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E-grāmata: Risk and Uncertainty Reduction by Using Algebraic Inequalities

(Professor, Dept. of Mechanical Engineering and Mathematical Sciences, Oxford Brookes University, UK)
  • Formāts: 208 pages
  • Izdošanas datums: 02-Jun-2020
  • Izdevniecība: CRC Press
  • Valoda: eng
  • ISBN-13: 9781000076400
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Risk and Uncertainty Reduction by Using Algebraic InequalitiesPalielināt
  • Formāts: 208 pages
  • Izdošanas datums: 02-Jun-2020
  • Izdevniecība: CRC Press
  • Valoda: eng
  • ISBN-13: 9781000076400
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This book covers the application of algebraic inequalities for reliability improvement and for uncertainty and risk reduction. It equips readers with powerful domain-independent methods for reducing risk based on algebraic inequalities and demonstrates the significant benefits derived from the application for risk and uncertainty reduction.

Algebraic inequalities:

• Provide a powerful reliability-improvement, risk and uncertainty reduction method that transcends engineering and can be applied in various domains of human activity

• Present an effective tool for dealing with deep uncertainty related to key reliability-critical parameters of systems and processes

• Permit meaningful interpretations which link abstract inequalities with the real world

• Offer a tool for determining tight bounds for the variation of risk-critical parameters and complying the design with these bounds to avoid failure

• Allow optimising designs and processes by minimising the deviation of critical output parameters from their specified values and maximising their performance

This book is primarily for engineering professionals and academic researchers in virtually all existing engineering disciplines.

