The Assay of Spatially Random Material

Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Clad in Crane Feathers' in R. Brown The point of a Pin'. van Gulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics.

Contenu
1 Introduction.- 1.1 The Nature of the Problem.- 1.2 Examples of Spatially Random Material.- 1.3 Notes.- 2 Deterministic Design I: Conceptual Formulation.- 2.1 Relative Mass Resolution.- 2.2 Response Functions.- 2.3 Point-Source Response Sets.- 2.4 The Convexity Theorem: Complete Response Sets.- 2.5 Relative Mass Resolution and the Concept of Expansion.- 2.6 Example: Pu Assay With One Detector.- 2.7 Example: Pu Assay With Two Detectors.- 2.8 Example: Coincidence Measurements.- 2.8.1 Formulation of the Assay Problem.- 2.8.2 One-Detector Assay Systems.- 2.8.3 Two-Detector Assay Systems.- 2.8.4 Multi-Detector Assay Systems: High-Order Coincidences.- 2.9 Computation of the Relative Mass Resolution.- 2.9.1 Intuitive Derivation.- 2.9.2 Exploiting Symmetry of the Response Set.- 2.9.3 Summary of the Algorithm for Evaluating the Relative Mass Resolution.- 2.10 Example: Pu Assay With Four Detectors.- 2.11 The Inclusion of Statistical Uncertainty.- 2.11.1 Single-Detector systems.- 2.11.2 Multi-Detector Systems.- 2.12 Example: Radioactive Waste Assay.- 2.13 Summary.- 2.13.1 The Deterministic Measure of Performance.- 2.13.2 Statistical Uncertainty.- 2.14 Notes.- 3 Deterministic Design II: General Formulation.- 3.1 Motivation.- 3.2 Unconstrained Spatial Distributions.- 3.3 Constrained Spatial Distributions.- 3.3.1 Evaluation of the Relative Resolution.- 3.3.2 Fundamental Response Sets.- 3.3.3 Inclusion of Statistical Uncertainty.- 3.3.4 Convexity and Compactness of $$\tilde C$$(h, u).- 3.4 Example: Simple Constrained Distributions.- 3.5 Example: Constrained Normal Distributions.- 3.6 Example: Meteorological Measurements.- 3.7 Example: Assay of a Pulmonary Aerosol.- 3.7.1 Formulation.- 3.7.2 Unconstrained Spatial Distributions.- 3.7.3 Constrained Spatial Distributions.- 3.8 Example: Thickness Measurement.- 3.8.1 Formulation.- 3.8.2 A Minimization Problem: Choosing Q.- 3.8.3 Resolution Without Statistical Uncertainty.- 3.8.4 Including Statistical Uncertainty.- 3.9 Example: Enrichment Assay.- 3.10 Auxiliary Parameters.- 3.11 Example: Variable Matrix Structure.- 3.12 Variable Spatial Distributions and Auxiliary Parameters.- 3.13 Constrained Time-Varying Distributions.- 3.14 Example: Flow-Rate Measurement.- 3.15 Notes.- 4 Probabilistic Interpretation of Measurement.- 4.1 Probability Density of the Measurement.- 4.1.1 Single Measurement of a Single Source Particle.- 4.1.2 Single Measurement of Identical Source Particles.- 4.1.3 Multiple Measurements of Identical Source Particles.- 4.1.4 Multiple Measurements of Nonidentical Source Particles.- 4.1.5 Example: Medical Whole-Body Assay.- 4.2 Probability Density of the Total Source Mass.- 4.2.1 Bayes' Rule.- 4.2.2 The Poisson Distribution.- 4.2.3 The Likelihood Function.- 4.2.4 Statistical Uncertainty.- 4.2.5 Example: Pu Assay With One Detector.- 4.2.6 Example: Assay of Aerosol Particles in the Lungs.- 4.3 Bayes' Decision Theory.- 4.3.1 General Formulation.- 4.3.2 Binary Decisions.- 4.3.3 Minimum Probability of Error.- 4.3.4 Quadratic Penalty.- 4.3.5 Biased Quadratic Penalty.- 4.3.6 Example: Plutonium Assay With One Detector.- 4.3.7 Estimating Continuous Parameters.- 4.4 Neyman-Pearson Decision Theory.- 4.4.1 Threshold Detection.- 4.4.2 Maximum Likelihood Estimation.- 4.5 Direct Probabilistic Calibration.- 4.5.1 General Formulation One Detector.- 4.5.2 General Formulation Multiple Detectors.- 4.5.3 Example: U Prospecting Assay of a Thick Deposit.- 4.6 Summary.- 4.7 Notes.- 5 Probabilistic Design.- 5.1 Motivation.- 5.2 Relative Error Criterion.- 5.2.1 General Considerations.- 5.2.2 Example: U Prospecting Assay of a Thin Deposit.- 5.3 Minimum Variance Criterion.- 5.3.1 Rao-Cramer Inequality.- 5.3.2 Examples.- 5.4 Probabilistic Expansion.- 5.4.1 Definition of the Overlap Function.- 5.4.2 Example: Overlap Functions.- 5.4.3 Reduction Theorems: Inequalities for the Overlap Function.- 5.5 Notes.- 6 Adaptive Assay.- 6.1 Motivation.- 6.2 Sequential Analysis.- 6.2.1 Formulation.- 6.2.2 Example: Batch Assay.- 6.2.3 Example: Flow Rate Measurement.- 6.3 Adaptive Barrel Assay.- 6.3.1 The Model.- 6.3.2 The Design Algorithm.- 6.3.3 The Interpretation Algorithm.- 6.3.4 Implementation.- 6.4 Adaptive Assay of a Pulmonary Aerosol.- 6.4.1 The Model.- 6.4.2 The Design Algorithm.- 6.4.3 The Interpretation Algorithm.- 6.4.4 Implementation.- 6.5 Adaptive Assay of a Uranium Deposit.- 6.5.1 The Model.- 6.5.2 The Design Algorithm.- 6.5.3 The Interpretation Algorithm.- 6.5.4 Implementation.- 6.6 Notes.- 7 Some Directions For Research.- 7.1 Overview.- 7.2 Nonlinearity in Mass.- 7.3 Non-Convex Response Sets.- 7.4 Asymptotic Designs.- 7.5 Malfunction Isolation.- 7.6 Adaptive Assay: Advanced Concepts.- 7.6.1 Global Optimization.- 7.6.2 Sequential Analysis of Adaptive Assay.- 7.7 Notes.- Author Index.