Interpolation of Hilbert and Sobolev spaces: quantitative estimates and counterexamples

This paper provides an overview of interpolation of Banach and Hilbert spaces, with a focus on establishing when equivalence of norms is in fact equality of norms in the key results of the theory. (In brief, our conclusion for the Hilbert space case is that, with the right normalisations, all the key results hold with equality of norms.) In the final section we apply the Hilbert space results to the Sobolev spaces Hs(Ω) and tildeHs(Ω), for s in R and an open Ω in R^n. We exhibit examples in one and two dimensions of sets Ω for which these scales of Sobolev spaces are not interpolation scales. In the cases when they are interpolation scales (in particular, if Ω is Lipschitz) we exhibit examples that show that, in general, the interpolation norm does not coincide with the intrinsic Sobolev norm and, in fact, the ratio of these two norms can be arbitrarily large.

The STEP scale: a measure of short-term export performance improvement

This article is a direct response to a recent observation in the literature that managers appear to be short-term oriented in their assessment of the performance of an export venture (Madsen 1998). On the basis of a cross-national survey of exporting firms, the authors present a three-dimensional scale for assessing managerial judgment of short-term export performance (i.e., the STEP scale). The three dimensions are (1) satisfaction with short-term performance improvement, (2) short-term exporting intensity improvement, and (3) expected short-term performance improvement. The scale presents evidence of reliability as well as convergent, discriminant, and nomological validity, and it reveals factorial similarity and factorial equivalence across both samples. The authors outline managerial and public policy implications that stem from the scale and identify avenues for further export marketing research.

Representation errors and retrievals in linear and nonlinear data assimilation

This article shows how one can formulate the representation problem starting from Bayes’ theorem. The purpose of this article is to raise awareness of the formal solutions,so that approximations can be placed in a proper context. The representation errors appear in the likelihood, and the different possibilities for the representation of reality in model and observations are discussed, including nonlinear representation probability density functions. Specifically, the assumptions needed in the usual procedure to add a representation error covariance to the error covariance of the observations are discussed,and it is shown that, when several sub-grid observations are present, their mean still has a representation error ; socalled ‘superobbing’ does not resolve the issue. Connection is made to the off-line or on-line retrieval problem, providing a new simple proof of the equivalence of assimilating linear retrievals and original observations. Furthermore, it is shown how nonlinear retrievals can be assimilated without loss of information. Finally we discuss how errors in the observation operator model can be treated consistently in the Bayesian framework, connecting to previous work in this area.

Observational and energetics constraints on the non-conservation of potential/Conservative Temperature and implications for ocean modelling

This paper seeks to elucidate the fundamental differences between the nonconservation of potential temperature and that of Conservative Temperature, in order to better understand the relative merits of each quantity for use as the heat variable in numerical ocean models. The main result is that potential temperature is found to behave similarly to entropy, in the sense that its nonconservation primarily reflects production/destruction by surface heat and freshwater fluxes; in contrast, the nonconservation of Conservative Temperature is found to reflect primarily the overall compressible work of expansion/contraction. This paper then shows how this can be exploited to constrain the nonconservation of potential temperature and entropy from observed surface heat fluxes, and the nonconservation of Conservative Temperature from published estimates of the mechanical energy budgets of ocean numerical models. Finally, the paper shows how to modify the evolution equation for potential temperature so that it is exactly equivalent to using an exactly conservative evolution equation for Conservative Temperature, as was recently recommended by IOC et al. (2010). This result should in principle allow ocean modellers to test the equivalence between the two formulations, and to indirectly investigate to what extent the budget of derived nonconservative quantities such as buoyancy and entropy can be expected to be accurately represented in ocean models.

Metacognition and proofreading: the roles of aging, motivation, and interest

Objectives: The current study examined younger and older adults’ error detection accuracy, prediction calibration, and postdiction calibration on a proofreading task, to determine if age-related difference would be present in this type of common error detection task. Method: Participants were given text passages, and were first asked to predict the percentage of errors they would detect in the passage. They then read the passage and circled errors (which varied in complexity and locality), and made postdictions regarding their performance, before repeating this with another passage and answering a comprehension test of both passages. Results: There were no age-related differences in error detection accuracy, text comprehension, or metacognitive calibration, though participants in both age groups were overconfident overall in their metacognitive judgments. Both groups gave similar ratings of motivation to complete the task. The older adults rated the passages as more interesting than younger adults did, although this level of interest did not appear to influence error-detection performance. Discussion: The age equivalence in both proofreading ability and calibration suggests that the ability to proofread text passages and the associated metacognitive monitoring used in judging one’s own performance are maintained in aging. These age-related similarities persisted when younger adults completed the proofreading tasks on a computer screen, rather than with paper and pencil. The findings provide novel insights regarding the influence that cognitive aging may have on metacognitive accuracy and text processing in an everyday task.