Fourier uniqueness in even dimensions

In recent work, methods from the theory of modular forms were used to obtain Fourier uniqueness results in several key dimensions (d=1,8,24), in which a function could be uniquely reconstructed from the values of it and its Fourier transform on a discrete set, with the striking application of resolving the sphere packing problem in dimensions d=8 and d=24. In this short note, we present an alternative approach to such results, viable in even dimensions, based instead on the uniqueness theory for the Klein–Gordon equation. Since the existing method for the Klein–Gordon uniqueness theory is based on the study of iterations of Gauss-type maps, this suggests a connection between the latter and methods involving modular forms. The derivation of Fourier uniqueness from the Klein–Gordon theory supplies conditions on the given test function for Fourier interpolation, which are hoped to be optimal or close to optimal.

Elements of causality

A standard part of studies of black holes, and in fact of mathematical general relativity, is causality theory, which is the study of causal relations on Lorentzian manifolds. An essential issue here is understanding the influence of energy conditions on the causality relations. The highlights of such studies include the incompleteness theorems, known also as singularity theorems, of Penrose, Hawking and Geroch, the area theorem of Hawking, and the topology theorems of Hawking and others. The aim of this chapter is to provide an introduction to the subject, with a complete exposition of those topics which are needed for the global treatment of the uniqueness theory of black holes. In particular we provide a coherent introduction to causality theory for metrics which are twice differentiable.

Uniqueness properties of meromorphic functions in the light of three shared sets

We use the notion of weighted sharing of sets to improve a number of results in the uniqueness theory of meromorphic functions sharing three sets. We have also presented a new idea in the formation of one of the range sets in connection with the three shared set problems, which ultimately yields a better result than many existing result in some sense.

Existential loneliness: An attempt at an analysis of the concept and the phenomenon

Background: According to ethical guidelines, healthcare professionals should be able to provide care that allows for the patients’ values, customs and beliefs, and the existential issues that are communicated through them. One widely discussed issue is existential loneliness. However, much of the debate dealing with existential loneliness concludes that both the phenomenon and the concept are quite vague. Aim: To clarify what constitutes existential loneliness, and to describe its lived experiences. A further aim was to provide a definition of existential loneliness that can function as a tool for identifying the phenomenon and for differentiating it from other kinds of loneliness. Method: A literature review including theoretical and empirical studies. Different search strategies were used to gather the articles included in the study. The analysis of the empirical studies had an interpretative approach. The articles were also analysed with the aim of providing a definition of existential loneliness. This was done by means of criteria such as language, uniqueness, theory and usefulness. This study is not empirical and does therefore not require an ethics review. Results: The analysis resulted in two main characteristics. The first one was perceiving oneself as inherently separated (disconnected) from others and from the universe. The second one brings out emotional aspects of EL, such as isolation, alienation, emptiness and a feeling of being abandoned. The empirical findings were divided into two categories: experiences of EL and circumstances in which EL arises. A definition of EL is also suggested, based on the two main characteristics identified. Conclusion: In order to meet patients’ needs, it is an ethical duty for healthcare staff to be able to recognise experiences of EL, that is, to communicate with the patients about these experiences in an appropriate manner. This in turn demands knowledge about existential issues and skills to deal with them.