Principles of Mathematics in Operations Research

(Rick Simeone) #1

  • 1 Introduction

    • 1.1 Mathematics and OR

    • 1.2 Mathematics as a language

    • 1.3 The art of making proofs

      • 1.3.1 Forward-Backward method

        • 1.3.2 Induction Method

        • 1.3.3 Contradiction Method

        • 1.3.4 Theorem of alternatives





    • Problems

    • Web material



  • 2 Preliminary Linear Algebra

    • 2.1 Vector Spaces

      • 2.1.1 Fields and linear spaces

      • 2.1.2 Subspaces

      • 2.1.3 Bases



    • 2.2 Linear transformations, matrices and change of basis

      • 2.2.1 Matrix multiplication

      • 2.2.2 Linear transformation



    • 2.3 Systems of Linear Equations

      • 2.3.1 Gaussian elimination

      • 2.3.2 Gauss-Jordan method for inverses

      • 2.3.3 The most general case



    • 2.4 The four fundamental subspaces

      • 2.4.1 The row space of A

      • 2.4.2 The column space of A

      • 2.4.3 The null space (kernel) of A

      • 2.4.4 The left null space of A

      • 2.4.5 The Fundamental Theorem of Linear Algebra



    • Problems

    • Web material



  • 3 Orthogonality X Contents

    • 3.1 Inner Products

      • 3.1.1 Norms

      • 3.1.2 Orthogonal Spaces

      • 3.1.3 Angle between two vectors

      • 3.1.4 Projection

      • 3.1.5 Symmetric Matrices

      • 3.2 Projections and Least Squares Approximations

        • 3.2.1 Orthogonal bases

        • 3.2.2 Gram-Schmidt Orthogonalization

        • 3.2.3 Pseudo (Moore-Penrose) Inverse

        • 3.2.4 Singular Value Decomposition

        • 3.3 Summary for Ax = b

        • Problems

        • Web material





    • 4 Eigen Values and Vectors

      • 4.1 Determinants

        • 4.1.1 Preliminaries

        • 4.1.2 Properties



      • 4.2 Eigen Values and Eigen Vectors

      • 4.3 Diagonal Form of a Matrix

        • 4.3.1 All Distinct Eigen Values

        • 4.3.2 Repeated Eigen Values with Full Kernels

        • 4.3.3 Block Diagonal Form



      • 4.4 Powers of A

        • 4.4.1 Difference equations

        • 4.4.2 Differential Equations



      • 4.5 The Complex case

      • Problems

      • Web material



    • 5 Positive Definiteness

      • 5.1 Minima, Maxima, Saddle points

        • 5.1.1 Scalar Functions

        • 5.1.2 Quadratic forms



      • 5.2 Detecting Positive-Definiteness

      • 5.3 Semidefinite Matrices

      • 5.4 Positive Definite Quadratic Forms

      • Problems

      • Web material





  • 6 Computational Aspects Contents XI

    • 6.1 Solution of Ax = b

      • 6.1.1 Symmetric and positive definite

      • 6.1.2 Symmetric and not positive definite

        • 6.1.3 Asymmetric





    • 6.2 Computation of eigen values

    • Problems

    • Web material

    • 7 Convex Sets

      • 7.1 Preliminaries

      • 7.2 Hyperplanes and Polytopes

      • 7.3 Separating and Supporting Hyperplanes

      • 7.4 Extreme Points

      • Problems

      • Web material



    • 8 Linear Programming

      • 8.1 The Simplex Method

      • 8.2 Simplex Tableau

      • 8.3 Revised Simplex Method

      • 8.5 Farkas' Lemma 8.4 Duality Theory Ill

      • Problems

      • Web material



    • 9 Number Systems

      • 9.1 Ordered Sets

      • 9.2 Fields

      • 9.3 The Real Field

      • 9.4 ' The Complex Field

      • 9.5 Euclidean Space

      • 9.6 Countable and Uncountable Sets

      • Problems

      • Web material





  • 10 Basic Topology

    • 10.1 Metric Spaces

    • 10.2 Compact Sets

    • 10.3 The Cantor Set

    • 10.4 Connected Sets

    • Problems

    • Web material

    • 11 Continuity XII Contents

      • 11.1 Introduction I

      • 11.2 Continuity and Compactness

      • 11.3 Uniform Continuity

        • 11.4 Continuity and Connectedness

        • 11.5 Monotonic Functions



      • Problems

      • Web material

      • 12 Differentiation

        • 12.1 Derivatives

        • 12.2 Mean Value Theorems

        • 12.3 Higher Order Derivatives

        • Problems

        • Web material



      • 13 Power Series and Special Functions

        • 13.1 Series

          • 13.1.1 Notion of Series

          • 13.1.2 Operations on Series

          • 13.1.3 Tests for positive series



        • 13.2 Sequence of Functions

        • 13.3 Power Series

        • 13.4 Exponential and Logarithmic Functions

        • 13.5 Trigonometric Functions

        • 13.6 Fourier Series

        • 13.7 Gamma Function

        • Problems

        • Web material





    • 14 Special Transformations

      • 14.1 Differential Equations

      • 14.2 Laplace Transforms

      • 14.3 Difference Equations

      • 14.4 Z Transforms

      • Problems

      • Web material





  • Solutions

  • Index

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