PG II Year Syllabus 2018-19(III SEM)

DEPARTMENT OF MATHEMATICS 

OSMANIA UNIVERSITY 

M.Sc. Mathematics

 MM301                                                                                                                     Semester III

      Paper-I Complex Analysis 

 UNIT-I
 Regions in the Complex Plane -Functions of a Complex Variable - Mappings -Mappings by the Exponential Function- Limits - Limits Involving the Point at Infinity - Continuity -Derivatives - Cauchy–Riemann Equations -Sufficient Conditions for Differentiability - Analytic Functions - Harmonic Functions -Uniquely Determined Analytic Functions - Reflection Principle - The Exponential Function -The Logarithmic Function -Some Identities Involving Logarithms -Complex Exponents -Trigonometric Functions -Hyperbolic Functions

 UNIT-II 
Derivatives of Functions w(t) -Definite Integrals of Functions w(t) - Contours -Contour Integrals - Some Examples -Examples with Branch Cuts -Upper Bounds for Moduli of Contour Integrals –Anti derivatives -Cauchy–Goursat Theorem -Simply Connected Domains- Multiply Connected DomainsCauchy Integral Formula -An Extension of the Cauchy Integral Formula -Liouville’s Theorem and the Fundamental Theorem of Algebra -Maximum Modulus Principle

 UNIT-III
 Convergence of Sequences - Convergence of Series - Taylor Series -Laurent Series -Absolute and Uniform Convergence of Power Series- Continuity of Sums of Power Series - Integration and Differentiation of Power Series - Uniqueness of Series Representations-Isolated Singular Points - Residues -Cauchy’s Residue Theorem - Residue at Infinity - The Three Types of Isolated Singular Points - Residues at Poles -Examples -Zeros of Analytic Functions -Zeros and Poles -Behavior of Functions Near Isolated Singular Points

 UNIT-IV
Evaluation of Improper Integrals -Improper Integrals from Fourier Analysis - Jordan’s Lemma - Indented Paths - - Definite Integrals Involving Sines and Cosines - Argument Principle -Rouche ́’s Theorem -Linear Transformations -The Transformation w = 1/z - Mappings by 1/z -Linear Fractional Transformations -An Implicit Form -Mappings of the Upper Half Plane

Text: James Ward Brown, Ruel V Churchill, Complex Variables with applications.

MM302                                                                                                                 Semester-III  
 Paper-II Functional Analysis

 Unit –I
 NORMED LINEAR SPACES: Definitions and Elementary Properties, Subspace, Closed Subspace, Finite Dimensional Normed LinearSpaces and Subspaces, Quotient Spaces, Completion of Normed Spaces.

 Unit-II 
HILBERT SPACES: Inner Product Space, Hilbert Space, Cauchy-Bunyakovsky-Schwartz Inequality, Parallelogram Law, Orthogonality, Orthogonal Projection Theorem, Orthogonal Complements, Direct Sum, Complete Orthonormal System, Isomorphism between Separable HilbertSpaces.

 Unit-III 
LINEAR OPERATORS: Linear Operators in Normed Linear Spaces, Linear Functionals, The Space of Bounded Linear Operators, Uniform Boundedness Principle,Hahn-Banach Theorem, Hahn-Banach Theorem for Complex Vector and Normed Linear Space, The General Form of Linear Functionals in Hilbert Spaces.

Unit-IV
 FUNDAMENTAL THEOREMS FOR BANACH SPACES AND ADJOINT OPERATORS IN HILBERT SPACES: Closed Graph Theorem, Open Mapping Theorem, Bounded Inverse Theorem, Adjoint Operators, Self-Adjoint Operators, Quadratic Form, Unitary Operators, Projection Operators.

Text Book: A First Course in Functional Analysis-Rabindranath Sen, Anthem Press An imprint of Wimbledon Publishing Company.

