Paper 1 Functional Analysis-M301
Course Outcomes
1.Facility with the main, big theorems of functional analysis.
2.Ability to use duality in various contexts and theoretical results from the course in concrete situations.
3.Capacity to work with families of applications appearing in the course, particularly specific calculations needed in the context of Baire Category.
4.Be able to produce examples and counterexamples illustrating the mathematical concepts presented in the course.
Normed Spaces - Banach Spaces - Further properties of normed spaces - Finite dimensional normed spaces and sub spaces - compactness and finite dimension - linear operators - Bounded and continuous linear operators. [2.2, 2.3, 2.4, 2.5, 2.6 and 2.7].
Linear functional – normed spaces of operators – Dual space – Inner product space-Hilbert Space – Further Properties of Inner product Spaces – Orthogonal complements and direct sums – Orthogonal sets and sequences. [ 2.8, 2.10, 3.1, 3.2, 3.3 and 3.4]
Series related to Orthonormal Sequences and sets – Total Orthonormal sets and sequences – Representation of Functions on Hilbert spaces – Hilbert – Adjoint Operator-Self-Adjoint,unitary and normal operators. [3.5, 3.6, 3.8, 3.9 and 3.10]
Hahn-Banach Theorem - Hahn-Banach Theorem for Complex Vector Spaces and Normed Spaces –Adjoint Operator- Reflexive Spaces- Category Theorem - Uniform Boundedness Theorem - Open Mapping Theorem - Closed Linear Operators – Closed Graph Theorem.
[4.2, 4.3, 4.5, 4.6, 4.7, 4.12 and 4.13]
Introductory Functional Analysis with Applications by Erwin Kreyszig, John Wiley and sons, NewYork.
Functional Analysis by B.V.Limaye 2nd Edition..
Introduction to Topology and Modern Analysis by G.F.Sinmmons. Mc.Graw-Hill International Edition.
Course Outcomes:
1.Explain divisibility, primes, and the Euclidean algorithm using the Fundamental Theorem of Arithmetic.
2.Apply arithmetical and multiplicative functions to compute Dirichlet products and Möbius inversions.
3.Solve problems involving congruences using Euler–Fermat theorem, Lagrange’s theorem, and the Chinese remainder theorem.
4.Analyze quadratic residues using Legendre and Jacobi symbols and apply the Quadratic Reciprocity Law.
The Fundamental Theorem of Arithmetic: Divisibility- GCD- Prime numbers, Fundamental theorem of arithmetic- the series of reciprocal of the primes- The Euclidean algorithm.
(Page No. 13 - 23)
Arithmetical Functions and Dirichlet Multiplication: The functions ϕ(n), µ(n) and a relation connecting them- Product formula for ϕ(n) - Dirichlet product- Dirichlet inverse and Mobius inversion formula -The Mangoldt function ∧(n)- Multiplicative functions and Dirichlet multiplication- The inverse of a completely multiplicative function- Liouville’s function λ(n)- The divisor functions σα(n).
(Page No. 24-39 & 46-51)
Congruences: Properties of congruences- Residue classes and complete residue system- Linear congruences-Reduced residue systems and Euler-Fermat theorem- Polynomial congruence modulo p - Lagrange’s theorem- Application of Lagrange’s theorem- Chinese remainder theorem and its applications.
(Page No. 106-120 & 126-128)
Quadratic Residues and The 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-The Jacobi symbol.
(Page No. 178-190 & 201-203)
Introduction to Analytic Number Theory by Tom M. Apostol. Narosa publishing house
Number Theory by Joseph H. Silverman.
Theory of Numbers by K.Ramchandra.
Elementary Number Theory by James K Strayer.
Elementary Number Theory by James Tattusall.
Course Outcomes:
1.Apply propositional logic, Boolean algebra, and proof techniques to analyze logical statements and circuits.
2.Use combinatorial principles to compute permutations, combinations, and apply the inclusion–exclusion principle.
3.Solve recurrence relations using substitution, generating functions, and characteristic roots.
