Eigenvalue problems associated with Korn's inequalities in the theory of elasticity

Interest in the possible applications of a priori inequalities in linear elasticity theory motivated the present investigation. Korn's inequality under various side conditions is considered, with emphasis on the Korn's constant. In the "second case" of Korn's inequality, a variational approach leads to an eigenvalue problem; it is shown that, for simply-connected two-dimensional regions, the problem of determining the spectrum of this eigenvalue problem is equivalent to finding the values of Poisson's ratio for which the displacement boundary-value problem of linear homogeneous isotropic elastostatics has a non-unique solution.

Previous work on the uniqueness and non-uniqueness issue for the latter problem is examined and the results applied to the spectrum of the Korn eigenvalue problem. In this way, further information on the Korn constant for general regions is obtained.

A generalization of the "main case" of Korn's inequality is introduced and the associated eigenvalue problem is a gain related to the displacement boundary-value problem of linear elastostatics in two dimensions.

On certain discrete inequalities and their continous analogues

In a 1955 paper, Ky Fan, Olga Taussky, and John Todd presented discrete analogues of inequalities of Wirtinger type, and by taking limits they were able to recover the continuous inequalities. We generalize their techniques to mixed and higher derivatives and inequalities with weight functions in the integrals. We have also considered analogues of inequalities of Müller and Redheffer and have used these inequalities to derive a necessary and sufficient condition on ordered pairs of numbers so that the first number is the square norm of the k^th derivative of some periodic function and the second number is the square norm of the m^th derivative of the same periodic function.

Energy inequalities and error estimates for axisymmetric torsion of thin elastic shells of revolution

The problem motivating this investigation is that of pure axisymmetric torsion of an elastic shell of revolution. The analysis is carried out within the framework of the three-dimensional linear theory of elastic equilibrium for homogeneous, isotropic solids. The objective is the rigorous estimation of errors involved in the use of approximations based on thin shell theory.

The underlying boundary value problem is one of Neumann type for a second order elliptic operator. A systematic procedure for constructing pointwise estimates for the solution and its first derivatives is given for a general class of second-order elliptic boundary-value problems which includes the torsion problem as a special case.

The method used here rests on the construction of “energy inequalities” and on the subsequent deduction of pointwise estimates from the energy inequalities. This method removes certain drawbacks characteristic of pointwise estimates derived in some investigations of related areas.

Special interest is directed towards thin shells of constant thickness. The method enables us to estimate the error involved in a stress analysis in which the exact solution is replaced by an approximate one, and thus provides us with a means of assessing the quality of approximate solutions for axisymmetric torsion of thin shells.

Finally, the results of the present study are applied to the stress analysis of a circular cylindrical shell, and the quality of stress estimates derived here and those from a previous related publication are discussed.

Changes in social inequalities in disability-free life expectancy in Southern Europe: the case of the Basque Country

Background: Health expectancy is a useful tool to monitor health inequalities. The evidence about the recent changes in social inequalities in healthy expectancy is relatively scarce and inconclusive, and most studies have focused on Anglo-Saxon and central or northern European countries. The objective of this study was to analyse the changes in socioeconomic inequalities in disability-free life expectancy in a Southern European population, the Basque Country, during the first decade of the 21st century. Methods: This was an ecological cross-sectional study of temporal trends on the Basque population in 1999-2003 and 2004-2008. All-cause mortality rate, life expectancy, prevalence of disability and disability free-life expectancy were calculated for each period according to the deprivation level of the area of residence. The slope index of inequality and the relative index of inequality were calculated to summarize and compare the inequalities in the two periods. Results: Disability free-life expectancy decreased as area deprivation increased both in men and in women. The difference between the most extreme groups in 2004-2008 was 6.7 years in men and 3.7 in women. Between 1999-2003 and 2004-2008, socioeconomic inequalities in life expectancy decreased, and inequalities in disability-free expectancy increased in men and decreased in women. Conclusions: This study found important socioeconomic inequalities in health expectancy in the Basque Country. These inequalities increased in men and decreased in women in the first decade of the 21st century, during which the Basque Country saw considerable economic growth.

The impact of economic crises on social inequalities in health: what do we know so far?