 Reference: 1. Introductory Functional Analysis- E.Kreyzig- John Wilely and sons, New York, 2. Functional Analysis, by B.V. Limaye 2nd Edition. 3. Introduction to Topology and Modern Analysis- G.F.Simmons. Mc.Graw-Hill International Edition.

MM –303 A                                                                                                           Semester -III 
 Paper-III (A) Discrete Mathematics 

UNIT- I 
LATTICES: Partial Ordering – Lattices as Posets – some properties of Lattices – Lattices as Algebraic Systems – Sublattices, Direct products and Homomorphisms – some special Lattices – Complete, complemented and distributive lattices. (Pages 183-192, 378-397 of [1])

UNIT- II
BOOLEAN ALGEBRA: Boolean Algebras as Lattices – Boolean Identities – the switching Algebra – sub algebra, Direct product and homomorphism – Join irreducible elements – Atoms (minterms) – Boolean forms and their equivalence – minterm Boolean forms – Sum of products canonical forms – values of Boolean expressions and Boolean functions – Minimization of Boolean functions – the Karnaugh map method. (Pages 397 – 436 of [1])

 UNIT- III
 GRAPHS AND PLANAR GRAPHS : Directed and undirected graphs – Isomorphism of graphs – subgraph – complete graph – multigraphs and weighted graphs – paths – simple and elementary paths – circuits – connectedness – shortest paths in weighted graphs – Eulerian paths and circuits – Incoming degree and outgoing degree of a vertex - Hamiltonian paths and circuits – Planar graphs – Euler’s formula for planar graphs. (Pages 137-159, 168-186 of [2])

UNIT- IV 
TREES AND CUT-SETS: Properties of trees – Equivalent definitions of trees - Rooted trees – Binary trees – path lengths in rooted trees – Prefix codes – Binary search trees – Spanning trees and Cut-sets – Minimum spanning trees (Pages 187-213 of [2])

Text Books:- [1] J P Tremblay and R. Manohar: Discrete Mathematical Structures with applications to Computer Science, McGraw Hill Book Company [2] C L Liu : Elements of Discrete Mathematics, Tata McGraw Hill Publishing Company Ltd. New Delhi. (Second Edition).

MM – 304A                                                                                                                  Semester III  Paper IV A   Operations Research       

 Unit I 

Formulation of Linear Programming problems, Graphical solution of Linear Programming problem, General formulation of Linear Programming problems, Standard and Matrix forms of Linear Programming problems, Simplex Method, Two-phase method, Big-M method, Method to resolve degeneracy in Linear Programming problem, Alternative optimal solutions. Solution of simultaneous equations by simplex Method, Inverse of a Matrix by simplex Method, Concept of Duality in Linear Programming, Comparison of solutions of the Dual and its primal.

 Unit II
 Mathematical formulation of Assignment problem, Reduction theorem, Hungarian Assignment Method, Travelling salesman problem, Formulation of Travelling Salesman problem as an Assignment problem, Solution procedure. Mathematical formulation of Transportation problem, Tabular representation, Methods to find initial basic feasible solution, North West corner rule, Lowest cost entry method, Vogel's approximation methods, Optimality test, Method of finding optimal solution, Degeneracy in transportation problem, Method to resolve degeneracy, Unbalanced transportation problem.

Unit III 
Concept of Dynamic programming, Bellman's principle of optimality, characteristics of Dynamic programming problem, Backward and Forward recursive approach, Minimum path problem, Single Additive constraint and Multiplicatively separable return, Single Additive constraint and Additively separable return, Single Multiplicatively constraint and Additively separable return.

Unit-IV
 Historical development of CPM/PERT Techniques - Basic steps - Network diagram representation - Rules for drawing networks - Forward pass and Backward pass computations - Determination of floats - Determination of critical path - Project evaluation and review techniques updating.

 Text Books: [1] S. D. Sharma, Operations Research.
 [2] Kanti Swarup, P. K. Gupta and Manmohan, Operations Research.
 [3] H. A. Taha, Operations Research – An Introduction.