There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations

On robust estimation of the location parameter

The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

Introduction.- Measure and Probability Theory.- Banach and Hilbert Spaces.- Optimization Theory.- Measures of Information and Uncertainty.- Bayesian Inverse Problems.- Filtering and Data Assimilation.- Orthogonal Polynomials and Applications.- Numerical Integration.- Sensitivity Analysis and Model Reduction.- Spectral Expansions.- Stochastic Galerkin Methods.- Non-Intrusive Methods.- Distributional Uncertainty.- References.- Index.

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Introduction to Uncertainty Quantification

With the development and implementation of performance-based earthquake engineering, harmonization of performance levels between structural and nonstructural components becomes vital. Even if the structural components of a building achieve a continuous or immediate occupancy performance level after a seismic event, failure of architectural, mechanical or electrical components can lower the performance level of the entire building system. This reduction in performance caused by the vulnerability of nonstructural components has been observed during recent earthquakes worldwide. Moreover, nonstructural damage has limited the functionality of critical facilities, such as hospitals, following major seismic events. The investment in nonstructural components and building contents is far greater than that of structural components and framing. Therefore, it is not surprising that in many past earthquakes, losses from damage to nonstructural components have exceeded losses from structural damage. Furthermore, the failure of nonstructural components can become a safety hazard or can hamper the safe movement of occupants evacuating buildings, or of rescue workers entering buildings. In comparison to structural components and systems, there is relatively limited information on the seismic design of nonstructural components. Basic research work in this area has been sparse, and the available codes and guidelines are usually, for the most part, based on past experiences, engineering judgment and intuition, rather than on objective experimental and analytical results. Often, design engineers are forced to start almost from square one after each earthquake event: to observe what went wrong and to try to prevent repetitions. This is a consequence of the empirical nature of current seismic regulations and guidelines for nonstructural components. This review paper summarizes current knowledge on the seismic design and analysis of nonstructural building components, identifying major knowledge gaps that will need to be filled by future research. Furthermore, considering recent trends in earthquake engineering, the paper explores how performance-based seismic design might be conceived for nonstructural components, drawing on recent developments made in the field of seismic design and hinting at the specific considerations required for nonstructural components.

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Performance-based seismic design of nonstructural building components: The next frontier of earthquake engineering

Probabilistic seismic hazard analyses were conducted to estimate both ground motion and fault displacement hazards at the potential geologic repository for spent nuclear fuel and high‐level radioactive waste at Yucca Mountain, Nevada. The study is believed to be the largest and most comprehensive analyses ever conducted for ground‐shaking hazard and is a first‐of‐a‐kind assessment of probabilistic fault displacement hazard. The major emphasis of the study was on the quantification of epistemic uncertainty. Six teams of three experts performed seismic source and fault displacement evaluations, and seven individual experts provided ground motion evaluations. State‐of‐the‐practice expert elicitation processes involving structured workshops, consensus identification of parameters and issues to be evaluated, common sharing of data and information, and open exchanges about the basis for preliminary interpretations were implemented. Ground‐shaking hazard was computed for a hypothetical rock outcrop at ‐...

Probabilistic seismic hazard analyses for ground motions and fault displacement at Yucca Mountain, Nevada

The damage to non structural masonry infills in RC frame structures induced by earthquake ground motions represents a considerable source of economic losses and a serious threat to human lives. Although measures for the prevention of infill damage are to some extent included in modern seismic design codes, an effective design procedure has not yet been achieved. Hence, the objective of this research is to identify, through numerical investigations and a review of previous experimental findings, the performance of masonry infills in RC frame structures designed following European code provisions, with reference to different design parameters, including building height, level of design seismic loading, ductility class, masonry typology, and infill density. The introduction of possible improvements to the current design procedure is envisaged, with the aim of achieving enhanced infill damage control through the limitation of inter-story drifts. In particular, as the first part of a wider research program, th...

Damage Control for Clay Masonry Infills in the Design of RC Frame Structures

Over forty years ago average-case error was proposed in the applied mathematics literature as an alternative criterion with which to assess numerical methods. In contrast to worst-case error, this ...

Timothy Sullivan

Papers

Introduction to Uncertainty Quantification

Performance-based seismic design of nonstructural building components: The next frontier of earthquake engineering

Probabilistic seismic hazard analyses for ground motions and fault displacement at Yucca Mountain, Nevada

Damage Control for Clay Masonry Infills in the Design of RC Frame Structures

Bayesian Probabilistic Numerical Methods