Since 2008, Western countries are going through a deep economic crisis whose health impacts seem to be fundamentally counter-cyclical: when economic conditions worsen, so does health, and mortality tends to rise. While a growing number of studies have presented evidence on the effect of crises on the average population health, a largely neglected aspect of research is the impact of crises and the related political responses on social inequalities in health, even if the negative consequences of the crises are primarily borne by the most disadvantaged populations. This commentary will reflect on the results of the studies that have analyzed the effect of economic crises on social inequalities in health up to 2013. With some exceptions, the studies show an increase in health inequalities during crises, especially during the Southeast Asian and Japanese crises and the Soviet Union crisis, although it is not always evident for both sexes or all health or socioeconomic variables. In the Nordic countries during the nineties, a clear worsening of health equity did not occur. Results about the impacts of the current economic recession on health equity are still inconsistent. Some of the factors that could explain this variability in results are the role of welfare state policies, the diversity of time periods used in the analyses, the heterogeneity of socioeconomic and health variables considered, the changes in the socioeconomic profile of the groups under comparison in times of crises, and the type of measures used to analyze the magnitude of social inequalities in health. Social epidemiology should further collaborate with other disciplines to help produce more accurate and useful evidence about the relationship between crises and health equity.

We could make a meal out of nothing: contested explanations of inequalities in health

This paper reports on the qualitative findings from a comparative study of public health and lifestyles in South East England and Northern France, regions with similar geographic and economic characteristics. Data from health surveys showed that both countries had an increasing BMI with age, particularly in Northern France. This was despite the finding that the percentage eating fresh fruit and vegetable at least five days a week in Northern France increased with age (from well over 50% to over 90%) compared to around 50% to around 75% in South East England. Qualitative data on health inequalities and how they could be addressed were gathered by focus groups sampling from five tiers using the Townsend Index for comparability (14 in England with 106 participants overall; 13 in France with 143 participants). Both had about two thirds women participants, with a preponderance of middle aged and older people. There was a striking difference in the salience of diet between the two countries; in the French data it was raised only 14 times, whereas in England there were 165 occurrences, and these were often distinguished by their use of narrative. Older respondents contrasted the pressures on families today and the expense of fresh fruit and vegetables with their own childhood or childrearing, when cheap meals could be created using skills which have now been lost. These data therefore provide further evidence that providing food is a moral activity.

How well does the Continuous Plankton Recorder (CPR) sample zooplankton? A comparison with the Longhurst Hardy Plankton Recorder (LHPR) in the northeast Atlantic

The Continuous Plankton Recorder (CPR) survey has collected data on basin- scale zooplankton abundance in the North Atlantic since the 1930s. These data have been used in many studies to elucidate seasonal patterns and long-term change in plankton populations, as well as more recently to validate ecosystem models. There has, however, been relatively little comparison of the data from the CPR with that from other samplers. In this study we compare zooplankton abundance estimated from the CPR in the northeast Atlantic with near-surface samples collected by a Longhurst-Hardy Plankton Recorder (LHPR) at Ocean Weather Station India (59 degree N, 19 degree W) between 1971 and 1975. Comparisons were made for six common copepods in the region: Acartia clausi, Calanus finmarchicus, Euchaeta norvegica, Metridia lucens, Oithona sp. and Pleuromamma robusta. Seasonal cycles based on CPR data were similar to those recorded by the LHPR. Differences in absolute abundances were apparent, however, with the CPR underestimating abundances by a factor of between 5 and 40, with the exception of A. clausi. Active avoidance by zooplankton is thought to be responsible. This avoidance is species specific, so that care must be taken describing communities, as the CPR emphasises those species that are preferentially caught, a problem common to many plankton samplers.

Comparison of Zooplankton Sampling Performance of Longhurst-Hardy Plankton Recorder and Bongo Nets

Data on the abundance and biomass of zooplankton off the northwestern Portuguese coast, separately estimated with a Longhurst-Hardy Plankton Recorder (LHPR) and a Bongo net, were analysed to assess the comparative performance of the samplers. Zooplankton was collected along four transects perpendicular to the coast, deployments alternating between samplers. Total zooplankton biomass measured using the LHPR was significantly higher than that using the Bongo net. Apart from Appendicularia and Cladocera, abundances of other taxa (Copepoda, Mysidacea, Euphausiacea, Decapoda larvae, Amphipoda, Siphonophora, Hydromedusae, Chaetognatha and Fish eggs) were also consistently higher in the LHPR. Some of these differences were probably due to avoidance by the zooplankton of the Bongo net. This was supported by a comparative analysis of prosome length of the copepod Calanus helgolandicus sampled by the two nets that showed that Calanus in the LHPR samples were on average significantly larger, particularly in day samples. A ratio estimator was used to produce a factor to convert Bongo net biomass and abundance estimates to equate them with those taken with the LHPR. This method demonstrates how results from complementary zooplankton sampling strategies can be made more equivalent.