Earth–outer space interactions challenge conventional legal structures through dynamics that transcend jurisdictional boundaries and temporal scales. International law historically operates through specific spatiotemporal assumptions: geometric space, chronometric time, and cartographic politics. These elements structure how legal authority is conceptualised and enacted. This study recognizes the interconnectedness between Earth and outer space, positioning legal thought and practice within planetary and cosmic contexts. This integrative framework moves beyond anthropocentric and state-centric paradigms to address the indeterminate nature of multifaceted systems. The research employs an interdisciplinary methodology that integrates legal theory and doctrine, systems engineering, and systems science to analyse emergent phenomena such as orbital debris dynamics. The study concludes that addressing Earth–outer space interactions effectively requires not merely integrating existing legal regimes but reconceptualizing core legal concepts to align better with complex, multi-scalar and emergent dynamics.

Self-working card tricks have been a staple for budding magicians and entire books have been written about them, e.g. Fulves [1]. Lately selfworking card tricks have become common fodder for social media. In this note, we generalise two such card tricks (from [2] and [3]) that are based on the Australian shuffle.

Mental health, like physical health, represents an important resource for participating in politics. We bring new insights from six surveys from five different countries (Britain, Germany, the Netherlands, Switzerland, and the United States) that combine diversified questions on mental health problems and political participation. Unlike previous research on depression, we find only limited evidence for the Resource Hypothesis that mental health problems reduce political participation, except in the case of voting and only in some samples. Instead, we find mixed evidence that mental health problems and their comorbidity (experiencing multiple problems) are associated with increased political participation. Our study leads us to more questions than answers: are the measures available in public opinion surveys appropriate for the task? Do general survey samples adequately capture people with mental disorders? And is the assumption that poor mental health reduces political participation wrong?

We developed a method called “component decomposition” to extract the pattern of each component of the sample from the multiple powder X-ray diffraction data. Using the component decomposition and the Direct Derivation Method™, we analyze the behavior of phase transitions of trehalose during the changes in temperature and humidity. Because we do not require databases or standard samples, this method is a powerful tool for the quantification of polymorphs in samples containing multiple polymorphs.

For integers n, , let Sk (n) denote the power sum 1k + 2k +… + nk, with S0 (n) = n. As is well known, Sk (n) can be represented by a polynomial in n of degree k + 1 with constant term equal to zero, i.e. ${S_k}\,(n)\, = \,\mathop \sum \nolimits_{j\, = \,1}^{k\, + \,1} \,{a_{k,j}}{n^j}$, for certain rational coefficients ak,j. In 1973, in the popular science magazine Kvant*, appeared an article by Vladimir Abramovich [1] in which, among other things, the author derived the following minimal recurrence relation which expresses in a very compact way the connection between the power sums Sk (n) and Sk-1 (n):

(1)

where the polynomial $S_{k\, - \,1}^*\,(n)$ is constructed by replacing in Sk-1 (n) the term nj with the expression ${{{n^{j + 1}} - n} \over {j + 1}}$ for each j = 1, 2, … , k. Note that $S_{k\, - \,1}^*\,(n)$ has degree k + 1, as it should be.

At university I was intrigued by a historian friend and his constant quest for gobbets – short, apposite quotations that he avidly collected for possible use in future essays. It strikes me that there might be analogous nuggets of mathematics and I offer the following five short, self-contained items both for amusement and for possible classroom use. Each contains a surprising element and I have deliberately left a few loose ends to encourage further exploration.

It is well known that the real geometric series ${\sum\limits_{n = 0}^\infty{a{k^n} = a + ak + a{k^2} + }}$ … converges to a definite sum if the common ratio, k, is such that |k| < 1, the sum being ${a}\over{1-k}$. For example, if a = 1 and ${k ={{1}\over{2}}}$ we obtain the series ${{1} + {{1}\over{2}} + ... + {{1}\over{2}^n} + ...}$, whose partial sums are ${{1}, {{3}\over{2}}, {{7}\over{4}}, ..., 2 -{{1}\over{2}^n}}$, …, and these are clearly approaching the value 2 as n becomes larger and larger. As n → ∞, ${2} - {{1}\over{2}^n}{\unicode{x2192}} \,\,2$, in agreement with the formula ${{a}\over{1-k}} {=} {{1}\over{1}-{1\over2}}=2$.

In order for accused persons with disabilities to be able to access justice on an equal basis with others, equality of outcomes is important. However, in the past century, the limited approach to legal aid which focuses on processes has continually been applied by criminal justice system actors in response to legal aid challenges faced by accused persons with disabilities in Kenya. The major dilemma facing this approach is its emphasis on steps to be taken as opposed to the end result or goal. This paper seeks to explain that a shift towards an outcomes approach to legal aid for accused persons with disabilities has the potential of supporting innovation in Kenya’s criminal justice system and help close access to justice gaps that may exist. To achieve this paradigm shift, the African Disability Protocol has been employed as it promotes an integral development of legal aid justice that requires states parties to put in place specific outcomes-based laws for promoting the rights of accused persons in access to justice.

We present some less known variations of the the Vecten configuration and give purely geometric proofs for them. It is unlikely that these variations (and even proofs?) are new, probably just well-hidden in the literature. If a reader happens to know references for the variations discussed (or other geometric proofs), please let the authors know. At [1] the reader can find a dynamic webpage on our topic.