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Often it is more instructive to know 'what can go wrong' and to understand 'why a result fails' than to plod through yet another piece of theory. In this text, the authors gather more than 300 counterexamples - some of them both surprising and amusing - showing the limitations, hidden traps and pitfalls of measure and integration. Many examples are put into context, explaining relevant parts of the theory, and pointing out further reading. The text starts with a self-contained, non-technical overview on the fundamentals of measure and integration. A companion to the successful undergraduate textbook Measures, Integrals and Martingales, it is accessible to advanced undergraduate students, requiring only modest prerequisites. More specialized concepts are summarized at the beginning of each chapter, allowing for self-study as well as supplementary reading for any course covering measures and integrals. For researchers, it provides ample examples and warnings as to the limitations of general measure theory. This book forms a sister volume to René Schilling's other book Measures, Integrals and Martingales (www.cambridge.org/9781316620243).
We are interested in the rate of convergence of a subordinate Markov process to its invariant measure. Given a subordinator and the corresponding Bernstein function (Laplace exponent), we characterize the convergence rate of the subordinate Markov process; the key ingredients are the rate of convergence of the original process and the (inverse of the) Bernstein function. At a technical level, the crucial point is to bound three types of moment (subexponential, algebraic, and logarithmic) for subordinators as time t tends to ∞. We also discuss some concrete models and we show that subordination can dramatically change the speed of convergence to equilibrium.
An energy resource is the first step in the chain that supplies energy services (for a definition of energy services, see Chapter 1). Energy services are largely ignorant of the particular resource that supplies them; however, often the infrastructures, technologies, and fuels along the delivery chain are highly dependent on a particular type of resource. The availability and costs of bringing energy resources to the market place are key determinants to affordable and accessible energy services.
Energy resources pose no inherent limitation to meeting the rapidly growing global energy demand as long as adequate upstream investment is forthcoming – for exhaustible resources in exploration, production technology, and capacity (mining and field development) and, by analogy, for renewables in conversion technologies.
Hydrocarbons and Nuclear
Occurrences of hydrocarbons and fissile materials in the Earth's crust are plentiful – yet they are finite. The extent of the ultimately recoverable oil, natural gas, coal, or uranium is the subject of numerous reviews, yet still the range of values in the literature is large (Table 7.1). For example, the range for conventional oil is between 4900 exajoules (EJ) for reserves to 13,700 EJ (reserves plus resources) – a range that sustains continued debate and controversy. The large range is the result of varying boundaries of what is included in the analysis of a finite stock of an exhaustible resource, e.g., conventional oil only or conventional oil plus unconventional occurrences, such as oil shale, tar sands, and extra-heavy oils.
We examined reported outbreaks of foodborne shigellosis in the USA from 1998 to 2008 and summarized demographic and epidemiological characteristics of 120 confirmed outbreaks resulting in 6208 illnesses. Most reported foodborne shigellosis outbreaks (n = 70, 58%) and outbreak-associated illnesses (n = 3383, 54%) were restaurant-associated. The largest outbreaks were associated with commercially prepared foods distributed in multiple states and foods prepared in institutional settings. Foods commonly consumed raw were implicated in 29 (24%) outbreaks and infected food handlers in 28 (23%) outbreaks. Most outbreaks (n = 86, 72%) were caused by Shigella sonnei. Targeted efforts to reduce contamination during food handling at multiple points in the food processing and distribution system, including food preparation in restaurants and institutional settings, could prevent many foodborne disease outbreaks and outbreak-related illnesses including those due to Shigella.
Recent papers have debated whether the negative correlation between measures of firm asset growth and subsequent returns is of little importance since it applies only to small firms, is justified as compensation for risk, or is evidence of mispricing. We show that the asset growth effect is pervasive, and evidence to the contrary arises due to specification choices; that one measure of asset growth, the change in total assets, largely subsumes the explanatory power of other measures; that the ability of asset growth to explain either the cross section of returns or the time series of factor loadings is linked to firm idiosyncratic volatility (IVOL); that the return effect is concentrated around earnings announcements; and that analyst forecasts are systematically higher than realized earnings for faster growing firms. In general, there appears to be no asset growth effect in firms with low IVOL. Our findings are consistent with a mispricing-based explanation for the asset growth effect in which arbitrage costs allow the effect to persist.
Molecular mixing measurements are reported for a high-Schmidt-number (Sc ~ 103), small-Atwood-number (A ≈ 7.5 × 10−4) buoyancy-driven turbulent Rayleigh–Taylor (RT) mixing layer in a water channel facility. Salt was added to the top water stream to create the desired density difference. The degree of molecular mixing was measured as a function of time by monitoring a diffusion-limited chemical reaction between the two fluid streams. The pH of each stream was modified by the addition of acid or alkali such that a local neutralization reaction occurred as the two fluids molecularly mixed. The progress of this neutralization reaction was tracked by the addition of phenolphthalein – a pH-sensitive chemical indicator – to the acidic stream. Accurately calibrated backlit optical techniques were used to measure the average concentration of the coloured chemical indicator. Comparisons of chemical product formation for pre-transitional buoyancy- and shear-driven mixing layers are given. It is also shown that experiments performed at different equivalence ratios (acid/alkali concentrations) can be combined to obtain a mathematical relationship between the coloured product formed and the density variance. This relationship was used to obtain high-fidelity quantitative measures of the degree of molecular mixing which are independent of probe resolution constraints. The dependence of molecular mixing on the Schmidt and Reynolds numbers is examined by comparing the current Sc ~ 103 measurements with previous Sc = 0.7 gas-phase and Pr = 7 (where Pr is the Prandtl number) liquid-phase measurements. This comparison indicates that the Schmidt number has a large effect on the quantity of mixed fluid at small Reynolds numbers Reh < 103. At larger Reynolds numbers, corresponding to later times in this experiment, all mixing parameters indicated a greater degree of molecular mixing and a decreased Schmidt number dependence. Implications for the development and quantitative assessment of turbulent transport and mixing models appropriate for RT instability-induced mixing are discussed.
Martingales are a key tool of modern probability theory, in particular, when it comes to a.e. convergence assertions and related limit theorems. The origins of martingale techniques can be traced back to analysis papers by Kac, Marcinkiewicz, Paley, Steinhaus, Wiener and Zygmund from the early 1930s on independent (or orthogonal) functions and the convergence of certain series of functions, see e.g. the paper by Marcinkiewicz and Zygmund which contains many references. The theory of martingales as we know it now goes back to Doob and most of the material of this and the following chapter can be found in his seminal monograph from 1953.
We want to understand martingales as an analysis tool which will be useful for the study of Lp- and almost everywhere convergence and, in particular, for the further development of measure and integration theory. Our presentation differs somewhat from the standard way to introduce martingales – conditional expectations will be defined later in Chapter 22 – but the results and their proofs are pretty much the usual ones. The only difference is that we develop the theory for σ-finite measure spaces rather than just for probability spaces. Those readers who are familiar with martingales and the language of conditional expectations we ask for patience until Chapter 23, in particular Theorem 23.9, when we catch up with these notions.