Decision Making Under Uncertainty and Constraints. A Why-Book

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Preface Contents Applications to Arts Baudelaire's Ideas of Vagueness and Uniqueness in Art: Algorithm-Based Explanations 1 Formulation of the Problem 2 What Is Beauty—Birkhoff's Approach and Its Algorithm-Related Formalization 3 How This Explains the Need for Vagueness 4 Why Uniqueness: An Algorithmic Explanation References Applications to Biosciences Selfish Gene Theory Explains Oedipus Complex 1 Oedipus Complex: A Brief Reminder 2 Our Explanation References Applications to Education How to Teach Advanced Highly Motivated Students: Teaching Strategy of Iosif Yakovlevich Verebeichik 1 Introduction 2 How Mathematics (And Other Disciplines) is Usually Taught: A Brief Reminder 3 What Verebeichik Wanted and What He Therefore Did 4 Let Us Summarize Reference Why 70/100 Is Satisfactory? Why Five Letter Grades? Why Other Academic Conventions? 1 Why 70/100 Is Satisfactory? 2 Why 40-40-20 Proportion for Research, Teaching, and Service: First Explanation 3 Why 40-40-20 Proportion for Research, Teaching, and Service: Second Explanation 4 Why 50-30-20 Proportion for Research Universities: Two Explanations 5 Why Five Letter Grades 6 Why Excellent Is Usually Close to 90 7 How to Allocate Grades to Tests, Homeworks, etc. References Shall We Ignore All Intermediate Grades? 1 Formulation of the Problem 2 Analysis of the Problem and the Resulting Conclusion Reference Why normal infinityinfty is a Reasonable Symbol for Infinity 1 Formulation of the Problem 2 Our Explanation 3 Real-Life Examples of an normal infinityinfty-Like Trajectory References What Is 1/0 from the Practical Viewpoint: A Pedagogical Note 1 Formulation of the Problem 2 Practical Problem 3 Idealized Setting 4 Realistic Setting Reference Historical Diversity Through Base-10 Representation of Mayan Math 1 Introduction 2 Base-10 Representation 3 Addition 4 Subtraction 5 Multiplication 6 Discussion References Why Base-20, Base-40, and Base-60 Number Systems? 1 Formulation of the Problem 2 Analysis of the Problem and the Resulting Explanation References Why Chomsky Normal Form: A Pedagogical Note 1 Formulation of the Problem: Why Chomsky Normal Form? 2 Analysis of the Problem and the Resulting Explanation Reference How to Best Write Research Papers: Basic English? Sophisticated English? 1 Formulation of the Problem 2 Towards Formulating the Problem in Precise Terms 3 So Which Level Is Optimal: Towards the Answer References Applications to Engineering How to Select Typical Objects 1 Formulation of the Problem 2 Main Idea and How We Can Implement It 3 Resulting Algorithm References Why Homogeneous Membranes Lead to Optimal Water Desalination: A Possible Explanation 1 Formulation of the Problem 2 Towards a Precise Formulation of the Problem 3 Formulation of the Problem in Precise Terms and the Resulting Explanation References Fault Detection in a Smart Electric Grid: Geometric Analysis 1 What Is a Smart Electric Grid 2 How the Grid of Sensors Can Detect Faults 3 Let Us Describe This Situation in Precise Terms 4 Research Question 5 Our Answer Reference Applications to Geosciences Why Geological Regions? 1 Formulation of the Problem 2 Our Idea 3 How Can We Measure Approximation Accuracy: Usual Case 4 Geosciences are Different 5 The Resulting Explanation: Formulation of the Result 6 Proof Reference Applications to Machine Learning Why, in Deep Learning, Non-smooth Activation Function Works Better Than Smooth Ones 1 Formulation of the Problem 2 Our Explanation References Why Residual Neural Networks 1 Formulation of the Problem 2 Our Explanation Reference Why Semi-supervised Learning Makes Sense: A Pedagogical Note 1 Formulation of the Problem 2 An Explanation References How to Gauge the Quality of a Multi-class Classification When Ground Truth Is Known with Uncertainty 1 Formulation of the Problem 2 Notations: Traditional Approach 3 Realistic Approach: Formulation of the Problem 4 Our Solution References An AlphaZero-Inspired Approach to Solving Search Problems 1 Introduction 2 Setups for Solving Search Problems 3 Examples for the Satisfiability Problem 4 Solvers Based on Setups 5 Concluding Remark References Applications to Physics Fuzzy Techniques, Laplace Indeterminacy Principle, and Maximum Entropy Approach Explain Lindy Effect and Help Avoid Meaningless Infinities in Physics 1 Formulation of the Problem 2 Let Us Use Fuzzy Techniques: A Straightforward Approach and Why It Does Not Work in This Case 3 Let Us Add Common Sense to Mathematics 4 What About Probabilistic Case 5 Why Strong Lindy Effect 6 Application to Physics: How to Avoid Physically Meaningless Infinite Values References Dimension Compactification Naturally Follows from First Principles 1 Formulation of the Problem 2 Explanation Based on the Second Law of Thermodynamics 3 Explanation Based on Geometry and Symmetries References Is Our World Becoming Less Quantum? 