DEPARTMENT OF MATHEMATICS
OSMANIA UNIVERSITY
M.Sc. Mathematics
MM 101 Semester-I
Paper-I: Abstract Algebra
Unit-I
Automorphisms - Conjugacy and G - sets - Normal series Solvable groups - Nilpotent groups. (Pages 104 to 128 of [1])
Unit-II
Structure theorems of groups: Direct product - Finitely generated abelian groups - Invariants of a nite
abelian group - Sylow's theorems - Groups of orders p2, pq . (Pages 138 to 155)
Unit-III
Ideals and homomorphisms - Sum and direct sum of ideals, Maximal and prime ideals - Nilpotent and nilideals - Zorn's lemma (Pages 179 to 211).
Unit-IV
Unique factorization domains - Principal ideal domains - Euclidean domains - Polynomial rings over UFD -Rings of Fractions.(Pages 212 to 228)
Text Book:
Basic Abstract Algebra by P.B. Bhattacharya, S.K. Jain and S.R. Nagpaul.
Reference:
[1] Topics in Algebra by I.N. Herstein.
[2] Elements of Modern Algebra by Gibert& Gilbert.
[3] Abstract Algebra by Je rey Bergen.
[4] Basic Abstract Algebra by Robert B Ash.
MM 102 Semester - I
Paper - II: Mathematical Analysis
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 - De nition and Existence of the Integral - Properties of the integral - Integration of vector valued functions - Recti able curves.
Unit-IV
Sequences and series of functions: Uniform convergence - Uniform convergence and continuity - Uni-form 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.
MM 103 Semester - I
Paper - III: Ordinary and Partial Differential Equations
Unit-I
Existence and Uniqueness of solution of dydx = 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 coeffcients.
Unit-II
Partial Differential Equations of order two with variable coeffcients - Canonical form - Classifi cation 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 - Recurrence relations - Laplace's defi nite integrals for Pn(x)- Rodrigue's formula.
Unit-IV
Bessels Functions: Bessel's equation and its solution - Bessel function of the rst 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.
MM 104 Semester - I
Paper - IV: Elementary Number Theory
Unit-I
The Fundamental Theorem of arithmetic: Divisibility, GCD, Prime Numbers, Fundamental theorem
of Arithemtic, the series of reciprocal of the Primes, The Euclidean Algorithm.
Unit-II
Arithmetic function and Dirichlet Multiplication, The functions (n), (n) and a relation connecting them, Product formulae for (n), Dirichlet Product, Dirichlet inverse and Mobius inversion formula and Mangoldt function (n), multiplication function, multiplication function and Dirichlet multiplication, Inverse of a completely multiplication function, Liouville's function (n), the divisor function is (n)
Unit-III
Congruences, Properties of congruences, Residue Classes and complete residue system, linear congruences conversion, reduced residue system and 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:
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.
MM 105 Semester - I
Paper - V: Discrete Mathematics
Unit-I
Mathematical Logic: Propositional logic, Propositional equivalences, Predicates and Quanti ers, 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. Divi-
sion algorithm, Modular arithmetic, Solving congruences, applications of congruences. Recursion: Proofs by mathematical induction, recursive de finitions, structural induction,generalized induction, recursive algorithms.
Unit-III
Counting: Basic counting principle, inclusion - exclusion for two - sets, pigeonhole principle, permutations and combinations, Binomial coe cient 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 de finitions, graph terminology, types of graphs, representing graphs, graph isomorphism, connectivity of graphs, Euler and Hamilton paths and circuits, Dijkstras algorithm to a nd shortest path, planar graphs Eulers formula and its applications, graph coloring and its applications. Trees: Trees de nitions properties of trees, applications of trees BST, Hanman Coding, tree traversals: pre - order, in - order, post- order, pre x, in x, post x notations, spanning tress DFS, BFS, Prims, Kruskals algorithms.
Text Book:
Discrete Mathematics and its Applications by 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