Preface xiii
Author xvii
Chapter 1 Fundamental Concepts Related to Risk and Uncertainty Reduction by Using Algebraic Inequalities
1(20)
1.1 Domain-Independent Approach to Risk Reduction
1(3)
1.2 A Powerful Domain-Independent Method for Risk and Uncertainty Reduction Based on Algebraic Inequalities
4(6)
1.2.1 Classification of Techniques Based on Algebraic Inequalities for Risk and Uncertainty Reduction
8(2)
1.3 Risk and Uncertainty
10(11)
Chapter 2 Properties of Algebraic Inequalities: Standard Algebraic Inequalities
21(18)
2.1 Basic Rules Related to Algebraic Inequalities
21(1)
2.2 Basic Properties of Inequalities
21(2)
2.3 One-Dimensional Triangle Inequality
23(1)
2.4 The Quadratic Inequality
24(1)
2.5 Jensen Inequality
25(1)
2.6 Root-Mean Square--Arithmetic Mean--Geometric Mean--Harmonic Mean (RMS-AM-GM-HM) Inequality
26(1)
2.7 Weighted Arithmetic Mean--Geometric Mean (AM-GM) Inequality
27(1)
2.8 Holder Inequality
28(1)
2.9 Cauchy--Schwarz Inequality
29(1)
2.10 Rearrangement Inequality
30(3)
2.11 Chebyshev Sum Inequality
33(1)
2.12 Muirhead Inequality
34(1)
2.13 Markov Inequality
35(1)
2.14 Chebyshev Inequality
36(1)
2.15 Minkowski Inequality
36(3)
Chapter 3 Basic Techniques for Proving Algebraic Inequalities
39(30)
3.1 The Need for Proving Algebraic Inequalities
39(1)
3.2 Proving Inequalities by a Direct Algebraic Manipulation and Analysis
39(3)
3.3 Proving Inequalities by Presenting Them as a Sum of Non-Negative Terms
42(2)
3.4 Proving Inequalities by Proving Simpler Intermediate Inequalities
44(1)
3.5 Proving Inequalities by Substitution
45(1)
3.6 Proving Inequalities by Exploiting the Symmetry
45(2)
3.7 Proving Inequalities by Exploiting Homogeneity
47(2)
3.8 Proving Inequalities by a Mathematical Induction
49(3)
3.9 Proving Inequalities by Using the Properties of Convex/Concave Functions
52(4)
3.9.1 Jensen Inequality
54(2)
3.10 Proving Inequalities by Using the Properties of Sub-Additive and Super-Additive Functions
56(2)
3.11 Proving Inequalities by Transforming Them to Known Inequalities
58(4)
3.11.1 Proving Inequalities by Transforming Them to an Already Proved Inequality
58(1)
3.11.2 Proving Inequalities by Transforming Them to the Holder Inequality
59(1)
3.11.3 An Alternative Proof of the Cauchy-Schwarz Inequality by Reducing It to a Standard Inequality
60(1)
3.11.4 An Alternative Proof of the GM-HM Inequality by Reducing It to the AM-GM Inequality
60(1)
3.11.5 Proving Inequalities by Transforming Them to the Cauchy--Schwarz Inequality
61(1)
3.12 Proving Inequalities by Segmentation
62(2)
3.12.1 Determining Bounds by Segmentation
64(1)
3.13 Proving Algebraic Inequalities by Combining Several Techniques
64(1)
3.14 Using Derivatives to Prove Inequalities
65(4)
Chapter 4 Using Optimisation Methods for Determining Tight Upper and Lower Bounds: Testing a Conjectured Inequality by a Simulation -- Exercises
69(26)
4.1 Using Constrained Optimisation for Determining Tight Upper Bounds
69(2)
4.2 Tight Bounds for Multivariable Functions Whose Partial Derivatives Do Not Change Sign in a Specified Domain
71(2)
4.3 Conventions Adopted in Presenting the Simulation Algorithms
73(3)
4.4 Testing a Conjectured Algebraic Inequality by a Monte Carlo Simulation
76(4)
4.5 Exercises
80(3)
4.6 Solutions to the Exercises
83(12)
Chapter 5 Ranking the Reliabilities of Systems and Processes by Using Inequalities
95(12)
5.1 Improving Reliability and Reducing Risk by Proving an Abstract Inequality Derived from the Real Physical System or Process
95(1)
5.2 Using Algebraic Inequalities for Ranking Systems Whose Component Reliabilities Are Unknown
95(5)
5.2.1 Reliability of Systems with Components Logically Arranged in Series and Parallel
96(4)
5.3 Using Inequalities to Rank Systems with the Same Topology and Different Component Arrangements
100(3)
5.4 Using Inequalities to Rank Systems with Different Topologies Built with the Same Types of Components
103(4)
Chapter 6 Using Inequalities for Reducing Epistemic Uncertainty and Ranking Decision Alternatives
107(12)
6.1 Selection from Sources with Unknown Proportions of High-Reliability Components
107(3)
6.2 Monte Carlo Simulations
110(3)
6.3 Extending the Results by Using the Muirhead Inequality
113(6)
Chapter 7 Creating a Meaningful Interpretation of Existing Abstract Inequalities and Linking It to Real Applications
119(20)
7.1 Meaningful Interpretations of Abstract Algebraic Inequalities with Applications to Real Physical Systems
119(7)
7.1.1 Applications Related to Robust and Safe Design
124(2)
7.2 Avoiding Underestimation of the Risk and Overestimation of Average Profit by a Meaningful Interpretation of the Chebyshev Sum Inequality
126(2)
7.3 A Meaningful Interpretation of an Abstract Algebraic Inequality with an Application for Selecting Components of the Same Variety
128(2)
7.4 Maximising the Chances of a Beneficial Random Selection by a Meaningful Interpretation of a General Inequality
130(4)
7.5 The Principle of Non-Contradiction
134(5)
Chapter 8 Optimisation by Using Inequalities
139(14)
8.1 Using Inequalities for Minimising the Deviation of Reliability-Critical Parameters
139(1)
8.2 Minimising the Deviation of the Volume of Manufactured Workpieces with Cylindrical Shapes
140(2)
8.3 Minimising the Deviation of the Volume of Manufactured Workpieces in the Shape of a Rectangular Prism
142(2)
8.4 Minimising the Deviation of the Resonant Frequency from the Required Level for Parallel Resonant LC-Circuits
144(2)
8.5 Maximising Reliability by Using the Rearrangement Inequality
146(5)
8.5.1 Using the Rearrangement Inequality to Maximise the Reliability of Parallel-Series Systems
146(4)
8.5.2 Using the Rearrangement Inequality for Optimal Condition Monitoring
150(1)
8.6 Using the Rearrangement Inequality to Minimise the Risk of a Faulty Assembly
151(2)
Chapter 9 Determining Tight Bounds for the Uncertainty in Risk-Critical Parameters and Properties by Using Inequalities
153(16)
9.1 Upper-Bound Variance Inequality for Properties from Different Sources
153(3)
9.2 Identifying the Source Whose Removal Causes the Largest Reduction of the Worst-Case Variation
156(1)
9.3 Increasing the Robustness of Electronic Products by Using the Variance Upper-Bound Inequality
157(1)
9.4 Determining Tight Bounds for the Fraction of Items with a Particular Property
158(1)
9.5 Using the Properties of Convex Functions for Determining the Upper Bound of the Equivalent Resistance
159(3)
9.6 Determining a Tight Upper Bound for the Risk of a Faulty Assembly by Using the Chebyshev Inequality
162(2)
9.7 Deriving a Tight Upper Bound for the Risk of a Faulty Assembly by Using the Chebyshev Inequality and Jensen Inequality
164(5)
Chapter 10 Using Algebraic Inequalities to Support Risk-Critical Reasoning
169(12)
10.1 Using the Inequality of the Negatively Correlated Events to Support Risk-Critical Reasoning
169(2)
10.2 Avoiding Risk Underestimation by Using the Jensen Inequality
171(5)
10.2.1 Avoiding the Risk of Overestimating Profit
171(2)
10.2.2 Avoiding the Risk of Underestimating the Cost of Failure
173(1)
10.2.3 A Conservative Estimate of System Reliability by Using the Jensen Inequality
174(2)
10.3 Reducing Uncertainty and Risk Associated with the Prediction of Magnitude Rankings
176(5)
References 181(4)
Index 185
Michael T. Todinov, PhD, has a background in mechanical engineering, applied mathematics and computer science. Prof.Todinov pioneered reliability analysis based on the cost of failure, repairable flow networks and networks with disturbed flows, domain-independent methods for reliability improvement and risk reduction and reducing risk and uncertainty by using algebraic inequalities.
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