Interpretation of Algebraic Inequalities: Practical Engineering Optimisation and Generating New Knowledge

Cover
Half Title
Title Page
Copyright Page
Dedication
Table of Contents
Preface
Author
Chapter 1 Fundamental Approaches in Modelling Real Systems and Processes by Using Algebraic Inequalities. The Principle of Non-contradiction for Algebraic Inequalities
	1.1 Algebraic Inequalities and Their General Applications
	1.2 Algebraic Inequalities as a Domain-Independent Method for Reducing Uncertainty and Optimising the Performance of Systems and Processes
	1.3 Forward Approach to Modelling and Optimisation of Real Systems and Processes by Using Algebraic Inequalities
	1.4 Inverse Approach to Modelling and Generating New Knowledge by Interpretation of Inequalities
	1.5 The Principle of Non-contradiction for Algebraic Inequalities
	1.6 Key Steps in the Interpretation of Algebraic Inequalities
Chapter 2 Basic Algebraic Inequalities
	2.1 Basic Algebraic Inequalities Used for Proving Other Inequalities
		2.1.1 Basic Properties of Algebraic Inequalities and Techniques for Proving Algebraic Inequalities
		2.1.2 Cauchy–Schwarz Inequality
		2.1.3 Convex and Concave Functions: Jensen Inequality
		2.1.4 Root-Mean Square–Arithmetic Mean–Geometric Mean–Harmonic Mean (RMS–AM–GM–HM) Inequality
		2.1.5 Rearrangement Inequality
		2.1.6 Chebyshev’s Sum Inequality
		2.1.7 Muirhead’s Inequality
	2.2 Algebraic Inequalities That Permit Natural Meaningful Interpretation
		2.2.1 Symmetric Algebraic Inequalities Whose Terms Can Be Interpreted as Probabilities
		2.2.2 Transforming Algebraic Inequalities to Make Them Interpretable
		2.2.3 Inequalities Based on Sub- and Super-additive Functions
			2.2.3.1 Proof
		2.2.4 Bergström Inequality and Its Natural Interpretation
		2.2.5 A New Algebraic Inequality Which Provides Possibility for a Segmentation of Additive Factors
	2.3 Testing Algebraic Inequalities by Monte Carlo Simulation
Chapter 3 Generating Knowledge about Physical Systems by Meaningful Interpretation of Algebraic Inequalities
	3.1 An Algebraic Inequality Related to Equivalent Properties of Elements Connected in Series and Parallel
		3.1.1 Elastic Components and Resistors Connected in Series and Parallel
		3.1.2 Thermal Resistors and Electric Capacitors Connected in Series and Parallel
	3.2 Constructing a System with Superior Reliability by a Meaningful Interpretation of Algebraic Inequalities
		3.2.1 Reliability of Systems with Components Logically Arranged in Series and Parallel
		3.2.2 Constructing a Series–Parallel System with Superior Reliability through Interpretation of an Algebraic Inequality
		3.2.3 Constructing a Parallel–Series System with Superior Reliability through Interpretation of an Algebraic Inequality
	3.3 Selecting the System with Superior Reliability through Interpretation of the Inequality of Negatively Correlated Events
Chapter 4 Enhancing Systems Performance by Interpretation of the Bergström Inequality
	4.1 Extensive Quantities and Additivity
	4.2 Meaningful Interpretation of the Bergström Inequality to Maximise Electric Power Output
	4.3 Meaningful Interpretation of the Bergström Inequality to Maximise the Stored Electric Energy in Capacitors
	4.4 Aggregation of the Applied Voltage to Maximise the Energy Stored in Capacitors
	4.5 Meaningful Interpretation of the Bergström Inequality to Increase the Accumulated Elastic Strain Energy
		4.5.1 Increasing the Accumulated Elastic Strain Energy for Components Loaded in Tension
		4.5.2 Increasing the Accumulated Elastic Strain Energy for Components Loaded in Bending
Chapter 5 Enhancing Systems Performance by Interpretation of Other Algebraic Inequalities Based on Sub-Additive and Super-Additive Functions
	5.1 Increasing the Absorbed Kinetic Energy during a Perfectly Inelastic Collision
	5.2. Ranking the Stiffness of Alternative Mechanical Assemblies by Meaningful Interpretation of an Algebraic Inequality
	5.3 Interpretation of Inequalities Based on Single-Variable Super- and Sub-Additive Functions
		5.3.1 General Inequalities Based on Single-Variable Super- and Sub-Additive Functions
		5.3.2 An Application of Inequality Based on a Super-Additive Function to Minimise the Formation of Brittle Phase during Solidification
		5.3.3 An Application of Inequality Based on a Super-Additive Function to Minimise the Drag Force Experienced by an Object Moving through Fluid
		5.3.4 Light-weight Designs by Interpretation of an Algebraic Inequality Based on a Single-Variable Sub-Additive Function
		5.3.5 An Application of Inequality Based on a Sub-Additive Function to Maximise the Profit  from an Investment
	5.4 Increasing the Mass of Substance Deposited during Electrolysis and Avoiding Overestimation of Density through Interpretation of an Algebraic Inequality
	5.5 Generating New Knowledge about the Deflections of Elastic Elements Arranged in Series and Parallel
Chapter 6 Optimal Selection and Expected Time of Unsatisfied Demand by Meaningful Interpretation of Algebraic Inequalities
	6.1 Maximising the Probability of Successful Selection from Suppliers with Unknown Proportions of High-Reliability Components
	6.2 Increasing the Probability of Successful Accomplishment of Tasks by Devices with Unknown Reliability
	6.3 Monte Carlo Simulations
	6.4 Assessing the Expected Time of Unsatisfied Demand from Users Placing Random Demands on a Time Interval
Chapter 7 Enhancing Decision-Making by Interpretation of Algebraic Inequalities
	7.1 Meaningful Interpretation of an Algebraic Inequality Related to Ranking the Magnitudes of Sequential Random Events
	7.2 Improving Product Reliability by Increasing the Level of Balancing
		7.2.1 Ensuring More Uniform Load Distribution among Components
		7.2.2 Ensuring Conditions for Self-Balancing
		7.2.3 Reducing the Variability of Risk-Critical Parameters
	7.3 Assessing the Probability of Selecting Items of the Same Variety to Improve the Level of Balancing
	7.4 Upper Bound of the Probability of Selecting Each Component from Different Variety
	7.5 Lower Bound of the Probability of Reliable Assembly
	7.6 Tight Lower and Upper Bound for the Fraction of Faulty Components in a Pooled Batch
	7.7 Avoiding an Overestimation of Expected Profit
		7.7.1 Avoiding the Risk of Overestimating Profit  through Interpretation of the Jensen’s Inequality
		7.7.2 Avoiding Overestimation of the Average Profit through Interpretation of the Chebyshev’s Sum Inequality
		7.7.3 Avoiding Overestimation of the Probability of Successful Accomplishment of Multiple Tasks
Chapter 8 Generating New Knowledge by Interpreting Algebraic Inequalities in Terms of Potential Energy
	8.1 Interpreting an Inequality in Terms of Potential Energy
	8.2 A necessary Condition for Minimising Sum of the Powers of Distances
	8.3 Determining the Lower Bound of the Sum of Squared Distances to a Specified Number of Points in Space
	8.4 A Necessary Condition for Determining the Lower Bound of Sum of Distances
	8.5 A Necessary Condition for Determining the Lower Bound of the Sum of Squares of Two Quantities
	8.6 A General Case Involving a Monotonic Convex Function
References
Index