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Interpretation of Algebraic Inequalities

Practical Engineering Optimisation and Generating New Knowledge

- Available for pre-order. Item will ship after October 25, 2021

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## Book Description

This book demonstrates how interpreting abstract inequalities can optimise engineering design processes, with applications in mechanical engineering, materials science, electrical engineering, reliability engineering and risk management.

Algebraic inequalities have been extensively used in mathematics, as a tool for characterisation of reliability functions and to reduce risk and uncertainty. These mathematical formulas can handle deep uncertainty associated with design variables and control parameters. Through interpreting algebraic inequalities, engineers can, for example, correctly predict the outcome of uncertain sequences, which helps choose materials and processes for manufacturing. This means that costs and profits can be correctly ascertained, optimising the process. The book will demonstrate how this can be put into practice through covering the algebraic inequalities suitable for interpretation in different contexts and describing how to apply this knowledge to enhance system performance.

Including example problems and solutions, alongside exemplar MATLAB code, this book will be of interest to engineers and students in the field of optimisation, engineering design, reliability engineering and risk management.

## Table of Contents

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 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-additive and super-additive functions 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 3. GENERATING KNOWLEDGE ABOUT REAL 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 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 a capacitor 4.5 Meaningful interpretation of the Bergström inequality to increase the accumulated strain energy 4.5.1 Increasing the accumulated strain energy for components loaded in tension 4.5.2 Increasing the accumulated strain energy for components loaded in bending 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-additive and sub-additive functions 5.3.1 General inequalities based on single-variable super-additive 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 An application of inequality based on a super-additive function to maximise the accumulated elastic strain energy 5.3 Increasing the mass of substance deposited during electrolysis and avoiding overestimation of density through interpretation of an algebraic inequality 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 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 7.3 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 components of different variety 7.5 Lower bound of the probability of reliable assembly by interpretation of the Chebyshev's sum inequality 7.6 Tight lower and upper bound for the fraction of faulty components in a pooled batch by interpretation of algebraic inequalities 7.7 Avoiding underestimation of risk and overestimation of profit by meaningful interpretation of an algebraic inequality 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 by interpretation of an algebraic inequality 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 by a meaningful interpretation of an inequality 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

## Author(s)

### Biography

Professor Michael Todinov works on mechanical engineering, applied mathematics and computer science. After receiving a DEng from the University of Birmingham, he built a reputation working on reliability and risk, flow networks, probability and probabilistic fatigue and fracture. He is Professor of Mechanical Engineering at Oxford Brookes University, UK.