دانلود کتاب Risk and Uncertainty Reduction by Using Algebraic Inequalities

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Cover Half Title Title Page Copyright Page Dedication Table of Contents Preface Author Chapter 1: Fundamental Concepts Related to Risk and Uncertainty Reduction by Using Algebraic Inequalities 1 1 Domain-Independent Approach to Risk Reduction 1 2 A Powerful Domain-Independent Method for Risk and Uncertainty Reduction Based on Algebraic Inequalities 1 2 1 Classification of Techniques Based on Algebraic Inequalities for Risk and Uncertainty Reduction 1 3 Risk and Uncertainty Chapter 2: Properties of Algebraic Inequalities: Standard Algebraic Inequalities 2 1 Basic Rules Related to Algebraic Inequalities 2 2 Basic Properties of Inequalities 2 3 One-Dimensional Triangle Inequality 2 4 The Quadratic Inequality 2 5 Jensen Inequality 2 6 Root-Mean Square–Arithmetic Mean–Geometric Mean–Harmonic Mean (RMS-AM-GM-HM) Inequality 2 7 Weighted Arithmetic Mean–Geometric Mean (AM-GM) Inequality 2 8 Hölder Inequality 2 9 Cauchy-Schwarz Inequality 2 10 Rearrangement Inequality 2 11 Chebyshev Sum Inequality 2 12 Muirhead Inequality 2 13 Markov Inequality 2 14 Chebyshev Inequality 2 15 Minkowski Inequality Chapter 3: Basic Techniques for Proving Algebraic Inequalities 3 1 The Need for Proving Algebraic Inequalities 3 2 Proving Inequalities by a Direct Algebraic Manipulation and Analysis 3 3 Proving Inequalities by Presenting Them as a Sum of Non-Negative Terms 3 4 Proving Inequalities by Proving Simpler Intermediate Inequalities 3 5 Proving Inequalities by Substitution 3 6 Proving Inequalities by Exploiting the Symmetry 3 7 Proving Inequalities by Exploiting Homogeneity 3 8 Proving Inequalities by a Mathematical Induction 3 9 Proving Inequalities by Using the Properties of Convex/Concave Functions 3 9 1 Jensen Inequality 3 10 Proving Inequalities by Using the Properties of Sub-Additive and Super-Additive Functions 3 11 Proving Inequalities by Transforming Them to Known Inequalities 3 11 1 Proving Inequalities by Transforming Them to an Already Proved Inequality 3 11 2 Proving Inequalities by Transforming Them to the Hölder Inequality 3 11 3 An Alternative Proof of the Cauchy-Schwarz Inequality by Reducing It to a Standard Inequality 3 11 4 An Alternative Proof of the GM-HM Inequality by Reducing It to the AM-GM Inequality 3 11 5 Proving Inequalities by Transforming Them to the Cauchy-Schwarz Inequality 3 12 Proving Inequalities by Segmentation 3 12 1 Determining Bounds by Segmentation 3 13 Proving Algebraic Inequalities by Combining Several Techniques 3 14 Using Derivatives to Prove Inequalities Chapter 4: Using Optimisation Methods for Determining Tight Upper and Lower Bounds: Testing a Conjectured Inequality by a Simulation – Exercises 4 1 Using Constrained Optimisation for Determining Tight Upper Bounds 4 2 Tight Bounds for Multivariable Functions Whose Partial Derivatives Do Not Change Sign in a Specified Domain 4 3 Conventions Adopted in Presenting the Simulation Algorithms 4 4 Testing a Conjectured Algebraic Inequality by a Monte Carlo Simulation 4 5 Exercises 4 6 Solutions to the Exercises Chapter 5: Ranking the Reliabilities of Systems and Processes by Using Inequalities 5 1 Improving Reliability and Reducing Risk by Proving an Abstract Inequality Derived from the Real Physical System or Process 5 2 Using Algebraic Inequalities for Ranking Systems Whose Component Reliabilities Are Unknown 5 2 1 Reliability of Systems with Components Logically Arranged in Series and Parallel 5 3 Using Inequalities to Rank Systems with the Same Topology and Different Component Arrangements 5 4 Using Inequalities to Rank Systems with Different Topologies Built with the Same Types of Components Chapter 6: Using Inequalities for Reducing Epistemic Uncertainty and Ranking Decision Alternatives 6 1 Selection from Sources with Unknown Proportions of High-Reliability Components 6 2 Monte Carlo Simulations 6 3 Extending the Results by Using the Muirhead Inequality Chapter 7: Creating a Meaningful Interpretation of Existing Abstract Inequalities and Linking It to Real Applications 7 1 Meaningful Interpretations of Abstract Algebraic Inequalities with Applications to Real Physical Systems 7 1 1 Applications Related to Robust and Safe Design 7 2 Avoiding Underestimation of the Risk and Overestimation of Average Profit by a Meaningful Interpretation of the Chebyshev Sum Inequality 7 3 A Meaningful Interpretation of an Abstract Algebraic Inequality with an Application for Selecting Components of the Same Variety 7 4 Maximising the Chances of a Beneficial Random Selection by a Meaningful Interpretation of a General Inequality 7 5 The Principle of Non-Contradiction Chapter 8: Optimisation by Using Inequalities 8 1 Using Inequalities for Minimising the Deviation of Reliability-Critical Parameters 8 2 Minimising the Deviation of the Volume of Manufactured Workpieces with Cylindrical Shapes 8 3 Minimising the Deviation of the Volume of Manufactured Workpieces in the Shape of a Rectangular Prism 8 4 Minimising the Deviation of the Resonant Frequency from the Required Level for Parallel Resonant LC-Circuits 8 5 Maximising Reliability by Using the Rearrangement Inequality 8 5 1 Using the Rearrangement Inequality to Maximise the Reliability of Parallel-Series Systems 8 5 2 Using the Rearrangement Inequality for Optimal Condition Monitoring 8 6 Using the Rearrangement Inequality to Minimise the Risk of a Faulty Assembly Chapter 9: Determining Tight Bounds for the Uncertainty in Risk-Critical Parameters and Properties by Using Inequalities 9 1 Upper-Bound Variance Inequality for Properties from Different Sources 9 2 Identifying the Source Whose Removal Causes the Largest Reduction of the Worst-Case Variation 9 3 Increasing the Robustness of Electronic Products by Using the Variance Upper-Bound Inequality 9 4 Determining Tight Bounds for the Fraction of Items with a Particular Property 9 5 Using the Properties of Convex Functions for Determining the Upper Bound of the Equivalent Resistance 9 6 Determining a Tight Upper Bound for the Risk of a Faulty Assembly by Using the Chebyshev Inequality 9 7 Deriving a Tight Upper Bound for the Risk of a Faulty Assembly by Using the Chebyshev Inequality and Jensen Inequality Chapter 10: Using Algebraic Inequalities to Support Risk-Critical Reasoning 10 1 Using the Inequality of the Negatively Correlated Events to Support Risk-Critical Reasoning 10 2 Avoiding Risk Underestimation by Using the Jensen Inequality 10 2 1 Avoiding the Risk of Overestimating Profit 10 2 2 Avoiding the Risk of Underestimating the Cost of Failure 10 2 3 A Conservative Estimate of System Reliability by Using the Jensen Inequality 10 3 Reducing Uncertainty and Risk Associated with the Prediction of Magnitude Rankings References Index