Functional Analysis-M301
Paper-I
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.
Unit I
Normed Space.
BanachSpaces.
Further properties of normed spaces.
Finitedimensional normed spaces and subspaces
compactness and finite dimension.
linear operators.
Bounded and continuous linear operators.
Unit II
Linear Functionals.
normed spaces of operators.
Dualspace-Inner product space-Hilbert Space.
Fur-ther Properties of Inner product Spaces.
Orthogonal complements and direct sums.
Orthogonal sets and sequences.
Unit III
Series related to Ortho normal Sequences and sets.
Total Orthonormal sets and sequences.
Representation of Functions on Hilbert spaces.
Hilbert-Adjoin tOperator,Self-Adjoint,unitary and normal operators.
Unit-IV
Hahn-BanachTheorem.
Hahn-BanachTheorem for Complex Vector Spaces and Normed Spaces.
Adjoint Operator.
Category Theorem.
Uniform Boundedness Theorem.
Open Mapping Theorem.
Closed Linear Operators, Closed Graph Theorem.
TextBook:
Introductory Functional Analysis with Applications by Erwin Kreyszig, JohnWiley and sons,NewYork.
References:
[1] Functional Analysis by B.V.Limaye 2nd Edition.
[2] Introduction to Topology and Modern Analysis by G.F.Sinmmons.Mc.Graw-Hill International Edition.
General Measure & Integration-M302
Paper-II
Course Outcomes
1.Demonstrate understanding of the basic concepts underlying the definition of the general Lebesgue integral
2.Demonstrate familiarity with a range of examples of these concepts
3.Prove the basic results of measure theory and integration theory
4.Demonstrate understanding of the statement and proofs of the fundamental integral convergence theorems, and their applications
Unit I
Measure spaces.
Measurable functions-Integration.
General Convergence theorem.
Unit II
Signed measures.
The Radon-Nikodym theorem.
Unit III
Outer measure and measurability.
The Extension theorem.
The Product measure.
Unit-IV
Inner measure.
Extension by sets of measure zero.
Caratheodory outer measure.
TextBook:
Real Analysis(Chapters11,12)ByH.L.Royden,Wiley.
Linear Algebra-M303
Paper-III
Course Outcomes
1.Solving linear equations, working with matrices, in particular eigenvalues and eigenvectors, and applying the techniques to real life problems like graph theory, computer science, electronics and applied mathematics. Spectral theorems, prevalent in many branches of mathematics.
2.Use computational techniques and algebraic skills essential for the study of systems of linear equations, matrix algebra, vector spaces, eigenvalues and eigenvectors, orthogonality and diagonalization. (Computational and Algebraic Skills
3.Use visualization, spatial reasoning, as well as geometric properties and strategies to model, solve problems, and view solutions, especially in R2 and R3 , as well as conceptually extend these results to higher dimensions. (Geometric Skills).
4.Use technology, where appropriate, to enhance and facilitate mathematical understanding, as well as an aid in solving problems and presenting solutions (Technological Skills)
Unit I
Elementary Canonical forms Introduction
Characteristic Values,Annihilating Polynomials.
Invariant Sub-spaces.
Simultaneous Triangulation and Simultaneous Diagonalization
Unit II
Direct sum Decomposition,Invariant Direc tsums.
The Primary DecompositionTheorem.
The Rational and Jordan Forms:Cyclic Subspaces and Annihilators.
Unit III
Cyclic Decompositions and the RationalForm.
TheJordanForm.
Computation of Invariant Factors.
Semi Simple Operators.
Unit-IV
Bilinear Forms:BilinearForms, Symmetric Bilinear Forms.
Skew-Symmetric Bilinear Forms.
GroupsPre-serving BilinearForms.
TextBook:
Linear AlgebrabyKennethHo manandRayKunze(2e)PHI
References:
[1] AdvancedLinearAlgebrabyStevenRoman(3e)
[2] LinearAlgebrabyDavidCLay
[3] LinearAlgebrabyKuldeepSingh
Operations Research-M304
Paper-IV
Course Outcomes
1.Define and formulate linear programming problems and appreciate. their limitations.
2.Solve linear programming problems using appropriate techniques and optimization solvers, interpret the results obtained and translate solutions into directives for action.
3.Conduct and interpret post-optimal and sensitivity analysis and explain the primal-dual relationship.
4.Develop mathematical skills to analyse and solve integer programming and network models arising from a wide range of applications
Unit I
Formulation of Linea rProgramming 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 o fAssignmentproblem.
Reduction theorem.
Hungarian Assignment Method.
Travelling sales man problem, Formulation of Travelling Sales man problem.
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,
Opti-malitytest, 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-Basicsteps.
Network diagram representation.
Rules for drawing networks.
Forwardpass andBack ward pass computations.
Determination of floats.
Determination of critical path.
Project evaluation and review techniques.
TextBooks:
[1] S.D.Sharma,OperationsResearch.
[2] KantiSwarup,P.K.GuptaandManmohan,OperationsResearch.
[3] H.A.Taha,OperationsResearch{AnIntroduction.
[4] G.I.Gauss,LinearProgramming.
Numerical Analysis-M305
Paper-V
Course Outcomes
1. Explain the least square method
2.Find the lagrange polynomial passing through the given points
3.Find the hermite polynomial passing through the given points
4.Find the cubic spline passing through the given points
Unit I
Transcendental and Polynomial Equations: Introduction.
Bisection Method-Iteration.
Methods Based on First Degree Equation:SecantMethod, Regula FalsiMethod, Newton-Raphson Method.
Iteration Methods Based on Second Degree Equation: Muller'sMethod, ChebyshevMethod, Multipoint IterationMethods.
Rate of convergence-Iteration Methods.
Unit II
System of Linear Algebraic Equations: Introduction.
DirectMethods: GaussEliminationMethod,Gauss Jordan Elimination Method, Triangularization Method, Cholesky Method,Partition Method.
Iteration Methods: Jacobi Iteration Method, Gauss Seidel Iteration Method, SOR Method.
Unit III
Interpolation and Approximation: Interpolation: Introduction.
Lagrange and Newton Interpolations.
Finite Di erence Operators-Interpolating Polynomials using FiniteDi erences.
Hermite Interpolations.
Piecewise and Spline Interpolation.
Approximation: Least Squares Approximation.
Unit-IV
Numerical Integration: Methods Based on Interpolation: Newton Cotes Methods.
Methods Based on Undetermined Coe cients: Guass Legendre Integration Methods-Composite Integration Methods.
Numerical Solution of ODE's: Introduction-Numerical Methods: EulerMethods-Midpoint Method- Single Step Methods: Taylor series method, Runge-Kutta Method.
Multi step Methods: Adam Bash forth Method-Adams Moulton Method.
Milne-Simpson Method.
Predictor Corrector Methods.
TextBooks:
[1] NumericalMethodsforScienti candEngineeringComputationbyM.K.Jain,S.R.K.Iyengar,R.K.
Jain, NewAgeInt.Ltd.,NewDelhi.