4.Analyze graph structures and apply graph algorithms for shortest paths, spanning trees, and traversals.
Propositional logic, Propositional Equivalences, Predicates and Quantifiers, Rules of Inference–
Valid Arguments in Propositional Logic.Rules of Inference for Quantified Statements.Introduction
to Proofs – Direct Proofs, Proofs by Contraposition, Proofs by Contradiction.NormalForms–Disjunctive Normal Form, Conjunctive Normal Forms, Principal Disjunctive Normal Form, Principal Conjunctive Normal Form. Boolean Algebra – Boolean Functions and Boolean Expressions, Identities of Boolean Algebra, Representing Boolean Functions. Logic Gates, Minimization of Circuits–K- maps. (1.1 to 1.3, 1.5 to 1.7, 10.1 to 10.4 of [1])
Elementary Combinatorics – Basics of Counting, Two Basic Counting Principles, Indirect Counting.Combinations and Permutations – Enumeration of Combinations and Permutations, Enumerating Combinations
and Permutations with Repetitions, Enumerating Permutations with Constrained Repetitions. Binomial Coefficients – Pascal’s Identity, Pascal’s Triangle. Multinominal Theorem, The Principle of Inclusion–Exclusion and its Applications. (2.1 to 2.8 of [2])
Recurrence Relations – Generating Functions of Sequences, Generating Function Models, Calculating Coefficients of Generating Functions. Solutions of Recurrence Relations, the Fibonacci Relation. Solving Recurrence Relations by Substitution and by Generating Functions, Method of Characteristic Roots. Solution of Inhomogeneous Linear Recurrence Relations, the Method of Undetermined Coefficients: Solving Nonlinear Recurrence Relations. (3.1 to 3.6 of [2])
Graphs – Graphs and Graph Models, Graph Terminology and Special Types of Graphs, The Hand shaking Theorem, Representing Graphs and Graph Isomorphism. Connectivity, Euler and Hamiltonian Paths and Circuits,Shortest Path Problems, Dijkstra’s Algorithm,Planar Graphs,Euler formula. Trees – Introduction to Trees, Tree Traversal. Spanning Trees, DFS, BFS Algorithms, Minimum Spanning Trees. Prim’s and Kruskal’s Algorithms. (8.1 to 8.7, 9.1, 9.3, 9.5 of [1])
Discrete Mathematics and its Applications by Kenneth HRosen, Seventh Edition, Mc GrawHill Education (India)Private Ltd, New Delhi.
Discrete Mathematics for Computer Scientists & Mathematicians by JoeL.Mott, Abraham Kandel and Theodore P. Baker, Second Edition, Prentice Hall of India, Private Ltd, NewDelhi.
Elements of Discrete Mathematics by C L Liu and D P Mohapatra, Third Edition, The McGraw-Hill Companies.
Discrete and Combinatorial Mathematics by Ralph P. Grimaldi and B. V. Ramana, 5th Edition,PEARSON education.
Course Outcomes:
1.Formulate and solve linear programming problems using graphical and simplex-based methods.
2.Analyze primal–dual relationships and apply simplex variations to solve linear systems.
3.Apply optimization techniques to solve transportation, assignment, and travelling salesman problems.
4.Solve multistage decision problems using dynamic programming and Bellman’s optimality principle.
Formulation of Linear Programming problems, Graphical solution of Linear Programming problem,Convex set,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,Revised Simplex Method,Concept of Duality in Linear Programming, Comparison of solutions of the Dual and its primal
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 method, Optimality test, Method of finding optimal solution, Degeneracy in transportation problem,
Method to resolve degeneracy, Unbalanced transportation problem. Mathematical formulation of Assignment problem, Reduction theorem, Hungarian Assignment Method, Travelling salesman problem, Formulation of Travelling Salesman problem as an Assignment problem, Solution procedure
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.
Operations Research by S.D.Sharma, 18th Revised Edition 2017, KedarNath Ram Nath Publications.
Operations Research – An Introduction by Hamdy A. Taha, 10th Edition.
Linear Programming by G.Hadley.