1 Formulation of the Problem 2 Analysis of the Problem and the Resulting Conclusion References Applications to Psychology and Decision Making As Complexity Rises, Meaningful Statements Lose Precision—But Why? 1 Formulation of the Problem 2 Analysis of the Problem References Why People Overestimate Small Probabilities? 1 Formulation of the Problem 2 How Do We Know Probabilities? 3 Which Outcomes Are Possible? 4 So What Can We Conclude About the Probability? 5 Which Probability Value Should We Select? References Why Ovals in Eliciting Intervals? 1 Need to Elicit Intervals 2 Eliciting Intervals Is not Easy 3 Why: The Question and Our Explanation References Why Moments (and Generalized Moments) Are Used in Statistics and Why Expected Utility Is Used in Decision Making: A Possible Explanation 1 Formulation of the Problem 2 Main Ideas Behind Our Explanation References Decision Making Under Uncertainty: Cases When We Only Know an Upper Bound or a Lower Bound 1 Formulation of the Problem 2 How This Problem Is Solved If We Know Both Bounds 3 What if We only Know the Lower Bound 4 What if We only Know the Upper Bound References Why Do People Become Addicted: Towards a Theoretical Explanation for Eyal's Experiment-Based Hook Model 1 Formulation of the Problem 2 Analysis of the Problem and the Resulting Explanation References Why Decimal System? Why Communities with More Than 150 Folks Tend to Split? New Consequences of the Seven Plus Minus Two Law 1 Formulation of the Problem 2 Seven Plus Minus Two Law: A Brief Reminder 3 Why Decimal System: A Possible Explanation 4 Why Communities with More Than 150 Folks Tend to Split: A Possible Explanation 5 Two Additional Related Phenomena References Lev Landau's Marital Advice Explained 1 Formulation of the Problem 2 Our Explanation References Why Too Much Interaction Between Different Parts of the Brain Leads To Unhappiness 1 Formulation of the Problem 2 Analysis of the Problem and Resulting Explanation References Applications to Religion Gödel's Proof of Existence of God Revisited 1 Formulation of the Problem 2 Intuitive Idea 3 Towards Formalizing This Idea 4 First (Preliminary) Result 5 It Is Good to Have Good Objects in All Possible Worlds 6 Second Result 7 What Is God? 8 Word of Caution: Shall We All Run to a Place of Worship? References Blessings, God, Sacrifices: Possible Rational Explanations of Biblical Ideas 1 Formulation of the Problem 2 Where Is God? 3 How Can We Characterize God? 4 Even God's Name Is Holy: What Does This Mean? 5 What Is a Blessing? 6 Sacrificing the Best Animals Verus Darwin 7 Fast-and-Feast 8 Conclusions and Future Work References General Computational Aspects Why Model Order Reduction 1 Formulation of the Problem 2 From Neural Networks to Model Order Reduction: Our Explanation References Bounding the Range of a Sum of Multivariate Rational Functions 1 Introduction 2 Bernstein Expansion 3 Bounds for the Range of a Sum of Rational Functions 3.1 The Naïve Bounds 3.2 Improved Bounds 4 Future Work References Fourier Transform and Other Quadratic Problems Under Interval Uncertainty 1 Formulation of the Problem 2 Class of Quadratic Expressions for Which the Range Can Be Feasibly Computed 3 Our Result 4 Facts from Calculus: Reminder 5 Let Us Apply These Facts to Our Problem 6 Resulting Algorithm: Non-degenerate Case 7 General Case 8 Homogeneous Case References B-Matrices and Doubly B-Matrices in the Interval Setting 1 Introduction 2 B-Matrices 2.1 Real B-Matrices 2.2 Interval B-Matrices 3 Doubly B-Matrices 3.1 Real Doubly B-Matrices 3.2 Interval Doubly B-Matrices References Commonsense ``And''-Operations 1 Why ``and''-Operations 2 How Usual ``and''-Operations are Obtained 3 Need to go Beyond the Usual ``and''-Operations 4 Our Solution 5 Discussion References Extension to Multidimensional Problems of a Fuzzy-Based Explainable and Noise-Resilient Algorithm 1 Introduction 1.1 A Subsection Sample 1.2 The Algorithm 2 Results 3 Discussion 4 Conclusions References Additional Spatial Dimensions Can Help Speed Up Computations 1 Computations and Space Dimensions: How They Are Related and What Are the Remaining Open Problems 2 First (Naive) Idea and Why It Is Naive 3 First Approach: How Actual Compactification Affects Computations 4 Second Approach: How It Affects Computations References