λ(n).
the divisor function is σα(n).
Unit-III
Congruences, Properties of congruences.
Residue Classes and complete residue system.
linear congruences conversion, reduced residue system.
Euler Fermat theorem, polynomial congruence modulo P.
Lagrange’s theorem, Application of Lagrange’s theorem.
Chinese remainder theorem and its application.
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. Chapters 1, 2, 5, 9.
References:
[1] Number Theory by Joseph H. Silverman.
[2] Theory of Numbers by K.Ramchandra.
[3] Elementary Number Theory by James K Strayer. [3] Elementary Number Theory by James Tattusall.
Discrete Mathematics-M105
Paper - V
Course Outcomes
1.Students completing this course will be able to express a logic sentence in terms of predicates, quantifiers, and logical connectives.
2.To apply the rules of inference and methods of proof including direct and indirect proof forms, proof by contradiction, and mathematical induction.
3.To use tree and graph algorithms to solve problems.
4.To evaluate Boolean functions and simplify expressions using the properties of Boolean algebra.
Unit-I
Mathematical Logic: Propositional logic, Propositional equivalences,
Predicates and Quantifiers, Rule of inference, direct proofs, proof by contraposition, proof by contradiction.
Boolean Algebra: Boolean functions and its representation, logic gates, minimizations of circuits by using Boolean identities and K - map.
Unit-II
Basic Structures: Sets representations, Set operations, Functions, Sequences and Summations.
Division algorithm.
Modular arithmetic.
Solving congruences, applications of congruences.
Recursion: Proofs by mathematical induction, recursive definitions, structural induction,generalized induction, recursive algorithms.
Unit-III
Counting: Basic counting principle, inclusion - exclusion for two - sets,
Pigeonhole principle.
Permutations and combinations, Binomial coefficient and identities.
Generalized permutations and combinations.
Recurrence Relations: introduction, solving linear recurrence relations, generating functions.
Principle of inclusion - exclusion.
Applications of inclusion - exclusion.
Relations: relations and their properties, representing relations, closures of relations, equivalence relations, partial orderings.
Unit-IV
Graphs: Graphs definitions, graph terminology, types of graphs, representing graphs, graph isomorphism, connectivity of graphs.
Euler and Hamilton paths and circuits.
Dijkstra’s algorithm to find shortest path.
planar graphs – Euler’s formula and its applications.
graph coloring and its applications.
Trees: Trees definitions – properties of trees, applications of trees – BST, Haffman Coding
tree traversals: pre - order, in - order, post - order, prefix, infix, postfix notations.
spanning tress – DFS, BFS, Prim’s.
Kruskal’s algorithms.
Text Book:
Discrete Mathematics and its Applicationsby Kenneth H. Rosen,
References:
[1] Discrete and Combinatorial Mathematicsby Ralph P. Grimaldi
[2] Discrete Mathematics for Computer Scientists by Stein, Drysdale, Bogart
[3] Discrete Mathematical Structures with Applications to Computer Scienceby J.P. Tremblay, R. Manohar
[4] Discrete Mathematics for Computer Scientists and Mathematicians by Joe L. Mott, Abraham Kandel, Theoder P. Baker