The goal of this work is to compare operators that are defined on probably varying Hilbert spaces. Distance concepts for operators as well as convergence concepts for such operators are explained and examined. For distance concepts we present three main notions. All have in common that they use space-linking operators that connect the spaces. At first, we look at unitary maps and compare the unitary orbits of the operators. Then, we consider isometric embeddings, which is based on a concept of Joachim Weidmann. Then we look at contractions but with more norm equations in comparison. The latter idea is based on a concept of Olaf Post called quasi-unitary equivalence. Our main result is that the unitary and isometric distances are equal provided the operators are both self-adjoint and have 0 in their essential spectra. In the third chapter, we focus specifically on the investigation of these distance terms for compact operators or operators in p-Schatten classes. In this case, the interpretation of the spectra as null sequences allows further distance investigation. Chapter four deals mainly with convergence terms of operators on varying Hilbert spaces. The analyses in this work deal exclusively with concepts of norm resolvent convergence. The main conclusion of the chapter is that the generalisation for norm resolvent convergence of Joachim Weidmann and the generalisation of Olaf Post, called quasi-unitary equivalence, are equivalent to each other. In addition, we specify error bounds and deal with the convergence speed of both concepts. Two important implications of these convergence notions are that the approximation is spectrally exact, i.e., the spectra converge suitably, and that the convergence is transferred to the functional calculus of the bounded functions vanishing at infinity.

Surveys are commonly tailored to produce estimates of aggregate statistics with a desired level of precision. This may lead to very small sample sizes for subpopulations of interest, defined geographically or by content, which are not incorporated into the survey design. We refer to subpopulations where the sample size is too small to provide direct estimates with adequate precision as small areas or small domains. Despite the small sample sizes, reliable small area estimates are needed for economic and political decision making. Hence, model-based estimation techniques are used which increase the effective sample size by borrowing strength from other areas to provide accurate information for small areas. The paragraph above introduced small area estimation as a field of survey statistics where two conflicting philosophies of statistical inference meet: the design-based and the model-based approach. While the first approach is well suited for the precise estimation of aggregate statistics, the latter approach furnishes reliable small area estimates. In most applications, estimates for both large and small domains based on the same sample are needed. This poses a challenge to the survey planner, as the sampling design has to reflect different and potentially conflicting requirements simultaneously. In order to enable efficient design-based estimates for large domains, the sampling design should incorporate information related to the variables of interest. This may be achieved using stratification or sampling with unequal probabilities. Many model-based small area techniques require an ignorable sampling design such that after conditioning on the covariates the variable of interest does not contain further information about the sample membership. If this condition is not fulfilled, biased model-based estimates may result, as the model which holds for the sample is different from the one valid for the population. Hence, an optimisation of the sampling design without investigating the implications for model-based approaches will not be sufficient. Analogously, disregarding the design altogether and focussing only on the model is prone to failure as well. Instead, a profound knowledge of the interplay between the sample design and statistical modelling is a prerequisite for implementing an effective small area estimation strategy. In this work, we concentrate on two approaches to address this conflict. Our first approach takes the sampling design as given and can be used after the sample has been collected. It amounts to incorporate the survey design into the small area model to avoid biases stemming from informative sampling. Thus, once a model is validated for the sample, we know that it holds for the population as well. We derive such a procedure under a lognormal mixed model, which is a popular choice when the support of the dependent variable is limited to positive values. Besides, we propose a three pillar strategy to select the additional variable accounting for the design, based on a graphical examination of the relationship, a comparison of the predictive accuracy of the choices and a check regarding the normality assumptions.rnrnOur second approach to deal with the conflict is based on the notion that the design should allow applying a wide variety of analyses using the sample data. Thus, if the use of model-based estimation strategies can be anticipated before the sample is drawn, this should be reflected in the design. The same applies for the estimation of national statistics using design-based approaches. Therefore, we propose to construct the design such that the sampling mechanism is non-informative but allows for precise design-based estimates at an aggregate level.

In the course of the COVID-19 pandemic, borders have become relevant (again) in political action and in people's everyday lives within a very short time. This was especially true for the inhabitants of border regions, whose cross-border life worlds were suddenly irritated by closed borders and police controls. However, the COVID-19 pandemic also led to an increased evidence of social, cultural, economic, health and mobility boundaries beyond national borders, which raised pressing questions about social inequalities. The authors shed light on these dynamics from the perspective of territorial borders, social boundaries and (dis)continuities in border regions through a variety of thematic and spatial approaches. The critical observations and scientific comments were made during the lockdown in April and May 2020 and provide insights into the events during the global pandemic.

In and with this thematic issue we would like to invite you to engage in productive boundary work and to critically examine the relationship between nature and culture in the Anthropocene. A few years ago, the term Anthropocene was proposed by Paul Crutzen as a term for the current geological epoch, in which humankind (the ‘anthropos’) is seen as the central driving force for global changes in ecological systems. This epoch is characterized by the blurring of boundaries between society and nature, science and politics, as well as by the increased drawing of boundaries between social groups, lifestyles, and the Global North and Global South. With this issue, we would like to give an impetus to explore boundary phenomena in the relationship between nature and society, which up to now have not been the focus of Border Studies. The challenges and problems of the Anthropocene require cross-border thinking and research that stimulates a new reflexivity and commitment, to which the multidisciplinary field of Border Studies can contribute.

The object of the current Thematic Issue is not to focus on the individuals (the cross-border commuters) but on the organization of the cross-border labor markets. We move from a micro perspective to a macro perspective in order to underline the diversity of the cross-border labor markets (at the French borders, for example) and shed light on the many aspects that impact cross-border supply or demand. Trying to understand the whole system that goes beyond the cross-border flows, the question we address in this thematic issue is about the organization of the labor markets: is the system organized in a cross-border way? Or do the borders still prevent a genuinely integrated cross-border labor market?